Handbook of mathematical analysis in mechanics of viscous fluids 9783319133447, 3319133446, 9783319133430, 9783319133454

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Yoshikazu Giga Antonín Novotný Editors

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Yoshikazu Giga • Antonín Novotný Editors

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids With 62 Figures and 3 Tables

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Editors Yoshikazu Giga Graduate School of Mathematical Sciences University of Tokyo Meguro-ku, Tokyo Japan

Antonín Novotný Université de Toulon IMATH, Toulon France

ISBN 978-3-319-13343-0 ISBN 978-3-319-13344-7 (eBook) ISBN 978-3-319-13345-4 (print and electronic bundle) https://doi.org/10.1007/978-3-319-13344-7 Library of Congress Control Number: 2017957054 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

In memoriam Mariarosaria Padula, our respected colleague and friend

Preface

Fluid mechanics has a long history since Greek philosopher Archimedes discovered his famous law of forces acting on bodies in motionless fluid. A good understanding of its principles and its mathematical formulation and properties is crucial in many branches of contemporary science and technology. Not only its modern foundation relies on mathematics but also many branches of mathematics were developed or even emerged through the research in fluid mechanics or through the mathematical formulation of problems of fluid mechanics. The examples include the theory of functions of one complex variable, topology, dynamical systems, differential equations, differential geometry, probability theory, and functional analysis, to name only a few. Mathematics has been always playing a key role in the research on fluid mechanics. Many imminent problems in various branches of mathematics have their origin or can be interpreted as problems of fluid mechanics although in many cases the community of pure mathematicians fails to notice this fact. The purpose of this handbook is to provide a synthetic review of the state of the art in the theory of viscous fluids, present fundamental notions, formulate problems of fluid mechanics representing the development of the theory during last several decades, and show the methods and mathematical tools for their resolution. Since the field of mathematical fluid mechanics is huge, it is impossible to cover all topics. In this handbook, we focus on mathematical analysis in mechanics of viscous Newtonian fluids. The first part consisting of two chapters is devoted to derivation of basic equations by physical modeling. The second part is devoted to mathematical analysis of incompressible fluids, while the third part is dealing with the mathematical theory of viscous compressible fluids. There are many topics that are not covered by the handbook. In particular, this is the case of numerical analysis of the equations which would deserve by itself an independent volume. The handbook reviews important problems and notions that marked the development of the theory. It explains the methods and techniques that may be used for their resolution. We hope that it will be useful not only to mathematicians who work on the future development of the theory but also to physicists and engineers who need to know the tools of mathematical analysis for developing applications.

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Part 1: Derivation of Basic Equations There are several ways to derive the basic equations. One of the most typical ways is to start from the balance equations for mass, momentum, and total energy incorporating to the process the second law of thermodynamics. This approach is discussed in the first chapter by J. Málek and V. Pr˚uša. The approach based on variational principles is discussed in the contribution by M.-H. Giga, A. Kirshtein, and C. Liu. Part 2: Incompressible Fluids The Navier-Stokes system is a conventional macroscopic model consisting of partial differential equations describing the motion of viscous incompressible Newtonian fluids. The modern mathematical analysis of these equations goes back to seminal works of J. Leray in 1933 and 1934. Leray introduced the notion of weak solutions for the Navier-Stokes equations and developed several key tools of functional analysis for their mathematical treatment. In his above mentioned papers, Leray formulated many imminent open problems. One of them, on the regularity of weak solutions, was named among the seven Millennium Prize Problems of the Clay Institute of Mathematics. The fundamental question is whether, in three space dimensions, the global in time (weak) solutions of the Navier-Stokes system emanating from the smooth initial data must be smooth. This problem is related to the question whether or not the system of Navier-Stokes equations is still a good model for fluid flows in regimes with large Reynolds numbers. Despite a lot of effort of excellent mathematicians, this problem is still open. Many mathematical tools and specific techniques, e.g., the theory of partial differential equations, theory of interpolation, and maximal regularity theory, have been developed or refined within the process of its investigation. Many important open problems have been formulated during its investigation, and some of them solved. The accelerated development of technology at the end of the last century gave rise to many new mathematical problems in fluid mechanics. We can name as examples free boundary problems and problems related to complex fluids. In all these problems, the Navier-Stokes equations play an important or even a crucial role. We shall include in the handbook these modern topics, as well as recent development in the problems on inviscid limits. In this part, we intend to provide key notions and tools for mathematical understanding of equations of incompressible fluids. We mainly discuss existence and uniqueness problems as well as behavior of solutions and different notions of their stability. When the flow is slow, or more precisely when the Reynolds number is small, it is convenient to consider a linearized version of Navier-Stokes equations called Stokes equations. The investigation of the Stokes system is not only extremely important for itself, but it is fundamental also for the development of the nonlinear theory. Well-posedness and regularity questions for the various initial-boundary value problems for the Stokes equations in various types of sufficiently smooth domains are discussed in the first chapter by M. Hieber and J. Saal. Similar

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problems in domains with less regular boundaries are investigated by S. Monniaux and Z. Shen. There are several long-standing problems opened by Leray for the stationary Navier-Stokes equations related to general inflow-outflow boundary conditions in the case of bounded domains and to velocity profiles at infinity in the case of unbounded domains. The recent development in the former case is reported in the contribution by M. V. Korovkov, K. Pileckas, and R. Russo, while the contribution of T. Hishida deals with the latter case in 3-D exterior domains. The contribution of G. P. Galdi and J. Neustupa is devoted to the stationary flows around a rotating body. After the short excursion to the stationary Navier-Stokes equations, the handbook continues by several chapters dealing with weak and strong solutions to the nonsteady Navier-Stokes equations. The existence of a weak solution in a smooth general domain is discussed by R. Farwig, H. Kozono, and H. Sohr. Self-similar solutions are introduced and investigated by H. Jia, V. Šverák, and T.-P. Tsai. The problem of existence of time periodic solutions is addressed in the contribution by G. P. Galdi and M. Kyed. Large time behavior of solutions is discussed by L. Brandolese and M. E. Schonbek. Since the Navier-Stokes equations have a regularizing property, it is interesting to investigate the structure of sets of “rough” initial data still allowing smooth solutions globally in time. These questions are addressed in the contribution by I. Gallager. Stability of some special solutions, e.g., of the so-called Lamb-Oseen vortex, is investigated by T. Gallay and Y. Maekawa. Some important exact solutions are constructed in the contribution by H. Okamoto. Asymptotic behavior of solutions near the boundary endowed with no-slip conditions is investigated in the chapter by Y. Maekawa and A. Mazzucato. Regularity and regularity criteria for weak solutions are discussed by G. Seregin and V. Šverák and by H. Beirão da Veiga, Y. Giga, and Z. Grujic from several different viewpoints. Behavior of solution of an ideal flow is discussed by T. Y. Hou and P. Liu. The Navier-Stokes flow coupled with other effects is increasingly important. A class of models for geophysical flows is discussed by J. Li and E. S. Titi. Equations for polymetric materials are investigated by N. Masmoudi, while nematic liquid crystal flows are discussed by M. Hieber and J. Prüss. Some problems for viscoelastic fluids are discussed by X. Hu, F.-H. Lin, and C. Liu. A problem dealing with two-phase flows is a typical free boundary problem. There are several chapters devoted to this topic by J. Prüss and S. Shimizu; by V. A. Solonnikov and I. V. Denisova; by G. Simonett and M. Willa; and by H. Abels and H. Garke. The first three chapters handle smooth solutions, while the latter chapter handles weak solutions which allow topological change of region occupied by the fluid. Finally, the classical free boundary value problem of water wave is discussed by D. Córdova and C. Fefferman. Part 3: Compressible Fluids The mathematical models that take into account the compressibility of the fluid and thermodynamical effects lead to a rich variety of systems of partial differential

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equations. Their mathematical character depends on the physical assumptions on transport phenomena (describing the viscous effects and the heat transport) and on constitutive laws (usually prescribing pressure and internal energy as functions of density and temperature), not speaking about the situations when these equations are coupled with systems describing other phenomena (e.g., in magnetohydrodynamics, multifluid models, and theory of chemically reacting mixtures). In these circumstances, the classical formulation may give rise to different nonequivalent weak formulations, which are all physically reasonable. This abundance of possibilities imposes severe restrictions on the material that can be treated in one handbook volume. In particular, as in the “incompressible part” we have restricted our subject of interest only to the so-called Newtonian fluids (when, loosely speaking, the stress tensor depends linearly on the gradient of velocity), and the heat flux is proportional to the gradient of temperature through the Fourier law. The resulting system is called Navier-Stokes-Fourier system. This system describes the most simple thermodynamically consistent model of fluid dynamics. If the pressure depends on density only, one obtains a simpler model communally called compressible Navier-Stokes equations in barotropic regime. Although simpler than the complete Navier-Stokes-Fourier system, it inherits most of its mathematical difficulties. The present handbook is exclusively treating the two above mentioned systems. The mathematical difficulties encountered in mathematical investigation of compressible Navier-Stokes equations are many fold: they are related to (1) the enormous range of scales of motion described by the system, (2) the absence of mechanisms preventing density to create a vacuum and the absence of dissipation in the mass conservation, (3) the mixed parabolic-hyperbolic (or elliptic-hyperbolic in the steady case) character of the underlying linearized equations, and (4) the nonlinear character of equations (which still allows, through a priori estimates, to prevent the field quantities of concentrations but does not defend them from oscillations). The history of mathematical investigation of the compressible Navier-Stokes equations is more recent than the history of investigation of the incompressible Navier-Stokes equations. It starts with the works of D. Graffi and J. Nash on the local in time existence of strong solutions in the 1960s and continues with global in time existence results for data close to the equilibrium in works of A. Matsumura and T. Nishida in the 1980s. The first results on the existence of weak solutions similar to those introduced for incompressible fluids by Leray in 1934 were established by P.-L. Lions only in 1998. Likewise, the stability analysis introduced for the Navier-Stokes equations by Prodi and Serrin in the 1960s was waiting for its “compressible” counterpart till 2012. Being much younger than its “incompressible” counterpart, the mathematical theory of compressible fluids undergoes still a tumultuous development, and the synthetic level of contributions in “compressible” part is therefore necessarily objectively less high than in the “incompressible” part. The handbook chapters report the state of the art of some parts of the theory at the end of 2016. It covers mostly the questions of well-posedness of solutions to

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various boundary and initial-boundary value problems to these equations (meaning the existence of strong and weak solutions, uniqueness or conditional uniqueness, stability, longtime behavior, conditional regularity, and qualitative properties of solutions) in one and several space dimensions. We have made a preference to cover in the handbook solely these topics knowing well that this is a subjective and nonexhausting choice. Various types of weak and strong solutions are introduced and discussed in the contribution of E. Feireisl. Qualitative theory of strong solutions has longer history and its solid foundations are nowadays relatively well established. These questions are treated in three chapters by R. Danchin; J. Burczak, Y. Shibata, and W. M. Zaja¸czkowski; and M. Kotschote. Related questions of blowup criteria and existence of strong solutions for small data with large oscillations are developed in two contributions by Z. Huang and Z. Xin and by J. Li and Z. Xin. The longtime behavior of strong solutions is investigated in the contribution of Y. Shibata and Y. Enomoto, while an overview of some results for free boundary value problems is given in the contribution by I. Denisova and V. A. Solonnikov. The existence, stability, and asymptotic behavior of solutions to the equations in one dimension, or of spherically and axially symmetric solutions, are treated in three chapters by A. Zlotnik, by Y. Qin, and by S. Jiang and Q. Ju. The chapter by A. Matsumura is a comprehensive introduction to waves in 1-D compressible fluids. The theory of weak solutions has a short history and is still in agitated development. The existence of different types of weak solutions, stability, longtime behavior, and weak-strong uniqueness principle are discussed by A. Novotný and by H. Petzeltová. The existence problem for weak solutions with degenerate viscosity coefficients is presented by D. Bresch and B. Desjardins. The contribution by P. I. Plotnikov and W. Weigant is devoted to the existence results in the case of critical adiabatic coefficients. A particular role in the theory of weak solutions is played by solutions belonging to the so-called intermediate regularity class. These solutions are introduced in the contribution by M. Perepelitsa. The contribution of Y. Sun and Z. Zhang on the conditional regularity of weak solutions provides a link between the theory of weak solutions on the one hand and strong solutions/blowup criteria on the other hand. Three chapters are devoted to the stationary solutions, written by S. Jiang and C. Zhou, by P. Mucha, M. Pokorný, and E. Zatorska, and by P. Mucha, M. Pokorný, and O. Kreml. Compressible Navier-Stokes equations are applicable to modeling of a large variety of fluid motions ranging from small-scale motions (as acoustic waves) to large-scale motions of planetary size. The specific regimes of some of these flows are described sufficiently by simplified models characterized by extreme values of nondimensional numbers (as Mach, Reynolds, Péclet, Strouhal, and other numbers). Some of the simplified models can be obtained rigorously from the compressible Navier-Stokes equations as singular limits of nondimensional numbers. Typically, the incompressible Navier-Stokes system is known to be a low Mach number limit of the compressible Navier-Stokes equations. Three handbook chapters by N. Jiang and N. Masmoudi, by Feireisl, and by R. Klein are devoted to the investigation of the

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singular limits involving low Mach number and related questions. These chapters can be viewed as a bridge between the “compressible” and “incompressible” part of the handbook. Finally, we have involved three chapters giving examples of the investigation of compressible Navier-Stokes equations coupled with other equations. The contributions by X. Blanc and B. Ducoment, by V. Giovangigli, and by D. Bresch, B. Desjardins, J.-M. Ghidaglia, E. Grenier, and M. Hillairet deal with and discuss compressible magnetohydrodynamics, chemically reacting mixtures, and various multifluid models, respectively. The original idea of this handbook started in early 2012 in a discussion of Yoshikazu Giga with Professor Mariarosaria Padula. The proposal was finalized at the end of September 2012. Unfortunately, Professor Padula passed away prematurely at the end of September 2012 and the plan was almost abandoned until the second editor has joined the project. We rewrote the proposal in early 2013 and invited several distinguished authors to participate at this enterprise. Most of the invited contributors, specialists, and leading experts in the field accepted our invitation. In spite of their academic duties and responsibilities, they accepted to work hard on their chapters. Without their deep engagement and generous investment, the project could never see the light of the day. Since the field of mathematical fluid mechanics is huge even after the restriction to viscous fluids, there are several important fields which are not covered in its table of contents. In spite of this fact, the volume of the handbook is already considerable. Since this work is not an encyclopedia, we believe that such selection is allowed. We hope that readers will find this handbook comprehensive and that it will help them to learn about typical problems and approaches in mathematical analysis in fluid mechanics and encourage them to go further. We believe that its review character and extended bibliography will help to improve the orientation in this vast subject especially among young scientists. We hope that the overview material and mathematical tools gathered in the handbook chapters will encourage applications in other fields of mathematics and even beyond mathematics. March 2018

Yoshikazu Giga Antonín Novotný

Acknowledgments

The idea of editing Handbook of Mathematical Analysis in Mechanics of Viscous Fluids began with a brief conversation in 2011 at the occasion of the International Congress on Industrial and Applied Mathematics in Vancouver between Hans Koelsch, one of the editors of Springer, and Yoshikazu Giga. Yoshikazu Giga and Mariarosaria Padula build up the preliminary proposal which was available in the middle of September 2012. Unfortunately, Professor Padula passed away on September 29, 2012, and the project was interrupted until Antonín Novotný joined the editorial team around the end of January 2013. With several aids from Springer team led by Achi Dosanjh, executive editor, we finally reached a concrete and explicit plan around fall of 2013 and adjusted the proposal by suggestions of several potential authors. Namely, our numerous discussions with Giovanni Paolo Galdi and Eduard Feireisl contributed a lot to establish the first global structure of the handbook. We thank them for their serious interest in our project. After Springer sent an official invitation to each of the potential contributors around September 2014, the project was gradually growing by contributions of authors and reports of referees. Our thanks go particularly to the authors for their deep, generous, and unselfish investment into the uneasy task of writing their chapters and to the referees for careful reading and critical remarks allowing to improve the quality of chapters. Last but not least, we would like to thank the production team at Springer, led by Achi Dosanjh, for the help, efficiency, and advice. We would like to thank the editorial assistant Saskia Ellis who coordinated the day-to-day interactions between the editors, the authors, and the production departments. Without the high level of commitment and devotion on her part, we would neither have had the bandwidth nor the patience to complete this work. Coeditors Yoshikazu Giga Antonín Novotný

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Contents

Volume 1 Part I Derivation of Equations for Incompressible and Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Josef Málek and Vít Pr˚uša 2 Variational Modeling and Complex Fluids . . . . . . . . . . . . . . . . . . . . . Mi-Ho Giga, Arkadz Kirshtein, and Chun Liu Part II

Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 73

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3 The Stokes Equation in the L -Setting: Well-Posedness and Regularity Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matthias Hieber and Jürgen Saal

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4 Stokes Problems in Irregular Domains with Various Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sylvie Monniaux and Zhongwei Shen

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5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail V. Korobkov, Konstantin Pileckas, and Remigio Russo 6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toshiaki Hishida 7 Steady-State Navier-Stokes Flow Around a Moving Body . . . . . . . . Giovanni P. Galdi and Jiˇrí Neustupa 8 Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reinhard Farwig, Hideo Kozono, and Hermann Sohr

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299 341

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9 Self-Similar Solutions to the Nonstationary Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hao Jia, Vladimír Šverák, and Tai-Peng Tsai

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Time-Periodic Solutions to The Navier-Stokes Equations . . . . . . . . . Giovanni P. Galdi and Mads Kyed

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11

Large Time Behavior of the Navier-Stokes Flow . . . . . . . . . . . . . . . . Lorenzo Brandolese and Maria E. Schonbek

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12

Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . Isabelle Gallagher

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Existence and Stability of Viscous Vortices . . . . . . . . . . . . . . . . . . . . . Thierry Gallay and Yasunori Maekawa

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Models and Special Solutions of the Navier-Stokes Equations . . . . . Hisashi Okamoto

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The Inviscid Limit and Boundary Layers for Navier-Stokes Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yasunori Maekawa and Anna Mazzucato

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Regularity Criteria for Navier-Stokes Solutions . . . . . . . . . . . . . . . . . Gregory Seregin and Vladimir Šverák

17

Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomas Yizhao Hou and Pengfei Liu

869

Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hugo Beirão da Veiga, Yoshikazu Giga, and Zoran Gruji´c

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Recent Advances Concerning Certain Class of Geophysical Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jinkai Li and Edriss S. Titi

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19

20

Equations for Polymeric Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nader Masmoudi

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Volume 2 21

Modeling of Two-Phase Flows With and Without Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 Jan W. Prüss and Senjo Shimizu

22

Equations for Viscoelastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 Xianpeng Hu, Fang-Hua Lin, and Chun Liu

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23

Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid Crystal Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 Matthias Hieber and Jan W. Prüss

24

Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . 1135 Vsevolod Alexeevich Solonnikov and Irina Vladimirovna Denisova

25

Stability of Equilibrium Shapes in Some Free Boundary Problems Involving Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221 Gieri Simonett and Mathias Wilke

26

Weak Solutions and Diffuse Interface Models for Incompressible Two-Phase Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267 Helmut Abels and Harald Garcke

27

Water Waves with or Without Surface Tension . . . . . . . . . . . . . . . . . 1329 Diego Córdoba and Charles Fefferman

Part III

Compressible Viscous Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1351

28

Concepts of Solutions in the Thermodynamics of Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353 Eduard Feireisl

29

Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stability, and Longtime Behavior . . . . . . . . . . 1381 Antonín Novotný and Hana Petzeltová

30

Weak Solutions for the Compressible Navier-Stokes Equations with Density Dependent Viscosities . . . . . . . . . . . . . . . . . . 1547 Didier Bresch and Benoît Desjardins

31

Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1601 P. I. Plotnikov and W. Weigant

32

Weak Solutions for the Compressible Navier-Stokes Equations in the Intermediate Regularity Class . . . . . . . . . . . . . . . . . 1673 Misha Perepelitsa

33

Symmetric Solutions to the Viscous Gas Equations . . . . . . . . . . . . . . 1711 Song Jiang and Qiangchang Ju

34

Local and Global Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Energy Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1751 Jan Burczak, Yoshihiro Shibata, and Wojciech M. Zaja¸czkowski

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Fourier Analysis Methods for the Compressible Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1843 Raphaël Danchin

36

Local and Global Existence of Strong Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Maximal Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1905 Matthias Kotschote

37

Local and Global Solvability of Free Boundary Problems for the Compressible Navier-Stokes Equations Near Equilibria . . . 1947 Irina Vladimirovna Denisova and Vsevolod Alexeevich Solonnikov

Volume 3 38

Global Existence of Regular Solutions with Large Oscillations and Vacuum for Compressible Flows . . . . . . . . . . . . . . . 2037 Jing Li and Zhou Ping Xin

39

Global Existence of Classical Solutions and Optimal Decay Rate for Compressible Flows via the Theory of Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2085 Yoshihiro Shibata and Yuko Enomoto

40

Finite Time Blow-Up of Regular Solutions for Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2183 Xiangdi Huang and Zhou Ping Xin

41

Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak Solutions for the Compressible Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2263 Yongzhong Sun and Zhifei Zhang

42

Well-Posedness and Asymptotic Behavior for Compressible Flows in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . 2325 Yuming Qin

43

Well-Posedness of the IBVPs for the 1D Viscous Gas Equations . . . 2421 Alexander Zlotnik

44

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2495 Akitaka Matsumura

45

Existence of Stationary Weak Solutions for Isentropic and Isothermal Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2549 Song Jiang and Chunhui Zhou

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46

Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2595 Piotr Bogusław Mucha, Milan Pokorný, and Ewelina Zatorska

47

Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2663 Ondˇrej Kreml, Piotr Bogusław Mucha, and Milan Pokorný

48

Low Mach Number Limits and Acoustic Waves . . . . . . . . . . . . . . . . . 2721 Ning Jiang and Nader Masmoudi

49

Singular Limits for Models of Compressible, Viscous, Heat Conducting, and/or Rotating Fluids . . . . . . . . . . . . . . . . . . . . . . 2771 Eduard Feireisl

50

Scale Analysis of Compressible Flows from an Application Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2827 Rupert Klein

51

Weak and Strong Solutions of Equations of Compressible Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2869 Xavier Blanc and Bernard Ducomet

52

Multi-Fluid Models Including Compressible Fluids . . . . . . . . . . . . . 2927 Didier Bresch, Benoît Desjardins, Jean-Michel Ghidaglia, Emmanuel Grenier, and Matthieu Hillairet

53

Solutions for Models of Chemically Reacting Compressible Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2979 Vincent Giovangigli

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3031

About the Editors

Yoshikazu Giga is professor at the Graduate School of Mathematical Sciences of the University of Tokyo, Japan. He is a fellow of the American Mathematical Society as well as of the Japan Society for Industrial and Applied Mathematics. Through his more than 200 papers and 2 monographs, he has substantially contributed to the theory of parabolic partial differential equations including geometric evolution equations, semilinear heat equations, as well as the incompressible Navier-Stokes equations. He has received several prizes including the Medal of Honor with Purple Ribbon from the Government of Japan.

Antonín Novotný is professor at the Department of Mathematics of the University of Toulon and member of the Institute of Mathematics of the University of Toulon, France. Coauthor of more than 100 papers and 2 monographs, he is one of the leading experts in the theory of compressible Navier-Stokes equations.

xxi

Contributors

Helmut Abels Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany Hugo Beirão da Veiga Dipartimento di Matematica, Università di Pisa, Pisa, Italy Xavier Blanc Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, F-75205 Paris, France Lorenzo Brandolese Institut Camille Jordan, Université Lyon 1, Villeurbanne, France Didier Bresch LAMA UMR 5127 CNRS Batiment le Chablais, Université de Savoie Mont-Blanc, Le Bourget du Lac, France Jan Burczak Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland OxPDE, Mathematical Institute, University of Oxford, Oxford, UK Diego Córdoba Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, Spain Raphaël Danchin Université Paris-Est Créteil, LAMA UMR CNRS 8050, France Irina Vladimirovna Denisova Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg, Russia BenoOıt Desjardins Fondation Mathématique Jacques Hadamard, CMLA, ENS Cachan, CNRS and Modélisation Mesures et Applications S.A., Paris, France Bernard Ducomet Département de Physique Théorique et Appliquée, CEA/DAM Ile De France, Arpajon, France Yuko Enomoto Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama, Japan

xxiii

xxiv

Contributors

Reinhard Farwig Department of Mathematics, Darmstadt University of Technology, Darmstadt, Germany International Research Training Group Darmstadt-Tokyo (IRTG 1529), Darmstadt, Germany Charles Fefferman Department of Mathematics, Princeton University, Princeton, NJ, USA Eduard Feireisl Evolution Differential Equations (EDE), Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, Czech Republic Giovanni P. Galdi Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, USA Isabelle Gallagher Department of Mathematics, Paris-Diderot University, Paris, France Thierry Gallay UMR 5582 – Mathematics Laboratory, Institut Fourier, Université Grenoble Alpes, Gières, France Harald Garcke Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany Jean-Michel Ghidaglia Centre de Mathématiques et de Leurs Applications (CMLA), Ecole Normale Supérieure Paris-Saclay, Centre National de la Recherche Scientifique, Université Paris-Saclay, Cachan, France Mi-Ho Giga Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan Yoshikazu Giga Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan Vincent Giovangigli CMAP-CNRS, Ecole Polytechnique, Palaiseau Cedex, France Emmanuel Grenier Unité de Mathématiques Pures et Appliquées, ENS Lyon, Lyon Cedex, France Zoran Gruji´c Department of Mathematics, University of Virginia, Charlottesville, VA, USA Matthias Hieber Angewandte Analysis, Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany University of Pittsburgh, Benedum Engineering Hall, Pittsburgh, PA, USA Matthieu Hillairet Institut Montpelliérain Alexander Grothendiek, UMR 5149 CNRS, Université de Montpellier, Montpellier, France Toshiaki Hishida Graduate School of Mathematics, Nagoya University, Nagoya, Japan

Contributors

xxv

Thomas Yizhao Hou Division of Engineering and Applied Science, Computing and Mathematical Sciences Department, California Institute of Technology, Pasadena, CA, USA Xianpeng Hu Department of Mathematics, City University of Hong Kong, Hong Kong, China Xiangdi Huang Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, China Hao Jia Institute for Advanced Studies, Princeton and Department of Mathematics, University of Minnesota, Chicago, IL, USA Song Jiang Institute of Applied Physics and Computational Mathematics, Beijing, China Ning Jiang School of Mathematics and Statistics, Wuhan University, Wuhan, China Qiangchang Ju Institute of Applied Physics and Computational Mathematics, Beijing, China Arkadz Kirshtein Department of Mathematics, Pennsylvania State University, University Park, PA, USA Rupert Klein FB Mathematik and Informatik, Institut für Mathematik, Freie Universität Berlin, Berlin, Germany Mikhail V. Korobkov Sobolev Institute of Mathematics, Novosibirsk, Russia Novosibirsk State University, Novosibirsk, Russia Matthias Kotschote Department of Mathematics and Statistics, University of Konstanz, Konstanz, Germany Hideo Kozono Department of Mathematics, Waseda University, Tokyo, Japan Japanese-German Graduate Externship Program, Japan Society of Promotion of Science, Tokyo, Japan Ondˇrej Kreml Institute of Mathematics, Czech Academy of Sciences, Praha, Czech Republic Mads Kyed Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany Jinkai Li Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China Jing Li Institute of Applied Mathematics, AMSS and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China

xxvi

Contributors

Fang-Hua Lin Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Chun Liu Department of Mathematics, Pennsylvania State University, University Park, PA, USA Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, USA Pengfei Liu Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA Yasunori Maekawa Department of Mathematics, Graduate School of Sciences, Kyoto University, Kyoto, Japan Josef Málek Faculty of Mathematics and Physics, Charles University in Prague, Praha 8 – Karlín, Czech Republic Nader Masmoudi Department of Mathematics, New York University in Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Akitaka Matsumura Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka, Japan Anna Mazzucato Mathematics Department, Eberly College of Science, The Pennsylvania State University, University Park, State College, PA, USA Sylvie Monniaux Aix-Marseille Université, CNRS, Centrale Marseille, Marseille, France Piotr Bogusław Mucha Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland Jiˇrí Neustupa Institute of Mathematics, Czech Academy of Sciences, Praha 1, Czech Republic Antonín Novotný Université de Toulon, IMATH, Toulon, France Hisashi Okamoto Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan Department of Mathematics, Gakushuin University, Toshima-ku, Tokyo, Japan Misha Perepelitsa Department of Mathematics, University of Houston, Houston, TX, USA Hana Petzeltová Department EDE, Mathematical Institute of the Academy of Sciences of the Czech Republic, Praha 1, Czech Republic

Contributors

xxvii

Konstantin Pileckas Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania P. I. Plotnikov Mathematical Department, Novosibirsk State University, Novosibirsk, Russia Siberian Division of Russian Academy of Sciences, Lavryentyev Institute of Hydrodynamics, Novosibirsk, Russia Milan Pokorný Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic Vít Pruša ˚ Faculty of Mathematics and Physics, Charles University in Prague, Praha 8 – Karlín, Czech Republic Jan W. Prüss Institut für Mathematik, Martin-Luther-Universität, HalleWittenberg, Halle, Germany Yuming Qin Department of Applied Mathematics, College of Science, Donghua University, Shanghai, China Remigio Russo Department of Mathematics and Physics, Second University of Naples, Caserta, Italy Jürgen Saal Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany Maria E. Schonbek Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA, USA Gregory Seregin Mathematical Institute, University of Oxford, Oxford, UK Zhongwei Shen University of Kentucky, Lexington, KY, USA Yoshihiro Shibata Department of Mathematics and Research Institute for Science and Engineering, Waseda University, Shinjuku-ku, Tokyo, Japan Senjo Shimizu Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, Japan Gieri Simonett Department of Mathematics, Vanderbilt University, Nashville, TN, USA Hermann Sohr Faculty of Electrical Engineering, Informatics and Mathematics, Department of Mathematics, University of Paderborn, Paderborn, Germany Vsevolod Alexeevich Solonnikov Laboratory of Mathematical Physics, St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia Yongzhong Sun Department of Mathematics, Nanjing University, Nanjing, China Vladimir Šverák School of Mathematics, University of Minnesota, Minneapolis, MN, USA

xxviii

Contributors

Edriss S. Titi Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel Department of Mathematics, Texas A&M University, College Station, TX, USA Department of Mathematics, University of California, Mathematics, Mechanical and Aerospace Engineering, Irvine, CA, USA Tai-Peng Tsai Department of Mathematics, University of British Columbia, Vancouver, BC, Canada W. Weigant Institute für Angewandte Mathematik, Universität Bonn, Bonn, Germany Mathias Wilke Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany Zhou Ping Xin The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, Hong Kong, China Wojciech M. Zaja¸czkowski Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, Warsaw, Poland Ewelina Zatorska Department of Mathematics, Imperial College London, London, UK Department of Mathematics, University College London, London, UK Zhifei Zhang School of Mathematical Sciences, Peking University, Beijing, China Chunhui Zhou Department of Mathematics, Southeast University, Nanjing, China Alexander Zlotnik Faculty of Economic Sciences, Department of Mathematics, National Research University Higher School of Economics, Moscow, Russia

Part I Derivation of Equations for Incompressible and Compressible Fluids

1

Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids Josef Málek and Vít Pr˚uša

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Balance of Mass, Momentum, and Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 2.2 Balance of Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Stress Power and Its Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stability of the Rest State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Compressible and Incompressible Viscous Heat-Conducting Fluids . . . . . . . . . . . . . 4.3 Compressible Korteweg Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Compressible and Incompressible Viscoelastic Heat-Conducting Fluids . . . . . . . . . . 4.5 Beyond Linear Constitutive Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Boundary Conditions for Internal Flows of Incompressible Fluids . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5 7 9 13 18 19 22 23 26 31 42 46 60 64 67 68 68

Abstract

The chapter starts with overview of the derivation of the balance equations for mass, momentum, angular momentum, and total energy, which is followed by a detailed discussion of the concept of entropy and entropy production. While the

Dedicated to professor K. R. Rajagopal on the occasion of his 65th birthday. J. Málek () • V. Pr˚uša Faculty of Mathematics and Physics, Charles University in Prague, Praha 8 – Karlín, Czech Republic e-mail: [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_1

3

4

J. Málek and V. Pr˚uša

balance laws are universal for any continuous medium, the particular behavior of the material of interest must be described by an extra set of material-specific equations. These equations relating, for example, the Cauchy stress tensor and the kinematical quantities are called the constitutive relations. The core part of the chapter is devoted to the presentation of a modern thermodynamically based phenomenological theory of constitutive relations. The key feature of the theory is that the constitutive relations stem from the choice of two scalar quantities, the internal energy and the entropy production. This is tantamount to the proposition that the material behavior is fully characterized by the way it stores the energy and produces the entropy. The general theory is documented by several examples of increasing complexity. It is shown how to derive the constitutive relations for compressible and incompressible viscous heat-conducting fluids (Navier-Stokes-Fourier fluid), Korteweg fluids, and compressible and incompressible heat-conducting viscoelastic fluids (Oldroyd-B and Maxwell fluid). Keywords

Continuum mechanics  Constitutive relations  Thermodynamics

2000 Mathematics Subject Classification. 76A02, 76A05, 74A15, 74A20

1

Introduction

Continuum mechanics and thermodynamics are based on the idea of continuously distributed matter and other physical quantities. Originally, continuum mechanics was equated with hydrodynamics, aerodynamics, and elasticity. The scope of the study was the motion of water and air and the deformation of some special solid substances. However, the concept of continuous medium has been shown to be useful and extremely viable even in the modelling of the behavior of much more complex systems such as polymeric solutions, granular materials, rock and land masses, special alloys, and many others. Moreover, the range of physical processes modelled in the continuum framework nowadays goes beyond purely mechanical processes. Processes such as phase transitions or growth and remodeling of biological tissues are routinely approached in the setting of continuum thermodynamics. Finally, the systems studied in continuum thermodynamics range from the very small ones studied in microfluidics (see, e.g., Squires and Quake [89]) up to the gigantic ones studied in planetary science; see, for example, Karato and Wu [48]. It seems that the laws governing the motion of a continuous medium must be extremely complicated in order to capture such a wide range of physical systems and phenomena of interest. This is only partially true. In principle, the laws

The authors were supported by the project LL1202 in the program ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic.

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

5

governing the motion of a continuous medium can be seen as reformulations and generalizations/counterparts of the classical physical laws (the laws of Newtonian physics of point particles and the laws of classical thermodynamics). Surprisingly, the continuum counterparts of the classical laws are relatively easy to derive. These laws are – in the case of a single continuum medium – the balance equations for the mass, momentum, angular momentum, and energy and the evolution equation for the entropy. The balance equations are supposed to be universally valid for any continuous medium, and their derivation is briefly discussed in Sect. 2. The critical and the most difficult part in formulating the system of governing equations for a given material is the specification of the response of the material to the given stimuli. The sought stimulus-response relation can be, for example, a relation between the deformation and the stress or a relation between the heat flux and the temperature gradient. The task of finding a description of the material response is the task of finding the so-called constitutive relations. Apparently, the need to specify the constitutive relation for a given material calls for an investigation of the microscopic structure of the material. The reader who is interested in examples of the derivation of constitutive relations from microscopic theories is referred to Bird et al. [4] or Larson [52] to name a few. However, the investigation of the microscopic structure of the material is not the only option. The constitutive relations can be specified staying entirely at the phenomenological level. Here the phenomenological level means that it is possible to deal only with phenomena directly accessible to the experience and measurement without trying to interpret the phenomena in terms of ostensibly more fundamental (microscopic) physical theories. Indeed, the possible class of constitutive relations is in fact severely restricted by physical requirements stemming, for example, from the requirement on Galilean invariance of the governing equations, perceived symmetry of the material, or the second law of thermodynamics. As it is apparent from the discussion in Sect. 4, such restrictions allow one to successfully specify constitutive relations.

2

Balance Equations

Before discussing the theory of constitutive relations, it will be convenient to briefly recall the fundamental balance equations for a continuous medium. The reader who is not yet familiar with the field of continuum mechanics and thermodynamics is referred to Truesdell and Toupin [93], Müller [62], Truesdell and Rajagopal [92], or Gurtin et al. [35] for a detailed treatment of balance equations and kinematics of continuous medium. Note that the same formalism can be applied even for several interacting continuous media which constitute the continuum approach to the theory of mixtures; see, for example, Samohýl [84] and Rajagopal and Tao [81]. The continuous body B is assumed to be a part of Euclidean space R3 . The motion of the body is described by the function  that maps the positions X of points at time t0 to their respective positions x at a later time instant t, such that x D .X ; t /. Concerning a suitable framework for the description of processes in a continuous medium, one can, in principle, choose from two alternatives.

6

J. Málek and V. Pr˚uša

Either one expresses the quantities of interest as functions of time and the initial position X or as functions of time and the chosen position x in space R3 . The former alternative is called the Lagrangian description, while the latter is referred to as the Eulerian description. The Lagrangian description is especially suitable for the description of change of form or shape, since in order to describe the change, a reference point is needed. (Change with respect to some initial state.) As such, the Lagrangian description is a popular choice for studying the motion of solids. On the other hand, the Eulerian description is not primarily focused on change with respect to some initial state, but on the rate of change. Naturally, the rate of change (the time derivative) can be captured by referring to the state of the material at the current time and in its infinitesimally small-time neighborhood. In other words, if one is interested in the instantaneous velocity field, then there is no need to know complete trajectories of the individual points from the possibly distant initial state. As such, the Eulerian approach is useful for the description of the motion of fluids, where one is primarily interested in the velocity field, and the knowledge of the motion (individual trajectories) is of secondary importance. (Note, however, that the Eulerian description can be advantageous even in the description of solids, in particular in problems that involve phenomena like fluid–structure interaction or growth; see, e.g., Frei et al. [29, 30].) In what follows a theory for fluids is of primary interest; hence the Eulerian description is used. The balance laws are derived by applying the classical laws of Newtonian physics and classical thermodynamics to a volume of the moving material V .t / Ddef .V .t0 /; t /, where V .t0 / is an arbitrarily chosen part of the continuous body B. The main mathematical tool is the Reynolds transport theorem; see, for example, Truesdell and Toupin [93]. Theorem 1 (Reynolds transport theorem). Let .x; t / be a sufficiently smooth function describing the motion of the body B, and let V .t0 / be an arbitrary part of B at the inital time t0 . Let .x; t / be a sufficiently smooth scalar Eulerian field, and let V .t / D .V .t0 /; t / be the volume transported by the motion . Then d dt

Z

Z .x; t / dv D

V .t/

V .t/



 d.x; t / C .x; t / div v.x; t / dv; dt

where d.x; t / @.x; t / Ddef C v.x; t /  r.x; t / dt @t

(1)

denotes the material ˇ time derivative, and v.x; t / is the Eulerian velocity field, ;t/ ˇ . v.x; t / Ddef @.X ˇ @t 1 X D

.x;t/

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

2.1

7

Balance of Mass, Momentum, and Angular Momentum

2.1.1 Balance of Mass Classical Newtonian physics assumes conservation of mass m, which can be written as dm D 0. The balance of mass is a generalization of this equation to the continuum dt mechanics setting. The mass mV .t/ of volume R V .t / is expressed in terms of the Eulerian density field .x; t / as mV .t/ Ddef V .t/ .x; t / dv. Using the notion of the density field, the Reynolds transport theorem, and the requirement dmV .t/ D0 dt

(2)

then immediately yield the integral form of the balance of mass Z V .t/



 d.x; t / C .x; t / div v.x; t / dv D 0; dt

(3)

where V .t / is an arbitrary volume in the sense that it is the volume obtained by tracking an arbitrarily chosen initial volume V .t0 /. Since the volume in (3) is arbitrary, and the considered physical quantities are assumed to be sufficiently smooth, the integral form of the balance of mass leads to the pointwise evolution equation d.x; t / C .x; t / div v.x; t / D 0: dt

(4)

Having (4), it follows that Reynolds transport theorem for the quantity .x; t / Ddef .x; t /w.x; t /, where w is an arbitrary Eulerian field, in fact reads d dt

Z

Z .x; t /w.x; t / dv D V .t/

.x; t / V .t/

dw.x; t / dv: dt

(5)

This simple identity will be useful in the derivation of the remaining balance equations. Note that the integral form of the balance of mass and the other balance equations as well can be reformulated in a weak form without the need to use the pointwise equation (4) as an intermediate step. See Feireisl [27] or Bulíˇcek et al. [8] for details.

2.1.2 Balance of Momentum Balance of momentum is the counterpart of Newton second law dp D F for dt point particles, where p D mv is the momentum of particle with mass m moving with velocity v. In continuum setting the momentum pV .t/ of volume V .t / is expressed in terms of the Eulerian density and the velocity field as R pV .t/ Ddef V .t/ .x; t /v.x; t / dv. The balance of momentum for V .t / then reads

8

J. Málek and V. Pr˚uša

dpV .t/ dt

D F:

(6)

The force F on the right-hand side of the continuum counterpart of the Newton second law consists of two contributions, F D F volume C F contact :

(7)

The first physical mechanism that contributes to the force acting on the volume V .t / is the specific body force b. This is the force that acts on every part of V .t /, hence the specific body force contributes to the total force F via the volume integral Z F volume Ddef

.x; t /b.x; t / dv:

(8)

V .t/

A particular example of specific body force is the electrostatic force or the gravitational force. The second contribution to the force F acting on the volume V .t / is the force F contact due to the resistance of the material surrounding the volume V .t /. Since this force contribution arises due to wading of the volume V .t / through the surrounding material, it acts on the surface of the volume V .t/. Consequently, it is referred to as the contact or surface force. The contact force contribution is a new physical mechanism that goes beyond the concept of forces between point particles, and it is the key concept in mechanics of continuous media. The total contact force F contact acting on V .t/ is assumed to take the form Z F contact Ddef

t.x; t; n.x; t // ds;

(9)

@V .t/

where t is the contact force density and n denotes the unit outward normal to the surface of volume V .t/. It is worth emphasizing that the formula for the contact force is a fundamental and nontrivial assumption concerning the nature of the forces acting in a continuous medium. Assuming that the contact force is given in terms of the contact force density t.x; t; n.x; t //, one can proceed further and prove that the contact force density is in fact given by the formula t.x; t; n.x; t // D T.x; t /n.x; t /;

(10)

where T.x; t / is a tensorial quantity that is referred to as the Cauchy stress tensor. The corresponding theorem is referred to as the Cauchy stress theorem, and the interested reader will find the proof, for example, in the classical treatise by Truesdell and Toupin [93]. Note that the proof of the Cauchy stress theorem could be rather subtle if one wants to work with functions that lack smoothness; see, for example, Šilhavý [87] and references therein for the discussion of this issue.

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

9

Using (10) and the formulae for the body force and the contact force, the equality D F volume C F contact that holds for arbitrary volume V .t / can be rewritten in the form Z Z Z d .x; t /v.x; t / dv D .x; t /b.x; t / dv C T.x; t /n.x; t / ds; dt V .t/ V .t/ @V .t/ (11) dp V .t / dt

which upon using the Reynolds transport theorem, the Stokes theorem, and the identity (4) reduces to Z Z dv.x; t / Œdiv T.x; t / C .x; t /b.x; t / dv; dv D (12) .x; t / dt V .t/ V .t/ denotes the material time derivative of the Eulerian velocity field. (The where dv.x;t/ dt divergence of the tensor field is defined as the operator that satisfies .div A/  w D div.A> w/ for an arbitrary constant vector field w.) Since the considered physical quantities are assumed to be sufficiently smooth and the volume V .t / can be chosen arbitrarily, the integral form of the balance of momentum (12) reduces to the pointwise equality .x; t /

dv.x; t / D div T.x; t / C .x; t /b.x; t /: dt

(13)

2.1.3 Balance of Angular Momentum Balance of angular momentum is for a single point particle a simple consequence of Newton laws of motion, and it is in fact redundant. In the context of a continuous medium, the balance of angular momentum however provides a nontrivial piece of information concerning the structure of the Cauchy stress tensor. In the simplest setting it reduces to the requirement on the symmetry of the Cauchy stress tensor T.x; t / D T> .x; t /;

(14)

See, for example, Truesdell and Toupin [93] for details.

2.2

Balance of Total Energy

The need to work with thermal effects inevitably calls for reformulating the laws of classical thermodynamics in continuum setting. Classical thermodynamics is essentially a theory applicable to a volume of a material wherein the physical fields are homogeneous and undergo only infinitesimal (slow in time) changes. This is clearly insufficient for the description of the behavior of a moving continuous medium. In particular, the concepts of energy and entropy as introduced in classical thermodynamics (see, e.g., Callen [15]) need to be revisited.

10

J. Málek and V. Pr˚uša

a

b h b

b

b

d

b

c

c

c

d d

a

f

a

d d

d d

d d c

e e k

k

b a

b

g

a

c

Fig. 1 Joule experiment (Original figures from Joule [47] with edits). (a) Overall sketch of Joule experimental apparatus. (b) Horizontal and vertical cross section of the vessel with a paddle wheel

A good demonstration of the (in)applicability of classical thermodynamics and its continuum counterpart is the analysis of the famous Joule experiment; see Joule [46]. In the experiment concerning the mechanical equivalence of heat, Joule studied the rise of temperature due to the motion of a paddle wheel rotating in a vessel filled with a fluid. The motion of the paddle wheel was driven by the descent of weights connected via a system of pulleys to the paddle wheel axis; see Fig. 1. The potential energy of the weights is in this experiment transformed to the kinetic energy of the paddle wheel and due to the resistance of the fluid in the vessel also to the kinetic and thermal energy of the fluid. Classical thermodynamics is restricted to the description of the initial and final state where the weights are at rest. At the beginning, the temperature is constant all over the vessel. This is the initial state. As the weights descend, the fluid moves, and its temperature rises. Once the weights are stopped, the temperature reaches, after some time, homogeneous distribution in the vessel. This is the final state. In the classical setting, it is impossible to say anything about the intermediate states since the descent of the weight induces substantial motion in the fluid and inhomogeneous distribution of the temperature in the vessel. (Recall that the energy, entropy, and temperature are in the classical setting defined only for the whole vessel.) On the other hand, the ambition of continuum thermodynamics is to describe the whole process and the time evolution of the spatial distribution of the physical quantities of interest. In order to describe the spatial distribution of the physical quantities of interest, one obviously needs to talk about specific internal energy e.x; t / and the specific entropy .x; t /. The internal energy or the entropy of the volume V .t / of the material is then obtained by volume integration of the corresponding densities. Besides the introduction of the specific internal energy e.x; t / and the specific entropy .x; t /, it is necessary to identify the energy exchange mechanisms in the continuous medium. Naturally, one part of the energy exchange is due to processes of mechanical origin.

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

11

In order to identify the energy exchange due to mechanical processes, it suffices to recall the same concept in classical Newtonian physics. The energy balance for a single particle of constant mass m is in the classical setting obtained via the multiplication of the Newton’s second law dtd .mv/ D F by the particle velocity v,   which yields dtd 12 mjvj2 D F  v. Repeating the same steps in the setting of continuum mechanics – see the pointwise equality (13) – yields Z .x; t / V .t/

dv.x; t /  v.x; t / dv D dt

Z Œdiv T.x; t / C .x; t /b.x; t /  v.x; t / dv; V .t/

(15) where v.x; t / is the Eulerian velocity field. Further, one needs to introduce a quantity describing the nonmechanical part of the energy exchange between the given volume of the material and its surroundings. It is assumed that this part of the energy exchange can be described using the energy flux j e and that the energy exchange through the volume boundary is then given by the means of a surface integral Z  @V .t/

j e .x; t /  n.x; t / ds;

(16)

where n denotes the unit outward normal to the surface of the volume. (The minus sign is due to the standard sign convention. If the energy flux vector j e points into the volume V .t / – that is, in the opposite direction to the unit outward normal – then the energy is transferred from the surrounding into the volume, and the surface integral is positive.) The energy flux j e is, in the simplest setting, tantamount to the heat flux j q .x; t /. Consequently, for the sake of clarity and specificity of the presentation, it is henceforward assumed that j e  j q . Using the heat flux j q , the nonmechanical energy exchange – the transferred heat – between the volume V .t / and its surrounding is given by the surface integral Z  @V .t/

j q .x; t /  n.x; t / ds:

(17)

Alternatively, one can also introduce volumetric heat source contribution to the energy exchange, Z .x; t /q.x; t / dv;

(18)

V .t/

but the volumetric contribution shall not be considered here for the sake of simplicity of presentation. (Similarly, volumetric contribution is not considered in the discussion on the concept of entropy and entropy production.)

12

J. Málek and V. Pr˚uša

Now one is ready to formulate the total energy balance for the volume V .t /. The total energy of the continuous medium in the volume V .t / is assumed to be given R by the volume integral V .t/ .x; t /etot .x; t / dV of the specific total energy etot . Further, the specific total energy is assumed to be given as the sum of the specific kinetic energy of the macroscopic motion 12 jvj2 and the specific internal energy e, 1 .x; t /etot .x; t / Ddef .x; t /e.x; t / C .x; t / jv.x; t /j2 : 2

(19)

(In the simplest setting, the internal energy represents the thermal energy, i.e., the energy of the microscopic motion; see Clausius [17].) The integral form of the balance law then reads Z Z d .x; t /etot .x; t / dv D .x; t /b.x; t /  v.x; t / dv dt V .t/ V .t/ Z ŒT.x; t /n.x; t /  v.x; t / ds C Z

@V .t/

 @V .t/

j q .x; t /  n.x; t / ds;

(20)

where the right-hand side contains all possible contributions to the energy exchange, namely, the mechanical ones (see the right-hand side of (15)) and the nonmechanical ones (see (17)). If the physical quantities of interest are sufficiently smooth, then (20) can be in virtue of the Reynolds theorem, the Stokes theorem, and the symmetry of the Cauchy stress tensor (14) reduced to the pointwise equation .x; t /

detot .x; t / D .x; t /b.x; t /  v.x; t / C div ŒT.x; t /v.x; t /  div j q .x; t /: dt (21)

Equation (21) can be further manipulated in order to get an evolution equation for the specific internal energy e only. Multiplying the balance of momentum (13) by v and subtracting the arising equation from the balance of total energy (21) yield, in virtue of the identity div.A> a/ D .div A/  a C A W ra, the equation .x; t /

de.x; t / D T.x; t / W D.x; t /  div j q .x; t /; dt

(22)

where the symmetry of the Cauchy stress tensor has been used.  The symbol  D stands for the symmetric part of the velocity gradient, D Ddef 21 rv C rv> , and   A W B Ddef Tr AB> . The term T.x; t / W D.x; t / in (22) is called the stress power and plays a fundamental role in thermodynamics of continuous medium. A few hints on its relevance are given in Sect. 3. Later, in the discussion concerning constitutive

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

13

relations for viscous fluids, it is shown that the stress power is one of the terms in the entropy production. Note that although equations (22) and (21) are, in virtue of the balance of momentum (13), equivalent for sufficiently smooth functions, they are different from the perspective of a weak solution; see Feireisl and Málek [28], Bulíˇcek et al. [8], and Bulíˇcek et al. [9] for details.

2.3

Entropy

Concerning the concept of entropy in the setting of continuum mechanics, two assumptions must be made. First, it is assumed, following the classical setting, that an energetic equation of state holds even for the specific internal energy and the specific entropy. Note that the particular form of the energetic equation of state depends on the given material; hence, the energetic equation of state is in fact a constitutive relation. In the simplest setting, the energetic equation of state reads

e.x; t / D e..x; t/; .x; t//;

(23)

which is clearly a straightforward generalization of the classical relation E D E.S; V /; see, for example, Callen [15]. A general energetic equation of state can however take a more complex form

e.x; t / D e..x; t /; y1 .x; t /; : : : ; ym .x; t //;

(24)

where y1 ; : : : ; ym , m 2 N, m  1 are other state variables. If the energetic equation of state is obtained as a direct analogue of a classical equilibrium energetic equation of state, then it is said that the system under consideration satisfies the assumption of local equilibrium and that the system is studied in the framework of classical irreversible thermodynamics. However, if one wants to describe the phenomena that go beyond the classical setting (see, e.g., Sect. 4), then the energetic equation of state must be more complex. In particular, the energetic equation of state can contain the spatial gradients as variables. In such a case, it is said that the system is studied in the framework of extended irreversible thermodynamics. Second, the energetic equation of state is assumed to hold at every time instant in the given class of processes of interest. This is again a departure from the classical setting, where the relations of type E D E.S; V / hold only in equilibrium or in quasistatic (infinitesimally slow) processes. In particular, in the current setting, it is

14

J. Málek and V. Pr˚uša

assumed that one can take the time derivative of the equation of state. For example, if the equation of state is (23), then ˇ ˇ @e.; / ˇˇ @e.; / ˇˇ de.x; t / d.x; t / d.x; t / D C dt @ ˇD.x;t/; D.x;t/ dt @ ˇD.x;t/; D.x;t/ dt (25) is assumed to be valid in the given class of processes of interest, no matter how rapid the time changes are or how large the spatial inhomogenities are.

2.3.1 Thermodynamic Temperature Besides the energetic equation of the state, one can, following classical equilibrium thermodynamics, specify the entropy as a function of the internal energy e and the other state variables y1 ; : : : ; ym , m 2 N, m  1, which leads to the entropic equation of state .x; t / D .e.x; t /; y1 .x; t /; : : : ; ym .x; t //:

(26)

The specific entropy  is assumed to be a differentiable function, and further it is assumed that ˇ ˇ @ .e; y1 ; : : : ; ym /ˇˇ >0 (27) @e eDe.x;t/;y1 Dy1 .x;t/;:::;ym Dym .x;t/ holds for any fixed m-tuple y1 ; : : : ; ym and point .x; t /. This allows one to invert (26) and get the energetic equation of state e.x; t / D e..x; t /; y1 .x; t /; : : : ; ym .x; t //:

(28)

Further, following classical equilibrium thermodynamics, the thermodynamic temperature  is defined as .x; t / Ddef

ˇ ˇ @e .; y1 ; : : : ; ym /ˇˇ : @ D.x;t/;y1 Dy1 .x;t/;:::;ym Dym .x;t/

(29)

2.3.2 Clausius-Duhem Inequality The last ingredient one needs in the development of thermodynamics of continuous medium is a counterpart of the classical Clausius inequality dS 

dQ ¯ ; 

(30)

where  is the thermodynamic temperature; see Clausius [18]. Recall that in the ideal situation of reversible processes, the inequality reduces to the equality

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

dS D

dQ ¯ : 

15

(31)

The generalization of the Clausius inequality is the Clausius-Duhem inequality d dt

Z

Z

j q .x; t /

.x; t /.x; t / dv   V .t/

@V .t/

.x; t /

Z  n ds C

.x; t / V .t/

q.x; t / dv; .x; t / (32)

which can be obtained from (30) by appealing to the same arguments as in the discussion of the concept of the energy. (Recall that the transferred heat is expressed in terms of the surface integral (17).) The last term in (32) accounts for entropy changes due to volumetric heat sources. As before (see the discussion in Sect. 2.2), the volumetric term shall not be henceforward considered for the sake of simplicity of presentation. In what follows, the Clausius-Duhem inequality is therefore considered in the form d dt

Z

Z

j q .x; t /

.x; t /.x; t / dv   V .t/

@V .t/

.x; t /

 n ds;

(33)

where V .t / is again an arbitrary volume in the sense that it is the volume obtained by tracking an arbitrarily chosen initial volume V .t0 /. Since the classical Clausius (in)equality (30) is the mathematical formalization of the second law of thermodynamics, the Clausius-Duhem inequality (33) can be understood as the reinterpretation of the second law of thermodynamics in the setting of continuum thermodynamics. As such the Clausius-Duhem inequality can be expected to play a crucial role in the theory which is indeed the case. It is convenient, for the sake of later reference, to rewrite the inequality in a slightly different form. Using the Stokes theorem, the Reynolds transport theorem, and the identity (5), the inequality (33) can be rewritten in the form   j q .x; t / d dv  0: .x; t / .x; t / C div dt .x; t /



Z V .t/

(34)

The first term in (34) is the change of the net entropy,  Z dSV .t/ d .x; t /.x; t / dv ; Ddef dt dt V .t/

(35)

while the other term is the entropy exchange – the entropy flux – with the surrounding of volume V .t/, Z J@V .t/ Ddef @V .t/

j q .x; t / .x; t /

 n ds:

(36)

16

J. Málek and V. Pr˚uša

The general statement of a balance law in continuum mechanics and thermodynamics is that the time change of the given quantity in the volume V .t / plus the flux J@V .t/ of the quantity through the boundary of V .t/ is equal to the production of the given quantity in the volume V .t/, dSV .t/ C J@V .t/ D „V .t/ : dt

(37)

Using this nomenclature, it follows that the left-hand side of (34) deserves to be denoted as the net entropy production in the volume V .t /. Further, it is assumed that the net entropy production can be obtained via a volume integral of the corresponding spatially distributed quantity .x; t /, Z „V .t/ Ddef

.x; t / dv:

(38)

V .t/

In such a case, the Clausius-Duhem inequality can be rewritten as „V .t/ D

dSV .t/ C J@V .t/  0: dt

(39)

Now it is clear that the Clausius-Duhem inequality in fact states that the net entropy production in the volume V .t/ is nonnegative, „V .t/  0:

(40)

Note that if the heat flux j q vanishes on the boundary of V .t /, then the flux term in (39) vanishes, and it follows that the net entropy of the volume V .t / increases in time. This is the classical statement concerning the behavior of isolated systems. Finally, using the concept of entropy production density (38), it follows that the localized version of (34) reads .x; t / D .x; t /

  j q .x; t / d  0: .x; t / C div dt .x; t /

(41)

The fact that (33) is indeed a generalization of (30) is best seen in the case of a vessel filled with a fluid with “almost” constant uniform temperature distribution .x; t / Ddef  that is subject to heat exchange with its surrounding. In such a case, time integration of (33) from t1 to t2 yields Z

Z

1 .x; t2 /.x; t2 / dv .x; t1 /.x; t1 / dv  V .t2 / V .t1 /

Z t2Z t1

@V .t/

 j q .x; t /  nds dt: (42)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

17

The left-hand side is the difference between the net entropy of the fluid in the vessel at times t2 and t1 , and the right-hand side is the heat exchanged with the surrounding in the time interval Œt1 ; t2 ; hence, one gets S .t2 /  S .t1 / 

Q ; 

(43)

which is tantamount to (30).

2.3.3 Entropy Production In the setting of continuum mechanics, it is assumed that the energetic equation of state is known, which means that a relation of the type (25) holds for the time derivative of the entropy. The knowledge of the energetic equation of state for the given material allows one to identify the entropy production mechanisms in the material, that is, the quantity .x; t / in (41). In other words, one can rewrite the lefthand side of (41) in terms of more convenient quantities than the entropy. Indeed, inspecting carefully (25) it is easy to see that (25) is in fact a formula for the material time derivative of the specific entropy d.x;t/ , dt  d.x; t / 1 de.x; t / pth .x; t / d.x; t / D  ; dt .x; t / dt Œ.x; t /2 dt

(44)

where the thermodynamic temperature .x; t / and the thermodynamic pressure pth .x; t / have been defined by formulae .x; t / Ddef pth .x; t / Ddef

ˇ @e.; / ˇˇ ; @ ˇD.x;t/;D.x;t/ ˇ ˇ 2 @e.; / ˇ Œ.x; t / : @ ˇD.x;t/;D.x;t/

(45a) (45b)

(These relations are counterparts of the well-known classical relations; see, e.g., Callen [15].) The time derivatives de.x;t/ and d.x;t/ are known from the balance laws (22) dt dt and (4), and using the balance laws in (44) yields  d.x; t / 1

D T.x; t / W D.x; t /  div j q .x; t / C pth .x; t / div v.x; t / : dt .x; t / (46) Finally, using (46) in (41) leads one, after some manipulation, to .x; t /

1 1 ŒT.x; t / W D.x; t / C pth .x; t / div v.x; t /  j q .x; t /  r.x; t / .x; t / Œ.x; t /2   j q .x; t / d  0: (47) D .x; t / .x; t / C div dt .x; t /

18

J. Málek and V. Pr˚uša

The expression on the left-hand side of (47), that is, .x; t / Ddef 

1 ŒT.x; t / W D.x; t / C pth .x; t / div v.x; t / .x; t / 1 Œ.x; t /2

j q .x; t /  r.x; t /

(48)

determines the entropy production in the given material; see the local version of the Clausius-Duhem inequality (41). In other words, in the continuum setting, one is able to explicitly evaluate, for the given material, the “uncompensated transformation” N Ddef dS 

dQ ¯ 

(49)

introduced by Clausius [18]. The Clausius-Duhem inequality (41) requires .x; t / to be nonnegative; hence (47) can be expected to impose some restrictions on the form of the constitutive relations. This is indeed the case, and this observation is the key concept exploited in the theory of constitutive relations discussed in Sect. 4. Furthermore, relations of the type (47) provide one an explicit criterion for the validity of the Clausius equality (31). In particular, a process in a material with energetic equation of state of the type (23) is reversible, that is, (31) holds, if the entropy production (48) vanishes. From (48) it follows that the entropy production vanishes if there are no temperature and velocity gradients in the material. This means that the process in the material is very close to an ideal reversible process if the gradients r and D are during the whole process kept extremely small. The other possibility is that the entropy production identically vanishes because of the particular choice of the constitutive relation for T and j q ; see Sect. 4.2.2 for details.

3

Stress Power and Its Importance

The second law of thermodynamics requires the entropy production to be a nonnegative quantity. However, such statement concerning the entropy production does not seem to be very intuitive. In what follows it is shown that the nonnegativity of the entropy production is in fact closely related to some fundamental direct observations concerning the qualitative behavior of real materials. Let us, for example, consider the entropy production for an incompressible homogeneous viscous heat nonconducting fluid. The incompressibility means that div v D 0 is required to hold in the class of considered processes, while the fact that a heat nonconducting fluid is considered translates into the requirement j q D 0. In such a case, the entropy production (48) reduces to

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

.x; t / D

1 T.x; t / W D.x; t /: .x; t /

19

(50)

(Details concerning the derivation of the entropy production in an incompressible fluid can be found in Sect. 4. Note that for an incompressible fluid, one gets T W D D Tı W Dı ; see the discussion in Sect. 4.2.3, in particular formula (119).) Since the thermodynamic temperature in (50) is a positive quantity, it follows that the requirement on the nonnegativity of the entropy production simplifies to T W D  0:

(51)

This means that the second law is satisfied provided that the stress power is a pointwise nonnegative quantity. The importance of the requirement concerning the nonnegativity of the stress power term T W D – and consequently the nonnegativity of the entropy production – is documented below by means of two simple mechanical examples. These two examples show that the seemingly esoteric requirement on the nonnegativity of entropy production leads to very natural consequences. First, it is shown that the pointwise positivity of the stress power is sufficient for establishing the fact that the drag force acting on a rigid body moving through an incompressible fluid decelerates the moving body; see Sect. 3.1. Second, it is shown that the pointwise positivity of the stress power is sufficient to establish the stability of the rest state of an incompressible fluid; see Sect. 3.2.

3.1

Drag

Let us first consider the setting shown in Fig. 2. A rigid body B moves with constant velocity U in an infinite domain, and it is assumed that no body force is acting on the surrounding fluid. Further, it is assumed that the velocity field in the fluid that is generated by the motion of the body decays sufficiently fast at infinity and that the fluid adheres to the surface of the body, that is, vj@B D U . The drag force is the projection of the force acting on the body to the direction parallel to the velocity U , Fig. 2 Body moving in a fluid

Flift

U B

Fdrag

20

J. Málek and V. Pr˚uša

U ˝U

F drag Ddef

Z

 Tn ds :

jU j2

(52)

@B

The key observation is that if the fluid is assumed to be a homogeneous incompressible fluid with symmetric Cauchy stress tensor, then the formula for the drag force (52) can be rewritten in the form Z F drag D 

 U T W D dv : jU j2 exterior of B

(53)

The derivation of (53) proceeds as follows. The balance of momentum (13) and the balance of mass (5) for the motion of the fluid outside the body B are 

dv D div T; dt

(54a)

div v D 0;

(54b)

and the boundary conditions read vj@B D U ;

(54c)

v.x; t / ! 0 as jxj ! C1:

(54d)

Note that the velocity field is considered to be a steady velocity field; hence, the D Œrv v. material time derivative is dv dt ˚ Let D x 2 R3 ; jxj < R n B denote the ball centered at origin with radius R (large number) with excluded body B and its boundary @B. Governing equations (54) hold in this domain. Now one can multiply (54a) by v and integrate over , Z

Z .div T/  v dv:

.Œrv v/  v dv D

(55)



Application of standard identities yields Z

  1 v  rjvj2 dv D 2

Z

  div T> v dv 

Z T W D dv:

(56)



The first term on the right-hand side can be in virtue of the Stokes theorem rewritten as a sum of integrals over the boundary of the ball and the boundary of the body

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

Z

  div T> v dv D

Z

 >  T v  n ds 

fx2R3 ; jxjDRg Z



D

Z

21



 T> v  n ds

@B

 >  T v  n ds  U 

Z

fx2R3 ; jxjDRg

Tn ds;

(57)

@B

where the boundary condition (54c) has been used. (Note that the last integral is taken over the surface of the body; hence, one needs to add the minus sign in order to compensate the opposite orientation of the surface of the body and the surface of .) Concerning the left-hand side of (56), integration by parts implies Z

  1 v  rjvj2 dv D 2

Z @

Z

1 jvj2 v  n ds  2

D f

x2R3 ;

Z

1 jvj2 div v dv 2

1 jvj2 v  n ds; 2 jxjDRg

(58)

where the boundary condition (54c) and the incompressibility, div v D 0, have been used. Summing up all partial results, it follows that (56) reduces to Z

Z  >  1 T v  n ds jvj2 v  n ds D fx2R3 ; jxjDRg 2 fx2R3 ; jxjDRg Z Z U  Tn ds  T W D dv: @B

(59)



Now one can take the limit R ! C1 and use the fact that the velocity vanishes sufficiently fast for R ! C1. This means that the surface integrals in (59) vanish for R ! C1 and that (59) reduces to Z

Z Tn ds D 

U @B

T W D dv;

(60)



which yields the proposition. The implication of formula (53) is, that if the stress power T W D is pointwise nonnegative, then the drag direction is opposite to the direction of the motion. (The drag force is acting against the motion of the body.) This is definitely a desirable outcome from the point of everyday experience, and, as it has been shown, this outcome is closely related to the second law of thermodynamics.

22

3.2

J. Málek and V. Pr˚uša

Stability of the Rest State

Let denote a vessel occupied at time t0 by a homogeneous incompressible viscous fluid, and let no external body force act on the fluid. The Cauchy stress tensor is again assumed to be symmetric. Further, it is assumed that the fluid adheres to the surface of the vessel, vj@ D 0:

(61)

The everyday experience is that whatever has been the initial state of the fluid, the fluid comes, after some time, to the state of rest, that is, v D 0 in . The question is whether it is possible to recover this fact for a general homogeneous incompressible viscous fluid. A good measure of the deviation from the rest state is the net kinetic energy of the fluid Z 1 Ekin Ddef .x; t /jv.x; t /j2 dv: (62) 2 In order to assess the stability of the rest state, one needs to find an evolution equation for a measure of the deviation from the rest state. This is an easy task if the measure of the deviation from the rest state is the net kinetic energy introduced in (62). The evolution equations for the motion of the fluid are 

dv D div T; dt

(63a)

div v D 0;

(63b)

and the multiplication of (63a) with v followed by integration over the domain yields Z

d dt



1  vv 2



Z .div T/  v dv:

dv D

(64)



(See identity (5).) The left-hand side is equal to the time derivative of the net kinetic energy (62). Concerning the right-hand side, one can use the identity div.A> a/ D .div A/  aCA W ra, the Stokes theorem, and the fact that v vanishes on the boundary to get Z

Z .div T/  v dv D 

T W D dv:

This implies that the evolution equation for the net kinetic energy reads

(65)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

dEkin D dt

23

Z T W D dv:

(66)



It follows that if the stress power T W D is pointwise positive, then the righthand side of (66) is negative, and the net kinetic energy decays in time. If one was interested in the rate of decay, and in the proof that the net kinetic energy decays to zero, then one would need further information on the relation between the Cauchy stress tensor and the symmetric part of the velocity gradient. (The details are not elaborated here; interested reader is referred to Serrin [86] for the detailed treatment of the Navier-Stokes fluid, T D pI C 2 D. Note that the net kinetic energy does not completely vanish in finite time.) The decay of the kinetic energy of the fluid contained in a closed vessel is definitely a desirable outcome from the point of view of everyday experience, and, as it has been shown, the decay is closely related to the second law of thermodynamics.

4

Constitutive Relations

The motion of a single continuous medium is governed by the following system of partial differential equations d.x; t / C .x; t / div v.x; t / D 0; dt dv.x; t / .x; t / D div T.x; t / C .x; t /b.x; t /; dt T.x; t / D T> .x; t /; .x; t /

de.x; t / D T.x; t / W D.x; t /  div j q .x; t / dt

(67a) (67b) (67c) (67d)

that are mathematical expressions of the balance law for the mass, linear momentum, angular momentum, and total energy. (See equations (4), (13), (14), and (22).) The unknown quantities are the density , the velocity field v, and the specific internal energy e. (For the first reading, the internal energy can be thought of as a proxy for the temperature field .x; t /.) These equations are insufficient for the description of the evolution of the quantities of interest. The reason is that the Cauchy stress tensor T and the heat flux j q act in (67) as additional unknowns a priori unrelated to the density , the velocity field v, and the specific internal energy e. Therefore, in order to get a closed system of governing equations, system (67) must be supplemented by a set of equations relating the stress and the heat flux to the other quantities. These relations are called the constitutive relations, and they describe the response of the material to the considered stimuli.

24

J. Málek and V. Pr˚uša

It is worth emphasizing that the constitutive relations are specific for the given material. For example, the Cauchy stress tensor is a quantity that describes how the given volume of the material is affected by its surrounding; see Sect. 2.1.2. Clearly, the interaction between the given volume of the material and its surrounding depends on the particular type of the material. Similarly, the heat flux describes the thermal energy transfer capabilities of the given material; hence, it is material specific. The fact that the constitutive relations are material specific means that they indeed provide an extra piece of information supplementing the balance laws. Consequently, the constitutive relations can be derived only if one appeals to physical concepts that go beyond the balance laws. Moreover, the constitutive relations are inevitably simplified and reduced descriptions of the physical reality, and as such they are designed to describe only the behavior of the given material in a certain class of processes. For example, the same material, say steel, can be processed by hot forging, and it can deform as a part of a truss bridge or move as a projectile. The mathematical models for the response of the material in these processes are however different despite the fact that the material remains the same. The specification of the constitutive relations therefore depends on two things – the material and the considered processes. The most convenient form of the constitutive relations would be a set of equations of the form T D T.; v; e/;

(68a)

j q D j q .; v; e/:

(68b)

This would allow one to substitute for T and j q into (67) and get a system of evolution equations for the unknown fields , v, and e. An example of a system of constitutive relations of the type (68) are the well-known constitutive relations for the compressible viscous heat-conducting Navier-Stokes fluid, T D pth .;  /I C  .div v/ I C 2 D; j q D  r ;

(69a) (69b)

where pth .;  / denotes the thermodynamic pressure determined by an equation of state. (See Sect. 4.2 for details.) Substituting (69) into (67) then leads to the wellknown Navier-Stokes-Fourier system of partial differential equations. Besides the constitutive relations, the system of balance laws must be supplemented with initial and boundary conditions. The boundary conditions can be seen as a special case of constitutive relations at the interface between two materials. As such the boundary conditions can be very complex. In fact, the processes at the interface can have, for example, their own dynamics, meaning that they can be governed by an extra set of partial differential equations at the interface. Unfortunately, discussion of complex boundary conditions goes beyond the scope of the present contribution, and the issue is only briefly touched in Sect. 4.6. Consequently, overall the discussion of boundary conditions is mainly restricted

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

25

to the claim that the considered boundary conditions are the standard ones such as the no-slip boundary condition for the velocity field. The reader should be however aware of the fact that the specification of the boundary conditions is far from being trivial and that it again requires additional physical insight into the problem. The early efforts concerning the theory of constitutive relations for fluids (see, e.g., Rivlin and Ericksen [83], Oldroyd [65], and Noll [64] or the historical essay by Tanner and Walters [90]) were mainly focused on the specification of the Cauchy stress tensor in terms of the kinematical variables. These early approaches were based almost exclusively on mechanical considerations, such as the symmetry of the material and the requirement of the invariance of the constitutive relations with respect to the change of the observer. Thermodynamic considerations were rarely the leading theme and were mainly reduced to the a posteriori verification of the nonnegativity of the stress power. Since then thermodynamics started to play an increasingly important role in the theory of constitutive relations; see, for example, Coleman [19] and Truesdell and Noll [91]. This paradigm shift went hand in hand with the development of the field of nonequilibrium thermodynamics; see, for example, de Groot and Mazur [34], Glansdorff and Prigogine [32], Ziegler [97], and Müller [62]. In what follows a modern thermodynamics based approach to the phenomenological theory of constitutive relations is presented. The presented approach is essentially based on the work of Rajagopal and Srinivasa [78] (see also Rajagopal and Srinivasa [80]), who partially took inspiration from various earlier achievements in the field, most notably from the work of Ziegler [97] and Ziegler and Wehrli [98]. The reader who is interested into some comments concerning the precursors of the current approach is referred to Rajagopal and Srinivasa [78]. The advantage of the presented approach is that the second law of thermodynamics plays a key role in the theory of constitutive relations. Unlike in some other approaches, the second law is not used a posteriori in checking whether the derived constitutive relations conform to the second law. In fact, the opposite is true; the second law is the starting point of the presented approach. The constitutive relations for the Cauchy stress tensor, the heat flux, and the other quantities of interest are in the presented approach derived as consequences of the choice of the specific form for the internal energy e and the entropy production . If the entropy production  is chosen to be nonnegative, then the second law is satisfied. Since the relations of the type (68) are in the presented approach the consequences of the choice of e and , then it is clear that the arising constitutive equations cannot violate the second law. The second law is automatically built in the arising constitutive relations of the type (68). Further, the specification of two scalar quantities e and  is apparently much easier than the direct specification of the constitutive relations between the vectorial and tensorial quantities as in (68). This brings simplicity into the theory of constitutive relations. Two scalar quantities determine everything. Finally, the advocated approach is robust enough to handle complex materials. Nonlinear constitutive relations and constitutive relations for constrained materials as well as constitutive relations for materials reacting to the stimuli of various

26

J. Málek and V. Pr˚uša

origins (thermal, mechanical, electromagnetic, chemical) are relatively easy to work with in the presented approach. A general outline of the advocated approach is given in Sect. 4.1, and then it is shown how the general approach can be used in various settings. First, the approach is applied in a very simple setting, namely, the well-known constitutive relations for the standard compressible Navier-Stokes-Fourier fluid are derived in Sect. 4.2. This example should allow the reader to get familiar with the basic concepts. Then the discussion proceeds to more complex settings, namely, that of Korteweg fluid; see Sect. 4.3. Finally (see Sect. 4.4), the discussion of the approach is concluded by the derivation of constitutive equations for viscoelastic fluids.

4.1

General Framework

The main idea behind the presented approach is that the behavior of the material in the processes of interest is determined by two factors, namely, its ability to store energy and produce entropy. The energy storage mechanisms are specified by the choice of the energetic equation of state, that is, by expressing the internal energy e as a function of the state variables. (Other thermodynamic potentials such as the Helmholtz free energy, Gibbs potential, or the enthalpy can be used as well. The discussion below is however focused exclusively on the internal energy.) The entropy production mechanisms are specified by the choice of the formula for the entropy production . The derivation of the constitutive relations (68) from the knowledge of e and  then proceeds in the following steps: STEP 1: Specify the energy storage mechanisms by fixing the constitutive relation for the specific internal energy e in the form of the energetic equation of state e D e.; y1 ; : : : ; ym /

(70)

or in the form of the entropic equation of state (26). At this level it is sufficient to determine the state variables yi that enter (70). STEP 2: Find an expression for the material time derivative of the specific entropy . This can be achieved by the application of the material time derivative to (70), and by the multiplication of the result by .x; t /, which yields de.x; t / D dt ˇ d.x; t / @e.; y1 ; : : : ; ym / ˇˇ .x; t / ˇ @ dt D.x;t/;y1 Dy1 .x;t/;:::;ym Dym .x;t/

.x; t /

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

C

m X

.x; t /

iD1

27

ˇ dyi .x; t / @e.; y1 ; : : : ; ym / ˇˇ : ˇ @yi dt D.x;t/;y1 Dy1 .x;t/;:::;ym Dym .x;t/ (71)

The definition of the thermodynamic temperature (see (29)) and (71) then yield

d.x; t / de.x; t / D .x; t / dt dt ˇ m X @e.; y1 ; : : : ; ym / ˇˇ dyi .x; t / .x; t / ;  ˇ @yi dt yj Dyj .x;t/; D.x;t/ iD1

.x; t /.x; t /

(72)

. which is the sought formula for d dt STEP 3: Identify the entropy production. Use the balance equations and kinematics to write down (derive) the formulae for the material time derivative of the state variables dyidt.x;t/ , i D 1; : : : ; m, and substitute these formulae into (72). Rewrite in the form the equation for .x; t / d.x;t/ dt

.x; t /

m 1 X d.x; t / j .x; t /  a˛ .x; t /; C div j .x; t / D dt .x; t / ˛D1 ˛

(73)

where j ˛ .x; t /  a˛ .x; t / denotes the scalar product of vector or tensor quantities, respectively. The right-hand side of (73) is the entropy production, where each summand is supposed to represent an independent entropy-producing mechanism. The quantities j ˛ .x; t / are called the thermodynamic fluxes, and the quantities a˛ .x; t / are called the thermodynamic affinities. The affinities are usually the spatial gradients of the involved quantities, for example, r or D, while the fluxes are, for example, the heat flux j q or the Cauchy stress tensor T. (As a rule of thumb, the fluxes are the quantities that appear under the divergence operator in the balance laws (67); see also Rajagopal and Srinivasa [78] for further comments.) The quantity j .x; t / is called the entropy flux, and in standard j .x;t/

q . cases, it is tantamount to .x;t/ STEP 4* (Linear nonequilibrium thermodynamics): The second law of thermodynamics states that the entropy production .x; t / Ddef .x; t / d.x;t/ C dt div j .x; t / is nonnegative; see Sect. 2.3.3. Referring to (73), it follows that

.x; t / D

m 1 X j .x; t /  a˛ .x; t / .x; t / ˛D1 ˛

(74)

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J. Málek and V. Pr˚uša

must be nonnegative. A simple way to fulfill this requirement is to consider linear relations between each pair of j ˛ .x; t / and a˛ .x; t /, that is, j ˛ .x; t / D ˛ a˛ .x; t /;

(75)

where ˛ , ˛ D 1; : : : ; m, are nonnegative constants. Alternatively, one can consider cross effects by assuming that the relations between j ˛ .x; t / and a˛ .x; t / take the form j ˛ .x; t / D

m X

˛ˇ aˇ .x; t /;

(76)

ˇD1

where ˛ˇ is a symmetric positive definite matrix. This is essentially the approach of linear nonequilibrium thermodynamics; see de Groot and Mazur [34] for details. STEP 4 (Nonlinear nonequilibrium thermodynamics): Since the linear relationships between the fluxes j ˛ .x; t / and affinities a˛ .x; t / can be insufficient for a proper description of the behavior of complex materials, an alternative procedure is needed. In particular, the procedure should allow one to derive nonlinear constitutive relations of the type j i D j i .a1 ; : : : ; am / or vice versa. Here it comes to the core of the approach suggested by Rajagopal and Srinivasa [78]. As argued by Rajagopal and Srinivasa [78], one first specifies function  in one of the following forms  D a1 ;:::;am .j 1 ; : : : ; j m /;

(77a)

 D j1 ;:::;jm .a1 ; : : : ; am /:

(77b)

or

Note that the state variables can enter the constitutive relations (77) as well, but they are not, for the sake of compactness of the notation, written explicitly in (77). Formula (77a) or (77b) is a constitutive relation that determines the entropy production ,  Ddef  . Since the constitutive function  is – up to the positive factor  – tantamount to the entropy production, it must be nonnegative, which guarantees the fulfillment of the second law of thermodynamics. Further,  should vanish if the fluxes vanish. Other restrictions concerning the formula for the entropy production can come from classical requirements such as the material symmetry and the invariance with respect to the change of the observer. Moreover, the assumed form of the entropy production must be compatible with the already derived form of the entropy production (74). Consequently, the following equation must hold

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

j1 ;:::;jm .a1 ; : : : ; am / 

m 1 X j .x; t /  a˛ .x; t / D 0; .x; t / ˛D1 ˛

29

(78)

and similarly for a1 ;:::;am .j 1 ; : : : ; j m /. In terms of the constitutive function , this reduces to the requirement a1 ;:::;am .j 1 ; : : : ; j m / 

m X

j ˛ .x; t /  a˛ .x; t / D 0;

(79a)

j ˛ .x; t /  a˛ .x; t / D 0;

(79b)

˛D1

or to j1 ;:::;jm .a1 ; : : : ; am / 

m X ˛D1

respectively, depending whether one starts with (77a) or (77b). Relations (79) can be rewritten in the form A .J /  J  A D 0;

(80a)

J .A/  J  A D 0;

(80b)

> >

0 and J D j 1 : : : j m are vectors in RM , where A Ddef a1 : : : am M 0 > m that contain all the affinities and fluxes, respectively. Here the tensorial quantities such as the Cauchy stress tensor T are understood as column vectors >

T11 T12 : : : T33 . Having fixed a formula for the constitutive function  via (77a) or (77b), respectively, the task is to determine the fluxes j 1 ; : : : ; j m or affinities a1 ; : : : ; am that are compatible with the single constraint (80a) or (80b), respectively. If the fluxes/affinities are vectorial or tensorial quantities, then the single constraint is not sufficient to fully determine the sought relation between the fluxes and the affinities. (This means that in general there exist many constitutive relations of the type j i D j i .a1 ; : : : ; am / or vice versa such that (80a) or (80b) holds.) Rajagopal and Srinivasa [78] argued that the choice between the multiple constitutive relations that fulfill the constraint (80a) or (80b) can be based on the assumption of maximization of the entropy production. The assumption simply requires that the sought constitutive relation is the constitutive relation that leads to the maximal entropy production in the material and that is compatible with other available information concerning the behavior of the material. In more operational terms, the maximization of the entropy production leads to the following problem. Given the function A .J / and the values of the state variables and the values of the affinities A, the corresponding values of the fluxes J are those which maximize A .J / subject to the constraint (80a),

30

J. Málek and V. Pr˚uša

and alternatively to other possible constraints. (Note that since  is a positive quantity, then the task to maximize  with respect to the fluxes is indeed tantamount to the task of maximizing the entropy production  with respect to the fluxes.) A similar reformulation can be made for J .A/; it suffices to switch the role of the fluxes and affinities. The constrained maximization problems arising from the assumption on the maximization of the entropy production read max A .J /

(81)

J 2J

o n 0 where J Ddef J 2 RM ; A .J /  J  A D 0 , and max J .A/

(82)

A2A

o n 0 where A Ddef A 2 RM ; j .A/  J  A D 0 . Assuming that A or J are smooth and strictly convex with respect to their variables, then the corresponding values of J and A, respectively, are uniquely determined and can be found easily by employing the Lagrange multipliers; see Rajagopal and Srinivasa [78]. For example, if the starting point is the constitutive function A .J /, then the auxiliary function for the constrained maximization reads ˆ.J / Ddef A .J / C ` ŒA .J /  J  A, where ` is a Lagrange multiplier. The condition for the value / of the flux J that corresponds to maximal entropy production is @.J D 0 @J which leads to equation AD

1 C ` @A .J / ; ` @J

(83)

where 1C` D `

A .J / @A .J / J @J

:

(84)

Relation (83) is the sought relation between J and A. Consequently, if one accepts the assumption on the maximization of the entropy production, then the relations between the fluxes and affinities are due to the strict convexity of A .J / indeed uniquely determined by the choice of the constitutive function A .J /. Moreover, if the constitutive function A .J / is quadratic in J , then one recovers the result known from linear nonequilibrium thermodynamics, namely, that the fluxes are linear functions of the affinities; see (76). Apparently, a similar procedure can be followed if the starting point is J .A/ instead of A .J /. STEP 5: Relations (75) or (76) or (83) are the sought constitutive relations.

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

31

The rest of the present contribution is focused mainly on linear relations between the fluxes and affinities; hence, a quadratic ansatz for the constitutive function  is predominantly used in the rest of the text. This means that STEP 4 is rarely referred to in its full complexity and that one basically stays in the framework of STEP 4*. The reason is that the following text is mainly focused on the most difficult step of the advocated approach which is the correct specification of the functions e and . This is a nontrivial task in the development of the mathematical model, since the correct specification of these functions requires physical insight into the problem. However, the key physical arguments concerning the choice of the functions e and  can be easily documented in the simplest possible case of a quadratic ansatz; generalization of the quadratic ansatz to a more complex one is then a relatively straightforward procedure. Also note that once the functions e and  are specified, the derivation of the constitutive relations is only a technical problem not worth of lengthy discussion.

4.2

Compressible and Incompressible Viscous Heat-Conducting Fluids

The application of the outlined procedure is first documented in a very simple case of the derivation of constitutive relations for the compressible Navier-Stokes-Fourier fluid (see Sect. 4.2.1), which is a simple model for a compressible viscous heatconducting fluid. The next section (see Sect. 4.2.2) is devoted to reduced models that are variants of the compressible Navier-Stokes-Fourier fluid model. Finally, the incompressible counterparts of the former models are discussed in Sect. 4.2.3.

4.2.1 Compressible Navier-Stokes-Fourier Fluid STEP 1: The specific internal energy e is assumed to be a function of the specific entropy  and the density , e D e.; /:

(85)

This is a straightforward generalization of the classical energetic equation of state E D E.S; V / known for the equilibrium thermodynamics; see, for example, Callen [15]. (For the sake of clarity of the notation, the spatial and temporal variable is omitted in (85). If written in full, the energetic equation of state should read e.x; t / D e..x; t /; .x; t //. The same approach is applied in the rest of this section.) STEP 2: Taking the material derivative of (85) yields 

@e d de @e d D  : @ dt dt @ dt

(86)

32

J. Málek and V. Pr˚uša

Recalling the definition of the temperature (see (29)) and introducing the thermodynamic pressure pth through  Ddef

@e ; @

pth Ddef 2

@e ; @

(87a) (87b)

which are again the counterparts of the classical relations known from equilibrium thermodynamics, one can rewrite (86) as 

d de pth d D  : dt dt  dt

(88)

STEP 3: Using the evolution equation for the internal energy (67d) and the balance and d which yields of mass (67a), one can in (88) substitute for de dt dt 

d D T W D  div j q C pth div v: dt

(89)

The right-hand side can be further rewritten as T W D  div j q C pth div v D Tı W Dı C .m C pth / div v  div j q ;

(90)

where m Ddef

1 Tr T 3

(91)

denotes the mean normal stress, and 1 .Tr T/ I; 3 1 D  .Tr D/ I 3

Tı Ddef T 

(92)

Dı Ddef

(93)

denote the traceless part of T and D, respectively. Since the thermodynamic temperature is positive, one can, after some manipulation, rewrite (89) as    jq 1 r d D Tı W Dı C .m C pth / div v  j q  C div  dt    which is the evolution equation for  in the desired form (73).

(94)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

33

The rationale for using the traceless parts of T and D in (94) is the following. The quantities D and div v that appear in the first and the last term on the righthand side of (89) are not independent quantities. (Recall that Tr D D div v.) However, the fluxes one would like to identify when writing (89) in the form (73) should be independent quantities. Therefore, it is necessary to split D to mutually independent quantities Dı and div v, which requires one to split the Cauchy stress tensor in a similar manner. A closer inspection of (94) indicates that there exist three “independent” entropy-producing mechanisms in the material. The first one, j q  r , is  entropy production due to heat transfer; the second one, .m C pth / div v, is entropy production due to volume changes; and the last entropy production mechanism, Tı W Dı , is due to isochoric processes such as shearing. STEP 4: The entropy production  – that is, the right-hand side of (94) – takes the form  3 1X 1 r Tı W Dı C .m C pth / div v  j q  : D j  a˛ D  ˛D1 ˛  

(95)

Using the flux/affinity nomenclature, the fluxes are the traceless part of the Cauchy stress tensor Tı , the mean normal stress plus the thermodynamic pressure m C pth , and the heat flux j q . The affinities are Dı , div v, and r . . However, the positive (Strictly speaking the last affinity is not r but  r  1 factor  is of no importance.) The entropy production is a positive quantity if the constitutive relations are chosen as follows, Tı D 2 Dı ;

(96a)

m C pth D Q div v;

(96b)

j q D  r ;

(96c)

where > 0, Q > 0, and > 0 are given positive functions of the state variables  and . This choice of constitutive relations follows the template specified in (75). (Note that since Dı is a symmetric tensor, then Tı is also symmetric; hence, the balance of angular momentum (67c) is satisfied.) The Q and are called the shear viscosity, the bulk viscosity, and the coefficients , , heat conductivity. Traditionally, the bulk viscosity is written in the form 3 C 2 Q D ; 3 where  is a function of the state variables such that Q remains positive.

(97)

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J. Málek and V. Pr˚uša

STEP 5: It follows from (96) that the constitutive relation for the full Cauchy stress tensor T D Tı C mI is T D 2 Dı C Q .div v/ I  pth I

(98)

which can be in virtue of (97) rewritten as T D pth I C 2 D C .div v/I:

(99a)

This is the standard formula for the Cauchy stress tensor in the so-called compressible Navier-Stokes fluid. The constitutive relation for the heat flux reads j q D  r ;

(99b)

which is the standard Fourier law of thermal conduction. The coefficients in the constitutive relations can be functions of the state variables  and . This choice of variables is however inconvenient in practice. Solving the equation (87a) for the entropy, one can see that the entropy could be written as a function of the temperature and the density  D f .; /:

(100)

If this is possible, then the coefficients can be rewritten as functions of the temperature and the density, pth ? . ; / D pth .f .; /; / ;

(101a)

? . ; / D .f .; /; / ;

(101b)

?

 . ; / D  .f .; /; / ;

(101c)

? . ; / D .f .; /; / ;

(101d)

?

e . ; / D e .f .; /; / :

(101e)

Consequently, the final set of the governing differential equations arising from the balance laws (67) and the constitutive relations (99) reads

 

d D  div v; dt

(102a)

dv D div T C b; dt

(102b)

de ? . ; / D T W D  div j q ; dt

(102c)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

T D pth ? . ; /I C 2 ? . ; /D C ? . ; /.div v/I; j q D  ? . ; /r :

35

(102d) (102e)

These are the standard governing equations for the so-called compressible NavierStokes-Fourier fluids. If necessary, the evolution equation for the internal total energy (102c) can be reformulated as a balance of total energy, that is, 

d dt



1 e ? . ; / C jvj2 2

 D b  v C div .Tv/  div j q :

(103)

Note that if the internal energy as a function of the temperature and the density takes the simple form e ? D cV  , where cV is a constant referred to as the specific heat capacity, then the evolution equation for the internal energy (102c) takes the well-known form d D T W D  div j q : cV (104) dt

4.2.2 Other Linear Models for Compressible Fluids Referring to (95) one can notice that the entropy production  can be written in the form 

 r D J  A: (105)  D Tı ; m C pth ; j q  Dı ; div v;  If the constitutive relation for any of the quantities Tı , m C pth , j q is trivial, that is, if Tı D 0, m C pth D 0 or j q D 0, then the corresponding term in the entropy production vanishes. Such models are referred to as the reduced models. For example, Tı D 0 corresponds to an inviscid compressible heat-conducting fluid (Euler-Fourier fluid). On the other hand, if any of the quantities Dı , div v, r vanishes, then the corresponding entropy production also vanishes regardless of the particular constitutive equation for Tı , mCpth , or j q , respectively. The processes in which Dı , div v, or r vanishes are the motion with a velocity field in the form v.x; t / Ddef a.t /  x C b.t /, where a and b are arbitrary time-dependent vectors, the isochoric motion, that is, volume-preserving motion, and isothermal process, that is, a process with no temperature variations, respectively. Going back to the constitutive relations, the entropy production completely vanishes if the constitutive relations take the form Tı D 0;

m D pth ;

and

j q D 0:

(106)

This means that T D pth ? . ; /I;

and

j q D 0;

(107)

36

J. Málek and V. Pr˚uša

which are the constitutive equations for a compressible Euler fluid. The corresponding system of partial differential equations reads d D  div v; dt dv  D rpth ? . ; / C b; dt   1 d  e ? . ; / C jvj2 D  div .pth ? . ; /v/ C b  v: dt 2

(108a) (108b) (108c)

There is a hierarchy of models between the compressible Euler fluid (108) with no entropy production and the compressible Navier-Stokes-Fourier fluid (102) in which all entropy production mechanisms are active. These models are, for example, the Euler-Fourier fluid where the constitutive relations read Tı D 0;

m D pth ;

and

j q D  ? . ; /r ;

(109)

the fluid where thermodynamic pressure coincides (up to the sign) with the mean normal stress, that is, the fluid with constitutive relations m D pth ;

j q D  ? . ; /r.x; t /;

Tı D 2 ? . ; /Dı ;

(110)

which leads to T D pth ? . ; /I C 2 ? . ; /Dı , or the fluid dissipating due to volume changes but not due to shearing, that is, the fluid with constitutive relations Tı D 0;

m D pth ? . ; / C ? . ; / div v;

j q D  ? . ; /r :

(111)

Finally, one can observe that if one deals with the ideal gas, that is, with the fluid given by constitutive relations (102) and e ? . ; / D cV  , where cV is a constant, then one can in some cases consider isothermal processes. (The temperature is a constant .x; t / D  ? .) Indeed, if the initial temperature distribution is uniform, and if the fluid undergoes only small volume changes, and if the entropy production mechanisms in the fluid are weak, then the terms T W D D pth div v C Q .div v/2 C 2 Dı W Dı ;

(112)

and div j q do not significantly contribute to the right-hand side of (104). In such a case, and under favorable boundary conditions for the temperature, the evolution equation (104) for the internal energy/temperature is of no interest, and one can focus only on the mechanical part of the system, which leads to the compressible Navier-Stokes equations

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

d D  div v; dt   dv Q div v C b; Q C r ./  D r p f th ./ C div .2 ./D/ dt

37

(113a) (113b)

? ? Q with ./ Q > 0 and 2 ./ Q C 3./ > 0. Here p f Q Ddef th ./ Ddef pth . ; /, ./ ? ? ? ? Q . ; /, and ./ Ddef  . ; /. Note that genuine isothermal processes are in fact impossible in a fluid with nonvanishing entropy production due to mechanical effects. Once there is entropy production due to the term T W D, kinetic energy is lost in favor of thermal energy, and the temperature must change; see (102c) or (104). The isothermal process is in this case only an approximation. Other simplified systems that arise from (113) are the compressible Euler equations

d D  div v; dt dv  D r p f th ./ C b; dt

(114a) (114b)

the compressible “bulk viscosity” equations d D  div v; dt 

(115a) !

Q dv 2 ./ Q C 3./ D r p f div v C b: th ./ C dt 3

(115b)

and the compressible “shear viscosity” equations d D  div v; dt dv Q C b:  D r p f th ./  div .2 ./D/ dt

(116a) (116b)

4.2.3 Incompressible Navier-Stokes-Fourier Fluid Incompressibility means that the density of any given material point X is constant in time. In terms of the Eulerian density field .x; t /, this requirement translates into the requirement of vanishing material time derivative of .x; t /, d D 0: dt

(117)

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J. Málek and V. Pr˚uša

(Note that the density can still vary in space, i.e., the density assigned to different material points can be different. In such a case, the material is referred to as inhomogeneous material.) Using the balance of mass (67a), it follows from (117) that div v D 0;

(118)

which means that the material can undergo only isochoric – that is, volume preserving – motions. In this sense, the incompressible material can be understood as a constrained material, meaning that the class of motions possible in the given material is constrained by the requirement div v D 0. If one takes into account the constraint div v D 0, then the term T W D D Tı W Dı C m div v

(119)

in the evolution equation for the internal energy (67d) reduces to Tı W Dı . (Recall that m denotes the mean normal stress, m Ddef 31 Tr T, and that Tr D D div v.) This implies that the mean normal stress cannot be directly present in the entropy production in the form of a flux/affinity pair, because the corresponding flux div v identically vanishes. This observation is easy to document by inspecting the entropy production formula (95) for compressible heat-conducting fluids, where the constraint div v D 0 would imply that .m C pth / div v D 0. However, the specification of the constitutive relations is based on the presence of the quantities of interest in the formula for the entropy production in the form of a flux/affinity product. In this context one could say that the mean normal stress cannot be specified constitutively. (This means that the mean normal stress m is not – in contrast to the traceless part Tı of the Cauchy stress tensor – given by a constitutive relation in terms of other variables such as the kinematical quantities.) If thought of carefully, this is a natural consequence of the concept of incompressibility. The mean normal stress in an incompressible fluid must incorporate the force mechanism that prevents the fluid from changing its volume. (In mathematical terms this means that it must depend on the Lagrange multiplier enforcing the incompressibility constraint (118).) As such it cannot be specified without the knowledge of boundary conditions. Think, for example, of a vessel completely filled with an incompressible fluid at rest. The value of the mean normal stress in the vessel cannot be determined without the knowledge of the forces acting on the vessel walls. This makes the normal stress in an incompressible fluid a totally different quantity than the mean normal stress in a compressible fluid. In a compressible fluid (see, e.g., (96b)), the mean normal stress is a function of the thermodynamic pressure pth and the velocity field, and the thermodynamic pressure is a function of the density and the temperature. Therefore, the mean normal stress in a compressible fluid is fully specified in terms of the local values of the state variables and the velocity field. On the other hand, the mean normal stress in an incompressible fluid is not a function of the local values of the state variables and the velocity field; it constitutes another unknown in the governing equations. Unfortunately, the simplest model for an incompressible viscous fluid is the Navier-Stokes fluid where the Cauchy stress tensor is given by the formula

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

T D pI C 2 D;

39

(120)

where p is called the “pressure.” This could be misleading since the quantity called “pressure” also appears in the compressible Navier-Stokes fluid, albeit the pressure in the compressible Navier-Stokes fluid is the thermodynamic pressure, which is a totally different quantity than the pressure p in (120). Further, the mean normal stress m in the incompressible Navier-Stokes fluid model (120) is m D p; hence it differs from the “pressure” p by a mere sign. Consequently, the notions of the mean normal stress and the “pressure” are often used interchangeably, which is a bad practice since these notions do not coincide if one deals with incompressible fluid models that go beyond (120). Concerning the compressible Navier-Stokes fluid model (102d), the mean normal stress is given by the formula m D pth C 3C2 div v; hence the mean normal stress 3 differs from the thermodynamic pressure by more than a sign. On the other hand, the mean normal stress coincides with the thermodynamic pressure if one adopts so-called Stokes assumption 3 C 2 D 0. In such a case, one is allowed to use interchangeably terms thermodynamic pressure and mean normal stress, which is otherwise not correct. The subtleties concerning the notion of the pressure, Lagrange multiplier enforcing the incompressibility constraint, thermodynamic pressure, and the mean normal stress can be of importance if one, for example, wants to talk about viscoelastic materials with “pressure”-dependent coefficients. The different contexts in which the term pressure is used are discussed in detail by Rajagopal [76] (see also comments by Huilgol [43] and Dealy [22]), and this issue is not further commented in the present treatise. The reader should be however aware of the fact that the notion of “pressure” becomes complicated in particular if one deals with non-Newtonian fluids. Concerning the possible approaches to the derivation of constitutive relations for incompressible materials, one can either opt for the strategy outlined in Sect. 4.1 and enforce the incompressibility constraint via an additional Lagrange multiplier. Examples of such approaches can be found in Málek and Rajagopal [54] (incompressible Navier-Stokes fluid) and Málek et al. [58] (incompressible viscoelastic fluids). This approach allows one to find a relation between the Lagrange multiplier enforcing the incompressibility constraint and the mean normal stress. However, this relation is usually of no interest in the final constitutive relations. Therefore, a different and more pragmatic approach is adopted in the rest of the text. The constitutive relations are derived in a standard manner with the constraint (118) in mind, which leads to a constitutive relation for the traceless part of the Cauchy stress tensor Tı . As shown above, the incompressibility constraint makes the search for a constitutive relation for the mean normal stress pointless; hence one can leave the final formula for the full Cauchy stress tensor in the from

T D mI C Tı ;

(121)

40

J. Málek and V. Pr˚uša

where m is understood as a new unknown quantity to be solved for. (In (121) one can further exploit the fact that the incompressibility constraint (118) allows one to write D D Dı .) In particular, if the outlined approach is applied in the case of the constitutive relations derived in Sect. 4.2.1, one gets T D mI C Tı D mI C 2 ? . ; /Dı ; hence T D mI C 2 ? . ; /D;

(122a)

where the relation D D Dı has been exploited. The constitutive relation for the heat flux remains formally the same, j q D  ? . ; /r :

(122b)

These are the constitutive relations for the incompressible Navier-Stokes-Fourier fluid. Note that unlike in the majority of the literature, the constitutive relation for the Cauchy stress tensor is written as (122a) and not as T D pI C 2 ? . ; /D:

(123)

This notation emphasizes the different nature the thermodynamic pressure pth in compressible fluids, which is frequently denoted as p, and the “pressure” p Ddef m in incompressible fluids. The governing equations for the incompressible Navier-Stokes-Fourier fluid then take the form d D 0; dt

(124a)

div v D 0;

(124b)

dv D rm C div .2 ? . ; /D/ C b; (124c) dt   1 d e ? . ; / C jvj2 D div .mv C 2 ? . ; /Dv C ? . ; /r / C b  v:  dt 2 (124d) 

The first equation in (124), the characteristics,

@ @t

C v  r D 0, implies that  is transported along

  .x; t / D 0 .X / D 0 1 .x; t / ;

(125)

where 0 is the density in the initial (reference) configuration. Consequently the density .x; t / is constant in space and time if the initial (reference) density 0 is uniform, that is, if 0 .X 1 / D 0 .X 2 /

(126)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

41

holds for all X 1 , X 2 2 B. If (126) does not hold, then the fluid at its initial (reference) state is inhomogeneous. This is why the system (124) is sometimes called inhomogeneous incompressible Navier-Stokes-Fourier equations. Obviously the incompressible Navier-Stokes-Fourier fluid model can be further simplified. Setting Tı D 0, one gets a model for a fluid without shear viscosity. Such a model can be referred to as inhomogeneous incompressible Euler-Fourier fluid model, and the governing equations reduce to d D 0; dt

(127a)

div v D 0;

(127b)

dv D rm C b; dt   1 d e ? . ; / C jvj2 D div .mv C ? . ; /r / C b  v:  dt 2 

(127c) (127d)

Note that equations (127a)–(127c) are not coupled with the temperature and can provide a solution for  and v independently of the energy equation. The fluid described by (127a)–(127c) is called the inhomogeneous incompressible Euler fluid. If the viscosity in (124) does not depend on the temperature, ? . ; / D ./, Q then energy equation again decouples from the system, and one can solve only the mechanical part of (124). This leads to inhomogeneous incompressible NavierStokes equations d D 0; dt

(128a)

div v D 0;

(128b)



dv D rm C div .2 ./D/ Q C b: dt

(128c)

If the density at initial (reference) configuration is uniform, then as a consequence of (126) and (125), one finds that .x; t / Ddef ? 2 .0; C1/

(129)

for all t > 0, x 2 B. System (124) then simplifies to the so-called homogeneous incompressible Navier-Stokes-Fourier equations div v D 0; dv D rm C div .2 ./D/ O C b; dt   1 2 d O C . O /r / C b  v; eO . / C jvj D div .mv C 2 ./Dv  dt 2 

(130a) (130b) (130c)

42

J. Málek and V. Pr˚uša

where ./ O Ddef ? . ; ? /, eO . / Ddef e ? . ; ? /, and . O / Ddef ? . ; ? /. Following the same template as above, one can introduce homogeneous incompressible Euler-Fourier fluids. If the viscosity does not depend on the temperature, ? Ddef . O ? / D ? .? ; ? /, then (130c) decouples from the rest of the system (130), and one can solve only the mechanical part of (130) for the mechanical variables m and v. This leads to the classical incompressible Navier-Stokes equations div v D 0; 

(131a)

dv D rm C div .2 ? D/ C b; dt

(131b)

where the right-hand side of (131b) can be rewritten as rm C div .2 ? D/ C b D rm C ? v C b. Finally, if Tı D 0, then one ends up with the incompressible Euler equations div v D 0; 

4.3

(132a)

dv D rm C b: dt

(132b)

Compressible Korteweg Fluids

Another popular model describing the behavior of fluids is the model introduced by Korteweg [51]. The model has been developed with the aim to describe phase transition phenomena, but it is also used for the modelling of the behavior of some granular materials. The key feature of the model is that the Cauchy stress tensor includes terms of the type r ˝ r, which is natural given the fact that the model is designed to take into account steep density changes that can occur, for example, at a fluid/vapour interface. In what follows it is shown how to derive the model using the thermodynamical procedure outlined in Sect. 4.1. The reader interested in various applications of the model and the details concerning the derivation of the model is referred to Málek and Rajagopal [55] and Heida and Málek [38]. Since the density gradient is the main object of interest, for later use, it is convenient to observe that the balance of mass (67a) implies d .r/ D r . div v/  Œrv> r: dt

(133)

STEP 1: Consider the constitutive equation for the specific internal energy – or the specific entropy  – in the form  D .e; Q ; r/

or

e D eQ .; ; r/:

(134)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

43

The presence of the density gradient r in the formula for the internal energy/entropy means that in this specific example, one goes, for the first time in this text, beyond classical equilibrium thermodynamics. Moreover, the energetic equation of the state is considered in a special form e Ddef eQ .; ; r/ D e.; / C

 jrj2 ; 2

(135)

where  is a positive constant. This choice is motivated by the fact that one expects the density gradient r to contribute to the energy storage mechanisms and the fact that the norm of the gradient r is the simplest possible choice if one wants the internal energy to be a scalar isotropic function of r. STEP 2 and 3: Applying the material time derivative to (135), multiplying the result by  and using the evolution equation for the internal energy (22), the balance of mass (4) and identity (133) yield @e d @Qe DTWDdiv j q C2 div v C .r/r . div v/C .rv/W.r ˝ r/: @ dt @ (136) Introducing the thermodynamic pressure and temperature via 

 Ddef

@Qe ; @

K Ddef 2 pth

@Qe ; @

(137a) (137b)

one ends up with a formula for the material time derivative of the entropy 

  d  K D .Tı C  .r ˝ r/ı / W Dı C m C pth C jrj2 div v  div j q dt 3 C div . .div v/ r/   div v;

(138)

where the relevant tensors have been split to their traceless part and the rest; see Sect. 4.2 for the rationale of this step. Notice however that this thermodynamic NSE K pressure pth is different from the thermodynamic pressure pth Ddef 2 @e for @ the Navier-Stokes fluid as they are related through K NSE D pth  pth

 jrj2 : 2

(139)

Equation (138) can be further rewritten as 

  d  K C jrj2   div v D .Tı C  .r ˝ r/ı / W Dı C m C pth dt 3   (140)  div j q  .r/ div v ;

44

J. Málek and V. Pr˚uša

and using the notation j qQ Ddef j q  r;

(141)

and applying the standard manipulation finally yields 

   j qQ 1 d D .Tı C  .r ˝ r/ı / W Dı C div dt      r r 2 K : C m C pth C jrj   div v  j q  C  .div v/ .r/  3   (142)

The affinities are in the present case Dı , div v, and r . Apparently, the last term on the right-hand side of (142) can be interpreted either as 

r .r/  div v 

(143)

  K C 3 jrj2   div v, or it can and grouped with the second term m C pth be read as Π.div v/ .r/ 

r 

(144)

. The question is left unresolved, and a and grouped with the third term j q  r  parameter ı 2 Œ0; 1 that splits the last term as  r r r div vCıŒ .div v/ .r/   .div v/ .r/  D.1ı/ .r/  ;    (145) is introduced. The arising terms are then grouped with the corresponding terms in (142). This yields    j d 1 .Tı C  .r ˝ r/ı / W Dı  D C div dt    C m C pth

K

 r C jrj2   C .1  ı/.r/  3 

 r  :  j q  ı .div v/ r  

 div v

(146)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

45

STEP 4: The entropy production  – that is, the right-hand side of (146) – takes the form D

3 1X j  a˛ ;  ˛D1 ˛

(147)

where the affinities are Dı , div v, and r . Requiring the linear relations between the fluxes and affinities in each term contributing to the rate of entropy production leads to Tı C  .r ˝ r/ı D 2 Dı ; K m C pth C

(148a)

2 C 3  r D div v; jrj2   C .1  ı/.r/  3  3 (148b) j q  ı .div v/ r D  r

(148c)

where > 0, 2 C3 > 0, and > 0 are positive constants. This choice of 3 constitutive relations follows the template specified in (75). STEP 5: It follows from (148) that the constitutive relation for the full Cauchy stress tensor T D Tı C mI is r K T D pth IC2 DC .div v/ I .r˝r/ C ./ I.1  ı/.r/  I  (149) that can be upon introducing the notation NSE   Ddef pth

 r jrj2   C .1  ı/.r/  2 

(150a)

rewritten as T D I C 2 D C  .div v/ I   .r ˝ r/ :

(150b)

Concerning the heat/energy flux, one gets j q D  r C ı .div v/ r:

(150c)

Note that if only isothermal processes are considered, then (149) reduces to ! jrj2 TD C  ./ I; C 2 D C  .div v/ I C  .r ˝ r/ C  2 (151) which is the constitutive relation obtained by Korteweg [51]. NSE pth I

46

4.4

J. Málek and V. Pr˚uša

Compressible and Incompressible Viscoelastic Heat-Conducting Fluids

Viscoelastic materials are materials that exhibit simultaneously two fundamental modes of behavior. They can store the energy in the form of strain energy; hence they deserve to be called elastic, and they dissipate the energy; hence they deserve to be called viscous. Since the two modes are coupled, the nomenclature viscoelastic is obvious. A rough visual representation of materials that exhibit viscoelastic behavior is provided by systems composed of springs and dashpots. The springs represent the elastic behavior, while the dashpots represent the dissipation. Such a visual representation is frequently used in the discussion of viscoelastic properties of materials; see, for example, Burgers [14] or more recently Wineman and Rajagopal [95]. The use of the visual representation as a motivation for the subsequent study of the constitutive relations for viscoelastic fluids is also followed in the subsequent discussion. A simple example of a spring–dashpot system is the Maxwell element consisting of a viscous dashpot and an elastic spring connected in series; see Fig. 3a. The response of the element to step loading is the following. (The step loading is a loading that is constant and is applied only over time interval Œt0 ; t1 ; otherwise the loading is zero.) Initially, the element is unloaded, and it is in an initial state; see Fig. 3a. Once the loading is applied at time t0 , the spring extends, and the dashpot starts to move as well; see Fig. 3b. When the loading is suddenly removed at some time t1 (see Fig. 3c), the spring, that is, the elastic element, instantaneously shrinks to its initial equilibrium length. On the other hand, the sudden unloading does not change the length of the dashpot at all. (The term “length of the dashpot” denotes the distance between the piston and the dashpot left wall.)

a

b

ls

ls

l0

l0

c

ls

l0

Fig. 3 Maxwell element. (a) Initial state. (b) Loaded state. (c) Unloaded state

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

47

This means that the instantaneous response to the sudden load removal is purely elastic, while the configuration (length) to which the material relaxes after the sudden load removal is determined by the length of the dashpot. Therefore, the unloaded configuration is determined by the behavior of the purely dissipative element.

4.4.1 Concept of Natural Configuration and Associated Kinematics Now the question is whether one can use this observation in the development of the constitutive relations for viscoelastic materials. It turns out that this is possible. One can associate these three states (initial/loaded/unloaded) of the Maxwell element with the three states of a continuous body: 1. the initial (reference) configuration 0 .B/, 2. the current configuration t .B/ that the body takes at time t > 0 during the deformation process due to the external load, 3. the natural configuration p.t/ .B/ that would be taken by the considered body at time t > 0 upon sudden load removal. The evolution from the initial configuration to the current configuration is then virtually decomposed to the evolution of the natural configuration (dissipative process) and instantaneous elastic response (non-dissipative process) from the natural configuration to the current configuration. Figure 4 depicts the situation. Referring back to the spring–dashpot analogue, the evolution of the natural configuration plays the role of the dashpot, while the instantaneous elastic response from the current configuration to the natural configuration plays the role of the spring.

current configuration

κt (B)

κ0 (B) elastic response

κp(t) (B) reference configuration dissipative response natural configuration

0

t

time

Fig. 4 Initial (reference), current and natural configurations associated with a material body

48

J. Málek and V. Pr˚uša

Derivation of constitutive relations for viscoelastic materials by appealing to the procedure outlined in Sect. 4.1 can then proceed as follows. Since the energy storage mechanisms in a viscoelastic fluid are determined by the elastic part – the spring – the plan is to enrich the constitutive relation for the internal energy (see (70)) by a state variable that measures the deformation of the elastic part (deformation from the natural to the current configuration). The next step of the thermodynamic procedure requires one to take material time derivative of the internal energy; hence a formula for the material time derivative of the chosen measure of the deformation of the elastic part is needed. This task requires a careful analysis of the kinematics of continuous media with multiple configurations. Recall that the key quantities in the kinematics of continuous media are the deformation gradient @ 0 .X ; t / ; @X

(152)

ˇ @ 0 .X ; t / ˇ ˇ : ˇ @t X D1

.x;t/

(153)

F.X ; t / Ddef

and the spatial velocity field

v.x; t / Ddef

0

(The subscript 0 recalls that the deformation gradient is taken with respect to the initial configuration.) Other quantities of interest are the left Cauchy-Green tensor and the right Cauchy-Green tensor B Ddef FF> , C Ddef F> F, the (spatial) velocity @v gradient L Ddef @x , and its symmetric part D. The material time derivative of F is then given by the formula dF D LF: dt

(154)

Within the framework consisting of three configurations 0 .B/, t .B/, and

p.t/ .B/, the standard kinematical setup is extended by the deformation gradient F p.t / describing the deformation between the natural and the current configuration and the deformation gradient G describing the deformation between the initial (reference) configuration and the natural configuration. Obviously (see Fig. 4), the following relation holds

F D F p.t / G:

(155)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

49

For later use, one also introduces notation B p.t / Ddef F p.t / F>

p.t / ;

(156a)

C p.t / Ddef F>

p.t / F p.t / ;

(156b)

for the left and right Cauchy-Green tensors for the deformation from the natural to the current configuration. The left Cauchy-Green tensor B p.t / is the sought measure of the deformation associated with the instantaneous elastic part of the deformation. The reason is the following. In the standard finite elasticity theory, one uses the left Cauchy-Green tensor B Ddef FF> in order to characterize the finite deformation of a body. In the current setting, the elastic response is the response from the natural to the current configuration; hence the left Cauchy-Green B p.t / tensor built using the relative deformation gradient F p.t / instead of the full deformation gradient F is used. Now one needs to find an expression for the material time derivative of B p.t / . (The knowledge of an expression for the time derivative is needed in the STEP 2 of the procedure discussed in Sect. 4.1.) The aim is to express B p.t / as a function of F p.t / and a rate quantity associated with the dissipative part of the response and a rate quantity associated with the total response. Noticing that (154) implies that L D dF F1 , a new tensorial quantity L p.t / is dt introduced through dG 1 G ; dt

(157)

 1 L p.t / C L>

p.t / : 2

(158)

L p.t / Ddef and its symmetric part is denoted as D p.t / , D p.t / Ddef

Relations (155), (154), and (157) together with the formula dG d  1  G D G1 G1 dt dt

(159)

then yield dG dF 1 dG1 G CF D LFG1  FG1 G1 D LF p.t /  F p.t / L p.t / ; dt dt dt dt (160) which implies dF p.t /

D

dB p.t / dt

D LB p.t / C B p.t / L>  2F p.t / D p.t / F>

p.t / :

(161)

50

J. Málek and V. Pr˚uša

This is the sought expression for the time derivative of the chosen measure of the deformation. As required, the time derivative is a function of F p.t / and the rate quantities D p.t / and L that are associated to the dissipative part of the response and the total response, respectively. Introducing the so-called upper convected time derivative through the formula O

A Ddef

dA  LA  AL> ; dt

(162)

where A is a second order tensor, it follows from (161) that O

B p.t / D 2F p.t / D p.t / F>

p.t / :

(163)

For later reference it is worth noticing that (162) implies O

I D 2D;

(164)

and that (161) implies

Since

d dt

d Tr B p.t / D 2B p.t / W D  2C p.t / W D p.t / : dt   det A D .det A/ Tr dA A1 , one can also observe that dt

   dB p.t / d 

ln det B p.t / D Tr B p.t / 1 D 2I W D  2I W D p.t / : dt dt

(165)

(166)

Finally, the upper convected derivative of the traceless part of B p.t / reads O



B p.t /

 ı

D

D

  d B p.t / ı dt

dB p.t / dt O

D B p.t / 



     L B p.t / ı  B p.t / ı L>

 1 dTr B p.t / 2 I  LB p.t /  B p.t / L> C Tr B p.t / D 3 dt 3

   2 2 2 B p.t / W D I C C p.t / W D p.t / I C Tr B p.t / D 3 3 3

D 2F p.t / D p.t / F p.t / > 

4.4.2

   2 2 2 B p.t / W D I C C p.t / W D p.t / I C Tr B p.t / D: 3 3 3 (167)

Application of the Thermodynamic Procedure in the Context of Natural Configuration The application of the thermodynamic procedure discussed in Sect. 4.1 in the setting of a material with an evolving natural configuration goes as follows:

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

51

STEP 1: The fact that the energy storage mechanisms are required to be related to the elastic part of the deformation (see Fig. 4) suggests that the internal energy e should be enriched by a term measuring the elastic part of the deformation. As it has been already noted, a suitable measure of the deformation is the left Cauchy-Green tensor B p.t / ; hence the entropic/energetic equation of state is assumed to take the form      D Q e; ; B p.t / or e D eQ ; ; B p.t / : (168) For the sake of simplicity of the presentation, the energetic equation of state is further assumed to take the form   

 e D eO ; ; Tr B p.t / ; det B p.t / Ddef e .; / C Tr B p.t /  3  ln det B p.t / ; 2 (169) where is a constant. The motivation for this particular choice comes from the theory of constitutive relations for isotropic compressible elastic materials; see for example Horgan and Saccomandi [41] and Horgan and Murphy [40] for a list of some frequently used constitutive relations in the theory of compressible elastic materials. STEP 2: Applying the material time derivative to (169), multiplying the result by  and using the balance of mass (67a) and the evolution equation for the internal energy (67d) together with the identities (165) and (166), one gets 

d M D T W D  div j q C pth div v  B p.t / W D C C p.t / W D p.t / dt C D W I  D p.t / W I

(170)

where M pth Ddef 2

 @Oe @e  D 2  Tr B p.t /  3  ln det B p.t / @ @ 2  

NSE D pth Tr B p.t /  3  ln det B p.t /  2

(171)

NSE Ddef 2 @e . denotes the “pressure”, and pth @ STEP 3: Splitting the tensors on the right-hand side of (170) into the traceless part and the rest yields



   d D Tı  B p.t / ı W Dı dt      

M C m C pth  Tr B p.t / C div v C C p.t / ı W D p.t / ı 3   Tr B p.t / C

 1 Tr D p.t /  div j q ; 3

(172)

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J. Málek and V. Pr˚uša

where the notation m Ddef 31 Tr T for the mean normal stress has been used. (See Sect. 4.2.1 for the rationale of the splitting.) The standard manipulation finally leads to   jq d  C div dt      

1 M  Tr B p.t / C div v D Tı  B p.t / ı W Dı C m C pth  3       Tr B p.t / r (173) C C p.t / ı W D p.t / ı C

 1 Tr D p.t /  j q  3  which is the sought formula that allows one to identify the entropy production.  The flux/affinity pairs are Tı  B p.t / ı versus Dı and so on. STEP 4: The entropy production – the right-hand side of (173) – is positive provided that the linear relations between the fluxes and affinities are   (174a) Tı  B p.t / ı D 2 Dı ; M  m C pth

2 C 3 Tr B p.t / C D div v; 3 3    

C p.t / ı D 2 1 D p.t / ı ; 



Tr B p.t / 3

 1 D

2 1 C 31 Tr D p.t / ; 3

j q D  r ;

(174b) (174c) (174d) (174e)

1 where > 0, 2 C3 > 0, 1 > 0, 2 1 C3 > 0, and > 0 are constants. This 3 3 choice of constitutive relations follows the template specified in (75). STEP 5: The first two equations in (174) allow one to identify the constitutive relation for the full Cauchy stress tensor T D mI C Tı ,

   2 C 3

 M .div v/ I  pth Tr B p.t / I  I; T D 2 Dı C B p.t / ı C IC 3 3 (175) which can be further rewritten as   M T D pth I C 2 D C  .div v/ I C B p.t /  I ; (176) or, in virtue of (171), as 

 NSE IC Tr B p.t /  3  ln det B p.t / I C 2 D C  .div v/ I T D pth 2   (177) C B p.t /  I :

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

53

The last step leading to the complete description of the material behavior is the elimination of D p.t / from the constitutive relations. Note that the Cauchy stress is given in terms of D, that is, the velocity field v, and the left Cauchy-Green tensor B p.t / . The evolution equation for B p.t / is (163), that is, O

B p.t / D 2F p.t / D p.t / F>

p.t / :

(178)

The equation expresses the time derivative of B p.t / as a function of L, F p.t / , and D p.t / . If the right-hand side of (163) can be rewritten as a function of B p.t / and L, then the time evolution of B p.t / would be given in terms of the other quantities directly present in the governing equations, and the process of specification of the constitutive relation would be finished. This can be achieved as follows. Summing (174c) and (174d) multiplied by the identity tensor yields    

C p.t /  I D 2 1 D p.t / C 1 Tr D p.t / I

(179)

and Tr D p.t / D

3

2 1 C 31



Tr C p.t / 3

 1 ;

(180)

which leads to  

C p.t /  I D 2 1 D p.t / C

3 1 2 1 C 31



Tr C p.t / 3

  1 I:

(181)

This is an explicit relation between D p.t / and C p.t / . Recalling the definition of C p.t / and B p.t / (see (156)) and noticing that Tr C p.t / D Tr B p.t / , it follows that the multiplication of (181) from the left by F p.t / and from the right by F>

p.t / yields



B2 p.t /



 B p.t / D

2 1 F p.t / D p.t / F>

p.t /

3 1 C 2 1 C 31



Tr C p.t / 3

  1 B p.t / : (182)

Substituting the formula just derived for F p.t / D p.t / F>

p.t / into (178) then gives   O 1 B p.t / C B2 p.t /  B p.t / D

3 1 2 1 C 31



Tr B p.t / 3

  1 B p.t / ;

(183)

which is the sought evolution equation for B p.t / that contains only B p.t / and L. The set of equations (67) supplemented with the constitutive equations for the heat flux (see (174e)), the Cauchy stress tensor (see (177)), and the evolution

54

J. Málek and V. Pr˚uša

equation for the left Cauchy-Green tensor B p.t / (see (183)) forms a closed system of equations for the density , the velocity field v, the specific internal energy e (or the temperature ), and the left Cauchy-Green tensor B p.t / . The constitutive relation for the Cauchy stress tensor (see (177)) can be further manipulated as follows. Introducing the notation NSE  Ddef pth C



 Tr B p.t /  3  ln det B p.t / ; 2

(184a)

  S Ddef B p.t /  I ;

(184b)

and

the equations (177) and (183) take the form T D I C 2 D C  .div v/ I C S;

(185)

and O

1 S C S2 C S D 2 1 D C

1 .Tr S/ .S C I/ : 2 1 C 31

(186)

Note that if 1 D 0, then (186) simplifies to O

1 S C S2 C S D 2 1 D;

(187)

hence the derived model could be seen as a compressible variant of the classical model for a viscoelastic incompressible fluid developed by Giesekus [31]. If the material is constrained in such a way that Tr D D Tr D p.t / D 0;

(188)

which means that the material is assumed to be an incompressible material and that the response from the initial to the natural configuration is assumed to be isochoric, it is clear that neither Tr B p.t / nor m can be specified constitutively. (These quantities correspond to the forces that make the material resistant to the volumetric changes; see Sect. 4.2 for a discussion.) However, formula (174a) still holds, and the only difference is that D D Dı ; hence   T D mI C Tı D mI C 2 D C B p.t / ı :

(189)

Similarly, (174c) is also valid; hence      1

C p.t /  Tr C p.t / I D 2 1 D p.t / ı ; 3

(190)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

55

  where, in virtue of (188), D p.t / D D p.t / ı . Multiplication of (190) by F>

p.t / from the right and F p.t / from the left and the evolution equation for the left Cauchy-Green tensor (163) then yield   O  1 Tr B p.t / B p.t / D 0 1 B p.t / C B2 p.t /  3

(191)

as the evolution equation for B p.t / .

4.4.3

Application of the Maximization of the Entropy Production: Constitutive Relations for Oldroyd-B and Maxwell Viscoelastic Fluids The last example is devoted to discussion of the derivation of incompressible Oldroyd-B and Maxwell models for viscoelastic fluids and their compressible counterparts. Unlike in the previous example, the full thermodynamic procedure based on the assumption of the maximization of the entropy production is now applied. The starting point is the formula for the entropy production derived in the previous section (see (173)) that can be rewritten in the form D

     1 

M  Tr B p.t / C div v Tı  B p.t / ı W Dı C m C pth  3   r : (192) C C p.t /  I W D p.t /  j q  

Using this formula one can continue with STEP 4 of the full thermodynamic procedure discussed in Sect. 4.1. STEP 4: At this point the constitutive function ,  D  , is chosen as   2 C 3  Ddef Q Dı ; div v; D p.t / ; r D 2 jDı j2 C .div v/2 3 C 2 1 D p.t / W C p.t / D p.t / C

jr j2 ; 

(193)

where the right Cauchy-Green tensor C p.t / is understood as a state variable and , 2 C3 , 1 , and are positive constants. Using the definition of the right 3 Cauchy–Green tensor (see (156b)) and the properties of the trace, it is easy to ˇ2 ˇ see that D p.t / W C p.t / D p.t / D ˇF p.t / D p.t / ˇ  0; hence the chosen constitutive function is nonnegative as required by the second law of thermodynamics. Further it is a strictly convex smooth function.

56

J. Málek and V. Pr˚uša

Introducing the auxiliary function ˆ for constrained maximization problem max

Dı ;div v;D p.t / ;r

  Q Dı ; div v; D p.t / ; r

(194)

subject to the constraint (192) as      ˆ Ddef Q Dı ; div v; D p.t / ; r C ` Q Dı ; div v; D p.t / ; r     

M  Tı  B p.t / ı W Dı  m C pth  Tr B p.t / C div v 3   r  C p.t /  I W D p.t / C j q  ; (195)  where ` is the Lagrange multiplier, the conditions for the extrema are   1 C ` @Q D Tı  B p.t / ı ; ` @Dı

1 C ` @Q M  Tr B p.t / C ; D m C pth ` @div v 3 Q   1 C ` @ D C p.t /  I ; ` @D p.t / jq 1 C ` @Q D : ` @r 

(196a) (196b) (196c)

(196d)

On the other hand, direct differentiation of (193) yields @Q D 4 Dı ; @Dı 2 @Q D .2 C 3/ div v; @div v 3   @Q D 2 1 C p.t / D p.t / C D p.t / C p.t / ; @D p.t / @Q r D 2 : @r 

(197a) (197b) (197c)

(197d)

Now it is necessary to find a formula for the Lagrange multiplier `. This can be done as follows. First, each equation in (196) is multiplied by the corresponding affinity, that is, by Dı , div v, D p.t / , and r, respectively, and then the sum of all equations is taken. This manipulation yields

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

57

"

# @Q @Q @Q @Q W Dı C W D p.t / C div v C  r @Dı @div v @D p.t / @r     

M  Tr B p.t / C div v D Tı  B p.t / ı W Dı C m C pth 3   r C C p.t /  I W D p.t /  j q  : (198) 

1C` `

The right-hand side is identical to Q D   (see (192)), while the term on the leftQ Consequently 1C` D Q , hand side reduces, in virtue of (193) and (197), to 2. ` 2Q which fixes the value of the Lagrange multiplier 1C` 1 D : ` 2

(199)

Inserting (197) and (199) into (196), one finally concludes that   Tı  B p.t / ı D 2 Dı ; M  m C pth

2 C 3 Tr B p.t / C D div v; 3 3    

C p.t /  I D 1 C p.t / D p.t / C D p.t / C p.t / ; j q D  r :

(200a) (200b) (200c) (200d)

Further, if (200c) holds, then it can be shown that the symmetric positive definite matrix C p.t / and the symmetric matrix D p.t / commute. (In order to prove that C p.t / and D p.t / commute, it suffices to show that the eigenvectors of C p.t / and D p.t / coincide, which is in virtue of (200c) an easy task. See the Appendix in Rajagopal and Srinivasa [77] for details.) If the matrices commute, then (200c) in fact reads  

C p.t /  I D 2 1 C p.t / D p.t / :

(201)

STEP 5: The first two equations in (200) allow one to identify the constitutive relation for the full Cauchy stress tensor T D mI C Tı ,   M I C 2 D C B p.t /  I C  .div v/ I: T D pth

(202)

Further, (201) can be rewritten as  

C p.t /  I D 2 1 F>

p.t / F p.t / D p.t / :

(203)

58

J. Málek and V. Pr˚uša > which upon multiplication by F>

p.t / from the right and by F p.t / from the left

yields a formula for F p.t / D p.t / F>

p.t / . This formula can be substituted into (163) that yields the evolution equation for the left Cauchy-Green tensor, O   1 B p.t / C B p.t /  I D 0:

(204)

The set of equations (67) supplemented with the constitutive equations for the heat flux (see (200d)), the Cauchy stress tensor (see (202)), and the evolution equation for the left Cauchy-Green tensor B p.t / (see (204)) forms a closed system of equations for the density , the velocity field v, the specific internal energy e (or the temperature ), and the left Cauchy-Green tensor B p.t / .   Introducing the extra stress tensor S by S Ddef B p.t /  I , equations (202) and (204) can be rewritten as M I C 2 D C S C  .div v/ I; T D pth

1 O S C S D 2 1 D:

(205a) (205b)

Further manipulation based on the redefinition of the extra stress tensor SQ Ddef S C 2 D allows one to rewrite system (205) as M I C SQ C  .div v/ I; T D pth

1 OQ 2 1 O D: S C SQ D 2 . 1 C / D C



(206a) (206b)

The expressions (206) suggest that the derived model can be seen as a compressible variant of the classical Oldroyd-B model for viscoelastic incompressible fluids developed by Oldroyd [65]. On the other hand, if the constitutive relation for the traceless part of the Cauchy stress tensor reads   Tı  B p.t / ı D 0;

(207)

which corresponds to D 0 in (200a), and the constitutive relation for the mean normal stress is M  m C pth

which corresponds to the choice Cauchy stress tensor reads

Tr B p.t / C D 0; 3

2 C3 3

(208)

D 0 in (200b), then the formula for the full

  M T D pth I C B p.t /  I :

(209)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

59

  Introducing the extra stress tensor S Ddef B p.t /  I , it follows that the system (205) reduces to M I C S; T D pth

1 O S C S D 2 1 D:

(210a) (210b)

This model can be denoted as a compressible variant of the classical Maxwell model for viscoelastic incompressible fluids; see, for example, Tanner and Walters [90]. Finally, if the fluid is assumed to be incompressible, then it is subject to constraint Tr D D 0:

(211)

However, then the same procedure as above can be still applied. One starts from (193) to (194) with the only modification that the terms with div v disappear, and one ends up with the same system of equations as those given in (200) (with the only exception that the second equation in (200b) is missing). The derived constitutive relations read



C p.t /

  Tı D 2 Dı C B p.t / ı ;   I D 2 1 C p.t / D p.t / ;

(212a) (212b)

j q D  r ;

(212c)

  T D mI C 2 Dı C B p.t / ı

(213)

which translates to

where the mean normal stress m is a quantity that cannot be specified via a constitutive relation. (See Sect. 4.2 for discussion.) Concerning the alternative approach based on the enforcement of the incompressibility constraint by an addition of an extra Lagrange multiplier to the maximization procedure, the interested reader is referred to Málek et al. [58]. Introducing the notation  Ddef m 

Tr B p.t / C ; 3

(214)

it follows that (213) can be rewritten as   T D I C 2 D C B p.t /  I :

(215a)

(Recall that in virtue of (211), one has D D Dı .) Further, the evolution equation for the left Cauchy-Green tensor B p.t / reads

60

J. Málek and V. Pr˚uša O   1 B p.t / C B p.t /  I D 0:

(215b)

  Defining the extra stress tensor as S Ddef B p.t /  I or as SQ Ddef 2 D C S, it follows that (215) can be converted into the equivalent form T D I C 2 D C S; 1 O S C S D 2 1 D;

(216a) (216b)

or into the following equivalent form Q T D I C S; 1 OQ 2 1 O D; S C SQ D 2 . 1 C / D C



(217a) (217b)

which are the frequently used forms of constitutive relations for the Oldroyd-B fluid, which is a popular model for viscoelastic fluids derived by Oldroyd [65]. (Oldroyd [65] has used formulae (217).) Further, if one formally sets D 0, then one gets the standard incompressible Maxwell fluid. Note however that the classical derivation by Oldroyd [65] is based only on mechanical considerations and that the compatibility of the model with the second law of thermodynamics is not discussed at all. Second, the present approach naturally gives one an evolution equation for the internal energy that automatically takes into account the storage mechanisms related to the “elastic” part of the deformation. If the internal energy is expressed as a function of the temperature and other variables, then the evolution equation for the internal energy leads, upon the application of the chain rule, directly to the evolution equation for the temperature. (See Sect. 4.2.1 for the same in the context of a compressible Navier-Stokes-Fourier fluid.) Moreover, the inclusion of the storage mechanisms in the internal energy is compatible with the specification of the constitutive relation for the Cauchy stress tensor. Again, such issues have not been discussed in the seminal contribution by Oldroyd [65] or for that matter by many following works on viscoelasticity. Obviously, more complex viscoelastic models can be designed by appealing to the analogy with more involved spring–dashpot systems; see, for example, Karra and Rajagopal [49, 50], Hron et al. [42], Pr˚uša and Rajagopal [72], or Málek et al. [59].

4.5

Beyond Linear Constitutive Theory

In the previous parts, the development of the constitutive equations has been based either on linear relationships between the affinities and fluxes or on postulating quadratic constitutive equations for the entropy production. In general, this

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

61

limitation to linear constitutive relations or to a quadratic ansatz for the entropy production is not necessary. In fact a plethora of nonlinear models can be developed in a straightforward manner following the method based on the maximization of entropy production. Their relevance with respect to real material behavior should be however carefully justified. A list of some popular simple nonlinear models for incompressible fluids is given below. Clearly, the list is not complete, and other models can be found in the literature as well. In particular, the list does not include nonlinear viscoelastic models at all. The first class of nonlinear models is the class of models where the Cauchy stress tensor is decomposed as T D mI C S;

(218)

where m is the mean normal stress, and the relation between the traceless extra stress tensor S and the symmetric part of the velocity gradient D takes the form of an algebraic relation f.S; D/ D 0;

(219)

where f is a tensorial function. This implicit relation is a generalization of the standard constitutive relation for the incompressible Navier-Stokes fluid (see Sect. 4.2.3) where S  2 ? D D 0:

(220)

Note that the standard way of writing algebraic constitutive relations for a nonNewtonian fluid is S D f.D/. However, it has been argued by Rajagopal [73,74] that the standard setting is too restrictive and that (219) should be preferred to S D f.D/. (See, e.g., Málek et al. [57], Pr˚uša and Rajagopal [71], Le Roux and Rajagopal [53], Perlácová and Pr˚uša [69], and Janeˇcka and Pr˚uša [45] for further developments of the idea. Mathematical issues concerning some of the models that belong to the class (219) have been discussed, e.g., by Bulíˇcek et al. [12] and Bulíˇcek et al. [13].) The implicit relation (219) opens the possibility of describing – in terms of the same quantities that appear in (220) – various non-Newtonian phenomena such as stress thickening or stress thinning, shear thickening or shear thinning, yield stress, and even normal stress differences. None of these important phenomena can be captured by the standard Navier-Stokes model (220). (The reader is referred to Málek and Rajagopal [54] or any textbook on non-Newtonian fluid mechanics for the discussion of these phenomena and their importance.) The mechanics of complex fluids is indeed an unfailing source of qualitative phenomena that go beyond the reach of the Navier-Stokes model. For interesting recent observations concerning the behavior of complex fluids, see, for example, the references in the reviews by Olmsted [66] and Divoux et al. [23].

62

J. Málek and V. Pr˚uša

Particular models that fall into class (219) are the models in the form S D 2 .D/D;

(221)

where the generalized viscosity .D/ is given by one of the formulae listed below. Model .D/ D 0 jDjn1 ;

(222)

where 0 is a positive constant and n is a real constant, is called Ostwald–de Waele power law model; see Ostwald [67] and de Waele [94]. Models 0  1

.D/ D 1 C

n ; .1 C ˛jDj2 / 2  n1  .D/ D 1 C . 0  1 / 1 C ˛jDja a ;

(223) (224)

are called Carreau model (see Carreau [16]) and Carreau–Yasuda model (see Yasuda [96]), respectively. Here 0 and 1 are positive real constants, and n and a are real constants. Other models with nonconstant viscosity are the Eyring models (see Eyring [26] and Ree et al. [82]), where the generalized viscosity takes the form .D/ D 1 C . 0  1 / .D/ D 0 C 1

arcsinh .˛jDj/ ; ˛jDj

arcsinh .˛1 jDj/ arcsinh .˛2 jDj/ C 2 ; ˛1 jDj ˛2 jDj

(225) (226)

and 0 , 1 , 2 , 1 , ˛1 , and ˛2 are positive real constants. Finally, the model named after Cross [21] takes the viscosity in the form 0  1 1 C ˛jDjn

(227)

.D/ D 1 C ˛jDjn1 ;

(228)

.D/ D 1 C and the model named after Sisko [88] reads

where 0 , 1 , and ˛ are positive real constants and n is a real constant. Another subset of general models of the type (219) are models where the generalized viscosity depends on the traceless part of the Cauchy stress tensor Tı D S, that is, models, where S D .S/D:

(229)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

63

Examples are the Ellis model (see, e.g., Matsuhisa and Bird [61]) 0

.S/ D

1 C ˛jSjn1

;

(230)

where 0 and ˛ are positive real constants and n is a real constant or the model proposed by Glen [33], .S/ D ˛jSjn1 ;

(231)

where ˛ is a positive real constant and n is a real constant, the model by Seely [85]  jSj

.S/ D 1 C . 0  1 / e

0

;

(232)

or the models used for the description of the flow of the ice, A

.S/ D 

jSj2 C 02

;  n1 2

(233)

See, for example, Pettit and Waddington [70] and Blatter [5]. Here 0 , 1 , 0 , and A are positive real constants, and n is a real constant. Particular parameter values for models (222), (223), (224), (225), (226), (227), (228) and (230), (231), (232), (233) can be found in the referred works and/or in the follow-up works. Other popular models are the models with activation criterion introduced by Bingham [3] and Herschel and Bulkley [39]. These models are usually (see, e.g., Duvaut and Lions [25]) written down in the form of a dichotomy relation jSj   , D D 0

and

jSj >  , S D

 D C 2 .jDj/D; jDj

(234)

where is a positive function and  is a positive constant. (Constant  is called the yield stress.) It is worth emphasizing that these models can be rewritten as DD

1 .jSj   /C S; 2 .jDj/ jSj

(235)

where x C D maxfx; 0g, which shows that the yield stress models also fall into the class (219). This observation can be exploited in the discussion of the physical and mathematical properties of the models; see Rajagopal and Srinivasa [79] and Bulíˇcek et al. [13]. Note that if (234) holds, then S cannot be considered as a function of D, while (235) gives a functional (continuous) dependence of D on S. Since the function at the right-hand side of (235) is tacitly supposed to be zero for S D 0, the alternative viewpoint given by (219) with a continuous tensorial function f defined on the

64

J. Málek and V. Pr˚uša

Cartesian product of S and D allows one to avoid description of the material via a multivalued or discontinuous function. See Bulíˇcek et al. [13] and Bulíˇcek et al. [10] for exploiting this possibility as well as the symmetric roles of S and D in (219) in the analysis of the corresponding initial and boundary value problems. Another advantage of the formulation (235) is that it allows one to replace, in a straightforward way, the constant yield stress  by a yield function that can depend on the invariants of both S and D,  D  .S; D/. Finally, the class of implicit relations (219) can be viewed as a subclass of models where the full Cauchy stress tensor T and the symmetric part of the velocity gradient D are related implicitly, f.T; D/ D 0:

(236)

This apparently subtle difference has significant consequences. Constitutive relations in the form (236) provide, contrary to (219), a solid theoretical background to incompressible fluid models where the viscosity depends on the pressure (mean normal stress); see Rajagopal [74] and Málek and Rajagopal [56] for an in-depth discussion. Such models have been proposed a long time ago by Barus [2], and the practical relevance of such models has been demonstrated by Bridgman [6] and Bridgman [7]. In the simplest settings, the model for a fluid with pressure dependent viscosity can take the form T D pI C 2 ref eˇ.ppref / D;

(237)

where pref – the reference pressure – and ref – the reference viscosity – are positive constants. If necessary, this model can be combined with power law type models. This is a popular choice in lubrication theory; see, for example, Málek and Rajagopal [56].

4.6

Boundary Conditions for Internal Flows of Incompressible Fluids

As it has been already noted, the specification of the boundary conditions is a nontrivial task. Since the boundary conditions can be seen as a special case of constitutive relations at the interface between two materials, thermodynamical considerations could be of use even in the discussion of the boundary conditions. A simple example concerning the role of thermodynamics in the specification of the boundary conditions for incompressible viscous heat nonconducting fluids is given below. The problem of internal flow in a fixed vessel represented by the domain has been discussed in Sect. 3.2, and it is revisited here in a slightly different setting. In Sect. 3.2 the boundary condition on the vessel wall has been the no-slip boundary condition

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

vj@ D 0:

65

(238)

In the present case, one enforces only the no-penetration boundary condition v  nj@ D 0;

(239)

where n denotes the unit outward normal to , and the question concerning the value of the velocity vector in the tangential direction to the vessel wall is for the moment left open. Introducing the notation u Ddef u  .u  n/ n

(240)

for the projection of any vector u to the tangent plane to the boundary, it follows that the product .Tv/  n can be rewritten as .Tv/  n D mv  n C S W .v ˝ n/ D S W .n ˝ v/ D .Sn/  v D .Sn/  v :

(241)

This observation is a consequence of the symmetry of the Cauchy stress tensor and the decomposition of the Cauchy stress tensor to the mean normal stress and the traceless part; see (218). The multiplication of the balance of momentum (63a) by v followed by integration over the domain yields Z

d dt



1  vv 2



Z .div T/  v dv:

dv D

(242)



The right-hand side can be, following Sect. 3.2, manipulated as Z

Z

Z

.div T/  v dv D

Z

.div Tv/ dv 

Z

T W D dv D 

T W D dv C

Tv  n ds: @

(243)

Consequently, the evolution equation for the net kinetic energy Ekin reads dEkin D dt

Z

Z T W D dv C

Tv  n ds;

(244)

@

which can be in virtue of (241) and the incompressibility condition div v D 0 further rewritten as dEkin D dt

Z

Z S W D dv 

s  v ds; @

(245)

66

J. Málek and V. Pr˚uša

where the notation s Ddef  .Sn/

(246)

has been used. Unlike in Sect. 3.2, one now gets a boundary term contributing to the evolution equation for the net kinetic energy. The boundary term is the key to the thermodynamically based discussion of the appropriate boundary conditions. The desired decay of the net kinetic energy is in the considered case guaranteed if one enforces pointwise positivity of the product T W D in and the pointwise positivity of the product s  v at @ . This restriction can be used to narrow down the class of possible relations between the value of the velocity in the tangential direction v and the projection of the stress tensor .Sn/ , where the relation between v and .Sn/ is the sought boundary condition. R Note that in the full thermodynamic setting, both the volumetric term T W D dv R and the boundary term @ s  v ds would appear in the entropy production for the whole system. Both the boundary term and the volumetric term have the flux/affinity structure. In linear nonequilibrium thermodynamics, the fluxes and affinities in the volumetric term are connected linearly via a positive (nonnegative) coefficient of proportionality in order to guarantee the validity of the second law of thermodynamics. Application of the same approach to the boundary term leads to a linear relation between s and v s D  v ;

(247)

where  is a positive constant. This is the Navier slip boundary condition; see Navier [63]. Moreover, one can identify two cases where the boundary term vanishes. If s D 0, then one gets the perfect slip boundary condition, and if v D 0, then one gets the standard no-slip boundary condition. Further, a general relation between s and v can take, following (219), the form of an implicit constitutive relation f .s; v / D 0;

(248)

which considerably expands the number of possible boundary conditions. In particular, the threshold-slip (or stick-slip) boundary condition that is usually described by the dichotomy relation jsj   , v D 0

and

jsj >  , s D

 v C  v  ; jv j

(249)

where  and  are positive constants, can be seen as a special case of (248). Indeed, (249) can be rewritten as v D

1 .jsj   /C s:  jsj

(250)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

67

(Note the similarity between the threshold-slip boundary condition (249) and the Bingham/Herschel-Bulkley model for fluids with the yield stress behavior (235). The formulation of the threshold-slip boundary condition in the form (250) can be again exploited in the mathematical analysis of the corresponding governing equations; see Bulíˇcek and Málek [11].) An example of more involved thermodynamical treatment of boundary conditions can be found in the study by Heida [37]. Concerning a recent review of the nonstandard boundary conditions used by practitioners in polymer science, the reader is referred, for example, to Hatzikiriakos [36].

5

Conclusion

Although the classical (in)compressible Navier-Stokes-Fourier fluid models have been successfully used in the mathematical modelling of the behavior of various substances, they are worthless from the perspective of modern applications such as polymer processing. Navier-Stokes-Fourier models are simply incapable of capturing many phenomena observed in complex fluids; see, for example, the list of non-Newtonian phenomena in Málek and Rajagopal [54], the historical essay by Tanner and Walters [90], or the classical experimentally oriented treatises by Coleman et al. [20], Barnes et al. [1], or Malkin and Isayev [60] to name a few. The need to develop mathematical models for the behavior of complex materials leads to the birth of the theory of constitutive relations. In the early days of the theory, constitutive relations have been designed by appealing to purely mechanical principles. This turns out to be insufficient as the complexity of the models increases. Nowadays, the mathematical models aim at the description of an interplay between various mechanisms such as heat conduction, mechanical stress, chemical reactions, electromagnetic field, or the interaction of several continuous media in mixtures; see, for example, Humphrey and Rajagopal [44], Rajagopal [75], Dorfmann and Ogden [24], and Pekaˇr and Samohýl [68]. In such cases the correct specification of the energy transfers is clearly crucial. Consequently, one can hardly hope that an ad hoc specification of the complex nonlinear constitutive relations for the quantities that facilitate the energy transfers is the way to go. A theory of constitutive relations that focuses on energy transfers and that guarantees the compatibility of the arising constitutive relations with the second law of thermodynamics is needed. The modern theory of constitutive relations outlined above can handle the challenge. The theory from the very beginning heavily relies on the concepts from nonequilibrium thermodynamics, and the restrictions arising from the laws of thermodynamics are automatically built into the derived constitutive relations. Naturally, the models designed to describe the behavior of complex materials in farfrom-equilibrium processes are rather complicated. In particular, the arising systems of partial differential equations are large and nonlinear.

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Fortunately, the available numerical methods for solving nonlinear partial differential equations as well as the available computational power are now at the level that makes the numerical solution of such systems feasible. This is a substantial difference from the early days of the mechanics and thermodynamics of continuous media; see, for example, Truesdell and Noll [91]. Since the quantitative predictions based on complex models are nowadays within the reach of the scientific and engineering community, they can actually serve as a basis for answering important questions in the applied sciences and technology. This is a favorable situation for a mathematical modeller equipped with a convenient theory of constitutive relations. The design of suitable mathematical models for complex materials undergoing farfrom-equilibrium processes does matter – from the practical point of view – more than ever.

6

Cross-References

 Concepts of Solutions in the Thermodynamics of Compressible Fluids  Equations for Viscoelastic Fluids  Multi-Fluid Models Including Compressible Fluids  Solutions for Models of Chemically Reacting Compressible Mixtures

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2

Variational Modeling and Complex Fluids Mi-Ho Giga, Arkadz Kirshtein, and Chun Liu

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Nonequilibrium Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Energetic Variational Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hookean Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Gradient Flow (Dynamics of Fastest Descent) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Basic Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Flow Map and Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Newtonian Fluids and Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Elasticity and Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Other Approaches to Elastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Generalized Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Complex Fluid Mixtures: Diffusive Interface Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Surface Tension and the Sharp Interface Formulation . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diffusive Interface Approximations (Phase Field Methods) . . . . . . . . . . . . . . . . . . . 4.3 Boundary Conditions in the Diffusive Interface Models . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 76 76 79 80 82 82 84 86 89 93 97 97 99 105 106 108 108

M.-H. Giga Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan e-mail: [email protected] A. Kirshtein () Department of Mathematics, Pennsylvania State University, University Park, PA, USA e-mail: [email protected] C. Liu Department of Mathematics, Pennsylvania State University, University Park, PA, USA Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_2

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Abstract

In this chapter, a general energetic variational framework for modeling the dynamics of complex fluids is introduced. The approach reveals and focuses on the couplings and competitions between different mechanisms involved for specific materials, including energetic contributions vs. kinematic transport relations, conservative parts vs. dissipative parts and kinetic parts vs. free energy parts of the systems, macroscopic deformation or flows vs. microscopic deformations, bulk effects vs. boundary conditions, etc. One has to notice that these variational approaches are motivated by the seminal works of Rayleigh (Proc Lond Math Soc 1(1):357–368, 1871) and Onsager (Phys Rev 37(4):405, 1931; Phys Rev 38(12):2265, 1931). In this chapter, the underlying physical principles and background, as well as the limitations of these approaches, are demonstrated. Besides the classical models for ideal fluids and elastic solids, these approaches are employed for models of viscoelastic fluids, diffusion, and mixtures.

1

Introduction

The focus of this chapter is on the mathematical modeling of anisotropic complex fluids whose motion is complicated by the existence of mesoscales or subdomain structures and interactions. These complex fluids are ubiquitous in daily life, including wide varieties of mixtures, polymeric solutions, colloidal dispersions, biofluids, electro-rheological fluids, ionic fluids, liquid crystals, and liquid-crystalline polymers. On the other hand, such materials often have great practical utility since the microstructure can be manipulated by external field or forces in order to produce useful mechanical, optical, or thermal properties. An important way of utilizing complex fluids is through composites of different materials. By blending two or more different components together, one may derive novel or enhanced properties from the composite. The properties of composites may be tuned to suit a particular application by varying the composition, concentration, and, in many situations, the phase morphology. One such composite is polymer blends [121]. Under optimal processing conditions, the dispersed phase is stretched into a fibrillar morphology. Upon solidification, the long fibers act as reinforcement and impart great strength to the composite. The effect is particularly strong if the fibrillar phase is a liquidcrystalline polymer [99]. Another example is polymer-dispersed liquid crystals, with liquid crystal droplets embedded in a polymer matrix, which have shown great potential in electro-optical applications [127]. Unlike solids and simple liquids, the model equations for complex fluids continue to evolve as new experimental evidences and applications become available [97]. The complicated phenomena and properties exhibited by these materials reflect the coupling and competition between the microscopic interactions and the macroscopic dynamics. New mathematical theories are needed to resolve the ensemble of microelements, their intermolecular and distortional elastic interactions, their coupling to hydrodynamics, and the applied electric or magnetic fields. The most

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common origin and manifestation of anomalous phenomena in complex fluids are different “elastic” effects [77]. They can be attributed to the elasticity of deformable particles; elastic repulsion between charged liquid crystals, polarized colloids, and multicomponent phases; elasticity due to microstructures; or bulk elasticity endowed by polymer molecules in viscoelastic complex fluids. These elastic effects can be represented in terms of certain internal variables, for example, the orientational order parameter in liquid crystals (related to their microstructures), the distribution density function in the dumbbell model for polymeric materials, the magnetic field in magnetohydrodynamic fluids, the volume fraction in mixture of different materials, etc. The different rheological and hydrodynamic properties will be attributed to the special coupling (interaction) between the transport (macroscopic fluid motions) of the internal variable and the induced (microscopic) elastic stress [115,116]. This coupling gives not only the complicated rheological phenomena but also formidable challenges in analysis and numerical simulations of the materials. The common feature of the systems described in this chapter is the underlying energetic variational structure. For an isothermal closed system, the combination of the first and second laws of thermodynamics yields the following energy dissipation law [6, 11, 39, 56]: d total E D ; dt

(1)

where E total is the sum of kinetic energy and the total Helmholtz free energy and  is the entropy production (here the rate of energy dissipation). The choices of the total energy functional and the dissipation functional, together with the kinematic (transport) relations of the variables employed in the system, determine all the physical and mechanical considerations and assumptions for the problem. The energetic variational approaches are motivated by the seminal work of Rayleigh [106] and Onsager [100, 101]. The framework, including Least Action Principle and Maximum Dissipation Principle, provides a unique, well-defined, way to derive the coupled dynamical systems from the total energy functionals and dissipation functions in the dissipation law (1) [67]. Instead of using the empirical constitutive equations, the force balance equations are derived. Specifically, the Least Action Principle (LAP) determines the Hamiltonian part of the system [2, 5, 50], and the Maximum Dissipation Principle (MDP) accounts for the dissipative part [11, 101]. Formally, LAP represents the fact that force multiplies distance is equal to the work, i.e., ıE D force  ıx; where x is the location and ı the variation/derivative. This procedure gives the Hamiltonian part of the system and the conservative forces [2, 5]. The MDP, by Onsager and Rayleigh [67, 100, 101, 106], produces the dissipative forces of the system, ı 12  D force  ı x: P The factor 12 is consistent with the choice of quadratic form for the “rates” that describe the linear response theory for longtime near-equilibrium dynamics [74]. The final system is the result of the balance of all these forces (Newton’s Second Law). Both total energy and energy dissipation may contain terms related to microstructure and those describing macroscopic flow. Competition between different parts

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of energy, as well as energy dissipation, defines the dynamics of the system. The main goal of this chapter is on describing the role of microstructures in the special coupling between the kinematic transport and the induced “elastic” stresses.

2

Nonequilibrium Thermodynamics

In this section, some basic thermodynamic principles and general relations between energy laws and differential equations are described. We first clarify notation of variations of the functionals [50, 54]. Let E D E. / be a functional depending on a function in a space H which is equipped with an inner product h ; iH . The variation ıE D ı E of a function E is defined as ı E. / D lim ŒE. h!0

C hı /  E. / =h;

where ı is a function so that C hı is a variation of . The quantity ı E is often called a directional derivative in the direction of ı at . It is formally the Gâteau derivative of E at in the direction of ı . If ı E can be written as ı E. / D hf; ı iH ; with some f for a big class of ı , we often write f by H_

ıE ı

or simply

ıE : ı

This quantity corresponds to the total derivative or the Fréchet derivative if the latter is well defined [55]. It is simply called the variational derivative . In this notation, denominator points to the function with respect to which the variation of the functional in the numerator is taken.

2.1

Energetic Variational Approaches

The first law of thermodynamics [56] states that the rate of change of the sum of kinetic energy K and the internal energy U can be attributed to either the work WP P done by the external environment or the heat Q: d P .K C U / D WP C Q: dt In other words, the first law of thermodynamics is really the law of conservation of energy. It is noticed the internal energy describes all the interactions in the system. In order to analyze heat, one needs to introduce the entropy S [56], which naturally leads to the second law of thermodynamics [39, 56]:

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77

dS P C DQ dt

where T is the temperature and S is the entropy.  is the entropy production which is always nonnegative and gives the rate of energy dissipation in irreversible systems. Subtracting the two laws, in the isothermal case when T is constant, one arrives at: d .K C U  T S/ D WP  ; dt where F D U  T S is called the Helmholtz free energy. We denote E total D K C F to be total energy of the system. If the system is closed, when work done by the environment WP D 0, the energy dissipation law of the system can be written as dE total D 2D: dt

(2)

The quantity D D 12  is sometimes called the energy dissipation. The dissipative law allows one to distinguish two types of systems: conservative (or Hamiltonian) and dissipative. The choices of the total energy components and the energy dissipation take into consideration all the physics of the system and determine the dynamics through the two distinct variational processes: Least Action Principle (LAP) and Maximum Dissipation Principle (MDP). To derive the differential equation describing the conservative system ( D 0), one employs the Least Action Principle (LAP) [2, 5], which says that the dynamics is determined as a critical point of the action functional (Remark 1 below). We RT RT give its equivalent form. We consider functionals 0 Kdt and 0 Fdt defined for a function x (the trajectory in Lagrangian coordinates, if applicable) depending on space time variables. The inertial and conservative force from the kinetic and free energies are, respectively, defined as RT ı 0 Kdt forceinertial D H _ ; ıx RT ı 0 Fdt : forceconservative D H _ ıx The space H is typically taken as the space time L2 space, L2x;t , i.e., L2x;t D L2 .0; T I L2x /, where L2x is the L2 space in the spatial variables. (These are called variational forces.) In other words, for all ıx, Z T Z T Kdt D hforceinertial ; ıxiL2x;t D hforceinertial ; ıxiL2x dt ı Z ı 0

0

T

Fdt D hforceconservative ; ıxiL2x;t D

Z

0

T 0

hforceconservative ; ıxiL2x dt:

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The LAP can be written as Z

T

ı 0

Kdt D ı

Z

T

Fdt

0

for all ıx. This gives the natural variational form (the weak form) of the forces with suitable test functions ıx. The strong form of the system (Euler-Lagrange equation) can be also written as a force balance (without dissipation). forceinertial D forceconservative :

(3)

The inertial force corresponds to the inertial term ma in Newton’s Second Law, where a is the acceleration and m is the mass. Note that if the variation is performed on a bounded domain and involves integration by parts, one has to assume specific boundary conditions to cancel the boundary terms, so that no boundary effects are involved. Remark 1. The standard approach [5] dictates to define the Lagrangian functional RT L D K  F and the action functional as A .x/ D 0 Ldt . The Euler-Lagrange D 0. equation in this case is H _ ıA ıx For a dissipative system . D 2D > 0/, according to Onsager [100, 101], the dissipation D is taken to be proportional to a “rate” xt raised to a second power. The Maximum Dissipation Principle (MDP) [67] implies that the dissipative force may be obtained by minimization of the dissipation functional D with respect to the above mentioned “rate.” Hence, through MDP, the dissipative force (linear with respect to the same rate function) can be derived as follows: ıD D hforcedissipative ; ıxt iHQ : In other words, Q _ıD=ıxt D forcedissipative : H Note that the test function in MDP is different from that in LAP before. Remark 2. It is important to note that although the limitation for the dissipation D to be quadratic in “rate” is rather restrictive, strong nonlinearities can be introduced through coefficients independent of the “rate.” When all forces are derived, according to the force balance (Newton’s Second Law, where inertial force plays role of ma): forceinertial D forceconservative C forcedissipative :

(4)

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Notation. For shorter notation, one can write Eq. (4) as H _ H_

ı

RT

0 F dt ıx

ı

RT 0

Kdt ıx

D

Q _ ıD with H D L2 .0; T I H Q /. CH ıxt

It is important to notice that Eq. (4) uses the strong form of the variational result. This might bring ambiguity in the original variational weak form, since the test functions may be in different spaces.

2.2

Hookean Spring

As a start, a simple ordinary differential equations (ODE) example of a dissipative system is considered here, which had been originally proposed by Lord Rayleigh [106]: the Hookean spring of which one end is attached to the wall and the other end to a mass m (see Fig. 1). Let x .t / be a displacement of the mass from the equilibrium. Consider that the spring has friction-based damping which is proportional to the velocity (relative friction to the resting media). Under these assumptions, KD

mxt2 ; 2

FD

kx 2 ; and 2

DD

 xt2 ; 2

where k is spring strength material parameter and  is damping coefficient. The energy dissipation law is clearly as follows: d dt



mxt2 kx 2 C 2 2



D  xt2 :

The corresponding action functional of the spring [50] in terms of the position x.t /: AD

Z 0

T



mxt2 kx 2  2 2

 dt:

Then the Least Action Principle, i.e., variation with respect to the trajectory x.t /, yields [50]:

Fig. 1 Spring attached to a wall on one end, with mass m on the other end

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Z ıA D 0

Z

T

Œmxt .ıx/t  kxıx dt D  D

T

.mxt t  kx/ ıx dt 0

ıA L2t _ ; ıx ıx



Z

T



D L2 .0;T /

0

ıA L2t _ ; ıx ıx

 dt: R

Here the space H with inner product is L2t D L2 .0; T / because here L2x is just R. On the other hand, the principle of maximum dissipation gives R_

ıD D  xt : ıxt

Indeed, looking at forces involved and formulating Newton’s Second Law (F D ma) for this system, one can get mxt t D kx   xt ; or equivalently mxt t C  xt C kx D 0;

(5)

D which is equivalent to the variational force balance (corresponding to (4)) L2t _ ıA ıx ıD R_ ıx for this example. t Looking at the explicit solution of (5), it is clear that the Hamiltonian part describes the transient dynamics, the oscillation near initial data, while the dissipative part gives the decaying longtime behavior near equilibrium.

2.3

Gradient Flow (Dynamics of Fastest Descent)

The energetic variational approaches have many different forms in practices and applications. Next look at the familiar example of gradient flow (dynamics of fastest descent): ıF.'/ (6)  C 't D 0; ı' where F is a general energy functional in terms of '. Here ' is a function of spatial variables with parameter time t, and the constant  > 0 is the dissipation rate which determines the evolution approaching the equilibrium. Such a system has been used in many applications both in physics and in mathematics; in particular, it is commonly used in both analysis and numerics to achieve the minimum of a given energy functional. It is clear that with natural boundary conditions (Dirichlet or Neumann), the solution of (6) satisfies the following energy dissipation law (by chain rule and integration by parts, if needed): Z d 1 F D j't j2 d x; dt   where  is a domain in a Euclidean space.

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On the other hand, one can put this in the general framework of energetic variational approaches. Notice that there is no kinetic energy in this system, indicating the nature of being the longtime near-equilibrium dynamics. Notation. When working on a bounded domain, one should consider a variation up to the boundary. Generally, for a functional E depending on a function defined in , the variation ı E is often of the form ı E. / D

Z

Z f ı dx C 

gı dS: @

Then, we denote f D

ıE ; ı

gD

ıE : ı@

Here f gives variational force inside the domain, while g is a kind of a boundary force. So if the boundary is taken into account, boundary forces should be also balanced. Unless mentioned otherwise, in this chapter, specific boundary conditions are taken to cancel the boundary effects (i.e., make boundary integralR equal to zero). R In the case of F D  W .'; rx '/ d x and D D 21  j't j2 d x, the variation leads to the following two variational derivatives: L2x _

ıF @W @W D r  C ; ı ' @r' @'

L2x _

ıD 1 D 't ; ı  't 

which after substitution in (4) yield equation (6). In this case, the boundary effects would be canceled out in case of homogeneous Dirichlet or Neumann boundary conditions. Remark 3. To derive implicit Euler’s time discretization scheme [8], one may consider minimization of the following functional: Z ( min ' nC1 given ' n



) ˇ ˇ2  nC1  1 ˇ' nC1  ' n ˇ C W ' ; r' nC1 d x:  2t

By introducing time discretization, one can avoid the two different variations and only take the variation with respect to ' nC1 . However, the scheme often fails in the case of  dependent on ', since it is unclear whether to take it explicit or implicit: explicit may cause stability issues and implicit will lead to a highly nonlinear system.

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3

Basic Mechanics

Before moving on to more complicated and realistic applications, it is important first to introduce some basic terminologies and concepts in continuum mechanics [36,59]. In particular, in this section, the relation between Eulerian (space reference) and Lagrangian (material reference) coordinates [119] is explored, and the variational techniques in terms of deformable medium are described. In this section, the boundary conditions are not in the focus of attention. However they may and should also be derived through the variational procedure with various specific boundary energy terms and dissipative terms.

3.1

Flow Map and Deformation Gradient

For a given velocity field u .x; t/, one can define the corresponding flow map (trajectory) x .X; t/ as xt D u;

x .X; 0/ D X:

(7)

In other words, x .X; t/ describes the position of a particle moving with velocity u and initial position X. Here x are the Eulerian coordinates, and X – the Lagrangian coordinates or initial configuration (see Fig. 2). Since the flow map should satisfy (7), its recovery is possible only if u .x; t/ has certain regularity properties, for instance, being Lipschitz in x [36]. In order to describe the evolution of structures or patterns (configurations), it is clear that one needs to consider the matrix of partial derivatives, the Jacobian matrix, the deformation gradient (or deformation tensor) [61]: F .X; t/ D

@x .X; t/ : @X

If one writes F by components .Fij /, our convention is Fij D

Fig. 2 A schematic illustration of a flow map x .X; t/. For t fixed x maps 0X to tx . For X fixed x .X; t/ is the trajectory of X

@xi : @Xj

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Then by chain rule: @Fij @ D @t @t



@xi @Xj

 D

@ @Xj



@xi @t

 D

X @ui @xk @ ui .x .X; t/ ; t / D ; @Xj @xk @Xj k

which in Eulerian coordinates will take the form as: @ @F Q D u .x .X; t/ ; t / D .rx u/ F: FQ t C .u  rx / FQ D @t @X Here FQ .x .X; t/ ; t / D F .X; t/ and rx denote the gradient. In Eulerian coordinates, FQ satisfies the following important identity: Q FQ t C .u  rx / FQ D .rx u/ F:

(8)

Remark 4. The form of (8) is directly related to the equation of vorticity w D curl u in inviscid incompressible fluids [94]: in two-dimensional cases, the solution of wt C .u  r/ w D 0 is expressed along the trajectory as w .x .X; t/ ; t / D w0 .X/; in three-dimensional case wt C .u  r/ w  .w  r/ u D 0, the solution becomes w .x .X; t/ ; t / D F w0 .X/. It is clear that the stretch term .w  r/u is the direct consequence of the deformation F , although F itself is absent from the original fluid equations. Remark 5. Incompressibility condition is actually a restriction on deformation det F D 1. By using Jacobi’s formula, 

0 D @t det F D det F  tr F

1





@X @u @t F D 1  tr @x @X

 D tr .rx u/ D rx  u;

which yields the usual incompressibility condition in conventional descriptions of fluids. Notice that the nonlinear constraint in Lagrangian coordinates becomes a linear one in Eulerian coordinates. (Here rx  u denotes the divergence of u.) Remark 6. F also determines the kinematic relations of transport of various physical quantities. The following formulations of kinematic relations describing transport of scalar quantities are expressed in Eulerian and Lagrangian coordinates as: 't C .u  rx / ' D 0 is equivalent to ' .x .X; t/ ; t / D ' .X; 0/ ; 't C rx  .'u/ D 0 is equivalent to ' .x .X; t/ ; t / D

' .X; 0/ : det F

(9) (10)

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3.2

Newtonian Fluids and Navier-Stokes Equations

Next the classic Newtonian fluids [36] are examined, and the Navier-Stokes equations are derived from the energetic variational approaches. Consider fluid with density  and velocity field u. Here the local mass conservation law is postulated, i.e., t C rx  .u/ D 0:

(11)

For fluids, one should consider the free energy depending only on the density  (the single most important characterization of the material being a fluid), which implies the following energy dissipation law: d dt

Z " 

3 ˇ ˇ  ˇ T ˇ2  2  juj ˇ ru C .ru/ ˇ 2 C ! ./ d x D  42 ˇ ˇ C    jr  uj 5d x; ˇ ˇ 2 2 3  2

#

Z

2

(12) ˇ ˇ P .ui;j Cuj;i /2 T ˇ2 ˇ where ˇ ruC.ru/ , ui;j D @ui =@xj . In general for matrix ˇ D i;j 2 2 p M , we write jM j D trMM T which is often called the Hilbert-Schmidt norm R R 2 or the Frobenius norm. Then K D  juj d x; F D  ! ./ d x; D D 2  ˇ ˇ  R ˇ ruC.ru/T ˇ2  1 2  ˇ ˇ C 2   13  jr  uj d x. The last being the viscosity contribu 2 tion [76], the relative friction between particles of the fluids. The constants  and  are called coefficients of viscosity ( is second viscosity coefficient), and !./ is free energy density. Since the rate in the dissipation is u D xt , one will have to take the variation with respect to the flow map x in the Lagrangian coordinates X. Since d x D .det F /X, and since (11) and (10) imply  .x .X; t/ ; t/ D 0 .X/ = det F .X; t/ with 0 .X/ D  .X; 0/, we observe that ı

ı

RT 0

RT 0

R T R 0 .X/ RTR .X; t/j2 det FdXdt D ı 0 12 0 .X/ jxt .X; t/j2 d Xdt Kdt D ı 0 12 det F jxt RT R RT R D 0 0 xt  ıxt d Xdt D  0 0 xt t  ıxd Xdt  RT R

d  dt u .x .X; t/ ; t /  ıx d xdt D 0 RT R D 0 Œ .ut C .u  r/ u/  ıx d xdt D h .ut C .u  rx /u/ ; ıxiL2x;t ;  RT R  Fdt D ı 0 ! det0F det F d Xdt       RT R

d Xdt ! det0F det0F C ! det0F det F tr F 1 @ıx D 0 @X  RT R

D 0 ! ./  C ! ./ .rx  ıx/d xdt  ˛  ˝

RT R D 0 rx ! ./   ! ./  ıx d xdt D r ! ./  !./ ; ıx L2 : x;t

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The normal component of the variation ıx  is assumed to be zero at the boundary @, which follows from no penetration boundary condition. This gives the following force terms expressed in the strong PDE form: L2x;t _

ı

RT

Kdt D  .ut C .u  r/ u/ ; ı x

0

L2x;t _

ı

RT

 Fdt D r ! ./   ! ./ ; ı x

0

The second (conservative) force term is exactly the gradient of the thermodynamic pressure. In the absence of the dissipation, from the force balance (3) with LAP, one obtains the compressible Euler equations [118]: 8 ˆ ˆ 0 is the Hookean constant and constant 3 is subtracted to null the energy of the nondeformed material. Here and hereafter  denotes the Laplace operator, i.e.,  D r  r. This model is closely related to neo-Hookean materials, where free energy may also depend on det F [33]. In [59] free energy density for linear

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elastodynamics depends on E, symmetric part   of displacement gradient, related to deformation gradient F by E D 12 F C F T  I . Remark 8. In case of incompressible elasticity , in order to enforce the nonlinear constraint det F D 1, we are tempted to consider the variation of action integral " under volume preserving diffeomorphism x" , i.e., det ddxX D 1. However, it is more convenient to introduce Lagrange multiplier 'ˇand consider variation of I.x/ under " no constraint of variation x" , i.e., x0 D x; ddx" ˇ"D0 D ıx: Then the variation of I .x/ D

Z

T



KF 

Z

0



 @x ' .X; t/ det  1 d X dt @X

yields the following incompressible elasticity equation: 8 2 C p1 , B can be continuously extended to a bounded operator s;p

sC1;p

from H0 ./ to H0

./n satisfying

krBgkH s;p  C . 0

s;p

diam ./ n diam ./ / .1 C /kgkH s;p ; 0 R R 0

for some C > 0, where H0 ./ D .H s;p .//0 for s < 0. The proof of Proposition 4 relies on the observations that @k .Bf /i consists of weakly singular and Calderón-Zygmund type operators. The assertion for s > 0 then follows from Calderón-Zygmund theory, whereas the cases for s < 0 follow from interpolation and a study of the adjoint kernels. Observe that bounded Lipschitz domains   Rn can be written as a finite union of star-shaped domains. More precisely, there exists m 2 N and fG1 ; : : : ; Gm g   such that S  \ Gi is star-shaped with respect to some ball Bj for j D 1; : : : ; m and D m iD1 . \ Gi /. Applying this procedure to the divergence problem yields the following result. Theorem 4. RLet 1 < p < 1,   Rn be a bounded Lipschitz domain and f 2 Lp ./ with  f D 0. Then there exists B W Cc1 ./ ! Cc1 ./n with div Bf D f: s;p

sC1;p

Moreover, B can be extended continuously from H0 ./ to H0 s > 2 C p1 .

./n provided

A complete proof of the above theorem in the case s  0 is due to Galdi [74]; for a detailed proof, see [76]. The general case s > 2 C p1 is due to Geissert, Heck, and Hieber; see [82]. Remarks 4. a) Note that Bf is defined for all f 2 Lp ./, whereas Bogovskii [25] and Galdi [76] constructed solutions to (14). Hence, B may be regarded as an extension to a solution operator to (14). However, Bf is a solution to (14) R only if  f D 0.

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b) Mitrea, Mitrea, and Monniaux [151] studied generalizations of Bogovskii-type integral operators on large classes of function spaces and various boundary conditions. c) Costabel and McIntosh followed a different approach and proved that Bogovskiitype integral operators are classical pseudo-differential operators of order 1 1 with symbols in the Hörmander class S1;0 .Rn /. This implies that the associated operators act as bounded operators in a wide range of function spaces, including Hölder, Hardy, Sobolev, Besov, and Triebel-Lizorkin spaces. For details, see [45]. s d) The setting of mixed estimates of type L1 .Bp;q / is investigated in [49]. e) Note that the above Theorem 4 does not hold for p D 1 or p D 1. For a detailed discussion of these questions, we refer here to the work of Bourgain and Brezis [39]. In addition, a different approach to (14) is presented there.

2.4

The Stationary Stokes Equation and Elliptic Estimates

In this section, the stationary Stokes equation 8 < u C rp D f in ; div u D 0 in ; : u D 0 on @;

(15)

is considered, where f 2 Lp ./, 1 < p < 1 and   Rn ; n  2, is a domain. It is the aim to establish estimates of the form kr 2 ukLp ./ C krpkLp ./  C kf kLp ./ ; which one refers to as elliptic estimates. To this end, for m 2 N, let D m;p ./ WD fu 2 L1loc ./ W D ˛ u 2 Lp ./ for all j˛j D mg, and let the homogeneous Sobolev b m;p ./ be defined as space W b m;p ./ WD D m;p ./=Pm1 : W

(16)

Here Pm1 denotes the space of all polynomials of order at most .m  1/. Note b m;p ./ becomes a Banach space, when equipped with kuk m;p that W WD b W ./ 1=p P p ˛ . j˛jDm kD ukLp ./ n If  D RC , one may use Fourier transform in tangential direction to obtain the following result. For a proof, see, e.g., [76], Sect. 4.3. Lemma 1. Let 1 < p < 1, m 2 N0 and f 2 W m;p .RnC /n . Then Eq. (15) admits a solution .u; p/ 2 D mC2;p .RnC /  D mC1;p .RnC /, and for all k 2 Œ0; m, there is a constant C > 0 such that

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133

kr 2 ukW k;p .RnC / C krpkW k;p .RnC /  C kf kW k;p .RnC / : < 1 for Moreover, if .v; q/ is another solution to (15) satisfying kvkb W kC2;p .RnC / D kp  qkb D 0. some k 2 Œ0; m, then ku  vkb W kC2;p W kC1;p Applying a localization procedure yields the following result for bounded domains. Proposition 5 ([76]). Let 1 < p < 1, m 2 N0 , f 2 W m;p ./ and   Rn be a bounded domain of class C mC2 . Then there exists a solution .u; p/ 2 W mC2;p ./  W mC1;p ./ of (15). Moreover, u is unique and p is unique up to an additive constant. Furthermore, for k 2 Œ0; m, there is a constant C > 0 such that kukW kC2;p ./ C krpkW k;p ./  C kf kW k;p ./ : Remark 1. Note that the constant C in Proposition 5 depends on . For certain applications it is important to know whether C is independent of the size of . In this context, the reader is referred to a result by Heywood [107] saying that for p D 2; m D 0, and n D 3, the above constant C does not depend on the size of @ or . Indeed, if   R3 is bounded and of class C 3 and .u; p/ 2 H 2 ./n  H 1 ./ is a solution of (15) with f 2 L2 ./n , then kr 2 ukL2 ./  C .kP;2 f kL2 ./ C krukL2 ./ /; where C depends on the C 3 -regularity of @, only. Here P;2 denotes the Helmholtz projection for L2 ./. For exterior domains   Rn , the situation reads as follows: Proposition 6 ([76]). Let 1 < p < 1; m 2 N0 and   Rn be an exterior domain of class C mC2 and f 2 W m;p ./n . Let .u; p/ 2 D 2;p ./n  D 1;p ./ be a solution of (15). Then, for all k 2 Œ0; m, there exists C > 0 such that kr 2 ukW k;p ./ C krpkW k;p ./  C .kf kW k;p ./ C kukLp .D/ /;

(17)

where D WD  \ BR for some sufficiently large R > 0. In particular, kr 2 ukW k;p ./ C krpkW k;p ./  C .kf kW k;p ./ C krukLp ./ /: For a proof of the above estimates, see [76], Chap. 5, as well as [42] and [146]. One finally notes that if p < n2 , then estimate (17) can be improved to kr 2 ukW k;p ./ C krpkW k;p ./  C kf kW k;p ./ for all k 2 Œ0; m.

(18)

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The Stokes Equation and Operator in L2 ./

In this section, the L2 -theory for the Stokes operator and the Stokes semigroup is presented by making use of form methods. To this end, let H and V be Hilbert spaces with V ,! H and denote by h; i the scalar product on H . Consider a sesquilinear form a W V  V ! C, which is continuous, i.e., ja.u; v/j  M kukV kvkV ;

u; v 2 V;

for some M > 0 and coercive, i.e., Re a.u; u/ C ıkuk2H  ˛kuk2V ;

u 2 V;

for some ı 2 R and some ˛ > 0. The space V is called the domain of the form a. Given such a form, one associates with a an operator A on H by setting D.A/ WD fu 2 V W there exists v 2 H such that a.u; '/ D hv; 'i for all ' 2 V g; Au WD v: The A is called the operator induced by the form a W V  V ! C. In the following, let   Rn be an open set, n 2 N, and assume that  ¤ ;. One then sets 1 ./ WD fu 2 Cc1 ./n W div u D 0g; Cc; 1 ./ L2 ./ WD Cc;

kkL2 ./

1 1 ./ H0; ./ WD Cc;

kkH 1 ./

;

;

1 1 and defines the form a W H0; ./  H0; ./ ! C by setting

a.u; v/ WD

n Z X j D1 

ruj  rvj dx;

1 u; v 2 H0; ./:

(19)

Then a is continuous, coercive, and symmetric, i.e., a.u; v/ D a.v; u/ for all u; v 2 1 ./. Based on this form, the Stokes operator on L2 ./ is defined as follows. H0; Definition 2. Let   Rn be open, n 2 N,  ¤ ;, and a be defined as in (19). Then the Stokes operator with Dirichlet boundary conditions A;2 in L2 ./ is defined as the operator induced by a. The following properties of the Stokes operator A;2 are consequences of the definition via the form.

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Proposition 7. Let   Rn ; n 2 N, be an open set. Then the Stokes operator A;2 has the following properties. a) A;2 is injective, self-adjoint, and densely defined. b) The numerical range W .A;2 / is contained in Œ0; 1/. c) A;2 generates a bounded analytic semigroup T;2 on L2 ./ of angle =2 satisfying kT;2 .z/kL.L2 .//  1;

z 2 †=2 ; and

tkA;2 T;2 .t /kL.L2 .//  C;

t >0

for some C > 0. The semigroup T;2 is called the Stokes semigroup on L2 ./. It is remarkable that the definition of the Stokes semigroup on L2 ./ does not need any assumptions on the open set   Rn . However, in general, no characterization of D.A;2 / in terms of suitable function spaces seems to be available in this setting. Note that if   Rn , n  2, is either Rn ; RnC or a domain with compact boundary of class C 2 , then it will be shown in Sect. 2.8.1 (see also [185]) that D.A;2 / D H 2 ./n \ H01 ./n \ L2 ./: For bounded domains, Poincaré’s inequality and Rellich’s theorem on compact embeddings imply the following result. Proposition 8. Let   Rn be a bounded domain. Then the following assertions hold true. a) The Stokes semigroup T;2 is exponentially stable, i.e., there exists ı > 0 such that kT;2 .t /kL.L2 .//  e ıt ;

t > 0:

2 2 b) 0 2 %.A;2 / and A1 ;2 W L ./ ! L ./ is compact. 2 c) The space L ./ has an orthonormal basis . j /  D.A;2 / consisting of eigenfunctions of A;2 corresponding to a sequence of eigenvalues .j / such that

0 < 1  2     ! 1: Consider now the case where   Rn is an exterior domain of class C 2 . Then, Proposition 6 implies the following global mapping properties of T;2 . Proposition 9. For an exterior domain   Rn of class C 2 , there exists a constant C > 0 such that for all f 2 L2 ./

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kT;2 .t /f kL2 ./  kf kL2 ./ ; t > 0, 1 krT;2 .t /f kL2 ./  C t  2 kf kL2 ./ ; t > 0, kA;2 T;2 .t /f kL2 ./  C t 1 kf kL2 ./ ; t > 0, 1 kr 2 T;2 .t /f kL2 ./  C t  2 kf kL2 ./ ; t > 1.

Note that the proof of the last assertion d) makes use of the elliptic estimate for the Stokes system given in Proposition 6. Observe also that for 0 < t  1 one has kr 2 T;2 .t /f kL2 ./  C t 1 kf kL2 ./ .

2.6

The Stokes Equation in the Half-Space: The Case 1 < p < 1

In this section, the Stokes equation is being considered in the half-space RnC WD fx 2 Rn W xn > 0g. In this case, several rather explicit representation formulas for the solution of the Stokes equation or the associated resolvent problem may be derived. The representation formulas due to Solonnikov [187], Ukai [202] and Desch, Hieber, and Prüss [56] will be discussed in the following in some detail. The reader is also referred to the representation formula given by McCracken [139] and to the monograph [76] by Galdi. Some words about notation are in order. Given x 2 Rn , the components of x will be written as x D .x 0 ; xn /, where x 0 2 Rn1 . Using this notation, one writes u D .u0 ; un / for functions and R D .R0 ; Rn / for operators. Similarly, F 0 denotes then the Fourier transformation with respect to the variable x 0 2 Rn1 . Consider the Stokes problem in the half-space RnC , which is given by 8 @t u  u C rp D f ˆ ˆ < div u D 0 ˆ uD0 ˆ : u.0/ D u0

in RC  RnC ; in RC  RnC ; on RC  @RnC ; in RnC :

(20)

Denote by Rj , j D 1; : : : ; n, and by Sj , j D 1; : : : ; n  1, the Riesz operators corresponding to the symbols i j =j j and i j =j 0 j, respectively. Furthermore, one puts R WD .R1 ; : : : ; Rn /;

S WD .S1 ; : : : ; Sn1 /:

(21)

In the following, solution formulas either for the instationary system (20) or for the corresponding resolvent problem 8 < u  u C rp D f in RnC ; div u D 0 in RnC ; : u D 0 on @RnC :

(22)

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137

will be discussed. Note that (22) is obtained by applying Laplace transformation to (20). Most of the explicit formulas below strongly rely on the orthogonality of x 0 and xn and therefore do not generalize to curved boundaries, at least not in such an explicit form.

2.6.1 The Green Tensor In the following, in order to simplify the presentation, assume that n D 3. Based on the fundamental solution G of the heat equation given by G.t; x/ WD

1 exp.jxj2 =4t/; .4t/3=2

x 2 R3 ; t > 0;

which is the Gaussian heat kernel, and on the fundamental solution E of the Laplace equation given by E.x/ WD

1 ; 4jxj

x 2 R3 n f0g;

(23)

the fundamental solution for (20) was constructed first by Solonnikov in [187]. In terms of the Green tensor, the solution .u; p/ of (20) with right-hand side f satisfying div f D 0 and f3 jx3 D0 D 0 has the following explicit representation Z tZ



 G.t  ; x  y/  G.t  ; x  y  / f . ; y/d dy

u.t; x/ D R3C

0

C4

2 X

Z @xj

p.t; x/ D 4

Z tZ rE.x  y/

0

j D1 2 X

x3Z R2

0

"Z @xj

j D1

R2

@x3 E.x y/

G.t  ; y z/fj . ; z/ d d zdy1 dy2 jy3 D0

0

0

R3C

#

Z tZ E.x  y/

R2

G.t  ; y  z /fj . ; z/ d d zdy;

Z tZ

Z C

R3C

R3C

@z3 G.t  ; y  z/fj . ; z/ d d zdy1 dy2 jy3 D0 ; (24)

where y  D .y 0 ; yn /. Formula (24) is probably the most classical tool in order to examine well-posedness and asymptotic properties of the instationary Stokes equation in R3C . Until today the formula (24) serves as a powerful tool in order to derive decay and regularity properties for the solution .u; p/ of (20). It is very remarkable that Solonnikov already proved maximal regularity estimates for (20) in [187] in 1977. This pioneering result implies in particular that p the associated Stokes operator generates an analytic semigroup on L .R3C / for 1 < p < 1. For a comprehensive study of the fundamental solution and Green functions for Stokes systems, the reader is also referred to Galdi’s monograph [76].

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There exists also a version of Green’s formula for the resolvent problem (22), which was constructed first by McCracken in [139]. Based on this representation of the resolvent, she proved resolvent estimates for (22), and hence the Stokes operator p is the generator of a bounded analytic C0 -semigroup on L .R3C / for 1 < p < 1.

2.6.2 Ukai’s Formula The representation formula derived by Ukai [202] in 1987 reduces the desired representation for the solution of the Stokes equation to the one of the heat equation combined with Riesz operators. In a first step of its derivation, one formally applies div to (20) with f D 0 and obtains that p D 0. Taking tangential Fourier transform, one obtains O xn / D 0; .@2n  j 0 j2 /p. ;

xn > 0; 0 2 Rn1 :

Hence, the integrable solution of the above equation must be of the form 0

p. O 0 ; xn / D e j jxn pO 0 . /;

(25)

with a certain, at this point still unknown, trace p0 . The key point in Ukai’s approach is given by the fact that in view of (25), the pressure p also solves the equation .@n C j 0 j/p. O 0 ; xn / D 0;

xn > 0; 0 2 Rn1 :

Setting z WD .@n Cj 0 j/Oun and applying @n Cj 0 j to the n-th line of (20), a calculation shows that z solves the equation 8 t > 0; xn > 0; < @t z  z D 0; t > 0; zjxn D0 D 0; : z.0/ D jr 0 jV1 u0 ; xn > 0;

(26)

V1 f WD S  f 0 C fn ;

(27)

where

and where jr 0 j denotes the pseudo-differential operator with symbol j 0 j. This yields z D jr 0 je tD V1 u0 ; where D denotes the Dirichlet Laplacian. Since uO n jxn D0 D 0, one may recover uO n as uO n .t; 0 ; xn / D

Z

xn 0

k. 0 ; xn  s/.F 0 e tD V1 u0 /.t; 0 ; s/ ds;

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

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0

where k. 0 ; y/ WD j 0 je j jy for y > 0. For a function f W RnC ! R, denote its trivial extension to Rn by ef , and for a function g W Rn ! R, denote its restriction to RnC by rg. Then un D Ue

tD

0 1

V1 u0 WD r.F /

Z R

.ek/.x 0 ; xn  s/e.F 0 e tD V1 u0 /.t; x 0 ; s/ ds:

A direct calculation yields Uf D rR0  S .R0  S C Rn /ef:

(28)

In order to derive a corresponding representation for u0 , one sets V2 f WD f 0 C Sfn :

(29)

Then, similarly as above, one sees that also V2 u satisfies a homogeneous heat equation with initial data V2 u0 . Consequently, u0 D e tD V2 u0  S un : Plugging the representation for un into the n-th line of (20), one may derive the corresponding representation formula for the pressure term. Summarizing, one obtains thus the following result. Theorem 5 (Ukai’s formula [202]). Let u0 2 Lp .RnC / and f D 0. Then the solution of (20) can be represented as un D Ue tD V1 u0 ;

u0 D e tD V2 u0  S un ;

0

pD

e jr jxn .@n e tD V1 u0 /jxn D0 ; jr 0 j

where V1 , U , and V2 are given as in (27), (28), (29). Since V1 , U , and V2 essentially consist of Riesz operators, many Lp -properties for the Stokes equation are reduced by Theorem 5 to the ones for the heat equation. In particular, the following corollary holds true. Corollary 2. The solution operator u0 7! u to Eq. (20) for t  0 with f D 0, where u is given as in Theorem 5, defines a bounded analytic C0 -semigroup T on p L .RnC / for 1 < p < 1.

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The semigroup T is called the Stokes semigroup. It is not difficult to show that the generator of T coincides with the Stokes operator ARnC ;2 introduced in Definition 2 provided p D 2.

2.6.3 The Formula of Desch, Hieber, and Prüss The following representation formula for the solution of the Stokes resolvent problem (22) due to Desch, Hieber, and Prüss [56] will be discussed in the sequel. This formula also relies on the representation (25) for the pressure term p. For u, the following Ansatz is made: 0

Z

0

1

uO . ; xn / D

k . 0 ; xn ; s/.fO 0 . 0 ; s/  i 0 p. O 0 ; s// ds;

(30)

kC . 0 ; xn ; s/.fO n . 0 ; s/  @n p. O 0 ; s// ds;

(31)

0 n

0

Z

1

uO . ; xn / D 0

where k˙ . 0 ; xn ; s; / WD

1 0 0 .e !.j j/jxn sj ˙ e !.j j/.xn Cs/ /; 2!.j 0 j/

and !.j 0 j/ WD

p  C j 0 j2 ;

 2 C n .1; 0:

Here k and kC represent the tangential Fourier transform of the kernel corresponding to the resolvent of the Laplacian with Dirichlet and Neumann boundary conditions, respectively. By construction, div u D 0. While u0 satisfies Dirichlet boundary conditions, un jxn D0 has to be enforced. This is the key step, since due to (25) and by taking the trace in (31), the trace of the pressure is given by pO 0 . ; xn / D 

!.j 0 j/ C j 0 j j 0 j

Z

1

0 e !.j j/s fOn . 0 ; s/ ds:

(32)

0

Plugging this term into (31) and (30) leads to the following representation formula for the Stokes resolvent problem. Theorem 6 (Desch, Hieber, Prüss [56]). Let  2 C n .1; 0, 0 2 Rn1 and xn > 0. Then the solution of (22) can be represented as un D .  D /1 fn C Kfn ; 0

F 0 p.; 0 ; xn / D e j jxn pO 0 ;

u0 D .  D /1 f 0  SKfn ;

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141

where pO 0 is given by (32), S is defined as in (21), and K is defined by its Fourier transform by

F 0 Kfn .; 0 ; xn / WD

e !.z/xn  e zxn !.z/  z

Z

1

0

e !.j j/s fOn . 0 ; s/ ds:

0

Remarks 5. a) Theorem 6 says that the resolvent of the Stokes problem may be represented as the resolvent of the Dirichlet-Laplacian plus an additive remainder term. b) Based on this formula, one may easily show that the Stokes operator generates a bounded analytic semigroup on Lp .RnC / for 1 < p < 1. c) A slightly different method in order to estimate the terms in Theorem 6, based on the H 1 -calculus for the operator jr 0 j, is performed in [167]. This approach generalizes also to partial slip boundary conditions. d) Considering the spaces L1 .RnC / and L1 .RnC /, the above formula of Desch, Hieber, and Prüß exploits an advantage compared to the other formulas. In fact, the above representation formula yields resolvent estimates for the Stokes resolvent problem in L1 .RnC /. Hence, one may deduce from this formula that the Stokes operator generates a bounded analytic semigroup on solenoidal subspaces of L1 .RnC /. More detailed information in this direction will be given in Sect. 4. Since the Riesz operators are unbounded in L1 , it seems not to be possible to obtain such a result by Ukai’s formula. Note, however, that Solonnikov proved in [194] also estimates for the solution u of the Stokes equation in spaces of bounded functions. His approach is based on the Green tensor (24). e) It was shown by Maekawa and Miura [141] that in the case of the half-space RnC , p there exists an isomorphism V W Lp .RnC / ! L .RnC / such the Stokes semigroup p n tA tA e on L .RC / can be represented as e D Ve tD V 1 , where D denotes the Dirichlet Laplacian on Lp .RnC /. Since the negative Dirichlet-Laplacian D admits an R-bounded H 1 -calculus on Lp .RnC / for 1 < p < 1, a kernel estimate of the remainder term yields the following result. Corollary 3 ([56]). The negative Stokes operator ARnC ;p admits an R-bounded p R1 D 0. H 1 -calculus on L .RnC / for 1 < p < 1 with RH1 -angle A Rn ;p C

Remark 2. Note that the existence of a bounded H 1 -calculus for ARnC ;p on p L .RnC / can be deduced also from Ukai’s formula. In fact, applying Laplace transformation to the formula for u given in Theorem 5 and taking into account that the negative Dirichlet-Laplacian D admits a bounded H 1 -calculus on Lp .RnC / for 1 < p < 1, the latter property can be transferred in this way to the Stokes operator.

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The Stokes Equation in Layers: The Case 1 < p < 1

Techniques similar to those described above for the half-space RnC work also for infinite layer domains of the form  WD Rn1  .R; R/; where R > 0. In fact, applying tangential Fourier transformation, explicit solution formulas can be derived in this case, too. These formulas will not be recalled here in detail, but note that their structure is similar to the one of the half-space. The interested reader may find more information on layer-like domains, e.g., in [158, 159]. An approach to asymptotically flat layers is provided in [9], and the boundedness of the H 1 -calculus for the Stokes operator in this situation was proved in [10]. p Sectoriality of the Stokes operator on L ./ was proved by Abe and Shibata in [6, 7] and by Abels and Wiegner in [15]. The boundedness of the imaginary powers for the negative of the Stokes operator in a layer within the Lp -framework was obtained by Abels in [8]. Only one characteristic result is formulated explicity in the following. It follows, for instance, from the results obtained in [10]. Defining the Stokes operator on a layer as A;p u WD P u;

1;p

u 2 D.A;p / WD W 2;p ./ \ W0 ./ \ Lp ./;

the following result holds true. Theorem 7. Let n  2, 1 < p < 1 and  D Rn1  .1; 1/. Then A;p p R;1 admits an R-bounded H 1 -calculus on L ./ with RH 1 -angle A D 0. ;p Furthermore, 0 2 .A;p /. For interesting results with respect to the spaces L1 ./ and L1 ./, where  denotes a layer, see Sect. 4.1.

2.8

The Stokes Equation in Lp ./ for 1 < p < 1 and for Domains with Compact Boundaries

2.8.1 The Stokes Equation with Dirichlet Boundary Conditions In the following, the Stokes equation on domains   Rn with smooth boundaries is investigated by means of a localization procedure. This setting is the most classical situation, and its analysis started with the pioneering results by Sobolevskii [184] and Ladyzhenskaja [135]. The procedure will be explained for the resolvent problem

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

8 < u  u C rp D f in ; div u D 0 in ; : u D 0 on @;

143

(33)

where  is assumed to lie in a suitable sector of the complex plane. For simplicity, in this section only, domains with compact boundaries will be investigated. By employing finite coverings, one then may include domains of the following type: given n  2, a domain   Rn is called a standard domain, if  coincides with Rn , RnC , a bounded domain, an exterior domain, or a perturbed half-space. The first step in the localization procedure consists of choosing a finite covering m n 2 .Uj /m j D1 of   R with boundary of class C and a partition of unity .'j /j D1 m 2 subordinate to .Uj /m j D1 . In other words, .Uj ; 'j /j D1 is an atlas for the C -manifold @. Multiplying (33) with 'j leads to a localized perturbed version in Uj . Since the Stokes equations are invariant under rotations and translations, and by choosing Uj sufficiently small, one may assume that the localized version of Eq. (33) is either an equation on Rn or on a bent half-space Hj WD fx 2 Rn W xn > hj .x 0 /g;

(34)

with a certain bending function hj W Rn1 ! R. One further transforms the localized system on Hj by v.x 0 ; xn / WD .u ı /.x 0 ; xn / WD u.x 0 ; xn C hj .x 0 //;

.x 0 ; xn / 2 RnC :

(35)

The resulting system for v then is an equation on RnC . Summarizing, by this procedure the Stokes resolvent problem on a domain is reduced to finitely many equations on RnC or Rn . A fundamental problem arising at this stage is the fact that the condition div u D 0 is not preserved, neither under multiplication with a cutoff function nor by transformation (35). This might be the reason why the first proofs of analyticity of the Stokes semigroup or other properties like bounded imaginary powers for the Stokes operator on domains rely on different methods. In order to investigate the Eq. (33) via a localization procedure and to overcome this difficulty, the following three strategies were established: Strategy 1. One may replace the transformation (35) by   v D T u WD u ı  0; : : : ; 0; .r 0 hj ; 0/  .u ı / :

(36)

Geometrically speaking, this transformation leaves the outer normal at the boundary invariant. This implies div v D 0, i.e., the above transformation is volume preserving. The price one has to pay for this is a lift of the boundary smoothness from C 2 to C 3 . This is due to the fact that r 0 hj appears in the transformation, and since one deals with a second-order system, one requires

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existence of third-order derivatives of the bending functions hj . The transformation (36) is utilized, e.g., in [24, 84, 160, 187]. Strategy 2. Suppose the divergence problem div w D g, wjxn D0 D 0 admits a solution given by a Bogovskii operator B W g 7! w, as investigated in Sect. 2.3. Then one may correct the lacking solenoidality by the term Qv WD v  B div v:

(37)

Note that Q is a projection onto solenoidal functions that keeps Dirichlet boundary values, which is not the case for the Helmholtz projection. Besides the appearance of a couple of additional perturbation terms in the transformed equations, the price to pay here is that one has to prove independently the existence of a Bogovskii type operator including sufficient regularity properties. This approach to the Stokes system is performed, e.g., in [84] and [83]. Strategy 3. It is also possible to work with an inhomogeneous divergence condition right from the beginning. In this case, the inhomogeneous Stokes systems in Rn and in RnC , where div u D 0 replaced by div u D g, have to be solved. Localizing the inhomogeneous equations produces then perturbation terms not only in the first n lines of (33) but also in the divergence condition. In this context, it is not possible to work on an operator theoretical level, since p the ground space of the Stokes operator is L ./. Instead, the full system including div u D g and the pressure term rp has to be handled. Another inconvenience is that one has to find a “good” conditions for the function space for the right-hand side g in the divergence condition. Considering, e.g., exterior domains in the strong 1;p Lp ./-setting, i.e., u 2 W 2;p ./ \ W0 ./, a sufficient regularity condition for g is that b 1;p ./: g D div u 2 W 1;p ./ \ W Estimating the additional perturbation terms leads to further technicalities (see, e.g., [68]). Of course, each one of the above strategies (1), (2), and (3) has its advantages and its disadvantages. If one is interested in results, where the regularity of the boundary is not so important, then a combination of strategies (1) and (2) might be an elegant and a convenient way. This was already demonstrated by Solonnikov in [187] (see also [84] and [83]), and this approach will be outlined in the following for the resolvent problem. For a characterization of the existence of a strong Lp -solution of the inhomogeneous Stokes equation by its data, see Theorem 11 below. In order to explain the strategy outlined above in more detail, one first considers the Stokes resolvent problem on  D H , where H is a bent half-space with h being a suitable bending function defined in (34). Since  is a C 3 domain, one may assume h 2 BUC 3 .Rn1 /. This implies that the transformation T defined in (36) is an isomorphism between the Sobolev spaces involved up to order two. In particular,

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145

it can be shown that T 2 Lis .Lp .H /; Lp .RnC // \ Lis .D.AH;p /; D.ARnC ;p //; where D.A;p / denotes the domain of the Stokes operator A;p on . Hence, one may define AT WD TAH;p T 1 ; p

which is an operator in L .RnC / with domain D.AT / D D.ARnC ;p /. Utilizing the smallness of r 0 h, one proves that B WD AT  ARnC ;p is a relatively bounded perturbation of the Stokes operator ARnC ;p . Depending on the property one wishes to verify, one then needs to have this property valid for AT as well as a perturbation result for this property. Concerning the generation of a bounded holomorphic semigroup or the property of maximal Lq -regularity, a standard perturbation argument for sectorial operators or maximal regularity may be applied (see [54, 133]). For the existence of a bounded H 1 calculus, one needs a more refined perturbation result as the one which is provided in [52]. Note that these properties are invariant under conjugation with isomorphisms. Consequently, these properties remains true for AH;p . In the next step, one sets u WD

m X

'j uj ;

j D1

p WD

m X

'j pj ;

j D1

where .uj ; pj / is the restricted bent half-space solution to data fj D j f that corresponds to Uj . To be precise, modulo rotation and translation, one has Uj \ Hj D Uj \ ;

Uj \ @Hj D Uj \ @:

Then u solves the perturbed Stokes resolvent problem 8 P ˆ f C m < u  u C rp D P j D1 .uj 'j  ruj r'j C pj r'j / in ; u in ; div u D m j D1 j r'j ˆ : uD0 on @: (38) It remains to prove that the remainder terms are of lower order so that they may be absorbed into the terms on the left-hand side. The perturbation term in the divergence condition can be handled by employing properties of the Bogovskii operator as explained in strategy 2. Whereas the terms uj 'j and ruj r'j are standard, a further difficulty is represented by the terms pj r'j . This relates to the fact that, a priori, only an estimate on rp of the form

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krpkp  C kf kp uniformly in the resolvent parameter  is available, but no suitable estimate for the pressure itself. Resolvent or H 1 -estimates for u, however, require a suitable decay estimate for p in . This is provided by the following pressure lemma (see [160, Lemma 13]). Lemma 2 ([160]). Let  2 .0; =2/, 1 < p < 1, and .u; p/ 2 D.A;p /  b 1;p ./ the unique solution of the Stokes resolvent problem (33) on Lp ./. Then W for each ˛ 2 .0; 1=2p 0 / and every bounded C 1;1 -domain G  , we have kpG kp  C jj˛ kf kp ;

 2 † ; jj  1;

(39)

with C > 0 independent of  and f and where pG D p 

1 jGj

Z

p

G

p dx 2 L0 .G/ D fg 2 Lp .G/I

Z g dx D 0g: G p

A sketch of the proof of the above lemma is as follows. Note that L0 .G/0 D p0 1;p 0 2 W0 .G/ of L0 .G/. Due to (4), for every  2 L0 .G/, there is a solution div D  such that k kW 1;p0 .G/  C kkp0 . Utilizing rpG D rp and the fact that p0

rpG .x/ D .I  P /u.x/

x 2 ;

where P denotes the Helmholtz projector on , one obtains Z

Z

Z

pG  dx D  

rpG  

Z

dx D

.D u/.I  P / dx 

Œ.D /1˛ u.D /˛ .I  P / dx:

D 

It is now straight forward to derive the estimate (39) from here. Taking into account 0 the fact that D has bounded imaginary powers on Lp ./ (see, e.g., [52]), it 0 follows by (8) that D..D /˛ / D H 2˛;p ./. The crucial point here is that for 0 2˛;p 0 ./, so that one may shift the part ˛ 2 .0; 1=2p 0 /, one has H 2˛;p ./ D H0 .D /˛ of D onto .I  P / without getting boundary terms. Collecting all the estimates for the perturbation terms, it is then straightforward to derive the desired resolvent or H 1 -estimates for A;p . Summarizing, one obtains the following result. Theorem 8. Let n  2, 1 < p < 1,  2 .0; / and assume that   Rn is a standard domain with boundary of class C 3 . Then there exists a unique solution .u; p/ of (33) satisfying

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147

1

jjkukLp ./Cjj 2 krukLp ./Ckr 2 ukLp ./CkrpkLp ./  C'0 kf kLp ./ ;  2 † ; where C > 0 is independent of f; u, and p. Furthermore, setting R./f WD u./, the set  n o   1 R./;  2 rR./; r 2 R./ I  2 †  L Lp ./; Lp .; Rn /

(40)

is R-bounded. Remarks 6. a) Note that the case n D 2 and small  for exterior domains and perturbed half-spaces requires special methods and techniques, which have been developed by Abels in [9]. b) The first assertion of Theorem 8 still holds true for standard domains of class C 1;1 . Theorem 8 implies the following important consequences for the Stokes operator and the Stokes equation. Theorem 9. Let n  2, 1 < p; q < 1, J D .0; T / for some T > 0 and assume that   Rn is a standard domain of class C 3 . Then the Stokes operator defined by A;p u WD P u;

1;p

D.A;p / WD W 2;p ./ \ W0 ./ \ Lp ./

(41)

p

admits maximal Lq -regularity on L ./. In particular, the solution u to the Cauchy problem u0 .t /  Ap u.t / D f .t/; t > 0;

u.0/ D u0 ;

satisfies the estimate ku0 kLq .J ILp .// C kAp ukLq .J ILp .//  C .kf kLq .J ILp .// C ku0 kX /; p

for some C > 0 independent of f 2 Lq .J I L .// and u0 2 X WD p .L ./; D.Ap //11=q;q . p Moreover, Ap generates a bounded analytic C0 -semigroup on L ./ and a)  .A;p / D .1; 0 if  is Rn , RnC or an exterior domain, b)  .A;p / D .1;  for some  D ./ > 0 provided  is bounded, 22=q p c) u0 2 X if and only if u0 2 Bp;q ./ \ L ./ and u D 0 on @. Setting r D .Id  P /.  Ap /1 , one obtains the following results for the Stokes equation (1) and its corresponding resolvent Eq. (33).

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Corollary 4. Given the assumptions of Theorem 9, the Stokes equation (1) admits a 1;p p unique solution .u; / 2 W 1;q .J I Lp .//\Lq .J I W 2;p ./\W0 ./\L .// q b 1;p L .W .//, and there exists a constant C > 0 such that kut kLq .J ILp .// C kukLq .J ILp .// C kr 2 ukLq .J ILp .// C krkLq .J ILp .//  C .kf kLq .J ILp .// C ku0 kX /: Corollary 5. Let 1 < p < 1,   Rn as in Theorem 9,  2 .0; /, and f 2 p 1;p p L ./. Then there exists a unique .u; / 2 W 2;p ./ \ W0 ./ \ L .//  b 1;p ./ satisfying (33) and a constant C > 0 such that W jjkukLp ./ C kr 2 ukLp ./ C krkLp ./  C kf kLp ./ ;

 2 † ; f 2 Lp ./:

The above Theorems 8 and 9 go back to several authors, who proved various assertions of the above theorems by different methods and approaches during the last decades. Some comments on key contributions are in order at this point. First results on maximal Lp -regularity estimates for the instationary Stokes system (1) go back to the pioneering work of Solonnikov; see [186] for  D Rn ; RnC and [187] for domains with compact boundaries. For a modern approach to his results, then also in the mixed Lq  Lp -context, based on the characterization of maximal Lp -regularity by the R-boundedness property of the resolvent, see the work of Geissert, Heck, Hieber, Schwarz and Stavrakidis [84], and [83]. For the half-space RnC , results on the existence and analyticity of the Stokes semigroup go back to [32, 56, 68, 139, 167, 202]. Giga and Sohr [97] proved for the first time global-in-time mixed Lq  Lp maximal regularity estimates for smooth exterior domains by combining a result on the boundedness of the imaginary powers of the Stokes operator with the DoreVenni theorem [62]. A different approach to maximal Lp -regularity of the Stokes equation (1) based on pseudo-differential methods was developed by Grubb and Solonnikov [101]. It will be discussed in some detail in Sect. 3.1. Concerning the existence and analyticity of the Stokes semigroup, a different proof for the case of bounded domains with smooth boundary is due to Giga [89]. His approach is based on Seeley’s theory on pseudo-differential operators rather than on localization methods. Borchers and Sohr [34] and Borchers and Varnhorn [35] were the first to prove that the Stokes semigroup on exterior domains is uniformly bounded. The fact that the Stokes semigroup on exterior domains is a bounded analytic semigroup is due to Giga and Sohr [78]. The maximal regularity results in the case n D 2 go back to Abels [9], who was in particular able to treat the case of small resolvent parameters. In [68] Farwig and Sohr also developed an Lp -approach to the Stokes equation on C 1;1 standard domains yielding the analyticity of the Stokes semigroup.

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The characterization of the space X  in terms of function spaces given in Theorem 9 c) is due to Amann (see [18], Thm. 3.4). The above theorem allows in particular to consider also the fractional powers of the Stokes operators. In fact, by the sectoriality of the negative Stokes operator, fractional powers .A;p /z x of A;p are well-defined for x 2 D.Ak;p /\R.Ak;p / and z 2 C such that jRezj < k (see, e.g., [17, 54, 133]). It follows from the abstract theory of sectorial operators that .A;p /z is linear, closed, injective, and densely defined and has dense range. An important tool in the investigation of nonlinear problems is the representation of the domain of the fractional powers of a sectorial operator in terms of suitable function spaces. In this context, the property of bounded imaginary powers introduced in Sect. 2.1 plays an important role. In fact, as described in Sect. 2.1, assuming this property for a sectorial operator on a Banach space X , one has ŒX; D.A/˛ D D.A˛ /;

˛ 2 .0; 1/:

(42)

The following result deals with the H 1 -calculus for the Stokes operator on

p L ./.

Theorem 10 ([160]). Let n  2, 1 < p < 1, and assume that   Rn is a standard domain of class C 3 . Then A;p admits an R-bounded H 1 -calculus p on L ./. In particular, A;p has bounded imaginary powers, and relation (42) p holds for A D A;p and X D L ./. The first proof of the boundedness of the imaginary powers of the Stokes operator on bounded domains with smooth boundaries goes back to Giga [91]. His proof is again based on Seeley’s theorem. The case of an exterior domain, due to Giga and Sohr, was treated in [98]. A first proof of the fact that ARnC ;p admits an R-bounded H 1 -calculus on p L .RnC / was given by Desch, Hieber, and Prüss in [56]. Moreover, the proof of the p existence of an R-bounded H 1 -calculus on standard domains in L ./ is due to Noll and Saal [160]. For the case n D 2, see the work of Abels [9]. p 1;p Theorem 10 implies that ŒL ./; D.A;p /1=2 D W0 ./ holds for standard domains. Thus, in particular, the norms k  kW 1;p ./

and

k  kLp ./ C k.A;p /1=2  kLp ./

(43)

are equivalent on D..A;p /1=2 /. In the theory of the equations of Navier-Stokes, it is often important to know that the norms kr  kLp ./

and

k.A;p /1=2  kLp ./

(44)

are equivalent on D..A;p /1=2 /. In the case of a bounded domain, this property follows immediately from the equivalence of the norms in (43), thanks to Poincaré’s

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inequality. In scaling invariant domains such as Rn , RnC , this property may be derived from (43) by a scaling argument (see [167]). On exterior domains and perturbed half-spaces, the equivalence of the norms in (44) cannot be obtained on the full scale for p. Here one needs to restrict the value of p to p 2 .1; n=2/, similarly to (18). Summarizing, one obtains the following proposition, which was proved first by Borchers and Miyakawa in [33], however, by different methods. Proposition 10. Let n  2 and assume that   Rn is a standard domain of class C 3 . Then the norms in (44) are equivalent on D..A;p /1=2 / provided a)  is bounded, Rn , or RnC and 1 < p < 1, b)  is an exterior domain or a perturbed half-space and 1 < p < n=2. Finally, consider the Stokes equation with inhomogeneous data of the form 8 .@t C !/u  u C r D f ˆ ˆ < div u D g ˆ uDh ˆ : u.0/ D u0

in RC  ; in RC  ; on RC  @; in :

(45)

where   Rn is a domain with compact boundary of class 3 and ! 2 R. In the following, the maximal Lq  Lp -regularity estimates for the solution .u; / of (45) are characterized by the following conditions on the data .f; g; h; u0 /. One may view this characterization for the solution of the Stokes equation as the counterpart of the characterization of the solution of the inhomogeneous parabolic boundary value problems subject to general boundary conditions in terms of the data given; see [55]. To this end, the set of conditions (D) is introduced: Condition (D): 22=q

a) f 2 Lq .RC I Lp .//; u0 2 Bp;q ./, P 1;p .// \ Lq .RC I H 1;p .//, div u0 D g.0/, b) g 2 H 1;q .RC I H 11=2p 21=p .RC I Lp . @// \ Lq .RC I Bp;p .@// and h.0/ D u0 on @ if c) h 2 Fq;p q > 3=2, P 1;p .// and h .0/ D .ju0 / on @. d) .gjh / 2 H 1;q .RC I H Then the following theorem holds: Theorem 11 ([165] Chapter 7). Let   Rn be a domain with compact boundary @ of class 3, let 1 < p; q < 1 and q ¤ 3; 3=2. Then there exists !0 2 R such that for each ! > !0 , there exists a unique solution .u; / of Eq. (45) within the class P 1;p .// u 2 H 1;q .RC I Lp .// \ Lq .RC ; H 2;p .// and  2 Lq .RC I H if and only if the data .u0 ; f; g; h/ satisfy the above condition (D).

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The proof of Theorem 11 is rather involved; see Sect. 7 of [165] for a very thorough study of the Stokes equation with inhomogeneous data. In addition, other types of boundary conditions as pure slip, outflow and free boundary conditions are studied there. For previous results in this direction, see also [19].

2.8.2 Energy Preserving Boundary Conditions In this section, various types of boundary conditions for the Stokes equation as well as inhomogeneous right-hand sides are considered. Boundary conditions different from the Dirichlet condition arise in many applications and in related quasilinear problems. For instance, considering a free boundary value problem, e.g., a free ocean surface, then there is no stress at the surface of the fluid. Hence, it is natural to impose the condition T .u; p/ D 0

(46)

at the free boundary. Here T .u; p/ D 2 D.u/  Ip denotes the stress tensor, D.u/ D 12 .ru C .ru/T / the deformation tensor, and  the outer normal. On the other hand, in many engineering applications, such as in the design of hydrophobic surfaces, the partial slip condition ˛u C .D.u// D 0;

  u D 0;

(47)

also called Navier condition, plays a fundamental role as a macroscopic model. Here ˛ 2 R is a parameter (that relates to the slip length) and v D v  .  v/ denotes the tangential part of a vector field v. For this reason, results related for the Stokes system 8 @t u  div T .u; p/ D f ˆ ˆ < div u D g ˆ B.u; p/ D h ˆ : u.0/ D u0

in RC  ; in RC  ; on RC  @; in ;

(48)

subject to a given boundary operator B.u; p/ will be discussed in the following. The conditions (46) and (47) with ˛ D 0 belong to the large class of so-called energy preserving boundary conditions. This notion arises from the fact that the kinetic energy balance related to (48) with homogeneous right-hand sides is given by 1d 2 dt

Z

juj2 dx C 2



Z

jD.u/j2 dx D 

Z Œu  T .u; p/ d  : @

Whenever B.u; p/ is chosen in such a way that the right-hand side vanishes, B.u; p/ D 0 is called an energy preserving boundary condition. Note that the constraint div u D 0 allows for a second variant. In fact, if one defines the rate

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of the rotation tensor by R.u/ D 12 .ru  .ru/T /, one still has div R.u/ D div D.u/ D

1 u: 2

Setting V .u; p/ WD 2 R.u/  Ip, one obtains the alternative form 1d 2 dt

Z

juj2 dx C 2



Z

jR.u/j2 dx D 

Z Œu  V .u; p/ d  @

of the kinetic energy balance. This yields another set of energy preserving boundary conditions in terms of the antisymmetric counterpart V of the stress tensor T . Examples of energy preserving boundary conditions for the case D 1 include B.u; p/ WD T .u; p/; free boundary (Neumann) condition, B.u; p/ WD .D.u// C .  u/; full slip boundary condition, B.u; p/ WD u; no slip boundary condition, B.u; p/ WD u C p; tangential velocity and pressure condition, B.u; p/ WD .R.u// C p; vorticity and pressure condition, B.u; p/ WD u C 2.@ u  /  p; outflow condition, B.u; p/ WD .R.u// C .u  /; adjoint full slip condition. To keep the notational cost moderate, a corresponding maximal regularity result for the solution of (48) is formulated here only for the first boundary operator B.u; p/ D T .u; p/, which is, however, nevertheless one of the most representative energy preserving boundary conditions. To this end, one introduces the following classes of data and solution spaces. For the class of data spaces, for 1 < p < 1, one sets Ff WD Lp ..0; T /; Lp .//; Fg WD H 1=2;p ..0; T /; Lp .// \ Lp ..0; T /; W 1;p .//; Fh WD Wp1=21=2p ..0; T /; Lp . @// \ Lp ..0; T /; Wp11=p .@//; Fu0 WD Wp22=p ./; and defines F as the class of all .f; g; h; u0 / 2 Ff  Fg  Fh  Fu0 such that div u0 D g.0/; if p  2, .D.u0 // D h .0/; if p > 3 and such that there exists  2 Wp11=2p ..0; T /; Lp . @// \ Lp ..0; T /; Wp11=p .@// with u0   D .0/ if p > 3=2, and b 1;p .//; .g; / 2 W 1;p ..0; T /; 0 W

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

153

where .g; / acts as a functional via Z

Z

.g; /. / WD

 d  @

g dx;

0

2 W 1;p ./:



For the corresponding classes of solution spaces, one puts Eu WD W 1;p ..0; T /; Lp .// \ Lp ..0; T /; W 2;p .//; n Ep WD p 2 Lp ..0; T /; W 1;p .// W pj@ 2 Wp1=21=2p ..0; T /; Lp . @// o \ Lp ..0; T /; Wp11=p .@// ; and E WD Eu  Ep : Then one obtains the following result. Theorem 12 ([36]). Let p 2 .1; 1/nf3=2; 3g, n  2, > 0, and assume that   Rn is a bounded domain with smooth boundary. Then (48) with B.u; p/ D T .u; p/ admits a unique solution .u; p/ 2 E if and only if the data satisfy .f; g; h; u0 / 2 F. In particular, the solution operator S W .f; g; h; u0 / 7! .u; p/ is an isomorphism between the corresponding classes of function spaces. Remark 3. The corresponding results remain valid for each of the other boundary operators B.u; p/ listed above. Even a much larger class is admissible, including, for instance, also partial slip-type conditions. The reader is referred to [36] for a comprehensive study of this topic. First results on other boundary conditions than Dirichlet conditions were derived by Miyakawa in [154] for classical Neumann conditions and by Giga in [90] for another first-order type condition. An approach to maximal regularity subject to partial slip-type boundary conditions for  D RnC was developed in [166, 167]. Its extension to standard domains can be found in [174, 180]. A first approach to the Neumann conditions related to the free boundary condition B.u; p/ D T .u; p/ was given by Solonnikov in [189–191]. Comprehensive approaches to this topic were also developed by Shibata and Shimizu in [177, 178], by Boyer and Fabrie in [40] and by Prüss and Simonett in [164,165]. Maximal regularity results for a wide class of boundary conditions, based on pseudo-differential operator methods, were derived in [101]. This approach will be discussed in more detail in Sect. 3.1. As already mentioned in Remark 3, an

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extensive approach to the class of energy preserving boundary conditions, as well as to further relevant classes, was derived in [36, 128]. This approach includes all boundary conditions discussed in the previously cited papers. Note that the characterization of the space for initial data u0 in Theorem 12 relies on Proposition 1 and that the general theory of the trace operator implies the the conditions on the boundary trace spaces given above are necessary. Observe finally that results corresponding to Theorem 12 are available on certain classes of domains with noncompact boundaries, too. For details, see the Sects. 2.7, 2.9 and 2.11.

2.9

The Stokes Equation in Lp ./ for 1 < p < 1 and for Domains with Noncompact Boundaries

In this section, one considers possibly unbounded domains, which are uniformly of class C k for some k 2 N. A key problem in the investigation of the Stokes operator in general unbounded domains is the fact that the Helmholtz decomposition for p Lp ./ into L ./ ˚ Gp ./ does not exist, in general. The reader is referred to the results by Maslennikova and Bogosvkii [147] described in Remark 1 b), where an example of an unbounded domain  with smooth boundary is constructed for which the Helmholtz decomposition exists only for certain values of p. In the following, one assumes that   Rn is a domain with uniformly C 3 -boundary and that the Helmholtz projection P exists for Lp ./. Then it is shown that the Stokes operator p Ap , defined as in (50) below, generates an analytic semigroup on L ./ and that the solution of the Stokes equation ut  u C r D f;

in   .0; T /;

div u D 0;

in   .0; T /;

u D 0; u.0/ D u0 ;

on @  .0; T /;

(49)

in 

satisfies the maximal Lq -Lp -regularity estimate. Theorem 13 ([83]). Let n  2, p; q 2 .1; 1/ and J D .0; T / for some T 2 .0; 1/. Assume that   Rn is a domain with uniform C 3 -boundary and p that the Helmholtz projection P exists for Lp ./. Let f 2 Lq .J I L .//. Then 1;q equation (49) with u0 D 0 admits a unique solution .u; / 2 W .J I Lp .// \ 1;p p b 1;p .//, and there exists a constant Lq .J I W 2;p ./\W0 ./\L .//Lq .J I W C > 0 such that kut kLq .J ILp .// C kukLq .J ILp .// C kr 2 ukLq .J ILp .// CkrkLq .J ILp .//  C kf kLq .J ILp .// :

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155

Assuming as in the above theorem that the Helmholtz projection P exists for p Lp ./, one may define the Stokes operator Ap in L ./ as 1;p

D.Ap / WD W 2;p ./ \ W0 ./ \ Lp ./;

(50)

Ap u WD P u for u 2 D.Ap /: p

Then maximal Lq -regularity for the Stokes operator A;p in L ./ u0 .t /  Ap u.t / D f .t/;

t > 0; (51)

u.0/ D u0 ;

holds true. Some comments about the strategy of the proof of Theorem 13 are in order. a) The above assumptions imply that one may choose balls Bj WD Br .xj / with centers xj 2  and C 3 -functions hj , (j D 1; 2; : : : ; N ) if  is bounded and j 2 N if  is unbounded, such that   [1 j D1 Bj ;

Bj  U .xj / if xj 2 @;

Bj   if xj 2 ;

where U .xj / are suitable neighborhhods of xj . Given the covering .Bj /, there exists a partition of unity 'j 2 Cc1 .Rn / satisfying supp 'j  Bj and 0  'j  1. Consider now

uQ WD

N X

'j uj ;

j D1

Q D

N X

'j  j ;

j D1

where .uj ; j / is the push forward of the solution .Ouj ; O j / to the resolvent problem on the half-space with a suitable right-hand side fOj . Note that uQ is not P solenoidal in general since div uQ D N j D1 .r'j /uj ¤ 0. In order to circumvent these difficulties, one therefore uses the modified ansatz u WD

1 X 

 'j uj  rvj ;

j D1

where vj is a weak solution to the Neumann problem v D div f

in ;

@v Df  @

on @;

(52)

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with f D 'j uj . Note that the existence and uniqueness of vj is guaranteed by the existence of the Helmholtz projection and by Theorem 2. By construction, one then obtains div u D

1 X

.div.'j uj /  vj / D 0I

j D1

however, the tangential component of u does not vanish at the boundary anymore, which leads to additional correction terms. b) The above strategy needs higher-order estimates for weak solutions of the above Neumann problem. In this context, the following lemma is important (see [83]). Lemma 3. Let p and  as above. Then, for k D 2; 3, there exists C > 0 such that b 1;p ./ of (52) for f 2 W k1;p ./ with   f D 0 on @, the weak solution v 2 W satisfies k X

  kr j vkp  C kf kp C kr  f kW k2;p :

(53)

j D1

For different approaches to the Stokes and Navier-Stokes equation on special domains with noncompact boundaries, e.g., domains with strip-like or cylindrical outlets at infinity or parabolically growing layers, the reader is referred to the works of Heywood [106], Solonnikov [188], and Pileckas [161–163]. In [14], Abels and Terasawa considered the reduced Stokes operator in unbounded domains with the additional assumption on  that the associated space for the pressure can be decomposed suitably. For related results, see also [13] and [173]. The following result, due to Geissert and Kunstmann, shows that under the present assumptions, the Stokes operator even admits a bounded H 1 -calculus on p L ./ for suitable values of p. Proposition 11 ([86]). Let   Rn be a domain with uniform C 3 -boundary and assume that the Helmholtz decomposition exists for Lp ./ for some p 2 .1; 1/. Then there exists ! 2 R such that Ap C ! admits a bounded H 1 -calculus on q L ./ for all q 2 .minfp; p 0 g; maxfp; p 0 g/. The proof relies on the maximal regularity Theorem 13 and an abstract result for H 1 -calculus in complemented subspaces. This section is being finished with a solution of a long outstanding question, whether or not the existence of the Helmholtz projection for Lp ./ is necessary for the Stokes operator being the generator of an analytic semigroup on Lp ./? A very recent result due to Bolkart, Giga, Miura, Suzuki, and Tsutsui [29] says that the existence of the Helmholtz projection for Lp ./ is not necessary for the

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157

Stokes operator being the generator of an analytic semigroup on Lp ./. More precisely, the following holds true. Theorem 14 ([29]). Let p 2 Œ2; 1/. Then there exists a sector-like domain   R2 with C 3 -boundary such that the solution operator T .t/ W u0 7! u.t; / of the p Stokes equation (49) in  with f D 0 defines an analytic semigroup on L ./, p while the Helmholtz decomposition for L ./ fails to exist. As written in the introduction, a characterization of those domains, even with smooth boundaries, for which the solution of the Stokes equation is governed by an p analytic semigroup on L ./ remains a challenging open problem until today.

2.10

The Stokes Equation in Noncylindrical Space-Time Domains

This subsection is motivated by applications of fluid flows in spatial regions with a moving boundary including moving obstacles. One hence considers the Stokes system 8 vt  v C rp ˆ ˆ < div v ˆ v ˆ : v.0/

D f in QT ; D 0 in QT ; S D 0 on t2.0;T / @.t /  ft g; D v0 in .0/ DW 0 ;

(54)

S on noncylindrical space-time domains of the form QT WD t2.0;T / .t/  ft g  RnC1 . Assume that the moving boundary, i.e., the evolution of the domain .t/, is determined by a level-preserving diffeomorphism W 0  .0; T / ! QT ;

. ; t / 7! .x; t / D

. ; t / WD . . ; t /; t /;

such that for each t 2 Œ0; T /, .; t / maps 0 onto .t/. More precisely, one assumes the following conditions on and , respectively: Let T 2 .0; 1, 0  Rn be a standard domain of class C 3 . The domains .t/, t 2 Œ0; T , shall be of the same type as 0 , i.e., f.t/gt2Œ0;T  is either a family of bounded domains, of exterior domains, or of perturbed half-spaces. Furthermore, it is assumed that (a) for each t 2 Œ0; T , .; t / W 0 ! .t/ is a C 3 -diffeomorphism with inverse 1 .; t /. (b) if Q0 WD 0  .0; T / and BC .Q0 / denotes the space of all bounded and continuous functions on Q0 , then 2 BC 3;1 .Q0 / WD ff 2 C .Q0 / W @kt Dx˛ f 2 BC .Q0 /; 1  2k C j˛j  3; k 2 N0 ; ˛ 2 Nn0 g. (c) det r . ; t/ 1, . ; t / 2 Q0 , (volume preserving). (d) if T D 1, then @kt .; t / ! @kt .; 1/ in BC 32k .0 /, k D 0; 1, for t ! 1.

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For t 2 Œ0; T / and 1 < p < 1, the Stokes operator in the time-dependent domain .t/ is given by

A.t/ WD P.t/ ; 1;p

D.A.t/ / WD W 2;p ..t // \ W0 ..t // \ Lp ..t //:

p

Here P.t/ W Lp ..t // ! L ..t // denotes the Helmholtz projection associated to p the Helmholtz decomposition Lp ..t // D L ..t // ˚ Gp ..t //. The following maximal regularity result for the Stokes equation on noncylindrical regions was obtained in [168]. Theorem 15 ([168]). Let n  2, 1 < p; q < 1, and T 2 .0; 1. Let the evolution of .t/, t 2 Œ0; T , be determined by a function satisfying the above assumptions. Then problem (54) has a unique solution t 7! .v.t /; p.t // 2 b 1;q ..t //, and t 2 Œ0; T . Furthermore, for T < 1, this solution D.A.t/ /  W satisfies the maximal regularity estimate Z

T 0

q

q

q

Œkvt .t /kLp ..t// C kv.t /kW 2;p ..t// C krp.t/kLp ..t//  dt  q  C .T / kv0 kI q .A / C 0

Z 0

T

q

kf .t/kLp ..t// dt



p

for all v0 2 I q .A0 / WD .L ..0//; D.A.0/ //11=q;q , and all f 2 Lq ..0; T /I Lp ..t ///. If 0 is bounded, then the above inequality is also valid for T D 1. Special cases of Theorem 15 were already considered earlier. First investigations of the solvability of Eq. (54) and the corresponding Navier-Stokes equations can be found in [170]. The L2 ..t //-situation for a family of bounded domains f.t/gt2Œ0;T  was considered by Inoue and Wakimoto in [120] (see also [24]). The periodic case for the Stokes equation, i.e., if t 7! .t/ is periodic, is covered by results given in [206]. Note that the assumptions on the evolution and regularity of .t/ differ in the existing literature. A rough sketch of the proof of Theorem 15 is as follows: first one transforms (54) to a nonautonomous Cauchy problem on the time-independent domain .0/ by utilizing properties of the diffeomorphism . The resulting system then can be handled by a result on nonautonomous Cauchy problems, which was derived in [168].

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

2.11

159

An Approach for General Domains

By the results described in Sect. 2.2, the Helmholtz decomposition for Lp ./ does not exist for general unbounded domains, unless p D 2. Farwig, Kozono, and Sohr presented in [65, 66] an approach to the Stokes equation for a large class of unbounded domains, more precisely for uniform C 1 -domains of type .˛; ˇ; K/ (see [65] for the definition). Their main idea is to replace the space Lp ./ by the space Q p ./, where L Q p ./ WD Lp ./ \ L2 ./ for 2  p < 1 and L Q p ./ WD Lp ./ C L2 ./ for 1 < p < 2: L kk Q p

p

Q  ./ WD fu 2 Cc1 ./ W div u D 0g L ./ . Moreover, let L Given a uniform C 1 -domain   Rn , n  2, of type .˛; ˇ; K/ and 1 < p < 1, Q p ./ can de decomposed uniquely as u D u0 C rp they showed that each u 2 L Q p ./ and satisfying with u0 2 L ku0 kLQ p ./ C krpkLQ p ./  C kukLQ p ./ : p

Q p ./ with range L Q  ./. In particular, PQ u WD u0 defines a bounded projection on L Setting Q p u WD PQ p u; A with ( Q p/ D D.A

D.Ap / \ D.A2 /;

2  p < 1;

D.Ap / C D.A2 /;

1 < p < 2;

1;p

p

where D.Ap / D W 2;p ./ \ W0 ./ \ L ./, one obtains the following result. Theorem 16 ([65, 66]). Let 1 < p < 1,  as above and J D .0; T / for some Q p generates an analytic semigroup on L Q p ./. Moreover, for all T > 0. Then A p Q  .//, there exists a unique u solving the inhomogenous Stokes f 2 Lp .J; L equation (49) with u0 D 0 and satisfying Q p uk p Q p ku0 kLp .J ILQ p .// C kA Q p .// L .J IL .//  C kf kLp .J IL for some C > 0.

160

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The Stokes Equation on Domains with Edges and Vertices

Whereas the Lp -theory for classical elliptic and parabolic problems on singular domains is well developed, corresponding results for the Stokes equation are very rare, in particular for the instationary case. For the stationary Stokes equation, there are the classical regularity results, which go back to Kondrat’ev [129], Kellogg and Osborn [125], Dauge [51], as well as to Maz’ya and Rossmann [150] and Grisvard [99]. For results concerning domains with conical boundary points, see the work of Deuring [57]. He also proved a negative result concerning the generation of an analytic semigroup in three dimensions in [58]. More recently, an approach to analytic regularity was presented by Guo and Schwab in [102]. Most of the approaches in the articles listed above rely on a thorough analysis of the Green tensor (see, e.g., [150]) or on the Kondrat’ev technique, that is, on the transformation onto a layer by introducing polar coordinates. This leads in a natural way to a treatment in weighted Sobolev or Hölder spaces, the so-called Kondrat’ev spaces. Dauge [51] presented a detailed analysis for the stationary Stokes system for a large class of two- and three-dimensional singular domains. Kellogg and Osborn [125] provided H 2 -regularity in two-dimensional convex domains. In [99], an approach to the stationary Stokes equation in two dimensions on polygonal domains is developed. The approach relies on the fact that in two space dimensions the stationary Stokes equation is equivalent to the biharmonic equation. In fact, since div u D 0, there is a vector potential such that D curl u D @1 u2  @1 u2 . Applying curl to the Stokes equation, one obtains the scalar equation 2 D 0. Note that this method does not generalize to three space dimensions or to the instationary counterpart. Hence, it is very restrictive. For a recent local well-posedness result of the contact line problem for twodimensional Stokes flow, see [208]. It seems that a general approach to the instationary Stokes equation on singular domains does not exist in the literature, even for domains having a simple structure such as wedges. Just in the case of reflecting boundary conditions or for their lower-order perturbations, well-posedness is assured. For example, partial slip (also called Navier)-type boundary conditions given by ˛u C .D.u// D 0;

uD0

on @

(55)

belong to this class. Here ˛ 2 R is a parameter related to the slip length. For many types of domains, the condition (55) can be reformulated (modulo lower-order perturbation terms) as ˛u    curl u D 0;

uD0

on @:

(56)

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161

This reformulation is a key step in this investigation, since the Stokes equation subject to (56) reduces then to a heat equation. This idea is outlined in the following for perfect slip conditions, that is, for the case ˛ D 0. The analysis of this special situation is justified by the fact that in many situations the term ˛u can be treated also by perturbation arguments. For the perfect slip condition, it is well known that the Helmholtz projection P and the Laplacian  commute. In fact, one has curl P u D curl .u  rp/ D curl u; which implies P .D.ps //  D.ps /. Here, ps denotes the Laplacian subject to perfect slip conditions. Relying on Gauß’ theorem, one shows that   curl 2 uj@ D 0 provided   curl j@ u D 0. This yields P u D P curl 2 u D curl 2 u D curl 2 .P u C rp/ D P u:

(57)

Thus, in the case of perfect slip boundary conditions, one has A;p D ps jLp ;

(58) p

which means that the Stokes operator is the part of the Laplacian in L ./. In other words, in order to prove well-posedness results for the Stokes equation subject to perfect slip boundary condtions, it suffices to consider the following resolvent problem for the Laplacian: 8 < u  u D f in ;   curl u D 0 on @; :   u D 0 on @:

(59)

Thus, once sectoriality (or maximal Lp regularity, or a bounded H 1 -calculus, respectively) is proved for ps , then this property immediately transfers to the Stokes operator subject to perfect slip boundary conditions. As mentioned above, employing suitable perturbation arguments, in a number of cases, this remains true even for the Stokes operator subject to partial slip-type boundary conditions (55). The above approach was used in [143] for three-dimensional wedges, i.e., on domains of the form G D fx D .x1 ; x2 ; x3 / 2 R3 W 0  x2  sx1 g; where s > 0. Theorem 17 ([143]). Let 1 < p < 1 and AG;p be the Stokes operator subject to perfect slip conditions defined as in (58). Then AG;p admits an R-bounded H 1 p R1 calculus on L .G/ with A < =2. G;p

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Remarks 7. a) The above approach relies heavily on the property (57) and hence does not extend to other relevant types of boundary conditions. In fact, the question whether the negative of the Stokes operator with Dirichlet conditions or p other than partial slip-type boundary conditions is a sectorial operator on L ./ for wegde-type domains  remains an open question until today. This fact is very unsatisfactory, in particular in regard to applications. Indeed, solvability of the Stokes equation on wedge-type domains, for instance, is a crucial ingredient for the well-posedness of the moving contact line problem, which has been an open problem for decades. Only a few results for very restrictive values of contact angles c are known (see, e.g., [191] for c D 0, [205] for c D =2). For very recent results in this direction in the twodimensional setting, see [208]. b) Results similar to the one described in Theorem 17 are available also on socalled weakly singular domains for the class of reflecting boundary conditions. For instance, cylindrical domains G D fx D .x1 ; x2 ; x3 / 2 R3 W x12 C x22  r; 0  x3  hg for r; h > 0 belong to this class. Imposing, e.g.,  ˇ .D.u// C .  u/ ˇx3 2f0;hg D 0; ˇ uˇfx 2 Cx 2 Drg D 0; 1

(60)

2

then a version of Theorem 12 remains valid for  D G. The reader is referred, e.g., to [128] for the precise definition of weakly singular domains and for more results in this direction.

3

Other Approaches to the Stokes Equation for 1 < p < 1

In this section, three further approaches to the Stokes equation are presented. They differ from the approaches described above in Sect. 2. The first one concerns the so-called reduced Stokes equation, whereas the second one is based on the theory of layer potentials. The last one relies on Muckenhoupt weights and on an extrapolation theorem due to Rubio de Francia.

3.1

The Reduced Stokes Equation and the Pseudo-differential Operator Approach

Consider the Stokes equation (1) with f D 0 on a domain   Rn . Applying div to the first equation yields the following Neumann problem for p: p D 0 in ;

@ p D   curl 2 u on @:

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

163

Let G W u 7! rp denote the operator mapping the velocity u to rp via the above Neumann problem for p. Then, replacing rp in the Stokes equation by Gu, the velocity u solves the system 8 < .@t   C G/u D u D : u.0/ D

0 in RC  ; 0 on RC  @; u0 in :

(61)

The set of Eq. (61) is called reduced Stokes system. It was first introduced by Grubb and Solonnikov in [100, 101] in order to solve the Stokes equation on domains with smooth boundaries by a pseudo-differential operator approach. The advantage of system (61) in comparison to the usual Stokes equation is given by the fact that there is no condition on the divergence of u anymore. Hence, an approach in standard Lp -spaces is possible, in principle. However, the price to pay is that G is a nonlocal operator G. On the other hand, G is a singular Green operator, and thus the structure of (61) fits into the Boutet de Monvel calculus of pseudodifferential operators [101]. Thus formally, system (61) is equivalent to the Stokes equation. To see this, it remains to ensure that the condition div u D 0 can be recovered from a solution u of (61). In fact, one may apply div to (61), which yields .@t  /div u D 0; due to the properties of G. Next, applying the normal  to (61) and employing the relation  D curl 2  rdiv , one obtains 0 D   curl 2 u C   Gu D @ div u; again thanks to the properties of G. Thus div u solves a homogeneous Neumann problem for the heat equation and hence is constant. This constant, however, must be zero in view of   uj@ D 0 and the divergence theorem. This formal argumentation can be made rigorous in many concrete situations. In [100] and [101], Grubb and Solonnikov presented a comprehensive approach to the Stokes equations on bounded domains for inhomogeneous boundary conditions Bk .u; p/ D h of the type B1 .u; p/ WD u;

B2 .u; p/ WD T .u; p/;

B4 .u; p/ WD @ u  p;

B3 .u; p/ WD ŒD.u/ C .  u/;

B5 .u; p/ WD @ u C .u  /:

As before, the tangential part of u is denoted u D u  .  u/ of u; moreover D.u/ D 1=2.ruCruT / denotes the deformation tensor and T .u; p/ D 2D.u/Ip the stress tensor. Maximal and higher-order regularity for the inhomogeneous Stokes equation subject to either one of the above inhomogeneous boundary conditions, i.e.,

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8 < .@t   C G/u D Bk .u; p/ D : u.0/ D

f in RC  ; 0 on RC  @; u0 in ;

(62)

can be obtained following this approach. In order to avoid the introduction of trace classes and higher-order compatibility conditions, the main result on the Stokes equation is formulated in the following only for homogeneous boundary conditions and right-hand sides in mild regularity classes. For the general result, the reader is referred to [101, Theorem 7.5]. Setting Hk WD



L2 ./; k D 1; 3; 5; L2div ./; k D 2; 4;

where L2div ./ D fv 2 Lp ./I div v D 0g, the following result holds true. Theorem 18 ([100,101]). Let T 2 .0; 1/, J D .0; T /, k 2 f1; : : : ; 5g and assume that  is a bounded domain of class C 1 . Then for each pair ˚ .f; u0 / 2 L2 .J; Hk /  H 1 ./ \ Hk satisfying the compatibility condition .u0 / D 0 on @ if k D 1, there exists a unique solution ˚ b 1 .// .u; p/ 2 H 1 .J; Hk / \ L2 .J; H 2 .//  L2 .J; H of (62). In the case where k D 2; 4, one has pj@ 2 H 1=2 .J; L2 . @// \ L2 .J; H 1 . @//: Abels employed the reduced Stokes system approach in order to develop an Lp theory to the above system and also for the Stokes operator on asymptotically flat layers (see [9–11] and also Theorem 7). In [157], the Ansatz via the reduced Stokes system is used in order to prove maximal Lp -regularity for the Stokes equation on Lipschitz domains subject to partial slip-type boundary conditions (see also Theorem 22).

3.2

Lp -theory for Lipschitz Domains via Double Layer Potentials

Consider a bounded Lipschitz domain   R3 . Then, due to the roughness of the boundary, an approach to the Stokes equation on  via localization as presented in the sections above is no longer possible. For this reason, various other approaches

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

165

have been developed in order to deal with the situation of Lipschitz domains, see, e.g., [60, 79, 153, 171, 200]. A common approach in this situation is the approach by layer potentials. To this end, let G.x; / be the fundamental solution to the Helmholtz equation in R3 ;  > 0:

.  /u D 0

An explicit representation for G may be given in terms of Hankel functions (cf. [172]). In three dimensions, G is given by p

e  jxj ; G.x; / D 4jxj

x 2 R3 ;  > 0:

Next, introducing the functions 1 @j @k .G.x; /  G.x; 0// and 

j k .x; / WD G.x; /ıj k  ˆk .x/ WD @k G.x; 0/;

j; k D 1; 2; 3;

a calculation shows that the pair .j k ; ˆk / is a fundamental tensor for the Stokes resolvent problem (33) on R3 nf0g. For a suitable h defined on @, say h 2 Lp . @/, the double layer potential for the Stokes system then is defined as Z vj .x; / WD .D h/j .x/ WD

3 3 X X

@ kD1

@i j k .y  x; /i .y/

iD1

  ˆj .y  x/k .y/ hk .y/ d  .y/; p.x/ WD

3 X

j D 1; 2; 3;

Z @i @k

i;kD1

C

G.y  x; 0/i .y/hk .y/ d  .y/ @

3 Z X kD1

G.y  x; 0/k .y/hk .y/ d  .y/: @

Then the pair .v; p/ solves the first two lines in Eq. (64) in R3 n @. Note, however, that v does not satisfy the boundary condition vj@ D h in Eq. (64), but it can be shown that 1 v./ D . I C K /h 2

on @;

(63)

for  > 0 and with an operator K 2 L.Lp . @//. Note that the trace of v in (63) is understood as a nontangential limit taken inside , cf. [123]. A crucial point now is

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M. Hieber and J. Saal

to show that  12 I C K is invertible on Lp . @/, since then the pair .w; p/, where  1 h; is a solution of w./ WD D  12 I C K 8 < w  w C rp D 0 in ; div w D 0 in ; : w D h on @:

(64)

This strategy was already used by Giga in [89] in the case of smooth boundaries. In this case, K is a compact operator, and K can be viewed as a lower-order perturbation for large . In the case of Lipschitz domains, one sets Z X n

hj ./ WD .

j k .x  y; /fk .y/ dy/j@ ;

j D 1; : : : ; n;

 kD1

and sees that a solution of (33) is given by .u; p/, where uj ./ D

Z X n

j k .x  y; /fk .y/ dy  wj ./;

j D 1; : : : ; n:

 kD1

This was proved first rigorously by Shen in [171] for the case  D 0. Theorem 19 ([171]). Let   R3 be a bounded graph Lipschitz domain and 3=2  p  3. Then, for each f 2 W 1;p ./, there is a unique solution 1;p p .u; p/ 2 W0 ./  L0 ./ of the stationary Stokes system (15) satisfying kukW 1;p C kpkp  C kf kW 1;p ; where C > 0 is independent of f , u, and p. The corresponding assertion for fixed  2 † and  2 .0; =2/ can be a carried out in an analogous way. On the other hand, it is not obvious how to derive uniform resolvent estimates for u, i.e., estimates of the form ku./kp  C kf kp ;

 2 † ; f 2 Lp ./:

This was known as Taylor’s conjecture (see [198]). An affirmative answer to Taylor’s conjecture was given by Shen in [172]. His result reads as follows. Theorem 20 ([172]). Let   R3 be a bounded graph Lipschitz domain. Then there exists " > 0 such that for all 3=2  " < p < 3 C ", the Stokes operator subject p to Dirichlet boundary conditions generates an analytic semigroup on L ./.

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

167

The strategy of Shen’s proof is to apply a Calderón-Zygmund argument in order to extrapolate sectoriality from L2 to Lp . For doing this, the main task is to derive a weak reverse Hölder estimate of type

1 jB.x0 ; r/ \ j C

Z

jujp dx

1=p

B.x0 ;r/\

1 jB.x0 ; 2r/ \ j

Z

2

juj dx

1=2 ;

B.x0 ;2r/\

for local solutions u of the Stokes resolvent problem, which is uniform in . Shen established this estimate in [172] based on an argument, which shows that  1 1  2 I C K is bounded on L2 by a constant independent of  2 † . The potential theoretical approach is standard in the theory of partial differential equations. However, there is a significant difference between smooth and nonsmooth domains. For smooth domains with compact boundary of class C 1C˛ , say, the operator K usually is compact. Then, invertibility follows easily by a Fredholm argument. However, compactness of K fails to be true in nonsmooth domains. Thus, invertibility of  12 I C K is much harder to prove in this setting and requires more subtle tools from Harmonic Analysis. The reader is referred to the classical papers [64, 123] for a potential theoretical approach to elliptic and parabolic equations on graph Lipschitz domains. p Recently, it was shown by Tolksdorf that the Stokes operator on L ./ for q bounded Lipschitz domains  admits maximal L -regularity provided the following condition on p is satisfied. Theorem 21 ([200]). Let   Rn be a bounded Lipschitz domain and n  3. Then there exists " > 0 such that for all 2n 2n "

0 such that jj kvkLp ./ CkvkW 2;p ./ CkkW 1;p .G/  C kf kLp ./ ;  2 †" [f0g; f 2 Lp ./: (84) As in the case of the classical Helmholtz projection, the existence of the hydrostatic Helmholtz projection is closely related to the unique solvability of the Poisson problem in the weak sense. In the given situation, the equation H  D divH f in G, subject to periodic boundary conditions, plays an essential role.

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Lemma 6 ([109]). Let p 2 .1; 1/ and f 2 Lp .G/. Then there exists a unique 1;p p  2 Wper .G/ \ L0 .G/ satisfying hrH ; rH iLp0 .G/ D hf; rH iLp0 .G/ ;

0

p0

1;p 2 Wper .G/ \ L0 .G/:

(85)

Furthermore, there exists a constant C > 0 such that kkW 1;p .G/  C kf kLp .G/ ;

f 2 Lp .G/:

(86)

The above Lemma 6 allows to define the hydrostatic Helmholtz projection Pp W 1;p p Lp ./ ! Lp ./ as follows: given v 2 Lp ./, let  2 Wper .G/ \ L0 .G/ be the unique solution of Eq. (85) with f D v. N One then sets Pp v WD v  rH ;

(87)

and calls Pp the hydrostatic Helmholtz projection. It follows from Lemma 6 that Pp2 D Pp and that thus Pp is indeed a projection. In the following one defines the closed subspace Xp of Lp ./ as Xp WD RgPp . This space plays the analogous role in the investigations of the primitive equations p as the solenoidal space L ./ plays in the theory of the Navier-Stokes equation. The hydrostatic Helmholtz projection Pp defined as in (87) allows then to define the hydrostatic Stokes operator as follows. In fact, let 1 < p < 1 and Xp be defined as above. Then the hydrostatic Stokes operator Ap on Xp is defined as (

Ap v WD Pp v; 2;p ./2 W divH vN D 0 in G; @z v D 0 on u ; v D 0 on b g: D.Ap / WD fv 2 Wper (88)

The resolvent estimates for Eqs. (82) and (83) given in Proposition 22 yield that Ap generates a bounded analytic semigroup on Xp . More precisely, one has the following result. Theorem 42 ([109]). Let 1 < p < 1. Then the hydrostatic Stokes operator Ap generates a bounded analytic C0 -semigroup Tp on Xp . Moreover, there exist constants C; ˇ > 0 such that kTp .t /f kXp  C e ˇt kf kXp ;

t > 0:

Recently, it was shown by Giga, Gries, Hieber, Hussein, and Kashiwabara [93] that Ap even admits an R-bounded H 1 -calculus on Xp of angle 0, which implies in particular maximal Lq  Lp -estimates for the solution of the hydrostatic Stokes equation and allows further to characterize the domains D.Ap / of the fractional powers Ap for 0 <  < 1 in terms of Sobolev spaces subject to the boundary conditions given. More precisely, one has the following results.

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195

Theorem 43 ([93]). Let p 2 .1; 1/. Then the operator Ap admits a bounded RH 1 -calculus on Xp with AR;1 D 0. p Combining this result with a characterization of the complex interpolation spaces ŒXp ; D.Ap / proved in [108] allows then to characterize the domains D.Ap / of the fractional powers Ap for 0 <  < 1 as follows. Corollary 7 ([93]). Let 1 < p < 1 and  2 Œ0; 1 with  … f1=2p; 1=2 C 1=2pg. Then 8 ˇ ˇ 2;p ˇ ˇ ˆ ˆ 0 such that

ke

tAp

kre

tAp

ke

6.6

tAp

Pp f kLq ./  C t Pp f kLq ./  C t

Pp div f kLq ./  C t



3 2



3 2



3 2







1 1  p q

for f 2 Lp ./; t > 0;

kf kLp ./ ;

1 1 1   p q 2

1 1 1   p q 2

kf kLp ./ ;

for f 2 Lp ./; t > 0;

kf kLp ./ ;

for f 2 Lp ./; t > 0:

The Stokes Operator in the Rotating Setting

Consider the linearization of Navier-Stokes equation with Coriolis force on all of R3 , i.e., consider the equation ut  u C e3  u C rp D 0;

in R3  .0; 1/;

div u D 0;

in R3  .0; 1/;

u.0; x/ D u0 .x/;

(89)

x 2 R3 ;

where denotes the viscosity coefficient of the fluid,  the speed of rotation, and e3 the unit vector in x3 -direction. This equation gained quite some attention due to its importance in geophysical flows. Taking Fourier transforms in (89) and

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denoting this equation by F(89), an explicit solution formula for F(89) was derived by different methods in [94] and [116]. Lemma 7. There exists a unique solution .Ou; p/ O of equation F(89), where uO for

D 1 is given by

uO .t; / D cos.

3 3 2 2 u0 . /; t /e j j t I ub0 . / C sin. t /e j j t R. /b j j j j

t  0; 2 R3 ;

and 0

0 B 3 R D R. / D @  j j 2 j j

3 j j

2  j j

0 1  j j

1 j j

1 C A:

0 p

One may deduce from Lemma 7 that the solution of Eq. (89) in L .R3 / for D 1 p is governed by a C0 -semigroup Tp on L .R3 /, which is explicitly given by

 3 3 2 2 Tp .t /f WD F 1 cos. t /e j j t I fO . / C sin. t /e j j t R. /fO . / ; j j j j t  0; f 2 Lp .R3 /: This semigroup is called the Stokes-Coriolis semigroup. Mikhlin’s theorem implies then the following result. Theorem 45. Let 1 < p < 1 and Tp be the Stokes-Coriolis semigroup defined as p above. Then Tp is a C0 -semigroup on L .R3 /, which may be represented as Q 3 t /I C sin.R Q 3 t /Re t f; Tp .t /f D Œcos.R

t  0; f 2 Lp .R3 /:

Q 3 denotes the operator associated to the symbol 3 . Here R j j Further information on the Stokes operator in the rotating setting can be found, e.g., in [43, 94, 95, 116]. The dispersive effect of the Coriolis force and its consequences concerning global well-posedness for the Navier-Stokes equations in the rotational framework was investigated in detail by Mahalov and Nicolaenko [142], by Chemin, Desjardin, Gallagher, and Grenier [43] as well as by Koh, Lee, and Takada in [127], by Iwabuchi and Takada in [121], and by Kozono, Mashiko, and Takada in [131].

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

7

197

Conclusion

This survey article is concerned with well-posedness questions and regularity properties of the Stokes equation in various classes of domains   Rn within the Lp -setting for 1  p  1. Taking the point of view of evolution equations, classical as well as modern approaches for obtaining well-posedness results for the Stokes equation in the strong sense are presented. Topics being discussed include the Helmholtz decomposition, the Stokes operator, the Stokes semigroup, and maximal Lp -regularity results for 1 < p < 1 via the theory of R-sectorial operators. Of concern are domains having compact or noncompact, smooth, or nonsmooth boundaries. In addition, the endpoints of the Lp -scale, i.e., p D 1 and p D 1, are considered, and recent well-posedness results for the case p D 1 are described. Results on Lp  Lq -smoothing properties of the associated Stokes semigroups and on various variants of the Stokes equation complete this article.

8

Cross-References

 Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value

Problem  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Large Time Behavior of the Navier-Stokes Flow  Local and Global Existence of Strong Solutions for the Compressible Navier-S-

tokes Equations Near Equilibria via the Maximal Regularity  Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid

Crystal Flows  Models and Special Solutions of the Navier-Stokes Equations  Recent Advances Concerning Certain Class of Geophysical Flows  Regularity Criteria for Navier-Stokes Solutions  Self-Similar Solutions to the Nonstationary Navier-Stokes Equations  Stokes Problems in Irregular Domains with Various Boundary Conditions  Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains  The Inviscid Limit and Boundary Layers for Navier-Stokes Flows  Time-Periodic Solutions to the Navier-Stokes Equations  Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

References 1. K. Abe, Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions. Acta Math. 211, 1–46 (2013) 2. K. Abe, Y. Giga, The L1 -Stokes semigroup in exterior domains. J. Evol. Equ. 14, 1–28 (2014)

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4

Stokes Problems in Irregular Domains with Various Boundary Conditions Sylvie Monniaux and Zhongwei Shen

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Linear Dirichlet-Stokes Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Nonlinear Dirichlet-Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Linear Neumann-Stokes Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Nonlinear Neumann-Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hodge Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Hodge-Laplacian and the Hodge-Stokes Operators . . . . . . . . . . . . . . . . . . . . . . 4.2 The Nonlinear Hodge-Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Robin Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Robin-Hodge-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Robin-Hodge-Stokes Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Nonlinear Robin-Hodge-Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

208 210 210 216 223 223 225 226 226 233 234 235 240 244 245 246 246

Abstract

Different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in R3 , such as Dirichlet, Neumann, Hodge, or Robin boundary conditions, are presented here. The situation is a little different from the case of smooth domains. The analysis of the problem involves a good comprehension

S. Monniaux () Aix-Marseille Université, CNRS, Centrale Marseille, Marseille, France e-mail: [email protected] Z. Shen University of Kentucky, Lexington, KY, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_4

207

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of the behavior near the boundary. The linear Stokes operator associated to the various boundary conditions is first studied. Then a classical fixed-point theorem is used to show how the properties of the operator lead to local solutions or global solutions for small initial data.

1

Introduction

The aim of this chapter is to describe how to find solutions of the Navier-Stokes equations 8 ˆ @ u  u C r C .u  r/u D 0 in .0; T /  ; ˆ < t div u D 0 in .0; T /  ; ˆ ˆ : u.0/ D u0 in ;

(NS)

in a bounded Lipschitz domain   R3 and a time interval .0; T / (T  1), for initial data u0 in a critical space, with one of the following boundary conditions on @: 1. Dirichlet boundary conditions: u D 0;

(Dbc)

also called “no-slip” boundary conditions, which can be also decomposed as a nonpenetration condition   u D 0 and a tangential part   u D 0 which model the fact that the fluid does not slip at the boundary; this is commonly used for a boundary between a fluid and a rigid surface; 2. Neumann boundary conditions: Œ.ru/ C .ru/>    D 0;

 2 .1; 1;

(Nbc)

which can be rewritten as T .u; / D 0 where T .u; / WD .ru/ C .ru/>   Id; if  D 0, (Nbc) becomes @ u D ; if  D 1, T1 .u; / is the Cauchy’s stress tensor so that (Nbc) can be viewed, for instance, as an absence of stress on the interface separating two media in the case of a free boundary; (Nbc) can be decomposed into its normal and tangential parts and can be rewritten in the following form:    .1 C /   @ u D ; .ru/ C .ru/>  tan D 0I (1) 3. Hodge boundary conditions:   u D 0;

  curl u D 0;

(Hbc)

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also called “absolute” boundary conditions (see [53, Section 9] or “perfect wall” condition (see [1]); they have been studied in, e.g., [4] and [24]; they are related to the more traditionally used “Navier’s slip” boundary condition:   u D 0;



  ru/> C .ru  tan D 0:

(2)

See discussion below (see also a detailed discussion in [35, Section 2]). 4. Robin boundary conditions:   u D 0;

  curl u D ˛ u;

˛ > 0I

(Rbc)

since   u D 0, u is a tangential vector field at the boundary, so it makes sense to compare it to the tangential part of the vorticity: it describes the fact that the fluid slips with a friction proportional to the vorticity. Remark that (Hbc) is recovered if ˛ D 0 and (Dbc) if ˛ D 1. In the boundary conditions above, .x/ denotes the unit exterior normal vector at a point x 2 @ (defined almost everywhere when @ is a Lipschitz boundary). As explained in [35, Section 2 and Section 6], the Hodge boundary conditions (Hbc) are close to the Navier’s slip boundary conditions (2). Indeed, if  is assumed to be smooth enough, say of class C2 , under the condition   u D 0, the following holds: 

  ru/> C .ru  tan D   curl u C 2 Wu

where W is the Weingarten map (also called the shape operator, see [45, Chapter 5]) on @ acting on tangential fields (see also [17, Section 3]). In particular, the term Wu is a zero-order term, depending linearly on the velocity field u and is equal to 0 on flat portions of the boundary. The strategy in this chapter to solve the Navier-Stokes equations with one of the boundary conditions described above is to find a functional setting in which the Fujita-Kato scheme applies, such as in their fundamental paper [20]. In all situations, the idea is to study the linear problem to prove enough regularizing properties of the Stokes semigroup so that the nonlinear problem can be treated via a fixed-point method. For the last two types of boundary conditions (Hbc) and (Rbc), the Navier-Stokes system is rewritten as follows: 8 ˆ @ u  u C r  u  curl u D 0 in .0; T /  ; ˆ < t div u D 0 in .0; T /  ; ˆ ˆ : u.0/ D u0 in :

(NS’)

This is motivated by the form of the boundary conditions and the fact that, for a smooth enough vector field u,

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.u  r/u D 12 rjuj2  u  curl u; so that (NS) becomes (NS’) with the pressure  replaced by the so-called dynamical pressure  C 12 juj2 (see, e.g., [24] or [4]). In this chapter,   R3 is a bounded, simply connected, Lipschitz domain. The chapter is organized as follows. In Sect. 2, the Dirichlet-Stokes operator is defined in 2 p the ˚ L setting and then in the L theory. Existence of a local 2solution of the system (NS); (Dbc) for initial values in a critical space in the L -Stokes scale is then shown. In Sect. 3, the previous proofs are adapted in the case of Neumann boundary ˚  ˚ conditions, i.e., for the system (NS); (Nbc) . In Sect. 4, the˚system (NS’); (Hbc) 3 3 is studied for initial conditions  in the critical space u 2˚ L .I R /Idiv u D 0 in ;   u D 0 on @ , whereas in Sect. 5, the system (NS’); (Rbc) is considered in a C1 domain.

2

Dirichlet Boundary Conditions

For a more complete exposition of the results in this section, as well as an extension to more general domains, the reader can refer to [34, 41] and [51]. The case where  is smooth was solved by Fujita and Kato in [20]. In [15], the case of bounded Lipschitz domains  was studied for initial data not in a critical space.

2.1

The Linear Dirichlet-Stokes Operator

2.1.1 The L2 Theory The following remarks about L2 vector fields on  will be used throughout this chapter. Remark 1. For   R3 a bounded Lipschitz domain, let u 2 L2 .I R3 / such that div u 2 L2 .I R/. Then   u can be defined on @ in the following weak sense in 1 H  2 .@I R/: for  2 H 1 .I R/, hu; ri C hdiv u; i D h  u; 'i@ ;

(3)

where ' D Trj@ , the right-hand side of (3) depends only on ' on @ and not on the choice of , its extension to . The notation h; iE stands for the L2 -scalar product on E. The following Hodge decomposition holds on vector fields: L2 .I R3 / is equal ?

to the orthogonal direct sum HD ˚ G, where ˚  HD D u 2 L2 .I R3 /I div u D 0 in ;   u D 0 on @

(4)

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

211

and G D rH 1 .I R/. This follows from the following theorem due to Georges de Rham [12, Chap. IV §22, Theorem 17’]; see also [55, Chap.I §1.4, Proposition 1.1]. 3 0 Theorem 1 (de Rham). Let T be a distribution in C1 c .I R / such that hT; i D 1 3 0 for all  2 Cc .I R / with div  D 0 in . Then there exists a distribution 1 0 0 S 2 C1 c .I R/ such that T D rS . Conversely, if T D rS with S 2 Cc .I R/ , 1 3 then hT; i D 0 for all  2 Cc .I R / with div  D 0 in .

Remark 2. In the case of a bounded Lipschitz domain   R3 , the space HD 3 coincides with the closure in L2 .I R3 / of the space of vector fields u 2 C1 c .I R / with div u D 0 in . Denote by J W HD ,! L2 .I R3 / the canonical embedding and P W L2 .I R3 / ! HD the orthogonal projection, called either Leray or Helmholtz projection. It is clear that PJ D IdHD. Define now the space VD D H01 .I R3 / \ HD : it is a closed subspace of H01 .I R3 /. The embedding J restricted to VD maps VD to H01 .I R3 /: denote it by J0 W VD ,! H01 .I R3 /. Its adjoint J00 D P1 W H 1 .I R3 / ! VD0 is then an extension of the orthogonal projection P. The space HD is endowed with the norm u 7! kuk2 and VD with the norm u 7! kruk2 . The definition of the Dirichlet-Stokes operator then follows. Definition 1. The Dirichlet-Stokes operator is defined as being the associated operator of the bilinear form: a W VD  VD ! R;

a.u; v/ D

3 X h@i J0 u; @i J0 vi: iD1

Proposition 1. The Dirichlet-Stokes operator AD is the part in HD of the bounded operator A0;D W VD ! VD0 defined by A0;D u W VD ! R, .A0;D u/.v/ D a.u; v/, and satisfies ˚  D.AD / D u 2 VD I P1 . D /J0 u 2 HD ; AD u D P1 . D /J0 u u 2 D.AD /; 2 3 where  D denotes the weak vector-valued Dirichlet-Laplacian in L .I R /. The operator AD is self-adjoint, invertible, AD generates an analytic semigroup of 1

contractions on HD , D.AD2 / D VD , and for all u 2 D.AD /, there exists  2 L2 .I R/ such that JAD u D J0 u C r and D.AD / admits the following description: ˚  D.AD / D u 2 VD I 9  2 L2 .I R/ W J0 u C r 2 HD :

(5)

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Proof. By definition, for u 2 D.AD / and for all v 2 VD , hAD u; vi D a.u; v/ D

n X

h@j J0 u; @j J0 vi

j D1

D

n X

2 H 1 h@j J0 u; J0 viH01

D

H 1 h./J0 u; J0 viH01

j D1

D VD0 hP1 ./J0 u; viVD : The third equality comes from thePdefinition of weak derivatives in L2 ; the fourth equality comes from the fact that nj D1 @2j D . The last equality is due to the fact that J00 D P1 . Therefore, AD u and P1 ./J0 u are two linear forms which coincide on VD ; they are then equal, which proves that A0;D D P1 ./J0 W VD ! VD0 . Moreover, the fact that u 2 D.AD / implies that AD u is a linear form on HD , so that the linear form P1 ./J0 u, originally defined on VD , extends to a linear form on HD (since VD is dense in HD by de Rham’s theorem). The fact that AD is selfadjoint and AD generates an analytic semigroup of contractions comes from the properties of the form a: a is bilinear, symmetric, sectorial of angle 0, and coercive 1

on VD VD . The property that D.AD2 / D VD is due to the fact that AD is self-adjoint, applying a result by J.L. Lions [29, Théorème 5.3]. To prove the last assertions of this proposition, let u 2 D.AD /. Then AD u 2 HD and P1 J .AD u/ D PJ .AD u/ D u. Moreover, if u 2 D.AD /, u belongs, in particular, to VD . Therefore, J0 u 2 H01 .I R3 / and ./J0 u 2 H 1 .I R3 /. The following identities take place in VD0 :   P1 J .AD u/  ./J0 u D P1 J .AD u/  P1 ./J0 u D AD u  AD u D 0: 0 By de Rham’s theorem, this implies that there exists p 2 C1 c .I R/ such that 1 3 Q J .AD u/  ./J u D rp: rp 2 H .I R /, which implies that p 2 L2 .I R/. t u

The relations between the spaces and the operators described above are summarized in the following commutative diagram:

VD

J0

d A0,D

HD

d J P=J

d

VD

H01 L2 d

P1 =J0

H −1

(−ΔΩ D)

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

213

In the case of a bounded Lipschitz domain   R3 , the following property of 3

D.AD4 / also holds; see [34, Corollary 5.5]. 3

Proposition 2. The domain of AD4 is continuously embedded into W01;3 .I R3 /. It has been proved by R. Brown and Z. Shen [7] that the domain of AD 3 1;p is embedded into W0 .I R3 / \ W 2 ;2 .; R3 / for some p > 3. The proof Proposition 2 uses the well-posedness result for the Poisson problem of the Stokes system [16, Theorem 5.6], similar to the corresponding result proved in [26] for the Laplacian.

2.1.2 The Lp Theory P. Deuring provided in [14] an example of a domain with one conical singularity such that the Dirichlet-Stokes semigroup does not extend to an analytic semigroup in Lp for p large, away from 2. M.E. Taylor in[54], however, conjectured that this should be true for p in an interval containing 32 ; 3 , which was indeed proved 12 years later by the second author in [51]. 1 3 Let C1 c; ./ denote the space of vector fields u 2 Cc .I R / with div u D 0 in  and p 3 Lp ./ D the closure of C1 c; ./ in L .I R /:

(6)

Note that if  is Lipschitz and p D 2, L2 ./ D HD . In view of Proposition 1, the Dirichlet-Stokes operator in the Lp setting for 1 < p < 1 is defined by AD;p D u C r;

(7)

with the domain n 1;p D.AD;p / D u 2 W0 .I R3 /I div u D 0 in  and  u C r 2

Lp ./

o for some  2 L ./ :

(8)

p

p

1 ./  D.AD;p /, the operator AD;p is densely defined in L ./ and Since Cc; 1 AD;p .u/ D P./u for u 2 Cc; ./. If p D 2, AD;p agrees with the DirichletStokes operator AD defined in the previous subsection. The following theorem was proved in [51].

Theorem 2. Let  be a bounded Lipschitz domain in R3 . Then there exists " > 0, depending only on the Lipschitz character of , such that AD;p generates p a bounded analytic semigroup in L ./ for .3=2/  " < p < 3 C ".

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It was in fact proved in [51] that if  is a bounded Lipschitz domain in Rd , p d  3, then AD;p generates a bounded analytic semigroup in L ./ for 2d 2d "

0 depends only on d and the Lipschitz character of . This was done by establishing the following resolvent estimate in Lp : k.AD;p C /1 f kLp .ICd /  Cp jj1 kf kLp .ICd /

(10)

d for any f 2 C1 c .I C / with div f D 0 in , where p satisfies (9),

˚   2 † WD z 2 C W  ¤ 0 and j arg.z/j <   ; and 2 .0; =2/. The constant Cp in (10) depends only on d , , p, and . It has long been known that if  is a bounded C2 domain in Rd , the resolvent estimate (10) holds for  2 † and 1 < p < 1 (see [21]). Consequently, the operator AD;p generates a bounded analytic semigroup in Lp for any 1 < p < 1, if  is C2 . The case of nonsmooth domains is much more delicate. As mentioned earlier, P. Deuring constructed a three-dimensional Lipschitz domain for which the Lp resolvent estimate (10) fails for p sufficiently large. This was somewhat unexpected. Indeed it was proved in [48] that the Lp resolvent estimate holds for 1 < p < 1 in bounded Lipschitz domains in R3 for any second-order elliptic systems with constant coefficients satisfying the Legendre-Hadamard conditions (the range is d2d  " < p < d2d C " for d  4). It is worth mentioning that C3 3 it is not known whether the range of p in Theorem 2 is sharp. The approach used in [51] to the proof of (10) is described below. Consider the operator T on L2 .I Cd /, defined by T .f / D u, where  2 † and u 2 H01 .I Cd / are the unique solution to the Stokes system: 8 u C r C u D f ˆ ˆ < div u D 0 ˆ ˆ : uD0

in ; in ;

(11)

on @:

Note that T is bounded on L2 .I Cd / and kT kL2 !L2  C . To show that T is bounded on Lp .I Cd / and kT kLp !Lp  C for 2 < p < d2d C ", a real 1 variable argument is used, which may be regarded as a refined (and dual) version of the celebrated Calderón-Zygmund lemma. According to this argument, which originated from [8] and was further developed in [49,50], one only needs to establish the weak reverse Hölder estimate:

4 Stokes Problems in Irregular Domains With Various Boundary Conditions



jujpd

1=pd

C



juj2

B.x0 ;r/\

for pd D

2d , d 1

1=2

215

(12)

B.x0 ;2r/\

whenever u 2 H01 .I Cd / is a (local) solution of the Stokes system: (

u C r C u D 0;

(13)

div u D 0

in B.x0 ; 3r/ \  for some x0 2  and 0 < r < c diam./. The extra " in the range of p is due to the self-improvement property of the weak reverse Hölder inequalities (see, e.g., [25]). To prove the estimate (12), the Dirichlet problem for the Stokes system (13) is considered in a bounded LipschitzR domain  in Rd , with boundary data u D f on @, where f 2 L2 .@I Cd / and @ f   D 0. The goal is to show that k.u/ kL2 .@/  C kf kL2 .@/ ;

(14)

where .u/ denotes the nontangential maximal function of u and is defined by n o .u/ .Q/ WD sup ju.x/j W x 2  and jx  Qj < C0 dist.x; @/ for any Q 2 @ (C0 > 1 is a large fixed constant depending on d and ). This, together with the inequality Z

juj

pd

1=pd

C

Z



j.u/ j2

1=2

;

@

which holds for any continuous function u in , leads to Z 

jujpd

1=pd

C

Z

juj2

1=2

:

(15)

@

The desired estimate (12) follows by applying (15) in the domain B.x0 ; t r/ \  for t 2 .1; 2/ and then integrating the resulting inequality with respect to t over .1; 2/. Finally, the nontangential maximal function estimate (14) is established by the method of layer potentials. The case  D 0 was studied in [11, 18], where the L2 Dirichlet problem as well as the Neumann type boundary value problems with boundary data in L2 for the system u C r D 0 and div u D 0 in a Lipschitz domain  was solved by the method of layer potentials, using the Rellich-type estimates: @u  krtan ukL2 .@/ : 2 @ L .@/

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Here @u is a conormal derivative and rtan u denotes the tangential derivative of @ u on @. The reader is referred to the book [27] by C. Kenig for references on related work on Lp boundary value problems for elliptic and parabolic equations in nonsmooth domains. In an effort to solve the L2 initial boundary value problems for the nonstationary Stokes equations @t u  u C r D 0 and div u D 0 in a Lipschitz cylinder .0; T /  , the Stokes system (13) for  D i with 2 R was considered by the second author in [47]. One of the key observations in [47] is that if  D i and 2 R is large, the Rellich estimates for the system (13) involve two extra terms j j1=2 kukL2 .@/ and j jkukH 1 .@/ , where H 1 .@/ denotes the dual of H 1 .@/. While the first term j j1=2 kukL2 .@/ was expected in view of the Rellich estimates for the Helmholtz equation  C i in [6], the second term j jku  kH 1 .@/ was not. Let @u @u D  : @ @ By following the general approach in [47], it was proved in [51] that if .u; / is a suitable solution of (13) in , then

k

@u kL2 .@/  krtan ukL2 .@/ C jj1=2 kukL2 .@/ C jjku  kH 1 .@/ @

(16)

holds uniformly in  for  2 † with jj  c > 0. As in the case of Laplace’s equation [56], the estimate (14) follows from (16) by the method of layer potentials. The reader is referred to [51] for the details.

2.2

The Nonlinear Dirichlet-Navier-Stokes Equations

˚  The system (NS); (Dbc) is invariant under the scaling u .t; x/ D  u.2 t; x/, ˚ 2 . t; x/ 2 .0; T /   ( > 0): if u is a solution of (NS); (Dbc) in .0; T /   for  ˚   the initial value u0 , then u is a solution of (NS); (Dbc) in 0; T2  1  for the initial value x 7! u0 .x/. ˚  The goal here is to find the so-called mild solutions of the system (NS); (Dbc) for initial values u0 in a critical space, in the same spirit as in [20]. 1

Lemma 1. The space D.AD4 / is a critical space for the Navier-Stokes equations. 1

Proof. The space D.AD4 / is invariant under the scaling u .x/ D u0 .x/ for 1 x 2 1 ,  > 0. Indeed, it suffices to check that ku k2 D  2 kuk2 and 1

1

kru k2 D  2 kruk2 and apply the fact that D.AD4 / is the interpolation space

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

217 1

(with coefficient 12 ) between HD , closed subspace of L2 .I R3 /, and VD D D.AD2 /, t u closed subspace of H01 .I R3 /. For T > 0, define the space ET by n 1 3 1 ET D u 2 Cb .Œ0; T I D.AD4 //I u.t / 2 D.AD4 /; u0 .t / 2 D.AD4 / for all t 2 .0; T  1

3

o

1

and sup kt 2 AD4 u.t /k2 C sup ktAD4 u0 .t /k2 < 1 t2.0;T /

t2.0;T /

endowed with the norm 1

1

3

1

kukET D sup kAD4 u.t /k2 C sup kt 2 AD4 u.t /k2 C sup ktAD4 u0 .t /k2 : t2.0;T /

t2.0;T /

t2.0;T /

The fact that ET is a Banach space is straightforward. Assumenow that u 2 ET ˚ and that .J0 u; p/ (with p 2 L2 .I R/) satisfy (NS); (Dbc) in H 1 .I R3 /: indeed, every term rp, @t J0 u, J0 u, and .J0 u  r/J0 u independently belongs to H 1 .I R3 /. Apply P1 to the equations and obtain   u0 .t / C AD u.t / D P1 .J0 u  r/J0 u ˚  since P1 rp D 0 and P1 ./J0 u D A0;D u. The problem (NS); (Dbc) is then reduced to the abstract Cauchy problem:   u0 .t / C A0;D u.t / D P1 .J0 u  r/J0 u u.0/ D u0 ;

u 2 ET ;

(17)

for which a mild solution is given by the Duhamel formula: u D ˛ C .u; u/;

(18)

where ˛.t/ D e tAD u0 and Z

t

.u; v/.t / D 0

   e .ts/AD  12 P1 .J0 u.s/  r/J0 v.s/ C .J0 v.s/  r/J0 u.s/ ds:

(19) The strategy to find u 2 ET satisfying u D ˛ C .u; u/ is to apply a fixed-point theorem. For that, ET needs to be a “good” space for the problem, i.e., ˛ 2 ET and .u; u/ 2 ET . The fact that ˛ 2 ET follows directly from the properties of the Stokes operator AD and the semigroup .e tAD /t0 . Proposition 3. The mapping  W ET  ET ! ET is bilinear, continuous, and symmetric.

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Proof. The fact that  is bilinear and symmetric is immediate, once it is proved that it is well-defined. For u; v 2 ET , let   f .t/ D  12 P1 .J0 u.t /  r/J0 v.t / C .J0 v.t /  r/J0 u.t / ;

t 2 .0; T /:

(20)

By the definition of ET and Sobolev embeddings, it is easy to see that .J0 u.t /  r/J0 v.t / C .J0 v.t /  r/J0 u.t / 2 L2 .I R3 / and .J0 u.t /  r/J0 v.t / C .J0 v.t /  r/J0 u.t /  C t  34 kukE kvkE T T 2 where C is a constant independent from t , which gives the following estimate: f .t/  C t  34 kukE kvkE T T 2

(21)

Therefore, Z

1 4

kAD .u; v/.t /k2  C

t 0

1

Z

t 0

and since obtained:

Rt

0 .t

1

3

 s/ 4 s  4 ds D

3

kAD4 e .ts/AD kL.HD / C s  4 kukET kvkET ds  1 3 .t  s/ 4 s  4 ds kukET kvkET ;

R1 0

1

3

.1  s/ 4 s  4 ds, the following estimate is finally

1

kAD4 .u; v/.t /k2  C kukET kvkET :

(22)

1

The proof of the continuity of t 7! AD4 .u; v/.t / on HD is straightforward once the estimate (22) is established. The proof of the fact that p 3 k t AD4 .u; v/.t /k2  C kukET kvkET 1

(23)

3

is proved the same way, replacing AD4 by AD4 and using the fact that 3

3

kAD4 e .ts/AD kL.HD /  C .t  s/ 4 and Z 0

t

3

3

1

.t  s/ 4 s  4 ds D t  2

Z 0

1

3

3

.1  s/ 4 s  4 ds:

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

219

It remains to prove the estimate on the derivative with respect to t of .u; v/. Rewrite f as defined in (20) as follows:   f .s/ D  12 P1 r  J0 u.s/ ˝ J0 v.s/ C J0 v.s/ ˝ J0 u.s/

(24)

where u ˝ v denotes the matrix .ui vj /1i;j 3 and the differential operator r acts on matrices M D .mi;j /1i;j 3 the following way: r M D

3 X

@i mi;j

iD1

 1j 3

:

For u; v 2 ET and s 2 .0; T /,  f 0 .s/ D 12 P1 r  J u0 .s/ ˝ J0 v.s/ C J0 u.s/ ˝ J v 0 .s/  C J v 0 .s/ ˝ J0 u.s/ C J0 v.s/ ˝ J u0 .s/ For all s 2 .0; T /, 5

1

s 4 kJ u0 .s/ ˝ J0 v.s/k2  ksJ u0 .s/k3 ks 4 J0 v.s/k6 1

1

1

 ksAD4 u0 .s/k2 ks 4 AD2 v.s/k2  kukET kvkET ; where the first inequality comes from the fact that L3  L6 ,! L2 , the second 1

inequality comes from the Sobolev embeddings D.AD4 / ,! L3 .I R3 / and 1

D.AD2 / ,! L6 .I R3 /, and the third inequality follows directly from the definition of the space ET . Of course the same occurs for the other three terms J0 u.s/˝J v 0 .s/, 1

J v 0 .s/ ˝ J0 u.s/, and J0 v.s/ ˝ J u0 .s/. Therefore, since AD 2 maps Vd0 to HD , 5

1

sup ks 4 AD 2 f 0 .s/k2  c kukET kvkET :

(25)

0 0 independent of such that k.u; v/kLp .0; ID.A1=4 //  Cp kukLp .0; ID.A1=4 // kvkL1 .0; ID.A1=4 // : D

D

(26)

D

If v 2 L1 .0; I VD /, the following improved estimate holds k.u; v/k

1

1 Lp .0; ID.AD4 //

 Kp 4 kukLp .0; ID.A1=4 // kvkL1 .0; IVD / ;

(27)

D

where Kp > 0 is a constant independent of . Proof. First, let M be the maximal regularity operator on HD : for all ' 2 Lp .0; I HD /, M' is defined by M'.t/ WD

Z

t

AD e .ts/AD '.s/ ds;

t 2 .0; /:

0

Since HD is a Hilbert space and AD generates an analytic semigroup in HD , the operator M is bounded on Lp .0; I HD / for all p 2 .1; 1/ and all > 0; see, e.g., [13]. Moreover, kMkL.Lp .0; IHD // is independent of . Then 1  3  AD4 .u; v/ D M AD 4 f

1

1

where f is defined by (24). For u 2 Lp .0; I D.AD4 / and v 2 L1 .0; I D.AD4 /, by Sobolev embeddings, J u ˝ J v C J v ˝ J u 2 Lp .0; I L3=2 .I R3 //, with the estimate kJ u ˝ J v C J v ˝ J ukLp .0; IL3=2 .IR3 //  C kukLp .0; ID.A1=4 // kvkL1 .0; ID.A1=4 // ; D

D

1

where the constant C depends only on the constant of the embedding D.AD4 / ,! 3   L3 .I R3 /. This implies that f 2 Lp 0; I P1 .W 1;3=2 / . Since D.AD4 / ,! 3     0 W01;3 .I R3 / (see Proposition 2), the embedding P1 W 1;3=2 .I R3 / ,! D.AD4 / 3

holds and therefore AD 4 f 2 Lp .0; I HD / with 3

kAD 4 f kLp .0; IHD /  C kukLp .0; ID.A1=4 // kvkL1 .0; ID.A1=4 // : D

Using the Lp maximal regularity result in HD gives (26).

D

222

S. Monniaux and Z. Shen 1

To prove (27), let u 2 Lp .0; I D.AD4 // and v 2 L1 .0; I VD /. Using the 1

embeddings D.AD4 / ,! L3 .I R3 / and VD ,! L6 .I R3 /, kJ u ˝ J v C J v ˝ J ukLp .0; IL2 .;R3 //  C kukLp .0; ID.A1=4 // kvkL1 .0; IVD / : D

As before, this implies that f 2 Lp .0; I VD0 /, and therefore Z

1 4

AD .u; v/.t / D

t 0

3  1  AD4 e .ts/AD AD 2 f .s/ ds;

t 2 .0; /:

Using the analyticity of the semigroup .e tAD /t0 in HD and Young’s inequality, 1

3

kAD4 .u; v/kLp .0; IHD /  C kt 7! t  4 kL1 .0; / kukLp .0; ID.A1=4 // kvkL1 .0; IVD / : D

t u Proof of Theorem 4. The proof is inspired by the method described in [39] (see also [2, Section 8]). Let p 2 .1; 1/, " > 0 to be chosen later and w WD u  v 2 1

1

Cb .0; T I D.AD4 //  Lp .0; T I D.AD4 //: w satisfies w D .u; w/ C .w; v/ D .w; u C v  2˛/ C 2.w; ˛/ D .w; u C v  2˛/ C 2.w; ˛  ˛" / C 2.w; ˛" / where ˛" .t / D e tAD u0;" , with u0;" 2 VD satisfying the estimate ku0;"  u0 kD.A1=4 / D

1

 ". Using Lemma 2, w is estimated in Lp .0; I D.A 4 // as follows: kwkLp .0; ID.A1=4 //

  1  kwkLp .0; ID.A1=4 // Cp .ku C v  2˛kL1 .0; ID.A1=4 // C "/ C Kp 4 ku0;" kVD D    p " C g" . / kwkLp .0; ID.A1=4 // ; 1

where g" . / D ku C v  2˛kL1 .0; ID.A1=4 // C 4 ku0;" kVD ! 0. This shows that D

t!0

choosing " > 0 small enough, there exists > 0 such that kwkLp .0; ID.A1=4 //  1 kwkLp .0; ID.A1=4 // ; in other terms, w D 0 on Œ0; / (recall that w is continuous 2 on Œ0; T /). If D T , then it was proved that u D v on Œ0; T /. If < T , by continuity, w. / D 0 also holds. The previous reasoning can be iterated on intervals of the form Œk ; .k C 1/ / to prove ultimately that w D 0 on Œ0; T / (remark again that all constants Cp ; Kp ; p appearing in the estimates above are independent of ). t u

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

3

223

Neumann Boundary Conditions

˚  In this section, the system (NS); (Nbc) is studied. The results proved in [37] will be only surveyed, the method to prove existence of solutions being similar to what has been done in Sect. 2.

3.1

The Linear Neumann-Stokes Operator

Before defining the Neumann-Stokes operator, the following integration by parts formula will be useful. Lemma 3. Let  2 R, u; w W  ! R3 , ;  W  ! R be sufficiently nice functions defined on the Lipschitz domain   R3 . Let L u D u C r.div u/ and define the conormal derivative:   @ .u; / D ru C .ru/>   

on @:

(28)

Then the following integration by parts formula holds: Z

Z





Z

I .ru; rw/   div w dx C

.L u  r/  w dx D  



@

@ .u; /  w d (29)

Z

Z

D

.L w  r/  u dx C 



  div w   div u dx



Z



(30)

.i;j i;j C i;j j;i /;

for  D .i;j /1i;j 3 and  D .i;j /1i;j 3 :

C @

 @ .u; /  w  @ .w; /  u d ;

where I .; / D

3 X

i;j D1

Recall that ru D .@i uj /1i;j 3 . The space L2 .I R3 / admits the following Hodge decomposition, dual to the one ˚  ? shown in Sect. 2: HN ˚ G0 , where G0 WD rI  2 H01 .I R/ and ˚  HN WD u 2 L2 .I R3 /I div u D 0 :

(31)

Following the steps of the previous section, define VN D H 1 .I R3 / \ HN and JN W HN ,! L2 .I R3 / the canonical embedding, PN D JN0 W L2 .I R3 / ! HN

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S. Monniaux and Z. Shen

the orthogonal projection, and JQ N W VN ,! H 1 .I R3 / the restriction of JN on VN and JQ N0 D PQ N W .H 1 .I R3 //0 ! VN0 , extension of PN to .H 1 .I R3 //0 . The Neumann-Stokes operator is defined as follows. Definition 2. Let  2 R. The Neumann-Stokes operator A is defined as being the associated operator of the bilinear form: a W VN  VN ! R;

Z

I .r JQ N u; r JQ N v/ dx

a .u; v/ D 

In the case where  2 .1; 1, the bilinear form a is continuous, symmetric, coercive, and sectorial. So its associated operator is self-adjoint, invertible and the negative generator of an analytic semigroup of contractions on HN . The following proposition is a consequence of the integration by parts formula (29), [37, Theorem 6.8] and [29, Théorème 5.3]. Proposition 4. Let  2 .1; 1. The Neumann-Stokes operator A is the part in HN of the bounded operator A0; W VN ! VN0 defined by .A0; u/.v/ D a .u; v/. The operator A is self-adjoint, invertible, A generates an analytic semigroup 1

of contractions on HN , D.A2 / D VN and for all u 2 D.A /, there exists  2 L2 .I R/ such that JN A u D JQ N u C r

(32)

and D.A / admits the following description: ˚  D.A /D u 2 VN I 9  2 L2 .I R/ W f DJQ N uCr 2 HN and @ .u; /f D 0 ; where @ .u; /f is defined in a weak sense for all f 2 .H 1 .I R3 //0 by h@ .u; /f

Z ; i@ D

.H 1 /0 hf; ‰iH 1

C

I .r JQ n u; r‰/ dx 

L2 h; div ‰iL2



for ‰ 2 H 1 ./ and

D Tr@ ‰.

Remark 3. If f 2 .H 1 .I R3 //0 , the quantity @ .u; /f exists on @ in the Besov 1 space B2;21 .@I R3 / D H  2 .@; R3 / according to [37, Proposition 3.6]. 2

Thanks to [37, Sections 9 & 10], a good description of the domain of fractional powers of the Neumann-Stokes operator A can be given. In particular, in [37, Corollary 10.6], it was established that 3

D.A4 / is continuously embedded into W 1;3 .I R3 /:

(33)

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

3.2

225

The Nonlinear Neumann-Navier-Stokes Equations

The results in Sect. 3.1 allow to prove a result similar ˚to Theorem 3 for the Navier-Stokes system with Neumann boundary conditions (NS); (Nbc) . As in the 1

previous section, it is not difficult to see that D.A4 / ,! L3 .I R3 / is a critical space for the system. For T 2 .0; 1, following the definition of ET in Sect. 2, define n 1 3 1 FT D u 2 Cb .Œ0; T I D.A4 //I u.t / 2 D.A4 /; u0 .t / 2 D.A4 / for all t 2 .0; T  3

1

1

o

and sup kt 2 A4 u.t /k2 C sup ktA4 u0 .t /k2 < 1 t2.0;T /

t2.0;T /

endowed with the norm 1

1

3

1

kukFT D sup kA4 u.t /k2 C sup kt 2 A4 u.t /k2 C sup ktA4 u0 .t /k2 : t2.0;T /

t2.0;T /

t2.0;T /

The same tools as in 2.2 apply, so the following result can be proved (see [37, Theorem 11.3]). 1

Theorem 5. Let   R3 be a bounded Lipschitz domain and let u0 2 D.A4 /. Let ˇ and be defined by ˇ.t / D e tA u0 ;

t  0;

and for u; v 2 FT and t 2 .0; T /, Z

t

.u; v/.t / D 0

  e .ts/A . 12 PN / .JN u.s/  r/JQ N v.s/ C JN v.s/  r/JQ N u.s/ ds W

1

(i) If kA4 u0 k2 is small enough, then there exists a unique u 2 F1 solution of u D ˇ C .u; u/. 1

(ii) For all u0 2 D.A4 /, there exists T > 0 and a unique u 2 FT solution of u D ˇ C .u; u/. A comment here may be necessary to link ˚  the solution u obtained in Theorem 5 and a solution of the system (NS); (Nbc) . If u 2 FT , then u0 2 HN and .JN u  r/JQ N u 2 L2 .I Rn /. Moreover, if u satisfies the equation u D ˇ C .u; u/, then u is a mild solution of   A u D u0  PN .JN u  r/JQ N u 2 HN :

226

S. Monniaux and Z. Shen

Going further,   JN PN .JN u  r/JQ N u D .JN u  r/JQ N u  rq where q 2 H01 .I R/ satisfies q D div .JN u  r/JQ N u/ 2 H 1 .I Rn /: Therefore, by definition of A , there exists  2 L2 .; R/ such that JQ n u C r D JN .A u/ D JN u0  .JN u  r/JQ N u C rq and at the boundary, .u; / satisfies (Nbc) in the weak sense as in Proposition 4. Since q 2 H01 .I R/, .u;˚  q/ satisfies  also (Nbc). This proves that .u;   q/ is a solution of the system (NS); (Nbc) . 1 The uniqueness is true in a larger space than FT : for each u0 2 D.A 4 /, there is ˚  1 at most one u 2 Cb .Œ0; T /I D.A 4 //, mild solution of the system (NS); (Nbc) . For a more precise statement, see [37, Theorem 11.8].

4

Hodge Boundary Conditions

Most of the results presented here are proved thoroughly in [36] for the linear theory and [35] for the nonlinear system. The linear Hodge-Laplacian on Lp -spaces is first studied and then the Hodge-Stokes operator before applying the properties of this operator to prove the existence of mild solutions of the Hodge-Navier-Stokes system in L3 . Some recent developments/improvements can be found in [30].

4.1

The Hodge-Laplacian and the Hodge-Stokes Operators

We denote by H the space L2 .I R3 /. Let

and

˚  WT WD u 2 H I curl u 2 H; div u 2 L2 .I R/ and   u D 0 on @ ; ˚  WN WD u 2 H I curl u 2 H; div u 2 L2 .I R/ and   u D 0 on @ ;

(subscript T is for “tangential” and N for “normal”) both endowed with the scalar product hhu; viiW WD hcurl u; curl vi C hdiv u; div vi C hu; vi ; where h; iE denotes the L2 .E/-pairing.

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

227

Remark 4. As in Remark 1 for a bounded Lipschitz domain  and a vector field w 2 H satisfying curl w 2 H , define   w on @ in the following weak sense in 1 H  2 .@I R3 /: for  2 H 1 .I R3 /, hcurl w; i  hw; curli D h  w; 'i@

(34)

where ' D Trj@  , the right-hand side of (34) depends only on ' on @ and not on the choice of , its extension to . Remark 5. In the case of smooth bounded domains, i.e., with a C1;1 boundary or convex, the spaces WT and WN are contained in H 1 .I R3 / (see, e.g., [3, Theorems 2.9, 2.12, and 2.17]). This is not the case if  is only Lipschitz. The Sobolev embedding associated to 1 the spaces WT;N is as follows: WT;N ,! H 2 .I R3 / with the estimate   kukH 1=2  C kuk2 C kcurl uk2 C kdiv uk2 ;

u 2 WT;N I

(35)

see, for instance, [9] or [32, Theorem 11.2] where it was proved moreover that if u 2 WT;N ; then u has an L2 trace at the boundary @ W uj@ D .  u/ C .  u/   2 L2 .@I R3 /;   and kuj@ kL2 .@IR3 /  C kuk2 C kcurl uk2 C kdiv uk2 :

(36) (37)

Remark 6. If  is of class C1 , the previous result applies also if u 2 Lp .I R3 / with curl u 2 Lp .I R3 /, div u 2 Lp .I R/, and   u D 0 on @ (or   u D 0 on @) if p 2 .1; 1/ (see [32, Theorem 11.2], where it was proved that if  is only Lipschitz, it is also true for p in a range around 2). Remark 7. The Helmholtz projection P˚ W L2 .I R3 / ! HD defined in Sect. 2 (after Remark 2) maps also WT to the space u 2 WT I div u D 0 DW VT . 3 in Sect. 3 (before Definition 2) The projection PN W L2 .I ˚ R / ! HN defined  maps also WN to the space u 2 WN I div u D 0 DW VN . On WT  WT , we define the following form: bT W WT  WT ! R;

bT .u; v/ D hcurl u; curl vi C hdiv u; div vi;

where h; i denotes either the scalar or the vector-valued L2 -pairing. Similarly, we define bN W WN  WN ! R;

bN .u; v/ D hcurl u; curl vi C hdiv u; div vi:

228

S. Monniaux and Z. Shen

Proposition 5. The Hodge-Laplacian operators BT and BN , defined as the associated operators in H of the forms bT and bN , satisfy ˇ n o ˇ   curl u D 0 on @ D.BT;N / D u 2 WT;N I rdiv u 2 H; curl curl u 2 H and ˇˇ .div u/ BT;N u D  u;

u 2 D.BT;N /:

(38)

Proof. Let u 2 WT;N and v 2 H01 .I R3 /  WT;N . Then bT;N .u; v/ D H 1 hrdiv u C curl curl u; viH 1 D H 1 hu; viH 1 0

0

so that BT;N u D u in H 1 .I R3 /. The proof of Proposition 5 is described now in the case of bT defined on WT WT . The case of bN defined on WN  WN can be proved with the same arguments (using PN instead of P in what follows). Let D be the space  ˚ D WD u 2 WT I rdiv u 2 H; curl curl u 2 H and   curl u D 0 on @ : If u 2 D, then BT u D u 2 H and therefore u 2 D.BT /. Conversely, assume that u 2 D.BT /. Then .Id  P/BT u 2 H satisfies for all v 2 WT h.Id  P/BT u; vi D hBT u; .Id  P/vi D bT .u; v/  bT .u; Pv/ D hdiv u; div vi D WT0 hrdiv u; viWT ; so that rdiv u D .Id  P/BT u 2 H . Then curl curl u D BT u C rdiv u 2 H . It remains to prove that   curl u D 0 on @. Remark that it makes sense to consider the tangential part of w WD curl u on the boundary @ since it was just proved 1 that curl w 2 H , and, therefore, thanks to (34),   w 2 H  2 .@I R3 /. For all 1 ' 2 H 2 .@I R3 / \ L2tan .@I R3 /, there exists  2 H 1 .I R3 / such that j@ D '. In that case,  2 WT , and therefore hrdiv u C curl curl u; i D hBT u; i D bT .u; / D hdiv u; div i C hcurl u; curl i D hrdiv u C curl curl u; i  H 1=2 .@/ h  curl u; 'iH 1=2 .@/ : 1

It proves that H 1=2 .@/ h  curl u; 'iH 1=2 .@/ D 0 for all ' 2 H 2 .@I R3 / \ L2tan .@I R3 /, and then   curl u D 0 on @. t u Since the forms bT;N are continuous, bilinear, symmetric, coercive, and sectorial, the operators BT;N generate analytic semigroups of contractions on H ; BT;N is

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229

1=2

self-adjoint and D.BT;N / D WT;N . The following property will be useful in the next section; it links BT and BN , as shown in [43, Proposition 2.2]. Lemma 4. For u 2 H such that curl u 2 H , the following commutator property occurs for all " > 0: curl .1 C "BT /1 u D .1 C "BN /1 curl u:

(39)

Proof. Let u 2 H such that curl u 2 H . Let u" D .1 C "BT /1 u and w" D .1 C "BN /1 curl u. Step 1: curl u" 2 D.BN /. By (38), it holds curl u" 2 H , curl curl u" 2 H , div .curl u" / D 0 2 H 1 ./,   curl u" D 0 on @, and div .curl u" / D 0 on @. To prove that curl u" 2 D.BT /, it remains to show, thanks to (38), that curl curl .curl u" / 2 H . This is due to the fact that curl curl .curl u" / D curl .u" /

in H 1 .; R3 /:

Since u" D BT .1 C "BT /1 u D

 1 u  u" "

and curl u" ; curl u 2 H , the claim follows. Step 2: curl u" D w" . By Step 1, curl u" 2 D.BN /. Moreover, in the sense of distributions,   .1 C "BN /.curl u" / D curl u"  "curl u" D curl u"  "u" D curl u since u"  "u" D .1 C "BT /.1 C "BT /1 u D u. Therefore, curl u" D .1 C "BN /1 curl u D w" which proves the claim.

t u

To prove that the operators BT;N extend to Lp -spaces, it suffices to prove that their resolvents admit L2  L2 off-diagonal estimates. This was proved in, e.g., [36, Section 6] (see also [30]). Proposition 6. There exist two constants C; c > 0 such that for any open sets E; F  R3 such that dist .E; F / > 0 and for all t > 0, f 2 H and u D .Id C t 2 BT;N /1 .1lF f /;

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it holds k1lE uk2 C tk1lE div uk2 C t k1lE curl uk2  C e c

dist .E;F / t

k1lF f k2 :

(40)

Proof. Start by choosing a smooth cutoff function  W R3 ! R satisfying  D 1 on k E,  D 0 on F , and krk1  dist .E;F . Then define  D e ı where ı > 0 is to be / chosen later. Next, take the scalar product of the equation: u  t 2 u D 1lF f;

u 2 D.BT;N /

with the function v D 2 u. Since  D 1 on F and kuk2  k1lF f k2 , it is easy to check then that k uk22 C t 2 k div uk22 C t 2 k curl uk22    k1lF f k22 C 2˛krk1 t 2 k uk2 k div uk2 C k curl uk2 and therefore, using the estimate on krk1 and choosing ı D

dist .E;F / , 4kt

k uk22 C t 2 k div uk22 C t 2 k curl uk22  2k1lF f k22 : Using now the fact that  D e ı on E, p dist .E;F / k1lE uk2 C tk1lE div uk2 C t k1lE curl uk2  2e  4k t k1lF f k2 ; which gives (40) with C D

p 2 and c D

1 . 4k

t u

With a slight modification of the proof, it can be shown that for all 2 .0; /, 3 there exist two constants C; c > 0 such that ˚ for any open sets E; F  R such that dist .E; F / > 0, and for all z 2 † D ! 2 C n f0gI j arg zj <   , f 2 H and u D .zId C BT;N /1 .1lF f /; it holds 1

1

1

jzjk1lE uk2 C jzj 2 k1lE div uk2 C jzj 2 k1lE curl uk2  C e c dist.E;F /jzj 2 k1lF f k2 : (41) Following [31] and [10] (see also [30]), there exist Bogovski˘ı-type operators Ri , Ti , i D 1; 2; 3, and K1;2 ; L1;2 such that for all p 2 .1; 1/, R1 W Lp .I R3 / ! W 1;p .I R/; R2 W Lp .I R3 / ! W 1;p .I R3 /;

1;p

T1 W Lp .I R3 / ! W0 .I R/; 1;p

T2 W Lp .I R3 / ! W0 .I R3 /;

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

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1;p

R3 W Lp .I R/ ! W 1;p .I R3 /;

T3 W Lp .I R/ ! W0 .I R3 /;

K1;2 W Lp .I R3 / ! W 1;p .I R3 /;

1;p

and L1;2 W Lp .I R3 / ! W0 .I R3 /

satisfying R2 curl u C rR1 u D u  K1 u

8 u 2 Lp .I R3 / with curl u 2 Lp .I R/ and curl K1 u D 0 if curl u D 0;

R3 div u C curl R2 u D u  K2 u;

8 u 2 Lp .I R3 / with div u 2 Lp .I R/ and div K2 u D 0 if div u D 0;

T2 curl u C rT1 u D u  L1 u;

(42)

(43)

8 u 2 Lp .I R3 / with curl u 2 Lp .I R/;

  u D 0 on @ and curl L1 u D 0 if curl u D 0; (44) T3 div u C curl T2 u D u  L2 u;

8 u 2 Lp .I R3 / with div u 2 Lp .I R/;

  u D 0 on @ and div L2 u D 0 if div u D 0:

(45)

With these potential operators (at this point, only the relations (43) and (45) are needed) and (41), it is easy to prove that (see, e.g., [30])  ; 2 uniformly in z 2 †

(46) ˚  p where HD WD u 2 Lp .I R3 / s.t. div u D 0 and   u D 0 on @ and Gp WD rW 1;p .I R/ are defined for p 2 .1; 1/; if p D 2, then HD2 D HD and G2 D G defined in Sect. 2. With the same reasoning, one can prove that p

z.zId C BT /1 is bounded in HD and in Gp for p 2

6 5

 ; 2 uniformly in z 2 †

(47) ˚  p 1;p where HN WD u 2 Lp .I R3 / s.t. div u D 0 and Gp;0 WD rW0 .I R/ are defined for p 2 .1; 1/; if p D 2, then HN2 D HN and G2;0 D G0 defined in Sect. 3. p

z.zId C BN /1 is bounded in HN and in Gp;0 for p 2

6 5

˚  1 Proposition 7. The resolvents z.zIdCB † are T;N / ; z 2 ˚    uniformly bounded in Lp .I R3 / for all p 2 q00 ; q0 , where q0 WD min 6; 3 C " (" > 0 depends on @). Proof. By [19, Theorems 11.1 and 11.2], the projections defined in Sect. 2 and Sect. 3   P and PN extend to bounded projections on Lp .I R3 / for p 2 .3 C "/0 ; 3 C " ; (48)

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where " > 0 depends on @ (and .3 C "/0 D 3C" < 32 ); if  is of class C1 , then 2C" p p " D 1. This means in particular that HD coincides with thespace L ./ defined  0 0 ˚in (6) for all p12 .3 C "/ ; 3 C " . Therefore for all pp 2 q0 ;3 2 , the resolvents z.zId C BT;N  / ; z 2 † are uniformly bounded in L .I R /. The same result for all p 2 2; q0 is obtained by duality. t u tBT;N /t0 extend to bounded analytic semigroups Corollary 1. The semigroups  0  .e p 3 on L .I R / for p 2 q0 ; q0 and satisfy

p p t div .e tBT;N f /  Cp kf kp t curl .e tBT;N f /  C 0 kf kp p p p t rdiv .e tBT;N f /  Kp kf kp t curl curl .e tBT;N f /  K 0 kf kp p p p

(49) (50)

for all f 2 Lp .I R3 /. Proof. The (49) and (50) in the corollary above come from the fact that  estimates  p for p 2 q00 ; q0 , the negative generators BT;N of the semigroups .e tBT;N /t0 satisfy ˚ p D.BT;N / D u 2 Lp .I R3 /I div u 2 W 1;p .I R3 /; curl u 2 Lp .I R3 /;  curl curl u 2 Lp .I R3 /;   u D 0 and   curl u D 0 on @ p

BT;N u D  u;

(51)

p

u 2 D.BT;N /:

This can be proved the same way we proved Proposition 5, (case p D 2) using the fact that P and PN are bounded in Lp .I R/. t u Remark 8. Let w 2 L2 .I R3 / such that curl w 2 L2 .I R3 / and   w D 0 on @. 1 Then   curl w D 0 in H  2 .@/. If the operator BT is restriced on HD and the operator BN on HN , the following Hodge-Stokes operators AT and AN defined by n o D.AT / D u 2 HD \ WT I curl curl u 2 L2 .I R3 / and   curl u D 0 on @ AT u D curl curl u

for u 2 D.AT /

and n o D.AN / D u 2 HN \ WN I curl curl u 2 L2 .I R3 / ; AN u D curl curl u for u 2 D.AN /

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

233

are obtained. The construction of the Hodge-Stokes operators is strongly related to the particular Hodge boundary conditions. As seen in Sect. 2 and 3, this doesn’t hold in general. The reason behind this is that the Helmholtz projection P commutes with BT and PN commutes with BN . Remark 8 ensures that if u 2 D.AT / as defined above,   curl curl u D 0 on @, so that curl curl u 2 HD . The properties (46) and (47), together with a duality argument and the fact   that the projections P and PN are bounded on Lp .I R3 / for p 2 .3 C "/0 ; 3 C " , prove p that .e tAT /t0 extends to an analytic semigroup on HD (its generator is denoted p tAN by AT;p ) and .e /t0 extends  analytic semigroup on HN (its generator  6 to an is denoted by AN;p ) for all p 2 5 ; q0 . Moreover, the estimates (49) and (50) are   valid if BT;N is replaced by AT;N for all p 2 65 ; q0 .   p Lemma 5. If u 2 HD3 and curl u 2 L3 .I R3 /, then u 2 HD for all p 2 3; q0 . Proof. Thanks to the relation (42),   u D Pu D P R2 curl u C K1 u since PrR1 u D 0. The mapping properties of R2 and K1 show that R2 curl u C K1 u 2 L3 .; R3 / \ L6 .; R3 /, which proves the claim of the lemma. This has been done in, e.g., [35, Sections 3 and 4]. t u Remark 9. One can actually prove that the operator AT;p generates an analytic p p semigroup in HD for all p 2 .1; 3 C "/. The same holds for AN;p on HN . See [30] for more details. Remark 10. In [54], M.E. Taylor conjectured that the Dirichlet-Stokes operator  p generates an analytic semigroup in HD for p 2 .3 C "/0 ; 3 C " , which was proved in [51]. The question of optimality of this range is still open; the counterexample provided by P. Deuring in [14] is for p > 6. We see here that, for the Hodge-Stokes operator, one can allow all p 2 .1; 3 C "/.

4.2

The Nonlinear Hodge-Navier-Stokes Equations

  The nonlinear Hodge-Navier-Stokes system (NS’); (Hbc) 8 @t u  u C r  u  curl u D 0 in .0; T /  ; ˆ ˆ ˆ ˆ < div u D 0 in .0; T /  ; ˆ   u D 0;   curl u D 0 on .0; T /  @; ˆ ˆ ˆ : u.0/ D u0 in ;

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is considered for initial data u0 in the critical space HD3 in the abstract form:   u0 .t / C AT;p u.t /  P u.t /  curl u.t / D 0;

u0 2 HD3 :

(52)

The idea to solve (52) is to apply the same method as in Sect. 2 and 3. With the properties of the Hodge-Stokes semigroup listed in the previous subsection (and more particularly Lemma 5), the following existence result for (52) is almost immediate. For T 2 .0; 1, define the space GT by n 3.1Cı/ /I curl u 2 C..0; T /I L3 .; R3 // GT D u 2 Cb .Œ0; T /I HD3 / \ C..0; T /I HD o   p ı with sup ks 2.1Cı/ u.s/k3.1Cı/ C k s curl u.s/k3 < 1 0 0 such that (55) holds follows from the closed graph theorem since fu 2 H I curl u 2 H g is complete for the norm kuk2 C kcurl uk2 . 2. Assume now that g 2 L2tan .@I R3 /. Let w 2 H such that curl w 2 H and (54) holds. Since   g 2 L2 .@I R3 /, we can approach it in L2 .@I R3 / by a sequence .'n /n2N of vector fields 'n 2 H 1=2 .@; R3 /. In particular, 'n   ! .  g/   D g

in L2 .@I R3 / as n ! 1:

By assertion 1, for each n 2 N, there exists wn 2 H such that curl wn 2 H satisfying h'n  ; i@ D hcurl wn ; i  hwn ; curl i

for all  2 WT :

Thanks to the estimate (55), it is immediate that wn ! w n!1

and

curl wn ! curl w n!1

in H:

Let now  2 H 1 .I R3 /. For " > 0, let " D .1 C "BT /1 . Then " 2 WT , and thanks to Lemma 4, " !  "!0

and

curl " D .1 C "BN /1 curl  ! curl  "!0

in H:

This implies also that   " !    "!0

in H 1=2 .@I R3 /:

Therefore, for all " > 0 and n 2 N h  " ; 'n i@ D h'n  ; " i@ D hcurl wn ; " i  hwn ; curl " i : First take the limit as " goes to 0 and obtain (recall that 'n 2 H 1=2 .@I R3 /) H 1=2 h

 ; 'n iH 1=2 D hcurl wn ; i  hwn ; curl i :

Since  2 H 1 .; R3 /, the first term of the latter equation is also equal to h'n  ; i@ . Taking the limit as n goes to 1 yields hg; i@ D hcurl w; i  hw; curl i which proves the claim made in 2.

t u

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Remark 12. If  is of class C1 , one can prove that Lemma 6 is also valid in Lp 0 instead of L2 for all p 2 .1; 1/, identifying the dual of Lp with Lp (noting that q0 defined in Proposition 7 is equal to 1). Proof of (53). For the time being, denote by D˛ the set on the right-hand side of (53). Let u 2 D˛ : u D curl curl u C rdiv u 2 H , and for all v 2 WT \ H 1 .I R3 /, hu; vi D hcurl curl u; vi  hrdiv u; vi D hcurl u; curl vi C h  curl u; vi@ C hdiv u; div vi D hcurl u; curl vi C hdiv u; div vi C ˛hu; vi@ D b˛ .u; v/: The second equality comes from the integration by parts formula. In the third equality, the characterization of elements in D˛ has been used. Thanks to the density of WT \ H 1 .I R3 / in WT , this proves the inclusion D˛ D.B˛ / and that B˛ u D u for u 2 D˛ . Conversely, let u 2 D.B˛ /. Let  D B˛ u 2 H , g D ˛ u. Since uj@ 2 L2tan .@I R3 /, Lemma 6 shows the existence of w 2 H with curl w 2 H such that ˛ u D   w on @. Therefore, the boundary value g D ˛ u satisfies the conditions of [33, Theorem 1.2] with p D 2. Then there exists a unique uQ satisfying 8 < uQ 2 WT ; curl curl uQ 2 H; div uQ 2 H 1 ./; Qu D  2 H; :   curl uQ D g 2 H 1=2 .@I R3 /;

(58)

For all v 2 WT , integrating by parts, hcurl uQ ; curl vi C hdiv uQ ; div vi D hQu; vi  h  curl uQ ; vi@ D h; vi  hg; vi@ D hB˛ u; vi  h˛ u; vi@ D b˛ .u; v/  ˛hu; vi@ D hcurl u; curl vi C hdiv u; div vi : The second equality comes from the fact that uQ is the solution of (58). The third equality is a simple reformulation of the previous line using the notations introduced before. The fourth equality uses the fact that B˛ is the operator associated with the form b˛ . Finally, the last equality comes directly from the definition of b˛ . Therefore, we proved that v D u  uQ 2 WT and satisfies curl v D 0 and div v D 0. Since  is simply connected, this proves that v D 0, or equivalently u D uQ , and then that u 2 D˛ from which follows the inclusion D.B˛ / D˛ . Ultimately, it has been proved that D.B˛ / D D˛ . t u

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

239

As in the case of Proposition 6, Gaffney-type estimates hold. Proposition 8. There exist two constants C; c > 0 such that for any open sets E; F  R3 such that dist .E; F / > 0 and for all t > 0, f 2 H and u D .Id C t 2 B˛ /1 .1lF f /; it holds p dist .E;F / k1lE uk2 Ctk1lE div uk2 Ct k1lE curl uk2 Ct ˛ k1lE ukL2 .@IR3 /  C e c t k1lFf k2 : (59) Proof. The proof goes as in the case ˛ D 0 (Proposition 6 for BT ). Choose a smooth cutoff function  W R3 ! R satisfying  D 1 on E,  D 0 on F , and krk1  k . Then define  D e ı where ı > 0 is to be chosen later. Next, take the dist .E;F / scalar product of the equation: u  t 2 u D 1lF f;

u 2 D.B˛ /

with the function v D 2 u. Since  D 1 on F and kuk2  k1lF f k2 , it is easy to check then that k uk22 C t 2 k div uk22 C t 2 k curl uk22 C t 2 ˛k uk2L2 .@IR3 /    k1lF f k22 C 2˛krk1 t 2 k uk2 k div uk2 C k curl uk2 and therefore, using the estimate on krk1 and choosing ı D

dist .E;F / , 4kt

k uk22 C t 2 k div uk22 C t 2 k curl uk22 C t 2 ˛k uk2L2 .@IR3 /  2k1lF f k22 : Using now the fact that  D e ı on E, p p dist .E;F / k1lE uk2 Ctk1lE div uk2 Ct k1lE curl uk2 Ct ˛ k1lE ukL2 .@IR3 /  2e  4k t k1lFf k2 ; which gives (59) with C D

p 2 and c D

1 . 4k

t u

As before, with a slight modification of the proof, it can be shown that for all 3

2 .0; / there exist two constants C; c > 0 such that ˚ for any open sets E; F  R such that dist .E; F / > 0 and for all z 2 † D ! 2 C n f0gI j arg zj <   , f 2 H and u D .zId C B˛ /1 .1lF f /;

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it holds 1

1

jzjk1lE uk2 C jzj 2 k1lE div uk2 C jzj 2 k1lE curl uk2 1

C jzj 2

p

1

˛ k1lE ukL2 .@IR3 /  C e c dist.E;F /jzj 2 k1lF f k2 :

(60)

With the same arguments as for the Hodge-Laplacian, the analogue of 7   Proposition and Corollary 1 can be obtained, as well as (51) for B˛ : for all p 2 q00 ; q0 :  ˚ z.zId C B˛ /1 ; z 2 † is uniformly bounded in Lp .I R3 /I

(61)

.e tB˛ /t0 extends to a bounded analytic semigroup on Lp .I R3 /I

(62)

p t div .e tB˛ f /  Cp kf kp ; p

p t curl .e tB˛ f /  C 0 kf kp I p p

(63)

t rdiv .e tB˛ f /  Kp kf kp ; p

t curl curl .e tB˛ f /  K 0 kf kp : p p

(64)

Moreover, if  is of class C1 , the following description of B˛;p , the negative generator of .e tB˛ /t0 in Lp .I R3 / holds: ˚ D.B˛;p / D u 2 Lp .I R3 /I div u 2 W 1;p .I R3 /; curl u 2 Lp .I R3 /;  curl curl u 2 Lp .I R3 /;   u D 0 and   curl u D ˛ u on @ B˛;p u D u;

(65)

u 2 D.B˛;p /;

To prove that, the result in Remark 6 has been used, as well as the solvability of (58)  in Lp for p in the interval .3 C "/0 ; 3 C " D .1; 1/ in that case ([33, Theorem 1.2] is also valid in this range of p).

5.2

The Robin-Hodge-Stokes Operator

From now on, assume that  is of class C1 . Let p 2 .1; 1/. Let g 2 Lp .I R3 /, with div g D 0. By Remark 1 (also valid for p 2 .1; 1/ with the obvious 1=p changes), it holds   g 2 Bp;p .@/ and also   g satisfies the condition h  g; 1liB 1=p .@/ D 0. By [19, Corollary 9.3], the problem 1=p B .@/ p;p

p 0 ;p 0

q 2 W 1;p ./;

q D 0 in ;

@ q D   g on @

(66)

has a unique (modulo constants) solution satisfying moreover krqkp . k  gkB 1=p .@/ : p;p

(67)

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

241

Consider the operator p W D.B˛;p / ! W 1;p ./;

u 7! q

where q is the solution of (66) with g D curl curl u. Lemma 7. For p 2 .1; 1/, u 2 D.B˛;p /, the following estimate holds   krp ukp . ˛ kcurl ukp C kdiv ukp :

(68) 0

1=p

Proof. Let p 2 .1; 1/ and u 2 D.B˛;p /. Let ' 2 Bp0 ;p0 .@/. Let ˆ 2 W 1;p ./, so that ˆj@ D ' (recall that p1 D 1  p10 ). Thanks to the description of D.B˛;p / given by (65) and the formula (34) (also valid in Lp ), there holds 1=p

Bp;p .@/

h  curl curl u; 'iB 1=p

p 0 ;p 0

.@/

D hcurl curl u; rˆi D h  curl u; rˆi@ D ˛ hu; rˆi@ D ˛ hcurl w; rˆi ;

where w 2 Lp .I R3 / with curl w 2 Lp .I R3 / is determined by Lemma 6, 2 (for g D u; see Remark 6). Therefore by Remark 11, k  curl curl ukB 1=p .@/  C kcurl wkp  C kukLp .@IR3 / p;p    C kukp C kcurl ukp C kdiv ukp : Since  is bounded, kukp can be estimated in terms of kcurl ukp and kdiv ukp , which gives (68). t u Next result links the operator p and B˛;p with the Robin-Hodge-Stokes resolvent problem for z 2 † : 8
0 if p  2. Proof. Let z 2 † . By Proposition 9,   .zId C A˛;p / D Id  rp .zId C B˛;p /1 .zId C B˛;p /: Lemma 7 and (63) imply that for all f 2 Lp .I R3 /,  krp .zId C B˛;p /1 f kp . ˛ kcurl .zId C B˛;p /1 f kp  C kdiv .zId C B˛;p /1 f kp  C p˛jzj kf kp : p

This proves that, for jzj large enough (jzj  4C 2 ˛ 2 ), zId C A˛;p W D.A˛;p / ! HD is invertible with  1 .zId C A˛;p /1 D .zId C B˛;p /1 Id  rp .zId C B˛;p /1 and z.zId C A˛;p /1

p

L.HD /

 2 z.zId C B˛;p /1 L.Lp .IR3 // . 1:

Moreover, the same reasoning gives p jzj curl .zId C A˛;p /1

p

L.HD ILp .IR3 //

p  2 jzj curl .zId C B˛;p /1 L.Lp .IR3 // . 1

(72)

and curl curl .zId C A˛;p /1

p

L.HD ILp .IR3 //

2 curl curl .zId C B˛;p /1 L.Lp .IR3 // .1 (73) p

To prove that zId C A˛;p W D.A˛;p / ! HD is invertible if z 2 † with jzj  4C 2 ˛ 2 , proceed by induction. The assertion is proved for p  2 (the range is obtained A˛;2 is self-adjoint in HD ). Assume first that  1 < p  2 by duality since p p 2 2; 92 , so that D.A˛;2 / ,! HD by Lemma 8. Let z 2 † with jzj  4C 2 ˛ 2 and let ! D z C 8C 2 ˛ 2 . There holds ! 2 † and j!j  8C 2 ˛ 2  jzj  4C 2 ˛ 2 . p Therefore, for f 2 HD ,! HD , .zId C A˛;2 /1 f D .!Id C A˛;p /1 f C 8C 2 ˛ 2 .!Id C A˛;p /1 .zId C A˛;2 /1 f;

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which gives .zId C A˛;2 /1 f  C˛ kf kp ; p p

and this proves that zId C A˛;p W D.A˛;p / ! HD is invertible with the norm of its inverse controlled by a constant depending on ˛. For any p  2, the previous procedure can be iterated using again Lemma 8 valid for all p  2. Estimates of the form (72) and (73) are straightforward. Eventually, the result claimed in Theorem 7 is obtained for p  2. As mentioned earlier, the case 1 < p  2 is obtained by duality. t u

5.3

The Nonlinear Robin-Hodge-Navier-Stokes Equations

  The nonlinear Robin-Hodge-Navier-Stokes system (NS’); (Rbc) 8 ˆ ˆ @t u  u C r  u  curl u D 0 in ˆ ˆ < div u D 0 in ˆ   u D 0;   curl u D ˛ u on ˆ ˆ ˆ : u.0/ D u0 in

.0; T /  ; .0; T /  ; .0; T /  @; ;

for initial data u0 is considered in the critical space HD3 in the abstract form:   u0 .t / C A˛;p u.t /  P u.t /  curl u.t / D 0;

u0 2 HD3 :

(74)

Recall that C1 domains  are considered here. The idea to solve (74) is to apply the same method as in previous sections. With the properties of the Robin-Hodge-Stokes semigroup listed in particular in Theorem 7, the following existence result for (74) is almost immediate. For T 2 .0; 1, define the space HT by n HT D u 2 Cb .Œ0; T /I HD3 /I curl u 2 C..0; T /I L3 .; R3 // o p with sup k s curl u.s/k3 < 1 0 0. Hence, k are separated from zero.

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

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Now take in (12)  D Jk2 , where  is an arbitrary vector field from H ./. l Passing again to the limit as kl ! 1 yields the integral identity Z



 v  r v   dx D 0

8 2 H ./:

(15)



Hence, v 2 H ./ is a weak solution of the Euler equation 8  v  r v C rp D 0; ˆ ˆ < div v D 0; ˆ ˆ : v D 0;

x 2 ; (16)

x 2 ; x 2 @:

The function p in (16) belongs to the space W 1;s ./, where s 2 Œ1; 2/ for n D 2 and s 2 Œ1; 3=2 for n D 3. Since v D 0 on @, it can be proved, using the equations (16), that the pressure p is equal to some constants pO j on the connected components j of the boundary @. More precisely, it was proved in [30, Lemma 4] and independently in [2, Theorem 2.2] that the following equalities p.x/ji D pO i ;

pO i 2 R; j D 0; 1; : : : ; N:

(17)

hold. Multiply the Euler system (16) by A and integrate the obtained equality over . Integrating by parts and using (17), we obtain Z 



 vr vA dx D 

Z p An dS D 

N X iD0

@

b pi

Z An dS D  i

N X

pO i Fi :

(18)

i D0

If N D 0 or Fi D 0; i D 0; 1; : : : ; N (the condition (4) is satisfied), then (18) gives Z



 v  r v  A dx D 0:

(19)



The last relation contradicts (14). Therefore, the assumption is wrong and the norms of all possible solutions w./ to the operator equation (8) are uniformly bounded with respect to  2 Œ0; 1. Thus, by the Leray-Schauder theorem, equation (7) has at least one solution. An analogous conclusion is obtained when all constants b p j are equal: pO 0 D pO 1 D : : : D pO N :

(20)

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Indeed, in virtue of (3), N X iD0

pO i Fi D b p0

N X

Fi D 0;

iD0

and from (18) again follows (19). However, in the general case, one cannot claim that all constants pO i are equal. Amick [2] exhibited a solution to problem (16), for which equalities (20) are not valid. Let  D fx 2 R2 W 1 < jxj < 2g be annulus on the plane, 2 C 1 .Œ1; 2/; 0 .1/ D 0 .2/ D 0, and 00 2 L2 ..1; 2//. A solution of the Euler problem (16) is defined by  v.x/ D

x2 jxj

0

x1 .jxj/;  jxj

0

 .jxj/ 2 H ./;

Zjxj p.x/ D

j

0

.s/j2 ds: s

(21)

1

It is easy to see that p.x/jjxjD1 D 0, and p.x/jjxjD2 D

R2 j 1

3

0

.s/j2 ds > 0. s

An Existence Theorem in the General Planar Case

In this section the problem (2) is studied in the general case. For the two-dimensional domains, the result reads as follows. Theorem 1. Assume that   R2 is a bounded domain with C 2 -smooth boundary @. If f 2 W 1;2 ./ and a 2 W 3=2;2 .@/ satisfies condition (3), then problem (2) admits at least one weak solution u. Remark 1. It is well known (see [47]) that under the hypotheses of Theorem 1, 3;2 every weak solution u of problem (2) is more regular, i.e., u 2 W 2;2 ./ \ Wloc ./. Generally speaking, the solution is as regular as the data allow; in particular, u is C 1 -smooth when f, a, and @ are C 1 -smooth. Similar result holds for the 3D axially symmetric case (see Theorem 6). Moreover, for the axially symmetric case, also the existence theorem for an exterior domain could be proved (see Theorem 7). Below (Sect. 3.4) the main ideas of the proof of Theorem 1 are shown. In order to make it easer, consider the case when @ has only two connected components of the boundary and assume that f D 0. Some needed auxiliary results are formulated in the subsections below.

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

3.1

257

Properties of Sobolev Functions and an Analog of the Morse-Sard Theorem for Functions from W 2,1 .R2 /

Recall some classical differentiability properties of Sobolev functions. Working with such functions, we always assume that the “best representatives” are chosen. If w 2 L1loc ./, then the best representative w is defined by ( 

w .x/ D

R lim Br .x/ w.z/d z; if the finite limit existsI

r!0

0

otherwise;

R R 1 w.z/d z, Br .x/ D fy W jy  xj < rg is a ball where Br .x/ w.z/d z D meas.B r .x// Br .x/ of radius r centered at x. Lemma 1 (see Proposition 1 in [12]). Let 2 W 2;1 .R2 /. Then the function is continuous, and there exists a set A such that H1 .A / D 0 and the function is differentiable (in the classical sense) at each x 2 R2 n A . Furthermore, the R classical derivative at such points x coincides with r .x/ D lim Br .x/ r .z/d z, r!0 R and lim Br .x/ jr .z/  r .x/j2 d z D 0. r!0

Hausdorff measure, i.e., Here and henceforth, denote by H1 the one-dimensional 1 1 P S diamFi W diamFi  t; F  Fi . H1 .F / D lim H1t .F /, where H1t .F / D inf t!0C

iD1 1;q

i D1

It is well known that for functions w 2 Wloc ./,   R2 , H1 -almost all points x 2  are the Lebesgue points, i.e., the above limit exists H1 -almost everywhere in . The next theorem has been proved recently by J. Bourgain, M. Korobkov, and J. Kristensen [6] (see also [7, 35] for multidimensional case). The statement (i) of this theorem is the analog for Sobolev functions of the classical Morse-Sard Theorem. Theorem 2. Let   R2 be a bounded domain with Lipschitz boundary and W 2;1 ./. Then

2

(i) H1 .f .x/ W x 2  n A & r .x/ D 0g/ D 0; (ii) for every " > 0, there exists ı > 0 such that H1 . .U // < " for any set U   with H11 .U / < ı; in particular, H1 . .A // D 0; (iii) for every " > 0, there exists an open set V  R with H1 .V / < " and a function g 2 C 1 .R2 / such that for each x 2  if .x/ … V , then x … A and .x/ D g.x/, r .x/ D rg.x/ ¤ 0; (iv) for H1 –almost all y 2 ./  R, the preimage 1 .y/ is a finite disjoint family of C 1 -curves Sj , j D 1; 2; : : : ; N .y/. Each Sj is either a cycle in  .i.e., Sj   is homeomorphic to the unit circle S1 / or a simple arc with endpoints on @ .in this case Sj is transversal to @/.

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Some Facts from Topology

Below some topological definitions and results will be needed. By continuum we mean a compact connected set. The connectedness is understood in the sense of general topology. A subset of a topological space is called an arc if it is homeomorphic to the unit interval Œ0; 1. A locally connected continuum T is called a topological tree, if it does not contain a curve homeomorphic to a circle or, equivalently, if any two different points of T can be joined by a unique arc. This definition implies that T has topological dimension 1. A point C 2 T is an endpoint of T (resp., a branching point of T ), if the set T nfC g is connected (resp., if T nfC g has more than two connected components). Let us shortly present some results from the classical paper of A.S. Kronrod [45] concerning level sets of continuous functions. Let Q D Œ0; 1  Œ0; 1 be a square in R2 , and let f be a continuous function on Q. Denote by Et a level set of the function f , i.e., Et D fx 2 Q W f .x/ D tg. A connected component K of the level set Et containing a point x0 is a maximal connected subset of Et containing x0 . By Tf denote a family of all connected components of level sets of f . It was established in [45]  that Tf equipped by a natural topology is a one-dimensional topological tree. The convergence in Tf is defined by the following rule: Tf 3 Ci ! C iff sup dist.x; C / ! 0.) Endpoints of this tree are the components C 2 Tf which do x2Ci

not separate Q, i.e., Q n C is a connected set. Branching points of the tree are the components C 2 Tf such that Q n C has more than two connected components (see [45, Theorem 5]). By results of [45, Lemma 1], see also [51] and [62], the set of all branching points of Tf is at most countable. The main property of a tree is that any two points could be joined by a unique arc. Therefore, the same is true for Tf . Lemma 2 (see Lemma 13 in [45]). If f 2 C .Q/, then for any two different points A 2 Tf and B 2 Tf , there exists a unique arc J D J .A; B/  Tf joining A to B. Moreover, for every inner point C of this arc, the points A; B lie in different connected components of the set Tf n fC g. Remark 2. The assertion of Lemma 2 remains valid for level sets of continuous functions f W  ! R, where  is a multi-connected bounded domain of type (1), provided f  j D const on each inner boundary component j with j D 1; : : : ; N . Indeed, f can be extended to the whole 0 by putting f .x/ D j for x 2 j , j D 1; : : : ; N . The extended function f will be continuous on the set 0 which is homeomorphic to the unit square Q D Œ0; 12 .

3.3

Euler Equation

Most of the results of this section are obtained under the following assumptions. (E) Let   R2 be a bounded domain of type (1) with Lipschitz boundary. Assume that v 2 W 1;2 ./ and p 2 W 1;s ./, s 2 Œ1; 2/, satisfy the

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

259

Euler equations (

 v  r v C rp D 0; div v D 0;

for almost all x 2 , and let Z v  n ds D 0; i D 0; 1; : : : ; N;

(22)

(23)

i

where i are connected components of the boundary @. If instead of (23) the solution v satisfies the homogeneous boundary conditions vji D 0; i D 0; 1; : : : ; N;

(24)

it will be said that v satisfies the condition Eı . Under the conditions (E), it is easy to see that there exists a stream function 2 W 2;2 ./ such that r D .v2 ; v1 / (note that by the Sobolev embedding jvj2 theorem, is continuous in ). Denote by ˆ D p C the total head pressure 2 1;s corresponding to the solution .v; p/. Obviously, ˆ 2 W ./ for all s 2 Œ1; 2/. By direct calculations one easily gets the identity    @v2 @v1  rˆ   v2 ; v1 D !r in ; (25) @x1 @x2 where ! denotes the corresponding vorticity: ! D @2 v 1  @1 v 2 D  . Since the stream lines in our case coincide with the level sets of , from (25), in the case of smooth functions ; ˆ, the classical Bernoulli law follows immediately: The total head pressure ˆ is constant along any stream line. But the Sobolev case is more delicate: now the stream function 2 W 2;1 ./ 1 is not C -smooth, and the total head pressure ˆ belongs to the spaces W 1;q ./ with q < 2, but functions of this space need not to be continuous and they are well defined everywhere except for some “bad” set of H1 -measure zero (see, e.g., Theorem 1 of §4.8 and Theorem 2 of §4.9.2 in [13]). So the formulation of the Bernoulli law for solutions in Sobolev spaces has to be modulo negligible “bad” set Av of one dimensional Hausdorff measure zero. Such version of Bernoulli’s law was obtained in [34, Theorem 1] (see also [36, Theorem 3.2] for a more detailed proof). Theorem 3 (The Bernoulli law). Assume the conditions (E). Then there exists a set Av with H1 .Av / D 0 such that any point x 2  n Av is a Lebesgue point for v; ˆ, and for every compact connected set K  , the following property holds: if ˇ ˇ D const; (26) K

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then ˆ.x1 / D ˆ.x2 /

for all x1 ; x2 2 K n Av :

(27)

(Here, in order to define a Lebesgue point at x 2 @, the usual Sobolev extension of v; ˆ to the whole R2 is considered.) Remark 3. In particular, if v D 0 on @ .in the sense of trace/, then by the MorseSard Theorem 2, there exist constants 0 ; : : : ; N 2 R such that .x/  j on each component j , j D 0; : : : ; N (Indeed, if v D 0 on @, then r D 0 on @, and by Theorem 2 (i)–(ii), the image .@/ has zero H1 -measure. This implies, by continuity of , that  const on each connected subset of @.). Therefore, by the above Bernoulli law, the pressure p.x/ ˇ is constant on @. Note that p.x/ could take different constant values b p j D p.x/ˇj on different connected components j of the boundary @. This fact was already mentioned in Sect. 2 (see example (21)). Using the assertion of Remark 3, one could prove the following regularity result for the pressure. Theorem 4. Let the conditions .Eı / be satisfied. Then p 2 C ./ \ W 2;1 ./:

(28)

The proof of this theorem is based on the div–curl lemma with two cancelations (e.g., [10, Theorem II.1]) and classical results concerning the Poisson equation (see, e.g., [48, Chapter II]). Under .Eı /-conditions by Remarks 2 and 3, one can apply Kronrod’s results to the stream function . Define the total head pressure on the Kronrod tree T (see Sect. 3.2) as follows. Let K 2 T with diam K > 0. Take any x 2 K n Av and put ˆ.K/ D ˆ.x/. This definition is valid by Bernoulli’s law (see Theorem 3). Lemma 3. Assume that the conditions .Eı / are satisfied. Let A; B 2 T , diam A > 0; diam B > 0. Consider the corresponding arc ŒA; B  T joining A to B (see Lemma 2). Then the restriction ˆjŒA;B is a continuous function. Remark 4. The continuity of ˆjŒA;B was proved in [41, Lemma 3.5]. The proof relies on the fact that each Sobolev function is continuous (in classical sense) on almost all straight line. Note that the total head pressure ˆ.x/ itself is not necessary continuous function in the whole  since about the velocity field v, it is only known that v 2 W 1;2 ./. For x 2  denote by Kx the connected component of the level set fz 2  W .z/ D .x/g containing the point x. Under .Eı /-conditions by Remark 3, Kx \

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

261

@ D ; for every y 2 ./ n f 0 ; : : : ; N g and for every x 2 1 .y/. Thus, Theorem 2 (ii), (iv) implies that for almost all y 2 ./ and for every x 2 1 .y/, the equality Kx \ Av D ; holds, and the component Kx   is a C 1 – curve homeomorphic to the circle. Such Kx is called an admissible cycle. The next lemma was obtained in [36, Lemma 3.3]. Lemma 4. Let the conditions .Eı / be satisfied. Assume that there exists a sequence 1;q 1;q of functions fˆ g such that ˆ 2 Wloc ./ and ˆ * ˆ in the space Wloc ./ for all q 2 Œ1; 2/. Then there exists a subsequence ˆkl such that ˆkl jS converges to ˆjS uniformly ˆkl jS  ˆjS on almost all admissible cycles S .here, “almost all cycles” means cycles in preimages 1 .y/ for almost all values y 2 .//. In connection with Lemma 4, note that in [2] Amick proved the uniform convergence ˆk  ˆ on almost all circles. However, his method can be easily modified to prove the uniform convergence on almost all level lines of every C 1 smooth function with nonzero gradient. Such modification was done in the proof of Lemma 3.3 of [36]. Below assume (without loss of generality) that the subsequence ˆkl of Lemma 4 coincides with ˆk . Admissible cycles S satisfying the statement of Lemma 4 will be called regular cycles. Let  be a bounded domain with Lipschitz boundary. The function f 2 W 1;s ./ is said to satisfy a one-side maximum principle locally in  if ess sup f .x/  ess sup f .x/ x20

(29)

x2@0

holds for any strictly interior subdomain 0 ( 0  / with the boundary @0 not containing singleton connected components. (In (29) negligible sets are the sets of two-dimensional Lebesgue measure zero in the left esssup and the sets of onedimensional Hausdorff measure zero in the right esssup.) If (29) holds for any 0   (not necessary strictly interior) with the boundary @0 not containing singleton connected components, then f 2 W 1;s ./ satisfies a one-side maximum principle in  (in particular, we can take 0 D  in (29)). Using Lemma 4, it could be proved that the one-side maximum principle is inherited by the limiting solutions. Theorem 5. Let the conditions .E/ be satisfied. Assume that there exists a sequence 1;q 1;q of functions fˆ g such that ˆ 2 Wloc ./ and ˆ * ˆ in the space Wloc ./ for all q 2 Œ1; 2/. If all ˆ satisfy the one-side maximum principle locally in , then ˆ satisfies the one-side maximum principle in . Theorem 5 was obtained in [34, Theorem 2] (see also [36, Theorem 3.4] for the detailed proof).

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Note that some version of a local weak one-side maximum principle was proved by Ch. Amick [2] (see Theorem 3.2 and Remark thereafter in [2]).

3.4

Arriving at a Contradiction

To prove the solvability of problem (2), we follow the arguments described in Sect. 2. First repeating Leray’s argument of getting an a priori estimate by a contradiction, we arrive to the following assertion: Lemma 5. Assume that   R2 is a bounded domain with C 2 -smooth boundary @, f 2 W 1;2 ./, and the boundary value a 2 W 3=2;2 .@/ satisfies the necessary condition (3). Then, if problem (2) admits no weak solutions, then there exists a sequence of functions uk 2 W 1;2 ./, pk 2 W 1;q ./ and numbers k ! 0C, k ! 0 > 0 with the following properties: (E-NS) the norms kuk kW 1;2 ./ , kpk kW 1;q ./ are uniformly bounded for every q 2 Œ1; 2/, and the pairs .uk ; pk / satisfy the system of equations 8   ˆ ˆ k uk C uk  r uk C rpk D fk ; x 2 ; < div uk D 0; x 2 ; ˆ ˆ : uk D ak ; x 2 @;

with fk D

(30)

k k2 k k f, ak D a, and 2 

kruk kL2 ./ ! 1;

uk * v in W 1;2 ./;

pk * p in W 1;q ./

8 q 2 Œ1; 2/;

where the pair of functions v 2 W 1;2 ./, p 2 W 1;q ./ is a solution to the Euler system (16). (In this lemma uk D wk C Jk1 A, k D .k Jk /1 ; fk D k k2  2 f, where the objects Jk ; wk were defined in Sect. 2.) Assume, in what follows, that the conditions (E-NS) are satisfied. As it is shown in Sect. 2, if all the fluxes Fi are zero (see (4)), then the conditions (E-NS) lead to a contradiction, thereby proving that (2) is solvable. In this section the goal is to demonstrate that these conditions also lead to a contradiction in the general case when the boundary data satisfy only the necessary condition (3). This will justify the existence of Theorem 1. Assume for simplicity that @ consists of two connected components 0 and 1 . Moreover, suppose that f D 0. The pressure p is equal to constants on 0 and 1 : pj0 D pO 0 ;

pj1 D pO 1

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

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(see (17)). If pO 0 D pO 1 , then, as it is shown in Sect. 2, a contradiction arises with the equality (14), and the required a priori estimate follows. Assume that b p 0 ¤ pO 1 . Normalizing the pressure (and changing the numeration of the components i , if necessary), one can assume without loss of generality that pO 0 D 0;

pO 1 < 0:

(31)

Introduce the main idea of the proof in a heuristic way. It is well known that total head pressures ˆk D pk C 12 u2k (under above assumptions f D 0) satisfy the linear elliptic equation ˆk D !k2 C

1 div .ˆk uk /; k

(32)

where !k D @2 u1k  @1 u2k is the corresponding vorticity. By Hopf’s maximum principle, in a subdomain 0 b  with C 2 – smooth boundary @0 , the maximum of ˆk is attained at the boundary @0 , and if x 2 @0 is a maximum point, then the normal derivative of ˆk at x is strictly positive. It is not sufficient to apply this property directly. Instead we will use some “integral analogs” that lead to a contradiction by using the Coarea formula. Namely, we construct a set Ei   consisting of level lines of ˆk such that ˆk jEi ! 0 as i ! 1 and Ei separates the boundary component 0 (where ˆ D 0) from the boundary component 1 (where ˆ < 0). On the one hand, the length of each of these level lines is bounded from below by a positive constant (since they separate the boundary components), and R by the Coarea formula, this implies the estimate from below for Ei jrˆk j. On the other Rhand, elliptic equation (32) for ˆk and boundary conditions allow us to estimate Ei jrˆk j2 from above, and this asymptotically contradicts the previous one. Describe this heuristic idea in more details. From (32) and the mentioned Hopf theorem, one concludes that all ˆk satisfy the strong maximum principle globally in . Then by conditions (E-NS) and Theorem 5, the limiting total head pressure ˆ satisfies the weak maximum principle globally in , i.e., max b p j D ess sup ˆ.x/ D 0:

j D1;2

(33)

x2

Using the results of Kronrod (see Sect. 3.2), one can construct a decreasing sequence of domains with the following properties. Let T be a family of all connected components of level sets of . Take B0 ; B1 2 T , B0  0 , and B1  1 , and set ˛D

min

C 2ŒB1 ;B0 

ˆ.C / < 0:

(this minimum exists by Lemma 3). Let ti 2 .0; ˛/; ti C1 D 12 ti and ti is such that ˆ.C / D ti ) C 2 .B1 ; B0 / is a regular cycle:

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t

k

0

k

Ai0

n

t

k

k

t

hk

1

Wik (t )

Ai0 1 Sik (t)

Fig. 1 The case of annulus-type domain (here, Ai is denoted as A0i )

(See the definition of the regular cycles in the commentary to Lemma 4.) Note that the existence of such a sequence ti follows from the fact that   H1 fˆ.C / W C 2 ŒB1 ; B0  and C is not a regular cycleg D 0I see [41, Corollary 3.2]. The proof of this equality is based on the Coarea formula (see [41]). Denote by Ai an element from the set fC 2 ŒB1 ; B0  W ˆ.C / D ti g which is closest to 0 . Let Vi be a connected component of the set nAi such that 0  @Vi , i.e., @Vi D Ai [ 0 . Obviously, Vi  ViC1 (since ti C1 D 12 ti ). Note that Ai are regular cycles and, therefore, ˆk jAi  ˆjAi D ti . Take t 2 Œ 58 ti ; 78 ti . Let Wik .t / be the connected component of the set fx 2 Vi n V iC1 W ˆk .x/ > tg such that @Wik .t /  Ai C1 (see Fig. 1). Put Sik .t / D .@Wik .t // \ Vi n V iC1 . Then ˆk jSi k .t/ D t, @Wik .t / D Sik .t / [ Ai C1 . Since 2 ˆk 2 W2;loc ./, by the Morse-Sard theorem for almost all t 2 Œ 58 ti ; 78 ti , the level set Sik .t / consists of a finite number of C 1 -cycles; moreover, ˆk is differentiable at every point x 2 Sik .t / and rˆk .x/ ¤ 0. Such values t are called .k; i /-regular. By construction Z

Z rˆk  ndS D 

Si k .t/

Si k .t/

jrˆk jdS < 0;

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

265

where n outward with respect to Wik .t / normal to @Wik .t / (see Fig. 1). Indeed, Sik .t / is a subset of the level set fx 2  W ˆ.x/ D t g, and by construction the nonzero gradient rˆ.x/ is directed inside the domain Wik .t / for x 2 Sik .t /, i.e., rˆk .x/ D n. jrˆk .x/j The key step in the proof is the following estimate Lemma 6. For any i 2 N, there exists k.i/ 2 N such that the inequality Z jrˆk .x/jdS  C t Si k .t/

holds for every k k.i/ and for almost all t 2 Π58 ti ; 78 ti . The constant C is independent of t; k, and i . The proof of Lemma 6 is based on the integration of the equality (32) over the suitable subdomain k .t / with @k .t / D Sik .t /[hk , where the cycle hk D fx 2  W dist.x; 0 / D hk g lies near the boundary component 0 and the parameter hk is taken in such a way that Z

ˆ2k

2

ds <  ;

hk

ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ ˇ rˆk  n ds ˇ D ˇ !k u?  n ds ˇ < "; k ˇ ˇ ˇ ˇ hk

(34)

hk

Z

juk j2 ds < C" k2 ;

(35)

hk

where  and " are some fixed sufficiently small numbers and C" does not depend on k and  . For sufficiently large k k.i/ such hk can be found, using the weak convergences ˆk * ˆ, uk ! v from the assumptions (E-NS) and the boundary conditions kuk kL2 .@/ k , v  0, ˆ  0 on @ (see (303 ) and (163 )) When Lemma 6 is proved, the required contradiction can be obtained using the Coarea formula. For i 2 N and k k.i/, put [

Ei D

Sik .t /:

t2Π58 ti ; 78 ti 

By the Coarea formula (see, e.g., [50]), for any integrable function g W Ei ! R, the equality 7

Z8 ti Z

Z gjrˆk j dx D Ei

5 8 ti

Si k .t/

g.x/ d H1 .x/ dt

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holds. In particular, taking g D jrˆk j and using Lemma 6 yield Z

7

7

Z8 ti Z

jrˆk j2 dx D

Ei

5 8 ti

jrˆk j d H1 .x/dt 

Z8 ti

C t dt D C ti2 :

5 8 ti

Si k .t/

Now, taking g D 1 in the Coarea formula and using the Hölder Inequality, we get 7

Z8 ti

  H1 Sik .t / dt D

5 8 ti

Z jrˆk j dx Ei

0

Z

@

1 12 jrˆk j2 dx A



 12

meas.Ei /



p

 1 C ti meas.Ei / 2 :

Ei

By construction, for almost all t 2 Π58 ti ; 78 ti , the set Sik .t / is a smooth cycle and each set Sik .t / separates 0 from 1 . In Sik .t / separates Ai from AiC1 . Thus,  particular, H1 .Sik .t // C D min diam 0 ; diam 1 /. Hence, 7

Z8 ti

  1 H1 Sik .t / dt C ti : 4

5 8 ti

So, it holds p  1 1 C ti  C ti meas.Ei / 2 ; 4 or p  1 1 C  C meas.Ei / 2 : 4

(36)

  By construction meas.Ei /  meas Vi n ViC1  . But since  Vi  Vi C1 is a decreasing sequence of bounded sets, we have meas Vi n ViC1 ! 0 as i ! 1; therefore, inequality (36) gives the contradiction. Thus, our assumption is wrong; the norms of all possible solutions w./ to the operator equation (8) are uniformly bounded with respect to  2 Œ0; 1, and by the Leray-Schauder theorem, the equation (7) and equivalently the problem (2) has at least one solution. Hence, in the case when f D 0, Theorem 1 is proved. If f ¤ 0, then the maximum principle is not valid, and one has to consider two cases

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(a) Maximum of ˆ is attained on the boundary @: p 1 g D ess sup ˆ.x/: maxfpO 0 ; b x2

(b) Maximum of ˆ is not attained on @: p 1 g < ess sup ˆ.x/ maxfpO 0 ; b x2

(the case ess sup ˆ.x/ D C1 is also possible). x2

In the case (a) the proof is literally the same as above, while in the case (b) it can be proved that there exists a regular cycle F 2 T such that diam F > 0, F \ @ D ;, and ˆ.F / > ˇ, where ˇ D maxfpO 0 ; pO 1 g. For such F we consider the behavior of ˆ on the Kronrod arcs ŒBj ; F , j D 0; 1. The remaining part of the proof is similar to that of the proof for the case (a) with the following difference: F plays now the role which was played before by B0 , and the calculations become easier since F lies strictly inside . The main idea of the proof for a general multiply connected domain is the same as in the case of annulus-like domains (when @ D 1 [ 0 ). The proof has an analytical nature and unessential differences concern only well-known geometrical properties of level sets of continuous functions of two variables.

4

3D Axially Symmetric Case

First, specify some notations. Let Ox1 ; Ox2 ; Ox3 be the coordinate axis in R3 and D arctg.x2 =x1 /, r D .x12 C x22 /1=2 , z D x3 be the cylindrical coordinates. Denote by v ; vr ; vz the projections of the vector v on the axes ; r; z. A function f is said to be axially symmetric if it does not depend on . A vectorvalued function h D .hr ; h ; hz / is called axially symmetric if hr , h , and hz do not depend on . A vector-valued function h0 D .hr ; h ; hz / is called axially symmetric with no swirl if h D 0, while hr and hz do not depend on .

4.1

Bounded 3D Axially Symmetric Domains

The main result of this section is as follows. Theorem 6 ([37,41]). Assume that   R3 is a bounded axially symmetric domain of type (1) with C 2 -smooth boundary @ (Fig. 2). If f 2 W 1;2 ./, a 2 W 3=2;2 .@/ are axially symmetric and a satisfies condition (3), then (2) admits at least one weak axially symmetric solution. Moreover, if f and a are axially symmetric with no swirl, then (2) admits at least one weak axially symmetric solution with no swirl.

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x3

Fig. 2 Axially symmetric domain (N D 3)

0 1

3

2

The proof of Theorem 6 follows the same ideas as for the two-dimensional case, so it is not discussed here. (Some specific details for axially symmetric case could be found in the next section where the more complicated case of exterior domains is discussed.)

4.2

Exterior 3D Axially Symmetric Domains

This section is based on results of the paper [39]. Consider the Navier-Stokes problem   8 u C u  r u C rp D f ˆ ˆ ˆ ˆ ˆ < div u D 0 ˆ ˆ ˆ ˆ ˆ :

uDa lim

jxj!C1

in ; in ; on @;

(37)

u.x/ D u0

in the exterior domain of R3  D R3 n

N  S j D1

 Nj ; 

(38)

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

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Nj \ where i are bounded domains with connected C 2 -smooth boundaries i ,  N i D ; for i ¤ j , and u0 is a constant assigned vector.  Let Z (39) Fi D a  n dS; i D 1; : : : N; i

where n is the unit outward normal to @. Under suitable regularity hypotheses on  and a and assuming that Fi D 0;

i D 1; : : : N;

(40)

in the celebrated paper [49] of 1933, J. Leray showed that (37) has a solution u with finite Dirichlet integral: Z jruj2 dx < C1; (41) 

and u satisfies (374 ) in a suitable sense for general u0 and uniformly for u0 D 0. In the 1950s, the problem was reconsidered by R. Finn [16] and O.A. Ladyzhenskaya [46, 47]. They showed that the solution satisfies the condition at infinity uniformly. Moreover, condition (40) and the regularity of a have been relaxed by requiring N P jFi j to be sufficiently small [16] and a 2 W 1=2;2 .@/ [47]. i D1

In 1973 K.I. Babenko [4] proved that if .u; p/ is a solution to (37), (41) with u0 ¤ 0, then .u  u0 ; p/ behaves at infinity as the solutions to the linear Oseen system. In particular, u.x/  u0 D O.r 1 /;

p.x/ D O.r 2 /:

(42)

(See also [21]. Here the symbol f .x/ D O.g.r// means that there is a positive constant c such that jf .x/j  cg.r/ for large r.) However, nothing is known, in general, on the rate of convergence at infinity for u0 D 0. (For small kakL1 .@/ existence of a solution .u; p/ to (37) such that u D O.r 1 / is a simple consequence of Banach contractions theorem [69]. Moreover, one can show that p D O.r 2 /, and the derivatives of order k of u and p behave at infinity as r k1 , r k2 , respectively [75]; see also [21, 58].) One of the most important problems in the theory of the steady-state NavierStokes equations concerns the possibility to prove the existence of a solution to (37) without any assumptions on the fluxes Fi (see, e.g., [21]). To the best of our knowledge, the most general assumptions assuring the existence is expressed by N X iD1

max i

jFi j < 8  jx  xi j

(43)

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(see [67]), where Fi is defined by (39) and xi is a fixed point of i (see also [5] for analogous conditions in bounded domains). In the recent paper [39], the above question was solved for the axially symmetric case. Note that for axially symmetric solutions u of (37), the vector u0 has to be parallel to the symmetry axis. The main result is as follows. Theorem 7 ([39]). Assume that   R3 is an exterior axially symmetric domain (38) with C 2 -smooth boundary @, u0 2 R3 is a constant vector parallel to the symmetry axis, and f 2 W 1;2 ./ \ L6=5 ./, a 2 W 3=2;2 .@/ are axially symmetric. Then (37) admits at least one weak axially symmetric solution u satisfying (41). Moreover, if a and f are axially symmetric with no swirl, then (37) admits at least one weak axially symmetric solution with no swirl satisfying (41). Remark 5. It is well known (see, e.g., [47]) that under hypothesis of Theorem 7, 2;2 every weak solution u of the problem (37) is more regular, i.e., u 2 Wloc ./ \ 3;2 Wloc ./. Emphasize that Theorem 7 furnishes the first existence result without any assumption on the fluxes for the stationary Navier-Stokes problem in exterior threedimensional domains.

4.2.1 Extension of the Boundary Values The next lemma concerns the existence of a solenoidal extensions of boundary values. Lemma 7 (see, e.g., [39]). Let   R3 be an exterior axially symmetric domain (38). If a 2 W 3=2;2 .@/, then there exists a solenoidal extension A 2 W 2;2 ./ of a such that A.x/ D  .x/ for sufficiently large jxj, where  .x/ D 

N x X Fi 4 jxj3 iD1

(44)

and Fi are defined by (39). Moreover, the following estimate kAkW 2;2 ./  ckakW 3=2;2 .@/

(45)

holds. Furthermore, if a is axially symmetric (axially symmetric with no swirl), then A is axially symmetric (axially symmetric with no swirl) too.

4.2.2 Leray’s Argument “Invading Domains” Consider the Navier-Stokes problem (37) with f 2 W 1;2 ./ \ L6=5 ./ in the C 2 smooth axially symmetric exterior domain   R3 defined by (38). Without loss of generality, assume that f D curl b 2 W 1;2 ./ \ L6=5 ./. (By the HelmholtzWeyl decomposition, f can be represented as the sum f D curl b C r' with curl b 2

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

271

W 1;2 ./ \ L6=5 ./, and the gradient part is included then into the pressure term; see, e.g., [21, 47].) Below the proof of Theorem 7 will be discussed in the case u0 D 0: The proof for u0 ¤ 0 follows the same steps with minor standard modification. A function u is called a weak solution of problem (37) if w  A 2 H ./ and the integral identity  R R R  rw  r dx D  rA  r dx  A  r A   dx 





R

  R A  r w   dx  w  r w   dx







(46)



R

 R w  r A   dx C f   dx





holds for any  2 J01 ./. Here A is the extension of the boundary data constructed in Lemma 7. We shall look for the axially symmetric (axially symmetric with no swirl) weak solution of problem (37) and find this solution as a limit of weak solution to the Navier-Stokes problem in a sequence of bounded domain k that in the limit exhausts the unbounded domains  (this is the main idea of the “invading domain” method). Namely, consider the sequence of the boundary value problems 8 uk C .b uk  r/b uk C r pO k D f in k ; ˆ < b (47) div b uk D 0 in k ; ˆ : b uk D A on @k ; jxj < kg, 12 Bk0 

where k D Bk \  for k k0 , Bk D fx W

N S

N i . By 

i D1

Theorem 6, each problem (47) has an axially symmetric solution b uk D A C b wk with b wk 2 H .k /. To prove the assertion of Theorem 7, it is sufficient to establish the uniform estimate Z jrb wk j2  c: (48) 

Estimate (48) will be proved following a classical reductio ad absurdum argument of J. Leray and O.A. Ladyzhenskaia (see [47, 49]). Indeed, if (48) is not true, then there exists a sequence fb wk gk2N such that Z wk j2 : lim Jk2 D C1; Jk2 D jrb k!C1

k

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The sequence wk D b wk =Jk is bounded in H ./. Extracting a subsequence (if necessary), one can assume that wk converges weakly in H ./ and strongly in q Lloc ./ .q < 6/ to a vector field v 2 H ./ with Z

jrvj2  1:

(49)



It is easy to check that v 2 H ./ is a weak solution to the Euler equations, and for some p 2 D 1;3=2 ./ the pair .v; p/ satisfies the Euler equations almost everywhere: 8  ˆ ˆ v  r v C rp D 0 < div v D 0 ˆ ˆ : vD0

in ; (50)

in ; on @:

Adding some constants to p (if necessary) by virtue of the Sobolev inequality (see, e.g., [21] II.6), it may be assumed without loss of generality that kpkL3 ./  const:

(51) 2

Put k D .Jk /1 . Multiplying equations (47) by J12 D k2 , one sees that the pair k   wk C J1k A; pk D J12 pO k satisfies the following system: uk D J1k b k

8   ˆ k uk C uk  r uk C rpk D fk ˆ < div uk D 0 ˆ ˆ : uk D ak

in k ; in k ;

(52)

on @k ;

2

3;2 2;2 where fk D k2 f, ak D k A, uk 2 Wloc ./, pk 2 Wloc ./ (the interior regularity of the solution depends on the regularity of f 2 W 1;2 ./, but not on the regularity of the boundary value a; see [47]). Using the classical local estimates for ADN-elliptic problems (see [1, 73]), one could prove the following uniform estimate:

kuk kL6 .k / C krpk kL3=2 .k /  C;

(53)

where C does not depend on k. By construction, there holds the weak convergences 1;3=2 1;2 uk * v in Wloc ./; pk * p in Wloc ./ (the weak convergence in 1;2 1;2 Wloc ./ means the weak convergence in W .0 / for every bounded subdomain 0  ). As in the two-dimensional case, in conclusion, one can prove the following lemma.

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

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Lemma 8. Assume that   R3 is an exterior axially symmetric domain of type (38) with C 2 -smooth boundary @, and a 2 W 3=2;2 .@/, f D curl b 2 W 1;2 ./ \ L6=5 ./ are axially symmetric. If the assertion of Theorem 7 is false, then there exist v; p with the following properties. (E-AX) The axially symmetric functions v 2 H ./, p 2 D 1;3=2 ./ satisfy the Euler system (50) and kpkL3 ./ < 1. (E-NS-AX) Condition (E-AX) is satisfied, and there exist sequences of axially symmetric functions uk 2 W 1;2 .k /, pk 2 W 1;3=2 .k /, k D  \ BRk , Rk ! 1 as k ! 1, and numbers k ! 0C, such that estimate (53) holds, 2 the pair .uk ; pk / satisfies (52) with fk D k2 f, ak D k A (here A is solenoidal extension of a from Lemma 7), and kruk kL2 .k / ! 1;

1;3=2

1;2 uk * v in Wloc ./; pk * p in Wloc Z  D .v  r/v  A dx

./;

(54) (55)

 3;2 2;2 Moreover, uk 2 Wloc ./ and pk 2 Wloc ./.

4.2.3 Euler Equation in 3D Axially Symmetric Case (Exterior Domains) Suppose that the assumptions (E-AX) (from Lemma 8) are satisfied, and, for definiteness, assume that (SO)  is the domain (38) symmetric with respect to the axis Ox3 and j \ Ox3 ¤ ;; j \ Ox3 D ;;

j D 1; : : : ; M 0 ; j D M 0 C 1; : : : ; N:

(The cases M 0 D N or M 0 D 0, i.e., when all components (resp., no components) of the boundary intersect the axis of symmetry, are also allowed.) Denote PC D f.0; x2 ; x3 / W x2 > 0; x3 2 Rg, D D  \ PC , Dj D j \ PC . Of course, on PC the coordinates x2 ; x3 coincide with coordinates r; z. Then v and p satisfy the following system in the plane domain D: 8 @vz @vz @p ˆ C vr C vz D 0; ˆ ˆ ˆ @z @r @z ˆ ˆ 2 ˆ @p .v / @vr @vr ˆ ˆ <  C vr C vz D 0; @r r @r @z @v @v v vr ˆ ˆ ˆ C vr C vz D 0; ˆ ˆ r @r @z ˆ ˆ ˆ @.rvz / @.rvr / ˆ : C D0 @r @z (these equations are fulfilled for almost all x 2 D).

(56)

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1;2 There hold the following integral estimates: v 2 Wloc .D/, Z Z rjrv.r; z/j2 drd z C rjv.r; z/j6 drd z < 1: D

(57)

D

Also, the inclusions rp 2 L3=2 ./, p 2 L3 ./ can be rewritten in the following two-dimensional form: Z Z 3=2 rjrp.r; z/j drd z C rjp.r; z/j3 drd z < 1: (58) D

D

The next statement was proved in [30, Lemma 4] and in [2, Theorem 2.2]. Theorem 8. Let the conditions (E-AX) be fulfilled. Then 8j 2 f1; : : : ; N g 9 pO j 2 R W

p.x/  pO j

for H2  almost all x 2 j :

(59)

In particular, by axial symmetry, p.x/  pO j

for H1  almost all x 2 M j :

(60)

Here and below the following convenient agreement is used: for a set A  R3 M WD A \ PC , and for B  PC denote by BQ the set in R3 obtained by rotation put A of B around Oz -axis. The following result gives more precise information about the constants from the previous theorem. Corollary 1 ([39]). Assume that the conditions (E-AX) are satisfied. Then ˆjj  0 whenever j \ Oz ¤ ;, i.e., pO 1 D    D pO M 0 D 0; where pO j are the constants from Theorem 8. This phenomenon is connected with the fact that the symmetry axis can be approximated by stream lines (see Theorem 10 below), where the total head pressure is constant according to the Bernoulli law (see Theorem 9 below). To formulate the last result, some preparation is needed. Below without loss of generality, assume that the functions v; p are extended to the whole half-plane PC as follows: v.x/ WD 0;

x 2 PC n D;

p.x/ WD pO j ; x 2 PC \ DN j ; j D 1; : : : ; N:

(61) (62)

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275

Obviously, the extended functions inherit the properties of the previous ones. 1;3=2 1;2 .PC /, p 2 Wloc .PC /, and the Euler equations (56) are fulfilled Namely, v 2 Wloc almost everywhere in PC . The last equality in (56) (which is fulfilled, after the above extension agreement, in the whole half-plane PC ) implies the existence of a continuous stream function 2;2 2 Wloc .PC / such that @ D rvz ; @r Denote by ˆ D p C .v; p/. From (57) we get Z

@ D rvr : @z

(63)

jvj2 the total head pressure corresponding to the solution 2

rjˆ.r; z/j3 drd z C

PC

Z

rjrˆ.r; z/j3=2 drd z < 1:

(64)

PC

By direct calculations one easily gets the identity vr ˆr C vz ˆz D 0

(65)

for almost all x 2 PC . The identities (61)–(62) mean that ˆ.x/  pO j

8x 2 PC \ DN j ; j D 1; : : : ; N:

(66)

Theorem 9 (Bernoulli law for Sobolev solutions [39]). Let the conditions (E-AX) be valid. Then there exists a set Av  PC with H1 .Av / D 0, such that every point x 2 PC n Av is a Lebesgue point for v; ˆ, and for any compact connected set K  PC , the following property holds : if ˇ ˇ D const; K

(67)

then ˆ.x1 / D ˆ.x2 / for all x1 ; x2 2 K n Av :

(68)

Theorem 9 is a space version of the above Theorem 3 (for the plane case). For the axially symmetric bounded domains, the result was proved in [37, Theorem 3.3]. The proof for exterior axially symmetric domains is similar: one has to overcome two obstacles. First difficulty is the lack of the classical regularity, and here the results of [6] have a decisive role (according to these results, almost all level sets of plane W 2;1 -functions are C 1 -curves; see Sect. 3.1). The second obstacle is the set where r .x/ D 0 ¤ rˆ.x/, i.e., where vr .x/ D vz .x/ D 0, but v .x/ ¤ 0. Namely, without assuming the boundary conditions (503 ), in general,

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the equality (67) even for smooth functions does not imply (68). For example, if vr D vz D 0 in the whole domain, v D r, then  const on the whole domain, while ˆ D r 2 ¤ const. Without the boundary assumptions, one can prove only that ˆ.r; z/ D f .r/ along every level set K of the stream function for some absolutely continuous function f .r/ (see [39, Lemma 4.5]). But the last equality together with the boundary conditions (503 ), (66) easily implies Theorem 9. For " > 0 and R > 0 denote by S";R the set S";R D f.r; z/ 2 PC W r "; r 2 C z2 D R2 g. From the assumptions (64) one gets Lemma 9. For any " > 0, there exists a sequence j > 0, j ! C1, such that S";j \ Av D ; and sup jˆ.x/j ! 0 as j ! 1:

(69)

x2S";j

One of the main results of this section is the following. Theorem 10. Assume that the conditions (E-AX) are satisfied. Let Kj be a sequence of continuums with the following properties: Kj  DN n Oz , jKj D const, and lim inf r D 0, lim sup r > 0. Then ˆ.Kj / ! 0 as j ! 1. Here we j !1 .r;z/2Kj

j !1 .r;z/2Kj

denote by ˆ.Kj / the corresponding constant cj 2 R such that ˆ.x/ D cj for all x 2 Kj n Av (see Theorem 9).

4.2.4 Obtaining a Contradiction From now on assume that the assumptions (E-NS-AX) (see Lemma 8) are satisfied. The goal is to prove that they lead to a contradiction. This implies the validity of Theorem 7. For simplicity assume that f D 0, N D 2 and M 0 D 1, i.e., the boundary @ splits into the two components @ D 1 [ 2 , where 1 \ Oz ¤ ;;

2 \ Oz D ;:

(70)

The main idea of the proof is similar to that for the two-dimensional case. Since every ˆk D pk C 12 juk j2 satisfies the linear elliptic equation ˆk D !k2 C

1 div .ˆk uk /; k

(71)

where !k D curl uk and !k .x/ D j!k .x/j, a contradiction is obtained by using some “integral analog” of Hopf’s maximum principle and the Coarea formula. Consider the constants pO j from Theorem 8 (see also Theorem 1). From the equality (55), the Euler equations (501 ), and the regularity assumptions (64), the identity follows

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

 D

N X

pO j Fj D b p 2 F2

277

(72)

j DM 0 C1

(since pO 1 D 0 by Theorem 1). Therefore, pO 2 ¤ 0. Further consider separately three possible cases. (a) The maximum of ˆ is attained at infinity, i.e., it is zero: 0 D ess sup ˆ.x/:

(73)

x2

(b) The maximum of ˆ is attained on a boundary component which does not intersect the symmetry axis: 0 < pO 2 D

max

j DM 0 C1;:::;N

b p j D ess sup ˆ.x/;

(74)

x2

(c) The maximum of ˆ is not zero and it is not attained on @: max

j DM 0 C1;:::;N

b p j < ess sup ˆ.x/ > 0

(75)

x2

(the case ess sup ˆ.x/ D C1 is not excluded). x2

4.2.5

The Case ess sup ˆ.x/ D 0. x2

Let us consider the case (73). By Theorem 1, pO 1 D ess sup ˆ.x/ D 0:

(76)

x2

Then pO 2 < 0. The arguments below are similar to the plane situation (see Sect. 3.4). Take the positive constant ıp D pO 2 . The first goal is to separate the boundary components where ˆ < 0 from infinity and from the singularity axis Oz by level sets of ˆ compactly supported in D. More precisely, for any t 2 .0; ıp / we construct a continuum A.t/ b PC with the following properties: M WD A \ PC ) lies in (i) The set M j (recall that for a set A  R3 by definition A a bounded connected component of the open set PC n A.t/; (ii) jA.t/  const, ˆ.A.t // D t; (iii) (monotonicity) If 0 < t1 < t2 < ıp , then the set A.t1 / [ M 1 lies in the unbounded connected component of the set PC n A.t2 / (in other words, A.t2 / [ M 2 lies in the bounded connected component of the set PC n A.t1 /; see Fig. 3).

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Fig. 3 The surface Sk .t1 ; t2 ; t / ( here A.t/ is denoted as A2 .t/)

For this construction, the results of Sect. 3.2 are used for the restrictions of the stream function on suitable compact subdomains of PC (see details in [39]). We have also one additional property (cf. with Lemma 4 for the plane case):   (iv) There exists a set T  .0; ıp / of full measure (i.e., meas .0; ıp / n T D 0) such that for all t 2 T the set A.t/ is a regular cycle, i.e., it is a C 1 -curve homeomorphic to the circle and ˆk .x/  ˆ.x/  t

on A.t/:

(77)

issue is to construct Let t1 ; t2 2 T and t1 < t 0 < t 00 < t2 . The very important  for sufficiently large k k0 and for almost all t 2 t 0 ; t 00 a C 1 -circle Sk .t / which separates A.t1 / from A.t2 / and satisfies ˆk jSk .t/  t . Moreover, the gradient of ˆk is directed toward 1 . For this purpose, for t 2 Œt 0 ; t 00  denote by Wk .t1 ; t2 I t / the bounded connected component of the open set fx 2 PC n A.t1 / W ˆk .x/ > tg such that @Wk .t1 ; t2 I t /  A.t1 / (see Fig. 3). This definition is valid since for sufficiently large k by the convergence (77), the estimates ˆk jA.t1 / > t;

ˆk jA.t2 / < t

8t 2 Œt 0 ; t 00 

hold. Now put Sk .t1 ; t2 I t / D .@Wk .t1 ; t2 I t // n A.t1 /:

(78)

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279

Clearly, ˆk  t on Sk .t1 ; t2 I t /. Moreover, Sk .t1 ; t2 I t / separates A.t1 / from A.t2 / because of the monotonicity property (iii) and (78) (see Fig. 3). Q denotes the set in R3 obtained by rotation of A Recall, that for a set A  PC , A around Oz -axis. By construction, for every regular value t 2 Œt 0 ; t 00  b .t1 ; t2 /, the set Sk .t1 ; t2 I t / is a C 1 -circle; consequently, SQ k .t1 ; t2 I t / is a torus, and Z

Z rˆk  n dS D 

SQ k .t1 ;t2 It/

jrˆk j dS < 0;

(79)

SQ k .t1 ;t2 It/

where n is the unit outward normal vector to @WQ k .t1 ; t2 I t /. Now formulate the key estimate. Lemma 10. For any t1 ; t2 ; t 0 ; t 00 2 T with t1 < t 0 < t 00 < t2 , there exists k D k .t1 ; t2 ; t 0 ; t 00 / such that for every k k and for almost all t 2 Œt 0 ; t 00 , the inequality Z

jrˆk j dS < Ft;

(80)

SQ k .t1 ;t2 It/

holds with the constant F independent of t; t1 ; t2 ; t 0 ; t 00 and k. Proof. Fix t1 ; t2 ; t 0 ; t 00 2 T with t1 < t 0 < t 00 < t2 . Below always assume that k is sufficiently large; in particular, the set SQ k .t1 ; t2 I t / is well defined for all t 2 Œt 0 ; t 00 . The main idea of the proof of (80) is quite simple: to integrate the equation ˆk D !k2 C

1 div .ˆk uk / k

(81)

over the suitable domain k .t / with @k .t /  SQ k .t1 ; t2 I t / such that the corresponding boundary integrals ˇ ˇ ˇ ˇ 

Z 

ˇ ˇ rˆk  n dS ˇˇ

(82)

@k .t/ nSQ k .t1 ;t2 It/

ˇ 1 ˇˇ k ˇ

Z 

ˇ ˇ ˆk uk  n dS ˇˇ

(83)

@k .t/ nSQ k .t1 ;t2 It/

are negligible. Such domain k .t / can be constructed because of the weak convergences ˆk * ˆ, uk ! v from the assumption (E-NS-AX) and the boundary conditions kuk kL2 .@/ k , v  0, ˆ  0 on @ (see (523 ) and (503 )).

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The following technical fact from the one-dimensional real analysis is needed. Lemma 11. Let f W S ! R be a positive decreasing function defined on a measurable set S  .0; ı/ with measŒ.0; ı/ n S  D 0. Then 4

Œf .t2 / 3 .t2  t1 / D 1: t1 ;t2 2S .t2 C t1 /.f .t1 /  f .t2 // sup

(84)

The proof of this fact is elementary (see, e.g., [39, Appendix]). For t 2 T denote by U .t/ the bounded connected component (the torus) of the Q By construction, U .t2 / b U .t1 / for t1 < t2 . set R3 n A.t/. From estimate (80), the isoperimetric inequality and from the Coarea formula, one can easily deduce Lemma 12. For any t1 ; t2 2 T with t1 < t2 , the estimate

4

meas U .t2 / 3  C

 t2 C t1

meas U .t1 /  meas U .t2 / t2  t1

(85)

holds with the constant C independent of t1 ; t2 . The R proof of this lemma is based on the same idea (Coarea formula for and jrˆk j2 ) as in Lemma 6 discussed for the plane case. The last estimate leads us to the main result of this subsection.

R

jrˆk j

Lemma 13. Assume that   R3 is an exterior axially symmetric domain of type (38) with C 2 -smooth boundary @ and f 2 W 1;2 ./ \ L6=5 ./, a 2 W 3=2;2 .@/ are axially symmetric. Then the assumptions (E-NS-AX) and (73) lead to a contradiction. Proof. By construction, U .t1 /  U .t2 / for t1 ; t2 2 T, t1 < t2 . Thus the just obtained estimate (85) contradicts Lemma 11. This contradiction finishes the proof of Lemma 13. t u

4.2.6

The Case ess sup ˆ.x/ > 0. x2

The cases (b) and (c), where ess sup ˆ.x/ > 0 (see (74) and (75)), are easily reduced x2

to the plane case, because now one can separate, by the level sets of ˆ, the region where ˆ is close to maximum both from infinity and from the singularity axis Oz and carry out all arguments in the constructed bounded plane domain.

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

4.3

281

A Simple Proof of the Existence Theorem in the Case Without Swirl

Here we discuss an alternative (and much more simple) approach to the existence result for the 3D exterior domain in the axially symmetric case without swirl. The proof is based on the idea of [36] of using some divergence identities for solutions to the Euler equations (see also [42]).

4.3.1 Some Identities for Solutions to the Euler System Let the conditions (E-AX) from Lemma 8 be fulfilled, i.e., the axially symmetric functions .v; p/ satisfy the Euler equations (50) and v 2 L6 .R3 /; rv 2 L2 .R3 /;

p 2 L3 .R3 /;

rp 2 L3=2 .R3 /;

r 2 p 2 L1 .R3 /

(these properties were discussed in Sect. 4.2.3; without loss of generality, assume that v is extended by zero into R3 n  and put p.x/ D pO j for x 2 j , j D 1; : : : ; N ). Assume also that (73) is valid, i.e., ˆ.x/  0:

(86)

First of all, discuss the integrability properties of these functions on half-plane PC . For any axially symmetric vector function g D .g ; gr ; gz /, the following equality jrgj2 D

jg j2 jgr j2 C C j@r gr j2 C j@z gr j2 C j@r g j2 r2 r2 2

2

Cj@z g j C j@r gz j C j@z gz j

(87)

2

holds. Thus, jgrr j  jrgj. Applying this formula to g D rp D .@r p; 0; @z p/ we get, by virtue of r 2 p 2 L1 .R3 /, @r p 2 L1 .R3 /: r

(88)

Hence @r p 2 L1 .PC /. Since p.r; z/ ! 0 for each z 2 R as r ! 1, the inclusion p.r; / 2 L1 .R/

(89)

is valid for each r > 0; moreover, Z1 Z

Z p.t; z/ d z D  R

@r p.r; z/ d z dr ! 0 t

R

as t ! 1:

(90)

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From the last formula and the inequality ˆ  0, it follows that jvj2 .r; / 2 L1 .R/

(91)

for each r > 0. Further, (87) and rv 2 L2 .R3 / imply jv j2 jvr j2 C 2 L1 .PC /: r r

(92)

From the Euler system (50), it follows by direct calculation that for any smooth function g, the following identity

 div Œp g C .v  g/v D p div g C .v  r/g  v

(93)

holds. Apply this formula two times for g D rer and g D 1r er , where er is the unit vector parallel to the r-axis. 2 2 (I) g D rer , div Œp g C .v  g/v this identity  D 2p C v C vr . Integrating over the q 2 2 3 3D infinite cylinder Ct D .x1 ; x2 ; x3 / 2 R W r D x1 C x2 < t yields

t2

Z



 p.t; z/ C vr2 .t; z/ d z D

R

“



r 2p C v 2 C vr2 d z dr;

(94)

Pt

where Pt D f.r; z/ 2 PC W r < t g. (II) g D 1r er , div Œp g C .v  g/v D r12 .v 2  vr2 /. Since there is an essential singularity at r D 0, one needs to integrate this identity over Ct0 t D  q .x1 ; x2 ; x3 / 2 R3 W r D x12 C x22 2 .t0 ; t / to obtain R

R

D

  R

p.t; z/ C vr2 .t; z/ d z  R p.t0 ; z/ C vr2 .t0 ; z/ d z

’ 1 Pt0 t

r

 .v 2  vr2 / d z dr;

where Pt0 t D f.r; z/ 2 PC W r 2 .t0 ; t /g. Since

R

R

 p.t; z/ C vr2 .t; z/ d z ! 0 as t ! C1 and “ PC

ˇ1  2 ˇ ˇ v  v 2 ˇ d z dr < 1; r r

(95)

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

it follows from the above formulas that

“ Z

  1 2 vr  v 2 d z dr; p.t; z/ C vr2 .t; z/ d z D r R

283

(96)

Pt1

where Pt1 D f.r; z/ 2 PC W r 2 .t; C1/g.

4.3.2 Proof of the Existence Theorem As in the beginning of Sect. 4.2.4, assume that the assumptions (E-NS-AX) (see Lemma 8) are satisfied, but now suppose, additionally, that all functions a; f; uk ; v have no swirl. Our goal is to receive a contradiction. This implies the validity of Theorem 7 for the case with no swirl. It turns out that a contradiction for this case could be obtained extremely easy. Consider the limit solution .v; p/ to the Euler equations (50) from Lemma 8 (E-AX). It is necessary to discuss only the case (73), since for other two cases (74)–(75), the arguments are carried outR for

bounded plane domains (see Sect. 4.2.6). From (73), (94), it follows that R p.t; z/R C vr2 .t; z/ d z  0 for all t > 0. But in the case v  0, the equality (96) implies R p.t; z/Cvr2 .t; z/ d z > 0 a contradiction. Remark 6. The similar idea was used in [36] to obtain the existence theorem for annulus-type plane domain under inflow conditions (the flux through the external boundary component is nonpositive) and in [42] to prove the Liouville theorem in R3 for the D-solution without swirl of the stationary Navier-Stokes system. Note that this result of [42] could be easily derived from the Liouville-type theorem for ancient solutions of nonsteady Navier-Stokes system in [33].

5

2D Axially Symmetric Case: Exterior Domain

5.1

Formulation of the Problem and Historical Review

Let  be an exterior domain of R2 :  D R2 n

N  S

 Nj ; 

(97)

j D1

where j  R2 , j D 1; : : : ; N; are bounded, simply connected domains with Nj \  N i D ; for i ¤ j . Look for a solution of the Lipschitz boundaries and  steady-state Navier-Stokes problem   8 u C u  r u C rp D f in ; ˆ ˆ < div u D 0 in ; (98) ˆ ˆ : uDh on @;

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satisfying the additional condition at infinity lim

jxj!C1

u.x/ D e1 ;

(99)

where for simplicity it is assumed that f vanishes outside a disk. The two-dimensional problem in an exterior domain is much harder than the above three-dimensional case (see Sect. 4.2). The main difficulty is to find a solution satisfying the condition at infinity (99). In 1933 J. Leray [49] proved that if the boundary data are sufficiently regular, f D 0, and the fluxes through every @i vanish Z h  n dS D 0; (100) Fi D @i

then problem (98) has a weak solution .u; p/ with finite Dirichlet integral Z

jruj2 dx < C1:

(101)



To show this, Leray introduced an elegant argument, known as invading domains method, which consists in proving first that the Navier-Stokes problem   8 C u  r uk C rpk u k k ˆ ˆ ˆ ˆ < div uk ˆ uk ˆ ˆ ˆ : uk

D0

in k ;

D0

in k ;

Dh

on @;

D e1

on @Bk

(102)

has a weak solution uk for every bounded domain k D  \ Bk , Bk D fx 2 R2 W jxj < kg, Bk c {, and shows that the following estimate holds Z

jruk j2 dx  c;

(103)

k

for some positive constant c independent of k. While (103) is sufficient to assure existence of a subsequence ukl which converges weakly to a solution u of (98) satisfying (101), it does not give any information about the behavior at infinity of the velocity u (Indeed, the unbounded function log˛ jxj .˛ 2 .0; 1=2/) satisfies (101).), i.e., we do not know whether this limiting solution u satisfies the condition at infinity (99). That means that the limiting solution u does not “remember” about the boundary value e1 despite the fact that this boundary value was used in the construction of ukl * u (cf. with Sect. 4.2.2 for 3D case).

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

285

In 1961 H. Fujita recovered, by means of a different method, Leray’s result. Nevertheless, due to the lack of a uniqueness theorem, the solutions constructed by Leray and Fujita are not comparable, even for very small . The solution to (98) constructed by the invading domains method is called Leray’s solution, while any solution satisfying (101) is called D-solution. Only 40 years after Leray’s paper, D. Gilbarg and H.F. Weinberger [25] were able to show that the velocity u in Leray’s solution is bounded, p converges uniformly to a constant at infinity, and there is a constant vector uN such that Z2 lim

r!C1

N 2d D 0 ju.r; /  uj

(104)

0

(here .r; / denote polar coordinates with center at O). Moreover, they proved that if uN D 0, then the convergence is uniform and ru decays at infinity as r 3=4 log r. In the subsequent paper [26], the same authors proved that a bounded D-solution meets the same asymptotic properties as the Leray solution (see also [2]). One of N To the most difficult and unanswered questions is the relation between e1 and u. point out the difficulties of the problem, recall that even the linear Stokes problem 8 ˆ ˆ u C rp ˆ ˆ ˆ < div u ˆ ˆ ˆ ˆ ˆ :

u lim

jxj!C1

D

0

in ;

D

0

in ;

D

h

on @;

(105)

u.x/ D e1 ;

does not have, in general, a solution. Indeed, the solutions of the homogeneous problem 8 v C rQ D 0 ˆ ˆ ˆ ˆ ˆ ˆ div v D 0 < ˆ ˆ ˆ ˆ ˆ ˆ :

v D 0 v.x/ D 0; lim jxj!C1 jxj

in ; in ; on @;

spans a two-dimensional linear space C, and (105) is solvable if and only if the data satisfy the following compatibility condition (Stokes’ paradox) Z @

.h  e1 /  Œ.rv C rv> /  n  QndS D 0;

8v 2 C

(106)

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(see [8, 23]). Let us observe, by the way, that this is not surprising. Indeed, the natural solution to (105)1;2;3 should behave at infinity as the fundamental solution to (105) .u D O.log r//, and the addition of (105)4 makes (105) overdetermined. Therefore, (106) appears to be quite natural. Unexpectedly, in 1967 R. Finn and D.R. Smith [17] discovered the existence of a solution to (98), (99) without any compatibility relation between h and ¤ 0, for  sufficiently large. They also showed that .u  e1 ; p/ behaves at infinity as the fundamental solution of the linear Oseen system (see also [22]). In particular, taking also into account the results in [11, 72], one obtains the following behavior: u1  D O.r 1=2 /; ru D O.r 1 log2 r/;

u2 D O.r 1 log r/; p D O.r 1 log r/;

(107)

and outside a parabolic “wake region” around axis e1 , the decay is more rapid; in particular, ! D @1 u2  @2 u1 behaves according to !.x/ D O.e c.x1 jxj/ /

(108)

for some absolute constant c. R. Finn and D.R. Smith called a solution .u; p/ to (98), (99) physically reasonable provided u  e1 D O.r 1=4" / for some positive ". D.R. Smith [72] proved that a physically reasonable solution satisfies (107) and D.C. Clark [11] that (107) implies (108). More recently, V. I. Sazonov [71] showed that a D-solution such that u  e1 D o.1/, with ¤ 0, is physically reasonable (see also [21, 24]). Notice that nothing is currently known about the asymptotic behavior, in general, for D 0 or for arbitrary . Later, in 1988, problem (98), (99) was taken up by C.J. Amick [3] under the assumption f D 0. He proved that if h D 0, then any D-solution is bounded and converges to uN in the sense of (104). Moreover, he considered a particular but physically interesting class of solutions u D .u1 ; u2 / such that u1 is an even function of x2 and u2 is an odd function of x2 : u1 .x1 ; x2 / D u1 .x1 ; x2 /;

u2 .x1 ; x2 / D u2 .x1 ; x2 /

(109)

in the symmetric domain .x1 ; x2 / 2  , .x1 ; x2 / 2 :

(110)

Using Leray’s argument C.J. Amick showed that for symmetric solutions the convergence of u at infinity is uniform; moreover, if @ is regular enough and h D 0, then u is nontrivial, i.e., u ¤ 0 whenever ¤ 0. (Amick assumes  to be of class C 3 . Recently, this result has been extended to Lipschitz domains [68].) These last results rule out the Stokes paradox for the nonlinear case for symmetric domains and homogeneous boundary data. A clear exposition of Amick’s results, as well as the results outlined above, can be found in [22]. For an exterior domain

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

287

condition (100) has been replaced in [66] by the weaker assumption that the sum P jFi j is sufficiently small. An interesting approach to the existence of solutions i

to (98)–(99) with D 0 and small data has been recently proposed by M. Hillairet and P. Wittwer [28]. Finally, in the recent paper [61] mentioned by the authors, the problem (98), (99) with D 0 was considered in exterior plane domains symmetric with respect to both coordinate axes and a solution was found in the class of vector fields v 2 C0 satisfying the following symmetry conditions: v1 .x1 ; x2 / D v1 .x1 ; x2 / D v1 .x1 ; x2 /; v2 .x1 ; x2 / D v2 .x1 ; x2 / D v2 .x1 ; x2 /:

(111)

It is proved in [61] that if data h, f 2 C0 satisfy only natural regularity assumptions, then (98) has a D-solution in C0 which converges uniformly to zero at infinity. The flux of the boundary value h over @ in this case is arbitrary. All abovementioned results (except [61]) were proved either under the condition that all fluxes Fi are equal to zero (see (100)) or assuming that fluxes Fi are “small.” N another relevant problem in the theory of the Besides the relation between e1 and u, stationary Navier-Stokes equations is to ascertain whether a solution to problem (98) exists without any restriction on the fluxes Fi . For exterior plane domains, this problem, in general, is unsolved until now (solutions of the problem for bounded plane and 3D axially symmetric domains as well as for 3D axially symmetric exterior domains were discussed above in Sects. 3 and 4). In the last paper [40] it is proved for arbitrary fluxes Fi the existence of a Dsolution to problem (98) for exterior plane domains in the case when only Amick’s symmetric conditions (109)–(110) are satisfied and every i intersects the x1 – axis, i.e., \ i (112) Ox1 ¤ ; for all i D 1; : : : ; N:

5.2

Formulation of the Main Result

Theorem 11 ([40]). Let   R2 be a symmetric exterior domain (97), (110), (110), (112) with multiply connected Lipschitz boundary @ consisting of N disjoint components j , j D 1; : : : ; N . Assume that f is a symmetric .in the sense ofR (109)/ distribution such that the corresponding linear functional H ./ 3  7! f   

is continuous .with respect to the norm k  kH ./ / and h is a symmetric field in W 1=2;2 .@/. Then problem (98) admits at least one symmetric weak solution u. The following estimate   kruk2L2 ./  c khk2W 1=2;2 .@/ C khk4W 1=2;2 .@/ C kfk2 is valid.

(113)

288

M.V. Korobkov et al.

Here the total flux FD

Z h.x/  n.x/ dS D

N X

Fi

(114)

i D1

@

is not required to be zero or small. By what was said before, if f has a compact support, then the solution converges uniformly at infinity to a constant vector ˛e1 ; moreover, for ˛ ¤ 0, it behaves at large distance according to (107), (108). However, it is not known whether this solution satisfies the condition at infinity (99). The proof of Theorem 11 is based on Leray’s method of invading domains. The needed a priori estimate is obtained using the special extension of the boundary value satisfying the Leray-Hopf inequality (cf. with (9)) which is obtained by applying a new inequality of Poincaré type (see Lemma 16) that could be useful also in other contexts.

5.3

Some Estimates for Plane Functions with Finite Dirichlet Integral

Lemma 14. Let  be the exterior domain (97), v 2 D./. Then the following inequality Z 

jv.x/j2 dx  c jxj2 log2 jxj

Z

jrv.x/j2 dx

(115)



holds. Inequality (115) is well known (e.g., [47]). As it follows from (115), functions v 2 D./ do not have to vanish at infinity. The next assertion gives some information about the behavior of a function of D./ as jxj ! 1. Lemma 15. Let  be the exterior domain (97), v 2 D./. Then 1 lim sup r!1 log r

Z2 0

jv.r; /j2 d  2

Z

jrv.x/j2 dx:

(116)



Inequality (116) is proved in [26] (see Lemma 2.1). Lemma 16 ([40]). Let  be the exterior domain (97), v 2 D./,  > 0, ˛ 2 .0; 1/, R R0 > 1. Then the following inequality

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow ˛ jx Z 1j

Z Rn.R ; R /

jv.x1 ; x2 /j2 dx1 dx2  c jxj2

0

Z

jrv.x/j2 dx

289

(117)



holds. The constant c in (117) depends only on R0 ; , and ˛.

5.4

Construction of the Extension of the Boundary Value

Below only the construction of the extension of the boundary value is given. Other details of the proof can be found in [40]. Let 2 C 1 .R/ be a nonnegative function such that 0  .t /  1,  .t / D

1; t 1; 0; t  0;

and  2 C 1 .R/ be a monotone function on RC with .t/ 0 > 0,  .t/ D

jt j˛ ; jt j 3R0 ; 1; jt j  2R0 ;

where ˛ 2 .0; 1/. Let C D fx 2  W x2 > 0g and  D fx 2  W x2 < 0g. Set   C .x/ D x2 .x1 / C .1  .x1 //ı.x/ ;

x 2 C ;

where  2 C 1 .R/ is a monotone function with 8 < 1; jt j 2R0 ; .t / D 3 : 0; jt j  R0 ; 2 and ı.x/ is the regularized distance from the point x 2  to @ D

N S

@j . Notice

j D1

that ı.x/ is infinitely differentiable function in R2 n@ and the following inequalities a1 d .x/  ı.x/  a2 d .x/; jD ˛ ı.x/j  a3 d 1j˛j .x/ hold. Here d .x/ D dist.x; @/ is the Euclidean distance from x to @ (see [74]). Let " 2 .0; 1/ be an arbitrary number. In the domain C , define the cutoff function    ".x1 / C .x; "/ D " ln : C .x/

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M.V. Korobkov et al.

Obviously, ( C .x; "/ D

0; ".x1 / < C .x/; 1 1; C .x/ < "e  " .x1 /:

Define b.x/ D

1 1 r ln jxj D 2 2



x 1 x2 ; jxj2 jxj2

 :

The vector field b.x/ satisfies the symmetry conditions (109). Moreover, it is well known that the flux of b.x/ over a closed curve  is equal to 1, Z b.x/  n.x/ d  D 1; 

if and only if the domain bounded by  contains the point x D 0. Here n is unit vector of outward (with respect to the domain bounded by  ) normal to  . Otherwise the flux is equal  to zero.  .j / .j / Let x D x1 ; 0 2 j ; j D 1; : : : ; N . Put   b.j / .x/ D Fj b x  x .j / : Then Z

b.j / .x/  n.x/ dS D Fj ;

j

Z

b.j / .x/  n.x/ dS D 0; i ¤ j:

i

In the domain C , the functions b.j / .x/ could be represented in the form b.j / .x/ D

Fj ? .j / r 'C .x/; 2

.j /

.j /

'C .x/ D arctg

x 1  x1 ; x 2 C ; j D 1; : : : ; N; x2

 @ @ .j / . Notice that j'C .x/j  =2 for x 2 C and j D ; where r D @x2 @x1 1; : : : ; N . Define 

?

.j /

 Fj ?  .j / r C .x; "/'C .x/ 2  Fj  .j / .j / 'C .x/r ? C .x; "/ C C .x; "/r ? 'C .x/ : D 2

BC .x; "/ D

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

291

.j /

Then div BC .x; "/ D 0, and, since C .x; "/ D 1 in the neighborhood of @C , it follows that ˇ ˇ .j / BC .x; "/ˇ

@C

Fj ? .j / ˇˇ r 'C .x/ˇ : @C 2

D

Lemma 17. Let j D 1; : : : ; N . Then for every ı > 0, there exists " D ".ı/ such that the following inequality Z ˇZ  ˇ  ˇ ˇ .j / jru.x/j2 dx u.x/  r u.x/  BC .x; "/ dx ˇ  ı ˇ C

8 u 2 HS ./

(118)

C

holds. HS ./ is the subspace of functions from H ./ satisfying the symmetry conditions (109). The proof of the Leray-Hopf inequality (118) is based on Lemmas 14–16 and is true only for functions u in HS ./, i.e., satisfying the symmetry conditions (109). For an arbitrary function u 2 H ./, this inequality can be wrong. Define 8   .j / .j / ˆ BC;1 .x1 ; x2 ; "/; BC;2 .x1 ; x2 ; "/ ; x 2 C;" ; < B.j / .x; "/D ˆ : B .j / .x ; x ; "/; B .j / .x ; x ; "/; x 2  ; C;1

1

2

C;2

1

2

;"

and B.x; "/ D

N P

B.j / .x; "/:

j D1

The vector field B is symmetric, Z divB D 0;

B  ndS D Fj ; j D 1; : : : ; N:

j

ˇ Let h1 .x/ D h.x/  B.x; "/ˇ@ . Then Z h1 .x/  n.x/dS D 0; j

j D 1; : : : ; N:

(119)

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M.V. Korobkov et al.

If h 2 W 1=2;2 .@/, then obviously h1 2 W 1=2;2 .@/ and   ˇ kh1 kW 1=2;2 .@/  c khkW 1=2;2 .@/ C kBˇ@ kW 1=2;2 .@/ h

 c khkW 1=2;2 .@/ C

N X

Fj2

1=2 i

 ckhkW 1=2;2 .@/ :

j D1

Because of condition (119), there exists a function H 2 H ./ such that supp H.x; "/ is contained in a small neighborhood of the boundary @, divH D 0;

H.x; "/j@ D h1 .x/;

H 2 L4 ./; rH 2 L2 ./;

kHkL4 ./ C krHkL2 ./  ckh1 kW 1=2;2 .@/  ckhkW 1=2;2 .@/ : Moreover, H.x; "/ satisfies Leray-Hopf’s inequality, i.e., for every ı > 0, there exists " D ".ı/ such that Z ˇZ  ˇ  ˇ ˇ u.x/  r u.x/  H.x; "/ dx ˇ  ı ju.x/j2 dx ˇ 

8 u 2 H ./



holds (see [47]). The function H.x; "/ is not necessarily symmetric. However, its boundary value is symmetric and, therefore, H.x; "/ can be symmetrized defining the function e "/ as follows: H.x;

 Q 1 .x; "/ D 1 H1 .x1 ; x2 ; "/ C H1 .x1 ; x2 ; "/ ; H 2

 1 Q 2 .x; "/ D H2 .x1 ; x2 ; "/  H2 .x1 ; x2 ; "/ : H 2 Put e "/: A.x; "/ D B.x; "/ C H.x; Lemma 18.

(i) The vector field A.x; "/ is symmetric, div A.x; "/ D 0;

ˇ A.x; "/ˇ@ D h.x/;

(ii) A 2 L4 ./, rA 2 L2 ./, kAkL4 ./ C krAkL2 ./  ckhkW 1=2;2 .@/ :

(120)

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

293

(iii) For every ı > 0, there exists " D ".ı/ such that the inequality Z ˇ ˇZ   ˇ ˇ u  r u  A dx ˇ  ı jruj2 dx ˇ 

8 u 2 HS ./



holds. The constant c in (120) depends on " and tends to infinity as " ! 0. This inequality is used with sufficiently small but fixed ". Remark 7. If the domain  and the data are symmetric with respect to both coordinate axes, the existence of a weak solution which also satisfies symmetry conditions (111) can be proved. In this case the solution satisfies the condition at infinity (99) with D 0: lim v.jxj; / D 0

jxj!1

uniformly in , i.e., lim

.x1 ;x2 /!1

v.x1 ; x2 / D 0

(see [61]).

6

Conclusion

The first paper devoted to the existence of solutions to the stationary NavierStokes problem without smallness assumptions on data was that of J. Leray [49], under the sole hypothesis that the fluxes through any connected component of the boundary vanish. The question whether this condition could be removed was by then a fundamental open problem in the mathematical theory of incompressible fluid dynamics and was the object of researches of several outstanding mathematicians. A comprehensive account of attempts devoted to give an answer to this question is contained in the book of G.P. Galdi [21]. Recently, the problem has been solved for (i) two-dimensional bounded domains [36, 41]; (ii) two-dimensional exterior axially symmetric domains and symmetric data [40, 61]; and (iii) three-dimensional bounded and exterior axially symmetric domains and symmetric data [37, 39, 41]. However, it remains much to do in order to get a complete picture of the flow of an incompressible fluid under adherence boundary conditions. Among the still open problems of particular interest are the following: (i0 ) to remove the symmetry assumptions required in [37, 39, 41]; (ii0 ) to determine the behavior at infinity of the solutions found in [39, 40], also under symmetry assumptions; and (iii0 ) to prove or

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disprove the Liouville theorem in the class of D-solutions vanishing at infinity in the three-dimensional case [21, 42] (see also [33]).

7

Cross-References

 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows  Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions  Steady-State Navier-Stokes Flow Around a Moving Body Acknowledgements The research of K. Pileckas leading to these results has received funding from the Lithuanian-Swiss Cooperation Programme to reduce economic and social disparities within the enlarged European Union under project agreement No. CH-3-SMM-01/01. The research of M. V. Korobkov was partially supported by the Russian Foundation for Basic Research (Grant No. 14-01-00768-a), by the Grant of the Russian Federation for the State Support of Researches (Agreement No. 14.B25.31.0029), and by the Dynasty Foundation.

References 1. S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. 17, 35–92 (1964) 2. Ch.J. Amick, Existence of solutions to the nonhomogeneous steady Navier–Stokes equations. Indiana Univ. Math. J. 33, 817–830 (1984) 3. C.J. Amick, On Leray’s problem of steady Navier–Stokes flow past a body in the plane. Acta Math. 161, 71–130 (1988) 4. K.I. Babenko, On stationary solutions of the problem of flow past a body. Mat. Sb. 91, 3–27 (1973). English translation: math. SSSR Sbornik. 20, 1–25 (1973) 5. W. Borchers, K. Pileckas, Note on the flux problem for stationary Navier–Stokes equations in domains with multiply connected boundary. Acta Appl. Math. 37, 21–30 (1994) 6. J. Bourgain, M.V. Korobkov, J. Kristensen, On the Morse–Sard property and level sets of Sobolev and BV functions. Rev. Mat. Iberoam. 29(1), 1–23 (2013) 7. J. Bourgain, M.V. Korobkov, J. Kristensen, On the Morse–Sard property and level sets of W n;1 Sobolev functions on Rn . Journal fur die reine und angewandte Mathematik (Crelles Journal) 2015(700), 93–112 (2015). https://doi.org/10.1515/crelle-2013-0002 8. I.-D. Chang, R. Finn, On the solutions of a class of equations occurring in continuum mechanics with applications to the Stokes paradox. Arch. Ration. Mech. Anal. 7, 388–401 (1961) 9. M. Chipot, K. Kaukalyt˙e, K. Pileckas, W. Xue, On nonhomogeneous boundary value problems for the stationary Navier–Stokes equations in 2D symmetric semi-infinite outlets. Anal. Appl. (2015, to appear). https://doi.org/10.1142/S0219530515500268 10. R.R. Coifman, J.L. Lions, Y. Meier, S. Semmes, Compensated compactness and Hardy spaces. J. Math. Pures App. IX Sér. 72, 247–286 (1993) 11. D.C. Clark, The vorticity at infinity for solutions of the stationary Navier–Stokes equations in exterior domains. Indiana Univ. Math. J. 20, 633–654 (1971) 12. J.R. Dorronsoro, Differentiability properties of functions with bounded variation. Indiana U. Math. J. 38(4), 1027–1045 (1989) 13. L.C. Evans, R.F. Gariepy, in Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics (CRC Press, Boca Raton, 1992)

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14. R. Farwig, H. Morimoto, Leray’s inequality for fluid flow in symmetric multi-connected twodimensional domains. Tokyo J. Math. 35, 63–70 (2012) 15. R. Farwig, H. Kozono, T. Yanagisawa, Leray’s inequality in general multi-connected domains in Rn . Math. Ann. 354, 137–145 (2012) 16. R. Finn, On the steady-state solutions of the Navier–Stokes equations. III. Acta Math. 105, 197–244 (1961) 17. R. Finn, D.R. Smith, On the stationary solution of the Navier–Stokes equations in two dimensions. Arch. Ration. Mech. Anal. 25, 26–39 (1967) 18. H. Fujita, On the existence and regularity of the steady-state solutions of the Navier-Stokes theorem. J. Fac. Sci. Univ. Tokyo Sect. I. 9, 59–102 (1961) 19. H. Fujita, On stationary solutions to Navier–Stokes equation in symmetric plane domain under general outflow condition. Pitman research notes in mathematics, in Proceedings of International Conference on Navier–Stokes Equations, Varenna. Theory and Numerical Methods, vol. 388 (1997), pp. 16–30 20. G.P. Galdi, On the existence of steady motions of a viscous flow with non–homogeneous conditions. Le Matematiche 66, 503–524 (1991) 21. G.P. Galdi, in An Introduction to the Mathematical Theory of the Navier–Stokes Equations, ed. by C. Truesdell, vol I, II revised edition. Springer Tracts in Natural Philosophy, vol. 38, 39 (Springer, New York, 1998) 22. G.P. Galdi, Stationary Navier–Stokes problem in a two-dimensional exterior domain, in Handbook of Differential Equations, Stationary Partial Differential Equations, ed. by M. Chipot, P. Quittner, vol. 1 (Elsevier, 2003) 23. G.P. Galdi, C.G. Simader, Existence, uniqueness and Lq estimates for the Stokes problem in an exterior domain. Arch. Ration. Mech. Anal. 112, 291–318 (1990) 24. G.P. Galdi, H. Sohr, On the asymptotic structure of plane steady flow of a viscous fluid in exterior domains. Arch. Ration. Mach. Anal. 131, 101–119 (1995) 25. D. Gilbarg, H.F. Weinberger, Asymptotic properties of Leray’s solution of the stationary twodimensional Navier–Stokes equations. Russ. Math. Surv. 29, 109–123 (1974) 26. D. Gilbarg, H.F. Weinberger, Asymptotic properties of steady plane solutions of the Navier– Stokes equations with bounded Dirichlet integral. Ann. Scuola Norm. Pisa (4) 5, 381–404 (1978) 27. J.G. Heywood, On the impossibility, in some cases, of the Leray-Hopf condition for energy estimates. J. Math. Fluid. Mech. 13, 449–457 (2011) 28. M. Hillairet, P. Wittawer, On the existence of solutions to the planar Navier–Stokes system. J. Differ. Equ 255(10), 2996–3019 (2013) 29. E. Hopf, Ein allgemeiner Endlichkeitssatz der Hydrodynamik. Math. Ann. 117, 764–775 (1941) 30. L.V. Kapitanskii, K. Pileckas, On spaces of solenoidal vector fields and boundary value problems for the Navier–Stokes equations in domains with noncompact boundaries. Trudy Mat. Inst. Steklov. 159, 5–36 (1983). English Transl.: Proc. Math. Inst. Steklov. 159, 3–34 (1984) 31. K. Kaulakyt˙e, On nonhomogeneous boundary value problem for the steady Navier–Stokes system in domain with paraboloidal and layer type outlets to infinity. Topol. Methods Nonlinear Anal. 46(2), 835–865 (2015) 32. K. Kaulakyt˙e, K. Pileckas, On the nonhomogeneous boundary value problem for the Navier– Stokes system in a class of unbounded domains. J. Math. Fluid Mech. 14(4), 693–716 (2012) 33. G. Koch, N. Nadirashvili, G. Seregin, V. Sverak, Liouville theorems for the Navier–Stokes equations and applications. Acta Math. 203(1), 83–105 (2009) 34. M.V. Korobkov, Bernoulli law under minimal smoothness assumptions. Dokl. Math. 83, 107– 110 (2011) 35. M.V. Korobkov, J. Kristensen, On the Morse-Sard theorem for the sharp case of Sobolev mappings. Indiana Univ. Math. J. 63(6), 1703–1724 (2014). https://doi.org/10.1512/iumj.2014. 63.5431

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36. M.V. Korobkov, K. Pileckas, R. Russo, On the flux problem in the theory of steady Navier– Stokes equations with nonhomogeneous boundary conditions. Arch. Ration. Mech. Anal. 207(1), 185–213 (2013). https://doi.org/10.1007/s00205-012-0563-y 37. M.V. Korobkov, K. Pileckas, R. Russo, Steady Navier–Stokes system with nonhomogeneous boundary conditions in the axially symmetric case. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 14(1), 233–262 (2015). https://doi.org/10.2422/2036-2145.201204_003 38. M.V. Korobkov, K. Pileckas, R. Russo, Steady Navier–Stokes system with nonhomogeneous boundary conditions in the axially symmetric case. Comptes rendus – Mécanique 340, 115–119 (2012) 39. M.V. Korobkov, K. Pileckas, R. Russo, The existence theorem for the steady Navier–Stokes problem in exterior axially symmetric 3D domains (2014). arXiv: 1403.6921, http://arxiv.org/ abs/1403.6921 40. M.V. Korobkov, K. Pileckas, R. Russo, The existence of a solution with finite Dirichlet integral for the steady Navier–Stokes equations in a plane exterior symmetric domain. J. Math. Pures Appl. 101(3), 257–274 (2014). https://doi.org/10.1016/j.matpur.2013.06.002 41. M.V. Korobkov, K. Pileckas, R. Russo, Solution of Leray’s problem for stationary NavierStokes equations in plane and axially symmetric spatial domains. Ann. Math. 181(2), 769–807 (2015). https://doi.org/10.4007/annals.2015.181.2.7 42. M.V. Korobkov, K. Pileckas, R. Russo, The Loiuville theorem for the steady-state Navier– Stokes problem for axially symmetric 3D solutions in absence os swirl. J. Math. Fluid Mech. 17(2), 287–293 (2015) 43. M.V. Korobkov, K. Pileckas, V.V. Pukhnachev, R. Russo, The flux problem for the Navier– Stokes equations. Russ. Math. Surv. 69(6), 1065–1122 (2014). https://doi.org/10.1070/ RM2014v069n06ABEH004928 44. H. Kozono, T. Janagisawa, Leray’s problem on the Navier–Stokes equations with nonhomogeneous boundary data. Math. Zeitschrift. 262, 27–39 (2009) 45. A.S. Kronrod, On functions of two variables. Uspechi Matem. Nauk (N.S.) 5, 24–134 (1950, in Russian) 46. O.A. Ladyzhenskaya, Investigation of the Navier–Stokes equations in the case of stationary motion of an incompressible fluid. Uspekhi Mat. Nauk. 3, 75–97 (1959, in Russian) 47. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible flow (Gordon and Breach, New York, 1969) 48. E.M. Landis, Second Order Equations of Elliptic and Parabolic Type (Nauka, Moscow, 1971, in Russian) 49. J. Leray, Étude de diverses équations intégrales non linéaire et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933) 50. J. Malý, D. Swanson, W.P. Ziemer, The Coarea formula for Sobolev mappings. Trans. AMS 355(2), 477–492 (2002) 51. R.L. Moore, Concerning triods in the plane and the junction points of plane continua. Proc. Nat. Acad. Sci. U.S.A. 14(1), 85–88 (1928) 52. H. Morimoto, A remark on the existence of 2D steady Navier–Stokes flow in bounded symmetric domain under general outflow condition. J. Math. Fluid Mech. 9(3), 411–418 (2007) 53. H. Morimoto, H. Fujita, A remark on the existence of steady Navier–Stokes flows in 2D semiinfinite channel involving the general outflow condition. Mathematica Bohemica 126(2), 457– 468 (2001) 54. H. Morimoto, H. Fujita, A remark on the existence of steady Navier–Stokes flows in a certain two-dimensional infinite channel. Tokyo J. Math. 25(2), 307–321 (2002) 55. H. Morimoto, H. Fujita, Stationary Navier–Stokes flow in 2-dimensional Y-shape channel under general outflow condition, in The Navier–Stokes Equations: Theory and Numerical Methods. Lecture Note in Pure and Applied Mathematics (Morimoto Hiroko, Other), vol. 223 (Marcel Decker, New York, 2002), pp. 65–72 56. H. Morimoto, Stationary Navier–Stokes flow in 2-D channels involving the general outflow condition, in Handbook of Differential Equations: Stationary Partial Differential Equations, ed. by M. Chipot, vol. 4, Ch. 5 (Elsevier, Amsterdam/London, 2007), pp. 299–353

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57. S.A. Nazarov, K. Pileckas, On the solvability of the Stokes and Navier–Stokes problems in domains that are layer-like at infinity. J. Math. Fluid Mech. 1(1), 78–116 (1999) 58. S.A. Nazarov, K. Pileckas, On steady Stokes and Navier–Stokes problems with zero velocity at infinity in a three-dimensional exterior domains. J. Math. Kyoto Univ. 40, 475–492 (2000) 59. J. Neustupa, On the steady Navier–Stokes boundary value problem in an unbounded 2D domain with arbitrary fluxes through the components of the boundary. Ann. Univ. Ferrara. 55(2), 353–365 (2009) 60. J. Neustupa, A new approach to the existence of weak solutions of the steady Navier-Stokes system with inhomoheneous boundary data in domains with noncompact boundaries. Arch. Ration. Mech. Anal. 198(1), 331–348 (2010) 61. K. Pileckas, R. Russo, On the existence of vanishing at infinity symmetric solutions to the plane stationary exterior Navier–Stokes problem. Math. Ann. 343, 643–658 (2012) 62. C.R. Pittman, An elementary proof of the triod theorem. Proc. Am. Math. Soc. 25(4), 919 (1970) 63. V.V. Pukhnachev, Viscous flows in domains with a multiply connected boundary, in New Directions in Mathematical Fluid Mechanics. The Alexander V. Kazhikhov Memorial Volume, ed. by A.V. Fursikov, G.P. Galdi, V.V. Pukhnachev (Birkhauser, Basel/Boston/Berlin, 2009), pp. 333–348 64. V.V. Pukhnachev, The Leray problem and the Yudovich hypothesis. Izv. vuzov. Sev.-Kavk. region. Natural Sciences. The Special Issue “Actual Problems of Mathematical Hydrodynamics” (2009, in Russian), pp. 185–194 65. R. Russo, On the existence of solutions to the stationary Navier–Stokes equations. Ricerche Mat. 52, 285–348 (2003) 66. A. Russo, A note on the two-dimensional steady-state Navier–Stokes problem. J. Math. Fluid Mech. 52, 407–414 (2009) 67. R. Russo, On Stokes’ problem, in Advances in Mathematica Fluid Mechanics, ed. by R. Rannacher, A. Sequeira (Springer, Berlin/Heidelberg, 2010), pp. 473–511 68. A. Russo, On symmetric Leray solutions to the stationary Navier–Stokes equations. Ricerche Mat. 60, 151–176 (2011) 69. A. Russo, G. Starita, On the existence of solutions to the stationary Navier–Stokes equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7, 171–180 (2008) 70. L.I. Sazonov, On the existence of a stationary symmetric solution of the two-dimensional fluid flow problem. Mat. Zametki. 54(6), 138–141 (1993, in Russian). English Transl.: Math. Notes. 54(6), 1280–1283 (1993) 71. V.I. Sazonov, Asymptotic behavior of the solution to the two-dimensional stationary problem of a flow past a body far from it. Math. Zametki. 65, 202–253 (1999, in Russian). English transl.: Math. Notes. 65, 202–207 (1999) 72. D.R. Smith, Estimates at infinity for stationary solutions of the Navier–Stokes equations in two dimensions. Arch. Ration. Mech. Anal. 20, 341–372 (1965) 73. V.A. Solonnikov, General boundary value problems for Douglis-Nirenberg elliptic systems, I. Izv. Akad. Nauk SSSR, Ser. Mat. 28, 665–706 (1964); II, Trudy Mat. Inst. Steklov. 92, 233–297 (1966). English Transl.: I, Amer. Math. Soc. Transl. 56(2), 192–232 (1966); II, Proc. Steklov Inst. Math. 92, 269–333 (1966) 74. E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970) 75. V. Šverák, T-P. Tsai, On the spatial decay of 3-D steady-state Navier–Stokes flows. Commun. Partial Differ. Equ. 25, 2107–2117 (2000) 76. A. Takeshita, A remark on Leray’s inequality. Pac. J. Math. 157, 151–158 (1993) 77. I.I. Vorovich, V.I. Yudovich, Stationary flows of a viscous incompres-sible fluid. Mat. Sbornik. 53, 393–428 (1961, in Russian)

6

Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions Toshiaki Hishida

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Asymptotic Structure of the Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Existence of Flows in L3,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Asymptotic Structure of the Navier-Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

300 304 309 319 324 334 336 336

Abstract

Consider the stationary Navier-Stokes flow in 3D exterior domains with zero velocity at infinity. What is of particular interest is the spatial behavior of the flow at infinity, especially optimal decay (summability) observed in general and the asymptotic structure. When the obstacle is translating, the answer is found in some classic literature by Finn; in fact, the optimal summability is Lq with q > 2 and the leading profile is the Oseen fundamental solution. This presentation is devoted to the other cases developed in the last decade, mainly the case where the obstacle is at rest, together with several remarks even on the challenging case where the obstacle is rotating. The optimal summability for those cases is L3;1 (weak-L3 ) and the leading term of small solutions being in this class is the homogeneous Navier-Stokes flow of degree .1/, which is called the Landau solution. In any case, the total net force is closely related to the asymptotic structure of the flow. An insight into the homogeneous Navier-Stokes flow of

T. Hishida () Graduate School of Mathematics, Nagoya University, Nagoya, Japan e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_6

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degree .1/, due to Šverák, plays an important role. It would be also worthwhile finding a class of the external force, as large as possible, which ensures the asymptotic expansion of the flow at infinity.

1

Introduction

Let  be an exterior domain in R3 occupied by a viscous incompressible fluid, where a compact set R3 n  is identified with an obstacle (rigid body) and the boundary @ of  is assumed to be sufficiently smooth. Given external force f D .f1 .x/; f2 .x/; f3 .x//> , the stationary motion of the fluid is described by the velocity u D .u1 .x/; u2 .x/; u3 .x//> and pressure p D p.x/ which obey the Navier-Stokes system  u C rp C .u    !  x/  ru C !  u D f;

div u D 0

in 

(1)

subject to the boundary conditions uDC!x u!0

on @;

(2)

as jxj ! 1;

(3)

where (2) is the usual no-slip condition in which the flow attains the rigid motion in the sense of trace, while (3) is understood from the class of solutions, mostly either pointwise or summability. All vectors are throughout column ones and ./> denotes the transpose of vectors or matrices. To understand the Eq. (1) of momentum, one should start with nonstationary Navier-Stokes system in a time-dependent region exterior to a moving body and make a suitable change of variables to reduce the problem to an equivalent one in the reference frame attached to the moving body (see Galdi [26] for the details). The translational velocity  and angular velocity ! are then in general time dependent; however, both are assumed to be constant vectors, and the flow is assumed to be stationary in the reference frame. The main issue one would like to address here is the spatial behavior of solutions at infinity. It turns out that the rate of decay (3) is controlled by the total net force (momentum flux) M D M .; !; f / Z Z D ŒT .u; p/  u ˝ .u    !  y/  .!  y/ ˝ u  d y C f .y/ dy @



(4) associated with (1), which is written as the divergence form  div ŒT .u; p/  u ˝ .u    !  x/  .!  x/ ˝ u D f;

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301

where T .u; p/ D ru C .ru/>  pI

(5)

is the Cauchy stress tensor, I is the 3  3 identity matrix, and  stands for the outward unit normal to @. One has nonzero force M ¤ 0 in general; however, if in particular u, p, and ru decay sufficiently fast at infinity, then integrating the equation above over fx 2 I jxj < g and letting  ! 1 yield M D 0. This simple observation suggests that M is related to the decay structure of the flow. One may also refer to [47, section 6] in the context of the self-propelled motion of a rigid body. In his celebrated paper [61], Leray showed the existence of at least one solution in the class of finite Dirichlet integral ru 2 L2 ./ to (1), (2) and (3) (when  D ! D 0) without any smallness condition on the data. The argument relies on compactness together with a priori estimate arising from structure of the NavierStokes system. Note that this structure is kept for the case  C !  x ¤ 0 as well. His theorem thus provides even large solutions, most of which would be unstable. From the viewpoint of stability, solutions of the Leray class do not give us enough information about the asymptotic behavior at infinity. In fact, the only thing one knows is u 2 L6 ./; however, mathematical analysis developed so far requires better decay property such as ju.x/j  C jxj1 or u 2 L3;1 ./ (as well as smallness) of the stationary flow u to show its stability, where L3;1 denotes the weak-L3 space (see [6, 7, 31, 38–40, 44, 46, 49, 55, 63, 65, 70], and the references therein). When  D 0, the summability L3;1 of stationary flows observed in general is actually optimal unless assuming any specific condition such as symmetry. As compared with this case, better summability of stationary flows for the case  2 R3 n f0g mentioned in the next paragraph is helpful in the proof of stability of such flows (under smallness conditions); indeed, there is no need to analyze the full linearized operator; in other words, analysis of the Oseen semigroup is enough, while that is not the case when  D 0 (unless using an interpolation technique due to [76]). The interest is focused on optimal decay/summability at infinity of the flow together with its asymptotic structure. This was addressed in a series of papers by Finn [22–24], in which the rigid body was assumed to be translating with velocity  2 R3 n f0g; however, ! D 0. In this case the essential step is to analyze the asymptotic behavior of solutions at infinity to the Oseen system  u C rp    ru D f;

div u D 0;

in :

(6)

The Oseen fundamental solution possesses anisotropic decay structure with paraboloidal wake region behind the body. In fact, the flow decays faster outside wake than inside, and consequently the summability near infinity is better like u 2 Lq with q > 2 than the case  D 0. Finn [23], Farwig [15], and Shibata [70] proved the existence of small Navier-Stokes flow which exhibits the same decay structure with wake as mentioned above; actually, the leading profile of the flow is the Oseen fundamental solution, and its coefficient is given by the force M .; 0; f /

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(see (4)), provided f is of bounded support. Such a flow was called physically reasonable solution by Finn. Furthermore, Babenko [1], Galdi [25], and Farwig and Sohr [21] showed that any solution of the Leray class without restriction on the magnitude becomes a physically reasonable solution (see Galdi [28, Theorem X.8.1]). This is a contrast to the case  D 0; indeed, when the translation of the body is absent, one has no result on the asymptotic behavior of large solutions of the Leray class except for Choe and Jin [13], in which some pointwise decay rates of axisymmetric solutions of that class were deduced. Later on, Galdi and Silvestre [33], Galdi and Kyed [29], and Kyed [57–59] generalized the results mentioned above for purely translational regime to the case   ! ¤ 0. In fact, the presence of translation still implies fine decay/summability at infinity of the flow past a rotating obstacle except the case where ! 2 R3 n f0g is orthogonal to , and, as a consequence, the leading profile of the Navier-Stokes flow is described in terms of the linear part, in which a remarkable role of axis of rotation can be also observed. Such a role was discovered first by Farwig and Hishida [18, 19] for the flow around a purely rotating obstacle, and it will be explained later. This presentation studies the other case where the translation of the body is absent ( D 0/. In this case the existence of solutions, which decay like ju.x/j  C jxj1 ;

jru.x/j  C jxj2

.jxj ! 1/

or u 2 L3;1 ./;

ru 2 L3=2;1 ./;

for small external forces was proved by [7, 56, 69] (! D 0) and by [17, 27] (! ¤ 0). For the case ! D 0, Deuring and Galdi [14] clarified that the leading profile was no longer the Stokes fundamental solution. Indeed, the rate jxj1 of decay yields the balance between the linear part and nonlinearity since u  u  ru  jxj3 (formally). This observation would suggest that a sort of nonlinear effect is involved in the leading term of the flow. Nazarov and Pileckas [68] first derived asymptotic expansion under smallness conditions, but the leading term was less explicit. It was explicitly found much later by Korolev and Šverák [50] (case ! D 0). When the nonlinearity is balanced with the linear part, it is reasonable to expect that the self-similar solution would be a candidate of the leading term. Since the stationary Navier-Stokes system ( D ! D 0) is invariant under the scale transformation u .x/ D u. x/;

p .x/ D 2 p. x/;

> 0;

(7)

in R3 n f0g

(8)

a smooth solution fu; pg to  u C rp C u  ru D 0;

div u D 0

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

303

is called (stationary) self-similar solution if u .x/ D u.x/;

8 > 0 8x 2 R3 n f0g

p .x/ D p.x/;

or, equivalently,   1 x ; u.x/ D u jxj jxj

1 p.x/ D p jxj2



 x ; jxj

8x 2 R3 n f0g;

(9)

that is, u and p are homogeneous of degree .1/ and .2/, respectively. Landau [60] derived its exact form under the assumption of axisymmetry (see (78)), in order to describe jets from a thin pipe (see also Tian and Xin [75] and Cannone and Karch [9]). Finally, Šverák [74] has characterized completely the set S of all self-similar solutions as follows: S is parameterized by vectorial parameter as n o S D fUb ; Pb gI b 2 R3

(10)

whose member fUb ; Pb g is symmetric about the axis Rb and satisfies  Ub C rPb C Ub  rUb D bı;

div Ub D 0

in D0 .R3 /

(11)

across the origin (see also [2, 9]), where ı denotes the Dirac measure. In other words, every self-similar solution must have its own axis of symmetry, and the set S eventually agrees with the family of solutions computed by Landau. This is the reason why the self-similar solution is often called the Landau solution. The proof of Šverák [74] is closely related to geometric properties of S2 (unit sphere). Based on his profound insight, he and Korolev [50] proved that the leading term of asymptotic expansion of solutions to (1) with .; !; f / D .0; 0; 0/ (without assuming (2)), which decay like jxj1 at infinity, is given by the specific Landau solution UM with label M D M .0; 0; 0/ (see (4)), provided lim supjxj!1 jxjju.x/j is small enough, where the error term satisfies the pointwise estimate like jxj2C" for " > 0 arbitrarily small (but the smallness of u depends on "). The result was extended to small time-periodic solutions (case  D ! D 0) with period T > 0 by Kang, Miura, and Tsai [48], where the leading term is the Landau solution RT Ub with label b D T1 0 M , that is, the time average of the force (4). Note that the Landau solution must be useful to describe the local behavior related to singularity/regularity as well; indeed, it was proved by Miura and Tsai [64] that the leading term of point singularity like jxj1 at x D 0 of the Navier-Stokes flow is also given by a Landau solution provided it is small enough. Later on, Farwig and Hishida [19] studied the case where the body is purely rotating with angular velocity ! 2 R3 n f0g and proved that the leading term of solutions to (1) (with  D 0, f D 0), which are small in L3;1 ./, is another ! ! Landau solution Ub with label b D . j!j  M / j!j , whose axis of symmetry is

304

T. Hishida

parallel to the axis of rotation along which the flow is largely concentrated, where M D M .0; !; 0/ (see (4)). The solution enjoys better summability if and only if !  M D 0, while so does the solution for the case where the body is at rest if and only if the full force M vanishes. Thus, one is able to find out the effect of rotation, which was not clear until the study of [18, 19]. They considered solutions in L3;1 ./ rather than pointwise decay properties, and the error term was estimated in terms of summability. Their result was then refined by Farwig, Galdi, and Kyed [16] in the sense that the asymptotic expansion with error term satisfying a pointwise estimate (as in [50]) was deduced even for solutions of the Leray class with the energy inequality under appropriate smallness of !. Most part of this presentation is devoted to the case ! D 0, but key points for the purely rotating case ! 2 R3 nf0g and the remarkable difference between those cases are also explained. One specifies a class of the external force f , which ensures the asymptotic expansion of the Navier-Stokes flow u 2 L3;1 ./ as long as it is small enough. Since the class of f is rather large, it seems difficult to deduce pointwise estimates of the error term; instead, it is estimated in terms of summability as in [19]. Before the analysis of the Navier-Stokes flow, it is also worthwhile showing the asymptotic expansion of the Stokes flow with general external force as above (see Theorem 1). This presentation consists of six sections. After some preliminaries in the next section, asymptotic structure for the linearized system is studied in sect. 3. At the end of sect. 3, a few crucial facts which interpret why the axis of rotation is preferred for the case ! 2 R3 n f0g are mentioned. In sect. 4 the existence theorem (Theorem 2) for small solutions in L3;1 ./ is provided. It is based on the linear theory (see Theorem 3). The result is due to Kozono and Yamazaki [56], but one carries out linear analysis in a different way, which can be applied to the other cases  C !  x ¤ 0 (see [17, 71]). The asymptotic expansion of the Navier-Stokes flow obtained in Theorem 2 is studied in sect. 5 (see Theorem 4), in which one needs a bit more decay property of the external force than assumed in Theorem 2. The final section summarizes what is done and raises several open questions about the related issues. Although this is a survey article, the complete proof of Theorems 1, 3, and 4 will be presented.

2

Preliminaries

In this section some function spaces are introduced and notation is fixed. Let B be the open ball in R3 centered at the origin with radius  > 0. For sufficiently large  > 0, we set  D  \ B , where  is the exterior domain under consideration. Let D be a smooth domain in R3 , such as the exterior domain , whole space 3 R , or a bounded domain. By C01 .D/ one denotes the class of smooth functions with compact support in D. For 1  q  1 the usual Lebesgue spaces are denoted by Lq .D/ with norm k  kq;D . To introduce the Lorentz space (for details, see Bergh and Löfström [3]), given measurable function f on D, set

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

mf .t / WD jfx 2 DI jf .x/j > t gj ;

305

t > 0;

where j  j stands for the Lebesgue measure. Then mf ./ is monotonically nonincreasing, right continuous, and measurable. It is well known that f 2 Lq .D/, 1  q < 1, if and only if Z

1

˚

t mf .t /1=q

0

q dt < 1: t

With this in mind, one denotes by Lq;r .D/ the vector space consisting of all measurable functions f on D which satisfy Z

1

˚

t mf .t /1=q

0

r dt t

1=r 0

Note that Lq;r0 .D/  Lq;r1 .D/ if r0  r1 I

Lq;q .D/ D Lq .D/;

the latter of which is obvious as mentioned above. Each of finite quantities (12) is a quasi-norm; however, by the use of the average function, it is possible to introduce a norm k  kq;r;D , which is equivalent to that, unless q D 1 (see [3]). Then Lq;r .D/ equipped with k  kq;r;D (1 < q < 1; 1  r  1) is a Banach space, called the Lorentz space; in particular, Lq;1 .D/ is well known as the weak-Lq space, in which C01 .D/ is not dense. As a typical function in this space, recall that jxj˛ 2 L3=˛;1 .R3 / as long as 0 < ˛  3. One also has the weak Hölder inequality ([7, Lemma 2.1]): let 1 < p  1; 1 < q < 1 and 1 < r < 1 satisfy 1=r D 1=p C 1=q, and let f 2 Lp;1 .D/, g 2 Lq;1 .D/, then fg 2 Lr;1 .D/ with kfgkr;1;D  kf kp;1;D kgkq;1;D

(13)

where L1;1 .D/ D L1 .D/. Let 1 < q < 1 and 1  r  1. The Lorentz spaces can be also constructed via real interpolation   Lq;r .D/ D L1 .D/; L1 .D/ 11=q;r :

(14)

This together with the reiteration theorem in the interpolation theory ([3, 5.3.1]) implies that   Lq;r .D/ D Lq0 ;r0 .D/; Lq1 ;r1 .D/ ;r

(15)

306

T. Hishida

provided 1 < q0 < q < q1 < 1;

1=q D .1  /=q0 C =q1 ;

1  r0 ; r1 ; r  1:

Then one knows

kf kq;r;D  C kf k1

q0 ;r0 ;D kf kq1 ;r1 ;D

(16)

for all f 2 Lq0 ;r0 .D/ \ Lq1 ;r1 .D/  Lq;r .D/. For fixed f 2 Lp;1 .D/, the map g 7! fg is bounded from Lq;1 .D/ to Lr;1 .D/ by (13), where 1 < p  1, 1 < q < 1, and 1 < r < 1 satisfy 1=r D 1=pC1=q. Hence, the interpolation (15) leads to kfgkr;s;D  kf kp;1;D kgkq;s;D

(17)

for f 2 Lp;1 .D/ and g 2 Lq;s .D/, where p; q; r are the same as above and 1  s  1. For 1 < q < 1;

1 < r  1;

1=q 0 C 1=q D 1;

1=r 0 C 1=r D 1;

(18)

the duality relation 0

0

Lq;r .D/ D Lq ;r .D/ 0

holds, in particular, Lq;1 .D/ D Lq ;1 .D/ . In what follows, the same symbols for vector and scalar function spaces are adopted as long as there is no confusion. The abbreviations k  kq D k  kq; and k  kq;r D k  kq;r; are used for the exterior domain  under consideration. P q1 .D/ be the One needs the homogeneous Sobolev space. For 1 < q < 1, let H completion of C01 .D/ with respect to the norm kr./kq;D . For D D R3 one has P q1 .R3 / D fu 2 Lq .R3 /I ru 2 Lq .R3 /g=R: H loc When 1 < q < 3, one may take the canonical representative elements to adopt P q1 .R3 / D fu 2 Lq .R3 /I ru 2 Lq .R3 /g; H where 1=q D 1=q  1=3, together with the embedding estimate kukq ;R3  C krukq;R3 : Let 1 < q0 < q < q1 < 1;

1=q D .1  /=q0 C =q1 ;

1  r  1;

(19)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

307

and define   1 P q1 .D/; H P q1 .D/ P q;r .D/ D H H 0 1

;r

;

which is independent of the choice of fq0 ; q1 g, with norm kr./kq;r;D . Note that 1 P q;r C01 .D/ is dense in H .D/ unless r D 1. For D D R3 the embedding relations ([56]) 1 P q;r .R3 / ,! Lq ;r .R3 /; H

kukq ;r;R3  C krukq;r;R3 ;

1 P 3;1 .R3 / ,! L1 .R3 / \ C .R3 /; H

(20)

kuk1;R3  C kruk3;1;R3 ;

hold provided 1 < q < 3, 1=q D 1=q  1=3 and 1  r  1. Let 1 < q < 3, 1=q D 1=q  1=3 and 1  r  1. Let   R3 be the exterior domain. For every u 2 L1loc ./ satisfying ru 2 Lq;r ./, there is a constant k D k.u/ such that u C k 2 Lq ;r ./ with ku C kkq ;r  C krukq;r where C > 0 independent of u (see [7, Theorem 5.9]). By taking the canonical 1 P q;r ./, one has the characterization ([34, 52, 56]) representative element of u 2 H 1 P q;r ./ D fu 2 Lq ;r ./I ru 2 Lq;r ./; uj@ D 0g; H

(21)

kukq ;r  C krukq;r :

(22)

together with

P 1 ./ ,! One can also take the canonical representative element of u 2 H 3;1 1 L ./ \ C ./, which goes to zero for jxj ! 1 and satisfies uj@ D 0 as well as kuk1  C kruk3;1 :

(23)

1 P q;r For fq; rg satisfying (18), the space H .D/ is defined as the dual space of 1 1 P q .D/ D H P q;q .D/. The duality theorem for interpolation and set H spaces ([3, 3.7.1]) implies that

P 10 0 .D/, H q ;r

  1 P q1 .D/; H P q1 .D/ P q;r .D/ D H H 0 1

(24)

;r

for q; q0 ; q1 ; r satisfying (19) with 1 < r  1. For r D 1, one also defines P 1 .D/, 1 < q < 1, as the dual space of the completion of C 1 .D/ with H q;1  0  O 10 .D/ ¤ H P 10 .D/ . respect to the norm kr./kq 0 ;1;D , which is denoted by H q ;1

q ;1

Then (24) holds for r D 1 as well (see [3, p.55]). Let 1 < q < 1 and 1  r  1,

308

T. Hishida

1 P q;r then there exists a constant C > 0 such that for every f 2 H .D/, one can take q;r F 2 L .D/ satisfying

div F D f;

kF kq;r;D  C kf kHP q;r 1 .D/ I

see [51, Lemma 2.2] and [56, Lemma 2.2]. Let D  R3 be a bounded domain. Then Lp;r .D/  Lq;s .D/ for all 1 < q < p < 1; r; s 2 Œ1; 1: Both embeddings 1 1 P q;r P q;r .D/ ,! Lq;r .D/ ,! H .D/ H

(25)

are compact, where 1 < q < 1 and 1  r  1 (see [3, 3.14.8]). For the same fq; rg, one also has 1 P q;r .D/ D fu 2 Lq;r .D/I ru 2 Lq;r .D/; uj@D D 0g; H

together with the Poincaré inequality kukq;r;D  C krukq;r;D :

(26)

Finally, consider the boundary value problem for the equation of continuity div w D f

in D;

wD0

on @D;

where D is a bounded domain in R3 with Lipschitz boundary @D. Given R f being in a suitable class, say f 2 Lq .D/, with compatibility condition D f D 0, there are a lot of solutions, some of which were found by many authors (see Galdi [28, Notes for Chapter III]). Among them a particular solution discovered by Bogovskii [4] is useful to recover the solenoidal condition in a cutoff procedure on account of some fine properties of his solution. By the following lemma, the operator f 7! his solution w (called the Bogovskii operator) is well defined, and its properties are summarized. For the proof, see Borchers and Sohr [8], Galdi [28], as well as Bogovskii [4]. Lemma 1. Let D  R3 be a bounded domain with Lipschitz boundary. Then there is a linear operator B W C01 .D/ ! C01 .D/3 such that, for 1 < q < 1 and k  0 integer, kr kC1 Bf kq;D  C kr k f kq;D with some C D C .D; q; k/ > 0 and that Z div Bf D f

f .x/dx D 0;

if D

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

309

where the constant C is invariant with respect to dilation of the domain D. P qk .D/ to By continuity, B is extended uniquely to a bounded operator from H P qk .D/ is the completion of C 1 .D/ with respect to the norm P qkC1 .D/3 , where H H 0 k kr ./kq;D . Furthermore, by real interpolation, it is extended uniquely to a bounded k kC1 P q;r P q;r .D/ to H .D/3 , where 1  r  1 and operator from H   k P qk .D/; H P qk .D/ P q;r .D/ D H H 0 1

;r

with q0 ; q1 and satisfying (19).

3

Asymptotic Structure of the Stokes Flow

Let us start with asymptotic structure of the simplest case, that is, the exterior Stokes flow without external force  u C rp D 0;

div u D 0;

in ;

(27)

where nothing is imposed at the boundary @. Note that the result below does not depend on the boundary condition on @. Since (27) admits polynomial solutions, it is reasonable to impose a growth condition, for instance, u.x/ D o.jxj/

as jxj ! 1; q;r

or

u=.1 C jxj/ 2 L ./

or

ru 2 Lq;r ./

for some q 2 .1; 1/; r 2 Œ1; 1;

(28)

for some q 2 .1; 1/; r 2 Œ1; 1;

to exclude polynomials except constants. Note that the growth condition on the pressure is not needed here since it is controlled through the Eq. (27) by the velocity (but that is not the case in Theorem 1 below; see Remark 1). Then the asymptotic structure is described in terms of the Stokes fundamental solution E.x/ D

1 8



1 x˝x IC jxj jxj3

 ;

Q.x/ D

x ; 4jxj3

(29)

to be precise (Chang and Finn [11]), for every solution to (27) subject to (28), there are constants u1 2 R3 and p1 2 R such that Z

T .u; p/ d  C O.jxj2 /;

u.x/ D u1 C E.x/ @

Z

p.x/ D p1 C Q.x/  @

(30) T .u; p/ d  C O.jxj3 /;

310

T. Hishida

as jxj ! 1. For the proof, there are two methods. One is to employ a potential representation formula (as in, for instance, [18]), and the other is a cutoff technique for reduction to the whole space problem. This section takes the latter way since it works for the Navier-Stokes system as well (see sect. 5). It will be also clarified which condition on the external force f ensures that the Stokes fundamental solution is still the leading profile at infinity for  u C rp D f;

div u D 0

in :

(31)

For simplicity, let us consider smooth solution to (31) for smooth external force. If f .x/ D O.jxj3 / or f D div F with F .x/ D O.jxj2 /, it is formally balanced with the second derivative of E.x/ and thus the situation would be delicate (the equality .jxj1 log jxj/ D jxj3 suggests that one could not expect even the rate jxj1 of decay for the former case). In order to make this point clear, one needs the following lemma on asymptotic structure of the volume potentials E f and Q f . Lemma 2. Let f 2 C 1 .R3 / and assume that there are constants ˛ > 3 and C > 0 such that jf .x/j 

C .1 C jxj/˛

8x 2 R3 :

Then 8 3 < ˛ < 4; < O.jxj˛C2j /; j j f .y/ dy C O.jxj2j log jxj/; ˛ D 4; r .E f /.x/ D r E.x/ : R3 O.jxj2j /; ˛ > 4; 8 Z 3 < ˛ < 4; < O.jxj˛C1 /; 3 .Q f /.x/ D Q.x/  f .y/ dy C O.jxj log jxj/; ˛ D 4; (32) : R3 O.jxj3 /; ˛ > 4; Z

as jxj ! 1, where j D 0; 1. Proof. One can split the error term as Z

f.r j E/.x  y/  r j E.x/gf .y/ dy R3

Z

f.r j E/.x  y/  r j E.x/gf .y/ dy  r j E.x/

D jyj2jxj

Z f .y/ dy jyjjxj=2

.r j E/.x  y/f .y/ dy

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

311

It is easy to see that jI2 j C jI4 j  C jxj˛C2j and that jI3 j  C jxj˛  C jxj˛

Z Z

jyj2jxj

dy jx  yj1Cj dy D C jxj˛C2j : jx  yj1Cj

jyxj3jxj

One also finds Z

Z

1

jI1 j  C jyj 0 such that jf0 .x/j 

C .1 C jxj/˛

8x 2 :

(36)

For every smooth solution of class u; p; ru 2 Lsloc ./ to (31) subject to (28), there is a constant u1 2 R3 such that u.x/ D u1 C E.x/M0 C v0 .x/ C v1 .x/;

(37)

for jxj  3R with Z



 T .u; p/ C F  d  C

M0 D @

Z f0 .y/ dy; 

8 3 < ˛ < 4; < O.jxj˛C2j /; j r v0 .x/ D O.jxj2j log jxj/; ˛ D 4; : O.jxj2j /; ˛ > 4;

(38)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

313

(j D 0; 1) as jxj ! 1 and v1 2 Ls . n B3R /;

rv1 2 Ls . n B3R /;

where s 2 .3=2; 3 is defined by 1=s D 1=s  1=3. Here, R > 0 is taken such that R3 n   BR . If in particular F D 0, then v1 is absent from (37). Proof. First of all, note that the boundary integral of M0 can be understood as 1=s h.T .u; p/ C F /; 1i@ in the sense of normal trace .T .u; p/ C F / 2 Hs .@/ s 1 since T .u; p/ C F 2 Lloc ./ and div .T .u; p/ C F / D f0 2 L ./  Lsloc ./. Let us reduce the problem to the one with vanishing flux at the boundary @. Without loss one may assume 0 2 int .R3 n /. We introduce the flux carrier Z

1 ; z.x/ D ˇ r 4jxj

ˇD

  u d;

(39)

@

for given solution fu; pg. Since Z   z d  D ˇ;

div z D 0;

in R3 n f0g;

z D 0

(40)

@

the pair fQu; pg with uQ D u  z fulfills also (31) subject to Z   uQ d  D 0:

(41)

@

By using a cutoff function 2 C01 .B3R I Œ0; 1/;

.x/ D 1

.jxj  2R/;

kr k1 

C R

(42)

and the Bogovskii operator B (Lemma 1) in the domain AR D fx 2 R3 I R < jxj < 3Rg;

(43)

one sets v D .1 

/Qu C BŒQu  r ;

D .1 

/p;

(44)

3 where the Bogovskii term R is understood as its zero extension to the whole space R . It should be noted that AR uQ  r dx D 0 follows from (41). Then the pair fv; g obeys

 v C r D g C .1  /f0 C div ..1 

/F /;

div v D 0

in R3

(45)

314

T. Hishida

for some function g 2 C01 .AR /. Here, one does not need any exact form of g and what is important is structure of the Eq. (45). When either u.x/ D o.jxj/ or u=.1 C jxj/ 2 Lq;r ./, it is obvious that v 2 S 0 .R3 /. Under the alternative assumption ru 2 Lq;r ./ in (28), one has rv 2 S 0 .R3 /, which implies v 2 S 0 .R3 / ([12, Proposition 1.2.1]). Going back to the Eq. (45) yields r 2 S 0 .R3 / and, therefore, 2 S 0 .R3 / by the same reasoning. Let P s1 .R3 / ,! Ls .R3 /; v1 2 H

1 2 Ls .R3 /;

P s1 .R3 / obtained be the solution to (33) with the external force div ..1  /F / 2 H in Lemma 3. We then find

v.x/ D E fg C .1  /f0 g .x/ C v1 .x/ C Pv .x/; (46)

.x/ D Q fg C .1  /f0 g .x/ C 1 .x/ C P .x/; with some polynomials Pv and P ; however, from (28) it follows that Pv must be a constant vector, which is denoted by u1 . Thus, one concludes from Lemma 2 that u.x/ can be represented as u.x/ D uQ .x/ C z.x/ D E.x/M0 C v0 .x/ C v1 .x/ C u1 with

.jxj  3R/

Z M0 D

fg C .1 

/f0 g.y/ dy;

R3

where r j v0 .x/ behaves like the remainder of (32) since r j z.x/ D O.jxj2j / (see (39)). Let  > 3R. From Z



rz C .rz/>

jyjD

one can deduce Z fg C .1 

y 

d D

ˇ 4

Z y d D 0

(47)

jyjD

Z /f0 g.y/ dy D 

B

div fT .v; / C .1  Z

/F g dy

B

y d  jyjD Z Z .T .u; p/ C F / d  C f0 .y/ dy: D

D

.T .u; p/ C F /

@



Letting  ! 1 leads to Z

Z

M0 D

.T .u; p/ C F / d  C @

which concludes (37). This completes the proof.

f0 .y/ dy; 

t u

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

315

Remark 1. Because of less information about div F , one cannot say anything about the polynomial P in (46) unless assuming the behavior of p at infinity. If in particular F D 0 so that 1 D 0, then P must be a constant. In fact, one knows that both .E h/ and r.Q h/ belong to Lr .R3 / for every r 2 .1; 1/ because so does h WD g C .1  /f0 . By going back to (31), one finds that P is a constant, which we denote by p1 . As a consequence, 8 3 < ˛ < 4; < O.jxj˛C1 /; p.x/ D p1 C Q.x/  M0 C O.jxj3 log jxj/; ˛ D 4; : O.jxj3 /; ˛ > 4;

(48)

as jxj ! 1. Theorem 1 immediately implies the following corollary. Corollary 1. In addition to the assumptions of Theorem 1 with s D 3=2, suppose either u 2 L3 ./ or ru 2 L3=2 ./. Then M0 D 0.

Remark 2. For the Stokes system in n-dimensional exterior domains, either u 2 Ln=.n2/ ./ or ru 2 Ln=.n1/ ./ yields M0 D 0 (under suitable assumptions on the external force). Consider (31) subject to uj@ D 0. Corollary 1 then tells us that the condition P1 u 2 H 3=2;1 ./ is an optimal class observed in general even if f D div F with F 2 C01 ./. The following corollary claims the uniqueness of solutions in this class. 3=2;1 P1 Corollary 2. Let fu; pg 2 H ./ be a solution to (27) subject to 3=2;1 ./  L uj@ D 0. Then fu; pg D f0; 0g.

Proof. Theorem 1 with f D 0 can be applied. Since u 2 L3;1 ./, one has (37) with u1 D 0 as well as v1 D 0. One also knows (48) with p1 D 0 (or even more directly, the same procedure as in the proof of Theorem 1 with use of p 2 L3=2;1 ./ leads to the same expansion). As a consequence, u.x/ D O.jxj1 /;

fru.x/; p.x/g D O.jxj2 /

(49)

as jxj ! 1. Let 2 C 1 .Œ0; 1/I Œ0; 1/ satisfy .t/ D 1 for 0  t  1 and

.t/ D 0 for t  2, and set  .x/ D .jxj=/ for  > 0 large enough and x 2 R3 . Since the local regularity theory for the Stokes boundary value problem together with a bootstrap argument yields fu; pg 2 H 2 .2 /  H 1 .2 /, one can multiply (27) by  u to obtain

316

T. Hishida

Z

jruj2  dx C 

Z

Z .ru  r  /  u dx   2jyj:

(57)

Proof. A brief sketch will be presented here. Let us take the Taylor formula (with respect to y) G.O.at /x  y; t / D G.x; t / C G.x; t /

.O.at /x/  y C .remainder/ 2t

and consider each term multiplied by O.at /> . To get (57), the point is rapid decay due to oscillation ˇZ ˇ ˇ ˇ

0

1



cos at sin at



ˇ ˇ C G.x; t / dt ˇˇ  jajjxj3

and a part of non-oscillating terms comes to the leading profile ˆ.x/. The term H .O.at /x  y; t / can be discussed similarly although the computation is more complicated. The details are found in [18, section 4]. The others (54), (55) and (56) are much easier (see [18, (2.11)] and [47, (6.23)]). t u By the same splitting of the volume potential as in the proof of Lemma 2, one can make use of Lemma 4 to conclude the following asymptotic expansion. In the proof, the dominant term is I1 for the region jyj < jxj=2, in which (57) is employed.

318

T. Hishida

Lemma 5. Let ! D ae3 with a 2 R n f0g. Suppose f satisfies the same condition (with ˛ > 3) as in Lemma 2. Then Z u.x/ D

.x; y/f .y/ dy R3

enjoys 8 3 < ˛ < 4; < O.jxj˛C2 /; 2 f .y/ dy E.x/e3 C O.jxj log jxj/; ˛ D 4; u.x/ D e3  : R3 ˛ > 4; O.jxj2 /; 



Z

(58)

as jxj ! 1, where E.x/ is the Stokes fundamental solution (29). By means of harmonic analytic method developed by [20], it is possible to deduce exactly the same well-posedness for (53) as in Lemma 3, where the constants in (34) and (35) are independent of a 2 R n f0g. It was done by [43, Theorem 2.1], [17, Proposition 3.2]. By using this together with Lemma 5, one can follow the proof of Theorem 1 (with a bit more care of the flux carrier, see [47, section 6]) under the same assumptions on the external force to obtain the asymptotic expansion of solutions to (51), in which the leading term is given by  V .x/ WD

 ! !  M0 E.x/ ; j!j j!j

! D e3 ; j!j

(59)

where Z M0 D

Z ŒT .u; p/ C u ˝ .!  y/  .!  y/ ˝ u C F  d y C

@

f0 .y/ dy: 

Since e3  .e3  y/ D 0, the third term of the boundary integral does not contribute to the coefficient e3  M0 . This was proved under the no-slip condition uj@ D !  x by Farwig and Hishida [18], who deduced not only the leading term but also the second one although they restricted their consideration to the simple case where f D div F with F 2 C01 ./. The leading term (59) satisfies V C r… D .e3  M0 /e3 ı;

div V D 0

in D0 .R3 /, where ….x/ D .e3  M0 /x3 =.4jxj3 /. But the pair fV; …g enjoys  V C r…  .!  x/  rV C !  V D .e3  M0 /e3 ı;

div V D 0

(60)

as well, since .e3  x/  rV  e3  V D 0:

(61)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

319

Note that (61) holds for all vector fields which are symmetric about Re3 (x3 -axis). In fact, because such vector fields must be of the form V D .W .r; x3 / cos ; W .r; x3 / sin ; V3 .r; x3 //> in cylindrical coordinates r; ; x3 , one finds .e3  x/  rV D @ V D e3  V . As long as V .x/  1=jxj near the origin, (61) holds also in D0 .R3 /, that is, hV ˝ .e3  x/  .e3  x/ ˝ V; r i D 0 for all 2 C01 .R3 /. Hence, the leading term (59) satisfies (60) in D0 .R3 /.

4

Existence of Flows in L3,1

Consider the problem (1) and (2) with  D ! D 0, that is,

 u C rp C u  ru D f; uD0

div u D 0

in ;

on @:

(62) (63)

Before proceeding to study of asymptotic structure of the Navier-Stokes flow, one should establish the existence of (small) solutions having optimal asymptotic behavior at infinity. The existence of solutions decaying like jxj1 was proved by Finn [23], Galdi and Simader [35], Novotny and Padula [69], and Borchers and Miyakawa [7]. Indeed, such a pointwise estimate is fine, but one may take another way by the use of function spaces, so that the proof becomes easier. Among some function spaces which are able to catch homogeneous functions of degree .1/, the weak-L3 space is probably the simplest one. Actually, Kozono and Yamazaki [56] succeeded in showing the existence of a unique Navier-Stokes flow in L3;1 ./ whenever the external force is small in a sense. Since the next section studies the asymptotic structure of solutions in L3;1 ./, one will provide the existence theorem due to [56]. It was known ([7, 53, 54]) that either u 2 L3 ./ or ru 2 L3=2 ./ necessarily yields M .0; 0; f / D 0 (see (4)), although those Lebesgue spaces are invariant under the scale transformation (7). Hence, one has no chance to find the NavierStokes flow belonging to L3 ./ in generic situation. This is a nonlinear counter part of Corollary 1 (for the Stokes flow) and can be also interpreted by asymptotic expansion in the next section. For the Stokes boundary value problem (31) subject to uj@ D 0, one has the P q1 ./ if and only if well-posedness in the class u 2 H  3  n D < q < 3 .D n: space dimension/; n1 2

(64)

320

T. Hishida

see Borchers and Miyakawa [5], Galdi and Simader [34], and Kozono and Sohr [51, 52]. To be precise, the condition q > 3=2 is necessary for solvability and it is consistent with Corollary 1, while for uniqueness one needs q < 3; in fact, the proof P 1 ./ because the constant u1 cannot of Corollary 2 does not work when u 2 H 3 be excluded in (37). Kozono and Yamazaki [56] clarified that all things for both Stokes and Navier-Stokes systems work well if replacing L3 ./ (resp. L3=2 ./) by L3;1 ./ (resp. L3=2;1 ./) for u (resp. ru). When n  4, the Lq -theory is enough to construct (small) Navier-Stokes flow u 2 Ln ./ with ru 2 Ln=2 ./ because n < n2 < n in this case. n1 The existence theorem due to [56] now reads as follows. P 1 ./ with Theorem 2. There is a constant  > 0 such that for every f 2 H 3=2;1 kf kHP 1 ./ < , problem (62) and (63) admits a unique solution 3=2;1

1 P 3=2;1 ./ ,! L3;1 ./; u2H

p 2 L3=2;1 ./;

kfru; pgk3=2;1 C kuk3;1  C kf kHP 1

; 3=2;1 ./

(65)

in the sense that hru; r'i  hp; div 'i  hu ˝ u; r'i D hf; 'i P 1 ./), where h; i stands for holds for all ' 2 C01 ./ (and, therefore, all ' 2 H 3;1 duality pairings. Remark 4. It is an open question whether the small solution fu; pg constructed 3=2;1 P1 in Theorem 2 is unique in the class H ./ without assuming 3=2;1 ./  L smallness, in other words, whether fu; pg coincides with other large solutions fv; qg in this class. The difficulty seems to stem from unremovable singularity like 1=jx  x0 j of the velocity being in L3;1 ./. If such a singular behavior is ruled out for large solutions v by assuming additionally v 2 L3 ./ C L1 ./, then the answer is affirmative (provided kuk3;1 is small enough that is accomplished by (65) when kf kHP 1 ./ is still smaller). This interesting uniqueness criterion was proved 3=2;1 by Nakatsuka [67] (see also [66]). Once the following linear theory is established, it is straightforward by using a simple contraction argument with the aid of (13) to show Theorem 2 (whose proof may be omitted). P 1 ./, problem (31) subject to (63) admits a Theorem 3. For every f 2 H 3=2;1 unique solution 1 P 3=2;1 u2H ./ ,! L3;1 ./;

p 2 L3=2;1 ./;

kfru; pgk3=2;1 C kuk3;1  C kf kHP 1

3=2;1 ./

;

(66)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

321

in the sense that hru; r'i  hp; div 'i D hf; 'i P 1 ./), where h; i stands for holds for all ' 2 C01 ./ (and, therefore, all ' 2 H 3;1 duality pairings. 1 1 P q;r P q;r The well-posedness in the class H ./  Lq;r ./ for every f 2 H ./ was established first by Konozo and Yamazaki [56] when fq; rg satisfies

fq; rg D f3=2; 1gI

fq; rg 2 .3=2; 3/  Œ1; 1I

fq; rg D f3; 1g;

which is a generalization of (64) (case q D r). Indeed Theorem 3 is just one of those cases, but it is the most important case to solve the nonlinear problem. Later on, Shibata and Yamazaki [71] proved the well-posedness not only in the class above but in the sum of function spaces 1 1 P q;r P 3=2;1 u2H ./ C H ./;

p 2 Lq;r ./ C L3=2;1 ./

even for the other cases fq; rg 2 .1; 3=2/  Œ1; 1I

q D 3=2 and r 2 Œ1; 1/:

This result suggests that ru and p do not decay faster than jxj2 in general. In [71] they discussed the Oseen system (6) as well as the Stokes system to study the relation between solutions to (1) with  ¤ 0 and  D 0 (i.e., the behavior for the limit  ! 0). The well-posedness in the class above for (51) was proved by Farwig and Hishida [17] when the obstacle is purely rotating. It was generalized by Heck, Kim, and Kozono [37] when taking both translation and rotation of the obstacle into account. As a result, one has Theorem 2 for the Navier-Stokes boundary value problem (1) and (2) even if ! ¤ 0 provided the data ( C !  x in (2) as well as f ) are small enough; in fact, the case   ! D 0 is reduced to [17], while the other case   ! ¤ 0 is reduced to [37] by using the Mozzi-Chasles transform ([32], [28, Chapter VIII]). The pointwise estimate like ju.x/j  C jxj1 for (1) and (2) with ! ¤ 0 was successfully deduced by Galdi [27] and Galdi and Silvestre [32]. For the proof of Theorem 3, a sort of duality argument was employed in [56], but this way is not taken here; instead, a parametrix is constructed as in [71]. The latter method was also adopted in [17] and [37] since the argument of [56] does not seem to work because of lack of homogeneity of the equation with  C !  x ¤ 0. Also, one cannot use any continuity argument since C01 ./ is not dense in L3=2;1 ./. P 1 ./, one intends to construct directly a solution with the use Given f 2 H 3=2;1 of solutions in the whole space (Lemma 3) and in a bounded domain (Lemma 6 below). Let D be a bounded domain in R3 with smooth boundary @D, and consider the boundary value problem

322

T. Hishida

 u C rp D f;

div u D 0

uD0

in D;

on @D:

(67) (68)

The following lemma is due to Cattabriga [10], Solonnikov [73], Kozono and Sohr [51], and Kozono and Yamazaki [56]. In (69) below, the Poincaré inequality (26) is involved. 1 P q;r .D/, problem (67) Lemma 6. Let 1 < q < 1 and 1  r  1. For every f 2 H and (68) admits a solution 1 P q;r .D/; u2H

p 2 Lq;r .D/; (69)

kfru; u; p  pgkq;r;D  C kf kHP 1 ; q;r.D/

with p WD

1 jDj

R D

p dx, in the sense that hru; r'i  hp; div 'i D hf; 'i

holds for all ' 2 C01 .D/, where h; i stands for duality pairings. The solution is unique up to an additive constant for p. One is in a position to show Theorem 3. Proof of Theorem 3. Since the uniqueness is already known by Corollary 2, one will show the existence part. Fix R > 0 so large that R3 n   BR5 . Take functions

; 1 2 C 1 .R3 I Œ0; 1/ satisfying 

.x/ D

1; jxj  R  3; 0; jxj  R  2;



1 .x/ D

0; jxj  R  5; 1; jxj  R  4;

P 1 ./, it is easily and set A D fx 2 R3 I R  4 < jxj < R  1g. Given f 2 H 3=2;1 seen that 1 P 3=2;1 f 2H .R /; 1 P 3=2;1

1 f 2 H .R3 /;

kf kHP 1

3=2;1 .R /

k 1 f kHP 1

 kf kHP 1

3 3=2;1 .R /

3=2;1 ./

; (70)

 C kf kHP 1

; 3=2;1 ./

for the latter of which (20)2 is used. Consider (67) and (68) with f in the bounded domain D D R , and let fu0 ; p0 g be the solution obtained in Lemma 6 subject to R p dx D 0. Consider also (33) with f replaced by 1 f , and let fu1 ; p1 g be the 0 R solution obtained in Lemma 3. Set ‰f WD .1  /u1 C u0 C BŒ.u1  u0 /  r ; …f WD .1  /p1 C p0 ;

(71)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

323

where B is the Bogovskii operator (see Lemma 1) in the bounded domain A. Since R .u u 1 0 /r dx D 0, one has div ‰f D 0. Then it follows from (34), (35), (69), A and (21), Lemma 1, and (70) that 1 P 3=2;1 .‰f; …f / 2 H ./  L3=2;1 ./;

(72)

kfr‰f; …f gk3=2;1 C k‰f k3;1  C kf kHP 1

3=2;1 ./

and that .‰f; …f / is a solution to  ‰f C r…f D f C Rf;

div ‰f D 0;

‰f j@ D 0

(73)

where Rf D 2r  r.u1  u0 / C . /.u1  u0 /  BŒ.u1  u0 /  r   .r /.p1  p0 /; P 1 .R / and satisfies which is in L3=2;1 .R / ,! H 3=2;1 kRf k3=2;1;R  C kf kHP 1

3=2;1 ./

For every

:

(74)

2 C01 ./, one finds jhRf; ij  kRf k3=2;1;R k k3;1;R

which combined with k k3;1;R  C k k1  C kr k3;1 (see (23)) implies that P 1 ./ with Rf 2 H 3=2;1 kRf kHP 1

3=2;1 ./

 C kRf k3=2;1;R :

Actually, one has even kRf kHP 1

3=2;1 ./

 C kRf kHP 1

3=2;1 .R /

:

(75)

In fact, with the use of a fixed function ' 2 C01 .R / with '.x/ D 1 .x 2 A/, one observes jhRf; ij D jh'Rf; ij  kRf kHP 1

3=2;1 .R /

 C kRf kHP 1

3=2;1 .R /

k' kHP 1

3;1 .R /

kr k3;1

for every 2 C01 ./, owing to (23) as well as Rf D 0 outside A. This implies (75).

324

T. Hishida

P 1 ./ ! H P 1 ./ is a compact operator. Now it turns out that R W H 3=2;1 3=2;1 P 1 ./, and then by (74) In fact, suppose ffj g is a bounded sequence in H 3=2;1 the sequence fRfj g is bounded in L3=2;1 .R / and, therefore, converges in P 1 .R / along a subsequence on account of the compact embedding (25). Then H 3=2;1 P 1 ./ by virtue of (75). it is also convergent in H 3=2;1 P 1 ./ fulfills .1 C One next shows that 1 C R is injective. Suppose f 2 H 3=2;1 P 1 ./. Since f D Rf 2 L3=2;1 .R / which vanishes outside R/f D 0 in H 3=2;1 A, one has f D 0 in  n A. It thus suffices to show that f D 0 in A. In view of (72) and (73), it follows from Corollary 2 that f‰f; …f g D f0; 0g. Hence, by (71) one observes fu1 ; p1 g D f0; 0g;

jxj  R  1I

fu0 ; p0 g D f0; 0g;

jxj  R  4

which shows that both fu1 ; p1 g and fu0 ; p0 g can be regarded as solutions to v C r D f;

div v D 0;

in BR I

vj@BR D 0

3=2;1 P1 and belong to H .BR /. It follows from uniqueness assertion of 3=2;1 .BR /  L Lemma 6 that u1 D u0 and that p1 D p0 C c for some constant c. One goes back to R (71) to see that 0 D …f D .1  /.p0 C c/ C p0 ; however, the side condition R p0 dx D 0 yields c D 0, so that p1 D p0 . After all, one finds fu1 ; p1 g D f0; 0g, yielding f D 0 in A. By the Fredholm alternative, 1 CR is bijective, and, therefore, the pair

u D ‰.1 C R/1 f;

p D ….1 C R/1 f

provides the desired solution, which enjoys (66) on account of (72). The proof is complete. u t

5

Asymptotic Structure of the Navier-Stokes Flow

This section is devoted to a precise look at the profile of solutions for jxj ! 1 obtained in Theorem 2. In order to make essential points clear, it would be better to avoid things caused by less local regularity of solutions. In what follows, let us consider smooth solutions which are of class fu; pg 2 L3;1 ./  L3=2;1 ./. If the solution enjoyed slightly faster decay property than described in Theorem 2, such as uru D O.jxj˛ / with ˛ > 3 or u˝u 2 Ls ./ with s  3=2, then one could regard the nonlinear term as the external force and use Theorem 1 to see that the leading profile would be still the Stokes fundamental solution. But that is not the case here. As mentioned in sect. 1, the balance between the linear part and nonlinearity implies that the leading term of asymptotic expansion would be a self-similar solution.

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

325

Given b 2 R3 n f0g, Landau [60] (see also [9, 75]) found a nontrivial exact solution to (8), which satisfies axial symmetry about Rb as well as homogeneity (9). Set x D jxj ;  D .1 ; 2 ; 3 /T 2 S2 (unit sphere). When b is parallel to e3 , the Landau solution is of the form U .x/ D



c3  1 1 2  C e 3 ; jxj .c  3 /2 c  3

(76)

4 .c3  1/ P .x/ D jxj2 .c  3 /2 with parameter c 2 .1; 1/ [ .1; 1/, and it satisfies U C rP C U  rU D ke3 ı;

div U D 0

in D0 .R3 /

(see (11)), where k is given by k D k.c/ D

  8c cC1 2 2 2 C 6c :  3c.c  1/ log 3.c 2  1/ c1

(77)

This calculation was done by Cannone and Karch [9, Proposition 2.1] (see also Batchelor [2, p.209]). The function k./ is monotonically decreasing on each of intervals .1; 1/ and .1; 1/ and fulfills k.c/ ! 0

.jcj ! 1/I

k.c/ ! 1

.c ! 1/I

k.c/ ! 1

.c ! 1/:

Hence, for every b 2 R3 nf0g parallel to e3 , there is a unique c 2 .1; 1/[.1; 1/ such that k.c/e3 D b. Since the Navier-Stokes system (8) is rotationally invariant, the Landau solution fUb ; Pb g for general b 2 R3 n f0g is given by rotation of (76). b D e3 . Then one finds Let O 2 R33 be an orthogonal matrix that fulfills O jbj

2 1 b c.O /3  1 C Ub .x/ D ; jxj fc  .O /3 g2 c  .O /3 jbj 4 fc.O /3  1g Pb .x/ D jxj2 fc  .O /3 g2

(78)

for x D jxj ;  2 S2 . Since kUb k1;S2 C kPb k1;S2 D O.jcj1 / for jcj ! 1 or, equivalently, jbj ! 0, one observes kUb k3;1;R3 C kPb k3=2;1;R3 ! 0

.b ! 0/:

(79)

326

T. Hishida

When b D 0, one may understand fU0 ; P0 g D f0; 0g. As proved by Šverák [74], for each b 2 R3 the Landau solution (78) is the only solution to (11) which is smooth in R3 n f0g and possesses the homogeneity (9) (however, without assuming axisymmetry), and the family of Landau solutions covers all of possible self-similar solutions to (8). This fact as well as the Eq. (11) itself is essential in the proof of Theorem 4 below, while the exact form (78) is not really needed except for (79). Given Navier-Stokes flow u 2 L3;1 ./, the aim is to clarify how a specific Landau solution is singled out from the set S (see (10)). One also takes care of the external force, to which less attention has been paid in previous literature except [48]. Concerning that, the situation is the same as in Theorem 1 for the Stokes P 1 ./ yields the balance between the Landau flow, that is, the class f 2 H 3=2;1 solution and the error term. Thus, one needs slightly more decay property of f . In this presentation the class of the external force is a bit larger than the one in [48], for pointwise decay of F is not assumed below. The main result reads Theorem 4. Let f D f0 C div F with F 2 Ls;1 ./ \ L3=2;1 ./ \ C 1 ./

(80)

for some s 2 .1; 3=2/. Suppose f0 2 C 1 ./ satisfies (36) for some ˛ > 3. Let fu; pg be a smooth solution of class u 2 L3;1 ./;

p 2 L3=2;1 ./;

3=2;1

ru 2 Lloc

./

(81)

to (62). Set Z

Z ŒT .u; p/  u ˝ u C F   d  C

M D @

f0 .y/ dy

(82)



and q D max f3=.˛  1/; sg 2 .1; 3=2/;

q D max f3=.˛  2/; s g 2 .3=2; 3/;

where s 2 .3=2; 3/ is defined by 1=s D 1=s  1=3 (note that 1=q D 1=q  1=3). There is a constant  > 0 such that if kuk3;1 C jM j <  ;

(83)

then uUM 2 Lr ./;

frurUM ; p PM g 2 Lr ./;

8r 2 .q; 3=2/;

(84)

where fUM ; PM g 2 S denotes the Landau solution with label given by (82), see (10), and r 2 .q ; 3/ is defined by 1=r D 1=r  1=3.

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

327

Even if F satisfies F 2 L3=2 ./ \ C 1 ./

(85)

in place of (80), the conclusion above still holds true, in which (84) is replaced by u  UM 2 L3;3=2 . n B3R /;

fru  rUM ; p  PM g 2 L3=2 . n B3R /;

(86)

where R > 0 is taken large enough. Proof. Since T .u; p/  u ˝ u  F 2 Ltloc ./ for every t 2 .1; 3=2/, the boundary integral of (82) makes sense by the same reasoning as in Theorem 1. Set ˇ D R   u d . One assumes 0 2 int .R3 n / without loss and uses the flux carrier @ 2

z.x/ given by (39), which fulfills not only (40) but also z  rz D r jzj2 . So the pair uQ D u  z;

pQ D p 

jzj2 2

belongs to the class (81) and obeys Qu C r pQ C uQ  r uQ D f  u  rz  z  ru;

div uQ D 0

in 

as well as vanishing flux condition (41). Fix R0 > 0 such that R3 n   BR0 . Let R 2 ŒR0 ; 1/ be the parameter to be determined later. One takes v D .1 

/Qu C BŒQu  r ;

D .1 

/p; Q

(87)

by using the cutoff function (42) together with the Bogovskii operator B (Lemma 1) in the domain AR (see (43)). One then finds  v C r C v  rv D h;

div v D 0

in R3

(88)

with h WD g C .1 

/f0 C div f.1 

/.F  z ˝ u  u ˝ z/g;

g 2 C01 .AR /;

where the exact form of g is not needed as in the proof of Theorem 1. One is going to show that Z fg C .1  R3

/f0 g.y/ dy D M;

(89)

328

T. Hishida

(see (82)) and that kvk3;1;R3  C kuk3;1 C

C jˇj : R

(90)

Let  > 3R. By taking 

Z jyjD

 Z y ˇ 2 jzj2 z˝z d D y d D 0 2  32 2 5 jyjD

as well as (47) into account, one finds Z fg C .1 

/f0 g.y/ dy

B

Z D

div fT .v; /  v ˝ v C .1 

/.F  z ˝ u  u ˝ z/g dy

B

Z ŒT .Qu; p/ Q  uQ ˝ uQ C F  z ˝ u  u ˝ z

D jyjD

y d 

Z

y d  jyjD Z Z ŒT .u; p/  u ˝ u C F   d  C f0 .y/ dy: D

D

ŒT .u; p/  u ˝ u C F 

@



Letting  ! 1 leads to (89). To show (90), set vu WD .1  /u C BŒu  r  and vz WD .1  /z C BŒz  r . Since the map u 7! vu is bounded from Lq ./ to Lq .R3 / for every q 2 .1; 1/, the real interpolation implies that kvu k3;1;R3  C kuk3;1 :

(91)

Fix 2 .3; 1/ arbitrarily. It then follows from (42), the Gagliardo-Nirenberg and Poincaré inequalities together with dilation invariance of the estimate of the Bogovskii operator (due to Borchers and Sohr [8], see Lemma 1) that 13=

3=

kBŒz  r k1;AR  C kBŒz  r k ;R3 krBŒz  r k ;R3  CR13= krBŒz  r k ;AR  CR13= kz  r k ;AR  C kzk1;AR D

C jˇj : R2

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

329

Thus, jxjjvz .x/j  C jˇj=R for all x 2 R3 and thereby kvz k3;1;R3 

C jˇj ; R

which together with (91) concludes (90). Let fU; P g D fUM ; PM g be the Landau solution whose label is given by (82). To regularize fU; P g around x D 0, one may follow the same cutoff procedure as in (87): V D .1  One then observes

R AR

/U C BŒU  r ;

U r

Z

Z div . U / dx D

jxjDR

AR

‚ D .1 

/P:

dx D 0 because

x  U d D R

Z jxjD"

x  U d  D O."/ "

." ! 0/:

The same reasoning as in (91) implies that kV k3;1;R3  C kU k3;1;R3 :

(92)

The pair fV; ‚g obeys  V C r‚ C V  rV D H;

div V D 0

in R3

(93)

for some function H 2 C01 .AR / with Z H .y/ dy D M:

(94)

R3

In fact, by using a test function 2 C 1 .R3 / satisfying .x/ D 1 .jxj  3R/ and

.x/ D 0 .jxj  4R/, one sees from (11) with b D M that Z

Z H .y/ dy D 

AR

Z

.T .U; P /  U ˝ U / jyjD3R

y d 3R

.T .U; P /  U ˝ U /.r / dy

D 3R 0, where if ˛ 2 .3; 4/; t D 3=.˛  1/; t D 3=.˛  2/ t 2 .1; 3=2/ is arbitrary; 1=t D 1=t  1=3 if ˛  4: Since w1 ; rw1 ; #1 2 L .BL / for every 2 .1; 1/, which follows from g C .1  /f0 C H 2 L .R3 / for such , we obtain w1 2 Lt ;1 .R3 / \ L3;1 .R3 /;

frw1 ; #1 g 2 Lt;1 .R3 / \ L3=2;1 .R3 /:

(102)

p 1  jzj 2 L3=2;1 .R3 / \ L1 .R3 /, which combined with p One observes 1  juj 2 L3;1 .R3 / and (13) imply that .1 

/.z ˝ u C u ˝ z/ 2 L ;1 .R3 /;

8 2 .1; 3:

(103)

It thus follows from (80) that ˚ div .1 

 1 1 P 3=2;1 P s;1 .R3 / \ H .R3 /: /.F  z ˝ u  u ˝ z/ 2 H

(104)

Let 1 1 P s;1 P 3=2;1 .R3 / \ H .R3 / ,! Ls ;1 .R3 / \ L3;1 .R3 /; w2 2 H

#2 2 Ls;1 .R3 / \ L3=2;1 .R3 /;

(105)

be the solution to (33) with the external force (104) obtained in Lemma 3. Then one finds w0 WD w1 C w2 2 Lq ;1 .R3 / \ L3;1 .R3 /; rw0 2 Lq;1 .R3 / \ L3=2;1 .R3 /;

(106)

#0 WD #1 C #2 2 Lq;1 .R3 / \ L3=2;1 .R3 /; 3 ; sg, where 1=q D 1=q  1=3. with q D maxf ˛1 Given w 2 Lq ;1 .R3 / \ L3;1 .R3 / (see (99)), the velocity part of the unique solution to (33) with the external force 1 1 P 3=2;1 P q;1 .R3 / \ H .R3 /  div .v ˝ w C w ˝ V / 2 H

332

T. Hishida

obtained in Lemma 3 is denoted by T w. Then problem (95) is rewritten as w D w0 C T w

(107)

and the right-hand side returns to the class (99) on account of (106). By (35) together with (98) (and the similar one in terms of kwkq ;1;R3 ), the map w 7! w0 C T w is contractive from Lq ;1 .R3 / \ L3;1 .R3 / into itself provided that kvk3;1;R3 C kV k3;1;R3 < 2

(108)

with a suitable small constant 2 D 2 .q/ > 0. One thus gets a fixed point w 2 Lq ;1 .R3 /\L3;1 .R3 / with rw 2 Lq;1 .R3 /\L3=2;1 .R3 /. Since the pressure associated with T w belongs to Lq;1 .R3 / \ L3=2;1 .R3 /, so does the pressure associated with the fixed point because of (106). In view of (90), (92) with U D UM , and (79), the parameter R 2 ŒR0 ; 1/ is first fixed so that jˇj=R is small enough, and then it is possible to take a suitable constant  > 0 such that both (97) and (108) are accomplished under the condition (83). Finally, consider the case when F satisfies (85) in place of (80), then ˚ div .1 

 1 P 3=2 /.F  z ˝ u  u ˝ z/ 2 H .R3 /

on account of (103). Hence, one has 1 P 3=2 .R3 / ,! L3;3=2 .R3 /; w2 2 H

#2 2 L3=2 .R3 /

(see (20)), instead of (105). On the other hand, (101) yields w1 2 L3;3=2 .R3 n BL /;

frw1 ; #1 g 2 L3=2 .R3 n BL /

for some L > 0, which implies w1 2 L3;3=2 .R3 /;

frw1 ; #1 g 2 L3=2 .R3 /

(109)

by the same reasoning as in (102). One thus obtains w0 2 L3;3=2 .R3 /;

frw0 ; #0 g 2 L3=2 .R3 /

(110)

instead of (106). For the proof of (86), it suffices to find a solution w 2 L3;3=2 .R3 /;

frw; #g 2 L3=2 .R3 /

to (95). Given w 2 L3;3=2 .R3 /, one observes 1 P 3=2 .R3 /  div .v ˝ w C w ˝ V / 2 H

(111)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

333

with kdiv .v ˝ w C w ˝ V /kHP 1 .R3 /  C .kvk3;1;R3 C kV k3;1;R3 /kwk3;3=2;R3 3=2

P 1 .R3 / ,! L3;3=2 .R3 /. By by (17). Therefore, the term T w in (107) belongs to H 3=2 virtue of (110), the rest of the proof of existence of a solution of class (111) is the same as above. The proof is complete. t u t 1 Remark 5. R div F 2 L R./ \ Lloc ./ for some t > 1, then the R If, in addition, equality  f .y/ dy D  f0 .y/ dy C @ F  d  is justified, so that (82) is equal to M .0; 0; f / given by (4).

Remark 6. If F is absent (so that f D f0 ) and if ˛  4, the exponent q is chosen arbitrarily in the interval .1; 3=2/ (as close to 1 as one wishes) and the small constant  depends on the choice of q. Finally, let us consider the Navier-Stokes system around a rotating obstacle  u C rp C u  ru  .!  x/  ru C !  u D f;

div u D 0

in ;

(112)

where ! D ae3 with a 2 Rnf0g. As mentioned in [74, section 3], a scaling argument with (7) works for the case ! D 0 to see that, if solutions are asymptotically homogeneous of degree .1/, then their leading terms are the Landau solutions. But (112) is no longer invariant under the transformation (7) unless ! D 0. One thus needs another heuristic observation, which is based on knowledge of the linearized system (51). The point is the asymptotic expansion (58), which yields the leading term (59) of the linearized flow. Let u 2 L3;1 ./ be the solution to the NavierStokes system (112). In view of features of (59), it is reasonable to expect that the leading term, denoted by U , still keeps symmetry about the axis of rotation (i.e., Re3 ) as well as homogeneity of degree .1/ and that the quantity e3  M controls the rate of decay; here, M D M .0; !; f / is given by (4), or Z

Z ŒT .u; p/  u ˝ .u  !  y/  .!  y/ ˝ u C F   d y C

M D @

f0 .y/ dy 

(113) when the external force is of the form f D f0 C div F satisfying the same assumptions as in Theorem 4. One can also expect, as in (60), that the leading term U together with some scalar field P solves  U C rP C U  rU  .!  x/  rU C !  U D .e3  M / e3 ı;

div U D 0 (114)

in D0 .R3 /; however, this is reduced to  U C rP C U  rU D .e3  M / e3 ı;

div U D 0

(115)

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because U satisfies (61) under the symmetry about Re3 . Hence, U is a self-similar solution to (8), and it should be a Landau solution U.e3 M /e3 . This observation can be justified along the same way as in the proof of Theorem 4, in which (94) and (96)2 should be replaced by Z Z H .y/ dy D .e3  M /e3 ; e3  fg C .1  /f0  H g.y/ dy D 0: (116) R3

R3

Then one can use Lemma 5 to obtain (101)1 for w1 , which combined with the result of [20] implies (102)1 =(109)1 for w1 . This was done by Farwig and Hishida [19] for (112) with no external force under the no-slip boundary condition uj@ D !  x. But their result can be extended to the case where the external force satisfies the same conditions as in Theorem 4 without assuming any boundary condition on @ (see [47, section 6] for the flux carrier). Note that the corresponding Landau pressure P.e3 M /e3 is not solely the leading term of the associated pressure unlike Theorem 4 for ! D 0. This is because (116)2 is not sufficient to get faster decay of the pressure, so that (101)2 for #1 is not available. In addition to P.e3 M /e3 , however, one can use (32)2 to take the leading term of #1 given by (100) so that one gets Z P.e3 M /e3 C Q.x/ 

fg C .1  R3

/f0  H g.y/ dy

D P.e3 M /e3 C Q.x/  fM  .e3  M /e3 g; which is the leading term of the pressure, where Q.x/ is the fundamental solution (29). This was found by Farwig, Galdi, and Kyed [16]. In [16] the authors deduced the asymptotic expansion of solutions of the Leray class satisfying the energy inequality, which eventually decay like jxj1 (see [30]), as long as they are small enough.

6

Conclusion

The asymptotic structure at infinity as well as existence of 3D exterior stationary Navier-Stokes flows being in L3;1 (weak-L3 ) is discussed when the obstacle is at rest. The class L3;1 is critical from both of the following points of view (one is essential, while the other would be technical). On the one hand, it is optimal summability of generic flows in the sense that better summability than L3;1 necessarily implies the vanishing total net force, M .0; 0; f / D 0 (see (4)). On the other hand, it is difficult to conclude the stability of flows with worse summability than L3;1 (even if they are small enough) as long as one adopts the mathematical analysis developed until now. The Stokes fundamental solution, which is the leading profile of the Stokes flow, is no longer the leading profile of the Navier-Stokes flow being in L3;1 on account of the balance between the linear part and nonlinearity. The correct leading term

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

335

is the Landau solution whose label is the net force M .0; 0; f / of given NavierStokes flow provided it is small enough. The reason comes essentially from Šverák’s observation on structure of the set that consists of all homogeneous Navier-Stokes flows of degree .1/. One finds a contrast with the case where the obstacle is translating, in which the leading profile is described in terms of the linear part, that is, the Oseen fundamental solution. When the obstacle is rotating with constant angular velocity !, the leading term is still a Landau solution; however, its label ! ! is given by . j!j  M / j!j with M D M .0; !; f /, which follows from a decay structure of the associated fundamental solution. One is able to specify a condition on the external force (which is not necessarily of bounded support) such that the conclusions above hold true. Several open questions about the related issues are in order. The results above (except the case where the obstacle is translating) require smallness of the NavierStokes flow in L3;1 because a perturbation argument is adopted. Asymptotic structure of large solutions of this class is much more involved and remains open. When ! D 0, as pointed out by Šverák [74, section 3], once one establishes the asymptotic expansion with homogeneous leading term of degree .1/ without any smallness, it turns out by a scaling argument that the leading term must be a Landau solution. If the external force f has less decay property (i.e., ˛ is close to 3 and s is close to 3=2, or even F 2 L3=2 ./, in Theorem 4), it is then hopeless to find out the second term after the leading one (a Landau solution) in the asymptotic expansion. For the simple case f D 0, however, one can ask what the second term is. It is probably homogeneous of degree .2/. As compared with the 3D problem, there are many open problems concerning exterior stationary Navier-Stokes flows in 2D (see Galdi [28, Chapter XII] for the details). The most difficult case is that the obstacle is at rest (unless assuming any symmetry), where the linearization method can no longer work because of the Stokes paradox. No one knows the asymptotic structure of the Navier-Stokes flow even if it is small enough; however, a remarkable conjecture based on numerical verification has been recently proposed by Guillod and Wittwer [36]. When the obstacle is translating, the problem is less difficult on account of decay structure of the 2D Oseen fundamental solution, which is the leading profile of the NavierStokes flows without restriction on the magnitude as in 3D (see Smith [72] and Galdi [28, Theorem XII.8.1]). But the stability/instability of such flows is far from clear. The case when the obstacle is rotating in 2D has been much less studied. As for the linearized problem, it was found by Hishida [45] that the oscillation caused by rotation of the obstacle leads to the resolution of the Stokes paradox and that the leading term of the flow at infinity involves the profile x ? =jxj2 whose coefficient is (not the net force but) the torque, where x ? D .x2 ; x1 /> . Very recently, Higaki, Maekawa, and Nakahara [41] showed that asymptotic structure of small NavierStokes flow around a slowly rotating obstacle exhibits the same profile as above. Such a structure has still remained open unless imposing smallness on the angular velocity. Note that the pair

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u.x/ D

cx ? ; jxj2

p.x/ D

c 2 2jxj2

.c 2 R/;

(117)

is a self-similar solution to the Navier-Stokes system (8) in R2 n f0g and that it also satisfies (112) with f D 0 in R2 n f0g since the last two terms in the left-hand side vanish, that is, ax ?  ru C au? D 0 (by following the standard notation in R 2D). By Theorem 2 of Šverák [74, section 5], under the zero flux condition S1  u d  D 0, the homogeneous Navier-Stokes flow of degree .1/ in 2D must be either the circular flow (117) or a particular Jeffery-Hamel flow (whose component tangent to the circle S1 vanishes). When the fluid region is in particular the exterior of a rotating disk, one refers to interesting papers by Hillairet and Wittwer [42] and by Maekawa [62]: the former finds the Navier-Stokes flow which is close to the solution (117) with large jcj and whose leading profile is also given by x ? =jxj2 , and the latter successfully proves the L2 stability of the solution (117) provided jcj is sufficiently small. One can expect that this latter result would hold for small Navier-Stokes flow constructed in [41].

7

Cross-References

 Self-Similar Solutions to the Nonstationary Navier-Stokes Equations  Steady-State Navier-Stokes Flow Around a Moving Body  Time-Periodic Solutions to the Navier-Stokes Equations

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7

Steady-State Navier-Stokes Flow Around a Moving Body Giovanni P. Galdi and Jiˇrí Neustupa

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Early Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Function-Analytic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Finn’s Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Babenko’s Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 A General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Geometric and Functional Properties for Large Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Steady Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Time-Periodic Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Stability and Longtime Behavior of Unsteady Perturbations . . . . . . . . . . . . . . . . . . . . . . 10.1 Spectrum of Operator A,T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 A Semigroup, Generated by the Operator A,T . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Existence and Uniqueness of Solutions of the Initial–Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Attractivity and Asymptotic Stability with Smallness Assumptions on v0 . . . . . . 10.5 Spectral Stability and Related Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

342 344 346 349 351 353 363 365 366 367 367 368 372 375 376 381 386 387 392 395 398 406

G.P. Galdi () Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, USA e-mail: [email protected] J. Neustupa Institute of Mathematics, Czech Academy of Sciences, Praha 1, Czech Republic e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_7

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11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

412 412 413

Abstract

In this chapter we present an updated account of the fundamental mathematical results pertaining the steady-state flow of a Navier-Stokes liquid past a rigid body which is allowed to rotate. Precisely, we shall address questions of existence, uniqueness, regularity, asymptotic structure, generic properties, and (steady and unsteady) bifurcation. Moreover, we will perform a rather complete analysis of the longtime behavior of dynamical perturbation to the above flow, thus inferring, in particular, sufficient conditions for their stability and asymptotic stability.

1

Introduction

The motion of a rigid body in a viscous liquid represents one of the most classical and most studied chapters of applied and theoretical fluid mechanics. Actually, the study of this problem, at different scales, is at the foundation of many branches of applied sciences such as biology, medicine, and car, airplane, and ship manufacturing, to name a few. The dynamics of the liquid associated to these problems is, of course, of the utmost relevance and, already in very elementary cases, can be quite intricate or even, at times, far from being obvious. For example, consider a rigid sphere of radius R, moving by constant translatory motion with speed v0 and entirely immersed in a surrounding liquid, of kinematic viscosity . Then, it is experimentally observed (see [67]) that if Re WD v0 R= . 200, the flow is steady, stable, and axisymmetric. However, if 200 . Re . 270, this flow loses its stability, and another stable, steady, but no longer axisymmetric flow sets in, as evidenced by the loss of rotational symmetry of the wake. It is worth emphasizing the loss of symmetry of the flow, in spite of the symmetry of the data. Moreover, if 270 . Re . 300, the steady flow is unstable, and the liquid regime becomes oscillatory, as shown by the highly organized time-periodic motion of the wake behind the sphere. The remarkable feature of this phenomenon is that the unsteadiness of the flow arises spontaneously, even though the imposed conditions are time independent (constant speed of the body). Another significant example is furnished if now the sphere, instead of moving by a translatory motion, rotates with constant angular velocity, !0 , along one of its diameters. Here, again in view of the symmetry of the data, one would guess that, at least for “small” values of j!0 j (more precisely, of the dimensionless number j!0 j R2 =), the flow of the liquid is steady with streamlines being circles perpendicular to and centered around the axis of rotation. Actually, this is not the case, unless the inertia of the liquid is entirely disregarded. In fact, though the flow is steady, due to inertia, the sphere behaves like a “centrifugal fan,” receiving the liquid near the poles and throwing it away at the equator; see [12, 85]. Already from these brief considerations, one can fairly deduce that a rigorous mathematical study of the motion of a viscous liquid around an obstacle presents a

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plethora of intriguing problems of considerable difficulties, beginning with the very existence of steady-state solutions under general conditions on the data and their uniqueness going through more complicated issues such as analysis of steady and time-periodic bifurcation and longtime behavior of time-dependent perturbations. It is the objective of this chapter to address some of these fundamental problems, as well as point out certain outstanding questions that still await for an answer. In real experiments the liquid occupies, of course, a finite (though “sufficiently large”) spatial region. However, “wall effects” are irrelevant for the occurrence of the basic phenomena of the type described above. Therefore, in order not to spoil their underlying causes, it is customary to formulate the mathematical theory of the motion of a body in a viscous liquid as an exterior problem. This corresponds to the assumption that the liquid fills the entire three-dimensional space outside the body. It should be remarked that this assumption, though simplifying on one hand, on the other hand adds more complication to the mathematical analysis, in that classical and powerful tools valid for bounded flow are no longer available in this case. As it turns out, most of the questions that we shall analyze require, for their answers, a somewhat detailed analysis of the solutions at large distance from the body. From a historical viewpoint, the mathematical analysis of the steady flow of a viscous liquid past a rigid body may be traced back to the pioneering contributions of Stokes [105], Kirchhoff [71], and Thomson (Lord Kelvin) and Tait [106] in the mid- and late 1880s. However, it was only in the 1930s that, thanks to the farreaching and genuinely new ideas introduced by Jean Leray [81], the investigation of the problem received a substantial impulse. Leray’s results, mostly devoted to the existence problem, were further deepened, extended, and completed over the years by a number of fundamental researches due, mostly, to O.A. Ladyzhenskaya, H. Fujita, R. Finn, K.I. Babenko, and J.G. Heywood. It is important to observe that the efforts of all these authors were directed to the study of cases where the body is not allowed to spin. The more general and more complicated situation of a rotating body became the object of a systematic study only at the beginning of the third millennium, with the basic contributions, among others, of R. Farwig, T. Hishida, M. Hieber, Y. Shibata, the authors of the present paper, and their associates. The main goal of this review chapter is to furnish an up-to-date state of the art of the fundamental mathematical properties of steady-state flow of a NavierStokes liquid past a rigid body, which is also allowed to rotate. Thus, existence, uniqueness, regularity, asymptotic structure, generic properties, and (steady and unsteady) bifurcation issues will be addressed. In addition, a rather complete analysis of the longtime behavior of dynamical perturbation to the above solutions will be performed to deduce, in particular, sufficient conditions for their stability and asymptotic stability as well. With the exception of part of the last section, Sect. 10, this study will be focused on the case when the translational velocity, v0 , of the body is not zero and its angular velocity is either zero or else has a nonvanishing component in the direction of v0 . The reason for such a choice lies in the fact that under these assumptions, the mathematical questions listed above have a rather complete answer. On the

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other side, if one relaxes these assumptions, the picture becomes much less clear. The interested reader is referred to [45, §§ X.9 and XI.7] for all main properties known in this case. Furthermore, for the same reason of incompleteness of results, only three-dimensional flow will be considered. An update source of information regarding plane motions can be found, for example, in [45, Chapter XII], [36], and [56]. Finally, other significant investigations are left out of this chapter, such as the motion of the coupled system body–liquid (i.e., when the motion of the body is no longer prescribed, but becomes part of the problem), as well as the very important case when the body is deformable, for which the reader is referred to [38] and [5,44], respectively. The plan of the chapter is as follows. After collecting in Sect. 2 the main notation used throughout, in Sect. 3 it is provided the mathematical formulation of the problem. Section 4 is dedicated to existence questions. There, one begins to recall classical approaches and corresponding results due to Leray, Ladyzhenskaya, and Fujita. Successively, improved findings obtained by the function-analytic method introduced by Galdi are presented, based on the degree for proper Fredholm maps of index 0. Regularity and uniqueness questions of solutions are next addressed in Sects. 5 and 6, respectively. Section 7 is dedicated to the (spatial) asymptotic behavior, beginning by recalling the original results of Finn and Babenko when the body is not spinning to the more recent general contributions of Galdi and Kyed and Deuring and their associates, valid also in the case of a rotating body. Successively, in Sect. 8, one investigates the geometric structure of the solution manifold for data of arbitrary “size.” In particular, it is shown that, generically, the number of solutions corresponding to a given (nonzero) translational velocity and angular velocity is finite and odd. Sect. 9 is devoted to steady and time-periodic (Hopf) bifurcation of steady-state solutions. There, it is provided necessary and sufficient conditions for this type of bifurcation to occur. In the final section, Sect. 10, one analyzes the longtime behavior of time-dependent perturbations to a given steady state, providing, as a special case, sufficient conditions for attractivity and asymptotic stability. These results can be, roughly speaking, grouped in two different categories. The first one is where one assumes that the unperturbed steady state is “small in size.” In the second one, instead, one makes suitable hypothesis on the location in the complex plane of eigenvalues of the relevant linearized operator (spectral stability). In conclusion to this introductory section, it is worth emphasizing that throughout this chapter, there have been highlighted a number of intriguing unsettled questions that still need an answer and represent as many avenues open to the interested mathematician.

2

Notation

The symbols N, Z and R, C stand for the sets of positive and relative integers and the fields of real and complex numbers, respectively. We also put NC WD N\.0; 1/, RC WD R \ .0; 1/.

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Vectors in R3 will be indicated by bold-faced letters. A base in R3 is denoted by fe 1 ; e 2 ; e 3 g  fe i g and the components of a vector v in the given base by v1 , v2 , and v3 . Unless stated otherwise, the Greek letter ˝ will denote a fixed exterior domain of R3 , namely, the complement of the closure, B, of a bounded, open, and simply connected set of R3 . It will be assumed ˝ of class C 2 , and the origin O of the coordinate system fO; e i g is taken in the interior of B. Also, d is the diameter of 1 B, so that, setting BR WD fx 2 R3 W .x12 Cx22 Cx32 / 2 < Rg, R > 0, one has B  Bd . For R  d , the following notation will be adopted ˝R D ˝ \ BR ; ˝ R D ˝  ˝R ; where the bar denotes closure. One puts ut WD @u=@t, @1 u WD @u=@x1 , and, for ˛ a multi-index, one denotes by D ˛ the usual differential operator of order j˛j. For j˛j D 2 one shall simply write D 2 . q Given an open and connected set A  R3 ; Lq .A/, Lloc .A/, 1  q  1; m;q 0;q W m;q .A/; W0 .A/ .W 0;q  W0  Lq ), W m1=q;q .@A/, m 2 NC [ f0g, stand for the usual Lebesgue, Sobolev, and trace space classes, respectively, of real or complex functions. (The same font style will be used to denote scalar, vector, and tensor function spaces.) Norms in Lq .A/, W m;q .A/, and W m1=q;q .@A/ are indicated by k:kq;A , k:km;q;A , and k:km1=q;q.@A/ . The scalar product of functions u; v 2 L2 .A/ will be denoted by .u; v/A . In the above notation, the subscript A will be omitted, unless confusion arises. As customary, for q 2 Œ1; 1 one lets q 0 D q=.q  1/ be its Hölder conjugate. By D m;q .˝/, 1 < q < 1, m 2 NC , one denotes the space of (equivalence classes of) functions u such that jujm;q WD

X Z j˛jDm

jD ˛ ujq

 q1

< 1;

˝

m;q

and by D0 .˝/ the completion of C01 .˝/ in the norm j  jm;q . Moreover, setting D.˝/ WD fu 2 C01 .˝/ W div u D 0g; D01;2 .˝/ is the completion of D.˝/ in the norm j  j1;2 . By D01;2 .˝/ [D01;2 .˝/], one denotes the normed dual space of D01;2 .˝/ [D01;2 .˝/] and by h; i [Œ; ] the associated duality pairing. By Hq .˝/ it is indicated the completion of D.˝/ in the norm Lq .˝/, and one simply writes H .˝/ for q D 2. Further, P is the (Helmholtz-Weyl) projection from Lq .˝/ onto Hq .˝/. Notice that, since ˝ is a sufficiently smooth exterior domain, P is independent of q. If M is a map between two spaces, by D .M /, N .M /, and R .M /, one denotes its domain, null space, and range, respectively, and by Sp .M / its spectrum.

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In the following, B is a real Banach space with associated norm k  kB . The complexification of B is denoted by BC WD B C i B. Likewise, the complexification of a map M between two Banach spaces will be indicated by MC . For q 2 Œ1; 1, Lq .a; bI B/ is the space of functions u W .a; b/ 2 R ! B such that ! q1 Z b

a

q

ku.t /kB dt

< 1; if q 2 Œ1; 1/ I ess supku.t /kB < 1; if q D 1: t2.a;b/

Given a function u 2 L1 .; I B/, u is its average over Œ; , namely, Z  1 u WD u.t / dt: 2  Furthermore, one says that u is 2-periodic, if u.t C 2/ D u.t /, for a.a. t 2 R. Set n W22;0 .˝/ WD u 2 L2 .; I W 2;2 .˝/ \ Do01;2 .˝// and ut 2 L2 .; I H .˝// W u is 2 -periodic with u D 0 with associated norm Z kukW2

2;0



WD 

kut .t /k22 dt

1=2

Z



C 

ku.t /k22;2 dt

1=2 :

One also defines n o H2;0 .˝/ WD u 2 L2 .; I H .˝// W u is 2-periodic with u D 0 : Finally, C , C0 , C1 , etc., denote positive constants, whose particular value is unessential to the context. When one wishes to emphasize the dependence of C on some parameter , it will be written C ./.

3

Formulation of the Problem

Suppose one has a rigid body, B, moving by prescribed motion in an otherwise quiescent viscous liquid, L, filling the entire space outside B. Mathematically, B will be taken as the closure of a simply connected bounded domain of class C 2 . For the sake of generality, a given velocity distribution is allowed on @B, due, for example, to a tangential motion of the boundary wall or to an outflow/inflow mechanism, as well as it is assumed that on L is acting a given body force. (The presence in the model of a body force other than gravity – whose contribution can be always incorporated in the pressure term – could be questionable on physical grounds. However, from the mathematical point of view, it might be useful in consideration of extending the results to more general liquid models, where now the “body force” would represent the contribution to the linear momentum equation

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of other appropriate fields.) In order to study the motion of L under these circumstances, it is appropriate to write its governing equations in a body-fixed frame, S, so that the region occupied by L becomes time independent. One thus gets vt C .v  V /  rv C ! v D v  rp C f

)

div v D 0

in ˝ .0; 1/:

(1)

(For the derivation of these equations, we refer to [38, Section 1, eq. (1.15)].) In these equations, v; p are absolute velocity and pressure fields of L, respectively,  and  its (constant) density and kinematic viscosity, and f is the body force acting on L. Moreover, V WD  C ! x; with  and !, in the order, velocity of the center of mass and angular velocity of B in S. Finally, ˝ WD R3 nB is the time-independent region occupied by L that will be assumed of class C 2 . (Several peripheral results continue to hold with less or no regularity at all. This will be emphasized in the assumptions occasionally.) The system (1) is endowed with the following boundary condition v D v C V at @˝ .0; 1/;

(2)

with v a prescribed field, expressing the adherence of the liquid at the boundary walls of the body, and asymptotic conditions lim v.x; t / D 0; t 2 .0; 1/;

jxj!1

(3)

representative of the property that the liquid is quiescent at large spatial distances from the body. Throughout this paper it shall be assumed that the vectors  and ! do not depend on time. This assumption imposes certain limitations on the type of motion that B can execute with respect to a fixed inertial frame. Precisely [45], the center of mass of B must move with constant speed along a circular helix whose axis is parallel to !. The helix will degenerate into a circle when   ! D 0, in which case the motion of the body reduces to a constant rotation. Without loss of generality, we set ! D ! e 1 , !  0, and  D v0 e with e a unit vector. As indicated in the introductory section, one is only interested in the case when the motion of the body does not reduce to a uniform rotation. For this reason, unless otherwise stated, it will be assumed v0 ¤ 0 and e  e 1 ¤ 0:

(4)

By shifting the origin of the coordinate system S suitably (Mozzi-Chasles transformation) and scaling velocity and length by v0 e  e 1 and d , respectively,

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one can then show that (1) can be put in the following form in the shifted frame S 0 (see [45, pp. 496–497]): vt C v C  @1 v C T .e 1 x rv  e 1 v/ D v  rv C rp C f div v D 0 v D v C V at @˝ .0; 1/ I

9 > = > ;

in ˝ .0; 1/

lim v.x; t / D 0; all t 2 .0; 1/;

jxj!1

(5) ! d2 where T WD ,  8 v d ˆ < 0 e  e 1 ; if ! ¤ 0;   WD ˆ : v0 d ; if ! D 0 .e  e 1 /; 

(6)

and V WD e 1 C

T e 1 x: 

(7)

Of course, all fields entering the equations in (5) are regarded as nondimensional. Observe also that, in the rescaled length variables, the diameter of B becomes 1. In order to simplify the presentation, the origin of the coordinate system S 0 will be supposed to lie in the interior of B. Finally, we notice that, in view of (4), it follows  ¤ 0. Since all results presented in this chapter are independent of whether  ? 0, it will be assumed throughout  > 0. Of particular relevance to this chapter are time-independent solutions (steadystate flow) of problems (5), (6), and (7), which may occur only when f and v are also time independent. From (5), one thus infers that these solutions must satisfy the following boundary value problem: v C  @1 v C T .e 1 x  rv  e 1 v/ D v  rv C rp C f div v D 0

) in ˝ (8)

v D v C V at @˝ I

lim v.x/ D 0:

jxj!1

The primary objective of this chapter is to provide an updated review of some fundamental properties of solutions to (6), (7), and (8). The latter include existence, uniqueness, regularity, asymptotic structure, generic properties, and steady and unsteady bifurcation issues. Moreover, a rather complete analysis of the longtime behavior of dynamical perturbation to these solutions will be performed that will lead, in particular, to a number of stability and asymptotic stability results, under various assumptions.

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349

Existence

The starting point is the following general definition of weak (or generalized) solution for problems (6), (7), and (8) [79]. Definition 1. Let f 2 D01;2 .˝/ and v 2 W 1=2;2 .@˝/. A vector field v W ˝ ! R3 is a weak solution to problems (8)–(7) if the following conditions hold: (a) v 2 D 1;2 .˝/ with div v D 0. (b) v satisfies the equation .rv; r'/C  .@1 v; '/ C T .e 1 x  rv  e 1 v; '/ C  .v  r'; v/ D hf ; 'i; for all ' 2 D.˝/.

(9)

(c) v D v CZV at @˝ in the trace sense.

(d) lim R2 R!1

jvj D 0. @BR

(Formally, (9) is obtained by taking the scalar product of both sides of (8)1 by ' and integrating by parts over ˝. Since D 1;2 .˝/  W 1;2 .˝R /, R > 1, condition (c) is meaningful.) Remark 1. If f 2 W01;2 .˝ 0 /, for all bounded ˝ 0 with ˝ 0  ˝, then to every weak solution, one can associate a suitable corresponding pressure field. More precisely, there exists p 2 L2loc .˝/ such that .rv; r /C  .@1 v; / C T .e 1 x  rv  e 1 v; / C  .v  r ; v/ D .p; div / C Œf ; ; for all 2 C01 .˝/, where Œ;  stands for the duality pairing D01;2 $ D01;2 . Notice that this equation is formally obtained by dot-multiplying both sides of (8)1 by and integrating by parts over ˝. The proof of this property, based on the representation of elements of D01;2 vanishing on D01;2 , is given in [45, Lemma XI.1.1]. The next step is the construction of a suitable extension, U , of the boundary data. The crucial property of such extension is condition (10) given below. In fact, as will become clear later on, this allows one to obtain the fundamental a priori estimate for the existence result. Now, the validity of (10) is related to the magnitude of the flux through the boundary @˝, ˚ , of the field v : Z ˚ WD

v  n; @˝

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with n unit outer normal to @˝. For simplicity, it will be assumed ˚ D 0, even though all main results continue to hold also when j˚ j is sufficiently “small.” The reader is referred to Open Problem 4.2 for further considerations about this issue. The existence of the appropriate extension of the boundary data is provided by the following result whose proof can be found in [45, Lemma X.4.1] Lemma 1. Let v 2 W

1=2;2

Z .@˝/;

v  n D 0: @˝

Then, for any > 0, there exists U D U . ; v ; V; ˝/ W ˝ ! R3 with bounded support such that: (i) U 2 W 1;2 .˝/. ii) U D v C V at @˝. (iii) div U D 0 in ˝. Furthermore, for all u 2 D01;2 .˝/, it holds that j.u  rU ; u/j  juj21;2 :

(10)

Finally, if kv k1=2;2.@˝/  M; for some M > 0; then kU k1;2  C1 kv C Vk1=2;2.@˝/

(11)

where C1 D C1 . ; M; ˝/. Remark 2. In view of the above result, it easily follows that the existence of a weak solution is secured if there is u 2 D01;2 .˝/ satisfying  .ru; r'/ C  .@1 u; '/ C T .e 1 x  ru  e 1 u; '/ C  .u  r'; u/ C Œ.U  r'; u/  .u  rU ; '/  .rU ; r'/ C  .@1 U  U  rU ; '/ C T .e 1 x  rU  e 1 U ; '/ D hf ; 'i; for all ' 2 D.˝/. (12) In fact, setting v WD u C U ; one gets at once that conditions (a)–(c) of Definition 1 are met. Moreover, since, by Sobolev theorem D01;2 .˝/  L6 .˝/ (e.g., [45, Theorem II.7.5]), from [45, Lemma II.6.3], it follows lim

R!1

1 3

R2

Z @BR

juj D 0; for all u 2 D01;2 .˝/;

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so that also requirement (d) is met, even with a better order of decay. In view of all the above, we may equally refer to both v and u as “weak solution.”

4.1

Early Contributions

Classical approaches and results to the existence of weak solutions due, basically, to Jean Leray [81], Olga A. Ladyzhenskaya [79], and Hiroshi Fujita [29] will be now presented and summarized. Besides their historical relevance and intrinsic interest, these results will also provide a further motivation for the entirely distinct approach – recently introduced in [41, 47]–that will be described in Sect. 5.

4.1.1 Leray’s Contribution In his famous pioneering work on the steady-state Navier-Stokes equations [81, Chapitres II & III], Leray shows that for any sufficiently regular f and v ; with ˚ D 0; there is at least one corresponding solution .v; p/ to (8)1;2;3 –(7), which, in addition, satisfies v 2 D 1;2 .˝/. (As a matter of fact, Leray requires f  0; [81, §3 at p. 32]. However, for his method to go through, the weaker assumption of a “smooth” f would suffice.) It is just in this weak sense that Leray interprets the condition at infinity (8)4 . (As noticed earlier on, this condition can be expressed in a sharper, though still weak, way; see Remark 2.) Leray’s construction, basically, consists in solving the original problem (8)1;2;3 –(7) on each elements of an increasing sequence of bounded domains f˝k gk>1 with ˝ D [1 kD1 ˝k ; under the further condition v D 0 on the “fictitious” boundary @Bk (“invading domains” technique). In turn, on every ˝k ; a sufficiently smooth solution, vk ; to the system (8) is determined by combining Leray-Schauder degree theory with a uniform bound on the Dirichlet integral jvk j21;2 . (It should be observed that, even though the demonstration provided by Leray is presented in the language of LeraySchauder fixed-point theorem, such a result was not yet available at that time; see [83, 84].) The latter is crucial, in that it allows Leray to select a subsequence that, uniformly on compact sets, converges to a solution of the original problem, meant in a suitable integral sense. It must be emphasized that in order to obtain the above bound, the property (10) of the extension is crucial. (Notice that a bound on vk can also be obtained by an alternative method, based on a contradiction argument; see [81, Chapitre II, §III]. Even though the latter is more general than the one based on the existence of an extension satisfying (10) (see [48, Introduction]), however, it does not necessarily provide a uniform bound independent of k and, therefore, is of no use in the context of the “invading domains” technique.) An important feature of the solution constructed by Leray is that it could be shown to satisfy the so-called generalized energy inequality juj21;2  .u  rU ; u/ .rU ; ru/ C  .@1 U  U  rU ; u/ CT .e 1 x  rU  e 1 U ; u/  hf ; ui  0;

(13)

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formally obtained by setting '  u in (12) and replacing “D” with “.” A more familiar form of (13) can be obtained if f and v have some more regularity. For example, if in addition f 2 L2 .˝/ and v 2 W 3=2;2 .@˝/; then it can be shown that (13) is equivalent to the following one (see [45, Theorem XI.3.1(i)]):  2kD.v/k22 C

Z

˚ @˝

  .v C V/  T .v; p/  .v C V/2 v  n  hf ; vi  0; 2

(14)

where T .v; p/ D rv C .rv/>  p I; I the identity matrix, is the Cauchy stress tensor. It is worth emphasizing that (14) would represent the energy balance for the motion .v; p/; provided one could replace “” with “D.” The inequality sign in the above formulas is, again, a consequence of the little information that this solution carries at large spatial distances. For the same reason, the uniqueness question is left out.

4.1.2 Ladyzhenskaya’s Contribution Ladyzhenskaya was the first to introduce the definition and the use of the term “generalized (or weak) solution” as currently used, for steady-state Navier-Stokes problems [79, p. 78]. Her construction still employs the “invading domains” technique utilized by Leray, but the way in which she proves the existence of the solution on each bounded domain ˝k of the sequence is somewhat simpler and more direct. More precisely, Ladyzenskaya considers (12) with T D 0 and v  0 and shows that it can be equivalently rewritten as a nonlinear equation in the Hilbert space D01;2 .˝k /: M.u/ WD u C  A.u/ D F

(15)

where F is prescribed D01;2 .˝k / and A is a (nonlinear) compact operator. (The extension to the case T ¤ 0 would be straightforward.) Therefore, the operator M; defined on the whole of D01;2 .˝k /; is a compact perturbation of a homeomorphism. Moreover, using arguments similar to Leray’s, one can show that every solution to (15) is uniformly bounded in D01;2 .˝k /; for all  2 Œ0; 0 ; arbitrary fixed 0 > 0. Then, by the Leray-Schauder degree theory, it follows that (15) has a weak solution, uk 2 D01;2 .˝k /; for the given F . Since juk j1;2 is uniformly bounded in k; Ladyzhenskaya shows that a subsequence can be selected converging to a weak solution in the sense of Definition 1; see Remark 2. It is worth emphasizing that, if ˝ is an exterior domain, the operator A is not compact (see [47, Proposition 80]) so that the “invading domains technique” is indeed necessary for the argument to work. Moreover, if u is merely in D01;2 .˝/; with ˝ exterior domain, the very equation (22) would not be meaningful in such a case. Finally, it is important to observe that Ladyzhenskaya’s solution, as Leray’s, satisfies only the generalized energy inequality (13), and, again, the uniqueness question is left open because of the little asymptotic information carried by functions from D01;2 .˝/.

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4.1.3 Fujita’s Contribution Fujita’s approach to the existence of a weak solution [29] is entirely different from those previously mentioned. In fact, it consists in adapting to the time-independent case the method introduced by Eberhard Hopf for the initial value problem [64]. The method referred to above is the by now classical Faedo-Galerkin method. (Also, strictly speaking, Fujita considers the case T D 0; even though the extension of his method to the more general case presents no conceptual difficulty.) As is well known, the idea is to look first for an “approximate solution” to (12), uN ; in the manifold M.N / spanned by the first N elements of a basis of D01;2 .˝/. This is a finite-dimensional problem whose solution, at the N -th step, is found by solving a suitable nonlinear equation. Fujita solves the latter by means of Brouwer fixedpoint theorem [29, Lemma 3.1], provided j˚j is “small enough,” and then shows that juN j1;2 is uniformly bounded in N . With this information in hand, one can then select a subsequence fuN 0 g that in the limit N 0 ! 1 converges (in a suitable sense) to a vector u 2 D01;2 .˝/ satisfying (12); see also [45, Theorem X.4.1]. The advantage of Fujita’s approach, besides being more elementary, resides also in the fact that the solution is constructed directly in the whole domain ˝. However, also in this case, solutions satisfy only the generalized energy inequality, and their uniqueness is also left out.

4.2

A Function-Analytic Approach

The most significant aspect of solutions constructed by the above authors is that their existence is ensured for data of arbitrary “size,” provided only the mass flux through the boundary is not too large. (Notice that, of course, kv k1=2;2;@˝ arbitrarily “large” and j˚j “small” are not, in general, at odds.) However, as emphasized already a few times, these solutions possess no further asymptotic information at large distances other than that deriving from the fact that v 2 L6 .˝/; consequence of the of the property v 2 D 1;2 .˝/ and Sobolev inequality; see Remark 2. With such a little information, it is, basically, hopeless to show fundamental properties of the solution that are yet expected on physical grounds, such as (i) balance of energy equation, namely, (14) with the equality sign, and (ii) uniqueness for “small” data. The main objective of this subsection is to show that, in fact, this undesired feature can be removed by using another and completely different approach. The approach, introduced in [41, 47], consists in formulating the original problem as a nonlinear equation in a suitable Banach space and then using the mod 2 degree for proper Fredholm maps of index 0 to show just under the conditions on the data stated in Definition 1 the existence of a corresponding weak solution possessing “better” properties at “large” distances. As a consequence, one proves that these weak solutions satisfy, in addition, the requirements (i) and (ii) above. Moreover, this abstract setting shows itself appropriate for the study of other important properties of solutions, including generic properties, and steady and time-periodic bifurcation; see Sects. 8 and 9.

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In order to give a precise statement of the main results, it is appropriate to introduce the necessary functional setting. To this end, let R.u/ WD e 1 x  ru  e 1 u and set  ˚ X .˝/ D u 2 D01;2 .˝/ W @1 u; R.u/ 2 D01;2 .˝/ ;

(16)

where @1 u 2 D01;2 .˝/ means that there is C > 0 such that j.@1 u; '/j  C j'j1;2 ; for all ' 2 D.˝/; and, therefore, by the Hahn-Banach theorem, @1 u can be uniquely extended to an element of D01;2 .˝/ that will still be denoted by @1 u. Analogous considerations hold for R.u/. It can be shown [47, Proposition 65] that when endowed with its “natural” norm kukX WD juj1;2 C j@1 uj1;2 C jR.u/j1;2 ; X .˝/ becomes a reflexive, separable Banach space. Obviously, X .˝/ is a strict subspace of D01;2 .˝/. The primary objective is to prove existence of weak solution in the space X .˝/. In this respect, one observes that all classical approaches mentioned earlier on furnish weak solutions in D01;2 .˝/ which embeds only in L6 .˝/; see Remark 2. The fundamental property of X .˝/; expressed in the following lemma, is that it embeds in a much “better” space. Lemma 2. Let ˝  R3 be an exterior domain and assume u 2 D01;2 .˝/ with @1 u 2 D01;2 .˝/. Then, u 2 L4 .˝/; and there is C1 D C1 .˝/ > 0 such that 1

3

4 4 juj1;2 : kuk4  C1 j@1 uj1;2

(17)

Thus, in particular, X .˝/  L4 .˝/. Proof. Obviously, if u  0; there is nothing to prove, so one shall assume u 6 0. The proof for an arbitrary exterior domain is somewhat complicated by several technical issues; see [41, Proposition 1.1 with proof on pp. 8–13]. However, if ˝  R3 ; it becomes simpler and will be sketched here. (The inequality proved in [41, Proposition 1.1] is, in fact, weaker than (17). However, one can apply to Eq. (1.31) of [41] almost verbatim the argument given in the current proof after (23) and show the stronger form (17).) For a given g 2 C01 .R3 /; consider the following problem '  @1 ' D g C rp; div ' D 0; in R3 ;

(18)

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where > 0. By [45, Theorem VII.4.1], problem (18) has at least one solution such that ' 2 Ls1 .R3 / \ D 1;s2 .R3 / \ D 2;s3 .R3 /; @1 ' 2 Ls3 .R3 / p 2 Ls4 .R3 / \ D 1;s3 .R3 /

(19)

for all s1 > 2; s2 > 4=3; s3 > 1; s4 > 3=2; which satisfies the estimate

1=4 j'j1;2  C kgk4=3 ;

(20)

with C D C .s1 ; : : : ; s4 /. Using (18)–(19) and recalling that by the Sobolev inequality D01;2 .R3 /  L6 .R3 /; one shows after integration by parts .u; g/ D .u; '  @1 '  rp/ D .ru; r'/  .u; @1 '/:

(21)

The following identity is valid for all u 2 D01;2 .R3 / with @1 u 2 D01;2 .R3 / and 6 2 D01;2 .R3 / with @1 2 L 5 .R3 / and can be shown by the arguments from [41, pp. 12–13] .u; @1 / D h@1 u; i: In view of (19), we may use the latter in (21) to get .u; g/ D .ru; r'/ C h@1 u; 'i; which implies j.u; g/j  .juj1;2 C j@1 uj1;2 / j'j1;2 :

(22)

Replacing (20) into this latter inequality, one finds  3  1 j.u; g/j  C 4 j@1 uj1;2 C  4 juj1;2 kgk 4 : 3

Since g is arbitrary in C01 .R3 /; it follows that u 2 L4 .R3 / and, furthermore,  3  1 kuk4  C 4 j@1 uj1;2 C  4 juj1;2 ; for all > 0.

(23)

By a simple calculation we show that the right-hand side of (23) as a function of

attains its minimum at

D juj1;2 =.3 j@1 uj1;2 /;

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which once replaced in (23) proves (17), provided j@1 uj1;2 ¤ 0. To show that this is indeed the case, suppose the contrary. Then .@1 u; '/ D 0 for all ' 2 D.R3 /; so that there is p 2 D 1;2 .R3 / such that @1 u D rp in R3 ; see, e.g., [45, Lemma III.3.1]. From div u D 0; we deduce p D 0 in R3 in the sense of distributions, which, by the property of p; in turn furnishes @1 u D rp  0; and this contradicts the fact that  u 2 L6 .R3 /. The proof is thus completed. One is now in a position to state the following general existence result. Theorem 1. For any  ¤ 0; T  0; f 2 D01;2 .˝/; and v 2 W 1=2;2 .@˝/ with ˚ D 0; there exists at least one weak solution, v; to (8)–(7) that in addition satisfies v  U 2 X .˝/; with U given in Lemma 1. Moreover, v obeys the estimate     kv  U kX  C1 jf j1;2 C jf j31;2 C C2 kv C Vk1=2;2;@˝/ C kv C Vk31=2;2.@˝/ (24) where C1 D C1 .; T ; ˝/ and C2 D C2 .; T ; ˝; M /; whenever kv k1=2;2.@˝/  M . A full proof of Theorem 1 is given in [47, Theorem 86(i)]. Here it shall be reproduced the main ideas leading to the result, referring the reader to the cited reference for all missing details. The first step is to write (12) as a nonlinear equation in the space D01;2 .˝/. To reach this goal, for fixed ; T ; one defines the generalized Oseen operator O W u 2 X .˝/ 7! O.u/ 2 D01;2 .˝/

(25)

where hO.u/; 'i WD .ru; r'/ C h@1 u; 'i C T hR.u/; 'i; ' 2 D01;2 .˝/:

(26)

Likewise, one introduces the operators N and K from X .˝/ to D01;2 .˝/ as follows: hN.u/; 'i WD .u  r'; u/ hK.u/; 'i WD Œ.U  ru; '/ C .u  rU ; '/

' 2 D01;2 .˝/:

(27)

(The dependence of the relevant operators on the parameters  and T will be emphasized only when needed; see Sects. 8 and 9.) Finally, let F denote the uniquely determined element of D01;2 .˝/ such that, for all ' 2 D01;2 .˝/; hF ; 'i WD .rU ; r'/  .@1 U  U  rU ; '/  T .R.U /; '/ C hf ; 'i:

(28)

In view of Lemma 1 and Lemma 2 and with the help of Hölder Inequality, it is easy to show that the operators O; N; and K and the element F are well defined.

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Setting L WD O C K;

(29)

the objective is to solve the following problem: For any F 2 D01;2 .˝/; find u 2 X .˝/ such that L .u/ C N.u/ D F :

(30)

It is plain that, if this problem is solvable, then v D u C U is a weak solution satisfying the statement of Theorem 1. The strategy to solve the above problem consists in showing that the map M WD L C N W X .˝/ 7! D01;2 .˝/ is surjective. To reach this goal, one may use a very general result furnished in [47], based on the mod 2 degree of proper C 2 Fredholm maps of index 0 due to Smale [103]. More precisely, from [47, Theorem 59(a)], it follows, in particular, the following. Proposition 1. Let Z; Y be Banach spaces with Z reflexive. Let L W Z 7! Y and N W Z 7! Y and set M D L C N . Suppose: (i) M is weakly sequentially continuous (i.e., if zn ! z weakly in Z; then M .zn / ! M .z/ weakly in Y ). (ii) N is quadratic (i.e., there is a bilinear bounded operator B W Z Z 7! Y such that N .z/ D B.z; z/ for all z 2 Z). (iii) L maps homeomorphically Z onto Y . (iv) The Fréchet derivative of N is compact at every z 2 Z. (v) There is  W RC 7! RC mapping bounded set into bounded set, with .s/ ! 0 as s ! 0; such that kzkZ   .kM .z/kY /: Then M is surjective. This proposition will be applied with Z  X .˝/; Y  D01;2 .˝/; L  L; and N  N. With this in mind, one begins to show the following. Lemma 3. The operator N is quadratic, and M WD L CN is weakly sequentially continuous. Proof. The first property is obvious, since N .u/ D  B.u; u/ where, for wi 2 X .˝/; i D 1; 2;

(31)

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hB.w1 ; w2 /; 'i WD .w1  r'; w2 /; all ' 2 D01;2 .˝/.

(32)

Suppose next uk ! u weakly in X .˝/; one has to show that M.uk / ! M.u/ weakly in D01;2 .˝/. This amounts to prove that lim hM.uk /; 'i D hM.u/; 'i; for all ' 2 D.˝/.

k!1

(33)

In fact, on one hand, being D01;2 .˝/ reflexive [45, Exercise II.6.2], the generic linear functional acting on W 2 D01;2 .˝/ is of the form hW ; 'i; for some ' 2 D01;2 .˝/. On the other hand, it is jM.uk /j1;2  C0 ; C0 > 0 independent of k; as is at once established from (26)–(27) and the uniform boundedness of kuk kX . Now, to show (33), it is observed that the latter implies that there is M1 > 0 independent of k; such that juk j1;2  C: Thus, along a subsequence fuk 0 g; lim .ruk 0 ; r'/ D .ru; r'/ I

k 0 !1

lim .R.uk 0 /; '/ D .R.u/; '/;

k 0 !1

lim .@1 uk 0 ; '/ D .@1 u; '/ I

k 0 !1

for all ' 2 D.˝/.

(34)

Moreover, by the embedding D01;2 .˝/  W 1;2 .˝R /; R > 1; Rellich compactness theorem, and Cantor diagonalization method, one can also show lim kuk 0  uk4;˝R D 0 I for all R > 1;

k 0 !1

(35)

see [45, Proposition 66] for details. The desired property (33) is then a simple consequence of (26), (27), (34), (35), and Hölder inequality.  The following result also holds. Lemma 4. Let u 2 X .˝/. Then, h@1 u; ui D hR.u/; ui D 0: Proof. If u 2 D.˝/; the proof is trivial, being a consequence of simple integration by parts. However, if u is just in X .˝/; the claim is not obvious since it is not known whether D.˝/ is dense in X .˝/. As a consequence, one has to argue in a different and more complicated way, especially to show the property for R. The proof becomes then lengthy, technical, and tricky. For this reason it will be

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omitted, and the reader is referred to [41, pp. 12–13] for the first property and to [47, Proposition 70] for the second one.  The above lemma is crucial for the next result – a particular case of that shown in [47, Proposition 78] – ensuring the validity of condition (iii) in Proposition 1. Lemma 5. The operator L WD O C K is a linear homeomorphism of X .˝/ onto D01;2 .˝/. Moreover, there is a constant C D C .; T ; ˝/ such that kukX  C jL .u/j1;2 :

(36)

Proof. Referring to the cited reference for a full proof, here only the leading ideas will be sketched. As shown in [76, Theorem 2.1] and [47], the generalized Oseen operator O is a homeomorphism of X .˝/ onto D01;2 .˝/; and, moreover, juj1;2 C j@1 uj1;2 C jR.u/j1;2  C jO.u/j1;2 : Therefore, by classical results on Fredholm operators, it is enough to show that (i) K is compact and (ii) N.L / D f0g. Let fuk g  X .˝/ be a bounded sequence and let ˝R contain the support of U . Observing that X .˝/  W 1;2 .˝R /; the Rellich compactness theorem implies that there is a subsequence of fuk g that is Cauchy in L4 .˝R /. Since by (27)1 and Hölder inequality, for all ' 2 D01;2 .˝/ jhK .uk 0 /; 'i  hK .uk 00 /; 'ij  2 kU k4 kuk 0  uk 00 k4;˝R j'j1;2 ; from Lemma 1(i), one infers (along a subsequence) lim 00

k 0 ;k !1

jK .uk 0 /  K .uk 00 /j1;2 D 0;

which proves (i). To show (ii), it must be shown that hO.u/ C K .u/; 'i D 0 for all ' 2 D01;2 .˝/ H) u D 0:

(37)

Since U 2 W 1;2 .˝/ is of bounded support with div U D 0 and u 2 D01;2 .˝/; by an easily justified integration by parts, we show .U  ru; u/ D 0. So that by replacing u for ' in (37) and using this property along with (26), (27)2 ; and Lemma 4, one deduces juj21;2  .u  rU ; u/ D 0: As a result, (37) is a consequence of the latter and of (10) in Lemma 1.



The following lemma guarantees condition (iv) in Proposition 1; see [47, Propositions 79].

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Lemma 6. The Fréchet derivative, N 0 .u/; of N is compact at each u 2 X .˝/. Proof. From (27)1 ; it follows that 1 N 0 .u/w D B.u; w/ C B.w; u/; with B defined in (32). Let fvk g  X .˝/ be such that kvk kX  C; with C independent of k 2 N; and so, by Lemma 2, one gets, in particular, kvk k4 C jvk j1;2  C1 ;

(38)

with C1 D C1 .˝/ > 0. Since X .˝/ is reflexive, there exist v 2 X .˝/ and a subsequence fvk 0 g  X .˝/ converging weakly in X .˝/ to v. As in the proof of Lemma 3, it can also be shown from (38) that (possibly, along another subsequence) lim kvk  vk4;˝R D 0; for all sufficiently large R ; 0 k

(39)

see also [47, Proposition 66]. From (32) and Hölder inequality, one finds jhB.u; vk 0 /  B.u; v/; 'ij D jhB.u; vk 0  v/; 'ij    kuk4;˝R kv  vk 0 k4;˝R C kuk4;˝ R kv  vk 0 k4;˝ R j'j1;2 ; for all sufficiently large R. Using (38) and (39) into this relation gives lim jB.u; vk 0 /  B.u; v/j1;2  C1 kuk4;˝ R ;

k 0 !1

where C1 > 0 is independent of k 0 . However, R is arbitrarily large, and so, by the absolute continuity of the Lebesgue integral, it may be concluded that lim jB.u; vk 0 /  B.u; v/j1;2 D 0:

k 0 !1

(40)

In a completely analogous way, one shows lim jB.vk 0 ; u/  B.v; u/j1;2 D 0:

k 0 !1

(41)

From (40) and (41), it then follows that the operator B.u; /; and hence N 0 .u/; is compact at each u 2 X .˝/. 

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In order to apply Proposition 1 to the operator M; it remains to show condition (v), which amounts, basically, to find “good” a priori estimates for the equation M.u/ D F . Lemma 7. There is a constant C > 0 such that all solutions u 2 X .˝/ to (30) satisfy kukX  C .jF j1;2 C jF j31;2 /:

(42)

Proof. Also using (26), (27)2 ; (10), and Lemma 4, one deduces hL .u/; ui D juj21;2  .u  rU ; u/ 

1 2

juj21;2 ; hF ; ui  jF j1;2 juj1;2 :

(43)

Moreover, it is easily checked that for all u 2 D.˝/; .u  grad u; u/ D 0:

(44)

Now, by Lemma 4, X .˝/  L4 .˝/; and so, by [45, Theorem III.6.2], one can find a sequence fuk g  D.˝/ converging to u in D01;2 .˝/ \ L4 .˝/. Since, by Hölder inequality, the trilinear form .u  grad w; v/ is continuous in L4 .˝/ D 1;2 .˝/ L4 .˝/; one may conclude that (44) continues to hold for all u 2 X .˝/; which gives hN .u/; ui D 0:

(45)

Thus, from this and (43), one obtains juj1;2  2jF j1;2 :

(46)

Since L .u/ D F  N .u/; from Lemma 5, it follows that kukX  C .jF j1;2 C jN .u/j1;2 /:

(47)

Moreover, by Lemma 2 1

3

2 2 1 jhN .u/; 'ij D j.u  r'; u/j  kuk24 j j'j1;2  C1 j@1 uj1;2 juj1;2 j'j1;2 ;

so that, by virtue of (46) and (47), one finds 3 1   2 kukX2 : kukX  C2 jF j1;2 C jF j1;2

Using Young’s inequality in the latter allows one to deduce the validity of (42), and the proof of the lemma is completed. 

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Proof of Theorem 1. The proof of the first statement follows from Proposition 1, and Lemma 3, and Lemmas 5–7. Furthermore, by property (11) of U and (28), one finds jF j1;2  jf j1;2 C C1 kv C Vk1=2;2.@˝/ ;

(48)

where C1 D C1 .; ; ˝; M / whenever kv k1=2;2.@˝/  M . Estimate (24) is then a consequence of Lemma 7 and (48).  Open Problem. Property (10) of the extension U is fundamental for the estimate (46). As mentioned earlier on, (10) is only known if the flux ˚ is of “small” magnitude. While by a procedure similar to [23, 48, 61] it is probably possible to show that such a condition on ˚ is also necessary for the existence of an extension with the above property, one may nevertheless wonder if a small j˚ j would indeed be necessary if the existence problem is approached by other methods. In this respect, by combining a contradiction argument of Leray with properties of the Bernoulli’s function in spaces of low regularity, in their deep work [72], Korobkov, Pileckas, and Russo have shown existence without restrictions on j˚ j; at least for flow and data that are axisymmetric along the direction of . Whether such a result is true in general remains open.

The following result shows an important property of weak solutions in the class X .˝/ and so, in particular, applies to those constructed in Theorem 1. Theorem 2. Let f 2 D01;2 .˝/ and v 2 W 1=2;2 .@˝/; and let v be a corresponding weak solution with v U 2 X .˝/. Then, v satisfies the energy equality, namely, (13) with the equality sign. If, in addition, f 2 L2 .˝/ and v 2 W 3=2;2 .@˝/; then the latter takes the form of the classical equation of energy balance: Z

  .v C V/2 v  n D hf ; vi 2 @˝ Proof. From (30), with F given in (28), one deduces  2kD.v/k22 C

˚

.v C V/  T .v; p/ 

(49)

L .u/; ui C hN .u/; ui D hF ; ui: Employing in this equation (43) and (45), one obtains juj21;2   .u  rU ; u/  hF ; ui D 0; which, recalling the definition of F in (28), shows that u obeys (13) with the equality sign. The second part of the theorem is shown exactly like in [45, pp. 770–771] and will be omitted. 

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Open Problem. The natural question arises whether any weak solution, v; corresponding to data satisfying merely the “natural” minimal conditions of Theorem 1, is such that v  U 2 X .˝/; and, in particular, obeys the equation of energy balance. In a remarkable work, [57] Heck, Kim, and Kozono have shown that this is indeed the case, at least when T D 0 (the body is not spinning), v  0; and f is assumed slightly more regular, namely, f 2 D01;2 .˝/. Whether this result continues to hold for T ¤ 0 is not known.

5

Regularity

It is expected that if the data f ; v and the boundary @˝ are sufficiently smooth, then the corresponding weak solution is smooth as well. In this respect, one has the following very general result about interior and boundary regularity. Theorem 3. Let v be a weak solution to (8)–(7). Then, if m;q

f 2 Wloc .˝/; m  0; where q 2 .1; 1/ if m D 0; while q 2 Œ3=2; 1/ if m > 0; it follows that mC2;q

v 2 Wloc

mC1;q

.˝/; p 2 Wloc

.˝/;

where p is the pressure associated to v in Remark 1. Thus, in particular, if f 2 C 1 .˝/;

(50)

v; p 2 C 1 .˝/:

(51)

then

Assume, further, ˝ of class C mC2 and v 2 W mC21=q;q .@˝/; f 2 W m;q .˝R /; for some R > 1 and with the values of m and q specified earlier. Then, v 2 W mC2;q .˝R /; p 2 W mC1;q .˝R /: Therefore, in particular, if ˝ is of class C 1 and v 2 C 1 .@˝/; f 2 C 1 .˝ R /;

(52)

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it follows that v; p 2 C 1 .˝ R /:

(53)

The proof of this result is rather complicated, and the interested reader is referred to [45, Theorems X.1.1 and XI.1.2]. However, if one assumes (50) [(52) and ˝ of class C 1 ], then the proof of (51) [(53)] can be obtained by classical results for the Stokes problem in conjunction with a simple bootstrap argument and will be reproduced here. To show this, one needs the following classical regularity results for weak solutions to the Stokes problem, a particular case of those furnished in [45, Theorems IV.4.1 and IV.5.1], to which the reader is also referred for their proofs. 1;q

q

Lemma 8. Let .w; / 2 Wloc .˝/ Lloc .˝/; 1 < q < 1; with div w D 0 in ˝; satisfy  .rw; r / D ŒF;   . ; div /; for all m;q

2 C01 .˝/. mC2;q

(54) mC1;q

Then, if F 2 Wloc .˝/; m  0; necessarily .w; / 2 Wloc .˝/ Wloc .˝/. Moreover, assume w 2 W 1;q .˝R / for some R > 1; and w D w at @˝. Then, if F 2 W m;q .˝R /; w 2 W mC21=q;q .@˝/; necessarily .w; / 2 W mC2;q .˝r / mC1;q Wloc .˝r /; for any r 2 .1; R/. With this result in hand, it can be proved that (50) implies (51). From Remark 1, the weak solution v and the associated pressure field p satisfy (54) with F WD  @1 v  T .e 1 x  rv  e 1 v/ C  v  rv C f : Then, by assumption, the embedding 1;2 D01;2 .˝/  Wloc .˝/  L6loc .˝/; 3=2

and the Hölder inequality one has that F 2 Lloc .˝/. From the first statement in 1;3=2 Lemma 8, it can then be deduced v 2 W 2;3=2 .˝/loc ; p 2 Wloc .˝/; and, moreover, 2;3=2 (v; p) satisfy (8)1 a.e. in ˝. Next, because of the embedding Wloc .˝/  1;3 Wloc .˝/  Lrloc .˝/; arbitrary r 2 Œ1; 1/; one obtains the improved regularity 1;s .˝/; for all s 2 Œ1; 3=2/. Using once again Lemma 8, one infers property F 2 Wloc 3;s 2;s v 2 Wloc .˝/ and p 2 Wloc .˝/ which, in particular, gives further regularity for F. By induction, one then proves the desired property v; p 2 C 1 .˝/. The proof of the boundary regularity is performed by an entirely similar argument and, therefore, will be omitted.

7 Steady-State Navier-Stokes Flow Around a Moving Body

6

365

Uniqueness

This section is dedicated to the investigation of the uniqueness property of weak solutions. Basically, the main known results depend on the summability and regularity assumptions made at the outset on the data f and v . The following theorem shows, in particular, that every solution in Theorem 1 is unique in its own class of existence, provided the size of the data is sufficiently restricted. Theorem 4. Assume vi ; i D 1; 2; are weak solutions with vi  U 2 X .˝/; corresponding to the same f 2 D01;2 .˝/; v 2 W 1=2;2 .@˝/. Then, there is C D C .; T ; ˝/ such that if jf j1;2 C kv C Vk1=2;2.@˝/ < C;

(55)

necessarily v1  v2 . Proof. Setting ui D vi  U ; i D 1; 2; with U given in Lemma 1, from (25)–(28) and the assumption, one finds L .ui / C N .ui / D F ; i D 1; 2:

(56)

Therefore, from Lemma 2, Lemma 7, and (48), one infers in particular     kui k4  C1 jf j1;2 Cjf j31;2 CC2 kv CVk1=2;2.@˝/ Ckv CVk31=2;2.@˝/ ;

(57)

where C1 D C1 .; T ; ˝/ and C2 D C2 .; T ; ˝; M /; whenever kv k1=2;2.@˝/  M . Arbitrarily fix the number M once and for all. Setting u WD u1  u2 ; from (56), one obtains L .u/ D B.u; u1 /  B.u2 ; u/;

(58)

where B is defined in (32). Thus, observing that   jB.u; u1 / C B.u2 ; u/j1;2  kuk4 ku1 k4 C ku2 k4 from (58), Lemma 2, and Lemma 6, one has, in particular,  

kuk4 1  C3 ku1 k4 C ku2 k4  0; with C3 D C3 .; T ; ˝/. The result then follows from this inequality and (57).



The natural question arises of whether the solutions constructed in Theorem 1 are unique in the class of weak solutions, that is, obeying just the requirements stated in Definition 1. The answer to this question is positive if f is assumed to possess

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“good” summability properties at large distances and v is sufficiently regular. Actually, this property is a particular consequence of the following result for whose proof the reader is referred to [45, Theorem XI.5.3], once one takes into account 6 that, by Sobolev inequality, L 5 .˝/  D01;2 .˝/ and that D01;2 .˝/  D01;2 .˝/. Theorem 5. Let f 2 L6=5 .˝/ \ L4=3 .˝/; v 2 W 5=4;4=3 .@˝/: Then, there exists C D C .˝; ; T / such that, if kf k6=5 C kv C Vk7=6;6=5.@˝/ < C;

(59)

v is the only weak solution corresponding to the above data.

Open Problem. In general, it is not known whether solutions of Theorem 1 are unique in the class of weak solutions, when f and v merely satisfy the assumptions of that theorem (and are sufficiently small).

In connection with this problem, it is worth remarking that in the special case T D 0 and v  0; and with f slightly more regular (namely, f 2 D01;2 .˝/), the result is shown in [57, Theorem 2.3].

7

Asymptotic Behavior

As shown in previous sections, some fundamental attributes of weak solution expected on physical grounds, such as verifying the energy balance and being unique for small data, can be established if one has enough information on their summability properties in a neighborhood of infinity, like the one provided by Theorem 1. However, there are other significant aspects that require a sharp pointwise knowledge of the solution at large distances, which in principle is not necessarily guaranteed just by the mild asymptotic information furnished in that theorem. These aspects include, for instance, the presence of a stationary, unbounded wake region “behind” the body and a “fast” decay of the vorticity outside the wake region, in support of boundary layer theory. Proving (or disproving) these properties constituted one of the most challenging questions in mathematical fluid dynamics since the pioneering chapter of Leray. The case T D 0 was eventually settled in the mid-1970s (about 40 years after Leray’s work), thanks to the effort of Robert Finn, Konstantin I. Babenko, and their collaborators. Their contributions will be briefly summarized in the following two subsections. The case T ¤ 0 presents much more difficulties and will be treated

7 Steady-State Navier-Stokes Flow Around a Moving Body

367

successively in Sect. 7.3, by means of a different approach, originally introduced in [39], that allows for a rather complete description of the pointwise asymptotic flow behavior also in that more general situation.

7.1

Finn’s Contribution

In the late 1950s/mid-1960s, in a series of remarkable papers [24–28], Robert Finn proved the following fundamental results. Let .v; p/ be any (sufficiently smooth) solution to (8)–(7) with T D 0 and with f of bounded support, such that jv.x/j  C jxj˛ ; some ˛ > 1=2 and all “large” jxj. Then the pointwise asymptotic structure of .v; p/ can be sharply evaluated. In particular, combining the integral representation of the solution, obtained via the (time-independent) Oseen fundamental tensor, E; along with a careful estimate of the latter, Finn showed that these solutions exhibit a paraboloidal “wake region,” R; with the property that the velocity field, v; inside R decays pointwise slower than it does outside R. More precisely, he proved that v [rv] admits an asymptotic expansion with E [rE] being the leading term. (Finn left open the question of the asymptotic behavior of the second derivatives of v [24], a problem that was finally solved another 40 years later by Deuring [13].) Finn called such solutions “physically reasonable” (PR) [27, Definition 5.1] and demonstrated their existence on a condition that the magnitude of the data is sufficiently restricted [27, Theorem 4.1]. Later on, one of his students, David Clark, showed that the vorticity field of any PR solution decays exponentially fast outside R and far from the body [10]. Thanks to its sharp asymptotic (and local regularity) properties, it is easy to show that any PR solution (regardless of the size of the data) is also weak, namely, v 2 D 1;2 .˝/. However, given that the latter is the only information that weak solutions carry in a neighborhood of infinity, the converse property is by no means obvious, to the point that some author even questioned its validity [59, p. 12]. All this seemed to cast profound doubts about the physical relevance of Leray’s weak solutions.

7.2

Babenko’s Contribution

The relation between weak and PR solutions was eventually addressed by Babenko [2]. Combining Lizorkin’s multipliers theory with anisotropic Sobolev-like inequalities and the representation formula for the solution employed by Finn, he was able to show that, if the body force f is of bounded support, every weak solution is, in fact, physically reasonable in the sense of Finn. Babenko’s paper can be divided into two main parts. In the first one, he shows, by a very elegant and straightforward argument, that any weak solution, v; corresponding to the given data must be in L4 .˝/; with corresponding pressure field in L2 .˝/. The second part of the paper is aimed to show that, actually, v 2 Lq .˝/; for any q 2 .2; 4. Once this property is established, then it is relatively simple to prove that the weak solution decays like jxj˛ for some ˛ > 1=2 and therefore is also PR. It must be noted that Babenko’s proof of these further summability properties has aspects that are not

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fully transparent. Also for this reason, a distinct and more direct proof of Babenko’s result was given later on by Galdi [35] and, successively and independently, by Farwig and Sohr [16].

7.3

A General Approach

It must be emphasized one more time that the results reported in the previous two subsections refer to the case T D 0; that is, the body is not spinning. If one allows T ¤ 0; then the detailed study of the asymptotic properties of a weak solution becomes even more complicated, for several reasons. In the first place, the linear momentum equation (8)1 contains a term that grows linearly fast at large spatial distances. As a consequence, the fundamental tensor of the linearized equations, T; is no longer the classical Oseen tensor E mentioned above, let alone a “perturbation” of it, but, rather, a much more complicated one; see [21, Section 2]. Thus, the representation of the solution that in both contributions of Finn and Babenko plays a fundamental role in the determination of the pointwise behavior becomes much more involved and, actually, useless for that matter. In fact, as shown in [21, Proposition 2.1], unlike E; the tensor T does not satisfy uniform estimates at large spatial distances. In view of these issues, in [39] Galdi introduced a completely different approach to the study of the asymptotic structure of a weak solution, that was further generalized and improved in [40, 42, 43, 46]. In this approach, the weak solution v is viewed as limit as t ! 1 along sequences of the (unique) solution, w.x; t /; to a suitable initial value problem. It can be shown that, in turn, w admits a somewhat simple space-time representation in terms of the Oseen fundamental solution to the time-dependent Oseen equation. This fact allows one to obtain a number of sharp spatial estimates for w uniformly in time, which are thus preserved in the limit t ! 1; and therefore continue to hold for the weak solution v. Referring to [45, §§X.6, X.8, XI.4, XI.6] for a full account of the (technically complicated and lengthy) proofs of all the above results, here it will only be provided an outline of the main steps of the procedure used in establishing them in the case T ¤ 0. The first step consists in determining sharp summability properties of a weak solution in a neighborhood of infinity, under appropriate hypothesis on the data. To this end, one can show the following result [45, Theorem XI.6.4]. Lemma 9. Assume, for some q0 > 3 and all q 2 .1; q0 ; that f 2 Lq .˝/ \ L3=2 .˝/; v 2 W 21=q0 ;q0 .@˝/ \ W 4=3;3=2 .@˝/: Then, every weak solution v to problems (8)–(7) corresponding to f ; v ; and the associated pressure field p (possibly modified by the addition of a constant; see also Remark 1) satisfies the following summability properties:

7 Steady-State Navier-Stokes Flow Around a Moving Body

v 2 Lr .˝/ \ D 1;s .˝/;

369

@v 2 Lt .˝/; p 2 L .˝/; @x1

for all r 2 .2; 1; s 2 .4=3; 1; t 2 .1; 1; and 2 .3=2; 1. If, in addition, f 2 W 1;q0 .˝/; v 2 W 31=q0 ;q0 .@˝/; then we have also v 2 D 2; .˝/; p 2 D 1; .˝/; for all 2 .1; 1. The next objective is to “translate” the above global asymptotic information into a pointwise one. For simplicity, it shall be assumed that f is of bounded support, which also implies, with the help of Theorem 3, that .v; p/ 2 C 1 .˝  / for sufficiently large . Thus, in the second step, one uses a standard “cutoff” procedure to rewrite (suitably) (8) in the whole space R3 . More specifically, let be a smooth function that is 0 in the neighborhood of @˝ that contains the support of f and 1 sufficiently far from it. Moreover, let Z 2 C01 .˝/ such that div Z D r  v in ˝. (Such a field Z exists, as shown in [45, Theorem III.3.3].) From (8) one can deduce that u WD v  Z and pQ WD p obey the following problem: u C  @1 u C T .e 1 x  ru e 1 u/ D  div . v ˝ div u D 0

v/ C r pQ C F c

9 = ;

in R3 (60)

where F c is smooth and of bounded support. At this point, the “classical” procedure would be to write the solution u in terms of the fundamental tensor solutions, T; associated with problem (60). However, as remarked earlier on, this would not lead anywhere due to the poor properties of T. Therefore, we argue differently. In the third step one performs a time-dependent change of coordinates which transforms (60) into a suitable initial value problem. To this end, for t  0; let 3 1 0 0 Q.t / D 4 0 cos.T t /  sin.T t / 5 ; 0 sin.T t / cos.T t / 2

and define y WD Q.t /  x; w.y; t / WD Q.t /  u.Q> .t /  y/; .y; t / WD p.Q Q > .t /  y/; > V .y; t / WD Q.t /  Π v.Q .t /  y/; H .y; t / WD Q.t /  F c .Q> .t /  y/:

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From (60) and Lemma 9, it then follows that 9 @w @w D w C   r  div ŒV ˝ V   H = in R3 .0; 1/; @t @y1 ; r wD0 lim kw.; t /  ukr D 0; all r 2 .2; 1/.

(61)

t!0C

Notice that equation (61)1 does not contain the linearly growing term. The solution to the Cauchy problem (61) has the following representation: Z

2

ejyzC t e 1 j =4t u.z/ d z w.y; t / D .4t/ Z t Z 3 R3    .y  z; t  /  Rr  ŒV ˝ V .z; / C H .z; / d z d ;  3=2

0

R

(62)

where  .; s/; .; s/ 2 R3 .0; 1/ is the well-known Oseen fundamental solution to the time-dependent Stokes system [45, §VIII.3]. In the final step one utilizes into (62) the summability properties for u and V obtained from Lemma 9 along with the classical pointwise estimates of  to produce a pointwise estimate for w.x; t /; [rw.x; t /] uniformly in t. As a result, by letting t ! 1 along sequences, the latter can be shown to provide analogous bounds for u.x/ [respectively, ru.x/], which means for the weak solution v.x/ [respectively, rv.x/] for all “large” jxj. Once the necessary asymptotic information on v is obtained, analogous estimates on the pressure field can be proved observing that from (8) it follows, for sufficiently large ; that p D r  G in ˝  ; @p D g on @˝  ; @n where G WD  v  rv; g WD Œv C .@1 v  v  rv/ C T .e 1 x  rv  e 1 v/  n j@˝  : The procedure just outlined is at the basis of the following result whose full proof is found in [45, Theorems XI.6.1–XI.6.3]. Theorem 6. Let v be a weak solution, corresponding to f of bounded support, and let p be the corresponding pressure field associated to v by Remark 1. Then, for any ı; > 0 and all sufficiently large jxj;

7 Steady-State Navier-Stokes Flow Around a Moving Body

371

  1 1 3=2Cı ; v.x/ D O jxj .1 C  s.x// C jxj   3=2 3=2 2C ; rv.x/ D O jxj .1 C  s.x// C jxj

(63)

p.x/ D p0 C O.jxj2 ln jxj/; for some p0 2 R; where s.x/ WD jxj C x1 . Remark 3. This theorem suggests, in particular, that outside any semi-infinite cone, C; whose axis coincides with the negative x1 axis, the decay is faster than inside C. This is the mathematical explanation of the existence of the wake “behind” the body, once one takes into account that the velocity of the center of mass of the body .v0 e) is directed along the positive axis x1 ( > 0). Remark 4. The fundamental tensor solution E.x; y/  fEij .x; y/g of the Oseen system (which is obtained by setting T D 0 and disregarding the nonlinear term v  rv in (8)1 ) is defined through the relations  Eij .x; y/ D ıij  

@2 @yi @yj



1 ˚.x  y/; ˚./ WD 4

Z

 2 .jjC1 /

0

1  e d :

Now, the first terms on the right-hand side of (63)1 and (63)2 are just (sufficiently sharp) bounds for E and rE at large jxj; respectively; see [45, Section VIII.3]. This is suggestive of the property that a  E and a  rE; for some suitable vector a; could be the leading terms in corresponding asymptotic expansions. Actually, if T D 0; this property is true, and one can show that, in such a case, the following formula holds for all sufficiently large jxj [45, Theorem X.8.1]: v.x/ D m  E.x/ C V.x/

(64)

where m is a constant vector coinciding with the total force, F; exerted by the liquid on the body and 

V.x/ D O jxj

3=2Cı

 ; arbitrary ı > 0:

Analogous estimate can be proven for rv.x/ [45, Theorem X.8.2]. If T ¤ 0; in [78, Theorem 1.1], Kyed has shown an asymptotic formula similar to (64) (and an analogous one for rv) where now m D .F  e 1 /e 1 and the quantity V is a “higher-order term” in the sense of Lebesgue integrability at large distances. A pointwise estimate (probably not optimal) for V is shown in [77, Theorem 5.3.1]. (An even more detailed asymptotic structure was first shown in [21] for solutions to the linearized (Stokes) problem and in the absence of translational motion.)

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This section ends with some important considerations concerning the asymptotic behavior of the vorticity field, $ WD curl v; of a weak solution, v. In this regard, one can prove the following theorem, due to Deuring and Galdi [14], that ensures that $ decays exponentially fast outside the wake region and sufficiently far from the body. Theorem 7. Under the same assumptions of Theorem 6, there are constants C; R > 0 such that j$.x/j  C jxj3=2 e.=4/ .jxjCx1 /=.1CR/

for all x 2 ˝ R :

It is worth stressing the importance of this estimate that agrees with the necessary condition supporting the boundary layer assumption, namely, that sufficiently far from the body and the wake, the flow is “basically potential.” As a matter of fact, in the case T D 0; one can prove a sharper result that provides a more accurate description of the asymptotic structure of the vorticity field. Precisely, in that case, one has, for all sufficiently large jxj;    $.x/ D r˚ m C O jxj2 e 2 .jxjCx1 /

(65)

where ˚.x/ D 

  e 2 .jxjCx1 / 4 jxj

and m is a constant vector denoting the total force exerted by the liquid on the body [3, 10]. Open Problem. In the case T ¤ 0; it is not known whether the vorticity admits an asymptotic expansion of the type (65), with an appropriate choice of the leading term.

8

Geometric and Functional Properties for Large Data

Theorem 1 shows that, for any set of data D WD .; T ; v ; f /; in the specified spaces, there exists at least one corresponding weak solution v with the further property that u WD vU 2 X .˝/; for a suitable extension field U . Also, Theorem 4 shows that this is, in fact, the only weak solution in that class, provided the data are suitably restricted, according to (55). Objective of this section is to analyze the geometric and functional properties of the solution manifold in the space X .˝/; corresponding to data of arbitrary magnitude in the class specified in Theorem 1. In order to make the presentation simpler, throughout this section, it is set v  0.

7 Steady-State Navier-Stokes Flow Around a Moving Body

373

To reach this goal, one begins to rewrite equation (30) in an equivalent way that emphasizes the dependence of the operator involved on the parameter p WD .; T /. One thus writes L .p; u/ for L .u/ and N .p; u/ for N .u/ with L and N defined in (25)–(27) and (29). Moreover, let H D H.p/ denote the uniquely determined member of D01;2 .˝/ such that hH; 'i WD .rU ; r'/.@1 U U rU ; '/T .R.U /; '/; ' 2 D01;2 .˝/:

(66)

Thus, for a given f 2 D01;2 .˝/; (30) can be written as M.p; u/ D f in D01;2 .˝/

(67)

where M W .p; u/ 2 R2C X .˝/ 7! L .p; u/ C N .p; u/ C H.p/ 2 D01;2 .˝/:

(68)

The solution manifold associated to (68) is defined next: ˚  M.f / D .p; u/ 2 R2C X .˝/ satisfying (67)–(68) for a given f 2 D01;2 .˝/ The main goal is then to address the following questions: (a) Geometric structure of the manifold M D M.f / (b) Topological properties of the associated level set S.p0 ; f / WD f.p; u/ 2 M.f /; p D p0 g; obtained by fixing also Reynolds and Taylor numbers. Clearly, “points” in S.p0 ; f / are solutions to equation (67) with a prescribed p D p0 or, equivalently, to equation (30). The following theorem collects the principal properties of the set S.p0 ; f /. Theorem 8. The following properties hold. (i) S.p0 ; f / is not empty. (ii) For any .p0 ; f / 2 R2C D01;2 .˝/; S.p0 ; f / is compact. Moreover, there is N D N .p0 ; f / 2 N such that S.p0 ; f / is homeomorphic to a compact set of RN . (iii) For any p0 2 R2C ; there is an open residual set O D O.p0 /  D01;2 .˝/ such that, for every f 2 O; S.p0 ; f / is constituted by a number of points,  D .p0 f /; that is finite and odd. (iv) The number  is constant on every connected component of O.

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Proof. As usual, only a sketch of some of the proofs of the above statements will be given while referring to the appropriate reference for whatever is missing. The statement (i) is a consequence of Theorem 1. The proofs of the other statements are based on two fundamental properties of the operator M WD L .p0 ; / C N .p0 ; /; namely, being (1) proper and (2) Fredholm of index 0. Now, Lemma 5 and Lemma 6 guarantee that the Fréchet derivative of M at every u 2 X .˝/ is a compact perturbation of a homeomorphism, which proves the Fredholm property. Properness means that if F ranges in a compact set, K; of D01;2 .˝/; all possible corresponding solutions u to M.u/ D F belong to a compact set, K  ; of X .˝/. To show that this is indeed the case, one observes that in view of the continuity of M; K  is closed, so that it is enough to show that from any sequence fun g  K  ; there is a subsequence (still denoted by fun g) and u 2 X .˝/ such that un ! u in X .˝/. Let F n D M.un /. Since fF n g  K; one deduces (along a subsequence) F n ! F in D01;2 .˝/; for some F 2 K.

(69)

Moreover, being fF n g bounded, by Lemma 7, it follows that fun g is bounded, and so there exists u 2 X such that un ! u weakly in X .˝/. By Lemma 3 and (69), the latter implies M.u/ D F; M.un / ! M.u/ in D01;2 .˝/:

(70)

One next observes that M.un /  M.u/ D L .un  u/ C N .un /  N .u/

(71)

and also, since N is quadratic (Lemma 3), N .un /  N .u/  B.un ; un /  B.u; u/ D B.un  u; u/ C B.u; un  u/ C B.un  u; un  u/  ŒN 0 .u/.un  u/ C N .un  u/: Replacing this identity in (71), one finds M.un /  M.u/  ŒN 0 .u/.un  u/ D M.un  u/:

(72)

In view of (70), the first term on the left-hand side of (72) tends to 0 as n ! 1. Likewise, since N 0 .u/ is compact, and hence completely continuous, also the second term on the left-hand side of (72) tends to 0 as n ! 1; so that properness follows from this and Lemma 7. The first property in (ii) is then a corollary of what has just been proven. As for the second one, we refer to [47, Theorem 93] for a proof. We next come to show statements (iii) and (iv). In this regard, since M is proper and Fredholm of index 0, by the mod 2 degree of Smale [103], it is enough to show that there exists F 0 2 D01;2 .˝/ with the following properties: (a) the equation

7 Steady-State Navier-Stokes Flow Around a Moving Body

375

M.u/ D F 0 has one and only one solution, u0 ; and (b) N.M0 .u0 // D f0g; see [41, Lemma 6.1]. Now, set F 0 D 0. From Lemma 7, it follows that the only solution to M.u/ D 0 is u0 D 0. Moreover, by Lemma 3, one finds N 0 .0/  0; so that  M0 .0/ D L and condition (b) is a consequence of Lemma 5. Remark 5. Taking into account that the set O in Theorem 8 is dense in D01;2 .˝/; from Theorem 8(iii), one deduces the following interesting property of weak solutions. Let  ¤ 0 and T  0 be arbitrarily fixed. Given f 2 D01;2 .˝/ and " > 0; there is g 2 D01;2 .˝/ with jf  gj1;2 < " such that the number of weak solutions given in Theorem 1 corresponding to the body force g (and v  0) is finite and odd. The next result furnishes a complete generic characterization of the manifold M.f /. Its proof, based on an infinite-dimensional version of the so-called parameterized Sard theorem [107, Theorem 4.L], is technically involved and lengthy. The interested reader is referred to [47, Theorem 88]. Theorem 9. The following properties hold. (i) There exists a dense, residual set Z  D01;2 .˝/ such that, for any f 2 Z; the solution manifold M.f / is a two-dimensional (not necessarily connected) manifold of class C 1 . (ii) For any f 2 Z; there exists an open, dense set P D P.f /  R2C such that, for each p 2 P; equation (67) has a finite number of solutions, n D n.p; f /. (iii) The integer n D n.p; f / is independent of p on every interval contained in P.

Open Problem. It is not known whether, in the physically significant case of vanishing body force f and boundary velocity v ; the number of corresponding steady-state solutions is generically finite.

9

Bifurcation

As pointed out in the introductory section, if the speed of the center of mass of the body, v0 ; reaches a critical value, it is experimentally observed, already in absence of rotation, that the characteristic features of the original steady-state flow of the liquid may change dramatically. The outcome could be either the onset of an entirely different steady-state flow or even of a time-periodic regime. The objective of this section is to provide necessary conditions and sufficient conditions for the occurrence of this phenomenon. More precisely, Sect. 9.1 will be concerned with

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time-independent problems, while Sect. 9.2 will deal with the time-periodic case. For the sake of simplicity, it will be assumed throughout v  0.

9.1

Steady Bifurcation

One is mainly interested in situations where bifurcation is generated by the “combined” action of translation and rotation of the body (provided the latter is not zero). To this end, it is convenient to use a different nondimensionalization for the equations (8)–(7), in order to introduce an appropriate bifurcation parameter. Precisely rescaling velocity with v0 and length with v0 =!; equations (8)–(7) become    v C  @1 v C e 1 x  rv  e 1 v  v  rv D rp C f in ˝ div v D 0 v D e 1 C e 1 x at @˝ I lim v.x/ D 0;

(73)

jxj!1

where now  WD v02 .e  e 1 /=.!/. Remark 6. Of course, the above nondimensionalization requires ! ¤ 0. However, for future reference, it is important to emphasize that all main results presented in this section continue to hold in exactly the same form also when ! D 0. With the notation introduced in the previous section (see (67) and (68) with p  ), the original equation (30) is equivalent to the following nonlinear equation M.; u/ WD L .; u/ C N .; u/ C H./ D f in D01;2 .˝/; u 2 X .˝/

(74)

Definition 2. Let u0 2 X .˝/ be a solution to (74) with  D 0 . The pair .0 ; u0 / .1/ is called a steady bifurcation point for (74), if there are two sequences fk ; uk g and .2/ fk ; uk g with the following properties: .i/

(i) fk ; uk g; i D 1; 2 solve (74) for all k 2 N. .i/ (ii) fk ; uk g ! .0 ; u0 / in R X .˝/ as k ! 1; i D 1; 2. .1/ .2/ (iii) uk  6 uk ; for all k 2 N.

One of the main achievements of this section is the proof that, under certain conditions that may be satisfied in problems of physical interest, bifurcation is reduced to the study of a suitable linear eigenvalue problem, formally analogous to that occurring in the study of bifurcation for flow in a bounded domain; see Theorem 11 and Remark 8.

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377

A necessary condition in order for .0 ; u0 / to be a bifurcation point is obtained as a corollary to the following result. Lemma 10. Let u0 2 X .˝/ be a solution to (74) with  D 0 and fixed f 2 D01;2 .˝/. If N.M 0 .0 ; u0 // D f0g;

(75)

namely, the (linear) equation

L .0 ; w/ C 0 B.u0 ; w/ C B.w; u0 / D 0 in D01;2 .˝/

(76)

has only the solution w D 0 in X .˝/; then there exists a neighborhood U .0 /; such that for each  2 U .0 / there is one and only one u./ solution to (74). Moreover, the map  2 U ! u./ 2 X .˝/ is analytic at  D 0 . (The prime means Fréchet differentiation with respect to u.) Proof. Consider the map F W .; u/ 2 U .0 / X .˝/ 7! M.; u/  f : Also using the fact that N .; / is quadratic (see (31)–(32)), it easily follows that F is analytic (polynomial, in fact) at each .; u/. Moreover, by assumption, F .0 ; u0 / D 0. Thus, the claimed property will follow from the analytic version of the implicit function theorem provided one shows that F 0 .0 ; u0 / is a bijection. Now, from (31)–(32),

F 0 .0 ; u0 / D M 0 .0 ; u0 /  L .0 ; / C 0 B.u0 ; / C B.; u0 / ; so that by Lemma 5 and Lemma 6, we infer that F 0 .0 ; u0 / is Fredholm of index 0 and the bijectivity property follows from the assumption (75).  From this result the following one follows at once. Corollary 1. A necessary condition for .0 ; u0 / to be a bifurcation point is that dim N.M 0 .0 ; u0 // > 0;

(77)

namely, the (linear) equation

L .0 ; w/ C 0 B.u0 ; w/ C B.w; u0 / D 0 in D01;2 .˝/ has a nonzero solution w 2 X .˝/.

(78)

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Remark 7. One can show that (77) is equivalent to the requirement that the linearization of (73) around .0 ; v0  u0 C U / corresponding to homogeneous data, namely, )   w C 0 @1 w C e 1 x  rw  e 1 w  v0  rw  w  rv0 D rp div w D 0 w D 0 at @˝ I

in ˝

lim w.x/ D 0;

jxj!1

(79) has a nontrivial solution .w; p/ 2 ŒD 2;2 .˝/ \ X .˝/ D 1;2 .˝/. In fact, saying that (78) has a nonzero solution w 2 X .˝/ means that there exists w 2 X .˝/  f0g such that (see (25)–(30) with  D T ))

 .rw; r'/ C 0 h@1 w C R.w/; 'i C .w  rv0 C v0  rw; '/ D 0;

(80)

for all ' 2 D01;2 .˝/. However, by the properties of v0 and w combined with the Hölder inequality, one shows G WD .w  rv0 C v0  rw/ 2 L4=3 .˝/; which, in turn, by classical results on the generalized Oseen equation [45, Theorem VIII.8.1] furnishes, in particular, w 2 D 2;4=3 .˝/. By embedding, the latter implies w 2 D 1;12=5 .˝/ \ L12 .˝/; so that G 2 L12=7 .˝/ which, again by [45, Theorem VIII.8.1], delivers w 2 D 2;12=7 .˝/ \ D 1;4 .˝/ \ L1 .˝/. Thus, G 2 L2 .˝/ and the property follows by another application of [45, Theorem VIII.8.1]. Notice that the asymptotic condition in (79) is achieved uniformly pointwise. The next objective is to provide sufficient conditions for .0 ; u0 / to be a bifurcation point. To this end, it will be assumed that, in the neighborhood of .0 ; u0 /; there exists a sufficiently smooth solution curve, that is, there is a map  2 U .0 / 7! u./ 2 X .˝/ of class C 2 (say), with u.0 / D u0 and satisfying (74) for the given f . Setting w WD u  u; one thus gets that w satisfies the equation F.; w/ WD L .; w/ C ŒB.w; u.// C B.u./; w/ C N .; w/ D 0:

(81)

Clearly, .0 ; u0 / is a bifurcation point for (74) if and only if .0 ; 0/ is a bifurcation point for (81). Since, as showed earlier on, F 0 .0 ; 0/  L .0 ; / C 0 ŒB.; u0 .// C B.u0 ; / is Fredholm of index 0, a classical result [107, Theorem 8.A] ensures that .0 ; 0/ is a bifurcation point provided the following conditions hold: (i) dim N .F 0 .0 ; 0// D 1; (ii) ŒF w .0 ; 0/.w1 / 62 R .F 0 .0 ; 0//; w1 2 N .F 0 .0 ; 0//; where the double subscript denotes differentiation with respect to the indicated variable. Condition (i) specifies in which sense the requirement of Corollary 1 must be met. In order to give a more explicit form to condition (ii), it is convenient to introduce the Stokes operator:

7 Steady-State Navier-Stokes Flow Around a Moving Body

379

Q 2 D1;2 .˝/; Q W u 2 D01;2 .˝/ 7! u 0

(82)

Q 'i D .ru; r'/; ' 2 D1;2 .˝/: hu; 0

(83)

with

As is well known, Q is a homeomorphism [45, Theorem V.2.1]. By a straightforward computation, one then shows that ŒF w .0 ; 0/.w1 / D 



1 Q P 0 // C B.u. P 0 /; w1 / ; w1 C 0 B.w1 ; u. 0

(with “” denoting differentiation with respect to ) and therefore condition (ii) is equivalent to the request that the equation

L.0 ; w/ C 0 B.u0 ; w/ C B.w; u0 /

1 Q P P D  w 1 C 0 B.w1 ; u.0 // C B.u.0 /; w1 / 0

(84)

has no solution. All the above is summarized in the following. Theorem 10. Suppose the solution set of the equation

L .0 ; w/ C 0 B.u0 ; w/ C B.w; u0 / D 0

(85)

is a one-dimensional subspace of X .˝/ and let w1 be a corresponding normalized element. If, in addition, equation (84) has no solution w 2 X .˝/; then .0 ; u0 / is a bifurcation point for (74). The assumptions of the result just proven admit a noteworthy conceptual P 0 / D 0. This happens, in particular, if u./ interpretation in the case when u. is constant in a neighborhood of 0 ; a circumstance that may occur by a suitable nondimensionalization of the original equation [34, Section VI]. To show the above, consider the operator L W w 2 X .˝/  D01;2 .˝/ 7! L.w/ D Q 1 Œ@1 w C R.w/ C B.v0 ; w/ C B.w; v0 / 2 D01;2 .˝/; where, as before, v0 WD u0 C U . The following lemma shows the fundamental properties of L. The proof is quite involved, and, for it, the reader is referred to [47, Lemma 111]. Lemma 11. Assume u0 2 L3 .˝/ \ L4loc .˝/. Then, the operator L is (graph) closed. Moreover, Sp.LC /\.0; 1/ consists, at most, of a finite or countable number

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of eigenvalues, each of which is isolated and of finite algebraic and geometric multiplicities, that can only accumulate at 0. (Of course, the assumption u0 2 L4loc .˝/ is redundant if u0 2 X .˝/. Also u0 2 L3 .˝/ is assured by Lemma 9 if f is suitably summable at large distances.) Combining Corollary 1, Lemma 11, and Theorem 10, one can then show the following. P 0 / D 0; with u0 2 L3 .˝/ \ L4 .˝/. Then, a necessary Theorem 11. Assume u. loc condition for .0 ; u0 / to be a bifurcation point for (74) is that 0 WD 1=0 is an eigenvalue for the operator LC . This condition is also sufficient if 0 is simple. Proof. With the help of (25), (26), and (27) and (30), one sees that condition (77) is equivalent to assuming that the following equation has a nonzero solution w1 2 X .˝/ Q 1 C 0 Œ@1 w1 C R.w1 / C B.v0 ; w1 / C B.w1 ; v0 / D 0: w Operating with Q 1 on both sides of the latter, one concludes that 0 must be an eigenvalue of LC ; which provides the first statement. Performing the same procedure on (85), it can be next shown that the first assumption in Theorem 10 is satisfied if and only if there is a unique (normalized) w1 2 X .˝/ such that L.w1 / D 0 w1 ; that is, 0 is an eigenvalue of LC of geometric multiplicity 1. Furthermore, operating P 0 / D 0; one gets again with Q 1 on both sides of (84) with u.

0 w  L.w/ D  20 w1 which, by the second assumption in Theorem 10, should have no solution, which means that the algebraic multiplicity of 0 must be 1 as well and the proof of the claimed property is completed.  Another interesting and immediate consequence of Lemma 11 and Theorem 11 is the following one. Corollary 2. Let u0 be a solution branch to (74) independent of  2 J; where J is a bounded interval with J  .0; 1/. Suppose, further, that u0 satisfies the assumption of the preceding theorem. Then, there is at most a finite number, m; of bifurcation points to (74) .k ; u0 /; k 2 J; k D 1;    ; m. Remark 8. It is significant to observe that the statements of Theorem 11 and Corollary 2 formally coincide with those of analogous theorems for steady bifurcation

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381

from steady solution to the Navier-Stokes equation in a bounded domain; see, e.g., [6, Section 4.3C]. However, in the latter case, L is a compact operator defined on the whole of D01;2 .˝/; whereas in the present case, L is a densely defined unbounded operator.

9.2

Time-Periodic Bifurcation

In spite of its great relevance and frequent occurrence in experimental fluid mechanics, time-periodic bifurcation in a flow past an obstacle has represented a long-standing and intriguing problem from a rigorous mathematical viewpoint. This situation should be contrasted with flow in a bounded domain where, thanks to the pioneering and fundamental contributions of Iudovich [66], Joseph and Sattinger [68], and Iooss [65], complicated time-periodic bifurcation phenomena, like those occurring in the classical Taylor-Couette experiment, could be framed in a rigorous mathematical setting. In order to understand the reason for this uneven situation and also provide a motivation for the approach presented here, it is appropriate to briefly describe what constitutes a rigorous treatment of the phenomenon of time-periodic bifurcation. Suppose, as will be in fact shown later on, that the relevant time-dependent problem can be formally written in the form ut C L.u/ D N .u; /;

(86)

where L is a linear differential operator (with appropriate homogeneous boundary conditions) and N is a nonlinear operator depending on the parameter 2 R; such that N .0; / D 0 for all admissible values of . Then, roughly speaking, timeperiodic bifurcation for (86) amounts to show the existence a family of nontrivial time-periodic solutions u D u. I t / of (unknown) period T D T . / (T -periodic solutions) in a neighborhood of D 0 and such that u. I / ! 0 as ! 0. Setting WD 2 t =T  ! t; (86) becomes ! u C L.u/ D N .u; /;

(87)

and the problem reduces to find a family of 2-periodic solutions to (87) with the above properties. If one now writes u D u C .u  u/ WD v C w; one gets that (87) is formally equivalent to the following two equations L.v/ D N .v C w; / WD N1 .v; w; /; ! w C L.w/ D N .v C w; /  N .v C w; / WD N2 .v; w; /:

(88)

At this point, the crucial issue to realize is that while in the case of a bounded flow, both “steady-state” component, v; and “oscillatory” component, w; may be taken in the same (Hilbert) function space [65, 66, 68]; in the case of an exterior flow,

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v belongs to a space with quite less “regularity” (in the sense of behavior at large spatial distances) than w does; see also [51]. For this basic reason, as emphasized for the first time only recently in [49, 50], in the case of an exterior flow, it is not appropriate (or even “natural”) to investigate the bifurcation problem for (87) in just one functional setting, as done, for example, in [99]; it is instead much more spontaneous to study the two equations in (88) in two different function classes. As a consequence, even though formally being the same as differential operators, the operator L in (88)1 acts on and ranges into spaces different than those the operator L in (88)2 does. With this in mind, (88) becomes L1 .v/ D N1 .v; w; / I ! w C L2 .w/ D N2 .v; w; /: The above ideas will be next applied to provide sufficient conditions for timeperiodic bifurcation in a viscous flow past a body. It will be assumed throughout T D 0, leaving the case T ¤ 0 as an open question. Set L1 W v 2 X .˝/ 7! L .0 ; v/ 2 D01;2 .˝/

(89)

with L .0 ; v/ defined in (76). From Lemma 10 and Remark 6, it follows that under the assumption N .L1 / D f0g;

(H1)

there exists a unique weak solution analytic branch vs ./ WD u./ C U to (8)–(7) in a neighborhood U .0 /; with vs .0 / D u0 C U . Thus, writing v D v.x; tI / C vs .xI /; from (5), one finds that v formally satisfies the (nondimensional) problem

 vt C .v  e 1 /  rv C vs ./  rv C v  rvs ./ D v  rp in ˝ R div v D 0 v D 0 at @˝ R; lim v.x; t / D 0; t 2 R: jxj!1

(90) The bifurcation problem consists then in finding sufficient conditions for the existence of a nontrivial family of suitably defined time-periodic weak solutions to (90), v.tI /;  2 U .0 /; of period T D T ./ (unknown as well), such that v.tI / ! 0 as  ! 0 . Following the general approach mentioned before, one thus introduces the scaled time WD ! t; split v and as the sum of its time average, v; over the time interval Œ; ; and its “purely periodic” component w WD v  v; and set WD   0 . In this way, problem (90) can be equivalently rewritten as the following coupled nonlinear elliptic-parabolic problem

7 Steady-State Navier-Stokes Flow Around a Moving Body

   v C 0 @1 v  v0  rv  v0  rv D rp C N 1 .v; w; / in ˝ div v D 0 v D 0 at @˝; lim v.x/ D 0

383

(91)

jxj!1

and   9 ! w  w  0 @1 w v0  rw  w  rv0 = in ˝2 D r' C N 2 .v; w; / ; div w D 0 w D 0 at @˝ .; /; lim w.x; t / D 0;

(92)

jxj!1

where N 1 WD  Œ@ 1 v  vs . C 0 /  rv  v  rvs . C 0 /

C 0 .vs . hC 0 /  v0 /  rviC v  r.vs . C 0 /  v0 /

(93)

N 2 WD Œ@1hw  vs . C 0 /  rw  w  rvs . C 0 / i  0 .vs . C 0 /  v0 /  rw C w  r.vs . C 0 /  v0 / h i C . C 0 / w  rv C v  rw C w  rw  w  rw ;

(94)

C . C 0 / v  rv C w  rw

and

with v0  vs .0 /. The next step is to rewrite (91), (92), (93), and (94) in the proper functional setting and to reformulate the bifurcation problem accordingly. To this end, one begins to introduce the operator

L2 W w 2 D.L2 /  H .˝/ 7!  P w C 0 .@1 w  v0  rw  w  rv0 / 2 H .˝/; D.L2 / WD W 2;2 .˝/ \ D01;2 .˝/: (95) The following result can be proved by the same arguments (slightly modified in the detail) employed in [50, Proposition 4.2]. ˚  Lemma 12. Let u0 WD v0  U 2 X .˝/. Then Sp.L2C / \ i R  f0g consists, at most, of a finite or countable number of eigenvalues, each of which is isolated and of finite (algebraic) multiplicity, that can only accumulate at 0. Consider, next, the time-dependent operator

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Q W w 2 W 22;0 .˝/ 7! !0 wt C L2 .w/ 2 H2;0 .˝/:

(96)

Again, by a slight modification of the argument used in the proof of [50, Proposition 4.3], one can show the following. Lemma 13. Let v0 be as in Lemma 12. Then, the operator Q is Fredholm of index 0, for any !0 > 0. Finally, one needs the functional properties of the quantities N i ; i D 1; 2; defined in (93)–(94), reported in the following lemma. The proof is, one more time, a slight modification of that given in [50, Lemma 4.5, and the paragraph after it] and will be omitted. Lemma 14. There is a neighborhood V.0; 0; 0/  R X .˝/ W that maps

2 2;0 .˝/

such

N1 W . ; v1 ; v2 / 2 V.0; 0; 0/ 7! P N 1 . ; v1 ; v2 / 2 D01;2 .˝/ N2 W . ; v1 ; v2 / 2 V.0; 0; 0/ 7! P N 2 . ; v1 ; v2 / 2 H2;0 .˝/ are analytic. Also in view of Lemmas 12–14, one then deduces that (91)–(94) can be put in the following abstract form L1 .v/ D N1 . ; v; w/ in D01;2 .˝/ I ! w C L2 .w/ D N2 . ; v; w/ in H2;0 : (97) Notice that the spatial asymptotic conditions on v in (91)4 are interpreted in the sense of Remark 2, while the one in (91)4 for w holds uniformly pointwise for a.a. t 2 R; see [50, Remark 3.2]. One is now in a position to give a precise definition of a time-periodic bifurcation point. Definition 3. The triple . D 0; v D 0; w D 0/ is called time-periodic bifurcation point for (97) if there is a sequence f. k ; !k ; vk ; wk /g  R RC D01;2 .˝/ W 22;0 with the following properties: (i) f. k ; !k ; vk ; wk /g solves (97) for all k 2 N. (ii) f. k ; vk ; wk /g ! .0; 0; 0/ as k ! 1. (iii) wk 6 0; for all k 2 N. Moreover, the bifurcation is called supercritical [resp. subcritical] if the above sequence of solutions exists only for k > 0 [resp. k < 0].

7 Steady-State Navier-Stokes Flow Around a Moving Body

385

The goal is to give sufficient conditions for the occurrence of time-periodic bifurcation in the sense specified above. This will be achieved by means of the general result proved in [50, Theorem 4.1]. With this in mind, one has to show that the assumptions of that theorem are indeed satisfied. In this regard, supported by Lemma 12, one supposes 0 WD i !0 is an eigenvalue of multiplicity 1 of L2C ; k 0 ; k 2 N  f0; 1g is not an eigenvalue of L2C :

(H2)

Next, consider the operator L2 . / WD L2  S; with  

S W w 2 Z 2;2 .˝/ 7! P @1 wv0 rwwrv0 0 vP s .0 /rwCwr vP s .0 / 2 H .˝/; where, as before, “” means differentiation with respect to . By [108, Proposition 79.15 and Corollary 79.16], one knows that for in a neighborhood of 0, there is a smooth map 7! . /; with . / simple eigenvalue of L2C . / and such that 0 D .0/. The following condition will be further assumed: < Œ.0/ P ¤ 0;

(H3)

which basically means that the eigenvalue . / must cross the imaginary axis with “nonzero speed” when  ! 0 . The general result proved in [50, Theorem 3.1] can be now applied to show the following time-periodic bifurcation result. Theorem 12. Suppose (H1)–(H3) hold. Then, the following properties are valid. (a) Existence. There are analytic families 

 v."/; w."/; !."/; ."/ 2 X .˝/ W 22;0 .˝/ RC R

(98)

satisfying (97), for all " in a neighborhood I.0/ and such that   v."/; w."/  " v1 ; !."/; ."/ ! .0; 0; !0 ; 0/ as " ! 0: (a) Uniqueness. There is a neighborhood U .0; 0; !0 ; 0/  X .˝/ W 22;0 .˝/ RC R such that every (nontrivial) 2-periodic solution to (97), .z; s/; lying in U must coincide, up to a phase shift, with a member of the family (98).

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G.P. Galdi and J. Neustupa

(a) Parity. The functions !."/ and ."/ are even: !."/ D !."/; ."/ D ."/; for all " 2 I.0/. Consequently, the bifurcation due to these solutions is either subcritical or supercritical, a two-sided bifurcation being excluded (unless  0).

Open Problem. Sufficient conditions for the occurrence of time-periodic bifurcation in the case when the body is also spinning (T ¤ 0) are not known.

10

Stability and Longtime Behavior of Unsteady Perturbations

In this section, v0 will denote the velocity field of a steady-state solution to (8). As usual,  is assumed to be positive. However, since the theories that will be described in this section have often been equally developed for both cases  D 0 and  6D 0; a number of cited results also concern the case when  D 0. The function v0 is supposed to satisfy v0 2 L3 .˝/;

@j v0 2 L3 .˝/ \ L3=2 .˝/

(for j D 1; 2; 3):

(99)

It follows from Lemma 9 that, if  ¤ 0; such a v0 exists for a large class of body forces f and boundary data. An associated pressure field is denoted by p0 . One is interested in the behavior of unsteady perturbations, .v0 ; p 0 / to the solution .v0 ; p0 /. Thus, writing v D v0 C v0 ; p D p0 C p 0 ; it follows from (5) that the functions v0 ; p 0 satisfy the equations v0t C v0 C  @1 v0 C T .e 1 x  rv0  e 1 v0 / D v0  rv0 C v0  rv0 C v0  rv0 C rp 0 0

div v D 0

9 > = > ;

in ˝ .0; 1/

(100)

and the conditions v0 D 0

at @˝ .0; 1/ I

lim v0 .x; t / D 0; all t 2 .0; 1/:

jxj!1

(101)

For simplicity, from now on the primes are omitted in the notation above. Thus, the formal application of the Helmholtz-Weyl projection P to the first equation in (100), as formulated in Lq .˝/ (1 < q < 1), yields the operator equation

7 Steady-State Navier-Stokes Flow Around a Moving Body

dv D Lv C Nv dt

387

(102)

in the space Hq .˝/. By suitably defining the domains of the operators L and N; it can be easily seen that (102) is, in fact, equivalent to (100) and (101). To this end, let Av WD P v; B1 v WD P @1 v 1;q

for v 2 D.A/ WD W 2;q .˝/ \ D0 .˝/; B2 v

WD P .e 1 x  rv  e 1 v/;

A;T v WD Av C B1 v C T B2 v for ( v 2 D.A;T / WD

1;q

W 2;q .˝/ \ D0 .˝/ if T D 0;  ˚ 1;q 2;q q v 2 W .˝/ \ D0 .˝/I e 1 x  rv 2 L .˝/ if T 6D 0:

Note that A  A0;0 and A;0  A C B1 are the classical Stokes and Oseen operators, respectively. Furthermore, let B3 v WD P .v0  rv C v  rv0 /; Lv WD A;T v C B3 v; Nv WD  P .v  rv/ for v 2 D.L/ WD D.A;T /. Obviously, the study of the stability of the solution .v0 ; p0 / is equivalent to that of the zero solution of problem (100), (101) or equation (102). The properties of the linear operator L and, especially, those of its “leading part” A;T play a fundamental role. Thus, the next two subsections will be concerned with a detailed analysis of these properties.

10.1

Spectrum of Operator A,T

The following notions and definitions from the spectral theory of linear operators will be relevant later on. Let X be a Banach space with norm k : k; X  be its dual, and T be a closed linear operator in X with a domain D.T / dense in X . (This guarantees that the adjoint operator T  exists.)

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– The symbols nul .T / and def .T / denote the nullity and the deficiency of T; respectively. If R.T / is closed, then nul .T / D def .T  / and def .T / D nul .T  / (see, e.g., Kato [69, p. 234]). – The approximate nullity of T; denoted by nul 0 .T /; is the maximum integer m (m D 1 being permitted) with the property that to each  > 0; there exists an m-dimensional linear manifold M in D.T / such that kT vk <  for all v 2 M ; kvk D 1. The approximate deficiency of T is denoted by def 0 .T / and defined as def 0 .T / WD nul 0 .T  /. Note that nul .T /  nul0 .T / and def .T /  def0 .T /; the equalities holding if the range R.T / is closed. ETC. On the other hand, if R.T / is not closed, then nul 0 .T / D def 0 .T / D 1. The identity nul 0 .T / D 1 is equivalent to the existence of a non-compact sequence fun g on the unit sphere in X such that T un ! 0 for n ! 1 (see [69, p. 233]). – T is called a Fredholm operator if both the numbers nul .T / and def .T / are finite. This implies, in particular, that R.T / is closed in X [108, Proposition 8.14(ii)]. Operator T is semi-Fredholm if the range R.T / is closed in X and at least one of the numbers nul .T / and def .T / is finite. Consequently, T is semiFredholm if and only if at least one of the numbers nul 0 .T / and def 0 .T / is finite. – The resolvent set Res.T / is the set of all  2 C such that R.T  I / D X and the operator T  I has a bounded inverse in X . Consequently, nul .T  I / D nul 0 .T  I / D def .T  I / D def 0 .T  I / D 0 for  2 Res.T /. Note that Res.T / is an open subset of C. – The point spectrum Spp .T / is the set of all  2 C such that nul .T  I / > 0. – The continuous spectrum Spc .T / is the set of all  2 C such that nul .T  I / D 0; R.T  I / is dense in X; but R.T  I / 6D X . (In this case, R.T  I / is not closed in X; which implies that def .T  I / D def 0 .T  I / D nul 0 .T  I / D 1.) – The residual spectrum Spr .T / is the set of all  2 C such that nul .T  I / D 0 and the range R.T  I / is not dense in X . The sets Spp .T /; Spc .T /; and Spr .T / are mutually disjoint and Spp .T / [ Spc .T / [ Spr .T / D Sp.T / D C X Res.T / (the spectrum of T ). – The essential spectrum Spess .T / is the set of all  2 C such that T  I is not semi-Fredholm. Both Sp.T / and Spess .T / are closed in C and Spess .T /  Sp.T /. Obviously, Spc .T /  Spess .T /. Any point on the boundary of Sp.T / belongs to Spess .T / unless it is an isolated point of Sp.T / (see [69, p. 244]). From [70] (if T D 0) and [102] (if T 6D 0), it follows that the operator A;T is closed in Hq .˝/ (1 < q < 1), and all  2 C with a sufficiently large real part belong to Res.A;T /. The effective shapes and types of spectra of the operator A;T ; for various values of  and T ; are described in [18] (in H .˝/; the case  D 0), [19] (in H .˝/; the general case  2 R), [22] (in Hq .˝/;  D 0), and [20] (in Hq .˝/;  2 R). The spectrum of A;0 ; as an operator in H .˝/; was studied by K.I. Babenko [4]. Babenko’s result says that Sp.A;0 / D Spc .A;0 / D ;0 ; where ;0 D f D ˛ C iˇ 2 CI ˛; ˇ 2 R; ˛  ˇ 2 =2 g

(103)

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389

for  6D 0. The set ;0 represents a parabolic region in C; symmetric about the real axis, which shrinks to the nonnegative part of the real axis if  ! 0. In fact, Sp.A/ ( Sp.A0;0 /) coincides with Spc .A/ and coincides with the interval .1; 0 in R; as mentioned, e.g., by O.A. Ladyzhenskaya in [80]. The spectrum of A;T for general T is studied in [20]. Notice that the case T 6D 0 is qualitatively different from the case T D 0; because the magnitude of the coefficient of the “new” term T e 1 x  rv becomes unbounded as jxj ! 1. Consequently, the operator T e 1 x  r cannot be treated as a lower-order perturbation of Stokes or Oseen operator. The main results in [20] read as follows. Theorem 13. Let 1 < q < 1;  6D 0 and ˝ D R3 . Then the spectrum of A;T ; as an operator in Hq .R3 /; satisfies the identities Sp.A;T / D Spc .A;T / D Spess .A;T / D ;T ; where ;T WD f D ˛ C iˇ C ikT 2 CI ˛; ˇ 2 R; k 2 Z; ˛  ˇ 2 =2 g: Note that ;T is a union of a family of overlapping solid parabolas, whose axes form an equidistant system of half-lines f 2 CI  D ˛ C kT i; ˛  0; k 2 Zg. All the parabolas lie in the half-plane Re   0; and their vertices are on the imaginary axis. Theorem 14. Let 1 < q < 1;  6D 0 and ˝  R3 be an exterior domain with the boundary of class C 1;1 . Then the spectrum of A;T lies in the left complex halfplane f 2 CI Re   0g and consists of the essential spectrum Spess .A;T / D ;T and possibly a set  of isolated eigenvalues  2 C X ;T with Re  < 0 and finite algebraic multiplicity, which can cluster only at points of Spess .A;T /. The set  of such isolated eigenvalues is independent of q 2 .1; 1/. Sketch of the proof of Theorem 13 (see [20] for the details). The proof develops along the following steps (a)–(f). (a) Using the definition of the adjoint operator, it can be verified that the adjoint operator A;T to A;T coincides with the operator A;T in Hq 0 .˝/; where 1=q C 1=q 0 D 1. (b) From [17, Theorem 1.1], one can deduce that there exist constants C4 > 0 and C5 > 0 such that if u 2 D.A;T / and f 2 Hq .˝/ satisfy the equation A;T u D f ; then kuk2;q C k.! x/  rukq  C4 kf kq C C5 kukq :

(104)

(c) If  2 CX;T ; then each solution of the resolvent equation .A;T I /u D f ; for f 2 Hq .˝/; satisfies the estimate kukq  C6 kf kq ;

(105)

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where C6 D C6 .; q/. This estimate is derived by means of the Fourier transform, and a subtle and rather technical application of the Mikhlin-Lizorkin multiplier theorem. Using inequalities (104) and (105), one can prove that R.A;T  I / is closed and the operator A;T  I is injective. The same statement also holds on the adjoint operator A;T in Hq 0 .˝/. As A;T  I is injective, the range R.A;T  I / is the whole space Hq .˝/. Consequently,  2 Res.A;T /. This shows that C X ;T  Res.A;T /. (d) We show that Spp .A;T / D ;. Assume that  2 ;T and u 2 D.A;T / satisfies the equation .A;T I /u D 0. Applying the Fourier transform F; this equation yields T .e 1   rb u/  .  i1 C jj2 /b u  T e1 b v D 0;

(106)

where b u D F.u/ and  D .1 ; 2 ; 3 / denotes the Fourier variable. The case 0 1 < q  2 is simpler, because b u is a function from Lq .˝/: if 1 ; r; and  denote the cylindrical coordinates in the space of Fourier variables, then one can calculate that e 1   rb u D @b u. Substituting this to (106), one obtains the equation T @b u  .  i1 C jj2 /b u  T e1 b v D 0: If O./ denotes the matrix of rotation about the 1 -axis by angle  and b w.1 ; r; / WD O./b u.1 ; r; /; then one arrives at the ordinary differential equation w  .  i1 C r 2 C 12 / b w D 0: T @ b This equation can be solved explicitly. The solution satisfies: b w.1 ; r;  C 2 .i1 Cr 2 C12 / . As b w is 2-periodic in variable  and 2/ D b w.1 ; r; / e Re  C r 2 C 12 D Re  C jj2 6D 0 for a.a.  2 R3 ; b w is equal to zero a.e. in R3 . It means that u is the zero element of Hq .˝/; which implies that it cannot be an eigenfunction and  therefore cannot be an eigenvalue. The case 2 < q < 1 is rather more complicated becauseb u is only a tempered distribution. Nevertheless, one can also arrive at the same conclusion, i.e., that any  2 ;T cannot be an eigenvalue of A;T . (e) The identity Spr .A;T / D ; can be proven by means of the duality argument:  2 Spr .A;T / would imply that  2 Spp .A;T /. However, the same considerations as in step (d), applied to the adjoint operator A;T ; show that Spp .A;T / D ;. (f) The identities Sp.A;T / D Spc .A;T / D Spess .A;T / follow from the facts that Spp .A;T / and Spr .A;T / are empty and Spc .A;T /  Spess .A;T /. The inclusion Sp.A;T /  ;T follows from item (c). The inclusion ;T  Spess .A;T / is proven in [20] so that  is assumed to be in ı;T (the interior of ;T ), and a concrete sequence fun g; such that k.A;T  I /un kq ! 0

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for n ! 1; is constructed on the unit sphere in Hq .˝/. The construction is quite technical, so the readers are referred to [20] for the details. Thus,  2 Spess .A;T /; which implies that ı;T  Spess .A;T /. The inclusion ;T  Spess .A;T / now follows from the fact that Spess .A;T / is closed. t u Sketch of the proof of Theorem 14 (see [20] for the details.) The proof is a consequence of the following steps (g)–(j). (g) One can show the same inequality as (104), applying the cutoff function technique and splitting the equation A;T u D f into an equation for the unknown u1 in a bounded domain ˝ (where  > 0 is sufficiently large) and an equation for the unknown u2 in the whole space R3 . Due to [17, Theorem 1.1], the function u2 satisfies (104), while u1 satisfies (104) because B1 u and T B2 u can be brought into the right-hand side and then one can apply the estimates of solutions of the Stokes problem in a bounded domain. The Lq -norms of B1 u and T B2 u over ˝ can be interpolated between kukq and ku1 k2;q ; and the norm ku1 k2;q can be absorbed by the left-hand side. Finally, the sum of the estimates of u1 (over ˝ ) and u2 (over R3 ) leads to (104). (h) The inclusion ;T  Spess .A;T /: assume that  2 ı;T . By analogy with (f), one can construct a sequence fun g on the unit sphere in Hq .˝/; such that k.A;T  I /un kq ! 0 for n ! 1. Since Spp .A;T / is not known to be empty, and it is necessary to show that  2 Spess .A;T /; it is important that the sequence fun g is non-compact in Hq .˝/. The details can be found in [20], where the functions un are defined so that they have compact supports Sn in ˝; and the intersection \1 nD1 Skn (where fSkn g is any subsequence of fSn g) is empty. (i) The opposite inclusion Spess .A;T /  ;T : if  2 Spess .A;T /; then, by definition, nul 0 .A;T I / D 1 or def 0 .A;T I / D 1. The latter means that nul 0 .A;T  I / D 1. Thus, nul 0 .A;T  I / may be assumed to be infinity; otherwise one can deal with the operator A;T instead of A;T . The identity nul 0 .A;T  I / D 1 enables one to construct, by mathematical induction, kun kq D 1; k.A;T  I /un kq ! 0 a sequence fun g in D.A  n ;T / satisfying  as n ! 1 and dist u I Ln1 D 1 for all n 2 N; where Ln1 denotes the linear hull of the functions u1 ; : : : ; un1 . Using a cutoff function technique, the functions un can be modified so that they are all supported for jxj >  (for sufficiently large ), and the modified functions (let us denote them e un ) are on the unit sphere in Hq .˝/ and satisfy k.A;T  I /e un kq ! 0 for n ! 1 as un can be considered to be a function from well. However, as suppe un  ˝; e D..A;T /R3 /; where .A;T /R3 denotes the operator A;T in Hq .R3 /. This yields the equality nul 0 ..A;T /R3  I / D 1; which implies, due to item (c) that  2 ;T . (j) The domain  2 C X ;T consists of points in Res.A;T / and possibly also of isolated eigenvalues of A;T with finite algebraic multiplicities, which may possibly cluster only at points of @;T . (See [69, pp. 243, 244].) Assume that

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 2 C X ;T is an eigenvalue of A;T with an eigenfunction u. Applying again an appropriate cutoff function technique and treating the equation .A;T  I /u D 0 separately in a bounded domain ˝ (for a sufficiently large ) and in the whole space R3 ; one can show that u is in W 2;s .˝/ for any 1 < s < 1. (This follows from estimates valid in a bounded domain and the result from item (c), implying that  in the resolvent set of .A;T /R3 .) Finally, multiplying the equation .A;T  I /u D 0 by u and integrating in ˝; one can show that Re  < 0. t u When q D 2; in [19] it is shown that if B is axially symmetric about the x1 axis, then Sp.A;T / D ;T . It means that the set of eigenvalues of A;T ; lying outside ;T ; is empty. The same statement for operator A;T in Hq .˝/ for general q 2 .1; 1/ follows from Theorem 14. The proof in [19] comes from the fact that an eigenfunction u; corresponding to a hypothetic eigenvalue ; is 2-periodic in the cylindrical variable, which is the angle ' measured about the x1 -axis. Then the proof uses the Fourier expansion of u in ' and splitting of the equation .A;T  I /u D 0 to individual Fourier modes. Open Problem. In the general case when body B is not axially symmetric, it is not known whether the set of eigenvalues of A;T in C X ;T is empty.

Finally, note that Sp.A0;T / can be formally obtained, by letting  ! 0 in ;T . Then ;T shrinks to a system of infinitely many equidistant half-lines. The spectrum of operator A0;T is studied in detail in [22].

10.2

A Semigroup, Generated by the Operator A,T

10.2.1 The Case T D 0 It is well known that the Stokes operator A generates a bounded analytic semigroup, eAt ; in Hq .˝/ [55]. The fact that the Oseen operator A;0  ACB1 also generates an analytic semigroup in Hq .˝/ was proved by T. Miyakawa [89]. The main tool is the inequality kB1 ukq   kAukq C C ./ kukq

(107)

for all u 2 D.A/ and  > 0; which implies that B1 is relatively bounded with respect to A with the relative bound equal to zero. Then the existence and analyticity of the semigroup e.ACB1 /t follow, e.g., from [69, Theorem IX.2.4]. The so-called Lr –Lq estimates of the semigroup eA;0 t play an important role in the analysis of stability of steady flow. They were first derived by T. Kobayashi and Y. Shibata in [70], whose main result is given next.

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Theorem 15. If 1 < r  q < 1; then there exists C D C .; q; r/ > 0 such that keA;0 t akq  C t

 32





1 1 r q

kakr

(108)

for all a 2 Hr .˝/ and t > 0. Moreover, if 1 < r  q  3; then jeA;0 t aj1;q  C t

 32



1 1 r q



 12

kakr

(109)

for all a 2 Hr .˝/ and t > 0. Sketch of the proof (see [70] for the details.) Following [70], the starting point is the following representation formula of the semigroup eA;0 t a D

1 2it

Z

!Ci1

@ .A;0  I /1 a d ; ! > 0: @

et !i1

In order to estimate .A;0  I /1 a; the Oseen resolvent problem .A;0  /a D f is split into the problem in the bounded domain ˝ (for sufficiently large ) and in the whole space R3 . The estimates in ˝ follow from the fact that the Oseen operator in ˝ has a compact resolvent and the spectrum (which coincides with the point spectrum) is in the left half-plane in C; with a positive distance from the imaginary axis. The estimates in R3 are obtained by means of the Fourier transform and the Mikhlin-Lizorkin multiplier theorem. The next step is the construction of a parametrix, which enables the authors to combine the estimates in ˝ and in R3 and obtain the estimates of j.A;0  I /1 aj2;r and jj k.A;0  I /1 akr in terms of C kakr in the exterior domain ˝. Then the limit procedure for ! ! 0C is considered. However, due to subtle technical reasons, the limit procedure works only in a norm over a bounded domain and one only gets the inequality 2 A;0 t k@m akrI ˝  C t 3=2 kakr t r e

(110)

for t  1n and a 2 Hr .˝/ with the support in ˝ ; where C D C .m; r; ; /. On the other hand, using the formula  u.x; t / D

1 4t

3=2 Z

2 =4t

ejxtyj

a.y/ d y:

R3

for solution of the unsteady Oseen equations 9 ut C u C  @1 u D 0 = div u D 0

;

in R3 .0; 1/

(111)

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with the initial condition u.x; 0/ D a.x/ (for x 2 R3 ), one can derive the estimate j

k@t r k u.t /kqI R3  C t

 32



1 1 r q



 k2

kakr

(112)

for all a 2 Hr .R3 / and t  1. The constant C on the right-hand side depends only on j; k; q; r; and . Finally, combining appropriately (110) with (112), one can obtain (108) and (109). 

10.2.2 The Case T 6D 0 The operator A;T  A C B1 C T B2 is the Oseen operator with the effect of rotation. T. Hishida [62] considered the case  D 0 and proved that A0;T  ACT B2 generates a C0 -semigroup eA0;T t in H .˝/. M. Geissert, H. Heck, and M. Hieber [53] also considered  D 0 and proved that A0;T generates a C0 -semigroup eA0;T t in Hq .˝/ for 1 < q < 1. The case  6D 0 was studied by Y. Shibata in [102], whose main finding is given next. Theorem 16. Let 1 < r  q < 1. The operator A;T generates a C0 -semigroup eA;T t in Hq .˝/. Moreover, it satisfies the same inequalities (108) and (109) as the semigroup eA;0 t . Sketch of the proof (see [102] for the details.) One begins to study the linear Cauchy problem, defined by the equations 9 ut C u C  @1 u C T .e 1 x  ru  e 1 u/ D rp = div u D 0

;

in R3 .0; 1/ (113)

and the initial condition u.x; 0/ D a.x/. The solution u.x; t / is expressed by means of the Fourier transform, and the solution of the corresponding resolvent problem with the resolvent parameter  (denoted by AR3 ;T ; ./) is expressed by the combined Laplace-Fourier transform. Then the estimates of kAR3 ;T ; ./kqI R3 and jAR3 ;T ; ./jm;qI R3 in terms of powers of jj and kakq are derived. The solution AR3 ;T ; ./ is then split into the part A1 ./; which “neglects” the term T .e 1 x  ru  e 1 u/ in (113), and A2 ./; which is a correction due to this term. While the estimates of A1 ./ are shown in a similar way and for the same values of  as in the proof of Theorem 15, the estimates of A2 ./ impose sharper restrictions on  and hold only for  2 CC WD f 2 CI Re  > 0g. However, remarkably enough, in [102], subtle estimates of A2 . C is/ (for  > 0 and s 2 R) are derived independent of  (for 0 <  < 0 ), provided a has a support in BR .0/ for R > 0. The essential role in the expression of the solution of (113) is played by the integral of A2 ./ on the line f D  C isI s 2 Rg; parallel to the imaginary axis. The estimates independent of  enable one to pass to the limit for  ! 0. Then the appropriate

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cutoff function procedure and the limit process for R ! 1 lead to an expression that confirms that u. : ; t / depends on the initial datum a through a C0 -semigroup. One can immediately observe from the shape of the spectrum of the operator A;T (see Theorems 13 and 14) that A;T is not a sectorial operator in Hq .˝/. Thus, unlike the case T D 0; the semigroup generated by A;T is only a C0 -semigroup and not an analytic semigroup. The idea used to derive estimates analogous to (108) and (109) is similar to that employed in the proof of Theorem 15. However, in contrast to the case T D 0 (when, expressing the solution by the line integral on a line parallel to the imaginary axis, one especially needs to control the behavior of the resolvent for the values of the resolvent parameter  near 0), the case T 6D 0 requires the control of the resolvent “uniformly” on the whole line. This is caused by the fact that as the line approaches the imaginary axis in the considered limit procedure, it approaches the spectrum of operator A;T not only in the neighborhood of 0 but in the neighborhood of the infinitely many points ik T ; k 2 Z. 

10.3

Existence and Uniqueness of Solutions of the Initial–Boundary Value Problem

This subsection presents a brief survey of results on the existence and uniqueness of weak and strong solutions to the initial–boundary value problem, consisting of equation (5) and the initial condition v.x; 0/ D a.x/

for x 2 ˝:

(114)

Referring to other chapters in this handbook for a detailed analysis, here only those results are recalled that are relevant to our study. The definition of the weak solution, in the unsteady case, is analogous to that provided for the steady state problems (7), (8). More precisely, v is called a weak solution to problem (5), (114) if: (i) v 2 L1 .0; T I H .˝// \ L2 .0; T I D 1;2 .˝//; for all T > 0: (ii) lim kv.t /  ak2 D 0. t!0C

(iii) v satisfies (5) in the sense of distributions.

10.3.1 The Case T D 0 In the absence of rotation, existence results can be found in many works. They mostly concern the Navier-Stokes equations, but their extension to the more general problem (5), (114) with T D 0 is rather straightforward. The first results on the global in time existence of weak solutions, v; assuming the initial velocity a 2 H .˝/; are due to J. Leray [82] (for ˝ D R3 ) and E. Hopf [64] (for arbitrary open set ˝  R3 ). A more recent and detailed presentation of these classical results can be found, e.g., in the book [104] or in the survey paper [37]. In particular, one shows the existence of a weak solution for any a 2 H .˝/

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and f 2 L2 .0; T I D01;2 .˝// (and v  0 at @˝). However, the uniqueness of such solutions in the same class of existence remains an open problem. The weak solution is known to be unique if, in addition, it is in Lr .0; T I Ls .˝//; where 2  r  1; 3  s  1; and 2=r C3=s  1. More precisely, if v1 and v2 are two weak solutions, with v1 in the class Lr .0; T I Ls .˝// above and v2 satisfying the so- called energy inequality (see (119) with v0  0 and s D 0), then v1 D v2 . As the solutions in the class Lr .0; T I Ls .˝// satisfy the energy inequality automatically, one can speak of “uniqueness in the class Lr .0; T I Ls .˝//.” Following [104], a weak solution v in the class Lr .0; T ; Ls .˝// with r; and s as above, is called a strong solution. In addition to be unique, strong solutions are also known to be “smooth” (= regular), provided that the body force f is either a potential vector field (and can be therefore absorbed by the pressure term) or “sufficiently smooth.” (See, e.g., [37] for more details.) In particular, if @˝ is of class C 2 and f 2 L2 .0; T I L2 .˝//; then the strong solution v belongs to C ..; T /I H .˝// \ L2 .; T I W 2;2 .˝// for any  2 .0; T /. (It depends on the regularity of the initial velocity a whether  D 0 can also be considered.) For initial velocity a and body force f in appropriate function spaces and of “arbitrary size,” strong solutions are known to exist in some time interval .0; T0 /; but it is not known whether one can take T0 D 1; in general. If, however, the size of the data is sufficiently restricted, then one can show T0 D 1. There exists a vast literature on the subject dealing with various types of domains and different choices of functions spaces for a and f ; starting from the pioneering and fundamental papers of A. A. Kiselev and O. A. Ladyzhenskaya, G. Prodi, and H. Fujita and T. Kato in the early 1960s and continuing with J.G. Heywood (1980), T. Miyakawa (1982), H. Amann (2000), and R. Farwig, H. Sohr, and W. Varnhorn (2009). Among these papers, especially [60] (by Heywood), [89] (by Miyakawa), and [1] (by Amann) deal with the Navier-Stokes problems in exterior domains. An important result concerning the length of the time interval .0; T0 /; where a strong solution exists without restriction on the “size” of the data, states (e.g., [1] or [59]) that if f   is, e.g., in L2 0; 1I L2 .˝/ ; then either T0 D 1 or else jv. : ; t /j1;2 ! 1 for t ! T0  .

10.3.2 The Case T 6D 0 The existence of a weak solution to the problem (5), (114) with T 6D 0 has been proven by W. Borchers [7]. It also follows from a more general result proven in [94] on the existence of weak solutions in domains with moving boundaries. Recall that the weak solution is a function in the same class as for the case T D 0; i.e., belongs to L1 .0; T I H .˝// \ L2 .0; T I W01;2 .˝//. Regarding the local in time existence of a strong solution to the problem (5), (114) with T 6D 0; only a few results are available. Below the contributions of T. Hishida [62], G. P. Galdi and A. L. Silvestre [40], and P. Cumsille and M. Tucsnak [11] are explained. They all concern the case when the motion of body B in the fluid reduces to the rotation and the translational velocity is zero. It means that the term  @1 v in the momentum equation (5)1 vanishes.

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T. Hishida [62] assumes that the body force f is zero and the initial velocity is in D.A1=4 / and proves the existence of a solution in the class C .Œ0; T0 ; D.A1=4 // \ C ..0; T0 I D.A// for certain T0 > 0. Recall that A denotes the Stokes operator. Hishida’s proof is based on a nontrivial generalization of the semigroup method, formerly used by Fujita and Kato [30]. G. P. Galdi and A. L. Silvestre [40] also deal with the case of the zero body force f . They assume that the initial velocity a in W 2;2 .˝/ satisfies div a D 0 and e 1 x  ra 2 L2 .˝/; and they obtain a solution in C .Œ0; T0 I W 1;2 .˝// \ C ..0; T0 /I W 2;2 .˝// for some T0 > 0. The proof is based on the construction of classical Faedo-Galerkin approximations in ˝R ; getting a solution in ˝R and letting R ! 1. The procedure is, however, not standard because of the “troublesome” term T e 1 x  rv whose influence has to be controlled. P. Cumsille and M. Tucsnak [11] consider the equations of motion of the viscous incompressible fluid around body B in a frame in which the velocity of the fluid vanishes in infinity and the body is rotating with a constant angular velocity about one of the coordinate axes. Thus, the domain filled in by the fluid is time dependent and it is denoted by ˝.t /. The authors consider a body force f locally square integrable from .0; 1/ to W 1;1 .R3 / and the no-slip boundary condition for the velocity on @˝.t/. The main theorem from [11] says that if the initial velocity a is in W01;2 .˝.0// and it is then there 0 > 0 and  divergence-free,   exists T1;2  a unique 2 2;2 strong solution u 2 L 0; T I W .˝.t // \ C Œ0; T I W .˝.t // such that 0 0   ut 2 L2 0; T0 I L2 .˝.t // . Moreover, either T0 can be extended up to infinity or the norm of u in W 1;2 .˝.t // tends to infinity for t ! T0 . In order to obtain a problem in a fixed exterior domain, the authors use a change of variables which coincides with the rotation in the neighborhood of body B; but it equals the identity far from the body. Then they solve the problem in the fixed exterior domain ˝. Using the relations between the solutions of the equations in the frame considered in [11] on the one hand and the body fixed frame on the other hand, the result of Cumsille and Tucsnak from [11] can be reformulated in terms of solution to the problem (5), (114), as follows: given a 2 H .˝/ \ W01;2 .˝/; there exists T0 > 0 and a unique solution v of the problem (5), (114), such that )     v 2 L2 0; T0 I W 2;2 .˝/ \ C Œ0; T0 I W 1;2 .˝/ ;   vt  T .e 1 x  rv  e 1 v/ 2 L2 0; T0 I L2 .˝/ :

(115)

Cumsille and Tucsnak’s result is applied in Sect. 10.5. Since it is also used in the case  6D 0; we note that following the proof in [11] and using the fact that the translation-related term  @1 v in the first equation in (5) can be considered to be a subordinate perturbation of v; the above formulated result can be extended to the case when it also includes the translation of B in the direction parallel to the axis of rotation. Consequently, the result also holds for the equations in (5) with the term  @1 v and the second inclusion in (115) can be modified:   vt   @1 v  T .e 1 x  rv  e 1 v/ 2 L2 .0; T0 I L2 .˝/ :

(116)

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10.4

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Attractivity and Asymptotic Stability with Smallness Assumptions on v0

Recall that v0 denotes (the velocity field of) a solution to problem (8) (i.e., a steadystate solution to problem (5)) and that its associated “perturbation” v satisfies (100) and (101). Since in studying longtime behavior the dependence of v on time is more relevant than the one on spatial variables, in the following considerations, v.x; t / is often abbreviated to v.t /. Thus, for example, the initial condition (114) may be written in the form v.0/ D a:

(117)

10.4.1 The Case T D 0 A number of results concern the longtime behavior of the unsteady perturbations v in the class of weak solutions. The first relevant contribution in this direction is due to K. Masuda [88], who assumes that v0 is continuously differentiable, rv0 2 L3 .˝/ along with the smallness condition which, according to our notation, yields sup  jxj jv0 .x/j < x2˝

1 : 2

(118)

The perturbed unsteady solution is supposed to satisfy the momentum equation in (5) with a perturbed body force. Thus, the corresponding perturbation v satisfies (100), (101), with an additional right-hand side f 0 in (100), representing the perturbation to the steady body force f. The Helmholtz-Weyl projected function Pf 0 is assumed to be in C 1 .Œ0; 1/I H .˝// \ L1 .0; 1I H .˝// and such that R tC1 R1 supt>0 t k.d =ds/Pf 0 .s/k22 ds C 0 s 1=2 k.d =ds/Pf 0 .s/k2 ds < 1. As for the class of perturbations, the author assumes that v is a weak solution to the problem (100) (with a nonzero f 0 on its right-hand side), (101), and (117), with initial data a 2 H .˝/; and satisfies the so-called strong energy inequality, namely, 1 kv.t /k22 2

 12 kv.s/k22 

Z s

t



.v. /  rv0 ; v. // C jv. /j21;2 C .f 0 . /; v. // d ; (119)

for a.a. s > 0 (including s D 0) and all t 2 Œs; T ; arbitrary T > 0. Notice that the latter is formally obtained by multiplying equation (100) by v and integrating over ˝ .s; t / and relaxing the equality sign to the inequality one. Under the above conditions, Masuda shows that there exists T > 0 such that v.t / becomes regular for t > T and decays at the following rate jv.t /j1;2  C t 1=4 ; kv.t /k1  C t 1=8 ; for all t > T :

(120)

The proof uses (119) and the assumptions on the integrability of f 0 to deduce, first, that v.t / is “small” for large t . Then, combining this with the estimates of A1=2 v;

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one shows that v.t / is regular and tends to zero for t ! 1 in the norm j : j1;2 . The rate of decay is calculated from the energy-type inequality, satisfied by vt . The author also generalizes these results to the case when the unperturbed solution v0 is time dependent. It should be noted that in [88], no assumption on the size of the initial perturbation a is used: it can be arbitrarily large. However, nothing can be said about the behavior of v.t / for t 2 .0; T /. If v0  0; the decay rates (120) are sharpened by J.G. Heywood in [60]. The above results have been further elaborated on by P. Maremonti in [86]. Maremonti studied the attractivity of steady as well as unsteady solutions v0 to problem (5) in the same class of weak solutions considered by Masuda with f 0  0. In particular, for the case v0 steady, he shows the following decay rates kvt .t /k2  C t 1 ; jv.t /j1;2  C t 1=2 ; kv.t /k1  C t 1=2 ; thus improving and extending the analogous finding of [88] and [60]. Instead of condition (118), the author assumes that the maximum of certain variational problem involving v0 is not “too large.” The latter condition is certainly satisfied if v0 is sufficiently regular and obeys (118). The somehow more complicated question of asymptotic stability of v in the L2 norm was first addressed by P. Maremonti in [87]. In particular, he shows that all v in the class of weak solutions, with a 2 H .˝/ and satisfying the strong energy inequality (119) with f 0  0; must decay to 0 in the L2 -norm, provided that the magnitude of v0 is restricted in the same way as specified in [86] and discussed earlier on. An important contribution to the studies of the asymptotic stability of the steady solution v0 was also made by T. Miyakawa and H. Sohr in [90]. The authors show that if the basic steady solution v0 of (5) is such that v0 2 L1 .˝/; rv0 2 L3 .˝/; and the smallness condition (118) is satisfied, and if, in addition, the perturbation f 0 to the body force f is in L2 .Œ0; T /I H .˝// for all T > 0 and in L1 .Œ0; 1/I H .˝// ; then the L2 -norm of each weak solution v to problem (100), (101) satisfying the energy inequality (119) tends to 0 for t ! 1. In [90] it is also shown that the class of such weak solutions is not empty, thus solving a problem left open in [87] and partially solved in [33]. Further results concerning the L2 -decay of the perturbation v.t / (as a weak solution to (100), (101)) for t ! 1 are provided in the paper [8] by W. Borchers and T. Miyakawa: the authors assume that the steady solution v0 of (5) is in L3 .˝/; rv0 2 L3 .˝/; the smallness condition (118) is satisfied, and the perturbation f 0 to the body   force f is 1;2 2 1 1 in Lloc .Œ0; 1/I H .˝// \ L .0; 1I H .˝// \ L 0; 1I D0 .˝/ . They show that then the L2 -norm of each weak solution v to problem (100), (101) obeying (119) tends to 0 for t ! 1. Moreover, if keLt ak2 D O.t ˛ / for some ˛ > 0; then kv.t /k2 D O .ln t /1=2 for any  > 0. (Here, eLt denotes the semigroup generated by operator L; see Sect. 10.5.1.) The results of [8] are generalized by the same authors to the case of n-space dimensions (n  3) in [9].

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Even sharper rates of decay of the norms kv.t /kr (2  r  1) and krv.t /kr (2  r  3) were obtained by H. Kozono f 0 to  in [75], provided  the perturbation 1  1 2 2 the body force is in L 0; 1I L .˝/ \C .0; 1/I L .˝/ and decays like t for t ! 1. Kozono does not use any condition of smallness of the basic flow v0 or its initial perturbation a; but needs v0 in Serrin’s class Lr .0; 1I Ls .˝// (2=r C3=s D 1; 3 < s  1). This implies that v0 is in fact a strong Solution, and it is in a suitable sense small for large t. Obviously, the only time-independent solution in the considered Serrin class is v0 D 0. There exists a series of results on stability of solution v0 in the class of strong unsteady perturbations, which, unlike the cited papers [8, 9, 60, 86, 88], and [75], provide an information on the size of the perturbations at all times t > 0 and not just for “large” t . However, on the other hand, the initial value of the perturbation is always required to be “small” as well as v0 is also supposed to be “sufficiently small” in appropriate norms. The first results of this kind come from the early 1970s of the twentieth century, and new results on this topic still appear. The next paragraphs contain the sketch of the main steps to obtain a result of the above type. Assume that v is a strong solution to problem (100), (101) in the time interval .0; T0 /; for some T0 > 0. Multiplying the first equation in (100) (where v0 D v) by v and integrating by parts over ˝; one obtains 1 d kvk22 C jvj21;2 D  .v  rv0 ; v/   jv0 j1;3=2 kvk26 2 dt  C72  jv0 j1;3=2 jvj21;2 :

(121)

(The norm kvk6 has been estimated by Sobolev’s inequality: kvk6  C7 jvj1;2 ; see, e.g., [45, p. 54].) Multiplying the first equation in (100) by Av  Pv and integrating over ˝; one obtains 1 d jvj21;2 C kAvk22 D  ..@1 v C v0  rv C v  rv0 C v  rv/ ; Av/ d x 2 dt   1  kAvk22 C 42 k@1 vk22 C kv0  rvk20 C kv  rv0 k22 C kv  rvk22 : (122) 4 The first term on the right-hand side can be absorbed by the left- hand side. The other terms on the right-hand side can be estimated by means of the inequalities 1=2 1=2   jvj1;6  C krvk1;2 D C jvj21;2 C jvj22;2  C kAvk22 C jvj21;2 ; where the first one follows from the continuous imbedding W 1;2 .˝/ ,! L6 .˝/ and the second one follows, e.g., from [45, pp. 322–323]. Thus, if YŒv is defined by the formula YŒv WD kAvk22 C jvj21;2 ;

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then k@1 vk22  22 jvj21;2 ; kv0  rvk22  kv0 k26 jvj1;2 jvj1;6  C jv0 j21;2 jvj1;2 YŒv1=2     kAvk22 C C ./ jv0 j21;2 C jv0 j41;2 jvj21;2 ; kv  rv0 k22  jv0 j21;3 kvk26  C72 jv0 j21;3 jvj21;2 ; kv  rvk22  kvk26 jvj21;3  C72 jvj31;2 jvj1;6  C jvj31;2 YŒv1=2  C jvj21;2 YŒv: Employing these inequalities into (122) and choosing, e.g.,  D 14 ; one gets   d jvj21;2 C kAvk22  C8 2 1 C jv0 j21;2 C jv0 j41;2 jvj21;2 C C9 2 jvj21;2 YŒv: dt (123) Adding the inequalities (121) (multiplied by 2) and (123) (multiplied by ˛ > 0) and passing everything to the left-hand side, one obtains  d  kvk22 C ˛ jvj21;2 dt  

C jvj21;2 2  2C72  jv0 j1;3=2  C8 2 ˛ 1 C jv0 j21;2 C jv0 j41;2  C9 2 ˛ jvj21;2

C kAvk22 ˛  C9 2 ˛ jvj21;2  0: This implies that    d  kvk22 C ˛ jvj21;2 C jvj21;2 2  2C72  jv0 j1;3=2  C8 2 ˛ 1 C jv0 j21;2 C jv0 j41;2 dt  

 

C9 2 kvk22 C ˛ jvj21;2 C kAvk22 ˛  C9 2 kvk22 C ˛ jvj21;2  0: (124) This inequality shows that if   2C72  jv0 j1;3=2 C C8 2 ˛ 1 C jv0 j21;2 C jv0 j41;2 < 2

(125)

and kvk22 C ˛ jvj21;2 is initially so small that   C9 2 kak22 C ˛ jaj21;2 ˚    < min 2  2C72  jv0 j1;3=2  C8 2 ˛ 1 C jv0 j21;2 C jv0 j41;2 I ˛

(126)

(recall that v.0/ D a), then kv.t /k22 C˛ jv.t /j21;2 is nondecreasing for t in some right neighborhood of 0. This consideration can be simply extended, by the bootstrapping

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argument, to the whole interval of existence of the strong solution v (let it be .0; T0 /) so that one obtains: kv.t /k22 C ˛ jv.t /j21;2 < kak22 C ˛ jaj21;2 for all t 2 .0; T0 /. This shows, among other things, that the norm kv.t /k1;2 cannot blow up when t ! T0 . Consequently, T0 D 1 and the inequality kv.t /k22 C ˛ jv.t /j21;2 < kak22 C ˛ jaj21;2 holds for all t 2 .0; 1/. Note that if C72  jv0 j1;3=2 < 1;

(127)

then one can choose ˛ > 0 so small that (125) holds. (˛ is further supposed to be chosen in this way.) Integrating inequality (124) with respect to t; one can also derive an information on the integrability of YŒv.t / and on the asymptotic decay of jv.t /j1;2 . Thus, also including the information on the uniqueness of strong solutions and using the result of [87], one obtains the following Lyapunov-type asymptotic stability of v0 in the W 1;2 -norm. Theorem 17. Suppose the steady solution v0 to the problem (8) satisfies conditions (99) and (127) and ˛ > 0 is chosen so that (125) holds. Then, if a 2 H .˝/ \ W01;2 .˝/ satisfies (126), problem (100), (101) with the initial condition v.0/ D a has a unique strong solution v on the time interval .0; 1/. Furthermore, there exists C10 > 0 such that this solution satisfies Z t   jv.s/j21;2 C ˛ kAv.s/k22 ds  kak22 C ˛ jaj21;2 kv.t /k22 C ˛ jv.t /j21;2 C C10 0

(128) for all t > 0 and lim kv.t /k1;2 D 0:

t!1

(129)

The noteworthy assumption in the above theorem is condition (127) of “sufficient smallness” of the solution v0 . The ideas of proof described previously and similar energy-type considerations have been applied to many other studies of stability or instability of steady-state solutions to Navier-Stokes and related equations. Concerning flows in exterior domains, the readers are referred, e.g., to [31, 32, 34, 58, 59]. A different approach, based on a representation of a solution by means of semigroups generated by the operators A;0 or L and on estimates of the semigroups, has been employed by H. Kozono and T. Ogawa [73], H. Kozono and M. Yamazaki [74], and Y. Shibata [100]. In particular, Kozono and Yamazaki [74] study the flow in an exterior “smooth” domain ˝ in Rn (n  3), under the assumption that the translational velocity of the moving body is zero, which in our notation means  D 0 in the first equation in (100). The steady-state solution v0 is supposed to belong to Ln;1 .˝/\L1 .˝/; and its gradient is supposed to be in Lr .˝/ for some

r;q r 2 .n; 1/. (The Lorentz-type space L .˝/ for 1 < r < 1 and 1  q  1

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  is defined by means of the real interpolation to be Hr0 .˝/; Hr1 .˝/ ;q ; where 1 < r0 < r < r1 < 1 and 1 <  < 1 satisfy 1=r D .1   /=r0 C =r1 ; see r;q [74]. It is shown in [9] that L .˝/ coincides with the space of all u 2 Lr;q .˝/ such that div u D 0 in ˝ in the sense of distributions and u  n D 0 on @˝ in the sense of traces.) Equations (100) are treated in the equivalent form (102). The operators L and N are defined in the introductory part of this section for n D 3; but the definition in the general case n 2 N; n  3 is analogous. In the considered case, L has the concrete form L D A C B3 . The operator L generates a quasi-bounded analytic semigroup in Hq .˝/ – this is shown in [74] by means of appropriate resolvent estimates which imply that operator L is sectorial. The strong solution is identified with the mild solution, which satisfies the integral equation v.t / D e

Lt

Z

1

aC

eL.ts/ Nv.s/ ds:

(130)

0

The solution of this equation is constructed as a limit of a sequence of approximations, which are defined by the equations v0 WD eLt a and j

Z

0

t

v .t / WD v .t / C

eL.ts/ Nvj 1 .s/ ds

.j D 1; 2; 3; : : : /:

0 n



1



1



The authors define Kj WD sup0 0 be fixed. The operator A C .1 C /B3s is self-adjoint in H .˝/. The spectrum of AC.1C/B3s consists of Spess .A C .1 C /B3s / D .1; 0 and at most a finite set of positive eigenvalues, each of whose has a finite multiplicity. Let the positive eigenvalues be 1  2      N; each of them being counted as many times as is its multiplicity. Let 1 ; : : : ; N be the associated eigenfunctions. They can be chosen in a way that they constitute an orthonormal system in H .˝/. – Denote by H .˝/0 the linear hull of 1 ; : : : ; N and by P 0 the orthogonal projection of H .˝/ onto H .˝/0 . Furthermore, denote by H .˝/00 the orthogonal complement to H .˝/0 in H .˝/ and by P 00 the orthogonal projection of H .˝/ onto H .˝/00 . Then H .˝/ admits the orthogonal decomposition H .˝/ D H .˝/0 ˚ H .˝/00 ; and the operator A C .1 C /B3s is reduced on each

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of the subspaces H .˝/0 and H .˝/00 . Moreover, it is positive on H .˝/0 and nonpositive on H .˝/00 .   – Since A C .1 C /B3s ;   0 for all  2 H .˝/00 \ D.A/; operator L satisfies    .A; / .L; / D .A C B3s /;  2 D 1C  1   .A; / D C12 jj21;2 ; .A C B3s C B3s /;   C 1C 1C for all  2 H .˝/00 \ D.A/; where C12 D =.1 C /. This inequality expresses the so-called essential dissipativity of L in space H .˝/00 . – All functions 1 ; : : : ; N belong to D01;2 .˝/ (the dual to D01;2 .˝/). The main result of the paper [93] is the following. Theorem 21. Suppose that the steady solution v0 to the problem (8) satisfies conditions (99), and let  > 0 be so large that jv0 j1;3=2;˝   18 . Moreover, assume (A) there exists a function ' 2 L1 .0; 1/\L2 .0; 1/ such that keLt i k2I ˝  '.t/ for all i D 1; : : : ; N and t > 0. Then there are positive constants ı; C13 ; C14 such that if a 2 H .˝/ \ W01;2 .˝/ and kak1;2  ı; the equation (102) with the initial condition v.0/ D a has a unique solution v on the time interval .0; 1/. The solution satisfies kv.t /k21;2

Z C C13 0

t



 jv.s/j21;2 C kAv.s/k22 ds  C14 kak21;2

(135)

(for all t > 0) and lim jv.t /j1;2 D 0:

t!1

(136)

The proof is based on splitting the equation (102) into an equation in H .˝/00 ; where L is essentially dissipative, and a complementary equation, where one uses the decay of the semigroup eLt following from assumption (A). Theorem 21 tells us that the question of stability of the steady solution v0 reduces to the L1 - and L2 -integrability of a finite family of certain functions in the interval .0; 1/; i.e., condition (A). In the paper [15], the authors consider the case ˝ D R3 and show that condition (A) is indeed satisfied under some assumptions on the spectrum of L. The latter amounts to assume that all eigenvalues of L have negative real parts, without any request on the essential spectrum of L. The important tool used in [15] is the fundamental solution of the Oseen equation in R3 and the estimates for the corresponding resolvent problem.

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Sufficient conditions for the stability of the null solution to (100), in terms of eigenvalues of L and when ˝ 6 R3 ; have been recently formulated in the paper [95]. Here, the author shows at first that condition (A) in Theorem 21 can be replaced by the following one (B) Given  > 0 there exist functions ' 2 L1 .0; 1/ \L2 .0; 1/ and H .˝/ \ D01;2 .˝/ such that kj  j k1;2   ˇ Lt ˇ ˇ e i ; j ˇ  '.t/

1; : : : ;

for j D 1; : : : ; N; for t > 0 and i; j D 1; : : : ; N:

N

2

(137) (138)

ˇ ˇ (Condition (B) reduces to the requirement that ˇ eLt i ; j ˇ  '.t/ if one chooses j D j for j D 1; : : : ; N .) In [95] it is further assumed that the steady solution v0 is in Lr .˝/ for all r 2 .2; 1 and @j v0 2 Ls .˝/; for j D 1; 2; 3 and for all s 2 . 43 ; 1. This assumption is fulfilled if the acting body force f is in Lq .˝/ for all q 2 .1; q0 ; where q0 > 3; see Lemma 9. Moreover, if f has a compact support, then v0 D E.x/  m C v00 .x/; where m is a certain constant vector, E is the Oseen fundamental tensor, and v00 .x/ is a perturbation which decays faster than E.x/ for jxj ! 1; see Theorem 6. This form of v0 is used in [95], where the main result states the following. Theorem 22. Let the conditions (C1 ) there exists ı > 0 and a0 > 0 such that all eigenvalues  of operator L satisfy Re  < maxfıI a0 .Im /2 g, (C2 ) 0 is not an eigenvalue of the operator Lext be fulfilled. Then the conclusions of Theorem 21 hold. Here, Lext denotes the operator L with the domain extended to D 2;2 .˝/\D01;2 .˝/. Condition (C1 ) implies that L has no eigenvalues with nonnegative real parts. Sketch of the proof of Theorem 22 (see [95] for the details.) The proof is based on showing that Œ(C1 ) ^ (C2 ) H) (B). The function eLt i is expressed by the formula Lt

e

i D .2i/

1

Z 

et .I  L/1 i d ;

(139)

where   is a curve in C X Sp.L /; which depends on a parameter  > 0. Recall that Sp.L / consists of the essential spectrum in the half-plane f 2 CI Re   0g; see (103), and at most a countable number of isolated eigenvalues. Curve   has three parts 1 ; 2 ; and 3 ; where 1 and 3 coincide with the half-lines arg./ D  ˙ ˛; respectively, for some fixed ˛ 2 .0; =2/ and large jj. Both the curves 1 and 3 lie in the half-plane f 2 CI Re  < 0g. Since Spess .L / touches the

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imaginary axis at point 0; the curve 2 WD f 2 CI  D as 2 C .s02  s 2 / C is for  s0  s  s0 g (where a > 0 and s0 > 0 are appropriate fixed positive numbers) extends into the half-plane f 2 CI Re  > 0g. If  ! 0C; then 2 approaches 20 WD f 2 CI  D as 2 C is for  s0  s  s0 g; and consequently,   approaches  0 D 1 [ 20 [ 3 . (Number a > 0 is chosen so that Sp.L / lies on the left from  0 ; with the exceptionof point 0.) In order to verify inequality (138) in condition (B), one has to estimate eLt i; j . Since one can prove that the range of L  jD.˝/ (the adjoint operator to L reduced to D.˝/) is dense in D01;2 .˝/; in order to satisfy (137), one can choose functions j in the form j WD L  0j ; where 0j 2 D.˝/ (for j D 1; : : : ; N ). Then Z  Lt    1 et .I  L/1 i ; j d  e i ; j D 2i   Z   1 t .I  L/1 i ; .L   I C I / 0j d  e D 2i Z Z     1 1 t 0 e et  .I  L/1 i ; 0j d : (140) D i ; j d  C 2i   2i   As the integrand in the first integral on the right-hand side depends on  only through et; one can consider the limit for  ! 0C and show that the integral equals the integral on the curve  0 . A simple calculation yields that the integral on  0 ; as a function of t; is in L1 .0; 1/ \ L2 .0; 1/. (Here, it is important that  0  f 2 CI Re   0g and  0 touches the imaginary axis only at the point 0.) The treatment of the second integral on the right-hand side of (140) is much more complicated. It is necessary to derive a series of estimates of u WD .L  I /1 i; which satisfies the equation .A C B1 C B3  I /u D i:

(141)

This equation can be treated as the perturbed Oseen resolvent equation with the resolvent parameter . Especially the estimates for  62 Sp.L / in the neighborhood of 0 (hence also in the neighborhood of Spess .L /) are very subtle. They finally enable  limit for  ! 0C and show that the integral of  one to pass to the 0 t 1 e  .I  L / i ; j on curve  0 is, as a function of t; in L1 .0; 1/ \ L2 .0; 1/. The factor  plays a decisive role because it allows one to control integrand for  on the critical part of curve  0 ; i.e., near  D 0. Note that the assumption on the nonzero translational motion of body B in fluid (i.e.,  6D 0) is important because it enables one to apply the theory of Oseen equation and to obtain appropriate estimates of function u .

the the the 

Remark 9. A result similar to Theorem 22 was stated by L.I. Sazonov [98]. There, the main theorem on stability claims that the steady solution v0 is asymptotically stable in the L3 -norm if L; as an operator in H3 .˝/; does not have eigenvalues in the half-plane Re  > 0. However, the proofs of the fundamental estimates of the

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Oseen semigroup as well as of the main theorem do not contain all the necessary details, which makes it difficult to assess the validity of author’s arguments.

10.5.2 The Case T 6D 0 The results of [93] are generalized to the case T 6D 0 (i.e., B is allowed to spin at constant rate) involving the rotational motion of body B; in the paper [34] by G. P. Galdi and J. Neustupa. The steady solution v0 is assumed to satisfy the properties v0 2 L3 .˝/; @j v0 2 L3 .˝/ \ L3=2 .˝/ (j D 1; 2; 3) and the estimate jrv0 .x/j  C jxj1 for x 2 ˝. The existence of such a solution is known for a large class of body forces f; provided  6D 0; see Lemma 9 and Theorem 6. The main theorem on stability of the zero solution of equation (102) is analogous to Theorem 21, that is why it is not repeated here. The presence of the term T B2 v in the operator L defined in equation (102) causes a series of new problems that one has to face and overcome. For example, unlike the case T D 0; the time derivative of v need not be an element of H .˝/. However, one can show that .d v=dt /  B1 v  T B2 v 2 H .˝/ and Z   dv d 1  B1 v  T B2 v  v d x D kvk22 ; dt dt 2 ˝ Z   dv d 1 2  B1 v  T B2 v  Av d x D  jvj : dt dt 2 1;2 ˝ These identities play an important role in the proof of the theorem on stability. Another important step is to show that the functions ri and i (i D 1; : : : ; N ) are square-integrable with the weight jxj2 in ˝. This enables one to estimate the norm kB2 vk2 by C jvj1;2 ; which is again a crucial property in the proof of the stability result. All details can be found in [52]. Open Problem. The question whether – in analogy with the case T D 0 and paper [93] – the stability of the zero solution of equation (102) can be determined by the location of the eigenvalues of operator L is open. The difficulties related to this problem are generated by the fact that now, being T 6D 0; the operator L is no longer sectorial. Thus, even if all the eigenvalues have negative real parts, one cannot express the eLt i by a formula similar to (139), where the curve   coincides with the half-lines arg./ D  ˙ ˛ (for some ˛ 2 .0; =2/ and large jj) and touches or intersects the half-plane CC only in a small neighborhood of 0. On the contrary, the curve   must lie at the right of infinitely many points ikT (k 2 Z) on the imaginary axis, and even if one formally passes to the limit  ! 0 in order to obtain a curve  0 in the half-plane f 2 CI Re   0g; then  0 must pass through the points ikT (k 2 Z). Consequently, the integral on the right-hand side of (139) cannot be treated and estimated in the same way as in the case T D 0.

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Conclusion

The chapter is an updated survey of important known qualitative properties of mathematical models of viscous incompressible flows past rigid and rotating bodies. The models are based on the Navier-Stokes equations. Greatest attention is paid to steady problems, as well as problems that are quasi-steady in the sense that the transformed equations describing the motion of the fluid around a rotating body are steady in the body-fixed frame. The presented results concern the existence, regularity, and uniqueness of solutions (see Sects. 4–6). Section 7 deals with the spatial asymptotic properties of steady solutions, like the questions of the presence of a wake behind the body and the decay of velocity and vorticity in or outside the wake in dependence on the distance from the body. Here, the case T 6D 0 (i.e., the case when the body rotates with a nonzero constant angular velocity) is much more difficult than the case T D 0; and the relevant results are therefore of a relatively recent date. The structure of the set of steady solutions for arbitrarily large given data is studied in Sect. 8 by means of tools of nonlinear analysis, like the theory of proper Fredholm operators, corresponding mod 2 degree, etc. One of the results asserts that, to a given nonzero translational velocity and angular velocity, the solution set is generically finite and has an odd number of elements. Sect. 9 analyzes sufficient and necessary conditions for bifurcations of steady or time-periodic solutions from steady solutions. The corresponding theorems provide a theoretical explanation of the well-known phenomenon, i.e., that the properties and shape of a steady solution may considerably change if some characteristic parameters of the flow field vary. The longtime behavior of unsteady perturbations of a given steady solution v0 is finally studied in Sect. 10. This section also brings some necessary results on the existence and uniqueness of solutions. The core of the section are (1) the results on the stability of v0 under the assumption that v0 is in some sense “sufficiently small” and (2) the results that do not need any condition of smallness of v0 . Instead, they use either an assumption on a “sufficiently fast” time decay of a certain finite family of functions related to v0 or an assumption on the position of eigenvalues of a certain associated linear operator. (Here, one has to overcome the difficulties following from the presence of the essential spectrum, having a nonempty intersection with the imaginary axis.) The readers find a series of references to related papers or books inside each section. The chapter also brings the formulation of altogether eight open problems that concern the discussed topics and represent a challenge for future research.

12

Cross-References

 Large Time Behavior of the Navier-Stokes Flow  Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes

Flow  Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions  Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

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Acknowledgements The authors acknowledge the partial support of NSF grant DMS-1614011 (G.P.Galdi) and the Grant Agency of the Czech Republic, grant No. 13-00522S, and Academy of Sciences of the Czech Republic, RVO 67985840 (J.Neustupa). This work was also partially supported by the Department of Mechanical Engineering and Materials Sciences of the University of Pittsburgh that hosted the visit of J. Neustupa in Spring 2015.

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8

Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains Reinhard Farwig, Hideo Kozono, and Hermann Sohr

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q q -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Stokes Operator on L q Q 2.1 L -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q q -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Stokes Resolvent on L 2.3 Maximal Regularity and Bounded Imaginary Powers of the Stokes Q q -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operator on L 2.4 The Stokes Operator with Robin Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . 3 The Navier-Stokes System in General Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . 3.1 Strong and Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Very Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Regularity of Mild, Weak, and Very Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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R. Farwig () Department of Mathematics, Darmstadt University of Technology, Darmstadt, Germany International Research Training Group Darmstadt-Tokyo (IRTG 1529), Darmstadt, Germany e-mail: [email protected] H. Kozono Department of Mathematics, Waseda University, Tokyo, Japan Japanese-German Graduate Externship Program, Japan Society of Promotion of Science, Tokyo, Japan e-mail: [email protected] H. Sohr Faculty of Electrical Engineering, Informatics and Mathematics, Department of Mathematics, University of Paderborn, Paderborn, Germany e-mail: [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_8

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Abstract

To solve the (Navier-)Stokes equations in general smooth domains   Rn , the Q q ./ defined as Lq \L2 when 2  q < 1 and Lq CL2 when 1 < q < 2 spaces L have shown to be a successful strategy. First, the main properties of the spaces Q q ./ and related concepts for solenoidal subspaces, Sobolev spaces, Bochner L spaces, and the corresponding Helmholtz projection and Stokes operator will be discussed. Then these concepts are used to construct and analyze very weak, weak, mild, and strong solutions to the instationary (Navier-)Stokes equations in general domains. In particular, the strategy allows to find weak solutions of the (Navier-)Stokes system satisfying the localized energy inequality and the strong energy inequality which are important in the context of Leray structure theorem and partial regularity results.

1

Introduction

One of the basic functional analytic tools in the analysis of the instationary NavierStokes system is the Stokes operator A D P  where P is the Helmholtz projection mapping a vector field onto its solenoidal part such that gradient fields will be canceled. To be more precise, let 1 < q < 1 and   Rn be a smooth domain like a bounded or exterior one. Then Pq W Lq ./ ! Lq ./;

u 7! u0 D Pq u;

is the uniquely defined bounded projection vanishing on gradient fields such that in the weak sense, div u0 D 0 and u0  N D 0 on @. Here N D N .x/ denotes the q exterior normal at x 2 @ and L ./ is the space of solenoidal Lq -vector fields 1 ./ D fu 2 with vanishing normal on @ defined as the closure in Lq ./ of Cc; q 1 n Cc ./ W div u D 0g. With its range R.Pq / D L ./ and its kernel N .Pq / D G q ./ where G q ./ is the homogeneous Sobolev space of gradient fields, q

G q ./ D frp 2 Lq ./n W p 2 Lloc ./g; the Helmholtz projection yields the direct and topological decomposition Lq ./n D Lq ./ ˚ G q ./;

1 < q < 1:

(1)

The gradient part rp in the Helmholtz decomposition u D u0 C rp of (1) is related to the Neumann problem p D div u in  with boundary condition N  rp D u  N on @ so that formally N  u0 D 0 on @. Rigorously, rp is the solution of a weak Neumann problem, i.e., of the variational problem .rp; r'/ D .u; r'/

0

for all r' 2 G q ./:

(2)

8 Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

421

Given Pq , the Stokes operator A D Aq (in the case of Dirichlet boundary conditions) is defined by 1;q

D.Aq / D W 2;q ./ \ W0 ./ \ Lq ./;

Aq u D Pq u:

With the help of the Stokes operator and the Helmholtz projection Pq , the instationary Navier-Stokes system ut  u C div .u ˝ u/ C rp D div u D uD u.0/ D

f 0 0 u0

in in on at

.0; T /  ; .0; T /  ; .0; T /  @; t D 0;

(3)

can be rewritten as the abstract evolution problem ut C Aq u C Pq div .u ˝ u/ D Pq f in .0; T / at t D 0: u.0/ D u0 q

(4)

in the space L ./. On the other hand, the pressure gradient is recovered from (4) via rp D .I  Pq /.f C u  div .u ˝ u//. Note that in this chapter the coefficient of viscosity  is put equal to 1. However, due to counterexamples, see Bogovski˘i and Maslennikova [9, 48]; the Helmholtz decomposition does not hold for all q > 1 when   R2 is an infinite cone with “smoothed vertex” at the origin and of opening angle larger than . Similar counterexamples can be found also for n > 2: This implies that the classical strategy considering the instationary Navier-Stokes system (3) as q an abstract evolution problem (4) solved by analytic semigroup theory in L ./ breaks down. Actually, as shown by Geißert, Heck, Hieber, and Sawada [37], the existence of the bounded Helmholtz projection, i.e., the well definedness of the weak Neumann problem (2), does not only imply the definition of the Stokes operator Aq D Pq , but even the maximal regularity of Aq , a fundamental property in the analysis of linear and nonlinear equations involving the Stokes problem; for details, we refer to chapter “ The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties”. A similar result due to Abels [1] even shows existence of so-called bounded imaginary powers of the Stokes operator supposing that the Helmholtz decomposition exists and pressure functions enjoy a certain extension property; see also [2]. On the other hand, Hilbert space methods allow for an L2 -approach to the Helmholtz decomposition and the Stokes operator A2 D P2  for any bounded and unbounded domain. Actually, A2 is a positive self-adjoint operator generating a bounded analytic semigroup e tA2 ; t  0; for details, see the monograph of Sohr [56, Ch. III.2]. Moreover, typical constants as in resolvent estimates, semigroup, and maximal regularity estimates are independent of the domain when q D 2.

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Since an Lq -theory with 1 < q ¤ 2 < 1 of the Stokes problem is necessary to deal with the nonlinear term u  ru in weak, strong, and very weak solutions of the Navier-Stokes problem, and, in particular, for the construction of weak solutions satisfying the localized energy inequality and the strong energy inequality, the idea Q q -spaces. This concept allows to combine elements of L2 - and Lq -theory is to use L to work locally with Lq -regularity and Lq -Sobolev spaces, but globally, as jxj ! 1, with L2 -theory. This handbook article is organized as follows. In Sect. 2, the Stokes operator Q q D PQ q  on the space L Q q ./ for the class of uniform C 2 -domains   Rn (see A Definition 1) will be investigated. The properties of the Helmholtz projection PQ q and Q q -spaces are recalled in Sect. 2.1. The Stokes resolvent of Sobolev spaces based on L is analyzed in Sect. 2.2 (cf. Theorem 2), and the main ideas of the proof are given. Crucial results on the maximal regularity of the Stokes operator (see Theorem 3) are Q q are mentioned; discussed in Sect. 2.3 where also bounded imaginary powers of A Q they are important to deal with domains of fractional powers of Aq and their relation Q q; ,  2 Œ0; 1/, to Sobolev spaces. The final Sect. 2.4 considers the Stokes operator A 3 on domains of uniform C type (see Assumption 1) with Robin and Navier slip boundary condition u  N D 0, .1   /u C .T .u; p/N / D 0 on @ (cf. (44)); here the subscript  denotes the tangential part of the Cauchy stress vector T .u; p/N along @. The nonlinear Navier-Stokes system is analyzed in Sect. 3. First, strong and mild Q q .//, where 2 C n D 1, are solutions u contained in Serrin’s class Ls ..0; T I L s q constructed; see Theorem 6 when n < q < 1 and Theorem 7 for the limit case when q D n, s D 1 in which the Fujita-Kato iteration has to be used. Then the focus will be on the theory of very weak solutions which is based on duality to strong solutions of the Stokes system; see Sect. 3.2 in general and Theorem 9 for the main result on existence and uniqueness. Q q -spaces was the theory of weak solutions in the The starting point for the use of L sense of Leray-Hopf: Although weak solutions satisfying the basic energy inequality can be constructed for any domain of R2 and R3 (cf. [56, Ch. V.3]), there was the open problem to get weak solutions satisfying the localized energy inequality (suitable weak solutions) and the strong energy inequality. The construction of such weak solutions is described in Sect. 3.3. The final Sect. 3.4 uses results from the previous parts of Sect. 3 to discuss the regularity of mild, weak, and very weak solutions.

2

The Stokes Operator on LQ q -Spaces

2.1

LQ q -Spaces

Definition 1. A domain   Rn is called uniform C k -domain, k 2 N0 D N [ f0g if the following holds: There are constants ˛; ˇ; K > 0 such that for all x0 2 @

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there exists a function h W B˛0 .0/ ! R of class C k (on the closed ball B˛0 .0/  Rn1 with center 0 and radius ˛) and a neighborhood U˛;ˇ;h .x0 / of x0 such that khkC k  K; h.0/ D 0 and, if k  1; h0 .0/ D 0I U˛;ˇ;h .x0 / W D f.y 0 ; yn / 2 Rn1  RW jy 0 j < ˛; jh.y 0 /  yn j < ˇg;  .x0 / W D f.y 0 ; yn / 2 Rn1  RW jy 0 j < ˛; h.y 0 /  ˇ < yn < h.y 0 /g U˛;ˇ;h

D  \ U˛;ˇ;h .x0 /; @ \ U˛;ˇ;h .x0 / D f.y 0 ; yn / 2 Rn1  RW h.y 0 / D yn g W here an orthogonal and a translational transform depending on x0 2 @ is used to map points .x 0 ; xn / 2  to y D .y 0 ; yn /-coordinates in Rn . The triple .˛; ˇ; K/ is called a type of  and, although not uniquely determined, will shortly be denoted by  D .˛; ˇ; K/. For a constant C in some estimate, we will write C D C . / if its dependence on  depends only on ˛, ˇ, and K. By analogy, we define uniform C k; -domains, k 2 N0 , 0 <  1. Note that bounded and exterior domains are uniform C k -domains as long as the boundary is smooth enough. Q q -spaces is as follows: Let The basic definition of L (

q 2 Q q ./ WD L ./ C L ./; 1 < q < 2 ; L Lq ./ \ L2 ./; 2  q < 1

(5)

Q q ./ equipped with the norm Obviously, L ˚  kukLQ q D inf ku1 kLq C ku2 kL2 W u D u1 C u2 ; u1 2 Lq ./; u2 2 L2 ./ when 1 < q < 2, but kukLQ q D maxfkukLq ; kukL2 g when 2  q < 1 is a Banach Q q 0 ./ where q 0 D q denotes the conjugate exponent to Q q .// D L space, and .L q1 q; cf. Bergh-Löfström and Triebel [8, 60]. By the latter duality, ( kukLQ q D sup

) hu; vi 0 Q q ./ : W0¤v2L kvkLQ q0

Q q ./ locally behave as Lq -functions but that It is easy to see that functions u 2 L their global behavior at space infinity is mainly given by L2 -functions. In particular, Q q ./ D Lq ./ with equivalent norms when  is bounded. L

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Spaces of Sobolev type are introduced analogously to the definition in (5): For k 2 N and 1  q  1, let WQ

k;q

( ./ WD

W k;2 ./ C W k;q ./;

1  q < 2;

W k;2 ./ \ W k;q ./;

2  q  1:

(6)

1;q Similarly, the spaces WQ 0 ./, 1 < q < 2, and 2  q < 1, based on the classical 1;q Sobolev spaces W0 ./ and W01;2 ./, are defined. Q q - and WQ k;q .-spaces have the following properties; for a proof, see The L Riechwald [50, Ch. 1], [51]:

• Let 1  q < r  1. Then kukLQ q  kukLQ r . Q p ./, v 2 L Q q . Then uv 2 L Q r ./ • Let 1  r; p; q  1, 1r D p1 C q1 and let u 2 L and kuvkLQ r  kukLQ p kvkLQ q . • Let m 2 N, 1  q  r < 1 and   Rn be a uniform C 2 -domain. Then m;q Q r ./ ./ ,! L WQ nq if either r  1 and mq > n, or r < 1 and mq D n, or r  nmq and mq < n. In general, the latter embedding estimate kukLQ r  ckukWQ m;q cannot be improved to a homogeneous estimate of kukLQ r by krukLQ q because of the L2 -norm inherent in Q q -spaces. the definition of L Q q ./ for a domain   Rn Concerning the Helmholtz decomposition of L 1; of uniform type C , 0 <  1, the following spaces are needed; see [21, Theorem 1.2] of the authors of this article: Let

( Q q ./ L

WD

q

L ./ C L2 ./;

1 0. Q q generates an analytic semigroup e t AQ q ; t  0; in (iii) The Stokes operator A Q q  ./ with bound L Q

ke t Aq f kLQ q   M e ıt kf kLQ q  ;

Q q  ./; t  0; f 2L

(13)

where M D M .q; ı;  / > 0. It is unknown whether the usual resolvent estimate for the infinitesimal generator Q q of the analytic semigroup e t AQ q holds uniformly in the resolvent parameter A  2 C as jj ! 0. Therefore, the semigroup may increase exponentially fast and the maximal regularity estimate in Theorem 3 below is stated only for finite time Q q has often to be replaced by I C A Qq intervals. For the same reason, the operator A in the following. Proof of Theorem 2. The proof of Theorem 2 consists of several steps. 1. Auxiliary L2 - and Lq -estimates in bounded domains Q q for bounded, uniform C 1;1 -domains with a constant 2. Resolvent estimates in L depending on  when 2 < q < 1 Q q for bounded, uniform C 1;1 3. Duality arguments to get resolvent estimates in L domains with a constant depending on  when 1 < q < 2

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4. Exhaustion of an unbounded, uniform C 1;1 -domain  by a sequence of bounded uniform C 1;1 -domains k and passage to the limit k ! 1 Step 1. As consequence of Definition 1, it is easily shown that there exists a covering of  by open balls Bj D Br .xj / of fixed radius r > 0 with centers xj 2 , such that with suitable functions hj 2 C 1;1 of type  B j  U˛;ˇ;hj .xj / if xj 2 @;

B j   if xj 2 :

(14)

Here j runs from 1 to a finite number N D N ./ 2 N if  is bounded, and j 2 N if  is unbounded. Related to the covering fBj g, there exists a partition of unity f'j g; 'j 2 C01 .Rn /; such that

0  'j  1; supp 'j  Bj;

and

N X

'j D 1 or

j D1

1 X

'j D 1 on :

j D1

(15) The functions 'j may be chosen so that jr'j .x/j C jr 2 'j .x/j  C uniformly in j and x 2  with C D C . /: Moreover, as an important implication, the covering fBj g of  may be constructed in such a way that no more than a fixed number N0 D N0 . / 2 N of these balls have a nonempty intersection. If  is unbounded, then  can be represented as the union of an increasing sequence of bounded uniform C 1;1 -domains k  ; k 2 N; 1  : : :  k  kC1  : : : ;

D

1 [

k ;

(16)

kD1

where each k is of the same type .˛ 0 ; ˇ 0 ; K 0 /; see Heywood [41, p. 652]. Without loss of generality, assume that ˛ D ˛ 0 ; ˇ D ˇ 0 ; K D K 0 : To deal with the pressure and the divergence condition, the following Lemma 1 on local results is needed; its proof is found for parts (i), (ii) in the monograph of Galdi [35, III, Theorems 3.1 and 3.2], [21, Lemma 2.1], and for (iii) in [31, Theorem 3.1, (i)]. Here H denotes a smooth set of type  (for each j  N or  .0/ \ Br .0/ such that B r .0/  U˛;ˇ;h .0/ and r D j 2 N) defined as H D U˛;ˇ;h ˚  R q r. / > 0. Also the space L0 .H / WD u 2 Lq .H / W H u dx D 0 of functions with vanishing mean value on H is indispensable. Lemma 1. Let 1 < q < 1. q

1;q

(i) There exists a bounded linear operator R W L0 .H / ! W0 .H / such that q div ı R D I on L0 .H / and a constant C D C . ; q/ > 0 such that

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kRf kW 1;q  C kf kLq .H / for all f 2 L0 .H /:

(17)

  q 1;q 2;q Moreover, R L0 .H / \ W0 .H /  W0 .H / and kRf kW 2;q  C kf kW 1;q .H / q 1;q for all f 2 L0 .H / \ W0 .H /. q (ii) There exists C D C . ; q/ > 0 such that for every p 2 L0 .H / kpkq  C krpkW 1;q D C sup

n jhp; div vij krvkq 0

1;q 0

W 0 ¤ v 2 W0

o .H / :

(18)

(iii) For any f 2 Lq .H /,  2 S" where 0 < " < 2 , let u; p satisfy the Stokes resolvent equation u  u C rp D f , div u D 0 in H, u D 0 on @H . Moreover, assume that supp u [ supp p  Br .0/: Then there are constants 0 D 0 .q;  / > 0, C D C .q; ";  / > 0 such that kukLq .H / C kukW 2;q .H / C krpkLq .H /  C kf kLq .H /

(19)

if jj  0 . Next, several results on Sobolev embedding estimates for a bounded C 1;1 -domain will be mentioned. Note that the short notation k  kq may be used for the classical Lq ./-norm k  kLq ./ when the underlying domain  is known from the context. Lemma 2. Let   Rn be a bounded C 1;1 -domain, 0 < "  1 and 1 < q < 1. Then there is a constant C" D C .q; ";  / > 0 such that for all u 2 W 2;q ./ krukLq  "kr 2 ukLq C C" kukLq ;   kukLq  "kr 2 ukLq C C" kr 2 ukL2 C kukL2 ; 1 kukW 2;q  kukLq C kAq ukLq  C1 kukW 2;q C1

(20) 2  q < 1;

.u 2 D.Aq //:

(21) (22)

2;q

Proof. The proofs of (20) and (21) are reduced to the case u 2 W0 .0 / with 0  using an extension operator, the norm of which is shown to depend only on q and  : A further tool is the trivial interpolation inequality kvkr  .1="/1= kvk2 C .1   /"1=.1/ kvkq ;

(23)

with  2 .0; 1/, 1r D 2 C 1 . q For (22), see [25, Lemma 3.1]. Actually, the proof uses the same ideas as that of the resolvent estimate for uniform C 1;1 -bounded domains; see Step 2. t u The estimates (9) and (22) will be used to change between a formulation of the Stokes problem in .u; p/ (cf. (11)) and the abstract one using the Stokes operator

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Q q . The same holds for the instationary Stokes problem formulated as (31) and (34) A below; see also (4) and (3) for the nonlinear case. Recall the resolvent estimate of the Stokes operator Aq D Pq  W D.Aq / ! q L ./, 1 < q < 1, on a bounded C 1;1 -domain (cf. Farwig-Sohr [31], Giga [38] q and Solonnikov [57]): For any  2 S" , 0 < " < 2 , and f 2 L ./, the Stokes resolvent problem u C Aq u D f , div u D 0 has a unique solution u 2 D.Aq / such that kukq C kAq ukq  C kf kq ;

C D C ."; q; / > 0I

(24)

however, the precise dependence of the constant C D C ."; q; / > 0 on  remains unclear. On the other hand, if q D 2, then the Stokes resolvent problem has a unique solution u 2 D.A2 / satisfying the estimate kuk2 C kA2 uk2  C kf k2

(25)

with constant C D 1 C 2= cos " independent of ; in particular, (25) even holds for general unbounded domains. Moreover, A2 is self-adjoint and hA2 u; ui D 1

kA22 uk2L2 D kruk2L2 for all u 2 D.A2 /, cf. [56]. Step 2. For  2 †" , 0 < "
0; (26) with a constant C D C .q; "; ı;  / > 0 depending on  only through  . As in Step 1, the finite partition of unity .'j /, the sets B j  Uj WD U˛;ˇ;hj .xj / if xj 2 @ and B j D Uj   if xj 2 , cf. (14), are used. Furthermore, define   2;q wj WD R .r'j /  u 2 W0 .Uj /, and choose a constant Mj D Mj .p/ such that q p  Mj 2 L0 .Uj /. Then the pair .'j u  wj ; 'j .p  Mj // is a solution of the local equation .'j u  wj /  .'j u  wj / C r.'j .p  Mj //

(27)

D 'j f C wj  2r'j  ru  .'j /u  wj C .r'j /.p  Mj /: Concerning the term wj , we apply a Sobolev embedding, Lemma 1(i), and the interpolation estimate (23) to get for M 2 .0; 1/ that kwj kLq .Uj /  M kukLq .Uj / C C kukL2 .Uj / :

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Moreover, kr 2 wj kLq .Uj /  C krukLq .Uj / . For p  Mj (18) and (21), imply that  n jhu; vij o 1;q 0 kp  Mj kLq .Uj /  C kf ; rukLq .Uj / C sup W 0 ¤ v 2 W0 .Uj / krvkq 0    C kf ; rukLq .Uj / C kukL2 .Uj / C M kukLq .Uj / : Finally, the estimate (19) with  replaced by  C 00 where 00  0 is sufficiently large such that j C 00 j  0 for jj  ı will be applied to the local resolvent equation (27). Combining these estimates, the local inequality k'j ukLq .Uj / C k'j r 2 ukLq .Uj / C k'j rpkLq .Uj /

(28)

   C kf kLq .Uj / C kukLq .Uj / C krukLq .Uj / C kukL2 .Uj / C M kukLq .Uj / with C D C .M; q; ı; ";  / > 0 is proved. Then raise each term in (28) to the qth power, take the sum over j D 1; : : : ; N , and use the crucial property of the integer N0 introduced in Step 1 to get the global estimate (29) kukLq ./ C kr 2 ukLq ./ C krpkLq ./    C kf kLq ./ C kukLq ./ C krukLq ./ C kukL2 ./ C M kukLq ./ with C D C .M; q; ı; ";  / > 0, jj  ı. For the proof of (29), also the reverse  P q 1=q  P 2 1=2 Hölder inequality  for the reals aj D kukL2 .Uj / valid j aj j aj for q  2 is exploited. With (20) and M sufficiently small, the terms krukLq ./ and kukLq ./ from the right-hand side in (29) can be removed by absorption. Then the term kukLq ./ is removed with the help of (21) implying that   kukq C kr 2 ukq C krpkq  C kf kq C kuk2 C kuk2 C kr 2 uk2 : Finally, this inequality is combined with the L2 -estimate (25) for jj  ı and (22) with q D 2 is applied. This proves the estimate (26) for 2  q < 1. q

q

Step 3. Let 1 < q < 2. Consider for f 2 L2 ./ C L ./ D L ./ and  2 S" , jj  ı, the resolvent equation (11), and its unique solution u 2 D.Aq / C Q q , Pq D PQ q . The proof D.A2 / D D.Aq /, rp D .I  PQ q /u. Note that Aq D A q0 is a simple duality argument using the resolvent estimate on L ./ from Step 2. To see that the constant in the resolvent estimate depends on  only through  , note that 

jhu; gij 0 sup I 0 ¤ g 2 Lq \ L2 kgkLq0 \L2



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431

q

defines a norm on L C L2 which is equivalent to the norm k  kLq CL2 with constants related to the norm of PQ q 0 and thus depending only on q and  . Step 4. Given the sequence of bounded subdomains j  , j 2 N, of uniform Q q ./ and fj WD PQ q f C 1;1 -type as in (16), let f 2 L j . Then consider the j

solution .uj ; rpj / of the Stokes resolvent equation uj  PQ q uj D uj  uj C rpj D PQ q fj ;

  rpj D I  PQ q uj

in j :

By Steps 2 and 3, there holds the uniform estimate kuj kLQ q  .j / C kuj kWQ 2;q . / C krpj kLQ q .j /  C kf kLQ q  ./ j

(30)

with jj  ı > 0, C D C .q; ı; ";  / > 0. Extending uj and rpj by 0 to vector fields on , there exist, suppressing subsequences, weak limits Q q  ./; in L

u D w lim uj j !1

rp D w lim rpj j !1

Q q ./n in G

solving (11) in , satisfying u 2 DQ q and the a priori estimates (12). Uniqueness is proved by an elementary duality argument and the above result of existence for both q 0 and q. Now Theorem 2 is proved. t u

2.3

Maximal Regularity and Bounded Imaginary Powers of the Stokes Operator on LQ q -Spaces

Theorem 3 ([20,25]). Let   Rn be a uniform C 2 -domain and let 1 < s; q < 1, Q q .// and an initial value 0 < T < 1. Given an external force f 2 Ls .0; T I L Q q / (for simplicity), there exists a unique vector field u 2 Ls .0; T I D.A Q q //\ u0 2 D.A q 1;s Q W .0; T I L .// solving the Cauchy problem Q q u D f; ut C A

u.0/ D u0 :

(31)

It can be represented by the variation of constants formula Q

u.t / D e t Aq u0 C

Z

t

Q

e .t /Aq f ./ d

for a.a. 0  t  T

(32)

0

and satisfies maximal regularity estimate   kukLs .0;T ID.AQ q // C kut kLs .0;T ILQ q /  C ku0 kD.AQ q / C kf kLs .0;T ILQ q / with a constant C D C .q; s; T;  / > 0.

(33)

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Proof of Theorem 3. The proof follows the lines of the proof of Theorem 2. The Q q /-estimate on the bounded domain H D H˛;ˇ;hIr replacing crucial point is an Ls .L Lemma 1 (iii) from Step 1 above. Lemma 3. Let 0 < T < 1; u0 2 D.Aq / and f 2 Ls .0; T I Lq .H // be given. Assume that u 2 Ls .0; T; D.Aq //, p 2 Ls .0; T I W 1;q .H // solves the system ut  u C rp D f;

div u D 0;

u.0/ D u0

(34)

and satisfies supp u0 [ supp u.t / [ supp p.t/  Br .0/ for a.a. t 2 Œ0; T . Then there is a constant C D C .q; s;  ; T / > 0 such that kut kLs .0;T ILq .H // C kukLs .0;T IW 2;q .H // C krpkLs .0;T ILq .H //

(35)

   C ku0 kW 2;q .H / C kf kLs .0;T ILq .H // : Proof. This estimate follows from [58, Theorem 1.1]; see also [57, Theorem 4.1, (4.2) and (4:210 )] and Maremonti-Solonnikov [47, Theorem 1.4]. A careful inspection of the proofs shows that the constant C in (35) depends only on  and on q; T ; actually, it suffices to assume the boundary regularity C 1;1 since only the boundedness of second-order derivatives of functions locally describing the boundary is used. t u For simplicity, let u0 D 0. Moreover, it suffices to consider the case s D q since for bounded as well as for unbounded domains, an abstract extrapolation argument shows that the validity of (33) with s D q immediately extends to all s 2 .1; 1/; see Amann [3, p. 191] and Cannarsa-Vespri [15, (1.12)], where A has to be replaced Q q C ıI with ı > 0. by A Step 2 of the proof (with s D q  2) analyzes the unique solution u 2 Lq .0; T I D.Aq // of (31) given by (32) (cf. Giga-Sohr [39, 57]). It remains to prove that u satisfies the estimate (33) with a constant C depending only on T; q and  : For this reason, use – as in Step 2 of the proof of Theorem 2 – the system of functions fhj g; 1  j  N , the covering of  by balls fBj g, and the partition of unity f'j g as well as the bounded sets Uj  Bj . On Uj , define w D R..r'j /  u/ 2 2;q Lq .0; T I W0 .Uj //, and let Mj D Mj .p/ be the constant depending on t 2 .0; T / q such that p  Mj 2 Lq .0; T I L0 .Uj //; see Lemma 1. Since div w D .r'j /  u and div wt D .r'j /  ut for a.a. t 2 .0; T /; , the pair .'j u  w; 'j .p  Mj // solves in Uj the local equation   .'j u  w/t  .'j u  w/ C r 'j .p  Mj / (36) D 'j f  wt C w  2r'j  ru  .'j /u C .r'j /.p  Mj /: Due to the estimates (23), (17), and (18) and Sobolev embeddings with embedding constants c D c.q; r;  / > 0 independent of j , and the equations wt D R..r'j /  ut / and rp D f  ut C u, for " 2 .0; 1/, there holds

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kwt kLq .Lq .Uj //  C kut kLq .L2 .Uj // C "kut kLq .Lq .Uj // ;   kr 2 wkLq .Lq .Uj //  C kukLq .Lq .Uj // C krukLq .Lq .Uj // ;

(37)

  kp  Mj kLq .Lq .Uj //  C kf; rukLq .Lq .Uj // C kut kLq .L2 .Uj // C "kut kLq .Lq .Uj // with C D C .q; T; ";  / > 0 for all j 2 N. Then an application of the local estimate (35)–(37) implies that k'j ut kLq .Lq .Uj // C k'j ukLq .Lq .Uj // C k'j r 2 ukLq .Lq .Uj // C k'j rpkLq .Lq .Uj //    C kf kLq .Lq .Uj // C kukLq .W 1;q .Uj // C kut kLq .L2 .Uj // C "kut kLq .Lq .Uj // with C D C .T; q; ";  / > 0. Raise this inequality to its qth power, sum over j D 1; : : : ; N , and exploit the crucial property of the number N0 to get the estimate q

kut ; u; r 2 u; rpkLq .Lq .// Z

T

Z

T

Z

D Z

0



 q q0

 CN0

j

X

q q0

 0

! ˇq ˇ X ˇq ˇ X ˇq ˇX ˇq ˇ X ˇ ˇ ˇ ˇ ˇ ˇ ˇ 2 ˇ 'j ut ˇ C ˇ 'j uˇ C ˇ 'j r uˇ C ˇ 'j rp ˇ dx dt ˇ N0

X

q q0

C "N0

j

j'j ut jq C

X

j

kf

j

q kLq .0;T ILq .Uj // C

j

X

j

j'j ujq C

X j

q kut kLq .0;T ILq .Uj //

X

j

j'j r 2 ujq C

j q kukLq .0;T IW 1;q .Uj // C

X

X

 j'j rpjq dx dt

j

 q kut kLq .0;T IL2 .Uj //

j

:

j

With a sufficiently small " > 0, the absorption principle and again the property of the number N0 help to simplify the above inequality to the Lq .0; T I Lq .//estimate kut kLq .0;T ILq / C kukLq .0;T ILq / C kr 2 ukLq .0;T ILq / C krpkLq .0;T ILq /    C kf kLq .0;T ILq / C kukLq .0;T ILq / C kut kLq .0;T IL2 /

(38)

where C D C .q;  / > 0. Here we also used the reverse Hölder inequality to deal with the sum of the terms kut kLq .0;T IL2 .Uj // . The term kukLq .0;T ILq / can be absorbed exploiting (21) with " > 0 sufficiently small. Finally, the L2 ./-maximal regularity estimate for kut kLq .0;T IL2 / (see [56, Ch.IV 2.5]) admits to absorb this term in (38) as well. Step 3 deals with the bounded domain case when 1 < s D q < 2. As in Step 3 of the proof of Theorem 2, the main tools are duality arguments and, in particular, the duality

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 0 0 0 0 Lq .0; T I Lq .// \ Lq .0; T I L2 .// D Lq .0; T I Lq .// C Lq .0; T I L2 ./ with a norm equivalence depending on q and  only. Step 4 considers the passage to the limit of solutions .uj ; rpj / in j to a weak limit identified as the unique solution .u; rp/ of (31) and (34), satisfying the a priori estimate (33). Now the proof of Theorem 3 is complete. t u Q q admits A further crucial property of the Stokes operator is the fact that I C A Q q /is ; s 2 R (see Kunstmann [42]) so that complex bounded imaginary powers .I C A interpolation methods can be applied to describe domains of fractional powers .I C Q q /˛ ; 1  ˛  1. For 0  ˛  1, let DQ q˛ D DQ q˛ ./ D D..I C A Q q /˛ / denote the A ˛ Q q /˛  k Q q . Q domain of the fractional power .I C Aq / equipped with the norm k.I C A L q ˛ Q  ./ in the norm k.I CA Q q /˛ k Q q . For 1  ˛ < 0, define DQ q as the completion of L L Qq As in Sects. 2.2 and 2.3, it is unknown whether similar results will hold for A Q q. instead of I C A Proposition 1. Let   Rn n  2; be a uniform C 1;1 -domain of type  , and let Q q ./. In particular, the Q q has a bounded H1 -calculus in L 1 < q < 1. Then I C A spaces DQ q˛ are reflexive and satisfy the duality relation .DQ q˛ / Š DQ q˛ 0 . Moreover,

DQ q˛ ; DQ qˇ



D DQ q ;

(39)

1=2 when 1  ˛  ˇ  1 and .1  /˛ C ˇ D  , 2 .0; 1/. Finally, DQ q D 1;q q Q  ./ with norm k.1 C A Q q /1=2  k Q q equivalent to k  k 1;q : WQ 0 ./ \ L L Q ./ W

The proof is found in [42, Theorem 1.1, Corollary 1.2] and [43]. These results imply the following Sobolev embedding and decay estimates   ([51, Proposition 3, Theorem 1]): Let n  3, 1 < q  r < 1, and ˛ WD n2 q1  1r > 0. Then kukLQ r ./  t AQ  e r f  r Q ./ L  t AQ  re r f  r Q ./ L  t AQ  e r PQ r div F  r Q ./ L

Q q /˛ uk Q q ;  C k.1 C A L ./  C e ıt t ˛ kf kLQ q ./ ; 1

 C e ıt t ˛ 2 kf kLQ q ./ ; 1

 C e ıt t ˛ 2 kF kLQ q ./ ;

0  ˛  1;

(40) (41) (42) (43)

Q q ./, or matrix field F 2 L Q q ./, respectively, for any for every u 2 DQ q˛ , f 2 L t > 0 and ı > 0; here C D C . ; r; q; ı/ > 0. Note that in (43) the operator e t AQ r PQ r div must be defined by duality to the operator re t AQ r 0 in (42).

8 Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

2.4

435

The Stokes Operator with Robin Boundary Condition

The Navier boundary condition or more general the Robin boundary condition reads as follows: u  N D 0;

.1   /u C .T .u; p/N / D 0

on @;

(44)

with  2 .0; 1 . Here N denotes the exterior normal, and the subscript  denotes the tangential part of a vector along @, i.e., u WD u  .u  N /N . Moreover, T D T .u; p/ D pI C 2D.u/;

D.u/ D

1 .ru C .ru/> / 2

are the Cauchy stress tensor and the symmetric part of the velocity gradient, respectively. In the limit  D 1, we are left with the Navier condition of pure slip. The analysis of the Stokes resolvent problem and of maximal regularity follows the Dirichlet case in Sects. 2.2 and 2.3. However, there are crucial differences. In Sect. 2.2, a sequence of bounded uniform C 1;1 -subdomains j of  exhausting  has been introduced; see (16); then the Dirichlet condition uj D 0 on @j yields in the weak limit the boundary condition u D 0 on @. For the Robin boundary condition, the sequence fj gj must be chosen more carefully. Assumption 1. A uniform C 3 -domain   Rn of type  D .˛; ˇ; K/ in Sect. 2.4 is assumed to have the following property: There exists a sequence fj gj 2N of bounded uniform C 3 -domains of type  D .˛; ˇ; K/ such that I I I

S j  j C1 for all j 2 N and  D 1 j D1 j , j WD @j \ @ 6D ; for all j 2 N,S j  j C1 for all j 2 N and @ D 1 j D1 j :

Note that typical layer-like domains or tubular-like domains with finitely and even countably many exits at infinity satisfy Assumption 1. The same holds for the smooth vertex domain from [9, 48]. To define the Stokes operator with Robin boundary condition, the Sobolev space ˚ W2;q ./ D u 2 W 2;q ./ W .1   /u C .T .u; p/N / D 0 2;q

 on @ ;

1 < q < 1, is needed. The boundary condition for the space W ./ is understood locally in the sense of usual traces. For a bounded domain , the domain of the q 2;q Stokes operator Aq; D Pq  is given by D.Aq; / D L ./ \ W ./: However, for general unbounded domains, its domain is defined as

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(

Q q; / D D.Aq; / \ D.A2; /; DQ q; D D.A D.Aq; / C D.A2; /;

2  q < 1; 1 < q < 2:

(45)

Then, with the help of the Helmholtz projection PQq , the Stokes operator with Navier boundary condition for a general uniformly smooth domain is given by Q q  ./ ! L Q q  ./: Q q; D PQ q  W DQ q;  L A

(46)

Q q; reads Now the main result on the Stokes resolvent problem for the operator A as follows: Q q; ). Let 1 < q < 1, 0 < " <  , ı > 0. Theorem 4 (Resolvent problem for A 2 n 3 Let   R , n  2, be a uniform C -domain of type  and let Assumption 1 be satisfied. Then the following assertions hold: Q q; . To be more precise, for (i) The sector †" is contained in the resolvent set of A q Q Q f 2 L  ./, there exists a unique u D . C Aq; /1 f of the abstract Stokes Q q; u D PQq f satisfying the estimate resolvent problem u C A kukLQ q ./ C kukWQ 2;q ./  C kf kLQ q ./

(47)

for all  2 †" with jj  ı, where C D C .q; "; ı;  / > 0. Q q ./ and  2 †" , the Stokes resolvent system in .u; p/ has (ii) For a given f 2 L Q q ./ defined by u D . C A Q q; /1 PQ q f a unique solution .u; rp/ 2 DQ q;  L Q and rp D .I  Pq /.f C u/, satisfying kukLQ q ./ C kukWQ 2;q ./ C krpkLQ q ./  C kf kLQ q ./

(48)

for jj  ı with C D C .q; "; ı;  / > 0. Q q  ./ is a densely defined closed Q q; W DQ q; ! L (iii) The Stokes operator A Q operator, and Aq; generates an analytic semigroup satisfying the estimate Q

ke t Aq; f kLQ q ./  C e ıt kf kLQ q ./

(49)

Q q  ./, t  0, where C D C .q; ı;  / > 0. for f 2 L Q q 0 ; , hA Q q; u; vi D Q 0q; D A (iv) For the adjoint operator, the duality relations A Q q 0 ; vi for all u 2 DQ q; , v 2 DQ q 0 ; , hold. hu; A The proof is similar to the proof of Theorem 2. One main difference concerns the resolvent estimate with Robin boundary condition in the set H (or for a bent half space) obtained by Shibata-Shimada [54] (cf. Lemma 1) (iii) above for the Dirichlet case. It is necessary to check that the proofs in [54] yield constants C D C .q; ı;  /;

8 Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

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see also Farwig-Rosteck [29, Prop. 2.4, Lemma 2.5]. The partition of unity for @ in the context of the Robin boundary condition also necessitates the analysis of nonhomogeneous Robin boundary conditions for bent half spaces. A second technical difference is the Friedrichs inequality replacing the Poincaré inequality. R 1;q Actually, if u 2 W0 .H / or u 2 W 1;q .H /, H u D 0, then kukLq .H /  C .q;  /krukLq .H / : In the case of vector fields u 2 W 1;q .H / satisfying u  N D 0 on @H , a similar estimate holds (cf. [29, Lemma 2.2] and [30]). Corollary 1. [[29], Corollaries 1.3, 1.4] Let 1 < q < 1, and let   Rn , n  2, be a domain satisfying Assumption 1. (i) The norms Q q;  k Q q ; k  k Q q C k.1 C A Q q; /  k Q q ; k  kWQ 2;q ./ ; k  kLQ q ./ C kA L ./ L ./ L ./ Q q; /  k Q q k.1 C A L ./ are equivalent on DQ q; with a constant depending on  only through  : (ii) The estimate Q

ke t Aq; ukLQ r ./  C t ˛ e ıt kukLQ q ./ ;

Q q  ./; t > 0; u2L

(50)

with a constant C D C .ı; q;  / > 0, holds true under the following conditions (a) and (b):   nq (a) q < n2 and q  r  n2q , where 0  ˛ D n2 q1  1r  1, (b) q  n2 and q  r, where 1  ˛  1  qr  0. Q q; ). Let 1 < q; s < 1, 0 < T < 1. Let Theorem 5 ( Maximal regularity for A n   R , n  2, be a domain satisfying Assumption 1. Then the following assertions hold: Q q  .// and u0 2 DQ q; , there exists a unique solution (i) For every f 2 Ls .0; T I L s Q Q q  .// of the evolution equation u 2 L .0; T I Dq; / with ut 2 Ls .0; T I L Q q; u D f; ut C A

u.0/ D u0 ;

satisfying the estimates Q q; uk s kut kLs .0;T ILQ q .// C kukLs .0;T ILQ q .// C kA Q q .// L .0;T IL    C kf kLs .0;T ILQ q .// C ku0 kDQ q; with a positive constant C D C . ; T; q; s/.

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Q q .// and every u0 2 DQ q; , the instationary (ii) For every f 2 Ls .0; T I L Stokes system in .u; p/ has a unique solution .u; rp/ 2 Ls .0; T I DQ q; /  Q q .// with ut 2 Ls .0; T I L Q q  .//, defined by ut C A Q q; u D PQ q f , Ls .0; T I G u.0/ D u0 , as well as rp.t/ D .I  PQ q /.f C u/.t /, satisfying kut kLs .0;T ILQ q .// C kukLs .0;T IWQ 2;q .// C krpkLs .0;T ILQ q .//    C kf kLs .0;T ILQ q .// C ku0 kDQ q; with a positive constant C D C . ; T; q; s/. The proof is similar to the proof in the Dirichlet case (cf. Theorem 3). The key lemma is the maximal regularity estimate for bent half spaces and the set H which are allowed to depend on the domain through  only. This result can be extracted from Shimada [55] where – starting with the half space case and nonhomogeneous boundary data as well as a nonzero divergence – the bent half space and finally the case H are analyzed (cf. Rosteck [52, Lemmata 3.9, 3.10 and 3.11]).

3

The Navier-Stokes System in General Unbounded Domains

Throughout this section, let   Rn , n  3, be a uniform C 1;1 -domain of type  . Then the instationary Navier-Stokes system (3) on a finite time interval Œ0; T / is solved for strong, mild, very weak, and weak solutions.

3.1

Strong and Mild Solutions

Using the variation of constants formula, mild solutions of (3) are given as solutions of the nonlinear integral equation u.t / D e

Qq t A

Z

t

u0 C

 Q  e .t /A PQ f ./  div .u ˝ u/. / d ;

0  t  T:

(51)

0

A mild solution u is called a strong solution if u is contained in Serrin’s class Q q .// where 2 C n D 1, q  n; in particular, u 2 Ls .0; T I Lq .//. Ls .0; T I L s q Theorem 6 (Strong Solutions). Let s > 2, q > n satisfy 2s C nq D 1; let f be given Q q=2 .// and, for simplicity, u0 2 DQ q . in the form f D div F with F 2 Ls=2 .0; T I L  Then there exists 0 < T  T such that the Navier-Stokes system (3) has a unique Q q .//. strong solution u 2 Ls .0; T  I L

8 Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

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The proof is a simple consequence of Banach’s fixed-point theorem. The crucial Q q .// is the estimate of the estimate to get in (51) a self-map on Ls .0; T  I L n 1 .0/ nonlinear term: By (43) and with ˛ D 2q D s 0 , u .t / D e t AQ q u0 , Z

.0/

ku.t /kLQ q  ku .t /kLQ q C C

t

0

 0 .t  /1=s kF . /kLQ q=2 C ku. /k2LQ q d

for a.a. t 2 Œ0; T /. Then the Hardy-Littlewood-Sobolev inequality implies for every 0 < T   T that   kukLs .0;T  ILQ q /  ku.0/ kLs .0;T  ILQ q / C C2 kF kLs=2 .0;T  ILQ q=2 / C kuk2Ls .0;T  ILQ q / with a constant C2 independent of T  . Choosing T  sufficiently small, Banach’s fixed-point theorem completes the proof. Next, the classical Fujita-Kato method will be modified to work for general unbounded domains (cf. [27]). For simplicity, let f D 0. Q n ./ be an initial velocity. Then Definition 2. Let 0 < T < 1 and let u0 2 L 1 n 1=2 1 Q Q n .// is called mild Fujitau 2 L .0; T I L .// with t ru.t / 2 L .0; T I L Kato solution to the Navier-Stokes system with initial velocity u0 if it solves the integral equation u.t / D e

Qn t A

Z

t

u0 

Q e .ts/An=2 PQ n=2 .u.s/  ru.s// ds

(52)

0

for almost all 0  t < T . Q n ./, and Theorem 7 (Mild Fujita-Kato Solutions). Let 0 < T < 1, u0 2 L n < q < 1. (i) There is a constant  D . ; q/ > 0 such that the condition Q

Q

sup t .1n=q/=2 ke t An u0 kLQ q ./ C sup t 1=2 kre t An u0 kLQ n ./  

0t 0 with the following property: If for a point t 2 .0; T / lim inf ı!0C

where ˛ D



r0 2 2 r0

C

3 q0

1 ı˛

Z

t

tı

r0 ku. /kL d  ; Q q0 ./

(87)

  1 , then t is a regular point in the sense that u 2

Q q ./). In particular, the left-side Serrin condition u 2 Lr .t  Lr .t  ı; t C ıI L q Q ı; t I L .// for some ı > 0 is sufficient for t to be a regular point. • There is a constant  D . ; T / with the following property: If for a point t 2 .0; T / and some > 0 the kinetic energy Ekin .t / D 12 ku.t /k2L2 satisfies the left-sided 12 -Hölder continuity jEkin .t /  Ekin .t  ı/j  ; ı 1=2 0 2 and Z.x/ D 0 for jxj < 1, and Z w.y; s/ D R3

ry

1 rz .z/  U0 .z; s/ d z: 4 jy  zj

(126)

One can show that W is differentiable in y and s, T periodic, divergence-free, U0  W 2 L1 .0; T I L2 .R3 //, and, for R0 , sufficiently large,

and

kW kL1 .0;T ILq .R3 //  C "0 ;

(127)

kW kL1 .0;T IL4 .R3 //  c.R0 ; U0 /;

(128)

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kLW kL1 .0;T IH 1 .R3 //  c.R0 ; U0 /:

(129)

Decompose U D W C V with the difference V satisfying a perturbed Leray system, LV C .W C V /  rV C V  rW C rP D R.W /;

div V D 0;

(130)

where the source term is R.W / WD LW C W  rW:

(131)

To get a solution of (130) satisfying the local energy inequality, consider mollified systems in R3 , LV C .W C " V /  rV C V  rW C rP D R.W /;

div V D 0;

(132)

where " , 0 < "  1, are standard mollifiers. Because Z

V  rV  W  kW kL1 Lq kV k2H 1 

1 kV k2H 1 8

(133)

for sufficiently small "0 and large R0 by (127), and 1 j.R.W /; V /j  .kLW kH 1 C kW k2L4 /kV kH 1  C2 C jjV jj2H 1 ; 8

(134)

where C2 D C .kLW kH 1 C kW k2L4 /2 , for both (130) and (132) one can obtain the a priori bound d 1 1 jjV jj2L2 C jjV jj2L2 C jjrV jj2L2  C2 : ds 2 2

(135)

Using this bound, one can first construct time-periodic Galerkin approximations V";k , k 2 N, for (132). They are uniformly bounded in the usual energy norms and converge weakly to a periodic solution V" of (132) as k ! 1 up to a subsequence, with kV" kL1 .0;T IL2 .R3 // C krV" kL2 .R3 Œ0;T /  C:

(136)

At this stage one can construct the pressure P" associated to V" for (132) by Riesz transforms as in [55, 70], with uniform bound kP" kL5=3 .R3 Œ0;T /  C kjV" j C jW jk2L10=3 .R3 Œ0;T / :

(137)

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499

One then passes limit " ! 0 to get a weak solution V; P of (130) satisfying the corresponding local energy inequality. Then U D W C V and P then become a suitable weak solution of the Leray equations (77). A DSS solution of the NavierStokes equations (14) is obtained by the similarity transform (76). An interesting scenario is the following. For the usual Leray-Hopf weak solutions, it is well known that the hypothetical singular set is contained in a compact subset of space-time. Such property can be called eventual regularity. The eventual regularity of local Leray solutions is unclear: if a -DSS solution v is singular at some point .x0 ; t0 /, it is also singular at .k x0 ; 2k t0 / for all integers k. Since v is regular if it is SS or if  is close to one, the possibility of a non-compact singular set for some local Leray solutions is suggested only by Theorem 12. When the initial data is self-similar, the above approach gives a third construction for self-similar solutions of (14). One has the following result: Theorem 13. Let u0 be a self-similar divergence-free vector field in R3 which is in L3loc away from the origin. Then, there exists a local Leray solution u to (14)–(15) which is self-similar and satisfies ku.t /  e t u0 kL2 .R3 /  C0 t 1=4 for any t 2 .0; 1/ and a constant C0 D C0 .u0 /.

6

Possible Nonuniqueness for Large Forward Self-Similar Solutions

As is well known, topological argument gives existence but not uniqueness of solutions. In fact in the case of steady Navier-Stokes equations, where the existence proof is similar in spirit, there is nonuniqueness [69] through nontrivial bifurcations. In [31, 32] the authors conjectured nonuniqueness for general large-scale-invariant initial data based on bifurcations from large-scale-invariant solutions. Consider a scale-invariant initial condition u0 which is smooth away from the origin. For   0, at first taken sufficiently small so that uniqueness holds, let u .x; t / D p1 t U . px t / be the unique scale-invariant solution to NSE with the initial data  u0 . The field U satisfies U C

x 1  rU C U  U  rU C rP D 0 2 2

(138)

1 in R3 , with jU .x/   u0 .x/j D o. jxj / as x ! 1. U can be viewed as a steady state to the time-dependent equation

@s U D U C

x 1  rU C U  U  rU C rP; 2 2

(139)

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with the boundary condition jU .x; s/  u0 j D o.

1 /; jxj

(140)

as x ! 1, in R3  Œ0; 1/. The linearization of Eq. (139) around the steady state U can be written as s D L ;

(141)

where L  WD  C

1 x  r C   U  r    rU C rP; 2 2

(142)

and the function P is chosen so that L  is divergence-free. Also, P is assumed to have a suitable decay at 1 (so that it is uniquely determined, perhaps up to a constant). Clearly, the behavior of the linearized equation (142) depends on the spectral property of the operator L . To study the spectrum of the linearized operator, one needs to set up some notations. Let X WD f 2 L2 \ L4 .R3 / W div  D 0g;

(143)

with the natural norm kkX WD kkL2 .R3 / C kkL4 .R3 / :

(144)

The choice of the space X is dictated by the need to preserve the boundary condition (140) for the perturbed solution and local regularity considerations. Define the domain of L as D WD f 2 X W @j ; @ij  2 X; and x  r 2 X g:

(145)

It was shown in [32] that the spectrum of L is contained in the set 1 † WD f 2 C W Re    g 4

(146)

and K  WD U  r C   rU C rP is relatively compact with respect to L. Thus the spectrum of L D L  K is the union of one part contained in † and some isolated eigenvalues in the set f 2 C W 

1 < Re g: 4

9 Self-Similar Solutions to the Nonstationary Navier-Stokes Equations

501

When  is small, one can solve Eq. (138) by a perturbation argument (and the solution is unique). Moreover, still for small , the set of eigenvalues of L can be shown to remain away from the imaginary axis, with Re < 0. It is not hard to show that as long as 0 is not an eigenvalue of L , one can use perturbation arguments to continue the solution curve U as a regular function of , thus there is no confusion in the definition of U and L . If one increases  further, some eigenvalues of L might cross the imaginary axis. The following crucial spectral assumption will play an important role: (A) For some   > 0, the operator L associated with forward self-similar solution p1t U px t has eigenvalues with positive real part, all of which lie to the

left of fz 2 C W Re z D 18 g. A few comments are in order on the condition that fz 2 C W Re z < 18 g: One of course expects the eigenvalues to have small real part when  has just crossed the critical value. The restriction on the size of the unstable eigenvalues plays a crucial role in the localization of the large forward self-similar solutions. Under spectral assumption (A) one has the following result. Theorem 14. Suppose that for some 1-homogeneous initial data u0 smooth away from origin and  > 0, the L , associated with the forward self-similar  operator  1 x solution u .x; t / D pt U pt as above, satisfies the spectral assumption (A), and then there exist two different solutions u and uQ  to the Navier-Stokes equations with the same initial data  u0 . uQ  is given by  1 x ; log t ; uQ  .x; t / D u .x; t / C p   t

(147)

k.; s/kX  C e ıs ; for all s  0:

(148)

and  satisfies

Remark. A proof will not be given here. The reader is referred to the paper [32]. The spectral assumption is slightly different, but the proof of Theorem 4.1 in [32] with obvious modifications works here as well. The main idea is to construct an unstable manifold around the steady state U , and then any nontrivial solution  on the unstable manifold will suffice. A large part of the work [32] deals with the spectral properties of L and their implications on the growth rate of the semigroup e L s . These properties are of course essential in the perturbation analysis around the steady state U . Due to slow decay at spatial 1, solutions u ; uQ  do not belong to the energy space. It is natural to localize these solutions to obtain nonuniqueness for LerayHopf weak solutions. This is done in [32], by perturbing u ; uQ  in L4 , so that one can preserve the singularity at the origin while cutting off the slowly decaying tail.

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In the end, one can obtain sharp nonuniqueness for Leray-Hopf weak solutions, provided spectral assumption (A) can be verified. An interesting feature of the perturbation argument is that the resulting equation contains critically singular scale-invariant lower-order terms, for which the classical well-posedness theory is not applicable. Consider, for example, @t    C a.x; t /  r C b.x; t/ D 0; where a.x; t / D

1 p A. px t / t

(149)

and b.x; t/ D 1t B. px t /. For simplicity, assume A; B 2 5=2

Cc1 .R3 /. Classical parabolic theory tells us if a 2 L5x;t ; b 2 Lx;t then Eq. (149) is well-posed in any Lp , p > 1. The coefficients just miss such space. The wellposedness (in Lp , say) is then not clear from the general parabolic theory. It is observed in [32] that the well-posedness of Eq. (149) in Lp is decided by the spectrum of the operator T  D  C

y  r  A  r  B: 2

(150)

3 g then More precisely if the spectrum of T is contained in f W Re  <  2p p Eq. (149) is well-posed in L . On the other hand if there are eigenvalues of T 3 with real part greater than  2p , Eq. (149) is ill-posed in Lp through appearance of nontrivial solution in the form u.x; t / D t  ‰. px t / with trivial initial data. Here ‰ is an eigenfunction corresponding to eigenvalue . Applying this observation to the Navier-Stokes equations, it is found that if the spectrum of L contains unstable eigenvalues but these eigenvalues are all contained in f W Re  < 18 g, then the perturbed Navier-Stokes would be ill-posed for 1homogeneous initial data, but still well-posed for initial data in L4 . More precisely one has the following theorem: Denote

Y D a 2 L1 .R3 /j div a D 0;    and sup .1 C jxj/1Cj˛j jr ˛ a.x/j < 1; for j˛j D 0; 1; 2: : x

equipped with the natural norm kakY WD

sup



 .1 C jxj/1Cj˛j jr ˛ a.x/j :

(151)

x2R3 ;j˛jD0;1;2

Theorem 15. Let a 2 Y be such that 18 > ˇ, which is the maximum of real part of any eigenvalue of L  K.a/. Denote a.x; Q t / D p1 t a. px t /. Let

9 Self-Similar Solutions to the Nonstationary Navier-Stokes Equations

x 1 Q b.x; t/ WD p b. p ; log t /; for x 2 R3 ; t 2 .0; 1/: t t

503

(152)

Suppose sup .kb.; s/kX C krb.; s/kX /

D

(153)

s2.1;0/

is sufficiently small depending on kakY ; ˇ. Let T D T .kakY ; ˇ; ku0 kL4 .R3 / / > 0 be sufficiently small. Then there exists a unique solution u 2 ZT to the generalized Navier-Stokes system with singular lower-order terms 9 @t u  u C aQ  ru C u  r aQ C bQ  ruC = in R3  .0; T /; (154) Cu  r bQ C u  ru C rp D 0 ; div u D 0 with initial data u.; 0/ D u0 2 L4 .R3 / in the sense that lim ku.; t /  u0 kL4 .R3 / D 0:

t!0C

(155)

In the above,

4 3 ZT WD u 2 L1 t Lx .R  .0; T // W

and

1 2

1

sup t 2 kru.; t /kL4 .R3 / < 1; : t2.0;T /



lim t kru.; t /kL4 .R3 / D 0 ;

t!0C

equipped with the natural norm kukZT WD sup



 1 ku.; t /kL4 .R3 / C t 2 kru.; t /kL4 .R3 / :

(156)

t2.0;T /

Moreover, u satisfies 1

C sup t 2 kru.; t /kL4x .R3 /  C .ˇ; kakY /ku0 kL4x .R3 / : kukL1 4 3 t Lx .R .0;T //

(157)

t2.0;T /

Using the above result, one can show Theorem 16. Assume the spectral condition (A) holds. Then there exist two different Leray-Hopf weak solutions which are smooth in R3  .0; 1/ with the same compactly supported initial data u0 2 C 1 .R3 nf0g/. The initial data obeys 1 u0 .x/ D O. jxj / bound near the origin.

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Conclusion

Classically, the study of self-similar solutions has played an important role in the understanding of singularities for many elliptic and parabolic equations. Backward self-similar singularities for Navier-Stokes equation can be excluded, as discussed in Sect. 3. However, many questions remain open. For example, one can combine the scaling symmetries with the rotational symmetries and consider solutions invariant by one-parameter subgroups of this larger group. Not much is known about the existence/nonexistence of such singularities. The study of forward self-similar solutions is closely related to the problem of optimal well-posedness results, as discussed in Sect. 4. The forward self-similar solutions seem to provide one of the natural ways to “test” the borderline between the regime where the viscosity term dominates and the regime where the nonlinear term dominates. It is conceivable that further study of these solutions can also shed light on the problem of uniqueness of the Leray-Hopf weak solutions with initial data in L2 .

8

Cross-References

 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes

Flow  Regularity Criteria for Navier-Stokes Solutions  Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions  Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

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Giovanni P. Galdi and Mads Kyed

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Existence of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Existence of Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Lq Estimates for the Linearized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Exterior Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Existence of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Existence of Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 New Approach to Lq Estimates for the Linearized Problem in the Whole Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Weighted Estimates for the Linearized Problem in the Whole Space . . . . . . . . . . . . 4.7 Lq Estimates for the Linearized Problem in Exterior Domains . . . . . . . . . . . . . . . . . 4.8 Existence of Lq Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Asymptotic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Flow in a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

510 513 515 515 520 523 527 529 532 533 534 537 539 545 548 550 560 565 572 575 575 576

The work of G.P. Galdi was partially supported by the NSF grant DMS-1614011 G.P. Galdi () Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, USA e-mail: [email protected] M. Kyed Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany e-mail: [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_10

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Abstract

The Navier-Stokes equations with time-periodic data are investigated with respect to solutions of the same period. In the physical terms, such a system models the flow of a viscous liquid under the influence of a time-periodic force. The three most relevant types of flow domains, from a physical point of view, are considered: a bounded domain, an exterior domain, and an infinite pipe. Methods to show existence of both weak and strong solutions are introduced. Moreover, questions regarding regularity, uniqueness, and asymptotic structure at spatial infinity of solutions are addressed.

1

Introduction

An object performing a time-periodic interaction with a viscous liquid is one of the most frequently occurring mechanical systems in nature. In such systems, the object exerts a time-periodic force on the liquid, and it can be expected that the resulting flow undergoes a motion of the same period. The mathematical analysis of such motions naturally leads to the study of time-periodic Navier-Stokes equations. If the region of flow is a domain   Rn , n  2, the Navier-Stokes equations governing the motion of a liquid subjected to a body force f W R   ! Rn can be written as 

@t u C u  ru D u  rp C f div u D 0

in R  ; in R  ;

(1)

where u W R   ! Rn denotes the Eulerian velocity field, p W R   ! R the pressure field, and  the constant coefficient of kinematic viscosity of the liquid. As is natural for time-periodic problems, the time axis is taken to be the whole of R. If the domain  has a boundary, that is,  ¤ Rn , a boundary condition u D u

on R  @

(2)

is added to the system. In case  is unbounded, an asymptotic value of the velocity field at spatial infinity lim u.t; x/ C u1 D 0

jxj!1

(3)

is prescribed. If the data f , u , and u1 are all time-periodic, say of period T > 0, the condition that the liquid undergoes a motion of the same period 8.t; x/ 2 R   W u.t C T; x/ D u.t; x/

(4)

completes the system. In the following, a mathematical analysis of (1), (2), (3), and (4) will be carried out for a fixed period T .

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Interaction with a moving boundary is often the driving force in a fluid flow. As typical examples, one may think of an object performing a motion in a fluid or the motion of a fluid in a container with the flow driven by momentum flux on the fluidstructure boundary due to an alteration of the surface geometry. In both cases, the domain  changes in time. In all relevant applications, a body force in the fluid is typically absent or, at the most, is potential-like and, therefore, can be absorbed in the pressure term in (1)1 . In order to mathematically investigate the corresponding equations of motion, however, such a system is usually rewritten in a fixed (in time) domain. In the new system, an artificial body force then appears. If the motion of the boundary, and hence the domain, is time-periodic, the artificial body force will be too. Consequently, the mathematical analysis of (1), (2), (3), and (4) in a fixed domain  is imperative for understanding more natural time-periodic fluid flows. In this article, basic questions concerning existence, regularity, uniqueness, and asymptotic properties of solutions to (1), (2), (3), and (4) will be addressed. The aim is to comprehensively introduce the most basic methods that can deliver answers to these questions. Focus will be on the following types of spatial domains: (i)  a bounded domain (Sect. 3). (ii)  an exterior domain, in which case a time-periodic velocity u1 .t / is prescribed at spatial infinity (Sect. 4). (iii)  a pipe, in which case a Poiseuille flow is prescribed at spatial infinity of each outlet (Sect. 5). Models based on bounded domains cover a large class of important physical systems. The motion of a liquid in a container with a time-periodic inflow (and outflow) is a classic example. The flow of a liquid in the gap between two rotating concentric spheres or cylinders is another. In fact, many intrinsic properties of viscous fluid flows are traditionally observed and studied in the setting of a timeperiodic flow in a bounded domain. The exterior domain case is just as interesting, as (1), (2), (3), and (4) in this case models a fluid flow past an object moving with time-periodic velocity u1 through a liquid. Also highly relevant from a physical point of view is the time-periodic fluid flow in a pipe. The cardiovascular system, for example, is essentially a fluid flow in a piping system with a prescribed timeperiodic flow rate. Generally, time-periodic fluid flows occurring in nature fall into one of the categories (i)–(iii). The investigation of time-periodic Navier-Stokes equations was initiated in a short note by Serrin [43], who suggested to study (1) as a dynamical system and identify a time-periodic solution as a periodic orbit. Serrin made the proposition that for time-periodic data f and any initial value, the solution u.t; x/ to the corresponding initial-value problem tends to a periodic orbit as t ! 1. However, as remarked by Serrin himself, the assumptions he makes are very stringent and, certainly at the time when the paper appeared, not sustained by any known result. The first complete results on existence of time-periodic solutions are due to Prodi [39] and Yudovich [53]. These authors considered the Poincaré map that takes an initial value into the state described by the solution to the initial-value problem

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at time T . A time-periodic solution can then be obtained via a fixed point of this map. The two originally proposed methods were subsequently used by several authors to build a mathematical foundation for the time-periodic Navier-Stokes equations. Of particular importance is the work [41] of Prouse, in which the celebrated method introduced by Hopf in [21] for the initial-value Navier-Stokes problem was adapted to the time-periodic setting for the first time. Although Prouse focused on weak solutions in bounded domains, the approach of Hopf based on a Galerkin approximation can be employed, as will be shown in the following, more broadly. The first to treat strong solutions were Kaniel and Shinbrot [23], followed shortly after by Takeshita [46]. Time-dependent domains were treated by Morimoto [38] and Miyakawa and Teramoto [37]. The first results for unbounded domains are due to Maremonti [34, 35], who treated the half- and whole-space problem. More general unbounded domains were investigated by Maremonti and Padula [36] by combining the Galerkin approximation with the “invading domain” technique. An important contribution was given by Kozono and Nakao [25], who, for the first time, proposed a direct representation formula for a time-periodic solution. Yamazaki [52] employed this formula to treat the case of a three-dimensional exterior domain. Further results for exterior domains were obtained by Galdi and Sohr [7] and Taniuchi [47]. Another direct representation formula was introduced by Kyed [27, 28] based on the Fourier transform on the locally compact abelian group R=T Z  Rn . This idea further lead to the concept of a time-periodic fundamental solution [30], maximal Lp regularity in Rn [29] for the linearization of (1), and an analysis by Lemarié-Rieusset [32] of time-periodic whole-space Navier-Stokes equations in critical spaces. Maximal Lp regularity was established in the twoand three-dimensional exterior domain by Galdi [11, 12] and Galdi and Kyed [14], respectively. Investigation of time-periodic Navier-Stokes equations in cylindrical domains was initiated by da Veiga [1] and continued by Galdi [9]. After the basic questions of existence, regularity, and uniqueness of solutions have been addressed, another critical issue emerges in the case of unbounded domains, namely, the inquiry into the asymptotic structure of solutions at spatial infinity. In the exterior domain case in particular, where u1 .t / in (3) describes the velocity of an object moving through a liquid, does the asymptotic structure of a solution reveal important physical properties. For u1 D 0, the leading term in an asymptotic expansion was identified by Kang, Miura, and Tsai [22] to be the same as the one found for the corresponding steady-state equations, namely, the Landau solution. Also for u1 ¤ 0, it has been shown [12, 14, 15, 27] that the leading term in the time-periodic case coincides with leading term found in the steady-state case. Other results concerning the asymptotic properties of time-periodic Navier-Stokes flow include a technique developed by Baalen and Wittwer [48] and the investigation by Silvestre [44] of flows with finite kinetic energy. The references given above concern results directly related to (1), (2), (3), and (4). Over the years, related systems have been investigated. Although out of the scope of this article, the system of equations governing the flow past a rigid body moving freely in a liquid under the action of a time-periodic force deserves mentioning. In order to mathematically investigate this problem, it is necessary to

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rewrite the equations of motion for the liquid, that is, the Navier-Stokes system, in a frame of reference attached to the body. The result is a system of time-periodic Navier-Stokes equations in a frame of reference that is not necessarily an inertial frame. This special type of time-periodic Navier-Stokes problem was investigated by Galdi and Silvestre [16,17], who extended a famous result of Weinberger [50,51] and Serre [42] to the time-periodic case. A general approach to time-periodic fluid flow problems was developed recently by Geissert, Hieber, and Nguyen [18]. A comprehensive treatment of time-periodic partial differential equations, including the Navier-Stokes equations, can be found in the books of Vejvoda [49] and Lions [33].

2

Notation

In the following,   Rn will always denote a domain, namely, an open connected set. Points in R   are generally denoted by .t; x/, with t being referred to as time and x as the spatial variable. Differential operators act only in the spatial variable unless otherwise indicated. In particular, @j D @xj for j D 1; : : : ; n. The notation BR refers to a ball in Rn centered at 0 with radius R > 0. Moreover, R B WD Rn n BR and BR1 ;R2 WD BR2 n BR1 . Additionally R WD  \ BR , R1 ;R2 WD  \ BR1 ;R2 , and R WD  \ B R . Einstein’s summation convention, that is, implicit summation over all repeated indices, is employed throughout. Constants in capital letters in the proofs and theorems are global, while constants in small letters are local to the proof in which they appear. For vector fields f ; g and second-order tensor fields F ; G on a domain   Rn , the notation 

 f ; g WD

Z

Z f  g dx D 

fi  gi dx;



 F ; G WD



Z

Z F W G dx D



Fij W Gij dx 

is used to denote their inner products. Classical Lebesgue spaces with respect to spatial domains are denoted by Lq ./ and with respect to time-space domains by Lq ..0; T /  /. If no confusion can arise, the norm is denoted by kkp in both cases. The norm in Lq ..0; T /  / is normalized so that  kf kLq ..0;T // WD

1 T

Z

T 0

Z

jf .t; x/jq dxdt

 q1 :



  The Lebesgue space Lq ./ is treated as the subspace of functions in Lq .0; T / that are time independent. This identification is made without furthernotification. With the normalization above, the Lq ./ norm coincides with the Lq .0; T /   norm on the subspace of time-independent functions.

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Classical Sobolev spaces are denoted by W m;q ./ and their norms by kkm;q . The subspaces of Sobolev functions vanishing on the boundary are denoted by kkm;q

m;q

m;q

W ./ WD C01 ./ . The dual space of the latter is denoted by W0 ./ WD  0 m;q 0 0 W0 ./ and its norm by kk1;2 . Sobolev spaces over time-space domains are introduced for j; k 2 N0 and q 2 Œ1; 1/ in the form     W j;k;q .0; T /   WD ff 2 L1loc .0; T /   j kf kj;k;q < 1g; 1 q1 0 X X q q kf kj;k;q WD @ k@˛t f kLq ..0;T // C k@ˇx f kLq ..0;T // A : j˛jj

(5)

0 0):

kvkX q

8 1 1 2 ˆ 2q C 4 krvk 4q C k@1 vkq C kr vkq ˆ 2 kvk 2q ˆ 4q ˆ ˆ ˆ ˆ 0 such that R n  BR0 , and recall for q 2 Œ1; 1/ that

hpi1;q

ˇZ ˇ WD krpkq C ˇ

R0

Z ˇ ˇ p.x/ dx ˇ; hpi0;1;q WD krpkq C

T 0

ˇZ ˇ ˇ

R0

ˇ ˇ p.t; x/ dx ˇdt (63)

1 .R  /, respectively. Classical and timedefines a norm on C01 ./ and C0;per periodic homogeneous Sobolev spaces can thus be introduced as the Banach spaces hi1;q

D 1;q ./ WD C01 ./

;

hi0;1;q

1;q 1 Dper .R  / WD C0;per .R  /

;

respectively. It is easy to verify that thelatter space coincides with the canonical  T -periodic extension of functions in Lq .0; T /I D 1;q ./ . Both characterizations, as well as the subspace 1;q

1;q Dper;? .R  / WD P? Dper .R  /;

are used in the following.

4.2

Existence of Weak Solutions

Existence of weak solutions to (61) in the case  is a three-dimensional exterior domain can be established without any restrictions on the “size” of the data. The approach employed below is sometimes referred to as the invading domain technique. It is based the Galerkin approximation used in the proof of Theorem 1.

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Firstly, a definition of a weak solution to (61) is needed. Compared to Definition 1 in the case a bounded domain , the concept of a weak solution in the exterior domain case has to incorporate also the decay property limjxj!1 u.t; x/ D 0 of the solution. The definition below is a simple extension of a similar definition in the steady-state case [10, Definition X.1.1].   Definition 2. Let   R3 be an exterior domain. Let f 2 L2per RI D01;2 ./ . A   vector field u 2 L2per RI D 1;2 ./ is called a weak time-periodic solution to (61) if div u D 0, u D u1 on R  @ in the trace sense, the identity Z

T

       u; @t ' C  ru; r' C .u  u1 /  ru; '  hf ; 'i dt D 0

(64)

0 1 .R  /, and u satisfies for almost all t 2 R the decay holds for all ' 2 C0;;per property Z 1 lim ju.t; x/j d .x/ D 0: (65) R!1 j@BR j @BR

In order to employ a Galerkin approximation to establish existence of a weak solution, it is necessary to first “lift” the non-homogeneous boundary values in (61), that is, find a suitable extension of the boundary values and subtract it from u to obtain an equivalent system with homogeneous boundary data. It is critical that the terms in which the extension appears in the new system can be suitably estimated in each step of the Galerkin approximation. A well-known method, which goes back to Leray and Hopf, can be modified to construct an appropriate extension. 1;2 Lemma 2. Let   R3 be an exterior domain of class C 0;1 . Let u1 2 Wper .R/.   1;2 2;2 For every " > 0, there is a vector field W " 2 Wper RI W ./ satisfying W " D u1 on @, div W " D 0 in ,

 C3 ku1 k1;2 ; kW " kWper 1;2 .RIW 2;2 .//

(66)

and 1 8' 2 C0;per .R  / W

Z

 

 '.t; x/  rW " .t; x/; '.t; x/ dx  "kr'.t /k22 : (67)

Proof. Let " 2 C01 .I R/ be a “cutoff” function with of @. It follows directly that

"

D 1 in a neighborhood

1  x3  u12 .t / W " .t; x/ WD r  @ " .x/  x1  u13 .t /A " .x/  x2  u11 .t / 0

" .x/

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satisfies all the desired properties except (67). To obtain (67), " needs to choose in t u a certain way. For example, one may choose " as in [10, Lemma III.6.2]. Remark 6. Lemma 2 can be extended to yield solenoidal extensions for a much larger class of time-periodic boundary values u .t; x/, .t; x/ 2 R@. For example, [10, Lemma X.4.1] can be modified to include time-periodic boundary values, which would then produce such a class. All results in this section continue to hold if the boundary condition u D u1 in (61) is replaced with u D u for a time-periodic vector field u that has a solenoidal extension with the properties from Lemma 2. With the lemma above, existence of a weak solution to (61) can be shown with the invading domain technique. The main idea is to apply a Galerkin approximation to show existence of a solution on the bounded domain  \ BR .0/. After securing a priori estimates independent on R, a weak solution is found by a limiting process R ! 1. 1;2 .R/ Theorem 7. Let   R3 be an exterior domain of class C 0;1 . If u1 2 Wper   1;2 2 and f 2 Lper RI D0 ./ , then there is a weak time-periodic solution u 2   L2per RI D 1;2 ./ to (61) in the sense of Definition 2.

Proof. Let W " be the extension field from Lemma 2. Existence of a solution to (61) on the u Dw C W " can be obtained by finding a solenoidal vector field  form 1;2 w 2 L2per RI D0; ./ satisfying Z T          w; @t ' C  rw; r' C w  rw; ' C .W "  u1 /  rw; ' dt 0 (68) Z T   e ; 'i dt  w  rW " ; ' C hf D 0 1 for all ' 2 C0;;per .R  /, where

    e ; 'i WD hf ; 'i  @t W " C .W "  u1 /  rW " ; '   rW " ; r' : hf

(69)

Let fRk g1 1 as k ! 1. Put k WD  \ BRk .0/. kD1  R be a sequence with Rk !   1;2 2 .k / to (68) with respect to test Since k is bounded, a solution wk 2 Lper RI W0; 1 functions ' 2 C0;;per .Rk / can be obtained as in the proof of Theorem 1. For this purpose, choose "  12  and employ (67) in order to deduce an identity equivalent to (11). By repeating the rest of the argument from the proof of Theorem 1, a solution wk that satisfies the equivalent to (15), which due to term w  rW " ; ' on the right-hand side in (68) becomes an inequality kwk kL2

1;2 per .RID0; .k //



2 

Z

T 0

e ; wk i dt; hf

10 Time-Periodic Solutions to The Navier-Stokes Equations

537

is then obtained. Provided the vector field is extended by 0 on R3 n k , it  wk 1;2  1 2 follows that fwk gkD1 is bounded in Lper RI D0; ./ . Consequently, there is a that converges weakly in this space to a vector field subsequence of fwk g1  kD1 1;2 2 w 2 Lper RI D0; ./ . To ensure that w satisfies (68), it suffices to verify that wk   converges strongly in L2per RI L2 .R / for any R > R0 . This will in particular   ensure convergence of the nonlinear term wk  rwk ; r' . The verification can   1;2 be done by deducing from (68) that @t wk is bounded in L1per RI D0; .R / , and   utilize that wk thereby lies in a space that embeds compactly into L2per RI L2 .R / . Finally, since  is a three-dimensional exterior domain, the decay property (65) 1;2 follows directly from the fact that w.t / 2 D0; ./ and W " .t / 2 W 2;2 ./; see [10, Lemma II.6.3]. Remark 7. A similar result is open in the two-dimensional case n D 2. Although the arguments in the proof of Theorem 7 that ensure existence of a vector field satisfying (64) are all valid also when n D 2, it is not clear if this field satisfies the decay property (65). The same problem has been open for the corresponding steady-state problem for decades. Recall that the steady-state problem is a special case of the time-periodic problem.

4.3

Existence of Strong Solutions

Compared to the bounded domain case, integrability properties are a more delicate matter in unbounded domains such as exterior domains, since they describe not only local regularity but also decay properties as jxj ! 1 of the solution. Consequently, different characterizations of strong solutions transpire. Below, a class (71), (72), and (73) of strong solutions is introduced that emerges from adaptation of the methods from Sect. 3.3 to the exterior domain case. By modification of the Galerkin approximation from the proof of Theorem 7, existence of a strong solution of this type to (61) for data f and u1 sufficiently restricted in “size“ is established in the three-dimensional case n D 3. Later, in Sect. 4.8, a different class of strong solutions based on Lq estimates for the linearization of (61) is treated. In comparison, the method based on Lq theory for the linearized problem yields better decay properties of the solution, while the method based on Galerkin approximation is more versatile when it comes to the admissible structure of u1 . 2 Theorem 8. Let   R3 be an exterior There is a constant  of class  domain  C . 1;2 2 2 2 "2 .; ; T / > 0 such that if f 2 Lper RI L ./ \ Lper RI D0 ./ and u1 2 1;2 Wper .R/ satisfy

kf kL2per .RIL2 .// C kf kL2

1;2 .// per .RID0

C ku1 k1;2  "2 ;

(70)

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G.P. Galdi and M. Kyed

then there is a solution .u; p/ to (61) that satisfies     2 u 2 L2per RI D 1;2 ./ ; ru 2 L1 per RI L ./ ;     r 2 u 2 L2per RI L2 ./ ; @t u 2 L2per RI L2 ./ ;     P? u 2 L2per RI L2 ./ ; p 2 L2per RI D 1;2 ./ :

(71) (72) (73)

Proof. Again, the “lifting” field W " from Lemma 2 is utilized. If w is a solution to 1 (68) for all ' 2 C0;;per .R  /, then u WD w C W " is a solution to (61). Since W " satisfies (71), (72), and (73), it suffices to verify that also w satisfies (71), (72), and (73). To obtain a vector field w with the desired properties, one proceeds as in the proof of Theorem 7 and employs a Galerkin approximation to first solve (68) on an ascending sequence of bounded domains k WD  \ BRk , limk!1 Rk ! 1. On each Rk , a solution wk with kwk kL2 .RID 1;2 .// bounded independently on k is per 0; thereby obtained. Furthermore, the argument from the proof of Theorem 3 can be reused to deduce that wk is a strong solution. One may verify, using, for example, a simple scaling argument, that the constants c0 and c1 in the proof of Theorem 3 are independent on k. By repeating the proof of Theorem 3 up till (37) and taking into consideration the additional terms in (68) containing the “lifting” field W " , which is not present in the proof of Theorem 3, one obtains the estimate krwk .t /k22



e k2 e k2 2  c0 kf C kf 1;2 L .0;T IL2 .// .// L2 .0;T ID 0

Z

t

C s0

Z



 kW " .s/k22;2 C ju1 .s/j2 krwk .s/k22 ds

t

C s0

krwk .s/k42

C

krwk .s/k62

(74)

 ds ;

e defined as in (69) and the constant c0 independent on k. Now take " WD "2 , with f with "2 still to be chosen. Then (66) and the “smallness” assumption (70) furnish krwk .t /k22



2

Z

t

 c1 "2 C s0

2

"2 krwk .s/k22

C

krwk .s/k42

C

krwk .s/k62

 ds ; (75)

again with the constant c1 independent on k. Based on the inequality (75), the , argument following (37) in the proof Theorem 3 yields that krwk kL1 2 per .RIL .// kr 2 wk kL2per .RIL2 .// , and k@t wk kL2per .RIL2 .// are bounded independently on k, provided "2 is chosen sufficiently small. At this point, it is therefore possible to let k ! 1. After possibly passing to a subsequence, one finds as a weak limit of fwk g1 kD1 in the spaces (71) and (72) a solenoidal vector field w. Clearly, u WD 1 w C W " satisfies (61) with respect to test functions ' 2 C0;;per .R  /. By a

10 Time-Periodic Solutions to The Navier-Stokes Equations

539

standard method from the well-known analysis of the corresponding initial-value problem, or alternatively described in Remark 2, the existence  by the approach  of a pressure p 2 L2per RI D 1;2 ./ that renders .u; p/ a solution to (61) follows.   It remains to show P? u 2 L2per RI L2 ./ . For this purpose, expand u into a   P i 2 2 T ht , and deduce from @t u 2 L2 Fourier series u.t / D per RI L ./ h2Z uh e that fhuh gh2Z 2 `2 .L2 .//. Consequently, fuh gh2Znf0g 2 `2 .L2 .//. Since the latter sequence  can be recognized as the Fourier coefficients of P? u, it follows that P? u 2 L2per RI L2 ./ . t u

4.4

New Approach to Lq Estimates for the Linearized Problem in the Whole Space

While the strong solutions to (61) established in Theorem 8 are locally very regular, little information is revealed about their integrability at spatial infinity, that is, the rate of decay as jxj ! 1. Only when the asymptotic behavior at spatial infinity is known can it be determined whether the fluid flow described by a solution is meaningful from a physical point of view. At the outset, no such information is available for the solution from Theorem 8. To gain more insight, global Lq estimates of solutions to an appropriate linearization of (61) in terms of the data are needed. Such estimates can be used to extract information from a solution to (61) directly or to establish, for example, by a fixed point argument, existence of a (strong) solution to (61) for which the decay as jxj ! 1 is better understood. An Lq theory for an exterior domain problem is usually derived via Lq estimates for the corresponding whole-space problem. In the following, the linearized time-periodic Navier-Stokes system in the whole-space 8 @t u  u  @1 u C rp D F ˆ ˆ ˆ ˆ ˆ ˆ < div u D 0 u.t C T; x/ D u.t; x/; ˆ ˆ ˆ ˆ ˆ ˆ : lim u.t; x/ D 0;

in R  Rn ; in R  Rn ; (76)

jxj!1

shall be investigated. Here,  0 is a constant. In the context of Lq estimates for (76), one has to distinguish between the two cases D 0 and ¤ 0. In the former case, the system is referred to as a timeperiodic Stokes problem and in the latter as a time-periodic Oseen problem. It is well known from the corresponding steady-state problem that Lq estimates for the Stokes and Oseen problem are different. From a physical point of view, the discrepancy is not surprising as the Oseen equations model a fluid flow past a moving body, which therefore should exhibit a wake region behind the body, whereas a Stokes flow describes a flow without a wake region around a stationary body .

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A simple but important step toward optimal Lq estimates for (76) lies in the decomposition of (76) by the complementary projections P and P? introduced in (62). If the velocity field and pressure term are expressed as u D Pu C P? u DW v C w;

p D Pp C P? p DW p C ;

one easily verifies that .v; p/ is a solution to the steady-state problem 8  v  @1 v C rp D PF ˆ ˆ ˆ < div v D 0 ˆ ˆ ˆ : lim v.x/ D 0;

in Rn ; in Rn ;

(77)

jxj!1

and .w; / a solution to the time-periodic problem 8 @t w  w  @1 w C r D P? F ˆ ˆ ˆ ˆ ˆ ˆ < div w D 0

in R  Rn ; in R  Rn ; (78)

w.t C T; x/ D w.t; x/; ˆ ˆ ˆ ˆ ˆ ˆ : lim w.t; x/ D 0: jxj!1

Since (78) resembles (76), not much insight seems to have been won by this decomposition. It turns out, however, that in the context of Lq estimates, system (78) has some remarkable characteristics. Since both the data P? F and the solution .w; / have vanishing time average over the period, that is, they are purely oscillatory, an analysis of (78) can be carried out in subspaces of Lq consisting entirely of oscillatory functions. As a result, much better Lq estimates materialize for .w; / than can be shown for a solution to the original problem (76). An optimal Lq theory for the time-periodic problem (76) can be established by combining these estimates for .w; / with well-known Lq estimates for the solution .v; p/ to the steady-state problem (77). Below, it is described how to establish the Lq estimates for .w; / by means of Fourier analysis. Alternatively, one can also establish the estimates by extending the method used in the proof of Theorem 6. In comparison, the approach below is more direct as it is based on a direct representation of the solution in terms of a Fourier multiplier. Moreover, it leads naturally to the concept of a fundamental solution for the time-periodic problem. The main idea is to reformulate (78) as partial differential equation on the locally compact abelian group G WD R=T Z  Rn and analyze the problem with the corresponding Fourier transform FG . q

Theorem 9. Let q 2 .1; 1/. For every F 2 Lper;? .R  Rn /, there is a solution 1;2;q

1;2;q

1;q

.w; / 2 Wper;? .R  Rn /  Wper;? .R  Rn /  Dper;? .R  Rn /

10 Time-Periodic Solutions to The Navier-Stokes Equations

541

to (78). The solution satisfies the estimate kwk1;2;q C kr kq  C4 P . ; T / kF kq ;

(79)

where P . ; T / is a polynomial in and T and C4 D C4 .; n; q/. If for some 1;2;r 1;r r 2 .1; 1/ .e w; e / 2 Wper;? .R  Rn /  Dper;? .R  Rn / is another solution, then wDe w and .t; x/ D e .t; x/ C d .t / for some T -periodic function d W R ! R. Some nomenclature from abstract harmonic analysis is needed to sketch a proof of Theorem 9. A topology and an appropriate differentiable structure on the group G WD R=T Z  Rn are inherited from R  Rn . More precisely, G becomes a locally compact abelian group when equipped with the quotient topology induced by the canonical quotient mapping q W R  Rn ! R=T Z  Rn ;

q.t; x/ WD .Œt ; x/:

(80)

The restriction … WD qjŒ0;T /Rn is used to identify G with the domain Œ0; T /  Rn ; … is clearly a (continuous) bijection. Via …, one can identify the Haar measure dg on G as the product of the Lebesgue measure on Œ0; T / and the Lebesgue measure on Rn . The Haar measure is unique up to a normalization factor, which in the following is chosen such that Z u.g/ dg D G

1 T

Z

T

Z u ı ….t; x/ dxdt:

0

Rn

For the sake of convenience, the symbol … in integrals of G-defined functions with respect to dxdt shall be omitted. By C 1 .G/ WD fu W G ! R j u ı q 2 C 1 .R  Rn /g;

(81)

the space of smooth functions on G is defined. For u 2 C 1 .G/, derivatives are defined by 8.˛; ˇ/ 2 Nn0  N0 W

ˇ

ˇ @t @˛x u WD @t @˛x .u ı q/ ı …1 : ˇ

(82)

It is easy to verify for u 2 C 1 .G/ that also @t @˛x u 2 C 1 .G/. The subspace C01 .G/ denotes the compactly supported smooth functions. With a differentiable structure available on G, the space of tempered distributions can be defined. For this purpose, recall the Schwartz-Bruhat space of generalized Schwartz functions (see, e.g., [2]) given by S.G/ WD fu 2 C 1 .G/ j 8.˛; ˇ;  / 2 Nn0  Nn0  N0 W ˛;ˇ; .u/ < 1g; 

˛;ˇ; .u/ WD sup jx ˛ @ˇx @t u.t; x/j: .t;x/2G

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G.P. Galdi and M. Kyed

˚ Equipped with the semi-norm topology of the family ˛;ˇ; j .˛; ˇ;  / 2 Nn0  Nn0   N0 , S.G/ becomes a topological vector space. The dual space S 0 .G/ equipped with the weak* topology is referred to as the space of tempered distributions on G. ˇ For a tempered distribution u 2 S 0 .G/, distributional derivatives @t @˛x u 2 S 0 .G/ are defined by duality as in the classical case. Similarly, tempered distributions on b are introduced. By associating each .k; / 2 Z  Rn with the G’s dual group G 2 character W G ! C; .t; x/ WD eix Ci T kt on G (it is standard to verify that all b D Z  Rn . By characters are of this form), one can characterize the dual group as G b WD fw 2 C .G/ b j 8k 2 Z W w.k; / 2 C 1 .Rn /g; C 1 .G/ b is introduced. The Schwartz-Bruhat space on the space of smooth functions on G b is given by the dual group G b j 8.˛; ˇ;  / 2 Nn0  Nn0  N0 W O˛;ˇ; .w/ < 1g; b WD fw 2 C 1 .G/ S.G/ ˇ

O˛;ˇ; .w/ WD sup j ˛ @ k  w.k; /j; .k; /2b G and equipped with the canonical semi-norm topology. The Fourier transform associated to the locally compact abelian group G is denoted by FG . It is explicitly given by b FG W L1 .G/ ! C .G/;

FG .u/.k; / WD

1 T

Z

T

Z

u.t; x/ eix i

2 T

kt

dxdt:

Rn

0

b is the product of the counting measure on Z and the Since the Haar measure on G n Lebesgue measure on R , the inverse Fourier transform is formally defined by 1 b F1 G W L .G/ ! C .G/;

F1 G .w/.t; x/ WD

XZ

w.k; / eix Ci

2 T

kt

d :

Rn

k2Z

b is a homeomorphism with F1 It is standard to verify that FG W S.G/ ! S.G/ G as the actual inverse, provided the Lebesgue measure d is normalized appropriately. b By duality, FG extends to a homeomorphism S 0 .G/ ! S 0 .G/. q 1;2;q 1;q Similar to the spaces Lper .R  Rn /, Wper .R  Rn /, and Dper .R  Rn /, Lebesgue and Sobolev spaces with respect to the domain G are defined by Lq .G/ WD C01 .G/ and

kkq

;

W 1;2;q .G/ WD C01 .G/

D 1;q .G/ WD C01 .G/

hi0;1;q

kk1;2;q

;

;

where kkq denotes the Lq -norm with respect to the Haar measure dg, the norm kk1;2;q defined as in (5), and the norm hi0;1;q as in (63). It is standard to verify

10 Time-Periodic Solutions to The Navier-Stokes Equations

543

that W 1;2;q .G/ D fu 2 Lq .G/ j kuk1;2;q < 1g. As in (62), the time-averaging projection P and its complement P? are introduced on functions defined on G. 1;2;q 1;q Thus, the subspaces W? .G/ WD P? W 1;2;q .G/, D? .G/ WD P? D 1;q .G/, and q q L? .G/ WD P? L .G/ can be defined. It is now possible to formulate (78) as a system of partial differential equations on G in a context that permits utilization of the Fourier transform FG . A representation formula for the solution in terms of a Fourier multiplier can then be obtained. The proof of Theorem 9 below is based on such a representation. Proof of Theorem 9. Since the topology and differentiable structure on G is inherited from R  Rn , the T -time-periodic problem (78) can be formulated equivalently as a system of G-defined vector fields (

@t w  w  @1 w C r D P? F

in G;

div w D 0

in G

(83)

with unknowns w W G ! Rn , W G ! R, and data F W G ! Rn . In this formulation, the periodicity condition is not needed anymore. Indeed, all functions defined on G are intrinsically T -time-periodic. Since (83) can be interpreted as a system of equations in S 0 .G/, the Fourier transform

be applied to obtain FG can a formula for w. An easy calculation shows that FG P? F D .1  ıZ .k//FG F , where ıZ denotes the Dirac delta distribution on Z, which is simply the function ıZ .0/ WD 1 and ıZ .k/ WD 0 for k ¤ 0. Formally, application of the Fourier transform FG in (83) yields    ˝ FG F ; I wD j j2 C i . 2 k  1 / j j2 T     1 i 1 i D FG  FG F D FRn  FR n F : j j2 j j2 F1 G



1  ıZ .k/

(84)

The Fourier multiplier b ! C; M .k; / W G

M .k; / WD

1  ıZ .k/   j j C i 2 k  1 T 2

(85)

b and M 2 C 1 .G/, b which can easily is bounded and smooth, that is, M 2 L1 .G/ be seen by observing that the numerator of M vanishes in a neighborhood of the only zero .0; 0/ of the denumerator. It can be shown that M is an Lq .G/ multiplier in the sense that the mapping f ! F1 G M FG Œf  extends from a mapping S.G/ ! S 0 .G/ into a bounded operator Lq .G/ ! Lq .G/. Since multiplier theorems like the ones of Mikhlin, Lizorkin, or Marcinkiewicz are only available in an Euclidean setting and not in the general setting of group multipliers, a proof has to rely on a so-called transference principle. Originally introduced by de Leeuw [4], a

544

G.P. Galdi and M. Kyed

transference principle for Fourier multipliers on local compact abelian groups makes it possible to study the properties of M via a corresponding multiplier defined on RnC1 . The transference principle (see, e.g., [5, Theorem B.2.1]) states that M is an b ! RnC1 such that Lq .G/ multiplier if there is a continuous homomorphism ˆ W G q nC1 M D m ı ˆ for some L .R / multiplier m. Moreover, the norm of the Lq .G/ operator corresponding to M coincides with the norm of the Lq .RnC1 / operator corresponding to m. To identify such an m in the particular case above, let be a “cutoff” function with 2 C01 .RI R/, ./ D 1 for jj  12 , and ./ D 0 for jj  1. Define m W R  R ! C; n

m.; / WD

1

T   2

j j2 C i .  1 /

:

(86)

  b ! RnC1 , ˆ.k; / WD 2 k; . Clearly, ˆ is a continuous Further, let ˆ W G T homomorphism and M D m ı ˆ. To show that m is a Lq .RnC1 / multiplier, a standard multiplier theorem can be applied. Since m is a rational function with nonvanishing denominator away from .0; 0/, it is easy to verify that all functions of type "

.; / ! 1"1    n"n "nC1 @"11    @"nn @nC1 m.; /

(87)

stay bounded as j.; /j ! 1. Consequently, it follows from Marcinkiewicz’s multiplier theorem (see, e.g., [45, Chapter IV, §6]) that m is an Lq .RnC1 / multiplier. A more careful analysis of the bounds obtained for the functions in (87) shows that the norm of the corresponding operator is bounded by a polynomial P . ; T / in

and T . It therefore follows that 8f 2 Lq .G/ W



kF1 G M FG Œf  kq  P . ; T /kf kq :

Now return to (84). Observe that I  ˝ 2 is the symbol of the Helmholtz-Weyl j j projection, that is, the projection PH W Lq .Rn /n ! Lq .Rn /n onto the subspace q L .Rn / of solenoidal vector fields. It is well known that PH is continuous on Lq .Rn / for any q 2 .1; 1/. The projection PH extends trivially to a continuous projection on Lq .G/. Now define .w; / by (84). Clearly, .w; / is a solution in the sense of distributions S 0 .G/ to (83). Moreover, 

   M .k; /F ŒP F  kwkq D F1   P . ; T / kPH F kq  P . ; T / kF kq : G H G q

The argument above can be repeated for @t w and @˛x w for j˛j  2. More precisely, one may verify that also .k; / ! kM .k; / and .k; / ! ˛ M .k; / are Lq .G/ multipliers for j˛j  2. It thus follows that kwk1;2;q  P . ; T / kF kq . Based on (84), it is standard to show kr kq  c kF kq . Since ıZ is the Fourier symbol of P,

10 Time-Periodic Solutions to The Navier-Stokes Equations

545

it follows directly from (84) that P? w D w and P? D . Thus a solution .w; / 2 1;2;q 1;q W? .G/D? .G/ to (83) that satisfies (79) is obtained. Via the canonical quotient 1 map q, the spaces C0;per .R  Rn / and C01 .G/ are isometrically isomorphic in the norm kk1;2;q . By the definition of Sobolev spaces as completions of theses spaces in 1;2;q 1;2;q the norm kk1;2;q , it follows that also W? .RRn / and W? .G/ are isometrically 1;q 1;q isomorphic. The same is clearly true for Dper;? .R  Rn / and D? .G/ as well q q n as for Lper;? .R  R / and L? .G/. It follows that .w ı q; ı q/ is a solution in 1;2;q

1;q

Wper;? .R  Rn /  Dper;? .R  Rn / to (78) that satisfies (79). It remains to establish the desired uniqueness property. It suffices to do so for the 1;r system (83). Assume .e w; e / is another solution in W?1;2;r .G/  D? .G/. Employ in (83) first the Helmholtz projection P and then the Fourier transform to deduce  

H 2 2 that i 2 F w  e w D 0. Since the polynomial j j k C j j 

i C i 2 k 1 G T  T

that supp F w  e w  f.0; 0/g.

1 vanishes only at .k; / D .0; 0/, it follows G

However, since P w  e w D 0 also ıZ .k/F w .k; / D 0, whence G w

e .0; 0/

… supp FG we w . Consequently, supp FG we w D ;. It follows that FG we w D 0 and thus w D e w. By (83), .t; x/ D e .t; x/ C d .t / for some T -periodic function d W R ! R. t u To complete the Lq estimate for a solution to (76), also the steady-state part .v; p) needs to be addressed. However, the Lq theory for the Stokes/Oseen system (77) is well known; see, for example, [10, Theorem IV.2.1 and VII.4.1]. As the steady-state Lq estimates for .v; p/ are completely decoupled from the time-periodic nature of (76), they are omitted here.

4.5

Fundamental Solution

The reformulation of the linear T -time-periodic system (76) on the group G carried out in the proof of Theorem 9 motivates the introduction of a time-periodic fundamental solution. In the setting of tempered distributions S 0 .G/, a fundamental solution to (76) or rather to the equivalent system on the group G (

@t u  u  @1 u C rp D F

in G;

div u D 0

in G;

(88)

can be defined as a tensor field 0

 TP

1 TP  TP 11 : : :  1n B :: : : :: C B : : C WD B : C 2 S 0 .G/.nC1/n @ TP : : :  TP A n1 nn 1TP : : : nTP

(89)

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G.P. Galdi and M. Kyed

that satisfies (

@t 

TP

ij

  TP ij  @1 

TP

ij

C @i jTP D ıij ıG ; @i 

TP

ij

D 0;

(90)

where ıij and ıG denote the Kronecker delta and delta distribution, respectively. For a sufficiently regular right-hand side, say F 2 S.G/n , a solution to (88) is then given by component-wise convolution over the group G with the fundamental solution:   u WD  TP  F : (91) p The ability to identify a solution in terms of such a direct expression offers many advantages. To the extent that pointwise information can be obtained for  TP , a similar type of information can be obtained for (91). In particular, knowledge of the asymptotic structure of  TP at spatial infinity can be used to analyze the pointwise behavior of u.t; x/ at as jxj ! 1. As already observed, the projection P can be expressed as Pf WD F1 G ıZ FG Œf  . From this expression, it is seen that P can be extended to a projection in the context of distributions P W S 0 .G/ ! S 0 .G/. The same is true for P? . Consequently, the idea from the previous section to use P and P? to decompose (90) into a steady-state and oscillatory part can be reused. As a result, it is possible to identify  TP as a sum of a well-known fundamental solution to the corresponding steady-state system and an oscillatory fundamental solution. It turns out that the oscillatory fundamental solution has significantly better decay properties as jxj ! 1. The structure of the fundamental solution differs depending on whether

D 0 or ¤ 0. This phenomenon is well known from the steady-state case, from which one may recall that a fundamental solution . ;  / 2 S 0 .Rn /nn S 0 .Rn /n to (



ij

 @1 

ij

C @i j D ıij ıRn ; @i 

ij

D 0;

(92)

in the Stokes case D 0 is given by the Stokes fundamental solution . Stokes ;  / and in the Oseen case ¤ 0 by the Oseen fundamental solution . Oseen ;  /; the pressure  is the same in two cases. Explicit expressions for both are well known; see, for example, [10, Chapter IV.2] and [10, Chapter VII.3] for the Stokes and Oseen fundamental solution, respectively. Theorem 10. Let n  2. Put (  WD

 Stokes

if D 0

(Stokes case);

Oseen

if ¤ 0

(Oseen case):



10 Time-Periodic Solutions to The Navier-Stokes Equations

547

Then  TP WD  ˝ 1R=T Z C  ? ; 

TP

(93)

WD  ˝ ıR=T Z ;

(94)

with 

?

WD

F1 G



  1  ıZ .k/ ˝ 2 S 0 .G/nn  I  2 j j2 C i 2 k 

j j 1 T

(95)

defines a fundamental solution  TP 2 S 0 .G/.nC1/n to (88) on the form (89) satisfying (90) and   nC2 8q 2 1; W  ? 2 Lq .G/nn ; n h n C 2

W @j  ? 2 Lq .G/nn 8q 2 1; nC1

(96) .j D 1; : : : ; n/;

8r 2 Œ1; 1/ 8" > 0 9C > 0 8jxj  " W k ? .; x/kLr .R=T Z/ 

(97) C ; jxjn

8r 2 Œ1; 1/ 8" > 0 9C > 0 8jxj  " W k@j  ? .; x/kLr .R=T Z/ 

C jxjnC1

(98) ; (99)

8q 2 .1; 1/ 9C > 0 8F 2 S.G/n W k

?

 F kW 1;2;q .G/  C kF kLq .G/ ; (100)

where 1R=T Z 2 S 0 .R=T Z/ denotes the constant 1. Proof. Apply in (90) first the projections P and P? and subsequently the Fourier transform to deduce P TP D  ˝ 1R=T Z ;

P? 

TP

D F1 G



  1  ıZ .k/ ˝ I    j j2 C i 2 k  1 j j2 T

and  TP D F1 G



i j j2



D F1 Rn



i j j2

 ˝ ıR=T Z :

It thus follows that . TP ;  TP / given by (93) and (94) defines a fundamental solution  TP 2 S 0 .G/.nC1/n to (88) on the form (89). Recall (85) and the observation made

548

G.P. Galdi and M. Kyed

b \ C 1 .G/ b to see that the right-hand in the proof of Theorem 9 that M 2 L1 .G/ 0 ? side in definition (95) of  is well defined. The properties (96), (97), (98), and (99) can be shown by a lengthy but direct computation and subsequent estimate of the inverse Fourier transform in (95); see [15, 30]. Finally, (100) is just a reiteration of the statement in Theorem 9. Remark 8. It is well known (see again [10, Chapter IV.2 and Chapter VII.3]) that both  Stokes and  Oseen have a pointwise decay rate as jxj ! 1 that is slower than jxjn . Estimate (98) therefore implies that the oscillatory part  ? of  TP decays faster than the steady-state part. Consequently, the asymptotic behavior as jxj ! 1 of a solution u to (88) given by (91) will be dominated by the steady-state part Pu.

4.6

Weighted Estimates for the Linearized Problem in the Whole Space

The properties obtained for the fundamental solution  TP in Theorem 10 can be  ˇ used to establish pointwise weighted estimates, with weights of type 1 C jxj , of solutions to the linearized system (76). Weighted estimates of this type are well known for the corresponding steady-state problem; see, for example, [10, Lemma V.8.2]. Thus, if a time-periodic solution is once again decomposed into a steadystate and oscillatory part, weighted estimates need only be established for the oscillatory part. As in the case of the Lq estimates in Theorem 9, better estimates in terms of decay at spatial infinity materialize for the oscillatory part. For ˇ 2 Œ1; 1/, let Xˇ denote the Banach space ˚  Xˇ WD ' 2 L1loc .Rn / j ŒŒ'ˇ < 1 ;

 ˇ ŒŒ'ˇ WD ess sup 1 C jxj j'.x/j: x2Rn

(101) It is illustrated below how to obtain  estimates for oscillatory  q-generalized solutions to (76) in spaces L1 WD P? L1 per RI Xˇ . These estimates can be per;? RI Xˇ augmented with Lq estimates. For this purpose, recall the interpretation from Sect. 4.4 of T -time-periodic vector fields as functions defined on the group G, and let 1

W?2

;1;q

˚  .G/ WD w 2 Lq .G/ j kwk 1 ;1;q < 1; Pw D 0 ; 2      1  12  kwk 1 ;1;q WD F jkj C j j C 1 FŒw  2

q

denote the Sobolev space of oscillatory functions with “half” a derivative in time and one derivative in space belonging to Lq .G/. The corresponding space 1

;1;q

2 Wper;? .R  Rn / is defined canonically via the quotient mapping q.

10 Time-Periodic Solutions to The Navier-Stokes Equations

549

Theorem 11. Let n  2, 2 Œ0; 1/ and ˇ 2 Œ1; n C 1. For every F D divG with nn , there is a solution .w; / to (76) with w 2 L1 G D L1 per;? RI Xˇ per;? RI Xˇ and     1   n r n 2 ;1;r .R  Rn /; 2 L1 ; 1 ; 1 W w 2 Wper;? 8r 2 max per;? RI L .R / ˇ (102) which satisfies kwkL1  C5 kG kL1 ; per .RIXˇ / per .RIXˇ /

(103)

    with C5 D C5 .ˇ; ; ; n/ and for all r 2 max ˇn ; 1 ; 1 r n r n kwk 1 ;1;r C k kL1  C6 kG kL1 per .RIL .R // per .RIL .R //

(104)

2

  w; e / is another solution with e w 2 L1 with C6 D C6 .r; ; ; n/. If .e RI X˛ for per;?   s n some ˛ 2 Œ1; 1/ and 2 L1 per;? RI L .R / for some s 2 .1; 1/, then .w; / D .e w; e /. Proof. From the integrability of the fundamental solution  ? stated in Theorem 10, recall (97), and the integrability of G , it follows that the convolution integral Z Z 1 T wi .t; x/ WD  @k  ? ij .t  s; x  y/ G kj .s; y/ dyds .i D 1; : : : ; n/ T 0 Rn (105) n is well-defined as an element of, say, L1 per .R  R /. From the definition (95) and the interpretation of T -time-periodic vector fields as functions defined on the group G, it is clear that w together with

WD

F1 G



 i j j j2



FG G ij



D

F1 Rn



 i j j j2



FRn G ij



is a solution to (76). It can be shown with the same approach as in the proof of 1

;1;r

2 Theorem 9 that w 2 Wper;? .R  Rn / and satisfies (104). It follows directly from n the definition of above that also satisfies (104). Since w 2 L1 per .R  R /, it is enough to establish the weighted estimate (103) for sufficiently large jxj. Consider for this purpose an jxj > 2 and decompose the integral in (105) as

1 wi .t; x/ D  T Z C

Z

T

Z

Z

Z

C 0

B 2jxj

Bjxj=2

C Bjxj=2;2jxj nB1 .x/

B1 .x/

@k  ? ij .t  s; x  y/ G kj .s; y/ dyds

DW I1 .t; x/ C I2 .t; x/ C I3 .t; x/ C I4 .t; x/:

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G.P. Galdi and M. Kyed

On the strength of estimate (99) in Theorem 10, it follows that Z jI1 .t; x/j  Bjxj=2

ˇ k@k  ? dy kG kL1 ij .; x  y/kL2per .R/ .1 C jyj/ per .RIXˇ /

Z

 c0 Bjxj=2

jx  yj.nC1/ .1 C jyj/ˇ dy kG kL1 per .RIXˇ /

 c1 jxjˇ kG kL1 ; per .RIXˇ / where the last inequality is valid due to the assumption that ˇ  nC1. The integrals I2 and I4 are estimated in a similar fashion. The summability property (97) from Theorem 10 is needed to deduce that @k  ? is integrable over B1 , which then leads to an estimate of I3 . It follows from the estimates of I1 , I2 , I3 , and I4 that jw.t; x/j  c2 jxjˇ kG kL1 for jxj > 2, which implies (103). The uniqueness property per .RIXˇ / can be obtained as in the proof of Theorem 9. t u The solution in Theorem 11 is called q-generalized since it possesses, at the outset, only “half” a derivative in time and one derivative in space belonging some 1

;1;q

2 Lq space, that is, it belongs to Wper .R  Rn /. Weighted estimates for strong   1;2;q solutions in Wper .R  Rn / corresponding to data F 2 L1 per;? RI Xˇ can be obtained in a similar fashion based on the estimates of the fundamental solution  ? available in Theorem 10.

4.7

Lq Estimates for the Linearized Problem in Exterior Domains

Consider the following linearization of (61) with homogeneous boundary values: 8 @t u  u  @1 u C rp D F ˆ ˆ ˆ ˆ ˆ ˆ div u D 0 ˆ ˆ ˆ < uD0 ˆ ˆ ˆ u.t C T; x/ D u.t; x/; ˆ ˆ ˆ ˆ ˆ ˆ : lim u.t; x/ D 0;

in R  ; in R  ; on R  @;

(106)

jxj!1

where  0 is a constant. The Lq estimates established in the whole-space case can be extended to solutions to the exterior domain problem (106). As in the whole-space case, the projections (62) can be used to decompose (106) into a steadystate and oscillatory problem. The following Lq estimate holds for the oscillatory problem:

10 Time-Periodic Solutions to The Navier-Stokes Equations

551

Theorem 12. Let   Rn (n D 2; 3) be an exterior domain of class C 2 . Let q q 2 .1; 1/ and 2 Œ0; 0 . For any vector field F 2 Lper;? .R  /, there is a 1;2;q

1;q

solution .u; p/ 2 Wper;? .R  /  Dper;? .R  / to (106) which satisfies kuk1;2;q C krpkq  C7 kF kq ;

(107)

1;2;r 1;r u; e p/ 2 Wper;? .R/Dper;? .R/ with C7 D C7 .q; ; ; 0 /. If r 2 .1; 1/ and .e is another solution, then e u D u and e p D p C d .t / for some T -periodic function d W R ! R.

A proof of Theorem 12 is given below. Beforehand, the Lq estimates for (106) that follow by combining Theorem 12 with well-known Lq estimates for the corresponding steady-state problem are manifested. Only the case ¤ 0 is included. A similar statement can be made in the case D 0; see also Remark 9. Corollary 1. Let   Rn (n D 2; 3) be an exterior domain of class C 2 and 2 .0; 0 . Let q 2 .1; 32 / if n D 2, and q 2 .1; 2/ if n D 3. Define q

q

X ./ WD fv 2 X ./ j div v D 0; v D 0 on @g;   Z D1;q ./ WD v 2 D 1;q ./ j p dx D 0 BR0

q

as subspaces of X ./ and D 1;q ./, defined in Sect. 4.1, respectively. Moreover, let r 2 .1; 1/ and 1;2;q;r

Wper;? .R  / ˚  1;2;q 1;2;r WD w 2 Wper;? .R  / \ Wper;? .R  / j div w D 0; w D 0 on @ equipped with the norm kwk1;2;q;r WD kk1;2;q C kk1;2;r . Additionally let 1;q;r Dper;? .R

  / WD 2

1;q Dper;? .R

 / \

1;r Dper;? .R

 / j

dx D 0 BR0

with norm hi0;1;q;r WD hi0;1;q C hi0;1;r and q;r

q



Z

Lper;? .R  / WD Lper;? .R  / \ Lrper;? .R  /

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G.P. Galdi and M. Kyed

with norm kkq;r WD kkq C kkr . Then the T -time-periodic Oseen operator q

1;2;q;r

AOseen W X ./ ˚ Wper;? .R  / 1;q;r

q;r

 D 1;q ./ ˚ Dper;? .R  / ! Lq ./ ˚ Lper;? .R  /;

(108)

AOseen .v C w; p C / WD @t w  .v C w/  @1 .v C w/ C r.p C / only on n, q, r, , , and . If q 2 is with kA1 Oseen  a3homeomorphism   k depending  1; 2 in the case n D 3 or q 2 1; 65 in the case n D 2, then kA1 Oseen k depends only on the upper bound 0 and not on itself. Proof. It is well known that the steady-state Oseen operator, that is, the Oseen operator from (108) restricted to time-independent functions, is a homeomorphism q as a mapping AOseen W X ./  D1;q ./ ! Lq ./; see, for example, [10, Theorem VII.7.1]. By Theorem 12, it follows that also the time-periodic Oseen operator is 1;2;q;r 1;q;r a homeomorphism as a mapping AOseen W Wper;? .R  /  Dper;? .R  / ! q;r Lper;? .R  /. Since clearly P and P? commute with AOseen , it further follows that AOseen is a homeomorphism in the setting (108). The dependency of kA1 Oseen k on the various parameters follows from [10, Theorem VII.7.1] and Theorem 12. t u Remark 9. In Corollary 1, the function space that yields maximal Lq regularity for the time-periodic Oseen operator, that is, the function space that is mapped q homeomorphically onto Lper .R/ by AOseen , is identified. As will be demonstrated later, this function space constitutes a suitable setting for investigation of the fully nonlinear problem in the Oseen case ¤ 0. More specifically, it is possible to choose the parameters in (108) in such a way that for any vector field u 2 q 1;2;q;r X ./ ˚ Wper;? .R  /, the corresponding nonlinear term u  ru belongs to the range of AOseen . The Stokes case D 0 is different. Although maximal Lq regularity for a certain range of exponents q can be obtained also in this case, one q can simply use [10, Theorem V.4.8] to identify the appropriate space X0 ./ that q is mapped homeomorphically onto L ./ by the steady-state Stokes operator and q use this space instead of X ./ in Corollary 1; the setting based on this space is not well suited for the investigation of the nonlinear problem when D 0. The weighted spaces introduced in Sect. 4.6 constitute a better alternative in this case. It is common practice to establish Lq estimates for an exterior domain problem by decomposing the solution into a solution to a bounded domain problem and a whole-space problem, respectively, and then employ the Lq theory available for these simpler cases. The decomposition is typically done by multiplying the solution with a “cutoff” function. In the case of the (linearized) Navier-Stokes system (106), this “cutoff” technique produces zero-order terms for the pressure on the “right-hand side” of the new equations. It is particularly challenging to estimate these terms. For

10 Time-Periodic Solutions to The Navier-Stokes Equations

553

this purpose, the following lemma, which was established for a two-dimensional and three-dimensional exterior domain in [12, 14], respectively, is needed: Lemma 3. Let  and be as in Theorem 12. Moreover, let R0 > 0 be a constant such that Rn n   BR0 and s 2 .1; 1/. There is a constant C8 D C8 .R0 ; ; s/ 1;2;r 1;r .R  /  Dper;? .R  / is a solution to (106) with data such that if .u; p/ 2 Wper;? R r F 2 Lper;? .R  / for some r 2 .1; 1/ and satisfying R p dx D 0, then 0

kp.t; /k n s;R0 n1   s1 1 s s  C8 kF .t; /ks C kru.t; /ks;R0 C kru.t; /ks; kru.t; /k 1;s;R R 0

0

(109)

for, a.e., t 2 R. Moreover, for every  > 0 with Rn n   B , there is a constant C9 D C9 .; ; s/ such that   krp.t; /ks;  C9 kF .t; /ks C kp.t; /ks;

(110)

for, a.e., t 2 R. Proof. For the sake of simplicity, the t -dependence of functions is not indicated in the proof. All norms are taken with respect to the spatial variables only. Consider an 1 arbitrary ' 2 C01 ./. Observe that for any 2 Cper .R/ holds Z

T

Z

Z @t u  r' dx 

0

which implies

R 

T

Z

dt D

div u  '  @t

dxdt D 0;



0

@t u  r' dx D 0 for, a.e., t . Moreover, Z

Z @1 u  r' dx D 



div u  @1 ' dx D 0: 

Hence it follows from (106)1 that p is a solution to the weak Neumann problem for the Laplacian: 8' 2

C01 ./

Z

Z

W

rp  r' dx D 

F  r' C u  r' dx: 

Recall that p2

L1loc ./

^

r

rp 2 L ./

Z ^

p dx D 0: R0

(111)

554

G.P. Galdi and M. Kyed

It is well known (see, e.g., [10, Section III.1]) that the weak Neumann problem for the Laplacian is uniquely solvable in the class (111). One can thus write p as a sum p D p1 C p2 of two solutions (in the class above) to the weak Neumann problem Z

8' 2 C01 ./ W

Z rp1  r' dx D

F  r' dx





and 8' 2 C01 ./ W

Z

Z rp2  r' dx D



u  r' dx; 

respectively. The a priori estimate 8q 2 .1; 1/ W krp1 kq  c0 kF kq

(112)

is well known. An estimate of p2 shall nowRbe established. Consider for this purpose an arbitrary function g 2 C01 .R0 / with R g dx D 0. The existence of a vector 0 field h 2 C01 .R0 / with div h D g and 8q 2 .1; 1/ W khk1;q  c1 kgkq is well known; see, for example, [10, Theorem III.3.3]. Let ˆ be a solution to the weak Neumann problem for the Laplacian: 8' 2

C01 ./

Z

Z

W

rˆ  r' dx D 

h  r' dx: 

By classical theory, such a solution exists with 8q 2 .1; 1/ W ˆ 2 C 1 ./ ^ krˆk1;q  c2 khk1;q  c3 kgkq : Since ˆ is harmonic in Rn n BR0 , an asymptotic expansion of ˆ implies rˆ.x/ D  n O jxj . The regularity and decay of ˆ ensure the validity of a computation (see [14] for the details) that yields Z p2 g dx D 

Z 

  ru W rˆ ˝ n  n ˝ rˆ d : @

Apply first Hölder’s inequality and then a classical trace inequality [10, Theorem II.4.1] to deduce ˇZ ˇ ˇ ˇ s ˇ p2 g dx ˇ  c4 kruks;@ krˆk s1 ;@ 

ns ns  c5 kruks;@ krˆk1; ns.n1/ ;R0  c6 kruks;@ kgk ns.n1/ ;R0 :

10 Time-Periodic Solutions to The Navier-Stokes Equations

Since

R R0

555

p2 dx D 0, it follows that

kp2 k

n n1 s;R0

D

sup g2C01 .R /;kgk D1 ns 0 ns.n1/

R

R

0

ˇZ ˇ ˇ ˇ ˇ p2 g dx ˇ  c7 kruks;@ : 

g dxD0

Another application of the trace inequality [10, Theorem II.4.1] now implies   s1 1 s s n kruk :  c C kruk kruk kp2 k n1 8 s;R0 s;R0 s;R 1;s;R 0

0

By Sobolev’s embedding theorem and (112), it can finally be concluded that n n kpk n1 s;R0  c9 krp1 ks;R0 C kp2 k n1 s;R0   s1 1 s s  c10 kF ks C kruks;R0 C kruks;R kruk1;s;R 0

0

and thus (109). To show (110), one can introduce an appropriate “cutoff” function

2 C 1 .Rn I R/ and analyze the weak Neumann problem satisfied by WD p in a similar manner; see again [14]. t u Also needed for the proof of Theorem 12 are the following embedding properties of time-periodic Sobolev spaces: n 1 Lemma 4. Let   R

(n  2/ be an exterior domain of class C and q 2 .1; 1/. Assume that ˛ 2 0; 2 and p0 ; r0 2 .q; 1 satisfy

8 2q ˆ ˆ r0  ˆ ˆ 2  ˛q < r0 < 1 ˆ ˆ ˆ ˆ : r0  1

if ˛q < 2; if ˛q D 2; if ˛q > 2;

8 nq ˆ p0  ˆ ˆ n  .2  ˛/q < p0 < 1 ˆ ˆ ˆ : p0  1

if .2  ˛/q < n; if .2  ˛/q D n; if .2  ˛/q > n; (113)

and that ˇ 2 0; 1 and p1 ; r1 2 .q; 1 satisfy 8 2q ˆ ˆ r1  ˆ ˆ 2  ˇq < r1 < 1 ˆ ˆ ˆ ˆ : r1  1

if ˇq < 2; if ˇq D 2; if ˇq > 2;

8 nq ˆ p1  ˆ ˆ n  .1  ˇ/q < p1 < 1 ˆ ˆ ˆ : p1  1

if .1  ˇ/q < n; if .1  ˇ/q D n; if .1  ˇ/q > n: (114)

556

G.P. Galdi and M. Kyed

Then 1;2;q 8u 2 Wper .R  / W

kukLrper0 .RILp0 .// C krukLrper1 .RILp1 .//  C kuk1;2;q : (115) t u

Proof. See [14].

On the strength of the estimates for the pressure in Lemma 3 and the embedding properties in Lemma 4, a proof of Theorem 12 can be provided: q

1 .R  / in Lper;? .R  /, it suffices to Proof of Theorem 12. By density of C0;per;? 1 consider only F 2 C0;per;? .R  /. The starting point will be the solution .u; p/ 2 1;2;2 1;2 .R  /  Dper;? .R/ from Theorem 8. By adding a function that depends Wper;? R only on time to p, one may assume without loss of generality that R p dx D 0. 0 The solution .u; p/ shall be decomposed by multiplication of a “cutoff” function. For this purpose, fix three constants 0 < R <  < R0 such that Rn n   BR . For convenience, the notation T for the time-domain Œ0; T / is used in the scope of the proof. Two fundamental estimates shall be established. To show the first one, a cutoff function 1 2 C 1 .Rn I R/ is introduced with 1 .x/ D 1 for jxj   and 1 .x/ D 0 for jxj  R . Let  L denote the fundamental solution to the Laplace operator in Rn and put

P W R  Rn ! R;

  V WD r L Rn r 1  u ;  P WD  L Rn Œ@t    @1 .r

w W R  Rn ! Rn ;

w.t; x/ WD

V W R  Rn ! Rn ;

W R  Rn ! R;

.t; x/ WD

1 .x/ u.t; x/ 1 .x/ p.t; x/

1

  u/ ;

(116)

 V .t; x/;

 P .t; x/:

Then .w; / is a solution to the whole-space problem 8 @t w  w C @1 w C r D ˆ ˆ < 1 F  2r 1  ru   ˆ ˆ : div w D 0

1u

C @1

1u

Cr

1p

in R  Rn ; in R  Rn : (117)

The precise regularity of .w; / is not important at this point. It is enough to observe that w and belong to the space of tempered time-periodic distributions S0per .R  Rn /, which is easy to verify from the definition (116) and the regularity of u and p. It is not difficult to show (see [28, Lemma 5.3]) that a solution w to (117) is unique in the class of distributions in S0per .R  Rn / satisfying Pw D 0. Consequently, w coincides with the solution from Theorem 9 corresponding to the right-hand side in (117) and therefore satisfies

10 Time-Periodic Solutions to The Navier-Stokes Equations

557

kwk1;2;s  c0 k 1 F  2r 1  ru   1 u C @1 1 u C r 1 pks    c1 kF ks C kuks;T C kruks;T C kpks;T for all s 2 .1; 1/. Clearly,   krV ks C kr 2 V ks  c2 kuks;T C kruks;T : Since u D w C V for x 2  , the estimates above imply kruks;T C kr 2 uks;T    c3 kF ks C kuks;T C kruks;T C kpks;T for all s 2 .1; 1/. For a similar estimate on u itself, first turn to (106) and apply Lemma 3 to deduce   k@t uks;T  c4 kF ks C kuks;T C k @1 uks;T C krpks;T    c5 kF ks C kuks;T C kruks;T C kpks;T : Since Pu D 0, Poincaré’s inequality yields kuks;T  c6 k@t uks;T . It thus follows that   kuk1;2;s;T  c7 kF ks C kuks;T C kruks;T C kpks;T

(118)

for all s 2 .1; 1/. Now a similar estimate for u over the bounded domain T   shall be established. To this end, a “cutoff” function 2 2 C 1 .Rn I R/ is introduced with 2 .x/ D 1 for jxj   and 2 .x/ D 0 for jxj  R0 . Let V be a vector field with 1;2;2 V 2 Wper;? .R  Rn /;

8s 2 .1; 1/ W kVk1;2;s Since

supp V  R  ;R0 ; div V D r 2  u;    c kuks;T;R0 C kruks;T;R0 C k@t uks;T;R0 : (119) Z

Z r

2  u dx D

;R0

div



R0

 2 u dx D

Z u  n d D 0; @R0

the existence of a vector field V with the properties above can be established by the same construction as the one used in [10, Theorem III.3.3]; see also [27, Proof of Lemma 3.2.1]. Now let w W R  Rn ! Rn ; W R  R ! R; n

w.t; x/ WD .t; x/ WD

2 .x/ u.t; x/ 2 .x/ p.t; x/:

 V.t; x/;

(120)

558

G.P. Galdi and M. Kyed

1;2;2 1;2 Then .w; / 2 Wper;? .R  R0 /  Dper;? .R  R0 / is a solution to the problem

8 @t w  w C r D 2 F C 2 @1 u  2r ˆ ˆ ˆ ˆ ˆ < C r 2 p C @t V  V

2

 ru  

2u

in R  R0 ;

ˆ div w D 0 ˆ ˆ ˆ ˆ : wD0

in R  R0 ; on R  @R0 :

By Theorem 6, it follows that kwk1;2;s  c8 k 2 F C 2 @1 u  2r 2  ru   2 u C r 2 p C @t V  Vks    c9 kF ks C kuks;TR0 C kruks;TR0 C kpks;T;R0 C kVk1;2;s for all s 2 .1; 1/. Since ;R0   , (119) and (118) can be combined to estimate   kuk1;2;s;T  c10 kF ks C kuks;T C kruks;T C kpks;T : In combination with (118), the estimate above implies   kuk1;2;s  c kF ks C kuks;TR0 C kruks;TR0 C kpks;TR0

(121)

for all s 2 .1; 1/. The estimate (121) has been established for all s 2 .1; 1/, but it is not actually known at this point whether the right-hand side is finite or not. At the outset, it is only known that the right-hand side is finite for s 2 .1; 2. A bootstrap argument can be used to show that it is also finite for s 2 .2; 1/. For this purpose, the embedding 1;2;s .R/ stated in Lemma 4 are employed to show the implication properties of Wper ns

8s 2 Œ2; 1/ W

1;2;s n1 u 2 Wper .R  / ) u; ru 2 Lper .R  /:

(122)

Now turn to estimate (109) of the pressure term. By Hölder’s inequality, Z

T 0



s1

sn  n1

1

s s kru.t; /ks; kru.t; /k1;s; R R 0

dt

0

Z  0

T

n.s1/s s.n1/n

kru.t; /ks;R

0

 s.n1/n s.n1/ dt

n1 n kuk1;2;s :

By Lemma 4, utilized this time with ˇ D 1, the right-hand side above is finite for 1;2;s all s 2 Œ2; 1/ provided u 2 Wper .R  /. Due to the normalization of the pressure p carried out in the beginning of the proof, Lemma 3 can be applied to infer from (109) that

10 Time-Periodic Solutions to The Navier-Stokes Equations

559 ns

8s 2 Œ2; 1/ W

1;2;s n1 u 2 Wper .R  / ) p 2 Lper .R  R0 /:

(123)

By (121) and the implications (122) and (123), it follows that 8s 2 Œ2; 1/ W

ns 1;2; n1

1;2;s u 2 Wper .R  / ) u 2 Wper

.R  /:

(124)

Starting with s D 2, the implication (124) can be bootstrapped a sufficient number 1;2;s of times to deduce u 2 Wper .R  / for any s 2 .2; 1/. It follows that the righthand side of (121) is finite for all s 2 .1; 1/. At this point, it is now possible to use interpolation and with Young’s inequality in (121) to show  (109) in combination  kuk1;2;s  c11 kF ks C kuks;TR0 for all s 2 .1; 1/. It then follows directly from (106) that   kuk1;2;s C krpks  c12 kF ks C kuks;TR0 (125) for all s 2 .1; 1/. 1;2;r 1;2;r 1;r If .U ; P/ 2 Wper;? .R  /  Wper;? .R  /  Dper;? .R  / (106) with homogeneous right-hand side, then U D rP D 0 follows by a duality argument. 1 More specifically, since for arbitrary ' 2 C0;per;? .R  / existence of a solution 0

0

1;2;r 1;r .R  /  Dper;? .R  / to (106) with ' as the right-hand side .W ; …/ 2 Wper;? has been established, the computation

Z

T

Z

0D

.@t U C @1 U  U C rP/  ' dxdt 0



Z

T

Z

Z

T

Z

U  .@t W C @1 W  W C r…/ dxdt D

D 0



U  ' dxdt 0



(126) is valid. It follows that U D 0 and in turn, directly from (106), that also rP D 0. Now return to the estimate (125). Owing to the fact that a solution to (106) with homogeneous right hand is necessarily zero, which was just shown above, a standard contradiction argument (see, e.g., [12, Proof of Proposition 2]) can be used to eliminate the lower-order term on the right-hand side in (125) to conclude kuk1;2;q C krpkq  c13 kF kq :

(127)

q

1 It is easy to verify that C0;per;? .R  / is dense in Lper;? .R  /. By a density 1;2;q

1;q

argument, the existence of a solution .u; p/ 2 Wper;? .R  /  Dper;? .R  / to q (106) that satisfies (127) therefore follows for any F 2 Lper;? .R  /. 1;2;r 1;r .R  /  Dper;? .R  / is another solution Finally, assume .e u; e p/ 2 Wper;? to (106) with r 2 .1; 1/. Applied to the difference .u  e u; p  e p/, the duality argument used in (126) yields u D e u and rp D re p. The proof of theorem is thereby complete. t u

560

4.8

G.P. Galdi and M. Kyed

Existence of Lq Strong Solutions

The question now arises as to whether or not existence of a solution to the fully nonlinear problem (61) can be established on the basis of the Lq estimates and corresponding function spaces from Sect. 4.7. Compared to the class of strong solutions introduced in Sect. 4.3, more information on the asymptotic structure at spatial infinity could be derived for such a solution. If the T -time-periodic velocity u1 .t / 2 Rn is directed along a single axis, say u1 .t / D u1 .t /e 1 , and its net RT motion over a period is non-zero, that is, 0 u1 .t / dt ¤ 0, the question can be RT answered affirmatively. If 0 u1 .t / dt D 0, the Lq estimates from Sect. 4.7 are not adequate. In this case, the fully nonlinear problem (61) can be treated in weighted function spaces based on the estimates introduced in Sect. 4.6. Below, the case RT 0 u1 .t / dt ¤ 0 is investigated more closely. The projections P and P? introduced in (62) can be employed to decompose u1 .t / into a constant WD Pu1 and a oscillatory part P? u1 . A linearization of (61) around leads to (106). In the case ¤ 0, Corollary 1 in combination with a fixed point argument can be used to show existence of a solution to the fully nonlinear problem (61) for data sufficiently restricted in “size.” For this purpose, it is convenient to put nq

1;2;q; nq

Wper;? 1;q;

1;2;

1;2;q

nq

.R  / WD Wper;? .R  / \ Wper;?nq .R  /;

nq

1;q

1;

nq

nq .R  /: Dper;?nq .R  / WD Dper;? .R  / \ Dper;?

The resolution of (61) then reads: 2 Theorem 13. Let   Rn (n D 2; 3) be an exterior domain of class

C . Assume 6 4 u1 .t / D u1 .t /e 1 with WD Pu1 > 0. If n D 3, let q 2 5 ; 3 . If n D 2, let   q 2 1; 65 . There is an 0 > 0 such that for all 2 .0; 0 , there is an "3 > 0 such q 1;1 that for all f 2 Lper .R  / and u1 2 Wper .R/ satisfying

kf kq C kP? f k

nq nq

C kP? u1 k1;1  "3 ;

(128)

there is a solution q

nq

1;2;q; nq

.u; p/ 2 X ./ ˚ Wper;?

1;q;

nq

.R  /  D 1;q ./ ˚ Dper;?nq .R  /

(129)

to (61).

Proof. Consider first the case n D 3 and q 2 65 ; 43 . In order to “lift” the boundary value u1 in (61), that is, rewrite the system as one of homogeneous boundary

10 Time-Periodic Solutions to The Navier-Stokes Equations 3q

1;2;q; 3q

values, a solution .W; …? / 2 Wper;?

561 1;q;

3q

.R  /  Dper;?3q .R  / to

8 W C r…? D W ˆ ˆ < div W D 0 ˆ ˆ : W D P ? u1

in R  ; in R  ;

(130)

on R  @;

is introduced. One can use standard theory for elliptic systems to solve (130) in T -time-periodic function spaces and obtain a solution that satisfies 8r 2 .1; 1/ W kWk1;2;r C kr…? kr  c0 kP? u1 k1;1 ;

(131)

where c0 D c0 .r; q; ; /. Furthermore, classical results for the steady-state Oseen q problem [10, Theorem VII.7.1] ensure existence of a solution .V; …s / 2 X ./  D 1;q ./ to 8 V  @1 V C r…s D 0 in ; ˆ ˆ < div V D 0 in ; (132) ˆ ˆ : V D on @; which satisfies 8r 2 .1; 2/ W kVkX r ./ C kr…s kr  c1 ;

(133)

where c1 D c1 .r; ; /. Focus will now be on finding a solution .u; p/ to (61) on the form u D v C V C w C W;

p D p C …s C C …? ;

(134)

q

where .v; p/ 2 X ./  D 1;q ./ is a solution to the steady-state problem 8 ˆ ˆ v  @1 v C rp D R1 .v; w; V; W/ < div v D 0 ˆ ˆ : vD0

in ; in ;

(135)

on @;

with R1 .v; w; V; W/ WD  v  rv  v  rV  V  rv  V  rV







 P w  rw  P w  rW  P W  rw  P W  rW



C P P? u1 @1 w C P P? u1 @1 W C Pf;

562

G.P. Galdi and M. Kyed 3q

1;2;q; 3q

and .w; / 2 Wper;?

1;q;

3q

.R  /  Dper;?3q .R  / a solution to

8 @t w  w  @1 w C r D R2 .v; w; V; W/ in R  ; ˆ ˆ < div w D 0 in R  ; ˆ ˆ : wD0 on R  @;

(136)

with







R2 .v; w; V; W/ WDP? w  rw P? w  rW P? W  rw P? W  rW  v  rw  v  rW  w  rv  w  rV  V  rw  V  rW  W  rv  W  rV CP? u1 @1 vCP? u1 @1 V

CP? P? u1 @1 w CP? P? u1 @1 W  @t W  W C @1 W C P? f: The systems (135) and (136) appear as the result of inserting (134) into (61) and subsequently applying first P then P? to the equations. Recalling the function spaces introduced in Corollary 1 to define the Banach space q

1;2;q;

q

3q

1;q;

3q

K .R  / WD X ./ ˚ Wper;? 3q .R  /  D 1;q ./ ˚ Dper;?3q .R  /; (137) one can obtain solutions .v; p/ and .w; / to (135) and (136), respectively, as a fixed point of the mapping q

q

N W K .R  / ! K .R  /;   N .v C w; p C / WD A1 Oseen R1 .v; w; V; W/ C R2 .v; w; V; W/ I q;

3q

3q .R  /. note that R1 .v; w; V; W/ 2 Lq ./ and R2 .v; w; V; W/ 2 Lper;? More specifically, one can show that N is a contracting self-mapping on ball of sufficiently small radius. For this purpose, let  > 0 and consider some q .v C w; p C / 2 K \ B . Suitable estimates of R1 and R2 in combination with a smallness assumption on "3 from (128) are needed to guarantee that N has the desired properties. Regarding the estimates, one can employ Hölder’s inequality, Sobolev embedding, and basic interpolation to obtain

kv  rvkq  kvk

2q 2q

 krvk2  c2 

3q3 q

kvk2X q  c2 

3q3 q

2 :

(138)

10 Time-Periodic Solutions to The Navier-Stokes Equations

563

This step requires q 2 65 ; 43 . The other terms in the definition of R1 can be estimated in a similar fashion to conclude in combination with assumption (128) that  3q3 1 1 3 kR1 .v; w; V; W/kLq ./  c3  q 2 C 4  C 2 C 2 C 2  C "3 C "3 2 C "3 : 3q

q

3q An estimate of R2 is required both in the Lper .R  / and Lper .R  / norm. Observe that

kP? u1 @1 vkLqper .R/  c4 kP? u1 k1 k@1 vkq  c4 1 "3 :

(139)

The other terms in R2 can be estimated, in part with the help of the embedding properties from Lemma 4, to obtain   kR2 .v; w; V; W/kLqper .R/  c5 1 "3  C 2 C "3 C "3 2 C  C "3 C "3 : 3q 3q .R  / estimate of R2 . For Lemma 4 can also be used to establish an Lper;? example,

kw  rwk

 c6 kwk

3q 3q

krwk

3q 3q

3q

3q .// L1 per .RIL

Lper .RIL1 .//

Lper .R/

 c7 2 ;

where Lemma 4 is utilized with ˛ D 0 and ˇ D 1 in the last inequality. For this utilization of Lemma 4, it is required that q  65 . Further note that kP? u1 @1 vk

3q 3q

 "3 kvkX q  "3 ;

3q which explains the choice of the exponent 3q in the setting of the mapping N . The rest of the terms in R2 can be estimated to conclude

   c8 2 C "3 C "3 2 C "3 C "3 :

kR2 .v; w; V; W/k

3q 3q Lper

.R/

Now choose 0  1 and deduce by Corollary 1; recall that kA1 Oseen k does not depend on , the estimate kN .v C w; p C /k

q K





1

 kAOseen k  kR1 k

Lq ./

C kR2 k

3q q; 3q

Lper



3q3  q

 C10

.R/

 1 3 2 C 1 "3  C 4  C 2 C "3 2 C "3 :

564

G.P. Galdi and M. Kyed

In particular, N becomes a self-mapping on B if  3q3  1 3 C10  q 2 C 1 "3  C 4  C 2 C "3 2 C "3  : One may choose "3 WD 2 and  WD to find the above inequality satisfied for sufficiently small . For such choice of parameters, one may further verify that N is also a contraction. By the contraction mapping principle, existence of a fixed point for N follows. This concludes the proof in the case  n D 3. The proof in the case n D 2 and q 2 1; 65 follows along the same lines. To ensure the existence of a solution to (132) satisfying an estimate like (133) with a constant independent of , one cannot use [10, Theorem VII.7.1], but can instead use [10, Theorem XII.5.1] to obtain a solution to (132) that satisfies 8r 2 .1; 6=5/ W kVkX r ./ C h…s i1;r  c9

2r2 r

jlog j1 ;

(140)

with c9 D c9 .r; ; /. The estimate corresponding to (138) in the case n D 2 reads kv  rvkq  c10 

3q2 q

kvk2X q  c10 

323 q

2 I

(141)



see, for example, [10, Lemma XII.5.4]. The rest of the proof in the case n D 2 follows by simple adjustments to the proof for n D 3 above. Remark 10. It is possible to establish higher-order regularity for the solution in Theorem 13 by a bootstrap argument based on the linear theory from Theorem 12. More specifically, if the data possesses higher-order regularity, say m;q f 2 Wper;? .R  /, one can put the nonlinear term u  ru on the right-hand side in (61) and iteratively apply Theorem 12 after taking partial derivatives on both sides. With such an argument, it is possible to establish a degree of regularity for .u; p/ ˇ q 1;2;q corresponding to the regularity of the data f , that is, @˛t @x u 2 X ./ ˚ Wper;? .R  / for j˛j C jˇj  m. For more details on such a result, see [28, Theorem 2.4]. Alternatively, higher-order regularity can be obtained via regularity theory for the initial-value problem as mentioned in Remark 5. Remark 11. The solution u in Theorem 13 possess enough summability at spatial infinity for an adaptation of the uniqueness argument from the proof of Theorem 2 to be carried out. More precisely, given a weak solution U in the sense of Definition 2, it is possible to insert u as a “test function” in the weak formulation (64) for U . In addition, after multiplication by U in the system (61) satisfied by u, the summability of both, in particular the latter, is adequate to integrate by parts. These are the two main steps in proof of Theorem 2. Provided therefore that both u and U satisfy an appropriate energy inequality corresponding to (61) and the data is sufficiently restricted in “size,” it can be shown that u D U . In other words, the strong solution in Theorem 13 can be shown to be unique in a class of weak solutions satisfying an energy inequality. See [28, Theorem 2.3] for more details on such a result.

10 Time-Periodic Solutions to The Navier-Stokes Equations

4.9

565

Asymptotic Structure

Important physical properties of a solution u to (61) are related to its asymptotic structure at spatial infinity. The asymptotic structure is best exposed by an asymptotic expansion u.t; x/ D A.t; x/ C R.t; x/ into an explicitly known leading term A and a remainder term R that decays faster to 0 as jxj ! 1 than A. The task of identifying such an expansion shall now be addressed for u1 .t / D u1 .t /e 1 , that is, for functions u1 directed along a single axis. The case Pu1 ¤ 0 is considered first. A strong solution in the class (129) is singled out for investigation. Theorem 13 yields existence of a solution in this class, so it is a reasonable starting point. By nature of the function space in (129), the steady-state part Pu of such a solution enjoys, at the outset, different Lq q summability properties than the oscillatory part P? u. In fact, since Pu 2 X ./ 1;2;q and P? 2 W .R  /, better spatial decay is available for P? u in the sense of summability, that is, the range of exponents q for which P? u.t; / 2 Lq ./ is lower than the range of exponents q for which Pu 2 Lq ./. This suggests that the leading term in an asymptotic expansion of u is dominated by Pu; a key observation that underpins the analysis below. Recall that the Oseen fundamental solution  Oseen introduced in Sect. 4.5 satisfies 8q0 2 Œ1; 2 W



Oseen

… Lq0 .Rn n Br / for any r > 0I

(142)

see, for example, [10, Chapter VII.3]. On the strength of this information, the theorem below yields an asymptotic expansion of a solution to (61) with the decay of the remainder term characterized in the sense of summability as described above. Theorem 14. Let   Rn (n D 2; 3) be an exterior domain of class C 2 . Assume 1 u1 .t / D u1 .t /e 1 with WD Pu1 > 0. Moreover, assume f 2 C0;per .R  /. 6 4  6 If n D 3, let q 2 5 ; 3 . If n D 2, let q 2 1; 5 . There is an "4 > 0 such that if 1 .R/ satisfies kP? u1 k1  "5 , then a solution .u; p/ to (61) in the class u1 2 Cper (129) satisfies u.t; x/ D  Oseen .x/  F C R.t; x/;

(143)

where F WD

1 T

Z

T

Z S.u; p/  n d C

0

@

1 T

Z

T

Z f .t; x/ dxdt

0

(144)

Rn

and  8q0 2

 nC1 ;1 W n

  q0 R 2 L1 per RI L ./ :

(145)

566

G.P. Galdi and M. Kyed

  Here, S.u; p/ denotes the Cauchy stress tensor S.u; p/ WD  ru C ru>  pI . Remark 12. As previously described, a solution to (61) describes the fluid flow around an object moving with constant velocity u1 . In physical terms, the quantity F in (144) equals the total force exerted upon the fluid by the moving object and the external force f combined over a full time-period. Remark 13. In light of (142), the decomposition (143) indicates that the velocity field u possesses the asymptotic structure of the Oseen fundamental solution  Oseen . From a physical point of view, this implies the existence of a wake behind the moving object; see again [10, Chapter VII.3]. Another interesting observation is that the leading term in the expansion (143) is independent of time, which suggests that oscillatory characteristics of the fluid flow appear only locally around the body. Proof of Theorem 14. Only the case n D 3 is considered. The case n D 2 can be treated by a similar approach. Since the data f and u1 are assumed to be smooth, 1 1 recall from Remark 10 that also u 2 Cper .R  / and p 2 Cper .R  /. Although not strictly necessary, the smoothness of the solution simplifies the proof as local regularity becomes a nonissue. Let 2 C01 .R3 I R/ be a “cutoff” function with

D 0 on B1 and D 1 on R3 n B2 . Put R .x/ WD Rx for some R > R0 . Since Z

Z r R .x/  u.t; x/ dx D

BR;2R

r R .x/  u.t; x/ dx 2R

Z

R .x/  div u.t; x/ dx D 0;

D 2R

1 .R  R3 / satisfying both supp U  R  BR;2R there is a vector field U 2 Cper and div U D r R  u. The construction of U is well known in the case of timeindependent vector fields; see [10, Theorem III.3.3]. The same construction can be applied in the above case of time-periodic vector fields; see [27]. Let e u WD R u  U and e p WD R p. Then

8 u  e u  @1e u C re p D P? u1 @1e u e u  re u C R f C h @te ˆ ˆ < ˆ ˆ :

in R  R3 ; in R  R3 ;

dive uD0

e u.t C T; x/ D e u.t; x/ (146)

1 .R  R3 / of bounded support supp h  R  B2R . Put v WD Pe u and with h 2 Cper w WD P?e u. At the outset q

3q

1;2;q; 3q

.v C w; p/ 2 X .R3 / ˚ Wper;?

1;q;

3q

.R  R3 /  D 1;q .R3 / ˚ Dper;?3q .R  R3 /: (147)

10 Time-Periodic Solutions to The Navier-Stokes Equations

567

However, it is possible to show that 8q0 2 .1; q W

q

1;2;q

v 2 X 0 .R3 /; w 2 Wper;?0 .R  R3 /:

(148)

For this purpose, put MR WD R re u and identify .e u; e p/ as a fixed point of the 2 mapping q

q

N W K .R  R3 / ! K .R  R3 /;   N .e u; e p/ WD A1 u  MRe u C R f C h ; Oseen P? u1 @1e q

where K is defined as in (137). Provided R is chosen sufficiently large and kP? u1 k1 sufficiently small, it can be shown, by estimates very similar to those made in the proof of Theorem 13, that N as mapping q

q

q

q

N W K 0 .R  R3 / \ K .R  R3 / ! K 0 .R  R3 / \ K .R  R3 / is a contraction for all q0 2 .1; q. By the contraction mapping principle, N therefore has a unique fixed point. Uniqueness of the fixed point implies (148). From the embedding properties in Lemma 4, it immediately follows from (148) that   q0 3 8q0 2 .1; 1/ W w 2 L1 per RI L .R / :

(149)

Focus is now shifted to v. Applying P to the system (146), one finds that v is a solution to the steady-state problem 8  v  @1 v C rp ˆ ˆ
0; 8q0 2 Œ1; 3/ W

q

0  Oseen 2 Lloc .R3 /

(152)

(see, e.g., [10, Chapter VII.3]), it follows as a consequence of (148) that the convolution on the right-hand side in (151) is well defined and thus a solution

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to (150). A standard uniqueness argument implies that this solution coincides with v. An asymptotic expansion of v can be derived from (151). To this end, recall (see again [10, Chapter VII.3]) that 8q0 2 .4=3; 1/ W

r Oseen 2 Lq0 .R3 n Br / for any r > 0;

8q0 2 Œ1; 3=2/ W

0 r Oseen 2 Lloc .R3 /:

(153)

q

Consequently, the summability properties from (148) in combination with Young’s inequality imply  

Oseen

 



 P P? u1 @1 w  v  rv  P w  rw 



D 

Oseen

i







 div P P? u1 w ˝ e 1  v ˝ v  P w ˝ w

 



Oseen D @k  ij  P P? u1 w ˝ e 1  v ˝ v  P w ˝ w

i

2 Lq0 .R3 /

jk

(154) for all q0 2 

Oseen

4 3

 ; 1 . Moreover, due to (153), it is standard to show

 P R f C h  

Oseen

Z  R3



P R f C h dx 2 Lq0 .R3 /

(155)

  for all q0 2 43 ; 1 . It remains to compute the integral in (155). For this purpose,

isolate P R f C h in (146), and compute Z



1 P R f C h dx D lim R!1 T R3

Z 0

T

Z BR

@te u  e u  @1e u

C re p  P? u1 @1e u Ce u  re u dxdt Z TZ

1  div S.e u; e p/ C div .e u  u1 / ˝ e u dxdt D lim R!1 T 0 BR Z TZ

1 D lim S.u; p/  n C .u  u1 / ˝ u  n d dt R!1 T 0 @BR Z TZ Z Z 1 T 1 S.u; p/  n d dt C f dxdt; D T 0 @ T 0  (156) where the change of sign in the last integral is due to n denoting the outer normal on @. The identity above together with (149), (154), and (155) concludes the proof. t u

10 Time-Periodic Solutions to The Navier-Stokes Equations

569

Under certain conditions, the asymptotic expansion in Theorem 14 can also be established with a pointwise decay estimate of the remainder term. Below, a sketch of a proof is given in the case Pu1 ¤ 0 and P? u1 D 0. The proof is based on the pointwise estimates of the fundamental solution in Sect. 4.5. The assumption P? u1 D 0 is equivalent to the requirement that u1 is constant. Only the case of a three-dimensional exterior domain is included. A similar result is not available in the two-dimensional case. Theorem 15. Let   R3 be an exterior domain of class C 2 . Assume u1 D e 1 1 with > 0 a constant. Let f 2 C0;per .R  /. Then a solution .u; p/ to (61) in the class (129) satisfies (143) with 3

kR.; x/k1  C jxj 2 C" :

8" > 0 9C > 0 8jxj > 1 W

(157)

p WD R p are introduced Proof. As in the proof of Theorem 14, e u WD R u  U and e and the system (146) investigated. As in the proof of Theorem 9, the interpretation of (146) as a system of partial differential equations on the group G is employed. One can proceed as in the proof of Theorem 14 to deduce (148), which implies that the convolution of the time-periodic fundamental solution  TP from Sect. 4.5 with the right-hand side in (146) is well defined in the classical sense and thus constitutes a solution to (146). By a uniqueness argument similar to the one made in the proof of Theorem 9, it therefore follows that e uD

TP

   e u  re u C R f C h :

Compared to the right-hand side in (146), the term P? u1 @1e u is missing above since u1 D e 1 implies P? u1 D 0. The structure of  TP identified in Theorem 10 now implies for v WD Pe u and w WD P?e u the identities 

vD

Oseen

wD

?







  v  rv  P w  rw C P R f C h ; 





(158) 

  v  rw  w  rv  P? w  rw C P? R f C h ;

(159)

where in the first identity, it is used that  Oseen ˝ 1R=T Z  F D  Oseen  PF and in the second the property  ?  P? F D  ?  F for all sufficiently regular F . Inspired by the proof in [10, Chapter X.8] of the asymptotic expansion of a solution to the corresponding steady-state problem, one can show Z 8" > 0 W

1 jrvj dx C T R3 nBR 2

Z 0

T

Z R3 nBR

jrwj2 dxdt  CR1C" :

(160)

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The above estimate is crucial. Together with the pointwise estimate (98) of  ? , it delivers the foundation for a pointwise estimate of all terms on the right-hand side of (158) and (159). The resulting estimates for v and w (see [15] for the details) are 8" > 0 W 8" > 0 W

jv.x/j  C jxj1C" ; (161)   3 2 jw.t; x/j  C jxj 2 C jxj 5 C" .kvkL1 .B jxj=2 / C kwkL1 .RB jxj=2 / / : (162)

Now insert (161) into (162). After two bootstrap iterations, it follows that 7 jw.t; x/j  C jxj 5 C" . A return to (159) then yields 12

jw.t; x/j  C jxj 5 C" I

8" > 0 W

(163)

see again [15] for the details. With this decay estimate for w, one can return to (158) and deduce as in [10, Theorem X.8.1] that Z 8" > 0 W

v.x/ D  Oseen .x/ 

R3



 3 P R f C h dx C O jxj 2 C" :

The theorem follows from (163), (164), and the computation made in (156).

(164) t u

The asymptotic structure of solutions to (61) changes significantly when Pu1 D 0. The canonical leading term in this case is no longer expressed in terms of the fundamental solution to the corresponding steady-state equation. In contrast to the case Pu1 ¤ 0, where the leading term, as was seen in Theorem 14 and Theorem 15, is expressed in terms of the Oseen fundamental solution  Oseen , the canonical leading term in the case Pu1 D 0 is not expressed in terms of the Stokes fundamental solution  Stokes . Instead, it is identified, in the three-dimensional case, as a so-called Landau solution; see below. In the purely steady-state case, the identification was made for the first time by Korolev and Sverak [24]. An extension of their result to the time-periodic case with u1 D 0 was made by Kang, Miura, and Tsai [22]. Both results hold for solutions u that are small in the norm of the weighted space X1 ./ introduced in (101). The Landau solution .U bLandau ; P b /, corresponding to a parameter b 2 R3 , is a solution in D0 .R3 / to (

U bLandau C U bLandau  rU bLandau C rP b D b ı; div U bLandau D 0

(165)

that is axially symmetric about the axis bR and .1/-homogeneous. Here, ı denotes the delta distribution. The Landau solution can be given explicitly. Assume for simplicity that b D k e1 , k 2 R, then

10 Time-Periodic Solutions to The Navier-Stokes Equations

0

x1 c jxj

2 B @ jxj

U bLandau .x/ D

x1 4 c jxj b P D jxj2 c 

571

1 1 x 1 C C

2 x1 e3 A jxj c  jxj x1

for x 2 R3 n f0g;

c  jxj

1 3

2 for x 2 R n f0g;

(166)

x1 jxj

where kD

  cC1 8 c 2 2 2 C 6c :  3c.c  1/ log 3.c 2  1/ c1

(167)

As one may easily verify, for each k 2 R n f0g, there exists a unique c 2 R with jcj > 1 so that .k; c/ satisfies (167). Hence, for each b 2 R3 nf0g, a Landau solution .U bLandau ; P b / to (165) is given by the expression above. Moreover, b D k e1 ! 0 as jcj ! 1. The Landau solution was originally constructed by Landau [31]. See [3] for the explicit computation of the expressions above. The asymptotic expansion at spatial infinity of a solution to (61) in the case u1 D 0 established by Kang, Miura, and Tsai [22] is given below. The isotropically weighted function spaces from Sect. 4.6 are employed to express the decay of the remainder term. For simplicity, the result is stated for a strong solution in 1;2;q Wper .R  / that is small in the norm of L1 per .RI X1 ./. While the assumption on the regularity of the solution can easily be reduced, the smallness assumption is critical. Theorem 16. Let   R3 be an exterior domain of class C 2 . Assume u1 D 0 1 .R  /. There is an "6 > 0 such that a solution .u; p/ to (61) in and f 2 C0;per 1;2;q

1;q

Wper .R  /  Dper .R  /, for some q 2 .1; 1/, with kukL1  "6 per .RIX1 .// satisfies u.t; x/ D U F Landau .x/ C R.t; x/;

(168)

  where F is given by (144) and R 2 L1 per RI Xˇ ./ for all ˇ 2 .1; 2/. Proof. As in the proof of Theorem 14, a cutoff function is applied to reduce the problem to that of determining the asymptotic expansion of a solution e u to the whole-space problem (146). Also as in the proof of Theorem 14, the velocity is decomposed into a steady-state part v WD Pe u and oscillatory part w WD P?e u. Since v satisfies the steady-state problem (150) with D 0, the expansion (168) can be shown for v as in [24]. The assumption kukL1  "6 is needed per .RIX  1 .//  1 for this step. It remains to show that w 2 Lper RI Xˇ ./ for all ˇ 2 .1; 2/. As in the proof of Theorem 15 (recall (159)), w can be expressed as the convolution

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of  ? with a vector nnfield of type div G C g, where g has compact support and G 2 L1 RI X by assumption. Estimates as in the proof of Theorem 15, 2 per;? ? based on the properties   of  from Theorem 10, can then be carried out to conclude 1 that w 2 Lper RI X2 . t u

5

Flow in a Pipe

Also highly relevant from a physical point of view are the equations governing time-periodic Navier-Stokes flows in infinite pipes. In many models, not least the model of a cardiovascular system, a time-periodic flow rate is prescribed. Below it is outlined how to adapt the methods from Sect. 3 to establish existence and uniqueness of strong time-periodic solutions for this particular model. For simplicity, a piping system with only two outlets whose cross sections are described by bounded domains S1  R2 and S2  R2 , respectively, is considered. The region of flow can then be written as a disjoint union  D 1 [ 2 [ 0

(169)

with 0  R3 bounded and 1 WD fx 2 R3 j x1 < 1; .x2 ; x3 / 2 S1 g; 2 WD fx 2 R3 j 1 < x1 ; .x2 ; x3 / 2 S2 g: A time-periodic Poiseuille flow shall be imposed at spatial infinity of each outlet. 1;2 Proposition 1. Let S  R2 be a bounded domain and ˆ 2 Wper .R/. The problem

8 in R  S; @t V D V C G ˆ ˆ ˆ ˆ < V D0 on R  @S; Z ˆ ˆ ˆ ˆ V .t; x/ dx D ˆ.t / : S

admits a unique solution .V; G/ in the class     V 2 L2per RI W 2;2 .S / ; @t V 2 L2per RI L2 .S / ;   V 2 Cper RI W01;2 .S / ; G 2 L2per .S /; which satisfies

(170)

10 Time-Periodic Solutions to The Navier-Stokes Equations

573

kV kL2per .RIW 2;2 .S// C k@t V kL2per .RIL2 .S // C kV kL1 .RIW 1;2 .S // C kGk2  C11 kˆk1;2 ; per

0

(171) with C11 D C11 .S /. Proof. See [9, Theorem 1.2]. Definition 3. Let .V; G/ be the solution in Proposition 1. Then .vp ; pp / defined by v p W R  S ! R3 ; pp W R  S ! R;

vp .t; x1 ; x2 ; x3 / WD V .x2 ; x3 /e 1 ; vp .t; x1 ; x2 ; x3 / WD G.t /x1

is called the time-periodic Poiseuille flow corresponding to flow rate ˆ and cross section S . A flow described by the velocity field of a Poiseuille flow is also called fully developed. It is the natural asymptotic value, from a physical point of view, to impose at spatial infinity for a flow through a pipe. Given a flow rate ˆ, the two time-periodic Poiseuille flows vp1 and vp2 corresponding to ˆ and cross sections S1 and S2 , respectively, are introduced. The full time-periodic Navier-Stokes system in the pipe  can then be written as 8 @t u C u  ru D u  rp in R  ; ˆ ˆ ˆ ˆ ˆ ˆ div u D 0 in R  ; ˆ ˆ ˆ ˆ ˆ < u D 0 on R  @; Z ˆ ˆ ˆ u.t; x/  n d .x2 ; x3 / D ˆ.t / for Œi D 1 ^ x1 < 1 and Œi D 2 ^ 1 < x1 ; ˆ ˆ ˆ Si ˆ ˆ ˆ   ˆ ˆ : lim u.t; x/  vpi .t; x/ D 0 .i D 1; 2/; u.t C T; x/ D u.t; x/: jxj!1

(172) In order to adapt the approach from Sect. 3 based on a Galerkin approximation to (172), the system must be rewritten as one of homogeneous boundary values and flow rate. For this purpose, a vector field that can “lift” those values is needed. This leads to the introduction of the following so-called flow-rate carrier: Proposition 2. Let   R3 be a (pipe) domain of type (169) with a C 2 -smooth 1;2 .R/. There is a T -time-periodic flow-rate carrier A with boundary and ˆ 2 Wper     8!   bounded W A 2 L2per RI W 1;2 .!/ ; @t A 2 L2per RI L2 .!/ ; div A D 0;

A D 0 on R  @;

(173) (174)

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G.P. Galdi and M. Kyed

Z A.t; x/  n d .x2 ; x3 / D ˆ.t /

for Œi D 1 ^ x1 < 1 and Œi D 2 ^ 1 < x1 ;

Si

(175) A D vpi in i ;

(176)

8!   bounded 9C12 > 0 W kAkL1 1;2 .!// C kAkL2 .RIW 2;2 .!// C k@t AkL2 .RIL2 .!//  C12 kˆk1;2 : per .RIW per per (177) t u

Proof. See [9, Section 1.3.2]. With the necessary “lifting” field at hand, one can show:

Theorem 17. Let   R3 be a (pipe) domain of type (169) with a C 2 -smooth 1;2 .R/ be a T -time-periodic flow rate; vp1 and vp2 boundary. Moreover, let ˆ 2 Wper Poiseuille flows corresponding to ˆ and cross sections S1 and S2 , respectively; and A a T -time-periodic flow-rate carrier satisfying (173), (174), (175), (176), and (177). There is a constant "7 > 0 such that if kˆk1;2  "7 ;

(178)

then there is .w; p/ with     1;2 2 2;2 ./ ; w 2 L1 per RI D0; ./ \ Lper RI W   p 2 L2per RI D 1;2 ./

  @t w 2 L2per RI L2 ./ ;

(179)

such that .u; p/ with u WD A C w is a solution to (172). Moreover, this solution satisfies kwkL1 1;2 .!// C kwkL2 .RIW 2;2 .!// C k@t wkL2 .RIL2 .!//  C13 kˆk1;2 ; per .RIW per per

(180)

where C13 D C13 ./. In addition, lim ku.t; x/  vpi .t; x/kL1 .i r /

r!1

.i D 1; 2/

(181)

u; e p/ with 1r WD fx 2 1 j x1 < rg and 2r WD fx 2 2 j r < x1 g. Finally, if .e is another solution in the class (179), then u D e u and rp D re p. Proof. One looks for a solution u on the form u D w C A with w satisfying

10 Time-Periodic Solutions to The Navier-Stokes Equations

575

8 @t w  w C rp C w  rw C A  rw C w  rA D f in R  ; ˆ ˆ ˆ ˆ ˆ div w D 0 in R  ; ˆ ˆ ˆ ˆ ˆ < w D 0 on R  @; Z ˆ ˆ w.t; x/  n d .x2 ; x3 / D 0 for Œi D 1 ^ x1 < 1 and Œi D 2 ^ 1 < x1 ; ˆ ˆ ˆ Si ˆ ˆ ˆ ˆ ˆ : lim w.t; x/ D 0; w.t C T; x/ D w.t; x/: jxj!1

(182) Since (182) is a system with homogeneous boundary values, flow-rate and asymptotic value at spatial infinity, one can initiate a Galerkin approximation. The same technique that was used in the proof of Theorem 3 can also be used here to first solve the problem on bounded domains of type k WD  \ fx 2 R3 j jx1 j < kg. The desired solution is then found as the limit k ! 1 of the resulting sequence in the same manner as in the proof of Theorem 8. For the details, see [9, Proof of Theorem 1.6]. t u

6

Conclusion

The time-periodic Navier-Stokes problem can be viewed as a generalization of the steady-state Navier-Stokes problem. Indeed, time-independent data and solutions are trivially time periodic. On the other hand, time-periodic solutions may also be viewed as solutions to an initial-value Navier-Stokes problem for some unspecified initial value. Both perspectives can be used to extend a number of fundamental results to the time-periodic problem. Without restrictions on the “size” of the data, existence of a weak solution can be Uniqueness of this solution is guaranteed  shown.  n when smallness in the space L1 RI L ./ is assumed. Existence and uniqueness per of a strong solution can be established when the magnitude of the data is sufficiently restricted. In the two-dimensional case, existence of a strong solution even follows without this restriction on the data. Moreover, a classical integrability condition ensures regularity of weak solutions. An appropriate Fourier transform can be utilized to show Lq estimates for the linearized equations in the whole space based on Fourier multipliers. These estimates can be extended to the cases of bounded and exterior domains. Finally, a fundamental solution can be identified and used to analyze the asymptotic structure of solutions.

7

Cross-References

 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions  Steady-State Navier-Stokes Flow Around a Moving Body  Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains  The Stoke Equation in the Lp -Setting: Well-Posedness and Regularity Properties

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Large Time Behavior of the Navier-Stokes Flow

11

Lorenzo Brandolese and Maria E. Schonbek

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Long-Time Behavior for the Navier-Stokes Equations in the Whole Space . . . . . . . . . . . 2.1 The Energy Decay Problem of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Small Data Approach and T. Kato’s Decay Results . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Masuda’s Approach and Nonuniform Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fourier Splitting Method and Decay Rates of Weak Solutions . . . . . . . . . . . . . . . . 2.5 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Rapidly Dissipative Solutions and the Role of Symmetries . . . . . . . . . . . . . . . . . . . 2.7 The Vorticity Approach and More on Rapidly Dissipative Solutions . . . . . . . . . . . 2.8 Asymptotic Behavior of Global Solutions in Scale-Invariant Spaces . . . . . . . . . . . 2.9 Decay in Weighted Spaces and Point-Wise Estimates . . . . . . . . . . . . . . . . . . . . . . . 2.10 Asymptotic Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Decay in Besov Spaces and Frequency Concentration Effects . . . . . . . . . . . . . . . . 2.12 Decay Characterization to Dissipative Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Non-decaying Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 The Role of External Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Decay Results in More General Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Decay in Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Decay in Exterior Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

580 582 582 583 585 587 590 593 594 596 597 601 606 608 611 612 613 614 614

In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Part I. Incompressible fluids. Unsteady viscous Newtonian fluids. Yoshikazu Giga and Antonn Novotný editors. Springer L. Brandolese is supported by the ANR project DYFICOLTI N. 36338 L. Brandolese () Institut Camille Jordan, Université Lyon 1, Villeurbanne, France e-mail: [email protected] M.E. Schonbek Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_11

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L. Brandolese and M.E. Schonbek

4 Convergence and Stability Results for Stationary Solutions . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Some Results on the Asymptotic Stability of Stationary Solutions of the Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 L2 -Perturbations of Infinite Energy Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The MHD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Boussinesq System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Nematic Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Quasi-Geostrophic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Incompressible Inhomogeneous NS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

618 618 621 626 626 627 629 633 636 637 637 638

Abstract

Different results related to the asymptotic behavior of incompressible fluid equations are analyzed as time tends to infinity. The main focus is on the solutions to the Navier-Stokes equations, but in the final section, a brief discussion is added on solutions to magnetohydrodynamics, liquid crystals, and quasi-geostrophic and Boussinesq equations. Consideration is given to results on decay, asymptotic profiles, and stability for finite and nonfinite energy solutions.

1

Introduction

This chapter focuses on the asymptotic behavior of solutions to dissipative incompressible fluid equations. The diffusion for these systems is given by the presence of a Laplacian or a fractional Laplacian. When a stationary solution exists, an important issue is to establish if as time goes to infinity, the evolutionary solutions tend to the corresponding stationary ones. The question at hand then is to investigate the asymptotic stability of the stationary solutions. If the original system has no external forcing terms, it is surmised that the stationary solution is zero. The stability is analyzed in Lp .Rn /; p  1, Sobolev, Lorentz, and Besov spaces. Emphasis is on solutions to the incompressible. In the last section of the chapter, a brief consideration is given to the decay of solutions corresponding to the quasigeostrophic equations, magnetohydrodynamics, , and liquid crystal systems. This small sample of Navier-Stokes-like equations illustrates how the methods developed in the study of large time behavior of basic fluid motions can be adapted to more complicated models. Asymptotic behavior of Leray’s weak solutions to the Navier-Stokes equations in Rn is the central theme of the first part of this chapter. Upper and lower bounds of rates of decay of the solutions are obtained, mostly when the corresponding system has zero external forces. The aim is to obtain optimal rates of decay, i.e., rates that coincide with the corresponding solutions of the underlying linear part: this is often achieved by establishing that the difference between the solution and its linear counterpart decays faster than the latter. Consideration is then given to several

11 Large Time Behavior of the Navier-Stokes Flow

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classes of strong solutions for which more detailed information can be obtained: point-wise decay, asymptotic profiles, etc. Once finite energy solutions have been investigated, various scenarios involving nonfinite L2 -norms for the Navier-Stokes flows are discussed. Restricting attention to finite energy solutions seems perfectly natural looking at the energy inequality, but there are some drawbacks; it does not cover, for instance, the dynamics of flows with vorticity ! in L1 .R2 /. This situation has recently attracted considerable interest. These are typically infinite energy flows, as the L2 condition for the velocity field would otherwise force the corresponding vorticity to have zero mean. The reader will find more on this topic in Th. Gallay and Y. Maekawa’s contribution ( “Existence and Stability of Viscous Vortices”, this volume) of the present handbook. A second drawback appears looking at the scale invariance of the Navier-Stokes equations: u 7! u , where u .x; t / D u.x; 2 t /. The function space associated P 1 .Rn //, with the energy inequality is X D L2 ..0; 1/; L2 .Rn // \ L2 ..0; 1/; H but ku kX is independent of  only in dimension 2. Starting with the classical contributions of H. Fujita and T. Kato [65, 99] a large amount of research has been developed regarding the constructions of solutions in function spaces respecting the scale invariance. These solutions in general have infinite energy. As soon as u0 satisfies a suitable smallness condition, (that can be dropped in the 2D case), the solutions are global-in-time. The construction of such solutions often provides valuable information on their decay as t ! 1. I. Gallagher’s ( “Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value Problem”, this volume) contribution in this handbook presents a detailed account on well-posedness issues in scale-invariant spaces. Here the focus is only on a few aspects of the large time decay results that follow from the so-called Kato’s method. A closely related issue to the scaling invariance is self-similarity, that is, solutions of the Navier-Stokes equations, like those constructed by Y. Giga and T. Miyakawa [80] or by M. Cannone, Y. Meyer, and F. Planchon [32]. Such solutions that are left-invariant by the natural scaling, u D u , play a major role, e.g., in the description of the large time behavior of flows that are asymptotically homogeneous, in the study of stability of stationary flows, in the understanding of axisymmetric flows, etc. Results on self-similar profiles are discussed on brief terms only as the reader will find a detailed account in H. Jia, V. Šverák, and T.P. Tsai’s contribution ( “Self-Similar Solutions to the Nonstationary Navier-Stokes Equations”, this volume) to this handbook. In Sect. 3 attention is turned on large time decay results for Navier-Stokes flows in domains other than the whole Rn . The presentation of this topic will be more succinct than in the case of the whole space. In particular, the focus is on the classical cases of bounded or exterior domains with smooth boundaries. In Sect. 4 results are reviewed on the asymptotic stability of stationary solutions. Recent developments on this topic are discussed: for certain solutions to the NavierStokes equations (possibly of infinite energy), results are presented that analyze the stability under arbitrary large L2 -perturbations. This is the case, for instance, of stationary solutions associated with a time-independent external force, suitably

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small in some rough norm. The classical one-parameter family of Landau solutions are asymptotically stable in this sense, at least when the parameter belongs to an appropriate range. In particular, results from [97] and its extensions in [98] and M. Hieber and J. Saal ( “The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties”, this volume) that deal with the asymptotics of such flows are discussed. The last section considers decay for other diffusive models. Among them, the quasi-geostrophic equations are an interesting example of a simplified model arising in geophysics, with a diffusive term given by a fractional power of the Laplacian. The other models discussed (MHD, inhomogeneous NS, system, nematic liquid crystal equations) require a more involved analysis since for them the Navier-Stokes system is strongly coupled with one or more additional equations.

2

Long-Time Behavior for the Navier-Stokes Equations in the Whole Space

2.1

The Energy Decay Problem of Weak Solutions

The first part of this chapter is centered on solutions to the Cauchy problem for the incompressible Navier-Stokes equations: ut  u C .u  r/u C rp D f;

.x; t / 2 Rn  .0; 1/; n  2;

div u D 0;

(NS)

u.x; 0/ D u0 .x/: As usual, u D u.x; t / represents the velocity of the fluid, p D p.x; t/ is the pressure, and f D f .x; t/ is the external force. The classical theory of viscous, incompressible fluid flows is governed by these equations. A large area of modern research is devoted to deducing the qualitative behavior of their solutions. The results on existence and asymptotic behavior depend on the initial conditions and either boundary requirements or conditions when jxj ! 1. Also adequate constraints need to be stipulated for the forcing term f . For simplicity, the viscosity coefficient is set equal to one, as this can be always achieved by an appropriate rescaling. All the results remain valid in the presence of a coefficient  > 0 in the diffusive term. What will change is the dependence on  for estimates. In the sequel the Navier-Stokes equations are referred to by NS. The literature related to the NS equations is too vast to attempt a complete list of references. The results in this chapter are restricted to the analysis related to the asymptotic behavior. The closing remark of the 1934 pioneering paper on the NS equations by J. Leray [118] (reproduced in the figure above), where global solutions were constructed for the first time, states: N.B. I do not know if W(t) goes necessarily to 0 when t grows indefinitely.

11 Large Time Behavior of the Navier-Stokes Flow

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N . B. J’ignore si W(t) tend nécessairement vers o quand t augmente indéfiniment.

Here W .t/ refers to the L2 .Rn /-norm of the weak solution to NS that Leray constructed, under the only assumption that u0 is in L2 .Rn / and divergence-free. The question in the N.B. was addressed in the particular case f  0 and for n D 3. Since then, this question has been studied not only for the decay in L2 .R3 / but for many other spaces and nonzero forcing terms.

2.2

Small Data Approach and T. Kato’s Decay Results

The first answer to Leray’s Nota Bene stated above was given by T. Kato ([99, Theorem 40 ]), showing decay to zero of solutions with datum in L2 .R3 / and zero forcing term. The results in [99] are very deep and helped to lay the groundwork for decay and existence questions for the solutions to the Navier-Stokes equations. Before stating the results from [99] the following notation is recalled (note that the notations used will not distinguish between function space of scalar and vector fields): 1. L2 D L2 .Rn / D fv 2 L2 .Rn / W div v D 0g, where the divergence of the vector field v is taken in the distributional sense. 2. P denotes the Leray projector, the orthogonal projector mapping L2 onto L2 . This projector extends to more general spaces, including the Lp spaces, 1 < p < 1. 3. PLn D fv 2 Ln .Rn I Rn /; div v D 0g. Once local existence of solutions with datum in PLn had been established in [99], it is shown that they can be extended globally, provided the datum is small in Ln .Rn /. Next the decay in Lq .Rn / spaces is analyzed for the solution and the first derivatives. The decay rates obtained coincide with the rates for solutions to the underlying Stokes flow. The two main results in this paper state: Theorem 1 ([99]). Let u0 2 PLn . Then there is T > 0 and a unique solution such that 1

n

t 2  2q u 2 BC .Œ0; T /I PLq /; n  q  1; n

t 1 2q Du 2 BC .Œ0; T /I PLq /; n  q < 1:

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L. Brandolese and M.E. Schonbek

The notation BC refers to bounded continuous functions. The Lp -norms will be denoted simply by k  kp . If ku0 kn <  with  sufficiently small, then the time T from the above theorem can be taken to be infinity and the following rates of decay are valid: 1

n

n

kukq  C t  2 C 2q ; n  q  1;

kDukq  C t .1 2q / ; n  q < 1:

(1)

Furthermore, Kato also established the following variant for data with better integrability properties: Theorem 2 ([99]). Let the initial data u0 2 PLn \ Lp ; 1 < p < n. There exists  > 0 such that if ku0 kn  , then the solution constructed in Theorem 1 is global. Moreover, for any finite q  p n 1

1

t 2 . p  q / u 2 BC .Œ1; 1/I PLq /;

n 1

1

t 2 . p  q /C1/ Du 2 BC .Œ1; 1/I PLq /;

provided the exponents of t are smaller than 1; otherwise, they be should replaced by an arbitrary number smaller than 1. Moreover, ku.t /kp ! 0 as t ! 1. The proof of these theorems is based on the construction of a sequence of approximating solutions represented in the mild form. Recursive bounds combined with a passage to the limit in this sequence yield the decay rates. For the argument to work, Kato needs at each step a uniform bound which is only possible if either the data or the time of existence is sufficiently small. The results in [99] can easily be extended to solution with nonzero forcing terms with appropriate decay conditions. A short sketch of Kato’s arguments is presented here. Sketch of Kato’s proof. Rewrite the NS solution as ut C Au C Pr  .u ˝ u/ D 0; where Au D Pu. In the whole space, the projector P commutes with the Laplacian, and A boils down to the operator  when applied to divergence-free vector fields. The solution in mild form can be represented by Z

u D e tA u0 C Gu;

t

with Gu.t / D 

e .ts/A Pr  .u ˝ u/.s/ ds:

0

Let ˛; ˇ;  > 0 and   ˛ C ˇ < m, then kD  Gu.t /k n  C

Z 0

t

1

.t  s/ 2 .C˛Cˇ/ kuk n kDuk ˇn ds;

Construct a sequence of the form

 D 0; 1:

(2)

11 Large Time Behavior of the Navier-Stokes Flow

u0 D u.x; 0/;

585

umC1 D a0 C Gum :

This allows to obtain the inductive estimates, for any 0 < ı < 1, 1

n

t 2 .1ı/ um 2 BC .Œ0; 1I PL ı /; kum k nı  Km ; 1

n

0

t 2 Dum 2 BC .Œ0; 1I PL ı /; kum k nı  Km : Setting K0 D K00 D C ku0 kn ; it is possible to show that KmC1  K0 C CKm Km0 ;

0 KmC1  K00 C CKm Km0 :

The last estimates for a small time interval or for data with sufficiently small norm yield K00 C CKm Km0  K, with K a fixed constant. If n  q  1,  D nq , ˛ D ı, and ˇ D 1, it follows that kumC1 kq  ke tA u0 kq C CKm Km0

Z

t

1

n

ı

1

n

.t  s/ 2 .1Cı q / s .1 2 / ds  Kt  2 .1 q / :

0

Passing to the limit as m ! 1 gives the conclusion of the theorem. The last claim of the second Theorem, applied to n D 3 and p D 2, answers Leray’s questions, at least for small solutions. Kato concludes his paper by showing how his results, in fact, apply to Leray’s weak solutions, that eventually become small, at least in dimension  4.

2.3

Masuda’s Approach and Nonuniform Decay

The same year that Kato’s decay results were established, K. Masuda [128] proved the nonuniform decay for weak solutions with large data in L2 .Rn / and appropriate nonzero forcing terms. His method applies to more general domains. Before stating Masuda’s main result, it is recalled that, by definition, a weak solution to NS is a P 1 .Rn //, which satisfies function u 2 Cw .Œ0; 1; L2 .Rn // \ L2 .RC ; H   Z t  @ C hru.s/; r .s/i C h.u.s/  r/u.s/; .s/i ds u.s/; @s 0 Z t D hf .s/; .s/i ds C hu0 ; .0/i; t > 0; (3)

hu.t /; .t /i C

0

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L. Brandolese and M.E. Schonbek

for all 2 C .Œ0; 1/; C1 .Rn //, where C1 .Rn / is the set for smooth and compactly supported functions with zero divergence. Here h; i denotes the L2 -inner product. For u0 2 L2 .Rn / and all f 2 L1loc .RC ; L2 .R2 //, a weak solution can be constructed, in an arbitrarily large time interval .0; T /, satisfying the energy inequality: ku.t /k22 C 2

Z

t

kru.r/k22 dr  ku0 k22 C 2

0

Z

t

hf; ui dr;

for all t  0:

(4)

0

On the other hand, the so-called strong energy inequality is the variant of (4) that reads ku.t /k22 C2

Z

Z

t s

kruk22



ku.s/k22

t

C2

hf; ui;

for s D 0, almost all s > 0

s

and all t  s.

(5)

Inequality (5) is only known to be true in dimensions n D 2; 3; 4 (or in any dimension, but for bounded domains). When available, such inequality provides useful additional information on the energy. Next, Masuda’s result is stated. Theorem 3 ([128]). Let u0 2 L2 .Rn / and the forcing term f with the projection Pf 2 L1 .RCR; L2 .R2 //. Let u be a weak solution to the Navier-Stokes equations in 1 Rn such that 0 kruk22 < 1. Then Z

tC1

t

ku.s/k22 ds ! 0;

as t ! 1:

In particular, if ku.t /k2 is monotonically decreasing for large enough t (when f D 0, this is the case at least for the weak solutions constructed by Leray in the case n D 3, and those obtained by Leray’s method for 2  n  4), then ku.t /k2 ! 0. Ideas from Masuda’s proof. The first step consists in showing that limt!1 k.I C A/˛ uk2 D 0. Here A D P and ˛ D .n2/=4. The crucial estimate of Masuda’s proof follows from an interpolation inequality and reads, with ˇ D 1=.1 C 2˛/, Z

Z

tC1 t

ku.s/k22



tC1

˛

k.I C A/ t

u.s/k22

ˇ Z

tC1

k.I C A/ t

1ˇ

u.s/k22

1ˇ :

The second integral on the right turns out to be bounded. This is used to obtain that R tC1 limt!1 t ku.s/k22 ds D 0. The decay is then immediate, since, when the L2 norm is decreasing, it converges to a constant, and this constant must be zero by the above integral limit. t u

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587

Masuda’s result that ku.t /k2 ! 0 not only applies to Leray’s solutions but also to a class of weak solutions constructed by O. Ladyzhenskaya [114], whose energy decays monotonically for large enough times. For data exclusively in L2 , decay without a uniform rate is the best that can be expected. Even at the level of the underlying linear part, solutions to the heat equations, with data in L2 alone, cannot decay at a uniform algebraic rate. For such n solutions, this can be seen considering the rescaled data u˛0 .x/ D ˛ 2 u0 .˛x/. Such initial values have all the same L2 -norm, and it is easy to show that, for all T > 0, ke T  u˛0 k2 D 1: ˛!0 ku˛0 k2 lim

Thus, no better decay rates can be expected for solutions to the Navier-Stokes equations. Specifically for any ˇ > 0, > 0, and T > 0, one can find (see [147]) u0 such that ku0 k2 D ˇ and ku.T /k2  1  : ku0 k2 For exterior domains the lack of decay uniformity for L2 was established by Hishida in [93].

2.4

Fourier Splitting Method and Decay Rates of Weak Solutions

For weak solutions to the NS equations, M.E. Schonbek [146] obtained algebraic decay for the Cauchy problem with large datum in L1 .Rn / \ L2 .Rn /; n  3, and f D 0. The Fourier splitting method used in [146] was first introduced to obtain the algebraic rates of decay to parabolic conservation laws [145]. Moreover, this method serves for a wide class of diffusive systems that satisfy an appropriate integral energy inequality. The method is based on the fact that the low frequencies of the data play a significant role in the decay. This fact will be discussed in more detail in Sect. 2.12. In the sequel Fourier splitting method will be referred by FS. A brief formal description of the FS technique is given following the steps in [149]. For a rigorous proof, FS can be applied to a sequence of approximating solutions and the decay follows by passing to the limit. Specifically FS is a technique used to determine the decay rates of solutions to an integral differential energy inequality of the form d dt

Z Rn

2

juj dx  C0

Z

jD m uj2 dx; Rn

x 2 Rn ; t > 0;

(6)

where C0 > 0 and m > 0 are independent on u and on t . (Note that when m D 0, exponential decay follows.) In the sequel v./ O denotes the Fourier transform of v. For the FS method, the following set Sm D Sm .t / needs to be defined:

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L. Brandolese and M.E. Schonbek

( Sm D  2 Rn W jj 



n C0 .t C 1/

 2m1 ) :

Theorem 4. Let u D u.x; t / satisfy the integral differential inequality (6), and let u0 2 L2 .Rn /. If there exist r > 0 and A > 0 so that jOu.; t /j  A for all jj < r, then for some constant K > 0 depending on r, A, and L2 data of the solution, it follows that n

1: If 1 < 2m ) ku.t /k22  K.t C 1/ 2m : 2: If 1 D 2m ) ku.t /k22  K.t C 1/n ln.t C 1/: 3: If 1 > 2m ) ku.t /k22  K.t C 1/n : Sketch of the formal proof. By Plancherel’s theorem inequality (6) can be rewritten as Z Z d jOuj2 d  C0 jj2m jOuj2 d; dt Rn Rn Split the frequency space on the right-hand side of the last inequality into integrals over Sm and Smc , and drop the term that comes from the low frequencies: d dt

Z

jOuj2 d  

Rn

n t C1

Z

jOuj2 d D  c Sm

n t C1

Z

jOuj2 d  C Rn

n t C1

Z

jOuj2 d:

Sm

Note that t can be taken as large as necessary so that for all jj 2 Sm one has jj < r. Hence, by the point-wise bound assumption on uO .; t /, it follows that  Z d n .t C 1/n juj2 dx  Cn .t C 1/n1 2m : n dt R Integrating in time the last inequality yields the conclusion of the theorem. FS can also be applied for differential inequalities of the form d dt

Z

juj2 dx  C Rn

Z

jD m uj2 dx C g.t /;

t > 0;

(7)

Rn

for appropriate functions g.t /. The technique can be used for solutions to systems such as Navier-Stokes, magnetohydrodynamics, and several other diffusive equations. In the particular case of the NS and MHD equations, inequality (7), if there is no external force, will be satisfied with m D 1 and g.t / D 0. For estimates of higher-order derivatives and/or solutions of systems with nonvanishing forcing terms, nonzero functions g.t / will appear; see [150, 155]. Other diffusive systems

11 Large Time Behavior of the Navier-Stokes Flow

589

may involve different values of m and g.t /. See Sect. 5.4 for applications of FS to fractional diffusion operators. Ideas for the formal decay proof of solutions .u; p/ to NS. The above FS argument is completed by suitable point-wise estimates on the Fourier transform uO . Namely, using

1

b

b  C jj ku.t /k22 ju  ruj C jr pj  C jjjuuj one gets 2

jOu.; t /j  e jj t jOu0 j C c

Z

t

2 .ts/

e jj

0

jj ku.s/k22 ds

(8)

In [146] for integrable initial data, the point-wise bound for jOu.; t /j  C jj1 n1 was then obtained, yielding the nonoptimal decay rate ku.t /k22  C .t C 1/ 2 . p 2 After this preliminary decay result, better rates (assuming u0 2 L \ L .Rn /, for some 1  p < 2) were obtained by R. Kajikiya and T. Miyakawa [96] for dimensions n  2: The improvement in [96] was based on using spectral theory for self-adjoint operators. For dimensions n  3, optimal decay rates were also obtained by M.E. Schonbek [147], by combining the FS method with improved frequency estimates. In two dimensions, the FS method as used in [146] only gave logarithmic decay. Its variant proposed by M. Wiegner [167], consisting in applying the Fourier splitting argument to the integral equation, gave the optimal decay for dimensions n  2. Furthemore, Wiegner’s result extends to give uniform decay rates for solutions arising from initial data u0 that do not need to belong to any Lp -space, with p 6D 2, but for which the decay of the heat equation is prescribed. Notation and preliminaries, that will be needed, are recalled before giving the main results in [167]. Wiegner assumed the initial data u0 and the external force f belong to the following class of functions: D˛n Df.u0 ; f / 2 LW ku0 .t /k22 C.1Ct /2 kf .t/k22  C .1Ct /˛ ; some C > 0; ˛ > 0g; (9) where L D L2 .Rn /n  L1 .RC ; .Rn /n / and u0 .t / is the solution to the heat equation with data u0 and forcing term f . The main results of [167] are described in the following Theorem: Theorem 5 ([167]). Let n  2. Let u be a weak solution to the NS equations with data and forcing terms .u0 ; f /. Suppose the strong energy inequality (5) holds. Let .u0 ; f / 2 D˛n . Then 1. ku.t /k22  C .1 C t /˛N ; with ˛N D minf˛; n2 C 1g 2. ku.t /  u0 .t /k22  h˛ .t /.1 C t /d ; where d D

n 2

C 1  2 maxf1  ˛; 0g, and

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L. Brandolese and M.E. Schonbek

8 ˆ ˆ < .t/ for ˛ D 0 with .t/ & 0; h˛ .t / D C ln2 .1 C .t// for ˛ D 1; ˆ ˆ :C for ˛ ¤ 0; 1:

as t ! C1:

As pointed out in [167], the assumption on the validity of the strong energy inequality (5) can be relaxed by requiring that weak solutions are suitably approximated by sequences of solutions verifying such inequality. Since this is known to be true also when n  5, Wiegner’s decay result applies in any dimension. R Remark 1. If Ru0 2 L1 \ L2 , then u0 D 0, see [130]. If one additionally assumes that .1 C jxj/ju0 .x/j dx < C1, then the solution of the heat equation satisfies ku0 .t /k2 D O.t .nC2/=4 / as t ! C1 (see [131]). By Wiegner’s theorem applied with f  0, one can construct a weak solution arising from u0 such that ku.t /k2 D O.t .nC2/=4 / as t ! C1. As observed above, this is true in any space dimension n  2. As it will be seen in the next section, this is the best decay rate that can be achieved for generic solutions of NS. The FS method for the Navier-Stokes was later extended to get the decay of the derivatives [150, 155]. For the extension of the FS method, the main tool was an energy estimate of the form d dt

2.5

Z

m

2

Z

jD uj dx  C Rn

Rn

jD mC1 uj2 C G.t / dx; x 2 Rn ; t > 0:

Lower Bounds

Throughout this section assume that f  0. As the solutions decay for large time, the influence of the linear terms supersedes the one of the nonlinear ones. To streamline the understanding of the asymptotics, both upper bounds and lower bounds of decay need to be studied. This gives better information on the closeness of the Navier-Stokes and their underlying linear counterpart solutions. The lower rates of decay were first considered by Schonbek in [148]. In this paper, conditions were introduced to show when solutions to the heat and NavierStokes equations have similar lower and upper bounds of rates of decay. The following notation will be used in the sequel, for all k; l D 1; : : : ; n: Z 1Z

Z bk;l D

Rn

yl u0;k .y/ dy;

ck;l .u/ D ck;l D 0

Rn

ul uk dy ds:

R The above integrals are convergent provided, e.g., u0 2 L2 .Rn / and Rn .1 C jyj/ju0 .y/jdy < 1, at least for weak solutions satisfying R the energy inequality in its strong form (5). Indeed, these assumptions imply that Rn u0 .x/ dx D 0, and then

11 Large Time Behavior of the Navier-Stokes Flow

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a simple computation (see [131]) shows that the solution u0 .t / of the heat equation in Rn with data u0 and zero forcing term satisfies ku0 .t /k2  C .1Ct /.nC2/=4 . Next one has just to apply the time decay estimate on ku.t /k2 provided by Theorem 5 to conclude. In [152] lower and upper rates of decay are studied for solutions to the MHD system. The results in [152] add a hypothesis on the solutions that was missed in [148] and corrects a gap. The ideas and hypothesis from [148, 152] give the basis for the more precise results by Miyakawa-Schonbek in [133]. The main theorem in [133] applies to weak solutions in all space dimensions and states: R Theorem 6 ([133]). Let u0 2 L2 .Rn /, such that Rn .1 C jyj/ju0 .y/jdy < 1. Let u be a weak solution of NS arising from u0 such that ku.t /k2 D O.t .nC2/=4 / as t ! C1 (see Remark 1 for the existence of such a solution). The following dichotomy holds: (i) If fbk;l g D 0 and if there exists c 2 R such that fck;l g D fcık;l g (where ık;l D 1 or 0 is the Kronecker symbol), then lim t

nC2 4

t!1

ku.t /k2 D 0:

(10)

(ii) Otherwise, (i.e., if fbk;l g 6D 0 or if fck;l g is not a scalar multiple of the identity matrix), then there exists c 0 > 0 such that ku.t /k2  c 0 t  In particular, if

R

nC2 4

:

(11)

xk u0;l .x/ dx 6D 0 for some k; l 2 f1; : : : ; ng, then (11) follows.

Remark 2. In view of the fact that the uniqueness of the weak solutions is a major open problem, a priori one cannot exclude the possible situation that for some initial data one could have several weak solutions, possibly with different values of ck;l and with different behavior for t ! C1. Ideas of the proof. Let u be a weak solution of NS starting from u0 , written in its integral form, as in (2). The integral formulation is indeed valid in the classical setting of weak solution (as well as in more general settings) as thoroughly discussed in [53, 116]. Writing this equation component-wise (j D 1; : : : ; n) leads to Z tZ

Z uj .t / D Here,

Rn

Et .xy/.u0 /j .y/ dy

0

Rn

Fl;j;k .xy; t s/ul .y; s/uk .y; s/ dy ds: (12)

592

L. Brandolese and M.E. Schonbek n

2 =4t

Et .x/ D .4 t/ 2 e jxj

is the heat kernel, and F denotes the kernel of the operator e t Pdiv. The components of F are given by (see [131]) Z

1

Fl;j;k .x; t / D @l Et .x/ıj k C

@l @j @k Es .t / dy ds:

(13)

t

Under the hypothesis of Theorem 6, in [63] Fujigaki and Miyakawa established that Z

nC2

lim t 4 uj .t /C@l Et ./ yl u0;j .y/ dy t!1

Rn

Z tZ

C Fl;j;k .x; t /

Rn

0



.ul uk /.y; s/ dy ds D 0 2

(14)

Let bl D .b1;l ;    ; bnl / and Fl;k D .Fl;1;k ; : : : ; Fl;n;k /. The key observation in [133] is that k@l Et ./bl CFlk .; t /ckl k2 D c 0 t .nC2/=4 , for some c 0  0 depending on u, with c 0 D 0 if and only if Z xk u0;l .x/ dx D 0

8 k; l,

and

Rn Z 1Z 0

Rn

(C) .uk ul /.x; t / dx dt D c ık;l

for some constant c  0 and 8 k; l:

Applying the triangle inequality to u.t / D .@l Et bl C Fl;k ck;l / C .u.t / C @l Et bl C Fl;k ck;l / and the limit (14) yields ku.t /k2 D c 0 t

nC2 4

 o.t

nC2 4

/: t u

This achieves the proof.

In [133], the authors also considered the case of flows arising from data u0 2 L2 n such that ke t u0 k2  C .1 C t / 4 : this is typically the case when u0 2 L1 \ L2 , even though the integrability is not always necessary. In this case they showed that n

n

ku.t /k2  ct  4 ; if and only if ke t ak2  ct  4 for t ! 1: The proof follows using triangle inequalities in one direction for ke t ak2 and for ku.t /k2 in the other, combined with upper bounds of decay for the heat kernel and the estimate (14).

11 Large Time Behavior of the Navier-Stokes Flow

2.6

593

Rapidly Dissipative Solutions and the Role of Symmetries

Let u.x; t / be the solution to NS with datum u0 . By Theorem 5, ku.t /k22  C .1 C t /˛ with 0 < ˛  .n C 2/=2/, provided such decay holds for the solution e t u0 of the heat equation. As discussed in Sect. 2.5, the restriction on the exponent ˛ is now known to be generically optimal. See [148] and, more recently, in [71, 72, 133] with different methods. Accordingly with Wiegner’s decay result, any solution of the free Navier-Stokes equation (NS) in Rn will be called rapidly dissipative provided that lim t

nC2 2

t!C1

ku.t /k22 D 0:

(15)

Even though Theorem 6 provides some sort of necessary and sufficient condition for solutions to be rapidly dissipative, the existence of such solutions (different from the identically zero one) is RnotRan obvious consequence of this theorem. Indeed, the 1 algebraic condition .C/ on 0 .uj uk /.y; s/ dy ds is difficult to check, as it involves the solution itself and not just the initial datum. However, exceptional flows which decay much faster do exist. For example, it has been known for a long time that, in dimension n D 2, there exists such an explicit solution of the Navier-Stokes equations with radial vorticity. This condition on the vorticity implies that the velocity field is rotationally invariant and the nonlinearity has potential form (i.e., r  .u ˝ u/ D rp), so that u is also a solution to the homogeneous heat equation. It was pointed out for the first time in [148] (see also [63, 133]) that inside this class of flows, one can find nonzero solutions with exponential decay at infinity in the L2 -norm. Specifically, let !0 D curl u0 be a radial function in R2 . Let !.t/ D curl u.t / and assume that ru0 and !0 satisfy !0 2 L1 , ru0 2 L1 , !./ O D 0 for jj  ı, some ı > 0. An easy computation shows that u0 2 L1 \ L2 and that u satisfies 1 u.x; t / D 2 r



x2 x1

Z

r

s!.s; t /ds;

(16)

0

The velocity remains rotationally invariant for all time and in particular it fulfills condition .C/, ensuring that Part (i) of Theorem 6 holds. But for this flow, it is established in [148] that ku.; t /k1  C exp.ı 2 t /;

and

ku.; t /k1  ct:

The exponential decay of the L2 -norm follows by interpolation. Similar flows with exponential decay exist in higher even dimension and a general method for their construction is described in [152]. All these solutions, sometimes called generalized Beltrami flows, turn out to solve simultaneously (NS) and the heat equation. As discussed in [152], it seems impossible to adapt these examples to the n D 3 case or for any other odd dimensions. One difficulty is related to the fact that one is faced with a topological obstruction

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in attempting to construct nontrivial divergence-free vector fields invariant under rotations. Besides generalized Beltrami flows, a few other exact solutions of the Navier-Stokes equations are known (see, e.g., the review paper [164]). However, no examples of rapidly dissipative solution in R3 has been known until Brandolese’s construction [16] in 2001 (see also [18]). The basic idea of [16] is to consider solutions invariant under a suitable discrete subgroup of rotations. For example, a weak solution arising from 0

1 2 x1 .x32  x22 /e jxj 2 C B u0 .x1 ; x2 ; x3 / D @ x2 .x12  x32 /e jxj A : 2 x3 .x22  x12 /e jxj

(17)

will inherit the symmetry property of the datum (the parity conditions and the discrete rotational invariance), and the energy decays considerably faster than predicted by Wiegner’s theorem (the decay will be O.t 9=2 / as t ! C1 for the above example). This behavior is due to the fact that such symmetries force both the velocity and vorticity to have a few additional vanishing moments for all time. The persistence of higher-order vanishing moments of the vorticity is also closely related to the persistence during the evolution of orthogonality-type relations, like Z 8 t  0;

.uj uk /.x; t / dx D c.t /ıj;k

(18)

which is even stronger than what is actually needed in condition .C/. Such orthogonality relations are essential to prevent the spatial spreading effects (the instantaneous loss of localization) on the solution described by Brandolese in [23]. Such type of results can be adapted in the case of flows in the half-space (see [64]) or some other domains with symmetries; see [85]. A complete picture of rapidly dissipative solution with symmetries, in dimension two and three, is presented in [17] by Brandolese, where the best decay rates, depending on the symmetry group, are fully classified.

2.7

The Vorticity Approach and More on Rapidly Dissipative Solutions

The analysis of Th. Gallay and C.E. Wayne [71, 72] (see also Sect. 2.10.2) provides a deeper insight of rapidly dissipative solutions. Their starting observation is that the decay property R (10) is left invariant by time translations of the solution u, but the hypothesis .1 C jxj/ju0 .x/j dx < 1 needed in Theorem 6 to achieve (10) is not. Thus, this hypothesis cannot be optimal. The work in [72] analyzes the two-dimensional NS equations. The vorticity is studied after being rescaled and the corresponding velocity is recuperated using the Bio–Savart law. The scaling, already considered in [34], is given by

11 Large Time Behavior of the Navier-Stokes Flow

D

595

x ; D log.1 C t /; t C1

yielding the new functions w.; / and v.; / defined by !.x; t/ D

 x 1 w ; log.1 C t / t C1 t C1

u.x; t / D

 x 1 v ; log.1 C t / : t C1 t C1

With this scaling the vorticity satisfies the equations in 2D @ w D Lw  .v  r /w;

(19)

1 Lw D  w C .  r /w C w 2

(20)

where L is defined by

The long-time asymptotic is analyzed in this new setting. Linearizing Eqs. (19) around the origin gives a countable set of isolated real eigenvalues corresponding to the time evolution generator: by working with vorticities in the weighted L2 -spaces 2

n

L .m/ D f W kf

k2L2 .m/

Z 

o m .1 C jj2 / 2 jf ./j d  < 1 ;

with sufficiently large m, the essential spectrum can be moved as far as needed to the left half plane. One can then establish the existence of invariant manifolds of finite dimension in the phase space of the vorticity equations. These manifolds are attractors for solutions in neighborhoods of the origin. Ordinary differential equations are obtained by restricting the vorticity partial differential equation to the invariant manifolds. The rate at which the solutions converge to the attracting manifolds is determined by the asymptotics of these ordinary differential equations. The work in [71] studies the three-dimensional NS equations in the spaces L2 .m/. The solution considered have small initial vorticity which decays algebraically as jxj ! 1. The same scaling of 2D is used. The vorticity equations in the new variables in 3D, with L defined in (20), read as @ w D Lw  .v  r /w C .w  r /v; The authors in [71] obtain an asymptotic development of the solution as t ! C1, in negative powers of t. Explicit computations are given for all the firstand second-order asymptotics. A strong stable manifold of the origin is constructed where the initial vorticity is supposed to be. For solutions with such initial vorticity, a new characterization of rapidly dissipative solutions (in the sense of Sect. 2.6) is obtained. Unlike in Theorem 6, no conditions on u0 are prescribed that are noninvariant under the flow. On the other hand, a smallness assumption is required on

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L. Brandolese and M.E. Schonbek

the initial vorticity, but this condition in the end is not really restrictive, as weak solutions constructed so far are eventually small due to to the energy . A by-product of their construction is that there exist nontrivial solutions of the NS equations with energy dissipating at an arbitrary fast algebraic decay rate. With appropriate changes, their method goes through in any space dimension.

2.8

Asymptotic Behavior of Global Solutions in Scale-Invariant Spaces

Kato’s theory described in Sect. 2.2 provides global solutions u 2 BC .Œ0; 1/, Ln .Rn //, satisfying ku.t /kn ! 0, assuming that ku0 kn is small enough. In fact, in the decay estimates (1), one could drop the smallness condition on ku0 kn , replacing it by the assumption that ku.t /kn does not blow up: in this case, however, the constants in those estimates would depend on the solution itself. The solutions constructed by T. Kato are in fact the only mild solutions in BC .Œ0; 1/; Ln .Rn // with u.0/ D u0 , by G. Furioli, P.-G. Lemarié-Rieusset, and E. Terraneo uniqueness theorem; see [116]. On the other hand, in the three-dimensional case, there are a few known examples of solutions, with a special geometrical structure, that are global-in-time, do belong to C .Œ0; 1/; L3 .R3 //, and arise from “large data” ( this means arbitrarily large in the L3 .R3 /-norm, or any other norm of possibly rougher spaces with the same scaling); see the work of Chemin and Gallagher in [43] and the references therein. A priori, one might think that such solutions do not vanish as t ! 1. However, I. Gallagher, D. iftimie, and F. Planchon [68] proved that all these solutions must decay in the L3 .R3 /-norm as t ! 1, no matter what the size of ku0 k3 is. Their proof uses the structure of the equation and in particular energy estimates on the difference v D u  w, where u is the a priori given solution, obtained through the decomposition of u0 as u0 D v0 C w0 , where w0 belongs to L3 .R3 / with small norm (in such a way that one can construct a global solution w from w0 via Kato’s method) and v0 2 L2 .R3 /. This idea is due to C. Calderón [28]. Moreover, such global solutions are stable, in the sense that if kQu0  u0 k3 is small enough, then the lifetime of the mild solution uQ 2 C .Œ0; 1/; L3 / arising from uQ 0 is infinite and, for all t  0, kQu.t /  u.t /k3 , remains bounded by a constant independent on t . These kinds of results in fact are not specific to the L3 .R3 / space; similar 1C 3 ;q conclusions hold true, e.g., in homogeneous Besov spaces BP p p .R3 / that share with L3 .R3 / the same scaling invariance; see [68]. Results in the same spirit, but with technically different proofs, are due to P. Auscher, S. Dubois, and Ph. Tchamitchian; see [4]. These authors essentially prove that an a priori global solution u belonging to the Koch-Tataru space [103] must decay to zero in the KochTataru norm as t ! 1 and is stable in such a space with an analytical dependence on the initial data. As the Koch-Tataru space is the largest scale-invariant space where the well-posedness has been established, the results of [4] encompass earlier stability results and are seemingly optimal. The regularity of Koch-Tataru solutions,

11 Large Time Behavior of the Navier-Stokes Flow

597

with decay estimates for higher-order derivatives generalizing estimates (1), has been obtained, e.g., by Germain, Pavlovic, and Staffilani [76]. Similar ideas can be used to construct infinite energy global solutions of NS in two dimensions, without putting any smallness assumptions. For example, if u0 1C 2 ;q belongs to the Besov space BP p p .R2 /, with p  2 and q > 2, then I. Gallagher 1C p2 ;q

and F. Planchon prove that there is a unique solution in C .Œ0; 1/; BP p 1C 2 ;q BP p p .R2 /-norm

solution decays to zero in the p; q), as shown by P. Germain [75].

2.9

/. This

(under additional restrictions on

Decay in Weighted Spaces and Point-Wise Estimates

2.9.1 Early Results The analysis of the solutions in weighted spaces provides useful information on their spatial behavior. In connection with the energy inequality, it is natural to ask whether weak solutions satisfy the following estimate: Z

2˛

2

Z tZ

jxj ju.x; t /j dx C

jxj2˛ jruj2 .x; s/ dx ds  c:

(21)

0

In the case of the Cauchy problem in R3 , this estimate was first established with 0  ˛  3=2 by M.E. Schonbek and T.P. Schonbek [151] for smooth solutions and then by C. He and Z. Xin [87] for weak solutions emanating from integrable data such that .1 C jxj/3=2 u0 2 L2 .R3 /. H.-O. Bae and B.J. Jin [6] improved this to ˛ < 52 , using Calderón-Zygmund-type inequalities, under slightly more stringent conditions on the decay of the datum. A similar conclusion, valid locally in time and with an additional weighted L2 estimate for u, was obtained in [19] by L. Brandolese. The bound ˛ < 52 , as pointed out by Brandolese in [23], is generically optimal, as any solutions satisfying estimate (21) with ˛ D 52 must satisfy orthogonality relations of the form (18). In the setting of strong solutions, there are many results establishing upper bounds for weighted Lp -norms of u or its derivatives. This issue is closely related to the problem of establishing point-wise decay estimates. Indeed, from a point-wise decay estimate, one immediately obtains weighted estimates by a straightforward integration. Deducing a point-wise estimate for large time from weighted estimates is less obvious; nevertheless, this can be done as will discussed below. Early results on space–time point-wise decay go back to G.H. Knightly [100], where he addresses such type of decay for solutions to the 3D NS equations with zero forcing term. The initial datum is assumed to be continuous and divergencefree and to fulfill the condition ju.x; 0/j  A minf1; jxjr g;

for some 1  r < 3:

(22)

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L. Brandolese and M.E. Schonbek

Provided A is sufficiently small, the corresponding solutions are shown to satisfy r

ju.x; t /j  C minf1; jxjr ; t  2 g;

x 2 R3 ;

t > 0:

These results were extended by Knightly in [101, 102] to any spatial dimension, where in addition he established that the solutions are space–time analytic. The case of nonzero spatial asymptotic behavior was also considered: If u satisfies ju.x; t /  u1 j D o.1/;

jru.x; t /j D o.x/;

jp.x; t/j D o.x/ as jxj ! 1;

where u1 is a constant, then it is shown that r

jr j u.x; t /j  C minf1; jxjr ; t  2 g j D 0; 1; r 2 Œ1; n/; n 1 1 jp.x; t/j  C t  2 minf1; jxj2r1 ; t r 2 g; r 2 Œ1; /: 2 Note that for the pressure decay, the result is only obtained when n  3. A different approach was used in S. Takahashi’s paper [162]. Using weighted equation techniques, Takahashi gets point-wise decay rates both in time and space, under the assumptions that the external force has an algebraic space–time decay rate, and the initial datum is zero. The solutions are assumed to be bounded in some weighted Lq;s -norms, with nq C 2s D 1 and q; s 2 Œ2; 1 (the limiting Serrin class), where Lq;s denotes the space of all u W Rn  .0; 1/ ! Rn such that Z 0

1 Z

ju.x; t /jq dx

 qs

dt

1s

< 1:

Rn

In [162] almost optimal uniform space–time decay estimates are established, that is, the estimates are close to the ones corresponding to solutions to the heat equations. For an outline of the previous work on space–time decay, the reader is referred to [162]. The work of C. Amrouche et al. [3] complements and extends the results in [162]. It considers nonzero initial data and zero forcing terms; the space– time decay is obtained for the derivatives of all orders in spaces of dimensions n  5. The proof combines the decay obtained for moments of the solutions u of NS in L2 in [151], the decay of derivatives of the solution [150, 155], and a Gagliardo-Nirenberg interpolation inequality that yields the decay in L1 -norm of k v.x; t / D .1 C jxj2 / 2 D ˛ u, for k  n2 . I. Kukavica’s paper [110] improves the decay rates computed in [3], obtaining rates similar to those that are available for n the heat equations up to the critical rates jxjn and t  2 . These critical rates were further improved to their optimal values by T. Miyakawa [131], as will be seen in Sect. 2.9.2. Point-wise estimates were used in [20] to put in evidence some strange concentration-diffusion effects in Navier-Stokes flows.

11 Large Time Behavior of the Navier-Stokes Flow

599

2.9.2

Miyakawa’s Point-Wise Space–Time Decay Result and Its Consequences T. Miyakawa [131] not only found the optimal point-wise decay rates in space– time but also got the optimal range of validity for these space–time decay estimates to hold. The slightly improved version below of his theorem (the improvement concerns only the smallness assumption, that in [131] was not scale invariant) is taken from Brandolese [18].

Theorem 7 (Miyakawa). Let 1    nC1 and let u0 be a divergence-free vector field such that .1 C jxj/ u0 .x/ 2 L1 .Rn /:

(23)

If  D n, n C 1 it is also assumed that sup .1 C jxj/ je t u0 .x/j < 1;

(24)

x2Rn ;t0

n

and in the case  D n, additionally it will be assumed that t 2 je t u0 .x/j is uniformly bounded in x and t . If supx2Rn jxj ju0 .x/j < , with  > 0 small enough, then there exist a constant C and a solution u of (NS) such that u.0/ D u0 (e.g., in the distributional sense) and ju.x; t /j  C .1 C jxj/ ;



ju.x; t /j  C .1 C t / 2 :

(25)

This theorem can also be seen as a persistence result of the point-wise decay condition .1 C jxj/ u0 2 L1 .Rn /. F. Vigneron addressed in [163] the related questions of the persistency (for strong solutions) of a condition like .1 C jxj/ u0 2 Lp .Rn /. In this case, the natural limitation on the exponent of the weight is  C pn < n C 1. The proof relies in writing the Navier-Stokes equation in its integral form, as in (12): u.t / D e t u0 

Z 1Z F .x  y; t  s/.u ˝ u/.y; s/ dy ds;

(26)

0

The kernel F is smooth outside the origin and satisfies the scaling invariance 1 F .x; t/ D t  2 .nC1/ F . px t ; 1/ and the bound jF .x; 1/j  C .1 C jxj/n1 . 1

Combining these two bounds, one sees that jF .x; t /j  jxj˛ t  2 .nC1˛/ for all 0  ˛  n C 1. Using such information one can perform the relevant estimates on the linear and bilinear terms that eventually lead to the desired result by a perturbation argument. Notice that the smallness condition on u0 is invariant under the natural scaling of the equation.

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n The solution of Theorem 7 is unique, e.g., in C .Œ0; 1Œ; L1  .R //, where  1 is the space of all functions f such that .1 C jxj/ f .x/ 2 L .Rn / and the continuity in t D 0 is defined in the distributional sense (as it is usually done in non-separable spaces). The simplest way of ensuring the validity of (24) is to assume u0 2 L1 .Rn / (if  D n) or .1 C jxj/u0 2 L1 .Rn / (if  D n C 1). However, these latter conditions instantaneously break down (unless there are symmetries), whereas condition (24) remains true during the evolution. For simplicity, focus is now centered on the case of fast decaying initial data, i.e., such that .1Cjxj/ u0 2 L1 with   nC1. From the point-wise estimates (25), one q readily deduces decay estimates in weak Lebesgue norm Lw of u. Next interpolating q q0 p the L -norm between two different weak norms, Lw and Lw , q < p < q 0 , leads to (the p D 1 case would require in fact the use of weak Hardy spaces; see [130]): n L1  .R /

1

n

ku.; t /.1 C jxj/˛ kp D O.t  2 .nC1˛ p / /;

(27)

valid for all 1  p  1 and ˛  0 such that ˛ C pn < n C 1, (or 0  ˛  n C 1 if p D 1). For more details on this argument, see the work by Bae and Brandolese [5]. For the corresponding lower bounds of these weighted norms, see Sect. 2.10.1. The arguments in [131] could be used to obtain estimates for the derivatives of any order: in agreement with what happens for the heat equation, an estimate for a spatial derivative of order b 2 Nn improves the above decay rate of u of an exponent jbj=2. I. Kukavica and J.J. Torres [111, 113] proposed a different approach to such kind of estimates. Namely, using a parabolic interpolation inequality, they proved that if u0 2 S.Rn / is an initial datum leading to a strong solution such that ku.t /k2 D O.t  /, for some   0, then (provided 0  ˛ < n C 1  d =p), 1 n

n

kD b u.; t /.1 C jxj/˛ kp D O.t  2 . 2 C2Cb˛ p / /:

(28)

Notice that the assumption u0 2 S.Rn / is enough to guarantee that one can take   .n C 2/=4, by Wiegner’s theorem (Theorem 5). Hence, for such data 1

n

kD b u.; t /.1 C jxj/˛ kp D O.t  2 .nC1Cjbj˛ p / /; for all 0  ˛ < n C 1  pn , b 2 Nn , and 1  p  1. When there is no weight (˛ D 0), a different approach leading to bounds of the form of (28) consists in deriving Gevrey-type estimates for the solutions, as done by M. Oliver and E.S. Titi, [139]. The key point ispto derive a differential inequality for the norms Gr D kAr e .t/ Ak22 , where A D  and appropriate choices of

D .t / that can be used to obtain explicit bounds on the radius of analyticity of the solution in time. An interesting feature of this approach is that it provides also some lower bounds for the L2 -large time behavior of the solutions and their derivatives.

11 Large Time Behavior of the Navier-Stokes Flow

2.10

601

Asymptotic Profiles

2.10.1 Asymptotic Profiles: The Case of Well-Localized Velocities A. Carpio [36] proved that strong solutions to NS admit an asymptotic expansion in terms of space–time derivatives of the heat kernel. Her results were improved by Y. Fujigaki and T. Miyakawa [62]. To present their asymptotic profile, some notation is needed: recall that F denotes the kernel of the operator e t Pdiv appearing in the integral formulation (IE) of the Navier-Stokes equation. More explicitly, F .x; t / D .Fl;j;k .x; t //, and the components of F are given by expression (13). Denote, as before, the heat kernel by E.x; t/ and Eh;k .t / D

Z Rd

.uh uk /.y; t / dy

the so-called energy matrix of the flow. Notice that to the notations introduced just before Theorem 6.

R1 0

(29) El;k .s/ ds D cl;k according

Theorem 8 ([62]). Let u0 be a divergence-free vector field such that .1 C jxj/u0 2 L1 .Rn / and u0 2 L1 .Rn /, giving rise to a unique global strong solution u (this is always the case if n D 2, or, e.g., n  3 and ku0 kn is small enough). Then, for any q 2 Œ1; 1 and j D 1; : : : ; n, one has

lim t

1 n 1 2 C 2 .1 q /

t!1

Z



uj .t / C @k E.t; / yk u0;j .y/ dy



Z 1



El;k .s/ ds D 0; C Fl;j;k .t; /

0

(30)

q

where the summation symbols over repeated subscripts have been omitted. The above limit does indeed describe the large time behavior of u, since by scaling reasons each one of the terms @k E.; t / and Fl;j;k .x; / decays exactly 1

n

1

as t 2 C 2 .1 q / . The expansion (30) corresponds to a first-order asymptotic profile. Putting more stringent assumptions on the localization of u0 (namely, under the assumptions of Theorem 7 with  D n C 1), then one can write a higher-order asymptotics for u, up to the order n that read as follows: uj .; t / D

X 1jj˛jm

C

.1/j˛j ˛ .@x E.x; / ˛Š

X jˇjC2pm1

Z

y ˛ u0;j dy

.1/jˇjCp p ˇ .@t @x /Fl;j;k pŠˇŠ

Z

1

s p y ˇ .ul uk /.y; s/ dy dsCl.o.t. ; 0

602

L. Brandolese and M.E. Schonbek

where the neglected terms are of lower order in the Lq -norm .1  q  1/, as t ! 1. Theorem 8 has several variants. One of these applies to weak solutions (in this case one only assumes u 2 L2 .Rn / and .1 C jxj/u0 2 L2 .Rn / where expansion (30) holds for 1  q  2. The special case q D 2 found an important application in Theorem 6 as already discussed. H. Y. Choe and B.J. Jin [44] extended (30) by allowing weight terms in the expansion. One drawback of the asymptotic expansions like the above or that in Eq. (30) is that any point-wise information on the velocity field is lost when computing the Lq norm. However, L. Brandolese and F. Vigneron [25] noticed that for mild decaying solutions (namely, the solutions obtained in Theorem 7 with  > .n C 1/=2), the following point-wise asymptotic profile holds as t ! 1: u.x; t / D e t u0 .x/ C rh.x/ C r.x; t /

(31)

Here, rh.x/ is a time-independent homogeneous gradient field of degree .n C 1/, smooth for x 6D 0. The remainder term r.x; t / is a lower-order term as x goes to infinity: actually it satisfies lim

jxj t; p t

jxjnC1 r.x; t / D 0:

!1

In fact, the second term in the right-hand side of (31) is almost explicit (it depends, however, on the solution), as  Z 1 X  ıh;k x h xk  h.x/ D cn  Eh;k .s/ ds: n jxjn jxjnC2 0 h;k

t The homogeneous function h.x/ p is closely related to the kernel K of e P. Indeed, d K.x; t / D h.x/ C jxj ‰.x= t /, where ‰ is a Gaussian-like decaying function. p Profile (31) follows taking both t ! 1 and jxj= t ! 1 in a more general formula in Brandolese and Vigneron [25, Theorem 1.7] that will not be reproduced here. Expansion (31) and its variants have several consequences. As noticed in [25], it can be used to derive the corresponding lower bounds for the weighted estimates (27), generalizingp those of Sect. 2.5. Indeed, integrating (31) in portions of the conic region jxj  A t, with A >> 1 one easily deduces 1

n

k.1 C jxj/˛ u.t /kp   t  2 .nC1˛ p / ;

(32)

for all 1  p < 1 and ˛  0 such that ˛ C pn < d C1 (or ˛  nC1 when p D 1). Inequality (32) extends earlier lower bounds established in particular cases (namely, for p D 2 and 0  ˛  2, see [6, 71, 148], or 1  p  1 and ˛ D 0, see [63]). The above constant   0 is strictly positive as soon as rh.x/ 6 0, and this is what

11 Large Time Behavior of the Navier-Stokes Flow

603

happens generically. In fact, accordingly with [148] R 1 and [133] (see also Theorem 6), rh.x/  0 if and only if the matrix .cl;k / D . 0 El;k .s/ ds/ is a scalar multiple of the identity. Of course, for symmetric solutions rh.x/ can be identically zero. An expansion similar to (31) is also meaningful for fixed t and jxj ! 1. In this case the result of [25] asserts that for mild decaying solutions (possibly defined only locally in time) u.x; t / D e t u0 .x/ C H .x/ W K.t/ C ot .jxjn1 /; 

ıh;k n jxjn

as jxj ! C1;

(33)



xh xk is homogeneous of degree .d C 1/, cn is jxjnC2 Rt an absolute constant, K.t/ D 0 Eh;k .s/ ds, the symbol “:” stands for a double summation on h; k D 1; : : : ; n, and the ot .jxjn1 / notation denotes a function R.x; t / such that limx!1 jxjnC1 R.x; t / D 0 for all fixed t . Thus, if u0 is well localized (say u0 2 S.Rn /), then

where H .x/ D cn r



u.x; t /  H .x/ W K.t/;

as x ! 1

for all fixed t such that the strong solution is defined. This expansion can be used to establish point-wise lower bounds as jxj ! 1. I. Kukavica and E. Reis [112] proposed a variant to (31), involving a higher-order expansion.

2.10.2 Asymptotic Profiles: The Case of Poorly Localized Velocities M. Cannone, C. He, and G. Karch [31] established an analogue of the expansion (30) for strong solutions with slower decay, arising from initial data small in Ln .Rn / and n belonging to the weak Lebesgue space Lw˛ .Rn /, for some ˛ 2 Œ1; n/. Let us recall p for 1 < p < 1 the space Lw D Lp;1 is normed by 1

kvkp;1 D sup rjfx 2 Rn W jv.x/j > rgj p : r>0

The results in [31] yield the second terms of the asymptotic expansions of the solutions to NS in Rn . Their result applies to the class of global solutions satisfying the following estimates, for 1 < ˛ < n: n ˛

1

sup ku.t /kn=˛;1 < 1; ku.t /kp < C .1 C t / 2 . n  p / ; kru.t /kp t>0 1

n ˛

1

 C t  2 .1 C t / 2 . n  p / :

(34)

One of the main results in [31] is Theorem 9 ([31]). Let u be a solution of NS satisfying (34). If 1 < ˛ < 1 C n2 , then for any p 2 Œ1; 1 satisfying p > n=.2˛/

604

lim t

L. Brandolese and M.E. Schonbek n ˛ 1 n1 2 . n  p /C 2

t!1

Z tZ



t

u.t /  e u0 C 0

Rn



e .t / Pr  .Qu ˝ uQ /.y; /dyd D 0 p

for all t > 0, where uQ D e  u0 D E.; / u0 . Remark 3. Estimates (34) hold for the unique mild solution u0 arising, e.g., from small data in u0 2 Ln=˛;1 \ L1 .Rn /. The authors of [31] complete their theorem studying the case ˛ > 1 C n2 : in this case they obtain essentially the same profile as in Theorem 8. The proof follows by writing the solution in the appropriate integral form and using an adaptation of the method in [63] by Fujigaki and Miyakawa.

2.10.3 Asymptotic Profiles: The Self-Similar Case The two-dimensional case is considered first. When the initial vorticity !0 D r  u0 is in L1 .R2 /, by the classical results of M. Ben Artzi and T. Kato (see, e.g., [7]), there is a unique global solution ! 2 C .Œ0; 1/; L1 .R2 // \ C ..0; 1/; L1 .R2 // of the vorticity equation @t ! C .u  r/! D !;

x 2 R2 ; t > 0

(35) 1

arising from !0 . Such solution is also known to decay as k!.t/kp D O.t 1 p / as t ! 1, for 1  p  1. The simplest explicit self-similar solutions of the Navier-Stokes equation, in 2D, are the Oseen vortices: !Oseen .x; t / D

˛  jxj2 e 4t ; 4 t

uOseen .x; t / D

jxj2 ˛ x? .1  e  4t /; 2 2 jxj

x 2 R2 ;

t > 0: (36)

RThe constant parameter ˛ 2 R is the total circulation of the vortex, ˛ D !.x; t/ dx. Such solutions play a key role in the description of the large time behavior of solutions to the vorticity equation with infinite energy, as illustrated by Theorem 10 below. This theorem was first established, by Y. Giga and Kambe [78], under the additional assumption that k!0 k1 is small. Such smallness assumption was subsequently relaxed by A. Carpio in [35] and finally completely removed by Th. Gallay and C.E. Wayne in [73]. See also [77, Chapter 2]. Theorem 10 ([73]). Let ! 2 L1 .R2 /. Then the R solution ! to (35) behaves as t ! 1 like the Oseen vortex with circulation ˛ D !0 : 1

lim t 1 p k!.; t /  !Oseen .; t /kp D 0

t!1

and

.1  p  1/;

11 Large Time Behavior of the Navier-Stokes Flow 1

1

lim t 2  q ku.; t /  uOseen .; t /kq D 0

t!1

605

.2 < q  1/:

See also [72] for higher-order asymptotics. The main idea in [73] is the discovery of a new Lyapunov function: the relative entropy of the distribution of the vorticity with respect to the Gaussian distribution of the Oseen kernel. When p D 1, the above result provides no convergence rate to the Oseen vortex. However, at least when !0 is very well localized (e.g., when it belongs to a L1 .R2 / space with a Gaussian weight), Th. Gallay and L. M. Rodrigues did establish that such convergence holds at a logarithmic rate. See [70]. A consequence of Theorem 10 is that the Oseen vortices are the only self-similar and integrable solutions of the vorticity equation in 2D. However, plenty of selfsimilar solutions exist such that !.; t / 62 L1 .Rn /. As a matter of fact, Y. Giga and T. Miyakawa [80] proposed a general method, based on the analysis of the vorticity equation in Morrey spaces, for constructing nonstationary self-similar solutions to NS in R3 , and for two dimensions in [80], it is stated that: When the space dimension is two and the total variation of the initial measure is small, there is only one forward self-similar solution for a given total circulation [81]; moreover according to [78] the forward self-similar solutions describe large time behavior of the vorticity. Existence of many self-similar solutions therefore suggests that the large time behavior of the vorticity is much more complicated in the three-dimensional case compared with the two-dimensional case.

More direct constructions have been proposed by Cannone, Meyer, and Planchon [21, 29, 30, 32, 33, 129]; see also [116, Chap. 23]. Their constructions go through in any spatial dimension n  2: basically, one only has to choose a vector field u0 .x/ in Rn , homogeneous of degree 1, satisfying some mild smallness and regularity assumption on the sphere Sn1 to guarantee the existence and uniqueness of a selfsimilar solution emanating from u0 . In [142], F. Planchon provided a condition implying that a solution u.x; t / of the Navier-Stokes equations in Rn has a self-similar behavior for large time. Roughly, 1C n ;1 P p p .Rn / (for enough in the Besov spaces B his results assert that if u0 is small p p some p > n), and is such that t e t u0 . t / ! v in Lp .Rn / as t ! 1, then the weak limit v0 .x/ D lim!1 u0 .x/ does exist, next v D e  v0 , and u.; t / ' 1 p V . p t / as t ! 1, where p1 t V . px t / is the self-similar solution of (NS) arising t from v0 . As observed in [142], it would not be enough to assume that the rescaled data u0 .x/ converge to get the same conclusion. A by-product of Planchon’s results is that if u0 .x/  jxj1 .x=jxj/ as x ! 1, where  is in L1 on the unit sphere with small enough L1 -norm, then the corresponding solution is asymptotically self-similar. If it is only known that ju0 .x/j  jxj1 (with > 0 small enough), a quite different situation occurs. In this case, it is still possible to construct a unique global solution u.x; t / such that p ju.x; t /j  C .jxj C t /1 . However, Th. Cazenave, F. Dickstein, and F. Weissler, in [40], showed that the large time behavior for such solution can be chaotic: They proved that if a sequence of dilates n u0 .n / converges weakly to a solenoidal

606

L. Brandolese and M.E. Schonbek

vector p field z./, when n ! 1, then, for tn D 2n , the discretely rescaled solution p tn u. tn ; tn / will converge as n ! 1 toward a limit depending on z (namely, S.1/z, where S.t / is the Navier-Stokes flow). Now, if tn0 D .0n /2 ! 1 is another time sequence, it can happen that 0n u0 .0n / converges to a vector field z0 ./ different from z./. In this case, the solution u has at least two different asymptotic behaviors along different sequences. In [40], the authors succeeded in constructing initial data u0 leading to solutions of the Navier-Stokes equation with infinitely many different asymptotic behaviors, along different time sequences.

2.11

Decay in Besov Spaces and Frequency Concentration Effects

2.11.1 Miyakawa’s X q Spaces The asymptotic profile in Theorem 8 describes the long-time behavior of u in the Lq norm, 1  q  1. In [132], T. Miyakawa addressed the problem of establishing the analogue result for q < 1. As the Lq spaces are no longer suitable for this purpose, he introduced the spaces X q : 8 q ˆ 1  q  1; ˆ 0, such that

lim inf

t!1

k.E.t/  E.t/ Q /u.t /k2 Q 1 :  ; where .t / D ˇt 1 ; and .t Q / D ˇt ku.t /k2 (37)

Further if ˛  0, then ˛

lim inf t 2 ku.t /kBP 2 C˛;1 > 0: t!1

2

Here, fE W   0g denotes the resolution of the identity of the Stokes operator. The proof of Theorem 12 is done in several short lemmas and propositions that use properties of Besov spaces and the spectral family E.t/ . The decay result of Theorem 12 is first shown for weak solutions that satisfy 0 < lim inf t  ku.t /k2  lim sup t  ku.t /k2 < 1: t!1

(38)

t!1

The second theorem of [158] shows that condition (38) is equivalent with u0 2 2 BP 2;1 . For solutions with datum as in Theorem above or satisfying (38), the work in [158] improves the results in [132] and of [133] as follows: 1. A lower bound of decay is obtained for Rany  2 .0; 54 . 2. If  D 5=4, the condition from [133]: .1 C jxj/ju0 .x/jdx < 1 is no longer required.  52 1 3. By the continuous embedding BP 1;1 ,! BP 2;1 , the lower bound obtained in Theorem 12 0 < lim inf ku.t /k t!1

5

2 BP 2;1

;

improves the lower bound in [132] given by 0 < lim inft!1 ku.t /kBP 1 : 1;1

Notice that (37) provides information on the long-time behavior of frequencies. In a series of papers (see [157] and the references therein), Z. Skalák gave new insight on the large time behavior of the solutions. He showed that a certain frequency concentration effect occurs for weak solutions with zero external forces,

608

L. Brandolese and M.E. Schonbek

satisfying the strong energy inequality (5). We first recall the notation given in [158], 1=2 2 let a D limt!C1 kAkukuk , and define 2 8 p p ˆ ˆ > 0; Ka; D BpaC .0/ if  a > 0; ˆ ˆ :Bp .0/ if a D 0

where Br .0/ denotes a ball centered at the origin with radius r. In [157] it is shown that frequencies tend to concentrate on an arbitrary small annulus (a > 0) or ball .a D 0/ as t ! 1. In particular it is established that for such weak solutions, which are nonzero for times larger than some T  0 R lim

t!C1

2.12

R3 nKa;

R

Ka;

jj4˛ jOu.; t /j2 d

jj4˛ jOu.; t /j2 d

D 0:

(39)

Decay Characterization to Dissipative Equations

It has been frequently remarked that the long-time behavior in L2 of solutions and derivatives to diffusive systems is influenced by the low frequencies of the initial data. This suggested the idea of finding the order of ƒs u0 ./ at the origin, by 1 comparing it to the powers f ./ D jjr , [11], where, as usual, ƒ D ./ 2 . For this comparison the notions of decay character and decay indicator were introduced by Bjorland and Schonbek in [11] and its use and applications extended by Niche and Schonbek in [136, 137]. The decay character and the Fourier splitting method yielded upper and lower bounds for decay of solutions and derivatives to appropriate dissipative nonlinear equations, incompressible and compressible. We recall the definitions of decay indicator and decay character . For details see [11] and [136].

b

Definition 1 ([11, 136]). Let u0 2 L2 .Rn /; B./ D f W jj  g; s  0: The decay indicator Prs .u0 /; r 2  n2 C s; 1 corresponding to ƒs u0 is defined by Prs .u0 / D lim 2rn !0

Z

jj2s jOu0 ./j2 d  B./

provided the limit exists. Definition 2 ([136]). For u0 2 L2 .Rn /, the decay character of ƒs u0 is defined by 8 

n s ˆ ˆ < the unique r 2  2 C s; 1 ; if 0 < Pr .u0 / c > 0 and 0 < ˛  1 and P ./ 2 O.n/. Let v be a solution to vt D Lv; v.x; 0/ D v0 :

(40)

With the notion of decay character in hand, the following result can be obtained : Theorem 13 ([11, 136]). Let v0 2 H s .Rn /; s  0, have decay character rs .v0 / D rs . Let v.t / be a solution to (40) with data v0 . Then there exist constants C1 ; C2 ; C ; Cm > 0, so that 1

n

rs 2 . n2 C s; 1/ ) C1 .1 C t / ˛ . 2 Crs Cs /  kv.t /k2HP s  C2 .1 C 1 n  t / ˛ . 2 Cr Cs / ;

1. If 2. If 3. If

rs D 1 ) kv.t /k2HP s  Cm .1 C t /m ; r0 D  n2 ) kv.t /k22  C .1 C t / ;



8m > 0; 8 > 0;

The above theorem follows combining energy inequalities with estimates resulting from the decay and an appropriate Fourier splitting. For details when s D 0, see [11]; when s > 0, see [136]. P s of solutions to (40), it is Remark 4. For s > 0, to determine the decay in H necessary first to establish how the decay character of ƒs u0 and that of u0 are related. Specifically if u0 2 H s .Rn /; s > 0, it can be shown that rs .u0 / D s C r  .u0 /, provided  n2  r  .u0 /  1. For a proof the reader is referred to [136]. The recent paper [22] explores further the relation between the behavior of vO 0 near  D 0 and the decay estimates from above and below. For example, under the same assumptions as before on L, with the additional condition that the symbol M./ is homogeneous of degree 2˛, it is proved therein that

610

L. Brandolese and M.E. Schonbek

Z 8 2 ˆ ˆ  jvO 0 ./j2 d > 0 lim inf < !0C Zjj 2 ˆ ˆ lim sup  jvO 0 ./j2 d < 1 :

 ˛ ” v0 2 AP 2 2;1 ; ” .1 C t /

jj

!0C



. kv.t /k2 . .1 C t / ˛ :

(41)

P 2 n Here AP 2 2;1 is a specific subset of the Besov space B2;1 .R / that can be defined through suitable size estimates on the dyadic blocks of frequency decompositions (see [22] for more details). The notation . means that the inequality holds up to a multiplicative constant independent on t. When an L2 function satisfies any of the above equivalent conditions (41), the exponent  2 .0; 1/ is uniquely determined. This exponent  D  .v0 / was coined the Besov character of v0 . It is related to (the improvement made in [22] of) Bjorland-Schonbek’s notion of “decay character,” by the relation  .v0 / D r  .v0 / C n2 .

2.12.2 Application to the Decay for Navier-Stokes The decay estimates for the linear system are used to obtain the decay rates for appropriate nonlinear diffusive systems. In [11] it was used for solutions to the NS equations with zero force. This allows, in particular, to restate the decay results Wiegner’s Theorem 5 in the following form: Theorem 14 (See [11]). Let u be as in Theorem 5 (with forcing term f  0). Assume that u0 2 L2 .Rn / has a decay character r  D r  .u0 /. Then, 1. If  n2 < r  < 1  n2 , then there are constants C1 ; C2 ; C3 > 0 so that C1 .1 C t /r

 n 2

 ku.t /k22  C2 .C3 C t /r

 n 2

:

n

2. If r   1  n2 , then ku.t /k22  C .1 C t / 2 ˇ where ˇ D min.r  ; 1/. 3. If r  D  n2 and n D 3; 4, then ku.t /k22 decays slower than any algebraic decay rate for large time. The first conclusion of the above theorem can be made more precise. In fact, necessary and sufficient conditions for the validity of upper–lower decay estimates can be prescribed. Theorem 15 (See [22]). Let u0 and u as in Theorem 5 (with forcing term f  0). Let 0 <  < 5=4. The three following properties are equivalent: (i) lim inf  !0

2

Z

2

jOu0 ./j d > 0 and lim sup  jj

!0

2

Z jj

jOu0 ./j2 d < 1;

11 Large Time Behavior of the Navier-Stokes Flow

611

(ii) u0 2 AP 2 2;1 , (iii) .1 C t / . ku.t /k2 . .1 C t / . The proof relies on the analysis of the linear system described in the previous subsection and on Z. Skalák’s version of the “inverse Wiegner’s theorem” [159]. Applications of the decay character for other diffusive equations will be revisited in Sect. 5.4.

2.13

Non-decaying Velocities

When the energy inequality Z 0

t

ku.t /k22 C 2

Z

t 0

kruk22 ds  ku0 k22

(42)

is not available, the tools described in Sect. 2.4 are no longer effective. The case of solutions with non-decaying velocities attracted considerable attention. For example, if one drops the L2 -condition on the initial datum (which implies some sort of decay at infinity), replacing it with the more general condition u0 2 L2uloc .R3 / (the space of uniformly locally-L2 vector fields), then it is still possible to construct weak solutions satisfying a local energy inequality, defined at least on some finite time interval: under mild additional condition, such solutions can be extended to global weak solutions (see P.G. Lemarié-Rieusset’s book [116] for a detailed account) but nothing is known on their large time behavior. Some results on the large time behavior of solution arising from L1 .R2 / initial data are briefly described here. In any space dimension n  2, if u0 2 L1 .Rn /, then it is possible to construct a unique local in time mild solution belonging to the space BUC of bounded and uniformly continuous functions, with the corresponding pressure satisfying the natural condition p D ./1 .r  .u ˝ u//:

(43)

In the two-dimensional case, such solution is in fact global in time. In this case, the main issue is to establish some a priori bound on the growth of the velocity for large times. The problem of its behavior at infinity was first addressed by Y. Giga, S. Matsui, and Y. Sawada [79], who established a double exponential bound for the growth of ku.t /k1 and Y. Sawada and Y. Taniuchi who improved this bound to a simple exponential-in-time growth; see [144]. Their proof relies on the maximum principle for the vorticity equation and splitting the frequencies into low and higher parts. More recently, using in a deeper way the structure of the Navier-Stokes equations, and the local energy inequality, S. Zelik (see [170], or Th. Gallay’s review [69]) proved that the L1 .R2 /-norm of u grows at most linearly in time:

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L. Brandolese and M.E. Schonbek

ku.t /k1  C ku0 k1 .1 C C t ku0 k21 /: In this bound, the constant explicitly depends on the viscosity coefficient (is set D 1 throughout for simplicity) and breaks down in the limit as  ! 0. However, when one also assumes that the initial vorticity is in L1 , then these bounds turn out to be independent on the viscosity coefficient.

2.14

The Role of External Forces

The results described in Sects. 2.2, 2.3, and 2.4 are valid with nonzero external forces provided they decay sufficiently fast in L2 . In [146], it is shown that if n

kf .t/k2  C .1 C t / 2 1 and f 2 L1 ..0; 1/; W 1;1 .Rn // then the energy decays at the same rate as for f D 0. In this case, the FS method includes an extra term that decays much faster and hence is negligent. A natural question is what happens in the case that the external forces are slowly decaying. The surmise is that the forcing terms will steer the decay to a slower rate. In [138], T. Ogawa, S.V Rajopadhye, and M.E. Schonbek studied the uniform and nonuniform decay for fluids with slowly decaying forcing terms. They looked for the weakest possible assumptions on f so that a weak solution u, constructed by the techniques in [26, 118], that complies with the strong energy inequality (5), and satisfies ku.t /k2 ! 0 as t ! 1, with or without rate. Technical assumptions on the force are needed, which include the assurance that f is in the dual of the space of the weak solution. The main assumption .A1/ is quite technical. Instead of giving the exact terms of .A1/, only some typical forces included under assumption are depicted here: 2n

1. f 2 L1 ..0; 1/; L2 / and f 2 L2 ..0; 1/; L nC2 / .n  3/. 2. If f belongs to appropriate weighted space, then a weaker condition such as jxjf 2 L2 ..0; 1/; L2 / will suffice. For details see [138]. For uniform and nonuniform decay, a generalization of the strong energy inequality (5) is needed: Proposition 1 (Existence and generalized energy inequality [138]). Let u0 2 L2 P 1 / and f satisfy assumption .A1/. There exists u 2 L1 ..0; T /; L2 / \ L2 ..0; T /; H for all T > 0, a weak solution of NS with datum u0 , satisfying (5). For any E.t/ 2 C 1 .RI RC / with E.t/  0 and 2 C 1 .RI C 1 \ L2 /, u satisfies the generalized energy inequality (GE):

11 Large Time Behavior of the Navier-Stokes Flow

E.t /k .t / Z

t

C2 s

Z

u.t /k22

 E.s/k .s/

ˇ E. /ˇh

0

u.s/k22

Z C s

t

E 0 . /k . / u. /k22 d

ˇ . / u. /; . / u. /i  kr . / u. /k22 ˇd

t

E. / .jhu  ru. /; . /

C2

613

. / u. /ij

s

C jhf . /; . /

. / u. /ij/ d

(44)

where 0  s < t < 1, for almost all s and all t. Nonuniform decay. The Fourier splitting method, combined with inequality (44), yields the desired decay. As in [128], for the nonuniform decay, one decomposes the solution into low- and high-frequency modes and considers them separately. 2 Choose D ./ D e jj : ku.t /k22 D k u.t /k22 C k.1  /u.t /k22 D L.t / C H.t / The decay of L.t / follows by technical Sobolev inequalities. For the decay of H.t /, a modified FS technique is used. Choose .t / D 1  . Split the frequency space into Rn D S.t / [ S.t /c ;

with S.t / D f W jj < G.t /g:

˛ . Then .E.t /; G/ satisfies E 0 .t /  Choose E.t / D .t C 1/˛ ; G 2 .t / D 2.tC1/ 2G 2 .t /E.t / D 0: Here ˛ depends only on the dimension. Note that the third term on the right-hand side of (44) vanishes since 0 D 0. Combining the second term with the diffuse fourth yields that it is only necessary to estimate the solution on a small ˛ ball of shrinking radius G.t / D p2.tC1/ . Assumption A1 allows the term involving the force to be estimated by a Hardy inequality. Uniform decay. For the uniform decay, more assumptions on the force are needed. Here one uses directly an FS technique combined with the generalized inequality (44). For details see [138]. For the influence of external forces on the point-wise decay, the reader may consult the results by Bae and Brandolese in [5].

3

Decay Results in More General Domains

The literature of this subject is huge; the presentation in this chapter will be restricted to a few of the many relevant results. H. Sohr’s book [160] is a classical reference for this topic. The reader is also referred to the work by R. Farwig, H. Kozono, and J. Sohr [56] for more recent developments about domains with non-

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compact boundaries. The contributions in this handbook by M. Geißert, M. Hieber, J. Saal ( “The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties”, this volume), and S. Monniaux and Z. Shen ( “Stokes Problems in Irregular Domains with Various Boundary Conditions”, this volume) give also upto-date results.

3.1

Decay in Bounded Domains

For bounded domains existence and stability have been understood at several levels. One advantage over unbounded domains is the validity of the Poincaré inequality. In two and three dimensions, it is known since Leray and Hopf’s work [117, 118] that the L2 -norm of the solution and its corresponding enstrophy decays at least exponentially as time tends to infinity. In [61] Foias and Saut show that the decay is exactly exponential, in the sense that the limit of the ratio of the enstrophy over the energy tends to an eigenvalue of the Stokes operator. This result serves as basis to obtain asymptotic expansion of the solution to NS as t ! 1. More information on the asymptotic behavior of solutions in bounded domains can be found in [52, 160]. An original approach is that of [134], where S. Neˇcasová and P. Rabier make a careful analysis of the relationship between the time decay of solutions of the Navier-Stokes system in a bounded domain and the time decay of the external force f . They show that the decay properties of the solution depend upon the type and amount of decay of f and upon the function spaces to which the data belong, but not on f or of the initial data themselves. The “type” (exponential, power-like, etc.) and the “amount” of the decay of f are defined in a rigorous way by considering the class of functions  D .t / such that, for some s > 0, e s f belongs to an Lp .RC ; Lq .// space. In the same way, the decay of the solution is measured by the validity of conditions of the form e s u 2 Lp .RC ; Lq .//. A consequence of the main result of [134] is that the solution will always inherit power-like decays of the external forces, but only part of their exponential decays, in a sense made precise in [134].

3.2

Decay in Exterior Domains

Exterior domain problems are used to investigate the behavior of fluids around an obstacle. One natural question to ask is what is the influence of the obstacle in the decay rates. Focus here is centered on the fundamental work of Borchers and Miyakawa [12–15]. Historically the analysis begins with the results by Heywood in [89, 90], where the question was to compare the behavior of the stationary and the nonstationary solutions to the NS equations. In [114] Ladyzhenskaya states that it seems impossible to prove the a priori estimates required to investigate for all t > 0 a nonstationary flow past an object which begins at t=0 as a perturbation of one of these stationary solution. An important element that allowed to overcome

11 Large Time Behavior of the Navier-Stokes Flow

615

the obstacles specified in [114] is the work by Finn [58–60] on stationary flows, in exterior domains. Briefly, in [90] the following stability result was established: Let u D u.x; t / be the solution to NS in an exterior domain  R3 outside of a bounded domain with smooth boundary †. Suppose the forcing term is only xdependent and small. Let w D w.x/ be the solution to the corresponding stationary problem. Suppose that u D w on @; t  0. In addition there are some appropriate somewhat involved conditions on u0 D u.x; 0/ and w; for details see [90]. If A is any bounded subset of , then lim krv.t /  rwkL2 ./ D 0; lim kv.t /  w.x/kL2 .A/ D 0:

t!1

t!1

This result was extended by Masuda [127] giving the first L2 -rate of decay result for nonstationary solutions to NS in an exterior domain . The domain  is as in Heywood’s papers [89, 90], and v; w satisfy the conditions (1) and (2) above. Then in [127] it is shown that 1

sup jv.x; t /  w.x/j  M t  8 ;

for t  T0 > 0; and some constant M:

x2

As Heywood points out in [88] It was recognized by Masuda that this order of decay might be improved through the use of sharper estimates for solutions of the Stokes problem. The decay results in [127] were ameliorated in [88]. The optimal rates with less conditions on the exterior region were finally obtained in a series of papers by W. Borchers and T. Miyakawa [12–14]. These results involve a deep analysis of properties of Ar , (the Stokes operator in Lr ) with fractional powers. In particular a fundamental Sobolev embedding theorem for fractional powers of the Stokes operators in exterior domains is obtained. The notation used is as follows: Au D Ar u D Pr u; u 2 D.Ar / D Lr ./r \ H01;r ./ \ H 2;r ./: Here  is a domain in Rn , Lr ./ are solenoidal vector functions in Lr and, Pr is the bounded projection from Lr ./ into Lr ./r . The exterior domain  under consideration is the complement of a finite number of bounded domains with C 1 boundaries. The forcing term is zero. The main theorems for solutions v.x; t /; x 2 ; t > 0 obtained in [12] are as follows: Theorem 16 ([12]). Let n  3. Let the initial data u0 2 L2 ./, and let v 0 .t / be the solution to the Stokes problem with data u0 and v 0 .t /j@ D 0; v 0 ! 0 as jxj ! 1. Then (i) There is a weak solution v of NS which satisfies kv.t /k2 ! 0 as t ! 1. (ii) If in addition kv 0 .t /k2 D O.t ˛ / as t ! 1, for some ˛ > 0, then kv.t /k2 D O.t ˇ / as t ! 1 with ˇ D minfa; n4  g, for some 2 .0; 14 /. n 1 (iii) kv.t /  v 0 .t /k2 D o.t  4 C 2 / / as t ! 1.

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L. Brandolese and M.E. Schonbek

(iv) If in addition kv 0 .t /k2 D O.t ˛ / as t ! 1, for some ˛ > 0, then kv.t /  v 0 .t /k2 D O.t  / ast ! 1,  D n4  12 C ˛, if ˛ < 12 and 0 <  < n4 if ˛  12 . If a weak solution satisfies the energy inequality (5) with f D 0, then v has all the previous properties. Main ideas for the proof : The spectral decomposition of the self-adjoint operator A2 given by Z

1

A2 D

dE 0

is used to obtain a lower bound for the L2 -norm of the gradient of the solution to NS. krvk22

D

1 2

kA2 vk22

Z

Z

1

D 0

kEz vk22

1

dz  

kEz vk22 dz  .kvk22  kE vk22 / (45)

This approach depicts Fourier splitting using spectral theory. The idea was already used in [96]. Estimate (45) combined with the energy inequality (5) yields kv.t /k22

Z C s

Z

t

. /kv. /k22

d 

kv.s/k22

t

C s

. /kE v. /k22 d :

(46)

Bounds for the Stokes operator with fractional powers combined with (46) yield an inequality for kE v.t /k2 where a Gronwall estimate can be applied. This provides the decay expected for hypothesis (i) and (ii). Cases (iii) and (iv) are more involved: in particular it is necessary to analyze w D v.t /  v 0 .t /. As stated in [12], if u0 2 Lr \L2 one can take ˛ D . nq  n2 /=2 and thus ˛ D ˇ. The final theorem in [12] gives decay in Lq , q < 2 for solutions with data in u0 2 Lr \L2 , r 2 .1; n=n1 and r < 2n . The decay obtained is kv.t /kq D o.t  /; as t ! 1, with  D . nq  nq /=2. nC2 Papers [13, 14] improve the results in [12]. The main difference in [13] is that less conditions are required on the data to get the decay. In [13] the comparison is with the solutions of the underlying heat equation. One of the main Theorems in [13] states: Theorem 17 ([13]). Let the initial data u0 2 Lr ./ \ L2 ./ for some 1 < r < 2; then there is a weak solution v of NS such that kv.t /k2 D O.t  /; as t ! 1, where  D 12 . nq  n2 /: The same holds for solutions which satisfy the energy inequality (5) (here f D 0).

11 Large Time Behavior of the Navier-Stokes Flow

617

The results in [14] involve arbitrary unbounded domains in dimensions n D 2; 3; 4 where the Poincaré inequality might not hold. The methods used are based on the ones in the former two papers. Remark 5. The last Theorem in [12] includes the main result of Maremonti [125]. Moreover, this theorem also ameliorates the work of Galdi and Maremonti [67] and Maremonti [126]. The work of Borchers–Miyakawa, [12–14], dealt with solutions to NS for dimensions n  3. The work of Kozono and Ogawa [106, 107] investigates questions related to decay in two dimensions. Specifically in [107] solutions to NS are studied for initial data in L2 ./. Uniqueness and existence of a global strong solution u D u.x; t / are first established. Decay rates obtained for u and du 2 Lp ; 2  p < 1 are optimal in the sense that they coincide with the rates of dt the solutions to the underlying linear equations. The L1 rates for u and ddtu differ from the optimal one by a logarithmic multiple. Rates of decay for the L2 -norms of A˛ u, 0 < ˛  1 and of A˛ . ddtu /, 0 < ˛ < 1 are also obtained. The results on decay in [107] follow by an analysis of the operator .A2 C /˛ ; 0 < ˛  12 . The main difficulty, that the authors have to handle, is that A2 in unbounded domains in R2 is not boundedly invertible. The work in [107] extends the one in [106]. In [107] for data with additional conditions, new rates are obtained for u; ddtu ; in Lp ; 2  p  1 and in L2 for A˛ u; A˛ . ddtu /. Furthermore, decay rates in Lp ; 1 < p < 2; are derived for solutions with data in Lp \ L2 ; 1 < p < 2. In [108] Kozono and Ogawa analyze in exterior domains in Rn ; n  3, the stability of the stationary solutions and first derivatives in Lp ./, with appropriate p depending on conditions on the data and on the stationary solutions. The decay in weighted spaces also attracted a considerable attention. For a nice review of recent results, the reader is referred to the introduction of the paper [86]. In this paper, C. He and T. Miyakawa address the problem of the time decay of weighted norms of weak and strong solutions to the NS equations in a 3D exterior domain. They prove that weak solutions satisfy, for all t  0: k jxj˛ u.t /k22 C

Z 0

t

k jxj˛ ru. /k22 d  c

.1 < ˛
1, every w0 2 L2 .R3 /, and for every T > 0 the perturbed problem has a weak solution in the usual energy space which satisfies the strong energy inequality kw.t /k22

Z

t

C 2.1  K.c// s

kr ˝ w. /k22 d 6 kw.s/k22

for almost all s > 0 including s D 0 and all t > s. Moreover, lim kw.t /k2 D 0:

t!1

622

L. Brandolese and M.E. Schonbek

Ideas of the proof. The energy inequality follows after multiplying the equation by w and integrating. Note that the convective term can be estimated by a Hardy-type inequality of the form ˇ ˇZ ˇZ ˇ ˇ ˇ ˇ .w  r/vc  wdx ˇ D ˇ ˇ ˇ ˇ 3 R

R3

ˇ ˇ P 1 .R3 /3 ; w  .w  r/vc dx ˇˇ 6 K.c/kr ˝wk22 ; 8 w 2 H

where K D K.c/ > 0; limjcj!1 K.c/ D C1, and limjcj!C1 K.c/ D 0. Hence, there exists c0 > 1 such that K.c/ < 1 for all c satisfying jcj > c0 > 1. The next step for the stability is the study of the linearized equation. For this consideration is given to the linearized operator L which satisfies   zt C Lz D zt  z C P .z  r/vc C P .vc  r/z D 0: It is shown that both operators L and L are infinitesimal generators of strongly continuous semigroups of linear operators on L2 .R3 / which are holomorphic in a sector fs 2 C W jArg sj < g for a certain D .c/ > 0, provided jcj is sufficiently large. The estimates on L allow to get the decay of ke Lt z0 k2 , for z0 2 L2 .R3 /. With this information in hand, the decay of kw.t /k2 follows now from an analysis that involves the integral form of the solution. t u The stability, under L2 -perturbations, of infinite energy solutions V .x; t / other than the Landau ones, is investigated in [98] by G. Karch, D. Pilarczyk, and M. Schonbek. One main assumption for the solutions V D V .x; t / under consideration in [98] is that the corresponding convective term satisfies a Hardytype inequality. More precisely it is required that – V D V .x; t / is a global-in-time solution to NS with an external force f D f .x; t/ and an initial datum V .x; 0/ D V0 .x/. – there exists a Banach space .X ; k  kX / such that this solution satisfies – V ˇ R 2 Cw .Œ0; 1; X /, ˇ – ˇ R3 .gr/hV .t/dx ˇ 6 K supt>0 kV .t/kX kr˝gk2 kr˝hk2 ; some K > 0; P 1 .R3 /; K supt>0 kV .t/kX < 1. – 8 g; h 2 H Remark 6. Spaces X that satisfy the Hardy-type inequalities as depicted above include the Lebesgue and weak Lebesgue spaces L3 .R3 / and L3;1 .R3 /. Other possible choices are X D ff 2 L1loc .R3 / W kf k D supx2X jxjjf .x/j < 1g, O < 1g. or X D PM2 .R3 / D fv 2 S 0 .R3 / W ess sup2R3 jj2 jv./j  

P 1 .R3 / . The following stability Let XT D Cw Œ0; T ; L2 .R3 / \ L2 Œ0; T ; H result is established in [98].

11 Large Time Behavior of the Navier-Stokes Flow

623

Theorem 19 ([98]). Let V D V .x; t/ be a solution to NS satisfying the conditions listed above, with data V0 2 X . Then – 8 w0 2 L2 .R3 /, there exists a global-in-time distributional u D u.x; y/ solution to NS system with data u0 D V0 C w0 and force f .x; t/. – The difference w D u.x; t /  V .x; t/ 2 XT ; 8 T > 0, and w satisfies Z  t

kr ˝ w. /k22 d 6 kw.s/k22 ; kw.t /k22 C 2 1  K sup kV .t/kX t>0

s

for almost all s > 0, including s D 0 and all t > s – ku.t /  V .t /k2 ! 0 as t ! 1. Comments regarding the proof. For the algebraic decay, the high and low frequencies of the L2 of norm of w are analyzed separately. For the high frequencies, the decay is obtained establishing a generalized energy inequality (similar to the one from Proposition 1) combined with a modified Fourier splitting. The low frequencies decay followed by technical energy estimates. The stable solutions V in [98] include the Landau solutions [97]. Other examples of asymptotically stable solutions V , under arbitrary large L2 -perturbations, are suitable stationary Navier-Stokes equations and Leray-type self-similar solutions. The results in [98] give only nonuniform stability in L2 . For algebraic rates of decay, additional conditions had to be added on the solutions V . This is the task done in M. Hieber and J. Saal ( “The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties”, this volume) by T. Hishida and M. Schonbek. To describe the results of M. Hieber and J. Saal ( “The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties”, this volume), we first define the following space and energy inequality: VD

V 2 L1 .RC I Ln;1 .Rn // \ Cw .Œ0; 1/ W Ln;1 .Rn /; div V D 0;

where Cw .Œ0; 1/I Ln;1 .Rn // consists of all weak*-continuous functions with values in Ln;1 .Rn /. Let w be the weak solutions of wt  w C .w  r/w C .w  r/V C .V  r/w C r D 0;

(51)

div w D 0; w.x; 0/ D w0 .x/; : which satisfies the energy inequality given by kw.t /k22 C 2

Z

t s

krwk22 d  kw.s/k22 C 2

Z

t

jhV ˝ w; rwijd s

for a.e. s  0, including s D 0, and all t  s:

(52)

624

L. Brandolese and M.E. Schonbek

With these definitions in hand, the following result of M. Hieber and J. Saal ( “The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties”, this volume) is recalled: Theorem 20. (M. Hieber and J. Saal ( “The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties”, this volume)). Let n=3,4. Let w0 2 L2 .Rn \ Lq .Rn /; q 2 Œ1; 2/, suppose V .x; t / 2 V satisfies kV k D supt kV .t /k3;1  ı with ı small enough. Let u be the solution to NS with data w0 C V , w D u  V be a weak solution to the perturbed problem given by (51) satisfying (52). Then 1. Let 1  q < 2 and w0 2 Lq \ L2 ., then every weak solution w.t / with (52) satisfies n 1

1

kw.t /k2  C .1 C t / 2 . q  2 / as t ! 1 R 2. Let w0 2 L1 \ L2 and jyjju0 .y/jdy < 1. Given " > 0 arbitrarily small, there is a constant ı1 D ı1 ."/ 2 .0; ı such that if kV k  ı1 , then every weak solution w.t / with (52) satisfies n

1

kw.t /k2  C .1 C t / 4  2 C" as t ! 1 The proof of the theorem combines the Fourier splitting method, with an analysis of large time behavior of solutions to the initial value problem for the linearized equations: @t u  u C V  ru C u  rV C rp D 0; div u D 0;

(53)

u.; s/ D f: This system is studied in Rn  .s; 1/, where s  0 is the given initial time; recall that V is time-dependent and the system is nonautonomous. The energy ku.t /k22 is split into time-dependent low- and high-frequency regions in the Fourier side. The decay rate is determined from the low-frequency part, and the argument is based on analyzing the integral equation in terms of the evolution operator T .t; s/ in L2 for all t  s  0. That is the equation given by Z u.t / D T .t; s/u.s/ 

t

T .t; /P .u  ru/. / d

(54)

s

which provides a unique solution to (53). Technical difficulties prevent the use of standard techniques, but Yamazaki’s idea of using real interpolation [169] is very useful in the study the operator T .t; s/.

11 Large Time Behavior of the Navier-Stokes Flow

625

4.2.2

Two Dimensional Case: On Flows with Radial Energy Decomposition The work by C. Bjorland and C. Niche [9] analyzes the behavior of infinite energy solutions u satisfying ut C u  ru C rp  u D u  rv  v  ru;

(55)

r  u D 0; u.0/ D u0 2 L2 .R2 / where v is an a priori given vector field which satisfies ( ˛ D 0; if 2 <  < 1  12  ˛2 C 1 ˛ ; where kr vkL .R2 /  C t ˛ D 0 or ˛ D 1 if  D 1:

(56)

System (55) can be obtained, for example, as the difference of two solutions to the Navier-Stokes equations: u D w  v. In [9] the the corresponding data for w is decomposed as w0 D u0 C v0 , where u0 2 L2 .R2 / has finite energy while v0 D v.; 0/, with v an a priori given solution of NS (possibly of infinite energy), that satisfies (56). If e t u0 decays in the L2 -norm at an algebraic rate, then it is claimed in [9] that the solution u D w  v of (55) remains in L2 and decays at the same algebraic rate (if this rate is not too large). This process would allow to obtain the asymptotic behavior of solutions with infinite energy. However, the proof main result of that paper seems to be built upon an incorrect Gronwall inequality [9, p.674]. This erratum was pointed to the authors of the present review paper by an anonymous referee. As a result, the main theorem of [9] does not seem to be proved successfully any longer. We therefore reformulate it as an interesting open problem: Let u be a global solution to (55), v satisfy (56), and assume: 1

(i) 8 t0 > 0; 9 Ct0 > 0; such that 8 T > t0 supt0 tT ku.t /k2  Cto .1 C T / 2 : (ii) For some  2 Œ0; 1; ke t u0 k22  C .1 C t / . Can one deduce from the above that 8 t  t0 ; ku.t /k2  CQ to .1 C t / ? Assumption (i) is satisfied in many natural situations. If positively answered, the above question would have several consequences, including an improved decay for 2 1;q solutions arising from data in w0 D v0 C u0 2 BP rr .R2 / or arising from data with radial energy decomposition. See [9] for a more detailed discussion. Remark 7. An anonymous referee pointed out to us that a partial solution of the open problem can be found in the paper by Y. Maekawa [124]. In [124] the vector field v.t / is assumed to be a Kozono-Yamazaki solution of the Navier-Stokes equations with data v0 2 L2;1 ./ and kv0 kL2;1 ./  1, where  D R2 or is an exterior domain. Such solutions satisfy 1

sup kv.t /kL2;1 ./ C sup t 4 kv.t /kL4 ./  1 t>0

t>0

626

L. Brandolese and M.E. Schonbek

The results obtained in [124] analyze the stability of the solutions. The main result obtained is the following: Theorem 21. There exists a constant ı > 0 such that for any u0 2 L2;1  ./ C kkL2;1

L2 ./

of the form kkL2;1

u0 D v0 C w0 ; kv0 kL2;1 ./  ı; w0 2 L2 ./

;

if u.t / is the solution to the NS equations with data u0 , then the perturbation w.t / D u.t /  v.t / satisfies lim kw.t /kL2;1 ./ D 0;

t!1

and moreover lim kw.t /kL2 ./ D 0:

t!1

5

Other Models

5.1

The MHD Equations

In plasma physics, the magnetohydrodynamics (MHD) equations describe the interactions between a magnetic field and a fluid made of moving electrically charged particles. In nondimensional form, these equations can be written in the following way: 8   1 S @u ˆ 2 ˆ u C .u  r/u  S .B  r/B C r p C jBj D ˆ ˆ ˆ @t 2 Re ˆ ˆ ˆ ˆ < @B 1 C .u  r/B  .B  r/u D B @t R m ˆ ˆ ˆ ˆ ˆ div u D div B D 0 ˆ ˆ ˆ ˆ : u.0/ D u0 and B.0/ D B0 :

(MHD)

Here the unknowns are the velocity field u of the fluid, the pressure p, and the magnetic field B, all defined in Rd .d  2/. The constants Re > 0 and Rm > 0 are, respectively, the Reynolds number and the magnetic Reynolds number; moreover, S D M 2 =.Re Rm /, where M is the Hartman number. From now on, unless otherwise stated, all these constants are assumed equal to 1. After rescaling u and B, it can be surmised that S D Re D 1. Many of the results established for the Navier-Stokes equations have their analogue counterpart for the MHD equations. For example, M.E. Schonbek,

11 Large Time Behavior of the Navier-Stokes Flow

627

T. Schonbek, and E. Suli [152] established that if u0 and B0 are in L2 .R3 / and satisfy suitable moment conditions, then weak solutions (at least those satisfying a strong energy inequality) satisfy c.1 C t /

nC2 4

 ku.t /k2 C kB.t /k2  C .1 C t /

nC2 4

;

where c  0,R but c is generically strictlyRpositive. In fact, c D 0 if and only if 1R 1R the matrices 0 .u ˝ u  B  B/ and 0 .u ˝ B  B ˝ u/ are multiple of the identity matrix. Nontrivial solutions inside this class with faster rates can be constructed in even dimension by adapting the classical construction of Beltrami flows (see [25, 152]) and in any dimension putting symmetry conditions, similar to those mentioned in Sect. 2.6. R. Agapito and M.E. Schonbek [2] addressed the limit case of infinite magnetic Reynolds number, in which case there is no diffusion in the second equation in (MHD). Under the conditional assumption that there exists a global strong solution .u; B/, with magnetic field B 2 L1 .R3  RC /, then, as time t goes to infinity, ku.t /k2 ! 0 and kB.t /k2 converges to a nonnegative constant. This result shows that the diffusion in the velocity is sufficient to prevent compensatory oscillations between the two energies of the velocity and the magnetic field. Moreover, in the presence of both fluid and magnetic diffusion, the energies of the velocity and the magnetic field decay to zero, but if the data are only assumed to be in L2 , there cannot be a uniform rate for the energy of the solutions. S. Weng [166] derived sharp bounds for moments and higher-order derivatives similar to (28) (both for u and B) using the approach of Kukavica and Torres, as mentioned in Sect. 2.9.2. His results hold true also for the Hall-MHD equation, extending earlier studies by D. Chae and M.E. Schonbek [42]. The Hall-MHD equations are obtained putting an additional r  ..r  B/  B/ term in the second equation in (MHD): this is the Hall term that takes into account the magnetic reconnection effects and is important when the magnetic shear is large.

5.2

The Boussinesq System

The Boussinesq system is a well-established model for studying the heat transfer in incompressible viscous flows. The Boussinesq approximation consists in neglecting density variations in the continuity equation and the heating process of viscous dissipation: the temperature variations are taken into consideration through an additional buoyancy forcing term acting on the fluid, leading to the following Cauchy problem: 8 @t  C u  r D  ˆ ˆ ˆ ˆ ˆ < @t u C u  ru C rp D u C ˇ e3 x 2 R3 ; t 2 RC (57) ˆ r  u D 0 ˆ ˆ ˆ ˆ : ujtD0 D u0 ;  jtD0 D 0 :

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Here uW R3  RC ! R3 is the velocity field. The scalar fields pW R3  RC ! R and  W R3  RC ! R denote, respectively, the pressure and the temperature of the fluid. Moreover, e3 D .0; 0; 1/, and ˇ 2 R is a physical constant related to the gravity. In what follows for simplicity, it is assumed that all these constants are equal to 1. To construct a global weak solution, it is only necessary to assume that u0 2 L2 .R3 / and, e.g., that 0 2 L2 .R3 /. See, e.g., R. Danchin and M. Paicu’s paper [51], where such construction is performed in the more general setting   0. Such solutions then satisfy the energy inequality Z t  ku.t /k22 C 2 (58) kru.s/k2 ds  C ku0 k C t 2 k0 k2 0

for all t  0 and some absolute constant C > 0. The above upper bound can be 1 2 3 /. improved under the slightly more stringent but natural p condition 0 2 L \ L .R 2 2 In this case one easily gets the bound ku.t /k2 D O. t / as t ! 1, and k.t /k2 D O.t 3=2 /. The latter bound looks optimal, as it agrees with that of the solution to linear heat equation e t 0 , but the bound of the velocity is physically strange, as one would not a priori expect that the energy can grow arbitrarily large. However, L. Brandolese and M.E. Schonbek [24] proved that for strong solutions p arising from small and well-localizedR data, as the matter of fact, ku.t /k22  t as Rt ! 1, provided that the mean 0 of the initial temperature is nonzero. When 0 D 0, the energy goes to zero, in the large time, accordingly to what happens in the NavierStokes case   0. The method of [24] consists in obtaining point-wise asymptotic profiles for the p velocity in the parabolic region jxj >> t. Namely, if ess sup jxj3 j0 .x/j < ;

k0 k1 < ;

x2R3

ess sup jxj ju0 .x/j <

(59)

x2R3

and > 0 is small enough, then one can construct a unique global solution to (57) satisfying suitable point-wise estimate. If in addition ju0 .x/j  C .1 C jxj/a and j0 .x/j  C .1 C jxj/b for some a > 3=2 and b > 3, then u.x; t / D e t u0 .x/ C

Z



 0 t rEx3 .x/ C R.x; t /

(60)

p where R.x; t / is a lower-order term with respect to t rEx3 .x/ for jxj >> t , R.x;t/ D 0. Such point-wise asymptotic profiles can be used to namely, lim p jxj !1 tjxj3 t

deduce upper and lower bound estimates for weighted Lp -norms: 1

3

1

3

c.1 C t / 2 .rC p 1/  k.1 C jxj/r u.t /kp  C .1 C t / 2 .rC p 1/ ;

(61)

for all r  0, 1 < p < 1 and r C p3 < 3. The constant c in the above lower bound R is strictly positive if and only if the mean of 0 6D 0. p The case r D 0 and p D 0 agrees with the energy growth phenomenon ku.t /k22  t mentioned before.

11 Large Time Behavior of the Navier-Stokes Flow

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It is interesting to observe that this result is specific to the whole space. Indeed, when studying the system (57) in the half-space R3C (complementing it with the boundary conditions u D  D 0 on @R3C  .0; 1/, then the energy growth estimate (61) is no longer true. In fact, P. Han and M.E. Schonbek [83] showed that ku.t /kL2 .R3 / ! 0 even if no cancellation of the temperature holds. C The only conditions to achieve this decay to zero of the energy of a weak solution are suitable size and decay conditions of the initial data, namely, u0 2 R L2 .R3C /, 0 2 L1 \ L2 .R3C /, R3 x3 j0 j < 1, and k0 k1 small enough. In the C exterior domain, to prevent energy growth, P. Han [82] put the decay assumption 1 ke t 0 k1  C .1 C t / 2 that however is not generic and requires cancellations. Under additional assumptions, decay of higher-order derivative can be obtained: see [83, 165]. One can view the energy growth result included estimates (61) as a physical limitation of the model (57) when studying heat transfers in unbounded domains for long time intervals. The correction proposed in [57] consists in replacing the constant vector ˇe3 with a term rG.x/ where G is a harmonic function behaving like 1=jxj as jxj ! 1, accordingly to Newton’s gravitational law. More precisely in the model used in [57], the self-gravitation of the fluid is neglected; hence, the origin of the gravitational force must be an object placed outside the fluid domain , and hence Z G.x/ D R3

1 m.y/ dy; with m  0; suppŒm R3 n ; jx  yj

where m denotes the mass density of the object acting on the fluid by means of gravitation. It follows that the forcing term G is a harmonic function in , and as stated above G.x/ 1=jxj as jxj ! 1. Motivated by the previous observations, this Boussinesq system (usually referred to as the Oberbeck-Boussinesq system) is considered in a domain  D R3 n K, exterior to a compact set K. The correction introduced by the forcing term G prevents the energy to become arbitrarily large. Furthermore, the solution is considered with zero boundary condition on @K and initial conditions u0 2 L2 ;  2 L1 \ L1 . Then u.t / ! 0 2 L2 and  ! 0 2 Lp ; 1 < p  1.

5.3

Nematic Liquid Crystals

The flow of nematic liquid crystals can be treated as slow-moving particles where the fluid velocity and the alignment of the particles influence each other. The hydrodynamic theory of liquid crystals was established by Ericksen [54, 55] and Leslie [119, 120] in the 1960s. As F.M. Leslie points out in his 1968 paper: liquid crystals are states of matter which are capable of flow, and in which the molecular arrangements give rise to a preferred direction. The discussion is first centered on

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the asymptotics of two simplified models defined in   .0; T /, where  is either a bounded domain of Rn or  D Rn , with n D 2 or n D 3. The models are given by ut C u  ru C r D 4u  r  .rd ˝ rd /; dt C u  rd D 4d  Hi .d /;

(62)

r  u D 0: Here, D .x; t / and u D u.x; t / are the fluid pressure and velocity, and d D d .x; t / is the director field representing the alignment of the molecules. The constant  > 0 is viscosity coefficient. The terms Hi , i D 1; 2 will determine which system is used. They are chosen to be H1 D f .d / D rF .d / D

1 .jd j2  1/d; or H2 D rd jrd j2 2

Depending on the choice of Hi ; i D 1; 2, the systems will be referred as (62) by LCD1 or LCD2, respectively. For LCD1 the forcing term rd ˝ rd is the stress tensor of the energy about the director field d , with energy of the form 1 2

Z

jrd j2 dx C

R3

Z F .d /dx; where F .d / D R3

1 .jd j2  1/2 ; 42

and  a constant. Remark 8. The F .d / is the penalty term of the Ginzburg-Landau approximation of the original free energy for the director field with unit length. When  ! 0, i.e., the penalty term disappears and the term rd jrd j2 should be recovered. There is a vast literature on the general behavior of liquid crystal systems. For background the reader is referred to [49, 91] and the references within. The asymptotic behavior of regular solutions to the flow of nematic liquid crystals was studied for bounded domains in [121, 168]. In [168] Hao Wu shows that, with suitable initial conditions, the velocity converges to zero and the direction field converges to the steady solution of (

d1 C f .d1 / D 0; x 2  d1 .x/ D d0 .x/; x 2 @:

(63)

In [168] a Łojasiewicz-Simon [156]-type inequality is used to derive the convergence when  is a bounded domain, with data satisfying u0 2 H01 ./; r  u0 D 0; d0 2 H 2 ./. The following additional conditions are stipulated: – If  R2 suppose limt!1 .ku.t /kH 1 C kd .t /  d1 kH 2 / D 0: – If  R3 suppose the viscosity constant is sufficiently large.

11 Large Time Behavior of the Navier-Stokes Flow

631

Then the unique global solution satisfies 

ku.t /kH 1 C kd .t /  d1 kH 2  .1 C t / 12 ; 8 t  0: Here  2 .0; 12 / is the constant from the Łojasiewicz-Simon-type inequality. Lack of compactness considerations does not allow to use similar arguments in the whole space R3 , and other methods need to be used. In [49] Dai, Qing, and M. Schonbek establish a nonoptimal decay rate for regular solutions in R3 to LCD1, with small initial data. Moreover, the initial director field is required to tend to a constant unit vector w0 , as the space variable tends to infinity: lim d0 .x/ D w0 :

(64)

jxj!1

This behavior at infinity of the initial director field allows to obtain the stability without needing the Liapunov reduction and Łojasiewicz-Simon inequality. Note that w0 is the simplest case of a steady solution to (63). The decay estimates in [49] were improved in [50] by Dai and M. Schonbek, where optimal rates in H m were obtained in the sense that they coincide with the rates of the underlying linear equations. The following initial conditions were supposed: u.x; 0/ D u0 .x/; r  u0 D 0;

d .x; 0/ D d0 .x/; jd0 .x/j D 1;

u0 2 H m .R3 / \ L1 .R3 /; d0  w0 2 H mC1 .R3 / \ L1 .R3 / for any integer m  1 with a fixed vector w0 2 S 2 , that is, jw0 j D 1. Theorem 22 ([50]). Assume the initial data .u0 ; d0 / satisfies the conditions above. Let .u; ; d / be the regular solution. There exist a small number 0 > 0 such that if ku0 k2H 1 .R3 / C kd0  w0 k2H 2 .R3 /  0 ; then for all m  0 and 2  p  1, 3

1

m

kD m .d .; t /  w0 /kp C kD m u.; t /kp  Cm .1 C t /. 2 .1 p /C 2 / ;

(65)

The constant Cm depends on initial data, , , and m. The proof of Theorem 22 first established decay for the director vector and then uses this decay to show that the forcing term “r  .rd ˝ rd /” decays sufficiently fast to be able to implement a modified Fourier splitting argument. The decay for solutions to the system LCD2, the system without the GinzburgLandau penalty term, was obtained by Shengquan Liu and Xinying Xu in [123]. Specifically the results in [123] deal with the decay of smooth solutions with data u0 satisfying

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u0 2 H m ; d0 2 H mC1 ; m  3; and ku0 k2 C krd k2  ; div u0 D 0: If m  3, solutions with data satisfying the last conditions are shown to decay at the rate: kr l uk22 C kr l rd k22  C .t C 1/

3C2l 2

; l D 1; 2

These rates are optimal for the velocity, but not for the director vector. The work in [123] was improved by Gao et al. [74], where for m  1 and under slightly stronger hypothesis, optimal decay rates were obtained for both the velocity and the director vector. Specifically in [74] the data is supposed to satisfy .u0 ; d0  w0 / 2 H m  H mC1 ; m  1; div u0 D 0 and ku0 kH 1 C kr.d  w0 /kH 2  , where w0 is a fixed constant unit vector. The methods used in both papers are similar; both used Fourier splitting and involved energy estimates. In particular clever energy estimates are introduced to handle the nonlinear term jrd j2 rd . For general information on the system LCD2 for bounded domains, the reader is referred to M. Hieber et al. [91]. As stated by the authors [91]: by means of the associated energy functional, [they] prove convergence of a solution to an equilibrium, whenever the solution is eventually bounded in the natural state space. Q-tensor model. A frequently used hydrodynamic model of nematic liquid crystals has the local configuration of the crystals represented by a Q-tensor D Q.t; x/ 2 R33 sym;0 , a symmetric traceless matrix, and the motion described through an Eulerian velocity field u D u.t; x/. Here .t; x/ 2 RC  R3 , and the system has the form @t Q C div .Qu/  S.ru; Q/ D Q  LŒ@F .Q/;

(66)

@t u C div .u ˝ u/ C rp D u C div †.Q/ div u D 0: The operator L denotes the projection onto the space of traceless matrices, and F is a special potential function. The tensors S and † both depend nonlinearly on the tensor Q. In [48], Dai, Feireisl, Rocca, Schimperna, and M. Schonbek investigate the existence and decay of solutions to (66) for data satisfying u0 2 L1 \ L2 .R3 I R3 /; div u0 3 D 0; Q0 2 L1 \ W 1;2 .R3 I R33 sym;0 /; jQ0 .x/j  r2 for a.e. x 2 R ;

For such solutions, it is shown under appropriate conditions on the potential function F that ku.t; /kL2 .R3 IR3 / C kQ.t; /kW 1;2 .R3 IR33 /  c.1 C t /˛ where ˛ D

15 . 28

This decay rate can be improved to

3 4

under the additional condition

11 Large Time Behavior of the Navier-Stokes Flow

633

F .Q/  jQj2 in Br1 ; some  > 0; and Br1 D a ball of radius r1 > 0: The decay rates in [48] are obtained by some quite technical modifications of the Fourier splitting technique. For more information on related issues of nematic liquid crystals, the reader is referred to [92] and the contributions in this handbook by M. Hieber and J. Saal ( “Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid Crystal Flows”, this volume).

5.4

Quasi-Geostrophic System

5.4.1 Decay in the Whole Plane Case Besides its intrinsic mathematical interest, the dissipative 2D quasi-geostrophic equation describes phenomena arising in meteorology and oceanography. The model can be derived from the general quasi-geostrophic equations by assuming constant potential vorticity and constant buoyancy frequency. See Constantin et al. [45] and Pedlosky [141]. The equations are given by t C .u  r/ C ./˛  D 0 .x; 0/ D 0 .x/

(67)

here ˛ 2 .0; 1; t > 0, and x 2 , with  D R2 , or  is an exterior domain of N with O an open connected bounded set in R2 . The variable the form  D R2 n O,  D .x; t / is a real scalar function, standing for the temperature of the fluid, and u is the velocity field, determined by the scalar function (the stream function) through u D .u1 ; u2 / D .

@ @ ; /; @x2 @x1

ƒ

D  ;

1

where ƒ D ./ 2

Decay in the whole space R2 . Constantin and Wu [46] showed that, for datum 0 in L2 \ L1 , the corresponding solution of (67) satisfies 1

k.t /k2  C .1 C t / 2˛ ;

t  0;

(68)

where C depends only on the L2 and L1 of the data. Their proof relies on a modified version of the Fourier splitting method developed in [153, 154]. The second main tool used was the retarded mollifiers method of Caffarelli, Kohn, and Nirenberg [26]. The authors of [46] proved that for generic initial data, the decay rate (68) is optimal. Remark 9. The results in [46] also work with  D T2 , a two-dimensional torus.

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Remark 10. The kind of modified FS used in [46] was also employed by Linghai Zhang in [171] to get optimal decay in two dimensions for solutions to the NavierStokes. The results in [46] were extended by M. Schonbek and T. Schonbek in [154] to obtain the decay of the derivatives: supposing that 0 2 L1 .R2 / \ H m .R2 /, then kƒˇ .t /k2  C .t C 1/

ˇC1 2˛

;

where the constant depends only on the L1 and H m norms of the data. O

0 ./ 1 2 Furthermore, if the Riesz potential of the data Iˇ 0 D jj ˇ 2 L .R /, it is shown p 2 in [154] that the L .R / of the solutions decay at a rate that depends on the order of the derivatives, ˛ and ˇ. Moreover, for data with Riesz potentials in L1 .R2 /, it was shown that there is a slightly better L2 decay rate toward A@ ƒˇ G˛ .t /, where R 2˛ A D R2 .Iˇ 0 /.x/ dx and G˛ .t /0 D e tƒ 0 . Using rather general point-wise estimates for the fractional derivative ƒ˛  and a positivity lemma, Córdoba and Córdoba [47] gave a proof of a maximum principle and used it to establish decay of solutions when 0 is in L1 \ Lp ; 1 < p < 1. More specifically, they showed that

k.t /kp  C1 .1 C C2 t/

p1 ˛p

;

t  0;

(69)

where C1 and C2 are explicit constants. Working along the same lines, Ju [95] obtained an improved maximum principle, for 0 in L2 \ Lp , with p  2 and a constant C ¤ 1, of the form 1

k.t /kp  k0 kp

C p2 1C t p2

! 2p 2p˛ :

When p > 2, this last inequality also gives decay. Note that for p D 2, i.e., 0 in L2 , this expression reduces to k.t /k2  k0 k2 . In [153], M. Schonbek and T. Schonbek established rates of decay that are for moments of the solutions to the QG equations and also obtain lower bounds on decay rates of the solutions. For data in Lp \ L2 ; p 2 Œ1; 2/ algebraic decay can be found in [135] by Niche and Schonbek. Using ideas from Kato algebraic decay for 2 large in Lq norms with large q is obtained in [135]. That is, provided 0 2 L 2˛1 , 2 then the solutions and first derivatives decay in Lq for q 2 Œ 2˛1 ; 1/. In the work of Carrillo and Ferreira [37–39], the asymptotics of the solutions to the 2DQG equations are carefully analyzed. In [39] it is shown that the solutions converge to an Oseen vortex-like solution in L1 .R2 /, for initial datum with appropriate decay at infinity. In [38] stability is studied in Lp and weak Lp spaces. The third paper [37] gives a fairly complete analysis of the asymptotic behavior in

11 Large Time Behavior of the Navier-Stokes Flow

635 2

certain Lp spaces. It is shown that solutions with datum in L1 .R2 / \ L 2˛1 .R2 / converge as time goes to infinity to a particular self-similar solution normalized by the mass. The convergence is algebraic in Lq ; 1  q  1 and it is for derivatives of all orders. Moreover, it is also shown that if the data is in L2=.2˛1/ .R2 /, then 2 ; 1. The decay is the solutions and derivatives tend to be zero in Lq , for q 2 Œ 2˛1 2 2 nonuniform in L and is algebraic for all the L -norms of the derivatives. As stated in Sect. 2.12, the decay character technique [136] can be applied for the solutions and derivatives of (67). Here only the results are stated and the reader is referred to [136] for more details. Theorem 23 ([136]). Let 0 2 L2 .R2 /, ˛ 2 .0; 1 with decay character r  D r  .0 /; then there exist constants Ci > 0; i D 1;    ; 5 so that 1

1





(i) – If r   1  ˛ ) C1 .1 C t / ˛ .1Cr /  k.t /k22  C2 .1 C t / ˛ .1Cr / : 1  – If r  2 .1  ˛/Œ1; 2; r   1 ) C3 .1 C t / ˛ .1Cr /  k.t /k22  C4 .1 C 1 t / ˛ .2˛/ : 1 – If r  > 1; r   2.1  ˛/ ) k.t /k22  C5 .1 C t / ˛ .2˛/ : (ii) If 0 2 H s .R2 /, 12 < ˛  1. For rs D rs .0 / there exist constants C such that 1  – If rs  1  ˛ ) k.t /k2HP s  C .t C 1/ ˛ .sC1Cr / I 1

– If rs  1  ˛ ) k.t /k2HP s  C .t C 1/ ˛ .sC2˛/ :

5.4.2 Decay in Exterior Domains Decay in exterior domains for the 2DQG equations is studied in the papers by T. Schonbek and L. Kosloff [104, 105]. The exterior domains considered are as follows:  D Rn n O where O is a nonempty open, bounded, and connected subset of Rn . For the particular case of the 2DQG equations, “n” is taken to be 2. In [104] the authors first generalize the Fourier transform introduced by Ramm [143] in three dimensions to n dimensions with n  2. This allows to obtain spectral representations of the Laplacian and fractional Laplacian in exterior domains and, moreover, opens new ways to investigate questions in such domains. In particular it is used in two dimensions to study the decay of solutions to the 2DQG equations. In [104] the QG Eqs. (67) are considered in the critical case when ˛ D 12 for solutions  2 C .Œ0; 1/; H 2 .// with data in L1 ./ \ L2 ./. In [105], the decay was extended to weak solutions with ˛ 2 .0; 1. The rates obtained were optimal, as they coincide with the rates of solutions to the underlying linear equations: 1

k.t /k22  C .t C 1/ ˛ ; with C depending only on k0 kL1 \L2 . In summary the work in [104, 105] combines three techniques: – A Fourier transform which can be used for exterior domains and gives a spectral representation for the Laplacian and the fractional Laplacian.

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– A localization of the operator ƒ, [27, 161]. – An interesting and quite involved modification to the Fourier splitting technique, using the spectral decomposition via the Fourier transform for exterior domains.

5.5

Incompressible Inhomogeneous NS

The motion of incompressible flows of mixing fluids with different densities, or of fluid flows with melted substances inside it, can be modeled by the following system: 8 ˆ @t  C r  .u/ D 0 ˆ ˆ ˆ 0, i.e., that there is no vacuum. It is then possible to prove local or global well-posedness results, putting some smalless assumption on u0 and a0 D ajtD0 . The early results in this direction by A.V. Kažihov, O. Ladjženskaja, and Solonnikov have been considerably improved and extended in the more recent works by H. Abidi, R. Danchin, G. Gui, M. Paicu, P. Zhang, and others (see [1] and the references therein). In [1], Abidi, Gui, and Zhang addressed the case of flows with viscosity independent on the density, i.e., ./ D constant > 0. They proved that any a priori given global solution (in suitable Besov spaces) must decay as t ! 1: ku.t /kL2 and kru.t /kL2 decay at the same rate as the solution to the heat equation, at least when the latter has an algebraic decay not exceeding a critical rate. In the two-dimensional case, H. Huang and M. Paicu [94] got similar decay results, namely, 1 2

ku.t /k2L2  C .1 C t / 2 . p 1/ ;

1 2

kru.t /k2L2  C .1 C t /1 2 . p 1/C

for Lions’ weak solution arising from u0 2 Lp .R2 / \ H 1 .R2 /, 0  1 2 L2 \ L1 .R2 /, in the case of a nonconstant viscosity. The main technical restriction in [94] is that the viscosity coefficient must remain close to a positive constant. See also the very recent paper by J.-Y. Chemin and P. Zhang [41], for an original

11 Large Time Behavior of the Navier-Stokes Flow

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construction of a class of “large,” yet global smooth solutions to the system (70) and the corresponding decay results.

6

Conclusion

The work in this chapter tries to summarize the main progress on decay of solutions to diffusive equations starting from the years after Leray’s 1934 pioneering paper on solution to the Navier-Stokes equations. The results presented analyze the decay of solutions to the Navier-Stokes, MHD, QG, Boussinesq, and liquid crystal systems, in particular showing that under appropriate initial conditions, the solutions decay at optimal rates, in the sense that their rates are the same as for their underlying linear counterpart. It is noted that for most of these systems, if the solutions to the underlying linear equations decay to zero, so do the solutions to the corresponding nonlinear system, one exception being the solutions to the Boussinesq system in the whole space, in the case that the temperature has nonzero initial mass. It is interesting to note that this is not so in the case of the half-space, provided the data is small. A question of interest is then if for large data in the half-space the solution will still decay to zero. The models other than the Navier-Stokes, considered within the chapter, had zero external forces. If forces are added which decay in Lp or Sobolev spaces sufficiently rapidly, then the methods used for systems with zero forces can be easily extended. A natural query then is how do the solutions to the diffusive systems behave in the presence of slowly decaying forces. Finally, a very interesting problem is to analyze the decay of solutions to the MHD with no magnetic diffusion. As pointed out in the chapter, there are some results provided specific bounds are supposed on the solutions. In conclusion what is shown by the results in this chapter is that due to the diffusion, the nonlinearity has limited influence as time tends to infinity. The solutions tend to behave as their linear part when the decay rates of the latter are low.

7

Cross-References

 Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value

Problem  Existence and Stability of Viscous Vortices  Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes

Flow  Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid

Crystal Flows  Self-Similar Solutions to the Nonstationary Navier-Stokes Equations  Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions  Steady-State Navier-Stokes Flow Around a Moving Body  The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

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Acknowledgements The authors thank the anonymous reviewers for some very helpful suggestions and corrections which served to improve the presentation of this chapter.

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143. A. Ramm, Scattering by Obstacles. Mathematics and Its Applications, vol. 21 (D. Reidel Publishing Co., Dordrecht, 1986) 144. O. Sawada, Y. Taniuchi, A remark on L1 solutions to the 2-D Navier–Stokes equations. J. Math. Fluid Mech. 9(4), 533–542 (2007) 145. M. Schonbek, Decay of solutions to parabolic conservation laws. Commun. Partial Differ. Equ. 5(5), 449–473 (1980) 146. M. Schonbek, L2 decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 88(3), 209–222 (1985) 147. M. Schonbek, Large time behaviour of solutions to the Navier–Stokes equations. Commun. Partial Differ. Equ. 11(7), 733–763 (1986) 148. M. Schonbek, Lower bounds of rates of decay for solutions to the Navier–Stokes equations. J. Am. Math. Soc. 4(3), 423–449 (1991) 149. M. Schonbek, The Fourier splitting method, in Advances in Geometric Analysis and Continuum Mechanics (Proceedings of a conference held at Stanford University, Stanford, 1993) (International Press, Cambridge, MA, 1995), pp. 269–274 150. M. Schonbek, Large time behaviour of solutions to the Navier–Stokes equations in H m spaces. Commun. Partial Differ. Equ. 20(1–2), 103–117 (1995) 151. M. Schonbek, Tomas P. Schonbek, On the boundedness and decay of moments of solutions to the Navier–Stokes equations. Adv. Differ. Equ. 5(7–9), 861–898 (2000) 152. M. Schonbek, T. Schonbek, E. Süli, Large-time behaviour of solutions to the magnetohydrody-namics equations. Math. Ann. 304 (4), 717–756 (1996) 153. M. Schonbek, T. Schonbek, Moments and lower bounds in the far-field of solutions to quasigeostrophic flows. Discret. Contin. Dyn. Syst. 13(5), 1277–1304 (2005) 154. M. Schonbek, T. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows. SIAM J. Math. Anal. 35(2), 357–375 (2003) 155. M. Schonbek, M. Wiegner, On the decay of higher-order norms of the solutions of Navier– Stokes equations. Proc. R. Soc. Edinb. Sect. A 126(3), 677–685 (1996) 156. L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. of Math. (2) 118(3), 525–571 (1983) 157. Z. Skalák, The large-time energy concentration in solutions to the Navier–Stokes equations in the frequency space. J. Math. Anal. Appl. 400(2), 689–709 (2013) 158. Z. Skalák, A note on lower bounds of decay rates for solutions to the Navier–Stokes equations in the norms of Besov spaces. Nonlinear Anal. 97, 228–233 (2014) 159. Z. Skalák, On the characterization of the Navier–Stokes flows with the power-like energy decay. J. Math. Fluid Mech. 16(3), 431–446 (2014) 160. H. Sohr, The Navier–Stokes Equations. Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks] (Birkhäuser Verlag, Basel, 2001). An elementary functional analytic approach 161. P. Stinga, J. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35(11), 2092–2122 (2010). https://doi.org/10.1080/ 03605301003735680. MR2754080 (2012c:35456) 162. S. Takahashi, A weighted equation approach to decay rate estimates for the Navier–Stokes equations. Nonlinear Anal. Ser. A: Theory Methods 37(6), 751–789 (1999) 163. F. Vigneron, Spatial decay of the velocity field of an incompressible viscous fluid in Rd . Nonlinear Anal. 63 4, 525–549 (2005) 164. C. Wang, Exact solutions of the Navier–Stokes equations—the generalized Beltrami flows, review and extension. Acta Mech. 81, 69–74 (1990) 165. S. Weng, Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system. arXiv:1412.8267 (2014) 166. S. Weng, Space-time decay estimates for the incompressible viscous resistive Hall-MHD equations. arXiv:1412.8267 (2014) 167. M. Wiegner, Decay results for weak solutions of the Navier–Stokes equations on Rn . J. Lond. Math. Soc. (2) 35(2), 303–313 (1987)

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Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value Problem

12

Isabelle Gallagher

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Critical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Plan of the Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Initial Value Problem in Critical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d 2.1 Wellposedness in H˙ 2 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d p

1

2.2 Wellposedness in B˙ p;1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Largest Critical Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Special Role of the Nonlinear Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Case of Two Space Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Global Wellposedness in L2 .R2 / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Global Wellposedness in Critical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Measure-Valued Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Examples of Large Data in Critical Spaces Giving Rise to a Global Solution . . . . . . . . . 4.1 Geometrical Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Anisotropic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Slow Variations in One Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Large-Time Behavior of Global Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Behavior of the Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Behavior of the Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Behavior at Blow-Up Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Blow-Up of Scale-Invariant Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Qualitative Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

648 648 649 650 651 651 655 657 659 660 660 661 662 664 664 664 666 669 669 670 672 672 679 681 681 681

I. Gallagher () Department of Mathematics, Paris-Diderot University, Paris, France e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_12

647

648

I. Gallagher

Abstract

In this paper the homogeneous, incompressible Navier-Stokes equations are considered, and a number of results are reviewed which are related to the scaling of the equations. More specifically the initial value problem is studied in scale-invariant function spaces, insisting on the special role of the “largest” scaleinvariant function space; the specificity of two space dimensions is recalled, in terms of the velocity field and the vorticity. Some examples of arbitrarily large initial data giving rise to a global solution are also provided, as well as a study of the long-time behavior of global solutions and their behavior at blow-up time (supposing such a time exists).

1

Introduction

1.1

The Equations

The initial value problem for the homogeneous, incompressible Navier-Stokes system writes

.NS/

8 < @t u C u  ru  u D rp div u D 0 : ujtD0 D u0 ;

in RC  Rd

where p D p.t; x/ and u D .u1 ; : : : ; ud /.t; x/ are, respectively, the pressure and velocity of an incompressible, viscous fluid in d space dimensions. The viscosity is chosen equal to one to simplify the notation, and all exterior forcing terms are neglected. In the sequel the physical situations d D 2 or 3 are considered. The reason for considering the equations in the whole space only, and not with physical boundary conditions, has to do with the critical nature of the study, as explained further down. As is well-known, the Navier-Stokes system enjoys two important features. First it formally conserves the energy, in the sense that any smooth solution, decaying to zero as jxj goes to infinity, satisfies the following energy equality for all times t  0: 1 ku.t /k2L2 .Rd / C 2

Z

t 0

kru.t 0 /k2L2 .Rd / dt 0 D

1 ku0 k2L2 .Rd / : 2

(1)

The energy equality (1) can easily be obtained (formally) by noticing that thanks to the divergence-free condition, the nonlinear term is skew-symmetric in L2 : one has indeed if u and p are smooth enough and decaying at infinity, 

 u.t /  ru.t / C rp.t/ju.t / L2 .Rd / D 0:

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

649

Second, (NS) enjoys a scaling invariance property: defining the scaling and translation operators, for any positive real number  and any point x0 of Rd , by 1  t x  x0   ; ;  2 

ƒ;x0 .t; x/ WD

(2)

if u solves (NS) with data u0 , then ƒ;x0 u solves (NS) with data ƒ;x0 u0 . This is the main property this chapter focusses on: the aim is to analyze the wellposedness of (NS) in critical spaces, meaning spaces invariant under the operators ƒ;x0 (see the coming Sect. 1.2 for a definition).

1.2

Critical Spaces

A family X0 of distributions defined on Rd is critical if 8 > 0; 8x0 2 Rd ; u0 2 X0 ” ƒ;x0 u0 2 X0

with

ku0 kX0 D kƒ;x0 u0 kX0 :

Similarly a family .XT /T >0 of spaces of distributions defined over .0; T /  Rd is scale invariant if for all T > 0 one has, with notation (2), 8 > 0; 8x0 2 Rd ; u 2 XT ” ƒ;x0 u 2 X2 T

with kukXT D kƒ;x0 ukX2 T : (3) Some examples of critical spaces follow. Defining (homogeneous) Sobolev spaces by Z kf kHP s .Rd / WD

jfO ./j2 jj2s d 

 12

Rd

where fO is the Fourier transform of f fO ./ WD

Z

f .x/e ix dx;

Rd

P d2 1 .Rd / is critical. Similarly the (larger) Lebesgue it is easy to see that H space Ld .Rd / is critical and so are the (yet larger again if p > d ) Besov d

1

p spaces BP p;q .Rd /, defined as follows, using the Littlewood-Paley decomposition.

Definition 1. Let O be a radial function in D.R/ such that .t O / D 1 for jt j 6 1 and .t O / D 0 for jt j > 2. For j 2 Z, Fourier truncations are defined by

b

  Sj f ./ WD O 2j .jj/ fO ./

and

j WD Sj C1  Sj :

650

I. Gallagher

For all p; q in Œ1; 1 and s in R with s < d =p (or s 6 d =p if q D 1), the s is defined as the space of tempered distributions f homogeneous Besov space BP p;q such that     js kf kBP p;q < 1: s .Rd / WD 2 kj f kLp .Rd /  q `

In all other cases of indexes s, the Besov space is defined similarly, up to taking the quotient with polynomials. An equivalent norm is given for all s 2 R by  1s=2    @ e f kLp Lq .RC ; d  / ; kf kBP p;q s .Rd /  k 

and when s < 0   s=2   e f kLp Lq .RC ; d  / : kf kBP p;q s .Rd /  k

(4)



Note that in dimension 2, the energy space L2 .R2 / is critical. The equation is said to be critical. On the opposite in dimensions 3 and higher, the equation is supercritical. Let us now define a solution to (NS). Definition 2. A vector field u is said a (scaled) solution to (NS) associated with the data u0 if it is a solution in the sense of distributions, belonging to a family of scale-invariant spaces. Remark 1. The pressure is not mentioned in the definition of a solution. This is due to the fact that it may be recovered from the velocity field by solving the equation p D div .u  ru/; as can be seen by applying the divergence operator to (NS). Remark that since one is considering translations and changes of scale in the space variable, it is natural to restrict our attention to the case of the whole space Rd . The (NS) system makes sense of course (in fact physically more sense) when set in a domain with boundaries, but scaling is less relevant in that setting so the focus is on the whole space here.

1.3

Plan of the Paper

Section 2 is devoted to the presentation of two general methods of solving (NS) in critical spaces: the first one, in Sect. 2.1, by an energy-type estimate and the other, in Sect. 2.2, by a fixed-point argument. In this chapter those techniques are developed

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

651

P d2 1 and the in two different functional settings, namely, in the Sobolev space H d

1

p Besov space BP p;1 , respectively, which are both scale-invariant spaces. One can then ask whether it is possible to solve (NS) in other scale-invariant spaces by such techniques and if so in “how many” such spaces. The answer is provided in Sect. 2.3, 1 where it is shown that there is a largest possible critical space, namely, BP 1;1 . Unfortunately (NS) is not wellposed in that space; however there is a slightly smaller space in which (NS) is wellposed, which is BMO1 ; that space is presented also in that paragraph. In Sect. 3 the special case of two space dimensions is studied, where the equations are critical and shown to be globally wellposed. Two frameworks are studied in particular: on the one hand, the case when the velocity lies in the energy space or in larger scale-invariant spaces (Sects. 3.1 and 3.2) and, on the other hand, measure-valued vorticity (Sect. 3.3). Contrary to Sect. 2, in this part not only scaling properties are used, but also the energy conservation for the velocity, or the transport of vorticity. This is important as it can be shown (examples are provided in Sect. 2.4) that some equations with the same scale invariance as (NS) blow up in finite time. In Sect. 4 some examples are provided showing that smallness of critical norms is not strictly necessary in order to prove the existence and uniqueness of global solutions. Section 5 is devoted to the large-time behavior of global solutions. Finally Sect. 6 studies the behavior of solutions at possible blow-up time, supposing such a time exists. As this text is intended as a survey rather than a research article, very few rigorous proofs are to be found here. Some results will be stated with no proofs at all, often due to their technical nature, and others will be presented with a rough sketch of proof, whose goal is to give a flavor of the methods involved. However precise references to the literature are given all along the text for the interested reader.

2

The Initial Value Problem in Critical Spaces

In this section some classical methods of solving the initial value problem for (NS) in critical spaces are reviewed. The first results in that direction go back to the pioneering works of J. Leray [68, 69] which will be referred to constantly in the sequel. However more modern versions of those results are presented, in particular the Fujita-Kato theorem in Sect. 2.1, the Cannone-Meyer-Planchon theorem in Sect. 2.2, and the Koch-Tataru theorem in Sect. 2.3. Some remarks on the special role of the nonlinear term are provided in Sect. 2.4, in particular in the construction of weak solutions.

2.1

d Wellposedness in H˙ 2 1

In this paragraph the proof of the following important result is sketched, originally due to H. Fujita and T. Kato [31].

652

I. Gallagher

P d2 1 .Rd / is a Theorem 1 ([31]). There is a constant c > 0 such that if u0 2 H divergence-free vector field satisfying ku0 k P

H

d 1 2 .Rd /

6 c;

(5)

then there is only one solution associated with u0 , which satisfies for all t  0 ku.t /k2 P H

Z d 1 2 .Rd /

t

kru.t 0 /k2

C

P H

0

d 1 2 .Rd /

dt 0 6 ku0 k2

P H

d 1 2 .Rd /

:

Without the smallness assumption (5), existence holds at least for a short time, time P d2 .Rd //. at which the solution ceases to belong to L2 .Œ0; T I H d

P 2 1 .Rd / on the (NS) Sketch of proof. One can write a (formal) energy estimate in H s d P .R /, system. Denoting by . j /HP s .Rd / the scalar product in H 1 C ku.t /k2 d 1 P 2 .Rd / 2 H

Z

t

kru.t 0 /k2

d 1 2 .Rd /

P H

0

ˇZ t ˇ C ˇ .u  ruju/ P

H

0

1 ku0 k2 d 1 P 2 .Rd / 2 H ˇ ˇ .t 0 / dt 0 ˇ:

dt 0 

d 1 2 .Rd /

Note that the pressure has disappeared, thanks to the fact that u is divergence-free which implies that .u j rp/ P

d 1 2 .Rd /

H

D 0:

Then writing ˇ ˇ.ajb/ P

H

d 1 2 .Rd /

ˇ ˇ 6 C kak

P H

d 2 2 .Rd /

krbk P

H

d 1 2 .Rd /

one infers 1 ku.t /k2 d 1 C P 2 .Rd / 2 H

Z 0

t

kru.t 0 /k2

P H

Z

CC 0

t

d 1 2 .Rd /

0

ku  ru.t /k P

H

dt 0 6

d 2 2 .Rd /

1 ku0 k2 d 1 P 2 .Rd / 2 H 0

kru.t /k P

H

d 1 2 .Rd /

If d D 2 then one can use the fact that u is divergence-free to write ku  ru.t 0 /kHP 1 .R2 / D kdiv .u ˝ u/.t 0 /kHP 1 .R2 / 6 C ku ˝ u.t 0 /kL2 .R2 /

(6) 0

dt :

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

653

where C is a constant which may change from line to line. Then by Hölder’s followed by Gagliardo-Nirenberg’s inequality, one finds ku  ru.t 0 /kHP 1 .R2 / 6 C ku.t 0 /k2L4 .R2 / 6 C ku.t 0 /kL2 .R2 / kru.t 0 /kL2 .R2 / : Similarly if d D 3 then by Sobolev’s embeddings, one has ku  ru.t 0 /k P  12 H

.R3 /

6 C ku  ru.t 0 /k

3

L 2 .R3 /

and then by Hölder’s inequality and Sobolev’s embeddings, again one infers ku  ru.t 0 /k P  12 H

.R3 /

6 C ku.t 0 /kL3 .R3 / kru.t 0 /kL3 .R3 / 6 C ku.t 0 /k P

1

H 2 .R3 /

kru.t 0 /k P

1

H 2 .R3 /

:

So in both cases, one finds finally ku  ru.t 0 /k P

H

d 2 2 .Rd /

6 C ku.t 0 /k P

H

d 1 2 .Rd /

kru.t 0 /k P

H

d 1 2 .Rd /

:

(7)

Returning to (6) one has therefore 1 ku.t /k2 d 1 C P 2 .Rd / 2 H

Z

t 0

kru.t 0 /k2

P H

Z

t

CC 0

ku.t 0 /k P

d 1 2 .Rd /

H

d 1 2 .Rd /

One concludes (formally) that as long as ku.t /k P

H

ku.t /k P

H

d 1 2 .Rd /

dt 0 6

6 ku0 k P

H

1 ku0 k2 d 1 P 2 .Rd / 2 H

kru.t 0 /k2

P H

d 1 2 .Rd /

d 1 2 .Rd /

d 1 2 .Rd /

dt 0 :

6 2c 6 1=2C then

6 c;

so a continuity argument gives the result in the case of small enough initial data. In the case when the initial data is not small, then one writes u D uL C v

with uL WD e t u0

and the equation for v is solved: 8 < @t v C v  rv C uL  rv C v  ruL  v D rp  uL  ruL div v D 0 : vjtD0 D 0:

in RC  Rd

654

I. Gallagher

P d2 1 .Rd / energy estimate gives formally, exactly as above, Again an H 1 C kv.t /k2 d 1 P 2 .Rd / 2 H Z t 6C kv.t 0 /k P Z

t

Z CC

0

t

krv.t 0 /k2

P H

0 d 1 2 .Rd /

kv.t 0 /k P

d 1 2 .Rd /

H

0

t

H

0

CC

Z

krv.t 0 /k P

H

d 1 2 .Rd /

krv.t 0 /k2

dt 0

P H

d 1 2 .Rd /

krv.t 0 /k P

d 1 2 .Rd /

H

d 1 2 .Rd /

kuL .t 0 /k2

d 1 2

P H

dt 0 kruL .t 0 /k P

H

.Rd /

d 1 2 .Rd /

dt 0

dt 0 ;

where in the last line the inequality kuL  ruL .t 0 /k P

H

d 2 2 .Rd /

6 C kuL .t 0 /k2

P H

d 1 2

has been used, which can be proved by a similar argument to (7) above. Now consider the largest time interval on which kv.t 0 /k P d2 1 d 6 2c 6 1=4C . Then H

.R /

on that time interval Z

kv.t /k2 P H Z

d 1 2 .Rd /

t

6 Z

P H

t

C

P H

d 1 2 .Rd /

kuL .t 0 /k4

P H

0

krv.t 0 /k2

0

kv.t 0 /k2

0

t

C

d 1 2

d 1 2 .Rd /

kruL .t 0 /k2

P H

dt 0

d 1 2 .Rd /

dt 0

dt 0 ;

 and Gronwall’s inequality again noting that kruLk

6Cku0k P

d P 2 1 .Rd // L2 .RC IH



H

d 1 2 .Rd /

gives the result as long as Z

T

kuL .t 0 /k4

0

P H

d 1 2

dt 0 6 c

for c small enough. It is easy to see that kuL k

P L4 .RC IH

(8) d 1 2

.Rd //

6 C ku0 k P

H

d 1 2 .Rd /

so one recovers again the fact that small data gives a global solution, but if ku0 k P d2 1 d is not small, then requirement (8) holds if T is small enough. H

.R /

This concludes the proof of Theorem 1.



Remark 2. The size of T su that (8) holds is actually a complicated function of the initial data: typically one can compute T as follows. One splits the initial data u0 ] into a small, high-frequency part u0 and a smooth, low-frequency part u[0 by setting

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . . ]

u0 D u0 C u[0 ; ]

If N is large enough, then ku0 k P

H

Z

T 0

ku[L .t 0 /k4

P H

]

uO 0 ./ WD 1jjN uO 0 ./:

d 1 2 .Rd /

d 1 2

.Rd /

655

is smaller than 1=8C . Then one writes

dt 0 6 T ku[L k4

P L1 .RC IH

6 T N 2 ku0 k4

P H

d 1 2

.Rd //

d 1 2 .Rd /

which may be made smaller than 1=8C if T is chosen small enough (with respect to N and ku0 k P d2 1 d ). H

.R /

Remark 3. In two space dimensions, the conservation of energy (1) implies that for P 1 .R2 // norm any initial data, the solution is global and unique since the L2 ..0; T /I H remains under control for all T > 0. More on this is provided in Sect. 3.

d

2.2

1

p Wellposedness in B˙ p;1

In this section a different method to prove a result similar to Theorem 1 is presented, using a fixed-point approach. The idea is to write (NS) under the following form (where P WD Id  1 rdiv denotes the projector onto divergence-free vector fields)   u.t / D uL .t / C B u; u .t / with

t

uL .t / WD e u0

Z and

t

B.a; b/.t / WD 

e .tt

0 /

Pdiv.a ˝ b/.t 0 / dt 0

0

and to look for a scale-invariant Banach space XT of distributions defined on Œ0; T  Rd such that uL belongs to X1 and kB.a; a/kXT 6 C kak2XT

(9)

for some constant C (independent of T since XT is scale invariant). If one can find such a space, then by the Banach fixed-point theorem, as long as kuL kXT 6

1 4C

(10)

there is a unique solution u in XT of size less than 1=2C . P d2 1 .Rd /, This procedure can be carried out in the context of the Sobolev space H d d P 2 1 .Rd //\L2 ..0; T /I H P 2 .Rd //. One choosing, for instance, XT WD L1 ..0; T /I H can implement this idea with the following space: fix p > d and define

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n o 1 d p XT WD u 2 C ..0; T /I Lp .Rd // = sup t 2 .1 p / ku.t /kLp .Rd / < 1 :

(11)

t2.0;T /

p

It is an exercise to check that XT is scale invariant, in the sense of (3). The fact that (9) holds is a consequence of Young’s inequalities. Indeed it is not difficult to see that ke t f kLp .Rd / 6

C t

d 1 1 2 .qp/

kf kLq .Rd /

j˛jD1

C

6 t

sup k@˛ e t f kLp .Rd /

and

kf kLq .Rd / :

1 d 1 1 2C 2 .qp/

Then for any couple .r; p/ of real numbers such that 1=r 6 2=p, one gets Z kB.u; u/.t /kLr .Rd / 6 C 0

t

1 t 0/

.t 

1 d 2 1 2C 2 .pr /

kP.u.t 0 / ˝ u.t 0 //k

p

L 2 .Rd /

dt 0 :

  Since the Leray projector is continuous over Lq Rd for all 1 6 q < 1, Hölder’s inequality gives Z

t

kB.u; u/.t /kLr .Rd / 6 C

1 .t 

0

t 0/

1 d 2 1 2C 2 .pr /

ku.t 0 /k2Lp .Rd / dt 0 :

It follows that 1

d

1

d

t 2 .1 r / kB.u; u/.t /kLr .Rd / 6 C t 2 .1 r / kuk2Xt

Z

dt 0

t 0

.t  t 0 /

1 d 2 1 2C 2 .pr /

d

t 01 p

6 C kuk2Xt ; p

since d < p < C1, and inequality (9) follows. Noticing that by (4) the space X1 d p 1

is nothing but the Besov space BP p;1 , the smallness condition (10) for T D 1 is in d

1

p . The following result therefore fact a smallness condition on the initial data in BP p;1 holds.

Theorem 2 ([13, 85]). Let d < p < 1 be given. There is a constant c > 0 such d

1

p that if u0 is a divergence-free vector field in BP p;1 .Rd / satisfying

ku0 k

d 1

p BP p;1 .Rd /

6 c;

(12) p

then there is only one solution to (NS) associated with u0 , which belongs to XT defined in (11) for all times T > 0.

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

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For large initial data, the idea is to find T small enough so that (10) holds. Unfortunately, linked to the fact that the space of smooth functions is not dense d

1

p .Rd /, that requires adopting a slightly smaller space for the initial data, in BP p;1 d

1

p .Rd / with finite q. Then the result is the following: namely, BP p;q

Theorem 3 ([13, 85]). Let d < p < 1 and q < 1 be given. If u0 is a divergenced

1

p .Rd /, then there is a time T and a unique solution to (NS) free vector field in BP p;q p associated with u0 , which lies in XT defined in (11).

Note that Theorems 2 and 3 follow a series of works generalizing Theorem [31] to the Lebesgue space Ld .Rd /; the interested reader can consult [50, 57, 78, 103] for more details. For the case of domains, we refer, for instance, to [28] (with no exterior force) and [29] (with an exterior force).

2.3

The Largest Critical Space

2.3.1 Definition of the Largest Critical Space One may want to try to implement the fixed-point method introduced above in the largest possible space for the initial data. Indeed choosing a larger space means actually shrinking the size of the initial data, so the larger the space in which one measures the data, the more likely one is to obtain a global unique solution. For P d2 1 .Rd / norm instance, if an initial data presents oscillations, then this makes its H d p 1 large, whereas its BP p;q .Rd / norm is small (because it measures negative regularity since p > d ). Typically if ' is a function in the Schwartz class S.Rd /, then the function x  '" .x/ WD '.x/ cos " satisfies k'" k P

H

d

d 1 2 .Rd /

 "1 2  1 whereas k'" k

d 1 p BP p;q

d

 "1C p 1 if " 1

and d < p. One may then wonder whether there is any end to the chain of function spaces in which one can implement the fixed-point algorithm. The answer is provided in the following proposition, due to Y. Meyer [78]: Proposition 1 ([78]). Any critical Banach space of tempered distributions embeds 1 in the space BP 1;1 .Rd /. Proof. The proof of that result is actually quite straightforward: let X be a scaleinvariant Banach space of tempered distributions and let f belong to X . Then denoting the duality bracket between S 0 .Rd / and S.Rd / by h; i and the Gaussian

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G.x/ WD

1 .4 /

d 2

e

jxj2 4

one has jhf; Gij 6 C kf kX : It suffices to use the invariance of X by scale and translation to find directly the result.  1 It follows that the Calderón space BP 1;1 .Rd / is the largest critical space in which a fixed-point algorithm may be implemented for (NS).

2.3.2 Illposedness in the Largest Critical Space 1 It is worth noticing that Theorem 2 just falls short of solving (NS) in BP 1;1 for 1C d

small data, since the initial data must be in BP p;1 p .Rd / for finite p. Actually it was 1 . More precisely P. Germain [45] proves proved later that (NS) is ill-posed in BP 1;1 that the map which associates with the data a solution of (NS) cannot be of class C 2 1 from BP 1;q to S 0 as soon as q > 2. This result is obtained by proving that the first iterate of the Picard fixed-point scheme Z  2 F 1 b P./e jj

1

Z

et

0 .jj2 j j2 j j2 /

.Ou. /  .  /Ou.  // d dt 0



0 1 1  BP 1;q to S 0 . The idea to prove that result is to construct is not bounded from BP 1;q sequences of functions which violate the boundedness inequality, which are closely related to the counterexample by Montgomery-Smith [79] and the counterexample 0 by Stein [96] on operators not bounded in L2 although their symbols are in S1;1 . A more general result is obtained independently by J. Bourgain and N. Pavlovi´c in [7]: they prove that initial data in the Schwartz class S that are arbitrarily small 1 1 in BP 1;1 can produce solutions arbitrarily large in BP 1;1 after an arbitrarily short time. This “norm inflation” result relies again on the study of the first iterate and a special choice of initial data. Other related results have been obtained since: for instance, Yoneda [105] shows 1 the solution map is discontinuous in BP 1;q for any q > 2 and Wang [102] obtains a 1 P norm inflation result in B1;q for any q  1.

2.3.3 The Space BMO1 In view of the results presented in the previous paragraph if one wants to implement the fixed-point algorithm in a scale-invariant space, one needs to choose the initial 1 data in a space strictly contained in BP 1;1 . To this day the best result in that direction is due to H. Koch and D. Tataru [62]. They are able to solve (NS) globally for initial data small enough in BMO1 , where

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

659

1

ku0 kBMO1 .Rd / WD sup t 2 ke t u0 kL1 .Rd / t>0

1

C sup x2Rd R>0

Z

d

R2

.0;R2 /B.x;R/

j.e t u0 /.t; y/j2 dydt

 12 :

If the initial data is not small, then the solution may be constructed (and is unique) on a small time interval, provided the data belongs to the closure of the Schwartz space for the BMO1 norm. Remark 4. The results on the initial value problem presented in Sects. 2.1, 2.2, and 2.3 above provide a unique solution in some function space XT , specially tailored to the space in which the initial data lies. However thanks to the regularizing effect of the Laplacian operator, it can be proved that those solutions are actually smooth and in fact analytic or even Gevrey. One can refer, for instance, to [2,15,48] and the references therein for more on the subject.

2.4

The Special Role of the Nonlinear Term

2.4.1 On the Initial Value Problem It is interesting to notice that none of the wellposedness results presented so far in the context of critical spaces use the specific structure of the nonlinear term, except for the Fujita-Kato theorem in two space dimensions (see Remark 3). Indeed one may check easily by reading X the proofs that those results hold as soon as the nonlinear term is of the type Qij .D/.ui uj / with Qij .D/ a Fourier multiplier of order 16i;j 6d

one. However it is possible to construct operators Qij such that some smooth initial data produce a solution to the associate equation blowing up in finite time. This was first performed in a one-dimensional model by Montgomery-Smith in [79]. The model was extended to two and three space dimensions, with the divergence-free condition, in [39]. Unfortunately in the example of [39], it is impossible to maintain the conservation of energy; on the other hand, it is worthwhile to notice that the blowing-up initial data of [39] actually leads to a global solution for (NS), thanks to a result in [18] (described in Sect. 4 further down). Recently T. Tao [99] was able to construct an example of an equation for which both the divergence-free condition and the energy inequality 1 ku.t /k2L2 .Rd / C 2

Z 0

t

kru.t 0 /k2L2 .Rd / dt 0 6

1 ku0 k2L2 .Rd / : 2

(13)

hold and for which blow-up in finite time can also hold for an open set of smooth initial data.

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Note also that D. Li and Ya. Sinai in [70] prove the blow-up in finite time of solutions to the Navier-Stokes equations for complex initial data. If one’s goal is to solve the Navier-Stokes equations in critical spaces, it is therefore crucial to go beyond the fixed-point algorithm and to use in a deeper way the structure of the nonlinear term (and not only the energy conservation, due to [99]). Sections 3 and 4 provide some examples of results where the structure of the nonlinear term is used.

2.4.2 Weak Solutions Except in two space dimensions (see the next paragraph), weak (distributional) solutions to (NS) are not constructed for initial data in critical spaces: they involve indeed the energy bound (1), which corresponds to initial data in L2 .Rd /. This is the reason why such solutions are not discussed in this survey. However it is interesting to know whether both theories coincide if the initial data lies both in L2 .Rd / and in a critical space such as described in this paragraph. Let us first recall the theory of weak solutions, which goes back to J. Leray [68]. Theorem 4 ([68]). Associated with any divergence-free vector field in L2 .R3 /, P 1 .R3 //, which there is a global in time solution in L1 .RC I L2 .R3 // \ L2 .RC I H satisfies the energy inequality (13). The question of weak-strong uniqueness is the following: assume the initial data lies in both in L2 .Rd / and in a critical space, then do all weak solutions coincide with the unique solution obtained by fixed point (for instance), as long as the latter exists? W. von Wahl proves the result for the critical space L3 .Rd / in [101] (see also [30, 63]). Generalizations may be found in [16, 40, 47].

3

The Case of Two Space Dimensions

This section gathers a number of results concerning the global wellposedness of the Navier-Stokes equations in two space dimensions. Section 3.1 recalls the wellknown Leray theorem in L2 , and Sects. 3.2 and 3.3 consist in extensions of that result to more general critical spaces.

3.1

Global Wellposedness in L2 .R2 /

As noted in Remark 3, the blow-up criterion of the Fujita-Kato theorem (Theorem 1 above) in dimension 2 joint with the energy equality (1) – which can be proved to be valid in this setting of strong solutions – implies that all solutions associated with an L2 initial data are in fact global. Theorem 5 ([69]). Associated with any divergence-free vector field in L2 .R2 /, P 1 .R2 //. there is a unique, global-in-time solution in L1 .RC I L2 .R2 // \ L2 .RC I H

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

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This theorem is proved by J. Leray in [69], along with a fundamental result of global wellposedness of weak solutions in L2 .Rd / in any space dimensions (with no uniqueness however) in [68], as recalled in Sect. 2.4.2.

3.2

Global Wellposedness in Critical Spaces

Following the Leray theorem of global, unique solutions in the energy space in dimension 2, one may ask whether all local-in-time results in critical spaces as described in Sect. 2 could be extended to global-in-time results in two space dimensions. This is far from being obvious a priori, since the key feature enabling this to be true in L2 .R2 / is the energy estimate, which fails in other scale-invariant 1C 2 spaces such as BP p;q p .R2 / for p > 2, or BMO1 .R2 /. In [40] the following theorem is nevertheless proved. Theorem 6 ([40]). Let r and q be two real numbers such that 2 6 r < 1 and 2 < 2 r 1 q < 1. Let u0 be a divergence-free vector field in BP r;q .R2 /. Then there exists a 2

1

r unique global solution to (NS) such that u 2 C .Œ0; 1/; BP r;q .R2 //. Moreover, if 2 2 C  1, then there exists a constant Cr;q such that r q

8t  0;

ku.t /k

2

r 1 2 .R / BP r;q

1C rC1 2

6 Cr;q ku0 k

2

r 1 2 .R / BP r;q

:

(14)

Theorem 6 is extended to initial data in the closure of the space of Schwartz functions for the BMO1 norm in [46]. Sketch of proof of Theorem 6. The proof relies on a method introduced by C. Calderón in [11] to prove the global existence of weak solutions in Lp . The idea is to split the initial data into 2

u0 D v0 C w0 ;

1 with v0 2 L2 .R2 / and w0 small in BP rQrQ;Qq .R2 /;

rQ > r; qQ > q:

It is known that there is a unique global solution w to (NS) associated with w0 thanks to Theorem 2 so it remains to solve the equation for v WD u  w: @t v  v C Pr  .v ˝ w/ C Pr  .w ˝ v/ C Pr  .v ˝ v/ D 0:

(15)

The idea is to first use a fixed-point argument to solve the equation locally in time in L2 .R2 / and then to check that the energy of v satisfies a global a priori bound in L2 .R2 /; that will show that the solution may be extended for all times in L2 .R2 /. Let us detail that part of the proof. Formally, one can multiply (15) by v and integrate over x and t (with t ranging from t0 > 0 to T ) to get, using the fact that v is divergence free (therefore here the structure of the nonlinear term is used)

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kv.T /k2L2 C 2

Z

T t0

krv.t /k2L2 dt C

Z

Z

T

R2

t0

.v  r/v  w dxds 6 kv.t0 /k2L2 :

(16)

It can be checked that the small solution w satisfies sup

p tkwkL1 < "0 ;

t

which allows to write by Hölder’s inequality, ˇZ ˇ ˇ ˇ

T t0

Z R

ˇ Z ˇ ˇ .v  r/vw dxdt ˇ 6 C "0 2

Z

T t0

krv.t /k2L2

T

dt C t0

kv.t /k2L2 t

! dt :

This yields the expected bound after applying the Gronwall Lemma. Note that the formal computation (16) is justified since one applies the energy inequality from a time t > t0 > 0, all terms are smooth and there is no difficulty in defining the various quantities. The local solution v may thus be extended globally, and the global existence result follows. The a priori bound (14) is obtained using a nonlinear real interpolation method which is not detailed here. 

3.3

Measure-Valued Vorticity

In dimension 2, the vorticity plays a crucial role. Defined in general by ! WD curl u, it satisfies in dimension 2 the transport-diffusion equation @t ! C u  r!  ! D 0:

(17)

In dimension 3 there is an additional stretching term !  ru on the left-hand side, which destroys the conservations that can be seen on (17), namely, the fact that any Lp .R2 / norm of ! is formally bounded by that of the initial data. The critical setting for the initial data in terms of the vorticity is L1 . Global existence for large data was proved for measure-valued vorticity !0 (see the works of G.-H. Cottet [26] and Y. Giga, T. Miyakawa and H. Osada [51]): define Z kkM WD sup

ˇ ˇ  d  ˇ  2 C0 .R2 /; kkL1 6 1 :

The initial velocity field u0 given by the Biot-Savart Law u0 .x/ D

1 2

Z Z

.x  y/? !0 .y/ dy ; jx  yj2

x 2 R2 ;

t >0;

is known to be in the Lorentz space L2;1 , which is strictly larger than L2 , but not all u0 2 L2;1 can be paired with a measure-valued vorticity.

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

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Uniqueness (and continuous dependence on the data) was proved in [51, 57] under a smallness assumption on the atomic part of the measure. The case of a large Dirac mass was solved in [41] using the following strategy: one can rewrite the vorticity equation (17) in self-similar variables and show that the Oseen vortices ˛G (˛ 2 R), with G.y/ WD

1  jyj2 e 4 ; 4

(18)

are the only equilibria of the rescaled equation. By compactness arguments, one then deduces that all solutions converge in L1 .R2 / to Oseen vortices as t ! C1; as a by-product one finds that (18) is the unique solution of (17) such that k!.; t /kL1 6 K for all t > 0 and !.; t / * ˛ı0 as t ! 0, where ı0 is the Dirac mass at the origin. Uniqueness in the case of general measures, with no smallness assumption, is proved in [33] (see [34] for another proof of the special case of a Dirac mass). The final result is the following: Theorem 7 ([33, 51, 57]). For any  2 M.R2 /, the vorticity equation (17) has a unique global solution ! 2 C 0 ..0; 1/; L1 .R2 / \ L1 .R2 // such that k!.; t /kL1 6 kkM for all t > 0 and !.; t / *  as t ! 0. This solution depends continuously on the initial measure  in the norm topology of M.R2 /, uniformly in time on compact intervals. Moreover, Z

!.t; x/dx D ˛ WD .R2 / ;

for all t > 0 ;

and  1 ˛  x   lim t 1 p !.t; x/  G p  p 2 D 0 ; t!1 t t L .R /

for all p 2 Œ1; 1 :

Let us give a very rough idea of the proof of the uniqueness result. Since previous works on the subject assume that the initial vorticity  either has a small atomic part [51, 57] or consists of a single Dirac mass [41], the idea is to decompose  into a finite sum of mutually singular Dirac masses and a remainder whose atomic part is arbitrarily small (depending on the number of terms in the previous sum). The idea is then to use the methods of [41] to deal with the large Dirac masses and the argument of [51, 57] to treat the remainder. The difficulty is of course that Eq. (17) is nonlinear so that the interactions between the various terms have to be controlled, but in the end, one can show that the solution ! also admits a natural decomposition into a sum of Oseen vortices and a remainder, and this enables one to conclude the proof.

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Examples of Large Data in Critical Spaces Giving Rise to a Global Solution

The results presented in Sect. 2 suggest that in order to obtain global solutions in critical spaces to (NS) in dimension 3, the initial data must be small in some critical space. Actually it is not necessarily so, and it is possible to construct initial 1 data as large as wanted (the size of the initial data is measured in BP 1;1 , which is meaningful due to Proposition 1) which nevertheless gives rise to a global solution. All results in that framework use in some way the structure of the nonlinear term: this is necessary as explained above in Sect. 2.4.

4.1

Geometrical Constraints

One can assume some additional geometrical constraints on the flow, which imply the conservation of quantities beyond scaling (namely, spherical, helicoidal, or axisymmetric conditions). One can refer, for instance, to [66, 75, 86], or [100] for such studies. One other possibility is to use the fact that the 2D equations are wellposed (as recalled in Sect. 3) to deduce results on the 3D case in some special situations for the initial data. An important example where a unique global in time solution exists for large initial data is the case where the domain is thin in the vertical direction (in three space dimensions): that is proved by G. Raugel and G. Sell in [88] (see also the paper [54] by D. Iftimie, G. Raugel, and G. Sell). The authors obtain the global existence of a strong solution for initial data which are allowed to have a large twodimensional part (the vertical mean of the initial data) and a small three-dimensional part. Another example of large initial data generating a global solution is obtained by A. Mahalov and B. Nicolaenko in [74]: in that case, the initial data is chosen so as to transform the equation into a rotating fluid equation (for which it is known that global solutions exist for a sufficiently strong rotation).

4.2

Anisotropic Oscillations

In [17] an example of periodic initial data is presented, which is strongly oscillating 1 and large in BP 1;1 but yet generates a global solution. Such an initial data is given by the following formula, writing uh;N for .u1;N ; u2;N /  h;N  .x1 ; x2 / cos.N x3 /; div h uh;N .x1 ; x2 / sin.N x3 / ; uN 0 .x/ WD N u 1 1 where kuh;N kL2 .T2 / 6 C .ln N / 4 , and its BP 1;1 norm is typically of the same size. One assumes that .x1 ; x2 ; x3 / lies in the three-dimensional torus T3 , where Td is the d -dimensional torus. The main idea is to prove a global existence result under a (nonlinear) smallness assumption on the first iterate of the Picard scheme instead of

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

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a smallness condition directly on e t u0 . That result will not be stated here but rather its counterpart in the whole space, which is proved in [18]. Theorem 8 ([18]). Let p 23; 1Πbe given. There is a constant C0 such that the P 12 .R3 / be a divergence-free vector field. Suppose following result holds. Let u0 2 H that        (19) P e t u0  re t u0  6 C01 exp C0 ku0 k2BP 1 ; E

1;2

where kf kE WD kf kL1 .RC IBP 1 / C

X

1;1

    2j kj f .t/kL1 

j 2Z

L2 .RC Itdt/

:

Then there is a unique, global solution to (NS) associated with u0 , satisfying     P 12 \ L2 RC I H P 32 : u 2 Cb RC I H Condition (19) is a nonlinear smallness condition on the initial data. The proof of Theorem 8 consists in writing the solution u (which exists for a short time at least), as u D e t u0 C R and in proving a global wellposedness result for the perturbed Navier-Stokes equation satisfied by R thanks to the smallness condition (19). We shall not enter into any detail here. Now let us give an example of large initial data satisfying the assumptions of Theorem 8. Theorem 9 ([18]). Let  2 S.R3 / be a given function, and consider two real numbers " and ˛ in 0; 1Œ. Define  x   . log "/ 3 x2 3 '" .x/ WD  x : cos ; ; x 1 3 "1˛ " "˛ 1

Then for any p > 3, there is a constant C > 0 such that for " small enough, the smooth, divergence-free vector field u0;" .x/ WD .@2 '" .x/; @1 '" .x/; 0/ satisfies 1

1

C 1 . log "/ 3 6 ku0;" kBP 1;1 6 C . log "/ 3 ; 1

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I. Gallagher

and ˛

ke t u0;"  re t u0;" kE 6 C " 3 :

(20)

In particular for " small enough, the vector field u";0 generates a unique, global solution to .NS/. The proof of this result, which will very briefly be sketched, relies heavily on the special structure of the nonlinear term. One starts by noticing that estimating e t u0;"  re t u0;" boils down to estimating e t u10;" @1 e t u10;" C e t u20;" @2 e t u10;" and e t u10;" @1 e t u20;" C e t u20;" @2 e t u20;" : But e t u10;" @1 e t u10;" C e t u20;" @2 e t u10;" D

1 2 . log "/ 5 e t f" e t g" 2 "

e t u10;" @1 e t u20;" C e t u20;" @2 e t u20;" D

and

1 2 . log "/ 5 e t fQ" e t gQ " ; "2˛

where f , fQ , g, and gQ are smooth functions and   x3 x2 f" .x/ WD e i " f x1 ; ˛ ; x3 : " 1 The conclusion comes from the fact that for any functions f and g in BP 1;2 \ 1 3 P .R /, one has the interpolation-type inequality H

 23   13  kP.e t f e t g/kE 6 C kf kBP 1 kgkBP 1 kf kHP 1 kgkHP 1 1;2

1;2

   along with the estimate, for 2 0; 3 1  p1 and p  1, ˛

kf" kBP  6 C " C p : p;1

4.3

Slow Variations in One Direction

In [19], the global wellposedness of the two-dimensional equation is used to prove a global existence result for large data which is slowly varying in one direction.

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

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Theorem 10 ([19]). Let v0h D .v01 ; v02 / be a horizontal, smooth divergence-free vector field on R3 and let w0 be a smooth divergence-free vector field on R3 . Then there exists a positive "0 such that, if 0 < " 6 "0 , the initial data u"0 .x/ WD .v0h C "wh0 ; w30 /.x1 ; x2 ; "x3 /

(21)

generates a unique, global solution u" of (NS). Remark 5. One can check that the initial data in (21) may be chosen as large as wanted: indeed if .f; g/ are in S.R2 /  S.R/ and if h" .x/ WD f .x1 ; x2 /g."x3 / then if " is small enough there holds kh" kBP 1;1 1 .R3 / 

1 1 kf kBP 1;1 1 .R2 / kgkL .R/ : 4

Sketch of proof of Theorem 10. The idea of the proof of Theorem 10 is the following: one defines the solution v h D .v 1 ; v 2 / to the two-dimensional Navier-Stokes equation associated with the data v0h , which is known to be global and unique thanks to Theorem 5. Next one solves the linear transport equation @t w" C v h  rw"  h w"  "2 @23 w" D .r h p; "2 @3 p/ with initial data w0 , where h WD @21 C @22 and r h WD .@1 ; @2 /. Finally one defines the approximate solution h h 3 uapp " WD .v C "w" ; w" /.t; x1 ; x2 ; "x3 /

and the proof of the result is complete if it can be proved that R" WD u"  uapp " exists globally in time, where u" is the solution (defined a priori on a finite time interval) associated with u0;" . This is possible due to the fact that R" solves 8 app app < @t R" C R"  rR"  R" C u"  rR" C R"  ru" D F"  rq" div R" D 0 : R"jtD0 D 0 where the force F" depends on v h and w" and can be proved to be small: this point is linked to the fact that all errors made between the 2D and the 3D equation involve partial derivatives in the x3 direction, which all produce a factor ". This implies that R" exists globally and concludes the proof.  Using the language of the weak compressible limit or fast rotating fluids, the case studied in Theorem 10 may be qualified as a “well-prepared” case. Indeed

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the initial data converges uniformly for x3 in any compact subset of R to a twodimensional vector field which generates global smooth solutions. The result of [19] is generalized in [20] to the “ill-prepared” initial data   1 u0;" .x/ WD v0h .x1 ; x2 ; "x3 /; v03 .x1 ; x2 ; "x3 / " as soon as (writing e ajD3 j for the Fourier multiplier e aj3 j in Fourier space) ke ajD3 j v0 kHP 4 6 ; for some a > 0 and for small enough: then for " small enough, u0;" generates a global smooth solution u" of (NS) on T2  R. The result [20] is further generalized in [82]. The result of [19] is also extended, in [21], to the case when one adds to the data (21) any vector field giving rise to a global solution. Theorem 11. Let u0 , v0h D .v01 ; v02 /, and w0 be three smooth divergence-free vector fields defined on R3 , satisfying P 12 .R3 / and generates a unique global solution to the Navier• u0 belongs to H Stokes equations; • v0h .x1 ; x2 ; 0/ D w30 .x1 ; x2 ; 0/ D 0 for all .x1 ; x2 / 2 R2 : Then there exists a positive number "0 depending on u0 and on norms of v0h and w0 such that for any " 2 .0; "0 , there is a unique, global solution to the Navier-Stokes equations with initial data u0;" .x/ WD u0 .x/ C .v0h C "wh0 ; w30 /.x1 ; x2 ; "x3 /: P 12 divergence-free vector Remark 6. Let u0 be any element of the (open) set G of H fields generating a global smooth solution to (NS), and let N be an arbitrarily large number. Then for any smooth divergence-free vector field f h (over R2 ) and 1 scalar function g (over R) satisfying kf h kBP 1;1 1 .R2 / kgkL .R/  4N , and such that g.0/ D 0, Theorem 11 implies that there is "N depending on u0 and on norms of f h and g such that u0 C .f h ˝ g; 0/.x1 ; x2 ; "N x3 / belongs to G, where f h ˝ g.x/ D .f 1 .x1 ; x2 /g.x3 /; f 2 .x1 ; x2 /g.x3 //. Since "N only depends on norms of f h and g, that implies that for any  2 Œ1; 1, the initial data u0 C .f h ˝ g; 0/.x1 ; x2 ; "N x3 / also belongs to G. One concludes that through u0 passes an uncountable number of segments of length N included in G. The proof of Theorem 11 begins, like the proof of Theorem 10, by defining an approximate solution associated with the data u0;" . In this case it is of the form h h 3 uapp " .t; x/ WD u.t; x/ C .v C "w" ; w" /.t; x1 ; x2 ; "x3 /

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

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where u is the global solution associated with u0 , and v h and w" are defined exactly as in the proof of Theorem 11. The fact that v0h .x1 ; x2 ; 0/ D w30 .x1 ; x2 ; 0/ D 0 is of crucial importance, as it can be proved that under that assumption, v h .t; x1 ; x2 ; 0/ and w" .t; x1 ; x2 ; 0/ are small. This means that the support in x3 of v h .t; x1 ; x2 ; "x3 / and w" .t; x1 ; x2 ; "x3 / lies essentially at x3  O.1="/, and since u does not depend on ", it follows that the supports in x3 of u and v h .t; x1 ; x2 ; "x3 / C ".wh" ; w3" /.t; x1 ; x2 ; "x3 / are essentially disjoint. So this sum can be proved to be an approximate solution to (NS).

5

Large-Time Behavior of Global Solutions

In this section the large-time behavior of global solutions is studied in two and three space dimensions: results on the velocity are presented first and then on the vorticity.

5.1

Behavior of the Velocity

In this section the large-time behavior of global solutions (whatever their initial size) is analyzed, in scale-invariant spaces. The link between large-time behavior in space and time is developed in another chapter (see [8], for instance). In two space dimensions, it is known since the work of Wiegner [104] that finite energy solutions decay to zero in L2 for large times (see also [77] and [92] among others). Here strong solutions (as constructed in Sect. 2) are studied, in three space dimensions. The main result is the following (where VMO1 .R3 / is the closure of the space of Schwartz class functions for the BMO1 .R3 / norm). Theorem 12 ([1]). Let u0 2 VMO1 .R3 / give rise to a unique, global solution u belonging to C .Œ0; C1Œ; BMO1 / (constructed, for instance, by a fixed-point argument). Then lim ku.t /kBMO1 D 0:

t!C1

This result follows the work [36] where the Besov setting is considered. Here the P 12 is sketched. proof of the easier case when the initial data lies in H P 12 setting. This proof may be found in [35], and it is based Sketch of proof in the H P 12 .R3 /. on the following remark: assume that u0 belongs to L2 and not only to H Then it can be proved, by a weak-strong uniqueness property as described in Sect. 2.4.2, that the solution remains in L2 .R3 / for all times and satisfies the energy P 1 .R3 //, inequality (13). By interpolation between L1 .RC I L2 .R3 // and L2 .RC I H 1 P 2 .R3 //. For all "0 > 0, one can therefore it is known that u belongs to L4 .RC I H find a time t0 such that ku.t0 /k P 12 3 6 "0 , and Theorem 1 then implies the result. H .R /

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P 12 .R3 / alone, one can use the method introduced in [11] Now if u0 belongs to H (in the context of building weak, infinite energy solutions) and recalled in Sect. 3.2, which consists in decomposing u0 D v0 C w0 P 12 .R3 / and v0 2 L2 \ H P 12 .R3 /. This may be done, for instance, with w0 small in H by cutting off the low frequencies of u0 , since the resulting vector field v0 will be P 12 .R3 /, while the part with the small frequencies w0 can be made as in L2 \ H small as needed provided the cutoff is small enough. Then one starts by solving the Navier-Stokes system with data w0 . This produces a global solution denoted w, which satisfies for all t  0 kw.t /k P

1

H 2 .R3 /

6 kw0 k P

(22)

1

H 2 .R3 /

thanks to Theorem 1. Then v WD u  w satisfies the equation @t v C v  rv C v  rw C w  rv  v D rp;

vjtD0 D v0 :

One writes the formal energy estimate kv.t /k2L2 .R3 / C2

Z

t 0

krvk2L2 .R3 /

0

dt 6

ˇZ t Z ˇ

kv0 k2L2 .R3 / C2 ˇˇ

ˇ ˇ .v  rw/  v.t / dxdt ˇ ; R3 (23)

0

0

0ˇ

and a product estimate along with (22) implies that ˇ ˇZ t Z Z t ˇ ˇ 0 0ˇ ˇ .v  rw/  v.t / dxdt ˇ 6 C kw0 k P 12 3 krv.t 0 /k2L2 .R3 / dt 0 : ˇ H .R / 3 R

0

If C kw0 k P

1

H 2 .R3 /

0

6

1 2

one concludes from (23) that the energy of v remains bounded 1

P 2 .R3 // so as above there for all times. One can deduce that v belongs to L4 .RC I H is a time t0 > 0 for which kv.t0 /k P 12 3 6 kw0 k P 12 3 . In particular one infers H .R /

H .R /

that ku.t0 /k P 12 3 6 2kw0 k P 12 3 which concludes the proof since kw0 k P 12 3 H .R / H .R / H .R / can be chosen arbitrarily small. 

5.2

Behavior of the Vorticity

In this section a result on the large-time behavior of the vorticity in two space dimensions is presented, in scale-invariant spaces. As recalled in Sect. 3.3, the Oseen vortex (18) plays an important role in the dynamics of the vorticity equation (17) in two space dimensions: it is the solution

12 Critical Function Spaces for the Well-Posedness of the Navier-Stokes. . .

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with a single Dirac mass as initial data. It also describes the long-time asymptotics of solutions to (17), as stated in the following result due to [41]. Recall the notation (18) for the Gaussian function. Theorem 13 ([41]). For all initial data !0 C .R I L1 .R2 // satisfies

2 M.R2 /, the solution !

   ˛  x   lim !  G p  ; with t!1  t t L1 .R2 /

2

Z ˛D

d !0 :

Sketch of proof. Let us briefly sketch the proof of this result. The idea is to use selfsimilar variables x yDp ; t

 D log

t T

and to write   x t 1 !.t; x/ D w p ; log t T t

and

1 u.t; x/ D p v t



x t p ; log T t

 :

Then w satisfies @ w C v  ry w D Lw;

with

1 L WD y C y  ry C 1: 2

Oseen vortices are equilibria of the rescaled system. The interest of this formulation is that it can be shown that the operator L has better spectral properties than the Laplacian, when acting on weighted function spaces. In particular the trajectories in L1 are compact, in the sense that the set w. / 0 is relatively compact in L1 .R2 / if w0 2 L1 .R2 /. The next important step of the proof consists in proving that the !-limit set of w0 is reduced to one point ˛G: this Liouville-type theorem is proved by exhibiting a new Lyapunov functional, other than the L1 norm. This functional is based on the relative entropy of w 

Z H .w/ WD

w.y/ log R2

w.y/ G.y/

 dy;

which is strictly decreasing on the trajectories of the equation, except on the stationary solutions. By the La Salle principle, the !-limit set of w0 can only be one point, namely, the stationary solution ˛G. The convergence result follows by rescaling back to the original variables. 

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Behavior at Blow-Up Time

In this section the space variable is taken in R3 and the following question is addressed: suppose that there do exist solutions blowing up in finite time; what is their behavior at blow-up time? In this section it is assumed that blowing-up solutions do exist.

6.1

Blow-Up of Scale-Invariant Norms

A natural question is to ask whether at blow-up time T  , scale-invariant norms of the solution blow up. The fixed-point methods described in Sect. 2 provide some information as to what types of norms should blow up at T  . For instance, it can be proved (see, for instance, [68]) that solutions can be continued as long as they lie in Lp ..0; T /I Lq .R3 // with 2=p C 3=q D 1 and p < 1. This is often known as the “Ladyzhenskaya-Prodi-Serrin criterion.” J. Leray also proves in [68] that for any q > 3, there is a constant C such that ku.t /kLq .R3 / 

C 1

3

.T   t / 2 .1 q /



We refer also, for instance, to [49] for a discussion in the case of domains. The much harder question (as it is not a consequence of the fixed-point method) concerns the limiting case p D 1 can be stated as follows: if the initial data belongs to some critical space X and if the solution blows up at time T  , then does sup ku.t /kX D 1‹ This corresponds, for instance (when X D L3 ), to the 0 0 be fixed, and choose s 2 .0; T  .u0;c // such that, writing U1 WD NS.1 /, Apc  kU1 .s/k

p 3 1

p BP p;p

< "=2:

Then defining un WD NS.u0;n / and tn WD 21;n s, one can show that Apc  kun .tn /k

p 3 1

p BP p;p

p

 kU1 .s/k

3 p 1 P p;p B

p

 kU1 .s/k

3 p 1 P p;p B

J X p C .ƒj;n Uj /.tn / C wJn .tn / C rnJ .tn / 3 1 C .n; s/ p BP p;p

j D2

C Ck

J X

p

ƒj;n Uj .tn / C wJn .tn / C rnJ .tn /k

3 1

q BP q;q

j D2

C .n; s/

since q  p, where .n; s/ ! 0 as n ! 1. Choosing J large enough so that p

C kwJn .tn / C rnJ .tn /k

3 1

q BP q;q

6 "=2;

for sufficiently large n, one finds that J q X  .ƒj;n Uj /.tn / j D2

3 1

q BP q;q

. "  .n; s/:

But orthogonality arguments (see the proof of [37, Lemma 3.6]) show that

(30)

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I. Gallagher J X

q

k.ƒj;n Uj /.tn /k

j D2

3 q 1 BP q;q

Dk

J X

.ƒj;n Uj /.tn /k

j D2

q 3 1

q P q;q B

C ".J; n/

(31)

where for each J , ".J; n/ ! 0 when n ! 1. In particular for j D j0 , (30) and (31) imply kj0 k

3 1

q BP q;q

D kUj0 .0/k

3 1

q BP q;q

."

since tn ! 0 as n ! 1, and hence j0  0 which proves the result.

6.1.4 Proof of Theorem 14,(3) This part is based on a backward uniqueness argument similar to that in [27] (see also [37, 58]). However in order to implement this argument, some positive regularity on the solution near blow-up time needs to be recovered. This is the main difficulty in the proof and the purpose of the next statement. We shall not prove that statement here as it is rather involved, but the main idea, which can also be found in [22], is the use of “self-improving bounds”: it is well-known that the Duhamel term is in some sense more regular than the linear heat flow, and this fact can be iterated up to positive regularity. Proposition 2 (Positive regularity at blow-up). For u0 divergence free belonging 3 p 1 with 3 < p < 1, define the associate solution u WD NS .u0 / on Œ0; T  Œ. to BP p;p 3

1

p If T  < 1 and if u belongs to L1 .Œ0; T  I BP p;p /, then there exist v; w defined  on Œ0; T Œ such that

u D v C w in XT  as defined in (11) and such that moreover, for some " 2 .0; T  /, v 2 L1 .ŒT   "; T  I Lp .R3 // and

w 2 L3 ..0; T  /I L3 .R3 //:

Let us apply Proposition 2, to u D NS .u0 /: as T  < 1, moreover v 2 Lp .ŒT   "; T    R3 /: Fix any R > 0 and set Q";R .x/ WD f.y; t / 2 R  R3 = jy  xj < R; t 2 ŒT   "; T  g: As p > 3, for fixed "; R > 0 kukL3 .Q";R .x// . kvkLp .Q";R .x// C kwkL3 .Q";R .x// ! 0

as

jxj ! 1:

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This (and a similar property on the pressure) is the key to the “"-regularity” theory for “suitable weak solutions.” Thanks to the Caffarelli-Kohn-Nirenberg theory [10], one can conclude as in [58] that u is a suitable weak solution and is smooth at and near the time T  outside of some large compact set K R3 . Hence if u.t / ! 0 in S 0 as t % T  , one can conclude that actually u.x; T  /  0 for all x 2 K c , and backward uniqueness and unique continuation applied to the vorticity ! WD r  u as in [27] allow us to conclude that in fact u.t /  0 for some t 2 .0; T  /; one can refer to [58] for more details, including the statements of the backward uniqueness and unique continuation results. Therefore T  D 1 by small data results, contrary to assumption, which proves the result, hence the theorem. 

6.2

Qualitative Behavior

In this section a number of other results concerning the behavior of solutions at possible blow-up time are collected, without proofs.

6.2.1 On the Size of the Singular Set Let the singular set for (NS) be defined as follows (in three space dimensions): ˚  S .u/ WD .t; x/ 2 RC  R3 = u is not bounded in a neighborhood of .t; x/ : It can be proved by a rather classical (in the theory of parabolic equations) bootstrap argument that if u is bounded in a (parabolic) space-time ball of radius R, then it is C 1 in the space-time ball of radius R=2. Caffarelli, Kohn, and Nirenberg prove in [10] that the singular set of a suitable weak solution (a weak solution satisfying a certain generalization of the conservation of energy) to (NS) has parabolic Hausdorff dimension at most 1, meaning that the singular set must be smaller than a curve in space-time. Note that this result is generalized in [89] to relate the Hausdorff dimension to the Ladyzhenskaya-Prodi-Serrin criterion recalled above. Note also that the theory of partial regularity for suitable weak solutions holds for general, bounded or unbounded domains.

6.2.2 Minimal Blowing-Up Solutions Assuming that there do exist blowing-up solutions, it is interesting to characterize the set of such initial data and in particular the set of “minimal” initial data giving rise to a solution blowing up in finite time. This is the object of the following theorem, due to [91]. ˚ Theorem 16 ([91]). Define  WD inf ku0 k P

1

H 2 .R3 /

 j T  .u0 / < C1 . Then there

P 12 .R3 / such that T  .u0 / < 1 and ku0 k exists u0 in H P

1

H 2 .R3 /

1 2

D . Moreover the

P .R3 /, up to the invariances of the equation set of such minimal u0 is compact in H (namely, space translations and dilations).

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This theorem was later generalized to larger scale-invariant spaces using profile decomposition techniques as presented above in Sect. 6.1 (see [37, 38]). Note that in [87], a similar result is proved in spaces that are not scale invariant; in that case the compactness holds up to translations only, with no change of scale.

6.2.3 Directions of the Vorticity In [24], the authors prove that the solution remains smooth as long as the following holds: denote by .t; x/ the direction of vorticity .t; x/ D curl u.t; x/ and by '.t; x; y/ the angle between .t; x/ and .t; y/. Then there are constants 0 > 0 and  > 0 if the magnitude of .t; x/ and .t; y/ is larger than  then j sin '.t; x; y/j 

jx  yj  

In other words, in regions of high vorticity, one must have good control on the direction of the vorticity. One can refer to [6, 52, 106] for related results and more references.

6.2.4 Specializing Components of the Velocity Field A number of articles are devoted to understanding under what minimal possible conditions one can ensure that blow up in finite time occurs. Many results are of the following type: if T  < 1

then

ku3 kLp .Œ0;T  ILq .R3 // D 1

or if T  < 1

then

k@j u3 kLp .Œ0;T  ILq .R3 // D 1

with various relations (which are not scale invariant) between p and q. One can refer, among other references, to [14, 64, 81, 83, 94, 107]. In [65], the authors are able to prove the following scale-invariant criterion: if

T < 1

then

k@3 ukLp .Œ0;T  ILq .R3 // D 1 with

3 2 C D 2; q 2 Œ9=4; 3: p q

Finally in [23], J.-Y. Chemin and P. Zhang prove that if the initial vorticity belongs 3 P 12 .R3 /) then to L 2 .R3 / (which is slightly stronger than u0 2 H if

T < 1

then

ku3 k

P Lp .Œ0;T  IH

1C 2 p 2

.R3 //

D1

with p 2 .4; 6/:

The main idea of the proof is to write an equation for the horizontal vorticity, and an equation for u3 , and to use the structure of the equation as much as possible to eliminate quadratic nonlinearities in the equation on the vorticity and to trade

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off some vertical derivatives for horizontal ones, thanks to the divergence-free condition.

7

Conclusion

This text is a (far from exhaustive) presentation of some results that can be proved on the initial value problem for the homogeneous, incompressible Navier-Stokes system in the whole space, using critical function spaces – meaning spaces invariant through the scaling of the system. Particular emphasis is made on the resolution of the initial value problem, on large-time asymptotics, as well as on the behavior of solutions at the possible blow-up time. For more on the Navier-Stokes equations in that setting, the interested reader can refer, for instance, to the books [3, 12, 25, 42, 67, 78, 95].

8

Cross-References

 Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak

Solutions for the Compressible Navier-Stokes Equations  Equations for Viscoelastic Fluids  Finite Time Blow-Up of Regular Solutions for Compressible Flows  Fourier Analysis Methods for the Compressible Navier-Stokes Equations  Global Existence of Regular Solutions with Large Oscillations and Vacuum for

Compressible Flows  Large Time Behavior of the Navier-Stokes Flow  Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes

Flow  Recent Advances Concerning Certain Class of Geophysical Flows  Regularity Criteria for Navier-Stokes Solutions  Self-Similar Solutions to the Nonstationary Navier-Stokes Equations  Time-Periodic Solutions to the Navier-Stokes Equations  Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical

Cases

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Existence and Stability of Viscous Vortices

13

Thierry Gallay and Yasunori Maekawa

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Stability of Lamb-Oseen Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Local Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Large Reynolds Number Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Lamb-Oseen Vortices in Exterior Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Axisymmetric Vortex Rings and Filaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Existence and Stability of Burgers Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Existence and Uniqueness of Asymmetric Burgers Vortices . . . . . . . . . . . . . . . . . . . 4.2 Two-Dimensional Stability of Asymmetric Burgers Vortices . . . . . . . . . . . . . . . . . . 4.3 Three-Dimensional Stability of Burgers Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

688 693 694 699 701 705 709 710 715 718 724 725 725

Abstract

Vorticity plays a prominent role in the dynamics of incompressible viscous flows. In two-dimensional freely decaying turbulence, after a short transient period, evolution is essentially driven by interactions of viscous vortices, the archetype of which is the self-similar Lamb-Oseen vortex. In three dimensions, amplification of vorticity due to stretching can counterbalance viscous dissipation and produce stable tubular vortices. This phenomenon is illustrated in a famous model

T. Gallay () UMR 5582 – Mathematics Laboratory, Institut Fourier, Université Grenoble Alpes, Gières, France e-mail: [email protected] Y. Maekawa Department of Mathematics, Graduate School of Sciences, Kyoto University, Kyoto, Japan e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_13

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originally proposed by Burgers, where a straight vortex tube is produced by a linear uniaxial strain field. In real flows, vortex lines are usually not straight, and can even form closed curves, as in the case of axisymmetric vortex rings which are very common in nature and in laboratory experiments. The aim of this chapter is to review a few rigorous results concerning existence and stability of viscous vortices in simple geometries.

1

Introduction

Since the pioneering work of Helmholtz [24], vorticity has been widely recognized as a quantity of fundamental importance in fluid dynamics, especially for turbulent flows. According to a famous quote by Küchermann [28], “vortices are the sinews and muscles of fluid motions.” Intuitively, vorticity describes the local rotation of fluid particles at a given point. In the Eulerian representation, if u.x; t / denotes the velocity of the fluid at point x D .x1 ; x2 ; x3 / 2 R3 and time t 2 R, the vorticity is the vector !.x; t / D curl u.x; t / D r ^ u.x; t /. Under the evolution given by the Navier-Stokes equations, the vorticity satisfies @t !.x; t / C .u.x; t /; r/!.x; t /  .!.x; t /; r/u.x; t / D !.x; t /;

(1)

where  > 0 is the kinematic viscosity of the fluid, i.e., the ratio of the viscosity to the fluid density. In the incompressible case considered here, the velocity field satisfies div u.x; t / D 0 and is thus entirely determined by the vorticity distribution up to an irrotational flow. The Biot-Savart law is a reconstruction formula that expresses u in terms of !, depending on the geometry of the fluid domain and the boundary conditions. In the whole space R3 , if the vorticity distribution is sufficiently localized, the Biot-Savart formula reads Z 1 .x  y/ ^ !.y; t / u.x; t / D  dy: (2) 4 R3 jx  yj3 In viscous fluids, vorticity is usually created within boundary layers near walls or interfaces or in the vicinity of a stirring device. Once produced, vorticity can be substantially amplified by the local strain in the fluid, through a genuinely threedimensional mechanism that is often referred to as “vortex stretching.” A Taylor expansion of the velocity field at a given point x0 reveals that 1 u.x; t / D u.x0 ; t / C !.x0 ; t / ^ .x  x0 / C .Du.x0 ; t //.x  x0 / C O.jx  x0 j2 /; 2   where Du D 12 .ru/C.ru/> is the deformation tensor, whose eigenvalues 1 ; 2 ; and 3 are called the principal strains at x0 . Incompressibility implies that 1 C 2 C 3 D 0, so that two generic situations may occur. If two principal strains (say, 1 and 2 ) are negative and the third one is positive, vorticity gets amplified at x0 in the direction of the principal strain axis corresponding to 3 . That stretching

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mechanism can compensate the viscous dissipation and result in the formation of stable vortex filaments, a typical example being the Burgers vortex [2] which will be studied in Sect. 4. In contrast, if two principal strains are positive at x0 , the stretching effect leads to the formation of vortex sheets, which are also commonly observed in turbulent flows although they undergo the Kelvin-Helmholtz instability at high Reynolds numbers. Vortex sheets play a prominent role in interfacial motion and boundary layer theory, and the interested reader is referred to the chapter entitled  “The Inviscid Limit and Boundary Layers for Navier-Stokes Flows” for further information. In the present chapter, emphasis is put on vortex tubes or filaments, for which vorticity is essentially concentrated along a curve with no endpoints in the fluid. In general, the curve will evolve with time, because it is advected by the flow. According to Helmholtz’s first law, the total circulation of such a vortex filament is constant along its length and is also independent of time as long as the viscous effects can be neglected. This very important quantity, often denoted by , can be defined as the flux of the vorticity vector through any cross section of the vortex tube or equivalently (in view of Stokes’ theorem) as the circulation of the velocity along any closed curve enclosing that tube. The ratio ˛ D = of the total circulation to the kinematic viscosity is a dimensionless quantity, sometimes referred to as the circulation Reynolds number, which measures the strength of the vortex and plays a crucial role in stability issues. Since vortex filaments have no endpoints, they must either extend to the fluid boundary, or to infinity, or form closed curves. In the simple situation, already considered in [24], where all vortex lines are straight and parallel to each other, the velocity and vorticity fields take the particular form 0 1 1 0 u1 .x1 ; x2 ; t / 0 A; u.x; t / D @u2 .x1 ; x2 ; t /A ; (3) !.x; t / D @ 0 !.x1 ; x2 ; t / 0 where x D .x1 ; x2 / 2 R2 and ! D @1 u2  @2 u1 . Here the coordinates have been chosen so that the third axis coincides with the direction of the vortex filaments. The evolution equation for the scalar vorticity !.x; t/ is @t !.x; t/ C u.x; t /  r!.x; t/ D !.x; t /;

(4)

where u D .u1 ; u2 / satisfies @1 u1 C @2 u2 D 0. If the vorticity distribution is sufficiently localized, the two-dimensional velocity field u.x; t / is given by the 2D Biot-Savart law u.x; t / D

1 2

Z R2

.x  y/? !.y; t / dy; jx  yj2

(5)

where x ? D .x2 ; x1 / and jxj2 D x12 C x22 . Equations (4) is just an advectiondiffusion equation for the scalar quantity !; hence (by the maximum principle) no

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amplification of vorticity can occur in the two-dimensional case. As a consequence, all localized vortex structures will eventually spread out and decay, since there is nothing to counterbalance the effect of viscosity. A typical example is provided by the Lamb-Oseen vortex, an exact self-similar solution to (4) of the form      x  x !.x; t/ D ; u.x; t / D p v G p ; (6) G p t t t t where the vorticity and velocity profiles are explicitly given by G./ D

1 jj2 =4 e ; 4

v G ./ D

 1 ?  jj2 =4 1  e ; 2 jj2

 2 R2 :

(7)

R R Note that R2 G./ d D 1, so that R2 !.x; t/ dx D  for all t > 0, in agreement with the general definition of the total circulation . The Lamb-Oseen vortex plays a distinguished role in the dynamics of the two-dimensional vorticity equation (4), for two main reasons. First, it deserves the name of fundamental solution, in the sense that it is the unique solution of (4) with initial data !0 D ı0 , where ı0 denotes the Dirac measure at the origin. Next, it describes to leading order the longtime asymptotics of all solutions of (4) with integrable initial data and nonzero circulation [19]. If self-similar variables are used, the Lamb-Oseen vortex becomes a stationary solution of some rescaled equation, and its stability properties can then be studied using spectral theory and other standard techniques. This analysis is presented in Sect. 2 below and serves as a model for further existence and stability results in more complex situations. Another relatively simple and mathematically tractable situation is the axisymmetric case without swirl, where the velocity field is invariant under rotations about a given axis and under reflections by any plane containing the axis. Here all vortex lines are circles centered on the symmetry axis and normal to it. Using cylindrical coordinates .r;  ; z/, so that r represents the distance to the symmetry axis and z the position along the axis, the velocity and vorticity fields are given by u.x; t / D ur .r; z; t / er C uz .r; z; t / ez ;

!.x; t / D ! .r; z; t / e ;

(8)

where er ; e ; and ez denote unit vectors in the radial, toroidal, and vertical directions, respectively. As in the two-dimensional case, the vorticity vector has only one nonzero component ! , which satisfies the evolution equation  ur !  (9) @t ! C u  r!  ! D  !  2 ; r r where u  r D ur @r C uz @z and  D @2r C 1r @r C @2z denotes the Laplace operator in cylindrical coordinates. The velocity u D .ur ; uz / can be expressed in terms of the axisymmetric vorticity ! by solving the linear elliptic system 1 @r ur C ur C @z uz D 0; r

@z ur  @r uz D ! ;

(10)

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in the half-plane D f.r; z/ 2 R2 j r > 0; z 2 Rg, with boundary conditions ur D @r uz D 0 at r D 0. Explicit formulas for the axisymmetric Biot-Savart law exist, see e.g., [6, 16], but are more involved than in the two-dimensional case. The analogue of the Lamb-Oseen vortex for axisymmetric flows is the solution of (9) with a vortex filament as initial data. This means that ! .; ; 0/ D ı.Nr;Nz/ where ı.Nr;Nz/ denotes the Dirac measure located at some point .Nr; zN/ 2 . The existence of a global solution to (9) with such initial data was recently shown by Feng and Šverák [6], and uniqueness can be established using, in particular, the approach presented in Sect. 2; see [17]. The reader is referred to Sect. 3 below for up-to-date results on existence of axisymmetric vortex rings. The third and final case considered here is a famous model for vortex filaments in turbulent flows, originally proposed by Burgers [2]. It is assumed that the velocity field has the form u.x; t / D us .x/ C v.x; t /, where us .x/ is a stationary straining flow of the form 1  1 x1 us .x/ D @2 x2 A D Mx; 3 x3 0

0 1 1 0 0 M D @ 0 2 0 A ; 0 0 3

(11)

where 1 C 2 C 3 D 0 and 1 ; 2 < 0, 3 > 0. According to the discussion above, the strain (11) describes to leading order the deformation rate of any smooth, incompressible velocity field near the origin, at a given time. Burgers model is crude in the sense that it assumes that the strain us .x/ is independent of time and extends all the way to infinity in space, which is certainly not realistic in turbulent flows. Nevertheless, the model is interesting because it clearly illustrates the vortexstretching effect, which in the present case produces a family of stationary solutions that can be compared with observations in experiments. If u.x; t / D us .x/Cv.x; t /, the vorticity equation (1) can be written in equivalent form @t !.x; t / C .v.x; t /; r/!.x; t /  .!.x; t /; r/v.x; t / D L!.x; t /;

(12)

where L is the linear operator defined by L! D !  .Mx; r/! C M!:

(13)

As div v D 0 and curl v D !, the Biot-Savart law (2) can be used to reconstruct the time-dependent velocity field v from the vorticity distribution !. In addition to the Laplacian, the linear operator L includes an advection term that depends linearly on the space variable x and a zero order term involving the strain matrix M whose main effect is to amplify the third component !3 while attenuating !1 and !2 . The Burgers vortex is a stationary solution of (12) which results from the balance between the amplification of vorticity due to stretching and the dissipation due to

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viscosity. In the axisymmetric case where 1 D 2 D =2 and 3 D  > 0, it has the explicit form    p !.x/ D  G x = ; 

r v.x/ D 

  G p v x = ; 

(14)

where  is the total circulation and 1 0 G./ D @ 0 A ; G./ 0

0 1  2 1  jj2 =4 @ v ./ D 1e 1 A : 2jj2 0 G

(15)

The striking similarity with the corresponding expressions (6) and (7) for the LambOseen vortex is of course not an accident. Indeed, if the pair !.x; t / and v.x; t / is a solution of (12) that is two-dimensional in the sense that @3 ! D @3 v  0 and !1 D !2  0, then the pair !.x; t/; u.x; t / defined by Lundgren’s transformation [31]     1 1 x 1 x 1 !3 p ; log. t / ; u.x; t / D p v p ; log. t / ; !.x; t/ D t t  t t  (16) satisfies the two-dimensional vorticity equation (4). In other words, the twodimensional solutions of Eq. (12), which includes an axisymmetric linear straining field, are in one-to-one correspondence with those of the two-dimensional vorticity equation (4), via a self-similar change of variables. This observation plays a crucial role both in Sect. 2, where stability of the Lamb-Oseen vortex is studied and in Sect. 4 where the corresponding results for the axisymmetric Burgers vortex are presented. There is however an important difference between both situations: although the Burgers vortex is a two-dimensional stationary solution of (12), there is no reason to restrict the stability analysis to perturbations in the same class. Quite the contrary, the Burgers vortex can be a relevant model for tubular structures in turbulent flows only if one can prove stability with respect to general threedimensional perturbations, and this is a difficult problem that has no counterpart in the two-dimensional case; see Sect. 4 for a detailed discussion. In the asymmetric case where 1 ¤ 2 , Burgers vortices still exist, but their profiles satisfy a genuinely nonlinear equation, and explicit formulas such as (14) and (15) are no longer available. Thus, even existence of such stretched vortices is a challenging mathematical question, which will also be discussed in Sect. 4. More generally, all existence, uniqueness, and stability results available for the axisymmetric Burgers vortex are expected to remain true in the asymmetric case too, although rigorous proofs are not always available. Remark 1. Although physical constants are useful for dimensional analysis and important for comparison with experiments, they often hinder the mathematical analysis by making formulas needlessly complicated. In Sects. 2 and 4 below,

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dimensionless variables and functions are systematically used, and this amounts to setting  D  D 1 in all formulas. In particular, the total circulation of a vortex coincides with the circulation Reynolds number and will be denoted by ˛.

2

Stability of Lamb-Oseen Vortices

This section is devoted to the stability analysis of the family of Lamb-Oseen vortices (6). These are self-similar solutions of the two-dimensional vorticity equation (4), and their properties are mostp conveniently studied if the equation itself is written in self-similar variables  D x= t and D log.t / [18]. Assuming  D 1 and setting     1 x 1 x !.x; t/ D w p ; log.t / ; u.x; t / D p v p ; log.t / ; (17) t t t t one obtains for the rescaled vorticity w.; / and the rescaled velocity v.; / the following evolution equation @ w.; / C v.; /  rw.; / D Lw.; /;

(18)

where L is the linear operator defined by L D C

  r C 1: 2

(19)

The change of variables (17) coincides with Lundgren’s transformation (16), except that it is used here in the opposite way: starting from the two-dimensional vorticity !.x; t/ and velocity u.x; t /, one obtains the rescaled quantities w.; / and v.; / whose physical meaning is not immediately obvious. In addition, the rescaled equation (18) looks more complicated than the original vorticity equation (4) because the Laplace operator  is replaced by the Fokker-Planck operator L. However, from a mathematical point of view, the rescaled equation (18) has several advantages which greatly simplify the analysis. In particular, the operator L has (partially) discrete spectrum when considered in appropriate function spaces, and that observation is crucial for the stability analysis of the Lamb-Oseen vortex presented below. Moreover, the associated semigroup e L has nice confinement properties, as a consequence of which it is possible to use compactness methods to investigate the longtime behavior of solutions to the rescaled vorticity equation (18); see [19]. Due to scale invariance, the Biot-Savart law (5) is not affected by the change of variables (17). This means that the rescaled velocity v.; / can be reconstructed from the rescaled vorticity w.; / through the formula v.; / D K2D  w.; / ;

where

K2D ./ D

1 ? : 2 jj2

(20)

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By construction, for any ˛ 2 R, the Lamb-Oseen vortex w D ˛G, v D ˛v G is a stationary solution of (18). The dynamical relevance of this family of equilibria is demonstrated by the following global convergence result. Theorem 1 ([19]). For any initial data w0 2 L1 .R2 /, the rescaled vorticity equation (18) has a unique global solution w 2 C 0 .Œ0; 1/; L1 .R2 //. This solution satisfies kw. /kL1 .R2 /  kw0 kL1 .R2 / for all  0 and Z lim kw. /  ˛GkL1 .R2 / D 0;

!1

where

˛ D R2

w0 ./ d:

(21)

Theorem 1 shows that Lamb-Oseen vortices describe, to leading order, the longtime behavior of all solutions of the two-dimensional Navier-Stokes equations with integrable initial vorticity and nonzero total circulation ˛. Similar conclusions were previously obtained for small solutions [22] and for large solutions with small circulation [3]. The first step in the proof consists in showing that the original vorticity equation (4) is globally well posed in L1 .R2 /, that the L1 norm R of the solutions is nonincreasing in time and that the total circulation ˛ D R2 ! dx is a conserved quantity [1]. Since the change of variables (17) leaves the L1 norm invariant, the same conclusions hold for the rescaled vorticity equation (18) too. Then, in view of the confinement properties of the linear semigroup e L , one can show that the solutions of (18) are not only bounded but also relatively compact in the space L1 .R2 /. Finally, using appropriate Lyapunov functions [19] or monotonicity properties based on rearrangement techniques [8], one can prove that the omega-limit set in L1 .R2 / of any solution of (18) is included in the family of Lamb-Oseen vortices. As the total circulation is conserved, the omega-limit set is in fact reduced to the singleton f˛Gg, which proves (21). The interested reader is referred to [19, 23] for details. The global convergence result (21) is very general, but the proof sketched above is not constructive and does not yield any estimate on the time needed to reach the asymptotic regime described by the Lamb-Oseen vortex. Explicit estimates of the convergence time can however be obtained if the vorticity has a definite sign [19] or is strongly localized [15]. In the rest of this section, emphasis is put on local stability results, for which explicit bounds are also available.

2.1

Local Stability Results

Theorem 1 strongly suggests, but does not really prove, that the Lamb-Oseen vortex ˛G is a stable equilibrium of the rescaled vorticity equation (18) for any ˛ 2 R. Stability can be established by considering solutions of the form w D ˛G C w, Q v D ˛v G C v. Q The perturbations satisfy the evolution equation Q C vQ  r wQ D .L  ˛ƒ/w; Q @ w

(22)

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where L is given by (19), and ƒ is the nonlocal linear operator defined by ƒw Q D v G  r wQ C vQ  rG;

with vQ D K2D  w: Q

(23)

It is possible to prove that the perturbation equation (22) is globally well posed in Q D 0 is a stable equilibrium, but at this level of the space L1 .R2 / and that the origin w generality, little can be said about the longtime behavior of the solutions. However, more precise stability results can be obtained if one assumes that the vorticity is sufficiently localized in space. Given m 2 Œ0; 1, let m W Œ0; 1/ ! Œ1; 1/ be the weight function defined by 8 ˆ if m D 0; ˆ 1. In that case, it is useful to introduce the closed subspace  ˇ Z ˇ 2 2 w./ d D 0 ; (27) L0 .m/ D w 2 L .m/ ˇ R2

which happens to be invariant under the action of both linear operators L and ƒ. To study the stability of the origin w Q D 0 for the perturbation equation (22), it is useful to compute the spectrum of the linearized operator L  ˛ƒ in the (complexified) Hilbert space L2 .m/. In the simple case where ˛ D 0, the spectrum is explicitly known: Proposition 1 ([18]). For any m 2 Œ0; 1, the spectrum of the linear operator (19) in the weighted space L2 .m/ defined by (24) and (25) is 

m .L/ D

ˇ n kˇ 1 mo ˇ ˇ  2 C ˇ Re./   [  ˇk 2 N : 2 2 2

(28)

Moreover, if m > k C 1 for some k 2 N, then k D k=2 is an isolated eigenvalue of L, with (algebraic and geometric) multiplicity k C 1.

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It follows in particular from Proposition 1 that L has purely discrete spectrum in L2 .m/ when m D 1. This is easily understood if one observes that 1 .jj2 / D 2 e jj =4 D .4/1 G./1 , and that G 1=2 L G 1=2 D  

jj2 1 C : 16 2

(29)

The formal relation (29) implies that the operator L in L2 .1/ is unitarily equivalent to the harmonic oscillator   jj2 =16 C 1=2 in L2 .R2 /, the spectrum of which is the sequence .k /k2N , where k D k=2 has multiplicity k C 1. If m < 1, the discrete part of the spectrum persists because the corresponding eigenfunctions decay rapidly at infinity. In addition, any  2 C such that Re./ < .1  m/=2 is an eigenvalue of L in L2 .m/ with infinite multiplicity [18]; hence, the spectrum m .L/ also includes the closed half-plane Hm D f 2 C j Re./  .1  m/=2g. In the more interesting case where ˛ ¤ 0, the spectrum of L  ˛ƒ in L2 .m/ cannot be computed explicitly. However, upper bounds on the real part of the spectrum are sufficient for the stability analysis, and such estimates can be obtained by combining the following three observations. Observation 1. The operator ƒ is a relatively compact perturbation of L in L2 .m/, for any m 2 Œ0; 1. This is intuitively obvious, because ƒ is a first-order differential operator whose coefficients decay to zero at infinity, whereas L involves in particular the Laplace operator . By Weyl’s theorem, the essential spectrum [25] of L  ˛ƒ in L2 .m/ does not depend on ˛ and hence coincides with the closed half-plane Hm D f 2 C j Re./  .1  m/=2g by Proposition 1. It thus remains to locate isolated eigenvalues of L  ˛ƒ outside Hm . Observation 2. The isolated eigenvalues of L  ˛ƒ in L2 .m/ do not depend on m. Indeed, if w 2 L2 .m/ satisfies .L  ˛ƒ/w D w for some  2 C n Hm , one can show that w decays sufficiently fast at infinity so that w 2 L2 .1/ [19]. This means that isolated eigenvalues of L  ˛ƒ can be located by considering the particular case m D 1, where the spectrum is fully discrete and consists of a sequence of eigenvalues .k .˛//k2N with Re.k .˛// ! 1 as k ! 1. Observation 3. The operator ƒ is skew symmetric in L2 .1/, namely, hƒw1 ; w2 i C hw1 ; ƒw2 i D 0;

for all

w1 ; w2 2 D.ƒ/  L2 .1/;

(30)

where D.ƒ/  L2 .1/ is the (maximal) domain of the operator ƒ, and h; i denotes the scalar product in L2 .1/, which (up to an irrelevant factor) can be written in the form Z hw1 ; w2 i D G./1 w1 ./w2 ./ d: (31) R2

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To prove (30) one decomposes ƒ D ƒ1 C ƒ2 , where ƒ1 w D v G  rw and ƒ2 w D .K2D  w/  rG. If w1 ; w2 2 L2 .1/ belong to the domain of ƒ, then Z   hƒ1 w1 ; w2 i C hw1 ; ƒ1 w2 i D G 1 w2 v G  rw1 C w1 v G  rw2 d R2

Z D

R2

G 1 v G  r.w1 w2 / d D 0;

because the vector field G 1 v G is divergence-free. Moreover using the identity rG D  12 G and the Biot-Savart law (20), one obtains Z   1 hƒ2 w1 ; w2 i C hw1 ; ƒ2 w2 i D  .  v1 /w2 C .  v2 /w1 d 2 R2 Z Z  .  /? .  /? 1  w1 ./w2 ./ d d D 0; C D  4 R2 R2 j  j2 j  j2 because the last integrand vanishes identically. This proves (30). One can also show that the operator ƒ is not only skew symmetric but also skew adjoint in L2 .1/; see [34]. The observations above lead to the following spectral stability result for the Lamb-Oseen vortex in the space L2 .m/. Proposition 2 ([19]). For any ˛ 2 R and any m 2 Œ1; 1, the spectrum of the linearized operator L  ˛ƒ in the space L2 .m/ satisfies ˇ n o ˇ (32)

m .L  ˛ƒ/   2 C ˇ Re./  0 : Moreover, if m  2, then ˇ 1 ˇ :  2 C ˇ Re./   2

(33)

ˇ n o ˇ [  2 C ˇ Re./  1 :

(34)



m .L  ˛ƒ/  f0g [ Finally, if m  3, then 

m .L  ˛ƒ/  f0g [

1  2



Proof. As before let Hm D f 2 C j Re./  .1  m/=2g. By Observation 1 above, if m  1, the essential spectrum of L  ˛ƒ is included in the half-space H1 . Assume that  2 C n H1 is an isolated eigenvalue of L  ˛ƒ, and let w 2 L2 .m/ be a nontrivial eigenfunction associated with . Then w 2 L2 .1/ by Observation 2, and using Observation 3 one finds Re./hw; wi D Reh.L  ˛ƒ/w; wi D hLw; wi  0;

(35)

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because L is a nonpositive self-adjoint operator in L2 .1/, and ƒ is skew symmetric. This contradicts the assumption that Re./ > 0; hence, the whole spectrum of L  ˛ƒ in L2 .m/ is contained in the half-space H1 , as asserted in (32). As LG D ƒG D 0, it is clear that 0 is an eigenvalue of L  ˛ƒ for any m  0 and any ˛ 2 R. If m > 1, one can write L2 .m/ D RG ˚ L20 .m/, where L20 .m/ is the hyperplane defined in (27), and this decomposition is left invariant by both operators L and ƒ. Now, the same argument as above shows that, if m  2, the spectrum of the operator L  ˛ƒ acting on L20 .m/ is contained in the half-plane H2 , because L  1=2 on L20 .1/. This proves (33). Finally, it is easy to verify that L.@i G/ D  12 @i G for i D 1; 2, and differentiating the identity v G  rG D 0, one finds that ƒ.@i G/ D 0 for i D 1; 2. This means that 1=2 is an eigenvalue of L  ˛ƒ for any m  0 and any ˛ 2 R. As above, if m > 2, one has the invariant decomposition L2 .m/ D f˛G j ˛ 2 Rg ˚ fˇ1 @1 G C ˇ2 @2 G j ˇ1 ; ˇ2 2 Rg ˚ L200 .m/ ; where L200 .m/ D



ˇ Z ˇ w 2 L20 .m/ ˇ

R2

i w./ d D 0 for i D 1; 2 :

(36)

As L  1 on L200 .1/, the same argument shows that the spectrum of the operator L  ˛ƒ acting on L200 .m/ is contained in the half-plane H3 , if m  3. This proves (34).  The linear operator L  ˛ƒ is the generator of a strongly continuous semigroup in the space L2 .m/ for any ˛ 2 R and any m 2 Œ0; 1 [18]. The following linear stability result is a natural consequence of Proposition 2 and its proof. Proposition 3 ([19]). For any ˛ 2 R and any m > 1, there exists a positive constant C such that ke .L˛ƒ/ kL2 .m/!L2 .m/  C;

for all  0:

(37)

Moreover, if m > 2, then ke .L˛ƒ/ kL2 .m/!L2 .m/  C e  =2 ; 0

0

for all  0:

(38)

for all  0:

(39)

Finally, if m > 3, then ke .L˛ƒ/ kL2

2 00 .m/!L00 .m/

 C e  ;

When studying the stability of the Lamb-Oseen vortex, there is no loss of generality in considering perturbations with zero total circulation. RIndeed, if w D ˛G C w Q for some w Q 2 L2 .m/ with m > 1, then defining ˛Q D R2 w./ Q d one

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can write w D .˛ C ˛/G Q C .w Q  ˛G/, Q where by construction w Q  ˛G Q 2 L20 .m/. Thus perturbations with nonzero circulation of the vortex ˛G can be considered as perturbations with zero circulation of the modified vortex .˛ C ˛/G. Q As the total circulation is a conserved quantity, the subspace L20 .m/ is invariant under the evolution defined by the full perturbation equation (22). By Proposition 3, the linear semigroup e .L˛ƒ/ is exponentially decaying in L20 .m/ if m > 2, and using that information, it is routine to deduce the following asymptotic stability result, which is the main outcome of this section. Proposition 4 ([19]). Fix ˛ 2 R and m 2 .2; 1. There exist positive constants  and C such that, for all w Q 0 2 L20 .m/ satisfying kw Q 0 kL2 .m/  , the rescaled vorticity equation (18) has a unique global solution w 2 C 0 .Œ0; 1/; L2 .m// with initial data w0 D ˛G C w Q 0 . Moreover, the following estimate holds kw. /  ˛GkL2 .m/  C kw0  ˛GkL2 .m/ e  =2 ;

0:

(40)

If m > 2, the codimension 3 subspace L200 .m/ is also invariant under the evolution defined by the full perturbation equation (18). As a consequence, if w Q0 2 L200 .m/, the solution of (18) given by Proposition 4 satisfies w. /  ˛G 2 L200 .m/ for all  0. If m > 3, one can then use (39) to conclude that kw. /  ˛GkL2 .m/  C kw0  ˛GkL2 .m/ e  ;

0:

As is shown in [11, 19], if ˛ ¤ 0, the assumption that w0 has vanishing first-order moments does not really restrict the generality, because this condition can always be met by a suitable translation of the initial data. Remark 2. If m D 1, one can show that the Lamb-Oseen vortex ˛G is uniformly stable for all ˛ 2 R in the sense that the constants  and C in Proposition 4 do not depend on ˛ [11]. This is in sharp contrast with what happens for shear flows, such as the Poiseuille flow in a cylindrical pipe or the Couette-Taylor flow between two rotating cylinders. In such examples, the laminar stationary flow undergoes an instability, of spectral or pseudospectral nature, when the Reynolds number is sufficiently large. In contrast, a fast rotation has rather a stabilizing effect on vortices, as the analysis below reveals.

2.2

Large Reynolds Number Asymptotics

Proposition 2 above gives uniform estimates on the spectrum of the linearized operator L  ˛ƒ, which are sufficient to prove stability of the Lamb-Oseen vortex for all values of the circulation parameter ˛ 2 R. However, such estimates do not describe how the spectrum changes as the circulation parameter varies. The most relevant regime for turbulent flows is of course the high Reynolds number limit

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where j˛j ! 1, which deserves a special consideration. As the essential spectrum of L  ˛ƒ in the space L2 .m/ does not depend on m, it is most convenient to work in the limiting space X D L2 .1/, equipped with the scalar product (31). In that space, as was already mentioned, the spectrum of L  ˛ƒ is discrete and consists of a sequence of eigenvalues .k .˛//k2N with Re.k .˛// ! 1 as k ! 1. It follows from Proposition 1 that k .0/ D k=2, for any k 2 N, and the goal of this section is to investigate the behavior of the real part of k .˛/ as j˛j ! 1. The starting point of the analysis is the determination of the kernel of the skewsymmetric operator ƒ. Let X0  X denote the closed subspace containing all radially symmetric functions. If w 2 X0 , the associated velocity field v D K2D  w satisfies   v./ D 0, and it follows that ƒw D 0, hence X0  ker.ƒ/. On the other hand, it was already observed that ƒ.@i G/ D 0 for i D 1; 2. The following result asserts that the kernel of ƒ does not contain any more elements: Lemma 1 ([34]). ker.ƒ/ D X0 ˚ fˇ1 @1 G C ˇ2 @2 G j ˇ1 ; ˇ2 2 Rg. In view of Lemma 1, the subspace ker.ƒ/  X is invariant under the action of both operators L and ƒ, and the orthogonal complement ker.ƒ/? is invariant too because L is self-adjoint and ƒ is skew adjoint. Inside ker.ƒ/, the spectrum of L  ˛ƒ  L does not depend on the circulation parameter ˛, and consists of all negative integers in addition to the double eigenvalue 1=2. In fact, for any n 2 N, the eigenfunction corresponding to the eigenvalue n is the radially symmetric Hermite function n G. The only difficult task is therefore to study the spectrum of L?  ˛ƒ? , which is defined as the restriction of L  ˛ƒ to the orthogonal complement ker.ƒ/? . That spectrum does depend in a nontrivial way upon the parameter ˛. It happens that the real parts of all eigenvalues converge to 1 as j˛j ! 1, which is of course compatible with the uniform bounds given by Proposition 2. This phenomenon illustrates the stabilizing effect of fast rotation on Lamb-Oseen vortices. Two natural quantities can be introduced to accurately measure the effect of fast rotation. For any ˛ 2 R, one can define the spectral lower bound ˇ n o ˇ †.˛/ D inf Re.z/ ˇ z 2 spec.L? C ˛ƒ? / ; or the pseudospectral bound  ‰.˛/ D

1

sup k.L?  ˛ƒ?  i / kX!X

1 :

2R

In the definition of †.˛/, the sign of the linearized operator has been changed to obtain a positive quantity. Although the spectral and pseudospectral bounds are of rather different nature, there is a simple one-sided relation between them: Lemma 2. For any ˛ 2 R one has †.˛/  ‰.˛/  1.

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Proof. Fix ˛ 2 R. By Lemma 1, one has ker.ƒ/?  L200 .m/, hence †.˛/  1 by (34). On the other hand, if .L  ˛ƒ C /w D 0 for some  2 C and some w 2 ker.ƒ/? such that hw; wi D 1, then .L  ˛ƒ C i Im.//w D  Re./w, hence Re./  k.L?  ˛ƒ? C i Im.//1 k1  ‰.˛/: This proves that †.˛/  ‰.˛/. Finally, the proof of Proposition 2 shows that the operator L?  ˛ƒ? C 1 is m-dissipative in the sense of [26]. This in particular implies that k.L?  ˛ƒ?  i /1 k  1 for all  2 R, hence ‰.˛/  1. The stabilizing effect in the large Reynolds number limit is qualitatively illustrated by the following result: Proposition 5 ([34]). One has ‰.˛/ ! 1 and †.˛/ ! 1 as j˛j ! 1. The proof given in [34] actually shows that †.˛/ ! 1 as j˛j ! 1 but can be easily modified to yield the stronger conclusion that ‰.˛/ ! 1. For the stability analysis of the Lamb-Oseen vortex ˛G, the divergence of the spectral bound means that the decay rate in time of perturbations in ker.ƒ/? becomes arbitrarily large as j˛j ! 1. On the other hand, using the divergence of the pseudospectral bound, one can show that the basin of attraction of the Lamb-Oseen vortex, in the weighted space L2 .1/, becomes arbitrarily large as j˛j ! 1. It should be emphasized, however, that the argument used in [34] is nonconstructive and does not provide any explicit estimate on the quantities ‰.˛/ and †.˛/ for large j˛j. In fact, there are good reasons to conjecture that †.˛/ D O.j˛j1=2 / and ‰.˛/ D O.j˛j1=3 / as j˛j ! 1. First of all, extensive numerical calculations performed by Prochazka and Pullin [40, 41] indicate that †.˛/ D O.j˛j1=2 / as j˛j ! 1. Next, the conjecture is clearly supported by rigorous analytical results on model problems [9]. In particular, for the simplified linear operator L  ˛ƒ1 where the nonlocal part ƒ2 has been omitted, it can be proved that ‰.˛/ D O.j˛j1=3 / as j˛j ! 1 [4]. The same result holds for the full linearized operator L  ˛ƒ restricted to a smaller subspace than ker.ƒ/? , where a finite number of Fourier modes with respect to the angular variable in polar coordinates have been removed [5]. The general case is still under investigation [7]. Assuming that the conjecture above is true, it is worth noting that the pseudospectral bound ‰.˛/ and the spectral bound †.˛/ have different growth rates as j˛j ! 1. This reflects the fact that the linearized operator L  ˛ƒ becomes highly nonself-adjoint in the fast rotation limit. Indeed, for self-adjoint or normal operators, it is easy to verify that the spectral and pseudospectral bounds always coincide.

2.3

Lamb-Oseen Vortices in Exterior Domains

As was already mentioned, the Lamb-Oseen vortex plays a double role in the dynamics of the Navier-Stokes equations in the whole space R2 : it is the unique solution of the system when the initial vorticity is a Dirac measure, and it describes

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the longtime asymptotics of all solutions for which the vorticity distribution is integrable and has nonzero total circulation. The proofs given in [19] demonstrate that both properties are closely related, due to scale invariance. Now, if the fluid is contained in a two-dimensional domain  R2 and satisfies (for instance) no-slip boundary conditions on @ , scale invariance is broken, and there is no simple relation anymore between the Cauchy problem for singular initial data and the longtime asymptotics of general solutions. Both questions are interesting and, at present time, largely open. In this section, the relatively simple case of a two-dimensional exterior domain is considered, where a few results concerning the longtime behavior of solutions with nonzero circulation at infinity have been obtained recently. Let  R2 be a smooth exterior domain, namely, an unbounded connected open set with a smooth compact boundary @ . The Navier-Stokes equations in with no-slip boundary conditions can be written in the following form: 8 < @t u C .u  r/u D u  rp ; div u D 0 ; for x 2 ; t > 0; (41) u.x; t / D 0 ; for x 2 @ ; t > 0; : u.x; 0/ D u0 .x/ ; for x 2 ; where p denotes the ratio of the pressure to the fluid density. The vorticity ! D @1 u2  @2 u1 still satisfies the simple evolution Eq. (4) (with  D 1), but the assumption that u D 0 on @ translates into a nonlinear, nonlocal boundary condition for !, which is very difficult to handle. So, whenever possible, it is preferable to work directly with the velocity formulation (41). If the initial velocity u0 belongs to the energy space ˇ o n ˇ L2 . / D u 2 L2 . /2 ˇ div u D 0 in ; u  n D 0 on @ ; where n denotes the unit normal on @˝, it is well known that system (41) has a unique global solution satisfying the energy identity 1 ku.; t /k2L2 . / C 2

Z 0

t

kru.; s/k2L2 . / ds D

1 ku0 k2L2 . / ; 2

t  0:

That solution converges to zero in L2 . / as t ! 1 [38], which means that the longtime behavior of all finite energy solutions is trivial. However, as in the whole plane R2 , one can consider flows with nonzero circulation at infinity: I ˛ D lim

R!1 jxjDR

.u1 dx1 C u2 dx2 / ¤ 0;

in which case the kinetic energy is necessarily infinite, and the longtime behavior may be nontrivial. To construct such solutions, it is convenient to introduce a smooth cutoff function  W R2 ! Œ0; 1 such that  vanishes in a neighborhood of R2 n and .x/ D 1

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whenever jxj is sufficiently large. For technical reasons, one also assumes that  is radially symmetric and nondecreasing along rays. The truncated Oseen vortex u .x; t / D

 jxj2 1 x?   4.1Ct/ 1  e .x/; 2 jxj2

x 2 R2 ;

t  0;

(42)

is a divergence-free velocity field which vanishes identically in a neighborhood of R2 n and coincides with the Lamb-Oseen vortex (with unit circulation) far away from the origin. In particular u … L2 . /. The corresponding vorticity distribution   !  D @1 u2  @2 u1 reads: !  .x; t / D

 jxj2 jxj2 1 1 1   4.1Ct/ e  4.1Ct/ .x/ C 1  e x  r.x/; 4.1 C t / 2 jxj2

(43)

R and satisfies !  .x; t / dx D 1 for all t  0. Of course the velocity field u is not an exact solution of the Navier-Stokes equations (41) (unless D R2 and   1), but the following result shows that it is a globally stable asymptotic solution. Theorem 2 ([14,35]). Fix q 2 .1; 2, and let  D 1=q1=2. There exists a constant  > 0 such that, for all initial data of the form u0 D ˛u .; 0/ C v0 with j˛j   and v0 2 L2 . / \ Lq . /2 , the Navier-Stokes equations (41) have a unique global solution which satisfies ku.; t /  ˛u .; t /kL2 . / C t 1=2 kru.; t /  ˛ru .; t /kL2 . / D O.t  / ;

(44)

as t ! C1. Moreover, if q D 2, then ku.; t /  ˛u .; t /kL2 . / ! 0 as t ! C1. Several comments are in order. Existence and uniqueness of global solutions to the Navier-Stokes equations (41) for a class of infinite energy initial data including those considered in Theorem 2 were established by Kozono and Yamazaki in [27]. The novelty here is the description of the longtime asymptotics for a specific family of solutions, corresponding to spatially localized perturbations of the truncated Oseen vortex. Theorem 2 is a global stability result, in the sense that arbitrary large perturbations v0 of the vortex can be considered. There is, however, a limitation on the size of the circulation parameter ˛, which is probably of technical nature. To remove that restriction, it seems rather natural to use the vorticity formulation and the nice properties of the linearized operator established in Sect. 2.1, but this is difficult in the present case because the boundary condition for the vorticity is very awkward. The parameter q in Theorem 2 measures the spatial decay of the initial perturbations v0 to the Oseen vortex and is directly related to the decay rate in time (called ) of the corresponding solutions. If q < 2, then  > 0, and it is shown in [14] that the constant  depends only on q and not on the domain . The limiting case where q D 2 was treated in [29, 35]. The main original ingredient in the proof of Theorem 2 is a logarithmic energy estimate that is worth discussing briefly. For solutions of the Navier-Stokes

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equations (41) of the form u.x; t / D ˛u .x; t / C v.x; t /, the perturbation v satisfies @t v C ˛.u ; r/v C ˛.v; r/u C .v; r/v D v C ˛R  rq;

div v D 0; (45)

where the source term R D u @t u measures by how much the truncated vortex u fails to be an exact solution of (41). Taking into account the uniform bounds kru .; t /kL1 .R2 / 

b ; 1Ct

kR .; t /kL2 .R2 / 

 ; 1Ct

which hold for some positive constants b;  depending only on the cutoff , a standard energy estimate yields the differential inequality 1 d bj˛j j˛j kv.t /k2L2 . / C krv.t /k2L2 . /  kv.t /k2L2 . / C kv.t /kL2 . / ; 2 dt 1Ct 1Ct (46) which predicts a polynomial growth of the L2 norm kv.t /kL2 . / . This naive estimate can be substantially improved if one observes that the truncated Oseen vortex u decays like jxj1 as jxj ! 1 and thus nearly belongs to the energy space. The optimal result is: Proposition 6. There exists a constant K > 0 such that, for any ˛ 2 R and any v0 2 L2 . /, the solution of (45) with initial data v0 satisfies, for all t  1, Z

kv.t /k2L2 . / C

t 0

  krv.s/k2L2 . / ds  K kv0 k2L2 . / C˛ 2 log.1Ct /C˛ 2 log.2Cj˛j/ : (47)

In the proof of Theorem 2, the logarithmic bound (47) is combined with standard energy estimates for a fractional primitive of the velocity field to prove that v.; t / converges to zero in L2 . / as t ! 1; see [12,14]. The optimal decay rate in (44) is then obtained by a direct study of small solutions to the perturbation equation (45). Although Theorem 2 is established using the velocity formulation of the NavierStokes system, it is instructive to see what it implies for the vorticity distribution !. Assume, for instance, that the initial vorticity !0 D @1 .u0 /2  @2 .u0 /1 is sufficiently localized so that Z .1 C jxj2 /m j!0 .x/j2 dx < 1;

for some m > 1. By Hölder’s inequality thisR implies that !0 2 L1 . /. Then denoting v0 D u0  ˛u .; 0/ where ˛ D !0 .x/ dx, it follows that v0 2 L2 . / \ Lq . /2 for any q 2 .1; 2/ such that q > 2=m [18]. In particular, if

13 Existence and Stability of Viscous Vortices

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j˛j  , the conclusion of Theorem 2 holds. Moreover, the vorticity satisfies Z

j!.x; t/  ˛!  .x; t /j dx D O.t  log t /;

as

t ! 1;

(48)



where !  .x; t / is defined in (43); see [12]. In both convergence results (44) and (48), one can replace the truncated Oseen vortex by the original Lamb-Oseen vortex for which   1, because the additional error converges to zero like O.t 1 / as t ! 1.

3

Axisymmetric Vortex Rings and Filaments

When restricted to axisymmetric flows without swirl, the three-dimensional NavierStokes equations bear some similarity with the two-dimensional situation considered in the previous section. The only nonzero component of the vorticity vector satisfies Eq. (9), which can be written in the equivalent form  !  @t ! C @r .ur ! / C @z .uz ! / D  .@2r C @2z /! C @r : r

(49)

The analogy is most striking if one introduces the related quantity  D ! =r, which satisfies the advection-diffusion equation @t  C u  r D  C

2 @r ; r

(50)

where u  r D ur @r C uz @z and  D @2r C 1r @r C @2z . Equation (50) is considered in the half-plane D f.r; z/ 2 R2 j r > 0; z 2 Rg, with homogeneous Neumann boundary conditions on @ . It is clear from (50) that .r; z; t / obeys the parabolic maximum principle, and this provides a priori estimates on the solutions which imply that the Cauchy problem for the axisymmetric Navier-Stokes equations is globally well posed, without any restriction on the size of the initial data. The first results in this direction were obtained by Ladyzhenskaya [30] and by Ukhovskii and Yudovich [45], for finite energy solutions. Recently, it was shown in [16] that the vorticity equation (49) is globally well posed in the scale invariant space L1 . ; dr dz/, equipped with the norm Z Z j! .r; z/j dr dz D j.r; z/j r dr dz: k! kL1 . / D



The proof follows remarkably the same lines as in the two-dimensional case and, in particular, uses very similar function spaces. Solutions constructed in this way have infinite energy in general, but if the initial vorticity !0 decays somewhat faster at infinity than what is necessary to be integrable, the velocity field becomes square integrable for all positive times.

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It is also possible to solve the Cauchy problem for Eq. (49) in the more general situation where the initial vorticity is a finite measure. Global existence and uniqueness are established in [16] assuming that the total variation norm of the atomic part of the initial vorticity is small compared to the viscosity parameter. The general case is open to the present date, but interesting results have been obtained for circular vortex filaments, which correspond to the situation where the initial vorticity is a Dirac measure. The resulting solutions can be considered as the analogue of the family of Lamb-Oseen vortices in R2 . These solutions cannot be written in explicit form, but small-time asymptotic expansions can be computed which involve the two-dimensional profiles G and v G defined in (7). If the circulation parameter  is small compared to the viscosity , the results of [16] imply the existence of a unique global solution to (49) with initial vorticity !0 D ı.Nr;Nz/ , for any .Nr; zN/ 2 . For larger circulations, the following existence result was recently established by Feng and Šverák: Proposition 7 ([6]). Fix  > 0, .Nr; zN/ 2 , and  > 0. Then the axisymmetric vorticity equation (49) has a nonnegative global solution such that ! .t / * ı.Nr;Nz/ as t ! 0. Moreover, this solution satisfies, for all t > 0, Z Z ! .r; z; t / dr dz  ; r 2 ! .r; z; t / dr dz D  rN 2 : (51)



Needless to say, the assumption  > 0 does not restrict the generality, because the corresponding result for  < 0 can be obtained by symmetry. Proposition 7 is proved by a very general approximation argument, which provides global existence without any restriction on the size of the circulation parameter but does not imply uniqueness and does not give any precise information on the qualitative behavior of the solution for short times. Using more sophisticated techniques, a more accurate result can be established: Theorem 3 ([17]). Fix  2 R, .Nr; zN/ 2 , and  > 0. Then the axisymmetric vorticity equation (49) has a unique global mild solution ! 2 C 0 ..0; 1/; L1 . / \ L1 . // such that sup k! .t /kL1 . / < 1 ;

and

! .t / *  ı.Nr;Nz/

as t ! 0:

(52)

t>0

In addition, there exists a constant C > 0 such that the following estimate holds: Z ˇ ˇ ˇ! .r; z; t / 

as long as

p

p ˇ .rNr /2 C.zNz/2 ˇ  t rN  4t e log p ; ˇ dr dz  C jj 4t rN t

(53)

t  rN =2.

Since the existence of a global solution to (49) satisfying (52) is already asserted by Proposition 7, the main contributions of Theorem 3 are the uniqueness of that

13 Existence and Stability of Viscous Vortices

707

solution and its asymptotic behavior as t ! 0, as described in (53). The first step in the proof is a localization estimate, which can be established using a Gaussian upper bound on the fundamental solution of the “linear” Eq. (49) for ! , where the velocity field u D .ur ; uz / is considered as given. It is found that, for any  > 0, there exists a constant C > 0 such that   C jj .r  rN /2 C .z  zN/2 ; j! .r; z; t /j  exp  t .4 C /t

.r; z/ 2 ;

t > 0: (54)

R Moreover ! .r; z; t / dr dz converges to  as t ! 0. The second step consists in introducing self-similar variables, in the spirit of (17). The rescaled vorticity f and velocity v are defined by  f ! .r; z; t / D t



 r  rN z  zN p ;p ;t ; t t

 u.r; z; t / D p v t



 r  rN z  zN p ;p ;t ; t t

and the following dimensionless quantities are introduced: r  rN RD p ; t

z  zN Z D p ; t

p D

t ; rN

˛D

 : 

The evolution equation for the new function f .R; Z; t/ reads tft C ˛ .@R .vR f / C @Z .vZ f // D Lf C @R



f  ; 1 C R

(55)

where as in (19) L D @2R C @2Z C

R Z @R C @Z C 1: 2 2

Note that Eq. (55) lives in the time-dependent domain where 1 C R > 0, but using the homogeneous Dirichlet boundary condition, one can extend the rescaled vorticity by zero outside that domain and consider it as defined on the whole plane R2 . In the small time limit  ! 0, the system formally reduces to the equation for perturbations around Oseen vortex, which was studied in detail in Sect. 2.1, and the proof of Theorem 3 consists in showing that this intuition is indeed correct. The Gaussian bound (54) provides a uniform control on the solution of (55) in the weighted space Xt defined by the norm kf .t/k2Xt D

Z

f .R; Z; t/2 e .R 1CR>0

2 CZ 2 /=4

dR dZ;

t > 0;

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which coincides when t D 0 with the norm of the space L2 .1/ introduced in (25). A compactness argument, as in the proof of Theorem 1, can then be invoked to show that f .R; Z; t/ necessarily converges to the Oseen vortex profile as t ! 0: lim kf .t/  GkXt D 0;

where

t!0

G.R; Z/ D

1 .R2 CZ 2 /=4 : e 4

(56)

The final step is an energy estimate which shows that, for some positive constants C and ı,

t

d kf .t/k2Xt  ıkf .t/k2Xt C C  2 j log j2 ; dt

(57)

when t > 0 is sufficiently small. The differential inequality (57) relies on the spectral properties of the linear operator L in the space L2 .1/, which were established in Sect. 2.1. As Xt ,! L1 .R2 /, it immediately implies estimate (53) in Theorem 3. Moreover, a similar argument applied to the difference f1  f2 of two solutions of (49) satisfying (52) leads to the conclusion that f1  f2 , which yields uniqueness. Theorem 3 shows that the two-dimensional Lamb-Oseen vortex naturally appears in the axisymmetric case too, where it describes the short-time behavior of solutions arising from vortex filaments as initial data; see (53). However, the longtime asymptotics are very different in both situations, as can be seen from the following result: Proposition 8 ([16]). Assume that the initial vorticity !0 2 L1 . / is nonnegative and has finite impulse: I D

Z

r 2 !0 .r; z/ dr dz < 1 :

(58)



Then the unique global solution of (49) satisfies ˇ ˇ 2 2 ˇ ˇ2 p p I  r Cz ˇ ˇ D 0: 4 lim sup t ! .r t ; z t ; t /  p r e ˇ t!1 .r;z/2 ˇ 16 

(59)

In particular k! .t /kL1 . / D O.t 2 / as t ! 1. Proposition 8 applies in particular to the vortex rings constructed in Proposition 7 and Theorem 3. It shows that the longtime asymptotics are described, to leading order, by a self-similar solution of the linearized equation obtained by setting u D 0 in (49). This is in sharp contrast with what happens in the two-dimensional case.

13 Existence and Stability of Viscous Vortices

4

709

Existence and Stability of Burgers Vortices

As was mentioned in the introduction, the Burgers vortex is a simple but important model in fluid mechanics, describing the balance between the dissipation due to the viscosity and the vorticity stretching through the action of a background straining flow. By rescaling variables in a suitable manner (see, e.g., [20]), one can assume without loss of generality that the rates of the linear strain in (11) have the following form 1 D 

1C ; 2

2 D 

1 ; 2

3 D 1:

(60)

Here  2 Œ0; 1/ is a free parameter that represents the asymmetry of the strain, and the case  D 0 corresponds to an axisymmetric strain. The Burgers vortex with circulation ˛ and asymmetry  is a two-dimensional stationary vorticity field of the form !;˛ D .0; 0; !;˛ /> . In view of Eq. (12) and (13), this means that the third component !;˛ depends only on the horizontal variable x D .x1 ; x2 / 2 R2 and satisfies the following elliptic problem in R2 : Z L !  .v; r/! D 0;

v D K2D  !;

! dx D ˛;

(61)

R2

where K2D .x/ D x ? =.2jxj2 /, and L is the two-dimensional differential operator defined by L D  C

1C 1 x1 @1 C x2 @2 C 1 D L C M; 2 2

M D

x1 x2 @1  @2 : 2 2 (62)

When  D 0, Eq. (61) has the explicit solution ! D ˛G, which is the classical axisymmetric Burgers vortex [2] with circulation ˛. Note that ˛G is in fact the unique solution of (61) in the space L1 .R2 /, as can be deduced from Theorem 1. Due to its simple explicit expression, the axisymmetric Burgers vortex is often used for comparison with experiments. However, the vortex tubes observed in real flows or numerical simulations usually exhibit an elliptical core region, rather than a circular one, because the local strain due to the background flow is not axisymmetric in general. It is therefore important to propose a model which takes into account the asymmetry of the strain in an appropriate way and allows one to understand its influence on the shape of the vortex tubes. This motivates the study of the Burgers vortex in the general case where the asymmetry parameter  is nonzero [39, 41, 42]. In that situation, solutions of (61) cannot be written in explicit form and have to be constructed by a rigorous mathematical argument. The aim of this section is to give an overview of the mathematical results available by now about the existence of asymmetric Burgers vortices (Sect. 4.1) and their stability with respect to two- or three-dimensional perturbations (Sects. 4.2 and 4.3).

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4.1

Existence and Uniqueness of Asymmetric Burgers Vortices

Since an explicit representation is no longer available for asymmetric Burgers vortices, existence of such solutions is the first question to address. One of the key observations is that, as the asymmetry parameter  in (60) is increased, the localizing effect due to the linear strain becomes weaker in the x2 direction. This phenomenon is illustrated by the shape of the function p

1  2  1C x 2  1 x 2 x 2 R2 ; (63) e 4 1 4 2; 4 R which solves the equation L G D 0 in R2 with R2 G dx D 1. The form of G indicates that asymmetric Burgers vortices, if they exist, still have a Gaussian decay at infinity but with a rate that becomes slower as  increases. Therefore, the function space L2 .1/ defined in (25) has to be modified in an appropriate way to allow for a general asymmetry parameter  2 Œ0; 1/. In view of (63), it is rather natural to introduce the function G .x/ D

G .x/ D

1    1 jxj2 e 4 ; 4

x 2 R2 ;

(64)

and the weighted L2 spaces 

2

L .1I / D

L20 .1I /

 D

Z ˇ ˇ 2 f 2 L .R / ˇ kf kL2 .1I/ WD 2

2

ˇ Z ˇ f 2 L .1I / ˇ

dx jf .x/j 0 such that, for all ˛ 2 R with j˛j  R0 ./, there exists an asymmetric Burgers vortex !;˛ 2 Pe L2 .1I / solving (61) and satisfying k!;˛  ˛G  w1 kW 1;2 .1I/ 

C ./ : 1 C j˛j

(73)

Here Pe is the even projection defined by Pe D ˚n2Z P2n , and the constants R0 ./; C ./ satisfy lim R0 ./ D lim C ./ D 1:

!1

!1

The conclusion of Theorem 5 was first established in [21] assuming 0   12 , in which case estimate (73) holds in the stronger norm of W 1;2 .1/ and not just in W 1;2 .1I /. The result was then extended to the larger range  2 Œ0; 12 / in [32], using again the space W 1;2 .1/, and the general case where  2 Œ0; 1/ was finally settled in [33]. The basic strategy in [21, 32, 33] is to construct a solution of (61) as a perturbation of the leading order approximation ˛G C w1 , in such a way that estimate (73) holds. To explain this idea more precisely, it is convenient to introduce the perturbation w.1/ D !;˛  ˛G  w1 , which has to solve the system   L  ˛ƒG  ƒw1 w.1/ D .K2D  w.1/ ; r/w.1/ C f ;

Z

w.1/ dx D 0; R2

f D L w1 C .K2D  w1 ; r/w1 : (74)

13 Existence and Stability of Viscous Vortices

713

To show that w.1/ is of order O.j˛j1 / as j˛j ! 1, the key observation is that the source term f in (74) also belongs to the range of ƒG . Indeed, a similar argument as in [21, Proposition 3.1] implies the existence of a unique h 2 .I  P0 /Pe W 1;2 .1/ satisfying ƒG h D f . Then f is decomposed as f D ƒG h D 

  1 1 L  ˛ƒG  ƒw1 h C L  ƒw1 h ; ˛ ˛

and Eq. (74) can thus be reduced to the following system for w.2/ D w.1/ C ˛ h : 

L ˛ƒG ƒw1  1 h ˛

Z  w D .K2D  w ; r/w C F;˛ ; w.2/ dxD0; ˛ R2    F;˛ D L  ƒw1 h C .K2D  h ; r/h : ˛ (75)



.2/

.2/

.2/

 / in L20 .1I 0/ as j˛j ! 1. It is clear that the source term ˛ F;˛ is of order O. j˛j  Since ˛ ƒh is a lower order perturbation that becomes small in the regime where j˛j 1, the crucial step to establish (73) is to prove the invertibility of the operator

L  ˛ƒG  ƒw1 ;

in the space L20 .1I /;

(76)

together with a uniform estimate for its inverse when j˛j 1. This is an easy task if  2 Œ0; 12 / is small, because one can then work in the space L20 .1/ instead of L20 .1I / and consider the operator in (76) as a small perturbation of the more familiar operator L  ˛ƒG , which has been thoroughly studied in Sect. 2. As was mentioned in (30), the operator ƒG is skew symmetric in L2 .1/, namely, hƒG f; giL2 .1/ C hf; ƒG giL2 .1/ D 0;

for all f; g 2 W 1;2 .1/:

Moreover, it follows from (29) that L is a self-adjoint operator in L2 .1/ with compact resolvent, which satisfies the lower bound L  0 in L2 .1/ and L  12 in L20 .1/. These two observations yield the uniform lower bound   1 h  L C ˛ƒG f; f iL2 .1/  kf k2L2 .1/ ; 2

f 2 L20 .1/ \ D.L/;

(77)

which in turn implies that k.L C ˛ƒG /1 kL2 .1/!L2 .1/  2 for all ˛ 2 R. A 0 0 straightforward perturbation argument then gives a uniform estimate on .L C ˛ƒG C ƒw1 /1 in L20 .1/ if  is sufficiently small; see [21, Proposition 2.1]. For larger values of , the term ƒw1 is not a small perturbation anymore, and the invertibility of the operator L  ˛ƒG  ƒw1 in L20 .1I / is not known for arbitrary ˛ 2 R. However, if j˛j is sufficiently large (depending on ), one can exploit the stabilizing effect that was already discussed in Sect. 2.2 for the simpler

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T. Gallay and Y. Maekawa

operator L  ˛ƒG ; see Proposition 5. For the modified operator L  ˛ƒG with  2 Œ0; 1/, one has the following estimates lim k.I  P0 /.L C ˛ƒG /1 Pe kL2 .1I/!L2 .1I/ D 0; 0

j˛j!1

0

lim kP0 .L C ˛ƒG /1 .I  P0 /Pe kL2 .1I/!L2 .1I/ D 0; 0

j˛j!1

(78)

0

which are proved in [32] for  2 Œ0; 12 / and in [33] for arbitrary  2 Œ0; 1/. Roughly speaking, this means that the nonradially symmetric elements of Pe L2 .1I / are strongly attenuated under the action of .L C ˛ƒG /1 when j˛j is large. In addition, since the function w1 belongs to P2 L2 .1/ C P2 L2 .1/, one has the identity P0 ƒw1 P0 f D 0

hence

P0 ƒw1 f D P0 ƒw1 .I  P0 /f;

(79)

for all f 2 Pe W 1;2 .1I /. Combining (78) and (79), it is possible to show that ƒw1 is a relatively small perturbation of L  ˛ƒG in Pe L20 .1I / if j˛j is sufficiently large (depending on ), and this implies the invertibility of L  ˛ƒG C ƒw1 in Pe L20 .1I 0/ and provides a uniform bound for the inverse when j˛j 1. When  2 Œ0; 12 /, the resolvent estimates (78) also hold in the smaller space 2 L0 .1/, instead of L20 .1; /, and are substantially easier to prove because one can then use the convenient property that ƒG is skew symmetric; see [21, 32]. However, when  2 Œ 12 ; 1/, the Burgers vortex does not belong to L2 .1/ anymore, as can be seen from the shape of the function G in (63). One is then forced to work in a wider space such as L2 .1I /, where the operator ƒG is no longer skew symmetric. The key idea in [33] to overcome this difficulty is to construct explicitly a bounded and invertible operator T so that ƒG T , the right action of T on ƒG , becomes skew symmetric. With this skew-symmetrizer T , the equation .L  ˛ƒG /w D f is written in the equivalent form .L  ˛ƒG T /T 1 w D L .T  I /T 1 w C f: Using the skew symmetry of ƒG T , one can show that the operator L  ˛ƒG T satisfies resolvent bounds similar to (78), and the additional term L .T  I / can be considered as a relatively small perturbation. This implies the invertibility of L  ˛ƒG T C L .T  I /, hence of L  ˛ƒG . The argument also yields uniform estimates on the inverse .L  ˛ƒG /1 , which in turn make it possible to treat ƒw1 as a relatively small perturbation when j˛j is sufficiently large (depending on ), thus concluding the proof of Theorem 5. The uniqueness of asymmetric Burgers vortices in a suitable class of functions is also available for some range of parameters .; ˛/.

13 Existence and Stability of Viscous Vortices

715

Theorem 6 (Uniqueness). (i) Case 0   12 and ˛ 2 R: There exist 0 2 .0; 12 / and 0 > 0 such that, if  2 Œ0; 0˚ and ˛ 2 R, there exists at most one asymmetric Burgers vortex !;˛

in the set f 2 L2 .1/ j kf  ˛GkW 1;2 .1/  0 . (ii) Case 0   < 1 and j˛j 1: For all  2 Œ0; 1/, there exists 0 D 0 ./ > 0 such that, for any ˛ 2 R with˚ j˛j  0 , there exists at most one asymmetric

Burgers vortex !;˛ in the set f 2 L2 .1/ j kf  ˛G kL2 .1/  0 . (iii) Case 0   < 1 and j˛j 1: For all  2 Œ0; 1/ and all > 0, there exists R0 .; /  R0 ./ such that, for any ˛ 2 R with j˛j  R0 .; /, there exists at most one asymmetric Burgers vortex !;˛ in the set

f 2 Pe L2 .1; / j kf  ˛G  w1 kW 1;2 .1I/  :

˚

Here R0 ./ is as in Theorem 5, and R0 .; / satisfies lim R0 .; / D 1.

!1

The statement (i) of Theorem 6 is proved in [21] using the uniform estimate (77) for the inverse of L  ˛ƒG in L20 .1/, while (iii) is established in [32, 33] using the stabilization effect at large circulations described in (78). The uniqueness in the case (ii) is obtained in [20] in the more general framework of the polynomially weighted spaces L2 .m/. The key point in the proof of (ii) is an estimate for the inverse 2 L1  in L0 .m/ when m is large enough, which enables to apply the Banach fixed point theorem when j˛j is sufficiently small. Remark that existence of asymmetric Burgers vortices is also established in [20, 21, 32, 33] for all three cases (i), (ii), and (iii) above, whereas uniqueness for the parameter regions not covered by Theorem 6 is an interesting but difficult question, which is essentially open.

4.2

Two-Dimensional Stability of Asymmetric Burgers Vortices

In the parameter regions where existence and uniqueness have been established, the next important issue is stability. Since the Burgers vortex itself is a two-dimensional vorticity field, it is possible to study its stability within the class of purely twodimensional flows, and this is the point of view adopted in this subsection. The axisymmetric case where  D 0 was already discussed in detail in Sect. 2; hence, the main focus here will be on the asymmetric case  ¤ 0. The evolution equations for the perturbations are obtained from system (12), where  D 1 and 1 ; 2 ; and 3 are as in (60), by expanding the vorticity vector !.x; t / around the stationary Burgers vortex !;˛ .x/ D .0; 0; !;˛ .x//> . When the vorticity field w.x; t / D .0; 0; w.x; t //> of the perturbation is two-dimensional, the problem is reduced to the following equations for the scalar function w: 8   ˆ < @t w  L  ƒ!;˛ w C .v; r/w D 0; Z ˆ w0 dx D 0: : wjtD0 D w0 ; R2

v D K2D  w;

t > 0; (80)

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Here the operators L and ƒf are defined by (62) and (68), respectively. As can be expected, the properties of the linearized operator L  ƒ!;˛ play a crucial role in the stability analysis. It is not difficult to show that L generates a C0 semigroup in the polynomially weighted space L2 .m/ for m < 1 and an analytic semigroup in the Gaussian weighted space L2 .1I /. In fact, the semigroup e tL has the following explicit representation  tL  e  f .x/    Z  1C et jx1  y1 j2 jx2  y2 j2 1 D p  f y1 e 2 t ; y2 e 2 t dy; exp  4a .t / 4a .t / 4 a .t /a .t / R2 where a .t / D .1  e .1C/t /=.1 C  /. Since ƒ!;˛ is a relatively compact perturbation of L , the full linearized operator L  ƒ!;˛ is also the generator of a C0 (or analytic) semigroup, and the main concern is the longtime behavior of that semigroup. The following results have been established in (essentially) the same parameter regions as in Theorem 6. Proposition 9 (Linear stability). (i) Case 0   12 and ˛ 2 R: There exists 1 2 .0; 12 / such that, for all  2 Œ0; 1  and all ˛ 2 R, ke t.L ƒ!;˛ / f kL2 .1/  C kf kL2 .1/ e 

1 2 t

;

t  0;

(81)

for all f 2 L20 .1/. Here C is a universal constant independent of  2 Œ0; 1  and ˛ 2 R. (ii) Case 0   < 1 and j˛j 1: For all  2 Œ0; 1/, there exists 1 ./ > 0 such that, if j˛j  1 ./, then ke t.L ƒ!;˛ / f kL2 .m/  C kf kL2 .m/ e 

1 2 t

;

t  0;

(82)

for all f 2 L20 .m/, m > 3. Here C depends only on  2 Œ0; 1/ and m. (iii) Case 0   < 12 and j˛j 1: For all  2 Œ0; 12 /, there exists R1 ./  R0 ./ such that, if j˛j  R1 ./, then ke t.L ƒ!;˛ / f kL2 .1/  C kf kL2 .1/ e 

1 2 t

;

t  0;

(83)

for all f 2 L20 .1/. Here C depends only on  and ˛, while R0 ./ is as in Theorem 5. The statement (i) of Proposition 9 is proved in [21], where L  ƒ!;˛ is regarded as a small perturbation of the simpler operator L  ˛ƒG for which, as recalled in (77), the stability estimate is obtained uniformly in ˛ using the skew symmetry of ƒG in L2 .1/. The case (ii) follows from the analysis developed in [20]. In fact, as is mentioned in the next subsection, the three-dimensional stability is the main concern of [20], but the class of perturbations considered there includes purely

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two-dimensional flows. In case (ii) the asymmetric Burgers vortex !;˛ is of order O.j˛j/ in L2 .1I /, and the operator L ƒ!;˛ is handled as a small perturbation of L , for which complete information on the spectrum and the associated semigroup is available. Case (iii) is treated in [32], using in an essential way the stabilizing effect at large circulations described in (78). The restriction  2 Œ0; 12 / in (iii) is due to the fact that, when   12 , the operator L  ƒ!;˛ has to be analyzed in the space L2 .1; /, where ƒG is no longer skew symmetric. So far this difficulty could not be overcome for the stability problem, although existence and uniqueness of Burgers vortices were established by constructing a suitable skew symmetrizer, as explained in Sect. 4.1. It should be emphasized here that, in all cases (i), (ii), and (iii), the temporal decay estimate for the semigroup e t.L ƒ!;˛ / involves the exponent  1 , which 2 is known to be optimal. Indeed, by differentiating the identity L !;˛  .K2D  !;˛ ; r/!;˛ D 0 with respect to x2 , one observes that @2 !;˛ is an eigenfunction of L  ƒ!;˛ for the eigenvalue  1 . Numerical results due to Prochazka and Pullin 2 1 [41] indicate that  2 is actually the largest eigenvalue of L  ƒ!;˛ in L20 .1; / for any  2 Œ0; 1/, but a mathematical proof of this conjecture is still missing, except in the three cases stated in Proposition 9. The semigroup e t.L ƒ!;˛ / has standard parabolic smoothing properties. Nonlinear stability with respect to small initial perturbations can thus be obtained by analyzing the integral equation associated with (80): w.t / D e t.L ƒ!˛; / w0 

Z

t

  e .ts/.L ƒ!˛; / K2D  w.s/; r w.s/ ds;

(84)

0

and applying the conclusions of Proposition 9. This gives the following result: Theorem 7 (Local 2D stability). (i) Case 0   12 and ˛ 2 R: There exists  > 0 such that, for all  2 Œ0; 1  and all ˛ 2 R, the following statement holds. For all initial data w0 2 L20 .1/ such that kw0 kL2 .1/  , Eq. (80) admits a unique solution w 2 C 0 .Œ0; 1/I L20 .1//, which satisfies kw.t /kL2 .1/  C kw0 kL2 .1/ e 

1 2 t

;

t  0:

(85)

Here the constant C is independent of  2 Œ0; 1  and ˛ 2 R, while 1 is as in Proposition 9. (ii) Case 0   < 1 and j˛j 1: For all  2 Œ0; 1/, there exists  D ./ > 0 such that, for any ˛ 2 R with j˛j  1 ./, the following statement holds. For all initial data w0 2 L20 .m/, m > 3, such that kw0 kL2 .m/  , Eq. (80) admits a unique solution w 2 C 0 .Œ0; 1/I L20 .m//, which satisfies kw.t /kL2 .m/  C kw0 kL2 .m/ e 

1 2 t

;

t  0:

Here C depends only on  and m, while 1 ./ is as in Proposition 9.

(86)

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(iii) Case 0   < 12 and j˛j 1: For all  2 Œ0; 12 / and any ˛ 2 R with j˛j  R1 ./, there exists  D .; ˛/ > 0 such that the following statement holds. For all initial data w0 2 L20 .1/ with kw0 kL2 .1/  , Eq. (80) admits a unique solution w 2 C 0 .Œ0; 1/I L20 .1//, which satisfies kw.t /kL2 .1/  C kw0 kL2 .1/ e 

1 2 t

;

t  0:

(87)

Here C depends only on  and ˛, while R1 ./ is as in Proposition 9. As in Proposition 9, the statement (i) of Theorem 7 is proved in [21], while (ii) follows from the results of [20]. The case (iii) is obtained in [32]. It should be emphasized here that the basin of attraction in the case (i) is uniform in the circulation number ˛, as a consequence of the linear estimate (81) in Proposition 9.

4.3

Three-Dimensional Stability of Burgers Vortices

The stability analysis becomes more complicated when the perturbations are three dimensional, because the vorticity field is no longer a scalar quantity, and vortex stretching terms already appear in the linearized operator. The problem is highly nontrivial even in the axisymmetric case  D 0, where Rossi and Le Dizès [43] have shown that the linearized operator does not have any eigenfunction with nontrivial dependence upon the vertical variable. Numerical evidence of linear stability with exponential decay of the perturbations was obtained by Schmid and Rossi [44], but their analysis also reveals the occurrence of short-time amplification for generic solutions. A mathematical understanding of the underlying mechanisms, leading to a rigorous explanation of these observations, is an important and challenging question, for which significant progress has been made in recent years. Starting from the vorticity equation (12), with  D 1 and 1 ; 2 ; and 3 as in (60), it is easy to write the evolution equation for perturbations w D !  !;˛ , where !;˛ D .0; 0; !;˛ /> is the Burgers vortex with circulation ˛. The result is 8   ˆ < @t w  L  ƒ!;˛ w C .v; r/w  .w; r/v D 0; v D K3D  w; t > 0; Z ˆ D w ; r  w D 0; w0;3 .x; x3 / dx D 0; x3 2 R; wj : tD0 0 0 R2

(88) where K3D is the kernel of the Biot-Savart law (2). Here the operator L is given by (13) with  D 1, namely, 0

L C @23  x3 @3 

B L D @ L C @23  x3 @3 

1

3C 2 3 C ; 2 A

L C @23  x3 @3

(89)

13 Existence and Stability of Viscous Vortices

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and L is the two-dimensional differential operator (62). On the other hand, the operator ƒ!;˛ is defined by ƒ!;˛ w D .K3D  !;˛ ; r/w C .K3D  w; r/!;˛  .w; r/K3D  !;˛  .!;˛ ; r/K3D  w:

(90)

R The divergence-free condition and the zero mass condition R2 w3 .x; x3 / dx D 0 are preserved under the evolution defined by (88). Note that, at least formally, a divergence-free vector field w D .w1 ; w2 ; w3 /> always R R satisfies the identity d w .x; x / dx D 0. This means that the condition 2 3 3 R2 w0;3 .x; x3 / dx D 0 dx3 R in (88) is a natural requirement on the initial data, which does not restrict the generality; see [13, Section 1] for a detailed discussion. Since the Burgers vortex itself is essentially a two-dimensional flow, it is natural to choose a functional setting that allows for purely two-dimensional perturbations and more generally for perturbations which do not decay to zero as jx3 j ! 1. For this purpose, the following function spaces are introduced in [13, 20]: X .m/ D BC .RI L2 .m//;

X0 .m/ D BC .RI L20 .m//;

(91)

as well as X.m/ D X .m/  X .m/  X0 .m/. Here BC .RI L2 .m// denotes the space of all bounded and continuous functions from R into L2 .m/, which is a Banach space equipped with the norm kkX.m/ D supx3 2R k.; x3 /kL2 .m/ , and BC .RI L20 .m// is the closed subspace defined in a similar way. Since the leading order term L in (88) contains the dilation operator x3 @3 , one cannot expect that the solutions will be continuous in time in the uniform topology of X.m/. To restore continuity in time, it is convenient to work in Xloc .m/, which is the very same space X .m/ equipped with the weaker topology induced by the countable family of seminorms kkXn .m/ D supjx3 jn k.; x3 /kL2 .m/ , for n 2 N. For vector-valued functions, the space Xloc .m/ is defined in a similar way by endowing X.m/ with the localized topology. Using these notations, the local stability results available so far can be summarized as follows: Theorem 8 (Local 3D stability). (i) For all  2 Œ0; 1/ and all  2 .0; 1 /, there exist  D ./ > 0 and 2 2 .; / 2 .0; 1 ./ such that, for any ˛ 2 R with j˛j  2 .; /, the following statement holds. For all initial data w0 2 X.m/, m > 3, with r  w0 D 0 and kw0 kX.m/  , Eq. (88) admits a unique solution w 2 L1 .RC I X.m// \ C 0 .Œ0; 1/I Xloc .m//, which satisfies kw.t /kX.m/  C kw0 kX.m/ e t ;

t  0:

Here C depends only on ˛, and 1 ./ is as in Proposition 9.

(92)

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(ii) Let  D 0. For all m 2 .2; 1 and all ˛ 2 R, there exists  D .m; ˛/ > 0 such that the following statement holds. For all initial data w0 2 X.m/ with r  w0 D 0 and kw0 kX.m/  , Eq. (88) admits a unique solution w 2 L1 .RC I X.m// \ C 0 .Œ0; 1/I Xloc .m//, which satisfies t

kw.t /kX.m/  C kw0 kX.m/ e  2 ;

t  0:

(93)

Here C depends only on m and ˛. The statement (i) of Theorem 8 is proved in [20], using estimates on the semigroup e tL generated by the operator L . In view of (89), one has the representation

e

tL

 w D

e

 3C 2 t tS

e

w1 ; e

 3 2 t tS

e

w2 ; e

tS

> w3

;

(94)

where S is the differential operator defined by S D L C @23  x3 @3 . Since the operators L and @23  x3 @3 act on different variables, it is possible to obtain the following explicit formula 

 1 e tS f .x/ D p 4a1 .t /

Z

   jx3 e t  y3 j2  tL e  f .; y3 / .x/ dy3 ; exp  4a1 .t / R (95)

where a1 .t / D .1e 2t /=2 and e tL is the two-dimensional semigroup encountered in Sect. 4.2. Useful estimates for the semigroup e tS in X .m/ are established in [20], together with elementary spectral properties of the generator S . Note that it is possible to take  D 1 in estimate (92), as can be shown using some arguments 2 borrowed from [13]. The statement (ii) of Theorem 8 is established in [13]. Remarkably, as in the twodimensional case, the local stability holds for all values of the circulation number t ˛, and moreover the rate of convergence e  2 is uniform in ˛. Although the result of (ii) is stated only in the purely axisymmetric case  D 0, by a standard perturbation argument, it is also possible to prove local stability of the asymmetric Burgers vortex !;˛ if the asymmetry parameter  is sufficiently small, depending on j˛j. The proof of (ii) in [13] is based on the analysis of the linearized operator L  ˛ƒG and its associated semigroup, where L and ˛ƒG are shorthand notations for the operators L and ƒ!;˛ , respectively, when  D 0. Since ƒG is a lower order perturbation, it is not difficult to construct the semigroup e t.L˛ƒG / in X.m/, but the main problem is to control the longtime behavior. The following result is the key achievement of [13].

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Proposition 10 (Axisymmetric linear stability). For all m 2 .2; 1 and all ˛ 2 R, one has t

ke t.L˛ƒG / fkX.m/  C e  2 kfkX.m/ ;

t  0;

(96)

for all f 2 X.m/, where C depends only on m and ˛. Moreover, if r  f D 0 then r  e t.L˛ƒ/ f D 0. The proof of Proposition 10 in [13] is based on two important observations: (I) As an effect of vortex stretching, the vertical derivatives of the velocity and vorticity fields decay exponentially as t ! 1, so that the longtime asymptotics are governed by a two-dimensional vectorial system. (II) When restricted to two-dimensional solutions, the linearized operator L  ˛ƒG has symmetry properties which imply uniform stability for all values of the circulation parameter. In the rest of this section, both mechanisms are explained in some detail for the more general semigroup e t.L ƒ!;˛ / , where 0   < 1. Proposition 10 is stated and proved in [13] in the axisymmetric case  D 0, but the arguments are robust and can be used to establish linear stability in the asymmetric case too. Property (I) above is due to a very specific dependence of the operator L  ƒ!;˛ upon the vertical variable x3 . Indeed, using the definition in (89), it is straightforward to verify that Œ@3 ; L  D @3 , where ŒA; B D AB  BA denotes the commutator of A and B. Moreover, since the Burgers vortex !;˛ is a twodimensional stationary solution, one has Œ@3 ; ƒ!;˛  D 0. At the level of the semigroup, these identities imply that @k3 e t.L ƒ!;˛ / D e kt e t.L ƒ!;˛ / @k3 ;

t  0;

(97)

for all integer k 2 N. Since the semigroup e t.L ƒ!;˛ / grows at most exponentially in time, at a rate that depends only on  and ˛, Eq. (97) shows that the k th order vertical derivative of any solution to the linearized equation @t w D .L  ƒ!;˛ /w decays exponentially as t ! 1, if k 2 N is large enough. By a simple interpolation argument, it follows that any expression involving at least one vertical derivative of the solution becomes negligible in the longtime regime, which means that one can restrict the analysis to the two-dimensional vectorial system obtained by disregarding the vertical dependence of all quantities under consideration. More precisely, in view of (90), the operator ƒ!;˛ can be decomposed as .2/ .3/ .4/ ƒ!;˛ w D ƒ.1/ !;˛ w C ƒ!;˛ w  ƒ!;˛ w  ƒ!;˛ w;

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where, with the notations rh D .@1 ; @2 /> and wh D .w1 ; w2 /> , ƒ.1/ !;˛ w D .K2D  !;˛ ; rh /w;

ƒ.2/ !;˛ w D .K3D  w; r/!;˛ ;

ƒ.3/ !;˛ w D .wh ; rh /K3D  !;˛ ;

ƒ.4/ !;˛ w D !;˛ @3 K3D  w:

(98)

The discussion above motivates the following decomposition of ƒ!;˛ into 2D and 3D parts:   .u;˛ ; rh /wh  .wh ; rh /u;˛ ƒ!;˛ w D C R;˛ w; ƒ!;˛ w3 (99)   .4/ R;˛ w D .K3D  w/h  K2D  w3 ; rh !;˛  ƒ!;˛ w: To derive (99) one uses the fact that !;˛ D .0; 0; !;˛ /> is a two-dimensional vorticity field, so that K3D  !;˛ D .u;˛ ; 0/> , with u;˛ D K2D  !;˛ . The operator ƒ!;˛ , defined in (68), is artificially produced by adding and subtracting the term .K2D  w3 ; rh /!;˛ in the right-hand side. As is shown in [13, Proposition 4.5], all terms in the operator R;˛ involve at least one derivative with respect to x3 and hence play a negligible role in longtime asymptotics. Therefore, the problem is now reduced to the analysis of the simpler operator L;˛ defined by L;˛ w D L w 



.u;˛ ; rh /wh  .wh ; rh /u;˛ ƒ!;˛ w3



D A;˛ w C .@23  x3 @3 /w;

where A;˛ w D



A;˛;h wh A;˛;3 w3



0

L  B D @ L 



3C w1 2  3 w2 2

 .u;˛ ; rh /w1 C .wh ; rh /u;˛;1

1

C  .u;˛ ; rh /w2 C .wh ; rh /u;˛;2 A :   L  ƒ!;˛ w3 (100)

The crucial observation here is that the (vectorial) operator A;˛ acts only on the horizontal variable, so that the semigroup e tL;˛ generated by L;˛ D A;˛ C @23  x3 @3 can be expressed in terms of the 2D semigroup e tA;˛ in the same way as in (95). As a consequence, the longtime behavior of the semigroup e t.L ƒ!;˛ / in X.m/ can be deduced from the spectral analysis of the two-dimensional operator A;˛ acting on L2 .m/ WD L2 .m/2  L20 .m/. This leads to the following criterion [13]: Stability criterion: Let  2 Œ0; 1/, ˛ 2 R, and m > 2. If the stability estimate ke tA;˛ kL2 .m/!L2 .m/  C e t ;

t  0;

(101)

, then ke t.L ƒ!;˛ / kX.m/!X.m/  C 0 e t holds for all holds for some  2 .0; 1 2 t  0.

13 Existence and Stability of Viscous Vortices

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The key observation (II) concerns the structure of the 2D operator A;˛ . From the definition (100), it is apparent that the horizontal component wh D .w1 ; w2 /> and the vertical component w3 are completely decoupled under the action of A;˛ . Furthermore, the third component A;˛;3 D L  ƒ!;˛ acting on w3 is exactly the linearized operator at the Burgers vortex considered in Sect. 4.2, when only twodimensional perturbations are allowed. Proposition 9 (iii) thus provides the desired stability estimate for the semigroup generated by A;˛;3 , uniformly for all ˛ 2 R if the asymmetry parameter  is small enough. One of the main contributions of [13] is the analysis of the horizontal component A;˛;h , which also has a nice structure that allows to obtain a stability estimate for all ˛ 2 R, at least if  D 0. The argument is as follows. Since the operator wh 7! .wh ; rh /u;˛  .u;˛ ; rh /wh is a relatively compact perturbation of the second-order operator L , a standard perturbation argument reproduced in [13, Proposition 3.4] shows that the longtime behavior of the semigroup e tA;˛;h in L2 .m/2 (with m > 1 sufficiently large) is determined by the eigenvalues of the generator A;˛;h in a Gaussian weighted space such as L2 .1I /2 . As usual, when  2 Œ0; 12 /, one can use L2 .1/2 instead of L2 .1I /2 . To locate the eigenvalues of A;˛;h , the following identities play a crucial role: xh  A;˛;h wh D .L  2/xh  wh  2rh  wh  .x1 w1  x2 w2 /  .u;˛ ; rh /xh  wh C .wh ; rh /xh  u;˛ ;

(102)

rh  A;˛;h wh D .L  1/rh  wh  .u;˛ ; rh /rh  wh : Here the notation xh D x D .x1 ; x2 /> is used. When  D 0 the two-dimensional velocity field u;˛ D ˛v G satisfies xh  u;˛ D 0; hence, the first identity in (102) becomes substantially simpler. If wh 2 L2 .1/2 \D.L/ is a nontrivial eigenfunction of A;˛;h with eigenvalue  2 C, one has the obvious relations wh D A;˛;h wh ;

xh  wh D xh  A;˛;h wh ;

rh  wh D rh  A;˛;h wh ;

which can be combined with (102) to obtain valuable information on . Indeed, assume for simplicity that  D 0. If rh  wh 6 0, the identity rh  wh D .L  1/rh  wh  ˛.v G ; rh /rh  wh implies that Re./  3=2, in view of the spectral properties of L established in Proposition 1 and the fact that the operator ! 7! .v G ; r/! is skew symmetric in L2 .1/; see (30). If rh  wh  0 and xh  wh 6 0, one has the relation xh  wh D .L  2/xh  wh  ˛.v G ; rh /xh  wh ; which implies that Re./  2 by the same argument. Finally, if xh  wh  0, the eigenvalue equation reduces to

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wh D .L  32 /wh  ˛.v G ; rh /wh C ˛w? h h;

h.xh / D

jx j2 1  h4 .1  e /; 2jxh j2

and a simple energy estimate leads to the conclusion that Re./  3=2 in all cases. As a consequence, when  D 0, one has the desired stability estimate 3

ke tA;˛;h wh kL2 .m/2  Cm;˛ e  2 t kwh kL2 .m/2 ;

t  0;

(103)

for all ˛ 2 R, if m > 1 is sufficiently large. Note that the constant in (103) depends on ˛ and becomes large as j˛j ! 1, which may be related to the short-time amplification phenomenon observed numerically by Schmid and Rossi [44]. By a perturbation argument, it is easy to show that the stability estimate also holds if the asymmetry parameter  is nonzero and small, but it is unclear whether the smallness assumption on  is uniform with respect to the circulation parameter ˛. This interesting question is answered affirmatively by Theorem 7 if the perturbations are restricted to purely two-dimensional flows. In the general case, what is missing so far is a precise information on the eigenvalues of the two-dimensional operator A;˛;h , especially in the regime where j˛j 1.

5

Conclusion

As can be seen from the results reviewed in the previous sections, the mathematical theory of viscous vortices has reached a certain level of maturity, but many interesting questions remain open to the present date. In the simple case of a single, axisymmetric, straight vortex tube, there are explicit formulas for the vorticity and velocity profiles, and the stability with respect to two-dimensional perturbations is fully understood for all values of the total circulation (Sect. 2.1), including the large Reynolds number limit where additional stabilization occurs (Sect. 2.2). In the presence of a non-axisymmetric strain, existence of stretched vortices is known for all values of the circulation and asymmetry parameters (Sect. 4.1), but uniqueness and stability results are not completely satisfactory, except perhaps in the large circulation limit where asymptotic symmetrization and stabilization are observed (Sect. 4.2). When arbitrary three-dimensional perturbations are allowed, local stability of the axisymmetric Burgers vortex is well understood for all values of the total circulation (Sect. 4.3), but less is known in the asymmetric case, and the question is essentially open for the self-similar Lamb-Oseen vortex, due to the lack of stretching in the vertical direction. In real fluids, however, one usually observes the interaction of several vortex tubes, none of which is perfectly straight, and vorticity is also created near the boundaries. All these discrepancies from the ideal situation considered above give rise to difficult mathematical questions, which are essentially open. The rigorous theory of curved vortex filaments in viscous flows is still in its infancy, except perhaps in the axisymmetric case without swirl where existence and uniqueness of

13 Existence and Stability of Viscous Vortices

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vortex rings can be established (Sect. 3). Interaction of vortices has been studied so far only in the weakly coupled regime where the distance between the vortex centers is much larger than the size of the vortex cores [10, 36, 37]. Stronger interactions, such as vortex merging (in two dimensions) or reconnection of vortex tubes (in three dimensions), play a crucial role in the dynamics of turbulent flows, but a rigorous description of these phenomena seems completely out of reach. Finally, there are no mathematical results yet concerning the interaction of viscous vortices with rigid boundaries, although the existence of self-similar vortices in two-dimensional exterior domains can be established at least for small values of the circulation parameter (Sect. 2.3).

6

Cross-References

 Large Time Behavior of the Navier-Stokes Flow  Models and Special Solutions of the Navier-Stokes Equations  Self-Similar Solutions to the Nonstationary Navier-Stokes Equations  The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

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14. Th. Gallay, Y. Maekawa, Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity. Anal. PDE 6, 973–991 (2013) 15. Th. Gallay, L.M. Rodrigues, Sur le temps de vie de la turbulence bidimensionnelle. Ann. Fac. Sci. Toulouse Math. 17, 719–733 (2008) 16. Th. Gallay, V. Šverák, Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations. Conflu. Math. 7, 67–92 (2015) 17. Th. Gallay, V. Šverák, Uniqueness of axisymmetric viscous flows originating from circular vortex filaments. https://arxiv.org/abs/1609.02030 18. Th. Gallay, C.E. Wayne, Invariant manifolds and the long-time asymptotics of the NavierStokes and vorticity equations on R2 . Arch. Ration. Mech. Anal. 163, 209–258 (2002) 19. Th. Gallay, C.E. Wayne, Global stability of vortex solutions of the two dimensional NavierStokes equation. Commun. Math. Phys. 255, 97–129 (2005) 20. Th. Gallay, C.E. Wayne, Three-dimensional stability of Burgers vortices: the low Reynolds number case. Physica D 213, 164–180 (2006) 21. Th. Gallay, C.E. Wayne, Existence and stability of asymmetric Burgers vortices. J. Math. Fluid Mech. 9, 243–261 (2007) 22. Y. Giga, T. Kambe, Large time behavior of the vorticity of two dimensional viscous flow and its application to vortex formation. Commun. Math. Phys. 117, 549–568 (1988) 23. M.-H. Giga, Y. Giga, J. Saal, Nonlinear Partial Differential Equations. Asymptotic Behavior of Solutions and Self-Similar Solutions. Progress in Nonlinear Differential Equations and Their Applications, vol. 79 (Birkhäuser, Boston, 2010) 24. H. von Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 5, 25–55 (1858) 25. D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lectures Notes in Mathematics, vol. 840 (Springer, Berlin/New York, 1981) 26. T. Kato, Perturbation Theory for Linear Operators. Grundlehren der mathematischen Wissenschaften, vol. 132 (Springer, Berlin/New York, 1966) 27. H. Kozono, M. Yamazaki, Local and global unique solvability of the Navier-Stokes exterior problem with Cauchy data in the space Ln;1 . Houst. J. Math. 21, 755–799 (1995) 28. D. Küchermann, Report on the I.U.T.A.M symposium on concentrated vortex motion in fluids. J. Fluid Mech. 21, 1–20 (1965) 29. D. Iftimie, G. Karch, C. Lacave, Self-similar asymptotics of solutions to the Navier-Stokes system in two-dimensional exterior domain. J. Lond. Math. Soc. 90, 785–806 (2014) 30. O. Ladyzhenskaya, Unique solvability in the large of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 7, 155–177 (1968, in Russian) 31. T. Lundgren, Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25, 2193 (1982) 32. Y. Maekawa, On the existence of Burgers vortices for high Reynolds numbers. J. Math. Anal. Appl. 349, 181–200 (2009) 33. Y. Maekawa, Existence of asymmetric Burgers vortices and their asymptotic behavior at large circulations. Math. Mod. Methods Appl. Sci. 19, 669–705 (2009) 34. Y. Maekawa, Spectral properties of the linearization at the Burgers vortex in the high rotation limit. J. Math. Fluid Mech. 13, 515–532 (2011) 35. Y. Maekawa, On asymptotic stability of global solutions in the weak L2 space for the twodimensional Navier-Stokes equations. Anal. 35, 245–257 (2015) 36. C. Marchioro, On the inviscid limit for a fluid with a concentrated vorticity. Commun. Math. Phys. 196, 53–65 (1998) 37. C. Marchioro, Vanishing viscosity limit for an incompressible fluid with concentrated vorticity. J. Math. Phys. 48(1–16), 065302 (2007) 38. K. Masuda, Weak solutions of Navier-Stokes equations. Tohoku Math. J. 36, 623–646 (1984) 39. H.K. Moffatt, S. Kida, K. Ohkitani, Stretched vortices–the sinews of turbulence; largeReynolds-number asymptotics. J. Fluid Mech. 259, 241–264 (1994)

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40. A. Prochazka, D.I. Pullin, On the two-dimensional stability of the axisymmetric Burgers vortex. Phys. Fluids 7, 1788–1790 (1995) 41. A. Prochazka, D.I. Pullin, Structure and stability of non-symmetric Burgers vortices. J. Fluid Mech. 363, 199–228 (1998) 42. A.C. Robinson, P.G. Saffman, Stability and structure of stretched vortices. Stud. Appl. Math. 70, 163–181 (1984) 43. M. Rossi, S. Le Dizès, Three-dimensional temporal spectrum of stretched vortices. Phys. Rev. Lett. 78, 2567–2569 (1997) 44. P.J. Schmid, M. Rossi, Three-dimensional stability of a Burgers vortex. J. Fluid Mech. 500, 103–112 (2004) 45. M. Ukhovskii, V. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech. 32, 52–61 (1968)

Models and Special Solutions of the Navier-Stokes Equations

14

Hisashi Okamoto

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Early Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Bateman and Burgers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hiemenz and Homann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Jeffery and Hamel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Nonstationary Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Riabouchinsky and Proudman-Johnson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Unimodality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Generalized PJ Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Coupled Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Oseen’s Self-Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Leray’s Self-Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 von Kármán’s Swirl Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Role of the Convection Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 PJ Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Examination by 2D Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Constantin-Lax-Majda and Its Sisters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Generalizations of the Constantin-Lax-Majda Equations . . . . . . . . . . . . . . . . . . . . 9 Vortex Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

730 731 732 735 735 741 743 743 748 750 750 752 753 755 758 759 762 765 766 771 773 774 774

H. Okamoto () Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan Department of Mathematics, Gakushuin University, Toshima-ku, Tokyo, Japan e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_14

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Abstract

Some aspects of the roles of exact solutions of the Navier-Stokes equations are considered with special emphasis on their well posedness. Many of them are classical examples, but they are looked at from a modern viewpoint of partial differential equations. Also considered are model equations describing motion of incompressible viscous fluid. Demonstrations are given for importance of both nonlinearity and nonlocality alike by mathematical analyses and numerical experiments.

1

Introduction

Difficulty of the three-dimensional nonstationary Navier-Stokes equations is well known. When people encounter a difficult problem, they often make a simpler model and wish to understand the nature of the difficulty by analogy. Endeavors of this kind have been in the scene of mathematical physics for considerably long time, and models proposed are numerous. In the present chapter, the author lists some of them and discusses open problems related to them. Attempts to extract essence of basic mechanism in fluid mechanics by simplification are old. Considering not general flows but only irrotational flows may be classified as such an attempt. Stokes’s analysis by dropping nonlinear terms may be regarded as such. But simplifications often lead to conclusions which contradict experimental data. In that case, the model must be suitably modified. A famous saying, which is often attributed to Albert Einstein, reads: everything should be made as simple as possible, but not simpler. This applies well to the study of fluid mechanics, too. In the present chapter, linearization is not considered as a model, since difficulty comes from the nonlinear terms. Accordingly a model worth being studied should contain some sort of nonlinearity. In what follows modeling not only implies inventing new equations but also includes analyzing simple special cases—the exactly solvable solutions. With this principle, the author considers below those models which interested him for the last 20 years. The author regrets that he is able to discuss only a small number of models and their solutions, due to limitation of his knowledge. He also regrets that only incompressible fluid is considered, and compressible fluid is excluded. In the present survey, the following symbols are adopted: .x; y; z/ denotes a point in the three-dimensional (3D) Euclidean space R3 . .x1 ; x2 ; x3 / is often used in place of .x; y; z/. u D .u1 ; u2 ; u3 / and v denote the velocity field. In this chapter, the polar coordinates in the plane are denoted by r and  . The Navier-Stokes equations are written as follows: @u 1 C .u  r/u D 4u  rp C f ; @t  div u D 0;

(1) (2)

14 Models and Special Solutions of the Navier-Stokes Equations

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where   0 is the kinematic viscosity and  > 0 is the density of the fluid. f D .f1 ; f2 ; f3 / is an external force. 4 denotes the Laplace operator and r denotes the vector differential operator nabla. In the present chapter,  is assumed to be a positive constant.  is either a positive constant or zero. The Euler equations, which are obtained by setting  D 0 in (1), are considered in the present chapter only when they help us understand the Navier-Stokes equations. For the Euler equations, the survey by Chae [18] is recommended (see also [53]). Vorticity, curl u, is denoted by !. In two dimensions (2D), it is often useful to write equations by a stream function defined by u D . y ;  x /. Here and hereafter, the subscripts imply differentiations. In 2D, vorticity ! becomes a scalar and satisfies 4 D !. Often used are the Navier-Stokes equations in vorticity form: @! C .u  r/!  .!  r/u D 4! C curl f : @t

(3)

Here ! D curl u is the vorticity. In 2D, the vorticity equations (3) becomes @f2 @f1 @!  : C .u  r/! D 4! C @t @x1 @x2

(4)

Here the vorticity ! is a scalar function. The vorticity equation is associated with the Biot-Savart law: 1 v.x/ D  4

Z

.x  y/  !.y/ R3

3

jx  yj

  dy1 dy2 dy3 D curl 41 ! :

(5)

This is a singular integral operator of order 1. Its precise analytical properties such as the domain of definition are not necessary in the following discussions. With (5) the equation (3) is regarded as closed in !.

2

Some Early Models

Until Helmholtz [62] began the quest for rotational flows, irrotational flows were the main focus. After his work, the importance of vorticity became gradually recognized. However, if the viscosity is neglected, the distribution of vorticity is not determined by the Euler equations themselves. To determine it, it is necessary to know the consequence of added small viscosity. This singular perturbation character of the Navier-Stokes equations is of utmost importance (see, for instance, [28, 89, 135]), but our understanding toward this direction is not enough. This is where models of some kind are needed. Before introducing modern models, it would be helpful to look at papers of historical importance. A brief (incomplete) historical survey is given in this section. If the reader wishes to read history of fluid mechanics, von Kármán [149], Anderson [2], Tokaty [144],

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Darrigol [38], and Truesdell [145] are recommended. The last two are particularly interesting.

2.1

Bateman and Burgers

Bateman [6] in 1915 came up with a scalar differential equation ut C uux D uxx , where the subscripts imply differentiation and  is a positive constant, and says: The possibility of the solution of the equations of motion of a viscous fluid becoming discontinuous when the viscosity approaches the value zero may perhaps be illustrated by a consideration of @u @2 u @u Cu D  2: @t @x @x

Thus Bateman’s view is quite modern already in 1915. Note that, before Bateman, Lanchester is said to have predicted a discontinuity of fluid motion. That seems to have partly motivated Bateman’s attempt. Bateman’s equation was later rediscovered by Burgers in [14], which caused a considerable impetus to the research of fluid mechanics. This forms a curious contrast with Bateman’s paper, which was forgotten shortly after its publication. Invention must come at the right time: It may be neglected if it appears too early. In fact, long before Bateman, Forsyth [46] discusses the Burgers equation simply as a mathematical exercise, but he does not seem to have noted the potential importance of the equation. Burgers considered in [14] the following coupled equations: dU D P  U  dt

Z

1

v.t; x/2 dx;

(6)

0

@ 2 @v @2 v D Uv C  2  v @t @x @x

.0 < x < 1/;

(7)

where  is the viscosity and P is a prescribed parameter playing a role of a driving force. Burgers says [15] as follows: “U will be the analogue of the primary or mean motion in the case of a liquid flowing through a channel; in the model it is a function of the time only. The other variable v represents the secondary motion; when it differs from zero we shall say there is turbulence in the system, even in a case where v should be independent of the time.” Burgers’s philosophy is clear. The Navier-Stokes equations are too difficult for mathematicians at that time, and he wanted something which he could handle more easily. This idea or strategy will appear repeatedly from many mathematicians in later years, knowingly or unknowingly of Burgers. The Burgers turbulence is even now a target of turbulence study; see [8], for instance. Its truncated version was considered in [90] to relate it with some statistical dynamics. In view of its historical value, some solutions, which are the author’s reconstruction of the results in [101], are now presented. By a certain nondimensionalization,

14 Models and Special Solutions of the Navier-Stokes Equations

733

(6) and (7) are transformed into the following form: U  Ut D 1  R

Z

1

v.t; x/2 dx;

1 vxx  2vvx C U v; R

vt D

0

(8)

where R is the Reynolds number. These equations are considered with the Dirichlet boundary condition v D 0 at x D 0 and x D 1. They have a trivial solution U D R; v  0. Linearization at this trivial solution leaves us the following equations: Ut D 

U ; R

vt D

1 vxx C Rv: R

Accordingly nontrivial stationary solutions bifurcate at R D ; 2; 3;    . Then steady states of (8) are computed by a spectral method (see, for instance, [139]): Set v.x/ 

N X

a.n/ sin nx;

nD1

and use the Galerkin method. The branch of steady states emanating from R D  constitutes a pitchfork as is shown in Fig. 1a. The limit as R ! 1 of the bifurcating solutions is computable: Our observation (see Fig. 1b) shows that for any small  > 0, v.x/ ! Ax in 0  x < 1  , where A is a constant. (Note that the solutions on the opposite side of the pitchfork consists of v.1  x/.) The solution has a sharp boundary layer at x D 1 which looks like a shock wave in the inviscid case. Also, our numerical experiment clearly shows that U tends to a certain finite value U1 as R ! 1. Then, in 0  x  1  , it holds that 2vvx C U1 v D 0. Therefore U1 D 2A. If R is set to be 1, the equation on 1.5

1.8 1.4 1.2

0.5

a(1)

v

0

1 0.8 0.6

-0.5

0.4

-1 -1.5

R= 4 R= 6 R = 10 R = 20 R = 50

1.6

1

0.2 0

5

10

R (a)

15

20

25

0

0

0.2

0.4

x (b)

0.6

0.8

1

Fig. 1 (a) Bifurcation diagram (solutions from R D  only). The broken line denotes the trivial solutions. (b) Solutions for 4  R  50. N D 1000

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H. Okamoto

the left of (8) becomes Z

1 2

0

p

Z

1

A2 : 3

x 2 dx D

v DA

1D Therefore A D

2

0

3  1:7320508, which yields

1 a.1/ D 2

Z

1

p Z v.x/ sin x dx D 3

0

p

1

x sin x dx D

0

3 : 

Namely, at R D 1 it holds that p 2 3  1:102657: a.1/ D 

(9)

The numerically computed values of a.1/ are 1:0930

for

R D 100;

1:09786

for R D 200:

Suppose that a.1/  ˛ C

ˇ R

holds true. Then, by extrapolation, ˛ is given by ˛ D .200  1:09786  100  1:0930/=100 D 1:10272; which nicely agrees with (9). In this way, the asymptotic behavior as R ! 1 of the steady state can be computed rather easily in Burgers’s system, and the boundary layer can be analyzed rigorously. Such a case is not common: One needs a lot of hard analysis in many other singular perturbation problems: See, e.g., [130, 135]. Burgers also considers the equation of v alone, i.e., vt C vvx D vxx ; which is used in many textbooks on partial differential equations. If  D 0, then, as is well known, solutions may blow up in finite time, but all the solutions remain smooth for 0  t < 1 if 0 <  and v0 2 H 1 : See Kato [73]. Here and hereafter, H m denotes the usual Sobolev spaces of L2 type (m D 0; 1; 2;    ). Many people expect that the situation is the same for (1) and the Euler equations. Namely, solutions of the Euler Equations may blow up, but the Navier-Stokes equations do not admit a blow-up solution. However it seems to the author that

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we are far from the solution of this famous conjecture. See [18, 48, 86, 89], for instance.

3

Exact Solutions

The equations considered in this section cannot be called models. In a broad sense, any solution of the Navier-Stokes equations which is reduced to solving an ordinary differential equation (ODE) (or a system of ODEs) or to an explicit analytic expression is called exact. The Poiseuille flow and the Couette flow are typical examples. If the reader wishes to find an exact solution for his/her purpose, reading [9] is recommended. References [42, 50, 128] are also interesting sources of exact solutions. In [128] Rosenblatt quotes the following words as Poincaré’s: These [exact] solutions are the only tear by which we can try to penetrate into a place thus far considered unaffordable.

Although the author could not find this quotation in Poincaré’s writings, there seems to be no reason to doubt Rosenblatt. Rosenblatt continues to say that “Indeed, their importance surpasses well those of pure existence theorems whose applicability is almost impossible to see in a given case”. Wang [152] is a more recent survey. Some exact solutions, such as Poiseuille’s and Couette’s, are obtained by solving linear differential equations. Also, harmonic functions provide us with irrotational 2D flows. Such exact solutions are not considered in the present chapter, since many examples of that sort are known: See, for instance, [75, 81, 97, 150]. Except for a few examples, our attention is focused on those which are reduced to nonlinear differential equations.

3.1

Hiemenz and Homann

Such a nonlinear exact solution seems to be discovered first by Hiemenz [63]. He considered a flow in the half plane y > 0 in the situation that the flow impinges on the wall y D 0. He started his analysis by positing .x; y/ D xf .y/, where is the stream function and f is a function of y only. (Note here that the symbol f has nothing to do with the external force f .) Assuming that the flow is stationary and that curl f  0 and substituting the stream function of the above form into the 2D Navier-Stokes equations, he found that all the equations are satisfied if the following single ODE of fourth order is satisfied: f 0000 C ff 000  f 0 f 00 D 0: As for the boundary conditions, he took f .0/ D f 0 .0/ D 0 and f 0 .1/ D constant. The two boundary conditions at y D 0 are reflections of the nonslip boundary conditions. The boundary condition at y D 1 is a consequence of the assumption that the velocity vector u tends to an irrotational flow .˛x; ˛y/ with a constant ˛.

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H. Okamoto

The equation above can be integrated once to obtain f 000 C ff 00  .f 0 /2 D constant: After certain nondimensionalization, the following equation is obtained: f 000 Cff 00 .f 0 /2 D 1

f .0/ D f 0 .0/ D 0; f 0 .1/ D ˙1: (10) The constant on the right-hand side of the differential equation is 1 because of the condition f 0 .1/ D ˙1. (Here it is assumed that f 00 decays to zero more quickly than f .y/ approaches infinity, whence f .y/f 00 .y/ ! 0 as y ! 1. Also assumed is that f 000 .y/ ! 0 as y ! 1.) There is no solution of (10) for f 0 .1/ D 1 [115], but there is a solution for 0 f .1/ D 1. The existence was proved by Weyl [156]. Later Hartman [61] proved that the solution is unique under the condition that 0 < f 0 .y/ < 1 everywhere. It is known (see [130]) that f .y/  y  0:6482 C o.1/ as y ! 1 and that f 0 .y/ D 1 C o.1/ as y ! 1, where o.1/ is actually exponentially small. In order to compute f numerically, the shooting method as follows is useful: Take a large L > 0 and ˇ > 0. Then compute f 000 C ff 00  .f 0 /2 D 1 with an initial condition f .0/ D f 0 .0/ D 0 and f 00 .0/ D ˇ, and choose a ˇ such that f 0 .L/ D 1. Our experiment shows that ˇ  1:2325892. The streamlines constructed by D xf .y/ are plotted in Fig. 2. Since the origin is the stagnation point (u D 0 there), Hiemenz’s flow and its variations are sometimes called stagnation-point flows. Hiemenz’s solution is one of the so-called self-similar solutions, and his idea is frequently used; see the discussion below. Though it is interesting, the reader should not forget about its limitation. For instance, if f 0 .1/ D 1, there is no solution. This can be well understood if the reader looks at Fig. 2: If the directions of the flow are reversed, then small vortices will be created near the origin, but a closed streamline cannot be represented by the ansatz D xf .y/ or D yf .x/. Hiemenz’s study was generalized by Homann [64] to a 3D axisymmetric flow. In the cylindrical coordinates, he set .ur ; u ; uz / D .rf 0 .z/; 0; 2f .z// to obtain an equation similar to Hiemenz’s. After nondimensionalization, it becomes .0 < y < 1/;

y=5

Fig. 2 Streamlines of Hiemenz’s flow

x = -5

x=0

x=5

14 Models and Special Solutions of the Navier-Stokes Equations

f 000 C 2ff 00 C 1  .f 0 /2 D 0

737

.0 < z < 1/;

which should be solved with the boundary condition f .0/ D f 0 .0/ D 0 and f 0 .1/ D ˙1. He constructed his exact solution from this f . The solution exists if f 0 .1/ D 1 (see [61, 156]) and there is no solution if f 0 .1/ D 1 (see [150]). The ansatz employed by Hiemenz and Homann can be applied to flows in other geometries, too. For instance, Berman [10] considered stationary flows contained in a domain bounded by two infinite planes with suction at the boundaries. If the planes are located at y D ˙h, then the following equation is obtained:  2 f 000 C ff 00  f 0 D constant

.h < y < h/;

and the following boundary conditions are imposed: .u; v/.x; h/ D .0; v0 /;

.u; v/.x; h/ D .0; v0 /;

where v0 is a constant. If v0 < 0, fluid is injected into the flow domain, while if v0 > 0, suction takes place. The boundary conditions are rewritten as f .h/ D v0 ;

f 0 .˙h/ D 0;

f .h/ D v0 ;

After a certain nondimensionalization, by introducing the Reynolds number RD

hv0 ; 

the problem becomes   f 0000 C R ff 000  f 0 f 00 D 0 f .1/ D 1;

f .1/ D 1;

.1 < y < 1/; 0

f .˙1/ D 0;

(11) (12)

For each R 2 R there exists a solution of this boundary value problem: Let fR denote them. Proof of existence of a solution of (11) and (12) is not difficult; see [96] and the references therein. As R ! C1, fR0 displays sharp boundary layers at x D ˙1 in the following sense: • for any small  > 0, fR0 .y/ converges to 1 in jyj  1   as R ! C1; • but it keeps fR0 .˙1/ D 0. This is shown numerically in Fig. 3 (the author’s own computation) and is proved mathematically by McLeod [96]. As R ! 1, fR tends to a sinusoidal function: lim fR .y/ D  sin

R!1

y : 2

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H. Okamoto

0

Fig. 3 Berman’s problem. Graphs of fR0 .y/ with R D 10; 20; 50; 100

R = 10 R = 20 R = 50 R = 100

-0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1

Fig. 4 Solutions fR0 .y/ of (11) and (13)

-0.5

0

0.5

1

0

0.5

1

y

1 R = 20 R = 100 R = 1000

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -1

-0.5

y

The case where the two bounding walls are elastic membranes was considered by Brady and Acrivos [11], where they consider (11) with f .˙1/ D 0;

f 0 .˙1/ D 1:

(13)

As in the case of Berman’s problem, there exists a family of solutions, denoted by fR , for all R 2 R. fR exhibits boundary layers at y D ˙1 as R ! 1: See Fig. 4. As R ! 1 it holds that fR0 .y/ !  cos y: If axisymmetric flows of the form .ur ; u ; uz / D .rf 0 .z/; 0; 2f .z// are considered, then the problem becomes f 0000 C Rff 000 D 0 f .1/ D 1;

f .1/ D 1;

.1 < z < 1/; 0

f .˙1/ D 0:

(14) (15)

14 Models and Special Solutions of the Navier-Stokes Equations 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 -2 -1

R = -10 R = -50 R = -100 R = -500

-0.5

0

z (a)

0.5

2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 1 -1

739

R = -5 R = -10 R = -100 R = -500 -0.5

0

z (b)

0.5

1

Fig. 5 Solutions of (14) and (16) at R D 10; 50; 100; 500. (a): fR0 . (b): fR00

or f .˙1/ D 0;

f 0 .˙1/ D 1:

(16)

Numerical solutions and mathematical analysis are given in [102], where related references are presented. The solutions have boundary layers as R ! C1 just as in the case of Berman or Brady-Acrivos. However, it is curious that there appears an interior layer at z D 0 at R 0: See Fig. 5. Note, in the case of (16), that for any small  > 0, fR00 ! 2 in   z  1 and fR00 ! 2 in 2  z  . Or, fR0 converges to 2jzj  1 uniformly in z 2 Œ1; 1. This asymptotic property can be proved rigorously: See [102]. The dynamical systems generated by ftxx D fxxxx C R Œffxxx  fx fxx 

(17)

ftxx D fxxxx C Rffxxx

(18)

or

with either Berman’s boundary condition, the Brady-Acrivos boundary condition, or their mixture offer us interesting problems. See [42, 103, 115, 142, 153]. There are many other directions of generalization. For instance, flows impinging obliquely on the wall can be considered. This solution was discovered independently by three people. See Stuart [140], Tamada [141], and Dorrepaal [41]. A similar problem was considered by Wang [151]: he considered an axisymmetric flow impinging onto a cylinder. He assumed that the outside of the cylinder b D f.r; z/ I b < r < 1; 1 < z < 1 g is occupied by a fluid, where b is the radius of the cylinder. If the viscosity is neglected, the following axisymmetric irrotational flow is possible in this domain:

740

H. Okamoto



b2 ur D ˛ r  r

 ;

u  0;

uz D 2˛z;

where ˛ is a constant. This suggests the following ansatz ur D ˛b 1=2 f . /;

uz D 2˛zf 0 . /;

u  0;

D .r=b/2 :

Then the Navier-Stokes equations are reduced to the following:   f 000 . / C f 00 . / C R 1 C f . /f 00 . /  f 0 . /2 D 0

.1 < < 1/;

(19)

where R is the Reynolds number defined by RD

˛b 2 : 2

Our problem is then to solve (19) with the following boundary conditions: f .1/ D f 0 .1/ D 0;

f 0 .1/ D 1:

(20)

Some numerical examples are shown in Fig. 6. Numerical experiments convince us that for each R > 0 there exists at least one solution of the boundary value problem above. But no proof was known until Yumoto [158] supplied one in 2015. Proof is not trivial even for a small R > 0. Indeed, suppose that R D 0 in (19). The resulting equation f 000 . / C f 00 . / D 0 has no solution under the boundary condition (20). This is a phenomena similar to the Stokes paradox about a flow past an infinitely long cylinder. Thus we cannot apply a perturbation argument for small R. Those are results for flow impinging onto a cylinder perpendicularly. A case of oblique impinging was considered by [111]. 1.2

Fig. 6 Solutions of (19) taken from [115]. R D 10:0; 1:0; 0:1; 0:01

R = 10

1

f '(η)

R=1

0.8

R = 0.1

0.6

R = 0.01

0.4 0.2 0

0

2

4

6

8

10

12

η

14 Models and Special Solutions of the Navier-Stokes Equations

3.2

741

Jeffery and Hamel

Another class of exact solutions was discovered by Jeffery [69] and independently by Hamel [60]. (Von Kármán [148] claims that he and one of his pupils discovered the same solution independently.) These solutions are concerned with flows in an infinite sector defined with a constant ˛ 2 .0; / as f.r;  / I 0 < r < 1; ˛ <  < ˛g, and the origin is either a source or a sink. In this case, the polar coordinates are convenient, and the Navier-Stokes equations become   u2 @ur 1 @p ur 2 @u C .u  r/ur  D C  4ur  2  2 ; @t r  @r r r @   1 @p @u ur u u 2 @ur C .u  r/u C D C  4u  2 C 2 ; @t r r @ r r @ 1 @.rur / 1 @u C D 0; r @r r @

(21)

(22)

(23)

where .u  r/ and 4 are defined, respectively, by .u  r/ D ur

u @ @ C ; @r r @

4D

@2 1 @2 1 @ C C : @r 2 r @r r 2 @ 2

Jeffery and Hamel substituted  .ur ; u / D

 f ./ ;0 : r

(24)

Then (21), (22) and (23) are satisfied if and only if .f 00 C 4f / C f 2 D constant:

(25)

Definition 1. The flow which is defined by (24) and (25) is called a Jeffery-Hamel flow. If g is defined by g D  1 f , then g 00 C 4g C g 2 D b; where b is a constant to be determined. The following integral Z

Z

˛

˛

rur .r;  / d D ˛

f ./ d ˛

(26)

742

H. Okamoto

represents the flux of fluid through the circular arc: r fixed, ˛ <  < ˛. Suppose that this flux is predetermined, say, Q. Then

Z

˛

g. / d D ˛

Q .DW R; say/: 

(27)

Since Q= is dimensionless, it is here denoted by R and is called the Reynolds number. (Though it is called the Reynolds number, this R can take any real number whatsoever.) Now the problem is to find g and b satisfying (26) and (27) for a given R. Since this problem looks very simple, the following theorem may look curious: Theorem 1. For a given R, positive, zero, or negative, there exist infinitely many solutions of (26) and (27) with g.˙˛/ D 0. This curious theorem was proved in [47]. More specifically for each R > 0 and positive integer N , there is a solution which has N inflows and N  1; N , or N C 1 outflows. As Fig. 7a indicates, this flow shows at high Reynolds number a laminar boundary layer, which is amenable to mathematical analysis and rigorous proof of existence. In the case of a solution in Fig. 7a, g. /=R tends to, for each ˛ <  < ˛, a nonzero constant as R ! 1, while the boundary condition g.˙˛/ D 0 is maintained. See [137]. Other properties of the Jeffery-Hamel flows can be found in [4, 113, 129] Jeffery-Hamel flows do not have a nonstationary counterpart. However, it has a 3D axisymmetric counterpart, namely, a flow in a cone. This was studied by [1]. Its quite nontrivial streamlines might be interesting to the reader.

Fig. 7 (a) Boundary layers appearing in the Jeffery-Hamel flow. ˛ D =4; R D 200. The thick line consists of the end points of the velocity vectors constructed on the arc r D constant. (b) Jeffery-Hamel flow with two inflows and two outflows. R D 30

θ=α

θ = -α

(a)

(b)

14 Models and Special Solutions of the Navier-Stokes Equations

4

743

Nonstationary Exact Solutions

The solutions in the preceding section are stationary. The present section is devoted to nonstationary exact solutions.

4.1

Riabouchinsky and Proudman-Johnson

In 1924 Riabouchinsky considered in [126] a nonstationary solution of the 2D Navier-Stokes equations of the form D xf .t; y/ in the half plane 0 < y. (His notation differs from ours. The author changed it since he wishes to use notation common to Hiemenz’s and others.) Then f satisfies ftyy  ffyyy C fy fyy D fyyyy :

(28)

His boundary conditions at the wall y D 0 are f .t; 0/ D 0 and fy .t; 0/ D a, where a is a constant. Namely, the wall is an expanding or contracting elastic membrane. (Boundary condition of this kind p after Brady and Acrivos [11].) He  is now called found that f D ˇ 1  e ˇy with ˇ D a= is a solution. (Note that a > 0 is necessary for this solution to exist.) Unfortunately his analysis stopped there, and his paper seems to have been forgotten. It would be interesting to study the stability of his solution under some boundary conditions. For instance, is 1  e y a stable solution of the following initial-boundary value problem? ftyy  ffyyy C fy fyy D fyyyy ;

f .t; 0/ D 0; fy .t; 0/ D 1;

f .t; 1/ D 1; fy .t; 1/ D 0:

(29)

What happens if a < 0? Riabouchinsky says nothing about this case. It is naturally expected that the situation is quite similar to that in Hiemenz’s flow. Since f ’s defining domain is unbounded, the answer does not seem to be easy. In 1962, Proudman and Johnson [125] considered a laminar boundary layer near a stagnation point and arrived at (28). They actually set F .t; y/ D f .t; y/ and obtained Ftyy C FFyyy  Fy Fyy D Fyyyy

.0 < y < 1/:

(30)

The no-slip boundary condition at the wall requires F .t; 0/ D Fy .t; 0/ D 0. If it is assumed, as Hiemenz did in the case of stationary flows, that the flow tends to an irrotational flow u D .˛x; ˛y/, where ˛ is a constant, then it holds that Fy .t; 1/ D ˛; Fyy .t; 1/ D 0. The equation (30) can be integrated once and the following equation is obtained: Fty C FFyy  .Fy /2 D Fyyy C constant:

744

H. Okamoto

The constant is ˛ 2 by the boundary condition at infinity. Here again, it is assumed that the decay of Fyy is faster than the growth of F . After nondimensionalization one obtains Fty C FFyy  .Fy /2 D Fyyy  1

.0 < y < 1/

(31)

with F .t; 0/ D Fy .t; 0/ D 0; Fy .t; 1/ D ˙1. By defining u D Fy , (31) is further rewritten as Z y 2 F .t; y/ D u.t; / d .0 < y < 1/ (32) ut C F uy  u D uyy  1; 0

with u.t; 0/ D 0; u.t; 1/ D ˙1. Their analysis was noted by many authors and (28) is now called the ProudmanJohnson equation (abbreviated to PJ henceforth). They seems to be interested in the flow near the rear end of a cylinder. If we magnify the neighborhood of the stagnation point at the rear part of the cylinder, its wall may be replaced by the straight wall. In that case, Fy .t; 1/ D 1 is imposed as a boundary condition. However, our analysis should begin with the simpler case. Thus, we first consider the case where the boundary condition Fy .t; 1/ D 1 is employed. Then Hiemenz’s solution FF 00  .F 0 /2 D F 000  1;

F .0/ D F 0 .0/ D 0; F 0 .1/ D 1:

(33)

is a steady state of (31). It is very natural to expect the stability of the solution, i.e., that solutions of (31) near Hiemenz’s solution (33) converges to it as t ! 1. In order to verify the asymptotic stability, the author performed the following simple numerical experiment. Let y run in 0  y  10 and replace (32) with the following: ut C F uy  u2 D uyy  1 .0 < y < 10/;

u.t; 0/ D 0; u.t; 10/ D 1; Z y F .t; y/ D u.t; / d :

(34)

0

Then this initial-boundary value problem is discretized by the central finite difference scheme. The result is shown in Fig. 8. If the initial data is below (33) (Hiemenz’s solution), the solution increases, as t increases, to the steady states. If the initial data u.0; y/ is above (33), then the solution decreases to the steady states (see Fig. 8a). Based on these and other experiments, the author believes that (30) admits a time-global solution converging to its unique steady state if u.0; x/  1 everywhere. However, he is unable to supply a mathematical proof. If the initial data u.0; y/ is greater than unity partly in the interval 0 < y < 1, global existence seems to be no longer guaranteed and the author cannot tell what happens in this case. Note that the one-dimensional (1D) dynamical system

14 Models and Special Solutions of the Navier-Stokes Equations 0 -0.2

u

t=0

-0.4

u

-0.6 -0.8 -1

0

2

4

y (a)

6

8

10

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

745

t=0

0

2

4

y (b)

6

8

10

Fig. 8 Solutions of (34).  D 1; 0  y  10; (a) u.0; y/ D 3y=.10 C 2y/. (b) u.0; y/ D sin.3:5y=10/

100

u

80 60 40 20

y=0

y = 200

Fig. 9 Solution of (34).  D 1. 0  y  200. u.0; y/ > 1 partly in 0  y  200

w P D w2  1 has two equilibria, w D 1 and w D 1. The former is unstable and the latter is stable. w blows up if w.0/ > 1. In view of this, the restriction u.0; x/  1 may not be very far from the best criterion of stability. If the boundary condition at infinity is u.t; 1/ D 1, then there is no steady state satisfying juj  1. How about the asymptotic behavior of u as t ! 1? This is closely related with the study of Proudman and Johnson [125]. Figure 9 shows an experiment. Here the computation is carried out in 0  y  200. The initial value u.0; y/ D y for 0  y  2 and u.0; y/ D 4  y for 2  y  3, and u.0; y/  1 for 3  y  200. maxy u.t; y/ increases rather quickly. Since the peak of the graph moves continuously rightward, even 0  y  200 is not long enough, and it is not easy to decide whether the blowup occurs or not.

746

H. Okamoto

Remark 1. In the present chapter, the author uses the algorithm in [25] for the computation of a blow-up solution. Although it is not very sophisticated, it has a merit that it is easy to implement and convergence of the blow-up time is guaranteed theoretically in the simplest case. It may be interesting to compare this conjecture with the blow-up result of [43], where ut C F uy  u2 D uyy .0 < y < 1/ is considered (see (2.3) of [43]). Namely, the inhomogeneous term 1 is lacking. The authors of [43] proved that there exist solutions which blow up in finite time. Note that this equation arises from the boundary layer approximation. Proving the global existence of solutions of (32) seems to be challenging, since F is unbounded and the domain is semi-infinite. Some other open problems are associated with it and its generalizations. Those will be discussed later in this chapter. The PJ equation in a finite interval was used by [24] to show that a similarity solution of the Navier-Stokes equation can blow up in finite time. This paper is important in that it suggested such a possibility for the first time. It is known (see Fujita and Kato [49] or Taylor [143]) that the 2D Navier-Stokes equations admit no blowup if the initial value belongs to L2 and if only those solutions satisfying the energy inequality 1 ku.t /k2L2 C  2

Z

t 0

kru.s/k2L2 ds 

1 ku.0/k2L2 2

(35)

are admitted. Our similarity solution is unbounded in the spatial variable x and does not belong to L2 , whence it does not satisfy the energy inequality. Therefore the usual theory does not apply to the similarity solutions. Well posedness of these solutions requires an independent study. Therefore [24] opened a window to a new class of mathematical problems. Now, as [35] shows, the authors of [24] seem to have realized that their “blow-up” solution of the 2D Navier-Stokes equations is not a real singularity but is actually a numerical artifact. The author and Sh¯oji [120] also independently realized this possibility. The problem is to study a self-similar flow given by D xF .t; y/ in an infinite rectangle 0 < y < 1; x 2 R with the no-slip boundary conditions at both ends of the rectangle (y D 0 and y D 1). The equation can be written as ut C F uy  u2 D uyy  ;

Z

y

F .t; y/ D

u.t; / d

.0 < y < 1/;

(36)

0

where is an unknown constant. The boundary conditions u.t; 0/ D 0 and u.t; 1/ D R1 0 are imposed. The boundary condition F .t; 1/ D 0 is rewritten as 0 u.t; x/ dx D 0, whence u must evolve with this integral constraint. The constant should be so R1 chosen that 0 u.t; y/ dy D 0 is maintained for all t .

14 Models and Special Solutions of the Navier-Stokes Equations

747

A finite difference method which replaces the differentiation with a central difference yields a numerical solution with good accuracy if  is not too small. Numerical experiments tell us that for initial short time the function u D Fy develops sharp boundary layers at both ends of the interval. See Fig. 10. Since this time interval is short, it is necessary to choose a sufficiently small time increment t . If t is not small enough, a spurious blowup may possibly result. As Fig. 11 shows, Fyy can be considerably large, so large that one might jump into a wrong conclusion, but eventually it begins to decrease and converges to zero as t ! 1. A mathematical proof that no blowup exists in (36) was first provided by [22]:

150

t = 0.5

100

u

50 0 -50 t

y=0

-100 -150

y=1

t=0 0

0.2

0.4

(a)

y (b)

0.6

0.8

1

A Fig. 10 F .0; y/ D 16 w.16w4  8w2 C 1/ C Bwy 2 .1  y/2 , where w D y  12 , A D 1000; B D 0:01A;  D 0:01. (a) graph of u D Fy in 0  t  0:5. (b) profiles of u.t; / at t D 0 and t D 0:5

16000 14000 12000 10000 8000 6000 4000 2000 0

0

0.5

1

1.5

2

2.5

t

3

3.5

4

4.5

Fig. 11 kFyy .t; /kL1 as a function of t . The same initial data as in Fig. 10

5

748

H. Okamoto

Theorem 2. Any one of the following three boundary conditions is considered: u.t; 0/ D u.t; 1/ D 0, or ux .t; 0/ D ux .t; 1/ D 0, or the periodic boundary condition. Then for all the initial value u.0; / 2 L2 , the solution of (36) exists globally in time, and it tends to zero as t ! 1. Since no conserved quantity is available for the PJ equation, a certain trick is needed for proving this theorem: See [22]. Something different happens if one considers the PJ equation (30) with nonhomogeneous boundary conditions, say, f .t; 0/ D 0 ;

fxx .t; 0/ D 0 ;

f .t; 1/ D 1 ;

fxx .t; 1/ D 1 :

Numerical experiments in [56] suggest that a solution can blow up if the constants 0 ; 1 ; 0 ; 1 are suitably chosen. There seems to be no mathematical proof, however. Since the nonlinear term is not altered, and since the boundary conditions are linear, this is rather curious. It is also curious that the solution is predicted (see [56]) to blow up with the rate of 2, i.e., ku.t /k  constant  .T  t /2

.t ! T /:

(37)

Note that, since the nonlinear term is quadratic, one may well expect that it blows up with the rate 1, i.e., ku.t /k  constant  .T  t /1 :

4.2

Unimodality

Here a numerical result in [76, 77] is briefly reported. This is about a stationary solution of ftxx C ffxxx  fx fxx D

1 .fxxxx  sin kx/ R

.0 < x < 2/

with the periodic boundary condition. Here R > 0 is the Reynolds number, and k can be any positive integer. f D k 4 sin kx is an exact steady state, which is stable if R > 0 is small enough. If R is increased, it loses stability and a new branch of steady states bifurcates. The new solutions, which are stable, have k peaks and k bottoms. However, a curious thing happens if R is increased further. If the Reynolds number R is large enough, the corresponding solution has one and only one peak and one bottom. See Fig. 12. In fact, the computation in [76, 77] shows that there is a family of solutions which are asymptotically f .x/ D k sin x C o.1/

as R ! 1:

14 Models and Special Solutions of the Navier-Stokes Equations

a(1)

2

2

1.5

1.5

1

1

0.5

0.5

f (x)

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

0

20

40

R (a)

60

80

100

749

-2

R = 3.65 R=5 R = 100 -3

-2

-1

0

x (b)

1

2

3

Fig. 12 (a) The bifurcation diagram. k D 2. (b) The steady states at R D 3:65; 5; 100, respectively

A computation by a singular perturbation analysis shows that

f .x/ D k sin x C

1 h.x/ C O.R2 / R

as R ! 1;

where h is a solution of h0000  hh000 C h0 h00 D 0: (Note that h is independent of k.) Making a paraphrase, it may be said that the external force sin kx yields a solution  k sin x at large Reynolds numbers. Since this solution has one and only one peak, [76] called it a unimodal solution. A very oscillatory (k 1) external force can produce a unimodal solution at R 1. As [79] suggests, this peculiar phenomena can be related to the inverse cascade theory of 2D turbulence [80]. According to it, the following picture is possible. If the energy is input in the (intermediate) range of wave numbers, it is, by nonlinearity, transferred to modes of higher and lower wave numbers. In 3D turbulence, this is almost exclusively toward higher wave numbers. Since the viscosity works more strongly for higher wave numbers than for lower wave numbers, the energy will be dissipated in high wave numbers. In 2D turbulence, however, the energy is transferred to lower wave numbers as well as to higher wave numbers. Therefore 2D structures of large size are often observed in otherwise turbulent motions. The existence of a unimodal solution may be considered as one of the consequences of the inverse cascade theory. The unimodal solutions are not restricted to the PJ equation, but appear in many 2D Navier-Stokes flows, too: See [76, 79].

750

4.3

H. Okamoto

Generalized PJ Equation

The following equation was introduced by [122] and was named the generalized Proudman-Johnson equation: ftxx C ffxxx  afx fxx D fxxxx ;

(38)

where a is a real parameter. They describe a variety of flows of similarity form. . Indeed, suppose first that m is a positive integer greater than 1, and set a D  m3 m1 Then an m-dimensional axisymmetric flow represented by ur D 

r fz .t; z/; m1

uz D f .t; z/;

q 2 , is reduced to (38). Second, suppose where z D xm ; r D x12 C x22 C    C xm1 that 1  a < 3. Then a flow around an infinite wedge (sector) of angle .a C 1/=2 can be described by a stationary solution of (38). This was considered mathematically by [156]. (This paper by Weyl, which seems to be little read now, is very interesting and is highly recommended.) Third, if differentiated twice, the Burgers equation ft C ffx D fxx becomes (38) with a D 3. Finally a model equation proposed by Hunter and Saxton [66] leads to (38) with a D 2 and  D 0 by differentiating their equation once. Accordingly many equations are described by (38) by varying a. Equation (38) is also related to the equation in [13] and the one called b-equation in [44, 45]. Well-posedness of (38) is only partly known and some conjectures exist. It will be discussed later in Sect. 8. Hamada [59] considered a problem similar to [56] for the generalized PJ equation for 1  a. His analysis suggests that the blow-up rate may be 2 for a D 1, but is equal to 1 for any 1 < a. This discontinuity is interesting, but the author does not know why this may happen, nor does he know whether this can be rigorously proved.

4.4

Coupled Equations

Another generalization of the PJ equation is obtained in 3D flows. This was done in Lin [84]. Although his coupled equations are quite general, the author believes that the coupled equations proposed by Grundy and McLaughlin [57] represent most important characters in 3D similarity solutions. In what follows, analysis shall be applied on a pair of equations, which are equivalent to those in [57]. These are introduced by [103, 160] and are derived from the velocity of the following form: u D f .t; x/  g.t; x/;

v D yfx .t; x/;

w D zgx .t; x/:

14 Models and Special Solutions of the Navier-Stokes Equations

751

If these are substituted into the 3D Navier-Stokes equations, then ftxx D fxxxx C .fx C gx /fxx  .f  g/fxxx ;

(39)

gtxx D gxxxx  .fx C gx /gxx  .f  g/gxxx :

(40)

These equations are to be satisfied in 0 < x < 1, say. As a boundary condition, the no-slip conditions are employed: f .t; 0/ D g.t; 0/ D f .t; 1/ D g.t; 1/ D 0; fx .t; 0/ D gx .t; 0/ D fx .t; 1/ D gx .t; 1/ D 0:

(41)

With a suitable nondimensionalization, it may be assumed that  D 1. Note that if g.0; x/  0, then g  0 for all t  0, and only (39) with g  0 survives, which is nothing but the PJ equation. Also note that if g D f , then they are reduced to (18)—the nonstationary Homann equation. The author believes that the solutions in general of (39) and (40) blow up in finite time. Numerical experiments for (39) and (40) are reported in [160], where some convincing numerical evidence for blowup can be found. This is most easily observed in the following special case. Put g.t; x/ D f .t; 1  x/. Then the coupled equations are reduced to a single, nonlinear, nonlocal equation: ftxx .t; x/ D fxxxx .t; x/ C Œfx .t; x/  fx .t; 1  x/ fxx .t; x/  Œf .t; x/  f .t; 1  x/ fxxx .t; x/:

(42)

Integrating once yields ftx .t; x/ D fxxx .t; x/ C fx .t; x/2  f .t; x/fxx .t; x/ C f .t; 1  x/fxx .t; x/ C c; (43) where c is independent of x. Or, by putting u.t; x/ D f .t; x/ and f .t; x/ D x Rx u.t; s/ ds, 0 ut .t; x/ D uxx .t; x/Cu.t; x/2 f .t; x/ux .t; x/Cf .t; 1x/ux .t; x/Cc:

(44)

The boundary condition u D 0 is imposed at x D 0; 1, and c is chosen so that the R1 integral constraint 0 u.t; x/ dx D 0 is maintained. If the term f .t; 1  x/ux .t; x/ is absent, the solution exists globally (Theorem 2). A solution of (44), however, can blow up in finite time if its initial data is large, as is shown in Fig. 13. Therefore it can be said that the added nonlocal term induces a blowup. A mathematical proof is, however, lacking. The system (39) and (40) with nonhomogeneous boundary conditions has quite a large number of steady states; see [103]. This suggests that the dynamics is rich. It is, however, yet to be explored.

752

H. Okamoto

800000

|| f xx ||

600000

400000

200000

0

0

t

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Fig. 13 Solution of (42). The time evolution of kfxx .t /k1 . fxx .0; x/ D 100.3x 2  1/

5

Oseen’s Self-Similar Solutions

Carl Wilhelm Oseen is most famous for his contributions to low Reynolds number fluid flows and his theory of liquid crystal. However, Oseen’s self-similar solutions of the Navier-Stokes equations are intriguing as well. They are discovered by [124] and were studied by [113] from the viewpoint of the bifurcation theory.   With the notation in [113], Oseen’s assumption is this: u D y ;  x , where the stream function is of the following form: D f . / C C , where C is a constant, f is a function of one variable, and and  are defined by

D B log r C  ;

 D log r  B ;

where B is another constant: r and  are the polar coordinates. It is assumed that is smooth except at the origin, that the gradient of is a single-valued function of , and that the Navier-Stokes equations are satisfied in R2 n f0g. Set f . / D R 2 ˛0 C g. / with a constant ˛0 so that 0 g. / d D 0. Then define U by g 0 . / D .1 C B 2 /U . /. This assumption leads us to the following problem: Find a 2periodic function U D U . / such that Z

2

U . /d D 0;

U 00 C AU D U 2 

0

1 2

Z

2

U . /2 d ; 0

where A is a parameter depending on B and the Reynolds number: AD

4 C R : .1 C B 2 /

(45)

14 Models and Special Solutions of the Navier-Stokes Equations

(a)

(b)

753

(c)

Fig. 14 Streamlines of Oseen’s spiral flows. (a) R D 0; A D 0:1. (b) R D 0; A D 0:6, (c) a streamline in (a)

The Reynolds number is defined by RD

1 

Z

2

rur .r;  / d ; 0

which is independent of r hence is a constant. Once U is obtained, the stream function is easily obtained by conversely tracing the process above: See [113]. Two samples are drawn in Fig. 14. Here the Reynolds number R is zero, and all the streamlines are closed. For instance, in the case of (a), streamlines are mutually similar to the closed curve depicted in (c). If A ! 1, Oseen’s flow exhibits a curious interior layer, which was analyzed by [68]. The interior layer appears along a logarithmic spiral. This is one of the examples in fluid mechanics in which exact analysis of singular perturbation is possible. On the other hand, the existence of a global branch of nontrivial solutions of (45) is proved by [67]. Oseen [124] considered also a 3D version of his flows, which is also studied by Rosenblatt [128]. Related works are [30, 92, 107].

6

Leray’s Self-Similar Solutions

In his famous thesis [83], Leray introduced many revolutionary ideas such as weak derivative, mollifier, etc. His self-similar solution, which appeared also in the paper, seems to have been hidden by these great ideas, but it seems to the author that Leray was attached to the idea that a singular solution can be constructed in his selfsimilar form. The following ansatz (47) reduces the 3D nonstationary Navier-Stokes equations to equations which describe “stationary solutions.” Indeed, suppose that T > 0 is given and that U ./ and P ./ . D .1 ; 2 ; 3 // satisfies the following equations: 4U ./  rP ./ D .U  r/U C U C .  r/U ;

div U D 0:

(46)

754

H. Okamoto

Then Leray defines u and p through U and P as p 

U u.t; x/ D p 2.T  t /

!

x

 ; p.t; x/ D P p 2.T  t/ 2.T  t /

x

!

: p 2.T  t / (47) It is easy to see that (47) satisfies the 3D Navier-Stokes equations. Leray expected that the solution blows up at t D T . However, he could not find a nontrivial solution of (46). He seems to have had an idea that singularities of the Navier-Stokes equations are strongly connected to turbulence. Since no singularity has been found up until now, his idea about turbulence is not directly supported by the mathematics/physics community. Instead, turbulent motion of incompressible fluids may be better explained by the languages of the theory of dynamical systems and/or statistical physics. Our present knowledge on dynamical systems seems to be far away from good understanding of turbulence. The author admits that some turbulent motion of fluid is well explained by a chaotic dynamical system. However, the phenomena of turbulence are so wide ranged and would not be explained only by a few simple theories. Long after Leray, [105] proved that if the solution of (46) satisfies that U 2 L3 , then U  0. Therefore a nontrivial U is difficult to find. The result by Tsai [146] is more decisive: he proved that there is no U which makes u satisfy the energy inequality (35). Accordingly proving blowup by Leray’s plan turned out to be impossible. These important results, however, do not imply that no self-similar solution exists at all. If the energy inequality is abandoned, there are nontrivial solutions. In fact [112] found many exact solutions of (46), which of course do not belong to L2 or L3 . One of his examples is U D . .2 /; 0; 0/;

P D p0 1 ;

where p0 is a constant. Then all the conditions are met once satisfies 00 .2 / D p0 C .2 / C 2 0 .2 /, whose general solutions are 

2 p0 C A exp 2 2





2 C B exp 2 2

Z

1 2



s2 exp  2

 ds;

.A; B W constants/:

In particular, let us consider a flow in the half space 2 > 0 and suppose that the velocity field is bounded and that .2 / ! 0 as 2 ! 1. Then p0 D A D 0, and one may set B D 1 without losing generality. Then the following asymptotic expansion holds true:

.2 / 

1 1 3  3 C 5   2 2 2

.2 ! 1/:

14 Models and Special Solutions of the Navier-Stokes Equations

755

p Note that satisfies .0/ D 2 . Note also that u.t; / belongs to L1 for all t < T . The asymptotic behavior as t ! T of u D .u; 0; 0/ is

u.t; x; y; z/ !

8  ˆ 0 is small or is zero. Note that the external force is absent, whence this growth of ! is caused spontaneously by nonlinear terms. Although there is a possibility that the vorticity becomes infinite in finite time, it seems to be a majority opinion that it eventually decreases to zero if  > 0. However, if  is small, the vorticity can be very large for quite a long time interval. It is of utmost importance to understand how and why this occurs. When the author was a student, he often heard that the viscous term suppresses the vorticity and that the convection term is neutral to its increase. Therefore, it was said, the stretching term was responsible for the increase of the vorticity. The author does not know when this was said first or who did, but most people seem to consider the “proposition” that the convection term is neutral, to be valid [28, 89].

14 Models and Special Solutions of the Navier-Stokes Equations

759

The author (and independently by other people, too) considered that this “proposition” was not substantiated by evidence. To examine it, he came up with the following trick. Suppose that the convection term is completely neutral. Then blowup or global existence of a solution would be determined by the stretching term alone. Therefore the following conjecture seems to be plausible. The equation obtained from (56) by dropping the convection term, i.e., @!  .!  r/u D 4!: @t

(57)

has the same property if only the global existence/blowup is concerned. Namely, if all the solutions of (56) exist globally in time, then the same is true for (57) and vice versa. (Equation (57) as it stands is not suitable for our analysis, since it destructs the divergence-free condition. See the 2D example below.) In the following pages, this hypothetical proposition will be examined through examples.

8.1

PJ Equation

The author began some study in [122] about the PJ equation with the periodic boundary condition. In favor of compatibility with [122], the PJ equation with f in place of f is considered. Thus, with  > 0, ftxx C ffxxx  fx fxx D fxxxx :

(58)

Note that this can be written as !t C f !x  fx ! D !xx

.! D fxx /:

It would therefore be appropriate to call ffxxx a convection term and fx fxx a stretching term. Now (58) admits no blow-up solution, as is proved in [22]. Then consider an equation where the convection term in (58) is deleted: ftxx  fx fxx D fxxxx

.0 < x < 1/:

(59)

If integrated, one obtains 1 ftx  .fx /2 D fxxx C constant 2

.0 < x < 1/:

If u D 12 fx is introduced, the following equation comes out: ut D uxx C u2 C constant

.0 < x < 1/:

(60)

760

H. Okamoto

The periodic or Dirichlet boundary condition on f imposes u on the following condition Z

1

u.t; x/ dx D 0:

(61)

0

In the case of the periodic boundary condition, the constant in (60) can be easily expressed by u if it is integrated: ut D uxx C u2 

Z

1

u.t; x/2 dx

.0 < x < 1/:

(62)

0

Therefore our problem has been reduced to solving (62) with the integral constraint (61) and the periodic boundary condition. In the case of the Dirichlet boundary condition, it is required to solve (60) and (61) with the Dirichlet boundary condition u.t; 0/ D u.t; 1/ D 0. Since the PJ equation is globally well-posed (see Theorem 2), the same must be true for (60) and (61) (and (61) and (62), too) according to the “proposition” above. If the integral constraint does not exist, the nonlinear heat equation with the periodic boundary condition admits blow-up solutions. Whether the integral constraint (61) is strong enough to prevent the solution from blowing up is our question. It is actually not so strong and (61) and (62) admits blow-up solutions. Since the maximum principle, which plays a prominent role in a scalar heat equation, does not hold true for (61) and (62), it cannot be used for proving blowup. However, the following theorem was proved in [122]. Theorem 3. Consider (61) and (62) with the periodic boundary condition. Suppose that the initial function u0 satisfies Z

1

u0 .x/ dx D 0; 0

1 3

Z

1

u0 .x/3 dx 

0

1 2

Z

1

u0;x .x/2 dx > 0:

0

Then the solution blows up in finite time. Proof is carried out by the so-called convexity argument, see [122]. See also [12,13]. Although (62) is a heat equation with an integral constraint and is just a small modification from a nonlinear heat equation ut D uxx C u2 , their behavior near the time of blowup is quite different. For ut D uxx C u2 , u.t; x/ is monotonically increasing in t near the blow-up time in a neighborhood of the blow-up point at least. However, for (62), this is true for only one specific x, and u.t; x/ eventually decrease for other x’s. See Fig. 16, where a blow-up profiles are computed numerically. Here the initial data is u.0; x/ D 1000 sin.2x/. Blowup occurs at t  0:00149. It is compared with the solution of ut D uxx C u2 , which blows up at t  0:00104 for the same initial data. Its asymptotic behavior is analyzed in [12,13] by an asymptotic method, but there are still something yet to be proved rigorously.

14 Models and Special Solutions of the Navier-Stokes Equations

761

u

u

t

x=0

t

x=0

x=1

(a)

x=1

(b)

Fig. 16 u.0; x/ D 1000 sin.2x/. (a) The blow-up solution of (61) and (62). (b) the same for ut D uxx C u2

Now the PJ equation is globally well posed (Theorem 2), while solutions may blow up once the convection term is deleted. One interpretation of this fact is that blowup is suppressed by the convection term. This does not imply that any convection term can prevent a solution from blowing up: See (44), for instance. In order to suppress blowup, the convection term should be a proper one. This example suggests that the convection term can play an important role in global well-posedness. (A similar view is presented in [82], too, in the author’s interpretation.) From this the author was naturally led to the following conjecture on the generalized PJ equation. Namely, if a is small, (38) is globally well posed; and if jaj is large, it admits a blowup. Chen and the author [23] proved the following theorem: Theorem 4. Suppose that  > 0 and 3  a  1. Then (38) is well-posed globally in time under the periodic boundary condition. If 1 < a < 3 or 1 < a and if  > 0, the author expects that large solutions blow up in finite time. For this conjecture, some numerical experiments were performed in [59, 122] and they seem to support the conjecture, but the author does not know a mathematical proof. Note that (38) tends to (59) as a ! 1. Indeed, if f is replaced by fQ =a and multiply (38) by a, the equation becomes 1 fQtxx C fQ fQxxx  fQx fQxx D  fQxxxx : a

(63)

762

H. Okamoto

Table 1 Cases of blowup and global existence >0

1 < ˛ < 3 Blow-up (?)

D0

1 < ˛ < 1 Blow-up

3  ˛  1 Global existence

1  ˛ < 1 Global existence

1 < ˛ < C1 Blow-up(?)

1  ˛ < C1 Blow-up(?)

By letting a ! 1 one obtains (59). Since it admits blow-up solutions, the conjecture above complies with this observation. Hamada [59] studied numerically blow-up solutions with the boundary condition in [56]. The situation changes if the viscosity is neglected. In this case, blow-up solutions exist in wider range of a. The results in [23,59,114,122] are summarized in Table 1. Here the question mark (?) implies that it has not been proved, though there is numerical evidence. Something different happens if the same equation is considered in the whole real line with a special discrete values in a D .n C 3/=.n C 1/ .n D 1; 2;    / and if one admits solutions which are bounded but not necessarily decay as x ! ˙1. See [26]. Here is a remark about the unimodality: If the convection term is removed from the PJ equation, then the unimodal solution does not appear. This was demonstrated numerically in [116].

8.2

Examination by 2D Euler Equations

Another interesting example is provided by the numerical experiment in [109], which is about 2D flow. Its vorticity !D

@v @u  @x @y

is a scalar and satisfies, if the external force is zero, @! @! @! Cu Cv D 4!: @t @x @y Therefore , which is defined by  D

@! @! ; @y @x

 ;

satisfies the following equations: @ C .u  r/  .  r/u D 4: @t

(64)

14 Models and Special Solutions of the Navier-Stokes Equations

763

If this is supplemented by  D 4u

or

u D .4/1 :

then (64) is closed in . Note that u 7! ! is of order 1, while u 7!  is of order 2. Accordingly, although it has a formal similarity to the 3D vorticity equations (56), (64) is more regularized than the 3D vorticity equations. Now, in the 2D vector equation (64), .ur/ may naturally be called a convection term. To verify its importance, it is cut off to obtain @  .  r/u D 4: @t

(65)

In this equation, the divergence-free conditions div u D 0 and div  D 0 are no longer maintained. Therefore there are two ways: to solve (65) without the divergence-free condition or to modify it into a divergence-free form. Accordingly (65) is supplemented by either u D .4/1 ;

(66)

  u D P .4/1  ;

(67)

or

where P is the Helmholtz projection, namely, it is the orthogonal projection in the Hilbert space L2 onto the subspace of all the divergence-free vector fields. In [109] numerical computation was carried out for (64) and (65) with  D 0 in the square 0  x  2; 0  y  2 with the periodic boundary condition. For computation, use has been made of the pseudo-spectral method with 1024  1024 modes with de-aliasing by the 2=3 rule. E.t /; Q.t /, and P .t / are defined, respectively, as ku.t; /k2L2 ;

k!.t; /k2L2 ;

k.t; /k2L2 :

(68)

Then they increase as are shown in Fig. 17. (Note that E.t / and Q.t / are invariants for (64).) The difference of the solid line and the broken line may be so interpreted that the growth of  is restricted by the incompressibility. In view of the theorem of Beale, Kato, and Majda [7], the following quantity Z t 7! 0

t

k.s; /kL1 ds

(69)

of the solution of (65) and (66) is plotted in Fig. 18. It shows a considerable growth of the integral (69). Whether it really blows up in finite time remains to be solved.

H. Okamoto 0.5 0.48 0.46 0.44 0.42 0.4 0.38 0.36 0.34 0.32 0.3

1.1 1 0.9 0.8

Q(t)

E(t)

764

0.7 0.6 0.5 0.4 0.3

0.5

1

1.5

t

2

P(t)

0

2.5

3

3.5

0

0.5

1

1.5

t

2

2.5

3

3.5

10 9 8 7 6 5 4 3 2 1 0 0

0.5

1

1.5

t

2

2.5

3

3.5

Fig. 17 Plot of E.t/; Q.t / and P .t/ taken from [109]. !.0; x; y/ D sin x sin y C cos y .0  x  2; 0  y  2/ with the periodic boundary conditions. Solid lines are for (65)(66). Broken lines are for (65)(67), Dashed-broken lines are for the Euler equations (64)

Fig. 18 Plot of (69) as a function of t: Taken from [109]

4000 3500 3000 2500 2000 1500 1000 500 0

0

1

2

3

t

4

5

6

14 Models and Special Solutions of the Navier-Stokes Equations

8.3

765

Constantin-Lax-Majda and Its Sisters

In 1985, Constantin, Lax, and Majda [31] proposed the following model equation: @! D !  H!; @t

(70)

where ! is a function of t > 0 and x 2 R and H denotes the Hilbert transform (in the x variable). The equation (70) is now called the Constantin-Lax-Majda equation. It is called a 1D model for the 3D Euler equations. Their motivation for (70) seems to be to show a possibility that the Euler equations admit blow-up solutions. Their paper stimulated quite many papers pursuing in a similar line. Burgers derived his equation based on an analogy from the Navier-Stokes equations (1), but (70) is an analog of the vorticity equation (3). ! D curl u in R3 implies the Biot-Savart law (5). In one dimension, there is only one nontrivial singular integral operator with homogeneous kernel sending L2 into itself. That is the Hilbert transform. There is, up to a constant multiplication, one and  2 1=2 d or only one homogeneous operator of order 1 in one dimension. That is  dx 2 d H dx . In this sense, the Biot-Savart equation is replaced by the equation vx D H!. If further the convection term is neglected (recall the “proposition” stated at the beginning of Sect. 8), then (70) is obtained. Definition of the Hilbert transform is as follows:

1 

Hu.x/ D

Z

C1

1

u.y/ dy; xy

where Cauchy’s principal part should be taken in the integral. If applied to 2periodic functions, it should be defined as Hu.x/ D

Z

1 2

2

u.y/ cot 0

xy dy: 2

Accordingly vx D H! can be written as either v.x/ D

1 

Z

C1

!.y/ log jx  yj dy 1

or v.x/ D

1 

Z

2 0

ˇ x yˇ ˇ ˇ !.y/ log ˇsin ˇ dy: 2

766

H. Okamoto

The equation (70) admits a lot of solutions which blow up in finite time. Indeed any solution is represented by the following explicit formula: !.t; x/ D

2!0 .x/.2  tH!0 .x// C 2t !0 .x/H!0 .x/ : 4  4t H!0 .x/ C t 2 Œ!0 .x/2 C .H!0 .x//2 

This formula tells us that the solution does not blow up if !0 does not vanish anywhere and that it blows up in a neighborhood of x0 such that !0 .x0 / D 0 and H!0 .x0 / ¤ 0. That an analog of the Euler equations has many blow-up solutions is quite encouraging to those who believe the conjecture that the Euler equations have blow-up solutions. However, the story is not that simple, since blow-up does not necessarily occur if one introduces a model which looks a little closer to the Euler equations as will be shown below.

8.4

Generalizations of the Constantin-Lax-Majda Equations

The Constantin-Lax-Majda equation (70) can be generalized in many ways. One way is to apply a similar idea in two dimensions. In this way lie models of what are called the quasi-geostrophic flows [32, 91, 110]. The author cannot review this dynamically moving area, however. A viscosity term may be added to (70). Schochet [136] considered !t D !  H! C !xx ;

(71)

where  > 0 is a constant. He constructed an explicit solution which blows up in finite time. Thus, the singularity of the Constantin-Lax-Majda equation cannot be deleted by simply adding !xx . Curiously his solution blows up earlier than the solution of (70) with the same initial data. Thus the added viscosity “urges” blowup. Although his observation is interesting, the reader should note that he only constructed a certain example of blowup. General criteria for blow-up were not sought in [136]. Wegert and Vasudeva Murthy [155] considered 1=2  d2 !t D !  H!    2 !: dx

(72)

They proved that the blow-up time is decreasing in   0; thus the strange phenomena observed by Schochet in (71) dissolve in (72). With an arbitrary ˛  1, Sakajo [131] considered ˛=2  d2 !; !t D !  H!    2 dx

(73)

14 Models and Special Solutions of the Navier-Stokes Equations

767

and gave a sufficient condition P for blowup. He assumed that ! is expanded into the Fourier-sine series ! D 1 nD1 An .t / sin nx and all the coefficients An .t / are nonnegative. He observed that this class of functions is invariant with respect to (73). Namely, if a solution starts from this class, then it remains there afterward. He proved the following: Theorem 5. Suppose that ˛ > 1 and 0 <   A21 e 3˛1 , where A1 is the first Fourier-sine coefficient and e is the base of the natural logarithm. Then the solution blows up in finite time. If ˛ D 1, then 0 <   A1 =.2e/ is sufficient. He proved in the next paper [132] the following: Theorem 6. Suppose that ˛  1 and that !.0; x/ is odd in x and !.0; / 2 H ˛ . Suppose also that all the Fourier coefficients are nonnegative. If 2 > k!.0/kH ˛ , then the solution exists globally in time. Thus, the solution of (73) exists globally in time if  is large enough and blows up in finite time if  is small enough. The author does not know how one can get rid of the restriction about the signs of the Fourier coefficients. Its study seems to be an interesting challenge. Another model is obtained by adding a convection term. This was first proposed by De Gregorio [39, 40]. His model is !t C v!x  vx ! D 0;

vx D H!;

(74)

where ! is a function of t > 0 and x 2 R and v is determined by vx D H! with the Hilbert transform H. After having proposed this equation, he claimed that the solution of (74) exists globally in time, but no mathematical proof was provided. The author tried unsuccessfully to prove De Gregorio’s claim. Up until now, no proof of global existence is obtained for (74), although numerical computation in [118] seems to support the global existence. The following equation !t C ˛v!x  vx ! D 0;

vx D H!

(75)

was proposed in [118] and was called the generalized Constantin-Lax-Majda equation. Here ˛ is a real parameter. Reference [118] also derived !t C v!x D 0;

vx D H!;

(76)

which is obtained from (75) by replacing t with t =˛ and letting ˛ tend to 1. It was proved in [118] that (76) is well-posed globally in time. Since blow-up is admitted if ˛ D 0, the author expected that (75) admits a blowup if ˛ is small and no blow-up is admitted if j˛j is large. This conjecture is not true as it stands. In fact, Castro and

768

H. Okamoto

Cordoba [16] proved a theorem which may be interpreted as follows: some solutions of (75) may blow up in finite time if 1 < ˛ < 0. (A different proof in the case of 1 < ˛ < 0 was provided by [119].) It is not difficult to see that the Lp -norm of ! is conserved if ˛ D p for 1  p < 1. Thus, curiously, blow-up may accompany a conserved quantity for some ˛ < 0. Now it seems natural to expect the following conjecture. Conjecture A. There exists an ˛0 such that solutions of (75) exist globally in time if ˛0 < ˛  1 (or ˛0  ˛  1) and that blow-up occurs if 1 < ˛  ˛0 (or 1 < ˛ < ˛0 ). This conjecture was put forth in [119]. Right now the author does not know the identity of ˛0 . The numerical computation in [119] seems to indicate that 0 < ˛0 < 1 but it is not very decisive. If viscosity is added, the equation becomes !t C ˛v!x  vx ! D !xx ;

vx D H!:

(77)

Since the L2 -norm of ! is conserved if  D 0 and ˛ D 2, (77) with ˛ D 2 may be regarded as a model for a two-dimensional turbulence. Numerical computation by [93] seems to be quite encouraging to study (77) in this direction. The following few equations are neither exact solutions of the Navier-Stokes equations nor models of fluid motion. However, because of their resemblance to models to the Navier-Stokes equations, the author considers them to be worth studying. In 1981 in a completely different context, Satsuma [133] invented, among others, ut C vx u C vux D uxx ;

(78)

where v is defined as Z

C1

v.t; x/ D k 1

.x  y/ 1 coth u.t; y/ dy 2ı 2ı

with parameters k 2 R and ı > 0. This is a diffusion equation with a nonlinear, nonlocal convection term. He remarked that as ı ! 0 with k D 2ı, (78) tends to Z wt C .I  2w/wx D wxx

x

w.t; x/ D

u.t; s/ ds; 1

R where I D R u.t; s/ ds. He says that this is essentially the Burgers equation. He also remarked that as ı ! 1 (78) tends to ut  k Œu  Hux D uxx :

(79)

14 Models and Special Solutions of the Navier-Stokes Equations

769

Here the constant k is inessential, and one may set k D 1 or k D 1 by a scale change. Satsuma and Mimura [134] generalized the study of [133]. Among others they constructed a solution of (79) which blows up in finite time. Since their construction of a blow-up solution uses a method of integrable systems (a method of linearization), the general question about sufficient condition for blowup was left unanswered. Matsuno [94] considered an integrable system which is an generalization of (79). One example by [134] is the following one: u.t; x/ D

4c.x 2 C c 2 / ; .x 2  c 2 C 2t /2 C 4c 2 x 2

(80)

where c is an arbitrary constant. This solution blows up at t D c 2 =.2/ at x D 0 as is shown in Fig. 19. Other examples of blow-up solutions for (79) are found in [52]. Later [157] discovered a linearization method which is different from that in [133, 134]. Finally for some class of initial data, Wegert [154], together with [157], gave a necessary and sufficient condition for blow-up. Note that (77) with ˛ D 1 is !t  Œv!x D !xx ;

vx D H!:

(81)

On the other hand, the Satsuma-Mimura equation (79), so-called by [154, 157], can be written as ut  Œuvx D uxx ;

v D Hu:

(82)

It is not difficult to prove that (81) with  > 0 is well posed globally in time, for sufficiently smooth initial data. The blow-ups which were proved by [16] in the case of  D 0 disappear for  > 0. 0

Fig. 19 Profiles of (80). c D 1;  D 1

–10 –20 –30

u

–40 –50 –60 –70 –6

–4

–2

0

x

2

4

6

770

H. Okamoto

Quite independently from these works, Baker, Li, and Morlet [3] proposed ut C Hu  ux D uxx

(83)

ut C ŒHu  ux D uxx :

(84)

and

These are obtained by partly replacing the convection term of the Burgers equation uux D .u2 =2/x with a nonlocally defined function. Note that (84) is equivalent to the Satsuma-Mimura equation (79) (and (82)). Without knowing [134], they showed by constructing special solutions that (84) have blow-up solutions. On the other hand, they showed that solutions of (83) exist globally in time if  > 0. It is so because ku.t; /kL1 is non-increasing in t for (83). These two equations (83) and (84) were combined in the following way by Morlet [100]: ut C ŒHu  ux C .1  /Hu  ux D uxx ;

(85)

ut C Hu  ux C Hux  u D uxx ;

(86)

or

where 2 Œ0; 1 is a parameter. There she proved that it is globally well posed if D 1=2 and  > 0. She also showed that a singularity may develop if  > 0 and 0 < < 1=3 hold. However, she left other cases as open problems. This problem of Morlet was later solved in inviscid case by [21]. Indeed, it proved that some solutions of (85) blowup for all 2 .0; 1 if  D 0. It also considered t C .  H /x D D;

(87)

where  is unknown and D is a certain operator representing dissipation. Then [21] showed how it can be viewed as a one-dimensional model for a surface quasigeostrophic flow. Neither [100] nor [21] studied the cases of ı < 0; 1 < ı. Cordoba et al. [34] considered (85) with  D 0 and D 0, i.e., ut C Hu  ux D 0

(88)

and obtained a fairly general sufficient condition for blowup. They proved an inequality concerning the Hilbert transform, which soon turned out to be useful for other purposes, too. (Note here the similarity and difference between (88) and (76).) Here it would be probably desirable to write something for quasi-geostrophic flows, which have received much attention recently. However, the author regrets to say that he is ignorant of recent progresses in this vigorously moving topic. Instead, he wish the reader to see [20, 33, 110] and references therein. Chae’s survey [18],

14 Models and Special Solutions of the Navier-Stokes Equations

771

too, contains an account for the quasi-geostrophic equations. The model equation for quasi-geostrophic flow arises in meteorology, and there are many related model equations in this research field. See [91], where the reader can find many interesting models.

9

Vortex Tube

If vorticity vanishes except in a tubelike domain, the flow is called a vortex tube. A vortex tube is called a vortex filament if the tube’s cross section is infinitesimally small. If the vortex filament is straight, it is enough to consider it in a plane perpendicular to the filament, and it may be called a point vortex. It is a stationary solution of the 2D Euler equations with the vorticity being equal to the delta function, if it is viewed in a widely generalized way. If the vorticity ! is a C 1 p 2 function of r D x C y 2 only, then ! D !.r/ is a stationary solution of the 2D Euler equations. (Note that r! and u is mutually perpendicular in this case.) If !.r/ tends to the delta function, then the corresponding vortex tube tends to a vortex filament. (These remarks apply to the case where the fluid occupies the whole space. The problem becomes very complicated if the boundary wall exists. See for instance, [58, 138].) A point vortex cannot be stationary if viscosity exists. Oseen considered in [123] the following solution of the 2D Navier-Stokes equations: !.t; x; y/ D

  2 A x C y2 ; exp  2t 4t

where A is an arbitrary constant. This is an exact solution of the Navier-Stokes equations with the delta function as its initial value. (The nonlinear term vanishes identically.) The point vortex is thus smoothed by the viscosity and the vorticity distribution is given by the Gaussian distribution. Then came the following observation by Burgers (explained near the end of [15]): Suppose that there exists an irrotational axisymmetric flow (called a straining flow) .u; v; w/ D a.x; y; 2z/, where a is a positive constant. If a vortex tube is located on the z-axis, it diffuses by the viscosity as was noted by Oseen. However, if the straining flow exists, too, then there can be a stationary distribution of the vorticity. Specifically Burgers’s idea is realized by the velocity of the following form: u D ax C U .t; x; y/;

v D ay C V .t; x; y/;

w D 2az:

(89)

Its vorticity is ! D Vx  Uy . The Navier-Stokes equations are reduced to !t C .ax C U /!x C .ay C V /!y  2a! D 4!; Z y 1 !.t; ; / dd ; U D 2 R2 .x  /2 C .y  /2

(90) (91)

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1 V D 2

Z R2

x !.t; ; / dd : .x  /2 C .y  /2

(92)

Setting ! D exp.B.x 2 C y 2 //, B is calculated as B D a=.2/. Thus 

a.x 2 C y 2 / ! D exp  2

 (93)

is a stationary solution. This steady state, called the Burgers vortex, is proved by [55] to be stable if the Reynolds number is small. Later the restriction on the Reynolds number was removed by [51]. See also ([54]; Gallay and Maekawa,  “Existence and Stability of Viscous Vortices”, this volume). Burgers’s vortex can be generalized in many ways. A notable example is the nonsymmetric Burgers vortex by [127]. The best way to learn about this topic is to read papers by Maekawa [87, 88]. p Consider now a special case where ! is a function of t and r D x 2 C y 2 only. Then the nonlinear terms of (90) vanish and the vorticity is governed by !t  a.r!r C 2!/ D 4!;

(94)

where 4D

@2 1 @ : C 2 @r r @r

! which is given by (93) is a steady state of (94). If a is regarded as a given constant, (94) can be integrated. The method of integration was discovered by Lundgren [85] and later independently by Kambe [71, 72]. Thus far, a has been assumed to be a constant. However, the scheme above gives an exact solution even if a is a function of t only. This has been noted by many, but the most remarkable is Moffatt [99], where he considered a case of a.t / D A=.T t / with A and T positive constants. Ohkitani and the author considered in [108] !t  k!.t/kp .r!r C 2!/ D 4!;

(95)

where !.t/ denotes !.t; / and k kp denotes the Lp norm. They proposed this equation as a model for interaction of vortex tubes. The following theorem was proved in [108]: Theorem 7. Let 1 < p < 1 and assume that !.0; / 2 L1 \ Lp . Then the solution of (95) is smooth for all t and is bounded in L1 \ Lp . The paper also presented a self-similar solution which blows up in finite time in the case of p D 1. A general blow-up criteria was given by [104]:

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Theorem 8. Let p D 1 and assume that !.0; / 2 L1 \ L1 and some other nice condition (see [104]). Then the solution of (95) is smooth for all t > 0 and is bounded in L1 \ L1 if k!.0/k1  . If k!.0/k1 > , then the solution blows up in finite time. Thus there exists a marked contrast between the L1 -norm with the Lp -norm (p < 1). The author considered in [117] the following generalization of (95): !t  k!.t/k˛p .r!r C 2!/ D 4!;

(96)

where the exponent ˛ is a parameter  0. He proved: Theorem 9. Let 1 < p < 1 and assume that 0  ˛ < p=.p  1/. Then the solution of (96) is smooth for all t > 0. If p=.p  1/  ˛, then a solution blows up if the initial data is large enough. Thus, a complete classification of blow-up/global existence was obtained in the case of axisymmetric flows. The author does not know a result for general (not necessarily axisymmetric) 2D flow, i.e., (90), (91), and (92). Some other unsolved problems are also listed in [117].

10

Conclusion

The differential equations in the present chapter are much simpler than the NavierStokes equations. Nevertheless, their well-posedness is not completely known. Since numerical experiments are relatively easy for those models, they may be used for further study. Modifications are not difficult and may serve as challenges in future. This is particularly true for solutions at large Reynolds numbers. Singular perturbation analysis, such as [78, 135], may be a clue to deeper understanding of the Navier-Stokes equations. Other exact solutions can be found in [36], which later became a source of some papers on dynamical systems including [37, 98]. See also [106]. The present handbook also contains articles which study self-similar solutions; see (Hou and Liu,  “Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric Euler Equations”, this volume; Šverák et al.,  “Self-Similar Solutions to the Nonstationary Navier-Stokes Equations”, this volume). Generalizations to magnetohydrodynamics or compressible fluid motion are possible, and the author is sure that there may be interesting examples to be explored. In the theory of water waves, some exact solutions are known, and they shed light on this difficult problem. See, for instance, [121].

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Also interesting are problems related to the two-dimensional Boussinesq equations for convective motion of fluid. Papers [27, 70] recently came to the author’s attention. These papers and those referred therein may help the reader. See also [18]. If applications to geophysics are included, very many models have been proposed. The author is unable to review this area, which is so dynamically moving right now. The reader is referred to the article by Titi and Li in the present monograph (Titi and Li,  “Recent Advances Concerning Certain Class of Geophysical Flows”, this volume), the monograph [91], or a review article [74], or [20]. Some fundamental aspects of self-similar solutions are discussed in [54], which is highly recommended.

11

Cross-References

 Existence and Stability of Viscous Vortices  Recent Advances Concerning Certain Class of Geophysical Flows  Self-Similar Solutions to the Nonstationary Navier-Stokes Equations  Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric Euler

Equations Acknowledgements The author is partially supported by JSPS KAKENHI 24244007. Part of the present chapter was written while the author was a visitor in 2014 to School of Mathematics, University of Minnesota. He would like to express his deepest gratitude for its generous support to him.

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115. H. Okamoto, Mathematics of the Navier–Stokes Equations (University of Tokyo Press, 2009) (in Japanese) 116. H. Okamoto, A nonlinear boundary-value problem with an integral constraint, in Proceedings of “Far-From-Equilibrium Dynamics”, RIMS Kokyuroku Bessatsu B31, (2012), 93–115 117. H. Okamoto, Blow-up problems in the strained vorticity dynamics and critical exponents. J. Math. Soc. Jpn. 65, 1079–1099 (2013) 118. H. Okamoto, T. Sakajo, M. Wunsch, On a generalization of the Constantin–Lax–Majda equation. Nonlinearity 21, 2447–2461 (2008) 119. H. Okamoto, T. Sakajo, M. Wunsch, Steady-states and traveling wave solutions of the generalized Constantin–Lax–Majda equation. Disc. Cont. Dyn. Syst. 34, 3155–3170 (2014) 120. H. Okamoto, M. Sh¯oji, Boundary layer in unsteady two-dimensional Navier-Stokes equations, in Recent Development in Domain Decomposition Methods and Flow Problems, ed. by H. Fujita et al. Gakuto International Series, Mathematical Sciences and Applications, vol. 11 (Gakkotosho, Tokyo, 1998), pp. 171–180 121. H. Okamoto, M. Sh¯oji, The Mathematical Theory of Bifurcation of Permanent Progressive Water-Waves. (World Scientific, 2001) 122. H. Okamoto, J. Zhu, Some similarity solutions of the Navier–Stokes equations and related topics. Taiwan. J. Math. 4, 65–103 (2000) 123. C.W. Oseen, Über Wirbelbewegung in einer reibenden Flüssigkeit. Arkiv Mate. Astro. Fysik 77 (1912) No. 14 124. C.W. Oseen, Exakte Lösungen der hydrodynamischen Differentialgleichungen. I. Arkiv Mate. Astro. Fysik 20(14), 1–14 (1927–1928); ibid. II., No. 22 125. I. Proudman, K. Johnson, Boundary-layer growth near a rear stagnation point. J. Fluid Mech. 12, 161–168 (1962) 126. D. Riabouchinsky, Quelques considérations sur les mouvements plans rotationnels d’un liquide, Comp. Rendus Hebd. Acad. Sci. 179, 1133–1136 (1924) 127. A.C. Robinson, P.G. Saffman, Stability and structure of stretched vortices. Stud. Appl. Math. 70, 163–181 (1984) 128. A. Rosenblatt, Solutions exactes des équations du mouvement des liquides visqueux (Gauthier-Villars, 1935) 129. L. Rosenhead, The steady two-dimensional radial flow of viscous fluid between two inclined plane walls. Proc. R. Soc. Lond. A 175, 436–467 (1940) 130. L. Rosenhead (ed.), Laminar Boundary Layers: An Account of the Development, Structure, and Stability of Laminar Boundary Layers in Incompressible Fluids, Together with a Description of the Associated Experimental Techniques (Oxford University Press, 1963) 131. T. Sakajo, Blow-up solutions of Constantin–Lax–Majda equation with a generalized viscosity term. J. Math. Sci. Univ. Tokyo 10, 187–207 (2003) 132. T. Sakajo, On global solutions of Constantin–Lax–Majda equation with a generalized viscosity term. Nonlinearity 16, 1319–1328 (2003) 133. J. Satsuma, Exact solutions of a nonlinear diffusion equation. J. Phys. Soc. Jpn. 50, 1423– 1424 (1981) 134. J. Satsuma, M. Mimura, Exact treatment of nonlinear diffusion equations with singular integral terms. J. Phys. Soc. Jpn. 54, 894–900 (1985) 135. H. Schlichting, K. Gersten, Boundary-Layer Theory. Transl. C. Mayes, 8th rev. (Springer, 2000) 136. S. Schochet, Explicit solutions of the viscous model vorticity equation. Commun. Pure Appl. Math. 39, 531–537 (1986) 137. J. Serrin, On the mathematical basis for Prandtl’s boundary layer theory: an example. Arch. Rat. Mech. Anal. 28, 217–225 (1968) 138. J. Serrin, The swirling vortex. Phil. Trans. R. Soc. A 271, 325–363 (1970) 139. J. Shen, T. Tao, L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications (Springer, 2011) 140. J.T. Stuart, The viscous flow near a stagnation point when the external flow has a uniform vorticity. J. Aerosp. Sci. 26, 124–125 (1959)

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141. K. Tamada, Two-dimensional stagnation point flow impinging obliquely on a plane wall. J. Phys. Soc. Jpn. 46, 310–311 (1979) 142. C.L. Taylor, W.H.H. Banks, M.B. Zaturska, P.G. Drazin, Three dimensional flow in a porous channel. Quart. J. Mech. Appl. Math. 44, 105–133 (1991) 143. M. Taylor, Partial Differential Equations III, 2nd edn. (Springer, 2011) 144. G.A. Tokaty, A History and Philosophy of Fluid Mechanics (G.T. Foulis & Co., Henley-onThames, 1971; Dover, 1994) 145. C.A. Truesdell, Rational Fluid Mechanics, 1687–1765, L. Euler’s Opera Omnia Ser. II, vol. 12 (O. Füssi, Lausanne, 1954), pp. 1–125 146. T.-P. Tsai, On Leray’s self-similar solutions of the Navier–Stokes equations satisfying local energy estimates. Arch. Rat. Mech. Anal. 143, 29–51 (1998) 147. T. von Kármán, Über laminar und turbulente Reibung, Z. Angew. Math. Mech. 1, 233–252 (1921) 148. T. von Kármán, The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, 615–683 (1940) 149. T. von Kármán, Aerodynamics (Cornell University Press, 1954; Dover, 2004) 150. R. von Mises, K.O. Friedrichs, Fluid Dynamics (Springer, 1971) 151. C.-Y. Wang, Axisymmetric stagnation flow on a cylinder. Quart. Appl. Math. 32, 207–213 (1974) 152. C.Y. Wang, Exact solutions of the steady-state Navier–Stokes equations. Ann. Rev. Fluid Mech. 23, 159–177 (1991) 153. E.B.B. Watson, W.H.H. Banks, M.B. Zatsurska, P.G. Drazin, On transition to chaos, in twodimensional channel flow symmetrically driven by accelerating walls J. Fluid Mech. 212, 451–485 (1990) 154. E. Wegert, On the global solvability of the diffusion equation of Satsuma and Mimura. Z. Anal. Anwen. 9, 313–318 (1990) 155. E. Wegert, A.S. Vasudeva Murthy, Blow-up in a modified Constantin–Lax–Majda model for the vorticity equation. Z. Anal. Anwen. 18, 183–191 (1999) 156. H. Weyl, On the differential equations of the simplest boundary-layer problems. Ann. Math. 43, 381–407 (1942) 157. L. von Wolfersdorf, On the linearization of the Satsuma–Mimura diffusion equation. Math. Nach. 145, 243–254 (1990) 158. E. Yumoto, On the existence of axi-symmetric exact solutions of the Navier–Stokes equations. Master’s thesis at RIMS, Kyoto University (2015) 159. P.J. Zandbergen, D. Dijkstra, Von Kármán swirling flows. Ann. Rev. Fluid Mech. 19, 465–491 (1987) 160. J. Zhu, Numerical study of stagnation-point flows of incompressible fluid. Jpn. J. Indust. Appl. Math. 17, 209–228 (2000)

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

15

Yasunori Maekawa and Anna Mazzucato

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Inviscid Limit Problem Without Physical Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Inviscid Limit Problem with Physical Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Case of Slip-Type Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Case of No-Slip Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Non-characteristic Boundary Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Prandtl Equations for the Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this chapter is to review recent progress on the mathematical analysis of this problem in each category. Y. Maekawa () Department of Mathematics, Graduate School of Sciences, Kyoto University, Kyoto, Japan e-mail: [email protected] A. Mazzucato Mathematics Department, Eberly College of Science, The Pennsylvania State University, University Park, State College, PA, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_15

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Introduction

Determining the behavior of viscous flows at small viscosity is one of the most fundamental problems in fluid mechanics. The importance of the effect of viscosity, representing tangential friction forces in fluids, is classical and has been recognized for a long time. A well-known example is the resolution of D’Alembert’s paradox concerning the drag experienced by a body moving through a fluid, which is caused by neglecting the effect of viscosity in the theory of ideal fluids. For real flows like water and air, however, the kinematic viscosity is a very small quantity in many situation. Physically, the effect of viscosity is measured by a nondimensional quantity, called the Reynolds number Re WD UL , where U and  L are the characteristic velocity and length scale in the flow, respectively, and  is a kinematic viscosity of the fluid. Therefore, when U and L remain in a fixed range, the limit of vanishing viscosity is directly related to the behavior of high-Reynolds number flows. Hence, the theoretical treatment of the inviscid limit has great importance in applications and has been pursued extensively in various settings. The aim of this chapter is to give an overview of recent progress in the inviscid limit problem for incompressible Newtonian fluids, although some results are also available in the important cases of compressible fluids or non-Newtonian fluids. The governing equations for incompressible homogeneous Newtonian fluids are the Navier-Stokes equations: @t u C u  ru C rp  D u C f  ;

div u D 0 ;

u jtD0 D u0 : (1)

Here u D .u1 ;    ; un /, n D 2; 3, and p  are the unknown velocity field and unknown pressure field, respectively, at time t and position x D .x1 ;    ; xn / 2 . In what follows, the domain   Rn will be either the whole space Rn or a domain with smooth boundary. Although many results can be stated for arbitrary dimension n, the physically relevant dimensions are n D 2; 3. The external force f  D .f1 ;    ; fn / is a given vector field, which will be typically taken as zero for simplicity, and u0 D .u0;1    ; u0;n / is a given initial velocity field. Standard notation will be used throughout for derivatives: @t D @t@ , @j D @x@j , r D .@1 ;    ; @n /, Pn Pn Pn 2    D j D1 @j , u  r D j D1 uj @j , and div f D j D1 @j fj . The symbol  represents the kinematic viscosity of the fluid, which is taken as a small positive constant. The equations (1) are closed by imposing a suitable boundary condition, which will be specified in different context (see Sect. 3). The vorticity field is a fundamental physical quantity in fluid mechanics, especially for incompressible flows, and it is defined as the curl of the velocity field. Denoting !  D curl u , one has !  D @1 u2  @2 u1

.n D 2/ ;

!  D r  u ;

.n D 3/:

(2)

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Above, we have identified the vector !  D $ k, where k D .0; 0; 1/, with the scalar $, and called the latter also !  with abuse of notation. For the reader’s sake, we recall the vorticity equations: @t !  C u  r!  D !  C !   ru :

u D K Œ!  ;

(3)

where K stands for the Biot-Savart kernel in the domain . The last term on the right is called the vorticity stretching term and it is absent in two space dimensions. This term depends quadratically in !  , and its presence precludes establishing longtime existence of Euler solutions in three space dimensions. By formally taking the limit  ! 0 in (1), the Navier-Stokes equations are reduced to the Euler equations for incompressible flows: @t u C u  ru C rp D f;

div u D 0 ;

ujtD0 D u0 :

(4)

The velocity field u of the Euler flows will be often written as u0 in this chapter. Broadly speaking, a central theme of the inviscid limit problem is to understand when and in which sense the convergence of the Navier-Stokes flow u to the Euler flow u is rigorously justified. Mathematically, the inviscid limit is a singular perturbation problem since the highest-order term u is formally dropped from the equations of motion in the limit. For such a problem, in many cases, the main issue is establishing enough a priori regularity for the solutions such that convergence is guaranteed via a suitable compactness argument. This singular perturbation for the Navier-Stokes equations provides a challenging mathematical problem, because of the fact that the nonlinearity contains derivatives of the unknown solutions and because of the nonlocality arising from the pressure term. Indeed, even when the flow is two dimensional and the fluid domain is the whole plane R2 , the study of the inviscid limit problem becomes highly nontrivial if given data, such as initial data, possess little regularity. It should be emphasized here that working with non-smooth data has an important motivation not only mathematically but also in applications, for typical structures of concentrated vorticities observed in turbulent flows, such as the vortex sheets, vortex filaments, vortex patches, are naturally modeled as flows with certain singularities. The problem for singular data but under the absence of physical boundaries will be discussed in Sect. 2 of this chapter. The inviscid limit problem becomes physically more important and challenging in the presence of a nontrivial boundary, where the viscosity effects are found to play a central role, in general, no matter how small the viscosity itself is, depending on the geometry and on the boundary conditions for the flow. Recent developments in the mathematical theory for this case will be reviewed in Sect. 3. The main obstruction in analyzing the inviscid limit in this context arises from the complicated structure of the flow close to the boundary. Indeed, due to the discrepancy between the boundary conditions in the Navier-Stokes equations and in the Euler equations, a boundary layer forms near the boundary where the effects of viscosity cannot be neglected even at very low viscosity. The size and the stability property of the

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boundary layer crucially depend on the type of prescribed boundary conditions and also on the symmetry of the fluid domain and of the flows, which are directly connected to the possibility of the resolution of the inviscid limit problem. These are discussed in details in Sects. 3.1, 3.2, and 3.3. The concept of a viscous boundary layer was first introduced by Prandtl [126] in 1904 under the no-slip boundary condition, that is, assuming the flow adheres to the boundary. Since then, the theory of boundary layers has had a strong impact in fluid mechanics and has also initiated a fundamental approach in asymptotic analysis for singular perturbation problems in differential equations. The reader is referred to [129] for various aspects of the boundary layer theory in fluid mechanics. The basic idea is that the fluid region can be divided into two regions: a thin layer close to the boundary (the so-called boundary layer) where the effect of viscosity is significant and the region outside this layer where the viscosity can be neglected and thus the fluid behaves like an inviscid flow. Aspfound by Prandtl, the thickness of the boundary layer is formally estimated as O. /, at least for no-slip boundary conditions, which is a natural scale given the parabolic nature of the Navier-Stokes equations. In the case of noslip boundary conditions, the fundamental equations describing the boundary layer are the Prandtl equations, which will be the focus of Sect. 3.4. Interestingly, the problem becomes most difficult in the case that the no-slip boundary condition , the most classical and physically justified type of boundary condition, is imposed, due to the fact that large gradients of velocity can then form at the boundary, which may propagate in the bulk, giving rise to a strong instability mechanism for the layer at high frequencies. The rigorous description for the inviscid limit behavior of NavierStokes flows is still largely open in many relevant situations, and several recent works in the literature have tackled this important problem. In this chapter, standard notations in mathematical fluid mechanics will be used for spaces of functions and p 1 vector fields. For example, the spaces C0; ./ and L ./ are defined as 1 C0; ./ D ff 2 C01 ./n j div f D 0 in g ; 1 ./ Lp ./ D C0;

kf kLp ./

;

1 < p < 1:

The Bessel potential spaces H s .Rn /, the Sobolev spaces W s;p ./, s  0, 1  p  1, and the space of Lipschitz continuous functions Lip .Rn / are also defined as usual. The n-product space X n will be often written as X for simplicity.

2

Inviscid Limit Problem Without Physical Boundary

This section is devoted to the analysis of the inviscid limit problem for the NavierStokes equations when the fluid domain has no physical boundary. In this case the effect of the boundary layer is absent at least from the physical boundary, and the problem is more tractable and has been analyzed in various functional settings. Typically the Navier-Stokes flow is expected to converge to the Euler flow in the inviscid limit. Then the main interest here is the class (regularity) of solutions for

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which this convergence is verified and its rate of convergence in a suitable topology. To simplify the presentation, only the Cauchy problem in Rn will be the focus in this section, and therefore the external force in (1) or (4) will be taken as zero. To give an overview of known results, it will be convenient to classify the solutions depending on their regularity as follows: (I) Regular solutions (II) Singular solutions: (II-1) Bounded vorticity  Vortex patch (II-2) Vortex sheet  Vortex filament  Point vortices Since the external force is assumed to be zero, the above classification is essential for the initial data. Then, the typical case of “(I) regular solutions” is that the initial data u0 and u0 belong to the Sobolev space H s .Rn / with s > n2 C 1, which is embedded in the space C 1 .Rn /. On the other hand, the category “(II) singular solutions” corresponds to the case when the initial data are less regular than the C 1 class. (I) Regular solutions. The verification of the inviscid limit in Rn for smooth initial data (e.g., better than the C 1 class) is classical. Indeed, it is proved in [58, 118] for R2 , in [74, 134] for R3 , and in [39] for a compact manifold without boundary of any dimension. The following result is given in [110, Theorem 2.1] and provides the relation between the regularity of solutions and the rate of convergence. For simplicity, the result is stated here only for the case u0 D u0 in (1) and (4), though the case u0 ¤ u0 is also given in [110]. Theorem 1. Let u0 D u0 2 H s .Rn / with some s > n2 C 1. Let u 2 Cloc .Œ0; T  /I H s .Rn // be the (unique) solution to the Euler equations, where T  > 0 is the time of existence of the solution. Then, for all T 2 .0; T  /, there exists 0 > 0, such that for all  2 Œ0; 0 , there exists a unique solution u 2 C .Œ0; T I H s .Rn // to the Navier-Stokes equations. Moreover, it follows that lim ku  ukL1 .0;T IH s / D 0 ;

!0

ku  ukL1 .0;T IH s0 /  C .T /

ss 0 2

;

for s  2  s 0  s. Here, C depends only on u and T . Since the time T  in Theorem 1 is the time of existence for the Euler flow, the inviscid limit holds on any time interval in two space dimensions. The estimate in 0 H s .Rn / for the case s 0 D s  2 is obtained by a standard energy method, and the case s  2 < s 0 < s is derived from interpolation. The convergence in H s .Rn / is more delicate, and a regularization argument for the initial data is needed in the proof. (II) Singular solutions. (II-1) Bounded vorticity  Vortex patch: The Euler equations are uniquely solvable, at least locally in time, when the initial vorticity is

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bounded [152] or nearly bounded [142, 153]. Hence, it is expected that the NavierStokes flow converges to the Euler flow in the vanishing viscosity limit also for such class of initial data. This class includes some important solutions, called vortex patches, which are typically vorticity fields defined as characteristic functions of bounded domains with smooth boundary. Theorem 1 cannot be applied for this class of solutions, since the condition that the vorticity be bounded is not enough to ensure the local-in-time H s regularity of the velocity for s > n2 C 1. The next result is established in [27] for the two-dimensional case, where the effect of the singularity appears in the rate of convergence. Theorem 2. Let u0 D u0 be an L2 perturbation of a smooth stationary solution to the Euler equations (see [27], Definition 1.1 for the precise definition). If in addition !0 D curl u0 2 L1 .R2 / \ L2 .R2 /, then 1

ku  ukL1 .0;T IL2 /  C k!0 kL1 \L2 .T / 2 exp.C k!0 kL1 \L2 T / : Thus, although the inviscid limit is still verified in L1 .0; T I L2 .R2 // even when the initial vorticity is merely in L1 \L2 , the upper bound on the rate of convergence in Theorem 2 decreases in time. This decrease, in fact, reflects the lack of Lipschitz regularity for the velocity field of the Euler flow, e.g., u 2 L1 .0; T I Lip.R2 //. On the other hand, the Lipschitz regularity of the velocity is ensured for a class of vortex patches. Then, as mentioned below, the rate of convergence can be estimated uniformly in time for this class. First, the definition of vortex patches is given as follows. p

Definition 1. Let n D 2; 3 and 0 < r < 1. A vector field u 2 L .Rn /, 2 < p < 1, is called a C r vortex patch if the vorticity ! D curl u has the form ! D A !i C Ac !e ;

(5)

where A  Rn is an open set of class C 1Cr and !i , !e are compactly supported C r functions (vector fields when n D 3). Here, A and Ac denote the characteristic functions of A and Ac D Rn n A, respectively, and the condition !i  n D !e  n is assumed on @A, which is always valid when n D 2. In Definition 1, the condition !i  n D !e  n is assumed on @A so that the divergence-free condition div ! D 0 is satisfied in the sense of distributions, which is necessary since ! D curl u. The simplest vortex patch in the two-dimensional case is the constant vortex patch introduced in [103], where !i D 1, !e D 0, and A is a bounded domain with C 1Cr boundary. In the three-dimensional case, the constant vortex patch cannot exist because of the requirement div ! D 0. The classical reference for constant vortex patches is [26], where it is proved that if the initial vorticity is a C r constant vortex patch, then it remains to be a C r constant vortex patch for all time; see also [15]. Moreover, the velocity is bounded

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in L1 .0; T I Lip .Rn // for all T > 0. The regularity of the vorticity field up to the boundary of general two-dimensional vortex patch is shown in [35, 67]. The result for the two-dimensional case is extended to the three-dimensional C 1Cr vortex patches by [46] but for a bounded time interval, since, due to possible vortex stretching mechanism, the existence of the Euler solutions is only local in time. The reader is also referred to [68] for a Lagrangian approach proving the regularity up to the boundary and to [37] for the result in a bounded domain, rather than in the whole space. Useful references about the study of vortex patches can be found in the recent paper [132]. The first result for the inviscid limit problem pertaining to vortex patches is given in [30], where it is proved that the Navier-Stokes flows in R2 starting from a constant vortex patch converges to the constant vortex patch of the Euler flows with the same initial data. In [30] the convergence is shown in the topology of L1 .0; T I L2 .R2 //, 1 T > 0, for the velocity fields, with convergence rate .T / 2 . The authors of [30] p proved in [31] the convergence of the vorticity fields in L .R2 /, 2  p < 1. The optimal rate of convergence is then achieved for constant vortex patches in R2 by the results in [1], as stated below. Theorem 3. Let u0 D u0 be a constant C r vortex patch in R2 . Then there exists C > 0 depending only on the initial data such that Ct

3

ku .t /  u.t /kL2  C e C e .t / 4 .1 C t / ;

t > 0:

3

The rate .t / 4 is optimal in the sense that, if u0 D u0 is a circular vortex patch, i.e., A is a disk, then an explicit computation leads to the following bound from above and below (see [1, Section 4]): 3

3

C .t / 4  ku .t /  u.t /kL2  C 0 .t / 4 ;

0 < t  1 :

The proof of Theorem 3 in [1] is based on an energy method combined with the Littlewood-Paley decomposition, i.e., a dyadic decomposition in the Fourier variables. In [1], the authors used the bound u 2 L1 .0; T I Lip .R2 // uniformly in  > 0, which was obtained in [33] and extended in [34] for general C r vortex patches in Rn , to study the vanishing viscosity limit. The reader is also referred to [64] for the extension of [33] and to [65] for the analysis when the constant patch has a singularity at its boundary. Later it was pointed out in [110] that the uniform Lipschitz bound of u itself is not necessary and the optimal rate is proved only under a Lipschitz bound of u. More precisely, in [110, Theorems 3.2, 3.4], the inviscid limit is verified for general C r vortex patches in Rn in terms of Besov s , s 2 R, 1  p; q  1, as follows. spaces Bp;q ˛ Theorem 4. Let u0 D u0 be a C r vortex patch. Assume that curl u0 2 BP 2;1 .Rn /, 1 n  0 < ˛ < 1 and that u 2 L .0; T I Lip .R //. Then any weak solution u to the

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Navier-Stokes equations with initial data u0 satisfies the estimate ku .t /  u.t /kL2  C .t /

1C˛ 2

;

0 0 (see [56]for the precise definition 3 of the space M 2 .R3 /). Vortex filaments possess an infinite energy in general. Although the vortex filament can belong to the class of functional solutions to the Euler equations introduced by [25], due to the strong singularity and underlying vortex stretching mechanism in three-dimensional flows, there seems to be no general existence result for vortex filaments as solutions to the Euler equations. Alternatively, the self-induction equation (localized induction approximation) and its significant generalization have been used to understand the dynamics of vortex filaments; that dynamics is out of the scope of this chapter and the interested reader is referred to [104, Chapter 7] and references therein. On the other hand, the vortex filament belongs to an invariant space for the three-dimensional NavierStokes equations. Hence, a general theory is available for the unique existence of solutions to the Navier-Stokes equations with vortex filaments as initial data under smallness condition on the scale-invariant norm [56]. There are two typical vorticity distributions of vortex filaments: circular vortex rings with infinitesimal cross section and exactly parallel and straight vortex filaments with no structural variation along the axis. The first one, called vortex ring for simplicity here, corresponds to an axisymmetric flow without swirl. Then the vorticity field of the vortex ring becomes a scalar quantity and is expressed as a Dirac measure supported at a point .r; z/ 2 .0; 1/  R in cylindrical coordinates. Recent

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results [41] show global existence of solutions to the Navier-Stokes equations with vortex ring as initial data, and in fact their uniqueness can be also proved in the class of axisymmetric flows without swirl; a more detailed overview about this topic is available in another chapter of this handbook; see [44]. The inviscid limit problem for vortex rings is attempted in [107], where the initial vortex ring is slightly regularized depending on the viscosity and the radius of the initial vortex ring is assumed to tend to infinity as the viscosity goes to zero. The result of [107] is significantly extended in [17], where the radius of the initial vortex ring can be taken independently of the viscosity. A more precise description of the result in [17] is given as follows. The velocity of the axisymmetric flow without swirl is expressed as u D .ur ; 0; uz / in cylindrical coordinates .r; ; z/, and the vorticity field is identified with a scalar quantity, !  D @z ur  @r uz . Then the evolution of !  in cylindrical coordinates is obtained from the equation @t !  C .ur @r C uz @z /!  

ur  ! D r



1 ! @r .r@r !  / C @2z !   2 r r

 :

(7)

Let  ˚ †.r0 ;z0 / .l/ D .r; z/ 2 .0; 1/  R j jr  r0 j2 C jz  z0 j2 < l 2 : Then a typical case of the result stated in [17, Theorem 1.1] leads to the next theorem. Theorem 5. Assume that a sequence of initial vorticities f!0 g0 0 and a 2 R are constants independent of  2 0; 14 . Then there exists a sequence f.r .t /; z .t //g0 0 and 4 f 2 BC .Œ0; 1/  R/, lim r .t / D 1;

lim z .t / D

!0

!0

at ; 4

 at ; 0 < t  T; lim j log j ! .t /f drdz D 2af 1; !0 4

†.r .t/;z .t// .D /   1 1  2 where D D C  exp j log j ; 0 <  < 1: 2 (9) 

Z







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Here !  is the (unique) solution to (7) with initial data !0 , and C > 0 is a constant independent of . Above BC stands for the space of bounded, continuous functions. Theorem 5 implies that when the vorticity is initially sharply concentrated in an annulus, then it remains concentrated during the motion even in the presence of small viscosity, and as  goes to zero, the support of the vorticity evolves via a constant motion. The support condition in (8) is related to the standard boundary layer thickness, i.e., of 1 order  2 . On the other hand, in (8), the vorticity is assumed to be logarithmically smaller than the standard scale. This is related to the fact that the velocity field for the circular vortex ring with infinitesimal cross section has a logarithmic singularity in the vertical component around the location of the ring (that is, at r D 1 in the setting of Theorem 5). In virtue of the additional smallness of order O.j log j1 / in (8), the vorticity is translated with finite speed in the inviscid limit, as described in (9). Next, the other typical class of vortex filaments, i.e., parallel straight vortex filaments, is discussed. By symmetry, these vortex filaments keep their shape under the motion, and then the problem is reduced to the motion of the points which are the intersection of the vortex filaments with the hyperplane fx3 D 0g. These are called point vortices, linear combinations of Dirac measures in R2 . For the inviscid flow, the motion of N point vortices is formally described by the following HelmholtzKirchhoff system: d 1 X .zi .t /  zj .t //? zi .t / D ˛j ; dt 2

jzi .t /  zj .t /j2

zi .0/ D xi :

(10)

j ¤i

Here i; j D 1;    ; N and each xi denotes the initial location of the i th point vortex with circulation ˛i 2 R. It is possible to realize the point vortices as functional solutions, introduced in [25], to the two-dimensional Euler equations. A further relation of point vortices with the Euler equations is shown by [108]. On the other hand, the two-dimensional Navier-Stokes equations are known to be globally well posed when the initial vorticity field is given by the point vortices of the form (11) below. In particular, the uniqueness of solutions is also available, P which is proved in [57, 76] under smallness condition on the total variation N i D j˛i j and in [42] without any smallness condition on the size of ˛i . The inviscid limit problem for point vortices is rigorously analyzed in [43, 105, 106] in the time interval in which the Helmholtz-Kirchhoff system is well posed and vortex collisions do not occur. The next result is due to [43, Theorem 2]. Theorem 6. Assume that the point vortex system (10) is well posed on the time interval Œ0; T . If the initial vorticity field is given by curl u0 D

N X iD1

˛i ı.  xi / ;

(11)

15 The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

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then the vorticity field !  D curl u of the solution to the Navier-Stokes equaN X ˛i ı.  zi .t // as  ! 0 in the sense of measures for tions (1) converges to iD1

all t 2 Œ0; T , where z.t / D .z1 .t /;    ; zN .t // is the solution of (10). Theorem 6 shows that the distribution of the vorticity field of the Navier-Stokes flow starting from (11) is described by the point vortex system (10) in the inviscid limit. A result similar to Theorem 6 was first obtained in [105,106], where the initial point vortices are slightly regularized depending on viscosity. Theorem 6 provides information on the location of viscous vortices, while little information is available about the shape of the viscous vortices, i.e., about the viscous profile of each point vortex. Since the vorticity field !  obeys the nonlinear heat convection equations 









@t ! C u  r! D ! ;

! jtD0 D

N X

˛i ı.  xi / ;

i D1

once !  is constructed by solving the above system, it is then decomposed as 

! D

N X

!i ;

(12)

iD1

where !i is the solution to the heat convection equations with initial data which is a Dirac measure supported at xi : @t !i C u  r!i D !i ;

!i jtD0 D ˛i ı.  xi / :

(13)

An interesting and important question is then to determine the correct asymptotic profile of each !i . It is worthwhile to recall here that, if the initial vorticity field is a Dirac measure supported at the origin, ˛ı.0/, then the unique solution to (1) is explicitly given by the Lamb-Oseen vortex with circulation ˛ [45]: ˛U  .t; x/ D ˛ G.x/ D

x? 2 jxj2



1  jxj2 e 4 : 4

jxj2

1  e  4t

 ;



 ˛ curl ˛U  .t; x/ D G t



x p t

 ; (14)

Hence, the function ˛i G, a Gaussian with total mass ˛i , is a natural candidate for the viscous profile of each !i . The next result ([43, Theorem 3]) establishes the asymptotic expansion of !i in the limit  ! 0 and shows that the asymptotic profile of each !i is indeed given by a two-dimensional Gaussian. Theorem 7. Assume that the point vortex system (10) is well posed on the time interval Œ0; T , and let !  D curl u be the vorticity field of the solution to (1) with

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initial data the vorticity of which is given by (11). If !  is decomposed as in (12), then the rescaled profiles wi , defined by !i .t; x/

  x  zi .t / ˛i  ; D wi t; p t t

(15)

satisfy the estimate max kwi .t /  GkL1  C

iD1; ;N

t ; d2

t 2 .0; T  ;

(16)

where d D min min jzi .t /  zj .t /j > 0. Here zi is the solution to a regularized t2Œ0;T  i¤j

point vortex system (see [43], Eq. (19)). In [43], the estimate (16) is proved in a stronger topology, that of a weighted L2 space that is embedded in L1 .R2 /. The behavior of zi in (15) is well approximated by zi in the limit  ! 0 with an exponential order [43, Lemma 2], and hence, Theorem 7 verifies the asymptotic expansion of the viscous vortices around zi .t /, t 2 .0; T . In order to show Theorems 6 and 7, one has to take into account the interaction between the viscous vortices in the inviscid limit. In particular, for each viscous vortex around zi , the velocity fields produced by the other vortices play a role in the background flow. The interactions result in a deformation of each viscous vortex. The analysis of this interaction is the key to prove Theorems 6 and 7, and in fact, it requires a detailed investigation of higher-order expansions due to the strong singularity of the flows. This approach is validated in a rigorous fashion in [43, Theorem 4]. One consequence is that each viscous vortex is deformed elliptically through the interaction with the other vortices.

3

Inviscid Limit Problem with Physical Boundary

This section is devoted to the analysis of a viscous fluid at very low viscosity moving in a domain  with physical boundaries. The behavior of the fluid is markedly influenced by the types of boundary conditions imposed. The case when rigid walls may move only parallel to itself, as in Taylor-Couette flows, will be the focus in this section. In this case, the fluid domain (assumed to be smooth) is fixed, and both the viscous, inviscid flows must satisfy the no-penetration condition at the boundary: u  n D 0;

(17)

where u is the fluid velocity and n is the unit outer normal to the domain. For ideal fluids, the no-penetration condition is the only one that can be imposed on the flow. It is often referred to, somewhat incorrectly, as a slip boundary condition, because

15 The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

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the fluid is allowed to slip, but no slip parameters are specified. For viscous fluids, there are several possible boundary conditions that are physically consistent. The simplest, and most difficult one from the point of view of the vanishing viscosity limit, is the no-slip boundary condition, where (17) is complemented with the following condition on the tangential fluid velocity at the boundary: utan D V;

(18)

where V is the velocity of the boundary and utan is the tangential component of the velocity. This is the boundary condition originally proposed by Stokes. If the friction force is prescribed at the boundary, then one obtains Navier friction boundary condition, which allow for slip to occur: u  n D 0;

ŒS .u/ n C ˛utan D 0;

(19)

where ˛  0 is the friction coefficient and S .u/ is the viscous stress tensor, which coincides for Newtonian fluids with a multiple of the rate of strain tensor .ru C ruT /=2. Above, given a vector field v on the boundary of , vtan means the component of v tangent to @, and if M is a matrix, M v denotes matrixvector multiplication. These boundary conditions are the ones originally proposed by Navier and derived by Maxwell in the context of gas dynamics. In the absence of friction (˛ D 0), the Navier boundary condition reduces to the condition that the tangential component of the shear stress be zero at the boundary. They are also called stress-free boundary conditions in the literature. The term ˛u can be replaced by Au, where A is a (symmetric) operator. In particular, if A is the shape operator on the boundary of the domain, then the generalized Navier boundary conditions reduce to the following [12, 54]: u  n D 0;

curl u  n D 0;

(20)

sometimes called slip-without-friction boundary conditions, as they reduce to (19) with ˛ D 0 on flat portions on the boundary. In two space dimensions, they are also referred to as free boundary conditions, given that the second condition reduces to curl u D 0 [89], that is, there is no vorticity production at the boundary. In higher dimensions, free boundary conditions lead to an overdetermined system, but are still compatible with the time evolution of the fluid since the boundary is characteristic if the initial data satisfies the same condition. Other types of slip boundary conditions based on vorticity can be imposed [13], but they have been studied less in the literature. Impermeability of the boundary implies that boundary conditions need to be imposed on a characteristic boundary, which complicates the analysis further. If the boundary is not characteristic, as is the case of a permeable boundary with injection and suction, where the normal velocity at the boundary is prescribed and non-zero, the zero-viscosity limit holds at least for short time [139], as boundary layers can

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be shown to be very weak (of exponential type). The case of a non-characteristic boundary is discussed in Sect. 3.3. As already mentioned, one of the main obstructions to establishing the vanishing viscosity limit in the presence of boundaries is the formation of a viscous boundary layer, where the behavior of the flow cannot p be approximated by that of an inviscid flow. A formal asymptotic analysis using  as small parameter leads to a reduced set of effective equations for the leading-order velocity term in the boundary layer, the so-called Prandtl equations (we refer to [129] for a historical perspective). Significant advances have been made in the analysis of these equations, which exhibit instabilities, possible blowup, and ill posedness. The analysis on the Prandtl equations and the connection between solutions to Prandtl equations and the validity of the vanishing viscosity limit will be reviewed in Sect. 3.4. The discussion here will be confined to the classical case of the vanishing viscosity limit for incompressible, Newtonian fluids, although some partial results are available in the important cases of compressible flows [131, 145, 150], MHD [148, 149], convection in porous media [82], and for non-Newtonian (secondgrade) fluids [19, 101]. In the ensuing discussion, the case of unsteady flows in bounded, simply connected domains or a half-space will be the main focus. The interesting case of exterior or multiply connected domains, such as flow outside one or more obstacles, brings in additional difficulties, for example, the infinite energy in the vorticity-velocity formulation of the fluid equations in two dimensions (see [69, 81, 130] and references therein). The introduction of slip makes the vanishing viscosity limit more tractable, essentially because the viscous boundary layer is weak compared to the outer Euler solution and it is possible to obtain a priori bounds on higher norms that are uniform in viscosity. The discussion starts, therefore, with slip-type boundary conditions in Sect. 3.1 and continues with the more challenging case of no-slip boundary conditions in Sect. 3.2. In the remainder of this section, the solution to the Navier-Stokes equations (1) is denoted by u , and the solution to the Euler equations (4) is written as u0 . Unless otherwise stated, it is also assumed that the initial data for the Navier-Stokes equations is ill prepared, that is, the initial velocity u .0/ is divergence free and satisfies the no-penetration condition at the boundary, but not necessarily viscous type, e.g., friction or no-slip boundary conditions. This assumption will allow, in particular, to take the same initial data for (1) and (4): u .0/ D u0 .0/ D u0 ; although this assumption can, and will at times, be relaxed. When the data is ill prepared, there is a corner-type singularity at t D 0, x 2 @ with two types of layers for the viscous evolution, an initial layer and the boundary layer. The initial layer is particularly relevant when the limit Euler flow is steady, as it affects vorticity production in the limit.

15 The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

3.1

797

Case of Slip-Type Boundary Condition

If the viscous boundary layer is of size much smaller than predicted by Prandtl asymptotic theory, one expects that the fluid will effectively slip at the boundary (see [87] for a review of experimental results). This situation is more likely the rougher the boundary is. In fact, it can be shown rigorously in certain situations that homogenization of the no-slip boundary condition on a highly oscillating boundary will give rise in the limit to slip boundary conditions (see [18,73,119] and references therein). With Navier friction boundary conditions, the vanishing viscosity limit holds in two- and three-space dimensions under additional regularity conditions on the initial data, even if vorticity is produced at the boundary and the boundary is characteristic. In two space dimensions, the problem can be studied in the vorticityvelocity formulation, as the Navier friction condition gives rise to a useful boundary condition for vorticity, namely: ! D .2  ˛/utan ;

(21)

where is the curvature of the boundary. The 2D Navier-Stokes initial value problem in vorticity-velocity formulation then reads 8 ˆ !t C u  r!  D !  ; ˆ ˆ ˆ 1, ensures, in particular, a uniform-in-time bound on the Lp norm of the vorticity and hence global existence and uniqueness of a strong solution to (22). In general, a solution of this problem is not a weak solution of (1), even if !0 2 Lp ./ \ L1 ./ for some p > 1, as u is not of finite energy. However, this is the case if  is a bounded domain. The zero-viscosity limit was first established for bounded initial vorticities and forcing in [28] and for unbounded forcing in [127]. This result was then extended to initial vorticities in Lp , p > 2, [99] and to initial vorticities in the Yudovich uniqueness class [77], when the forcing is zero. To be precise, the result in [99] is stated below, which applies to the most general class of initial data. Theorem 8. Let !0 2 Lp ./, p > 2, and let u0 D K Œ!0 . Let u be the unique weak solution of (1) with u .0/ D u0 . Then there exists a sequence k ! 0 and a distributional solution u0 of the Euler equations (4) with initial data u0 , such that uk ! u0 strongly in C .Œ0; T I L2 .// as k ! 1.

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The proof relies on a priori bounds on higher Sobolev norms for the velocity that are uniform in viscosity. These bounds in turn allow by compactness, via an AubinLions-type lemma, to pass to a limit along subsequences as  ! 0. Because compactness arguments are used, the proof does not give rates of convergence of u to u0 . Uniqueness of the limit can be guaranteed a posteriori if !0 is sufficiently regular. The key ingredient in establishing the vanishing viscosity limit is an a priori bound on the vorticity in L1 .Œ0:T ; Lp / uniform in viscosity (for e.g.,  2 .0; 1). This estimate in turn is achieved through the use of the maximum principle, which requires estimating the L1 -norm of the velocity at the boundary, except in the case of free boundary conditions. The condition p > 2 allows to obtain such an estimate via the Sobolev embedding theorem, but it is not expected to be sharp. A more natural condition would be p > 1, which would ensure passing to a limit in nonlinear terms. When  D R2C , the zero-viscosity limit in the energy norm as in Theorem 8 is justified by [125, Theorem 2] even in the case when the friction coefficient ˛ depends on the viscosity as ˛ D ˛ 0  ˇ with some constants ˇ 2 Œ0; 1/ and ˛ 0 > 0. Interestingly, the instability of the Prandtl boundary layer shown by [59] in the no-slip case (see Theorem 13 below) can be proved as well in this viscosity-dependent slip condition when ˇ D 12 ([125, Theorem 3]), for which the L2 convergence for velocity is valid. This result implies that there is an essential discrepancy between the validity of the zero-viscosity limit in the energy norm and the validity of the Prandtl boundary layer expansion. In three space dimensions, it is generally only possible to establish the vanishing viscosity limit for more regular Euler initial data, namely, u0 in the Sobolev space H s , s > 52 , and only for the time of existence of the strong Euler solution. Weak wild solutions are generically too irregular to allow passage to the vanishing viscosity limit even in the full space (this point will be revisited in Sect. 3.2). For regular initial data u0 , the results in Theorem 8 extend to three space dimensions [70]. In fact, since the maximum principle for the vorticity no longer holds due to vortex stretching, a direct energy estimate on the velocity is employed, and consequently, the approximating sequence of Navier-Stokes solutions uk can be taken in the Leray-Hopf class of weak solutions. Uniform bounds in viscosity in higher Sobolev norms H s , s > 52 , do not hold, due to the presence of a boundary layer, except in the case of free boundary conditions. Nevertheless, it is possible to show convergence of the Navier-Stokes velocity u to the Euler solution u0 uniformly in space and time, but utilizing so-called co-normal Sobolev spaces [111]. These are Sobolev spaces that, at the boundary, give control on tangential derivatives, defined as m Hco ./ D ff 2 L2 ./; Z ˛ f 2 L2 ./; j˛j  mg;

(23)

where Z ˛ is a vector-valued differential operator which is tangent to @. Such conormal spaces will be employed in studying the no-slip case as well in Sect. 3.2. Because of the Navier boundary condition, it is possible to obtain a bound in L1 ..0; T /  / uniform in viscosity on the full gradient of the Navier-Stokes velocity using co-normal spaces of high enough regularity. Then, this Lipschitz

15 The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

799

control on the Navier-Stokes solution allows to pass to the limit as viscosity vanishes by using compactness again. Set m m V m D ff 2 Hco ./; rf 2 Hco ./; div f D 0g

Then the convergence result in [111] is stated as follows: Theorem 9. Fix m 2 ZC , m > 6. Let u0 2 V m . Assume in addition that ru0 2 Wco1;1 . Let u 2 C .Œ0; T ; V m / be the strong solution of the Navier-Stokes equations (1) with initial data u0 and boundary conditions (19). Then, there exists a unique solution to the Euler equations (4) with initial data u0 and boundary condition (17), u0 2 L1 ..0; T /; V m /, such that ru0 2 L1 ..0; T /; W 1;1 / and such that ku  u0 kL1 ..0;T // ! 0;

as  ! 0:

Note that in Theorem 9 the high regularity of the initial data is needed to control the pressure at the boundary. One can interpret the uniform convergence of Theorem 9 in terms of boundary layer analysis. For simplicity the boundary is assumed to be flat, or the domain  is locally identified with the half-space RnC by writing a point x 2  as x D .x 0 ; z/ with x 0 2 Rn1 , where n D 2; 3 is the space dimension again, and z > 0. Thus, the boundary of  is identified locally with z D 0. Then, the following asymptotic expansion of the viscous velocity holds in the energy space L1 .Œ0; T ; L2 .// [71]: u .t; x/ D u0 .t; x/ C

p z U .t; x 0 ; p / C O./; 

(24)

where U is a smooth, rapidly decreasing boundary layer profile on RnC . Hence, the boundary layer has the same width as predicted by the Prandtl theory for the case of no-slip boundary condition, but small amplitude. By contrast, the amplitude of the boundary layer corrector to the Euler velocity can be of order one if no-slip boundary conditions are imposed. The validity of the expansion (24) implies, in particular, that in general one cannot expect strong convergence in high Sobolev norm, since then, by the trace theorem, the limit Euler solution would satisfy the Navier-slip boundary condition (the so-called strong zero-viscosity limit). For the case of slip without friction, it was shown in [12] that the boundary condition (20) is not necessarily preserved under the Euler evolution in three space dimensions if the boundary is not flat. The strong zero-viscosity limit does hold for free boundary conditions, hence for arbitrary smooth domains in two space dimensions, and for slip-without-friction boundary conditions on domains with flat boundary [9–11, 147]. In fact, only the initial data need to satisfy the stronger free condition at the boundary [14]. It is interesting to note that the main difficulty in dealing with non-flat boundaries comes from the nonvanishing of certain integrals in the energy estimate due to the convective term u  ru . In the no-slip case, the interaction

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between convection and the boundary layer is thought to be a main obstruction to the validity of the zero-viscosity limit. The last part of this subsection is about an approach to the zero-viscosity limit that yields rate of convergence in viscosity. p The point of departure is the expansion (24). If this expansion is valid, then u0 C V approximates the NavierStokes solution. One can, therefore, define an approximate Navier-Stokes solution uapprox in terms of an outer solution uou valid away from the walls and an inner solution uin valid near the walls (cf. [90, 143]). The parabolic nature of the NavierStokes equations suggests p that the viscous effects are felt in a thin layer close to the boundary of width  (this idea goes back to the original work of Prandtl; in fact, again the reader is referred to [129] for more details). For simplicity  is assumed to be a half-space, as in (24). To define the inner solution on a fixed domain independent of viscosity, it is convenient to introduce the stretched variable p Z D z= . If the zero-viscosity limit holds, one then expects that a regular asymptotic expansion for uin is valid, that is uin .t; x/ D

1 X



kC1 2

k .t; x 0 ; Z/;

kD0

where k is the kth order corrector to the outer flow. It should be stressed that the correctors are not assumed to be independent of viscosity and their amplitude is dictated by the equations of motions and by the boundary conditions. Similarly, the outer solution should have a regular expansion of the form uou .t; x/

D

1 X

k

 2 uk .t; x 0 ; z/;

kD0

with u0 independent of . In fact, u0 D u0 , the Euler solution. Consistency of the formal asymptotic expansion gives effective equations for the flow correctors, together with boundary and initial conditions, from the Navier-Stokes and Euler equations. The goal is then to derive the regularity of the correctors and their decay away from the boundary Z D 0 from the effective equations. The regularity and decay properties typically depend on compatibility conditions between the initial and boundary data. Using these properties, norm bounds on the error u  uapprox can then be obtained from the Navier-Stokes equations via energy estimates. This approach was used in [54] to establish the zero-viscosity limit and rates of convergence under generalized Navier boundary conditions on any smooth, bounded domain; see Theorem 10 below. Both [71] and [54] employ linear boundary correctors, another measure of the weakness of the boundary layer under Navier conditions, but the corrector in [54] is constructed using a covariant formulation and is coordinate independent. Under geodesic boundary normal coordinates in a tubular neighborhood of the boundary, it has an explicit form, which allows to prove uniform space-time bounds on the error u  u0 even close to the boundary.

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Theorem 10. Denote by a a tubular neighborhood of @ interior to  of width a > 0. Fix m > 6 and assume u0 2 H m ./. Let u be the unique, strong solution of (1) with generalized Navier boundary conditions and initial data u0 . Let u0 be the unique strong solution of (4) with the no-penetration boundary conditions and initial data u0 . Then 3

1

ku  u0 kL1 .Œ0;T ;L2 .//   4 ; 3

ku  u0 kL1 .Œ0;T ;H 1 .//   4 3

ku  u0 kL1 .Œ0;T a /   8  8.m1/ ;

3

9

ku  u0 kL1 .Œ0;T na /   4  8m ;

where is a constant independent of . In two space dimensions, under free boundary conditions, the use of correctors allows to study the boundary layer for the vorticity and obtain rates of convergence in Sobolev spaces [53]. Lastly, in the context of Navier-type boundary conditions, the zero-viscosity limit has been used as a mean to establish existence of solutions to the Euler equations (see, e.g., [152], [89, pp. 87–98], [7], and [91, pp. 129–131]). Whether it is also a selection mechanism for uniqueness of weak solutions in two space dimensions remains open.

3.2

Case of No-Slip Boundary Condition

The classical case of no-slip boundary conditions (18) is perhaps the most relevant in applications and the most challenging to study from a mathematical point of view. The main difficulty stems from the formation of a possibly strong boundary layer (of amplitude order one in viscosity) in flows at sufficiently high Reynolds numbers. It is experimentally observed (see, e.g., the classical experiments of flow around a solid sphere [141]) that laminar boundary layers, where the flow lines are approximately parallel to the boundary, destabilize and detach from the boundary, a phenomenon known as boundary layer separation. This layer separation is due to the presence of an adverse pressure gradient that leads to stagnation first and then flow reversal in the layer. In the unsteady case, the connection between the vanishing viscosity limit and the stability of the boundary layer has still not been completely clarified. In particular, there are no known analytical examples of unsteady flows where layer separation occurs. Nevertheless, a connection can be made in terms of vorticity production at the boundary. The mismatch between no penetration and no slip at the boundary leads potentially to the creation of large gradients of velocity in the layer, in particular normal derivatives of tangential components of the velocity at the boundary. While the creation of a boundary layer can be a purely diffusive effect, it is its interaction with strong inertial terms that is thought to lead to boundary layer separation. Therefore, one expects that in the context of the Navier-Stokes equations linearized around a nontrivial profile, i.e., Oseen-type equations, it should be possible to establish the zero-viscosity limit. This is indeed the case at least if the Oseen profile is regular enough and under some compatibility conditions between

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the initial and boundary data [2, 3, 96, 135, 136] (see also [52] for incompatible data for the Stokes equation). Whether the vanishing viscosity limit holds generically even for short time under no-slip boundary conditions is largely an open problem. An asymptotic ansatz for the velocity similar to (24), but taking into account the amplitude of the boundary layer profile, that is, postulating an expansion for the velocity of the form    1 z CO 2 ; u .t; x/ D u0 .t; x/ C t; x 0 ; p 

(25)

leads to the classical Prandtl equation, which will be discussed in more details in Sect. 3.4. One brief remark here is that the Prandtl equations are well posed only under strong conditions on the flow, such as when boundary and the data have some degree of analyticity [5, 23, 84, 95, 128] or the data is monotonic in the normal direction to the boundary [83, 122, 124]. The most classical result verifying (25) is [128] in the analytic functional framework, after the pioneering work of [5, 6]. The result of [128] is stated here only in an intuitive manner without introducing the precise definition of function spaces. Theorem 11. Suppose that given data and the Euler flow are analytic in all variables x D .x 0 ; z/. Then the Prandtl asymptotic expansion (25) is valid for a short time. The proof of [128] is based on the analysis of the integral equations for the NavierStokes equations with the aid of the Cauchy-Kowalewsky theorem. On the other hand, quantifying the production of vorticity by the boundary at finite viscosity and its interaction with convective terms is crucial in understanding the behavior of the viscous fluid near impermeable walls. The next result from [102] shows that the zero-viscosity limit is verified for a short time by using the vorticity formulation in the half plane, identified with R2C , as long as the initial vorticity stays bounded away from the boundary. Theorem 12. Let !0 D curl u0 D curl u0 be the initial vorticity for the Euler and Navier-Stokes flows. Assume that d0 WD dist .@R2C ; supp !0 / > 0: Define uapprox D u0 C uP , where uP is the boundary layer corrector, which satisfies a modified Prandtl equation. Then, 1

ku  uapprox kL1 ..0;T /R2 /  C  2 ; for some constant C > 0 independent of . The time T can be estimated from below as follows:

15 The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

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T  c minfd0 ; 1g; for some positive constant c which depends only on k!0 kW 4;1 \W 4;2 . 1 k Note that the class of natural test functions C0; .R2C / (or even C0; .R2C / for large k) is admissible for the initial data in Theorem 12. This case has been excluded in Theorem 11 due to the analyticity assumption in the entire half-space. In Theorem 12, the L1 choice for the norm in which to take the limit is more natural than the energy norm, as the vorticity-velocity formulation is used to obtain uniform bounds in viscosity and, as already remarked, the velocity obtained from the Biot-Savart law, is not in L2 unless the integral of vorticity is zero (see, e.g., [104]). Recently, the result of [102] is extended in [40] to the three-dimensional case, where the proof is based on a direct energy method. The data discussed in Theorems 11 and 12 have analytic regularity at least near the boundary. To remove the condition of analyticity is a big challenge. Recently, Prandtl expansion has been verified in [49] in a Gevrey space around a shear flow with a monotone and concave boundary layer profile. By virtue of the spectral instability shown in [60], the requirement of Gevrey regularity is considered to be optimal for the stability of the boundary layer at least at a linear level. If the analyticity is totally absent, one should not expect an asymptotic expansion of the form (25) to be valid in general, given the underlying strong instability mechanism at high frequencies. A classical instability result is given in [59] and recalled in Theorem 13 below, where the invalidity of (25) is shown when the initial boundary layer profile is linearly unstable for the Euler equations.

Theorem 13. Let Us .z/ be a smooth shear layer satisfying Us .0/ D 0, such that Us e1 D .Us ; 0/ is a linearly unstable stationary solution to the Euler equations. Let n be an integer, arbitrarily large. Then there exists ı0 > 0 such that the following statement holds. For every large s and sufficiently small , there exist T > 0 and v0 2 H s .R2C / \ L2 .R2C / such that lim T D 0 ;

!0

kv0 kH s .R2 /   n ; C

and the solution u to (1) in R2C (under the no-slip boundary condition) with the x2 initial data u0 .x/ D Us . p /e C v0 .x/ satisfies the estimate  1      e1 kL1 .R2 / D 1 ; lim kcurl u .T /  curl us T ; p C !0     1 e1 kL1 .R2 /  ı0  4 : lim ku .T /  us T ; p C !0  

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Here us .t; z/ is the smooth solution to the heat equations @t us  @2z us D 0, us jtD0 D Us , and us jzD0 D 0. 1

In the proof given in [59], the time T is of the order O. 2 j log j/. In particular, Theorem 13 implies that the expansion (25) may cease to be valid in general at least in this very short time. Theorem 13 is proved based on the instability of the shear profile Us for the Euler equations, but in the rescaled variables X D px . Recently, spectral instability has been established [60] even in the case for which Us is (neutrally) stable for the Euler equations. This result indicates the invalidity of Prandtl expansion in the Sobolev framework, no matter whether the boundary layer profiles possess a good shape, although in this case there is still no rigorous proof about the nonlinear instability in Sobolev classes. On the other hand, as it will be mentioned in Sect. 3.4.2, the invalidity of the asymptotic estimate (25) can be also derived from the high-frequency instability of the shear profile in the Prandtl equations, which is proved in [61]. Note that, however, the invalidity of (25) observed in [59] and [61] relies on the assumption that the boundary layer is formed already at the initial time (and thus, the initial data for the Navier-Stokes flows is also assumed to depend on the viscosity coefficient, more precisely, on the fast variable x2 p ). It is still open whether (25) can be disproved, or proved, in the case when the  Sobolev initial data of the Navier-Stokes flows is taken independently in , for in this case there is no boundary layer at the initial time and the layer forms only at a positive time. Mathematically, the main difficulty in the case of the no-slip boundary condition is the lack of a priori estimates on strong enough norms to pass to the limit, which in turn is due to the lack of a useful boundary condition for vorticity or pressure. Then, the other types of results found in the literature can be roughly divided into two groups: (I) conditional convergence results for generic flows under conditions on the flow that control the growth of gradients in the layer, such as Kato’s condition on the energy dissipation rate, discussed below; (II) convergence results for specific classes of flows, where some conditions as in (I) are valid automatically, such as parallel flows in pipes and channels discussed below. Again, for general initial conditions, the zero-viscosity limit is sought to hold on the interval of existence of the Euler solutions. Kato [75] realized that the vanishing of energy dissipation in a small layer near the boundary is equivalent to the validity of the zero-viscosity limit in the energy space. In fact, this condition is enough to pass to the limit in the nonlinear term in the weak formulation of the equations. Theorem 14 (Kato’s criterion). Let u be a Leray-Hopf weak solution of the threedimensional Navier-Stokes equations (1) with initial data u0 2 L2 ./,   R3 . Let

15 The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

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u0 be a strong solution of the Euler equations (4) with initial data u0 2 H s , s > 5=2 on the time interval Œ0; T . Assume u0 ! u0 strongly in L2 ./. Then u ! u0 strongly in L1 .Œ0; T ; L2 .//, i.e., the vanishing viscosity limit holds on Œ0; T  if and only if, for T 0  T , Z lim 

!0C

0

T0

kru .t /k2L2 .c  / dt D 0;

where c > 0 is a fixed, but arbitrary, constant and c  is a boundary strip of width c . Variants of Kato’s criterion have been established, involving only the partial gradient of the velocity field and allowing for non-zero boundary velocity [137,144] or involving the vorticity [78], for instance. If the Euler flow satisfies a sign condition at the boundary, namely, Oleinik’s monotonicity condition, that on a half plane reads u01 .x1 ; 0; t /  0, so that no back flow occurs, then it is enough for the vorticity to be not too negative in a Kato-type boundary layer (of width ) [29]. Unfortunately, it is not known whether flows generically satisfy Kato’s criterion at least for short time. In fact, Kato’s criterion cannot hold if boundary layer separation occurs by a result of [79, 80] stated below, where it is shown that the zero-viscosity limit holds if and only if vorticity accumulates only at the boundary as a co-normal distribution. Theorem 15. In the hypothesis of Kato’s criterion, the following are equivalent: (a) u ! u0 in L1 .Œ0; T ; L2 .//; (b) !  ! ! 0 C u0  n  weakly- in L1 .Œ0; T /; H 1 ./0 /. where n is the unit outer normal to the boundary and  is a Radon measure that agrees with surface area on @. In two space dimensions, condition (b) can be equivalently restated as !  ! ! 0  u0   

weakly- in L1 .Œ0; T /; H 1 ./0 /;

where  is the unit tangent vector to @ (obtained by rotating n counterclockwise by =2). It should be noted that the equivalence of (a) and (b) above is purely kinematic, in the sense that is a direct consequence of the validity of the limit and results on weak convergence of gradients. If dynamics is taken into account, for example, when the initial-boundary-value problem for vorticity is solvable, the convergence in (b) can often be improved to convergence in the sense of Radon measures on the closure of . In fact, it is enough that the vorticity be uniformly bounded in viscosity in L1 .Œ0; T /; L1 .// [80, Corollary 4.1]. This will be the case for parallel pipe and channel flows discussed later in this subsection. In this

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Y. Maekawa and A. Mazzucato

situation, one can interpret the extra measure on the boundary appearing in the limit as a vortex sheet due to a jump in velocity across the boundary even if there is no fluid outside the domain . A consequence of the theorem is that boundary layer separation cannot occur if the zero-viscosity limit holds, even though the convergence is in a relatively weak norm, the energy norm, because then the convergence of u to u0 is in the H 1 -Sobolev norm in the interior, and this strong convergence is incompatible with layer separation. The vanishing viscosity limit and associated boundary layer can be studied for special classes of flows that satisfy strong symmetry assumptions. In this situation, no additional assumptions are made on the flow, except assuming symmetry of the initial data, as the symmetry is preserved by the Navier-Stokes and Euler evolution, at least for strong solutions. (For a discussion of possible symmetry breaking in the context of weak solutions, the reader is referred to [8].) The classes of flows that can be studied are so-called parallel flows in straight infinite channels or straight infinite circular pipes. These flows can be thought of as generalization of the classical Poiseuille and Couette flows, but they are unsteady and generally nonlinear. In fact, the walls of the pipe or channel are allowed to move rigidly along itself, as in the classical Taylor-Couette case, so that the no-slip boundary condition for solutions to (1) takes the form (18) with V ¤ 0. Parallel channel and pipe flows were considered before in the context of the zero-viscosity limit by [144], who lists them as cases for which Kato’s criterion applies. In fact, it is easy to see that the criterion applies in the extension due to [137] if the boundary velocity V is not too rough. However, due to the symmetry in the problem, it is possible to obtain a detailed analysis of the flow in the boundary layer and quantify vorticity production even in the case of impulsively started and stopped boundary motions, where V is of bounded variation in time. It should be noted that the boundary layer is not weak p here, and, in fact, it has the width predicted by the Prandtl theory proportional to . But, because of symmetry, the flow stays laminar and the boundary layer never detaches. A similar analysis for truly nonlinear, symmetric flows, such as axisymmetric flows (without swirl) and helical flows, seems out of reach at the moment. In what follows, fer ; e g will denote the orthonormal frame associated to polar coordinates .r; / in the plane, and fer ; e ; ex g will denote the orthonormal frame associated to cylindrical coordinates .r; ; x/ in space. The symmetric flows that have been considered in the literature are: (i) Circularly symmetric flows: planar flows in a disk centered at the origin  D fx 2 C y 2 < Rg. The velocity is of the form u D V .t/e ;

(26)

using polar coordinates, where V .t/ is a radial function. The vorticity, which can be identified with a scalar for planar flows, is also radial.

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(ii) Plane-parallel flows: 3D flows in a infinite channel, with periodicity imposed in the x and y directions. The velocity takes the form u D .u1 .t; z/; u2 .t; x; z/; 0/

(27)

and is given on the domain  WD .0; L/2  .0; h/ where h is the width of the channel and u1 , u2 satisfy periodic boundary conditions in x and y. The boundary is identified with the set @ D Œ0; L2  Œ0; h (ii) Parallel pipe flows: 3D flows in a infinite straight, circular pipe, with periodicity imposed along the pipe axis, identified with the x-axis. The velocity is of the form u D u .t; r/e C ux .t; ; r/ex ;

(28)

using cylindrical coordinates on the domain  WD f.x; y; z/ 2 R3 j y 2 C z2 < R; 0 < x < Lg; where R is the radius of circular cross section and u , ux satisfy periodic boundary conditions in x. The boundary is identified with the set @ D f.y; z/ 2 R2 j y 2 C z2 D Rg  Œ0; h. For all these flows, the divergence-free condition is automatically satisfied. In the case of circular symmetry, the Navier-Stokes equations reduces to a heat equation and the Euler flow is steady, making this more of a pedagogical example. Both for the channel and pipe geometry, symmetry and periodicity ensure uniqueness of solutions to the Navier-Stokes equations and the Euler equations, in particular by forcing the only pressure-driven flow to be the trivial flow. The well posedness is global in time for both Euler and Navier-Stokes for sufficiently regular initial data. As an illustration, only parallel pipe flows will be discussed here, which is the most interesting case due to the effect of curvature of the boundary. The reader is referred to [16, 97, 98, 113] for the case of circularly symmetric flows and to [115, 117] for the case of channel flows. In parallel pipe flows, the velocity is independent of the variable along the pipe axis and, in any circular cross section of the pipe, it is the sum of a circularly symmetric, planar velocity field and a velocity field pointing in the direction of the axis. As in the case of plane-parallel flows, even though the flows are not planar, the Navier-Stokes and Euler equations reduce to a weakly nonlinear system in only two space variables, given, respectively, by (for simplicity it is assumed that the boundary is stationary)

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8  @u ˆ 1 ˆ ˆ  u C  2 u D 0 ˆ ˆ @t r ˆ ˆ ˆ @ux 1 @u <  ux C u x D 0 @t r @ ˆ 1  2 @p  ˆ ˆ ˆ D0  .u / C ˆ ˆ r @r ˆ ˆ :u D 0 ; i D ; x i

in .0; T /   ; in .0; T /   ; in .0; T /   ; on .0; T /  @ ;

and by 8 @u0 ˆ ˆ ˆ D0 in .0; T /   ; ˆ ˆ ˆ < @t0 0 1 @u @ux C u0 x D 0 in .0; T /   ; ˆ @t r @ ˆ ˆ ˆ ˆ 1 0 2 @p 0 ˆ : .u / C D 0 in .0; T /   : r @r The Navier-Stokes system is amenable to the analysis of the vanishing viscosity limit primarily because it is diffusion dominated and because the pressure is slaved to the velocity and drops out of the momentum equation. A detailed analysis of the boundary layer using techniques borrowed from semiclassical analysis was performed in [116], for ill-prepared data. There, in particular, convergence rates in viscosity for the L1 norm were derived by constructing a parametrix to a suitable associated linear problem and taking the corrector as the double layer potential associated to this problem. For well-prepared data, convergence rates in higher Sobolev norms were obtained by the use of flow correctors and effective equations in [62]. By the use of a different type of correctors, it is possible to obtain similar results for ill-prepared data and quantify production of vorticity at the boundary [55]. Similarly to the case of Navier boundary conditions, one defines an approximate Navier-Stokes solution uapprox as a sum of an outer solution uou and an inner solution uin . At zero order in viscosity, uou D u0 , the Euler solution, while uin is given by a smooth, radial cut-off supported in a collar neighborhood of the boundary times a corrector of the form

.t; x/ D .t; r/e C x .t; ; r/ex ;

(29)

using again cylindrical coordinates .r; ; x/, where and x satisfy weakly coupled parabolic systems. Then, the following convergence rates can be obtained [55, 62, 116]. Theorem 16. Assume u0 2 H k ./, k large enough (k > 4 suffices) and has symmetry (28). Then, the zero-viscosity limit holds on .0; T /, for all 0 < T < 1 and, in particular,

15 The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

(

1

809 3

ku  uapprox kL1 .0;T IL2 .// C  2 kru  ruapprox kL2 .0;T IL2 .//  T  4 ; 1 ku  u0 kL1 .0;T IL2 .//  T  2 ; (30)

where T is a constant independent of . In addition, N !  ! ! 0 C .u0  n/ weakly in L1 .0; T I M.//;

(31)

N is the space of Radon measures on  N and  is a measure supported where M./ on @, which on the boundary agrees with the normalized surface area. A complication over plane-parallel flows is that the effect of nonvanishing curvature cannot be neglected in the analysis. Furthermore, in cylindrical coordinates, the behavior of the solution near the axis cannot be controlled and in the same way as it can be controlled away from the axis, similar to the case of axisymmetric flows. To overcome this difficulty, a two-step localization, one near the boundary where curvilinear coordinates are used and the other near the axis where Cartesian coordinates and energy estimates are employed, is utilized. As a consequence, however, the error estimates for the approximate solution suffer from the loss of one derivative. In particular, the estimates for the correctors are not as sharp as in the case of a pipe with annular cross section. As seen below, there are other situations for which it is possible to pass to the limit due to the fact that the boundary layer is weak or absent. This is the case, for example, of flows outside shrinking obstacles, if the obstacle is shrinking faster than viscosity vanishes. For such flows, the local Reynolds number, built by taking the size of the obstacle as characteristic length, stays of order one, as already observed in [69]. In this context, the Navier-Stokes solution in the exterior of the obstacle is expected to converge to the Euler solution in the whole space. Most of the results concern flows in the plane, as the vorticity-velocity formulation is used to obtain the inviscid solution. At the same time, the fact that the exterior of compact obstacles is not simply connected in two space dimensions adds some technical difficulties, which are overcome by assuming that the circulation around the obstacles is zero. Let  be the scale of the obstacle. The vanishing viscosity limit was shown to hold in the exterior of one obstacle diametrically shrinking to a point in [69] by assuming the condition   C  for some positive constant C , which depends on the initial data for the Euler equations in R2 , u0 , and the shape of the obstacle, and assuming that the initial condition for the Navier-Stokes solution, u0 , extended by zero to the whole plane, converges to u0 in L2 .R2 /, with an optimal rate of p convergence of . (See [81] for the opposite situation of an expanding domain.) This result can be extended to the exterior of a finite number of fixed obstacles. It is interesting to ask whether a similar result holds in the setting of a porous medium, that is, if the domain for the viscous flow is the exterior of an array of particles. It is known that homogenization of the Navier-Stokes equations and Euler equations gives different filtration laws (e.g., Darcy or Brinkman) depending on the relative ratio of the particle size  and the inter-particle distance d , and the permeability of

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Y. Maekawa and A. Mazzucato

the homogenized medium is very different between the viscous and inviscid cases (see the discussion in [86, 120] and references therein). Therefore, it is relevant to study the joint limit of vanishing , d , . In [86], the limit was established under the condition that d >  and   A ; see Theorem 17 below. In this regime, one expects that the in the limit the Euler flow, defined in the whole plane, does not feel the presence of the porous medium. Below, for each , the domain  is set as the viscous fluid domain, which, for simplicity, is the exterior of a regular array of identical particles arranged in a square. Theorem 17. Given !0 2 Cc1 .R2 /, let u0 the solution of the Euler equations in the whole plane with initial condition u0 WD KR2 Œ!0 . For any ;  > 0, let d    and let u; be the solution of the Navier-Stokes equations in  with initial velocity u; 0 . Then, there exists a constant A depending only on the particle shape, such that if  A  ; d k!0 kL1 \L1 .R2 / and if !0 is supported in  , then for any T > 0 we have ;

sup ku 0tT

0

 u kL2 . /  BT

p



d

C

ku; 0

  u0 kL2 . /

(32)

where BT is a constant depending only on T , k!0 kL1 \W 1;1 .R2 / , and the particle shape. ; ; 2 It is then possible to construct pinitial data u0 such that u0 ! u0 in L , which establishes the limit with rate =d . Therefore, there is a ghost of the porous medium in the convergence rate. It should be noted that in the case of the DarcyBrinkman system, the equations for the boundary corrector are linear, and thus, the passage to the zero-viscosity limit is possible [63, 82].

3.3

Non-characteristic Boundary Case

One of the main difficulties in treating the vanishing viscosity limit for classical no-slip boundary conditions is the fact that the boundary is characteristic for the problem, that is, it consists of streamlines for both the viscous and inviscid flows. Hence, any attempt to control the flow in the interior from the boundary seems unsuccessful unless analyticity or monotonicity of the data is imposed. If non-characteristic boundary conditions are imposed, in particular, if the walls are permeable, then under certain conditions, the boundary layer is stable; hence there is no layer separation and one expects the vanishing viscosity limit to hold. This is the case when injection and suction rates are imposed at the boundary. For simplicity we describe the setup in the geometry of a (periodized) channel

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Œ0; L2  Œ0; h, where injection and suction are imposed at the top and bottom walls. The velocity at the boundary for Navier-Stokes is then given as ui .t; x/ D .0; 0; Ui .t; x1 ; x2 //;

i D top; bot;

(33)

where Ui  ai > 0 for some constants ai and top, bot refer to top and bottom of the channel. For Euler, one needs to specify the entire velocity and the inlet/outlet. These conditions are also appropriate when a domain is truncated, e.g., for computational reasons, when making a Galilean coordinate transformation. By correcting the velocity field, it was shown in [138] that the zero-viscosity limit holds with sharp rates of convergence of  1=4 in the uniform norm. In particular, there is only a stable boundary layer at the suction wall (the bottom) that is exponentially small. It is interesting to note that, differently than in the nonlinear case, for the Oseen equations, adding injection and suction at the boundary does not seem to change the size of the boundary layer (see [96]).

3.4

Prandtl Equations for the Boundary Layer

This subsection is devoted to an overview of the study of the Prandtl equations, introduced by Prandtl in 1904 in order to describe a viscous incompressible flow near the boundary at high Reynolds numbers [126]. The Prandtl equations are derived from the Navier-Stokes equations with no-slip boundary condition, and their derivation is briefly recalled here in the case that the fluid domain is the half plane R2C . The initial-boundary-value problem for the Navier-Stokes equations reads 8      ˆ ˆ @t u C u  ru C rp D u ; ˆ ˆ ˆ < div u D 0 ; ˆ ˆ ˆ ˆ ˆ :

t > 0 ; x 2 R2C ; t  0 ; x 2 R2C ;

(34)

x 2 R2C ;

u jtD0 D u0 ;

t > 0; x1 2 R:

u jx2 D0 D 0 ;

As is discussed in the previous sections, the equations in the limit  D 0 are the Euler equations, for which only the impermeability condition u2 D 0 on @R2C can be prescribed. Heuristically, such an incompatibility in the boundary condition leads to a fast change of the tangential component of the velocity field, which is u1 when the fluid domain is R2C . As a result, the derivative of u1 in the vertical direction tends to have a singularity near the boundary and forms a boundary layer. To study the formation of the boundary layer, Prandtl made the ansatz that u near the boundary has the following asymptotic form: u1 .t; x1 ; x2 /



uP1



x2 t; x1 ; p 

 ;

u2 .t; x1 ; x2 /

p uP2



x2 t; x1 ; p 

 : (35)

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p The thickness of the boundary layer O. / is coherent with the parabolic nature of the Navier-Stokes equations. The underlying assumption here is that the velocity u remains of order O.1/ in all @k1 u , k D 0; 1;    , in the limitp ! 0, and then the vertical component u2 is expected to be of the order O. / since the boundary condition for the normal component is preserved in the limit, and it is also compatible with the divergence-free condition. By formally substituting the ansatz (35) into the first equation of (34), the velocity profile uP1 and the associated pressure p P should obey the equations @t uP1 C uP  ruP1 C @1 p P  @22 uP1 D 0 ;

@2 p P D 0 ;

and uP1 must satisfy the no-slip boundary condition uP1 D 0 on @R2C . Here the x2 spatial derivatives are for the rescaled variables X1 D x1 and X2 D p , but in this  subsection @j will denote both

@ @xj

and @j D

@ @Xj

for notational ease. The rescaled

variables X will be also relabeled as x from now on. The vertical component uP2 is recovered from uP1 and the boundary condition uP2 D 0 on @R2C by virtue of the second equation (divergence-free condition) in (34), which yields uP2 .t; x/

Z

x2

D 0

@1 uP1 .t; x1 ; y2 /dy2 :

The velocity in the boundary layer has to match with the outer flow which is assumed to satisfy the Euler equations. This requirement leads to the following boundary condition (matching conditions) on uP1 and p P at x2 D 1: lim uP x2 !1 1

D uE ;

lim p P D p E ;

x2 !1

where uE .t; x1 / D u0 .t; x1 ; 0/ and p E .t; x1 / D p 0 .t; x1 ; 0/, and .u0 ; p 0 / is the solution to the Euler equations (4) in R2C . Since p P must be independent of x2 , because of the equation @2 p P D 0 in R2C , the matching condition on p P at x2 D 1 implies that pP D pE : That is, the pressure field is not one of the unknowns in the Prandtl equations. Collecting the above equations gives the Prandtl equations in R2C (in the spatial variables), which are a system of equations for the scalar unknown function uP1 : 8 P @ u C uP  ruP1  @22 uP1 D @1 p E ; ˆ ˆ ˆ t 1 ˆ Z x2 ˆ < uP2 D  @1 uP1 dy2 ; ˆ 0 ˆ ˆ ˆ ˆ : uP1 jtD0 D uP0;1 ; uP1 jx2 D0 D 0 ;

(36) lim uP x2 !1 1

D uE :

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The reader is referred to [92,109,129] for more details about the formal derivation of the Prandtl equations. Note that, by taking the boundary trace in the Euler equations, the data .uE ; p E / coming from the Euler flows in the outer region is subject to the Bernoulli law: @t uE C uE @1 uE C @1 p E D 0 :

(37)

The Prandtl equations are deceptively simpler than the original Navier-Stokes equations. In fact , due to the inherent instability of boundary layers, well posedness of (36) has been proven only in some specific situations (Sect. 3.4.1), while strong ill-posedness results are present in the literature (Sect. 3.4.2).

3.4.1 Well-Posedness Results for the Prandtl Equations The Prandtl equations are known to be well posed under some restricted conditions. This subsection gives a list of the categories in which the well posedness of the Prandtl equations holds, at least for short time. (I) Monotonic data. This category is the most classical in the theory of the Prandtl equations. The system (36) is considered for 0 < t < T and for .x1 ; x2 / 2 1 RC , where 1 is usually set as either f0 < x1 < Lg, T, or R. When 1 D f0 < x1 < Lg, an additional boundary condition has to be imposed on uP1 at the boundary fx1 D 0g: uP1 .t; 0; x2 / D uP1;1 .t; x2 / : The given boundary data uP1;1 also has to be compatible with the monotonicity. The basic assumption describing the monotonicity is @2 uP0;1 .x1 ; x2 / > 0 ;

x1 2 1 ; x2  0 ;

(38)

@2 uP1;1 .t; x2 / > 0 ;

t > 0 ; x2  0 :

(39)

As a compatibility condition, the outer flow uE and given data uP0;1 , uP1;1 must be positive for x2 > 0, and the solution uP1 is also expected to be positive for x2 > 0 together with its derivative in the x2 variable. The solvability of (36) in this class has been established by Oleinik and her coworkers, especially for the case 1 D f0 < x1 < Lg. (See [121–123]. The reader is also referred to [124] for more details and references.) The steady problem is solved in [121] for small L > 0, and this local existence result is extended in [114], where it is shown that the solution can be continued to the separation point. For the unsteady problem, unique solvability is proved in [122] for a short time if L is given and fixed, while for an arbitrary time if L is sufficiently small. The stability of the steady solutions is shown in [123]. A natural question arises, already present in the monograph [124], namely, under which condition the solutions exist globally in time without any smallness of L > 0. A significant contribution to this problem is given by the work [151], where the global existence of weak solutions to (36) is proved when the pressure gradient is

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Y. Maekawa and A. Mazzucato

favorable: @1 p E .t; x1 /  0 ;

t > 0 ; 0 < x1 < L :

(40)

The analysis in [122, 151] uses the classical Crocco transformation:  Dt;

 D x1 ;

D

uP1 .t; x1 ; x2 / ; uE .t; x1 /

w. ; ; / D

@2 uP1 .t; x1 ; x2 / ; uE .t; x1 /

which transforms the domain f.t; x1 ; x2 / j 0 < t < T ; 0 < x1 < L ; x2 > 0g, T > 0, into QT D f. ; ; / j 0 <  < T ; 0 <  < L ; 0 <  < 1g : Then the Prandtl equations for the case 1 D f0 < x1 < Lg are transformed into the system 8 1 E 1 1 1 2 ˆ < @ w C u @ w C A@ w  Bw D @ w P ˆ : wjD0 D w0 D @2 u1 jtD0 ; E u

in QT ;

wjD0 D w1 ; .w@ w/jD0 D

; @1 p E ; wj D 0 : D1 uE (41)

where A D .1  2 /@1 uE C .1  /

@t uE ; uE

B D @1 uE C

@t uE ; uE

w1 D

@2 uP1;1 uE

jx1 D0 :

The following result is proved in [151, Theorem 1.1]. Theorem 18. Assume that (38) and (39) hold together with compatibility conditions. If, in addition, the pressure condition (40) holds, then there exists a weak solution w 2 BV .QT / \ L1 .QT / to (41) such that for some C > 0 C 1 .1  /  w  C .1  /

in QT ;

and @2 w is a locally bounded measure in QT . In Theorem 18, the initial and boundary conditions are satisfied in the sense of the trace, and (41) is considered in the sense of distributions. The reader is referred to [151] for more on the properties and the regularity of weak solutions obtained in this theorem. As a consequence of Theorem 18, global existence of weak solutions to (36) follows. However, uniqueness and smoothness of weak solutions in the Crocco variables seem to be still unsettled. In particular, when L is not small enough, the global existence of smooth solutions to the Prandtl equations

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remains open even under the monotonicity conditions (38) and (39) and the pressure condition (40). The Crocco transformation has been a basic tool in the classical works [122– 124, 151]. Recently, an alternative, independent approach has been presented in [4, 112], where the crucial part of the proof is based on a direct energy method but for new dependent variables. In particular, the Crocco transformation  P  in not needed u1 in this new approach. The key new unknown is w D @2 in [4] and g D ! QP  P u1 in [112], where ! P D @2 uP1 and !Q P D @2 uQ P1 represent the vorticity ! P @2 !P fields of the Prandtl flow uP1 and of the background Prandtl flow uQ P1 , respectively. The function w is introduced in [4] in the analysis of the linearized Prandtl equations around uQ P1 , and the function g is analyzed in [112] for the nonlinear energy estimate. As explained in [112], for example, the crucial obstacle one meets in the energy j j estimate for the standard unknowns @1 uP1 or @1 ! P is the presence of the terms j j .@1 uP2 /! P or .@1 uP2 /@2 ! P , since they contain the highest-order derivatives in x1 and do not vanish after integration by parts. The new unknown is in fact chosen so that these crucial terms cancel. The development of the Prandtl theory in [4, 112] has had a significant impact in the field and has led to significant recent progress [50, 72, 83]. Very recently, the Prandtl equations have been studied in the three-dimensional half-space in [93, 94]. These works show that the monotonicity condition on the tangential velocities is not enough to ensure local well posedness, and a sharp borderline condition is found for well posedness/ill posedness. (II) Analytic data. The second classical category for local well posedness of the Prandtl equations is the space of analytic functions. Under analyticity of the initial data and of the outer Euler flow, local existence of the Prandtl equations has been proven in [128] after the pioneer work [5, 6], where analyticity is imposed on both variables x1 and x2 . Later, it was realized that the condition of analyticity in the vertical variable x2 can be removed, and the local well posedness is known to hold under only analyticity in the tangential variable [23, 24, 84, 95]. To be precise, a typical existence result available by now in this category is stated here. Let  2 R, ˛; ˇ; T > 0. The space H;˛ is the space of functions f .x1 ; x2 /, 2 periodic in x1 , such that the norm jf j;˛ D

X

sup hx2 i˛

j 2 x2 2RC

X

j

j@2 fO .k; x2 /je jkj ;

1

hx2 i D .1 C x22 / 2 ;

k2Z

is finite. Here fO .k; x2 / is the kth Fourier mode of f with respect to x1 . The space ;˛ Hˇ;T is the space of functions f .t; x1 ; x2 /, 2 periodic in x1 , such that the norm jf j;˛;ˇ;T D

X

j

sup j@2 f .t/jˇt;˛ C sup j@t f .t/j ˇt;˛

j 2 0tT

0tT

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 is finite. The space Hˇ;T is the space of functions f .t; x1 /, 2 periodic in x1 , such that the norm

jf j;ˇ;T D

X

sup j@it f .t/j ˇt

iD0;1 0tT

;˛ is finite. The spaces H;˛ and Hˇ;T are used for the boundary layer profiles, while  Hˇ;T is used for the Euler flows. The next theorem is proved in [24], where local solvability is obtained even with incompatible initial data, that is, uP0;1 jx2 D0 ¤ 0.

Theorem 19. Let uE 2 Hˇ00 ;T0 and uP0;1  uE jtD0 2 H0 ;˛ for some 0 ; ˇ0 ; T0 > 0 and ˛ > 12 . Then there exist  2 .0; 0 /, ˇ 2 .0; ˇ0 /, and T 2 .0; T0 / such that Prandtl equations (36) admit a unique solution uP1 in Œ0; T  of the form uP1 .t; x1 ; x2 /

D

2uP0;1 .x1 ; 0/ erfc

;˛ where uQ 2 Hˇ;T . Here erfc



x2 p 2 t





x2 p 2 t

1 Dp

t

Z



C uQ .t; x1 ; x2 / C uE .t; x1 / ;

1

x2

 2 y exp  2 dy2 . 4t

As in [5, 23, 95, 128], the proof given in [24] relies on the abstract CauchyKowalewsky theorem, which is applied to the integral equations associated with (36). The existence of solutions under a polynomial decay condition on uP0;1  uE jtD0 has been shown first in [84]. The proof of [84] is based on the direct energy method, rather than the use of the abstract Cauchy-Kowalewsky theorem and integral equations. Without monotonicity, one cannot expect global existence of smooth solutions of the Prandtl equations to hold in general even if the given data is analytic. Indeed, in this case, the existence of finite-time blowup solutions can be shown ([38]; see Sect. 3.4.2 below). However, the class of initial data for blowup solutions in [38] must have O.1/ size, and, hence, it is still not clear whether global existence holds for sufficiently small data or not. Recently an important progress has been achieved in this direction, and longtime well posedness is established in [72, 154] for small solutions in the analytic functional framework. In [154] the life span of 3 local solutions is estimated from below to be of order O.  4 / when the uniform 5 outer Euler flow uE D u is of the order O. 3 / and the initial data uP0;1 is O./ in a suitable norm measuring tangential analyticity. In [72], almost global existence is established if the smallness condition on the uniform Euler flow is removed and the 1 //, 0 <  1, when uP0;1 is life span is then estimated from below as O.exp.  log  O./. (III) Other categories: beyond analyticity or monotonicity. Without monotonicity or analyticity of the given data, the solvability of the unsteady Prandtl equations

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becomes a highly difficult problem even locally in time. There are a few classes of initial data that are not strictly included in the categories (I) and (II) above, but for which the Prandtl equations can be solved for a short time. (1) Gevrey class with a non-degenerate vorticity. As it will be seen in Sect. 3.4.2, the

Prandtl equations are ill posed in general in Sobolev spaces. This ill posedness is due to the instability at high frequencies for the tangential components, occurring when the monotonicity of the given data is absent. A key argument for this instability is given in [48], where it is proved that the linearization around the non-monotonic shear flow satisfying (45) for some a > 0 has a solution growing exponentially in  1 time with growth rate O jnj 2 for high tangential frequencies n. Although such a high-frequency instability yields the ill posedness in Sobolev spaces, there is still hope to obtain the well posedness for initial data whose nth Fourier mode in the  x1 variable decays in order O.e cjnj / for jnj  1 with some  > 12 , that is, the Gevrey class less than 2. A crucial difference between the Gevrey class 1 ( D 1, analytic functions) and the Gevrey classes m with m > 1 ( D m1 ) is that the latter class can contain compactly supported functions. The verification of the instability in [48] motivates the work of [50], where the local solvability is established for a set of initial data without monotonicity, but belonging to the Gevrey class 74 in the x1 variable. The key condition for the initial data in [50] is that the monotonicity is absent only on a single smooth curve but in a non-degenerate manner. More precisely, in [50], it is assumed that uE D p E D 0 and uP0;1 is periodic in x1 with Gevrey 74 regularity and that @2 uP0;1 .x1 ; x2 / D 0 iff x2 D a0 .x1 / > 0 with @22 uP0;1 .x1 ; a0 .x1 // > 0 for all x1 2 T: (42) Note that the condition (42) is a natural counterpart of (45). The crucial observation for the proof in [50] is that in the region where the monotonicity is absent, the flow is expected to be convex by virtue of the non-degenerate condition, while away from the curve of the critical points one can use the monotonicity of the flow. However, due to the nonlocal nature of the Prandtl equations, taking advantage of each in the different regions requires an intricate analysis, and this difficulty is overcome in [50] by introducing various kinds of energy. (2) Data with multiple monotonicity/analyticity regions. The flows in this class are

introduced in [83], where the local existence and uniqueness of the Prandtl equations are proved for initial data with multiple monotonicity regions, by assuming that the initial data is tangentially real analytic on the complement of the monotonicity regions. A typical example of the initial data in this category is @2 uP0;1 < 0 for x1 < 0 ; @2 uP0;1 > 0 for x1 > 0 ; uP0;1 is real analytic in x1 around x1 D 0 :

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Y. Maekawa and A. Mazzucato

That is, the monotonicity of the initial flow is lost around x1 D 0, but instead, the analyticity in the tangential variable is imposed there. Because of this complementary distributions of two totally different structures and the nonlocal nature of the problem, the methods developed in the categories (I) and (II) are not enough in constructing local solutions in this class. Indeed, there are several difficulties in this problem; the norms for the analyticity class and the monotonicity class are not compatible, and moreover, nontrivial analytic functions cannot have a compact support, which indicates the breakdown of standard localization arguments. The key observation in [83] is that one can in fact construct the analytic solution around x1 D 0 in a decoupled manner without using the lateral boundary conditions. On the other hand, by introducing a suitable extension of the data in the tangential direction, one can construct a monotone solution in the entire half plane. Finally, by virtue of the finite propagation property in the tangential direction, the analytic solution and the monotonic solution actually coincide with each other on some strip regions, which implies the existence of solutions to the original Prandtl equations. The construction of [83] reveals in some sense a possibility of localizing the Prandtl equations in the tangential direction. Moreover, the result of [83] indicates that, even at the point of separation, the flow can be stable at least locally in time and space, if the flow is analytic in the tangential variable around the separation point.

3.4.2 Ill-Posedness Results for the Prandtl Equations Although the local solvability of the unsteady Prandtl equations still remains open for general initial data in Sobolev spaces, several ill-posedness results have been reported in the literature. This subsection is devoted to give an overview on recent progress in this direction. (I) Ill posedness of the Prandtl equations in Sobolev spaces. When given data are not monotonic, the unsteady Prandtl equations are known to be ill posed in the sense of Hadamard. The ill posedness is triggered by the instability of non-monotonic shear flows at high frequencies. The first rigorous result for this instability is given in [48], where the linearization around a shear flow possessing a non-degenerate critical point is studied in detail. To be precise, let uP D .uP1 ; uP2 / be the solution to the Prandtl equations (36) for constant data uE D u 2 R and @1 p E D 0 (that is, the Euler flow u0 is a stationary shear flow u0 D .u01 .x2 /; 0/ and u D u01 .0/). In this case, uP is also a shear flow uP .t; x/ D .us .t; x2 /; 0/, and us is the solution to the heat equation: 8 < @t us  @22 us D 0 ; : us jtD0 D Us ;

us jx2 D0 D 0 ;

lim us D u :

(43)

x2 !1

Here Us is a given initial shear profile satisfying the compatibility conditions. Then, a natural question is whether or not one can construct a solution to the Prandtl equations around this shear flow. The key step to tackle this problem is to analyze the linearization around us :

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8 @t v1P C us @1 v1P C v2P @2 us  @22 v1P D 0 ; ˆ ˆ ˆ ˆ Z x2 ˆ < v2P D  @1 v1P dy2 ; ˆ 0 ˆ ˆ ˆ P ˆ : v1P jtDs D v0;1 ; v1P jx2 D0 D 0 ; lim v1P D 0 :

(44)

x2 !1

Here, t > s, x1 2 T, and x2 2 RC . The system (44) is uniquely solvable at least P locally in time, if the initial data v0;1 is analytic in the x1 variable, and then the P evolution operator T .t; s/, T .t; s/v0;1 WD v1P .t /, is shown to be locally well defined in the analytic functional framework; see [48, Proposition 1]. With this observation one can define the operator norm of T .t; s/ from H m1 to H m2 , where H m1 and H m2 are suitable Sobolev space in T  RC and the exponents m1 ; m2 denote the order of the Sobolev regularity; see [48] for the precise definition of H m . In [48], the given initial data Us in (43) is assumed to have a non-degenerate critical point: there is a > 0 such that Us0 .a/ D 0 ;

Us00 .a/ ¤ 0 :

(45)

Then the following ill posedness in the Sobolev class is given by [48, Theorem 1]. Theorem 20. If (45) holds, then there exists  > 0 such that for all ı > 0, sup ke .ts/ 0stı

p

j@1 j

T .t; s/kL.H m ;H m / D 1

for all m  0 ;

1  2 Œ0; / : 2

Moreover, there is a solution us to (43) and  > 0 such tat for all ı > 0 sup ke .ts/ 0stı

p

j@1 j

T .t; s/kL.H m1 ;H m2 / D 1

for all m1 ; m2  0 :

The key step in the proof in [48] is to construct an approximate solution to (44), 1

which grows in time with order e ıjnj 2 .ts/ , ı > 0, for a tangential frequency p 1 jnj  1. This growth rate O.jnj 2 / is responsible for the weight e .ts/j j@1 j in the statement of Theorem 20. The approximate solution is constructed as a singular perturbation from an explicit solution to the inviscid linearized Prandtl equations (dropping the viscous term @22 v1P in (44) and replacing us by Us ), for which the spectral problem has been studied in details in [66]. This instability mechanism, 1 bifurcating from the inviscid solution and resulting in the growth rate O.jnj 2 /, was first reported at a formal level in [32]. Due to the nature of the singular perturbation, however, the rigorous justification of this mechanism requires a highly delicate asymptotic analysis, and it is successfully completed by [48]. The result of [48] on the ill posedness for the linearized Prandtl equations is strengthened in [51,61], where it is shown that the solutions to the nonlinear Prandtl

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equations cannot be Lipschitz continuous with respect to the initial data in Sobolev spaces. Moreover, it is proved by [61] that in the Sobolev framework, one cannot expect a natural estimate of the asymptotic boundary layer expansion for the NavierStokes flows, if the leading term in the boundary layer is a non-monotonic shear layer flow as in [48]. Recently the ill posedness of the three-dimensional Prandtl equations is studied in [94], and it is revealed that there is a stronger instability mechanism in the threedimensional case even at the linear level. In particular, it is shown in [94] that, in contrast to the two-dimensional case, the monotonicity condition on tangential velocity fields is not sufficient for the well posedness of the three-dimensional Prandtl equations. (II) Blowup solutions to the Prandtl equations. In the absence of the monotonicity, it is known that the solution to the Prandtl equations can blow up in finite time. The formation of such singularity was observed numerically in [140] for data corresponding to an impulsively started flow past a cylinder. The existence of blowup solutions was also reported by [66] through numerical and asymptotic analysis for the inviscid Prandtl equations. The rigorous existence of finite-time blowup solutions is first given by [38] (see also [85]), which is stated as follows: Theorem 21. Let uE D p P D 0. Assume that the initial data uP0;1 is of the form uP0;1 .x1 ; x2 / D x1 b0 .x1 ; x2 / for some smooth b0 and that a0 .x2 / D @1 uP0;1 .0; x2 / is nonnegative, smooth, and compactly supported. Assume in addition that E.a0 / D

1 1 k@2 a0 k2L2 .RC /  ka0 k3L3 .RC / < 0 2 4

(46)

holds. Then there exist no global smooth solutions to (36). Remark 2. In Theorem 21 the condition of the compact support of a0 is not essential, and it can be replaced by a decay condition which ensures the boundedness of @1 uP1 .t; 0; x2 / in L1x2 .RC / as long as the solution exists. A simple example of the initial data satisfying the conditions of Theorem 21 is p

2

u0;1 .x1 ; x2 / D x1 e x1 f

x  2

R

;

(47)

where f is a (nontrivial) nonnegative smooth function with compact support and R > 0 is a sufficiently large number. Indeed, in this case the function a0 is given by a0 .x2 / D f . xR2 /, and the quantity E.a0 / is computed as E.a0 / D

1 R k@2 f k2L2 .RC /  kf k3L3 .RC / ; 2R 4

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which is negative when R > 0 is large enough. Since the initial data defined by (47) is analytic in x1 , in virtue of Theorem 19, there exists a unique solution to (36) at least for a short time. Theorem 21 shows that this local solution cannot be extended as a global solution and the proof in [38] implies that the blowup occurs for the quantity k@1 uP1 .t /kL1 .R2 / . The singularity formation observed in [140] is studied in C details by [47], in which numerical evidence is reported for the strong ill posedness of the Prandtl equations in the Sobolev space H 1 .RC /.

4

Conclusion

The inviscid limit problem of the Navier-Stokes flows is one of the most fundamental and classical issues in fluid mechanics, particularly in understanding the flows at high Reynolds numbers. Mathematically, the fundamental question here is whether or not the Navier-Stokes flows converge to the Euler flows in the zeroviscosity limit, by taking the effect of the boundary into account if necessary. Even when there is no physical boundary, the analysis of the inviscid limit is a challenging problem if one works with singular flows such as vortex sheets or filaments (Sect. 2). The rigorous understanding of these structures, in terms of the analysis of the Navier-Stokes equations at high Reynolds numbers, is still out of reach except for the case when the distribution of the possible singularities can be well specified in advance due to some additional prescribed symmetry. In the presence of physical boundary, the verification of the inviscid limit is far from trivial in general even when given data have enough regularity, e.g., in higherorder Sobolev spaces (Sect. 3). The main obstacle is the formation of the viscous boundary layer, the size and stability of which are crucially influenced by the type of the boundary conditions. If the boundary condition allows the flow to slip on the boundary or the boundary is non-characteristic, then the effect of the boundary layer is moderate, and the mathematical theory has been well developed by now in these categories (Sects. 3.1 and 3.3). However, if the no-slip boundary condition is imposed, the size of the boundary layer is at least O.1/ even at a formal level, and the underlying instability mechanism of the boundary layer leads to a serious difficulty in the analysis of the inviscid limit problem (Sect. 3.2). The general result in this research area is Kato’s criterion, which describes the condition for the convergence of the Navier-Stokes flows to the Euler flows in the energy space. This criterion can be confirmed under some symmetry conditions on both the fluid domain and the given data without assuming strong regularity of data such as analyticity. Without symmetry, the known results verifying the inviscid limit (the convergence in the energy space as well as the Prandtl boundary layer expansion) require, so far, the analyticity at least near the boundary, and the boundary is also assumed to be flat there. In understanding the formation of the boundary layer, mathematically the analysis of the Prandtl equations is a central issue (Sect. 3.4). However, the solvability of the Prandtl equations is available only for some restricted classes of given data including the monotonicity or the analyticity class (Sect. 3.4.1). In

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the general Sobolev framework, the strong instability and resulting ill posedness have been observed (Sect. 3.4.2). Moreover, without monotonicity, large solutions to the Prandtl equations may blow up in finite time, while the existence of smooth global solutions for small data, but in a wide fluid domain, remains open even under the monotonicity condition in general. Finally, even when the Prandtl equations are solvable, their solvability does not imply the validity of the inviscid limit, and in fact, there is a significant discrepancy between these two problems. This discrepancy actually indicates the limitation of Prandtl’s approach to understand boundary layer separation in physical situations, although the latter has been often discussed within the framework of the Prandtl equations. The mathematical understanding of the stability of the boundary layer and the analysis of the boundary layer separation needs a significant development of the present theory for the inviscid limit problem of the Navier-Stokes flows.

5

Cross-References

 Existence and Stability of Viscous Vortices Acknowledgements Y. Maekawa was partially supported by the Grant-in-Aid for Young Scientists (B) 25800079. A. Mazzucato was partially supported by the US National Science Foundation Grant DMS-1312727.

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Regularity Criteria for Navier-Stokes Solutions

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Gregory Seregin and Vladimir Šverák

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 "-Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 More About Scale-Invariant Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Type I Blowups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Bounded Ancient Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Liouville-Type Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

830 839 843 850 852 857 858 864 864 864

Abstract

In this chapter some of the known regularity criteria for the weak solution of the incompressible 3D Navier-Stokes equations are discussed. At present the problem of regularity of general solutions starting from smooth data is open, and all the criteria involve an assumption on a suitable quantity which is invariant under the scaling symmetry of the equations. Both interior regularity and boundary regularity are addressed. The methods developed by Scheffer and Caffarelli-Kohn-Nirenberg play an important role. Simple but important considerations based on dimensional analysis and the scaling symmetry are recalled, together with some heuristics. Connections between the Liouville-type

G. Seregin () Mathematical Institute, University of Oxford, Oxford, UK e-mail: [email protected] V. Šverák School of Mathematics, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_16

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theorems and Type I singularities are also discussed. Proofs of some statements which are not easily accessible in the literature are presented.

1

Introduction

The Navier-Stokes equations @t v C v  rv C rq  v D 0 div v D 0

(1)

in three space dimensions are considered. As usual, v denotes a velocity field and q denotes a pressure function, both defined on some open subset O of space-time R3  R. The equations can be formulated weakly for any vector field v 2 L2;loc .O/ as follows:  R   vi 'i dx dt D 0 O vi @t 'i  vi vj 'i;j R (2) O .vi ; i / dx dt D 0 for each smooth divergence-free vector field ' D .'1 ; '2 ; '3 / which is compactly supported in O and any (scalar) smooth compactly supported function  in O. Here and in what follows, the notation ; i is used for the i th spatial partial derivative, and summation over repeated indices is understood. Formulation (2) is local, and it is natural to ask about the local regularity of the solutions. Sometimes the term very weak solutions is used for this class of solutions. If the assumption v 2 L2;loc .O/ is not strengthened, not much is known about regularity. It is perhaps worth pointing out that the vector fields of the form v.x; t / D rh.x; t/, where h.x; t / D 0 and the dependence of t is “arbitrary” (with the understanding that it is still assumed that v 2 L2;loc .O/), are solutions of (2), and therefore one sees immediately that for such local solutions, the regularity in the time direction is, in general, limited. Even in the steady-state case (solutions independent of time), there are many open problems for the very weak solutions. In fact, even the following question seems to be open: if v D v.x/ is a vector field in the unit ball B  R3 with v 2 L2;loc .B/ satisfying the steady-state form of (2) and, in addition, jv.x/j 

C ; jxj

x 2 B n f0g ;

(3)

is v smooth in B? (The assumption (3) together with the equation give smoothness in B n f0g, the issues is smoothness at x D 0.) Note that condition (3) is invariant under the scaling symmetry of the equation v.x/ ! v .x/ D v.x/ ;  > 0. In that sense the condition (3) is “critical” as the information from it neither improves nor deteriorates as one goes to smaller scales. A slightly stronger condition v 2 L3 .B/ is often also considered critical, although it implies better behavior on

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smaller scales, as jjv jjL3 .B/ ! 0 for  & 0. This makes a huge difference for the regularity theory and is enough to prove smoothness by standard bootstrapping arguments, just as is the case when one has (3) for sufficiently small C ; see, for example, [21]. This simple example already illustrates some important features of the NavierStokes regularity theory (and regularity theory of many other PDEs). To obtain smoothness, one needs some information about the solution on small scales, and hence some “critical quantity” has to be controlled. A boundedness of a critical quantity is not enough, however. Instead, one needs “smallness on small scales” of at least some critical quantity. This is a feature of all known results, although the smallness condition may sometimes be somewhat hidden, as is the case with the condition v 2 L3 .B/ in the example above. Ideally, estimates giving the smallness at small scales should be derived from the equation. This is in fact the case in our above if one assumes the finiteness of the rate of energy dissipation R example R 2 jrvj dx. (Strictly speaking, the rate of dissipation is proportional to jDj2 dx, B t / is the deformation tensor, but for where D D 12 .rv R C.rv/ R our purposes here, one 2 can work with jrvj dx.) An estimate on the integral B jrvj2 dx immediately gives v 2 L6 .B/, which is more than what is needed for regularity. (The problem becomes “subcritical.”) Hence, from the point of view of the physical flows, the regularity problem for very weak solutions with only the assumption (3) is perhaps somewhat artificial. However, in the time-dependent case in three spatial dimensions, the finiteness of the rate of energy dissipation does not by itself give a bound on a critical quantity. It does bring more structure to the problem, but from many points of view, one is in the same situation as in the simpler model problem for very weak solutions just discussed. Another interesting model problem R is the problem of regularity of the weak solutions (with the assumption that B jrvj2 dx is finite) of the steady equation in spatial dimension n D 5. Important results for this problem have been obtained by Frehse and Ružiˇcka; see [11]. It was already discussed that the estimates based on energy and its dissipation play an important role. Smooth solutions of the Navier-Stokes equations in R3 t1 ; t2 Œ with suitable decay condition at the spatial infinity satisfy the energy inequality Z R3

1 jv.x; t /j2 dx C 2

Z tZ t1

jrvj2 dx dt 0  R3

Z R3

1 jv.x; t1 /j2 dx 2

t 2t1 ; t2 Œ: (4)

This inequality can also be localized; see (29) below. Importantly, for the natural initial value problems (both in R3 and in domains with boundaries), one can prove (following Leray) the existence of solutions which belong to the natural energy classes and satisfy the energy estimates (both local and global). In addition, quite general perturbation techniques, independent of the

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energy inequality, can be used to prove (following Kato [20], see also [4], [13] and [30] for references) local-in-time existence or global small data existence of solutions. (An optimal result in this direction is due to Koch-Tataru [23].) This is why the regularity theory has mostly focused not on the very weak solutions, but on two other classes of solutions: the suitable weak solutions and mild solutions . The first of these classes represents local solutions which belong to the “energy class” (see (26)) and satisfy the local energy inequality. The second class represents global solutions defined on R3 t1 ; t2 Œ (or perhaps on t1 ; t2 Œ, when the equations are considered in a domain   R3 , with appropriate boundary conditions), which are smooth for t 2t1 ; t2 Œ (open interval), and the study of regularity focuses on estimates for t % t2 . In that case one should think of t2 as the possible “first blow-up time,” and one often sets t1 ; t2 ŒD0; T Œ. With the mild solutions, one is essentially considering the equation as a Banach space-valued ODE, and one can think of the time interval 0; T Œ as the maximal time of existence for the solutions. The mild solutions will not be considered beyond the time when the smoothness is potentially lost (if such a time exists). If S .t / is the solution operator for the linear part of the equation @t v C rq  v D 0 div v D 0 vjtD0 D v0

(5)

with appropriate boundary conditions in case of a domain with boundary , in the sense that S .t /v0 D v.t / ;

(6)

then the mild solution of the Navier-Stokes initial value problem is defined by the usual Duhamel’s formula Z t v.t / D S .t /v0 C S .t  s/Œv.s/  rv.s/ ds ; (7) 0

in an appropriate Banach space. For a characterization of mild solutions not based on the Duhamel’s formula, see Proposition 3. Although the suitable weak solutions considered in this chapter will be mostly local in both space and time, it should be mentioned that there are theorems which guarantee that, under natural assumptions, global-in-space suitable weak solutions and mild solutions of natural initial value problems coincide (under some assumptions) if the initial datum is the same; see, for example, [30]. Strictly speaking, it has not been defined above in which sense the weak solutions should assume their initial datum at t D 0. One has of course the natural requirement that the first equation of (2) can be extended to test vector fields ' which do not vanish at the initial time, if appropriate terms are added, but usually a stronger condition is demanded, namely, that u.t / converges to u0 strongly in L2;loc as t ! 0.

16 Regularity Criteria for Navier-Stokes solutions

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Let us now make some heuristic considerations concerning possible singularity formation from smooth rapidly decaying initial data. As above, it will be assumed that one has a smooth solution v on some time interval 0; T Πwhich blows up as t % T . For simplicity the case  D R3 will be considered. An important role in the regularity theory is played by the scaling symmetry of the Navier-Stokes equations: v.x; t / ! v .x; t / D  v.x; 2 t / q.x; t/ ! q .x; t / D 2 q.x; 2 t / ;

(8)

where  > 0. As the notion of regularity is clearly invariant under this scaling, the conditions on regularity should be preferably formulated in terms of scale-invariant quantities. For example, if one knows that the Cauchy problem in R3 is globally well posed for initial data v0 with jjv0 jj2L2 C jjrv0 jj2L2 < 1 (as proved by Leray), one can minimize the sum of the two norms over the functions v0 .x/ D v0 .x/ with  > 0 and see that it is enough that jjv0 jjL2 jjrv0 jjL2 < 1 =2. In fact, as Kato showed, the smallness of jjv0 jj P 12 is sufficient, but this is not a direct consequence H of Leray’s result (although it is suggested by it and the scaling symmetry, if one takes the parabolic smoothing into account). The dimensional analysis (discussed in some detail in [3]) plays an important role. The dimension of the kinematic viscosity in the Navier-Stokes equation @t v C v  rv C rq=  v D 0 (where  denotes density) is dim D

length2 ; time

(9)

so if one normalizes to one, then dim t dim v R dim R3 jv.x; t /j2 dx R dim R3 jv.x; t /j3 dx

 length2 ;  length1 ; D length time  length ; D 1;

(10)

and similarly for other quantities. Hence, with the normalization D 1, all physical dimensions are just powers of length, and, changing the terminology somewhat, one can also say, following [3], that the dimension of a quantity is the corresponding exponent. In this terminology dim t dim v R 2 dim R3 jv.x; t /j dx R dim R3 jv.x; t /j3 dx and similarly for other quantities.

D 2; D 1 ; D 1; D 0;

(11)

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At a heuristic level, if one wishes to establish regularity, one needs to control v and its derivatives, which all are quantities of negative dimension. To be able to use the smoothing properties of the linear part of the equation, one needs to control a “good” quantity of negative dimension (subcritical case) or at least have some smallness condition on a quantity of dimension zero (critical case). The smallness condition can sometimes be “implicit.” For example, if a function is in Lp with p < 1, the Lp norm of its restriction to sets of sufficiently small measure is small. A typical example of “subcritical regularity” is the following statement: a bounded mild solutions of the Navier-Stokes equations is smooth, with bounds on the derivatives given, up to constants, by dimensional analysis. More precisely, if jjvjjL1 .R3 0;tŒ/  M , then jj@lt r k v.t /jjL1 .R3 / . M kC1C2l ;

t & M 2 :

(12)

This follows easily, for example, from estimates in [22]. Modulo the scaling symmetry, this statement is roughly equivalent to saying that mild solutions starting at time t D 0 which are of size of order 1 (in the sup-norm) will have all derivatives again of order 1 at times of order 1. The statement can be modified in various ways. One can work with different norms, and one can localize with some modifications. For the local version one has to keep in mind the solutions already mentioned earlier: v D rh.x; t/ ;

1 q D @t h.x; t/  jrh.x; t/j2 ; 2

(13)

where h is harmonic in x with arbitrary (measurable) dependence on t . Here, we consider q as a distribution, and if we wish that it be represented by a locally integrable function, the distributional derivative @t h must be represented by a locally integrable function. Solutions (13) show that for local solutions, we cannot expect full regularity in time. A typical example of a simple critical regularity statement is that a mild solution v 2 L5 .R3 0; T Œ/ is smooth, with p tjjv.t /jjL1  C .jjvjjL5 .R3 0;tŒ/ / ;

(14)

where C .  / is a suitable continuous function vanishing at 0. This can be derived from our previous “subcritical” L1 -estimate together with some simple scaling arguments; see below. Estimates for higher derivatives then follow from (12). Again, the statement can be modified in many ways and localized (with appropriate adjustments), and the best results in this direction are quite harder to prove than (14). See, for example, Theorem 2 below (due to Caffarelli-Kohn-Nirenberg [3]). Another example of local critical regularity statement is Corollary 1 below. The corollary follows from Theorem 6, which also illustrates how estimates of one scale-invariant

16 Regularity Criteria for Navier-Stokes solutions

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quantity may be employed together with the local energy bounds to obtain bounds on other scale-invariant quantities. One simple heuristics of regularity estimates is as follows. Assume that the maximum M .t/ of u.x; t / grows unboundedly. Let us consider tN such that M .t/  M .tN/ for t  tN. The function x ! ju.x; tN/j has very high “peaks” as M .t/ grows. Let us consider such a peak of large height M D M .tN/. Due to the viscosity effects, the peak should have some minimal width and also some minimal duration in time . This can be interpreted, for example, as a condition that jv.x; t /j 

M ; 2

when jx  xj N  and tN    t  tN:

(15)

Can one estimate ;  from below in terms of M ? Note that the dimensions of ;  , and M are, respectively, 1; 2, and 1. Hence, one expects, based on dimensional analysis,

&

1 ; M

&

1 : M2

(16)

This is what the parabolic smoothing gives: one cannot really expect much more unless some deeper (yet unknown) properties of the equation are used. One can now try to see if (15) and (16) lead to a contradiction with some known (or assumed) estimates. For example, estimate (14) can be obtained in this way. What is happening with the solution “outside the peaks”? Heuristically, one expects that the peaks need some “background,” i.e., that the solution cannot be close to zero even outside the peaks. If one, for example, assumes that the solution develops a singularity at .x; N tN/ (while being regular for t < tN), the best possible estimate one might hope for is 1 jv.x; t /j & p ; 2 jx  xj N C .tN  t /

.x; t / 2 A

(17)

where A is some “sufficiently large” set. While one can show (following Leray) that at a singular time tN one has jjv.t /jjL1 & p

1 ; .tN  t /

(18)

the spatial part of (17) seems to be much harder to establish. For example, it is not known how to rule out the unlikely scenario where .x; N tN/ is a singular point, but at the same time jv.x; t /j2 C jq.x; t/j .

1 F a.t /2Cı



x a.t /

 (19)

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3 for some function F W R p !0; 1Œ with exponential decay for x ! 1, small ı > 0 and a.t / & 0 at rate tN  t as t % tN. Note that in such a scenario, one would be able to obtain (by rescaling) singular solutions v with

Z tN Z t1



jv.x; t /j3ı1 C jq.x; t/j

3ı1 2



dx dt  "1

(20)

B.x;r/ N

for small "1 > 0 and small ı1 > 0. This should illustrate the significance of Theorem 1 below and also the significance of Corollary 1. Note that in the assumptions of Corollary 1 below, an assumption similar to (20) needs to be imposed on “all scales.” On the other hand, if one could establish (17) for sufficiently large sets A, it would be enough to impose such assumptions only on one scale, just as in Theorem 1. Theorem 1 below (which goes back to Scheffer [39–41] with later versions due to Caffarelli-Kohn-Nirenberg [3] and F.-H. Lin [32]) has a special role because one needs to impose smallness of a seemingly quite weak norm only on one scale. In some sense, this theorem can be considered as a step toward establishing (17). Results of this form originate in the pioneering work of De Giorgi on minimal surfaces (see, e.g., [15]) and is also closely related to partial regularity statement in the regularity theory for harmonic mappings and other elliptic and parabolic systems; see, for example, [10, 42]. In the theory of minimal surfaces and harmonic mappings, one has the monotonicity formulae which implies that suitable “small excess” conditions on one scale “propagate” to smaller scales. This is not known to be the case for the Navier-Stokes equation, which seems to be closer to the partial regularity theorems for more general elliptic systems in [10]. Why is the one-scale smallness condition on Z

tN tNt1

Z



3

jvj3 C jqj 2



dx dt

(21)

B.x;r/ N

enough to establish regularity? The key is that the energy flux can be estimated by this quantity and that the energy norm controls the L 10 -norm, which is a gain over 3 (the velocity part of) (21). Some work needs to be done to handle the pressure part, but that part is based on some quite standard estimates. In fact, the regularity gain from L3 to L 10 can be bootstrapped, at the costs of needing to use the smallness 3 condition on other scales, too. This is roughly behind the proof of Theorem 6. See also [16]. The bootstrapping enables to lower the exponent in the smallness condition from 3 to 52 C ı (at the cost of having to use smallness at all scales). Estimate (17) is concerned with a lower bound at a singularity. On the other hand, it is also natural to ask if one can expect some upper bounds. The first natural idea is to try to combine (16) with the energy estimate. In space dimension n D 2, this gives full regularity, as peaks described by (16) are not compatible with the energy estimate as M ! 1. In dimension n D 3, a similar consideration suggests at a heuristic level a bound on the Hausdorff dimension of the set of the possible

16 Regularity Criteria for Navier-Stokes solutions

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singular points, although the rigorous arguments used in [38–41] and [3, 29, 32] are based on different ideas. A possible upper bound at a potential singularity .x; N tN/, based on dimensional analysis, might be the opposite of (17), namely, 1 jv.x; t /j . p ; jx  xj N 2 C .tN  t /

.x; t / 2 A ;

(22)

for a suitable subset A of space-time. There are many other scale-invariant quantities which one can think about in this context, such as Z Z 1 jv.x; t /jp dx dt ; r > 0: (23) r 5p tNr 2 BxN ;r By definition, scale-invariant bounds persist at all scales and hence are preserved as one “zooms in” on the singularity by rescaling. In the limit of zero scale, one gets a solution defined in R3  .1; 0/, and scale-invariant bounds can guarantee that this solution is nontrivial. This is related to Liouville theorem discussed in Sects. 5 and 6. Roughly speaking, if one has a Liouville theorem and a scale-invariant bound, regularity can be obtained. Without the Liouville theorem, a smallness condition (which can be “hidden”) is needed. Boundary regularity for the Navier-Stokes equation presents new difficulties, with more open problems. Some of the results in this direction are also discussed below, especially in Sect. 7. In the rest of the chapter, a more detailed account of some of the important regularity results will be presented. As already discussed above, there are two main ways to study regularity: global and local ones. The global approach is essentially a consequence of proving the existence of solutions to the initial boundary value for the Navier-Stokes equation in classes of smooth functions. In the local approach, a sort of energy solutions is considered, the existence of which is relatively easy to prove. Then using methods of PDE’s theory, one can try to prove that the energy solutions are in fact smooth, i.e., the corresponding PDEs make solutions smoother. The effect is similar to what one has for, say, L1 -solutions to the heat equation that are in fact infinitely smooth. In the chapter, we are going to study local regularity mostly. To be more precise, consider the Navier-Stokes system in a space-time Qt1 ;t2 D t1 ; t2 Œ @t v C v  rv  v C rq D 0;

div v D 0:

(24)

Since viscosity > 0 is fixed all the time, for simplicity, it is assumed that D 1. As already emphasized above, the Navier-Stokes equations are invariant with respect to the following simple scaling: v.x; t / ! v.x; 2 t /;

q.x; t / ! 2 q.x; 2 t /

(25)

838

G. Seregin and V. Šverák

with any positive . Further considerations will be completely local, and, in view of (25), one may consider the system (24) in a canonical domain B  1; 0Œ, where B.x0 ; r/ is a ball of R3 centered at the point x0 with radius r and B.r/ D B.0; r/ and B D B.1/. The following abbreviations for parabolic balls will be used as well: Q.z0 ; r/ D B.x0 ; r/t0  r 2 ; t0 Œ, Q.r/ D Q.0; r/, Q D Q.1/. Further assumptions are related to the class of solutions to be considered. A reasonable class comes out from the energy estimate and is as follows: v 2 L2;1 .Q/ \ W21;0 .Q/:

(26)

This class is important because one can prove the existence of global weak solutions in it. Here, anisotropic Lebesgue and Sobolev spaces are exploited: Ls;l .Q/ WD Ll .1; 0I Ls .B//;

W21;0 .Q/ WD fjvj C jrvj 2 L2 .Q/g:

The class for the pressure may vary. The most convenient space for the pressure is q 2 L 3 .Q/: 2

(27)

Recall that a point z D 0 is called a regular point if there exists r > 0 such that v 2 L1 .Q.r//. Otherwise z D 0 is called a singular point. What is important here, as already mentioned above, is that, in a parabolic neighborhood of a regular point, any spatial derivatives of v are Hölder continuous (with respect to parabolic distance) with an exponent depending on a choice of the function space for the pressure. This is a consequence of the regularity theory for the Stokes system; see the paper [34]. Now, the problem of local interior regularity can be formulated in the following way. Problem IR. Let v and q satisfy (26), (27), and the Navier-Stokes system (24) in Q in the sense of distributions. Find minimal assumptions on v and q (if they are needed at all) under which the origin of Q, i.e., z D 0, is a regular point of the velocity v. The similar problem can be set up in the case of regularity up to the boundary. In this case, the Navier-Stokes system is considered in QC D B C   1; 0Œ, where B C .x0 ; r/ D fx 2 B.x0 ; r/ W x3 > x03 g, B C .r/ D B C .0; r/, and B C D B C .1/. The boundary condition on the flat part of the lateral boundary of QC should be imposed on v.x 0 ; 0; t / D 0

(28)

for any jx 0 j < 1 and any t 2  1; 0Œ. Here, x 0 D .x1 ; x2 /. Problem BR. Let v and q belong to energy classes (26) and (27) in QC and satisfy the Navier-Stokes system (24) in QC and the boundary condition (28). Find minimal

16 Regularity Criteria for Navier-Stokes solutions

839

assumptions on v and q (if they are needed at all) under which the origin of QC , i.e., z D 0, is a regular point of the velocity v. It is said that z D 0 is a regular point of v if there exists r > 0 such that v 2 L1 .QC .r//. The significant difference between interior and boundary regular points is that regularity near the boundary does not imply continuity of spatial derivatives of v. This is a consequence of the fact that a local version of a nonlocal problem has a certain limitation and takes place even for the Stokes system; see the corresponding examples in [19, 56].

2

"-Regularity

There are two important notions in the local regularity theory for the Navier-Stokes equations: suitable weak solutions and scale-invariant quantities. Here, it is a definition of suitable weak solutions in the case of interior regularity: Definition 1. It is said that the pair v and q is a suitable weak solution to the NavierStokes equations in Q if they satisfy (24), (26), (27), and, for a.a. t 2  1; 0Œ, the local energy inequality Z

2

Zt Z

2

' 2 jrvj2 dxds 

' .x; t /jv.x; t /j dx C 2 1 B

B

Zt Z

.jvj2 .@t ' 2 C ' 2 / C v  r' 2 .jvj2 C 2q//dxds

(29)

1 B

for any ' 2 C01 .B  1; 1Œ/. Examples of quantities that are invariant with respect to the scaling (25) are as follows: 1 A.r/ D sup t2r 2 ;0Œ r E.r/ D

1 r

Z

1 C .r/ D 2 r

2

jv.x; t /j dx; B.r/

Z

jvj3 dxdt;

Q.r/

jrvj2 dxdt;

H .r/ D

1 r3

Q.r/

Z Q.r/

D.r/ D

Z

1 r2

Z

3

jqj 2 dxdt: Q.r/

jvj2 dxdt;

840

G. Seregin and V. Šverák

The first three quantities are called the energy scale-invariant ones. Their important property is that if one of them is uniformly bounded with respect to r 2  1; 0Œ, so do others; see Sect. (4) and [46] for details. Now, we are in position to formulate the main statements of local interior "regularity theory. The first one is this: Theorem 1. There exist positive universal constants " and ck , k D 1; 2; : : :, such that for any suitable weak solution v and q, satisfying the additional smallness condition Z   3 jvj3 C jqj 2 dxdt < "; (30) Q

functions z D .x; t / 7! r k1 v.z/ are Hölder continuous in the completion of Q.1=2/, k D 1; 2; : : :. Moreover, the following estimates are valid: jr k1 v.z/j  ck

sup

(31)

z2QC .1=2/

for k D 1; 2; : : :. The importance of the above statement, proved by [3], is that all the constants in it are universal. This will be exploited many times in the future. In fact, Theorem 1 can be slightly improved in a way that shows better the role of the pressure. Proposition 1. Given M > 0, there are two positive constants ".M / and c1 .M / such that if a suitable weak solution v and q satisfies the additional assumptions Z

3

jqj 2 dxdt < M Q

and

Z

jvj3 dxdt < ".M /;

Q

then the function z D .x; t / ! v.x/ is Hölder continuous in the closure of Q.1=2/ with the estimate sup jv.z/j < c1 .M /: z2Q.1=2/

In fact, the statement of Proposition 1 is a scale-invariant version of Wolf’s regularity condition; [54] and [66] for an elementary proof.

16 Regularity Criteria for Navier-Stokes solutions

841

There are interesting consequences of Theorem 1. Among them is the famous Caffarelli-Kohn-Nirenberg condition of local regularity. Theorem 2. There exists a positive universal constant "1 with the following property. If a suitable weak solution v and q in Q satisfies the extra assumption sup E.r/ < "1 ;

(32)

0 0. Therefore, f is bounded and U is continuous in B5  Œ1; r 2  for any 0 < r < 1. On the other hand, since r k !, k D 2; 3, is bounded for t  r 2 < 0, r5k1 f , k D 2; 3, is bounded for t  r 2 < 0. Using the latter information, one can easily show that the equation for f holds in B5   1; 0Œ. It is easy to see that one can apply a weak maximum principle and thus the function f is bounded in B5   1; r 2 Œ with a bound independent of r. Passing r ! 0, we conclude that the vorticity ! of u is bounded in Q that implies that ru 2 L3 .Q/ and, by Theorem 2, z D 0 is a regular point. Regarding Problem BR, similar results can be proven; see [44,48]. But, first, one should modify the definition of suitable weak solutions. Definition 2. It is said that the pair v and q is a suitable weak solution to the NavierStokes equations if they satisfy (24), (26), and (27) in QC , boundary condition (28), and, for a.a. t 2  1; 0Œ, the local energy inequality Z

2

2

Zt Z

' .x; t /jv.x; t /j dx C 2 1 B C

BC

Zt Z

' 2 jrvj2 dxds 

.jvj2 .@t ' 2 C ' 2 / C v  r' 2 .jvj2 C 2q//dxds

(34)

1 B C

for any ' 2 C01 .B  1; 1Œ/. Then the main statement is as follows. Theorem 4. There exist positive universal constants " and c, such that for any suitable weak solution v and q, satisfying the additional condition

16 Regularity Criteria for Navier-Stokes solutions

843

Z 

 3 jvj3 C jqj 2 dxdt < ";

(35)

QC

functions z D .x; t / 7! v.z/ are Hölder continuous in the completion of Q.1=2/. Moreover, the following estimate is valid: sup jv.z/j  c:

(36)

z2Q.1=2/

Unfortunately, in contrast to Theorem 1, extension of Theorem 4 to spatial derivatives of v in general is not true: one can construct a counterexample showing that boundedness of the velocity v in a neighborhood of z D 0 does not imply local boundedness of rv; see [19, 56]. Theorem 5. There exists a universal constant "1 > 0 with the following property. If a suitable weak solution v and q in QC obeys the extra assumption Z

1 sup E .r/ D sup 0 3 and 2 3 C D 1: s l Then z D 0 is a regular point of v. Proof. Without loss of generality, one may assume M s;l .1/ < ı where ı is a positive given number. Indeed, if it is not so, one finds r0 > 0 such that M s;l .r0 / < ı. And by scaling, the cylinder Q.r0 / can be replaced with the unit cylinder Q. Consider the first case l > 3. Then, the result follows from the inequality 3

3

C .r/  c.M l;s .r// l < cı l

for any 0 < r < 1. In the second case, 2  l  3. Here, one can find s1 < s such that the pair s1 and l satisfies condition (38). Then, simply by Hölder inequality, M s1 ;l .r/  cM s;l .r/ < cı for any 0 < r < 1.

t u

As it has been mentioned above, our aim is a local regularity theory. For regularity results under the global Ladyzhenskaya-Prodi-Serrin condition, we refer the reader to papers [26, 36, 61]. In fact, one can prove a sort of Ladyzhenskaya-Prodi-Serrin condition even without the assumption that solutions should have the finite energy; see [47].

850

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G. Seregin and V. Šverák

Type I Blowups

Up to now, it is unknown whether or not there are initial data for which weak LerayHopf solutions to the Navier-Stokes equations blow up. However, on the local level, it is known that if a pair v and q is a suitable weak solution to the Navier-Stokes equations in Q and one of the main scale-invariant quantities like E.r/, A.r/, or C .r/ is uniformly in 0 < r < 1 small, then z D .x; t / D 0 is a regular point. Obviously, the next interesting question is to understand whether or not suitable weak solutions blow up at the origin z D 0 if one of the above quantities is just bounded, i.e., 

sup E.r/; sup A.r/; sup C .r/ < 1:

min

0 0

u

871

Singularity u

symmetric

r

u

ω1 < 0

u

ω1 < 0

Fig. 1 Vorticity and velocity fields in the numerical simulation [30]. (a) In the 3D space. (b) In the meridian plane

t .x; t / C u.x; t /x .x; t / D 0;

(2b)

where x, u.x; t /, w.x; t /, and .x; t / correspond to z, uz , w1 , and u21 in (1), respectively. To close the above one-dimensional system (2), one needs an appropriate modified Biot-Savart law that connects the velocity field u.x; t / to the vorticity field w.x; t /. In [21], Hou and Luo proposed and investigated the following model: ux D Hw;

x 2 Œ1; 1;

(3a)

where H is the periodic Hilbert transform, namely, ux .x/ D P:V:

1 2

Z

1 1

 w.y/ cotΠ.x  y/dy; 2

x 2 Œ1; 1:

(3b)

For odd vorticity field w, one can derive by direct integration that u.x/ D

1 

Z 0

1

ˇ ˇ ˇ tan. 2 x/  tan. 2 y/ ˇ ˇ dy; w.y/ log ˇˇ tan. 2 x/ C tan. 2 y/ ˇ

x 2 Œ0; 1:

(3c)

We refer to the above model (2) and (3) as the HL-model. The finite-time singularity of the HL-model from smooth initial conditions is proved very recently in [5]. In [6], Choi, Kiselev, and Yao proposed the following model: Z

1

u.x/ D x x

w.y/ dy; y

x 2 Œ0; 1;

(4)

and proved its finite-time singularity. We will refer to (2) and (4) as the CKY-model. By carefully examining the kernels in the Biot-Savart laws (3) and (4), one can see that the CKY-model can be viewed as a leading order approximation to the

872

T.Y. Hou and P. Liu

HL-model. See [5] for more about the connections of the two models to the Euler equations (1). The mechanism of the finite-time singularity of the two 1D models is roughly the following: the positive w.x; t / near the origin creates a compressing flow u.x; t / < 0 according to the Biot-Savart laws (3) and (4); this compressing flow produces a larger x .x; t / near the origin according to Eq. (2b), since .x; t / is transported by the velocity field u.x; t /; according to Eq. (2a), x .x; t / is the time derivative of w.x; t / along the characteristics, thus w.x; t / will in turn get larger and finally blow up in finite time. One can easily verify that the one-dimensional models (3) and (4) both enjoy the following scaling invariant property for  > 0,  > 0: w.x; t / !

1 x t w. ; /;   

u.x; t / !

 x t u. ; /;   

.x; t / !

 x t . ; /: 2  

(5)

The numerical simulations in [20, 30] suggest that the singular solutions to the HL- and the CKY-model both develop local self-similar structure around the singular point, which is similar to that observed for the 3D Euler equations (1) in [30]. We numerically investigate the stability of the self-similar profiles of the two 1D models in this work. It is proved recently in [20] that there exist a discrete family of analytic selfsimilar profiles for the CKY-model, corresponding to different leading orders of .x; t / at the origin, which is preserved by the flow. We will review this result briefly in Sect. 4. However, the method employed in [20] relies on the special structure of the Biot-Savart law (4) in the CKY-model and cannot be generalized to study the HL-model or the 3D Euler equations. In this work, we take another approach and study the evolution equations of the spatial profiles of the singular solutions through a dynamic rescaling formulation. By adding scaling terms to the model Eq. (2) according to the scaling invariant property (5) of the two models, we get the dynamic-rescaling equations, which are equivalent to the original dynamics (2) up to rescaling. After choosing certain normalization conditions, we get the evolution equations of the spatial profiles in the singular solutions of the two models. We numerically simulate the dynamic rescaling equations and observe that the solutions converge to their steady states for a range of initial data, which implies certain stability property of the self-similar singularity of the two models. Then we consider the linearization of the discretized dynamic rescaling equations at the steady state and compute the eigenvalues of the Jacobian matrix. This allows us to get detailed characterizations of the asymptotics of the spatial profiles in the singular solutions. Our interest in this lies in that this approach can be easily generalized to investigate the 3D axisymmetric Euler equations and the 2D Boussinesq equations. It is hoped that these calculations on the dynamics of the spatial profiles may lead to better understanding of the potential self-similar singularity for the 3D Euler equations. We comment that self-similarity plays an important role in the singularity formation of nonlinear partial differential equations, and considerable efforts have been

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

873

devoted to studying the self-similar profiles and scaling exponents in the singular solutions, since they characterize all the asymptotic behaviors of the solutions close to the time of singularity. For the stable self-similar singularity formation of the nonlinear Schrödinger equations, see [32,35,39]; for the semi-linear heat equations, see [13,14]; for a numerical study of the self-similar singularities in the free surface flow equations, see [3]; the self-similar profiles of the singular solutions to the aggregation equations are numerically investigated in [23]. The rest part is organized as follows: In Sect. 2, we give a brief review on the regularity results of the Euler equations. In Sect. 3, we derive the dynamic rescaling equations for the two 1D models (3) and (4). In Sect. 4, we first review the results about self-similar singularity of the CKY-model, which is obtained in [20], and then demonstrate the stability of these self-similar profiles using the dynamic rescaling equations. In Sect. 5, we determine the self-similar profiles and scaling exponents for the HL-model through direct numerical simulation and then study the dynamic rescaling equations to demonstrate the stability of the self-similar profiles. Concluding remarks are made in Sect. 6.

2

A Brief Review of the Search of the 3D Euler Singularity

The interest in the global regularity or finite-time blowup of the 3D Euler equations comes from several directions. Mathematically, the question has remained open for over 250 years and has a close connection to the Clay Millennium Prize Problem on the Navier-Stokes equations. Physically, the formation of a singularity in inviscid flows may signify the onset of turbulence in viscous flows, and it may provide a mechanism of energy transfer to small scales. Numerically, the resolution of nearly singular flows requires special numerical treatment, which presents a great challenge to computational fluid dynamics. Considerable efforts have been devoted to the study of the regularity properties of the 3D Euler equations. The main difficulty in the analysis lies in the presence of the nonlinear vortex stretching terms in the following vorticity-stream formulation: !t C .u  r/! D .!  r/u;

! D r  u;

(6)

and the lack of a regularization mechanism. Despite these difficulties, a few important partial results have been obtained, which have led to improved understanding of the regularity properties of the Euler equations. More specifically, the celebrated theorem of Beale, Kato, and Majda [2] and its variant [12] characterize the regularity of the solutions in terms of its maximum vorticity, asserting that a smooth solution RT u blows up at time T if and only if 0 k!.; t /kL1 dt D C1. The non-blowup criterion of Constantin, Fefferman, and Majda (CFM) [7] focuses on the geometric aspects of Euler flows instead and asserts that there can be no blowup at time T if the velocity field u.x; t / is uniformly bounded in an O.1/ region containing the maximum vorticity and R t the vorticity direction  D !=j!j is sufficiently “well behaved”; to be specific, 0 krk21 d is uniformly bounded for t < T .

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It has been shown in several works by the authors that the convection term .u  r/! in (6) may lead to dynamic depletion of vortex stretching. Motivated by the result of CFM [7], Deng, Hou, and Yu (DHY) [8, 9] obtained a sharper non-blowup condition by taking a Lagrangian point of view and putting the convection in the time derivative of !, d !.˛; t / D !t .x.˛; t /; t / C .u  r/!.x.˛; t /; t / D .!  r/u; dt and the resulting criterion uses only very localized information of the vortex lines. Assume that at each time t there exists some vortex line segment Lt on which the local maximum vorticity is comparable to the global maximum vorticity. Further we denote L.t / as the arclength of Lt , n the normal vector of Lt , and the curvature of Lt . If (1) maxLt .ju  j C ju  nj/  CU .T  t /A with A < 1, and (2) CL .T  t /B  L.t /  C0 = maxLt .j j; jr  j/ for 0  t < T , then the solutions of the 3D Euler equations u.x; t / remain regular up to t D T , provided that A C B < 1. In the critical case A C B D 1, if the scaling constants CU ; CL ; C0 satisfy some algebraic inequality f .CU ; CL ; C0 / > 0, then the solutions of the Euler equations remain regular up to t D T . Hou and Li in [18] considered the following 3D axisymmetric Navier-Stokes equations in the cylindrical coordinate: u1;t C ur u1;r C uz u1;z D 2u1 1;z C .@2r C .3=r/@r C @2z /u1 ; w1;t C ur w1;r C uz w1;z D .u21 /z C .@2r C .3=r/@r C @2z /w1 ; Œ@2r C .3=r/@r C @2z 1 D w1 :

(7a) (7b) (7c)

They proposed a 1D model on the axis r D 0, which is exact in the sense that one can obtain a solution to the Navier-Stokes equations based on a solution to the 1D model, Q u/z D .Qu/zz  2vQ uQ ; .Qu/t C 2.Q

(8a)

Q v/ .v/ Q t C 2. Q z D .v/ Q zz C .Qu/2  .v/ Q 2 C c.t /;

(8b)

where uQ D u1 and vQ D  1;z . They showed that part of the convection terms and the stretch terms cancel each other exactly, and .Qu2z C vQ z2 / in (8) satisfies Q u2z C vQ z2 /z D .Qu2z C vQ z2 /zz  2Œ.Quzz /2 C .vQ zz /2 : .Qu2z C vQ z2 /t C 2.Q

(9)

Global regularity of the 1D model (8) follows from the maximal principle, and this result reflects the potential stabilizing effects of the convection. In [16,28], a 3D model of the Navier-Stokes equations was proposed by dropping the convection terms in (7), and some partial regularity results were obtained:

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

875

@t u1 D .@2r C .3=r/@r C @2z /u1 C 2@z u1 ;

(10a)

@t w1 D .@2r C .3=r/@r C @2z /w1 C @z .u21 /;

(10b)

 .@2r C .3=r/@r C @2z /

(10c)

1

D w1 :

The finite-time singularity of this 3D model in the inviscid setting,  D 0, was proved in [17,22] with appropriate initial conditions and certain Neumann-Robin or Dirichlet-Robin boundary conditions. However, for the same initial conditions, no finite-time singularity was observed for the original 3D Euler equations (adding back the convection terms). These results further reveal the stabilizing effect of convection in 3D incompressible flows. Recall that the potential finite-time singularity obtained by Hou and Luo [30], see Fig. 1, takes place at the boundary .r; z/ D .1; 0/, which is a stagnation point of the flow. Thus the convection term vanishes there, and the possible stabilizing effect of convection is minimized. Besides, in the construction of the self-similar profiles for the CKY-model in [20], the fact that the singularity takes place at the stagnation point of the flow is crucial. Besides the analytical results mentioned above, there also exists a sizable literature focusing on the numerical search of a finite-time singularity for the 3D Euler equations. In [15, 37], Euler flows with swirls in the axisymmetric geometries were numerically studied, and finite-time singularity was reported. W. E and C.W. Shu [10] studied the potential finite-time singularities in the 2D Boussinesq equations with initial data completely analogous to those of [15, 37], and found no evidence of singular solutions. Kerr and his collaborators [4, 24, 25] did the famous computation of the Euler flows generated by a pair of perturbed antiparallel vortex tubes, and finite-time singularity was predicted. Hou and Li [19] repeated the computation in [24] using a pseudo-spectral method with higher resolution using the same initial condition except for the extra filtering which was not described in the [24]. They observed only double exponential growth in the maximum vorticity and did not see any evidence of the finite-time singularity reported in [24].

3

The Dynamic Rescaling Equations

The finite-time singularities of the two 1D models both take place at the origin, so we make the following self-similar ansatz to the local singular solutions:  x .x; t / D .T  t / ‚ ; .T  t /cl   x w.x; t / D .T  t /cw W ; .T  t /cl   x : u.x; t / D .T  t /cu U .T  t /cl c



(11a) (11b) (11c)

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Plugging these self-similar ansatz into Eqs. (2), (3), and (4), and matching the exponents of .T  t / for each equation, we get the following relation of the scaling exponents: cw D 1;

cu D cl  1;

c D cl  2:

(12)

And the profiles of the singular solution, U ./, W ./, and ‚./, satisfy the following equations, which are defined on  2 RC and will be referred as the selfsimilar equations: W ./ C cl W 0 ./ C U ./W 0 ./  ‚0 ./ D 0; 0

(13a)

0

.2  cl /‚./ C cl ‚ ./ C U ./‚ ./ D 0:

(13b)

The Biot-Savart laws of the HL-model (3) and the CKY-model (4) become UHL

1 D 

Z

1

0

Z

ˇ ˇ ˇ  ˇ ˇ W . /d ; ˇ ln ˇ  C ˇ

1

UCKY D  

(14a)

W . / d ;

(14b)

respectively. We consider smooth self-similar profiles U ./, W ./, and ‚./ in this work for the two models, which requires us to impose the following boundary conditions for the CKY-model: W .0/ D 0;

‚.0/ D ‚0 .0/ D 0;

(15)

and the following boundary conditions for the HL-model: W .2k/ .0/ D 0;

‚.0/ D 0;

‚.2kC1/ D 0;

k D 0; 1; 2 : : : :

(16)

In addition, the finite-time singularities of the two 1D models are point singularities, namely, w.x; t / and .x; t /, remain bounded away from the origin up to the singularity time. This requires us to impose the following decay conditions for the self-similar profiles at infinity: ‚./ D O. 12=cl /;

W ./ D O. 1=cl /;

U ./ D O. 11=cl /;

as

 ! 1: (17)

The self-similar Eq. (13) enjoy the following scaling invariant property: W ./ ! W

   ; 

‚./ ! ‚

   ; 

U ./ ! U

   : 

(18)

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

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A classification of self-similarity into the first and the second kind has been expounded in [1, 38]. In that classification, a self-similar solution is of the first kind if the scaling exponents are fixed by either dimensional analysis or symmetry. The exponents are typically rational numbers. If the scaling exponents are unknown, we call the self-similarity of the second kind. For a self-similar singularity of the second kind, typically, the self-similar profiles exist locally for a continuous set of scaling exponents. The determination of the scaling exponents involves solving a nonlinear eigenvalue problem by imposing appropriate far-field boundary conditions. Selfsimilar singularity of the second kind is in general difficult to track analytically. The self-similar solutions to the two 1D models that we consider in this work are of the second kind. In this work, we investigate the self-similar singularity of the models through the dynamic rescaling equations. To be specific, we add rescaling terms to the Eq. (2) and get wt .x; t / C u.x; t /@x w.x; t / C cl .t /x@x w.x; t / D x .x; t / C cw .t /w.x; t /; (19a) t .x; t / C u.x; t /@x .x; t / C cl .t /x@x .x; t / D c .t /.x; t /:

(19b)

The cl x@x terms correspond to stretching the solutions in the spatial direction, and the cw w and c  terms correspond to scaling the amplitude of the solutions. According to the scaling invariant property (5), we need to impose the following constraint on the scaling parameters: c .t / D cl .t / C 2cw .t /:

(19c)

Using the above relation, one can show that the dynamic rescaling system (19) is equivalent to the two 1D models (2). To be specific, assuming that .x; t /, w.x; t /, and u.x; t / are smooth solutions to the 1D models (2), then correspondingly, Q t / D exp .x;

Z

  Z t  Z t Z s   c .s/ds  exp cl .s/ds x; exp cw . /d ds ;

t

0

Z

0 t

w.x; Q t/ D exp Z

0 t

0

(20a)

0

  Z t  Z t Z s   cw .s/ds w exp cl .s/ds x; exp cw . /d ds ;

0

uQ .x; t/ D exp

0

(20b)

0

  Z t  Z t Z s   .cw .s/ C cl .s//ds u exp cl .s/ds x; exp cw . /d ds

0

0

0

0

(20c)

are a set of solutions rescaling equations (19).  Rto the dynamic t Let L.t / D exp 0 cl .s/ds . For the HL-model, the solutions (20) are defined on the domain ŒL.t /; L.t /, with Biot-Savart law scaled accordingly, 1 uQ .x/ D 

Z

L.t/ 0

  ˇ  ˇ ˇ tan x  tan y ˇ ˇ 2L.t/ 2L.t/ ˇ     ˇˇ dy: w.y/ Q log ˇˇ y x ˇ tan 2L.t/ ˇ C tan 2L.t/

(21)

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T.Y. Hou and P. Liu

For the CKY-model, the solutions (20) are defined on Œ0; L.t /, with Biot-Savart law, Z

L.t/

uQ .x/ D x x

w.y/ Q dy: y

(22)

As L.t / ! 1, the Biot-Savart laws (21) and (22) become uHL D

1 

uCKY D 

Z

1 0

1 x

Z

ˇ ˇ ˇx y ˇ ˇ dy; w.y/ log ˇˇ x Cyˇ

1 x

w.y/ dy: y

(23a) (23b)

We will use (23) to close the dynamics rescaling equations (19). We need to choose appropriate normalization conditions in (19) to fix the two rescaling parameters cl .t / and cw .t /. Our choice is to make the values of the leading order of w.x; t / and .x; t / at the origin remain constant. According to Eq. (2), for smooth velocity field u.x; t /, the leading order of .x; t / at x D 0 will not change since u.0; t/ D 0. The leading order of ‚./ actually determines the profiles of the singular solutions to (2), and it is showed in [20] that there exists a family of selfsimilar profiles, corresponding to different leading orders of the initial condition .x; 0/. In this work, we will focus on the case that the leading order of .x; 0/ is 2. And in this case, we choose cw .t / and cl .t / to make d wx .0; t/ D 0; dt

d xx .0; t/ D 0; dt

(24)

which according to (19), gives cl .t / D

2xx .0; t/ ; wx .0; t/

cw .t / D

xx .0; t/ C ux .0; t /: wx .0; t/

(25)

We remark that a similar formulation to (19) has been used to study the nonlinear Schrödinger equations in [26, 27, 29, 31, 34]. In those works, the dynamic rescaling formulation is used primarily as an approach to accurately solve the equations numerically. In this work, we employ this dynamic formulation to investigate the stability of spatial profiles in the singular solutions. The normalization conditions that we choose in (24) are also different from those employed in the study of nonlinear Schrödinger equations, which reflects the transport nature of our model equations. In [31], the normalization condition was chosen to be the conservation of some integral norms. And it was argued in [31] that normalization using local scaling factors may be numerically unstable. However, for our problem, at the steady state, the velocity field in (19), namely, cl x C u.x/, is positive near the origin, thus the information propagates away from the origin. Our normalization conditions at the origin (24) turn out to be stable according to our numerical results.

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

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We denote the steady state of the system (19) by w.x/; Q

Q .x/;

uQ .x/;

cQ l ;

cQ w ;

(26a)

then they satisfy the following equations: uQ .x/@x w.x/ Q C cQ l x@x w.x/ Q D Qx .x/ C cQ w w.x/; Q

(26b)

Q Q Q C cQ l x@x .x/ D .Qcl C 2Qcw /.x/: uQ .x/@x .x/

(26c)

Given a steady state of (19) and (26), one can correspondingly construct a set of solutions to the self-similar equations (13) using cl D Qcl =Qcw ;

W ./ D 

1 w./; Q cQ w

‚./ D

1 Q ./; cQ w2

U ./ D 

1 uQ ./: cQ w (27)

We remark that there exist some previous works that use the self-similar variables [13, 14] to investigate the asymptotic properties of self-similar singularities. They introduced D  ln.T  t /;

D

x ; .T  t /cl

(28a)

and made the following change of variables correspondingly: W .; / D .T  t /w.x; t /;

‚.; / D .T  t /2cl .x; t /;

U .; / D .T  t /cl 1 u.x; t /:

(28b)

With the above coordinates transformation (28), the model Eq. (2) become W C cl W .; / C U .; /W .; / C W .; /  ‚ .; / D 0;

(29a)

‚ C cl ‚ .; / C U .; /‚ W .; / C .2  cl /‚.; / D 0;

(29b)

the steady state of which corresponds to solutions to the self-similar Eqs. (13). The above system allows one to characterize the behavior of the solutions near the singularity point using linearization, and this approach has been taken to study, for example, the free surface flow (see [3] for a systematic study of the surface diffusion equations and see [13, 14] for the study of the semi-linear heat equations). However, this approach does not apply to the two 1D models that we are considering here, because the two models have self-similar solutions of the second kind, and the scaling exponent cl in (29) is unknown in advance. Determining the scaling exponent cl is actually one of the primary challenges in self-similar singularity of the second kind [11]. Another issue for the above formulation is that (29) is not stable because there exist unstable modes that correspond to temporal and spatial

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translations. Thus the formulation (29) cannot be used as a method to construct the self-similar profiles, which is our ultimate goal in investigating the stability of the self-similar profiles. We consider adding a perturbation ıw.x/ and ı.x/ to the steady state (26). We require that the perturbation agrees with the boundary conditions, symmetry, and leading order properties of the equations. To be specific, we require d2 ı.0/ D 0; dx 2

d ıw.0/ D 0: dx

(30)

The above conditions on the derivatives of the perturbations are in general difficult to impose, and for convenience, we introduce the following change of variables: w.x; t / D x w.x; O t /;

O t /; .x; t / D x 2 .x;

u.x; t / D x uO .x; t /:

(31)

With this change of variables, the Eq. (19) become w O t C .cw .t / C cl .t //w O C uO wO C .cl .t /x C uO .x; t /x/wO x D 2O C x Ox ; Ot C .2cw .t / C cl .t //O C 2OuO C .cl .t /x C uO .x; t /x/Ox D 0:

(32a) (32b)

Correspondingly, the Biot-Savart laws become Z

1

uO CKY .x/ D 

w.y/dy; O x

1 uO HL .x/ D 

Z

1

0

ˇ ˇ ˇx y ˇ x ˇ ˇ w.y/dy: log ˇ O y x Cyˇ

(32c)

The constraints (30) on the perturbations become O ı .0/ D 0;

ı w.0/ O D 0:

(32d)

O 0/ D w.0; Initially we choose .0; O 0/ D 1, then the normalization conditions (25) become cl .t / D 4;

cw .t / D uO .0; t/ C 2:

(32e)

We will use the dynamic rescaling system (32) in the next two sections to study the stable spatial profiles of the CKY and the HL-model separately.

4

The CKY-Model

In this section, we first review the results about the self-similar singularity of the CKY-model that are obtained in [20], and then demonstrate the stability of the self-similar profiles by studying the dynamic rescaling equations (32) using linearization.

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

4.1

881

Review of the Existence of Self-Similar Profiles for the CKY-Model

The self-similar singularity for the CKY-model was investigated in [20], where the existence of a discrete family of self-similar profiles was proved. The scaling exponents and self-similar profiles obtained there are consistent with that obtained from direct numerical simulation of the model. The main result is summarized in the following theorem (Theorem 1.1 in [20]). Theorem 1. There exist a discrete family of scaling exponents cl , such that the self-similar Eqs. (13) have analytic solutions U ./, W ./, and ‚./ with boundary conditions (15) and far-field conditions (17). This family of solutions corresponds to different leading orders of ‚./ at the origin, s  2, where dk s D minfk 2 N C j k ‚.0/ ¤ 0g: (33) d Moreover, W ./, U ./ 1 , and ‚./ 1 are analytic with respect to D  1=cl at

D 0. An important fact for the CKY-model is that the Biot-Savart law (4) in the selfsimilar equations can be written as a local relation with a decay constraint 

U ./ 

0

lim

!1

W ./ ; 

(34a)

U ./ D 0: 

(34b)

D

In [20], the authors first neglect the decay condition (34b), and then the self-similar Eq. (13) with (34a) become a nonlinear ODE system. This ODE system has a formal singularity at the origin  D 0 because the coefficients of the first order derivatives vanish. The power series method is employed to construct solutions to (13) ‚./ D

1 X kDs

‚k  k ;

W ./ D

1 X kD1

Wk  k ;

U ./ D

1 X

Uk  k :

(35)

kD1

And it is showed in [20] that for any cl > 2, and leading order of ‚./, s (33), the power series (35) can be uniquely determined (up to scaling), and they converge in a short interval near  D 0, namely, the self-similar equations have local solutions. After constructing the local solutions (35), the authors of [20] prove that the local self-similar profiles can be extended to C1 by solving the nonlinear ODE system (13) and (34a). The resulting self-similar profiles do not necessarily satisfy the decay condition (34b) for any cl > 2, and the decay condition actually determines the scaling exponent cl .

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For cl > 2, the authors of [20] define the following function of cl : U ./ : !C1 

G.cl / D lim

(36)

Then to prove the existence of self-similar profiles (13), one needs to find a root of G.cl /. By analyzing the nonlinear system (13), one can prove the following result [20]. Theorem 2. For cl > 2 and leading order of ‚./, s  2, construct the power series (35) with ‚s D 1 and extend the profiles to the whole RC by solving (13). Then G.cl / D lim

!1

U ./ < C1; 

(37)

and G.cl / is a continuous function of cl . With the above theorem, one can prove the existence of self-similar profiles using the intermediate value theorem. Namely, if one can find cll ; clr > 2 and show that G.cll / < 0;

G.clr / > 0;

(38)

then there exists some cl between cll and clr such that the self-similar profiles constructed using the power series (35) and extended to C1 satisfy the decay condition (34b). The authors in [20] prove (38) with the aid of numerical computation. They first obtain quantitative estimates of the decay of the coefficients in (35), from which one can estimate the truncation error in the power series (35). To control the roundoff error introduced in the floating point operation of a computer, they employ the interval arithmetic method [33], which uses computer-representable intervals instead of numbers to do arithmetic computation with appropriate rounding rules, thus allows one to rigorously control the roundoff error in the floating point operation. Then the forward Euler method is used to numerically integrate the nonlinear ODE system (13). With some short time a priori estimates and the interval arithmetic technique, one can solve the system to some fixed 0 and get rigorous estimates of the solutions at 0 . With these estimates the authors [20] further show that the limit lim!C1 U ./= is positive or negative and finally prove (38). See Lemma 4.1 in [20]. After constructing self-similar profiles satisfying (34b), one can make the change of variables D  1=cl and use some uniqueness argument to show that the selfsimilar profiles satisfy the desired condition (17) and are analytic with respect to . The resulting self-similar profiles have the following property: W ./ > 0;

U ./ < 0;

‚0 ./ > 0:

(39)

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

4.2

883

Stability of the Self-Similar Profiles

In this work, we further investigate the stability of the self-similar profiles of the CKY-model. We study the discretized dynamics of the rescaling equations (32) near their steady states using linearization. Recall that the dynamic rescaling equations (32) are defined on the whole real line RC . So to conduct numerical simulation, we first restrict the Eq. (32) to a fixed finite interval Œ0; M . Note that the velocity field (32c) uO depends on the w O on the whole RC , so we need to truncate w O to the fixed interval Œ0; M  to close the system. For this purpose, we introduce the following linear projection operator P : P ŒF .x/ D F .x/ 

x2 F .M /; M2

x 2 Œ0; M :

(40)

With the truncation operator P , the dynamic rescaling equations (32) become O  uO wO  .cl x C uO x/w O x ; w O t D P Œ2O C x Ox  .cl C cw /w Ot D .2cw  cl /O  2.cl C u/O  .cl x C uO x/Ox ; Z M w.y; O t /dy: uO .x; t / D 

(41a) (41b) (41c)

x

The projection operator P in (41a) guarantees that w.x; t / remains continuous and vanishes outside the domain Œ0; M . Based on the far-field decay conditions of these self-similar profiles (17), one can easily see that Z lim

1

M !1 M

W ./ d ! 0; 

(42)

which justifies the truncation and projection introduced in (41). The dynamic rescaling equations (41) are essentially an iteration scheme, and we need appropriate initial conditions so that the solutions converge to their steady states. For this purpose, we solve the steady-state equations (26) using a shooting method and use the numerical solutions as the initial conditions for (41). To be specific, we solve the Eq. (26) numerically as an ODE from x D 0 to M with initial O conditions w.0/ O D .0/ D 1 and select uO .0/ using a bisection method to make uO .M / D 0. In our computation, we choose M D 1:6  109

(43)

in the truncation (40). To discretize the spatial domain Œ0; M , we first divide it into the inner region Œ0; 16 and outer region Œ16; M . In the inner region, we use a uniform mesh of size h D 16=N1 and denote the nodal points as xi ,

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T.Y. Hou and P. Liu

xi D ih;

i D 0; 1;    ; N1 :

(44)

In the outer region, according to Theorem 1, the profiles have slow variation and are smooth with the transformed variable D  1=cl, so we will use a coarser mesh to reduce the computational cost. We consider a transformed variable  D x 1=4 , and for x 2 Œ16; 1:6  109 , we have  2 Œ5  103 ; 1=2. We put a uniform mesh of size hQ D 2N1 2 on the -space and get i D 1=2  .i  N1 /h;

i D N1 C 1; N1 C 2;    ;

99 N2 C N1 D N: 100

(45)

Then we map i back to the x-space and get the nodal points in the outer region, xi D .i /4 ;

N1 C 1;    ; N D N1 C

99 N2 : 100

(46)

In the discretization of the spatial derivatives in (41), we use the upwind scheme, namely, O i /  .x O i1 / .x Ox .xi /  ; xi  xi1

w O x .xi / 

w.x O i /  w.x O i 1 / : xi  xi 1

(47)

In the temporal direction, we use the forward Euler scheme and choose time step dt based on the CFL condition, dt D min i

xi  xi1 : uO .xi /xi C cl xi

(48)

With the above discretization and the choice of initial conditions, we observe that the solutions to (41) converge to their steady states. For the choice that N1 D 2000 and N2 D 4000 in (44) and (45), we obtain that at the steady state, cQ l D 4;

cQ w  1:0586:

(49)

According to (27), we get the corresponding self-similar profiles with cl D Qcl =Qcw  3:7785:

(50)

This scaling exponent cl (50) agrees with the one obtained in [20], which is 3:7967. Next we locate the maxima of the steady-state profile w./ Q (26), which is wQ max D w. Q 0 /, and rescale the maxima to .1; 1/. We then get the rescaled self-similar profile W ./ W ./ D

1 w Q max

w. Q 0 /:

(51)

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

a

b

Self-similar Prof iles of W

1

0.9

Exact Dynamic Rescaling

0.78

0.7

0.76

0.6

0.74

0.5

Zoom In

0.72

0.4

0.7

0.3

0.68

0.2

0.66

0.1 0

Self-similar Profiles of W

0.8

Exact Dynamic Rescaling

0.8

885

0.64 0

0.2

0.4

0.6

0.8

1

0.1

0.12

0.14

0.16

0.18

0.2

Fig. 2 The self-similar profiles obtained from (32) and [20]. (a) Self-similar profiles of W . (b) Profiles of W (zoom in)

The rescaled self-similar profile (51) is plotted in Fig. 2 together with the one obtained in [20]. We can see that there is very small discrepancy between the two profiles. The small difference in cl (50) and the resulting self-similar profile of W ./ (51) implies that the errors introduced by the truncation (40) and spatial discretization (47) are small. Next, we consider the stability of the steady states (26) of the system (32). We denote w O D .w O 0 ; wO 1 ;    wO N /;

uO D .Ou0 ; uO 1 ;    uN /;

O D .O0 ; O1 ;    ON /;

(52)

O i /. The equations (41) after the spatial where wO i D w.x O i /, uO i D uO .xi /, Oi D .x discretization (47) become an ODE system of w O and O , d .w; O O /T D F .w; O O / D .PF w .w; O O /; F  .w; O O //T ; dt

(53)

where P corresponds to the projection operator (40) and F w , F  correspond to the right-hand side of (41a) and (41b). F w and F  are given by

O i  uO i w O i  .cl xi C uO i xi / Fiw D .cw  cl /w

Oi  Oi 1 w O i  wO i 1 C 2Oi C xi ; xi  xi 1 xi  xi 1 (54)

Oi  Oi 1 Fi D .2cw  cl /Oi  2Oui Oi  .cl xi C uO i xi / : xi  xi 1 The velocity uO i and scaling parameter cw are functions of w O i,

(55)

886

T.Y. Hou and P. Liu

uO i D 

N 1 X j Di

cw D 2 C

w Oj C w O j C1 .xj C1  xj /; 2

N 1 X iD0

w Oi C w O iC1 .xi C1  xi /: 2

(56)

(57)

Denote the steady state as w O  ; O  , then we compute the Jacobian of F (53) with O at the steady state, namely, respect to .w; O /    O rw; : F w O ;  O O 

(58)

The entries of the Jacobian (58) can be computed from   xj C1  xj cl xi C uO i xi @Fiw ıij C w D cw  cl C uO i  Oi @w Oj xi  xi1 2   w Oi  w O i1 @Oui ; C w O i  xi xi  xi1 @w Oj   xi xi @Fiw ıij  D 2C ıj;i1 ; O x  x x  xi1 i i1 i @j ! O ai1 @Oui Oi  t het @Fi @c  w D 2Oi C 2Oi  xi ; @w Oj @wO j xi  xi1 @wO j   cl xi C uO i xi cl xi C uO i xi @Fi ıi;j C D 2cw  cl  2Oui  ıi 1;j ; O x  x xi  xi 1 i i1 @j

(59a) (59b)

(59c)

(59d)

where xj C1  xj 1 @cw D ; @wO j 2

(60a)

xj C1  xj xj  xj 1 @Oui D 1j i C 1j iC1 : @wO j 2 2

(60b)

F .w O  ; O  /, we compute its eigenvalues. With the above explicit formula of rw; O O For N1 D 1000, N2 D 2000, the first several eigenvalues of (58) are plotted in Fig. 3a. We can see that the maximal real part of the eigenvalues of rw; F .w ;   / is negative, Re.1 /  1:0745:

(61)

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

10

Distribution of Eigenvalues

b

8

6

6

4

4

2

2

0 −2

Distribution of Eigenvalues

10

8

Im(λ)

Im(λ)

a

0 −2

−4

−4

−6

−6

−8

−8

−10 −4

−10 −4

−3.5

−3

−2.5

−2

−1.5

−1

887

Re(λ)

−3.5

−3

−2.5

−2

−1.5

−1

Re(λ)

Fig. 3 The distribution of the first several eigenvalues of Jacobian matrix (58). (a) Coarse mesh. (b) Fine mesh

Then the classical dynamic system theory [36] implies that the discretized dynamics (53) of rescaling equations (41) are asymptotically stable near its steady state. To demonstrate the accuracy of our computation, we use a finer spatial mesh with N1 D 2000, N2 D 4000 in (44) and (45). We compute the eigenvalues of the Jacobian (58) for this refined system at its steady state, and the first several of them are plotted in Fig. 3b. We can see that after refinement, the first several eigenvalues of the Jacobian remain the same. This implies that the negativity of the eigenvalues of (58) is not a numerical discretization artifact, and the dynamic rescaling equations (19) has a stable steady state. Correspondingly, we obtain that the self-similar profiles (27) are asymptotically stable. To further demonstrate the stability of the resulting self-similar profiles, we introduce a reduced dynamics. Note that in our simulation of the system (41), at each time step, we compute the velocity field based on the vorticity and then update O and w O simultaneously using the velocity field for a short time step dt . To accelerate the convergence, we use the following strategy: at each time step, after we get the velocity uO , we update O according to (41b) until it converges to its steady state, which is equivalent to solving an ODE  .2cw  cl /O  2.cl C uO /O  .cl x C uO x/Ox D 0:

(62)

After we get the steady state O , we put it in (41a) and update w O for a small time step dt . The resulting dynamics is a reduced version of (41), which only evolves w O and always puts O in its steady state. We simulate the reduced dynamics and observe that the solutions converge to the steady state of (41). Again, at the steady state, we compute the Jacobian matrix of the reduced dynamics. We denote the Jacobian (58) as ! @F w @F w @w O @O @F  @F  @w O @O

:

(63)

888

b

Distribution of Eigenvalues

15 10

10

5

5

0

0

−5

−5

−10

−10

−15 −4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

Distribution of Eigenvalues

15

Im(λ)

Im(λ)

a

T.Y. Hou and P. Liu

−15 −4.5

−4

−3.5

Re(λ)

−3

−2.5

−2

−1.5

−1

Re(λ)

Fig. 4 The distribution of the first several eigenvalues of the Jacobian of the reduced dynamics at its steady state. (a) Coarse mesh. (b) Fine mesh

At the steady state of the reduced dynamics, one can show that the corresponding Jacobian matrix becomes @F w @F w  @w O @O



@F  @O

1

@F  : @wO

(64)

The proof of (64) is a direct calculation, and we omit the details here. The distribution of the first several eigenvalues of (64) on the coarse and fine meshes are plotted in Fig. 4. We can see that the real part of the eigenvalues of (64) are negative, and this implies that the reduced dynamics is also stable. Before we end this section, we want to comment that the stability that we demonstrate here does not mean that after a small perturbation in the initial conditions of the system (2), the perturbed solutions will finally converge to the original solutions (11). Actually, a small perturbation will result in a different singularity time T . However, our numerical results suggest that the normalized profiles in the singular solutions will converge, namely, the spatial profiles in the singular solutions are stable.

5

The HL-Model

In this section, we consider the self-similar singularity of the HL-model (3). Because of the nonlocal nature of the Biot-Savart law (3), the techniques employed in [20] to prove the self-similar singularity will not apply. For the HL-model, we first simulate the model (2) numerically and obtain the scaling exponents and self-similar profiles based on the singular numerical solutions. Then we solve the dynamic rescaling equations (32) and compare the profiles and exponents obtained from the steady state (27) of (19) with those obtained from direct numerical simulation of the

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

889

HL-model. Finally we demonstrate the stability of these self-similar profiles by analyzing (19) using linearization.

5.1

Numerical Simulation of the HL-Model

The singularity of the HL-model takes place near the origin, and to efficiently resolve the singularity of the solution, a very fine mesh is required. To save the computational cost, we employ a moving mesh method introduced in [30], which adaptively puts a certain portion of the grid points in the singularity region. We first construct an analytic mesh mapping function x.s/ with derivative given by .1s/2 ˇ xs .s/ D a C p e  ˛2 : ˛ 

(65)

x.s/ maps from s 2 Œ0; 1 to x 2 Œ0; 1, and a, ˛, ˇ are parameters that will be chosen dynamically. Note that xs .s/ is even at s D 1, thus this map (65) preserves the symmetry of the solutions at x D 1. And for ˛ small, xs .s/  a for s near 0 due to the fast decay of the exponential function, thus xs can also be viewed as even at s D 0, and correspondingly this map x.s/ also preserves the symmetry of the solutions at s D 0. We use a uniform mesh in the s-space with nodal points, sj D j h;

j D 0; 2;    N;

hD

1 ; N

(66)

which correspond to xj D x.sj /; j D 0;    N in the physical space. Preliminary simulation suggests the profile of w in the singular solutions to the HL-model (3), (2) is similar to that of the CKY-model and takes the shape of a bump. So we define the singular region S.t / as S.t / D fxj0 < x < x.2M .t//g;

(67)

where M .t/ is position at which the maximum of w is attained in the s-space. In our simulation of the HL-model (2) and (3), we choose the following initial condition: w.x; 0/ D sin.x/;

.x; 0/ D 1000  1000 cos.x/:

(68)

We initially choose a D 1, ˇ D 0, ˛ D 13 in (65) and will adjust the parameters to make sure that Œı1 ; ı2  portion of the nodal points (66) are placed in the singular region (67). To be specific, we fix ˛ D 1=3 and choose ı1 D 1=5, ı2 D 1=2. With this choice we can see that the exponential part of xs .s/ is small for s < 1=2 due to fast decay of exponential function, which means us .s/  a for s  1=2. At time t0 , if the portion of nodal points in the singular region drops below ı1 , we modify the parameters, and choose a based on

890

T.Y. Hou and P. Liu

aı2 D x.2M .t0 //;

(69)

which guarantees that x approximately maps Œ0; ı2  to S.t0 /. Besides, we need Z

1

x.1/ D a C 0

.1x.t //2 ˇ p e  ˛2 dt D 1; ˛ 

(70)

22a . and consequently ˇ can be determined as ˇ D erf.1=˛/ Every time we adjust the parameters, we reconstruct w.s; t / and .s; t / using interpolation in the s-space. Our numerical results show that the map (65) with appropriate parameters can resolve the singularity, and the solutions remain smooth with respect to s. In the s-space, based on (3), we have

Z i h 1 1 ux .x.s// D P:V: .x.s/  x.t // xs .t /dt; s 2 Œ0; 1; w.x.t // cot (71) 2 1 2   Z    1  1 1 .x.s/x.t //  cot .s  t / dt; w.x.t //xs .t / cot D 2 1 2 xs .s/ 2 (72) Z 1   1 C P:V: xs .s/ .s  t / dt: (73) w.x.t //xs .t / cot 2 2 1 We denote (72) as I1 and denote (73) as I2 . I2 is the Hilbert transform of w.x.s//xs .s/, and since we have a uniform mesh on the s-space, we can compute I2 efficiently using FFT. For the first part I1 , the integrand is continuous, and we have      xss .s/  1 cot .s  t / Dw.x.s// : lim w.x.t //xs .t / cot .x.s/x.t //  t!s 2 xs .s/ 2  (74) We use the trapezoidal rule to compute the integral in the first part I1 , which is known to have spectral accuracy for smooth periodic functions. In the s-domain, the Eq. (2) become t C u

s D 0; xs .s/

(75a)

wt C u

s ws D : xs .s/ xs .s/

(75b)

We use a 6-order finite difference scheme in the spatial direction to discretize s , ws , fs .xi / 

1 3 3 3 f .xi3 / C f .xi2 /  f .xi 1 / C f .xi C1 / 60h 20h 4h 4h

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

a

1

b

w ∞ = 103 w ∞ = 104 w ∞ = 105

0.9 0.8

w ∞ = 103 w ∞ = 104 w ∞ = 105

0.5 0.45

0.7 0.6

W(ξ)

W(ξ)

891

0.5 0.4

Zoom In

0.4 0.35

0.3 0.2 0.1 0

0.3 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

ξ

ξ

Fig. 5 The rescaled solutions W at different time steps close to singularity. (a) Solutions of w.x; t/. (b) Zoom in



3 1 f .xiC2 / C f .xiC3 /: 20h 60h

(76)

In the temporal direction, we use the fourth-order explicit Runge-Kutta method. The time step dt is chosen such that adt D 0:9h;

a D max j

u j: xs .s/

(77)

The simulation will be terminated once the minimal mesh size in the physical space gets below 1015 . We choose N D 2000 in our simulation, and the rescaled solution w.x; t / at different time steps corresponding to different kwk1 are plotted in Fig. 5. From the figure we can see that the rescaled solutions are very close at different time steps, which implies the solutions develop self-similar singularity. In our simulation, we track the maximum vorticity wmax .t / and the position at which the maximum is attained, l.t /. They are obtained in each time step using a 1 sixth-order polynomial interpolation. We also track dkwk at each time step, which dt according to (2), is equal to dkwk1 D x .l.t /; t /: dt

(78)

Based on (11), we have wmax .t /  C .T  t /cw ;

l.t /  C .T  t /cl :

(79)

Using the above ansatz, we first obtain the singular time T and cw using linear regression, 

1

d log.kw.x; t /k1 / dt



t T  ; cw cw

(80)

892

b

x 10−3 2

x 10−5 14 13

1.6

12

1.4

11

−1

1.8

d log w ∞ dt

1.2 1 0.8 0.6 Line fitting

8

6

0.2

where

d dt

t

d dt

9

8

7

6

0 74 3.

73 3.

73 3.

73 3.

5

73 3.

73

4

Fig. 6 The line fitting to determine cw and the singularity time T . (a) (b) The regression region

3.

2

1

3

73 3.

73 3.

73 3.

73

0

× 10−2

t

3.

73

75

73

72

71

70

69

68

67

66

65

64

63

62

61

60

59

58

57

74

3.

3.

3.

3.

3.

3.

3.

3.

3.

3.

3.

3.

3.

3.

3.

3.

3.

3.

3.

56

5

3.

55

9

7

0.4

3.

10

3.

d log w ∞ dt

−1

a

T.Y. Hou and P. Liu

× 10−2

log.kw.x; t/k1 and t.

log.kw.x; t /k1 / can be computed from d d 1 log.kw.x; t /k1 D kwk1 : dt kw.x; t /k1 dt

(81)

d dt

log.kw.x; t /k1 is plotted in Fig. 6a, and the line fitting region is plotted in Fig. 6b. The linear regression (80) in the fitting zone t 2 Œ0:0373; 0:0374 gives us cw  0:9378;

T  3:7444  102 :

(82)

We can see that the cw we obtain is very close to 1, as predicted by (12). With the singularity time T obtained from (80) and using (79), we can obtain cl from the following linear regression, where log.T t / and log.l.t // are explanatory and dependent variables, respectively, log.l.t //  log C C cl log.T  t /:

(83)

We plot log.l.t // over the line fitting zone t 2 Œ0:0373; 0:0374 in Fig. 7. From the figure we can see that the curve is very close to a straight line, which implies the validity the scaling law (79). And the linear regression gives cl  2:9800:

(84)

At time t D 0:0374 which is endpoint of the line fitting zone that we choose in (80) and (83), the solution w.s; t / is plotted in Fig. 8. From the figure, we can see that the solution remains smooth with respect to the variable s until close to the singularity time T , and this implies that the moving mesh method that we adopt from [30] can resolve the singularity of the solutions of the HL-model very well.

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

a

b −15.5

0

−16 −16.5 Line Fitting

logl(t)

logl(t)

−5 −10

893

−15

−17 −17.5 −18

−20

−18.5 −25 −12 −11 −10

−9

−8 −7 log(T − t)

−6

−5

−4

−10

−3

−9.8

−9.6

−9.4 log(T − t)

−9.2

−9

Fig. 7 The line fitting to determine cl . (a) log.l.t // and log.T  t /. (b) The regression region 16000 14000 12000

w(s)

10000 8000 6000 4000 2000 0 −2000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s

Fig. 8 The configuration of w with respect to s at t D 0:0374

5.2

Stability of the Self-Similar Profiles

In this subsection, we investigate the stability of the profiles in the HL-model using the dynamic scaling equations (32). In the spatial discretization, we first restrict the computational domain of (32) to Œ0; M  with M D 105 . Since the velocity field uO .x; t / (32c) depends on the w.x; O t/ on the whole RC , we need to truncate the solution w.x; O t / using the projection operator (40). To be specific, the Biot-Savart law (32c) becomes 1 uO .x/ D 

Z

M 0

ˇ ˇ ˇx  y ˇ y ˇ ˇ w.y/dy; log ˇ O x x yˇ

x 2 Œ0; M ;

(85)

and the equation for w.x; O t / (32a) becomes O  uO w O  .cl x C uO x/w O x : w O t D P Œ2O C x Ox  .cl C cw /w

(86)

894

T.Y. Hou and P. Liu

The truncation operator P preserves the symmetries of the HL-model at the origin. In the discretization of the spatial domain, we put N1 C 1 nodal points on the interval Œ0; 16, which is the inner region xi D ih;

i D 0; 1;    N1 ;

1 : 16N1

hD

(87)

And in the outer region, we introduce a map  D log x and denote 

1 i D log 16

 C .i  N1 /h;

i DN1 C 1;    N1 C N2 ;

log.M /  log hD N2

1 16

:

(88) Then we get the outer region nodal points xi D exp.i /;

i D N1 C 1;    N1 C N2 :

(89)

We denote the mesh size as hi D xiC1  xi :

(90)

We use the same notations (52) for the discretized dynamics as in the CKYmodel. In computing the spatial derivatives in Eq. (41), we use the upwind scheme, w O x .xi / 

O i1 / w.x O i /  w.x ; hi

O i 1 / O i /  .x .x Ox .xi /  : hi

(91)

In computing the velocity field using (3), we need to be careful since it involves singular integral operators. Our strategy is the following: we first interpolate w.x/ O using a piecewise linear function, and on each interval Œxi ; xi C1  we have w.x/ O Dw O i1 C

w Oi  w O i1 .x  xi1 /; hi

x 2 Œxi 1 ; xi :

(92)

With the above interpolation of w.x; O t /, we compute the velocity field according to (32c) explicitly. In another word, we only discretize the vorticity field, and do not discretize the singular kernel in the Biot-Savart law (3). To be specific, we denote 1 Si;j

2 Si;j

 ˇxj ˇ y3 y2 xi2 xi3  xj 1  C xj 1 log jxi  yj ˇˇ ; 3 2 3 2 yDxj 1  3  ˇxj ˇ y 1 x3 y2 x2 D ;  xj 1 C i C i xj 1 log jxi C yj ˇˇ 2hi xi 3 2 3 2 yDxj 1 1 D 2hi xi



(93)

(94)

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

3 Si;j

1 D 2hiC1 xi



y 3 xj C1 y 2 x3 x2  C i xj C1  i 3 x 2 3



895

ˇyDxj C1 ˇ log jxi  yj ˇˇ ; yDxj

(95) 4 Si;j

D

"

1 2hiC1 xi

2 xj C1 xi2 y 3 xj C1 2 xi3  y C C 3 2 3 2

!

#ˇyDxj C1 ˇ ˇ log jxi C yj ˇ : ˇ yDxj

(96) And by a direct calculation, we get uO .xi / D

X

1 Si;j

C

2 Si;j

C

3 Si;j

C

4 Si;j

j

xj C1 hi C1 xi xj 1 hi xi    C 6 6 6 6

 wO j : (97)

In the temporal direction, we use the forward Euler scheme, which is the same as that in the CKY-model and choose the time step dt according to the CFL condition (48). To simulate the dynamic rescaling equations (32), we need an appropriate initial condition. We cannot solve the steady-state equations (26) using the shooting method as we did for the CKY-model because of the nonlocal nature of the Biot-Savart law (32c). Considering the close connection of the two models, we use the self-similar profiles of the CKY-model as the initial condition to simulate dynamic rescaling equation of the HL-model. According to our numerical results, the solutions to (32) converge to a steady state. For N1 D 2000, N2 D 4000, we have the following scaling parameters at the steady state: cQ l D 4;

cQ w D 1:3516:

(98)

Then according to (27), we get a set of self-similar profiles with cl D 

cl D 2:9596: cw

(99)

This cl agrees with the one we obtained from the direct simulation of the model (84). The rescaled steady-state solution W (27) is plotted in Fig. 9, together with the self-similar profiles that we obtain from the direct simulation in the previous subsection. From the figure we can see that the error in the self-similar profile of w is small, which implies that the errors introduced in the truncation (85), (86) and discretization (47) are small. At the steady state, we compute the Jacobian (58) of the discretized dynamic rescaling system and compute its eigenvalues. The computation is exactly the same as that for the CKY-model (59) except in the Biot-savart law (60). For the HL-model, the derivative of uO i with respect w O j is given explicitly in (97).

896

T.Y. Hou and P. Liu

a

b

Self-similar Profiles of W

1

Self-similar Profiles of W Direct Simulation Dynamic Rescaling

0.5

0.9 0.8

0.45

0.7 0.6

0.4

0.5 0.4

Zoom In 0.35

0.3 0.2

Direct Simulation Dynamic Rescaling

0.1 0

0

0.2

0.4

0.6

0.8

0.3 0.1

1

0.12

0.14

0.16

0.18

0.2

Fig. 9 The self-similar profiles obtained from (32) and [20]. (a) Self-similar profiles of W . (b) Profiles of W (Zoom in)

a

b

Distribution of Eigenvalues

5 4

Distribution of Eigenvalues

4

3 2

2 Im(λ)

1 Im(λ)

6

0 −1

0 −2

−2 −3

−4

−4 −5 −3

−2.5

−2

−1.5 Re(λ)

−1

−0.5

−6 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 Re(λ)

Fig. 10 The distribution of the first several eigenvalues of the Jacobian. (a) Coarse mesh. (b) Fine mesh

For N1 D 1000, N2 D 2000, the distribution of eigenvalues of (58) is plotted in Fig. 10a. From the figure we can see that the real parts of the eigenvalues of the Jacobian (58) are negative, and this implies that the self-similar profiles of the HL-model are stable. Then we refine the mesh using N1 D 2000, N2 D 4000 and compute the corresponding Jacobian matrix at the steady state. The eigenvalues of the Jacobian are plotted in Fig. 10b. Again we find that the first several eigenvalues do not change much after the mesh refinement. Next, we consider the reduced dynamics of the HL-model, as we have done for the CKY-model in (64). The eigenvalues of the Jacobian for the reduced dynamics at its steady state on the coarse and fine mesh are plotted in Fig. 11. We can see that the real parts of the eigenvalues are negative, which implies the reduced dynamics is also stable. Besides, we can see that the gap between the eigenvalues of the Jacobian and 0 are larger for the reduced dynamics, which implies that the reduced dynamics has better stability.

17 Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric. . .

b

Distribution of Eigenvalues

8

6

4

4

2

2

0

0

−2

−2

−4

−4

−6

−6

−8 −5

−4.5

−4

−3.5

−3

−2.5

Re(λ)

−2

−1.5

Distribution of Eigenvalues

8

6

Im(λ)

Im(λ)

a

897

−8 −4

−3.5

−3

−2.5 Re(λ)

−2

−1.5

Fig. 11 The distribution of the first several eigenvalues of the Jacobian matrix at the steady state for the reduced dynamics. (a) Coarse mesh. (b) Fine mesh

6

Conclusion

In this work, the spatial profiles of the singular solutions for two 1D models of the 3D axisymmetric Euler equations were investigated. We derived the evolution equations of the normalized profiles of the solutions using dynamic rescaling. The stability of these profiles was demonstrated by analyzing their discretized dynamics using linearization. We can take the same approach to study the stability of the profiles in the singular solutions to the 3D axisymmetric Euler equations based on the scenario observed in [30]. The results for the 3D Euler equations will be presented in another work. It is hoped that these detailed calculations will help to analyze the potential singularity formation of the 3D Euler equations.

7

Cross-References

 Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value

Problem  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes

Flow  Self-Similar Solutions to the Nonstationary Navier-Stokes Equations

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Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

18

Hugo Beirão da Veiga, Yoshikazu Giga, and Zoran Gruji´c

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Vorticity Direction and Regularity: Super-Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Whole Space Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Domains with Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Vorticity Direction and Regularity: Breaking the Criticality . . . . . . . . . . . . . . . . . . . . . . . 3.1 Vortex Stretching and Locally Anisotropic Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Type I Blowup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

902 904 905 908 913 915 915 921 927 928 928

Abstract

It has been well documented – both in computational simulations of the 3D Navier-Stokes equations and the experiments with 3D incompressible turbulent fluid flows – that the regions of intense vorticity self-organize in coherent vortex structures, most notably, vortex filaments. One of the morphological characteristics of these structures is local coherence of the vorticity direction. The

H. Beirão da Veiga () Dipartimento di Matematica, Università di Pisa, Pisa, Italy e-mail: [email protected] Y. Giga Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan e-mail: [email protected] Z. Gruji´c Department of Mathematics, University of Virginia, Charlottesville, VA, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_18

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goal of this chapter is to review several avenues taken by the mathematical fluids community in order to study how the local coherence of the vorticity direction – a purely geometric condition – influences the possible formation of singularities in solutions to the 3D Navier-Stokes equations.

1

Introduction

Vortex stretching has been widely accepted as the principal physical mechanism behind the creation of small scales in turbulent (incompressible) fluid flows. This was already noted in G. I. Taylor’s fundamental paper “Production and dissipation of vorticity in a turbulent fluid” from 1937 [86]. The exact mechanics behind the process of locally anisotropic dissipation is not transparent. Taylor based his remarks on anisotropic dissipation mainly from the measurements of the wind tunnel turbulence past a uniform grid, concluding that “It seems that the stretching of vortex filaments must be regarded as the principal mechanical cause of the high rate of dissipation which is associated with turbulent motion.” Providing a rigorous argument in favor of this observation – directly from the mathematical model, the three-dimensional (3D) Navier-Stokes equations (NSE) – has been one of the central topics in the mathematical fluid mechanics ever since. In addition, computational simulations of 3D turbulent flows indicate that the regions of intense vorticity self-organize in coherent vortex structures, e.g., vortex sheets and/or vortex tubes (filaments). As the complexity of the flow increases, the vortex filaments appear to be the dominant structure [3, 61, 81, 82, 88]. On the analysis side, a thorough investigation of creation and dynamics of vortex tubes was presented in [30], including a dynamic estimate of a suitably defined scale of coherence of the vorticity direction field. More recent analytic and numerical studies on coherent structures include [38, 40, 59, 71]. Two local morphological signatures of the geometry exhibited by the regions of intense vorticity are coherence of the vorticity direction and existence of a sparse/thin direction. The precise mathematical characterizations of the above two concepts needed to formulate and prove rigorous, mathematical statements will be given in the subsequent sections. In particular, the degree of coherence of the vorticity direction will mostly be quantified by a suitable continuity condition (e.g., a Hölder continuity or the uniform continuity). At this point, for the sake of transparency, one should note that highly turbulent flows also feature a phenomenon of folding of vortex filaments producing sharp changes in the vorticity direction, which might lead to forming a vorticity direction discontinuity. However, the good news is that there are rigorous results in which the only assumption on the vorticity direction is a stipulation that it belongs to a certain, local, weighted space of functions of bounded mean oscillations containing geometrically spectacular singularities, i.e., discontinuities at which every point on the unit sphere can be a limit point (cf. Theorem 10).

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The foundational work in the study of the influence of the local coherence of the vorticity direction on dynamics of solutions to the 3D NSE was presented by P. Constantin in [25]. The starting point was a singular integral representation of the stretching factor in the evolution of the vorticity magnitude in which the kernel was depleted by the local coherence of the vorticity direction. Since the cancellations originate in the geometric structure of the vortex stretching term – in a sense the highest order nonlinear term in the vorticity formulation of the 3D NSE – the phenomenon has been referred to as geometric depletion of the nonlinearity. This inspired a sequence of works establishing various coherence of the vorticity direction-type regularity criteria for solutions to the 3D NSE, starting with a theorem [28] stating that as long as the vorticity direction is Lipschitz coherent – in the regions of intense vorticity – no finite time blowup can occur. The Lipschitz condition was later replaced by 12 -Hölder [12], and a spatiotemporal localization of the 12 -Hölder condition was obtained in [53] (for a different approach to localization, see [22]). A related work [56] features a family of local, hybrid geometric-analytic regularity criteria in which the vorticity magnitude is dynamically balanced against the coherence of the vorticity direction. Regarding the domains with boundary, the case of the slip boundary conditions was presented in [8, 11, 13] and the case of the no-slip boundary conditions in [9]. The primary geometric obstruction to the manifestation of the anisotropic dissipation in this setting is “crossing of vortex lines,” i.e., the vorticity direction exhibiting a spatial discontinuity at a (possible) singular time (cf. [58]). It is worth noting that, on one hand, all the aforementioned articles feature regularity classes that are either subcritical or critical with respect to the natural scaling of the 3D NSE. On the other hand, all the known a priori bounded quantities are supercritical, and, moreover, there is a “scaling gap” between any regularity criterion and the corresponding a priori bound. This is also true for all – more classical – analytic regularity classes (e.g., Ladyzhenskaya-Prodi-Serrin and BealeKato-Majda type) and is the manifestation of the supercriticality of the 3D NSE regularity problem. Another avenue to gaining insight into the possible singularity formation is to assume a criticality scenario and then identify the conditions breaking the criticality (leading to regularity of the solutions in view). At this point, there are two groups of results along this avenue in which coherence of the vorticity direction comes to the fore. One assumes the criticality in the form of the so-called type I blowup (at most scaling-invariant blowup) [45, 51], and the condition breaking the criticality is the continuous alignment condition, essentially, existence of the uniform modulus of continuity of the vorticity direction. The other one assumes a numerically and mathematical analysis-motivated geometric criticality scenario in which the transversal scale of a filament matches the small scale needed for the locally anisotropic diffusion to engage [17, 18, 54], and the condition breaking the criticality is boundedness of the vorticity direction in a logarithmically weighted local space of functions of bounded mean oscillations.

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It is also possible to utilize – working directly with the 3D NSE – a suitably defined “local existence of a sparse direction” within the regions of the intense fluid activity [54]. This is based on the concept of locally anisotropic diffusion. The sparseness is exploited within the realm of local-in-time spatially analytic solutions in L1 via the harmonic measure maximum principle; shortly, the existence of a local, sparse direction depletes the harmonic measure associated with the one-dimensional traces of the regions of high vorticity magnitude resulting in selfimproving bounds in the local-in-time solution scheme in L1 . The chapter is organized as follows. Section 2 reviews several instances featuring coherence of the vorticity direction-type critical and subcritical regularity classes: more precisely, the case of the spatial domain being the whole space, the case of the domains with boundary, and the spatiotemporal localization to an arbitrarily small parabolic cylinder. Section 3 concerns the two aforementioned criticality scenarios in which the criticality can be broken via suitable conditions on the regularity of the vorticity field. Section 4 contains the concluding remarks.

2

Vorticity Direction and Regularity: Super-Criticality

In this section, several types of regularity criteria for solutions to the 3D NSE involving a geometric information carried by the vorticity direction are presented. As in the case of analytic regularity criteria, these conditions are invariant with respect to the scaling of the equations and at a fixed “scaling distance” from the corresponding a priori bounds, illustrating supercriticality of the current state of the 3D NS problem. Consider the 3D NSE 8 ˆ ˆ ut   4u C .u  r/ u C rp D 0; < r uD0 in   .0; T /; (1) ˆ ˆ : u.x; 0/ D u0 .x/ in ; where the velocity u and the pressure p are the unknowns. The positive parameter  denotes the viscosity. The symbol  may denote the whole space R3 ; the halfspace R3C ; or an open, connected, bounded subset of R3 with a smooth boundary  D @. Further, Lp D Lp ./; 1  p  1 denotes the usual Lebesgue spaces equipped with norm k: kp , and H s is the standard L2 -based Sobolev space of order s. Arbitrary positive constants are denoted simply by c. The starting point of the mathematical theory of the 3D NSE is weak solutions considered in the well-known Leray-Hopf class; more precisely, for a divergencefree (in the sense of distributions) initial data u0 in L2 , one considers a variational/distributional divergence-free solutions to the 3D NSE belonging to the Leray class L1 .0; T I L2 .// \ L2 .0; T I H 1 .//. Note that the requirement that a divergence-free u belongs to L2 .0; T I H 1 .// per se does not guarantee that assuming the initial condition at time t D 0 makes sense; however, paired with the

18 Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

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variational formulation of the equations, it implies that u is almost everywhere (in time) equal to a continuous function, and after a modification on the set of measure zero, assuming the initial condition is meaningful. A Leray-Hopf (weak) solution u is strong if it belongs to L1 .0; T I H 1 .// \ L2 .0; T I H 2 .//: It is well known that strong solutions are smooth/regular, if data and domain are also smooth. In what follows, a leading role is played by the vorticity field ! D r  u D curl u. By applying the curl operator to both sides of Eq. (1), one gets the vorticity formulation of the 3D NSE, @! C .u  r/ !! D .!  r/ u: @t

(2)

Scalar multiplication of both sides of the above equation by !, integration over , and suitable integration by parts yield Z Z @! 1 d (3) k!k22 C kr!k22    ! d  D .!  r/ u  ! dx; 2 dt  @n  where it is assumed that u is divergence free and tangent to the boundary. The focus here is on the problem of global existence of smooth solutions, under suitable hypotheses on the vorticity direction  D !O D !=j!j. To be more precise, let .a; b/ denote †.a; b/, i.e., the amplitude of the angle between two nonzero vectors a and b. Frequently, ..x; t /; .y; t // will be denoted simply by .x; y; t /. One is searching for sufficient conditions on sin .x; y; t / that would guarantee the regularity of solutions.

2.1

The Whole Space Results

In the fundamental 1993 paper [28], P. Constantin and C. Fefferman proved that solutions to the evolution Navier-Stokes equations in the whole space are smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. More precisely, they obtained the following result. Theorem 1. Let be  D R3 ; and let u be a weak solution of (1) in Œ 0; T / with u0 2 H 1 .R3 / and r  u0 D 0. If sin .x; y; t /  g.t; x/ jx  yj for some g.; / 2 L12 .0; T I L1 .R3 //, then the solution u is regular on .0; T . Clearly, the result holds if g is a constant. Conditions on sin .x; y; t / are assumed for almost all x and y in ; and almost all t in .0; T /: Furthermore, in all the statements, one may verify (cf. [28]) that conditions on sin .x; y; t / are necessary only in a region where the vorticity at both points x and y is larger than an arbitrary fixed positive constant K and only for points x and y such that jx  yj < ı; for an arbitrary positive constant ı.

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In reference [12], by following an approach similar to that introduced in [28], the authors showed that regularity still holds in the whole space by replacing Lipschitz continuity by 12 Hölder continuity. The following theorem was proved. Theorem 2. Let ; u; and u0 ; be as in the previous theorem. Further, suppose that there exists ˇ 2 Œ1=2; 1 and g 2 La .0; T I Lb .//; where 3 1 2 C D ˇ a b 2



with

 4 a2 ;1 ; 2ˇ  1

(4)

such that sin .x; y; t /  g.t; x/ jx  yjˇ

(5)

holds in   .0; T /: Then, the solution u is regular on .0; T . In particular, the regularity result holds if sin .x; y; t /  c jx  yj1=2 :

(6)

In addition, the complementary case ˇ  12 is considered in [7]. This extension is useful since – as sketched at the end of this subsection – it helps better understanding of the whole chain of results. The conclusion is as follows: Theorem 3. Let u be a weak solution of (1) in Œ0; T / with u0 2 H 1 and r  u0 D 0. Let ˇ 2 Œ0; 1=2 and assume that sin .x; y; t /  cjx  yjˇ

(7)

holds in   .0; T /: Moreover, suppose that ! 2 L2 .0; T I Lr /;

(8)

where rD

3 : ˇ C1

(9)

Then the solution u is regular on .0; T . In particular sin .x; y; t /  c jx  yj1=2 is sufficient for regularity. The pair .2; r/ defined by Eq. (9) may be replaced by a wider family of parameters, via straightforward modifications in the proof.

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Note that in Theorem 2, assumption (6) alone is a sufficient condition for regularity. In fact, for ˇ D 12 the assumption on g.x; t / is simply g.x; t /  c; since a D b D 1: Similarly, in Theorem 3, when ˇ D 12 , assumption (7) alone is sufficient for regularity since then r D 2; and any weak solution satisfies ! 2 L2 .0; T I L2 .//. The above two statements are formally split into two families of sufficient conditions for regularity, namely, ˇ  12 and ˇ  12 . In Theorem 2, the advantage of assuming ˇ > 12 is counterbalanced by replacing in Eq. (5) the constant c by a function g 2 La .0; T I Lb .//. On the other hand, in Theorem 3, the penalizing condition ˇ < 12 is mitigated by assuming (8). This may give the wrong idea that the two families of results are relatively independent. On the contrary, the above formal separation is not substantial. In fact, the two families glue perfectly at the intersection point ˇ D 12 since the conclusion (namely, “condition (6) implies regularity”) is the same in both cases. On the other hand, a step-by-step analysis of the proofs given for each of the above two theorems shows that, inside each class, the results have the same “strength,” independently of the value of the parameter ˇ: Since the two families glue at point 12 , it follows that this is in fact just one family of closely connected results, all having an equivalent strength. The meaning of “strength” in this context is best explained via the scaling argument. This is now illustrated by showing that the sufficient conditions for regularity sin .x; y; t /  g.t; x/jx  yjˇ ; as ˇ goes from 12 to 1; enjoy the same strength. Assume that .u.x; t /; p.x; t // is a solution to the NSE in .0; C 1/  R3 . Then .u .x; t /; p .x; t /  . u. x; 2 t /; 2 p. x; 2 t / is a solution in the same domain. In particular, ! .x; t /  curl u .x; t / D 2 !. x; 2 t /: def

Set  .x; y; t / D †.! .x; t /; ! .y; t //: It readily follows that sin  .x; y; t / D sin . x; y; 2 t /: Assume now that the solution u.x; t / satisfies sin .x; y; t /  g.t; x/jx  yjˇ ; for some ˇ 2 Œ 12 ; 1 , where g 2 La .0; C1 I Lb .R3 //; and the exponents are defined by Eq. (4). It follows that sin  .x; y; t /  g .x; t /jx  yjˇ ; def

where the function g is given by g .x; t / D ˇ g. x; 2 t /: Consequently, 1

k g kLa .0; C1I Lb .R3 // D 2 k g kLa .0; C1I Lb .R3 // ; for all ˇ 2 Œ 12 ; 1 : The equivalence of the “strength” of the different sufficient conditions for regularity follows from the independence of the exponent 12 with

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respect to ˇ. The reader may verify that weaker (respectively stronger) sufficient conditions for regularity lead to larger (respectively smaller) exponents. It is worth noting that the above common strength is at the same level as the classical Ladyzhenskaya-Prodi-Serrin (L-P-S) integrability conditions for regularity. In fact, for ˇ D 0, condition (7) is superfluous, since it holds automatically. Furthermore, condition (8) simply reads ! 2 L2 .0; T I L3 .// – in other words – u 2 L2 .0; T I W 1; 3 .//, which is a regularity class [6] (W k;p denotes the classical Lp -based Sobolev space of order k). This class is in turn scaling equivalent to the classical scaling-invariant L-P-S condition u 2 L2 .0; T I L1 .//. These ruminations justify labeling the above family of ˇ-dependent results sharp results; namely, the regularity criteria in view are all scaling invariant, i.e., critical with respect to the natural scaling of the 3D NSE. Note that, in Theorem 2, the weak regularity implied by (4) for the coefficients g.t; x/ is fundamental to obtain sharp results. This is the reason why proving the “minimal regularity” for the coefficients g.t; x/ in this setting is given considerable attention. There are many interesting articles related to this topic. See, for instance, [10,14– 16, 20, 21, 26, 27, 29, 31, 35, 36, 51, 53, 56, 57, 62, 63, 74, 87], and references therein.

2.2

Domains with Boundary

The presentation will focus on the case of slip-type boundary conditions where the theory is more complete. Denote by T D p I C .ru C ruT / the stress tensor and by t D T  n the stress vector, where n denotes the exterior unit normal vector to the boundary. Further, consider the tangential component of t

.u/ D t  .t  n/ n:

(10)

The homogeneous Navier slip boundary conditions, see [70, 80], read u  n D 0;

.u/ D 0

(11)

on ; a related type of the boundary conditions that differ only by lower order terms (no derivatives) is the following: u  n D 0;

!nD 0

(12)

(sometimes, this type of boundary conditions is referred to as “absolute boundary conditions,” see, e.g., [23]). It should be noted that both conditions coincide when  D R3C ; and can be written in the simplified form @uj D 0; (13) @x3 where j D 1; 2: A natural framework to study the boundary condition (13) is the space ˚  V D v 2 ŒH 1 .R3C /2  H01 .R3C / W r  v D 0 : u3 D

18 Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

909

In reference [8], the author considered the NSE in Œ0; T /  R3C , endowed with the slip boundary condition, and obtained the following result. Theorem 4. Let u0 2 V be tangent to the boundary, and let u be a weak solution of (1) in Œ0; T /  R3C , endowed with the slip boundary condition (12) or, equivalently, with (11). Let ˇ 2 Œ0; 1=2 and assume that for almost all t 2 .0; T /, (7) holds for almost all .x; y/. Moreover, suppose that (8) holds, where r is given by (9). Then the solution u is regular on .0; T . If ˇ D 1=2, the assumption (8) is superfluous. The theorem is proved by appealing, separately, to the classical Dirichlet and Neumann Green functions in the half-space. This can be done for flat boundaries since the boundary conditions (12) separate in the Dirichlet and Neumann parts (13). Such favorable circumstance is no longer true if the boundary is not flat. However, it suggests trying to extend the result to non-flat boundaries by appealing to Green’s function theory. This was indeed achieved in [13] where the authors extended Theorem 4 to the case in which   R3 is an open, bounded set with a smooth boundary. Since the boundary is not flat, one appeals to the representation formulas for Green’s matrices derived in Solonnikov’s fundamental work [83, 84]. With the aid of these tools, novel local representation formulas for the velocity (in terms of the vorticity) were obtained which, in turn, lead to the key estimates on the vortex stretching term. The following is the main result presented in [13]. Theorem 5. Let   R3 be an open, bounded set with a smooth boundary, say of class C 3;˛ , for some ˛ > 0. Suppose that u0 2 H 1 ./, r  u0 D 0, u0 tangent to the boundary, and u is a weak solution to (1) under the boundary condition (12) on the interval Œ0; T /. In addition, suppose either that there exists ˇ 2 Œ1=2; 1 such that (5) holds, where g 2 La .0; T I Lb .// with a, b, and ˇ related by (4), or that there exists ˇ 2 .0; 1=2 such that (7) holds, where ! satisfies (8) with r given by expression (9). Then, the solution u is regular on .0; T . The proof of the above result is quite involved and that motivated a search for simpler proofs in the non-flat boundary setting. This goal was accomplished in reference [11] for the Constantin and Fefferman’s Lipschitz condition case ˇ D 1. The result is the following: Theorem 6. Let   R3 be an open, bounded set with a smooth boundary. Assume that u0 2 H 1 ./ is divergence free and tangent to the boundary. Let u be a weak solution to (1) under the boundary condition (12) on the interval Œ0; T /. In addition, assume that there exists g 2 La .0; T I Lb .//; where 2 3 1 C D a b 2

with a 2 Π4; 1 ;

(14)

910

H. Beirão da Veiga et al.

and also a positive ı.x; t/, such that sin .x; y; t /  g.t; x/ jy  xj;

(15)

for x; y 2 ; satisfying jy  xj < ı.x; t /: Then u is a regular solution on .0; T . Note that for ˇ D 1 (4) becomes (14). A sketch of the proof is given at the end of this subsection. Many other geometric regularity criteria can be found in the current literature. Among closely related results is Theorem 7 below (cf. [10]) in which the geometric regularity criterion is stated in terms of the local alignment of the vorticity and the velocity R direction fields; this is inspired by the fundamental role that the helicity, H D  u  ! dx, plays in the dynamics of 3D incompressible turbulent flows. Denote by ˛.x; t/ the angle between the velocity u.t; x/ and the vorticity !.t; x/ at the point .x; t /, def

O t //; ˛.x; t/ D †.Ou.x; t /; !.x; def

where, for each non-null vector v, vO D v=jvj. Further, set M .t/ D sup sin ˛.x; t/: x2 

Theorem 7. In addition to the conditions on the domain and the initial data, as well as the boundary conditions stated in the previous theorem, assume, for simplicity, that  is simply connected. Set !0 D curl u0 . Then, there is a constant C D C ./ such that if M .t/ 

C ; k!0 k2

(16)

for almost all t 2 .0; T /; then the solution u is regular on .0; T : It is worth noting that the smallness assumption in the theorem is consistent with the geometric structure of certain regions in turbulent flows. More precisely, in the coherent structures present in turbulent shear layers, there are regions where the helicity is large (of either sign), and also there are regions where the helicity is very small (see, e.g., [60] and the references therein).

2.2.1 Sketch of the Proof of Theorem 6 Set f D curl ! and g D ! in the identity Z

Z .curl f /  g dx D



Z f  .curl g/ dx C



.n  f /  g d : 

(17)

18 Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

911

By taking into account that the mixed product .n  curl !/  ! vanishes on the boundary of the domain , it readily follows that Z Z j curl ! j2 dx;  . !/  ! dx D 



where the identity  ! D curl curl ! was used. Consequently, scalar multiplication of (2) by !, followed by integration in , yields Z   1 d 2 2 k !k2 C k curl ! k2 D .!  r/ u  ! dx: (18) 2 dt  As div ! D 0 in  and ! n D 0 on , one has k r ! k22  c .k curl ! k22 C k ! k22 /, and the inequality ˇZ ˇ Z Z Z ˇ ˇ 1 1 d j!j2 dx C jr!j2 dx  c./ j!j2 dx C ˇˇ .!  r/u  ! dx ˇˇ 2 dt  4    (19) follows. On the other hand, integration by parts leads to Z .!r/ u ! dx D  

Z 

Z  .@j !k / !j .@j !j / !k uk dx C .! u/ .! n/ d :





(20)

Next, from assumption (15), by setting y D x C h; one can show that the estimate j !.x/  !.x C h/ j  g.t; x/ j h jj !.x/ j j !.x C h/ j holds for sufficiently small h. Thus, ˇ !.x C h/  !.x/ ˇˇ ˇ ˇ !.x/  ˇ  g.t; x/ j !.x/ j j !.x C h/ j: jhj

(21)

In particular, by letting h ! 0; one gets j !.x/  @j !.x/ j  g.t; x/ j !.x/ j2 ;

(22)

for each j D 1; 2; 3: It readily follows, by considering the expressions of the single components of !.x/  @j !.x/, that j !l @j !k  !k @j !l j  g.t; x/ j ! j2 ;

(23)

for k ¤ l. Furthermore, for k D l; the inequality (23) is obvious, and (23) holds for each triple of indices fi; j; k g: This inequality, together with the identity (20), leads to the estimate Z ˇZ ˇ ˇ ˇ g.t; x/ juj j!j2 dx; ˇ .!  r/u  ! dx ˇ  c 



912

H. Beirão da Veiga et al.

which – by Hölder’s inequality – yields ˇ ˇZ ˇ ˇ 6 k ! k2 k ! k6 ; ˇ .!  r/u  ! dx ˇ  k g k ˛6 k u k 2˛ 

for each ˛ 2 Œ0; 1. Hence, by appealing to (19), the embedding H 1 ./  L6 ./; and the interpolation inequality ˛ k u k2 6  k u k1C k u k1˛ 2 ; 6 2˛

one arrives at the following estimate,   1 1 d ˛ k u k1˛ k !k22 C kr!k22  c 1 C k g k26 k u k1C k ! k22 : 2 6 ˛ 2 dt 4

(24)

By invoking Hölder’s inequality once again, now with respect to the time variable, it transpires that RT  0

 ˛ k g.t / k26 k u.t / k1C dt  k u.t / k1˛ 2 6 ˛

(25) k g k2

4 L 1˛

.0; T

6 IL˛

.//

˛ k u k1C k u k1˛ : L1 .0; T I L2 .// L2 .0; T I L6 .//

By setting aD

4 ; 1˛

bD

6 ; ˛

one gets RT  0

 ˛ k g.t / k26 k u.t / k1C dt  k u.t / k1˛ 2 6 ˛

2 4

4

a k g k2La .0; T I Lb / k u kL2 .0; k u kLa 1 .0; T I L2 .// ; T I L6 .//

(26)

where the parameters a and b are related by (14). From (24), (26), and Gronwall’s lemma, it follows that k ! k2L1 .0; T I L2 .//  k !.0/k2 4 ˚   2 a4 exp c t C k g k2La .0; T I Lb / k u kL2 .0; k u kLa 1 .0; T I L2 .// : T I L6 .//

(27)

The boundedness of u in L1 .0; T I H 1 .// – and hence regularity on .0; T  – is now established.

18 Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

2.3

913

Localization

The goal of this subsection is to present several results illustrating localization of purely geometric and hybrid analytic-geometric regularity criteria for the 3D NSE to an arbitrarily small parabolic cylinder in the space-time. This is of interest as the possible formation of singularities in the 3D NSE is essentially a local phenomenon. In addition, the localization of the coherence of the vorticity direction regularity criteria was a key ingredient in the proof of 3D enstrophy cascade under the conditions consistent with the turbulence phenomenology and the geometry of vortex filaments [35]. It should be noted that the local nature of the regularity properties of solutions to the 3D NSE is known since the works of Scheffer [76] and Caffarelli, Kohn, and Nirenberg [19] (in the velocity formulation). Moreover, since the velocity is recovered from the vorticity by solving a Poisson equation with the right-hand side equal to curl !, the ellipticity indicates that the vorticity-type regularity criteria should also be local in nature. The main point of the material presented in this subsection was to show that it is possible to perform a localization in such a way that the leading order term in the localization of the vortex stretching term inherits both analytic (Calderon-Zygmund kernel and div-curl structure) and geometric cancellations present in the global vortex stretching term. Let  be a spatial domain of interest and T > 0. Fix a point .x0 ; t0 / in .0; T /, and let 0 < R < 1 be such that Q2R .x0 ; t0 / D B.x0 ; 2R/.t0 4R2 ; t0 / is contained in   .0; T /. For an r  R, let .x; t / D .x/ .t/ be a smooth cutoff function on Q2r .x0 ; t0 / satisfying jrj c  for some 2 .0; 1/; 0    1;  r c supp  .t0  .2r/2 ; t0 ; D 1 on Œt0  r 2 ; t0 ; j 0 j  2 ; 0   1: r

supp   B.x0 ; 2r/;  D 1 on B.x0 ; r/;

The following localization formula for the vortex stretching term was obtained in [53],  2 .x/.!  r/u  ! .x/ Z D c P:V: B.x0 ;2r/

j kl

@2 1  !l dy .x/ !i .x/ !j .x/ C LOT @xi @yk jx  yj (28)

where P:V: denotes the principal value of the integral, j kl is the Levi-Civita symbol,

j kl

8 ˆ ˆ M g. Then the localized enstrophy remains uniformly bounded up to t D t0 , i.e., Z sup t2.t0 R2 ;t0 /

j!j2 .x; t / dx < 1 and .x0 ; t0 / is not a singularity:

B.x0 ;R/

Denote by ;r the following  -Hölder measure of coherence of the vorticity direction  at a point .x; t /,

;r .x; t / D

sup y2B.x;r/;y¤x

  j sin  .x; t /; .y; t / j : jx  yj

Theorem 9. Let u be a Leray-Hopf solution on the space-time domain   .0; T /. 3 and Let 0 <  < 1; ˛ < 3 ; ˛ > C2 ıD

2 .2 C  /˛  3

18 Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

915

(the scaling invariance). Suppose that u is smooth in Q2R .x0 ; t0 /, and Z

Z

t0

˛

j!.x; t/j t0 .2r/2

B.x0 ;2r/

˛

;2r .x; t /dx

ı dt

is finite. Then the localized enstrophy remains uniformly bounded up to t D t0 , i.e., Z

j!j2 .x; t / dx < 1 and .x0 ; t0 / is not a singularity:

sup t2.t0 R2 ;t0 /

B.x0 ;R/

The case  D 12 ; ˛ D 2 and ı D 1, namely, Z

t0

t0 .2R/2

Z B.x0 ;2R/

j!.x; t/j2 21 ;2R .x; t /dx dt;

(30)

2

is included and is a scaling-invariant improvement over the 12 -Hölder coherence – the sup in x and t bound on 1 . For a comparison with what can be bounded, a 2 local spatiotemporal average of j!jjrj2 averaged over a ball around a spatial point moving with the fluid over a suitable interval of time is a priori bounded (cf. [30]).

3

Vorticity Direction and Regularity: Breaking the Criticality

This section presents two different criticality scenarios, one geometric and one analytic, in which a suitable condition on the vorticity direction, being in a logarithmically weighted, local space of functions of bounded mean oscillations and being uniformly continuous, respectively, suffices to break the criticality and prevent a possible formation of a singularity.

3.1

Vortex Stretching and Locally Anisotropic Diffusion

A geometric criticality scenario to be outlined in this subsection is based on the interplay between the vortex stretching and the locally anisotropic diffusion. The manifestation of locally anisotropic diffusion in this setting is sourced in local-in-time, spatially analytic smoothing effect exhibited by the 3D NSE in L1 . The idea is to split the spatial domain of interest in two parts, a suitably defined region of intense vorticity and the complement, and then show that if the region of intense vorticity is “sparse enough” (in a local, one-dimensional sense), then the harmonic measure associated to certain one-dimensional traces of the region is small enough to result in a self-improving bound on the L1 norm, yielding a uniform lower bound on the time step in the solution scheme and preventing a finite time blowup [54]. The precise meaning of the local, one-dimensional sparseness is as follows:

916

H. Beirão da Veiga et al.

Definition 1. Let x0 be a point in R3 , r > 0, S an open subset of R3 and ı in .0; 1/. The set S is linearly ı-sparse around x0 at scale r if there exists a unit vector d in S 2 such that jS \ .x0  rd; x0 C rd /j  ı: 2r Denote by t .M / the vorticity super-level set at time t ; more precisely, t .M / D fx 2 R3 W j!.x; t/j > M g: Then, the region of intense vorticity is defined to be tC.t/

1

 k!.t/k1 ;

c1 where .t / is the half step in the local-in-time solution scheme in L1 initiated at time t and c1 > 1 is an appropriate absolute constant. (More precisely, 1 1 .t / D , for a suitable absolute constant c0 > 1.) c0 k!.t/k1 The main result in [54] states that if the region of the intense vorticity is linearly ı-sparse (for an appropriate ı, while the direction of sparseness is allowed to vary 1 1 across the region) at the scale less or equal to , where c2 > 1 is an c2 k!.t/k 12 1 appropriate absolute constant, no blowup can occur. (It is worth mentioning that it is enough to assume the sparseness at a suitably chosen finitely many times; i.e., the condition is intermittent in time.) A key question is whether there are any indications that the scale of local linear sparseness needed for the local anisotropic diffusion to engage is in fact achieved in turbulent flows; this is now briefly addressed assuming that the region of intense vorticity comprises of vortex filaments. Let R0 be a suitable macroscale associated with the flow in view, e.g., the length of the cube side L in the case of Œ0; L3 -periodic boundary conditions. Computational simulations indicate that – intermittently in time – the longitudinal scale of the filaments is comparable to R0 . This, paired with the estimate on the decay of the volume of the region of intense vorticity, provides an indirect estimate on the transversal scale of the filament (a natural microscale in this setting). More precisely, it was shown in [24] that – starting with a bounded measure – the L1 norm of the vorticity is a priori bounded on any finite time interval. This, via Chebyshev’s inequality, implies that 1 

c3  k!.t/k1 ; Vol tC.t/ c1 k!.t/k1 which – in turn – yields the decrease of the transversal scale of the filaments of at c4 least ; here, c3 and c4 are suitable constants greater than 1. Consequently, 1 2 k!.t/k1 thinking in terms of vortex filaments, one arrives at criticality,

18 Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

1 1 c2 k!.t/k 12

1

vs:

c4

917

:

1 2 k!.t/k1

It is worth mentioning that, in addition to the computational evidence, there is also a mathematical evidence in favor of creation and persistence of the macroscalelong vortex filaments [36]. The main result states that, in the time average and with respect to a multi-scale spatial ensemble averaging procedure, the range of scales of positivity of the vortex stretching term extends from a suitable dynamic microscale all the way to the macroscale. Moreover, the microscale converges to 0 as the flow approaches the possible singular time; i.e., in the singular limit, the range of positivity of the vortex stretching term exhausts the full range of spatial scales. The multi-scale ensemble averaging procedure utilized was previously developed in the study of existence and locality of the energy and enstrophy cascades in 3D incompressible flows [32–35]. In the remainder of this subsection, it is indicated how a very mild condition on the vorticity direction transforms the above criticality scenario into a log subcritical scenario preventing the formation of singularities. p The plan is to get a uniform-in-time L log L-bound on w D 1 C j!j2 . This would impose an extra-log decay on the volume of the super-level sets breaking the criticality. For simplicity, let B.0; R0 / be the macroscale spatial domain of interest, and consider a solution that is smooth on .0; T / (i.e., T is the first possible singular time). It is plain that w satisfies the following partial differential inequality, @t w  4w C .u  r/w  !  ru 

! w

(31)

(for more details see [24]). Multiplying (31) by .1Clog w/ – where is a smooth cutoff function associated with the macroscale domain – yields (after a fair amount of work), Z I . / 

Z



Z !  ru 

.x/ w.x; / log w.x; / dx  I .0/ C c 0

 log w dx dt

x

C a priori bounded; for any in Œ0; T /I  denotes the vorticity direction. Without utilizing cancellations in the remaining term, namely, the div-curl structure of the vortex stretching term !  ru, it does not seem possible to obtain the desired localized L log L-bound on w. In order to exploit the cancellations efficiently, one needs several constructs and results from harmonic analysis which are now briefly reviewed; for more details and the references, see [18, 55]. Let f be a distribution. The maximal function of f is defined as Mh f .x/ D sup jf ht .x/j; x 2 Rn ; t>0

918

H. Beirão da Veiga et al.

where h is a fixed, normalized test function supported in the unit ball, and ht denotes t n h.=t /, and a distribution f is in the Hardy space H1 if kf kH1 D kMh f k1 < 1. The local maximal function is defined as, mh f .x/ D sup jf ht .x/j; x 2 Rn ; 0 2k t > tk

with k D 1=Mk . Since (32) is invariant under the above rescaling, it follows that uk is a mild solution of (32) in R3  .tk Mk2 ; 0.

924

H. Beirão da Veiga et al.

Step 2 (Compactness). Property (i) implies juk j  1 in R3  .tk Mk2 ; 0. A higher derivative estimate (see, e.g., [52]) yields .uk ; !k / ! .Nu; !/ N

.k ! 1/

locally uniformly in R3  .1; 0 – by taking a subsequence – for some .Nu; !/ N where uN is a bounded global mild solution in R3  .1; 0/. By (iii) one concludes that jNu.0; 0/j D 1. So far, neither (CA) nor type I condition are used. Step 3 (Characterization of the limit). Utilization of the continuous alignment condition (CA) will imply that uN is a constant, and since type I blowup is assumed, uN will have to be zero. Here is a sketch of the proof. Set k D !k =j!k j: Then (CA) implies jk .x; t /  k .y; t /j 

jx  yj Mk

! 0I

hence, N D !=j N !j N is independent of x. By the existence and uniqueness theory ([46]) of mild solutions N must be constant in time. Thus .Nu; !/ N is a two-dimensional flow. The following Liouville theorem implies that uN is a constant. Theorem 12. If u is a bounded mild solution of (32) in R2  .1; 0/, it must be a constant solution. There are several ways to prove this Liouville-type theorem (cf. [51, 65]). Step 4 (Contradiction). One has jNu.0; 0/j D 1 by Step 2, while in Step 3 it was shown that uN  0. This is a contradiction and T is not a blow-up time.

3.2.2

Continuous Alignment Condition: The Case of the Half-Space with the No-Slip Boundary Conditions The main step in obtaining a continuous alignment result in the case of the halfspace with the no-slip boundary conditions is proving a suitable Liouville-type theorem in this setting (the half plane with the no-slip boundary conditions). The major obstacle here is that the maximum principle is no longer an efficient tool in deriving a priori bounds on the vorticity. This stems from the fact that the no-slip boundary condition is effectively a source of production of the vorticity at/near the

18 Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

925

boundary that is not captured by the maximum principle. This is in contrast to the cases of the whole plane or the slip-type boundary conditions where there is no vortex shedding off the boundary. In the case of the no-slip boundary condition, it is beneficial to exploit suitable boundary conditions on the vorticity. Before explaining the results in some detail, various Banach spaces of nondecaying functions are introduced. Let  be a domain in Rn , n 2 N. Then, for k 2 N [ f0g and  2 .0; 1/ the spaces BC ./, C k ./, and C kC ./ are, respectively, defined by ˚  BC ./ D f 2 C ./ j kf k1 D sup jf .x/j < 1 ; k

x2

˚

˛

C ./ D f 2 BC ./ j r f 2 BC ./; j˛j  k; X  kr ˛ f k1 < 1 ; kf kC k D j˛jk

˚

C kC ./ D f 2 C k ./ j kf kC kC D kf kC k C

X

sup

j˛jDk x;y2; x¤y

 jr ˛ f .x/  r ˛ f .y/j 0g, and u D u.t; x/ D .u1 .t; x/; u2 .t; x// and p D p.t; x/ denote the velocity field and the pressure field, respectively. Theorem 13 ([45]). Let .u; p/ be a solution to (40) and (41) satisfying the following conditions.   (C1) sup ku.t /kC 2C C k@t u.t /kC  < 1 for some  2 .0; 1/. 1 0. Moreover, due to the local character of the weak compactness proof, the assumption (29) can be weakened by assuming the bound to hold locally in space, i1=2 R hR namely, K B 0 log2 0 01 dR dx < 1 for any compact set K of .

990

3.5

N. Masmoudi

Sketch of the Proof

To prove the existence of weak solutions, one has to regularize the system, construct a solution to the regularized system, and then pass to the limit and prove that the limit is a solution of the original system. The main step is the passage to the limit. We only sketch that part. The main step of the proof is the stability of weak solutions: Take .un ; n / a sequence of weak solutions with initial data .un0 ; 0n / satisfying, uniformly in n, the free energy bound and the log2 bound (29). Extracting a subsequence if necessary, one can assume that .un0 ; 0n / converges strongly to .u0 ; 0 / in n n L2 . /  L1loc . I L1 .B// and 0n log 0 10  0n C 0n 1 converges strongly to log 0 01  0 C 0 1 in L1 .  B/. Now, one can extract a subsequence such that un converges weakly to u in Lp ..0; T /I L2 . // \ L2 ..0; T /I H01 . // and n converges weakly to in Lp ..0; T /I L1loc .  B// for each p < 1. Then, one would like to prove that .u; / is still a solution of (8). The main difficulty is to pass n n to the limit in the nonlinear term in the second q ru R  appearing q equation of (8).  Qn Qn Qn 1=2 n n Now, introduce g D log where Q n D and f D 0

1

1

1

n

C a 1 and a > 1 is any constant. Extracting a subsequence if necessary, one can assume that g n and f n converge weakly to some g and f in Lp ..0; T /I L2loc .  B; dx 1 dR// for each p < 1. To prove that.u; / is a Qn

Qn

solution of (8), it will be enough to prove that .g n /2 D 1 log 1 converges   Q Q weakly to g 2 D 1 log 1 , namely, that g n converges strongly to g in L2 ..0; T /I L2 .  B; dx 1 dR//. Introduce the defect measure: Z

Œ.g n /2  g 2

D

1 dR:

B

It measures the lack of strong convergence of g n to g in L2 .dt dx 1 dR/. Qn Take ‚.t / D t 1=2 log1=2 .t / and recall that g n D ‚. 1 /. Next, one introduces the following defect measures ij ; ij0 and ˇij such that run g n ! rug C  ; ! Qn 0 Qn Qn n ! ru ‚ ‚0 ru 1

1

1

Qn

! C  0;

1

run Q n ! ru Q C ˇ; where  ;  0 2 L2 ..0; T /   B/, and ˇ 2 L2 ..0; T /  I L1 .B// are matrix valued.

20 Equations for Polymeric Materials

991

Passing to the limit in the equation satisfied by g n , one gets h i @t g C u:rg D divR  ri uj Rj g  ij Rj C rR U:ruR

1

C

divR

h

Qn

Qn

‚0

1

! C rR U R W  0

1

1 rR g

i

  jrR f n j2 log1=2 C log3=2 C

 ak

Qn

! ri unj

1



1

fn

1

‚0

Qn

2Ri Rj 1  jRj2

where Fn denotes the weak limit of Fn . Passing to the limit in the equation satisfied by Q n , one gets h i @t Q C u:r Q D divR  ru  R Q  ˇij Rj C divR

h

Q i

1 rR

 2akru W

1

1

Besides, Q n log





Qn 1

"Z

n

satisfies

Qn

Q n log

.@t C u :r/

4 B

Introduce N1n D

!# D run W  . Q n /

1

B

Z

Ri Rj : 1  jRj2

ˇ s ˇ2 ˇ ˇ Z Qn ˇ ˇ Ri Rj ˇ ˇ  2ak run r 1ˇ R ˇ 1  jRj2 1 B ˇ ˇ R B

Q n log



Qn 1



and N2n D

R B

1

log

Qn

! :

1

h  Q n log2

Qn 1



 2 log



Qn



1

C2 /1=2 . Hence, one gets

ı;

.@t C u:r/ Drun W . Q n /

Z  B

ˇ s ˇ2 ı; ˇ ˇ Qn ˇ ˇ ˇ ˇ r 1ˇ R ˇ 1ˇ ˇ

Z

Ri Rj  2ak run 1  jRj2 B

1

log

Qn 1

!ı; (32)

992

N. Masmoudi

where  D N1n and

Fn

ı;

Fn .1 C ıN2n /.1 C N1n /2

D lim lim

!0 ı!0

for any sequence Fn bounded in L1 . Combining the previous identities and using the renormalizing factor N , one gets !# "  Qn Qn   ı;

1 2 n n 0 Q .@t C u:r/ 4 D 4 ru W  C ru W  ‚. / 1 g  2g N N 1 1 2    Qn  3 Z n j2 log1=2 C log3=2 jr f R ı; 1 1 6 7 n 2 2jr gj2 C  4 2g5 1 44jrR f j R n N B f 2ak  4 N 2  4 N D

0

Z B

Qn

@run log

Z

4 X



1

!



C1

ı;

  2‚0

1

Qn 1



ij Rj rg  rR UR W .   0 /g

!



1

ri unj g A

Ri Rj 1  jRj2

1

Ai :

iD1

Introduce the unique a.e flow X in the sense of DiPerna and Lions of u, solution of @t X .t; x/ D u.t; X .t; x//

X .t D 0; x/ D x:

(33)

Hence, one gets

d .t; X .t; x//  C jruj2 4 .t; X .t; x// 4 dt N N

Z 2 jrR gj

CC 1C .t; X .t; x// 1 N N B where N satisfies .@t C u:r/ N1 D 0 and N2n  N: For almost all x, N .t; X .t; x// (which is constant in t) is bounded. Hence, one deduces that for almost all x, Z 0

T

Z N3 1 C B

1

 jrR gj2 C jruj2 .t; X .t; x// < 1: N

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Hence, by Gronwall lemma, one deduces that for a.e x, one has for all t < T ,

.0; x/ CT .x/

.t; x/  e 4 N N4 and since .0; x/ D 0 due to the initial strong convergence, one deduces that

.t;x/ 4 D 0 and hence D 0: Therefore, one deduces that the limit .u; / is still a N weak solution of the FENE model.

3.6

Open Problems

This section is concluded by some important open problems related to the FENE model and some of its extensions: • The zero diffusion limit in x. If one adds a center-of-mass diffusion term "x in the Fokker-Planck equation of (8) with a boundary condition for when x 2 @ in the case is a smooth bounded domain, then one can prove the global existence of weak solutions to the model (see [6] for an existence result for the FENE model with center-of-mass diffusion). A natural question is whether one recovers a weak solution of the unregularized system (8) when " goes to zero. The difficulty comes from the fact that the proof of existence without the centerof-mass diffusion term uses in a critical way the fact that we have a transport equation in the x variable. Adding viscosity in x destroys this nice structure and requires the use of a method that can make the bridge between the two proofs. • Relaxing the assumption (29). This extra bound was only used to give some extra control for the stress tensor and to justify the calculation in the limit. Can one prove the same existence result without it? • Other models. A natural question is whether one can extend the result on the FENE model to the Hookean model (where the system can be reduced to a macroscopic model). The main difficulty is that one does not know whether the extra stress tensor  is in L2 . Nevertheless, the strategy of this section can be used to prove global existence for the FENE-P model (see below and [62]). One should also mention the paper [7] where global existence to the Hookean-type bead-spring chain model is proved when center-of-mass diffusion is taken into account and the potential U .R/ grows faster than jRj2 when R goes to infinity. Moreover, in [19], global existence is proved for the Hooke model (which is equivalent to the macroscopic Oldroyd-B model) when the data is small in L1 . • Regularity in 2D. Many works on polymeric flows are motivated by similar known results for the Navier-Stokes system. In particular a natural question is whether one can prove global existence of smooth solutions to (8) in 2D. It should be pointed out that this is known for the co-rotational model [54, 61]. This seems to be a difficult problem, since one only has an L2 bound on  and that an L1 bound on  was necessary in the previously mentioned works. In particular a

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similar result is not known for the co-rotational Oldroyd-B model where one can prove Lp bounds on  for each p > 1. • Is system (8) better behaved than Navier-Stokes? One does not expect to prove results on (8) which are not known for Navier-Stokes since (8) is more complicated than Navier-Stokes. However, one can speculate that due to the polymer molecules and the extra stress tensor, system (8) may behave better than Navier-Stokes and that one can prove global existence of smooth solutions to (8) even if such result is not proved or disproved for the Navier-Stokes system.

4

The Navier-Stokes-Fokker-Planck Coupling

This section is about a different class of kinetic models. The particles in the system are described by a probability distribution f .t; x; m/ that depends on time t , macroscopic variable x 2 Rn , and particle configuration m 2 M . Here M is a smooth compact Riemannian manifold without boundary (see Sect. 4.3 for more general configuration spaces). The particles are transported by a fluid, agitated by thermal noise, and interact among themselves. This is reflected in a kinetic equation for the evolution of the probability distribution of the particles ([9]). The interaction between particles – a micro-micro interaction – is modeled in a mean-field fashion by a potential that represents the tendency of particles to favor certain coherent configurations. The interaction between particles occurs only when the concentration of particles is sufficiently high. Mathematically, this term is responsible for the nonlinearity of the Smoluchowski (Fokker-Planck) equation, and physically, it is responsible for nematic phase transitions. Because the particles are considerably small, and for smooth flows, the Lagrangian transport of the particles is modeled using a Taylor expansion of the velocity field. This gives rise to a drift term in the Smoluchowski equation that depends on the spatial gradient of velocity. It is a macro-micro term, and it causes mathematical difficulties in the regularity theory. The fluid is described by the incompressible Navier-Stokes system. The microscopic particles add stresses to the fluid. This is the micro-macro interaction and it is the most puzzling and important physical aspect of the problem. Indeed, while a macro-micro interaction can be derived, in principle, by assuming that the macroscopic entities vary little on the scale of the microscopic ones, the “scaling up” of the effect of microscopic quantities to the macroscopic level is more mysterious. The linear Fokker-Planck system coupled with Stokes equations was considered in [68]. The nonlinear Fokker-Planck equation driven by a time averaged NavierStokes system in 2D was studied in [17].

4.1

The Model

If M; the phase space of polymers, has a smooth Riemannian structure (e.g., rodlike model etc.), then we get a PDE system coupling the Navier-Stokes equation for the

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fluid velocity with a Fokker-Plank equation describing the evolution of the polymer density: 8 < :

@f @t

@v @t

C v  rv  v C rp D r   in  .0; T / C v  rf C divg .G.v; f /f /  g f D 0 in  .0; T / divv D 0 in  .0; T /;

(34)

R R R .1/ .2/ where ij D M ij .m/f .t; x; m/d mC M M ij .m1 ; m2 /f .t; x; m1 /f .t; x; m2 / d m. We denote G.v; f / D rg U C W where U D Kf is a potential given by Z K.m; q/f .t; x; q/ dq

U .t; x; m/ D

(35)

M

with a kernel K which is a smooth-, time-, and space-independent symmetric function K W M  M ! R. Moreover, the vector field W in the FokkerPlank equation depends on the spatial gradient of the velocity, i.e., W .t; x; m/ WD W .ru.t; x/; m/. This is a macro-micro interaction. The vector field W depends linearly on the macroscopic gradient of fluid velocity, given locally as

W .t; x; m/ D .W˛ .t; x; m//˛D1;:::;d WD

3 X

c˛ij .m/

i;j D1

 @ui .x; t / ˛D1;:::;d @xj

(36)

ij

where the smooth coefficients c˛ do not depend on time and x. For example, in the rodlike polymers, M D S2 and W .t; x; m/ is the projection of the vector rx u.t; x/m onto the tangent plane Tm S2 at m 2 S2 : W .t; x; m/ WD rx u.t; x/m  hrx u.t; x/m; mim: Here h; i is the standard inner product in R3 . To state the result from [18], we take D R2 : 0 ;r \ L2 .R2 / and f .0/ 2 W 1;r .H s /, for some Theorem 5. Take v.0/ 2 W 1C" R r > 2 and "0 > 0 and f0  0, M f0 2 L1 \ L1 . Then (34) has a global solution in 1;r 1;r v 2 L1 / \ L2loc .W 2;r / and f 2 L1 .H s //. Moreover, for T > T0 > loc .W loc .W 1 2";r 0, we have v 2 L ..T0 ; T /I W /.

4.2

The Doi Model

In the Doi model, polymers are idealized as rods of fixed length. The phase space is R 2 SN 1 . The micro-macro equation is the so-called Doi model (see (37)). In (37), R is the Laplacian on the sphere, and PR? is the orthogonal projection on the

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tangent space to the sphere at R, namely, PR? ruR D ruR  .R:ru:R/R and b is a parameter: 8 ˆ < @t u C .u  r/u  uh C rp D div; i divu D 0; @t C u:r D divR  PR? ru  R  R ˆ R R : ij D SN 1 N .Ri ˝ Rj / .t; x; R/dR C brk ul W Rk Rl Ri Rj dR;

(37)

Refer to [17, 18, 68] for mathematical results about the system. The free energy satisfies

Z @t

TN

juj2 C 2

Z

Z TN

 Z Z log  C1 D jruj2 C4

SN 1

Z jr TN

p

j2

SN 1

(38) Z Cb TN

Z rk ul W

SN 1

Rk Rl Ri Rj dR W ri uj (39)

To make sure that the free energy is dissipated, one has to assume that b >  NN1 . Global existence of weak solution was proved in [58]: R Theorem 6. Let u0 2 L2 .TN / and 0 such that SN 1 0 dR D 1 a.e in x and R R log 1 dRdx < 1. Assume that b >  NN1 . Then, there exists a global TN B weak solution .u; / to (37). One should note that this theorem also applies to the more general SmoluchowskiNavier-Stokes system studied in [17, 18].

4.2.1 Idea of the Proof of Theorem 6 As for the FENE model, the main difficulty is to prove the propagation of the compactness for a sequence .un ; n / of weak solutions. For simplicity, one can take b D 0. Extracting a subsequence, one can assume that un converges weakly to u in Lp ..0; T /I L2 .TN // \ L2 ..0; T /I H 1 .TN //, and n converges weakly to in Lp ..0; T /  TN I L1 .dR/ for each p < 1. Let H D .R C I /s for some s > N =2 C 1. Hence, H W L1 .SN 1 / ! L1 .SN 1 / and H n is bounded in all variables. As for FENE, one introduces the following defect measures: jH .

n



/j2 ! ; jrR H .

jr.un  u/j2 ! ; n



/j2 ! ;

n

run !

j n  j2 ! ˛

ru C ˇ

(40) (41)

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Using that Z

j n  j D j

N .R  R/.

n

Z 

/j  C

H 1 .R  R/H .

n



/

(42)

R 1 one deduces R that ˛  C dR is inR L .dt dx/. Moreover, one has as above  D N ˇij Ri Rj and   C ˛  C dR. Arguing as above, after applying the operator H to the equation of , one gets Z

Z

Cu:r

@t S

Z

h

D S

S

H rR PR? Œ.

n run 

i

ru/  R .H .

n

Z 

//

 S

(43) where g n denotes the weak limit of g n . Using Cauchy-Schwarz, the first term on the right-hand side can be estimated by C

Z S

 Z 1=2 i Z  1 Z  C jruj  C .1 C jruj2 / 

C : 4 S S S

1=2 h

(44)

R R R R Hence, one deduces that @t S Cu:r S  C .1Cjruj2 / S and since, S D 0 at time 0, one deduces that D 0 for all t and hence .u; / is a weak solution of (37).

4.3

Length Space

In (34), the configuration space for the polymer is a smooth manifold and the evolution of f is governed by a Fokker-Planck equation. A natural question is to try to extend this to cases where the configuration space M is less smooth, for instance, it is just a length space (see [10] for an introduction to these metric spaces). This is physically justified by the fact that the configuration space of the polymers is very complex and that different parts of the polymer may repulse each other (see [20,43]). In this case, one would like to replace the Fokker-Planck equation, which is a PDE, by a flow on the metric space P2 .M /, namely, the set of all Borel probability measures defined on M equipped with the Wasserstein distance (see [43]).

5

Closure Models

As described above, the micro-macro models present numerous challenges, simultaneously at the level of their derivation, the level of their numerical simulation, and that of their mathematical treatment. There are many macroscopic models called closure approximations that attempt to give a good approximation of these models. The advantage of these models is that they are easier to implement numerically. However, the disadvantage is that they are sometimes unable to describe all the physical properties of the original model (see [41, 42]). An approximate closure

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of the linear Fokker-Planck equation reduces the description to a closed viscoelastic equation for the added stress. This leads to well-known non-Newtonian fluid models such as the Oldroyd-B model that has been studied extensively. We refer to Guillopé and Saut [34, 35] who proved the existence of local strong solutions and FernándezCara, Guillén, and Ortega [29], [28], and [30] who proved local well-posedness in Sobolev spaces. In Chemin and Masmoudi [11], local and global well-posedness in critical Besov spaces were given. For global existence of weak solutions, one refers to Lions and Masmoudi [57]. One can also mention the work of Lin, Liu, and Zhang [52] where a formulation based on the deformation tensor is used to study the Oldroyd-B model. Global existence for small data was proved in [46, 48] and some non-blowup criteria were given in [47, 65]. Actually, the Oldroyd-B model is a macroscopic model that is exactly equivalent to the Hooke model in which each polymer is idealized as a linear “elastic dumbbell.”

5.1

The FENE-P Model

One of the most classical closure approximations of the FENE model is the one proposed by Peterlin, namely, the FENE-P model: 8 ˆ < @t u C .u  r/u  u C rp D div ; divu D 0; @t A C u:rA D ruA C A.ru/T  1trA.A/=b C Id ˆ A :  D .A/ D  Id: 1tr.A/=b

(45)

In (45), u is the velocity,  is the extra stress tensor due to the polymers, A is sometimes called the mean of the structure tensor, and  > 0 is the viscosity of @ui . Many other authors use the fluid. Here, one adopts the notation .ru/i;j D @x j the alternative convention. The system is considered in a domain that can be a bounded domain of RD , the whole space RD , or the torus TD . In the case the problem is considered in a bounded domain , one adds the Dirichlet boundary condition u D 0 on @ in which case there is no need to add a boundary condition for A. The second and third equations replace the Fokker-Planck equation in the FENE model and the expression of the stress tensor. Indeed, if oneR introduces the so-called mean of the structure tensor A which is given by Aij D B Ri Rj .t; x; R/dR, then A is a positive symmetric matrix, and multiplying the second equation of (8) by Ri Rj and integrating over B, one easily gets that A solves @t A C u:rA D ruA C A.ru/T  2

Z .R ˝ rU / .t; x; R/dR C 2ˇI: B

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If one chooses the constant appropriately, this becomes @t A C u:rA D ruA C A.ru/T   : Notice that for (8),  depends on the whole distribution function .t; x; R/. The FENE-P approximation consists is setting  D  .A/ D 1trA.A/=b  Id where b D R02 . It is worth noting that the FENE-P model has also a free energy that decays R  R in time [37]. Indeed, let H .t / D h D h1 .t; x/ C h2 .t; x/ C .b C D/  b  log bCD dx be given by h1 .t; x/ D  log.det A/;

h2 .t; x/ D b log.1  tr.A/=b/:

Using that @t det A D .det A/ tr.A1 @t A/, one gets .@t C u:r/h1 D tr.A1 / C

D : 1  tr.A/=b

(46)

Moreover, one has .@t C u:r/h2 D

2ru W A D tr.A/ C  1  tr.A/=b 1  tr.A/=b .1  tr.A/=b/2

(47)

and Z @t

juj2 D 2

Z

Z ru W   

jruj2 :

(48)



Adding (46), (47), and (48) yields the following formal decay of the free energy:  Z h.t; x/ juj2 C jruj2 C 2 2

 tr.A/ 1 2D 1 Ctr.A / D 0: C  2 .1  tr.A/=b/2 1  tr.A/=b

Z

@t

(49)

One can refer to Hu and Lelièvre [37] for this derivation. Recall that one had the following inequalities for positive symmetric matrices [37]: 2D tr.A/  C tr.A1 / 2 .1  tr.A/=b/ 1  tr.A/=b   log.det A/  b log.1  tr.A/=b/ C .b C D/ log



b bCD

 0:

(50) b Id . Notice that both terms vanish when .A/ D 0, namely, A D bCD Based on this decay, one can expect to construct global weak solutions such that u 2 L1 ..0; T /I L2 . // \ L2 ..0; T /I H01 . //,  2 L2 ..0; T /I L2 . //, A 2

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L1 ..0; T /  // and H .t / 2 L1 .0; T /. The fact that  is in L2 ..0; T /  // comes from the dissipation of the free energy that yields an L2 ..0; T /  // bound Db tr.A/ DCb on 1tr.A/=b . Indeed, one has

bD .b C D/ tr.A/  bCD tr. / D b 1  tr.A/=b and given a positive definite matrix A, the minimum of the left-hand side of (50) is attained when A D trD.A/ I since one always has that tr.A/tr.A1 /  D 2 , and hence, the free energy dissipation controls tr.A/ D2 Œ.D C b/tr.A/  Db 2 2D C D 2 :  2 .1  tr.A/=b/ 1  tr.A/=b tr.A/ b .1  tr.A/=b/2 tr.A/

(51)

Using this free energy estimate, one can prove the following existence of weak solutions result of the Leray type (see [62]): Theorem 7. Let u0 .x/ 2 L2 . / be a divergence free field and A0 .x/ a positive definite matrix function of x such that Z  log.det A0 /  b log.1  tr.A0 /=b/ C .b C D/ log.

b / < 1: bCD

(52)

Then, (8) has a global weak solution .u; A/ such that u 2 L1 .RC I L2 / \ P 1 /, A 2 L1 ..0; T /  //, and  2 L2 ..0; T /I L2 . //, and (49) holds L2 .RC I H with an inequality  instead of the equality. As for the FENE model, one can state two important open problems: Remark 3. (1) As mentioned above, a natural question is whether one can extend Theorem 7 to the Oldroyd-B model which corresponds to the Hookean spring model on the microscopic level. The main difficulty is that we do not know whether the extra stress tensor  is in L2 . One refers to [7] where global existence to the Hookean-type bead-spring chain model is proved when centerof-mass diffusion is taken into account, and the potential U .R/ grows faster than jRj2 when R goes to infinity. Moreover, in [19], global existence is proved for the Hooke model (which is equivalent to the macroscopic Oldroyd-B model) when the data is small in L1 . (2) Another natural question is whether one can prove global existence of smooth solutions and/or uniqueness of solutions to (45) or (53) in 2D. It should be pointed out that this is known for some co-rotational models [54,61]. This seems to be a difficult problem, since one only has an L2 bound on  and that an L1 bound on  was necessary in the previously mentioned works. In particular the

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similar result is not known for the co-rotational Oldroyd-B model where one can prove Lp bounds on  for each p > 1.

5.2

The Giesekus and the PTT Models

In the Giesekus model, the second equation in (45) is replaced by an equation on  : 8 ˆ < @t u C .u  r/u  u C rp D div; divu D 0;   ˆ :  @  C u:r  ru  .ru/T C  C ˛ C .D.u/ C  D.u//D2 D.u/ t 1 (53) where D.u/ D 12 .ru C .ru/T / is the symmetric part of ru. Here,  is a relaxation time, 1 > 0 is an extra viscosity, ˛ > 0 is a constant which measures the effect of the extra nonlinear term , and  is a constant that is typically in Œ0; 2 . In the case  D 0, one gets the upper-convective model and in the case  D , one gets the co-rotational model. In the Phan-Thien and Tanner (PTT) model, the second equation in (53) is replaced by .@t  C u:r  ru  .ru/T / C  C ˛tr. / D 21 D.u/

(54)

The main difference with the Giesekus model (53) is that the quadratic term ˛ is replaced by ˛tr. / . One can prove similar results to the one for the FENE-P model: Theorem 8. Take  D 0 and let u0 .x/ 2 L2 . / be a divergence free field. Let A0 .x/ D 1 0 .x/ C Id be a positive definite matrix function of x such that detA0  1, A0 2 L1loc . /, and Z trA0  d < C:

(55)



Then, (53) has a global weak solution .u;  / such that u 2 L1 .RC I L2 / \ P 1 /, tr. / 2 L1 ..0; T /I L1 . //, and  2 L2 ..0; T /I L2 . //. Moreover, L2 .RC I H the free energy bound holds with an inequality  instead of the equality.

6

Conclusion

Polymeric flows are very important models due to their many applications. Moreover, they provide a very rich family of interesting mathematical problems due to

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their important structures. To prove global existence of weak solutions (à La Leray), one has to use some recent developments in the theory of flows with Sobolev vector fields to prove the propagation of compactness and hence pass to the limit in the nonlinear term. The main structure behind this is the transport equation satisfied by the density of polymers. This transport structure is also shared by the mathematical analysis of non-Newtonian flows as well as compressible models. There are still many open problems concerning the study of global existence of weak solutions for many polymeric models. This is clearly an area that will see many new development in the future. Let us also mention problems related to asymptotic limit, namely, studying various limits and deriving new limit problems. Finally, the study of long time behavior should be also treated more.

7

Cross-References

 Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Equations for Viscoelastic Fluids  Low Mach Number Limits and Acoustic Waves  Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid

Crystal Flows  Variational Modeling and Complex Fluids

References 1. A. Arnold, J.A. Carrillo, C. Manzini, Refined long-time asymptotics for some polymeric fluid flow models. Commun. Math. Sci. 8(3), 763–782 (2010) 2. H. Bahouri, J.-Y. Chemin, Équations de transport relatives á des champs de vecteurs nonlipschitziens et mécanique des fluides. Arch. Ration. Mech. Anal. 127(2), 159–181 (1994) 3. J.W. Barrett, C. Schwab, E. Süli, Existence of global weak solutions for some polymeric flow models. Math. Models Methods Appl. Sci. 15(6), 939–983 (2005) 4. J.W. Barrett, E. Süli, Existence of global weak solutions to some regularized kinetic models for dilute polymers. Multiscale Model. Simul. 6(2), 506–546 (electronic) (2007) 5. J.W. Barrett, E. Süli, Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off. Math. Models Methods Appl. Sci. 18(6), 935–971 (2008) 6. J.W. Barrett, E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: finitely extensible nonlinear bead-spring chains. Math. Models Methods Appl. Sci. 21(6), 1211–1289 (2011) 7. J.W. Barrett, E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers II: Hookean-type bead-spring chains. Math. Models Methods Appl. Sci. 22(5) (2012, to appear) 8. R.B. Bird, R. Amstrong, O. Hassager, Dynamics of Polymeric Liquids, vol. 1 (Wiley, New York, 1977) 9. R.B. Bird, C. Curtiss, R. Amstrong, O. Hassager, Dynamics of Polymeric Liquids, Kinetic Theory, vol. 2 (Wiley, New York, 1987) 10. D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry. Volume 33 of Graduate Studies in Mathematics (American Mathematical Society, Providence, 2001)

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11. J.-Y. Chemin, N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33(1), 84–112 (electronic) (2001) 12. X. Chen, J.-G. Liu, Global weak entropy solution to Doi-Saintillan-Shelley model for active and passive rod-like and ellipsoidal particle suspensions. J. Differ. Equ. 254(7), 2764–2802 (2013) 13. L. Chupin, The FENE model for viscoelastic thin film flows. Methods Appl. Anal. 16(2), 217–261 (2009) 14. L. Chupin, Fokker-Planck equation in bounded domain. Ann. Inst. Fourier (Grenoble) 60(1), 217–255 (2010) 15. L. Chupin, Global existence results for some viscoelastic models with an integral constitutive law. SIAM J. Math. Anal. 46(3), 1859–1873 (2014) 16. P. Constantin, Nonlinear Fokker-Planck Navier-Stokes systems. Commun. Math. Sci. 3(4), 531–544 (2005) 17. P. Constantin, C. Fefferman, E.S. Titi, A. Zarnescu, Regularity of coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systems. Commun. Math. Phys. 270(3), 789–811 (2007) 18. P. Constantin, N. Masmoudi, Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D. Commun. Math. Phys. 278(1), 179–191 (2008) 19. P. Constantin, W. Sun, Remarks on Oldroyd-B and related complex fluid models. CMS 10, 33–73 (2012) 20. P. Constantin, A. Zlatos, On the high intensity limit of interacting corpora. Commun. Math. Sci. 8(1), 173–186 (2010) 21. P. Degond, M. Lemou, M. Picasso, Viscoelastic fluid models derived from kinetic equations for polymers. SIAM J. Appl. Math. 62(5), 1501–1519 (electronic) (2002) 22. P. Degond, H. Liu, Kinetic models for polymers with inertial effects. Netw. Heterog. Media 4(4), 625–647 (2009) 23. R.J. DiPerna, P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. (2) 130(2), 321–366 (1989) 24. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford, 1986) 25. Q. Du, C. Liu, P. Yu, FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul. 4(3), 709–731 (electronic) (2005) 26. W. E, T. Li, P. Zhang, Well-posedness for the dumbbell model of polymeric fluids. Commun. Math. Phys. 248(2), 409–427 (2004) 27. E. Feireisl, Dynamics of Viscous Compressible Fluids. Volume 26 of Oxford Lecture Series in Mathematics and Its Applications (Oxford University Press, Oxford, 2004) 28. E. Fernández-Cara, F. Guillén, R.R. Ortega, Some theoretical results for viscoplastic and dilatant fluids with variable density. Nonlinear Anal. 28(6), 1079–1100 (1997) 29. E. Fernández-Cara, F. Guillén, R.R. Ortega, Some theoretical results concerning nonNewtonian fluids of the Oldroyd kind. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26(1), 1–29 (1998) 30. E. Fernández-Cara, F. Guillén, R.R. Ortega, The Mathematical Analysis of Viscoelastic Fluids of the Oldryod Kind (2000) 31. J. Frehse, J. Málek, M. Ružiˇcka, Large data existence result for unsteady flows of inhomogeneous shear-thickening heat-conducting incompressible fluids. Commun. Partial Differ. Equ. 35(10), 1891–1919 (2010) 32. X. Gallez, P. Halin, G. Lielens, R. Keunings, V. Legat, The adaptive Lagrangian particle method for macroscopic and micro-macro computations of time-dependent viscoelastic flows. Comput. Methods Appl. Mech. Eng. 180(3-4), 345–364 (1999) 33. M. Grmela, H.C. Öttinger, Dynamics and thermodynamics of complex fluids. I and II. Development of a general formalism. Phys. Rev. E (3) 56(6), 6620–6655 (1997) 34. C. Guillopé, J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. 15(9), 849–869 (1990)

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60. H. Liu, J. Shin, Global well-posedness for the microscopic FENE model with a sharp boundary condition. J. Differ. Equ. 252, 641–662 (2012) 61. N. Masmoudi, Well-posedness for the FENE dumbbell model of polymeric flows. Commun. Pure Appl. Math. 61(12), 1685–1714 (2008) 62. N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows. J. Math. Pures Appl. (9) 96(5), 502–520 (2011) 63. N. Masmoudi, Regularity of solutions to the FENE model in the polymer elongation variable R (2011, in preparation) 64. N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows. Invent. Math. 191(2), 427–500 (2013) 65. N. Masmoudi, P. Zhang, Z. Zhang, Global well-posedness for 2D polymeric fluid models and growth estimate. Phys. D 237(10–12), 1663–1675 (2008) 66. J.-G. Oldryod, Non-newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. R. Soc. London Ser. A 245, 278–297 (1958) 67. H.C. Öttinger, Stochastic Processes in Polymeric Fluids: Tools and Examples for Developing Simulation Algorithms (Springer-Verlag, Berlin, 1996) 68. F. Otto, A.E. Tzavaras, Continuity of velocity gradients in suspensions of rod-like molecules. Commun. Math. Phys. 277(3), 729–758 (2008) 69. R.G. Owens, T.N. Phillips, Computational Rheology (Imperial College Press, London, 2002) 70. M. Renardy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations. Z. Angew. Math. Mech. 65(9), 449–451 (1985) 71. M. Renardy, An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal. 22(2), 313–327 (1991) 72. M. Renardy, Mathematical Analysis of Viscoelastic Flows. Volume 73 of CBMS-NSF Regional Conference Series in Applied Mathematics (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000) 73. M. Renardy, Global existence of solutions for shear flow of certain viscoelastic fluids. J. Math. Fluid Mech. 11(1), 91–99 (2009) 74. D. Saintillan, M.J. Shelley, Orientational order and instabilities in suspensions of selflocomoting rods. Phys. Rev. Lett. 99(5), 058102 (2007) 75. D. Saintillan, M.J. Shelley, Active suspensions and their nonlinear models. Comptes Rendus Physique 14(6), 497–517 (2013) 76. M.E. Schonbek, Existence and decay of polymeric flows. SIAM J. Math. Anal. 41(2), 564–587 (2009) 77. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edn. (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1995) 78. H. Zhang, P. Zhang, Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal. 181(2), 373–400 (2006) 79. L. Zhang, H. Zhang, P. Zhang, Global existence of weak solutions to the regularized Hookean dumbbell model. Commun. Math. Sci. 6(1), 85–124 (2008)

Modeling of Two-Phase Flows With and Without Phase Transitions

21

Jan W. Prüss and Senjo Shimizu

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 First Principles in the Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 First Principles on the Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Resulting Dynamic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Entropy and Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Equilibria as Critical Points of the Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Equilibria Which Are Maxima of Total Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Manifold of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Isothermal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Isothermal Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Available Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Equilibria as Critical Points of the Available Energy . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Analysis of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Transformation to a Fixed Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Strategies and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1008 1009 1010 1014 1019 1021 1023 1023 1024 1026 1027 1028 1028 1029 1030 1032 1032 1032 1039 1043 1043 1043 1044

J.W. Prüss () Institut für Mathematik, Martin-Luther-Universität, Halle-Wittenberg, Halle, Germany e-mail: [email protected] S. Shimizu Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, Japan e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_24

1007

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J. Prüss and S. Shimizu

Abstract

The purpose of this chapter is to explain the modeling of one-component twophase flows with moving interface in some detail. The models are derived from first principles, and some of the main structural properties are presented. In particular, the models are shown to be thermodynamically consistent, the equilibria are identified, and their thermodynamic stability properties are discussed. In addition, several analytical results in the incompressible case with phase transition are presented, which include topics like the short-time well-posedness, local semiflow, stability of equilibria, long-time existence, and convergence to equilibria.

1

Introduction

There is a large literature on isothermal incompressible Newtonian two-phase flows without phase transitions and also on the two-phase Stefan problem with surface tension modeling temperature-driven phase transitions. On the other hand, mathematical work on two-phase flow problems including phase transitions is rare. In this direction, there must be mentioned the papers by Hoffmann and Starovoitov [7, 8] dealing with a simplified two-phase flow model and Kusaka and Tani [10, 11] which are two-phase for temperature, but only one phase is moving. The papers of Di Benedetto and Friedman [3] and Di Benedetto and O’Leary [4] consider weak solutions of conduction-convection problems with phase change. However, none of these papers deals with models which are consistent with thermodynamics. The purpose of this chapter is to explain the modeling of one-component twophase flows with moving interface in some detail. As a guiding principle, it is assumed throughout that on the interface, there is no entropy production except that induced by surface heat flow. The chapter is organized as follows. In Sect. 2, models for two-phase flows with phase transitions are developed from first principles, e.g., conservation of mass, momentum, and energy in bulk, as well as conservation of energy on the interface. Surface mass and hence also surface momentum are excluded, here. The constitutive equations employed are fairly standard in the physical literature; they are based on the laws of Newton and Fourier. Section 3 is devoted to an entropy analysis, to determine the equilibria of the systems without external forces and heat sources and to show that the models are thermodynamically consistent. It is proved that the equilibria are precisely the critical points of the entropy functional with prescribed total mass and total energy and those which are thermodynamically stable are singled out. Section 4 deals with the isothermal case; here the total available energy is the guiding functional. Finally, in Sect. 5, some analytical tools and results are presented, which give a fairly complete picture of the behavior of the solutions in the incompressible case.

21 Modeling of Two-Phase Flows With and Without Phase Transitions

2

1009

Modeling

General references for the modeling are the monographs Drew and Passman [5] and Ishii and Hibiki [9], as well as the papers by Anderson et al. [1] and Bothe and Dreyer [2], as well as Chapter I of [13]. Suppose a (fixed) container  – a bounded domain in Rn with smooth boundary – is filled with a material which is present in two phases that occupy the regions 1 .t / and 2 .t / (Fig. 1). The interface .t / separating these two phases will depend on time t , but should not be in contact with the outer boundary @ of the container in order to avoid the contact angle problem. Then the so-called continuous phase 2 .t / is in contact with the outer boundary, while the disperse phase 1 .t / is not, which means that @1 .t / D .t / and @2 .t / D @ [ .t /. The outer unit normal of .t / w.r.t. 1 .t / will be denoted by  , it depends on p 2 .t / as well as on t ; the outer unit normal of  is called , it only depends on p 2 @. The Weingarten tensor L is defined by L WD r  , where r means the surface gradient, and the (.n  1/-fold) mean curvature H of .t / is defined by H D tr L D div   ; where div means the surface divergence on . In the sequel, the jump of a physical quantity  across  will be denoted by ŒŒ.p/ WD lim Œ.p C s .p//  .p  s .p//; s!0C

Fig. 1 A typical geometry

p 2 :

∂Ω Ω（t) ₁ Ω（t) ₁

v

Ω（t) ₂

Γ

Ω（t) ₁

1010

2.1

J. Prüss and S. Shimizu

First Principles in the Bulk

The first issue is the basic balance laws in the bulk.

2.1.1 Balance of Mass Let % > 0 denote the density and u the velocity in the bulk phases j D j .t /, u the velocity and V WD u   the normal velocity of , respectively. Note that % and u may jump across the interface  and that u is in general not a tangent vector field to . If there are no sources of mass in bulk, then conservation of mass is given by the continuity equation: @t % C div .%u/ D 0 in  n .t /:

(1)

If there is no surface mass on , the following jump condition holds. ŒŒ%.u  u /    D 0 on .t /:

(2)

The interfacial mass flux j , phase flux for short, is defined by means of: j WD %.u  u /   ;

1 ŒŒ j D ŒŒu   : %

i.e.,

(3)

Observe that j is well defined, as (2) shows. Phase transition takes place if j 6 0. On the other hand, if j  0, then u   D u   D V , and in this case the interface is advected with the velocity field u. Next, by the transport theorem for moving domains, d dt

Z

Z

Z

% dx D 1 .t/

%V d  C Z.t/

@t % dx Z 1 .t/

%V d  

D Z.t/ D

div.%u/ dx 1 .t/

Z

.%u    %u   / d  D  .t/

and in case u   D 0 on @ in the same way d dt

j d ; .t/

Z

Z % dx D 2 .t/

proving conservation of total mass, i.e.,

j d ; .t/

21 Modeling of Two-Phase Flows With and Without Phase Transitions

d dt

1011

Z % dx D 0:

(4)

.t/

A fluid is called completely incompressible if the densities are constant in the phases j . Then conservation of mass reduces to div u D 0

in  n .t /:

If only the latter property holds, the fluid is said to be incompressible. In case both phases are completely incompressible, %1 j1 .t /j C %2 j2 .t /j  %1 j1 .0/j C %2 j2 .0/j DW c0 : This implies ŒŒ%j1 .t /j D %2 jj  c0 I hence j1 .t /j is constant in the case of nonequal densities, i.e., the phase volumes are preserved. On the other hand, there is no preservation of phase volumes in general, if one or both phases are compressible or if the densities are constant and equal.

2.1.2 The Universal Balance Law Let  be any (mass-specific) physical quantity, J its flux, and f its sources. Then the balance law for  in the bulk reads @t .%/ C div.%u C J / D %f

in  n .t /;

(5)

and if there is a source f for  on the interface, one has ŒŒ.%.u  u / C J /    D f

on .t /:

(6)

Employing balance of mass and the definition of the phase flux j , this simplifies to %.@t  C u  r/ C div J D %f ŒŒj C ŒŒJ    D f

 n .t /;

in

on .t /:

The corresponding universal transport theorem becomes d dt

Z

Z

Z

% dx D 

@t .%/ dx  Z



ŒŒ%V d  

Z

.%f  div .%u C J // dx 

D 

ŒŒ%u    d  

(7)

1012

J. Prüss and S. Shimizu

Z

Z

D

%f dx C

ŒŒ.%.u  u / C J /    d 





Z



.%u C J /   d .@/ @

Z D

Z

Z

%f dx C 

.ŒŒj C ŒŒJ   / d  C 

g d .@/; @

with g D .%u C J /   on @. This yields the conservation law d dt

Z

Z

Z

% dx D 

%f dx C 

Z f d  C



g d .@/: @

In particular, if .f; f ; g/ D 0, then the total amount of  in  is conserved.

2.1.3 Balance of Momentum Let  denote the pressure, T the (symmetric) stress tensor, and let f be a force field, say gravity. Then balance of momentum reads, employing (5) with  D u and J D T , @t .%u/ C div .%u ˝ u/  div T D %f

in  n .t /:

Similarly, using (6), the following jump condition at the interface is valid. ŒŒ.%u ˝ .u  u /  T /    D div  T

on .t /:

Here T denotes the (symmetric) surface stress, a tensor field on . Using balance of mass and the definition of the phase flux j , these conservation laws may be rewritten as follows: %.@t u C u  ru/  divT D %f ŒŒuj  ŒŒT   D div T

in

 n .t /;

on

.t /:

(8)

By the surface divergence theorem, total conservation of momentum reads as d dt

Z

Z %u dx D



Z %f dx C



g d .@/ @

with g D .%uu    T / on @. Note that total momentum is in general not conserved as the boundary term g on @ need not be zero.

2.1.4 Balance of Energy Let  denote the (mass-specific) internal energy density, > 0 the absolute temperature, q the heat flux, and r an external (mass-specific) heat source. Then

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1013

with  D juj2 =2 C  and J D T u C q, the universal balance law (5) implies conservation of energy, which in the bulk reads @t

%

 n %  o juj2 C % C div juj2 C % u  div.T u  q/ 2 2

D %f  u C %r

in  n .t /: On the interface, (6) yields hh % 2

 ii juj2 C % .u  u /  T u C q   D .div T /  u C r

on .t /; (9)

where r denotes a heat source on . Using (1), (8), and the definition of the phase flux j , this conservation law may be rewritten as follows: 

%.@t  C u  r/ C div q  T W ru D %r  1 ŒŒ C ŒŒ ju  u j2  j  ŒŒT  .u  u / C ŒŒq    D r 2

in

 n .t /;

on

.t /: (10)

The total bulk energy is given by Eb .u; ; / WD

1 2

Z

%juj2 dx C n

Z % dx: n

For its time derivative we obtain from the universal balance law d Eb D dt

Z

Z .%f  u C %r/ dx C



Z g d .@/ C

@

fdiv T  u C r g d ; 

  where g D  . %2 juj2 C %/u    T u   C q   on @. In particular, if .f; r/ D 0 in , .u  ; q  ; T   u/ D 0 on @, as well as div T  u C r D 0 on , then d Eb .u; ; / D 0; dt which means that the total bulk energy is preserved.

2.1.5 The Entropy As is common in thermodynamics, .%; / D

.%; / C .%; /;

.%; / D @ .%; /;

(11)

1014

J. Prüss and S. Shimizu

where is the Helmholtz free energy density, here it is considered as given. means the (mass-specific) entropy density. Then the Clausius-Duhem equation holds in the bulk, which means @t .% / C div.% u/ C div .q= / D

1 1 S W ru  2 q  r C

%2 @%



div u

(12) in  n .t /;

where S WD T C  denotes the viscous stress tensor. Therefore, entropy is nondecreasing locally in the bulk provided the right-hand side of (12) is nonnegative. This gives the well-known requirements S W ru  0;

q  r  0;

(13)

and, since in general the last term will not have a sign, either div u  0, which corresponds to the incompressible case, or  D p.%; / WD %2 @% .%; /;

(14)

which is the famous Maxwell relation for compressible materials. Note that p should be an increasing function in both variables, % and . Hence the conditions, @% @

 0;

2@%

C %@2%

 0;

%; > 0;

are imposed in the compressible case. The total bulk entropy is defined by Z Nb .%; ; / D % .%; /dx: 

By the universal balance law, d Nb .%; ; / D dt

Z n



Z 1 1 S W ru  2 q  r dx C fŒŒ j C ŒŒq=    gd ; 

provided u   D q   D 0 on @. In particular, there is no entropy production on the interface if ŒŒ j C ŒŒq=    D 0

2.2

on :

First Principles on the Interface

Throughout, it is assumed that there is no surface mass and therefore also no surface momentum on . However, due to surface tension, surface energy and surface entropy have to be taken into account. A basic principle of the present approach is the conservation of energy and entropy across the interface.

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1015

2.2.1 The Universal Balance Law on the Interface Suppose  is a scalar physical quantity which also lives on  with surface density  and let J denote its flux. Thus, J is a tangent vector field to . The basic balance law for  reads D  C  div u C div J D f : Dt

(15)

Here D=Dt means the Lagrangian derivative with respect to the vector field u which moves , i.e., ˇ D d ˇ  .t; / D  .s C t; x.s C t; t; //ˇsD0 ; Dt ds with x.s C t; t; / the flow induced by the velocity field u , i.e., d x.s C t; t; / D u .s C t; x.s C t; t; //; ds

x.t; t; / D ;  2 .t /:

It should be emphasized that the velocity field u is in general not tangent to . The surface transport theorem then yields d dt

Z 

Z  d  D 

Z



D

D  C  div u Dt

 d Z

.div J  f / d  D  

f d ; 

by the surface divergence theorem. Therefore, conservation of the total amount of  in  is valid, which means d dt

Z



Z % dx C 

 d  D 0; 

provided .f; g/ D 0. Thus the balance law for  on  reads D  C  div u C div J D .ŒŒJ C ŒŒJ    /: Dt

(16)

The interface conservation law is first applied to:

2.2.2 Conservation of Energy on the Interface Here  D  and J D T u C q , where  denotes the surface energy density and q denotes the heat flux on the interface. Then balance of surface energy reads D  C  div u C div .q  T u / D f.div T /  u C r g; Dt

1016

J. Prüss and S. Shimizu

hence by (9) D  C  div u C div q D T W r u  r : Dt R The total surface energy E is defined by E D   d . By the conservation laws, this implies conservation of total energy E D Eb C E d d ED dt dt

Z

 %



juj2 C 2



 d  D 0;

Z dx C 

provided .f; r/ D 0 in , u   D q   D 0, and T   u D 0 on @.

2.2.3 Surface Entropy As in the bulk,  .  / D

 .  /

where the free surface energy Similarly, we decompose

C   .  /; 

 .  / D 

0  .  /;

is a given function of surface temperature  .

T D .  /P C S ; where denotes the coefficient of surface tension, P D I  ˝ the orthogonal projection onto the tangent bundle of , and S the interface viscous stress. Then surface force becomes div T D H  C r C div S : The first term in this decomposition is surface tension which acts in normal direction, the second is called Marangoni force which acts tangentially, and the last one is the viscous surface force induced by surface viscosity. The total surface entropy is given by Z N D

 d : 

With the surface transport theorem, d N D dt

Z  

Z

D

D

 C  div u Dt

.div q 



 div u

Z  d D 

 D  C   div u =  d  Dt

C T W r u  r /=  d 



Z D





S W r u =   q  r  = 2 C . 

 /div u = 

  r =  d :

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1017

Now one argues as in the bulk case. To ensure entropy production on the interface, the conditions S W r u  0;

q  r   0;

as well as 

D ;

must be valid, where the latter is the analogue of the Maxwell relation on the interface. Thus in the situation considered here, the free energy on the interface is the coefficient of surface tension, which acts as a negative surface pressure. For the total entropy N D Nb C N in , this finally yields Z  Z  Z d 1 1 d ND S W ru  2 q  r dx % dx C  d  D dt dt    Z  1 1 (17) S W r u  2 q  r  d  C    Z C fŒŒ j C ŒŒq=     r =  g d : 

The integrand of the last integral must also be nonnegative, as it is independent from that of the second integrand. This implies  

 1 ŒŒ C ŒŒ ju  u j2  j  ŒŒT  .u  u / 2  CŒŒq    =  C ŒŒ j C ŒŒq=     0;

which by employing the phase flux j and the projection P can be written as 

 T  1 1 2 2 ŒŒ  C ŒŒ 2 j  ŒŒ  C ŒŒ .   / C ŒŒjP .u  u /j  j =  2

2   ŒŒP S   P .u  u / 1 1   C ŒŒq    0  

As these three terms are independent, they must be nonnegative, separately. Setting 1 D  , 2 D  leads to  ŒŒq  

     1 1 1 1 1 1 C q2   2 ;     D q1  1  1  2 

1018

J. Prüss and S. Shimizu

hence both terms must be nonnegative, as they are independent, coming from different phases. Consequently,  qi   i

1 1  i 

  0;

i D 1; 2;

and similarly P Si i  P .ui  u /  0;

i D 1; 2:

If there is no entropy production on the interface, these terms must vanish; hence in the simplest case, i D 

and

P .ui  u / D 0;

i D 1; 2;

which is equivalent to ŒŒ  D 0;

 D

and

ŒŒP u D 0;

P u D P u

on

;

and finally   1 ŒŒ  C ŒŒ 2 j2  ŒŒT    =% j D 0: 2%

(18)

This is the generalized Gibbs-Thomson relation. It holds trivially if j  0, i.e., if there is no phase transition, and otherwise it reads

ŒŒ  C ŒŒ

1 2 j  ŒŒT    =% D 0: 2%2 

(19)

This is the Gibbs-Thomson relation. Taking it for granted, the balance of surface energy becomes D  C  div u C div q D S W r u C div u  .ŒŒ j C ŒŒq   /: Dt (20)

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1019

On the interface, the Clausius-Duhem equation reads 1 1 D

 C  div u C div .q =  / D S W r u  2 q  r  Dt  

(21)

 .ŒŒ j C ŒŒq=    /; showing that surface entropy production is nonnegative, locally, on . Note that in case .  ; q ; S / D 0 on , this equation reduces to the famous Stefan condition: ŒŒ j C ŒŒq    D 0:

  0 means  D  const ant and  D . In this case total surface energy becomes jj, and surface energy balance is trivial. Remark. If one would allow for entropy production on the interfaces besides those coming from friction and heat conduction, linear closing relations of the form  qi  i D ˛i

1 1  i 

 ;

P Si i D ˇi P .ui  u /; i D 1; 2;

and ŒŒ  C ŒŒ

T  1 1 2 j  ŒŒ  C ŒŒ .   / C ŒŒjP .u  u /j2  D  j ; 2 2

2

would be reasonable, where ˛i ; ˇi  0 may depend on ; i , and   0 on  .

2.3

Constitutive Laws

In the sequel it is assumed that there are no external sources for momentum and energy, i.e., .f; r/ D 0.

2.3.1

Constitutive Laws on the Outer Boundary q D0

2.3.2

and

u D 0:

(22)

Constitutive Laws in the Phases

.%; / D

.%; / C .%; /;

.%; / D @ .%; /;

T D 2s .%; /D C b .%; /.div u/I  I; q D d .%; /r :

DD

 1 ru C .ru/T ; 2 (23)

1020

J. Prüss and S. Shimizu

Here s is called shear viscosity, b bulk viscosity, and d is the coefficient of heat conduction or heat conductivity. The viscosities s ; b ; d are functions depending on .%; /, and on the phase, and hence may jump across the interface .t /. The second and the third equations are the classical Newton’s law and Fourier’s law. To meet the requirements (13) we assume s .%; /; d .%; / > 0;

b .%; / C 2s .%; /=n > 0;

%; > 0;

and in the compressible case also the Maxwell relation (14).

2.3.3

Constitutive Laws on the Interface  .  / D .  / C   .  /; ŒŒ  D 0; ŒŒP u D 0;

 .  / D  0 .  /;

 D ; P .u  u / D 0;

T D .  /P C 2s .  /D C b .  /.div u /P ;

(24)

  1 D D P r u C Œr u T P ; q D d .  /r  ; 2   1 ŒŒ  C ŒŒ 2 j2  ŒŒT    =% j D 0: 2% The coefficient of surface tension and the surface viscosities .s ; b / are functions of  , which are subject to .  /; s .  / > 0;

b .  / C

2s .  / > 0: n1

Recall the relation V WD u   D u   

1 j ; %

for the normal velocity of the interface. In case ŒŒ% ¤ 0, this implies ŒŒu D ŒŒ1=%j  ;

j D ŒŒu   =ŒŒ1=%;

V D ŒŒ%u   =ŒŒ%;

(25)

and if ŒŒ% D 0, we have ŒŒu D 0. This shows a fundamental difference between these cases: if the densities are not equal, then the phase flux enters directly the velocity jump on the interface, inducing what is called Stefan current. If the densities are equal, there is no Stefan current and the velocity field is continuous across the interface. On each side of the interface, one has the identity u D u C j  =%;

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1021

which, in view of the definition of the phase flux j , is equivalent to the conditions ŒŒP u D 0;

P .u  u / D 0;

ŒŒ%.u  u /    D 0:

The heat capacity  and the surface heat capacity  are defined as usual by .%; / D @ .%; / D  @2 .%; /;

 .  / D 0 .  / D   00 .  /;

respectively. Moreover, the latent heat l and the surface latent heat l are defined by l.%; / D ŒŒ .%; / D ŒŒ @ .%; /; The conditions @2

2.4

l .  / D    .  / D  0 .  /:

 0 as well as 00  0 will be needed for well-posedness.

The Resulting Dynamic Problem

Summarizing, the following initial-boundary value problem arises, in the absence of external forces and heat sources.

2.4.1 The General Case If the phase flux j is nontrivial, there results the following problem: @t % C div.%u/ D 0

in

 n .t /;

%.@t u C u  ru/  div S C r D 0

in

 n .t /;

uD0

on

@;

on

.t /;

on

.t /;

in

;

ŒŒu D ŒŒ1=%j  ŒŒ1=%j2   ŒŒT   D div T %.0/ D %0 ;

u.0/ D u0

(26)

where S D T C  and T D .  /P C S are defined above, %.@t C u  r /  div.d r / D S W ru  @ p div u

in

 n .t /;

on

@;

D 

on

.t /;

.0/ D 0

in

:

@ D 0

On the interface, the following equations hold.

(27)

1022

J. Prüss and S. Shimizu



D   div .d r  / D S W r u Dt C  0 .  /div u C l. /j C ŒŒd . /@  ŒŒ  C ŒŒ1=2%2 j2  ŒŒT    =% D 0 V D u   D u    j =%

on

.t /;

on

.t /;

on

.t /

(28)

.0/ D 0 : This system has to be supplemented with the constitutive laws for T and T from the previous subsection. Here the first system should be read as a problem for u and %, respectively, , the second as one for , while the last set determines  , the free boundary , and the phase flux j .

2.4.2 Without Phase Transition In case the phase flux j is absent, there results the following problem: @t % C div.%u/ D 0

in

 n .t /;

%.@t u C u  ru/  div S C r D 0

in

 n .t /;

uD0

on @;

ŒŒu D 0

on .t /;

ŒŒT   D div T

on .t /;

%.0/ D %0 ;

u.0/ D u0

in

(29)

;

where S D T C  and T D .  /P C S are defined above, %.@t C u  r /  div.d r / D S W ru  @ p div u

in

 n .t /;

on

@;

D 

on

.t /;

.0/ D 0

in

:

on

.t /;

on

.t /

@ D 0

(30)

On the interface, the following equations hold. 

D   div .d r  / Dt D S W r u C  0 .  /div u C ŒŒd . /@  V D u  

.0/ D 0 :

(31)

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1023

This is the model for non-isothermal two-phase flows with surface tension, surface viscosity, and Marangoni forces. The classical model is obtained by setting S D 0 and D const:

3

Entropy and Equilibria

3.1

The Entropy

As shown above, the total entropy Z

Z % dx C

ND 

 d  

satisfies Z

d . dt

Z % dx C



Z

 d / D



Z

1 1 f S W ru  2 q  r g dx 

C

f 

1 1 S W r u  2 q  r  g d   0  

Hence the negative total entropy is a Lyapunov functional for the problem. But it is even a strict one. To see this, assume that N is constant on some interval .t1 ; t2 /. Then d N=dt D 0 in .t1 ; t2 /; hence D D 0 and r D 0 in .t1 ; t2 /  . Therefore, is constant, which implies ŒŒd @  D 0, and then the interfacial boundary condition yields j D 0, provided ŒŒ  ¤ 0 on ; we assume this for the moment. This implies ŒŒu D 0; hence by Korn’s inequality, ru D 0 and then u D 0 by the no-slip condition on @. Hence V D 0, u D 0, and .@t ; @t u; @t %; D  =Dt / D 0, which means that the system is in equilibrium. Further, r D 0, i.e., the pressure is constant in the components of the phases. If one or both phases are compressible, then assuming pj to be strictly increasing in %, one concludes that % is constant in the components of j .t / as well. Actually % is even constant in each phase. To see this, employing Maxwell’s relation, rewrite the Gibbs-Thomson condition ŒŒ  C ŒŒ=% D 0 as @% .%

1 .%//

D @% .%

2 .%//:

Suppose %2 is known; then %1 is uniquely determined by %2 (and ) since @% .% j .%// is strictly increasing, for, by assumption, pj has this property. Since is continuous across the interface, the last relation shows that  and therefore % are constant in all of 1 , even if it is not connected. From this, one finally deduces by the Young-Laplace law ŒŒ D H that 1 is a ball if it is connected or otherwise a finite union of nonintersecting balls of equal radii, since 1 is bounded by assumption.

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J. Prüss and S. Shimizu

If, by chance, ŒŒ  D 0 on , or only on part of it, one is not allowed to use Korn’s inequality since u may have a jump across the interface. Nevertheless, u D 0 holds in this case as well, but the proof is a little more involved. For this the following lemma is needed (see [13] for a proof): Lemma 1. Suppose u 2 H22 . n / satisfies u D 0 on @ and P ŒŒu D 0 on . Then D D 0 implies u D 0 in . Having shown that u D 0, one may proceed as before, provided %1 ¤ %2 . Actually, there is a problem if ŒŒ% D ŒŒ  D 0; then one cannot conclude j D 0 which means that V may be nontrivial. This pathology is excluded in the sequel. It is absent anyway if in the modeling, some additional dissipative processes on the interface, like kinetic undercooling, are taken into account. If there is no phase transition, i.e., j  0, then ŒŒu D 0, and one obtains directly u  0 by Korn’s inequality. In this case one may conclude as above that the pressures are constant in the components of the phases; hence the densities are so as well, assuming as before that pj is increasing. The interface stress condition further shows that H is constant on each component of the interface, which implies that these components are spheres. But they may have differing sizes, as the GibbsThomson relation is no longer available. If phase transition is absent, constant temperature does no longer ensure that the spheres have equal size! Note that in this case, the disperse phase might contain bubbles of arbitrarily small radius. For such balls the inside pressure can be arbitrarily large, but then physically phase transition will occur. Therefore, from the physical point of view, the assumption j  0 is not reasonable; phase transitions have to be taken into account.

3.2

Equilibria as Critical Points of the Entropy

The next goal is to determine the critical points of the total entropy N under the constraints of given total mass M0 and given total energy E0 . With Z

Z % dx;

MD 

%.juj2 =2 C / dx C

ED 

Z  d ; 

the method of Lagrange multipliers then yields N0 C M0 C E0 D 0: The derivatives of the involved functionals are, with z D . ; v; #; # ; h/ , 0

Z

hN jzi D

Z f@% .% / C %@ #g dx 



hM0 jzi D

Z

Z  dx 





ŒŒ%h d ; 

fŒŒ% h  0 # C  H hg d ;

21 Modeling of Two-Phase Flows With and Without Phase Transitions

hE0 jzi D

Z

1025

f%u  v C %@ # C .juj2 =2 C  C %@% / g dx



Z

 

fŒŒ%juj2 =2 C %h  0 # C  H hg d :

Varying first # and # , this yields %@ C %@  D 0; and

0 C 0 D 0; hence @  D @ D  > 0 and 0 D  0 D  > 0 imply  D D 1= > 0 is constant. Next vary v to obtain u D 0 since  ¤ 0. Variation of  (when % is not a priori constant) implies similarly

C %@% C  C . C %@% / D 0; hence  D .

C %@% /= . As a consequence % is constant, since

0 < @% p.%; /=% D 2@% .%; / C %@2% .%; / D @% . .%; / C %@% .%; // in a phase where % is not a priori constant. In particular, if both phases are compressible, this yields ŒŒ C p=% D 0, which is the generalized Gibbs-Thomson relation at equilibrium. Finally, vary h to obtain ŒŒ%    H  ŒŒ% C .ŒŒ% C  H /= D 0; which by the definition of  and



D yields

H C ŒŒ%  D  ŒŒ% on the interface . This implies that H is constant; hence 1 consists of a finite number of balls with the same radius. If both phases are compressible, one may further conclude H D ŒŒp, which is the normal stress condition on the interface. In this derivation,  > 0 is assumed. If instead   0, then 0  0 as well; hence no information on  is obtained. However, the remaining conclusions are valid as before. In this case .  / is linear, and as there is no surface heat capacity, it makes sense then to ignore surface diffusion as well. In summary, the critical points of the total entropy with the constraints of given mass and prescribed total energy are precisely the equilibria of the system.

1026

3.3

J. Prüss and S. Shimizu

Equilibria Which Are Maxima of Total Entropy

Let an equilibrium e WD .%; u; ;  ; / be given, where the total entropy has a local maximum, w.r.t. the constraints M D M0 and E D E0 constant. Then D WD ŒN C M C E00 is negative semi-definite on the kernel of M0 intersected with that of E0 , where .; / are the fixed Lagrange multipliers found in the previous subsection. The kernel of M0 .e/ is easily seen to be characterized by the relation Z

Z  dx D ŒŒ%

h d ;



(32)



and that of E0 .e/ by Z

Z

Z

@% .% / dx C 

Z

.%= /# dx C . = / 

# d  D .ŒŒ%  C  H / 

h d : 

(33)

On the other hand, a straightforward but somewhat lengthy calculation yields: Z  hDzjzi D

%jvj2 dx C



Z 

Z

C . = / 

@2% .% / 2 dx C

#2 d  

Z 

Z

.%= /# 2 dx

(34)



.H0 h/h d :

As %, ,  , and @2% .% / D 2@%

C %@2% D Œ@% p.%/=%

are nonnegative, the form hDzjzi is negative semi-definite as soon as H0 is negative semi-definite. In the next section, it is shown that H0 D .n  1/=R2 C  ; where  denotes the Laplace-Beltrami operator on  and R means the radius of the equilibrium spheres. In the next step, necessary conditions for an equilibrium e to be a local maximum of entropy are derived: 1. Suppose that  is not connected, i.e.,  consists of a finiteP union of spheres k . Set . ; v; #; # / D 0, and let h D hk constant on k with k hk D 0. Then the constraints (33) and (34) hold and hDzjzi D

. /.n  1/ X 2 hk > 0; R2 k

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1027

hence D is not negative semi-definite in this case. Thus if e is an equilibrium with local maximal total entropy, then  must be connected; hence both phases are connected. This is related to the so-called Ostwald ripening effect. 2. Assume that  is connected andR %1 ¤ %2 are a priori constant. Then  D 0 and the first constraint (33) implies  h d  D 0. As H0 .h/ is negative semi-definite for functions with average zero, in this case, D is negative semi-definite. 3. Assume that  is connected and %1 D %2 DW % is constant. Then  D 0, but the first constraint gives no information. Setting v D 0, # D # constant, as well as h constant, D negative semi-definite on the kernel of E0 .e/ implies the condition l02 jj . /.n  1/ ;  R2 ..j%/ C  jj/

(35)

where l0 D l0 . / D  .%ŒŒ  C  H /. 4. If e is an equilibrium which (locally) maximizes the total entropy, it is generically not isolated. If the sphere  does not touch the outer boundary, one may move it inside of  without changing the total entropy. This fact is reflected in D by choosing  D # D # D 0 and h D Yj , the spherical harmonics for , which satisfy H0 Yj D 0. It turns out that in the completely incompressible case, an equilibrium is stable if and only if the total entropy at this equilibrium is locally maximal. Thus in case %1 ¤ %2 is a priori constant, an equilibrium is stable if and only if the interface is connected and in case %1 D %2 if in addition the stability condition (35) is satisfied with strict inequality. (Here the limiting case where in (35) equality holds is excluded.)

3.4

The Manifold of Equilibria

As shown above, the equilibria of the system (26), (27), (28) are zero velocities, constant pressures in the phases, and constant temperature, vanishing phase flux, and the disperse phase 1 consists of finitely many nonintersecting balls with the same radius if a phase transition is present. We call an equilibrium nondegenerate if the balls do not touch the outer boundary @ and also do not touch each other. This set will be denoted by E; it is a manifold, not connected in general, but it has infinitely many finite dimensional components. These are given by the number of spheres. The dimension of the component consisting of m spheres is m.n C 1/, where n comes from the center and 1 from the radius of a particular sphere. To prove that E is a manifold, it only has to be shown how a neighboring sphere is parameterized over a given one. In fact, assume that † D SR .0/ is centered at the origin of Rn . Suppose S   is a sphere that is sufficiently close to †. Denote by .y1 ; : : : ; yn / the coordinates of its center and let y0 be such that R C y0 corresponds to its radius. Then the sphere S can be parameterized over † by the distance function

1028

J. Prüss and S. Shimizu

ı.y/ D

n X

v 12 u0 u X n n X u 2 @ A t yj Yj  R C yj Yj C .R C y0 /  yj2 ;

j D1

j D1

j D1

where Yj are the spherical harmonics of degree one. Obviously, this is a real analytic parameterization. These considerations are summarized in: Theorem 1. (a) The total mass M and the total energy E are preserved for smooth solutions. (b) The negative total entropy N is a strict Lyapunov-functional except on the pathological points .%; / constant, ŒŒ% D ŒŒ  D 0. (c) The critical points of the entropy functional for prescribed total mass and total energy are precisely the equilibria of the system. (d) The nondegenerate equilibria are zero velocities, constant temperature, and constant pressures in the components of the phases, and the interface is a union of nonintersecting spheres which do not touch the outer boundary @. If phase transition takes place, then the spheres are of equal size. (e) If the total entropy at an equilibrium is locally maximal, then the phases are connected and, in addition, in the case of equal constant densities the stability condition (35) holds. (f) The set E of nondegenerate equilibria forms a real analytic manifold. This result shows that the models are thermodynamically consistent, hence are physically reasonable.

4

The Isothermal Case

Here problems (26), (27) and (28) are reformulated in the isothermal case, by setting D const and ignoring energy balance. In this case, the entropy has to be replaced by the available energy.

4.1

The Isothermal Problem

4.1.1 The General Case If the phase flux j is nontrivial, there results the following problem: @t % C div.%u/ D 0

in

 n .t /;

%.@t u C u  ru/  div S C r D 0

in

 n .t /;

uD0

on

@;

in

;

%.0/ D %0 ;

u.0/ D u0

(36)

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1029

where S D T C  and T D .  /P C S . On the interface, ŒŒu D ŒŒ1=%j  ŒŒ1=%j2   ŒŒT   D div T ŒŒ  C ŒŒ1=2%2 j2  ŒŒT    =% D 0 V D u   D u    j =%

on

.t /;

on

.t /;

on

.t /;

on

.t /

(37)

.0/ D 0 :

4.1.2 Without Phase Transition In case the phase flux j is absent, there results the following problem: @t % C div.%u/ D 0

in

 n .t /;

%.@t u C u  ru/  div S C r D 0

in

 n .t /;

uD0

on

@;

in

:

%.0/ D %0 ;

u.0/ D u0

(38)

On the interface, ŒŒu D 0

on

.t /;

ŒŒT   D div T

on

.t /;

V D u  

on

.t /

(39)

.0/ D 0 : In the case S D 0, i.e., T D P , this is the classical isothermal model for two-phase flows.

4.2

The Available Energy

The density of the mass-specific available energy ea is defined by ea D

1 2 juj C 2

.%/:

Denoting the flux of the total mass-specific energy e by e and that of the entropy  , in the phases, leads to %.@t C u  r/e C div e D 0; %.@t C u  r/ C div  D r= :

1030

J. Prüss and S. Shimizu

This yields by a simple computation %.@t C u  r/ea C div.e   / D r  % @t    r ; so the flux of the available energy is a WD e   . In particular, in the isothermal case where D const, hence also , this yields %.@t C u  r/ea C div a D r; and so with a D q  T u  q D T u and r D S W ru, Z Z Z  d

%ea dx C jj D  S W ru dx  S W r u  0: dt    This shows that the total available energy Z Ea D %ea dx C jj 

is nonincreasing, i.e., it is a Lyapunov functional for the isothermal problem. But it d is also a strict Lyapunov functional. In fact, if dt Ea D 0 in some interval .t1 ; t2 /, then D  0 on  n .t /, D D 0 on .t /, hence u D 0 by Lemma 1. This implies @t % D 0, @t u D 0 and  is constant in each component of the phases. Balance of normal stress and the Gibbs-Thomson relation on .t / yield ŒŒ D H

ŒŒ=% C ŒŒ  D 0;

and

which as in Sect. 2 implies that % is constant in the phases, and  consists of finitely many disjoint spheres of equal size, as the functions j .%/ C % j0 .%/ are strictly increasing. This shows that the system is in equilibrium.

4.3

Equilibria as Critical Points of the Available Energy

(i) Next the critical points of the available energy Ea with given total mass M D M0 are determined. The method of Lagrange multipliers yields E0a  M0 D 0 for some  2 R. With z D . ; v; h/ one has hM0 jzi D

Z

Z  dx  

hE0a jzi

Z

ŒŒ%h d  

f%u  v C .juj2 =2 C

D

C %@% /g dx



Z

.ŒŒ%ea h C H h/ d ;

 

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1031

hence Z

f%u  v C .juj2 =2 C 

C %@%

 /g dx

Z



.ŒŒ%ea  C H  ŒŒ%/h d  D 0

for all z:



Varying first v leads to u D 0, hence ea D , and then varying  yields  D C %@% is constant in . Finally, varying h implies ŒŒ%  C H D ŒŒ% D ŒŒ%  C ŒŒ%2 @% ; hence H D ŒŒ%2 @%  D ŒŒ, i.e., the Young- Laplace law, and ŒŒ  C ŒŒ=% D 0, which is the Gibbs-Thomson relation. This implies that the densities are constant in the phases; hence the pressures are so, as well, and finally the curvature H is constant. This shows that each critical point of Ea with constraint M D M0 is an equilibrium. (ii) Assume that the critical point .%; 0; / is a local minimum of the available energy Ea with constraint M D M0 . The kernel of M0 is given by Z Z  dx D ŒŒ% h d ; (40) 



as before and hDzjzi WD h.E00a  M00 /zjzi here becomes, after a short computation, Z Z Z 2 2 2 0  hDzjzi D %jvj dx C @% .% / dx  H0 hh d : 





Clearly, as % > 0 and @2% .% / D  0 .%/=% > 0, the first two terms are nonnegative.P Setting r D  D 0, h D hk constant on the m  1 components of , such that m kD1 hk D 0, shows as in Sect. 3 m

0  hDzjzi D 

.n  1/ X 2 hk ; R2 kD1

hence m D 1, i.e.,  is necessarily connected. Assuming this, Z  

H0 hh d   0

if hN D

1 jj

Z h d  D 0: 

Setting v D 0 and h D const yields with constraint (40)

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J. Prüss and S. Shimizu

Z 0  hDzjzi D 

Z D 

@2% .% / 2 dx  @2% .% / 2 dx 

.n  1/ R2 jj

Z

.n  1/ R2 jjŒŒ%2

2 h d



Z

2  dx

;



which shows that the condition .n  1/ R2 jjŒŒ%2

Z

% 

 0 .%/

dx  1

(41)

must hold for the equilibria densities. In the incompressible case, %1 ¤ %2 both constant,  D 0 shows that D is positive semi-definite if and only if  is connected.

4.4

Summary

These considerations are summarized in Theorem 2. (a) The total mass M is preserved along smooth solutions. (b) The total available energy Ea is a strict Ljapunov functional except on the pathological points where ŒŒ% D 0. (c) The critical points of the total available energy with prescribed mass are precisely the equilibria, which are the same as in the non-isothermal case. (d) If the total available energy at an equilibrium is locally minimal, then the phases are connected, and the stability condition (41) holds.

5

Analysis of the Model

The main ideas and tools employed for the analysis of problems (26), (27), (28) are outlined, concentrating on the case with phase transition, for j  0 we refer to the next chapters of this book.

5.1

Transformation to a Fixed Domain

A basic idea is to transform the problem to a domain with a fixed interface †, where .t / is parameterized over † by means of a height function h.t/. For this, one may employ the so-called Hanzawa transform which will be explained below. This transformation was introduced in the famous paper by Hanzawa [6] in connection with the classical Stefan problem. For the necessary geometric background, we

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1033

refer to the paper by Prüss and Simonett [12] and to the monograph by Prüss and Simonett [13], Chapter 2.

5.1.1 The Hanzawa Transform Assume that   Rn is a bounded domain with boundary @ of class C 2 and that    is a hypersurface of class C 2 , i.e., a C 2 manifold which is the boundary of a N 1 . Note that 2 typically bounded domain 1  . As above, one sets 2 D n is connected, while 1 may be disconnected. In the later case, 1 consists of finitely many components, since @1 D    by assumption is a manifold, at least of class C 2 . Recall that the second normal bundle N 2  is defined by N 2  D f.p;  .p/; L .p// W p 2 g: The hypersurface  can be approximated by a real analytic hypersurface †, in the sense that the Hausdorff distance of the second order normal bundles is as small as we please. More precisely, given > 0, there exists an analytic hypersurface † such that dH .N 2 †; N 2 /  . If > 0 is small enough, then † bounds a domain † † † 2 † 1 with 1   and then we set 2 D  n 1  . The C hypersurface † admits a tubular neighborhood, which means that there is a0 > 0 such that the map ƒ W †  .a0 ; a0 / ! Rn ƒ.p; r/ WD p C r† .p/ is a diffeomorphism from †  .a0 ; a0 / onto im.ƒ/, the image of ƒ. The inverse ƒ1 W im.ƒ/ ! †  .a0 ; a0 / of this map is conveniently decomposed as ƒ1 .x/ D .…† .x/; d† .x//;

x 2 im.ƒ/:

Here …† .x/ means the metric projection of x onto † and d† .x/ the signed distance from x to †; so jd† .x/j D dist.x; †/ and d† .x/ < 0 if and only if x 2 † 1 . In particular, im.ƒ/ D fx 2 Rn W dist.x; †/ < a0 g. The maximal number a0 is given by the radius r† > 0, defined as the largest number r such the exterior and interior ball conditions for † in  holds. In the following, let a0 D r† =2

and

a D a0 =3:

The derivatives of …† .x/ and d† .x/ are given by rd† .x/ D † .…† .x//;

@…† .x/ D M0 .d† .x//P† .…† .x//;

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J. Prüss and S. Shimizu

where, as before, P† .p/ D I  † .p/ ˝ † .p/ denotes the orthogonal projection onto the tangent space Tp † of † at p 2 †, and M0 .r/ D .I  rL† /1 , with L† the Weingarten tensor. Then jM0 .r/j  1=.1  rjL† j/  3

for all jrj  2r† =3:

If dist.; †/ is small enough, the map ƒ can be used to parameterize the unknown free boundary .t / over † by means of a height function h.t/ via .t / D fp C h.t; p/† .p/ W p 2 †g;

t 0

N by means of for small t  0, at least. Extend this diffeomorphism to all of  „h .t; x/ D x C .d† .x/=a/h.t; …† .x//† .…† .x// DW x C h .t; x/: Here  denotes a suitable cutoff function. More precisely, let  2 D.R/, 0    1, .r/ D 1 for jrj < 1, and .r/ D 0 for jrj > 2. ( may be chosen in such a way that 1 < j0 j1  3.) Note that „h .t; x/ D x for jd† .x/j > 2a, and …† .„h .t; x// D …† .x/;

jd† .x/j < a;

as well as d† .„h .t; x// D d† .x/ C .d† .x/=a/h.t; …† .x//;

jd† .x/j < 2a:

This yields „1 h .t; x/ D x  h.t; …† .x//† .…† .x//

for jd† .x/j < a;

in particular, „1 h .t; x/ D x  h.t; x/† .x/

for x 2 †:

Furthermore, @ „h D I C @h ;

.@ „h /1 D I  ŒI C @h 1 @h DW I  M1T .h/;

where @ WD @x denotes the derivative with respect to x 2 Rn , and @h .t; x/ D † .…† .x// ˝ M0 .d† .x//r† h.t; …† .x//  h.t; …† .x//M0 .d† .x//L† .…† .x// for jd† .x/j < a, h0 .t; x/ D 0 for jd† .x/j > 2a, and in general

21 Modeling of Two-Phase Flows With and Without Phase Transitions

@h .t; x/ D

1035

1 0  .d† .x/=a/h.t; …† .x//† .…† .x// ˝ † .…† .x// a

C .d† .x/=a/† .…† .x// ˝ M0 .d† .x//r† h.t; …† .x//  .d† .x/=a/h.t; …† .x//M0 .d† .x//L† .…† .x//: It is a matter of simple algebra to determine the inverse of @„h , to the result .@„h .t; x//1 D  I  hL† 

 0 h=a  ˝   ˝ r h M0 .d† C h/;   † † † † 1 C 0 h=a 1 C 0 h=a

where the obvious arguments are dropped. This implies M1 .h/ D M0 .d† C h/

 r† h ˝ † 1C

0 h=a

  hL† C

0 h=a † ˝ † : 1 C 0 h=a

Note that M1 .h/ depends linearly on r† h. On the interface we then have   M1 .h/ D M0 .h/ r† h ˝ †  hL† : In particular, @„h is invertible, provided M0 .d† C h/ D .I  .d† C h/L† /1 exists, and 1 C 0 h=a > 0. This certainly holds if jd† C hjjL† j  2=3

and

j0 j1 jhj=a  1=2;

which leads to the restriction jhj1  h1 WD a=2j0 j1 ; note that j0 j1 > 1. Observe that at this place, no restrictions on r† h are required. Next, @t „h .t; x/ D .d† .x/=a/@t h.t; …† .x//† .…† .x//;

N x 2 ;

hence the relation „1 h .t; „h .t; x// D x implies @t „1 h .t; „h .t; x// D m0 .h/@t h.t; …† .x//† .…† .x//; where m0 .h/.t; x/ D

.d† .x/=a/ : .1 C h.t; …† .x//0 .d† .x/=a/=a

With the Weingarten tensor L† and the surface gradient r† ,  .h/ D ˇ.h/.†  a.h//; a.h/ D M0 .h/r† h;

N x 2 ;

1036

J. Prüss and S. Shimizu

M0 .h/ D .I  hL† /1 ; ˇ.h/ D .1 C ja.h/j2 /1=2 ; and V D @t „h   D .†   /@t h D ˇ.h/@t h: The surface gradient of a function  on  is given by N r  D P .h/M0 .h/r† N DW G .h/; where N D  ı „h , the surface divergence of a vector field f on  becomes div f D trŒP .h/M0 .h/r† fN ; and the Laplace-Beltrami operator  reads  ' D trŒP .h/M0 .h/r† P .h/M0 .h/r† ': N Finally, the mean curvature H .h/ is given by H .h/ D ˇ.h/ftrŒM0 .h/.L† C r† a.h//  ˇ 2 .h/.M0 .h/a.h/jŒr† a.h/a.h//g; a differential expression involving second order derivatives of h only linearly. Hence H .h/ D C0 .h/ W r†2 h C C1 .h/; where C0 .h/ and C1 .h/ depend on h and r† h, provided jhj  h1 holds. The linearization of H .h/ at h D 0 is given by H0 .0/ D tr L2† C † : Here † denotes the Laplace-Beltrami operator on †.

5.1.2 The Transformed Problem Now define the transformed quantities by %.t; N x/ D %.t; „h .t; x//;

uN .t; x/ D u.t; „h .t; x//

in

n†;

.t; N x/ D .t; „h .t; x//;

N x/ D .t; „h .t; x// .t;

in

n†;

uN  .t; p/ D u .t; „h .t; p//;

jN .t; p/ D j .t; „h .t; p//

on †;

(42)

the pull back of .%; u; ; ; u ; j /. This way, the time varying regions n.t / are transformed to the fixed region n†. This transforms the general problems (26), (27), (28) to the following problem for .%; N uN ; ; N N ; uN  ; jN ; h/:

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1037

@t %N C G.h/  %N N u D m0 .h/@t h.†  r/%/ N

in

 n †;

in

 n †;

on

@;

%@ N t uN  G.h/  SN C G.h/N D %R N u .Nu; N ; h/ uN D 0 ŒŒ1=% N jN2  .h/ŒŒSN  .h/

(43)

C ŒŒ N  .h/ D G .h/. . N /P .h/CSN  /

ŒŒNu  ŒŒ1=% N jN  .h/ D 0 %.0/ N D %N 0 ;

on

†;

on

†;

uN .0/ D uN 0 :

where N %/.G.h/N SN D . ; N u C ŒG.h/NuT / C . N ; %/.G.h/ N  uN /I; SN  D  .  /P .h/.G .h/Nu C ŒG .h/Nu T /P .h/ C . N /.G .h/  uN  /P .h/; and N %/@ %. N ; N t N  G.h/  d . N ; %/G.h/ N N D %. N N ; %/R N .Nu; N ; h/

in

n†;

on

@;

N D N

on

†;

N .0/ D N0

in

;

@ N D 0 ŒŒ N  D 0;

(44)

as well as  . N /@t N  .G .h/jd . N /G .h/ N /  ŒŒ N . N ; / N jN N /G.h/ CŒŒd . ; N N   .h/ D SN  W G .h/Nu C . N /G .h/  uN  C R . N ; h/

on

†

N / ŒŒ . ; N C ŒŒ1=2 N2 jN2  ŒŒSN    =  N C ŒŒ= N  N  .h/ D 0

on

†;

ˇ.h/@t h  .Nuj / C jN = N D 0;

on

†;

N .0/ D N0 ;

h.0/ D h0 :

(45)

Here G.h/ and G .h/ denote the transformed gradient, respectively, the transformed surface gradient. More precisely, the relations T Œr ı „h D G.h/N D Œ.@ „1 N D .I  M1 .h//r N h / ı „h r 

and Œr  ı „h D .I  M1 .h//r N ;

1038

J. Prüss and S. Shimizu

as well as .r  u/ ı „h D .G.h/jNu/ D ..I  M1 .h//rjNu/ are valid. Furthermore, D  ı „h D @t N C uN   r† N  uN   M1 .h/r† N ; Dt and N  m0 .h/@t h.†  r/Nu; Œ@t u ı „h D @t uN C @NuŒ.@t „1 h / ı „h  D @t u hence Ru .Nu; N ; h/ D Nu  G.h/Nu C m0 .h/@t h.†  r/Nu: Similarly, Œ@t  ı „h D @t N  m0 .h/@t h.†  r/ N ; and so R .Nu; N ; h/ D Nu  G.h/ N C m0 .h/@t h.†  r/ N : In the same way one gets R . N ; h/ D Nu  r† N C uN   M1 .h/r† N C N 0 . N /G .h/  uN  : It is convenient to decompose the stress boundary condition into tangential and normal parts; here we set S D 0. For this purpose let P† D I  † ˝ † denote the projection onto the tangent space of †. Multiplying the stress interface condition with † =ˇ, we obtain N  H .h/ D .ŒŒSN .†  M0 .h/r† h/j† / ŒŒ1=% N jN2 C ŒŒ C 0 ˇ.M0 r† hjM0 r† N /

(46)

for the normal part of the stress boundary condition. Substituting this expression into the stress interface condition and then applying the projection P† yields, after some computation, P† ŒŒSN .†  M0 .h/r† h/ D .ŒŒSN .†  M0 .h/r† h/j† /M0 .h/r† h C . 0 =ˇ/M0 .h/r† N

(47)

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1039

for the tangential part. Note that the latter neither contains the phase flux nor the pressure jump nor the curvature.

5.2

Strategies and Results

In this section, some strategies and results for the models (26), (27), (28) are described. Throughout, Lp theory is employed, since it avoids higher-order compatibility conditions. Attention is restricted here to the incompressible case with phase transition in which viscous surface stress is neglected, i.e., T D P with > 0 constant. %.@t u C u  ru/  div T D 0 T D 2. /D  I; ŒŒ1=%j 2   ŒŒT  D H  u D 0 on @;

in  n .t /;

div u D 0

in  n .t /;

ŒŒu D ŒŒ1=%j  ujtD0 D u0

(48)

in :

%. /.@t C u  r /  div .d . /r /  2s jDj22 D 0 ŒŒ . /j  ŒŒd . /@  D 0; @ D 0 on @;

on .t /;

ŒŒ  D 0

in  n .t /; on .t /;

jtD0 D 0

ŒŒ . / D ŒŒ1=2%2 j 2 C ŒŒT   =% V D u   D u    j %

(49)

in :

on .t /; on .t /;

(50)

jt D0 D 0 : Assume throughout this section the Condition (H): ; %1 ; %2 > 0; j

%1 ¤ %2 ; are constant;

2 C 3 .0; 1/;

j .s/ D s

00 j .s/

j ; dj 2 C 2 .0; 1/; > 0;

j .s/ > 0;

dj .s/ > 0;

for s > 0; j D 1; 2:

In particular, these conditions ensure thermodynamical consistency, as shown before.

5.2.1 Local Well-Posedness To obtain local well-posedness, the transformed problem is rewritten in the abstract form: Lz D .N .z/; z0 /:

1040

J. Prüss and S. Shimizu

Here L is the principal linear part of the problem in question, and N is the remaining nonlinear part which is small in the sense that N collects all lower-order terms and contains only highest-order terms which carry a factor jr† hj which is small on small time intervals due to the choice of the Hanzawa transform. The variable z with initial value z0 collects all essential variables of the problem under consideration. The first step is to find function spaces E.J / and F.J /, J D .0; a/ or J D RC , such that L W E.J / ! F.J /E is an isomorphism. Here E denotes the time-trace space of E.J / where the initial value z0 should belong to. This is the question of maximal regularity. The second step then employs the contraction mapping principle to obtain local solutions. For this, estimates of the nonlinearity N are needed, eventually showing that N W E.J / ! F.J / is continuously Fréchet-differentiable, at least. This requires some smoothness of the coefficient functions in the constitutive laws. If these are real analytic, the interface will become instantaneously real analytic, which shows the strong regularizing effect, characteristic for parabolic problems. For (48), (49), (50), the following result holds (cf. [15, Theorem 3.2]): Theorem 3. Let p > n C 2, and let Condition (H) be valid. Assume the regularity conditions .u0 ; 0 / 2 ŒWp22=p . n 0 /nC1 ;

0 2 Wp32=p ;

where   Rn is a bounded domain with boundary @ 2 C 3 , the compatibility conditions div u0 D 0

in  n 0 ;

u0 D @ 0 D 0

on @;

P0 ŒŒu0  D P0 ŒŒ. 0 /.ru0 C Œru0 T /0  D ŒŒ 0  D 0 l. 0 /ŒŒu0  0 =ŒŒ1=% C ŒŒd . 0 /@0 u0  D 0

on 0 ;

on 0 ;

N and the well-posedness condition 0 > 0 on . Then there exists a unique Lp solution of the transformed problems (48), (49), (50) on some possibly small but nontrivial time interval J D Œ0; a.

5.2.2 The Local Semiflow Recall that the closed C 2 hypersurfaces contained in  form a C 2 manifold, which is denoted by MH2 ./. Define the state manifold SM as follows: n SM WD .u; ; / 2 Lp ./nC1  MH2 W .u; / 2 Wp22=p . n /nC1 ;  2 Wp32=p ; div u D 0 in ;

N > 0 in ;

u D @ D 0 on @;

ŒŒ  D ŒŒP u D P ŒŒ. /.ru C ŒruT / D 0 o l. /ŒŒu   =ŒŒ1=% C ŒŒd . /@  D 0 on 

on ; (51)

21 Modeling of Two-Phase Flows With and Without Phase Transitions

1041

(cf. [15, (6.1)]). Problem (48), (49), (50) induces a local semiflow on SM. Theorem 4. Let p > n C 2, and let Condition (H) be valid. Then problem (48), (49), (50) generates a local semiflow on the state manifold SM. Each solution .u; ; / exists on a maximal time interval Œ0; tC /, where tC > 0 depends on the initial value u0 ; 0 ; 0 .

5.2.3 Stability Analysis of Equilibria For the stability analysis of equilibria, it is natural to employ again the Hanzawa transform, where the reference manifold † now is the equilibrium interface  , a union of finitely many disjoint spheres contained in . As the linearized problem enjoys maximal Lp -regularity, an abstract result shows that the operator A associated with the linearized problem is the negative generator of a compact analytic C0 -semigroup. Therefore the spectrum of A consists only of countably many isolated eigenvalues of finite algebraic multiplicity. Thus, it is natural to study these eigenvalues and to apply the principle of linearized stability for the nonlinear problem. However, a major difficulty of this approach lies in the fact that the equilibria are not isolated in the state manifold but form a finite-dimensional submanifold E of SM. For the linearization of the transformed problem, this implies that the kernel of A is nontrivial, i.e., the imaginary axis is not in the resolvent set of A, and so the standard principle of linearized stability is not applicable. Fortunately, 0 is the only eigenvalue of A on i R and it is nicely behaved: the kernel N.A/ is isomorphic to the tangent space of E at this equilibrium, and 0 is semi-simple. This shows that 0 is normally stable if the remaining eigenvalues of L have positive real parts and normally hyperbolic if some of them have negative real parts; these are only finitely many. Therefore, one may employ what is called the generalized principle of linearized stability, a method which is adapted to such a situation and has been worked out recently for quasilinear parabolic evolution equations in [14]. So the stability analysis of equilibria proceeds in two steps. In the first step, the eigenvalues of A are analyzed and one finds necessary and sufficient conditions, which ensure that all eigenvalues of A except 0 have positive real parts; this is the normally stable case. In the normally hyperbolic case, the dimension of the unstable subspace of A is determined. It also turns out that 0 is semi-simple, the kernel of A is determined, and one can show that N.A/ is isomorphic to the tangent space of E. In the second step, the generalized principle of linearized stability is applied to the nonlinear problem. For this the implicit function theorem is used, as due to maximal regularity, there is no loss of regularity. For the problem (48), (49), (50), the following result holds (cf. [15, Theorem 5.2]). Theorem 5. Let p > n C 2, and let Condition (H) be valid. Then in the topology of the state manifold SM we have:

1042

J. Prüss and S. Shimizu

(i) .0;  ;  / 2 E is stable if and only if  is connected. (ii) Any solution starting in a neighborhood of a stable equilibrium converges to another stable equilibrium. (iii) Any solution starting and staying in a neighborhood of an unstable equilibrium converges to another unstable equilibrium.

5.2.4 Long-Time Behavior of Solutions In general, solutions in SM will exist on a maximal time interval Œ0; tC .z0 // which typically will be finite, due to several obstructions, such as missing a priori bounds, loss of well-posedness, or topological changes in the moving interface. However, if a solution does not develop singularities in a sense to be specified, then the solution exists globally, i.e., tC .z0 / D 1, and it converges in the topology of SM to an equilibrium. This essentially relies on a method using time weights to improve regularity and on compact Sobolev embeddings. Actually, one can characterize solutions which exist globally and converge as t ! 1. This only involves general properties of semiflows, relative compactness of bounded orbits, the existence of a strict Lyapunov functional (the negative entropy), and the results on stability of equilibria. Let .u; ; / be a solution in the state manifold SM with maximal interval Œ0; t /. By the uniform ball condition, we mean the existence of a radius r0 > 0 such that for each t, at each point x 2 .t /, there exists centers xi 2 i .t / such that Br0 .xi /  i and .t / \ BN r0 .xi / D fxg, i D 1; 2. Note that this condition bounds the curvature of .t / and prevents parts of it to touch the outer boundary @ and to undergo topological changes. Combining Theorems 4 and 5 with the Lyapunov functional and compactness, the following result is obtained. Theorem 6. Let p > n C 2, and let Condition (H) be valid. Suppose that .u; ; / is a solution of (48), (49), (50) in the state manifold SM on its maximal time interval Œ0; tC /. Assume there is constant M > 0 such that the following conditions hold on Œ0; tC /: (i) ju.t /jW 22=p ; j .t /jW 22=p ; j.t /jW 32=p  M < 1; p

p

p

(ii) .t / satisfies the uniform ball condition. N (iii) .t /  1=M on . Then tC D 1, i.e., the solution exists globally, and its limit set !C .u; ; /  E is non-empty. If further .0; 1 ; 1 / 2 !C .u; ; / with 1 connected, then the solution converges in SM to this equilibrium. Conversely, if .u.t /; .t /; .t // is a global solution in SM which converges to an equilibrium .0;  ;  / 2 E in SM as t ! 1, then (i), (ii) and (iii) are valid. For the proofs of the results in this section and for more details, one should consult [15] and [13].

21 Modeling of Two-Phase Flows With and Without Phase Transitions

6

1043

Conclusion

This chapter contains a thermodynamically consistent modeling of two-phase flows with phase transitions, in the compressible as well as in the incompressible case, non-isothermal as well as isothermal. Basic thermodynamical properties of the models are derived, the equilibria are identified, thermodynamical consistency is shown, and thermodynamical stability of equilibria is discussed. For the incompressible non-isothermal problem with phase transitions, some recent analytical results due to the authors are explained.

7

Cross-References

 Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible

Fluid Mechanics  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid

Crystal Flows  Stability of Equilibrium Shapes in Some Free Boundary Problems Involving

Fluids

List of Symbols d heat conductivity, 1020 div surface divergence on , 1009 e total mass specific energy, 1030 E0 given total energy, 1024 ea mass-specific energy, 1029 Ea total available energy, 1030 E set of non degenerate equilibrium, 1027 f force field, 1012 H (.n  1/-fold) mean curvature, 1009 j phase flux, 1010 l latent heat, 1021 l surface latent heat, 1021 L Weingarten tensor, 1009 M0 given total mass, 1024 N total surface entropy, 1016 N total entropy, 1023 P orthogonal projection onto , 1009 q heat flux, 1012

q heat flux on the interface, 1015 r external heat source, 1012 S viscous stress tensor, 1013 S interface viscous stress, 1016 T stress tensor, 1012 T surface stress tensor, 1012 u velocity, 1010 u velocity on , 1010 N total bulk entropy, 1014 V normal velocity of , 1010 Πjump of , 1009  internal energy density, 1012  surface energy density, 1015

entropy density, 1013  outer unit normal of t , 1009  pressure, 1012 Helmholtz free energy, 1013  free surface energy, 1016

1044

coefficient of surface tension, 1016 absolute temperature, 1012  surface temperature, 1016 b bulk viscosity, 1020 s shear viscosity, 1020 b surface (bulk) viscosity, 1020 s surface (shear) viscosity, 1020

J. Prüss and S. Shimizu

 Laplace-Beltrami operator on , 1026 % density, 1010 .%; /, 1020

.%; /, 1020  heat capacity, 1021  surface heat capacity, 1021

References 1. D.M. Anderson, P. Cermelli, E. Fried, M.E. Gurtin, G.B. McFadden, General dynamical sharpinterface conditions for phase transformations in viscous heat-conducting fluids. J. Fluid Mech. 581, 323–370 (2007) 2. D. Bothe, W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures. Acta Mech. 226, 1757–1805 (2015) arXiv:1401.5991v2 (2014) 3. E. DiBenedetto, A. Friedman, Conduction-convection problems with change of phase. J. Differ. Equ. 62(2), 129–185 (1986) 4. E. DiBenedetto, M. O’Leary, Three-dimensional conduction-convection problems with change of phase. Arch. Ration. Mech. Anal. 123(2), 99–116 (1993) 5. D.A. Drew, S.L. Passman, Theory of Multicomponent Fluids. Volume 135 of Applied Mathematical Sciences (Springer, New York, 1999) 6. E.-I. Hanzawa, Classical solutions of the Stefan problem. Tôhoku Math. J. (2) 33(3), 297–335 (1981) 7. K.-H. Hoffmann, V.N. Starovoitov, The Stefan problem with surface tension and convection in Stokes fluid. Adv. Math. Sci. Appl. 8(1), 173–183 (1998) 8. K.-H. Hoffmann, V.N. Starovoitov, Phase transitions of liquid-liquid type with convection. Adv. Math. Sci. Appl. 8(1), 185–198 (1998) 9. M. Ishii, T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow (Springer, New York, 2006) 10. Y. Kusaka, A. Tani, On the classical solvability of the Stefan problem in a viscous incompressible fluid flow. SIAM J. Math. Anal. 30(3), 584–602 (1999) (electronic) 11. Y. Kusaka, A. Tani, Classical solvability of the two-phase Stefan problem in a viscous incompressible fluid flow. Math. Models Methods Appl. Sci. 12(3), 365–391 (2002) 12. J. Prüss, G. Simonett, On the manifold of closed hypersurfaces in Rn . Discrete Contin. Dyn. Sys. A 33, 5407–5428 (2013) 13. J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics 105 (Birkhäuser, Basel, 2016) 14. J. Prüss, G. Simonett, R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems. J. Differ. Equ. 246(10), 3902–3931 (2009) 15. J. Prüss, S. Shimizu, M. Wilke, On the qualitative behaviour of incompressible two-phase flows with phase transition: the case of non-equal densities. Commun. Partial Differ. Equ. 39(7), 1236–1283 (2014)

Equations for Viscoelastic Fluids

22

Xianpeng Hu, Fang-Hua Lin, and Chun Liu

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Viscoelasticty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Polymeric Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 IMHD System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1046 1048 1051 1054 1058 1062 1066 1067 1067 1068

Abstract

This chapter aims at the mathematical theory of incompressible viscoelastic fluids and related complex fluid models. An energetic variational approach is employed to derive the hydrodynamics of complex fluids which focuses on the competition and coupling between different physical effects. Such a framework

X. Hu () Department of Mathematics, City University of Hong Kong, Hong Kong, China e-mail: [email protected] F.-H. Lin Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA e-mail: [email protected] C. Liu Department of Mathematics, Pennsylvania State University, University Park, PA, USA Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_25

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also provides guides to the corresponding analysis. This chapter includes those analytical results for both classical solutions with small initial data and weak solutions with large initial data for these systems.

1

Introduction

The study of elastic complex fluids is currently an active research area and, with its physical and biological applications, attracting more and more attentions of researchers from many different disciplines. While many materials found in nature can be categorized into different classes of complex fluids based on their mechanical or rheological properties, the understanding of their underlying properties has been relatively primitive, due largely to the associated complex and nonlinear phenomena under different environments. Much of hydrodynamic or rheological behavior of complex fluids can be attributed to their internal elastic properties resulting from their microscopic structures and their induced effects on their macroscopic flows. These couplings and competitions between the microscopic structures and macroscopic properties provide formidable challenges in analysis and numerical simulations. To mathematically describe an elastic complex fluid with both the microscopic and the macroscopic considerations, it has been an established practice that both Lagrangian and Eulerian coordinates should be used together for such a purpose. Let X be the initial position (or the Lagrangian coordinate) of the particle, and the orientation-preserving diffeomorphism x.X; t / W Rd  RC 7! Rd be the flow map (or the Eulerian coordinate) image of the particle X at t  0. The velocity field u.x; t / associated with the flow map is defined by dx.X; t/ D u.x.X; t /; t /; dt

x.X; 0/ D X:

(1)

A straightforward application of the chain rule to (1) yields @rX x.X; t / D rx u.x.X; t /; t /rX x.X; t /; @t

rX x.X; 0/ D I;

(2)

or in components d X @rXj xi .X; t / rxk ui .x.X; t /; t /rXj xk .X; t /; D @t

rXj xi .X; 0/ D ıij

kD1

for all 1  i; j  d , where ıij denotes the Kronecker delta. From (2) there follows @ det rX x.X; t / D divx u.x.X; t /; t // det rX x.X; t /; @t

det rX x.X; 0/ D 1: (3)

22 Equations for Viscoelastic Fluids

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The equation (3) particularly implies that the Jacobian determinant identity det rX x.X; t / D 1 in Lagrangian coordinates is equivalent to the incompressibility divx u D 0 in Eulerian coordinates. The density .x; t / of the fluid is governed by the conservation of mass: @t  C div.u/ D 0;

(4)

which, along with the equation (3), yields a relation between the density .x; t / and the reference density 0 .X /: .x.X; t/; t/ det rX x.X; t / D 0 .X / for all

t  0:

At the microscopic level, molecules are transported and may be even deformed by the macroscopic flow x.X; t /. In return, through an internal elastic stress, they affect on the momentum equation macroscopically through certain averaging (coarse graining) or a hydrodynamic process. The whole phenomenon can be nicely interpreted by the least action principle (LAP) or the minimum dissipation principle (MDP), which yields, in the form of conservation laws, the following: .@t u C u  ru/ D div ;

(5)

where  is the stress tensor,  is the density, and u is the velocity. Due to the elastic effect, the stress  in general contains two contributions: the Newtonian part n and the non-Newtonian (or elastic) part e  D n C e : The Newtonian stress n may be given as   1 n D 2 D.u/  divuI C divuI  P I d

with D.u/ D

ru C .ru/> ; 2

where d is the spatial dimension, P is the hydrodynamic pressure, and I is the identity matrix. Here,  is the shear viscosity and  D  C .2=d / is the bulk viscosity. For Newtonian flows, the elastic stress e vanishes, and the energy law for the system (4) and (5) with P ./ D  for  > 1 reads d dt

Z 

    Z  1 1 d 2 jruj2 C  C juj2 C  dxC  jdivuj2 dx D 0: 2  1 d

Comparing the relatively uniform form for the Newtonian stress, the sources for the elastic stress e are much broader. They can be the elasticity of deformable particles, elastic repulsion between charged liquid crystals, immersed elastic bound-

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aries, elasticity due to microstructures, Lorentz forces due to the magnetic fields, etc. Formally, the elastic stress e is transported through the flow as @t e C u  re D g.e ; u/:

(6)

Although the exact formulation of g.e ; u/ depends on the model, one of the common interests of these “elastic” effects is their energy variational formulation; see (9) and (33). As a prototype, the classical incompressible Oldroyd-B model [102] takes the form @t e C u  re D 

 1  e  ru  .ru/> C rue C e .ru/> ; We

(7)

where W e stands for the Weissenberg number and .ru/> is the transpose matrix of ru. The study of complex fluids had attracted many researchers over the years. There are many different approaches and models for these complicated materials. This chapter will describe several state-of-the-art analytical results arising from the studies of some elastic complex fluids including the viscoelastic fluids, the micromacro models of polymeric fluids, and the incompressible magnetohydrodynamics (IMHD). We attempt to focus on the analytical difficulties and to treat all investigated systems in a uniform way taking into account their common nature to present self-contained analytical results. Although all systems considered in this chapter share a similar energy dissipation mechanism, each individual system retains its own analytical difficulties as we shall demonstrate below. For the analytical development of other models in complex fluids, there is a list of references including a number of books and survey articles; see [12, 21–33, 36–41, 68–71, 81, 83, 109]. However, the cited works have been chosen on the basis of selectiveness rather than completeness to keep the bibliography concise. The list of references in this chapter is by no way comprehensive, and there are still many references in the literature for different directions of interesting research.

2

Viscoelasticty

The first topic of this chapter is a macroscopic description of incompressible elastic fluids. The elastic aspect of the fluid demonstrates a memory of the fluid of its past behavior, and this can be described via the flow map (1). Recall that X is the initial position of the particle, while x.X; t / is the current location of the particle X at time t . The elasticity may be represented by the matrix-valued deformation tensor Q F.X; t/ D

@x ; @X

(8)

22 Equations for Viscoelastic Fluids

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which describes the deformation of configurations or patterns. The notation F .x; t / defined later on is also used in the Eulerian coordinates. For the classical elasticity in the Lagrangian coordinates, the action functional between the kinetic and elastic energy is defined as ˇ ˇ  # Z 1Z " ˇ @x.X; t/ ˇ2 @x.X; t/ 1 ˇ ˇ A.x/ D 0 .X / ˇ dXdt; (9) ˇ W 2 @t @X Rd 0 where the orientation-preserving diffeomorphism x.X; t / W Rd  RC 7! Rd is Q D W @x.X;t/ the flow map and the strain energy function W .F/ depends on FQ @X through the principal invariants of the left Cauchy-Green tensor FQ FQ > . Using the functional (9), the least action principle gives rise to the so-called elastodynamics Q D 0: 0 xt t  divX WFQ .F/

(10)

In the Eulerian coordinates, F.x; t / is defined as Q F.x.X; t /; t / D F.X; t /: In terms of (2), the unknown tensor F satisfies the following transport equation @t F C u  rF D ruF;

(11)

which is the same equation as for the vorticity in the Euler equation and it is of hyperbolic nature. Let divF> is defined as @xi Fij . A direct calculation yields [78, 88] @t divF> C u  r.divF> / D 0;

(12)

by using the incompressibility of the fluid, i.e., divx u D 0. It yields, in particular, that divF> D 0 if it holds initially. For a Newtonian fluid, the constitutive relation in (5) is  D n D P I C D; where P is the pressure,   0 the viscosity, and D D .ru C .ru/> /=2 the symmetric part of the velocity gradient. To capture the elasticity part of a viscoelastic fluid, it is assumed that the elastic energy of the material is W .F/, and then the corresponding elastic stress becomes e D S .F/F>

with S .F/ D @F W .F/;

where e denotes the Cauchy-Green form and S .F/ takes the Piola-Kirchhoff form. In the case of linear (or Hookean) elasticity, the elastic energy W .F/ D 12 jFj2 and e D FF> . The symmetric tensor e D FF> satisfies the transport equation @t e C u  re  rue  e .ru/> D 0;

(13)

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and the Oldroyd system [102, 108] (without dampings or with infinite Weissenberg number) is recovered. Set  D 1 in (5); the momentum equation then takes the form ut C u  ru C rP D u C div.FF> /:

(14)

From (11) and (14), the following Oldroyd-B model is derived: 8 > ˆ ˆ D 0;

det F D 1;

and

curlF1 D 0;

(16)

22 Equations for Viscoelastic Fluids

1051

where the divergence operator div and the curl operator curl apply to each row of a matrix. The first two identities are due to the incompressibility of the fluid; see, for instance, (3), and the third one follows from the definition (8) of F. Indeed, F1 is of the form @X which is a gradient with respect to the Eulerian coordinates, and @x hence the identity curlF1 D 0 follows naturally. Moreover, the last identity above can be replaced by the Piola identity [14] Flk rl Fij D Flj rl Fik

for i; j; k D 1; 2; 3;

(17)

whose applications had been demonstrated in [78,113]. It will be demonstrated later on that quantities in (16) play key roles in proving global existence (at least for the case of smooth small data). From (12), and if divF> 0 D 0 is assumed, then for all time there holds that divF> D 0. Thus, F D r  where is a matrix. In the two-dimensional case,   @2 1 @2 2 FD @1 1 @1 2 Denoting D . 1 ; 2 /, clearly is a volume-preserving map, i.e., det.r / D 1. Thus, the Oldroyd system (15) can also be transformed into (

t C u  r D 0;

divu D 0;

(18)

ut C u  ru C rP D u  r   ;

P  P2 2 where r   D D  r (see [88]). This iD1 @1 i  i ; iD1 @2 i  i system is obviously related to the system of liquid crystal flows, phase boundary dynamics coupled with fluids, the Boussinesq system, and the incompressible magnetohydrodynamics; see [81]. The Boussinesq system is simpler and involves only a linear function of . Liquid crystal flows are much more complicated due to the quadratic nonlinearity of the stress and the presence of defect measures; see [59,62,82,91,92,115,116,120,121] for the recent progresses in both two dimensions and three dimensions. See also [83] for a very recent survey of liquid crystal flows.

2.1

Classical Solutions

The local existence and uniqueness of the system (15) and (18) can be easily obtained by the energy method. However, the global existence for small initial data is still difficult because of the lack of a damping mechanism on . One first demonstrates how to handle this difficulty in two dimensions. Introduce .x/ D

.x/  x and w D u  1 . Then w satisfies a parabolic equation wt  w D u  rw  rp   r

C

u : 

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From this and energy estimates, one may obtain a decay estimate for w, thus also for as u is known to decay. But the difficulty would be to show “errors” on the right-hand side of the above equation which is indeed of higher order. For the latter, one shows that the pressure p actually satisfies p D ru W .ru/>  div. r /  div ; where the last term could be harmful as it is order one. Here, it comes the identity det.I C r / D 1 which implies that div

D @1

2 @2

1

 @1

1 @2

2

as desired. One now demonstrates further how crucial these identities in (16) are in the analysis of incompressible viscoelasticity system (15) in three dimensions (see [57, 105] for the compressible case). Indeed, from a physical point of view, the elastic stress div.FF> / has two (mutually orthogonal) contributions: shear waves and pressure waves. In order to see this more precisely, one considers the linearization of div.FF> /, divE;

where

F D I C E:

In terms of .u; E/, the linearized system of (15) reads ( @t u C rP D u C divE @t E  ru D 0;

divu D 0:

(19)

Due to divE> D 0, one has divdivE D 0, and hence the pressure wave of the isotropic elastic materials has been eliminated due to the incompressibility. Moreover, since P D 0, the pressure P in (19) can be dropped temporarily to yield (

@t u  u  divE D 0; @t E  ru D 0;

divu D 0:

(20)

In order to investigate the shear wave of divE, that is, curldivE, one considers the equation for vorticity, ! D curlu: Since the Piola identity (17) implies in particular that in the linear level @j Eik D@k Eij ; and hence curl.curlE> /> D0;

22 Equations for Viscoelastic Fluids

1053

up to higher order terms. Taking curl in (20) (this also says that the pressure P in (19) can be eliminated), one thus obtains in the linear level ( @t !  !  div.curlE> /> D 0; (21) @t .curlE> />  r! D 0: Note that curldivE D div.curlE> /> ;

and

 D rdiv  curlcurl:

Taking one more time derivative of (21) yields a system of wave equations with damping (

@t t !  @t !  ! D 0; @t t curlE>  @t curlE>  curlE> D 0:

(22)

The parabolic-hyperbolic coupling system (22) is the backbone of the study for the incompressible viscoelasticity system (15). The roles that it has played are the same as that for compressible Navier-Stokes equations in [34]; for the latter, the decays of solutions of (22) were established by Ponce in [104]. Keeping in mind, the information on curlE> is good enough to control full information of E due to the constraint divE> D 0 and the Biot-Savart law (see [98]). Based on the similar arguments as above, Lin-Liu-Zhang [84, 88] and Lei-LiuZhou [78] (see also [20] and [85] for the initial-boundary problem) proved the following theorem: Theorem 1. Assume that u0 2 H 3 .Rd / and F0  I D E0 2 H 2 .Rd / and ku0 kH 3 C kE0 kH 2  " for some sufficiently small " > 0. Moreover the constraints (16) and (17) hold true initially. The Cauchy problem (15) has a unique global classical solution .u; F/. The arguments in [78, 88] explored the intrinsic coupling between the fluid and the elasticity, in particular the weak dissipation mechanism for a combined quantity between the velocity and the deformation gradient. The solvability in either bounded domains or exterior domains had been investigated in [48, 50, 85]. Various free boundary problems and the optimal decay rates of (15) have been discussed in [125] and [58, 60], respectively. Moreover, if an additional linear damping term is presented in the equation for F, the global existence of a small smooth solution was proved earlier by Guillopé and Saut [44]. Unlike the case of viscous Newtonian fluids, the large viscosity alone may not be sufficient to guarantee the regularity of the entire system (15). The latter is an interesting issue that worths further investigations.

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In [97], Liu and Walkington studied the mixture of a fluid with a viscoelastic solid by decomposing the deformation gradient in a polar decomposition as F D R.I C U /; where R is the rotation part and U is the symmetric part, and obtained the equations @t R C u  rR D W .u/RI

@t U C u  rU D R> D.u/R;

where D.u/, W .u/ are the symmetric and skew part of u, respectively. The special form of the equation for R allows one to get an approximate system for R and to generalize the tools for scalar transport equations [35] to the small strain case and eventually leads to the global existence of weak solutions of the system. This unconventional formulation was also used in the famous work [63] of F. John, where he applied the John-Nirenberg inequality to study nonlinear elasticity for the static small strain cases. Following the work of F. John for the stationary elasticity, the authors in [73,77] constructed a global-in-time classical solution which allows large rotations but small strains. The key strategy in [73, 74, 77] relies on the transition of the dissipation of the velocity to the weak dissipations of both the strain and the rotation and the separation of equations between the strain and the rotation. These efforts also shed some new light on the large Weissenberg number problem in viscoelastic fluids [65, 103]; the analytical results indicate that in the infinite Weissenberg number limit, the system possesses well posedness only for small strain or near-equilibrium data. As  D 0, the system (15) is referred as incompressible elastodynamics [112], for which a global-in-time classical solution had been established in [110–112] by Sideris and Thomases through the vector field method due to the dispersive structure of the incompressible elastodynamics since formally by letting  D 0 in (22) one obtains a system of wave equations (

@t t !  !D0; @t t curlE>  curlE> D0:

(23)

The nonlinear terms in (15) satisfy automatically the null conditions which make the global-in-time existence with small data possible in 3-D (and even in 2-D); see also [1, 56, 76]. The global existence of classical solutions of two-dimensional incompressible elastodynamics was discussed in [72,73,79]. The vanishing viscosity limit as  ! 0 is studied recently by Sideris and Jonov in [64].

2.2

Weak Solutions

In [95], by estimating the defect measure associated with approximate solutions, Lions and Masmoudi proved the global existence of weak solutions of the coro-

22 Equations for Viscoelastic Fluids

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tational Oldroyd-B model with a linear damping in  , where the co-rationality means that ru in the equation of e in (13) is replaced by its antisymmetric part .ru  .ru/> /=2. One of the advantages of the corotational Oldroyd-B model is the Lp estimate for the elastic stress for p > 1. See also [11,13] for an extension in this direction to allow a general diffusion term on velocity. As far as weak solutions of (15) are concerned, the key difficulty to show the convergence of approximate solutions is to verify the weak convergence of FF> at least in the sense of distributions, which requires a strong convergence of F in L2 . To have this strong convergence of F in L2 , it requires better understanding on both the oscillation phenomena and the concentration phenomena associated with approximate solutions. For instance, this problem in two dimensions can be explained as follows. In two dimensions, denoting F D .F1 ; F2 / where F1 and F2 are columns of F and then the second equation in (15) can be written as @t Fj C u  rFj D Fj  ru for j D 1; 2. Since divFj D 0 due to divF> D 0, the equation for Fj can be further rewritten as @t Fj C div.Fj ˝ u  u ˝ Fj / D 0; where .a ˝ b/ij D ai bj : Next, the pair .u; F/ is said to be a weak solution of (15) with Cauchy data provided that F; u; ru 2 L1loc .R2  RC / and for all test functions ˇ; 2 D.R2  RC / with div D 0 in D0 .R2  RC / Z R2

Z

1

Z

.Fj /0 ˇ.; 0/dx C

R2

0

for j D 1; 2, and Z Z u0 .; 0/dx C R2

Z

1

Z

D

0

1

.Fj ˇt C .Fj ˝ u  u ˝ Fj / W rˇ/dxdt D 0

Z R2

h

u  @t

C .u ˝ u  FF> / W r

i

(24)

dxdt (25)

ru W r dxdt: 0

R2

According to this formulation, Hu-Lin proved in [53] recently the following theorem. Theorem 2. Let ku0 kLp  ˛ for some p > 2, and assume that "0   for a sufficiently small  that may depend on ˛ and p. The Cauchy problem (15) with constraints (16) and (17) has a global weak solution .u; F/ which is actually smooth for positive time. Moreover, there exist a positive constant  that depends on p and a positive constant C that may depend on p and ˛ such that A.t/  C "0 ;

and

B.t /  C "0

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for all t 2 RC , where Z



ju.x; t /j2 C jF.x; t /  I j2 C .t /jru.x; t /j2

A.T / D sup R2

0tT

 P C .t /2 jP u.x; t /j2 dx Z

T

Z

C 0



 P 2 C .t /2 jrP uj P 2 dxdt; jruj2 C .t /juj

R2

and B.T / D kF  I k2L1 .R2 Œ0;T / : with .t / D minf1; tg. The weak-strong uniqueness is addressed in [60]. Formally speaking, Theorem 2 implies that near the equilibrium, both concentration and oscillation phenomena do not exist. To measure a possible oscillation phenomena, the authors in [53] introduce a quantity, which was called the effective viscous flux, G D ru  ./1 rPdiv.FF>  I /: This terminology was first introduced in the field of compressible Newtonian fluid mechanics [42, 94], and it is a combination of effects from both the velocity of the fluid and the deformation gradient. One can easily check from the first equation in (15) that P G D rP u;

(26)

where uP D @t u C u  ru is the material derivative of u and P is the projection to the divergence-free vector field. From (26), one would expect a bound of G in H 1 for positive time, which is better than either components of G that appeared to be, and hence a certain sort of cancelations occur. On the other hand, the L1 bound on F which is implied by the bound on B.T / in Theorem 2 is important, as it eliminates the possibility of concentrations as well. This uniform bound is obtained through an integration of the equation for F along the trajactories of the flow map. Very recently, the smallness assumption in Theorem 2 has been removed, and a global-in-time existence of weak solutions with finite energy was established in [55]. Theorem 3. Assume that u0 2 L2 .R2 / and F0 2 L2 .R2 / with the constraint (16) in D0 .R2 /. If the assumption (29) holds true, the Cauchy problem (15) admits a

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global-in-time weak solution .u; F/ with finite energy. Moreover, the solution .u; F/ satisfies: • The deformation gradient F is bounded in L3loc .Q/, and the hydrodynamic 3=2 pressure P is bounded in Lloc .Q/, here Q D R2  RC ; • The constraint (16) holds true in D0 .R2  RC /. Theorem 3 implies that the concentration and oscillation phenomena do not occur even for large initial data in the two-dimensional case. The control on the effective viscous fluxes as in [43, 53] along with a convexity argument (see [42, 94, 101]) prohibits the occurrence of oscillation phenomena. The new essential point in [55] is a decomposition of the symmetric Cauchy-Green tensor e D FF> div.FF> / D r…1 C .@1 …2 ; @2 …2 / C .@2 …3 ; @1 …3 /   …2 …3 ; D r…1 C div …3 …2

(27)

where 2 > 2 > 2…1 D jFj2 D jF> 1 j CjF2 j D tr.FF /;

2 > 2 2…2 D jF> 1 j jF2 j ;

> …3 D F> 1 F2 ;

> and F> i is understood to be .F /i . In terms of (27), the momentum equation reads

  …2 …3 O @t u C u  ru  u C r P D div …3 …2

(28)

with PO being the total pressure P  …1 . The critical improved integrabilities on …2 and …3 were achieved through the following five useful observations: • Taking curl of (28), one has @t curlu C curldiv.u ˝ u/  curlu D 2@1 @2 …2 C .@22  @21 /…3 ; h i and hence ./1 2@1 @2 …2 C .@22  @21 /…3 could be bounded in Lp for some p > 1 as the left hand side does. • Next we assume the quantity h i ./1 .@21  @22 /…2 C 2@1 @2 …3

is also bounded in Lp :

(29)

• Taking the divergence of (28) yields PO D divdiv.u ˝ u/ C .@21  @22 /…2 C 2@1 @2 …3

(30)

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due to the incompressibility divu D 0. Thanks to the assumption above and the regularity theory of elliptic equations, one concludes that the total pressure PO would be bounded then in Lp . • Applying the operator .@1 ; @2 / to (28), one obtains …2 D 2@t @1 u1 C @1 div.uu1 /  @2 div.uu2 /  2@1 u1 C .@21  @22 /PO due to the incompressibility divu D 0: According to the third observations and again the elliptic regularity, it yields that …2 is again bounded in Lp . • Applying the operator .@2 ; @1 / to (28) yields O …3 D @t .@2 u1 C@1 u2 /C @2 div.uu1 / C @1 div.uu2 /  .@2 u1 C @1 u2 / C 2@1 @2 P; and hence an estimate of …3 in Lp follows as before. Because of the incompressibility of fluids or equivalently the volume-preserving property of the flow map, det F D 1, one then can obtain an estimate of F in terms of …2 , …3 as follows: ˇ ˇ  ˇ 2 > 2ˇ > > C jF jFj2  C ˇjF> j  jF j  F j C 1 : ˇ 1 2 1 2 This inequality above, combined with improved integrability of …2 and …3 , yields a higher integrability of F as desired. The latter eliminates the possibility of concentrations.

3

Polymeric Fluids

The second topic discussed here is the macro-micro models of polymeric fluids. It couples fluids motion with kinetic-type equations on configuration spaces at different scales. During the manufacture of polymeric (or plastic) materials, liquids containing polymers are subjected to flows. Polymer molecules in the microscopic scale behave like springs and become stretched by the flow in the macroscopic scale, giving rise to a strong elastic behavior of polymeric fluids. Elastic stresses in polymeric (and other complex) fluids can then create strange flow behavior and sometimes undesirable instabilities not seen in Newtonian fluids. At a microscopic level, polymeric fluids usually can be viewed as materials that consist of beads joined by springs or rods, and the classical kinetic theory leads to a Fokker-Planck-Smoluchowski equation for the probability density .Q; x; t/ with a drift term depending on the spatial gradient of the velocity, where the function is the distribution function of molecule orientations and Q denotes the set of variables defining the coarse-grained microstructure whose lower moments provide the elastic stress for the moment transport. At the macroscopic level, assuming that the flow is incompressible, the velocity of the fluid is subject to

22 Equations for Viscoelastic Fluids

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the incompressible Navier-Stokes equation with an elastic stress that reflects the microscopic contribution of the polymer molecules to the overall macroscopic flow fields. A macro-micro model of polymeric fluids is described by 8 ˆ ˆ 0, and assume the potential function satisfies some decay assumptions at infinity. There exists a sufficiently small constant " such that if 1 ju0 j2L2 C 2 ju0 j2H s

C

Z

Z

R3



R3

kf0 k2H s



0 ln 0 C U 0 dQdx  "1Ca ;

C jQf0 j2H 4  ";

where a is a fixed small positive p constant, then (31) has a unique global classical solution .u; / with D M C M f > 0 and sup t0



ju.t /j2H s

C kf

.t/k2H s

C jQf

.t /j2H 4



Z t Z

jruj2 dx

C 0

R3

ˇ2 # ˇ ˇ2 ! ˇ  ˇ ˇ ˇ ˇ 1 1 C ˇˇrQ f C rQ Uf ˇˇ ds  C ": C ˇˇjQj rQ f C rQ Uf ˇˇ 2 2 H4 Hs The well posedness of (31) in bounded domains had been discussed in [113]. The large-time behavior of (31) is considered by He and Zhang in [49], where the solution obtained in Theorem 4 is shown to converge to zero as t goes to infinity. For FENE-type models, their known mathematical results are usually limited to the small time well posedness or the global well posedness with small data of classical solutions. Renardy in [107] proved the local existence for some special potential U D .1  jQj2 /1

22 Equations for Viscoelastic Fluids

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for some > 1. Jourdain-Lelievre-Le Bris [66] proved local existence as b 2 > 2 for a Couette flow by solving a stochastic differential equation; see also [119] by E-Li-Zhang for b > 6 and [67] for the exponential convergence to equilibrium via an entropy inequality. Local well posedness for b > 6 and b > 0 was proved by ZhangZhang in [127] and by Masmoudi in [99], respectively. Near the equilibrium, the global solution had also been established in [99] by using Hardy-type inequalities due to the diffusion of Fokker-Planck operators. The problem without size restriction in general is notoriously difficult and remains largely open. When ru in the equation is replaced by its antisymmetric > part S .u/ D ru.ru/ ; referred as the corotational model, the global classical 2 solution without any restriction on the size of solutions had been obtained by Lin-Zhang-Zhang [90] in two dimensions (see also [30, 113]). The key argument in [30, 90] is based on a priori estimates for incompressible Navier-Stokes equations by Chemin and Masmoudi [19]. On the other hand, in terms of weak formulations, Lions and Masmoudi in [96] proved the global existence of weak solutions of corotational micro-macro models; see also [8]. With the presence of a center-of-mass diffusion term x in the Fokker-Planck equation, the corotational assumption can be removed, and the global weak solutions have been constructed both for incompressible fluids in [3–5] and for compressible fluids in [2, 6, 7]. Masmoudi proved in [100] a remarkable global existence of weak solutions of FENE models as Leray’s solutions for incompressible Navier-Stokes equations. One of key observations in [100] is the improved L2 bound on the elastic stress  (see (34)) which comes from Hardy-type inequalities by H 1 norm in Q due 1

to the Fokker-Planck operator. An extra entropy-type Lx2 L log2 L bound on the density

R

Z R3

B

1C

0 log2

R B

0 dQ 0 1

0 log

2

0 dQ 0 1

 12 dx < 1

(35)

is needed in [100] to ensure the equi-integrability of the elastic stress and the weak stability. The argument in [100] is based on a renormalization procedure and the measurement of the defect measure associated with approximate solutions. Whether the extra bound (35) can be removed remains still an open problem. The transition from the micro-macro models such as (31) to the pure macroscopic model, that is, the mean field limit, is challenging not only in computations but also in analysis. For instance, using the Hilbert expansion, Wang-Zhang-Zhang verifies the mean field limit from the Doi-Onsager equation to the Ericksen-Leslie equation in [117] and from Landau-de Gennes equation to Ericksen-Leslie equation in [118]; see also [45]. Moreover, using the weak compactness argument, the diffusion limit from the Vlasov-Poisson-Fokker-Planck system to the PNP system in a bounded domain with reflection boundary conditions is discussed in [122].

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IMHD System

The third topic of this chapter is the following multidimensional incompressible magnetohydrodynamic fluids (IMHD): 8 ˆ @t u C u  ru  u C rP D .r  B/  B; ˆ ˆ ˆ 0; 1 > 0; 2˛4 C ˇ > 0; ˛1 C ˇ C ˛4 > 0; 1 .2˛4 C ˇ/  22 > 0: For recent results on Parodi-like relations, we refer to [3].

3.4

The Variational Approach

One may derive the Ericksen-Leslie equations also by a variational approach. The latter is due to Liu, Wu, and Xu [84] and is based on the least action and the maximal dissipation principle. The starting point start of a sketch of this approach is the flow map x D .X; t / W x X 0 ! t , where X and x denote the Lagrangian and Eulerian coordinates, respectively. Given a velocity field u.x; t /, the flow map solves the equation xt .x.X; t /; t / D u.x.X; t /; t /;

x.X; 0/ D X 2 X 0 :

@xi The deformation tensor F of x is given by F D . @X /1i;j 3 and satisfies j

@t F C u  rF D ruF:

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M. Hieber and J.W. Prüss

The kinematic transport of the director d is given by d .x.X; t /; t / D Ed0 .X /, where the deformation tensor E satisfies the transport equation @t E C u  rE D VE C .2˛  1/DE;

(25)

where ˛ is a constant. The total energy is given by eDe

ki n

i nt

C e ; where e

ki n

1 D 2

Z

2

u dx and e

i nt

Z D



.d; rd /dx: 

Consider the case of incompressible fluids, i.e., one has div u D 0, and the flow map is hence volume preserving. The least action principle says that the action functional A.x/ D

Z

T

.e ki n  e i nt /dt

0

minimizes among all volume-preserving flow maps x. Liu, Wu, and Xu considered in [84] the modified Oseen-Frank functional D 12 jrd j2 C f .d /, where f .d / D 1 .d 2  1/2 . Note that this choice of implies that the condition jd j2 D 1 is not 4"2 preserved for all t > 0. It follows from the least action principle (for the precise calculation of the term  below, see [84], Sect. 7.1) that @t u C u  r C rp D r  .rd Œrd T / C r ;

(26)

for  D  12 .1  2 =1 /.d  rf .d // ˝ d + 12 .1 C 2 =1 /d ˝ .d  rf .d //. Then (25) yields @t d C u  rd  Vd C

2 1 Dd D  .d  rf .d //: 1 1

(27)

The dissipation of this system is given by Z 1 D D ˛1 jd T Dd j2C ˛4 jruj2 C.˛5 C˛6 /jDd j2 C1 jN j2C.2 ˛2 ˛3 / : 2  The maximal dissipation principle asserts that the first order variation of the dissipation functional with respect to u is vanishing subject to the constraint div u D 0. Following the calculation of [84], Sect. 7.2, and taking account Parodi’s relation (22), one arrives at 0 D rp  div .rd Œrd T / C div  L : Combining this with equation (26) yields the first line of (21). Taking into account also equation (27), one obtains the Ginzburg-Landau approximated version of the Ericksen-Leslie system (21).

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

3.5

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Other Approaches

The classical isothermal Ericksen-Leslie equations can be derived in addition also by other approaches. A classical model that describes nematic phase transitions is the model proposed by Onsager [71]. Following Onsager, Maier and Saupe [68] introduced a slightly different model by replacing Onsager’s mean-field interaction potential by a modified interaction potential, which is nowadays called the Maier-Saupe potential. Since the Maier-Saupe potential is much easier to handle analytically compared to the original one, this potential was used often for analytical purposes in various textbooks; see, e.g., [7,13]. The free energy associated to MaierSaupe model is a Lyapunov functional, and the equilibrium points of this free energy have been characterized in [65]. Doi and Edwards [15] extended Onsager’s theory and presented a model describing the behavior of liquid crystal polymer flow. Of special interest here is the nowadays called Doi-Onsager equation. Recently, Wang, P. Zhang, and Z. Zhang presented in [87] a rigorous derivation of the Ericksen-Leslie equations starting from the Doi-Onsager equation by making use of the Hilbert expansion method. They also gave a rigorous derivation of the Ericksen-Leslie system starting from the Beris-Edwards system; see [88]. Han, Luo, Wang, P. Zhang, and Z. Zhang [27] derived the Ericksen-Leslie system from a newly developed Q-tensor system. One also would like to mention here a statistical approach to liquid crystal theory due to Seguin and Fried [79, 80]. All these approaches are fairly involved, and we refrain from giving details. One reason for this is also that, following the thermodynamical approach presented in the following Sect. 4, one obtains a rigorous derivation of the Ericksen-Leslie model in a very efficient, elegant, and transparent manner. This approach is based on the entropy principle and will be presented in detail in the following Sect. 4.

4

Thermodynamically Consistent Modeling of the Ericksen-Leslie Model

In this section, a self-contained presentation of a thermodynamically consistent modeling of liquid crystals is given. The approach presented does not only extend the classical Ericksen-Leslie model to the non-isothermal situation in a thermodynamically consistent way, but it also allows to exhibit the physical and mathematical beauty of this model. Below,   Rn always denotes a domain with C 1 -boundary.

4.1

First Principles

One begins with the balance laws of mass, momentum, and energy. They read as @t  C div.u/ D 0

in ;

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M. Hieber and J.W. Prüss

.@t C u  r/u C r D div S

in ;

.@t C u  r/ C div q D S W ru  div u u D 0;

(28)

in ;

q D0

on @:

Here  means density, u velocity,  pressure,  internal energy, S extra stress, and q heat flux. This immediately gives conservation of the total energy. In fact, we have .@t C u  r/e C div.q C u  S u/ D 0

in ;

where e WD juj2 =2 C  means the total mass-specific energy density (kinetic plus internal). The energy flux ˆe is given by ˆe WD q C u  S u. Integrating over  yields Z @t E.t / D 0;

E.t / D Eki n .t / C Ei nt .t / D

.t; x/e.t; x/dx; 

provided q DuD0

on @:

(29)

Hence, if (29) holds, total energy is preserved, independent of the particular choice of S and q.

4.2

Thermodynamics

Assume a given free energy of the form D .;  ; /, where  denotes the (absolute) temperature, and will be specified later. One then has the following thermodynamical relations: D

C 

D @

internal energy; entropy;

 D @  D  @2

(30) heat capacity

Later on, for well-posedness of the heat problem, one requires  > 0, i.e., to be strictly concave with respect to  2 .0; 1/. In the classical case, where depends only on  and  , one has the ClausiusDuhem equation .@t C u  r/ C div.q=/ D S W ru=  q  r= 2 C .2 @  /.div u/=

in :

Hence, in this case, the entropy flux ˆ is given by ˆ WD q= and the entropy production by

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

r WD S W ru  q  r= C .2 @

1093

 /.div u/:

Employing the boundary conditions (29), one obtains for the total entropy N by integration over  Z @t N.t / D

Z r.t; x/dx  0;

N.t / D



.t; x/ .t; x/dx; 

provided r  0 in . As div u has no sign, one requires  D 2 @ ;

(31)

which is the famous Maxwell relation. Further, as S and q are independent, this requirement leads to the classical conditions S W ru  0

and

q  r  0:

(32)

Summarizing, one sees that whatever one chooses for S and q, one always has conservation of energy, and the total entropy is non-decreasing provided (32),(31), and (29) are satisfied. Thus, these conditions ensure the thermodynamic consistency of the model. As an example for S and q, consider the classical laws due to Newton and Fourier which are given by S WD SN WD 2s D C b div u I;

2D D .ru C Œru T /;

q D ˛0 r :

In this case, (32) is satisfied as soon as s  0, 2s C nb  0 and ˛0  0 holds. Note that it does not matter at all whether s ; b ; ˛0 are constants or whether they depend on ; , or on other variables.

4.3

Nematic Liquid Crystals

For isotropic nematic liquid crystals, one assumes a free-energy density form D

.;  ; /;

with D

of the

1 jrd j22 : 2

Here, d means the orientation vector, also called the director, which should satisfy the condition jd j22 WD

n X j D1

dj2 D 1:

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M. Hieber and J.W. Prüss

Note that 2 D tr.rd Œrd T / is the first invariant of the matrix rd Œrd T , and for its last invariant, it holds det.rd Œrd T / D .detrd /2 D 0, as rd d D 0 by jd j2 D 1. One neglects spin energy below but takes into account transport of energy due to couple stress. This means that the energy flux is replaced by Dt D @t C u  r;

ˆe WD q C u  S u  …Dt d; where … has to be modeled. As constitutive laws, one employs S D SN C SE C SL ;

SE D rd Œrd T ;

q D ˛0 r  ˛1 .d  r /d: (33)

SN means the Newton stress introduced above, SE the Ericksen stress, and SL the Leslie stress which will be defined later. Assuming these two constitutive laws, one obtains the following balance of entropy, i.e., the Clausius-Duhem equation. A short computation gives .@t C u  r/ C div ˆ D r;

(34)

with ˆ D q=, and r D q  r= C 2s jDj22 C b jdiv uj2 C .2 @ C .@

 /div u

 /rd Œrd T W ru C .…  @ rd / W rDt d

C SL W ru C .div… C ˇd /  Dt d: for some scalar function ˇ. Note that d  Dt d D 0 as jd j2 D 1, hence ˇ 2 R can be chosen arbitrarily. For the entropy production r to be nonnegative, one requires s  0;

2s C nb  0;

˛0  0;

˛0 C ˛1  0:

Except for the last one, these conditions are the well-known conditions from fluid dynamics; see Sect. 4.2. The subsequent terms in the definition of r have no sign, hence one requires them to vanish, which yields the relations  D 2 @ ;

 D @

… D @ rd:

(35)

Finally, to obtain nonnegativity of the last two terms, in the simplest case, one may assume that the Leslie stress SL vanishes, and  Dt d D divŒ.@ /r d C ˇd;

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1095

for some  D .;  ; /  0. The condition jd j2 D 1 then requires ˇ D jrd j22 ; which leads to the equation .@t C u  r/d D divŒr d C jrd j22 d;

(36)

a nonlinear convection-diffusion equation for d . This is the basic equation governing the evolution of the director field d . With these assumptions, the entropy production reads as r D q  r= C 2s jDj22 C b jdiv uj2 C

1 2 jaj ;  2

where a D divŒr d C jrd j22 d D  Dt d D Pd divŒr d;

Pd D I  d ˝ d:

At the boundary @, energy should be conserved and entropy production should also be nonnegative, which by q  D 0 and u D 0 requires 0 D  ˆe D @ d  @t d: This is clearly valid if d satisfies the Neumann condition @ d D 0, which is physically reasonable.

4.4

Stretching and Vorticity

Observe that the equation (36) for d admits the solutions d D const, no matter how the velocity field and the temperature field are defined. In this case, the director field is not at all affected by the fluid dynamics. This means that the above model needs to be adapted to be physically meaningful. This can be done by introducing a so-called stretching stress. To introduce this stress, one follows Leslie [49]. Define the vorticity V according to 2V D ru  Œru T , and set n D V Vd C D Pd Dd   Dt d; where V ; D ;  are scalar functions of ;  ; and  > 0. Now one defines the stretch tensor SLst retch WD

D  V D C  V n˝d C d ˝ n: 2 2

(37)

This modification of the model does not change the entropy flux ˆ D q=, and the relevant entropy production becomes

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M. Hieber and J.W. Prüss

SLst retch W ru C Dt d  a D D

1 .jnj22 C  Dt d  n/ C a  Dt d  1 .jaj22 C .n C a/  .V Vd C D Pd Dd  a//: 

If one wants to keep the total entropy production at the same level as in the previous section, the simplest way to achieve this is to set n C a D 0, which yields the equation .@t d C u  rd / D div.r/d C jrd j22 d C V Vd C D Pd Dd:

(38)

This is the stretched equation for d . Note that it preserves the constraint jd j2 D 1. The entropy production is the same as before, and one has r D Œ˛0 jrj22 C ˛1 .d jr /2 = C 2s jDj22 C b jdiv uj2 C

1 jPd div.r/d j22 : 

R In particular, the total entropy N satisfies @t N.t / D  r.t; x/dx; and so N will be shown below to be a strict Lyapunov functional for the system, as soon as s > 0;

2s C nb > 0;

˛0 > 0;

˛0 C ˛1 > 0;

 > 0;

 > 0;

@  > 0:

 > 0;

(39)

and (40)

Note that no conditions on the new parameter functions D ; V are needed.

4.5

Additional Dissipation

One may add additional dissipative terms in the stress tensor of the form SLd i ss WD

P .n ˝ d C d ˝ n/  C

L C 2P .Pd Dd ˝ d C d ˝ Pd Dd / C 0 .Dd jd /d ˝ d; 2

(41)

whereas before 2D D ru C Œru T and 2V D ru  Œru T are the symmetric and antisymmetric parts of the rate of strain tensor ru. Note that the tensor SLd i ss is symmetric. Adding these terms to the stress tensor will be thermodynamically consistent provided their contribution to the entropy production ensures that the total entropy production remains nonnegative. A simple calculation yields

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1097

2 P .njPd Dd / C .L C P /jPd Dd j22 C 0 .Dd jd /2 :   (42) Hence, with n D a, the total relevant dissipation amounts to SLd i ss W ru D SLd i ss W D D 2

1 .jaj22 C 2P .njPd Dd / C 2P jPd Dd j22 / C L jPd Dd j22 C 0 .Dd jd /2  D

1 ja  P Pd Dd j22 C L jPd Dd j22 C 0 .Dd jd /2 ; 

hence, the total entropy production becomes r D Œ˛0 jr j22 C ˛1 .d jr /2 = C 2s jDj22 C b jdiv uj2 C

1 jPd div.r/d  P Pd Dd j22 C L jPd Dd j22 C 0 .Dd jd /2 : 

Note that the parameter functions j , j D 0; s; b; V; D; P; L, ˛0 ; ˛1 , and  for thermodynamical consistency are only subject to the requirements ˛0 ; ˛0 C ˛1  0;

s ; 2s C nb  0;

0 ; L  0;

 > 0:

(43)

Recall that all parameters functions are allowed to be functions of ;  ; . Remark 2. A more refined algebra shows that it is enough to require for the viscosities j s > 0;

2s C L  0;

2s C 0  0

in the incompressible case, and additionally 20 2s 0  .2s C 0 /. C b C 2 / 2 n n n in the compressible case. Remark 3. One would like to emphasize that in the case V D  , the parameters s ; 0 ; V ; D ; P ; L are in one-to-one correspondence to the celebrated Leslie parameters ˛1 ; : : : ; ˛6 given in (18). This shows that the above model contains the isotropic Ericksen-Leslie model as described in (21) as a special case.

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M. Hieber and J.W. Prüss

The General Model: Non-isothermal, Compressible Fluid, and Isotropic Elasticity

Summarizing the above considerations, the complete model may be represented as 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ
0; (47)

Finally, one uses in addition the following conditions

s > 0;  > 0;

2s C nb > 0; ˛0 > 0;  > 0; @  > 0;

˛0 C ˛1 > 0;

(48)

to identify the equilibria and to investigate their thermodynamic stability in Sect. 5.

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

4.7

1099

The General Compressible Non-isotropic Model D

Finally, the non-isotropic case is being discussed. Then the Ericksen stress tensor becomes SE D 

.;  ; d; rd /, and

@ Œrd T : @.rd /

Following the derivation in Sects. 4.3, 4.4, and 4.5, the energy and entropy fluxes now read again as ˆe WD q C u  S u  …Dt d;

ˆ D q=:

The equation for d becomes  Dt d D Pd a;

a D @i .r@i d /  rd ;

in the case without stretching, and  Dt d D Pd a C V Vd C D Pd Dd in the stretched case. The couple stress here is … D @rd , and the entropy production now reads as r D Œ˛0 jr j22 C ˛1 .d jr /2 = C 2s jDj22 C b jdiv uj2 C

1 jPd .a  P Dd /j22 C L jPd Dd j22 C 0 .Dd jd /2 : 

Summarizing, the complete model in the case of non-isotropic elasticity becomes 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
0 and ˛0 C ˛1 > 0. Hence,  D  is constant in . Next, by s > 0, 2s C nb > 0, we also have D D 0 in . By Korn’s inequality and the no-slip boundary condition for u, we hence obtain u D u D 0 in , t 2 .t1 ; t2 /. Therefore @t  D @t u D 0, which implies r D 0. Finally,  > 0 yields Dt d D 0 in , which implies that d satisfies the nonlinear eigenvalue problem div.a.x/r/d C a.x/jrd j22 d D 0 in ; jd j2 D 1 in ; @ d D 0 on @;

(53)

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1101

where a.x/ D ..t; x/; .t /; .t; x//, for each fixed t 2 .t1 ; t2 /. But, as the following lemma proved in [30] shows, this implies rd D 0 in . Hence, d D d is constant. Lemma 1 ([30]). Let q > n, a 2 Hq1 ./, a > 0 and suppose that d 2 Hq2 .I Rn / satisfies (53). Then d is constant in . Knowing that  and d are constant in  and r D 0, one sees that  D 2 @ .;  ; 0/ is constant; hence,  D  is constant, provided the function  7! .; ; 0/ is strictly increasing. This shows that we are at an equilibrium . ; u ;  ; d / 2 E with E D f. ; u ;  ; d / 2 .0; 1/  f0g  .0; 1/  Rn W jd j2 D 1g; the set of physical equilibria. In particular, the functional N is a strict Lyapunov functional. Observe that E forms an n C 1-dimensional manifold. If one takes into account conservation of mass and energy, Z M0 WD

Z dx D  jj;

E0 WD





.juj22 =2 C /dx D  " jj;

at an equilibrium, then the values of  and  are uniquely determined by  D M0 =jj;

 WD . ;  ; 0/ D E0 =M0 ;

whenever  7! .;  ; 0/ is strictly increasing, i.e., whenever  > 0.

5.2

Critical Points of the Total Entropy

(a) Consider the entropy functional N with constraints of prescribed mass M D M0 and energy E D E0 , as well as G.d / WD .jd j22  1/=2 D 0. Suppose one has a sufficiently smooth critical point .; u;  ; d / of N with ;  > 0, subject to the constraints. Then the method of Lagrange multipliers yields M ; E 2 R, and G 2 L2 ./ such that hN0 C M M0 C E E0 C G G 0 jzi D 0; where z D . ; v; #; ı/. We have hM0 jzi D

Z  dx; 

hG G 0 jzi D

Z G d  ıdx; 

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M. Hieber and J.W. Prüss

and

Z

0

hN jzi D

Œ.@ . // C @ # C @ rd W rı dx; 

as well as

Z

0

hE jzi D

Œu  v.@ .// C @ # C @ rd W rı dx: 

This yields the relation Z

0D

1 fŒ@ . / C M C E . juj22 C @ .//  C Œ@ C E @  #gdx 2  Z C fE u  v C Œ@ C E @  rd W rı C G d  ıgdx: 

One first varies # to obtain .@ C E @ / D 0, which by  > 0 and by the definition of ;  and  > 0 yields E D 1=. Hence,  is constant and E < 0. Next, varying v, we obtain u D 0, as E and  are not zero. Varying ı, which after an integration by parts, employing the boundary condition @ d D 0, implies div.r/d C G d D 0

in :

But then jd j2 D 1 implies G D jrd j22 , and d is a solution of the problem (53), which by Lemma 1 shows that d is constant. Finally, one varies  to the result that @ . / D  M is constant. As  D 2 @ is strictly increasing in the variable , this shows that  is constant in  as well. Therefore, the critical points of the entropy functional are precisely the equilibria of the problem. (b) Let H WD N00 C E E00 denote the second variation of N. Note that M00 D 0 and G D jrd j22 D 0. The identities 1 1 @ / D ; @ @ . /  @ @ ./ D 0;   @  1 1   @2 . /  @2 ./ D  ; @2  @2  D  2 ;     .@ 

imply Z hH zjzi D 

 @  

2 C

 2 # C jrıj22 dx  0; 2 

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1103

by ; ; @   0. This shows that the second variation of N at an equilibrium is negative semidefinite, which means that the equilibria are thermodynamically stable. (c) Summarizing, one obtains the following basic result. Theorem 2 ([31]). The general, compressible isotropic model described in Sect. 4.6 has the following properties: (i) (ii) (iii) (iv) (v)

Along smooth solutions, total mass M and energy E are preserved. Along smooth solutions, the total entropy N is non-decreasing. The negative total entropy is a strict Lyapunov functional. The condition jd j2 D 1 is preserved along smooth solutions. The equilibria are given by the set of constants E D f. ; 0;  ; d / W  ;  2 .0; 1/; d 2 Rn ; jd j2 D 1g: Here,  ;  are uniquely determined by the identities  D M0 =jj;

. ;  ; 0/ D E0 =M0 :

(vi) The equilibria are precisely the critical points of the total entropy with prescribed mass and energy. (vii) The second variation of N with given mass and energy at equilibrium is negative semidefinite. In particular, the model is thermodynamically consistent, and it is also thermodynamically stable. Remark 4. In the non-isotropic case, N is still a strict Lyapunov functional for the problem, but in that case, the set of equilibria is unknown. The same remark is valid if in the isotropic case, Dirichlet conditions for d are employed.

6

Return to the Isothermal Case

In the isothermal case, we set  D const and ignore the equation for the energy. In this case, instead of the total mass-specific energy e D juj22 =2 C , one employs the available energy ea , which is defined by ea D juj22 =2 C . One has the following balance of ea which is a direct consequence of balance of total energy and entropy .@t C u  r/ea C div.ˆe   ˆ / D  r   Dt   ˆ  r : In case  is constant, this reduces to .@t C u  r/ea C div.ˆe   ˆ / D ra ;

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M. Hieber and J.W. Prüss

with ra D 2s jDj2 C b jdiv uj2 C SLd i ss W ru C  jDt d j22 : Therefore, in the isothermal case, the total available energy Ea is a strict Lyapunov functional for the system, Z Z .t; x/ea .t; x/dx: @t Ea .t / D  ra .t; x/dx; Ea .t / D 



As a consequence, the equilibrium set is the same as in the non-isothermal case, dropping temperature, hence is a manifold of dimension n, and when one incorporates preserved mass, it is isomorphic to the unit sphere in Rn .

6.1

The General Isotropic Compressible Isothermal Ericksen-Leslie Model

In the above situation, the equations for the general compressible Ericksen-Leslie model in the isothermal situation read as 8 @t  C div.u/ D 0 in ; < (54) .@t C u  r/u C r D divS in ; : 2 .@t C u  r/d  V Vd  div.r/d D jrd j2 d C D Pd Dd in ; where

D

.; /,  D 2 @ ,  D @ , and

S D 2s D C b div u I  rd Œrd T C

D  V D C V n˝d C d ˝ n C SLd i ss ; 2 2 (55)

where n D V Vd C D Pd Dd   Dt d as before.

6.2

The Isothermal Simplified Ericksen-Leslie Model

If one further restricts to the incompressible case, where  > 0 is constant,  is constant, and moreover to the case where D D V D 0, and SLd i ss D 0, with  D s , one obtains the so-called isothermal simplified Ericksen-Leslie model which reads as 8 .@t C u  r/u C r D u  div .rd Œrd T / in ; ˆ ˆ < div u D 0 in ; 2 ˆ .@t C u  r/d  d D jrd j d in ; ˆ : u.0/ D u0 ; d .0/ D d0 in : Of course, we have to add suitable boundary conditions.

(56)

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1105

Remark 5. Note that the model (56) coincides with the model considered by Lin [55] and Lin and Liu [57]. Part II: Analysis of the Ericksen-Leslie Equations In the following four sections, well-posedness results for most of the models derived in Part I are being discussed. In particular, well-posedness results are presented for the simplified and penalized model, the isothermal simplified model, as well as for the general isotropic model both for incompressible and compressible fluids. As already written in the introduction, mainly results on strong solutions for the abovementioned models are being described. It seems that the results given in Theorems 9 and 14 below are the first wellposedness results on the Ericksen-Leslie equations dealing with general Leslie stress S L and without assuming additional conditions on the Leslie coefficients. They answer all the questions (a)–(e) raised in the introduction in a satisfactory way, provided homogeneous Neumann boundary conditions for d are assumed. They are proved by means of techniques involving maximal Lp -regularity.

7

Analysis of Simplified Penalized Models

7.1

The Simplified and Penalized Model

The first simplification of the full Ericksen-Leslie model described in (19) is going back to Lin [55] and Lin and Liu [57]. They consider a modification of the original system, where the microscopic alignment is governed by the heat flow of harmonic maps on spheres and where the macroscopic behavior of the fluid is given by the Navier-Stokes equations. They also provided a formulation of the original system in which the director does not need to have unit length. More precisely, they studied the system 8 ˆ ˆ ut  u C .u  r/u C r D < div u D ˆ dt C u  rd D ˆ : .u; d /jtD0 D

div .rd Œrd T / in .0; T /  ; 0 in .0; T /  ; .d  f .d // in .0; T /  ; .u0 ; d0 / in ;

(57)

subject to suitable boundary conditions and where f" .d / D

1 .jd j22  1/d: 4"2

This system was derived by free-energy (a) Setting k1 D k2 D k3 D 1 and k4 D 0 in the Oseen-Frank R this case, F reduces to the Dirichlet energy F D 12  jrd j2 ,

F

. In

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M. Hieber and J.W. Prüss

(b) Replacing the condition jd j22 D 1 by the penalization f" .d /. This means that the original free-energy functional F.d / is being replaced by the functional F" .d / WD

Z

1 Πjrd j2 C f" .d / dx;  2

(c) Assuming that the Leslie coefficients ˛i satisfy ˛1 D ˛5 D ˛6 D 0, (d) Eliminating all terms related to stretching effects, (e) All parameters are set constant. Lin and Liu proved in [57] the following results on weak and classical solutions for system (57) subject to the boundary conditions: uD0

and

d D db on @:

(58)

Theorem 3 ([57] Existence of global weak solutions). Let J D .0; T / for some T > 0,   R3 be a domain with boundary @ of class C 2 and " > 0. Assume 3=2 that u0 2 L2; ./, d0 2 H21 ./, and d0 j@ 2 H2 .@/. Then system (57) subject to (58) admits a global weak solution .u; d / satisfying 1 u 2 L2 .J I H2; .// \ L1 .J I L2; .//;

d 2 L2 .J I H22 .// \ L1 .J I H21 .//:

Concerning classical solutions, they proved the following result. Proposition 2 ([57] Well-posedness for large viscosity). Assume that   R3 1 is a bounded domain with boundary of class C 2 , " > 0 and u0 2 H2; ./ and 2 d0 2 H2 ./. Then there exists 0 D 0 .;  ; u0 ; d0 / such that system (57) subject to (58) admits a global classical solution .u; d / provided   0 . Remark 6. (a) Note that the result in the above proposition is not a result on large data as it appears at first glance. It is a result on small data, as a simple scaling argument shows. (b) The partial regularity result on suitable weak solutions for the Navier-Stokes equations due to Caffarelli, Kohn, and Nirenberg [5] is a classical result in the theory of the Navier-Stokes equations. Lin and Liu [58] proved a similar result for the system (57). More precisely, they showed that for sufficiently smooth initial and boundary data, there exists a suitable weak solution to (57) such that the one-dimensional parabolic Hausdorff measure of the singular set of this solution is zero. (c) Global strong well-posedness of the simplified and penalized system on bounded domains subject to Dirichlet boundary conditions was studied by Wu in [89]. In addition, a longtime behavior result using the Lojasiewicz-Simon approach is presented there. For related results, see also [37].

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1107

(d) Dai and Schonbeck [10] obtained global existence of strong solutions to the simplified and penalized system in the spaces H2N .R3 /  H2N C1 .R3 / for N  1 by requiring the smallness condition ju0 j2H 1 C jd0  db j2H 2  ; 2

2

where db is a constant unit vector. They also established time decay rates for the solution .u; d / of the form jr k u.t /jL2 C jr k .d  db /jL2  C .1 C t /

3C2k 4

;

t > 0;

for k D 0; 1; : : : ; N .

7.2

The Penalized Model with Stretching

(a) Lin and Liu [59] considered the Ericksen-Leslie model under the assumption that the energy functional is assumed to be of the form "

D

1 1 jrd j22 C 2 .jd j22  1/2 : 2 4"

Replacing F in equation (19) by " yields the so-called penalized EricksenLeslie system. Lin and Liu showed that under some further restrictions, and assuming that ˛1 ; ˛4 ; ˛5 C ˛6 > 0, 1 < 0, and 2 D 0, the penalized Ericksen-Leslie system defined on a bounded domain   R3 with smooth boundary @ and subject to Dirichlet boundary conditions admits a global weak solution .u; d / satisfying u 2 L2 .J I H21 .// \ L1 .J I L2 .//;

d 2 L2 .J I H22 .// \ L1 .J I H21 .//; 3=2

for u0 2 L2 ./ and d0 2 H21 ./ satisfying d0 j@ 2 H2 .@/. Here for T > 0, we set J D .0; T /. As indicated by the authors, the above conditions on the Leslie coefficients seem to have no physical meaning. (b) Coutand and Shkoller [9] considered a modification of system (57) in which the third line of (57) is replaced by @t d C u  rd  d  ru D .d  f" .d // in .0; T /  :

(59)

They proved local well-posedness for this system and gave as well a global existence result for small data within this setting. Note, however, that the presence of the stretching term d  ru causes the loss of the total energy balance and, moreover, the condition jd j2 D 1 in .0; T /   is not preserved. (c) In order to prevent the failure of the energy balance, Sun and Liu [82] introduced a variant of the above model, where the stretching term is included, and an additional component in the stress tensor is added in order to save the energy balance.

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M. Hieber and J.W. Prüss

(d) Wu, Xu, and Liu reconsidered in [84] the penalized Ericksen-Leslie model and where able to remove some of the above conditions on the Leslie coefficients. More precisely, the authors proved the existence of a unique, strong solution provided the viscosity is large enough. Moreover, they showed Lyapunov stability of local energy minimizers and convergence to equilibria as time goes to infinity.

7.3

Convergence of Solutions of the Penalized Equations for  D R3

It is a natural question to ask whether, up to a subsequence, weak limits of weak solutions .u" ; d" / of the penalized Ericksen-Leslie system are weak solutions of the original system (19). It was shown in [59] that this holds true under various additional assumptions on the so-called defect measure. It is a difficult task to decide whether this defect measure is identically zero. Concerning strong solutions, the situation is more clear. Indeed, it was shown by Hong, Li, and Xin in [36] that the family of strong solutions .u" ; d" / of the penalized system on  D R3 given by 8 ˆ .@t C u  r/u C r D div S ˆ ˆ < div u D 0 F ˆ .@ C u  r/d D div. @@rd /  rd t ˆ ˆ : u.0/ D u0 ; d .0/ D d0

F

in ; in ;  f" .d / in ; in ;

(60)

where S D SN C SE , converges to the strong solution .u; d / of the original system up to the maximal existence time. Here, given T > 0, a couple .u; d / is called a strong solution to system (60) on R3  .0; T /, if 2 u 2 L2 .0; T I H2; .R3 //; @t u 2 L2 .0; T I L2 .R3 //;

d0  d1 2 L2 .0; T I H23 .R3 //; @t d 2 L2 .0; T I H21 .R3 //; and if it satisfies equation (60) a.e. on R3  .0; T /. Here, d1 is a given constant vector of length one. Then the following holds true. Theorem 4 ([36]). Suppose .u; d / is a given strong solution of the non-penalized version of (60) in R3  .0; T  / with jd j2  1. Let .u" ; d" / be the unique strong solution of the penalized equation (60) in R3  .0; T" /, where T" is the maximal existence time for (60). Then, for sufficiently small ", T"  T  and for any T 2 .0; T  / .u" ; rd" / ! .u; rd / in L1 .0; T I L2 .R3 // \ L2 .0; T I H21 .R3 // and

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1109

lim supŒj.u" ; rd" /jL1 .H 1 / C j.u" ; rd" /jH 2;1 .R3 .0;T // < 1: 2

"!0

2

Furthermore, T  < 1 if and only if lim j.u" ; rd" /jLq .0;T  ILp .R3 // D 1:

"!0

for any q 2 Œ2; 1/; p 2 .3; 1 satisfying 2=q C 3=p D 1. The above theorem can be regarded as a blowup criterion in terms of Serrin type norms for strong solutions of the non-isotropic Ericksen-Leslie model with vanishing Leslie stress SL as well as stretching. It applies in particular to the simplified model which we will discuss in the following in more detail.

8

Analysis of the Isothermal Simplified Model

In the following, one discusses results for the isothermal simplified model given by (56) and subject to either Dirichlet boundary conditions u D 0;

d D db

on @;

(61)

on @:

(62)

or Neumann boundary conditions for d u D 0;

@ d D 0

The case of Robin boundary conditions for d , i.e., @t d C ˇ@ d D 0, will not be considered here.

8.1

Weak Solutions

In this subsection, a recent result due to Lin and Wang [63] on the existence of global weak solutions of (56) in three dimensions is described. More precisely, they proved the following result. Theorem 5 ([63]). Let   R3 be a bounded domain with smooth boundary or let  D R3 . Let u0 2 L2; ./ , and assume that d0 2 H21 .; S2 / satisfies d0 ./  S2C . Then there exists a global weak solution .u; d / W   Œ0; 1/ ! R3  S2 of problem (56) subject to (61) with db D d0 satisfying 1 .//; u 2 L1 .0; 1/I L2; .// \ L2 .0; 1/I H2;

as well as a global energy inequality.

d 2 L1 .0; 1/I H21 .; S2 //;

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M. Hieber and J.W. Prüss

The proof of Theorem 5 is based on blowup analysis of harmonic maps by Lin [56] and harmonic map heat flows by Lin and Wang [61].

8.2

Strong Solutions for the Case  D Rn

Consider the situation where  D Rn . In this case, inspired by the work of Tataru and Koch on the Navier-Stokes equation, Wang [85] proved the following result. Theorem 6 ([85]). Let  D Rn . Then there exists "0 > 0 such that if .u0 ; d0 / W Rn ! Rn  S2 with div u0 D 0 satisfies Œu0 BMO 1 .Rn / C Œd0 BMO 1 .Rn /  "0 ; then there exists a unique, global smooth solution .u; d / 2 C 1 .Rn  .0; 1/; Rn  S2 / of problem (56) satisfying in addition the estimate t 1=2 ku.t /k1 C krd .t /k1  C "0 ;

t > 0:

Remark 7. (a) Note that local well-posedness for initial data .u0 ; d0 / 2 R3 with u0 ; rd0 2 L3uloc .R3 /, the uniform locally L3 -space, satisfying small k.u0 ; rd0 /kL3;uloc .R3 / norms was obtained by Hinemann and Wang in [35]. (b) A criterion on possible blowup of local strong solutions for problem (56), (61) in analog to the Beal-Kato-Majda criterion for the Navier-Stokes equations, was obtained by Huang and Wang in [41]. It says the following: if the maximal existence time T  of a strong solution .u; d / for this problem satisfies 0 < T  < 1, then Z

T 0

.jr  uj1 C jrd j21 /dt D 1:

(c) Gao, Tao, and Yao [25] obtained global existence of strong solutions to the simplified system in the space H2N .R3 /  H2N C1 .R3 / for N  1 by requiring the smallness of ju0 j2H 1 C jd0  db j2H 2 , where db is a constant unit vector. They 2

2

also established time decay rates for the solution .u; d / assuming in addition that the data belong to L1 . (d) Fan and Li [19] gave regularity criteria for strong solutions in the whole space R3 of the following form: given T > 0, u0 2 H21 ; rd0 2 H21 with div u0 D 0, and jd0 j2 D 1, let .u; d / be the unique, local strong solution on R3  .0; T /. r / and rd 2 Lq .0; T I Lp / for certain values of If  2 L2=.2Cr/ .0; T W BP 1;1 r; p; q, then the solution .u; d / can be continued beyond T . Before presenting the quasilinear approach to the simplified system, key methods and techniques of this approach are collected in the following subsection.

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

8.3

1111

Background on Quasilinear Parabolic Evolution Equations

This section briefly recalls some results on abstract quasilinear parabolic problems vP C A.v/v D F .v/;

t > 0;

v.0/ D v0 ;

(63)

which will be essential for understanding the quasilinear approach presented in the following Sects. 8.4, 9.1, and 10.2. A convenient reference for this theory is the monograph by Prüss and Simonett [76], Chapter 5; see also [45, 75, 76] and [46]. Assume that .A; F / W V ! L.X1 ; X0 /  X0 and v0 2 V . Here, the spaces X1 ; X0 are Banach spaces such that X1 ,! X0 with dense embedding, and V is an open subset of the real interpolation space X; WD .X0 ; X1 /1=p;p ;

 2 .1=p; 1/:

One is mainly interested in solutions v of (63) having maximal Lp -regularity, i.e., v 2 Hp1 .J I X0 / \ Lp .J I X1 / DW E1 .J /; where J D .0; T /: The trace space of this class of functions is given by X WD X;1 . However, to see and exploit the effect of parabolic regularization in the Lp -framework, it is also useful to consider solutions in the class of weighted spaces 1 v 2 Hp; .J I X0 / \ Lp; .J I X1 / DW E1; .J /;

which means t 1 v 2 E1 .J /:

The trace space for this class of weighted spaces is given by X; . In the following approach, it is crucial to know that the operators A.v/ have the property of maximal Lp -regularity. Recall that an operator A0 in X0 with domain X1 has maximal Lp regularity, if the linear problem vP C A0 v D f;

t 2 J; v.0/ D 0;

admits a unique solution v 2 E1 .J /, for any given f 2 Lp .J I X0 / DW E0 .J /. It is known that in this case, maximal regularity also holds in the weighted case. Proposition 3. Let p 2 .1; 1/, v0 2 V be given and suppose that .A; F / satisfies .A; F / 2 C 1 .V I L.X1 ; X0 /  X0 /;

(64)

for some  2 .1=p; 1 . Assume in addition that A.v0 / has maximal Lp -regularity. Then there exist a D a.v0 / > 0 and r D r.v0 / > 0 with BN X ; .v0 ; r/  V such that problem (63) has a unique solution v D v.; v1 / 2 E1; .0; a/ \ C .Œ0; T I V /;

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M. Hieber and J.W. Prüss

on Œ0; a , for any initial value v1 2 BN X ; .v0 ; r/. In addition, t @t v 2 E1; .0; a/; in particular, for each ı 2 .0; a/, we have v 2 Hp2 ..ı; a/I X0 / \ Hp1 ..ı; a/I X1 / ,! C 1 .Œı; a I X / \ C 11=p .Œı; a I X1 /; i.e., the solution regularizes instantly. The next result provides information about the continuation of local solutions. Corollary 1. Let the assumptions of Proposition 3 be satisfied, and assume that A.v/ has maximal Lp -regularity for all v 2 V . Then the solution v of (63) has a maximal interval of existence J .v0 / D Œ0; tC .v0 //, which is characterized by the following alternatives: (i) Global existence: tC .v0 / D 1; (ii) lim inft!tC .v0 / distX ; .v.t /; @V / D 0; (iii) limt!tC .v0 / v.t / does not exist in X; . Next, assume that there is an open set V  X such that .A; F / 2 C 1 .V; L.X1 ; X0 /  X0 /:

(65)

Let E  V \ X1 denote the set of equilibrium solutions of (63), which means that v2E

if and only if

v 2 V \ X1 and A.v/v D F .v/:

Given an element v 2 E, assume that v is contained in an m-dimensional manifold of equilibria. This means that there is an open subset U  Rm , 0 2 U , and a C 1 function ‰ W U ! X1 , such that  ‰.U /  E and ‰.0/ D v ;  the rank of ‰ 0 .0/ equals m; and  A.‰.//‰./ D F .‰.//;

(66)

 2 U:

Suppose now that the operator A.v / has the property of maximal Lp -regularity, and define the full linearization of (63) at v by A0 w D A.v /w C .A0 .u /w/v  F 0 .v /w

for w 2 X1 :

(67)

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1113

After these preparations, one is able to formulate the following result on convergence of solutions starting near v , which is called the generalized principle of linearized stability. Proposition 4. Let 1 < p < 1. Suppose v 2 V \ X1 is an equilibrium of (63), and suppose that the functions .A; F / satisfy (65). Suppose further that A.v / has the property of maximal Lp -regularity, and let A0 be defined in (67). Suppose that v is normally stable, which means (i) (ii) (iii) (iv)

Near v the set of equilibria E is a C 1 -manifold in X1 of dimension m 2 N, The tangent space for E at v is isomorphic to N.A0 /, 0 is a semi-simple eigenvalue of A0 , i.e. N.A0 / ˚ R.A0 / D X0 ,  .A0 / n f0g  CC D fz 2 C W Re z > 0g.

Then v is stable in X , and there exists ı > 0 such that the unique solution v of (63) with initial value v0 2 X satisfying jv0  v j  ı exists on RC and converges at an exponential rate in X to some v1 2 E as t ! 1. The next result contains information on bounded solutions in the presence of compact embeddings and of a strict Lyapunov functional. Proposition 5. Let p 2 .1; 1/,  2 .1=p; 1/, N 2 .; 1 , with V  X; open. Assume that .A; F / 2 C 1 .V I L.X1 ; X0 /  X0 / and that the embedding X;N ,! X; is compact. Suppose furthermore that v is a maximal solution which is bounded in X;N and satisfies distX ; .v.t /; @V /  > 0; for all t  0:

(68)

Suppose that ˆ 2 C .V \ X I R/ is a strict Lyapunov functional for (63), which means that ˆ is strictly decreasing along nonconstant solutions. Then tC .v0 / D 1, i.e., v is a global solution of (63). Its !-limit set !C .v0 /  E in X is nonempty, compact, and connected. If, in addition, there exists v 2 !C .v0 / which is normally stable, then limt!1 v.t / D v in X .

8.4

The Quasilinear Approach for the Simplified Problem with Neumann Conditions

The main idea developed in [29] is to consider the problem not as a semilinear equation as in all of the previous approaches but as a quasilinear evolution equation. One thus incorporates the term div.rd Œrd T / into the quasilinear operator A given by

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M. Hieber and J.W. Prüss

 A P L.d / ; A.d / D 0 D 

where A denotes the Stokes operator, P the Helmholtz projection, D the NeumannLaplacian, and L is given by ŒL.d /h i WD @i dl hl C @k dl @k @i hl ; for which we employ the sum convention. Note that B.d /d D div.rd Œrd T /. Starting from this, a complete dynamic theory for problem (56) subject to (62) was developed in [29]. In fact, first by local existence theory for abstract quasilinear parabolic problems, the existence and uniqueness of a strong solution to problem (56), (62) on a maximal time interval is proved. Thus, this problem gives rise to a local semi-flow in the natural state space. Furthermore, the equilibria E of this problem are determined to be E D f.0; d / W d 2 Rn ; jd j2 D 1g; and the energy functional ED

1 2

Z 

Œjuj22 C jrd j22 dx

for problem (56); (62) is shown to be a strict Lyapunov functional. In addition, the equilibria are shown to be normally stable, i.e., for an initial value close to E; the solution of problem (56), (62) exists globally, and the solution converges exponentially to an equilibrium. More generally, a solution, eventually bounded on its maximal interval of existence, exists globally and converges to an equilibrium exponentially fast. Due to the polynomial character of the nonlinearities, we can even show that the solution of problem (56), (61) is real analytic, jointly in time and space. The approach is based on the theory of quasilinear parabolic problems described in the previous subsection (see [45, 46, 74, 76, 77]) and relies in particular on the maximal Lp -regularity property for the heat and the Stokes equation (see [33]). In order to formulate the main result for the case of bounded domains, one introduces a functional analytic setting as follows. We denote the principal variable by v D .u; d /; v belongs to the base space X0 defined by X0 D Lq; ./  Lq .I Rn /; where 1 < p; q < 1, and  indicate solenoidal vector fields. The regularity space will be X1 D fv 2 Hq2 ./2n \ X0 W u D @ d D 0 on @g:

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1115

We consider solutions of type 1 .J I X0 / \ Lp; .J I X1 /; v 2 Hp;

where J D .0; a/ with 0 < a  1 is an interval and  2 .1=p; 1 indicates a time weight. More precisely, k .Xl / v 2 Hp;

,

t 1 v 2 Hpk .Xl /;

k; l D 0; 1:

If 1   > 1=2 C 1=p C n=2q, the time trace space of this class is given by 2.1=p/ X; D fv 2 Bqp ./2n \ X0 W u D @ d D 0 on @gI

for brevity, we set X WD X;1 . The state manifold of the problem is defined by SM D fv 2 X W jd .x/j2 D 1 in g: The main result due to Hieber, Nesensohn, Prüss, and Schade [29] then reads as follows. Theorem 7 ([29]). Let J D .0; a/, 1 < p; q < 1, 1   > 1=2 C 1=p C n=2q, and assume s ; ;  > 0 are constant. Then the following assertions are valid. (i) Local Well-Posedness Let v0 2 X; . Then for some a D a.v0 / > 0, there is a unique solution 1 .J; X0 / \ Lp; .J I X1 /; v 2 Hp;

to system (56) subject to (62) on J . Moreover, v 2 C .Œ0; a I X; / \ C ..0; a I X /; i.e., the solution regularizes instantly in time. It depends continuously on v0 and exists on a maximal time interval J .v0 / D Œ0; t C .v0 //. Moreover, 1 t @t v 2 Hp; .J I X0 / \ Lp; .J I X1 /;

a < t C .v0 /;

and jd .; /j2  1, and Ea is a strict Lyapunov functional. Furthermore, Problem (56), (62) generates a local semi-flow in its natural state manifold SM. (ii) Stability of Equilibria Any equilibrium v 2 E of Problem (56), (62) is stable in X . Moreover, for each v 2 E there is " > 0 such that if v0 2 SM with jv0  v jX ;  ", then

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the solution v of the system (56), (61) with initial value v0 exists globally in time and converges at an exponential rate in X to some v1 2 E. (iii) Long-Time Behavior (a) Suppose that sup

jv.t /jX ; < 1:

t2.0;t C .v0 //

Then t C .v0 / D 1, and v is a global solution. (b) If v is a global solution, bounded in X; , then v converges exponentially in SM to an equilibrium v1 2 E of system (56), (62), as t ! 1. Remark 8. Note that part (i) of Theorem 7 also holds true in the case of Dirichlet boundary conditions as long as db is smooth enough. But even for a constant director, it is not clear what the set of equilibria will be in this case. Clearly, zero velocity and constant director are equilibria, and they are even asymptotically stable in SM. However, there is no longer a manifold of constant equilibria; the only constant one is d D db . There may be, however, many other equilibria, and if db is not constant, the situation is even more complicated. Remark 9. Li and Wang considered in [52] the system (56) subject to boundary conditions (62) and claimed the existence and uniqueness for both local strong and global strong solutions with small data. Note that their proof, however, contains serious defects, in particular about regularity properties of the heat equation. The same remark applies to their article [53] on the density dependent case.

8.5

Blowup in Finite Time in the Case of Dirichlet Boundary Conditions

It is an interesting question to ask whether smooth local solutions of the system (56) with either Dirichlet (61) or Neumann boundary conditions (62) develop singularities in finite time. In the case n D 2 and d is the heat flow of harmonic maps, solutions with finite time singularities are constructed in [8], which yields blow in the d -equation. Recently, Huang, Lin, Liu, and Wang [39] were able to construct examples, in the case of  being the unit ball of initial data .u0 ; d0 / having sufficiently small energy and d0 fulfilling a topological condition in the case of Dirichlet boundary conditions d D e D .0; 0; 1/ 2 S2 , for which one has finite time blowup of .u; d /. In order to describe their result, we state first a local existence result. To this end, let e D .0; 0; 1/ 2 S2 and let  D fx 2 R3 W jxj < 1g be the unit ball. 1 Lemma 2 ([60]). Let u0 2 Cc; .; R3 / and d0 2 fd 2 C 1 .; S2 / W d D e on @g. Then there exist T0 > 0 and a unique, smooth solution .u; d / 2 C 1 .Œ0; T0 /  I R3  S2 / to (56) subject to (61) with db D e.

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1117

Theorem 8 ([39]). 1 .; R3 / and d0 2 fd 2 C 1 .; S2 / W (a) There exists "0 > 0 such that if u0 2 Cc; d D e on @g are such that d0 is not homotopic to the constant map e W  ! S2 relative to @, and

Z 

.ju0 j22 C jrd0 j22 /  "2 ;

then the short time smooth solution .u; ; d / of (56) subject to (61) with db D e given in Lemma 2 blows up before T D 1. (b) There are examples of initial data .u0 ; d0 / satisfying the above assumptions. Here, for continuous maps f; g 2 C .I S2 / with f D g on @, we say that f is homotopic to g relative to @, if there exists a continuous map ˆ 2 C .Œ0; 1 I S2 / such that ˆ.; t / D f D g on @ for all t 2 Œ0; 1 and ˆ.; 0/ D f and ˆ.; 1/ D g in .

9

Analysis of the General Ericksen-Leslie Model: Incompressible Fluids

In this section, one considers the general non-isothermal Ericksen-Leslie model for incompressible fluids. First, one considers the non-isothermal Ericksen-Leslie system with general Leslie tensor SL and isotropic free-energy . ; /. The results described below in Sect. 9.1 have been obtained recently by the authors in [31]. The case of anisotropic elasticity, i.e., the case of the general Oseen-Frank functional F, is also discussed in the sequel, however, in the isothermal case and with vanishing Leslie stress tensor SL . Finally, again for the case of anisotropic elasticity, one considers the stationary case.

9.1

The Non-isothermal, Incompressible, and Isotropic Model

In the following, the incompressible, isotropic case is considered, including nonisothermal behavior as well as stretching. The model reads as follows. 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ
0;

˛ > 0;

0 ; L  0;

;  > 0;

;  C 2 @  > 0:

(72)

It is convenient below to write the equation for the internal energy as an equation for the temperature . It reads Dt Cdiv q D .S.1 @ =/SE / W ruCdiv.r/d Dt d C. @ /rd W rDt d: To formulate the main well-posedness result for this problem, one introduces a functional analytic setting as follows. Denote the principal variable by v D .u;  ; d /. Then v belongs to the base space X0 defined by X0 WD Lq; ./  Lq .I R/  Hq1 .I Rn /; where 1 < p; q < 1, and  indicate solenoidal vector fields, as before. The regularity space will be X1 WD fu 2 Hq2 .I Rn / \ Lq; ./ W u D 0 on @g  Y1 ; with Y1 WD f. ; d / 2 Hq2 ./  Hq3 .I Rn / W @  D @ d D 0 on @g:

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1119

One considers solutions within the class 1 v 2 Hp; .J I X0 / \ Lp; .J I X1 /;

where J D .0; a/ with 0 < a  1 is an interval and  2 .1=p; 1 indicates a time weight, as before. The time trace space of this class is given by 2.1=p/ ./n \ Lq; ./ W u D 0 on @g  Y; ; X; D fu 2 Bqp

where 2.1=p/ 1C2.1=p/ ./  Bqp .I Rn / W @  D @ d D 0 on @g; Y; D f. ; d / 2 Bqp

whenever the boundary traces exist. It satisfies 2.1=p/ 1C2.1=p/ ./nC1  Bqp ./n ,! C ./nC1  C 1 ./n ; X; ,! Bqp

provided 1 n C <   1: p 2q

(73)

For brevity we set X WD X;1 , as before. The state manifold of the problem is defined by SM D fv 2 X W .x/ > 0; jd .x/j2 D 1 in g: The fundamental well-posedness result regarding the general Ericksen-Leslie system reads then as follows. Theorem 9 ([31]). Let J D .0; a/, 1 < p; q < 1, 1   > 1=2 C 1=p C n=2q, and assume that 2 C 4 ..0; 1/  Œ0; 1// as well as ˛; j ;  2 C 2 ..0; 1/  Œ0; 1//, j D S; V; D; P; L; 0, and the positivity conditions (72). Then the following assertions are valid. (i) (Local Well-Posedness) Let v0 2 X; . Then for some a D a.v0 / > 0, there is a unique solution 1 .J; X0 / \ Lp; .J I X1 /; v 2 Hp;

of (69), (70), (71) on J . Moreover, v 2 C .Œ0; a I X; / \ C ..0; a I X /;

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i.e., the solution regularizes instantly in time. It depends continuously on v0 and exists on a maximal time interval J .v0 / D Œ0; t C .v0 //. Moreover, 1 t @t v 2 Hp; .J I X0 / \ Lp; .J I X1 /;

a < t C .v0 /;

and jd .; /j2  1, E.t /  E0 , and N is a strict Lyapunov functional. Furthermore, the problem (69), (70), (71) generates a local semi-flow in its natural state manifold SM. (ii) (Stability of Equilibria) Any equilibrium v 2 E of (69), (70), (71) is stable in X . Moreover, for each v 2 E there is " > 0 such that if v0 2 SM with jv0  v jX ;  ", and then the solution v of (69), (70), (71) with initial value v0 exists globally in time and converges at an exponential rate in X to some v1 2 E. (iii) (Long-Time Behavior) (a) Suppose that sup

Œjv.t /jX ; C j1=.t/jL1 < 1:

t2.0;t C .v0 //

Then t C .v0 / D 1, and v is a global solution. (b) If v is a global solution, bounded in X; and with 1= bounded, then v converges exponentially in SM to an equilibrium v1 2 E of (69), (70), (71), as t ! 1. It is very remarkable that the above theorem holds true without any structural assumptions on the Leslie coefficients, except for Condition 72. In particular, the above well-posedness results hold true without assuming Parodi’s relation (22), and no conditions for V ; D; P are needed. Remark 10. (a) A related class of models also dealing with the non-isothermal situation was presented by Feireisl, Rocca, and Schimperna [20] as well as by Feireisl, Frémond, Rocca, and Schimperna in [21]. Their model includes stretching as well as rotational terms and is consistent with the fundamental laws of thermodynamics. The equation for the director d , however, is given in the penalized form. They show that the presence of the term jrd j22 in the internal energy as well as the stretching term d ru gives rise, in order to respect the laws of thermodynamics, to two new non-dissipative contributions in the stress tensor S and in the flux q. It is interesting to note that these two new contributions coincide with the extra terms derived by Sun and Liu [82] by different methods. (b) In [21], it is moreover shown that this system admits a global weak solution for a natural class of initial data. For global weak solutions for the penalized general Ericksen-Leslie system, see also the work of Cavaterra, Rocca, and Wu in [6]. (c) Wu, Xu, and Liu reconsidered in [84] the isothermal penalized Ericksen-Leslie model. They proved, under certain assumptions on the data and the Leslie

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1121

coefficients, the existence of a unique, strong solution provided the viscosity is large enough. Moreover, they showed Lyapunov stability of local energy minimizers and convergence to equilibria as time goes to infinity. (d) Wang, P. Zhang, and Z. Zhang [86] proved local well-posedness of the isothermal general Ericksen-Leslie system as well as global well-posedness for small initial data under various conditions on the Leslie coefficients, which ensure that the energy of the system is dissipated.

9.1.1 Key Ideas of the Proof of Theorem 9 Some comments about the key ideas of the proof of Theorem 9 are in order. Recall that the parameter functions are having the regularity properties j ; ˛;  2 C 2 ..0; 1/  Œ0; 1//;

2 C 4 ..0; 1/  Œ0; 1//:

(74)

and that furthermore the positivity conditions (72) are assumed to be fulfilled. Step 1: Linearization: One linearizes equation (69) at an initial value v0 D Œu0 ; 0 ; d0 T and drop all terms of lower order. This yields the principal linearization 8 < L .@t ; r/v D f in J  ; (75) u D @  D @ d D 0 on J  @; : u D  D d D 0 on f0g  : Here J D .0; a/, v D Œu; ;  ; d T is the unknown, and f D Œfu ; f ; f ; fd T are the given data. The differential operator L .@t ; r/ is defined via its symbol L .z; i /, which is given by 3 0 i zR1 ./T Mu .z; / i  7 6 i T 0 0 0 7; L .z; i / D 6 4 0 0 m .z; / i z0 ba./ 5 iR0 ./ 0 i ba./ Md .z; / 2

(76)

with b D @ , and 1 D @ . One also introduces the parabolic part of this symbol by dropping pressure gradient and divergence, i.e., 2

3 Mu .z; / 0 i zR1 ./T L.z; i / D 4 0 m .z; / i z0 ba./ 5 : iR0 ./ i ba./ Md .z; / The entries of these matrices are given by m D z C ˛jj2 ; 2

a./ D   rd0 ;

Md D  z C jj C 1 a./ ˝ a./ D md .z; / C 1 a./ ˝ a./;

(77)

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M. Hieber and J.W. Prüss

D C  V D  V P0  ˝ d0 C .jd0 /P0 ; 2 2 D C V D  V C P /P0  ˝ d0 C . C p /.jd0 /P0 ; R1 D . 2 2

R0 D

Mu D z C s jj2 C 0 .jd0 /2 d0 ˝ d0 C a1 .jd0 /P0  ˝ d0 ; C a2 .jd0 /2 P0 C a3 jP0 j2 d0 ˝ d0 C a4 .jd0 /d0 ˝ P0 : Here P0 D Pd0 D I  d0 ˝ d0 , and aj are certain coefficients. Step 2: Maximal Lp -regularity: Let 1 < p; q < 1, and assume that (74) holds. Then (75) admits a unique solution v D Œu; ;  ; d T satisfying .u;  / 2 0 H 1p .J I Lq .//nC1 \ Lp .J I Hq2 .//nC1 ; P q1 .//;  2 Lp .J I H d 2 0 H 1p .J I Hq1 .//n \ Lp .J I Hq3 .//n ; if and only if .fu ; f / 2 Lp .J I Lq .//nC1 ;

fd 2 Lp .J I Hq1 .//n ;

1 f 2 0 H 1p .J I 0 H 1 q .// \ Lp .J I Hq .//:

In order to prove this, one sets J WD diag.I; 1=0 ; zI / and shows first that the symbol JN L is accretive for Re z > 0, i.e., the associated system is strongly elliptic. Observe that for proving this, one does not need any conditions for the coefficients D; V ; P; @ . Next, one performs a Schur reduction to reduce the above symbol to a symbol depending only for u. This implies that the resulting generalized Stokes symbol for .u; / is strongly elliptic. Then one may apply a result due to Bothe and Prüss [4] to prove maximal Lp regularity in the case of Rn . In a second step, one verifies the LopatinskiiShapiroo condition to obtain the corresponding result on the half space. A localization procedure finishes the proof of the above assertion. Step 3: Local existence: One rewrites Problem (69) as an abstract quasi-linear evolution equation of the form vP C A.v/v D F .v/;

t > 0; v.0/ D v0 ;

(78)

replacing Dt d appearing in the equations for u and  by the equation for d . Here v D .u;  ; d / and the Helmholtz projection P is applied to the equation for u. The base space will be X0 WD Lq; ./  Lq ./  Hq1 .I Rn /:

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

1123

Then, by quasilinear theory, one obtains that for some a D a.z0 / > 0, there is a unique solution 1 .J; X0 / \ Lp; .J I X1 /; v 2 Hp;

J D Œ0; a ;

of (78), i.e., (69) on J . Moreover, tŒ

d 1 .J I X0 / \ Lp; .J I X1 /; v 2 Hp; dt

and jd .t; x/j2  1, E.t /  E0 , and N is a strict Lyapunov functional. Furthermore, the problem (78) generates a local semi-flow in its natural state manifold SM. Step 4: Dynamics: The linearization of (69) at an equilibrium v D .0;  ; d / is given by the operator A D A.v / in X0 . This operator has maximal Lp regularity, it is the negative generator of a compact analytic C0 -semigroup, and it has compact resolvent. So its spectrum consists only of countably many eigenvalues of finite multiplicity, which have all positive real parts, hence are stable, except for 0. The eigenvalue 0 is semi-simple. Its eigenspace is given by N.A / D f.0; #; d/ W # 2 R; d 2 Rn gI N when ignoring the hence, it coincides with the set of constant equilibria E, constraint jd j2 D 1 and conservation of energy. Therefore each such equilibrium is normally stable, and one is in the position to apply the generalized principle of linearized stability, to prove the assertion (ii). Assertion (iii) then follows by techniques from quasilinear parabolic evolution equations.

9.2

The Isothermal Anisotropic Model with Vanishing Leslie Stress and Stretching in R3

The isothermal and anisotropic model on  D R3 has been considered by various authors in the case of vanishing Leslie stress and stretching, i.e., for V D D D 0 as well as SLd i ss D 0. In this case, the equations read as 8 ˆ .@t C u  r/u C r D div S ˆ ˆ ˆ ˆ div u D 0 <  F .@t C u  r/d D Pd div. @@rd /  rd ˆ ˆ ˆ jd j2 D 1 ˆ ˆ : u.0/ D u0 ; d .0/ D d0

in R3 ; in R3 ;  F ; in R3 ; in R3 ; in R3 ;

(79)

where S D SN C SE . Given T > 0, we denote here a couple .u; d / to be a strong solution to system (79) on R3  .0; T /, if

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M. Hieber and J.W. Prüss

u; rd 2 C .Œ0; T I H21 .R3 // \ L2 .0; T I H22 .R3 //;

@t u 2 L2 .0; T I L2 .R3 //;

@t d 2 L2 .0; T I H21 .R3 //; and if it satisfies equation (79) a.e. on R3 .0; T /. It is called an almost global strong solution to (79) if it is a strong solution for (79) for any T 2 .0; 1/. 1 Theorem 10 ([66]). Assume that u0 2 H2; .R3 / and that d0  db 2 H22 .R3 /, jd0 j2 D 1, where db is a constant unit vector. Then system (79) subject to boundary conditions (61) admits a unique, almost global strong solution provided

.ju0 jL2 C jrd0 jL2 /.jru0 jL2 C jr 2 d0 jL2 /  "0 ;

(80)

where "0 denotes a small positive constant depending only on k1 ; k2 ; k3 . The proof of Theorem 10 is based on a local existence result as well as on a blowup criterion for strong solutions due to Hong, Li, and Xin [36]. It says that under the assumptions, system (79) admits a unique, local strong solution on R3  .0; T / for some T > 0 depending only on the initial data and k1 ; k2 ; k3 . Energy estimates then yield the existence of a constant "0 such that if (80) holds, then sup .juj2H 1 C jrd j2H 1 / C

0st

2

2

Z

t 0

.juj2H 1 C jrd j2H 1 /ds  C .juj2H 1 C jrd j2H 1 / 2

2

2

2

for any t 2 .0; T / and where C is independent of T . For the formulation of various blowup criteria of Serrin or Beal-Kato-Majda type, we refer to [36]. Note that .ju0 j2 C jrd0 j2 /.jru0 j2 C jr 2 d0 j2 / is scaling invariant under the transformations u .x; t / D u.x; 2 t / and d .x; t / D d .x; 2 t /. Hence, the above result can be viewed as global existence result for strong solutions in a critical space.

9.3

The Stationary Case: Anisotropic Elasticity

The Oseen-Frank theory seeks vector fields d W  ! S2 that minimizes the energy functional Z F EF .d; rd / D .d; rd /dx; 

where F is defined as in (9). It can be shown that under Dirichlet boundary conditions for d , there always exists a minimizer d 2 H21 .; S2 / of E F . In addition, such as d solves the Euler-Lagrange equation

23 Modeling and Analysis of the Ericksen-Leslie Equations for Nematic. . .

d  Œdiv .

@ F @ F .d; rd // C .d; rd / D 0; @.rd / @d

jd .x/j2 D 1 in ;

1125

(81)

1=2

with boundary condition d D db 2 H2 .@/, or equivalently   @ F @ F Pd div . .d; rd //  .d; rd / D 0; @.rd / @d

jd .x/j2 D 1 in :

A fundamental result in this context due to Hardt, Lin, and Kinderlehrer [28] asserts the following. Theorem 11 ([28]). If d 2 H21 .; S2 / is a minimizer of EF , then d is analytic on nfsi ng.d /g for some closed set si ng.d /   whose Hausdorff dimension is smaller than one. It already has been noted that when k1 D k2 D k3 D 1 and k4 D 0, then D 12 jrd j2 D is the Dirichlet energy. Hence, in this case, the problem reduces to minimizing harmonic maps into S2 , and the equation (81) simplifies to F

d C jrd j22 d D 0;

jd j2 D 1 in ;

d D db on @:

For a comprehensive study of harmonic maps, we refer to the book by Lin and Wang [62].

10

Analysis of the General Ericksen-Leslie System: Compressible Fluids

Recall from Sect. 4.6 that the compressible non-isothermal and isotropic model reads as 8 ˆ in ; @t  C div.u/ D 0 ˆ ˆ ˆ ˆ Dt u C r D div S in ; ˆ ˆ < Dt  C div q D S W ru  div u C div.rd Dt d / in ; ˆ  Dt d  V Vd  divŒr d D jrd j22 d C D Pd Dd in ; ˆ ˆ ˆ ˆ d D 0 on @; u D 0; q  D 0; @ ˆ ˆ : .0/ D 0 ; u.0/ D u0 ; .0/ D 0 ; d .0/ D d0 in : (82) The unknown variables ; u;  ; d denote density, velocity, (absolute) temperature, and director, respectively, Dt D @t C u  r means the Lagrangian derivative, and Pd D I  d ˝ d . Here, isotropic means that the free-energy is a function only of ,  and D jrd j22 =2. These equations have to be supplemented by the thermodynamical laws for the internal energy , entropy , pressure , heat capacity

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M. Hieber and J.W. Prüss

, and Ericksen tension , according to D

C  ;

D @ ;

 D @ ;

 D 2 @ ;

 D @ ;

(83)

and by the constitutive laws 8 S ˆ ˆ ˆ ˆ S ˆ N < SLst retch ˆ ˆ ˆ S d i ss ˆ ˆ : L

D SN C SE C SL ; D D .ru C Œru T /=2; q D ˛0 r  ˛1 d .d  r /; D 2s D C b I div u; SE D rd Œrd T ; SL D SLst retch C SLd i ss CV V D D2 n ˝ d C D2 d ˝ n; n D V Vd C D Pd Dd   Dt d; D

 C2

P 

.n ˝ d C d ˝ n/ C L2 P .Pd Dd ˝ d Cd ˝ Pd Dd / C 0 .Dd jd /d ˝ d: (84)

All coefficients j; ˛j, and  are functions of ;  ; , in accordance with the principle of equi-presence. For thermodynamic consistency and well-posedness, one requires s ; 2s C nb > 0;

˛0 ; ˛0 C ˛1 > 0;

0 ; L  0;

;  > 0;

;  C 2 @ ; @  > 0:

(85)

It is convenient below to write the equation for the internal energy as an equation for the temperature  . It reads Dt  C div q D .S  SE / W ru C . @ =/SE W ru   @ div u C div.r/d  Dt d C . @ /rd W rDt d: Due to its complexity, not many well-posedness results for this system are known, so far. Consider first the simplified case.

10.1

The Isothermal Simplified Compressible Model

Consider the compressible simplified model which reads in the isothermal situation as 8 ˆ ˆ
0 and 3b C 2s > 0. Then the following result holds true. Theorem 12 ([42, 43]). Let   R3 be a bounded domain with smooth boundary. Assume that 0 2 C 0;1 .RC /, 0  %0 2 Hq1 ./, for some q 2 .3; 6/ and that u0 2 H22 ./ and that d0 2 H22 .; S2 / satisfies the compatibility condition Lau0  1=2 r..%0 //  d0  rd0 D %0 g for some 0 g 2 L2 ./. Then system (86) subject to boundary conditions (87) with db D d0 admits a unique, local, strong solution .%; u; d /. The local strong solution established above has been shown to be a global one, provided the initial data are small enough. For details, see [44, 51] and [78]. For a corresponding result in critical spaces defined on R3 , see [38]. Remark 11. For the case of  D R3 , Hu and Wu [38] proved the existence and uniqueness of a global, strong solution in critical Besov spaces provided the initial data are close to the equilibrium state .1; 0; d / for some constant d 2 S2 . There is also a global result for small data in the case of the so-called incompressible inhomogeneous case. This means that we keep the incompressibility condition div u D 0 which implies that the pressure  is a free variable, but % is allowed to vary with the flow, i.e., Dt % D 0. It reads as 8 ˆ Dt % D 0 ˆ ˆ ˆ ˆ @ u C .u  r/u   u C r D div.Œrd T rd / t s ˆ ˆ < .@t d C .u  r/d / D d C jrd j22 d ˆ div u D 0 ˆ ˆ ˆ ˆ jd j22 D 1 ˆ ˆ : .%; u; d /jtD0 D .%0 ; u0 ; d0 /

in ; in ; in ; in ; in ; in :

(88)

where   R3 denotes a bounded domain with smooth boundary. Here, % W   .0; 1/ denotes the density of the fluid. We assume that uD0

and d D db

on @;

(89)

where db is a given constant unit vector. The global existence of strong solutions for small data was established by Li in [50].

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Theorem 13 ([50]). Let   R3 be a bounded domain with smooth boundary @. 1 ./ Assume that %0 2 H21 ./ \ L1 ./, 0  %0  % in , u0 2 H22 ./ \ H2;0 with div u0 D 0, d0 2 H23 ./ with jd0 j2 D 1 in , d D db on @, and the 1=2 compatibility condition u0  rp0  div .rd0 ˝ d0 / D %0 g0 in  for some 1 2 .p0 ; g0 / 2 H ./  L ./. Then there exists "0 > 0 such that if 1=2

.j%0 u0 j2L2 C jrd0 j2L2 /.jru0 j2L2 C jd0 j2L2 /  "0 ; then equation (88) subject to (89) admits a unique, global strong solution .%; u; p; d /. For recent well-posedness results in critical Besov spaces concerning the incompressible inhomogeneous situation, see [11].

10.2

The General Isotropic Compressible Non-isothermal Model

It seems that the following Theorem 14 is the first well-posedness result for the general isotropic Ericksen-Leslie system subject to compressible fluids even in the isothermal situation. In order to formulate the result for the model (82) (83), (84), one introduces a functional analytic setting as follows. We denote the principal variable by v D .%; u;  ; d /. Then v belongs to the state space X defined by 22=p .I RnC1 /  Hq2 .I Rn /; X WD Hq1 ./  Bqp

where 1 < p; q < 1. The state manifold of the problem is defined by 12=p SM D fv 2 X W %;  > 0; jd j2 D 1 in ; div.r/d 2 Bqp .I Rn /

u D ˛0 @  C ˛1 .d j /@d  D @ d D 0 on @g;

(90) (91)

and the manifold of equilibria is given by E D fv D .% ; 0;  ; d / 2 R2nC2 W  ;  > 0; jd j2 D 1g: Then we have the following main result. Theorem 14 ([32]). Let 1 < p; q < 1, and assume 2 C 3 ..0; 1/  Œ0; 1//, 2 ˛; j ;  2 C ..0; 1/  Œ0; 1//, j D S; V; D; P; L; 0, as well as the positivity conditions (85). Then the following assertions are valid. (i) (Local Well-Posedness) The system (82), (83), (84) generates a local semi-flow in its natural state manifold SM, each solution exists on a maximal time interval Œ0; tC .v0 //. In

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addition, the total mass M and the total energy E are constant, and the negative total entropy N is a strict Lyapunov functional. (ii) (Stability of Equilibria) Any equilibrium v 2 E of (82), (83), (84) is stable in SM. Moreover, for each v 2 E, there is " > 0 such that if v0 2 SM with jv0  v jX  ", then the solution v of (82), (83), (84) with initial value v0 exists globally in time and converges at an exponential rate in SM to some v1 2 E. (iii) (Long-Time Behavior) (a) Suppose sup t2.0;tC .v0 //

Œjv.t /jX C jdiv.r/d jB 12=p C j1=%.t /jL1 C j1=.t/jL1 < 1: qp

Then tC .v0 / D 1, and v is a global solution. (b) If v is a global solution, bounded in X and with ;  uniformly bounded from below, then v converges exponentially in SM to an equilibrium v1 2 E of (82), (83), (84), as t ! 1. (c) Conversely, if v.t / is a global solution in SM which converges to an equilibrium v1 2 E, then (a) is valid on RC . The above result is proved in principle in a similar way as Theorem 9. However, the proof is now much more involved, due to the hyperbolic part of the system coming from the continuity equation. Local well-posedness and also the stability part are proven by introducing Lagrangian coordinates, which changes the boundary conditions into nonlinear ones, thereby leading to another complication. In addition, in contrast to the incompressible case, the compressible case one cannot use d 2 Hq3 ./, as the density % does not have enough regularity. Instead, one raises the time regularity of d by 1=2. Then the solution space in Lagrangian coordinates is given by 1 1 % 2 Hp; .J I Hq1 .//; .u;  / 2 Hp; .J I Lq .RnC1 / \ Lp; .J I Hq2 .RnC1 //;

while the director lies in 2 n 1=2 2 n 1 1 n d 2 Hp; .J I 0 H 1 q .I R // \ Hp; .J I Hq .I R // ,! Hp; .J I Hq .I R //;

in contrast to Sect. 9.1. Observe that the compatibility condition 12=p div.r/d 2 Bqp .I Rn /

is induced by the time trace of @t d resulting from the raised time regularity of d . In the incompressible case % D const, it means 32=p d 2 Bqp .I Rn /;

which is the same time trace space for d as in Sect. 9.1.

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Conclusion

Various aspects of modeling and analysis of the Ericksen-Leslie equations describing nematic liquid crystal flow are being discussed, both in the isothermal as well as in the non-isothermal situation. Of special interest is the development of thermodynamically consistent Ericksen-Leslie models in the general situation based on the entropy principle. The analytical understanding of the dynamics of the underlying system is then based on this principle and allows to determine the set of all equilibria. The well-posedness results given in Sects. 9 and 10 for the general Ericksen-Leslie system subject to homogeneous Neumann boundary condition for d give a rather complete understanding of the dynamics of the underlying system, both in the incompressible and also in the compressible situation. In particular, there exists a unique, global strong solution to the Ericksen-Leslie system provided the initial data are close to an equilibrium point in an appropriate norm and also the longtime behavior of the solution is determined. It is remarkable that for these wellposedness results no structural conditions on the Leslie coefficients are imposed and that in particular Parodi’s relation on the Leslie coefficients is not being assumed.

12

Cross-References

 Concepts of Solutions in the Thermodynamics of Compressible Fluids  Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value

Problem  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Large Time Behavior of the Navier-Stokes Flow  Local and Global Existence of Strong Solutions for the Compressible Navier-

Stokes Equations Near Equilibria via the Maximal Regularity  The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties  Variational Modeling and Complex Fluids

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Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible Fluid Mechanics

24

Vsevolod Alexeevich Solonnikov and Irina Vladimirovna Denisova

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linear Problem with and Without Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Explicit Solution of a Model Homogeneous Problem . . . . . . . . . . . . . . . . . . . . . . . 2.2 Linear Problem in a Half-Space Without Surface Tension . . . . . . . . . . . . . . . . . . . 2.3 Passage to the Case of Positive Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Linear Problem in a Bounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Proof of Local Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Auxiliary Propositions: Solvability of the Linearized Problem . . . . . . . . . . . . . . . . 3.2 Sketch of the Proof of Existence Theorem for the Nonlinear Problem . . . . . . . . . . 4 Global Solvability of the Free Boundary Problem Governing the Evolution of an Isolated Liquid Mass with Zero Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 An Interface Problem on the Motion of Two Immiscible Fluids . . . . . . . . . . . . . . . . . . . 5.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Local Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Global Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Equilibrium Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Linear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Stability and Instability of Equilibrium Figures: Sketch of the Proof of Theorem 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Stationary Free Boundary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Periodic Motion of the Fluid Above a Rigid Bottom . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Motion of the Fluid Partially Filling a Container . . . . . . . . . . . . . . . . . . . . . . . . . . .

1136 1144 1144 1148 1153 1155 1159 1159 1163 1166 1172 1173 1175 1179 1190 1190 1195 1200 1205 1205 1209

V.A. Solonnikov () Laboratory of Mathematical Physics, St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia e-mail: [email protected]; [email protected] I.V. Denisova Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg, Russia e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_27

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7.3 The Problem of Moving Contact Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The Problem on Filling a Plane Capillary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1212 1213 1215 1216 1216

Abstract

The chapter deals with unsteady motion of incompressible fluids with free boundaries and interfaces where the surface tension may be taken into account. Such flows are governed by free boundary problems for the Navier-Stokes system. For the problems on the motion of an isolated liquid mass in vacuum or in another fluid, local in time existence theorems are established in the Hölder and Sobolev-Slobodetskiˇı classes of functions. The proofs are based on the estimates of solutions of the corresponding linear problems in fixed domains obtained by the Schauder localization and on constructing explicit solutions of the model problems in the dual Fourier-Laplace space. The global solvability of some problems with small data is established by energy method. The solutions are proved to tend exponentially to those corresponding to equilibrium states. Evolutionary problem on the stability of a rigid rotation of a viscous capillary drop with a given angular momentum is considered. The solution of the problem is proved to be exponentially stable in time if energy functional is positive. Instability theorem is also stated. Some stationary free boundary problems are discussed. Among them there are problems on the periodic motion of a fluid above a rigid bottom and the motion of a fluid partially filling a container, the problem of filling a plane capillary with the contact angle equal to . Existence theorems for these problems are stated.

1

Introduction

The present chapter concerns problems of mathematical fluid mechanics governing the motion of a viscous incompressible fluid with free surfaces. As a rule, these surfaces are given only at the initial instant of time; their determination for t > 0 is part of the problem. Free boundary problems are well known for a long time, but the rigorous mathematical analysis of them has started relatively recently. Such problems govern the flow of the fluid above a rigid bottom [2, 4, 5, 36], in particular the flow with moving contact points and lines [37, 59]; the evolution of an isolated liquid mass [31, 50, 55, 56], including the evolution of rotating fluid with prescribed angular momentum [68, 71]; the motion of fluid partially filling a container [22, 23, 29, 34, 43, 50, 53, 74]; and finally the evolution of two immiscible fluids with a free interface [1, 8, 9, 12, 17–20, 24, 35, 44, 75]. This chapter mainly deals with evolutionary free boundary problems. In Sects. 2, 3, and 4, the problem concerning the evolution of an isolated liquid mass is considered, both with surface tension on the free boundary and without it. Sketches

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of proofs of local and global (in time) classical solvability of such problems are outlined under certain assumptions on the data. In Sect. 5, two-phase fluid motion is studied. The same results as in Sects. 2, 3, and 4 are obtained for the evolution of a drop in another liquid medium. A model problem in two half-spaces is studied in the Hölder spaces. Section 6 is devoted to the analysis of a rotating fluid, including the stability problem (it is worth noting that it has a long history). Finally, in Sect. 7, one can find discussion on stationary and quasi-stationary free boundary problems, including the problems with moving lines of the contact of a free surface with a rigid wall. For the sake of simplicity, only two-dimensional stationary problems of this type are considered. The statement of the free boundary problem governing the motion of an isolated liquid mass is as follows. At the initial instant, the fluid occupies a given bounded domain ˝0  R3 with the boundary  . For every t > 0, it is necessary to find t D @˝t , velocity vector field v.x; t / D .v1 ; v2 ; v3 /, and the function p, the deviation from the hydrostatic pressure, that satisfy the initial–boundary value problem for the Navier-Stokes system Dt v C .v  r/v  r 2 v C

1 rp D f; 

v.x; 0/ D v0 .x/;

r  v D 0;

x 2 ˝t ; t > 0;

(1)

x 2 ˝0 ;

T.v; p/n D  H n  pe n on t : (2)   Here, Dt  @t@ ; r  @x@1 ; @x@2 ; @x@3 ; r 2 D r r, f is the vector field of mass forces; v0 is the initial velocity; ıik is the Kronecker symbol; T.v; p/ D pI C S.v/ is the stress tensor; I is the identity matrix; S.v/ D rv C .rv/T is the doubled rate-ofstrain tensor with the components Sik .v/  @vi =@xk C @vk =@xi ; the superscript T denotes the transposition; ;  are the constants of kinematic viscosity and density of the fluid, respectively;  D ;  > 0 is the surface tension coefficient; n is the outward normal to ˝t ; pe is the exterior pressure function defined for x 2 R3 , t > 0; and H .x; t / is twice the mean curvature of t .H < 0 at the points where t is convex). A Cartesian coordinate system fxg is introduced in R3 . The centered dot denotes the Cartesian scalar product. The summation is implied from 1 to 3 with respect to repeated indexes expressed by Latin letters and from 1 to 2 with respect to repeated Greek indexes. The vectors and the vector spaces are marked by boldface letters. In order to exclude the loss of mass through the free surface, it is assumed that the boundary t is formed by points x.y; t / the radius vector x.y; t / of which is a solution of the Cauchy problem Dt x D v.x.y; t /; t /; xjtD0 D y;

y 2  ; t > 0;

(3)

so that t D fx.y; t /jy 2  g and ˝t D fx.y; t /jy 2 ˝0 g,  is given. It follows that Vn .x/ D v  n.x/;

x 2 t ;

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V.A. Solonnikov and I.V. Denisova

where Vn is the velocity of the evolution of t in the normal direction. Condition (3) completes system (1), (2) and enables one to get rid of the unknown boundary by transforming the Eulerian coordinates x 2 ˝t into the Lagrangian coordinates y 2 ˝0 according to the formula Z t u.y;  / d  Xu .y; t /; (4) x.y; t / D y C 0





where u.y; t / D v x.y; t /; t is the velocity vector field in the Lagrangian coordinates. In the case  > 0, it is convenient to introduce the new pressure function p1 D   3j˝0 j 1=3 2 , where R0 D and j˝j D mes˝. Then boundary condition (2) p R0 4 takes the form   2 n  pe n on t : T.v; p1 /njt D  H C (5) R0 It is clear that p1 D p for  D 0. Mapping (4) converts systems (1), (5), (3) into Dt u  ru2 u C

1 ru q D f.Xu ; t /; 

ru  u D 0 in QT D ˝0  .0; T /;

ˇ uˇtD0 D v0 in ˝0 ; ˘0 ˘ Su .u/n D 0

(6)

on GT    .0; T /;  2 n0  n  pe n0  n n0  Tu .u; q/n D  H .Xu / C R0 

on GT ;

(7)

where q D p1 .Xu ; t /, ru D Ar, A is the matrix of cofactors Aij to the elements R t @ui 0 j aij .y; t / D ıi C 0 @y dt of the Jacobian matrix of transformation (4), i; j D j 1; 2; 3; and ˘ k D k  n.n  k/ and ˘0 k D k  n0 .n0  k/ are projections of a vector k onto the tangent planes to t and  , respectively; finally, Tu .w; q/ D  qI C Su .w/; where Su .w/ is the tensor with the elements Su .w/ij D Aik @wj =@yk CAj k @wi =@yk : System (7) is equivalent to (5), if the angle between n and n0 is acute. The normal n.Xu / is connected with n0 by the relation nD

An0 jAn0 j

(8)

and H n D .t /x  .t /Xu ;

(9)

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1139

where .t / is the Laplace-Beltrami operator on t . In local coordinates fs1 ; s2 g on the surface  , it has the form 1 @ ˛ˇ p @ @2 @ g g  g ˛ˇ C hˇ ; .t / D p g @s˛ @sˇ @s˛ @sˇ @sˇ ˚ 2 ˚ 2 where g ˛ˇ ˛;ˇ D 1 is the inverse of the metric tensor g˛ˇ ˛;ˇ D 1 , g˛ˇ D

˚ 2 g D det g˛ˇ ˛;ˇD1 ;

@Xu .y.s/; t / @Xu .y.s/; t /  ; @s˛ @sˇ

hˇ D p

@  ˛ˇ p  g g : g@s˛

In the present chapter, the proof of classical solvability of the problem (6), (7) in a finite time interval is outlined. The statement of the existence theorem is prefaced by the definition of the Hölder spaces. Let ˝ be a domain in Rn ; T > 0; QT D ˝  .0; T /, and let ˛ 2 .0; 1/. The space C ˛;˛=2 .QT / means the class of the functions that are defined in QT and have finite norm .˛;˛=2/

jf jQT

.˛;˛=2/

D jf jQT C hf iQT

;

where jf jQT D

.˛;˛=2/

sup jf .x; t /j;

hf iQT

.x;t/2QT

hf

.˛/ ix;QT

D sup

.˛/

.˛=2/

D hf ix;QT C hf it;QT ;

sup jf .x; t/  f .y; t/jjx  yj˛ ;

x;y2˝ t2.0;T /

.ˇ/

hf it;QT D sup sup jf .x; t/  f .x; /jjt   jˇ ; ˇ 2 .0; 1/: x2˝ t;2.0;T /

.˛/

.˛/

.ˇ/

.ˇ/

Moreover, jf jx;QT D jf jQT C hf ix;QT , jf jt;QT D jf jQT C hf it;QT : Let Dxr D @jrj =@x1r1 : : : @xnrn ; r D .r1 ; : : : rn /; ri 2 N [ f0g; jrj D r1 C : : : C rn ; kC˛ Dts D @s =@t s ; s 2 N [ f0g, and k 2 N. The space C kC˛; 2 .QT / is defined by the norm     X kC˛; kC˛ kC˛; kC˛ 2 2 r s D jDx Dt f jQT C hf iQT ; jf jQT jrjC2s6k

where 

kC˛; kC˛ 2

hf iQT



D

X jrjC2sDk

.˛;˛=2/ hDxr Dts f iQT

C

X



1C˛

2 hDxr Dts f it;QT



:

jrjC2sDk1

kC˛ kC˛ 2 .QT / which The symbol CV kC˛; 2 .QT / denotes the subspace of C kC˛;

i kC˛ consists of functions f such that Dt f jtD0 D 0; i D 0; : : : ; 2 .

1140

V.A. Solonnikov and I.V. Denisova

The space C kC˛ .˝/; k 2 N [ f0g, is a set of functions f .x/; x 2 ˝, with the norm X .kC˛/ .kC˛/ jf j˝ D jDxr f j˝ C hf i˝ ; jrj6k

where .kC˛/

hf i˝

D

X

.˛/

hDxr f i˝ D sup

x;y2˝

jrjDk

X

jDxr f .x/  Dyr f .y/jjx  yj˛ :

jrjDk

The following Hölder semi-norms with ˛; 2 .0; 1/ will be useful in the sequel:  . ;1C˛/ j f j QT

D

. ;1C˛/ h f i QT

1C˛

2

C hf it;QT



;

where . ;1C˛/ h f i QT D

sup

sup

t;2.0;T / x;y2˝

jf .x; t/  f .y; t/  f .x; / C f .y; /j : jx  yj jt  j.1C˛ /=2

It is known (see [48]) that 

hf

. ;1C˛/ iQT

1C˛; 1C˛ 2

6 c1 hf iQT



:

(10)

. ;1C˛/

It is said that f 2 C . ;1C˛/ .QT / if jf jQT C jf jQT < 1: Finally, the space C ; .QT / consists of functions with finite norm . ;/

jf jQT

. /

./

 hf ix;QT C jf jt;QT ;

2 .0; 1/;

 2 Œ0; 1/;

where ( jf

./ jt;QT

D

./

jf jQT C hf it;QT

if  > 0;

jf jQT

if  D 0:

The Hölder spaces of functions given on smooth surfaces, in particular, on  and on GT , are introduced in a standard way, with the help of local maps and partition of unity. A vector-valued function is said to be an element of a Hölder space if all of its components belong to this space, and its norm is defined as the maximal norm of the components. The theorem of local solvability of the nonlinear problem was proved by Moghilevskiˇı and Solonnikov [30, 31]. Theorem 1 (Local existence for a single fluid). Let  2 C 3C˛ , f; Dx f 2 1C˛

C˛; 2 .R3T /, v0 2 C2C˛ .˝0 /, pe ; Dx pe 2 C . ;1C˛/ .R3T / \ C ˛;˛=2 .R3T /, for some

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1141

˛; 2 .0; 1/, < ˛, T < 1, R3T  R3  .0; T /. Moreover, let the following compatibility conditions ˇ ˘0 S.v0 /n0 ˇ D 0

r  v0 D 0;

be satisfied. Then there exists ".T / such that problem (6), (7) has a unique solution .u; q/: u 2 C2C˛;1C˛=2 .QT /, q 2 C . ;1C˛/ .QT /, rq 2 C˛;˛=2 .QT /, provided that   1C˛

˛; 2



˛;

C jDx fjR3

jfjR3

T

1C˛

2

T

 .2C˛/

C jv0 j˝0

.˛;0/ CjDx pe jR3 T

C  jH0 C

. ;1C˛/ C j Dx pe jR3 6 T

2 .1C˛/ j C R0 

. ;1C˛/

jpe jR3

T

".T /:

(11)

The surface t is of class C 3C˛ . Moreover, there holds the estimate .2C˛;1C˛=2/ jujQT

C

.2C˛/

C jv0 j˝0

.˛;˛=2/ jrqjQT

C  jH0 C

C

. ;1C˛/ jqjQT

n

  1C˛

˛; 2

6 c jfjR3

T

  1C˛

˛; 2

C jDx fjR3

T

o 2 .1C˛/ . ;1C˛/ . ;1C˛/ .˛;0/ j C jpe jR3 C jDx pe jR3 C jDx pe jR3 : T T T R0 (12)

Remark 1. If T tends to infinity, then " ! 0. Remark 2. If  D 0, it is sufficient to assume that  2 C 2C˛ . The surface t will belong to class C 2C˛ as well. The terms with  are absent in the estimate. Local existence theorem with  D 0 was obtained in the papers [48–50]. Along with (6), (7), a linearized problem is considered. It has the form Dt w  ru2 w C 1 ru s D f; wjtD0 D w0

ru  w D r

in QT ;

in ˝;

(13)

˘0 ˘ Su .w/n D ˘0 d on GT ; Z t Z t ˇ wˇ d D b C  B d n0  Tu .w; s/n  n0  .t / 0

on GT :

0

The unique solvability for problem (13) is proved on any finite time interval. Theorem 2 (Finite time solvability of the linearized problem). Let ˛; 2 .0; 1/,

< ˛, T < 1. It is assumed that  2 C 2C˛ and that u 2 C2C˛1C˛=2 .QT / satisfies the inequality  .2C˛;1C˛=2/  6ı (14) T C T =2 jujQT

1142

V.A. Solonnikov and I.V. Denisova

with some sufficiently small ı > 0. Moreover, let the following three groups of conditions be fulfilled: 1C˛

1C˛

1. f 2 C˛;˛=2 .QT /; r 2 C 1C˛; 2 .QT /; w0 2 C2C˛ .˝0 /; d 2 C1C˛; 2 .GT /, ˛ ˛ b 2 C . ;1C˛/ .GT /, r b 2 C ˛; 2 .GT /, B 2 C ˛; 2 .GT /; ˛;˛=2 2. there exist a vector g 2 C .QT / and a tensor G D fGik g3i;kD1 with Gik 2 C . ;1C˛/ .QT / \ C ;0 .QT / such that Dt r  ru  f D r  g;

gi D @Gik =@yk ;

i D 1; 2; 3;

(these relations are understood in a weak sense); 3. the compatibility conditions hold: r  w0 .y/ D r.y; 0/ D 0; y 2 ˝I

˘0 S.w0 .y//n0 D ˘0 d.y; 0/;

y 2:

Under all these assumptions, problem (13) has a unique solution .w; s/ with the properties w 2 C2C˛;1C˛=2 .QT /, s 2 C . ;1C˛/ .QT /, rs 2 C˛;˛=2 .QT /; moreover, this solution is subjected to the inequality

Nt 0 Œw; s

.˛; ˛ / .2C˛;1C˛=2/ . ;1C˛/ jwjQt 0 CjrsjQt 0 2 CjsjQt 0 6

  n ˛; ˛ 1C˛; 1C˛ . 2/ 2 c1 .t / jfjQt 0 CjrjQt 0 0



1C˛; .˛; ˛ / .2C˛/ . ;1C˛/ . ;0/ C jgjQt 0 2 C jw0 j˝0 C hGiQt 0 C jGjQt 0 C jdjGt 0

1C˛ 2



o .˛; ˛ / .˛; ˛ / . ;1C˛/ .1/ C jbjGt 0 C jr bjGt 0 2 C jbjGt 0 C  jBjGt 0 2 C Pt 0 Œujw0 j˝0 o n .1/  c1 .t 0 / F1 .t 0 / C Pt 0 Œujw0 j˝0 ; (15) where c1 .t 0 / is a monotone nondecreasing function of t 0 6 T , r is the 1˛ .˛;˛=2/ : tangential gradient on  , and Pt Œu D t 2 jrujQt C jrujQt Problem (13) with u D 0 is referred to as a linear problem: Dt v   v C 1 rp D f;

r  v D r in ˝;

t > 0;

vjtD0 D v0 in ˝; b  n D 0; ˘ Snj D b; Z t Z t ˇ 0 ˇ n  Tn  n   v  dt D b C  B dt 0 on  ; 0

0

where f; r; b; b; B; v0 are given functions, n is the outward normal to  .

(16)

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1143

Theorem 3 (Finite time existence for the linear problem). Let the representation formulas r Dr R

  Dt R  f  g D r  M  @Mik =@xk

and

(17)

hold and the compatibility conditions ˘ S.v0 .x//n D b.x; 0/;

r  v0 .x/ D r.x; 0/; x 2 ˝;

x 2;

(18)

be satisfied; moreover, it is assumed that 0 < < ˛ < 1, T < 1,  2 C 2C˛ , 1C˛ f; g 2 C ˛;˛=2 .QT /, r 2 C 1C˛; 2 .QT /, R 2 C 2C˛;0 .QT /, Dt R 2 C˛;˛=2 .QT /, 1Ca ˛ v0 2 C2C˛ .˝/, b 2 C1C˛; 2 .GT /, b  n D 0, B 2 C ˛; 2 .GT /, b 2 C . ;1C˛/ .GT /, ˛; ˛2 r b 2 C .GT / and the elements of the tensor M possess finite semi-norms . ;1C˛/ . / ; hMik ix;QT , i; k D 1; 2; 3: jMik jQT Then, problem (16) has a unique solution v 2 C2C˛;1C˛=2 .QT /; p 2 C . ;1C˛/ .QT /, rp 2 C˛;˛=2 .QT /, and the solution satisfies the inequality .2C˛;1C ˛2 /

jvjQT

  n ˛; ˛ 1C˛; 1C˛ .˛; ˛2 / . 2/ 2 . ;1C˛/ C jrpjQT C jpjQT 6 c13 .T / jfjQT C jrjQT 

1C˛; .˛; ˛ / .˛;0/ . ;1C˛/ . / C jgjQT 2 C jrrRjQT C jMjQT C hMix;QT C jbjGT

1C˛ 2



C jbjGT

o .˛; ˛ / .˛; ˛ / . ;1C˛/ .2C˛/ C jr bjGT 2 C jbjGT C  jBjGT 2 C jv0 j˝  c13 .T /F .T /;

(19)

where c13 .T / is a nondecreasing function of T . Remark 3. If the function b 2 C 1C˛; space C

1C˛; 1C˛ 2

.GT /, and   1C˛; 1C˛ 2

of (19) with jbjGT

1C˛ 2

.GT /,then 1C˛; 1C˛ 2



the norm jpjGT

the pressure p belongs to the

is estimated by the right-hand side

.

Problem (6), (7) was also studied in the anisotropic Sobolev-Slobodetskiˇı spaces l;l=2 W2 (see the definition below). Theorems similar to Theorems 1 and 2 were proved in [58]. An analog of Theorem 3 in L2 -norms was obtained in [57]. The space W2l .˝/, ˝  Rn , n 2 N, is equipped with the norm 0 kukW l .˝/ D @

11=2

X Z

2

0 0g, R21 D Here, D r  r is the Laplacean, R1 2 0 0 2 R  .0; 1/, x D .x1 ; x2 /; D @ =@x12 C @2 =@x22 . The given functions bi ; i D 1; 2; 3; B should vanish at t D 0 and decay sufficiently rapidly as jxj ! 1. By the Fourier transform with respect to the tangential space variables .x1 ; x2 / D x 0 and the Laplace transform with respect to t defined by the formula

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

fQ . ; x3 ; s/ D

Z1

est

0

Z

0

f .x; t/eix  dx 0 dt;

Re s > 0;

1145

D . 1 ; 2 /; (21)

R2

problem (20) is converted into the system of ordinary differential equations with the unknown functions vQ ; p: Q 

 i ˇ d2 vQ ˇ  s C C 2 vQ ˇ C pQ D 0; 2   dx3



 1 dpQ d2 vQ 3  s C 2 vQ 3 C C D 0;   dx3 dx32

ˇ D 1; 2;

dvQ 3 C i 1 vQ 1 C i 2 vQ 2 D 0; dx3

(22) x3 > 0;

completed with the boundary conditions e v!0; x3 !1

e p !0; x3 !1

ˇ ˇ dvQ ˇ C i ˇ vQ 3 ˇˇ D bQ ˇ ; ˇ D 1; 2; dx3 x3 D0 ˇ  ˇ dvQ 3 ˇˇ ˇ Q s  pQ C 2   2 vQ 3 ˇ D se b 3 C  B; ˇ x3 D0 dx3 x3 D0 



(23)

where 2 D 12 C 22 . The construction of an explicit solution of problem (22), (23) is started by calculating the determinant of the matrix of the algebraic system obtained by  replacing dxd3 with the parameter in system (22) [57]. It is equal to s C  2  2    2 2  2 ; hence, it has six roots with respect to . Three of them should be ignored since they generate solutions of (22) increasing as x3 goes q to infinity. The q s C 2 , j j D 12 C 22 , where remaining roots 1;2 D r, 3 D j j, r D  p j arg zj < =2 for 8z 2 C, generate decaying solutions. Since r is a double root, general solution of (22) has the form

0 1 0 1 a 1 C b 1 x3 c1   B a2 C b 2 x3 C rx B c 2 C j jx vQ 3 3 C C DB CB ; @ a 3 C b 3 x3 A e @ c3 A e pQ c4 a 4 C b 4 x3

x3 > 0:

(24)

This expression (24) is substituted into the Stokes system, and the constants in (24) are found by equating to zero the coefficients at different exponential functions. The exponent erx3 generates equations

1146

V.A. Solonnikov and I.V. Denisova



 s i 1 2 2 a4 D 0; C  r a1 C rb 1 C     s i 2 C 2  r 2 a2 C rb 2 C a4 D 0;     s r C 2  r 2 a3 C rb 3  a4 D 0  

which imply bˇ D 

i ˇ a4 ; r

ˇ D 1; 2;

b3 D 

a4 : 

(25)

To the function x3 erx3 correspond the equations s  i 1 C 2  r 2 b1 C b4 D 0   and i 1 b 1 C i 2 b 2  rb 3 D 0:

(26)

It follows that b 4 D 0, and substitution of (25) into (26) gives a4 D 0; hence, b i D 0. To define ai , the single equation i 1 a1 C i 2 a2  ra3 D 0 is left which yields two linearly independent solutions: a1 D 0, a2 D r, a3 D i 2 and a2 D 0, a1 D r, a3 D i 1 . In order to determine c i , the coefficients at ej jx3, it is necessary to solve the linear system i ˇ s cˇ C c 4 D 0; ˇ D 1; 2;   j j s c3  c 4 D 0;   i 1 c 1 C i 2 c 2  j jc 3 D 0: It has the solution c ˇ D i ˇ ;

c 3 D j j;

c 4 D s:

Thus, the general solution of problem (22) in the half-space is 1 0 1 0 1 0 r 0 i 1 vQ D C1 @ 0 A erx3 C C2 @ r A erx3 C C3 @ i 2 A ej jx3 ; i 1 pQ D C3 sej jx3 ;

j j

i 2 x3 > 0;

(27)

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1147

where Ci , i D 1; 2; 3, are arbitrary constants. They are found by substituting (27) in the boundary conditions (23) and solving the system obtained. The solution of (22), (23) can be written in the following form convenient for subsequent calculations: vQ D Werx3 C Ve1 ;

x3 > 0;

j jx3

pQ D .r C j j/ e

;

x3 > 0;

(28)

where 0

1 i 1 erx3  ej jx3 ; V D @ i 2 A  ; e1 D r  j j j j n

o 1 Q 2r C  2 ;  D bQ 30 s.r 2 C 2 / C ds .r C j j/P s

1 !1 W D @ !2 A ; !3

0

! D

bQ   i  f  !3 g ; r

!3 D

Q ds.r  j j/ bQ 30 sj j  ; .r C j j/P P

dQ D i 1 bQ 1 C i 2 bQ 2 ;

 D 1; 2;

(29)

 Q bQ 30 D bQ 3 C B; s

and

P D s 2 C 4s 2

r C  j j3 : r C j j

(30)

Lemma 1. If Re s D a > a0 > 0, 2 R2 , then the function P . ; s/ given by (30) satisfies the estimate   jP . ; s/j > c jsj2 C jsj 2 C j j3 ;

(31)

where constant c depends on a0 . It should be noted that the uniqueness of solution (28), (29) for a > a0 > 0 is guaranteed by inequality (31), i.e., by the fact that P is bounded away from zero. Hence, the transformation inverse to (21) is well defined, and the solution can be calculated as a function of the variables x, t. It is worth noting that estimate (31) implies 3 jsj1=2 C j j :

1=2 

jP . ; s/j > ca0

(32)

1148

V.A. Solonnikov and I.V. Denisova

2.2

Linear Problem in a Half-Space Without Surface Tension

The solution of (22) with the boundary conditions e p !0;

vQ !0; x3 !1





x3 !1

ˇ ˇ dvQ ˇ C i ˇ vQ 3 ˇˇ D aQ ˇ ; dx3 x3 D0

ˇ D 1; 2;  pQ C 2

ˇ dvQ 3 ˇˇ D aQ 3 ; (33) dx3 ˇx3 D0

is given by (28), where  D

o 1 n Q ; aQ 3 .r 2 C 2 / C 2dr P1

dQ D i 1 aQ 1 C i 2 aQ 2 ;

! D

aQ   i  f  !3 g ; r

!3 D

dQ .r  j j/  aQ 3 j j.r C j j/ ; P1

 D 1; 2;

(34) P1 D s.r C j j/ C 4r 2 :

Under the assumption Re s > a0 > 0, there hold c1 .jsj C j j2 /1=2 6 cRer 6 jrj 6 c2 .jsj C j j2 /1=2 ; c3 .jsj C 2 /3=2 6 jP1 . ; s/j 6 c4 .jsj C 2 /3=2 : Theorem 4 (Existence theorem for the linear homogeneous problem in a half1C˛ space). Let aˇ 2 CV 1C˛; 2 .R2T /, ˇ D 1; 2, a3 2 CV 1C˛;0 .R2T / \ C . ;1C˛/ .R2T / for some ˛; 2 .0; 1/, < ˛, and T > 0; R2T D R2  .0; T /. Then the original of problem (22), (33) has a solution .v; p/ with the properties V 2C˛;1C ˛2.RC /, p 2 CV . ;1C˛/ .RC /, rp 2 CV ˛;˛=2 .RC /, RC D R3  .0; T /: v2C C T T T T and hvi

.2C˛;1C ˛2 / RTC

.˛; ˛2 /

C hrpi 6c

RTC

X 2

C hpi 

RTC

1C˛; 1C˛ 2

haˇ iR2

ˇD1

. ;1C˛/

T

 0

.˛; ˛2 /

C hr a3 iR2

T

C

. ;1C˛/ ha3 iR2 T

;

(35)

where r 0 is the tangential gradient .@=@x1 ; @=@x2 /. 1C˛ 1C˛ If a3 2 CV 1C˛; 2 .R2T /, the pressure p 2 CV 1C˛; 2 .R2T / too, and the semi-norm 1C˛; 1C˛ 2

hpiR2

T

can be added to the left-hand side of (35).

The growth of the function p is bounded by cjxj as jxj ! 1.

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1149

Proof. The functions ai (that are assumed to decrease like power functions as jxj ! 1) can be extended by zero in the interval t < 0 and then with preservation of class in the half-axis t > T . The vector-field v can be viewed as a solution of the second boundary value problem for the heat equation C Dt v   v D f in R1 ;

ˇ vˇtD0 D 0 in

R3C ;

v ! 0;



jxj!1

@v ˇˇ D d.x 0 ; t / ˇ @x3 x3 D0

on

R21 ;

with f and d defined by 1 f D  rp;  @v3 ˇˇ dˇ D aˇ C  ; ˇ @xˇ x3 D0

2 X @v ˇˇ d3 D  : ˇ @x x3 D0  D1

ˇ D 1; 2;

(In the last relation, the solenoidality of v has been used.) The pressure p.x; t/ being a solution of the Dirichlet problem for the Laplace equation p D 0

in R3C ;

pjx3 D0 D a3  2d3   0 R x3 1 0 0 is given by p.x; t/ D  2 R2 jxy 0 j3 a3 .y ; t / C 2d3 .y ; t / dy . Well-known estimates for the parabolic Neumann problem and for the harmonic functions imply

hvi hpi

. ;1C˛/ RTC

.2C˛;1C˛=2/ RTC

Chrpi

n

 .˛;˛=2/

6 c hrpi

RTC

.˛;˛=2/ RTC

6c

1C˛; 1C˛ 2

C hdiR2



o

;

(36)

T

  o 1C˛; 1C˛ 2 . ;1C˛/ .˛;˛=2/ 0 ha3 iR2 Chr a3 iR2 Chd3 iR2 : T T T

˚

(37)

In view of (28), (34), the vector dQ is expressed in terms of the known functions aQ j , j D 1; 2; 3; as follows: ˇ dQ ˇ D aQ ˇ C i ˇ vQ 3 ˇx3 D0 D aQ ˇ C i ˇ !3 ; dQ 3 D 

2 X  D1

2 X ˇ i  vQ  ˇx3 D0 D  i  ! :  D1

ˇ D 1; 2;

1150

V.A. Solonnikov and I.V. Denisova

Taking the inverse Fourier-Laplace transform, one obtains @ dˇ D aˇ C  @xˇ

(

) 2 X @ 0 K1  a C K1  a3 ; @x  D1

ˇ D 1; 2;

 2 X @ 0 @ K2  a C K  a3 ; d3 D  @x @x 2  D1 where K  a means the convolution Z tZ .K  a/.x 0 ; t / D K.x 0  y 0 ; t  t 0 /a.y 0 ; t 0 /dy 0 dt 0 ; R2

0

y 0 D .y1 ; y2 /:

It follows from (34) that the Fourier-Laplace transform of the kernels Ki ; Ki0 is given by 2 2 2 f0 D  j j.r C j j/ ; K f0 D K f2 D r.r C / C .r  j j/ ; K f1 : f1D r  j j ; K K 1 2 P1 P1 rP1

The point-wise estimates of these kernels obtained in [65] (see Propositions 2.1–2.3) permit one to evaluate the norms of d in terms of a: 

1C˛; 1C˛ 2

hdiR2



6c

X 2

T

.1C˛;1C˛=2/

haˇ iR2

T

ˇD1

. ;1C˛/

C ha3 iR2

T

.˛;˛=2/

C hr 0 a3 iR2

:

T

t u

Together with (36), (37), this inequality yields (35). The next step is the analysis of inhomogeneous problem Dt v   v C ˇ vˇtD0 D v0

1 rp D f; 

r vDg

in R3C ;

v ! 0;

C ; in R1

jxj!1

(38)

ˇ @vˇ @v3 ˇˇ  C D bˇ .x 0 ; t /; ˇ D 1; 2I @x3 @xˇ ˇx3 D0 @v3 ˇˇ D b3 .x 0 ; t /; .x 0 ; t / 2 R21 ;  p C 2 ˇ @x3 x3 D0 

with given f; g; bi ; i D 1; 2; 3. Theorem 5 (Existence theorem for the linear nonhomogeneous problem in a half-space). Let ˛; 2 .0; 1/, < ˛, and T 2 .0; 1/, f 2 C˛;˛=2 .RTC /, bˇ 2

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1151

1C˛

C 1C˛; 2 .R2T /, ˇ D 1; 2, b3 2 C 1C˛;0 .R2T / \ C . ;1C˛/ .R2T /, v0 2 C2C˛ .R3C /, and let g 2 C 1C˛;1=2C˛=2 .RTC / have the form g D r  R; R 2 C2C˛;0 .RTC /; Dt R 2 C˛;0 .RTC /; Dt R  f D r  H;

H D fHik g3i;kD1 ;

Hik 2 C . ;1C˛/ .RTC /:

In addition, let the data decrease as power functions at infinity and the compatibility conditions ˇ @v03 @v0ˇ ˇˇ  C D bˇ .x 0 ; 0/; @xˇ @x3 ˇx3 D0 

r  v0 .x/ D g.x; 0/;

ˇ D 1; 2;

x 0 2 R2 : ˛

be satisfied. Then problem (38) has a unique solution .v; p/: v 2 C2C˛;1C 2 .RTC /, ˛ p 2 C . ;1C˛/ .RTC /, rp 2 C ˛; 2 .RTC /, and n ˛; ˛ .˛; ˛ / . / .2C˛;1C ˛2 / . ;1C˛/ . ;1C˛/ C hrpi C2 C hpi C 6 c hfi C2 C hHi C hvi C RT

RT

.˛; ˛2 /

Chrgi

RTC

C hrrRi

.˛/

T

C hDt Ri

x;RTC

. ;1C˛/

C hb3 iR2

RT

RT

.˛/

C

x;RTC

RT

  2 X 1C˛; 1C˛ 2 .2C˛/ hv0 iR3 C hbˇ iR2 T C ˇD1

(39)

o .˛; ˛ / C hr 0 b3 iR2 2 : T

1C˛ 1C˛ the semi-norm If b3 2 CV 1C˛; 2 .R2T /, the pressure p 2 CV 1C˛; 2 .R2T / too, and  

1C˛; 1C˛ 2

hpiR2

T

1C˛; 1C˛ 2

is estimated by the right-hand side of (39) with hb3 iR2

.

T

Proof. The solution of (38) can be found in the form v D w C w0 C u, p D q C g 0  Dt ˚ , where w is a solution of Dt w   w D f ˇ wˇtD0 D v0 ;

in R3  .0; T /;

f and v0 are extensions of f and v0 into R3 with preservation of class; w0 D r˚ , ˚ solves the Dirichlet problem for the Poisson equation ˚ D g  r  w  g 0 ˇ ˚ ˇx3 D0 D 0I

in R3C ;

1152

V.A. Solonnikov and I.V. Denisova

and the pair .u; q/ is a solution of a homogeneous problem (38) with the boundary data  0  1 ˇ  @ wˇ C w0ˇ @ w3 C w03 A ˇˇ C ; ˇ D 1; 2; aˇ D bˇ C  @ x3 D0 @x3 @xˇ !   ˇ @ w3 C w03 0 ˇ  g ˇ : a3 D b3  2 x3 D0 @x3 The vector field w0 can be viewed as a solution of the parabolic problem Dt w0   w0 D r.Dt ˚  g 0 / ˇ w0 ˇtD0 D 0;

w0ˇ jx3 D0 D 0;

in R3  .0; T /; @w03 jx D0 D g 0 jx3 D0 : @x3 3

ˇ D 1; 2;

It is clear that hwi hw0 i

.2C˛;1C˛=2/ RTC

˚ .˛;˛=2/ .2C˛/  ; 6 c hfi C C hv0 iR3 RT

.2C˛;1C˛=2/ RTC

˚

C

6 c hrDt ˚ i

.˛;˛=2/ RTC

C hg 0 i

.1C˛;1=2C˛=2/ 

:

RTC

The function Dt ˚ is given by Dt ˚.x; t / D 

Z

Z ry G.x; y/Dt .Rw/ dy D 

R3C

R3C

ry G.x; y/.rH w/ dy;

where G is the Green function for the Laplacian in R3C (with Dirichlet boundary condition). Arguing as in [11] (page 221), one can prove by (10) that hrDt ˚ i

.˛/ x;RTC

C hDt ˚ i

. ;1C˛/ RTC

o n .2C˛;1C˛=2/ . ;1C˛/ .˛/ 6 c hwi C C hHi C C hDt Ri C : RT

RT

x;RT

Moreover, from the interpolation inequality o n .˛/ . / jrDt ˚.x; t /j 6 c  ˛ hrDt ˚ iR3 C  1C hDt ˚iR3 ;

8 > 0;

t > 0;

it can be shown that hrDt ˚ i

.˛=2/ t;RTC

n o .˛/ . ;1C˛/ 6 c hrDt ˚ i C C hDt ˚ i C : x;RT

RT

Finally, applying (35) to .u; q/ and putting together all the above estimates, one arrives at the desired inequality (39). t u

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

2.3

1153

Passage to the Case of Positive Surface Tension

This subsection concerns the problem Dt v   v C 1 rp D f; ˇ vˇtD0 D v0

r vDg

in R3C ;

in

C R1 ;

v ! 0;

(40)

jxj!1

ˇ @vˇ @v3 ˇˇ C D bˇ .x 0 ; t /; ˇ D 1; 2I  @x3 @xˇ ˇx3 D0 

@v3 p C 2 C @x3

Z 0

t

ˇ v3 ˇx3 D0 d D b3 C  0

Z

t

B d ; 0

.x 0 ; t / 2 R21 :

Theorem 6 (Existence theorem for the linear problem in a half-space with  > 0). Under the assumption of Theorem 5, problem (40) with  > 0 and B 2 C ˛;˛=2 .R3T / has a unique solution .v; p/: v 2 C 2C˛;1C˛=2 .RTC /; rp 2 C ˛;˛=2 .RTC /; .˛;˛=2/ p 2 C . ;1C˛/ .RTC /, satisfying (39) with additional term hBiR2 in the rightT hand side.   1C˛; 1C˛ 2

The remark in Theorem 5 concerning the semi-norm hpiR2 T problem (40).

is also valid for

Proof. At first, (40) for the homogeneous equations, i.e., problem (20), is studied. As in [65], .v; p/ can be viewed after Fourier-Laplace transform (21) as a solution of problem (22), (33) with certain boundary data aQ . From (23), (28), (29), it is clear that aQ ˇ D bQ ˇ , ˇ D 1; 2; and aQ 3 D bQ 30 C

 2 !3 ; s

 Q bQ 30  bQ 3 C B; s

which is equivalent to j j3 Q 0 r  j j aQ 3 D bQ 30  b3 C i ˇ bQ ˇ ; P .r C j j/P i.e., r  j j M Q j j3 Q b3 C  2 i ˇ bQ ˇ C B; aQ 3 D bQ 3  P .r C j j/P P where M D 1s .P  j j3 /: In turn, the last equation can be written as a3 .x 0 ; t / D b3 .x 0 ; t / C 2Wˇ 

@bˇ @b3 CW  C V  B; @xˇ @xˇ

1154

V.A. Solonnikov and I.V. Denisova

with the kernels Wˇ , W , V defined by i ˇ j j ; WQ ˇ D P

r  j j WQ D  2 ; .r C j /P

M VQ D : P

The point-wise estimates of these kernels obtained in [63] (Propositions 2.1–2.3) enable one to prove that .˛/

. ;1C˛/

hr 0 a3 ix 0 ;R2 C ha3 iR2 T T 8 9   2 < 1C˛ X ˛ = 1C˛; .˛; / 2 .˛/ . ;1C˛/ 6 c hr 0 b3 ix 0 ;R2 C hb3 iR2 C hbˇ iR2 C hBiR2 2 : : ; T T T T ˇD1

.˛=2/

The semi-norm jr 0 ajt;R2 is evaluated by the semi-norms in the right-hand side of T the last inequality. Together with (35), this inequality yields the estimate (39) for .2C˛;1C˛=2/ .˛;˛=2/ hvi C with zero functions f, g, R, H, v0 and the additional term  hBiR2 . RT

T

The pressure function can be estimated with the help of (28), (29), i.e., pQ D pQ 0 ej jx3 , where pQ 0 D .r C j j/ D 

 dQ    2 bQ 3  2 .r C 2 /BQ  s C 2s 2  2rs C  2 : P P P

As in the preceding inequality, one has hpi

. ;1C˛/ RTC

.˛; ˛2 /

C hrpi

8
2ı; x0 2 ˝, dist.x0 ;  / > 2ı. The functions w D v, q D p satisfy the equations Dt w  r 2 w C 1 rq D f0 ;

r  w D r0

in B2ı D fjx  x0 j 6 2ıg;

where f0 D fCFŒv; p;

F D .r 2 vr 2 .v//C1 pr;

r 0 D r Cvr:

(42)

Similar to Sect. 2.2, one can set w D w.1/ C w.2/ with w.1/ , a solution of the Cauchy problem Dt w.1/  r 2 w.1/ D f0 .x; t /;

x 2 R3 ;

w.1/ .x; 0/ D w0 ;

where f0 and w0 are zero extensions of f0 and v0 into the domain R3 n B2ı , and w.2/ D r˚.x; t /, where ˚ is a solution of ˚ D r 0  r  w.1/ , i.e., Z ˚.x; t / D R3

  E.x  y/ r 0 .y; t /  r  w.1/ .y; t / dy;

E.z/ D 

1 : 4jzj

The vector field w.2/ can be viewed as a solution of the Cauchy problem   Dt w.2/  r 2 w.2/ D r Dt ˚  .r 0  r  w.1/ / ;

w.2/ .x; 0/ D 0:

0  r  w.1/ /. The function Dt ˚ is given by It is clear R that q D D0t ˚ C .r .1/ Dt ˚ D R3 E.x  y/Dt .r  r  w / dy. Since

Dt .r 0  r  w.1/ / D .Dt r  r  f  r  r 2 w.1/ / C .Dt v  f/  r  r  F;

1156

V.A. Solonnikov and I.V. Denisova

the formula for Dt ˚ takes the form Dt ˚.x; t / D 

Z

˚

  ry E.x  y/ r .M  rw.1/ /

R3

  1 C E.x  y/ r  .r 2 v  rp/  r  F dy:  Classical parabolic estimates yield o n o n .˛; ˛ / .2C˛;1C ˛2 / .˛; ˛ / .˛; ˛ / .2C˛/ .2C˛/ jw.1/ jR3 6 c jf0 jB2ı;T2 C jv0 jB2ı;T 6 c jfjB2ı;T2 C jFjB2ı;T2 C jv0 jB2ı;T ; T

.2C˛;1C ˛2 / jw.2/ jR3 T

o n .˛; ˛ / .2C˛;1C ˛2 / .˛; ˛ / 6 c jrDt ˚ jR3 2 C jw.1/ jR3 C jrr 0 jR3 2 ; T

T

T

where B~;T D B~  .0; T /. Moreover, by [11] and (10), . ;1C˛/

hDt ˚ iR3

T

.˛;˛=2/

C jDt ˚ jR3

T

T

1C˛

. 2 t;B2ı;T

C jrvj

 .˛; ˛ / . ;1C˛/ .2C˛;1C˛=2/ C jrDt ˚ jR3 2 6 c jMjB2ı;T C jw.1/ jR3

/

.˛; ˛2 / B2ı;T

C jgj

C jpj

T

1C˛

. 2 t;B2ı;T

/



C jv.x; t /jB2ı;T :

This and the preceding inequalities imply .2C˛;1C ˛2 /

jvjBı;T

. ;1C˛/

CjpjBı;T

. ;1C˛/

C jMjB2ı;T

n .˛; ˛ / .˛; ˛ / .1C˛; 1C˛ / .2C˛/ CjrpjBı;T2 6 c.T / jfjB2ı;T2 Cjv0 jB2ı CjrjB2ı;T 2 .˛; ˛ /

.

1C˛

2 C jgjB2ı;T2 C jrvjt;B2ı;T

/

.

1C˛

2 C jpjt;B2ı;T

/

o C jvjB2ı;T :

(43)

It is also needed to consider the solution in the neighborhood of a certain point x0 2 . Without loss of generality, it can be assumed that x0 D 0 and the x3 -axis is directed along the interior normal n.0/. Bı denotes now the intersection of the ball fjxj 6 ıg with ˝. For simplicity, it is assumed that  is flat near the origin and is contained in the plane x3 D 0. As above, the cut-off function  and w D v, q D p are introduced so that Dt w  r 2 w C 1 rq D f0 ; r  w D r 0 in B2ı ; t > 0;   @wˇ @w3 ˇˇ C D bˇ0 ; ˇ D 1; 2;  ˇ @x3 @xˇ x3 D0 Z t Z t ˇ @w3  q C 2 C 0 w3 d  ˇx3 D0 D b30 C  B 0d  ; @xˇ 0 0 wjtD0 D w0 ;

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1157

where f0 ; r 0 are given by formulas (42), and bˇ0 D bˇ .x 0 ; t /  .vˇ Dx3  C v3 Dxˇ /jx3 D0 ;

b30 D b3  C 2v3 Dx3 jx3 D0 ;

B 0 D B C 2r 0 v3  r 0  C v3 0 : Let w.1/ be defined as a solution of the Cauchy problem Dt w.1/  r 2 w.1/ D f0 .x; t /;

x 2 R3 ;

w.1/ .x; 0/ D w0 ;

where f0 and w0 are zero extensions of f0 and v0 into the domain R3C n B2ı and then into R3 with preservation of class. The function w.2/ is defined as r˚.x; t /, where Z ˚.x; t / D G.x; t /.r 0 .y; t /  r  w.1/ .y; t // dy; R3

G being the Green function, as in Theorem 5. Hence, w.2/ satisfies   Dt w.2/  r 2 w.2/ D r Dt ˚  .r 0  r  w.1/ / ; .2/

wˇ jx3 D0 D 0;

x3 > 0;

w.2/ .x; 0/ D 0;

.2/ @w3 ˇˇ D r 0  r  w.1/ jx3 D0 ; ˇ @x3 x3 D0

ˇ D 1; 2;

Finally, the couple .w.3/ ; s/ is introduced as a solution of (20) with the boundary data .a; A/ defined by (41) with w D w.1/ , w0 D w.2/ , bi D bi0 , B D B 0 . It can be easily seen that w D w.1/ Cw.2/ Cw.3/ , q D s Dt  C.r 0 r w.1/ /. The estimates of w.ˇ/ ; ˇ D 1; 2, and of Dt ˚ are the same as in interior domains B2ı . The estimates of w.3/ and s are the same as in Theorem 4. As a result, an inequality similar to (43) is proved: .2C˛;1C ˛2 /

jvjBı;T

. ;1C˛/

CjpjBı;T

.˛; ˛ /

 .˛; ˛ / .˛; ˛ / .1C˛; 1C˛ / .2C˛/ CjrpjBı;T2 6c.T / jfjB2ı;T2 Cjv0 jB2ı CjrjB2ı;T 2

. ;1C˛/

C jgjB2ı;T2 CjMjB2ı;T

C

2 X ˇD1

.˛; ˛2 /

1C˛

. 2 t;B2ı;T

C  jBj C jrvj b B 2ı;T

/

.1C˛; 1C˛ 2 /

jbˇ j b B 2ı;T

1C˛

C

. 2 / jpjt;B2ı;T

. ;1C˛/

Cjb3 j b B 2ı;T

.˛; ˛ /

Cjr 0 b3 j 2 b B 2ı;T

C jvjB2ı;T ;

(44)

b~;T D B~;T \ fx3 D 0g. where B Remark 4. If the surface  near the point x0 D 0 is not flat but given by the equation x3 D .x 0 / 2 C 2C˛ and .0/ D r.0/ D 0, then it is necessary to introduce the coordinate transformation “rectifying” the boundary zˇ D xˇ ;

ˇ D 1; 2;

z3 D x3  .z0 /:

1158

V.A. Solonnikov and I.V. Denisova

As shown in [48] (Theorem 4.2), this leads to the appearance of the terms containing higher-order derivatives of solution with coefficients proportional to positive powers of ı on the right-hand side in (44). If ı is small, then the norms of these terms can be absorbed by the norms on the left-hand side in (44). We omit these calculations. From (43), (44) and the interpolation inequalities, it follows that   1C˛

. / Y .v; p/ 6 c F .T / C jpjt;QT2 C jvjQT ;

(45)

where Y stands for the left-hand side in (19). As shown in [48], the norm of p can be estimated by analyzing the boundary value problem r 2 p.x; t/ D Dt r C r 2 r C r  f;

pjx2 D n  S.v/n  b;

x 2 ˝;

that is equivalent to r 2 h D r  g D r  .r  M/;

x 2 ˝;

hjx2 D r  b C n  S.v/nj ;

where h D p  r (see (5.43) in [48] and subsequent calculations). This enables one to eliminate the norm of p from the right-hand side in (45). As for the maximum modulus of v, it is easily estimated by applying the Gronwall inequality. This completes the proof of (19). In the case  D 0, estimate (19) is proved by the same arguments on the basis of Theorem 5. The solvability of problem (40) can be established by the following arguments [31]. First of all, by constructing some auxiliary functions, this problem is reduced to homogeneous system (16), i.e., system (16) with zero functions f, r, b, b and with BjtD0 D 0. Then, the operator R is introduced which makes correspond to arbitrary B 2 C ˛;˛=2 .GT /, such that BjtD0 D 0, a couple .v; p/ which solves homogeneous problem (16) with .I C M/B instead of B, v 2 C 2C˛;1C˛=2 .˝T /; . ;1C˛/ being finite. The operator M is continuous rp 2 C ˛;˛=2 .˝T /, the norm jpj˝T in C ˛;˛=2 .GT /; I is identity operator; in addition, the sum I C M is invertible. It is clear that .v; p/ D R.I C M/1 B is a solution of homogeneous system (16). The proof of these statements reduces to constructing R that reminds the construction of the regularizer in the theory of parabolic problems (see [27] and also [19]) and to estimating MB. The estimate has the form .˛;˛=2/ jMBjGt

6

.˛;˛=2/ ıjBjGt

Z

t

C c.ı/ 0

.˛;˛=2/

jBjG

with ı  1, that guarantees the invertibility of I C M.

d t u

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1159

These arguments are also applicable to other problems (e.g., see Sect. 5.2 and [9]). Remark 5. Along with (19), the solution of problem (16) satisfies the coercive estimate with reduced regularity with respect to time .2C˛/

sup jv.; t /j˝ t 0. It is necessary to show that there exists ".T / such that for 8m 2 N, the norms NT Œu.m/ ; q .m/  are uniformly bounded and that the sequence fu.m/ , q .m/ g1 mD1 converges to a solution of problem (6), (7). The proof of these facts is based on Lemmas 3, 4, and 5 applied to the right-hand sides of the systems Q .j C1/   ˙ rj2 w Q .j C1/ C Dt w

1 .j / .j 1/ .j / rj sQ .j C1/ D l1 .u.j / ; q .j / /  l1 .u ; q .j / / ˙ .j 1/

.j /

Q .j C1/ D l2 .u.j / /  l2 .u.j / / in QT ; C f.Xj ; t /  f.Xj 1 ; t /; rj  w ˇ .j / .j 1/ .j / Q .j C1/ /nj ˇ D l3 .u.j / /  l3 Q .j C1/ .y; 0/ D 0; y 2 ˝0 ; ˘0 ˘j Sj .w .u /; w Z t ˇ ˇ .j / Q .j C1/ d ˇ D l4 .u.j / ; q .j / / Q .j C1/ ; sQ .j C1/ /nj ˇ  n0  j .t / n0  Tj .w w 

.j 1/ .j / .u ; q .j / / l4

C

0

Z t 0

.j / l5 .u.j / /





.j 1/ .j / l5 .u /

Z

t

d C 

ˇ   P j 1 . / y d ˇ  pe .Xj ; t /n0  nj C pe .Xj 1 ; t /n0  nj 1 ;

 n0  P j . /

0

t 2 .0; T /; (57)

Q .j C1/ , sQ .j C1/ mean the differences u.j C1/  u.j / ; q .j C1/  q .j / , respectively, where w .k/ 1 6 j 6 m; the operators li are calculated by (48) with u D u.k/ , k 6 m; u.0/ D 0. The norms of the right-hand sides in (57) are estimated either by lower norms Q .j / and sQ .j / or by the leading part of their norms but with small coefficients of w including ı from inequality (14). In particular, by (51) .

.˛/

1C˛

jf.Xj ; t /f.Xj 1 ; t /jx;QT Chf.Xj ; t /f.Xj 1 ; t /it;QT2 .˛;

1C˛

where c depends on the norms jfjR3 2 T If one takes into account the estimate Q .j / jQT 6 jw Q .j / .; 0/j˝0 C jw

Z

/

.˛;

and jDx fjR3

/

1C˛

2

/

.

T

T

Q .j / .; t /j˝0 dt 6 jDt w 0

.˛/

Q .j / jx;QT ; 6 c.T CT =2 /jw

Z 0

T

Q .j / ; sQ .j /  dt; Nt Œw

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . . 0 one can deduce the boundedness of ˙mC1 .T / 

0 0 ˙mC1 .T / 6 ~˙mC1 .T / C c1

Z

T 0

PmC1 j D1

1165

Q .j / ; sQ .j / : NT Œw

0 Q .1/ ; sQ .1/  ˙mC1 .t / dt C NT Œw

with some ~ < 1. Finally, the Gronwall lemma applied to the inequality 0 ˙mC1 .T /

Z

T

6 c2 0

0 ˙mC1 .t / dt C

1 Q .1/ ; sQ .1/ ; NT Œw 1~

yields 0 NT Œu.mC1/ ; q .mC1/  6 ˙mC1 .T / C NT Œu.0/ ; q .0/  6

NT Œw.1/ ; sQ .1/  c2 T e : 1~

(58)

Q .1/ ; sQ .1/ / D .u.1/ ; q .1/ /, the inequality Since .w Q .1/ ; sQ .1/ .1  ~/1 e c2 T 6 ı; .T C T =2 /NT Œw holds due to (55) provided that " from (11) is small enough. Consequently, assumption (14) is also valid for .u.mC1/ ; q .mC1/ /. Thus, as follows from (58), NT Œu.j / ; q .j /  are uniformly bounded and the P1 Q .j / ; sQ .j /  is convergent. Hence, the sequence fu.j / ; q .j / g1 series j D1 NT Œw j D1 is convergent in itself with respect to the norm NT . Since the Hölder classes of functions are complete spaces, passing to the limit in system (56), one makes sure that the approximations .u.j / ; q .j / /, j 2 N, converge to a solution of problem (6), (7) for which the inequality (12) holds. It is obvious that the solution satisfies (14). One should show also the uniqueness of the solution obtained. To this end, it is assumed that there are two solutions of (6), (7): .u; q/ and .u0 ; q 0 /. The differences w D u  u0 , s D q  q 0 satisfy the system Dt w  ru2 w C

1 ru s D l1 .u0 ; q 0 /  l01 .u0 ; q 0 / C f.Xu ; t /  f.Xu0 ; t /; 

ru  w D l2 .u0 /  l20 .u0 / in QT ; ˘0 ˘ Su .w/njGT D l3 .u0 /  l03 .u0 /; Z t n0  Tu .w; s/njGT   n0  .t / w djGT D l4 .u0 ; q 0 /  l40 .u0 ; q 0 / wjtD0 D 0;

0

Z

t

C 0





l5 .u0 /  l50 .u0 / d C 

Z

t

P /  P 0 . //y d jGT n0  . .

0

 pe .Xu ; t /n0  n C pe .X ; t /n0  n0 : u0

1166

V.A. Solonnikov and I.V. Denisova

Repeating the above arguments, one arrives at the inequality Z

T

NT Œw; s 6 ~NT Œw; s C c3 jwjQT 6 ~NT Œw; s C c3

Nt Œw; s dt;

~ < 1;

0

whence it follows by the Gronwall lemma that w D 0, s D 0.

4

Global Solvability of the Free Boundary Problem Governing the Evolution of an Isolated Liquid Mass with Zero Surface Tension

The problem consists in the determination of a bounded domain ˝t , t > 0, filled with the fluid, together with the velocity vector field v.x; t / and the pressure function p.x; t/, x 2 ˝t , satisfying the second initial–boundary value problem without surface tension forces and external pressure for homogeneous NavierStokes system (1), (2), (3). At initial instant, the domain ˝0 and the vector field v0 are given. Mass forces are not taken into account, and the initial total momenta of the fluid vanish, i.e., Z

Z v0 .x/dx D 0; ˝0

˝0

v0 .x/  i .x/dx D 0;

i D 1; 2; 3;

(59)

where i .x/ D ei  x; eij D ıji : Elementary calculation shows that d dt

Z v.x; t /dx D ˝t

d dt

Z ˝t

v.x; t /  i .x/dx D 0I

hence, Z

Z v.x; t /dx D 0;

˝t

˝t

v.x; t /  i .x/dx D 0;

i D 1; 2; 3;

t > 0:

(60)

Finally it is assumed that the barycenter of the fluid is located at the origin, i.e., Z xdx D 0: (61) ˝t

The problem is treated in the Lagrangian coordinates y 2 ˝0 related to u the Eulerian coordinates x 2 ˝t by (4). Let L be the Jacobi matrix @X of @y T transformation (4). Since detL D 1, there holds A D L , where A is the matrix of cofactors of the elements of L. After mapping (4), homogeneous problem (1), (2), (3) and conditions (60), (61) go over into the system with the unknown functions u.y; t / D v.Xu ; t /, q.y; t/ D p.Xu ; t /:

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

Dt u  ru2 u C

1 ru q D 0; 

u.y; 0/ D v0 .y/; Z

ru  u D 0 in ˝0 ;

u.y; t /dy D

t > 0;

Tu .u; q/nj D 0;

y 2 ˝0 ;

Z

˝0

1167

(62)

Z u.y; t /  i .Xu ; t /dy D 0;

˝0

yi dy D 0;

i D 1; 2; 3;

˝0

where n.Xu / is calculated by (8), and the other notations are also the same as in (6), (7). Equations (62) are equivalent to Dt u  r 2 u C

1 rq D l1 .u; q/; 

u.y; 0/ D u0 .y/;

r  u D l2 .u/

(63)

ˇ q C n0  S.u/n0 ˇ D l4 .u/

Z

on  ;

t > 0;

Z

u.y; t /dy D 0; ˝0

t > 0;

y 2 ˝0 ;

ˇ ˘0 S.u/n0 ˇ D l3 .u/; Z

in ˝0 ;

˝0

u  i .y/dy D

˝0

u  .i .y/  i .Xu //dy;

i D 1; 2; 3;

where li are the operators calculated by formulas (48). Theorem 7 (Global solvability for a single fluid with zero surface tension [67]). If the surface  2 C 2C˛ , and the initial velocity v0 2 C2C˛ .˝0 / satisfies (59), ˇ ˘0 S.v0 /n0 ˇ D 0;

r  v0 D 0;

(64)

and .2C˛/

jv0 j˝0

6   1;

(65)

then problem (63) has a unique solution .u; q/ such that u.; t / 2 C 2C˛ .˝0 /, Dt u.; t / 2 C ˛ .˝0 /, q.; t / 2 C 1C˛ .˝0 /, and there exists ˇ > 0 such that .2C˛/

ju.; t /j˝0

.˛/

.1C˛/

C jDt u.; t /j˝0 C jq.; t /j˝0

.2C˛/

6 ceˇt jv0 j˝0

;

8t > 0:

(66)

Remark 6. For the problem with surface tension forces on the free boundary, a similar theorem is proved in [67]. In this case, it was assumed that the initial surface  2 C 3C˛ . One can find the main ideas of the proof in Sect. 6.3.

1168

V.A. Solonnikov and I.V. Denisova

Along with (63), the linear problem Dt v  r 2 v C

1 rp D 0; 

r  v D 0;

y 2 ˝0 ;

t > 0; (67)

v.y; 0/ D v0 .y/; y 2 ˝0 ; ˇ ˇ ˘0 S.v/n0 ˇ D 0; p C n0  S.v/n0 ˇ D 0; Z

Z v.y; t /dy D 0; ˝0

˝0

v  i .y/dy D 0;

i D 1; 2; 3;

is studied. Existence theorem for this problem is stated as follows. Theorem 8 (Global solvability for the linear problem). If v0 2 C 2C˛ .˝0 / satisfies orthogonality and compatibility conditions (59), (64), then problem (67) has a unique solution .v; p/ such that v.; t / 2 C 2C˛ .˝0 /, Dt v.; t / 2 C ˛ .˝0 /, p.; t / 2 C 1C˛ .˝0 /, 8t > 0, and for some a > 0 .2C˛/

.˛/

jv.; t /j˝0

.1C˛/

C jDt v.; t /j˝0 C jp.; t /j˝0

.2C˛/

6 ceat jv0 j˝0

:

(68)

Proof. It follows from the energy relation d kvk2˝0 C kS.v/k2˝0 D 0; dt where k  k˝ is the norm of L2 .˝/, and the Korn inequality ˇZ  ˇ kvk˝0 6 c kS.v/k˝0 C ˇ

˝0

3 ˇZ ˇ X ˇ ˇ v dx ˇ C ˇ iD1

˝0

ˇ ˇ v  i dx ˇ

(69)

that d kvk2˝0 C 2akvk2˝0 6 0; dt

a > 0;

which implies kv.; t /k2˝0 6 e2at kv0 k2˝0 :

(70)

The estimate of the higher-order norms of the solution follows from (70) and the local estimate .2C˛;0/

jvjQt

0 1;t0

.˛;0/

C jDt vjQt

0 1;t0

.1C˛;0/

C jpjQt

0 1;t0

6 ckvkQt0 2;t0 ;

(71)

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . . .l;0/

1169

.l/

where Qt1 ;t2 D ˝0 .t1 ; t2 /, jujQt ;t D supt2.t1 ;t2 / ju.; t /j˝0 : This estimate is proved 1 2 in the following way: let w D  .t /v, q D  .t /p, where 2 .0; 1 and  is a smooth monotone function of t equal to zero for t < t0  2 C =2 and to one for t t0  2 C and such that jDkt  .t /j 6 c k : It is clear that Dt w  r 2 w C

1 rq D vDt  ; 

T.w; q/n0 jy2 D 0;

r  w D 0;

y 2 ˝0 ;

t 2 .t0  2; t0 /;

w.y; t0  2/ D 0:

Making use of (46) and of an interpolation inequality for Hölder continuous functions, one obtains .2C˛;0/

jwjQt

0 2C ;t0

.˛;0/

.1C˛;0/

CjDt wjQt



.˛;0/

C jqjQt

0 2C ;t0

0 2C ;t0

.2C˛;0/

6 1  jvjQt

0 2C =2;t0

6 c 1 jvjQt

0 2C =2;t0

C c 3=2˛

sup t2.t0 2;t0 /

.2C˛;0/

 kv.; t /k˝0 :

.˛;0/

.1C˛;0/

Upon setting  D ı , f . / D 5=2C˛ .jvjQt 2C ;t CjDt wjQt 2C ;t CjqjQt 2C ;t /, 0 0 0 0 0 0 K D supt2.t0 2;t0 / kv.; t /k˝0 ; one is led to f . / 6 ı1 f

  2

C cı 3=2˛ K 6 ı12 f

  4

C cı 3=2˛ .1 C ı1 /K : : :

c.ı/K.1 C ı1 C ı12 C : : :/ 6

c.ı/ K; 1  ı1

where ı1 D ı25=2C˛ < 1: This implies .2C˛;0/

jvjQt

0 2C ;t0

.˛;0/

.1C˛;0/

C jDt wjQt

0 2C ;t0

C jqjQt

0 2C ;t0

6

c.ı/ 5=2˛ 1  ı1

sup t2.t0 2;t0 /

kv.; t /k˝0

which is equivalent to (71) for D 1. By (70), the last inequality yields .2C˛/

jv.; t0 /j˝0

.˛/

.1C˛/

CjDt v.; t0 /j˝0 Cjp.; t0 /j˝0

6 ceat0 kv0 k˝0 6 ceat0 j˝0 j1=2 jv0 j˝0 ;

which proves (68) for t > 2. In the case t 6 2, (68) is a consequence of (19).

t u

The solvability of the nonlinear problem in an infinite time interval is based on Theorem 8 and the following estimates of nonlinear terms (48). Theorem 9. Let functions u, q possess finite norm .2C˛;0/

Y .T /  Y .u; q/ D jujQT

.˛;0/

.1C˛;0/

C jDt ujQT C jqjQT

:

(72)

1170

V.A. Solonnikov and I.V. Denisova

If the function U.y; t / D

Rt 0

u.y;  /d satisfies the inequality .2C˛;0/

jUjQT

6 ı1 ;

ı1 > 0;

(73)

then the estimate .˛;0/

.1C˛;0/

Z.T / D jl1 jQT Cjl2 jQT

.˛;0/

.1C˛/

CjDt L2 jQT Cjl3 jGT

.1C˛/

Cjl4 jGT

  6 c ı1 Y .T /CY 2 .T / (74)

holds. An elementary proof is omitted. Proof (of Theorem 7). By Theorem 1 with  D 0, problem (63) is solvable in a finite time interval .0; T / with T fixed later. The solution is sought in the form u D u0 C u00 , q D q 0 C q 00 , where .u0 ; q 0 / is a solution to (67), (59), and .u00 ; q 00 / solves the problem Dt u00  r 2 u00 C

1 00 rq D l1 .u0 C u00 ; q 0 C q 00 /;   r  L2 .u0 C u00 /;

ˇ ˘0 S.u00 /n0 ˇ D l3 .u0 C u00 /;

r  u00 D l2 .u0 C u00 /

u00 .y; 0/ D 0;

(75) ˇ q 00 C n0  S.u00 /n0 ˇ D l4 .u0 C u00 /:

Problem (67), (59) is solvable in the infinite time interval t > 0, and in view of (68) .2C˛/

ju0 .; T /j˝0

.2C˛/

6 c1 eaT jv0 j˝0

:

In the arguments below, T is fixed such that c1 eaT 6

1 : 16

Problem (75) is solvable in the time interval .0; T / if  from (65) is sufficiently small. Indeed, the solution can be constructed by iterations. The nonlinear terms .2C˛;0/ in (48) can be estimated by (74) with ı1 D T jujQT ; this leads to   Y .u00 ; q 00 / 6 cY 2 .u0 C u00 ; q 0 C q 00 / 6 c Y 2 .u0 ; q 0 / C Y 2 .u00 ; q 00 /   6 c2 .T / N .v0 / C Y 2 .u00 ; q 00 / ; .2C˛/

where N .v0 / D jv0 j˝0 . It can be shown in the case of small  that the solution of (75) exists and satisfies Y .u00 ; q 00 / 6 2c2 N .v0 /:

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1171

Hence     1 N u.; T / 6 c1 eaT C 2c2  N .v0 / 6 N .v0 /; 8 if 2c2  < 1=16. Let the solution of (63) be found for t < kT and satisfy Nk 6

1 1 Nk1 6 : : : 6 k N0 ; 8 8

where Nk D N .u.; kT //; moreover, let inequality (73) with ı1  1 be valid for t 6 kT . For t 2 .kT; .k C 1/T /, the functions u; q are defined as the sum of solutions of (67), (59), and (75) on this time interval with the initial velocities u0k , u00k , where u0k D uk  u00k , uk D ujtDkT , u00k is subjected to the relations r  u00k D l2 .uk /; Z ˝0

u00k dy

Z D 0; ˝0

u00k

ˇ ˘0 S.u00k /ˇ D l3 .uk /; Z

 i .y/ dy D

˝0

   uk i .y/  i Xuk .y; t / dy;

N .u00k / 6 cı1 N .uk /: The construction of vector fields satisfying similar conditions can be found in [71]. Since Z Z u0k dy D 0; u0k  i .y/dy D 0; ˝0

˝0

inequalities    N u0k ; .k C 1/T 6 c1 eaT N .uk /; Yk .u; q/ 6 c3 N .uk /;   1 Nk .ukC1 / 6 c1 eaT C c3 ı1 N .uk / 6 N .uk / 8 are satisfied if ı1 is chosen small enough (here Yk means the norm (72) computed on the time interval .kT; .k C 1/T ). The same inequalities hold for t 2 .j T; .j C 1/T /, j < k. Hence, taking ˇ 2 .0; a/ such that eˇT 6 4, one obtains k X j D0

Yj2 e2ˇj T 6 c

k X e2ˇj T j D0

82j

N02 6 cN02 :

1172

V.A. Solonnikov and I.V. Denisova

This shows that Z sup jU.y; t /j 6 t n C 3, locally in time, when the interface is close to a flat configuration in the whole space [35], and in bounded domains [24], the solution becoming instantaneously real analytic. Global (in time) classical solvability of the problems without and with surface tension forces was analyzed by the authors in [12] and [18], respectively. The main existence theorems from these papers are cited here (Theorems 12 and 13). As in the case of one fluid, exponential estimates of solutions in terms of the data are proved. Existence of a global solution of the problem was also obtained in the Sobolev spaces Wp2;1 .˝ ˙ / for p > n C 2 in [24] and for p > n D 3 in [73]. In addition, global existence theorem for the problem governing two-phase fluid motion without surface tension was proved in L2 - setting in Sobolev-Slobodetskiˇı spaces [14].

5.1

Statement of the Problem

At the initial instant, let a fluid with viscosity  C > 0 and density C > 0 occupy a bounded domain ˝0C  R3 ; the surface @˝0C is denoted by  . And let a fluid with viscosity   > 0 and density  > 0 fill a domain ˝0 surrounding ˝0C . The boundary ˙  @.˝0C [  [ ˝0 / is a given closed surface, ˙ \  D ; (Fig. 1).

Fig. 1 Motion of two-phase fluid

1174

V.A. Solonnikov and I.V. Denisova

For every t > 0, one needs to find t D @˝tC , the velocity vector field v.x; t / D .v1 ; v2 ; v3 /, and the function p.x; t/, the deviation from the hydrostatic pressure, which solve the diffraction problem for the Navier-Stokes system: Dt v C .v  r/v   ˙ r 2 v C vjtD0 D v0 Œvjt 

˝0 [ ˝0C ;

in lim

x!x0 2t ; x2˝tC

1 rp D f; r  v D 0 ˙

v.x/ 

ŒT.v; p/njt D  H n

lim

x!x0 2t ; x2˝t

on

in

˝t˙ ; t > 0;

vj˙ D 0;

(76)

v.x/ D 0;

t :

(77)

Here, v0 is the initial velocity,  ˙ ; ˙ are the step functions of kinematic viscosity and density, T.v; p/ is stress tensor with the elements Tik .v; p/ D ıik p C ˙ Sik .v/; i; k D 1; 2; 3, ˙ D ˙ ˙ is the step function of dynamical viscosity,  > 0 is surface tension coefficient, n is the outward normal to ˝tC . The other notation is the same as in problem (1), (2). If  > 0, one introduces a new pressure function p1 D p in ˝tC and p1 D p C  R20 in ˝t . Then only the last boundary condition (77) is changed:  2  ŒT.v; p1 /njt D  H C n on t : R0

(78)

If  D 0, then p1  p in both domains. The liquids are assumed to be immiscible. A condition excluding the mass transportation through t implies that t consists of the points x.y; t /, the radius vector x.y; t / of which is a solution of the Cauchy problem (3). Hence, t D fx.y; t /jy 2  g, ˝t˙ D fx.y; t /jy 2 ˝0˙ g. Let QT˙ D ˝0˙  .0; T /; DT D [QT˙ ; ˝ D ˝0 [ ˝0C ; QT D ˝  .0; T /: One studies the problem which is equivalent to system (76), (78), (3) in Lagrangian coordinates (4): 1 ru q D f.Xu ; t /; ru  u D 0 in QT˙ ; ˙ ˇ ˇ uˇtD0 D v0 in ˝0 [ ˝0C ; uˇ˙.0;T / D 0; ˇ ˇ ŒuˇGT D 0; Œ˙ ˘0 ˘ Su .u/nˇGT D 0; GT    .0; T /;  ˇ 2  Œn0  Tu .u; q/nˇGT D  H .Xu / C n0  n on GT : R0

Dt u  ru2 u C

(79)

(80)

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

5.2

1175

Local Solvability

5.2.1 The Problem in a Bounded Domain Local classical solvability of the problem governing the motion of two fluids in the case of strictly positive  was obtained in [9, 16, 17]. In that papers, the surface ˙ was absent and the domain ˝ coincided with the whole space R3 . The result was obtained in Hölder spaces with power-like weights at infinity, but it is also valid in the case of a bounded domain, because the weighted Hölder spaces are equivalent to the ordinary ones in this case. The solvability “in the small” for the case without surface tension was proved in [12]. Theorem 10 (Local existence for two liquids). Let  2 C 3C˛ , ˙ 2 C 2C˛ , f, 1C˛

Dx f 2 C˛; 2 .QT /, v0 2 C2C˛ .˝0 [ ˝0C / for some ˛; 2 .0; 1/, < ˛, T < 1. In addition, the compatibility conditions are assumed to hold: ˇ ˇ r  v0 D 0; Œv0 ˇ D 0; Œ˙ ˘0 S.v0 /n0 ˇ D 0; h iˇ ˇ 1 ˇ ˘0 . ˙ r 2 v0  ˙ rq0 ˇ D 0; v0 ˇ˙ D 0;    ˇ 1 ˇ ˘˙ f.y; 0/   rq0 .y/ C   r 2 v0 .y/ ˇ D 0; y2˙ 

(81)

where q0 .y/ Dp1 .y; 0/ is a solution of the following diffraction problem:     1 2  r q .y/ D r  f.y; 0/  D B .v / v .y/ ; y 2 ˝0 [ ˝0C ; 0 t 0 0 ˙ h  iˇ 2  @v0 ˇ  n0 ˇ   H0 .y/ C Œq0 j D 2˙ ; y 2;  @n0 R0  @  h 1 @q iˇ ˇ 0 ˇ ˙ 2 ˇ D Œ n  r v  D n  r ; ˇ 0 0 0  ˙ @n0  @n0  @  ˇ ˇˇ   2 1 @q0 ˇˇ ˇ D n   r v C f D n  r : ˇ ˇ ˙ 0 ˙ tD0 ˙  @n˙ ˙ @n˙

(82)

ˇ (Here, H0 .y/ D n0  .0/y ˇ is twice the mean curvature of  ; B is the transpose of B DA  I; I is the identity matrix; ˘˙ b Db  n˙ .n˙  b/; n˙ is the outward normal to ˙.) Then for 8T < 1, there exists ".T / such that problem (79), (80) has a unique solution .u; q/: u 2 C2C˛;1C˛=2 .DT /, q 2 C . ;1C˛/ .DT /, rq 2 C˛;˛=2 .DT /, provided that .˛;

jfjQT

1C˛

2

/

.˛;

C jDx fjQT

1C˛

2

/

.2C˛/

C jv0 j[˝ ˙ C  jH0 C

2 .1C˛/ j 6 ".T /; R0 

(83)

1176

V.A. Solonnikov and I.V. Denisova

q being unique up to a function of time. The interface t is a surface of class C 3C˛ . Moreover, the estimate .2C˛;1C˛=2/

jujDT

.˛;˛=2/

. ;1C˛/

C jrqjDT C jqjDT n .˛; 1C˛ / 1C˛

2 .1C˛/ o .˛; / .2C˛/ : 6 c jfjQT 2 C jDx fjQT 2 C jv0 j[˝ ˙ C jH0 C j R0  (84)

holds. This theorem was proved in the same way as Theorem 1. In view of this analogy, the main difficulty of the proof consisted in constructing and estimating a solution to the problem with a plane interface [8, 16]. Remark 7. If  D 0, it is sufficient to assume in Theorem 10 that  2 C 2C˛ , and then the interface t is also of class C 2C˛ .

5.2.2 Model Problem in the Whole Space with Plane Interface The following linear problem with unknowns v D .v1 ; v2 ; v3 /; p is considered in two half-spaces: Dt v   ˙ v C

1 ˙ rp D f; r  v D g in R1 D R3˙  .0; 1/; ˙ ˇ R3 [ R3C ; v ! 0; vˇtD0 D v0 jxj!1

ˇ Œvˇx3 D0 D 0;

(85)

ˇ

 @vˇ @v3 ˇˇ  ˙ C D bˇ .x 0 ; t /; ˇ D 1; 2I @x3 @xˇ ˇx3 D0

ˇ Zt Zt ˇ ˇ 0 ˇ  p C 2 C  v3 ˇx3 D0 d D b3 C  B d  b30 @x3 ˇx3 D0 ˙ @v3

0

on R21 ;

0

where R3˙ D f˙x3 > 0g, x 0 D .x1 ; x2 /; 0 D @2 =@x12 C @2 =@x22 ;  ˙ ; ˙ > 0 are the step functions of viscosity and density in R3˙ , respectively; ˙ D  C C in R3C and ˙ D    in R3 . Let f; v0 D 0, g D 0 and let given functions bi ; i D 1; 2; 3; B vanish at t D 0 and decrease sufficiently quickly as jx 0 j ! 1. The Fourier-Laplace transform with respect to x 0 and t given by formula (21) converts problem (85) with f; v0 D 0, g D 0 into the system with unknown functions vQ ; pQ  i ˛ d2 vQ ˛  s  ˙ C 2 vQ ˛  ˙ pQ D 0; 2   dx3  1 dpQ d2 vQ 3  s  ˙ C 2 vQ 3  ˙ D 0; 2   dx3 dx3

˛ D 1; 2; (86)

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

dvQ 3 C i 1 vQ 1 C i 2 vQ 2 D 0; dx3

1177

˙ x3 > 0;

completed by the boundary conditions vQ ! 0; jx3 j!1

ˇ ŒQvˇx3 D0 D 0;

pQ ! 0; jx3 j!1

ˇ ˇ dvQ ˛ C i ˛ vQ 3 ˇˇ D bQ ˛ ; ˛ D 1; 2; dx3 x3 D0 ˇ

dvQ 3 ˇˇ Q s  pQ C 2˙   2 vQ 3C D s bQ 3 C  B; dx3 ˇx3 D0

 ˙



(87)

ˇ Q ˇx3 D0 D w QC  w Q , w Q ˙ D lim w. Q where 2 D 12 C 22 ; Œw x3 !0˙

By analogy with the model problem in a half-space (Sect. 2.1), the explicit solution of (86), (87) is found in the form vQ D We0˙ C V˙ e1˙ ; ˙

˙

˙x3 > 0; ˙ j jx3

pQ D  .r C j j/ e

;

˙x3 > 0;

(88)

where 0

0 1 1 ˙ !1 i 1 er x3  ej jx3 ˙ W D @ !2 A ; V˙ D @ i 2 A  ˙ ; e0˙ D er x3 ; e1˙ D ; r ˙  j j !3 j j n

o 1 Q .C C  /s C j jq  r ˙ q 0 C  j j3 ; ˙ D j jbQ 30 s.r ˙ q C j jq 0 /  As j jP q s ! D

bQ   i  fC  C C     .C   /!3 g ; C r C C  r 

!3 D

Q 0 j jbQ 0 s Asq 3  : Pq P

 D 1; 2;

(89)

In formulas (89), the following notation has been used:  Q bQ 30 D bQ 3 C B; s  ˚   4j js 2 2 j j C r C C  r  C C  .r C r  C j j2 / C  j j3 ; P D .C C /s 2 C q (90) Q D i 1 bQ 1 C i 2 bQ 2 ; A

q D C .r C C j j/ C  .r  C j j/;

q 0 D C .r C  j j/   .r   j j/:

1178

V.A. Solonnikov and I.V. Denisova

For the expression (90), the following lemma is valid. Lemma 6. If Re s D a > a0 > 0, 2 R2 , then the function P . ; s/ satisfies the estimates   jP . ; s/j > c jsj2 C jsj 2 C  j j3 ; ˇ @P ˇ   ˇ ˇ ˇ ˇ 6 c jsj3=2 C jsjj j C  2 ; @  ˇ @P ˇ   ˇ ˇ ˇ ˇ 6 c jsj C 2 ; @s ˇ @2 P ˇ   h jsj1=2 i ˇ ˇ C j j ; ˇ 6 c jsj 1 C ˇ @ 1 @ 2 j j

(91) ˇ @2 P ˇ  1=2 ˇ ˇ ; ˇ 6 c jsj C 2 ˇ @s@ 

 D 1; 2I (92)

ˇ ˇ ˇ

 @3 P ˇˇ jsj1=2  : ˇ6c 1C @s@ 1 @ 2 j j

The constants c depend on a0 . Inequalities (91), (92) with  > 0 were proved in [16]. Estimate (91) implies (32) which together with (92) allows one to apply the theorem on Fourier multipliers in the Hölder spaces to solution (88), (89) and prove its smoothness [16]. The case  D 0 was considered in [11]. As a result, one obtains the theorem where the following notation is introduced: DT3 D .R3 [ R3C /  .0; T /, RnT D Rn  .0; T /, n D 2; 3; .kC˛; kC˛ 2 /

jf jD 3

T

.kC˛; kC˛ /

.kC˛; kC˛ 2 /

 jf jR3 .0;T2 / C jf jR3

C .0;T /



;

.kC˛/

.kC˛/

jf j[R3  jf jR3 ˙



.kC˛/

C jf jR3

:

C

Theorem 11 (Solvability of the linear problem in two half-spaces). Let ˛; 2 .0; 1/; < ˛; and T 2 .0; 1/. It is also assumed that f 2 C˛;˛=2 .DT3 /, g 2 1C˛ 1C˛ C 1C˛; 2 .DT3 /, bˇ 2 C 1C˛; 2 .R2T /, ˇ D 1; 2, b3 2 C 1C˛;0 .R2T / \ C . ;1C˛/ .R2T /, v0 2 C2C˛ .R3 [ R3C /, and, moreover, g has the form g D r  R; R 2 C2C˛;0 .DT3 /; Dt R 2 C˛;0 .DT3 /; ŒR3 jx3 D0 D 0; Dt R  f D r  H; where r  H  conditions

P3

kD1

H D fHik g3i;kD1 ;

Hik 2 C . ;1C˛/ .DT3 /;

@Hik =@ k ; i D 1; 2; 3. In addition, the compatibility

ˇ r  v0 .x/ D g.x; 0/; Œv0 ˇx3 D0 D 0; ˇ

 @v0ˇ ˇˇ ˙ @v03 C D bˇ .x 0 ; 0/;   ˇ @xˇ @x3 x3 D0

ˇ D 1; 2;

x 0 2 R2 ;

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

h

fˇ .x; 0/ 

1179

iˇ 1 @p.x; 0/ ˇ ˙ C  v .x/ D 0; ˇ 0ˇ x3 D0 ˙ @xˇ

ˇ D 1; 2;

are assumed to hold, and p0 .x/ D p.x; 0/ must be a solution of the problem  ˇ 1 p0 D r  f C  ˙ g  Dt g ˇtD0 in R3 [ R3C ; ˙  h ˇ @v03 iˇˇ p0 ˇx3 D0 D 2˙  b3 jtD0 ; ˇ @x3 x3 D0 h 1 @p iˇ

ˇ 0 ˇ D f3 jtD0 C  ˙ v03 / ˇx3 D0 ˇ ˙  @x3 x3 D0 in a weak sense. Then, problem (85) with  > 0 has a solution with the properties v 2 ˛ C2C˛;1C 2 .DT3 /, p 2 C . ;1C˛/ .DT3 /, rp 2 C ˛;˛=2 .DT3 /, and .2C˛;1C ˛2 /

jvjD 3

.˛; ˛2 /

C jrpjD 3 C

T

T

. ;1C˛/

C hHiD 3

T

. ;1C˛/ hpiD 3 T



C jv0 j[R3 C jbˇ jR2

T

˙

.2C˛;0/

T

.1C˛; 1C˛ 2 /

.2C˛/

.˛; ˛ /

6 c jfjD 3 2 C jRjD 3

T

.1C˛;0/

C jb3 jR2

T

.˛;0/

C jDt RjD 3

T

. ;1C˛/

C hb3 iR2

T

.˛; ˛ / C jBjR2 2 : T

The vector field v is uniquely determined, while the pressure function p is defined in the class of functions of weak power growth in x up to a bounded function of time.

5.3

Global Solvability

Global solvability of problem (76), (78), (3) without taking surface tension into account was proved in [12]. Theorem 12 (Global existence theorem for two fluids,  D 0). Let f; rf 2 1C˛

C ˛; 2 .Q1 /, v0 2 C 2C˛ .˝0 [ ˝0C /, and  2 C 2C˛ for ˛; 2 .0; 1/, < ˛, and let the compatibility conditions (81), (82) with  D 0 be satisfied. Moreover, the data are assumed to be small enough, i.e.,

jv0 j

.2C˛/ [˝0˙

.˛;

1C˛

2

Cjeb0 t fjQ1

/

.˛;

1C˛

2

Cjeb0 t rfjQ1

/

Z

1

keb0 t fk˝ dt 6 "  1;

C

(93)

0

where b0 D minf C ;   g=.2c0 / and c0 is the constant from Korn inequality (101).

1180

V.A. Solonnikov and I.V. Denisova

Then problem (76), (78), (3) with  D 0 is uniquely solvable on an infinite time interval t > 0. The solution .v; p/ has the properties: v 2 C2C˛;1C˛=2 , p 2 C . ;1C˛/ , rp 2 C˛;˛=2 , the surface t lies in C 2C˛ -class. This means that for every t0 2 .0; 1/, the solution .u; q/ and its derivatives in the Lagrangian coordinates belong to the corresponding spaces over D.t0 ;t0 C0 /  [Q.t˙0 ;t0 C0 / for a small enough time interval .t0 ; t0 C 0 / and .2C˛;1C˛=2/ jujD.t ;t C / 0 0 0

Z1 C

b0 

e

.˛;˛=2/

C jrqjD.t

. ;1C˛/

0 ;t0 C0 /

b0 t

kf.;  /k˝ d C je

C jqjD.t

0 ;t0 C0

1C˛

2

.˛; fjQ1

/

 .2C˛/ b0 t0 jv0 j ˙ 6 c.ı;  /e 0 / [˝0

b0 t

C je

1C˛

2

.˛; rfjQ1

/

:

(94)

0

The following theorem is proved in [18] for f D 0 and in [13] for small f. Theorem 13 (Global existence theorem for two fluids,  > 0). Let the assumptions of Theorem 10 with q0 D p1 .x; 0/ hold, and let  be given for t D 0 by the equation jxj D R

 x  ;0 jxj

on the unit sphere S1 . In addition, one assumes that the inner domain ˝0C is close to a ball of the same volume and the data of problem (76), (78) are small enough, i.e., .˛;

1C˛

2

jebt fjQ1 jv0 j

.2C˛/ [˝0˙

/

.˛;

1C˛

2

C jebt rfjQ1 .3C˛/

C jr0 jS1

/

C kebt fkQ1 "; (95)

"  1;

where r0 .x=jxj/ D R.x=jxj; 0/  R0 , and R0 is the radius of the ball BR0 , j˝0C j D 4R03 =3. Then, problem (76), (78), (3) is uniquely solvable in the whole positive halfaxis ft > 0g, the solution .v; p1 / possesses the properties v 2 C2C˛;1C˛=2 , p1 2 C . ;1C˛/ , rp1 2 C˛;˛=2 , and the boundary t is given for every t by a function R.; t / of the class C 3C˛ : 

xh jx  h.t/j D R ;t jx  hj



.where h.t/ is a position of the barycenter of ˝tC at the instant t ). Moreover, t tends to a sphere SR0 .h1 / of radius R0 with center at a certain point h1 , and the pressure is defined up to a bounded function of time. This means that for any

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1181

t0 2 .0; 1/, the solution .u; q/ and its derivatives in the Lagrangian coordinates belong to the corresponding Hölder spaces over D.t0 ;t0 C0 /  [Q.t˙0 ;t0 C0 / for a sufficiently small time interval .t0 ; t0 C 0 /. Moreover, the estimate .2C˛;1C˛=2/

N.t0 ;t0 C0 / Œv; p1 ; r  jujD.t C

.3C˛/

sup t2.t0 ;t0 C0 /

jr.; t /jS1

0 ;t0 C0 /

.˛;˛=2/

C jrqjD.t

0 ;t0 C0 /

. ;1C˛/

C jqjD.t

0 ;t0 C0 /

1C˛

1C˛

˚ .˛; / .˛; / 6 cebt0 jebt fjQ1 2 C jebt rfjQ1 2

C kebt fkQ1 C jv0 j

.2C˛/ [˝0˙

.3C˛/ 

C jr0 jS1

;

(96)

where r.!; t / D R.!; t /  R0 , ! 2 S1 , holds. Theorems 12, 13 guarantee solution stability understood in the sense that the velocity vector field differs a little from zero as well as the pressure function is close to a step function for small enough initial data and mass forces as the time t tends to infinity. In addition, in the case of strictly positive surface tension, the limit interface as t ! 1 is a sphere SR0 .h1 / of the radius R0 ; however, the center h1 of the limit sphere may be displaced with respect to the origin, the barycenter of ˝0C , for arbitrarily small initial velocity and mass forces. This displacement will be evaluated by inequality (119) at the end of the Section. There will be also an estimate of the initial distance between the outer boundary and fluid interface sufficient for preventing the intersection of the surfaces in the future. The proof of Theorem 13 consists of several steps. It is based on an energy inequality of a local solution, which implies an exponential decay of the solution. Theorem 12 is demonstrated in a similar way.

5.3.1 Energy Estimate with Positive Surface Tension Coefficient At first, the exponential estimate for the solution of nonlinear problem (76), (78), (3) is proved in L2 by using the notion of generalized energy E introduced by Padula [32], [33]. Let the solution exist in the interval Œ0; T  (this is guaranteed by the local existence theorem (Theorem 10)). The barycenter trajectory of the drop ˝tC is given by Zt Z Z 1 1 h.t / D x dx D v.x;  / dx d : (97) j˝tC j j˝tC j ˝tC

0 ˝C 

Proposition 1 (Energy inequality,  > 0). It is assumed that v0 satisfies compatibility conditions (81). Moreover, let the deviation function r.!; t / of t from the sphere SR0 .t /  SR0 .h.t // D fjx  h.t/j D R0 g be such that jr.!; t /jS1 .0;T / C jrS1 r.!; t /jS1 .0;T / 6 ı1 R0  1:

(98)

1182

V.A. Solonnikov and I.V. Denisova

Then the classical solution of problem (76), (78), (3) defined in Œ0; T  is subject to the inequalities n o kv.; t /k2˝ Ckr.; t /k2W 1 .S /6 ce2bt keb fk2QT Ckv0 k2˝Ckr0 k2W 1 .S / ; 2

1

2

1

ZT 

o n  kv.;  /k˝Ckr.; t /kW 1 .S1 / d 6 c keb fkQT Ckv0 k˝Ckr0 kW 1 .S1 / 2

2

t 2 .0; T ; (99) (100)

0

with the constants b and c independent of T . Proof. One multiplies the first equation in the Navier-Stokes system (76) by ˙ v and integrates by parts over ˝t [ ˝tC . This gives Z Z  q q 1 1 d 2  k ˙ vk2˝ C k ˙ S.v/k2 ˙ D f  v dx C  HC n  v d : [˝t 2 dt 2 R0 ˝

t

It is known (see [54]) that, in view of (9), Z

Z v  H n d D 

 t

v  .t /x d D 

d jt j: dt

t

Due to fluid incompressibility,

R

v  n d D 0, one can write

t

Z q o 1 q  d n1 f  v dx: k ˙ vk2˝ C  jt j  4R02 C k ˙ S.v/k2 ˙ D [˝t dt 2 2 ˝ ˇ Since Œvˇt D 0, the norm kvkW 1 .˝/ coincides with kvkW 1 .˝  [˝ C / . Moreover, 2 t t 2 the vector field v satisfies the Korn inequality in ˝: kvkW 1 .˝/ 6 c0 kS.v/k˝ ; 2

(101)

ˇ because of the no-slip boundary condition vˇ˙ D 0 ( see, for instance, [55]). Consequently, q o  d n1 k ˙ vk2˝ C  jt j  4R02 C c1 kvk2W 1 .˝/ 6 c2 kfk2˝ : 2 dt 2

(102)

In order to apply the Gronwall inequality to the generalized energy mentioned above, one should add to the left-hand side the term of the type jt j  4R02 . To this end, an auxiliary solenoidal vector-valued function W.x; t / is constructed in ˝. It ˇ has the properties Wˇ˙ D 0,

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

ˇ   1 Wnˇt D m.x; t / r.!; t /r.t /  m.x; t /Qr; r D 4

1183

Z r.!; t / dS1 ; m > m0 > 0; S1

where m.x; t / is a known function (see [18]), and kW.; t /k˝ 6 ckr.; t /kS1 ;

kW.; t /kW 1 .˝/ 6 ckr.; t /kW 1=2 .S / ; 2

1

2

  kDt W.; t /kW 1 .˝/ 6 c kr.; t /kW 1=2 .S / C kv.; t /kW 1 .˝/ : 2

2

1

2

Multiplication of the first equation in the Navier-Stokes system by ˙ W and integration by parts over D ˝t [ ˝tC lead to d dt

Z

Z

˙

˙ v  .Dt W C .v  r/W/ dx

 v  W dx  D

(103)

D

Z C

˙ S.v/ W S.W/ dx   2

D

Z



HC

t

2  n  W dS D R0

Z f  W dx; D

where S.v/ W S.W/ D Sij .v/Sij .W/. Finally, by multiplying (103) by small > 0 and adding to (102), one obtains  q Z   d 1 2 ˙ 2 ˙  v  W dx C  jt j  4R0 C c1 kvk2W 1 .˝/ k  vkD C

2 dt 2 D Z

˙ v  .Dt WC.v  r/W/ dx C



D

Z D

Z  HC

  S1

˙ S.v/ W S.W/ dx 2

 2 rQ dS1 6 c."1 /kfk2˝ C "1 kWk2˝ : R0

(104)

As was proved in [54] (Theorem 3), under condition (98), the inequality jt j  4R02 > c3 krk2W 1 .S 2

1/

holds with the constant c3 independent of ı1 and R0 . In addition, the surface integral is positive [18]: Z   S1

 2 HC rQ dS1 > ckrk2W 1 .S / : 1 2 R0

One has employed here the known formula H ŒR0 C r D

1 rS r 2 rS1  p1  p ; R0 C r g g

(105)

1184

V.A. Solonnikov and I.V. Denisova

where rS1 r D r0 e C

g D .R0 C r/2 C jrS1 rj2 ; rS1  u D

1 0 r e' ; sin  '

1 d 1 du' .sin  u / C sin  d sin  d'

in the spherical coordinates. Thus, the function of generalized energy defined by Z q   1 E.t / D k ˙ vk2˝ C

˙ v  W dx C  jt j  4R02 2 ˝ satisfies the inequalities     c4 kvk2˝ C krk2W 1 .S / 6 E.t / 6 c5 kvk2˝ C krk2W 1 .S / 2

1

2

1

(106)

for sufficiently small . The sum of the terms in the left-hand side of (104) except the derivative Dt E.t / is denoted by E1 .t /. For small there exists a constant b > 0 such that E1 .t / > 2bE.t /: Hence, Dt E.t / C 2bE.t / 6 ckfk2˝ and from the Gronwall lemma it follows that Z t e2b.t / kf.;  /k2˝ d  E.t / 6 e2bt E.0/ C 

0

6 ce2bt kv0 k2˝ C kr0 k2W 1 .S / C keb fk2Qt 2



1

which implies (99), in view of (106). Easy calculations lead one to (100).

t u

Corollary 1. The coordinates of the barycenter of ˝tC satisfy the inequality jh.t/j 6

c j˝tC j1=2

n

o keb fkQ1 C kv0 k˝ C kr0 kW 1 .S1 / ; 2

8t 2 Œ0; T :

Proof. From formula (97) it follows that jh.t/j 6

1 C 1=2 j˝t j

Z 0

t

kv.;  /k˝C d ;

which together with (100) implies inequality (107).

(107)

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1185

5.3.2 The Proof of Global Existence Theorem for the Case  > 0 Proposition 2. Let the solution of the problem (76), (78), (3) be defined in the interval .0; T  and let the estimate .2C˛;1C˛=2/

N.0;T / Œv; p1   ju0 jDT

.˛;˛=2/

C jrq 0 jDT

. ;1C˛/

C jq 0 jDT

6

be satisfied, where .u0 ; q 0 / is a solution of the problem (76), (78), (3) dependent on the Lagrangian coordinates. Then  1C˛

1C˛

.˛; / .˛; / N.t0 0 ;t0 / Œv; p1 ; r 6 c1 .ı; 0 / jfj[Q0 2 C jrfj[Q0 2 0

Z C

t0

0

 kv.;  /k˝ C kr.;  /kW 1 .S1 / d ;



t0 20

2

(108)

where t0 2 .0; T , 0 2 .0; t0 =2/, 0 depends on  and on the constant ı in an inequality similar to (14)I [Q00 D [ Q.t˙0 20 ;t0 / . Proof. Let an arbitrary t0 2 .0; T  be fixed, 0 2 .0; t0 =2/, and let  .t / be a smooth monotone function of t such that ( 0 if t 6 t0  20 C =2;  .t / D 1 if t > t0  20 C ; ˇ ˇ ˝ ˛.˛=2/ 2 .0; 0 , and P  d .t /=dt , ˇP .t /ˇR 6 c 1 ; P .t / R 6 c 1˛=2 : The couple .w D v ; s Dp / satisfies the system Dt w C .v  r/w   ˙ r 2 w C

1 rs D f C vP ; ˙

r  w D 0 in ˝t [ ˝tC ; t > t0  20 ; ˇ wˇtDt0 20 D 0 in [˝ 0  ˝t0 20 [ ˝tC ; 0 20 ˇ Œwˇt D 0;

ˇ  2  ˇˇ n ; ŒT.w; s/nˇt D   H C R0 t

ˇ wˇ˙ D 0:

The Lagrangian coordinates are introduced according to the formula Z t 0 u.y 0 ;  / d Xu .y 0 ; t /; y 0 2 [˝ 0 ; t > t0  20 ; xDy C

(109)

t0 20

where u.y 0 ; t / D v.Xu .y 0 ; t /; t /. The functions w and s written in the Lagrangian coordinates satisfy the system Dt w   ˙ ru2 w C

1 ru s D f.X; t / C uP ; ˙

1186

V.A. Solonnikov and I.V. Denisova

ru  w D 0 in [ ˝ 0 ; t > t0  20 ; ˇ wˇtDt0 20 D 0 in [ ˝ 0 ; ˇ ˇ wj˙ D 0; Œwˇ 0 D 0; Œ˙ ˘00 ˘ Su .w/nˇ 0 D 0;  ˇ 2  0 ˇˇ Œn00  Tu .w; s/nˇ 0 D   H C n  n 0 ; R0 0

(110)

where  0 Dt0 20 , n00 is the exterior normal to  0 , ˘00 and ˘ are projections onto the tangential planes to  0 and t . By virtue of formulas (9) and (109), the last boundary condition in (110) can be written in the form Z t Z t ˇ ˇ 0 0 ˇ ˇ w  0 d D B.y 0 ;  / d ; y 0 2  0 ; Œn0  Tu .w; s/n  0   n0  .t / t0 20

t0 20

(111)

where 2 d ˇ P /y0  n00  nˇ 0 C P Œn00  Tu .u; q/nj 0 C   . /n00  . R0 dt Z  Z t ˇ P / P 0 / d 0 wˇ 0 : C   . /n00  . u.y 0 ;  0 /d 0   n00  . 

B.y 0 ;  / D

t0 20



Rt P 0 / d 0 has been used. Here the equality . /  .t / D   . The interval 0 is chosen so small that the inequality .20 C .20 / =2 / 6 ı

(112)

holds, where ı is a constant from the analog of (14). Then an analog to Theorem 2 for two-phase fluid can be applied to problem (110), (111). An inequality similar to (15) can be obtained with lower-order semi-norms of u in the right-hand side. (One can find the details in [13, 18].) In the following lemma, the estimates are given for the lower-order semi-norms of a function in terms of the higher-order ones and the integral in t of L2 -norm of the function. Lemma 7. Let u 2 C 0; subject to the inequality .

1C˛

huit;DT2

0

/

1C˛ 2

1=2

.DT0 /, T0 > 0, 0 <  < T0 . Then the function u is

. 1C˛ /

9

6 2 huit;D2T C c  ˛ 2 0

˛

Z

T0

ku.;  /k˝ d : 0

For an arbitrary u 2 C 2C˛;1C 2 .[QT˙0 /, 0 < 1 < d i amf˝g, the inequality

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

Z n .˛; ˛ / .2C˛;1C ˛2 / ˛ 9 hruiDT 2 6 c 1 huiDT C 1 2 0

0

T0

ku.;  /k˝ d

1187

o

(113)

0

is valid. .2C˛/

The norm supt2.t0 20 ;t0 / j  jS1

of the function r.; t / .t / defined on the .3C˛/

interface is evaluated by (113) in terms of jr.!; t / .t /j!;S1 .t0 20 ;t0 / with a small Rt coefficient and of t0020 krS1 r.;  /kS1 d . The norm of r in C 3C˛ .S1 / can be estimated by substituting relation (105) into boundary condition (80). Since r is a solution of a quasi-linear elliptic equation on the interface, for its norm, the inequality .3C˛/

jr.; t /jS1

o nˇ ˇ.˛/ .2C˛/ 6 c ˇr 0 Œq.; t /j 0 ˇ 0 C ju.; t /j[˝ 0 C kr.; t /kW 1 .S1 / 2

holds [26]. Summing all these arguments, one obtains the estimate  N.t0 20 C ;t0 / Œv; p1 ; r 6 c3 .ı/ ." C 0 /N.t0 20 C ;t0 / Œv; p1 ; r 2

~

C c



1C˛

2

.˛; jfj[Q0 0

/

Z Cjrfj[Q00 Cc."/

t0 t0 20







ku.;  /k˝ Ckr.;  /kW 1 .S1 / d 2

(114) with small " and some ~ > 0. The inequality (114) is rewritten by means of the function ˚. / D ~ N.t0 20 C ;t0 / Œv; p; r in the following way: ˚. / 6 ." C 0 /c4 ˚. =2/ C K;

c4 D c3 .ı/2~ ;

(115)

where  Zt0  1C˛

.˛; 2 / 0 Cjrfj[Q0 Cc."/ ku.;  /k˝ C kr.;  /kW 1 .S1 / d : K D c3 .ı/ jfj[Q0 2

0

t0 20

Let " C 0 D 2c14 . Then by iterations with =2,. . . , =2k , one deduces from (115) that ˚. / 6 2K in the limit as k ! 1. This estimate with D0 implies (108). t u

1188

V.A. Solonnikov and I.V. Denisova

Proof (of Theorem 13 on global existence). By Theorem 10, there exists a solution .v; p1 / of system (79), (80) in .0; T0 , T0 > 1, if " 6 ".T0 / in (83). For .v; p1 / estimate (84) holds; therefore   .˛; 1C˛ / .˛;0/ .2C˛/ .3C˛/ N.0;T0 / Œv; p1  6 c jfjQT 2 C jrfjQT C jv0 j˝0 C jr0 jS1 6 : 0

0

Then by Proposition 2, there exists 0 < T0 =2 such that inequality (112) is satisfied, and for .v; p1 /, T0 estimate (108) holds. The application of Proposition 1 yields n 1C˛

1C˛

.˛; / .˛; / C jebt rfj[Q0 2 C kebt fk[Q00 N.t0 0 ;t0 / Œv; p1 ; r 6c5 eb.t0 20 / jebt fj[Q0 2 0 0 o C kv0 k˝ C kr0 kW 1 .S1 / ; (116) 2

where j˝j is the measure of ˝, and t0 2 .20 ; T0 . For t0 D T0 , (116) implies the estimate .2C˛/

jv.; T0 /j˝0

n 1C˛

1C˛

.˛; / .˛; / 6 c6 ebT0 jebt fj[Q0 2 C jebt rfj[Q0 2 0 0 o .2C˛/ .3C˛/ C jv0 j˝0 C jr0 jS1 :

.3C˛/

C jr.; T0 /jS1 C kebt fk[Q00

One may choose T0 so large that the constant c6 ebT0 < 1=2 because of the smallness of " in assumptions (95). After fixing this value of T0 , one can make  still smaller, on the account of ", to satisfy the condition (112). It is worth noting that this does not change the value of 0 . Thus, it has been proved that for t D T0 the norm of the solution is smaller than at the initial instant. One can apply local existence theorem again and obtain the solution corresponding to the initial values v.; T0 /, r.; T0 /, at least on the interval .T0 ; 2T0 . While repeating our argument for obtaining the exponential estimates, it is necessary to make the transformation to the Lagrange coordinates according to the formula Zt .1/ Q .1/ ;  / d; y .1/ 2 ˝T0 ; t 2 .T0 ; 2T0 /: X D y C u.y (117) T0

In fact, it follows from the additivity of the integral that the formula (117) coincides with (4): ZT0 Zt X.y; t / D y C u.y;  / d C u.y;  / d; y 2 ˝0 ; t 2 .T0 ; 2T0 /; 0 .1/

Q ;  / D u.y;  /. because u.y

T0

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1189

The same remark applies to the radius vector of internal fluid barycenter, since the volume of the fluid is conserved: Zt

1 j˝C j

h.t / D h.T0 / C T0

ZT0 D 0

1 j˝C j

3 D 4R03

Z v.x;  / dx d ˝C

Zt

Z v.x;  / dx d C

T0

˝C

1 j˝C j

Z v.x;  / dx d ˝C

Zt Z v.x;  / dx d: 0 ˝C 

By Corollary 1, one can conclude that jh.t /j 6 a;

t 6 2T0 ;

(118)

where by (107) aD

c7 j˝0C j1=2

n

o kebt fkQ1 C kv0 k˝ C kr0 kW 1 .S1 / : 2

The solution of system (76), (78), (3) can be extended in this way with respect to t as far as necessary; moreover, it satisfies inequality (96). The limiting position of the barycenter is estimated from (118):   .2C˛/ .3C˛/ 6 2c8 ": jh1 j 6 a 6 c8 kebt fkQ1 C jv0 j ˙ C jr0 jS1 [˝0

(119)

Inequality (119) means that " is to be chosen so small that the initial distance between the surfaces  and ˙ is strictly larger than 2c8 " C ı1 R0 with ı1 ; R0 from (98), in order to avoid the intersection of these surfaces in the future. The uniqueness of a solution follows from the local existence and uniqueness theorem (Theorem 10). t u Remark 8. In the case of  D 0, one can estimate from above the distance between the solid boundary and the interface by evaluating the integral I  R1 ju.; t /j˝ C dt . In virtue of (94), 0 0

ju.; t0 /j˝ C 6 c.ı; 0 /eb0 t0 "; 0

(120)

1190

V.A. Solonnikov and I.V. Denisova

where " is the constant from condition (93). By using (84) with  D 0 and integrating (120) with respect to t0 > T20 , one arrives at T0 I 6 jujQC C T0 =2 2

Z

1

T0 =2

ju.; t /j˝ C dt 6 c 0

T0 1 " C c.ı; 0 / " 6 c9 ": 2 b0

(121)

Thus, if the initial distance between the surfaces is greater than c9 ", t will never intersect ˙ . An estimate similar to (121) is valid for problem (76), (78), (3) with " from (95), b0 D b, and the other constants from inequality (96).

6

Equilibrium Figures

6.1

Introduction

Problem (1), (2), and (3) with pe D 0, f D 0 governing the motion of an isolated mass of a viscous incompressible capillary fluid with the unit density has a solution, corresponding to a rigid rotation of the fluid with a constant angular velocity !, which is given by V.x/ D !3 .x/;

P .x/ D

!2 0 2 jx j C p0 ; 2

(122)

where 3 .x/ D e3  x D .x2 ; x1 ; 0/; x 0 D .x1 ; x2 /, p0 D const dependent on the mass of the fluid. The boundary condition reduces to a single equation that defines the domain F, called equilibrium figure of the rotating fluid:  H.y/ C

!2 0 2 jy j C p0 D 0; 2

y 2 G  @F;

where H is the doubled mean curvature of G. For simplicity, we assume that F is rotationally symmetric with respect to the x3 -axis, in this case (122) is a stationary solution of (1), (2), and (3). The construction of equilibrium figures homeomorphic to a ball or a torus can be found in [3]. The problem of stability of the solution (122) reduces to the analysis of a free boundary problem for the perturbations vr D v  V, pr D p  P (r means “relative”). The couple of perturbed velocity and pressure .v; p/ is a solution of (1), (2), and (3) with a small v0 and with 0 D @˝0 close to G. Since the solution of (1),(2), and (3) satisfies the relations Z Z Z v.x; t / dx D v0 .x/ dx; v.x; t /  i .x/ dx ˝t

˝0

Z D

˝0

˝t

v0 .x/  i .x/ dx;

i D 1; 2; 3;

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

1191

(conservation of momenta), where i D ei  x, ei D fıij gj D1;2;3 , it is natural to assume that Z Z Z Z v0 .x/ dx  V.x/ dx D 0; v0 .x/  i .x/ dx  V.x/  i .x/ dx; ˝0

Z

F

j˝0 j D jFj;

F

˝0

xi dx D 0;

i D 1; 2; 3:

˝0

Then Z

Z v.x; t / dx D 0; ˝t

˝t

Z v.x; t /  i .x/ dx D !

F

3  i .x/ dx;

(123)

Z xi dx D 0;

i D 1; 2; 3;

8t > 0:

˝t

Since S.3 / D 0 and r  3 D 0, the substitution of v D vr C V; p D pr C P into (1), (2), and (3) gives 8 ˆ Dt vr C .V  r/vr C .vr  r/V C .vr  r/vr  r 2 vr C rpr D 0; ˆ ˆ ˆ ˆ ˆ ˆ 0; r

t

ˆ ˆ vr jtD0 D v0  V in ˝0 ; ˆ ˆ ˆ ˆ ˆ 2 : T.vr ; pr /n D . H C !2 jx 0 j2 C p0 /n;

Vn D .vr C V/  n

(124)

on t :

It is convenient to write problem (124) in the coordinate system rotating about the x3 -axis with the angular velocity !. This is achieved by the change of the independent variables and unknown functions according to the formulas x D Z .!t /z;

w.z; t / D Z

1

.!t /vr .Z .!t /z; t /;

q.z; t / D pr .Z .!t /z; t /;

where 1 cos   sin  0 Z . / D @ sin  cos  0 A : 0 0 1 0

One multiplies (124) by Z

1

i .Z .!t /z/ D Z .!t /i .z/; Z

from the left and takes into account that

1

Z

Z

.!t / ˝t

3 .x/ dx D

˝t0

3 .z/ dz; Z

1

D Z T:

1192

V.A. Solonnikov and I.V. Denisova

This yields 8 ˆ Dt w C 2!.e3  w/ C .w  r/w  r 2 w C rq D 0; ˆ ˆ ˆ 0; t t ˆ wjtD0 D v0  V  w0 in ˝0 ; ˆ ˆ ˆ 2 : T.w; q/n0 D n0 . H 0 C !2 jz0 j2 C p0 /; Vn0 D w  n0

(125) on t0  @˝t0 ;

where Vn0 is the velocity of evolution of t0 in the direction of the normal n0 D Z 1 n to t0 and H 0 is the doubled mean curvature of t0 (see [68]). If t D fx D X . ; t /g, then t0 D fz D Z 1 X  Zg and Vn0 D Dt Z  n0 D Dt .Z

1

X/  Z

1

dZ 1 Z Z  n0 dt

n D Dt X  n C

D Vn  !3 .z/  n0 D .vr C V/  n  !3 .z/  n0 D vr  n D w  n0 : Equations (123) are converted into Z

Z ˝t0

w.z; t / dz C !

Z

˝t0

˝t0

3 .z/ dz D 0;

w.z; t /  i .z/ dz C ! 

!ıi3

Z F

(126)

Z ˝t0

Z 3 .z/  i .z/ dz D !

F

3 .z/  i .z/ dz

.z21 C z22 /dz;

i D 1; 2; 3;

j˝t0 j D j˝0 j D jFj;

Z

Z ˝t0

z dz D

z dz D 0:

(127)

˝0

Problem (125) can be reduced to a nonlinear problem in a fixed domain F by applying the Hanzawa transformation. Let t0 D fz D y C N.y/.y; t /; y 2 Gg;

(128)

where N is the normal to equilibrium figure surface G, .y; t / is the function of deviation of the free boundary t0 from G. It is given for t D 0 and unknown for t > 0. The Hanzawa transformation is given by z D y C N .y/ .y; t /  e .y/ W F ! ˝t0 ;

y 2 F;

(129)

where N .y/ is a smooth extension of N into F such that N .y/ D N.y/ near G (y is the point of G closest to y) and  is an extension of  satisfying the condition

24 Classical Well-Posedness of Free Boundary Problems in Viscous. . .

@ .y; t / D 0 for @N

y 2 G;

1193

(130)

and inequality (160). Hence,  is small for small , and transformation (129) is invertible. Introducing w0 .y; t / D w.z.y/; t /; q 0 .y; t / D q.z.y/; t / and then omitting the primes, one can rewrite (125) in the form 8 ˆ Dt w C 2!.e3  w/  r 2 w C rq D l1 .w; q; /; r  w D l2 .w; / in F; t > 0; ˆ ˆ ˆ ˆ ˆ 0. Assume to the contrary that (19) has a nontrivial solution .u; h/ for  > 0. Let 1;k denote the components of 1 and set †k D @1;k . By the divergence theorem, Z Z Z 0D div u dx D .uj† / d †k D  h d †k : 1;k

†k

†k

This shows that the mean values of h vanish for all components of †. As A† is positive semi-definite on functions which have mean value zero for each component of †, (20) implies  D 0. Hence there are no eigenvalues with nonnegative real part. Concerning the eigenvalue problem (18) one obtains, after taking the L2 inner product of the first line with #, integrating by parts, and employing the condition ŒŒd @ # D 0, the following relation

1238

G. Simonett and M. Wilke 1=2

0 D j.%  /1=2 #j2L2 ./ C jd r#j2L2 ./ :

(21)

This readily shows that all eigenvalues of (18) are real and nonpositive. (d) Suppose  D 0. Then (20) and (21) yield 1=2

1=2

j D.u/j2L2 ./ D jd r#j2L2 ./ D 0; hence # is constant and D.u/ D 0, and then u D 0 by Korn’s inequality and the no-slip condition u D 0 on @. This implies further that the pressures ˇ are constant in the components of the phases. From the relation ŒŒ ˇ† D k A†k hk , one concludes that A†k hk is constant for k D 1;    ; m, where hk D hj†k . The kernel of the linearization L is spanned by e D .0; 1; 0/, ej k D j j .0; 0; Yk /, with Yk the spherical harmonics of degree one for the spheres †k , j D 1;    ; n, k D 1;    ; m, and e0k D .0; 0; Yk0 /, where Yk0 equals one on †k and zero elsewhere. Hence the dimension of the null space N.L/ is m.nC1/C1. Next it will be shownP that  D 0 is semi-simple. So suppose .u; #; h/ is a P solution of L.u; #; h/ D j;k ˛j k ej k C k ˇk e0k C  e . This means  u C r D 0

in

 n †;

div u D 0

in

 n †;

uD0

on

@;

ŒŒu D 0

on

†;

on

†;

on

†;

ŒŒT † C  .A† h/† D 0 X X j ˛j k Y k C ˇk Yk0 .uj† / D j;k

(22)

k

and  d # D %  

in

 n †;

@ # D 0

on

@;

ŒŒ# D 0

on

†;

ŒŒd @ # D 0

on

†:

(23)

It is to be shown that .˛j k ; ˇk ;  / D 0 for all j; k. Integrating the divergence equation for u over 1;k yields Z

Z div u dx D

0D 1;k

.uj† / d †k D  †k

X k

Z ˇk †k

Yk0 d †k D ˇk j†k j;

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

1239

where the property that the spherical harmonics have mean value zero is employed. Therefore, ˇk D 0 for k D 1;    ; m. Taking the L2 -inner product of the equation for u with u, one obtains as in (19) 1=2

j D.u/j2L2 ./ D 0: This implies D.u/ D 0; hence u D 0 by Korn’s inequality and the no-slip boundary condition on @. This, in turn, yields 0 D .uj† / D

X

j

˛j k Y k :

j;k j

Thus ˛j k D 0 for all j; k, as the spherical harmonics Yk are linearly independent. Finally, integrating the equation for # yields Z

Z

.  j%/L2 ./ D 

Z

d # dx D 

ŒŒd @ # d †  †

d @ # d .@/ D 0; @

and hence  D 0. Therefore, the eigenvalue  D 0 is semi-simple. (e) The assertion follows from the fact that Te E  N.L/ and the relation dim N.L/ D dim Te E D m.n C 1/ C 1: t u

4

Nonlinear Stability of Equilibria

Suppose e D .0;  ; †/ 2 E is a nondegenerate fixed equilibrium. Choosing † as reference manifold and employing the Hanzawa transform to problem (1), (2) and (3), one obtains the nonlinear system %@t u   u C r D Fu .u; #; h; /

in

 n †;

in

 n †;

uD0

on

@;

ŒŒu D 0

on

†;

P† ŒŒ2 D.u/† D G .u; #; h/

on

†;

.ŒŒ2 D.u/† / j† / C ŒŒ C A† h D G .u; #; h/

on

†;

in

;

div u D Gd .u; h/

u.0/ D u0

(24)

1240

G. Simonett and M. Wilke

%  @t #  d # D F .u; #; h/

in

 n †;

@ # D 0

on

@;

ŒŒ# D 0

on

†;

ŒŒd @ # D G .#; h/

on

†;

#.0/ D #0

in

 n †;

(25)

and @t h  .uj† / D Fh .u; h/

on †; (26)

h.0/ D h0

on †;

where #,  ,  , d , and A† have the same meaning as in Sect. 3. The precise expressions for the nonlinearities will not be listed here, and the reader is referred to [30, Chap. 9] (and also to [17] for the isothermal case). It suffices to point out that the nonlinearities are C 1 in all variables and vanish, together with their firstorder derivatives, at .u; #; h; / D .0; 0; 0; c/, where c is constant in the phase components. To obtain an abstract formulation of problem (24), (25) and (26), we choose as the principal system variable z D .u; #; h/. The regularity space for z is ˚  E.a/WD .u; #; h/ 2 Eu .a/  E .a/  Eh .a/ W u; @ #D0 on @; ŒŒu ; ŒŒ# D0 on † ; where Eu .a/ D E .a/n ;

E .a/ D Hp1 .J I Lp .// \ Lp .J I Hp2 . n †//;

Eh .a/ D Wp21=2p .J I Lp .†// \ Hp1 .J I Wp21=p .†// \ Lp .J I Wp31=p .†// and J D .0; a/. Here the regularity for the height function h is accounted for as follows. Asserting that u 2 Hp1 .J I Lp .// \ Lp .J I Hp2 . n †//, one obtains by trace theory .ŒŒ2 D.u/† / j† / 2 Wp1=21=2p .J I Lp .†// \ Lp .J I Wp11=p .†//; .uj† / 2 Wp11=2p .J I Lp .†// \ Lp .J I Wp21=p .†//: Requiring that the function h in (24), (26) has the best possible regularity then amounts to † h 2 Wp1=21=2p .J I Lp .†// \ Lp .J I Wp11=p .†//; @t h 2 Wp11=2p .J I Lp .†// \ Lp .J I Wp21=p .†//;

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

1241

1=21=2p

which in turn results in h 2 Eh .a/, as Eh .a/ embeds into Wp

.J I Hp2 .†//. By 1=21=2p 11=p .J I Lp .†//\Lp .J I Wp .†//. Wp

the same reasoning one also has ŒŒ 2 The trace space X of E.a/ is given by

˚ X D .u; #; h/ 2 Wp22=p . n †/nC1  Wp32=p .†/ W u; @ # D 0 on @

 ŒŒu ; ŒŒ# D 0 on † :

Finally, let O P p1 . n †//g; E.a/ D fw D .z; / W z 2 E.a/;  2 Lp .J I H XO  D fw D .z; / W z 2 X ;  2 WP p12=p . n †/; ŒŒ 2 Wp13=p .†/g; P p1 . n †/ and WP p12=p . n †/ denote corresponding homogeneous spaces. where H

4.1

The Tangent Space at Equilibria

In this subsection, the structure of the state manifold SM in a neighborhood of a fixed equilibrium e D .0;  ; †/ will be studied. Toward this objective, observe that near e , the state manifold is described by ˚ SM D .u; #; h/ 2 X W div u D Gd .u; h/ in  n †;

  P† ŒŒ2 D.u/† D G .u; #; h/; ŒŒd @ # D G .#; h/ on † :

Associated to SM is the linear subspace ˚ SX D .u; #; h/ 2 X W div u D 0 in  n †;

 P† ŒŒ D.u/† D 0; ŒŒd @ # D 0 on † I

the boundary trace space 12=p

Y D Wp;0

. n †/  Wp13=p .†I T †/  Wp13=p .†/;

where 12=p Wp;0 .

 n †/ WD v 2

Wp12=p .



Z n / W

v dx D 0 ; n†

with T † the tangent bundle of †; the linear stationary boundary operator Bz D .div u; P† ŒŒ2 D.u/† ; ŒŒd @ # /;

1242

G. Simonett and M. Wilke

and the stationary boundary nonlinearity G .z/ D .Gd .u; h/; G .u; #; h/; G .#; h//:

With this notation one has SM D fz 2 X W Bz D G.z/ in Y g; SX D fz 2 X W Bz D 0 in Y g:

(27)

This structure will now be employed to parameterize SM over SX near .0; 0; 0/. This shows, in particular, that SX is isomorphic to the tangent space of SM at .0; 0; 0/ or, equivalently, to the tangent space of SM at e . It will be convenient to enlarge the system variable z by the pressure, setting 12=p 13=p w D .z; /, where  2 WP p . n †/ and ŒŒ 2 Wp .†/. Furthermore, including the normal component of the normal stress balance .ŒŒ2 D.u/† / j† / C ŒŒ C  A† h D G .u; #; h/ O . It is worthwhile to point out that here the leads to the extended operators BO and G O. pressure appears only linearly, i.e., it does not appear in G The differential operator A is defined by the expression for the operator L introduced in Sect. 3. Consequently, system (24), (25) and (26) can be restated as @t z C Aw D F.w/ in O .z/ Ow D G B

 n †;

on

(28)

†;

z.0/ D z0 ; where F.w/ WD .Fu .w/=%; F .z/=%  ; Fh .z//. It is important to note that, formally, O .0/; G O 0 .0// D 0, with F0 and G O 0 the Fréchet derivative of F .F.0/; F0 .0// D 0 and .G O , respectively. and G

4.2

Parameterization of SM.

In order to parameterize SM over SX , one solves the problem !z C A0 w D 0 O .z C zQ/ Ow D G B

in

 n †;

on

†;

(29)

where ! > 0 is sufficiently large and A0 w WD .. =%/u C .1=%/r; .d =%  /#; 0/;

w D .u; #; h; / 2 XO  :

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

1243

Given zQ 2 SX small, we are looking for a solution w 2 XO  . For this, the implicit function theorem will be employed. Obviously, for zQ D 0 one has the trivial solution O W X ! YO w D 0. Moreover, one notes that the first line in (29) yields h D 0. As G 13=p 0 1 O O O is of class C with .G.0/; G .0// D 0, where Y D Y  Wp .†/, it is to be shown that the linear problem !z C A0 w D 0

in

 n †;

O w D gO B

on

†;

(30)

admits a unique solution, for any given datum gO 2 YO . In fact, the propositions in the next subsection will do this job, up to lower order perturbations. Therefore, one may apply the implicit function theorem to find a ball BSX .0; r/ and a map

O W BSX .0; r/ ! XO  O O z/ is the unique solution of class C 1 with . .0/;

O 0 .0// D 0 such that w D .Q of (29) near zero. The map .id C / W BSX .0; r/ ! SM is surjective onto a O To see this fix neighborhood of zero, where means dropping the pressure  in . O any z 2 SM and solve the linear problem (30) with gO D G.z/ to obtain a unique w D .z; / 2 XO  . Let then zQ D z  z. If z is chosen small enough, zQ 2 BSX .0; r/ O z/ by uniqueness, yielding the representation z D zQ C .Qz/. and hence w D .Q Consequently, the map ˆ W BSX .0; r/ ! SM defined by ˆ.Qz/ D zQ C .Qz/

(31)

yields the desired parameterization. In conclusion, the following result has been obtained. Theorem 4. The state manifold SM can be parameterized via the map ˆ over the space SX . In particular, the tangent space Te SM at the equilibrium e , and equivalently the tangent space T0 SM at zero, is isomorphic to the space SX . Note that an equilibrium e1 2 E close to e 2 E in SM, respectively, z1 close to zero in X , decomposes as z1 D zQ1 C z1 D zQ1 C .Qz1 /; with zQ1 2 SX . This follows as Aw1 D F.w1 / D 0 at an equilibrium.

1244

4.3

G. Simonett and M. Wilke

Auxiliary Linear Elliptic Problems

For the application of the implicit function theorem in Sect. 4.2, the following results were employed. The first one concerns an elliptic transmission problem for the temperature and the second one a two-phase Stokes problem. Proposition 1. Let ! > 0 be large, %;  ; d > 0, and p > nC2. Then the problem %  !#  d # D 0

in

 n †;

@ # D 0

on

@;

ŒŒd @ # D g

on

†;

ŒŒ# D 0;

22=p

has a unique solution # 2 Wp

13=p

. n †/ if and only if g 2 Wp

.†/.

The assertion follows, for instance, from the results in [30, Sect. 6.5]. Proposition 2. Let ! > 0 be large, %;  > 0, and p > n C 2. Then the problem %!u   u C r D 0

in

 n †;

in

 n †;

on

@;

P† ŒŒ2 D.u/† D g

on

†;

.ŒŒ2 D.u/† j† / C ŒŒ D g

on

†;

div u D gd uD0 ŒŒu D 0;

has a unique solution u 2 Wp22=p . n †/;  2 WP p12=p . n †/; ŒŒ 2 Wp13=p .†/; if and only if 12=p

gd 2 Wp;0

. n †/;

.g ; g / 2 Wp13=p .†I T †  R/:

The assertion follows, for instance, from [30, Chap. 8].

4.4

Nonlinear Stability Analysis

In oder to analyze the stability properties of an equilibrium e D .0;  ; †/, the time-dependent variables are decomposed in the same way as in the previous section into z.t / D z.t / C zQ.t / and w.t / D w.t / C w.t Q /. The full problem (28) may then be decomposed into two systems, formally one for w and one for w, Q according to

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

.! C @t /z C Aw D F.w C w/ Q O .z C zQ/ Ow D G B z.0/ D .Qz0 /

in

 n †;

on

†;

in

;

1245

(32)

and @t zQ C Aw Q D !z

in

 n †;

Ow B Q D0

on

†;

zQ.0/ D zQ0

in

:

(33)

Adding these equations yields problem (28). One should think of this decomposition in the following way. The first part has a fast dynamics due to ! > 0 large and takes care of the stationary boundary conditions, while the second equation lives in the tangent space SX and carries the actual dynamics. There should be a word of caution. While for the initial value z0 as well as for z1 the decomposition z0 D zQ0 C .Qz0 / and z1 D zQ1 C .Qz1 / is employed, it no longer holds in the time-dependent case; in general z.t / ¤ .Qz.t //! In order to show stability and exponential convergence of solutions starting close to the equilibrium e D .0;  ; †/, the decomposition z D z C zQ C z1 is used, with the idea that z1 will be the limit of z.t / as t goes to infinity, and z; zQ are exponentially decaying. This means that the corresponding equations for z and zQ are shifted to .! C @t /z C Aw D F.w C w Q C w1 /  F.w1 / O .z C zQ C z1 /  G O .z1 / Ow D G B z.0/ D .Qz0 /  .Qz1 /

in

 n †;

on

†;

in

;

(34)

and Q D !z @t zQ C Aw Ow B Q D0 zQ.0/ D zQ0  zQ1

in

 n †;

on

†;

in

;

(35)

where z0 D zQ0 C .Qz0 / and z1 D zQ1 C .Qz1 /. It is convenient to remove the pressure Q from (35) by solving the weak transmission problem 1 .%1 r jr / Q ujr /L2 ./ C !.ujr /L2 ./ ; L2 ./ D .%  Q

ŒŒ Q D A† hQ C .ŒŒ2 D.Qu/† j† /

2 Hp10 ./; on †

1246

G. Simonett and M. Wilke

and insert it into (34) for w. Q The first problem may be written abstractly as L! w D N .w; zQ; zQ1 /;

t > 0;

z.0/ D .Qz0 /  .Qz1 /;

(36)

and with the Helmholtz projection P, the second one as the evolution equation @t zQ C LQz D !P z;

t > 0;

zQ.0/ D zQ0  zQ1 :

(37)

Here L W X1 ! X0 is the operator defined in Sect. 3. For further use the space F.a/ of data .fu ; f ; fh ; gd ; gu ; g /, F.a/ D F1 .a/  F2 .a/  F3 .a/  F4 .a/  F5 .a/  F6 .a/; is introduced, with F1 .a/ D Lp .J I Lp .//n ; F2 .a/ D Lp .J I Lp .//; F3 .a/ D Wp11=2p .J I Lp .†// \ Lp .J I Wp21=p .†//; 1 P 1 F4 .a/ D Hp1 .J I 0 H p .// \ Lp .J I Hp . n †//;

F5 .a/ D Wp1=21=2p .J I Lp .†//n \ Lp .J I Wp11=p .†//n ; F6 .a/ D Wp1=21=2p .J I Lp .†// \ Lp .J I Wp11=p .†//; 1

P 10 ./  is the dual of the homogeneous P p .// D ŒH where J D .0; a/ and 0 H p ;@ space P p10 ;@ ./ WD f 2 L1;loc ./ W r 2 Lp0 ./; D 0 on @g: H In addition, the function space Q E.a/ WD Hp1 .J I X0 / \ Lp .J I X1 /;

J D .0; a/;

(38)

will be used, with X0 and X1 as in Sect. 3. It follows from the results in Chapters 6 and 8 of [30] that O .L! ; tr/ 2 Isom.E.1; ı/; F.1; ı/  X /; provided ! is chosen sufficiently large, that is, .L! ; tr/ is an isomorphism from O E.1; ı/ onto F.1; ı/  X . Here the following notation is employed: z 2 E.1; ı/

,

e ıt z 2 E.1/;

O Q and similarly for F.1; ı/, E.1; ı/ and E.1; ı/.

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

1247

According to Theorem 3, L is the generator of an analytic C0 -semigroup with maximal Lp -regularity in X0 . Moreover, it follows from the same theorem that there is a number ı0 > 0 such that Re  .L/ \ .ı0 ; 0/ D ;. Let ı be chosen so that 0 < ı < ı0 . On shows that the function N is of class C 1 with respect to the variables O Q ı/  E.1; ı/, provided condition (4) holds, but .w; zQ/ in the function spaces E.1; merely continuous in zQ1 , unless one additional degree of regularity is imposed on the coefficients. The main theorem of this chapter is the following. Theorem 5. Let p > n C 2 and suppose that condition (4) holds. Then each equilibrium e D .0;  ; †/ 2 E is nonlinearly stable in the state manifold SM. Any solution with initial value close to e in SM exists globally and converges in SM to a possibly different stable equilibrium e1 2 E at an exponential rate. Proof. Based on the spectral properties of L derived in Theorem 3, let P s denote the projection onto the stable subspace X0s D P s X0 D R.L/ and P c the complementary projection onto the kernel N.L/ D X0c D P c X0 . Moreover, let Ls be the part of L in X0s . Let y D P s zQ, x D P c zQ and note that the equilibria over P c SX D P c X0 may be parameterized according to z1 D x1 C

.x1 / C .x1 C

.x1 //;

x1 2 X0c ;

by solving the nonlinear stationary problem Ls y D !P s P .x C y/ by the implicit function theorem. Finally, let y1 D P s to the equation for zQ, one obtains the problem @t y C Ls y D !P s P z;

t > 0;

.x1 /. Applying the projection

y.0/ D y0  y1 ;

and for x analogously @t x D !P c P z;

t > 0;

x.0/ D x0  x1 :

It is important to observe that Q ı/; Lp .RC ; ıI X0s /  Xs /: .@t C Ls ; tr/ 2 Isom.P s E.1;

(39)

Finally, the whole problem (36)–(37) may be rewritten as H.v; .x1 ; y0 // D 0, where v D .w; y; x; x0 / and

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G. Simonett and M. Wilke

3

2 6 6 H.v; .x1 ; y0 // D 6 6 4

L! w  N .v; x1 /; z.0/  .x0 C y0 / C .x1 C y1 /   @t y C Ls y  !P s P z; y.0/  y0 C y1 R1 x.t / C ! t P c P z.s/ ds R1 x0  x1 C ! 0 P c P z.s/ ds

7 7 7: 7 5

One shows that the mapping O Q Q H W E.1; ı/  P s E.1; ı/  P c E.1; ı/  X0c  .X0c  Xs / ! .F.1; ı/  X /  .Lp .RC I ı; X0s /  Xs /  Lp .RC I ı; X0c /  X0c is of class C 1 w.r.t .v; y0 /, continuous w.r.t. x1 , and differentiable w.r.t. x1 at x1 D 0. One notes that X0c D Xc D X1c . The Fréchet derivative Dv H.0; 0/ w.r.t. the variable v is given by the operator matrix 2

.L! ; tr/ 0 6  .@t C Ls ; tr/ Dv H.0; 0/ D 6 4  0  0

0 0 I 0

3 0 07 7: 05 I

(40)

Here the stars indicate bounded linear operators which, due to the triangular structure of the operator matrix, do not need to be computed explicitly, as the diagonal terms of this operator matrix are invertible. Therefore, by the implicit function theorem, see, for instance, [6, Theorem 15.1], there are balls BX0c .0; r/ and BXs .0; r/, and a continuous map O Q ı/  E.1; ı/  X0c ; T W BX0c .0; r/  BXs .0; r/ ! E.1;

T .x1 ; y0 / D .w; zQ; x0 /;

with T .0; 0/ D 0. Then .z; / WD .z C zQ C z1 ;  C Q C 1 / yields the unique solution of (28) such that z.t / ! z1 WD x1 C

.x1 / C .x1 C

.x1 // in X as t ! 1:

(41)

Q ı/ seemingly has less regularity than its One should observe that zQ WD x C y 2 E.1; counterpart z 2 E.1; ı/. However, a moment of reflection shows that any solution Q zQ 2 E.1; ı/ of problem (37) inherits the additional regularity zQ 2 E.1; ı/. It follows from the implicit function theorem that T is C 1 in y0 , but only continuous in x1 , unless more regularity for the parameter functions is required. Nonetheless, T is differentiable with respect to x1 at x1 D 0: The properties of T can be summarized as follows: given an equilibrium z1 D x1 C

.x1 / C .x1 C

.x1 // 2 E

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

1249

and an initial value y0 2 Xs , one determines with the help of the implicit function theorem a value x0 in X0c and a solution z of (28) with initial value .x0 ; y0 / such that z.t / convergences to z1 exponentially fast. Exponential convergence is obtained by setting up the implicit function theorem in a space of exponentially decaying functions. Next, the mapping S W BX0c .0; r/  BXs .0; r/ ! X0c  Xs ;

.x1 ; y0 / 7! .x0 ; y0 /

will be analyzed in more detail. It is worthwhile to point out that this mapping gives rise to the construction of a stable foliation for (28) (by invariant, locally stable manifolds) in a neighborhood of the fixed equilibrium e 2 E; see [31]. To complete the proof, the question which remains is whether S is surjective near .0; 0/. Indeed, surjectivity would imply that for any initial value .x0 ; y0 / in a sufficiently small neighborhood of .0; 0/ in X0c  Xs , there exists z1 2 E and a unique solution z to problem (28) which converges to z1 at an exponential rate in the topology of SM. To prove surjectivity of S , degree theory will be employed. For this purpose, define a map f W BX0c .0; r/BXs .0; r/ ! X0c by means of f .x1 ; y0 / D x0 .x1 ; y0 /. As has already been established, this map is continuous, and it is close to the identity. In fact, differentiating the relation H1 .T .x1 ; y0 /; T .x1 ; y0 // D 0 with respect to .x1 ; y0 / at .0; 0/, one obtains .D1 T1 .0; 0/; D2 T1 .0; 0// D 0: Here H1 denotes the first line of H and T1 the first component of T , respectively. This implies .D1 z.0; 0/; D2 z.0; 0// D 0: From the representation Z

1

P c P z ds;

f .x1 ; y0 / D x1  ! 0

one infers that for every " > 0 there is a constant  > 0 such that Z jf .x1 ; y0 /  x1 jX0c !

0

1

jP c P zjX0c ds ".jy0 jXs C jx1 jX0c /;

whenever j.x1 ; y0 /j ; with  r. In the following, let " D 1=3 be fixed. Here y0 only serves as a parameter, so we are in a finite dimensional setting and may employ the Brouwer degree, in particular its homotopy invariance. Define the homotopy h.; x; y0 / D f .x; y0 / C .1  /x, and consider the degree deg .h. ; ; y0 /; BX0c .0; r/; /;

.; y0 / 2 BX0c .0; =2/  BXs .0; =2/:

For  D 0 it is equal to one; hence it is equal to one for all  2 Œ0; 1 , provided there are no solutions of h.; x; y0 / D  with jxjX0c D r. To show this, suppose

1250

G. Simonett and M. Wilke

h.; x; y0 / D ; i.e.,   x D .f .x; y0 /  x/, and jxjX0c D r. Then by the above estimate r D jxjX0c jjX0c C jx  jX0c jjX0c C ".jy0 jXs C jxjX0c / < r; provided jjX0c < =2 and jy0 jXs < =2: Hence, deg.f .; y0 /; BX0c .0; r=2/; / equals one as well, showing that the equation f .x1 ; y0 / D  has at least one solution for each .; y0 / 2 BX0c .0; =2/BXs .0; =2/, i.e., the mapping is surjective near zero. This completes the proof of the theorem. t u Remark 2. It should be noted that the proof of surjectivity can be based on the inverse function theorem (in lieu of employing degree theory), provided the mappings involved are C 1 in all variables. This property can be ensured by asking for one more degree of regularity for the functions i and di ; i in condition (4).

5

Global Existence and Convergence

It has been shown in Sect. 2 that the negative total entropy is a strict Lyapunov functional for (1), (2) and (3). Therefore, the !-limit sets ˚ !.u;  ; / WD .u1 ; 1 ; 1 / 2 SM W  9 tn % 1 s.t. .u.tn /; .tn /; .tn // ! .u1 ; 1 ; 1 / in SM ; of solutions .u;  ; / in SM are contained in the manifold E  SM of equilibria. There are several obstructions for global existence: • Regularity: the norms of either u.t /, .t /, or .t / may become unbounded; • Geometry: the topology of the interface may change; or the interface may touch the boundary of ; or a part of the interface may contract to a point. Let    be a hypersurface. Then  satisfies a ball condition if there is a number r > 0 such that for each point p 2  there are balls B.xi ; r/  i such N i ; r/ D fpg for i D 1; 2. The subset MH2 .; r/  MH2 ./ consists, that  \ B.x by definition, of all hypersurfaces  2 MH2 ./ that satisfy the ball condition for a fixed radius r > 0. Let .u;  ; / be a solution of (1), (2) and (3) on its maximal existence interval Œ0; tC /. Then .t / is said to satisfy a uniform ball condition if there exists a number r > 0 such that .t / 2 MH2 .; r/ for all t 2 Œ0; tC /. Note that this condition bounds the curvature of .t / and prevents parts of .t / to shrink to points, to touch the outer boundary @, and to undergo topological changes. With this property, combining the local semiflow for problem (1), (2) and (3) with the corresponding Lyapunov functional (i.e., the negative total entropy), relative compactness of bounded orbits, and the convergence results from the previous section, one obtains the following global result.

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

1251

Theorem 6. Let p > n C 2 and suppose that condition (4) holds. Suppose that .u;  ; / is a solution of (1), (2) and (3) in the state manifold SM on its maximal time interval Œ0; tC /. Assume there are constants M; m > 0 such that the following conditions hold on Œ0; tC /: (i) ju.t /jW 22=p ; j.t /jW 22=p ; j.t /jW 32=p M < 1, p p p (ii) m .t /; (iii) .t / satisfies a uniform ball condition. Then tC D 1, i.e., the solution exists globally and the solution converges in SM to an equilibrium .0; 1 ; 1 / 2 E. On the contrary, if .u.t /; .t /; .t // is a global solution in SM which converges to an equilibrium .0; 1 ; 1 / in SM as t ! 1, then (i)–(iii) hold. Proof. It is well known that each  2 MH2 ./ admits a tubular neighborhood Ua WD fx 2 Rn W dist.x; / < ag of width a D a./ > 0 such that the signed distance function d W Ua ! R;

jd .x/j WD dist.x; /;

is well defined and d 2 C 2 .Ua ; R/. Here d .x/ < 0 iff x 2 1 \Ua by convention. One then defines a level function ' by means of ( ' .x/ WD

d .x/.3d .x/=a/ C sgn .d .x//.1  .3d .x/=a//;

x 2 Ua ;

ex .x/  in .x/;

x … Ua ;

where ex and in denote the exterior and interior component of Rn n Ua , respectively, and  is a smooth cut-off function with .s/ D 1 if jsj < 1 and .s/ D 0 if jsj > 2. The level function ' is then of class C 2 , ' .x/ D d .x/ for x 2 Ua=3 , and ' .x/ D 0 iff x 2 . Let MH2 .; r/ denote the subset of MH2 ./ which consists of all  2 MH2 ./ such that    satisfies the ball condition with fixed radius r > 0. This implies in particular that dist.; @/  2r and all principal curvatures of  2 MH2 .; r/ are bounded by 1=r. Furthermore, the level functions ' are well N The map defined for  2 MH2 .; r/ and form a bounded subset of C 2 ./. N ˆ W MH2 .; r/ ! C 2 ./;

ˆ./ D ' ;

is a homeomorphism of the metric space MH2 .; r/ onto ˆ.MH2 .; r//  N see [30, Sect. 2.4.2]. C 2 ./; Let s  .n  1/=p > 2. For  2 MH2 .; r/ one defines  2 Wps .; r/ if ' 2 Wps ./. In this case the local charts for  can be chosen of class Wps as well. A subset A  Wps .; r/ is said to be (relatively) compact, if ˆ.A/  Wps ./ is (relatively) compact. Finally, one defines distWps .1 ; 2 / WD j'1  '2 jWps ./ for 1 ; 2 2 MH2 .; r/.

1252

G. Simonett and M. Wilke 32=p

Suppose that the assumptions (i)–(iii) are valid. Then .Œ0; tC //  Wp .; r/ 32=p" is bounded, hence relatively compact in Wp .; r/. Thus .Œ0; tC // can be covered by finitely many balls with centers †k such that distW 32=p" ..t /; †j / ı p

for some j D j .t /; t 2 Œ0; tC /:

Let Jk D ft 2 Œ0; t / W j .t / D kg. Using for each k a Hanzawa transform „k , we see that the pull backs f.u.t; /; .t; // ı „k W t 2 Jk g are bounded 22=p 22=p" . n †k /nC1 , hence relatively compact in Wp . n †k /nC1 . in Wp 1 1 Employing now [30, Theorem 9.2.1], one obtains solutions .u ;  ;  1 / with initial configurations .u.t /; .t /; .t // in the state manifold SM on a common time interval, say .0; a , and by uniqueness one has .u1 .a/;  1 .a/ 1 .a// D .u.t C a/; .u C a/; .t C a//: Continuous dependence implies then relative compactness of f.u./; ./; .// W 0 t < tC g in SM; in particular tC D 1 and the orbit .u;  ; /.RC /  SM is relatively compact. The entropy is a strict Lyapunov functional; hence the limit set !.u;  ; / of a solution is contained in the set E of equilibria. By compactness, !.u;  ; /  SM is nonempty; hence the solution comes close to E. Finally, one may apply the convergence result Theorem 5 to complete the sufficiency part of the proof. Necessity follows by a compactness argument. t u

6

The Isothermal Problem

In this section the isothermal Navier-Stokes problem with surface tension (5) will be considered. It turns out that the main results for this problem concerning wellposedness and the stability analysis of equilibria parallel those in Sects. 2, 3 and 4 for problem (1), (2) and (3). A decisive difference is caused by the fact that, as temperature is neglected, the principles of thermodynamics do no longer apply. The basic local well-posedness result for problem (5) reads as follows. Theorem 7. Let p > n C 2 and suppose that @ 2 C 3 . Assume the regularity conditions u0 2 Wp22=p . n 0 /;

0 2 Wp32=p ;

and the compatibility conditions div u0 D 0 in  n 0 ;

u0 D 0 on @;

ŒŒu0 D 0; P0 ŒŒD.u/0 D 0 on 0 :

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

1253

Then there exists a number a D a.0 ; u0 / and a unique classical solution .u; ; / S of (5) on the time interval .0; a/. Moreover, M D t2.0;a/ ft g.t / is real analytic. Proof. The reader is referred to [17]; see also [30, Sects. 9.2 and 9.4].

t u

The state manifold for (5) is defined by SM WD

n

N n  MH2 W u 2 Wp22=p . n /n ;  2 Wp32=p ; .u; / 2 C ./ div u D 0 in  n ;

u D 0 on @;

o P ŒŒD.u/ D 0 on  :

Applying Theorem 7 and re-parameterizing the interface repeatedly, one shows once more that problem (5) generates a semiflow on SM. As in problem (1), (2) and (3), the pressure  is determined for each time t from .u; / by means of the weak transmission problem ˇ ˇ     1 % r ˇ r L2 ./ D %1 u  .ujr/u ˇ r L2 ./ ;

2 Hp10 ./;

ŒŒ D  H C .ŒŒ2D.u/ j / on : connected disjoint compoSuppose that S the dispersed phase 1 consists ofSm m nents, 1 D m  . Let  WD @ ,  D 1;k k 1;k kD1 kD1 k , and let Mk WD j1;k j denote the volume of 1;k . Then one shows exactly as in Sect. 2.3 that problem (5) preserves the volume of each individual phase component. The available energy for problem (5) is defined by ˆ0 .u; / WD

1 2

Z

%juj2 dx C  jj:

n

For the time derivative of ˆ0 , one obtains Z Z n o d % ˆ0 D ŒŒ juj2 C  H V d  %.@t uju/ dx  dt 2   Z n Z o % ŒŒ juj2 C  H V d  D  f%..ujr/uju/  .div T ju/g dx  2   Z Z ˚  D 2  jD.u/j22 dx  ŒŒ.T uj / C  H V d  Z



D 2 



 jD.u/j22 dx;

showing that the available energy is decreasing. Therefore, ˆ0 constitutes a Lyapunov functional for (5). As in Sect. 2.5, one shows that ˆ0 is a strict Lyapunov functional. The same arguments as in Sect. 2 also imply that the equilibria of (5) consist of zero velocities and constant pressures in the phase components and

1254

G. Simonett and M. Wilke

that the dispersed phase consists of a collection of nonintersecting balls in . Consequently, the set E of nondegenerate equilibria for (5) is given by E D f.0; †/ W † 2 Sg; where S is defined in (9). E defines a real analytic manifold of dimension m.n C 1/. In analogy to Sect. 2.7 one shows that the critical points of the energy functional ˆ0 under the constraints of Mk D M0;k constant correspond exactly to the equilibria of (5) and that all critical points are local minima of the energy functional with the given constraints. Theorem 8. The following assertions hold for problem (5). (a) The phase volumes j1;k j are preserved. (b) The energy functional ˆ0 is a strict Lyapunov functional. (c) The nondegenerate equilibria are zero velocities, constant pressures in the components of the phases, and the interface that is a finite union of nonintersecting spheres which do not touch the outer boundary @. (d) The set E of nondegenerate equilibria forms a real analytic manifold of dimension m.n C 1/, where m denotes the number of connected components of 1 . (e) The critical points of the energy functional for prescribed phase volumes are precisely the equilibria of the system. (f) All critical points of the energy functional for prescribed phase volumes are local minima. This result was first established in [17, Proposition 5.2]; see also [5, Theorem 3.1]. It should be observed that the assertions in Remark 2.3 do also apply to the isothermal case. S Suppose .0; †/ 2 E, with † D m kD1 †k and †k D @B.xk ; Rk /, is a fixed equilibrium for problem (5). In analogy to Sect. 3, and using the notation introduced there, one associates with system (5) the following linear problem %@t u  u C r D %fu

in

 n †;

in

 n †;

uD0

on

@;

ŒŒu D 0

on

†;

ŒŒT † C  .A† h/† D gu

on

†;

@t h  .uj† / D fh

on

†;

u.0/ D u0

in

;

h.0/ D h0

on

†;

div u D gd

(42)

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

1255

and the linear operator L,   L.u; h/ WD  %1 .u  r/; .uj† / ; 31=p

defined on X0 D Lp; ./  Wp

.†/ with domain

X1 D D.L/ D f.u; h/ 2 Hp2 . n †/n  Wp31=p .†/ W div u D 0 in  n †; u D 0 on @; ŒŒu ; P† ŒŒD.u/† D 0 on †g: Here  is again determined as the solution of the weak transmission problem .%1 rjr /L2 ./ D .%1 ujr /L2 ./ ;

2 Hp10 ./;

ŒŒ D A† h C .ŒŒ2D.u/† j† /

on †:

The principal variable in (42) is z D .u; h/, the dynamic inhomogeneities are f D .fu ; fh /, and the static ones are g D .gd ; gu /. The eigenvalue problem associated with L becomes %u  u C r D 0

in

 n †;

div u D 0

in

 n †;

uD0

on

@;

ŒŒu D 0

on

†;

ŒŒT † C  .A† h/† D 0

on

†;

h  .uj† / D 0

on

†:

(43)

Theorem 9. Let e 2 E be an equilibrium. Then the operator L has the following properties. (a) L generates a compact, analytic C0 -semigroup in X0 which has the property of maximal Lp -regularity. (b) The spectrum of L consists of countably many eigenvalues of finite algebraic multiplicity. (c) L has no eigenvalues  ¤ 0 with nonnegative real part. (d)  D 0 is a semi-simple eigenvalue of L of multiplicity m.n C 1/. (e) The kernel N.L/ of L is isomorphic to the tangent space Te E of the manifold of equilibria E at e . Hence, each equilibrium e 2 E is normally stable.

1256

G. Simonett and M. Wilke

Proof. The proof proceeds in the same way as the corresponding proof of Theorem 3, with the only difference that here all quantities and assertions relating to the temperature are dismissed. One verifies, for instance, that the kernel of L is j j spanned by the functions ej k D .0; Yk / with Yk the spherical harmonics of degree one for the spheres †k , j D 1;    ; n, k D 1;    ; m, and e0;k D .0; Yk0 /, where Yk0 equals one on †k and zero elsewhere. Hence the dimension of the null space N.L/ is m.n C 1/. t u The main theorem of this chapter concerning the stability of equilibria reads as follows. Theorem 10. Let p > n C 2. Then every equilibrium e D .0; †/ 2 E is nonlinearly stable in the state manifold SM. Any solution with initial value close to e in SM exists globally and converges in SM to a possibly different stable equilibrium e1 2 E at an exponential rate. Proof. The proof proceeds in the same way as the corresponding proof for Theorem 5, with the obvious modification that all quantities and assertions relating to the temperature are to be disregarded. t u Analogous assertions as in Theorem 6 hold for solutions .u.t /; .t // of the isothermal problem (5), again with the obvious modification that the temperature variable is dropped; see also [17, Theorem 7.1].

7

The Two-Phase Stokes Flow with Surface Tension

In this section the two-phase quasi-stationary Stokes problem with surface tension (6) will be considered. This problem is considerably easier to analyze than problem (1), (2) and (3) or problem (5). In fact, it turns out that in this case, the only system variable is the unknown hypersurface . In order to obtain this reduction, the two-phase Stokes problem u C r D 0

in

 n ;

div u D 0

in

 n ;

uD0

on

@;

ŒŒu D 0

on

;

on



ŒŒT  D g

(44)

11=p

will play an important role. One shows that (44) admits for each g 2 Wp unique solution .u; / (up to constants in the pressure) with regularity P p1 . n /; .u; / 2 Hp2 . n /  H

./ a

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

1257

P p1 denotes the homogeneous Sobolev space of order one. where 1 < p < 1, and H 11=p

./, let .u; / be the solution of (44). Then the For a given function g 2 Wp 11=p 21=p ./ ! Wp ./ is defined by Neumann-to-Dirichlet operator N W Wp N g WD .uj /: The following results hold for N . Proposition 3. Suppose @ 2 C 3 and  2 MH2 ./ consists of m components, Sm  D kD1 k . Then the operator N has the following properties. 1=2

(a) .N gjh/L2 ./ D .gjN R  h/L2 ./ , g; h 2 W2 ./. (b) .N gjg/L2 ./ D 2  jD.u/j22 dx, where .u; / is the solution of (44). (c) Let ek be the function which is one on k and zero elsewhere. Then 1=2

.N gjek /L2 ./ D 0 for each g 2 W2 ./ and each 1 k m: In particular, N ek D 0 for each k, and N g has mean value zero for each 1=2 function g 2 W2 ./. (d) N.N / D spanfe1 ;    ; em g: 1=2

Proof. Let g; h 2 W2 .†/ be given, and let .u; / denote the solution of (44) corresponding to g, and .v; q/ the solution corresponding to h, respectively. Then one obtains Z

Z

.N gjh/L2 ./ D

.ujh / d  D  

Z D

ŒŒ.ujT .v; q/ / d  

Z

div .T .v; q/u/ dx D 2 

 D.v/ W D.u/ dx 

and the assertions in (a)–(b) follow at once. Here, D.u/ W D.v/ D trace .D.u/D.v// denotes the Frobenius inner product of the (symmetric) matrices D.u/ and D.v/. 1=2

(c) Let g 2 W2 .†/ be given, and let .u; / be the solution of (44). By the divergence theorem Z

Z

.N gjek /L2 .†/ D

.uj / d k D k

div u dx D 0; 1;k 1=2

with 1;k the region enclosed by †k . Therefore, by density of W2 ./ in L2 ./, .gjN ek /L2 .†/ D 0 for all g 2 L2 .†/, and hence N ek D 0.

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(d) It remains to show that N.N /  spanfe1 ;    ; em g. Suppose g 2 N.N /. It then follows from part (b) that D.u/ D 0. Korn’s inequality and the no-slip boundary condition readily imply that u D 0 on , and hence  is constant on connected components of . This shows that gjk D ŒŒ jk is constant on each boundary component k , and hence g 2 spanfe1 ;    ; em g. t u By means of the Neumann-to-Dirichlet operator N , problem (6) can be reformulated as a geometric evolution equation V D  N H ;

.0/ D 0 :

(45)

In order to study problem (45), one may parameterize  over an analytic reference manifold † which is C 2 close to 0 . Problem (45) can then be cast as a quasilinear evolution equation @t h C A.h/h D F .h/;

h.0/ D h0 ;

(46)

where h.t/ denotes the height function which parameterizes .t /, that is, .t / D fq C h.t; q/† .q/ W q 2 †; t  0g: The resulting problem (46) is amenable to the theory of maximal Lp -regularity for quasilinear parabolic evolution equations; see, for instance, [30, Chapter 5] for a comprehensive account of this theory. The following basic well-posedness result holds true. 32=p

Theorem 11. Suppose p > n C 2. Then for each 0 2 Wp there is a number a D a.0 / and S a unique classical solution  D f.t / W t 2 .0; a/g for (45). Moreover, M D t2.0;a/ ftg  .t / is real analytic. Proof. The reader is referred to [30, Sect. 12.5]

t u

Suppose, as in the previous sections, Sthat the dispersed phase 1 consists Sm of m  . Let  WD @ ,  D disjoint connected components, 1 D m 1;k k 1;k kD1 kD1 k , and let Mk WD j1;k j denote the volume of 1;k . Then one shows, as in Sect. 2.3, that problem (45), or equivalently problem (6), preserves the volume of each individual phase component. Indeed, by Proposition 3(c) d j1;k .t /j D dt

Z

Z V d k D  k

N H d k D  .N H jek /L2 .†/ D 0: k

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

1259

The time derivative of the surface area j.t /j is given by Z Z d j.t /j D  V H d  D  .N H /H d  dt   Z D 2 jD.u/j22 dx; 

where .u; / is the solution of (44) with g D H . This shows that surface area is decreasing. Therefore, ˆ0 ./ D jj constitutes a Lyapunov functional for (45). As in Sect. 2.5, one shows that ˆ0 is a strict Lyapunov functional. An analogous argument as in Sect. 2 also implies that the equilibria of (6) consist of zero velocities and constant pressures in the phase components and that the dispersed phase consists of a collection of nonintersecting balls in . Consequently, the set E of nondegenerate equilibria for (45) is given by E D f† W † 2 Sg; where S is defined in (9). E gives rise to a real analytic manifold of dimension m.n C 1/. In analogy to Sect. 2.7, one also shows that the critical points of the area functional ˆ0 under the constraints of Mk D M0;k constant correspond to the equilibria of (45) and that all critical points are local minima of the area functional ˆ0 under the given constraints. Theorem 12. The following assertions hold for problem (45). (a) The phase volumes j1;k j are preserved. (b) The area functional ˆ0 is a strict Lyapunov functional. (c) Each nondegenerate equilibrium consists of a finite union of nonintersecting spheres which do not touch the outer boundary @. (d) The set E of nondegenerate equilibria forms a real analytic manifold of dimension m.n C 1/, where m denotes the number of connected components of 1 . (e) The critical points of the area functional for prescribed phase volumes are precisely the equilibria of the system. (f) All critical points of the area functional for prescribed phase volumes are local minima. In oder to analyze the stability properties of equilibria for the geometric evolution Sm equation (45), one may proceed as follows. Suppose † D kD1 †k 2 E is an equilibrium for (45). Choosing † as a reference manifold one shows that problem (45), or for that matter also problem (46), can be written as @t h C  N† A† h D G† .h/;

h.0/ D h0 ;

(47)

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where A† has the same meaning as in Sect. 4. The nonlinear function G† satisfies .G† .0/; G†0 .0// D 0. 21=p 31=p Let X0 WD Wp .†/, X1 WD Wp .†/, and set L W D.L/ D X1  X0 ! X0 ;

L WD  N† A† :

(48)

Theorem 13. The operator L has the following properties. (a) L generates a compact, analytic C0 -semigroup in X0 which has the property of maximal Lp -regularity. (b) The spectrum of L consists of countably many real eigenvalues of finite algebraic multiplicity. The spectrum is independent of p. (c) L has no positive eigenvalues. (d)  D 0 is a semi-simple eigenvalue of L of multiplicity m.n C 1/. (e) The kernel N.L/ of L is isomorphic to the tangent space T† E. Hence, the equilibrium † 2 E is normally stable. Proof. The assertions in (a)–(b) follow from standard arguments. (c) Suppose that  2 C,  ¤ 0; is an eigenvalue for L, that is, h C  N† A† h D 0

(49)

5=2

for some nontrivial function h 2 W2 .†/. Taking the inner product of (49) with A† h in L2 .†/ yields .hjA† h/L2 .†/ C  .N† A† hjA† h/L2 .†/ D 0:

(50)

As N† and A† are symmetric, this identity implies that  must be real; hence the spectrum of L is real. Suppose that  > 0. By Proposition 3(c), .N† A† hjek /L2 .†/ D 0 and consequently, .hjek /L2 .†/ D 0 as well, which implies .hjA† h/L2 .†/  0. As N† is positive semi-definite on L2 .†/, see Proposition 3(b), one concludes that h D 0, and then h D 0 by (49) as  > 0. .hjA† h/L2 .†/ D 0. This yields A†P (d) Suppose h 2 N.L/. Then A† h D m kD1 ak ek by Proposition 3(d). This implies h D h0 

m X .ak Rk2 =.n  1// ek ; kD1

with h0 2 N.A† /, where Rk denotes the radius of the sphere †k . As N.A† / is j spanned by the spherical harmonics Yk on †k , we see that dim N.L/ D m.nC1/. Next it will be shown that the eigenvalue 0 is semi-simple. Suppose L2 h D 0: Then

25 Stability of Equilibrium Shapes in Some Free Boundary Problems. . .

N† A† h D h0 C

m X

ak e k ;

1261

for some h0 2 N.A† / and ak 2 C:

kD1

Multiplying this relation with ej in L2 .†/, one obtains ak D 0 for all k. Taking the L2 .†/ inner product of the relation N† A† h D h0 with A† h yields .N† A† hjA† h/L2 .†/ D .h0 jA† h/L2 .†/ D 0 as A† is symmetric and h0 2 N.A† /. Therefore, N† A† h D 0, that is, h 2 N.L/. (e) The assertion follows as N.L/ and T† E are of the same dimension. t u Theorem 14. Let p > n C 2 and suppose that † is a (nondegenerate) equilibrium of (45). 32=p Then any solution of (47) starting close to 0 in Wp .†/ exists globally and 32=p .†/ at an exponential rate. Here, h1 converges to an equilibrium h1 in Wp corresponds to some 1 2 E. Proof. The proof of this result is based on the generalized principle of linearized stability for quasilinear parabolic equations introduced in [33]; see also Chapter 5 in the monograph [30]. t u The state manifold for (45) is defined by means of SM D MH2 ./. The main result of this section reads as follows. Theorem 15. Let p > n C 2. Suppose that .t / is a solution of (45), defined on its maximal existence interval Œ0; tC /. Assume there is a constant M > 0 such that the following conditions hold on Œ0; tC /: (i) j.t /jW 32=p M < 1; p (ii) .t / satisfies a uniform ball condition. Then tC D 1, i.e., the solution exists globally, and .t / converges in SM to an equilibrium 1 2 E at an exponential rate. The converse is also true: if a global solution converges in SM to an equilibrium, then (i) and (ii) are valid. Proof. The proof is similar to that of Theorem 6; see also [30, Sect. 12.5].

8

t u

Conclusions

In this chapter, the equilibrium states for the two-phase Navier-Stokes problem with heat-advection and surface tension (1), (2) and (3), the two-phase isothermal Navier-Stokes problem with surface tension (5), and the two-phase Stokes flow with

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surface tension (6) are characterized. It is shown that every equilibrium is normally stable and that every solution that starts close to an equilibrium exists globally and converges to a (possibly different) equilibrium at an exponential rate. Moreover, it is shown that the negative total entropy for (1), (2) and (3), the available energy for (5), and the surface area for (6) constitute strict Lyapunov functionals. This implies that solutions which do not develop singularities converge to an equilibrium in the topology of the state manifold SM.

9

Cross-References

 Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible

Fluid Mechanics  Modeling of Two-Phase Flows With and Without Phase Transitions  The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

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Weak Solutions and Diffuse Interface Models for Incompressible Two-Phase Flows

26

Helmut Abels and Harald Garcke

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Weak Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Two-Phase Flow Without Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Case with Surface Tension: Varifold Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Existence of Weak Solutions for a Navier-Stokes/Mullins-Sekerka System . . . . . 3 Diffuse Interface Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Models Based on the Volume Averaged Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Model Based on the Mass Averaged Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Analytic Results in the Case of Same Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Analysis for the Model with General Densities Based on the Volume Averaged Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Analysis for the Model with General Densities Based on the Mass Averaged Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Sharp Interface Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Models Based on a Volume Averaged Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Sharp Interface Expansions for the Lowengrub-Truskinovsky Model . . . . . . . . . . 4.3 Known Results on Sharp Interface Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1268 1271 1274 1276 1283 1285 1287 1292 1296 1304 1308 1313 1314 1319 1322 1323 1324 1324

Abstract

In two-phase flows typically a change of topology arises. This happens when two drops merge or when one bubble splits into two. In such a case the concept of

H. Abels () • H. Garcke Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany e-mail: [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_29

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classical solutions to two-phase flow problems, which describe the interface as a smooth hypersurface, breaks down. This contribution discusses two possible approaches to deal with this problem. First of all weak formulations are discussed which allow for topology changes during the evolution. Such weak formulations involve either varifold solutions, so-called renormalized solutions or viscosity solutions. A second approach replaces the sharp interface by a diffuse interfacial layer which leads to a phase field-type representation of the interface. This approach leads typically to quite smooth solutions even when the topology changes. This contribution introduces the solution concepts, discusses modeling aspects, gives an account of the analytical results known, and states how one can recover the sharp interface problem as an asymptotic limit of the diffuse interface problem.

1

Introduction

In the flow of immiscible fluids with interfaces in general, topological transitions like droplet breakup and coalescence occur. In such situations classical formulations based on an explicit parameterization break down as singularities will appear at points where the topology changes. This contribution discusses two approaches to deal with this issue. First of all, weak formulations of the two-phase flow problem for incompressible fluids are introduced which all allow for singularities in the geometry. The known results for the different approaches are stated, and the advantages and disadvantages of the different formulations are discussed. Secondly, diffuse interface methods provide an alternative way to allow for topological transitions. In these models quantities which in traditional sharp interface models are localized to the interfacial surface are now distributed over a diffuse interfacial region. For example, quantities like the density and the viscosity are suitably averaged in the diffuse interface, and the surface tension, which is supported on the interface in a sharp interface model, is now a distributed stress within the diffuse interfacial layer, cf. Fig. 1.

Hv

v

G(t)

W+(t)

W-(t) Fig. 1 Sharp versus diffuse interface models

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

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In the classical sharp interface approach for incompressible viscous flows, the Navier-Stokes equations have to hold in the two phases, described by disjoint open sets ˙ .t /, which are separated by a hypersurface .t /, which evolves in time. In this contribution slip is not allowed at the interface which leads to the fact that the tangential part of the fluid velocity does not jump at the interface and also it is assumed that no phase transitions occur which implies that also the normal part of the velocity does not jump at the interface and that the interface is transported with the fluid velocity. One hence obtains ŒvC  D 0; V D v  ; where v is the fluid velocity, Œ:C  denotes the jump across the interface .t /,  is a unit normal at the interface .t /, chosen as interior normal with respect to C .t /, and V is the normal velocity. In addition a tangential stress balance has to hold at the interface. In cases where the interface itself does not produce stresses at the interface, the normal stresses have to balance, i.e., on the interface it has to hold TC   T  D 0

,

ŒTC   D 0;

where TC and T are the values of the stress tensor on both sides of the interface. For a viscous incompressible fluid, the simplest choice for T is T D 2Dv  p Id, where Dv D 12 .rv C rvT /. In the following mainly the case with surface tension will be discussed, i.e., surface energy effects are taken into account and in this case the stress balance at the interface is given as the Young-Laplace law ŒTC   C  H  D 0; where  is the surface tension and H is the mean curvature, which is chosen to be the sum of the principal curvatures with respect to . For weak formulations it is often convenient to reformulate the fact that the interface is transported with the fluid velocity. Defining  as the characteristic function of one of the phases, one can formally rewrite the equation V D v   as @t  C v  r D 0 which for a velocity field that has zero divergence is formally equivalent to @t  C div.v/ D 0: Of course the last two equations need to be interpreted in a suitable weak sense, and different formulations will be discussed in Sect. 2. All these formulations will allow

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for singularities of the interface and in particular allow for topological transitions. The weak formulations mentioned above are: • approaches based on the theory of viscosity solutions, see [42, 69, 71], • methods which use the concept of renormalized solutions of transport equations, see [56, 57]. These approaches work for the case without surface tension. If surface tension effects are present, also the mean curvature has to be interpreted in a weak sense, and this can be done in the context of varifolds; see [2, 14, 16, 21, 58, 72] and Sect. 2.2. In diffuse interface models (which are also called phase field models), the sharp interface is replaced by an interfacial layer of finite width, and a smooth order parameter is used to distinguish between the two bulk fluids and the diffuse interface. The order parameter takes distinct constant values in each of the bulk fluids and varies smoothly across the thin interfacial layer. In the sharp interface case with surface tension, the total energy is given as Z 

 2 jvj dx C  Hd 1 ./ 2

where  is the domain occupied by the fluid,  is the mass density,  is the interface, and Hd 1 is the .d 1/-dimensional surface measure. The first term is the kinetic energy, and the second term accounts for interfacial energy. It is well known based on the work of van der Waals, Korteweg, Cahn, and Hilliard that interfacial energy and also related capillary forces can be modeled with the help of density variables which vary continuously across the interface. In these approaches the term  Hd 1 ./ is replaced by a multiple of F.'/ WD

Z  

 " 1 jr'j2 C .'/ dx 2 "

(1)

where " > 0 is a small parameter, ' is an order parameter taking the values ˙1 in the two phases, and is a double well potential which simplest form is .'/ D 1 .1  ' 2 /2 . One can now try to model the physics at the interface with the help 4 of ' and would obtain a new problem which should approximate the above sharp interface problem. As new energy one obtains Z 

.'/ 2 jvj dx C O 2

Z  

 " 1 jr'j2 C .'/ dx; 2 "

and the transport equation becomes @t ' C v  r' D m"

O > 0;

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

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with

D O

ıF ı'

where ıF is the first variation of F. ı' One obtains the law V D v   from the free boundary problem in the limit " ! 0 by choosing m"  "; see Sect. 4. It will turn out, see Sect. 3, that the term  H  d Hd 1 contributing to the stress balance at the interface will become a multiple of

r' which is a term which is distributed over the diffuse interfacial layer. In the simplest case, the momentum balance in the Navier-Stokes equation can be written as    r' r' @t v C v  rv C divT D O " div jr'j2 Id  ˝ jr'j jr'j r' r' ˝ jr'j corresponds to Id  ˝  which is a multiple of the where the term Id  jr'j classical interfacial stress tensor which is just the projection onto the tangent space, cf., weak formulation (17) below. Diffuse interface models for incompressible twophase flows were introduced and studied in [13, 23, 28, 37, 46, 51, 52].

2

Weak Formulations

In this section different notions of weak/generalized solutions of the two-phase flow of two incompressible, immiscible Newtonian fluids inside a bounded domain   Rd , d D 2; 3, are discussed. The fluids fill disjoint domains C .t / and  .t /, t > 0, and the interface between both fluids is denoted by .t / D @C .t /. It is assumed that .t / is compactly contained in , which means that one excludes flows, where a contact angle problem occurs. Hence  D C .t / [  .t / [ .t /. The flow is described using the velocity vW   .0; 1/ ! Rd and the pressure pW   .0; 1/ ! R in both fluids in Eulerian coordinates. Cases with and without surface tension at the interface are considered. Precise assumptions are made below. Under suitable smoothness assumptions, the flow is obtained as solution of the system ˙ @t v C ˙ v  rv  ˙ v C rp D 0

in ˙ .t/; t > 0;

(2)

div v D 0

in ˙ .t/; t > 0;

(3)

ŒvC 

on .t /; t > 0;

(4)

D0

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H. Abels and H. Garcke C Œ2DvC   C Œp  D  H 

on .t /; t > 0;

(5)

V Dv

on .t /; t > 0;

(6)

vD0

on @; t > 0;

(7)

in ;

(8)

vjtD0 D v0

together with ˙ .0/ D ˙ 0 . Here V and H denote the normal velocity and mean curvature of .t /, resp., taken with respect to the interior normal  of @C .t / D .t /,   0 is the surface tension constant ( D 0 means that no surface tension is present), and ˙ > 0 and ˙ > 0 are the (constant) densities and viscosities of the fluids, respectively. The equations (2) and (3) describe the conservation of linear momentum and mass for both fluids. Furthermore, (4) is a no-slip boundary condition at .t /, implying continuity of v across , (5) is the balance of forces at the boundary, (6) is the kinematic condition that the interface is transported with the flow of the mass particles, and (7) is the no-slip condition at the boundary of . Here exterior forces are neglected for simplicity. There are many results on well-posedness locally in time or global existence close to equilibrium states for quite regular solutions of this two-phase flow and similar free boundary value problems for viscous incompressible fluids, cf. Solonnikov [66,68], Beale [24,25], Tani and Tanaka [70], Shibata and Shimizu [64], Shibata and Shimizu [65] or Prüss and Simonett [59], and the references given there. These approaches are a priori limited to flows, in which the interface does not develop singularities and the domain filled by the fluid does not change its topology. In the following different notions of generalized solutions, which allow for singularities of the interface and which exist globally in time for general initial data, are discussed. A similar and more detailed discussion can be found in [2]. To this end, first a suitable weak formulation of the system above is needed. By multiplication of (2) with a divergence-free vector field ' and integration by parts using in particular the jump relation (6), one obtains Z

1

Z

Z



./v  @t ' dx dt  0

Z



1

.0 /v0  'jtD0 dx 

Z

C

Z

1

Z

./.v  rv/  ' dx dt C 0

 1˝

Z

2./Dv W D' dx dt 0



˛ H.t/ ; '.t / dt

D

(9)

0 1 for all ' 2 C.0/ .Œ0; 1//d with div ' D 0, where Dv D 12 .rvCrvT /, .x; t / D C .t/ .x/ for all x 2 , t > 0, 0 D C , A denotes the characteristic function 0 of a set A, .1/ D C , .0/ D  , .1/ D C , .0/ D  , and

˛ H.t/ ; '.t / WD

˝

Z

H .x; t /.x/  '.x; t / d Hd 1 .x/: .t/

(10)

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

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Here Hd 1 denotes the .d  1/-dimensional Hausdorff measure. Now, if v and  are sufficiently smooth, one obtains by choosing ' D v the energy inequality Z

..x; T //jv.x; T /j2 dx C Hd 1 ..T // 2  Z TZ Z .0 /jv0 j2 C dx C  Hd 1 .0 / 2./jDvj2 dx dt  2 0  

(11)

for all T > 0 (even with equality), where 0 D @C 0 . Here one uses d d 1 H ..t // D  dt

Z

H V d Hd 1 D hH.t/ ; v.t /i

(12)

.t/

which is due to (6), cf. [43, Equation 10.12]. More details for a more general model can be found in [13, Section 5]. Since ˙ ; ˙ > 0, (11) yields the a priori estimate v 2 L1 .0; 1I L2 .// and

Dv 2 L2 .  .0; 1//d d

(13)

for any sufficiently smooth solution of (2), (3), (4), (5), (6), (7), and (8). Here p Lp .M /, 1  p  1, denotes the usual Lebesgue space, Lloc .M / its local and Lp .M I X / its vector-valued analog for a given Banach space X . Moreover, if A R, then Lp .M I A/ consists of all f 2 Lp .M / with f .x/ 2 A for a.e. L2 ./

p

is the set of all weakly x 2 M . Finally, L ./ D f' 2 C01 ./d W div ' D 0g divergence free vector fields f 2 Lp ./d . As will be shown below, if  > 0, then (11) yields an a priori bound of  2 L1 .0; 1I BV .//; where BV ./ D ff 2 L1 ./ W rf 2 M./g denotes the space of functions with bounded variation, cf., e.g., [22, 39] and M./ D C0 ./0 is the space of finite Radon measures. In the case without surface tension, i.e.,  D 0, one only obtains that  2 L1 .Q/ is a priori bounded by one, where Q WD .0; 1/. This motivates to look for weak solutions .v; / lying in the function spaces above, satisfying (11) with a suitable substitute of (10), such that .v; / solve (9) as well as the transport equation @t  C v  r D 0 jtD0 D 0

in Q;

(14)

in 

(15)

for 0 D C in a suitable weak sense. Note that (14) is a weak formulation of (6), 0 cf. [56, Lemma 1.2].

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H. Abels and H. Garcke

Two-Phase Flow Without Surface Tension

Throughout this subsection, it is assumed that  D 0, i.e., no surface tension is present. Then the two-phase flow consists of a coupled system of the Navier-Stokes equation with variable viscosities and a transport equation for the characteristic function .t / D C .t/ . Then this is a special case of the so-called density-dependent Navier-Stokes equation, cf., e.g., Desjardins [36] and references given there. For given  it is not difficult to construct a weak solution of the Navier-Stokes equation (9) with the aid of a suitable approximation scheme (e.g., Galerkin approximation). New difficulties arise due to the mean curvature term hH.t/ ; :i, which depends nonlinearly on the normal of .t /. For the coupled system (9) together with (14) and (15), there are two different approaches. The essential difference is in which sense the transport equation is solved. One approach is due to Giga and Takahashi [42], who solved (14) and (15) in the sense of viscosity solutions, where the characteristic functions ..t /; 0 / are replaced by continuous level-set functions . .t /; 0 / such that ˙ 0 D fx 2  W

0 .x/

? 0g:

For simplicity they consider periodic boundary conditions, i.e.,  D Td . Since v is in general not Lipschitz continuous, the existence of a viscosity solution of (14) and (15) with .; 0 / replaced by continuous level-set functions . ; 0 / is not known. There are only a least super-solution C .t / and a largest sub-solution  .t / of the transport equation. Then one defines ˚ ˙ .t / D x 2  W

˙

 .x; t / ? 0 :

With this definition ˙ .t / are disjoint open sets, but the “boundary” .t / D Td n.C .t /[ .t // might have interior points and might have positive Lebesgue’s measure. Giga and Takahashi call this possible effect “boundary fattening.” With this definition they construct weak solutions of a two-phase Stokes flow, i.e., the convective term v  rv is neglected in (9), assuming that the viscosity difference jC   j is sufficiently small and C D  ; cf. [42] for details. This approach was adapted to the case of a Navier-Stokes two-phase flow by Takahashi [69] under similar assumptions and to a one-phase flow for an ideal, irrotational, and incompressible fluid by Wagner [71]. The other approach was established by Nouri and Poupaud [56] and Nouri et. al. [57] and is based on the results of DiPerna and Lions [38] on renormalized solutions of the transport equation (14) and (15) for a velocity field v with bounded divergence. Here  2 L1 .Q/ is called a renormalized solution of (14) and (15) if for all ˇ 2 C 1 .R/ which vanish near 0 the function ˇ./ solves (14) and (15) with initial values ˇ.0 /, cf. [38] for details. In particular, this implies that

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

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.t; x/ 2 f0 .x/ W x 2 g for almost all t > 0; x 2 . Due to [38, Theorem II.3], for every 0 2 L1 .Rd /, there is a unique renormalized solution of (14) and (15) under general conditions on v, which are weaker than the condition (13). Based on this notion the following result for the two-phase flow without surface tension holds true: Theorem 1 (Existence of Weak Solutions, [56, Theorem 1.1]). For every v0 2 L2 ./, 0 2 L1 .I f0; 1g/ there are v 2 L1 .0; 1I L2 .// \ L2 .0; 1I H01 ./d / and  2 L1 .QI f0; 1g/ that are a weak solution of the twophase flow (2), (3), (4) (5), (6), (7), and (8) without surface tension ( D 0) in the 1 sense that (9) holds true for all ' 2 C.0/ .  Œ0; 1//d with div ' D 0;  is the unique renormalized solution of the transport equation of (14) and (15), and (11) holds for almost all t > 0 with  D 0. 1 Here C.0/ .  Œ0; T //, T 2 .0; 1, is the space of all smooth 'W   Œ0; T / ! R with compact supp '    Œ0; T /. Moreover, H0k ./, k 2 N is the closure of compactly supported, smooth functions 'W  ! R in H k ./. The result was proved by Nouri and Poupaud [56] for the case of a bounded domain  with Lipschitz boundary. The authors even considered the case of a multiphase flow with more than two components. The result was extended to generalized Newtonian fluids of power-law type for a power-law exponent q  2d C 1 in [3]. d C2 In order to prove the latter theorem, a key step is to show strong compactness of the sequence k in Lp .QT /, 1  p < 1, where QT D   .0; T /; T > 0 and .vk ; k / is a suitably constructed approximation sequence. This is done by using the fact that Z Z p k .t; x/ dx D 0 .x/ dx kk .t /kLp ./ D 



if k are solutions of (14) and (15) with v replaced by vk and div vk D 0. Using that k *k!1 

in L1 .Q/;

rvk *k!1 rv

in L2 .Q/

for a suitable subsequence, one shows that  solves (14) and (15), cf. [3, Lemma 5.1]. Here * denotes the weak- convergence. Therefore p

k.t /kLp ./ D

Z

Z

p

.t; x/ dx D 



0 .x/ dx D kk .t /kLp ./ :

This implies strong convergence k !k!1  in Lp .QT /, 1  p < 1, for every T > 0. Based on this, one can pass to the limit in all terms in (9).

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Remark 1. Using the solution of Theorem 1, one can define the sets C .t / D fx 2  W .t / D 1g and  .t / D fx 2  W .t / D 0g. Then one knows that jC .t /j D jC 0 j and  n .C .t / [  .t // has Lebesgue measure zero. But, since only  2 L1 .Q/ is known, it is not clear whether ˙ .t / have interior points. In particular, it is not excluded that C .t / D  and int C .t / D ;. Therefore it is not immediately clear what the “interface” between both fluids should be. If one naively defines the interface as .t / D @C .t /, then .t / can have positive Lebesgue measure as in the result by Giga and Takahasi. It seems that by neglecting surface tension in the two-phase flow, one looses a “good control” of the interface between both fluids. At least the precise regularity of the interface seems to be unknown in general. Some results in this direction can be found in the contribution by Danchin and Mucha [34], where existence and uniqueness of more regular solutions for the inhomogeneous Navier-Stokes equation with discontinuous initial density are shown under several smallness assumptions.

2.2

Case with Surface Tension: Varifold Solutions

As discussed in the previous section, a deficit of the two-phase flow without surface tension is that there is no good information on the properties of the interface. As mentioned in the introduction, if  > 0, the energy equality (11) for sufficiently smooth solutions provides an a priori estimate of the interface: sup Hd 1 ..t //  0t 0, then .t / 2 BV ./ for all t > 0, and (16) gives an a priori estimate of  2 L1 .0; 1I BV .//:

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

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Conversely, if .t / D E 2 BV ./ for some set E D E.t /, then E is said to be of finite perimeter, and the following characterization holds, cf. [39, Section 5.7, Theorem 2]: Z hr.t /; 'i D @ E

 E  '.x/ d Hd 1 .x/;

where @ E is the reduced boundary of E, cf. [39, Definition 5.7],  E D @ E is countably .d  1/-rectifiable in the sense that @ E D

1 [

rE , jrE j

and

Kk [ N;

kD1

where Kk are compact subsets of C 1 -hypersurfaces Sk , k 2 N, Hd 1 .N / D 0, and E jSk is normal to Sk . Moreover, by [39, Section 5.8, Lemma 1] @ E  @ E and Hd 1 .@ E n @ E/, where @ E is the measure theoretic boundary of E consisting of all x 2  such that lim sup r!0

Ld .B.x; r/ \ E/ >0 rd

and

lim sup r!0

Ld .B.x; r/ n E/ > 0; rd

where Ld is the Lebesgue measure on Rd . Based on these properties, one can define the mean curvature functional of a set of finite perimeter E as Z hH@ E ; 'i  hHE ; 'i WD 

@ E

tr.P r'/ d Hd 1 ;

' 2 C01 ./d ;

(17)

where P D I  E .x/ ˝ E .x/ and C01 ./ is the closure of smooth functions 'W  ! R with compact support in . Note that tr.P r'/ corresponds to the divergence of ' along the “surface” @ E and that by integration by parts (17) coincides with the usual definition if @ E is a C 2 -surface, cf., e.g., Giusti [43, Chapter 10]. In the following it is assumed that C D  D 1 and  D Rd . Motivated by the considerations above, one defines weak solutions of the two-phase flow in the case of surface tension as follows: Definition 1 (Weak Solutions). Let  > 0. Then v 2 L1 .0; 1I L2 .Rd // \ L2 .0; 1I H01 .Rd /d /; d  2 L1 ! .0; 1I BV .R I f0; 1g//;

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are called a weak solution of the two-phase flow for initial data v0 2 L2 .Rd /, d 0 D C for a bounded domain C 0 R of finite perimeter if the following 0 conditions are satisfied: 1 (i) (9) holds for all ' 2 C.0/ .Rd  Œ0; 1//d with div ' D 0, where H.t/ is replaced by H.t/ defined as in (17). (ii)  is a the renormalized solution of (14) and (15). (iii) The energy inequality

1 kv.t /k22 C  kr.t /kM./ 2 Z tZ 1 C 2./jDvj2 dx d  kv0 k22 C kr0 kM./ 2 0 

(18)

holds for almost all t 2 .0; 1/. Unfortunately, the existence of weak solutions as defined above is open. The reasons are possible oscillation and concentration effects related to the interface, which cannot be excluded so far. One can hence not pass to the limit in the mean curvature functional (17) during an approximation procedure used to construct weak solutions. In order to demonstrate these effects, let Ek be a sequence of sets of finite perimeter such that k  Ek is bounded in BV ./ and let  D Rd . Then after passing to a suitable subsequence, one assumes that k !k!1 

in L1loc .Rd /;

rk *k!1 r

in M.Rd /;

jrk j *k!1

in M.Rd /:

But then the question arises how jrj and are related and whether lim hH"k ; i D hH ; i

k!1

(19)

holds. The continuity result due Reshetnyak, cf. [22, Theorem 2.39], gives a sufficient condition for (19): If lim jrk j.Rd / D jrj.Rd /;

k!1

(20)

then (19) holds. But in general (20) will not hold, for example, because of the following oscillation/concentration effects at the reduced boundary of E: (i) Several parts of the boundary @ Ek might meet. (ii) Oscillations of the boundary might reduce the area in the limit. (iii) There might be an “infinitesimal emulsion.”

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

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Fig. 2 Some possible oscillation/concentration effect

These effects are sketched in Fig. 2. It is an open problem how to exclude such kind of oscillation/concentration effects. This might even not be possible in general since the model might not describe the behavior of both fluids appropriately when, e.g., a lot of small-scale drops are forming. One way out of this problem is to define so-called varifold solution of a two-phase flow, which was first done by Plotnikov [58] in the case of d D 2 for shear-thickening non-Newtonian fluids. Here a general (oriented) varifold V on a domain  is simply a nonnegative measure in M.  Sd 1 /, where Sd 1 denotes the unit sphere in Rd . Here M.  Sd 1 / D C0 .  Sd 1 /0 is the space of finite Radon measures on   Sd 1 , where   Sd 1 is equipped with the product of the Lebesgue and .d  1/-dimensional Hausdorff measure. Moreover, C0 .  Sd 1 / is the closure of smooth, compactly supported functions 'W   Sd 1 ! R. By disintegration, cf. [22, Theorem 2.28], a varifold V can be

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decomposed in a nonnegative measure jV j 2 M./, and a family of probability measures Vx 2 M.Sd 1 /, x 2 , such that Z Z .x; s/ d Vx .s/ d jV j.x/ for all 2 C0 .  Sd 1 /: hV; i D 

Sd 1

Moreover, jV j corresponds to the measure of the “area of the interface,” and Vx defines a probability for the “normal at the interface” for jV j-a.e. x. The reduced boundary @ E of a set of finite perimeter induces naturally a varifold by setting jV j D jrE j and Vx D ıE .x/ for x 2 @ E; where ı denotes the Dirac measure at  2 Sd 1 . Hence the associated varifold VE is Z hVE ; i D

.x; E .x// d jV j.x/

for all

2 C0 .  Sd 1 /:



Now let Ek be a sequence of sets of finite perimeter as above. Then by the weak-

compactness of M.  Sd 1 /, there is a limit varifold V 2 M.  Sd 1 / such that hV; i D lim hVEk ; i k!1

for a suitable subsequence. Hence using C01 ./d , it follows

for all

2 C0 .  Sd 1 /

.s; x/ D tr..I  s ˝ s/r'.x// for ' 2

Z lim hHEk ; i D

k!1

tr..I  s ˝ s/r'.x// d V .s; x/ DW hıV; 'i

(21)

Sd 1

for all ' 2 C01 ./d . Here ıV 2 C01 .I Rd /0 defined as above is called the first variation of the generalized varifold V . Moreover, hrE ; 'i D  lim hrEk ; 'i k!1 Z Z E .x/  ' d jVEk j.x/ D D lim k!1 

s  '.x/ d V .x; s/:

Sd 1

Hence V can be used to describe the limit of HEk as well as the limit of rEk . Now a varifold solution of the two-phase flow is defined as follows: Definition 2 (Varifold solutions). Let  > 0. Then v 2 L1 .0; 1I L2 .Rd // \ L2 .0; 1I H01 .Rd /d /;  2 L1 .0; 1I BV .Rd / \ L1 .Rd  .0; 1/I f0; 1g//; d 1 // V 2 L1 ! .0; 1I M.  S

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1281

are called a varifold solution of the two-phase flow for initial data v0 2 L2 .Rd / and d 0 D C for a bounded domain C 0 R of finite perimeter if the following 0 conditions are satisfied: 1 (i) (9) holds for all ' 2 C.0/ .Rd  Œ0; 1//d with div ' D 0, where hH.t/ ; 'i is replaced by

Z tr..I  s ˝ s/r'.x// d V .s; x/;

hıV .t /; 'i D Rd Sd 1

' 2 C01 ./d :

(ii) The modified energy inequality 1 kv.t /k22 C  kV .t/kM.Sd 1 / 2 Z tZ 1 C 2./jDvj2 dx d  kv0 k22 C kr0 kM./ d 2 R 0

(22)

holds for almost all t 2 .0; 1/. (iii) The compatibility condition Z  hr.t /; 'i D

s  '.x/ d V .x; s/; Sd 1

' 2 C0 ./d;

(23)

holds for almost all t > 0. 0 Here L1 ! .0; T I X / denotes the space of weakly- measurable essentially bounded functions f W .0; T / ! X 0 .

Remark 2. (i) Let .Vx .t /; jV .t/j/, x 2 Rd , denote the disintegration of V .t / 2 M.Rd  Sd 1 / as described above. Then (23) implies that jr.t /j.A/  jV .t /j.A/ for all open sets A and almost all t 2 .0; 1/. Hence jr.t /j is absolutely continuous with respect to jV .t/j and Z

Z f .x/ d jr.t /j D Rd

Rd

f .x/˛t .x/ d jV .t /j;

f 2 C0 .Rd /;

for a jV .t /j-measurable function ˛t W Rd ! Œ0; 1/ with j˛t .x/j  1 almost everywhere. In particular, this implies supp rt  supp V .t / and kr.t /kM  kV .t/kM for almost all t 2 .0; 1/. Hence every varifold solution satisfies the energy inequality (18) for almost every t > 0.

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Moreover, if E.t/ D fx 2 Rd W .x; t / D 1g, t > 0, then (23) yields the relation (

Z Sd 1

s d Vx .t /.s/ D

˛t .x/ E.t/ .x/ if x 2 @ Et 0

else

for jV .t /j-almost every x 2 Rd and almost every t > 0. In other words, the expectation of Vx .t / is proportional to the normal  on the interface described by r and zero away from it. (ii) In general, it is an open problem whether V .t/ is a so-called countably .d  1/rectifiable varifold, which implies that up to orientation Vx .t / is a Dirac measure for jV .t/j-almost every x. Then V .t/ can naturally be identified with a countably .d 1/-rectifiable set – a “surface” – equipped with a density t  0. So far only a sufficient condition for the rectifiability of V .t / in terms of the first variation ıV .t/ is known, cf. [2, Section 4]. (iii) As noted above, the existence of weak solutions to the two-phase flow with surface tension is open. But a general property of varifold solutions is that a varifold solution is a weak solution if the energy equality holds, i.e., (18) holds with equality for almost every t > 0. See [3, Proposition 1.5] for details. Theorem 2 (Existence of Varifold Solutions, [3, Theorem 1.6]). Let  > 0, d D d 2; 3. Then for every v0 2 L2 .Rd / and 0 D C where C 0 R is a bounded 0

C 1 -domain, there is a varifold solution of the two-phase flow with surface tension  > 0 in the sense of Definition 2. In [3, Theorem 1.6] further properties are stated, which can be shown for the constructed varifold. Further and related results: The result was extend by Yeressian [73], where the existence of axisymmetric varifold solutions in the case of axisymmetric initial values in R3 was shown. In the case d D 2 and ˙ D ˙ D 1, existence of varifold solutions was also obtained by Ambrose et al. [21]. Their definitions and statements are slightly different; but the result is essentially the same. Moreover, they discuss possible defects in the surface tension functional. Earlier generalized solutions for the two-phase flow with surface tension were also constructed by Salvi [61]. But in the latter work, the meaning of the mean curvature functional is not specified and can be chosen arbitrarily within in a certain function space. Moreover, a Bernoulli free boundary problem with surface tension was discussed by Wagner [72]. Existence of varifold solutions was also obtained in [14] by a sharp interface limit of a diffuse interface model, which will be discussed in the next section. But the definition of varifold solution and their properties are slightly different. The limit system obtained in this sharp interface limit depends on the scaling of a mobility coefficient in the diffuse interface model. In one case the classical model (2), (3), (4), (5), (6), (7), and (8) is obtained; in another case the system studied in the next section is obtained.

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

2.3

1283

Existence of Weak Solutions for a Navier-Stokes/Mullins-Sekerka System

In this subsection we consider only the case C D  D 1 for simplicity. In this case an alternative model to the classical two-phase flow model (2), (3), (4), (5), (6), (7), and (8) is the following : @t v C v  rv  ˙ v C rp D 0

in ˙ .t /;

(24)

div v D 0

in ˙ .t /;

(25)

D 0

in ˙ .t /;

(26)

j.t/ D  H

on .t /;

(27)

ŒvC  D0

on .t /;

(28)

on .t /;

(29)

on .t /;

(30)

vD0

on @;

(31)

 @  r j@ D 0

on @;

(32)

in 

(33)

C Œ2DvC   C Œp D  H 

  v  V D mΠ r C 

vjtD0 D v0

for t > 0 together with ˙ .0/ D ˙ 0 . The system arises naturally as a sharp interface limit of the diffuse interface models discussed in Sect. 3.1 if the mobility coefficient m does not vanish in the limit. If m D 0 in the system above, then the equations for decouple from the rest of the system and can be deleted from the system. Then the system coincides with the classical model (2), (3), (4), (5), (6), (7), and (8). Here W   .0; 1/ ! R is a new quantity in the system and plays the role of a chemical potential associated to a free energy, which is  Hd 1 restricted to the interface .t /. Moreover, m > 0 is a mobility coefficient, which influences the strength of a (nonlocal) diffusion in the system. The system (26), (27), (30), and (32) for v D 0 is the so-called Mullins-Sekerka system (or two-phase Hele-Shaw system), which arises as sharp interface limit of the CahnHilliard equation, which models phase separation in a two-component mixture. It is well known that solutions of this system show the so-called Ostwald ripening effect in the long-time dynamics, which is the diffusion of mass from smaller droplets to larger droplets until finally one large droplet remains. This effect is also present in the full system (24), (25), (26), (27), (28), (29), (30), (31), (32), and (33). In the following a result on existence of weak solutions for the NavierStokes/Mullins-Sekerka system above is discussed. It is noted that sufficiently smooth solutions of (24), (25), (26), (27), (28), (29), (30), (31), (32), and (33) satisfy the following energy dissipation identity,

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H. Abels and H. Garcke

Z

d d 1 H ..t // dt  Z Z D  2./jDvj2 dx  m jr j2 dx;

d 1 dt 2

jv.t /j2 dx C 



(34)



where .0/ D  and .1/ D C as before. This identity can be verified by multiplying (24) and (26) with v, , resp., integrating and using the boundary and interface conditions (26), (27), (28), (29), (30), (31), and (32). This energy equality motivates the choice of solution spaces in the weak formulation and shows that the regularization introduced for m > 0 yields an additional dissipation term. In particular, one expects .; t / 2 H 1 ./ for almost all t 2 RC and, formally, using Sobolev inequality and (7), that H .; t / 2 L4 ..t // for d  3. This gives some indication of extra regularity properties of the interface in the model with m > 0 and is in big contrast to the classical model (the case m D 0), where no control of the mean curvature of .t / can be derived from the energy identity in a straightforward manner. The following result on existence of weak solutions of (24), (25), (26), (27), (28), (29), (30), (31), (32), and (33) was proved in [16]. Theorem 3 (Existence of Weak Solutions, [16, Theorem 1.1]). Let d D 2; 3, T > 0, let   Rd be a bounded domain with smooth boundary or let  D Td , let .0/ WD  , .1/ WD C and ; m > 0. Then for any v0 2 L2 ./, 0 2 BV .I f0; 1g/ there are v 2 L1 .0; T I L2 .// \ L2 .0; T I H01 ./d /;  2 L1 w .0; T I BV .I f0; 1g//;

2 L2 .0; T I H 1 .//; that satisfy (24), (25), (26), (27), (28), (29), (30), (31), (32), and (33) in the following sense: For almost all t 2 .0; T /, the phase interface @ f.; t / D 1g has a generalized mean curvature vector H.t / 2 Ls .I jr.t /j/d , cf. [60], with s D 4 if d D 3 and 1  s < 1 arbitrary if d D 2, such that Z TZ .v  @t ' C .v  r/v  ' C 2./Dv W D'/ dx dt 0

Z





Z

T

Z

'jtD0  v0 dx D  

H.t /  '.t / d jr.t /j dt 0

(35)



1 .// with 'jtDT D 0, holds for all ' 2 C 1 .Œ0; T I C0;

Z

T

Z

Z .@t

0



Z

T

C div. v// dx dt C

Z

Dm

r  r 0



0 .x/ .0; x/ dx 

dx dt

(36)

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

holds for all

2 C 1 .Œ0; T   / with

 H.t; :/ D .t; :/

r.; t / jrj.; t /

1285

jtDT D 0 and

Hd 1 -a.e. on @ f.t; :/ D 1g

(37)

holds for almost all 0 < t < T . Here Ls .I jr.t /j/ is the standard Lebesgue space with respect to the measure jr.t /j on , and the concept of generalized mean curvature for non-smooth phase interfaces is taken from [60] and can also be found in [16, Definition 4.4]. Remark 3. Equation (35) is the weak formulation of (24), (29), and (33). It is obtained from testing (24) with ' in ˙ .t /, integrating over C .t / [  .t / and using Gauss’ theorem, (29), and (33). Moreover, (36) is a weak formulation of (26), (30), (32), and C .0/ D C 0 . The conditions (25), (28), and (31) are included in the choice of the function spaces, namely, v.t / 2 H01 ./ for almost every t 2 .0; T /, and (27) is formulated in (37). The proof is essentially based on a compactness result of Schätzle [62] for .d  1/dimensional hypersurfaces with mean curvature given as the trace of an ambient Sobolev function in Wp1 .Rd / for p > d2 . For the application of this result, the bound of r 2 L2 .0; T I L2 .//d obtained from (34) is used. Such a control of the curvature of the interface is missing for the classical model (2) (3) (4) (5) (6) (7), and (8), which is one of the main reasons that existence of weak solutions to the latter system is open in general if  > 0.

3

Diffuse Interface Models

In diffuse interface models, a partial mixing of the two incompressible fluids in a thin interfacial region is assumed. In the following two fluids with mass densities  and C are considered. The mass balance equation for the two fluids in local form is given by J˙ D 0 @t ˙ C div b where b J˙ are the mass fluxes of the fluids C and . Introducing the velocities J˙ =˙ , one can rewrite the mass balance as v˙ D b @t ˙ C div.˙ v˙ / D 0:

(38)

The further modeling now crucially depends on the way how an averaged velocity v is defined. Precise choices of v will be given below.

1286

H. Abels and H. Garcke

The mass flux of the two fluids relative to the velocity v is denoted by J ˙  ˙ v J˙ D b and the mass balances are rewritten as @t ˙ C div.˙ v/ C div J˙ D 0

(39)

where J˙ are diffusive flow rates. Defining the total mass  D C C  one obtains @t  C div.v/ C div.JC C J / D 0:

(40)

One observes that the classical continuity equation does not hold if div.JCC J / ¤ 0. Considering a conservation of linear momentum with respect to the above velocity, one obtains @t .v/ C div.v ˝ v/ D div e T

(41)

where e T is the stress tensor which has to be specified by constitutive assumptions. It turns out that e T in general is not an objective tensor, i.e., the tensor is not invariant under a change of observer; see [20] for details. Rewriting (41) with the help of the mass conservation (40), one gets with e J D J C JC .@t v C v  rv/ D div e T C .div e J/  v D div.e T C v ˝e J/  e J  rv: The system now allows for an objective tensor, i.e., a tensor which is frame indifferent, TDe T C v ˝e J

(42)

J/  rv D div T @t v C .v C e

(43)

and one obtains

which is the classical formulation if e J D 0, which is equivalent to JC D J . The work [12] and [20] give more details concerning the objectivity of the mass– momentum system with e J ¤ 0. It remains to specify the averaged velocity v with the help of the individual velocities v and vC . Two choices are used in the literature.

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1287

The volume averaged velocity v D u v C uC vC where u and uC are the volume fractions of the two fluids and the mass averagedvelocity vD

 C v C vC :  

In the following the two modeling variants which result from different choices of the averaged velocity v are discussed.

3.1

Models Based on the Volume Averaged Velocity

In the interfacial zone, the total volume occupied by each fluid is no longer conserved. Insisting on a conservation of volume during the mixing process would lead to the necessity that when fluid C flows out of a region, an amount of fluid  of the same volume would have to enter this region. Defining the specific (constant) density of the unmixed fluid by Q˙ , one introduces the volume fraction u˙ D ˙ =Q˙

(44)

and the above discussion on the volume conservation leads to u C uC D 1

(45)

which states that the excess volume is zero. Multiplying (38) with 1=Q˙ , using u C uC D 1 and the definition of v as the volume averaged velocity gives     C C   C div 0 D @t C vC C v QC Q QC Q D @t .u C uC / C div v D div v: From (39) one derives J˙ D 0 @t u˙ C div.u˙ v/ C div e where one sets e J˙ D J˙ =Q˙ . Because of div =0 and u C uC D 1 we require, see ¯ [13], e J C e JC D J =Q C JC =QC D 0:

(46)

1288

H. Abels and H. Garcke

Taking the difference of these two equations gives for ' D uC  u , the equation @t ' C div.'v/ C div J' D 0

(47)

where J' D JC =QC  J =Q : It is also noted that (44) and (45) together with  D C C  give  D .'/ D QC

1C' 1' C Q ; 2 2

i.e.,  is an affine linear function of '. Using 0 .'/ D .QC  Q /=2, one obtains from (47) JD0 @t  C div.v/ C div e

(48)

where the relation .QC  Q /J' D 2 e J holds; compare (46). Motivated by the discussion in the introduction, one introduces a total energy density e.v; '; r'/ D

 2 jvj C f .'; r'/ 2

asthe sum of a kinetic  and a free energy. As an example one can take f .'; r'/ D O 2" jr'j2 C 1" .'/ , cf. (1). In an isothermal situation, the appropriate formulation of the second law of thermodynamics is given as the following dissipation inequality, see, e.g., [45], as follows d dt

Z

Z

Je   d Hd 1  0

e.v; '; r'/dx C V .t/

@V .t/

where V .t / is a test volume which is transported with the flow, described by v, and Je is a general energy flux which will be specified later. Using a transport theorem and the fact that the test volume is arbitrary, one obtains the local form, see [13, 50],  D WD @t e C div.ve/ C div Je  0:

(49)

One can now use the Lagrange multiplier method of Liu and Müller [50, 55] to derive constitutive relations which guarantee that the second law is fulfilled. Every field .'; v/ which fulfills the dissipation inequality (49), div v D 0 and (47) also fulfill  D D @t e C v  r' C div Je  .@t ' C v  r' C div J' /  0

(50)

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1289

where is a Lagrange multiplier which will be specified later. Using (43), (48) one obtains @t

   jvj2 jvj2 C div jvj2 v D  div e J C .div T  e J  rv/  v 2 2 2   1 2e T D div  jvj J C T v  T W rv: 2



Denoting by f;' and f;r' , the partial derivatives with respect to ' and r' one gets Dt f D f;' Dt ' C f;r'  Dt r' where Dt u D @t u C v  ru is the material derivative. Using Dt r' D rDt '  .rv/T r'

(51)

yields that (50) gives after some computations 

jvj2 D D div Je  e J C TT v  J' C f;r' Dt ' 2



C.f;'   div f;r' /Dt ' .T C r' ˝ f;r' / W rv C r  J'  0: Choosing the chemical potential as

D f;'  div f;r' and jvj2  TT v C J'  f;r' Dt ' 2

J Je D e

one ends up with the dissipation inequality .T C r' ˝ f;r' / W rv  r  J'  0: Often it is convenient, see, e.g., [45], to introduce an extra stress e S and the pressure p such that e S D T C p Id :

1290

H. Abels and H. Garcke

Here, T is the total stress tensor, and e S is a stress tensor from which the part stemming from the hydrostatic pressure is subtracted. It will turn out that e S contains viscous stresses and stresses that can be related to capillary forces. Due to the incompressible condition, the pressure p is still indeterminate; see also [45]. With the stress e S one obtains .e S C r' ˝ f;r' / W rv  r  J'  0 since div v D 0. The term S D e S C r' ˝ f;r' is the viscous stress tensor since it corresponds to irreversible changes of energy due to friction. One now considers specific constitutive assumptions. For a classical Newtonian fluid, one chooses S De S C r' ˝ f;r' D 2.'/Dv for some '-dependent viscosity .'/  0. The simplest form of the flux J' is of Fick’s type J' D m.'/r

where m.'/  0 in order to guarantee that the dissipation inequality is fulfilled. Choosing  f .'; r'/ D O

1 " jr'j2 C .'/ 2 "



gives in conclusion the following diffuse interface model J/  rv  div.2.'/Dv/ C rp D O " div.r' ˝ r'/; @t v C .v C e div v D 0;

O "

(52) (53)

@t ' C v  r' D div.m.'/r /;

(54)

0

(55)

.'/  O " ' D :

It is remarked that QC  Q QC  Q e J' D  m.'/r

JD 2 2 which gives that the term involving e J in the momentum equation vanishes for equal densities, i.e., if QC D Q . In the case of equal densities, one hence recovers the famous “model H ” discussed in Hohenberg and Halperin [46]. The model (52), (53), (54), and (55) was first derived in [13]. However, other diffuse interface models based on a volume averaged velocity were also studied in [29, 37].

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1291

For both models neither global nor local energy inequalities seem to be known. The model of Ding, Spelt, and Shu [37] is given by (52), (53), (54), and (55) with e J being zero which hence drops a term which is important for the dissipation inequality. Using the fact that the pressure can be redefined, there are a few reformulations of (52) which are convenient. Due to the identity

r' D r.f .'; r'//  div.r' ˝ f;r' / it is possible to redefine the pressure as follows pO D p  f .'; r'/ and one obtains instead of (52) @t v C .v C e J/  rv  div.2.'/Dv/ C r pO D r':  2 For the following one assumes that f .'; r'/ D O " jr'j C 2 situations it is more convenient to consider the formulation

.'/ "



(56)

, cf. (1). In some

@t v C .v C e J/  rv  div..'/D.v// C rp    r' r' 2 D div O "jr'j Id  ˝ : jr'j jr'j It turns out that "jr'j

2



r' r' Id  ˝ jr'j jr'j



in some sense converges in the sharp interface limit " ! 0 to a multiple of .Id  ˝ /ı where ı is a surface Dirac distribution concentrated on the interface and Id  ˝  is the projection onto the interface which is up to a factor of the relevant surface stress tensor, cf. (17) below. Moreover, using (48) one obtains that (52) is equivalent to @t .v/ C div.v ˝ .v Ce J//  div.2.'/Dv/ C rp D O " div.r' ˝ r'/:

(57)

Furthermore, in the same way as discussed above, the right-hand side of (57) can be 2 replaced by r' if p is replaced by p C " jr'j C .'/ and for the new pressure, 2 " one obtains @t .v/ C div.v ˝ .v C e J//  div.2.'/Dv/ C rp D r':

(58)

1292

3.2

H. Abels and H. Garcke

Model Based on the Mass Averaged Velocity

A model based on a mass averaged velocity was derived by Lowengrub and Truskinovsky [52]. They define the averaged velocity v as vD

 v C C vC : 

In this case the mass balance becomes @t  C div.v/ D 0;

(59)

which is obtained by adding both mass balances in (38). Defining the mass concentrations ˙ c˙ D  one now introduces the concentration difference c D cC  c ; as phase field variable. One now wants to model the mixing of two incompressible fluids and in the following assumes that the total density  depends only on the concentration difference c. Hence, it is assumed that there exists a function O W Œ1; 1 ! .0; 1/ such that  D .c/. O Adapting a model of a simple mixture, see [47], one obtains C  C D 1: QC Q This condition is just the assumption of zero excess volume which was discussed earlier. In this case the functional dependence between  and c D cC  c is given as (one has to use cC C c D 1)  .c/ O D

1 1 .1 C c/=QC C .1  c/=Q 2 2

1 :

(60)

However, in what follows one allows for a more general relation  D .c/. O Taking the difference of the mass balances (39) now yields (using ˙ D .c/c O and c D ˙ cC  c ) @t .c/ C div.c v/ C div j D 0

(61)

which is, using (59), equivalent to .@t c C v  rc/ C div j D 0

(62)

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1293

where j D JC  J . The equation (62) has to be supplemented with the momentum equation (41) which, using (59), can be rewritten as .c/.@t v C v  rv/ D div e T: As in Sect. 3.1 one requires a free energy inequality D WD @t e C div.ev/ C div je  0 where e.v; c; rc/ D

.c/ O O fO .c; rc/: jvj2 C .c/ 2

It turns out that for the mass averaged velocity, it is more convenient to work with a free energy density fO per unit mass. For solutions of the mass and momentum equations, one obtains, using Lagrange multipliers  and , @t e Cdiv.ve/Cdiv je  .Dt  C div v/ .Dt c Cdiv j/v.Dt vdiv e T/  0 which is equivalent to fO;c Dt c C fO;rc  .Dt rc/ C div je   Dt     div v  Dt c  div. j/ C r  j C div.e TT v/  e T W rv  0: Using the identity fO;rc  .Dt rc/ D div.fO;rc Dt c/  .div.fO;rc //Dt c  .rv/ W .rc ˝ fO;rc /; which follows using (51), one obtains Dt c .fO;c  div.fO;rc /   O0 .c/  / Crv W .  Id rc ˝ fO;rc  e T/ TT v/  0: Cr  j C div.je C fO;rc Dt c  j  e This is true for all solutions of the mass and momentum balance equations if

D

1 . div.fO;rc / C fO;c   O0 .c//; 

je D j  fO;rc Dt c C e TT v;

1294

H. Abels and H. Garcke

and rv W S  r  j  0; where SDe T C   Id Crc ˝ fO;rc : Interpreting   as the pressure, i.e., setting  D

p 

and making the specific choices S D .c/.rv C rvT / C .c/ div v Id j D m.c/r

leads the model of Lowengrub and Truskinovsky [52] Dt c  div.m.c/r / D 0 1  div.fO;rc / C fO;c  O0 .c/p D 

 @t  C div.v/ D 0 Dt v  div.2.c/Dv/  r..c/ div v/ C rp D  div.rc ˝ fO;rc /:

(63) (64) (65) (66)

One can now use Dt  D O0 .c/Dt c to rewrite (65) as div v D ./ O 2 O0 div.m.c/r /: Choosing 1 fO .c; rc/ D Q ."jrcj2 C .c// " gives that (64) and (66) become

D

Q "

0

.c/ 

p 0 Q " div.rc/ O .c/  2  

and Dt v  div.2.c/Dv/  r..c/ div v/ C rp D Q " div.rc ˝ rc/:

(67)

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1295

Sometimes also the following ansatz for fO is chosen O 1 fO .c; rc/ D ."jrcj2 C .c//; .c/ O " see, e.g., [1, 4]. In this case (64) and (66) are given as  D

O "

0

.c/  O " c C

1 0 O .c/p 

(68)

and Dt v  div.2.c/Dv/  r..c/ div v/ C rp D O " div.rc ˝ rc/: In the case of a simple mixture, see (60), one has .c/ O D

1 ˛ C ˇc

with ˇD

1 1 1 1  ; ˛D C : 2Q 2QC 2QC 2Q

One hence obtains O0 .c/ D 

ˇ D ˇ .c/ O 2 .˛ C ˇc/2

this then implies that (67) has the following simple divergence structure div.v  ˇm.c/r / D 0: In addition, (68) becomes

D

O "

0

.c/ 

O " c  ˇp: 

The major difference between the model studied in Sect. 3.1 which was based on a volume averaged velocity and the model studied in this section is that the model which is based on a mass balanced velocity leads to a velocity which in general is not divergence free and to a pressure-dependent chemical potential. Both facts make the analysis of this model much more involved, cf. Sect. 3.5 below. It is pointed out that both models reduce to “model H ” in the case that the two mass densities  and C are the same.

1296

3.3

H. Abels and H. Garcke

Analytic Results in the Case of Same Densities

In this subsection the mathematical results concerning existence and uniqueness of weak and strong solutions and partly their qualitative behavior for large times in the case that .c/  1: are discussed. In this case (52), (53), (54), and (55) as well as (63), (64), (65), and (66) reduce to the system @t v C v  rv  div.2.c/Dv/ C rp D " div.rc ˝ rc/; div v D 0;

(69) (70)

@t c C v  rc D div.m.c/r /;

D " c C

1 "

0

.c/;

(71) (72)

where one has chosen fO .c; rc/ D "jrcj2 C 1" .c/ and O D 1. The system is studied in   .0; T /, T 2 .0; 1, where   Rd , d D 2; 3, is a suitable domain, e.g., a bounded sufficiently smooth domain. It has to be closed by suitable initial and boundary conditions. The standard choice, which was done for most mathematical results, consists of vj@ D 0  @  rcj@ D  @  r j@ D 0

on @  .0; T /;

(73)

on @  .0; T /;

(74)

.v; c/jtD0 D .v0 ; c0 /

(75)

for suitable initial values .v0 ; c0 /. For all results mentioned in the following, it is assumed that W R ! .0; 1/ is sufficiently smooth, strictly positive and bounded. For existence of weak solutions, continuity of  is usually sufficient. But more smoothness is needed for higher regularity and uniqueness. In the following one assumes for simplicity that  D 1. However, the results will also be true for general positive constant . Before the analytic results are discussed, it is noted that every sufficiently smooth solution of (69), (70), (71), (72), (73), (74), and (75) on a suitable domain  (e.g., bounded with Lipschitz boundary) satisfies d E.c.t /; v.t // dt Z Z 2 D  2.c.x; t //jDv.x; t /j dx  m.c/jr .x; t /j2 dx 

(76)



for all t 2 .0; T /, where E.c; v/ D Ef ree .c/ C Eki n .v/ and " Ef ree .c.t // D 2 1 Eki n .v.t // D 2

Z

2

Z

jrc.x; t/j dx C 

Z





jv.x; t /j2 dx:

.c.x; t // dx; " (77)

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1297

This is a consequence of the energy dissipation inequality (50) integrated with respect to x 2  together with the boundary conditions (73) and (74). Alternatively, it follows from testing (69) with v, (71) with and (72) with @t c as well as integration by parts. Moreover, it is often useful to replace (69) by @t v C v  rv  div.2.c/Dv/ C rg D rc

(78)

with the new pressure g D p C 2" jrcj2 C "1 .c/, cf. (58). The analytic results often differ by their assumptions on the mobility m and the (homogeneous) free energy density . Therefore, a brief overview of these assumptions is given. It is always assumed that mW R ! Œ0; 1/ is sufficiently smooth and bounded. Most of the time it is assumed that the mobility coefficient m is nondegenerate, which means that m is strictly positive. In the case of a degenerate mobility, it is assumed that m.c/ D 0 if and only if c 2 fa; bg, where a; b 2 R represent the pure phases, which are a D 1; b D C1 in the derivation above. Moreover, a suitable behavior of m.c/ as c ! ˙1 is assumed in the degenerate case. A canonical example is m.c/ D m0 .1  c 2 / with m0 > 0. A mathematical advantage of the degenerate case is that it prevents the concentration c from leaving the physical interval Œ1; 1. But in most cases, one even assumes that m is a positive constant (e.g., m  1). A standard choice for is that W R ! R is a sufficiently smooth function satisfying suitable growth conditions for c ! ˙1. From the physical point of view, it should be of double well type, which in particular means that .c/  0 with equality if and only if c 2 f˙1g. A canonical example is c 2 R:

.c/ D .1  c 2 /2 ;

But choosing such a smooth free energy density has the mathematical disadvantage that there is no mechanism known, which prevents c from leaving the physical reasonable interval Œ1; 1 even if the initial value c0 attains only values in Œ1; 1. One possibility to ensure that c stays in Œ1; 1 is to choose as a singular free energy density, e.g., of the form .c/ D

c ..1 C c/ ln.1 C c/ C .1  c/ ln.1  c//  c 2 2 2

(79)

if c 2 Œ1; 1 and .c/ D C1 else. Here 0 < < c and 0 ln 0 WD 0 D lims!0C s ln s. Essential properties of this choice of are 0

.s/ !.1;1/3s!˙1 ˙1;

inf

s2.1;1/

00

.s/   c > 1:

Using these properties it is possible to prove existence of weak (or strong) solutions with c.x; t/ 2 .1; 1/ for almost every x 2 , t 2 .0; T /, in many situations if the mobility is nondegenerate. Instead of more general free energy densities with the latter properties can be considered. More details will be given below. Now the analytic results in the case of matched densities (i.e.,   const:) are discussed in more detail. A first result on existence of strong solutions, in the case

1298

H. Abels and H. Garcke

that  D R2 and is a suitably smooth double well potential, was obtained by Starovoitov [67]. More complete results were presented by Boyer [27] in the case of a shear flow in a periodic channel. More precisely, it is assumed that  D fx D .x 0 ; xd / 2 Rd W xd 2 .1; 1/g; d D 2; 3; with periodic boundary conditions with respect to x 0 2 Rd 1 and vjxn D˙1 D ˙Ue1 with U > 0. Moreover, either the mobility m is nondegenerate and is a suitable smooth potential or m is degenerate and D 1 C 2 , where 00 1 W .1; 1/ ! R is convex such that m 1 has a continuous extension on Œ1; 1 and 2 2 C 2 .Œ1; 1/. These assumptions are satisfied for as in (79) if

2 1 .c/ D 2 ..1 C c/ ln.1 C c/ C .1  c/ ln.1  c// and m.c/ D 1  c . In the case of nondegenerate mobility, the existence of global weak solutions, which are strong and unique if either d D 2 or d D 3 and t 2 .0; T0 / for a sufficiently small T0 > 0, was shown in [27]. Furthermore, in the degenerate case the existence of weak solutions with c.t; x/ 2 Œ1; 1 almost everywhere is proved. The system (69), (70), (71), and (72) was also briefly discussed by Liu and Shen [51]. In the case of a singular free energy density and for constant positive mobility, existence of weak solutions, strong well-posedness, and convergence for large times was proven in [5]. These results are now described in more detail. Assumption 1. Let   Rd be a bounded domain with C 3 -boundary and let C .Œ1; 1/ \ C 2 ..1; 1// such that 0

lim

s!˙1

00

.s/ D ˙1;

.s/  ˛

2

for all s 2 .1; 1/

for some ˛ 2 R. Furthermore, one assumes that  2 C 2 .Œa; b/ is a positive function. Finally, .s/ is extended by C1 if s … Œ1; 1. Definition 3 (Weak Solution). Let 0 < T  1. A triple .v; c; / such that v 2 BCw .Œ0; T I L2 .// \ L2 .0; T I H01 ./d /; c 2 BCw .Œ0; T I H 1 .//;

0

.c/ 2 L2loc .Œ0; T /I L2 .//; r 2 L2 .  .0; T //d

is called a weak solution of (69), (70), (71), (72), (73), (74), and (75) on .0; T / if Z

T

Z

Z v  @t

 0



Z

T

2.c/Dv W D 0



v0 

T

Z

jtD0 dx C



Z

C

Z

dx dt 

Z

T

Z

dx dt D

.v  rv/  0

rc  0



dx dt



dx dt

(80)

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . . 1 2 C.0/ .Œ0; T /  /d with div

for all Z

T

Z

0



Z

T



Z 

Z

Z

T

Z



m.c/r  r' dx dt 

0

Z

v  rc ' dx dt; 0

T

Z

0

' dx dt D 0

T

c0 'jtD0 dx C

D Z

D 0,

Z c@t ' dx dt 



0

1299

Z

T

(81)

Z

.c/' dx dt C



rc  r' dx dt 0

(82)



1 for all ' 2 C.0/ .Œ0; T /  /, and if the (strong) energy inequality

Z tZ E.v.t /; c.t // C t0



 2.c/jDvj2 C jr j2 dx d



 E.v.t0 /; c.t0 //

(83)

holds for almost all 0  t0 < T including t0 D 0 and all t 2 Œt0 ; T /. Note that for the weak formulation (80), we have used (78) instead of (69). L2 ./

, BCw .Œ0; T I X / is the space of Here L2 ./ D f' 2 C01 ./d W div ' D 0g all weakly continuous and bounded functions f W Œ0; T  ! X and L2loc .Œ0; 1/I X / the space of all strongly measurable f W Œ0; 1/ ! X such that f jŒ0;T  2 L2 .0; T I X / for all T < 1, where X is a Banach space. Furthermore, in the following BUC .I I X / denotes the space of all bounded and uniformly continuous f W I ! X if I  R is an interval. Due to (76) sufficiently smooth solutions satisfy (83) with equality for all 0  t0  t < T . Moreover, this estimate motivates v 2 L1 .0; T I L2 .// \ L2 .0; T I H01 ./d /; c 2 L1 .0; T I H 1 .//;

r 2 L2 .  .0; T //d :

As usual weak solutions are constructed by solving a suitable system, which approximates (69), (70), (71), (72), (73), (74), and (75) and satisfies the same kind of energy inequality. Then one passes to the limit using the bounds in the spaces above. To this end, one of the crucial points is to obtain a suitable bound on 0 .c/. To this end the assumptions on due to Assumption 1 play an essential role. If one 2 defines 0 .s/ D .s/ C ˛ s2 , then 0 2 C .Œ1; 1/ \ C 2 ..1; 1// is convex and satisfies 0 .s/

!s!˙1 ˙1:

1300

H. Abels and H. Garcke

If one replaces functional on

by

0

in Ef ree , one obtains a lower semicontinuous, convex

L2.m/ ./ WD ff 2 L2 ./ W

1 jj

Z f .x/ dx D mg; 

R 1 with m WD jj  c0 .x/ dx. Its subgradient plays an important role in the analysis of (71) and (72) and can be characterized as follows: Theorem 4 (Subgradient Characterization, [18, Theorem 4.3]). Let 0 be as above. Moreover, one sets 0 .x/ D C1 for x 62 Œ1; 1 and let E0 W L2.m/ ./ ! .1; C1 be defined as Z  E0 .c/ WD 

jrcj2 C 2

 0 .c/

dx

if c 2 H 1 ./ with c.x/ 2 Œ1; 1 almost everywhere and E0 .c/ D C1 else. Moreover, let @E0 be its subgradient with respect to the L2 -inner product. Then n D.@E0 / D c 2 H 2 ./ \ L2.m/ ./ W 0 0 .c/

2 L2 ./;

00 2 0 .c/jrcj

2 L1 ./; @  rcj@ D 0



and 0 0 .c/;

@E0 .c/ D  c C P0

(84)

where P0 W L2 ./ ! L2.0/ ./ is the orthonormal projection onto L2.0/ ./. Moreover, there is some C > 0 independent of c 2 D.@E0 / such that kck2H 2 ./ C k 00 .c/k2L2 ./ Z   00 2 2 2 C 0 .c.x//jrc.x/j dx  C k@E0 .c/kL2 ./ C kckL2 ./ C 1 :

(85)



The result was proven by Abels and Wilke in [18, Theorem 4.3]. Formally, one can obtain (85) by multiplying @E0 .c/ D  c C P0 00 .c/ by  c. This yields Z

Z @E0 .c/ c dx D

 

2

Z

j cj dx  

Z D



P0 . 

j cj2 dx C

Z



r „

D

0 0 .c// c 0 0 .c/

ƒ‚

dx

 rc dx  k ck2L2 ./ : …

00 2 0 .c/jrcj 0

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1301

Using regularity results for elliptic equations and Young’s inequality, one obtains (85) formally. These formal arguments are justified rigorously in the proof of [18, Theorem 4.3]. Using (85) together with the a priori estimates for c and from the energy inequality, one obtains a bound on r 2 c; 0 .c/ 2 L2loc .Œ0; 1/I L2 .//. Based on this one obtains: Theorem 5 (Existence of Weak Solutions, [5, Theorem 1]). Let m > 0 be independent of c. Then for every v0 2 L2 ./, c0 2 H 1 ./ with c0 .x/ 2 Œ1; 1 almost everywhere there is a weak solution .v; c; / of (69), (70), (71), (72), (73), (74), and (75) on .0; 1/. Moreover, if d D 2, then (83) holds with equality for all 0  t0  t < 1. Finally, every weak solution on .0; 1/ satisfies 1

r 2 c;

0

.c/ 2 L2loc .Œ0; 1/I Lr .//;

t2 1Ct

1 2

c 2 BUC .Œ0; 1/I Wq1 .//

(86)

where r D 6 if d D 3 and 1 < r < 1 is arbitrary if d D 2 and q > 3 is independent of the solution and initial data. If additionally c0 2 HN2 ./ WD fu 2 H 2 ./ W @  ruj@ D 0g and  c0 C 00 .c0 / 2 H 1 ./, then it holds c 2 BUC .Œ0; 1/I Wq1 .//. The inclusions r 2 c; 0 .c/ 2 L2loc .Œ0; 1/I Lr .// in (86) follows from a generalization of (85), Theorem 4, resp., for Lr ./ instead of L2 ./, cf. [5, Lemma 2]. For further regularity studies and uniqueness results, it is important that Theorem 5 provides c 2 BUC .ı; 1I Wq1 .// for some q > d and for all ı > 0 and ı D 0 for suitable initial data. This makes it possible to use a result on maximal regularity for an associated Stokes system with variable viscosity, cf. [5, Proposition 4], to conclude higher regularity for the velocity v in the case of small or large times and in the case d D 2, which is enough to obtain a (locally) unique solution. Then one obtains: Theorem 6 (Uniqueness, [5, Proposition 1]). Let m > 0 be independent of c, 0 < T  1, q D 3 if d D 3 and let q > 2 if d D 2. Moreover, assume that 1 v0 2 Wq;0 ./ \ L2 ./ and let c0 2 H 1 ./ \ C 0;1 ./ with c0 .x/ 2 Œ1; 1 for all x 2 . If there is a weak solution .v; c; / of (69), (70), (71), (72), (73), (74), and (75) on .0; T / with v 2 L1 .0; T I Wq1 .// and rc 2 L1 .  .0; T //, then any weak solution .v0 ; c 0 ; 0 / of (69) (70), (71), (72), (73), (74), and (75) on .0; T / with the same initial values and rc 0 2 L1 .  .0; T //d coincides with .v; c; /. 1Cj

For the following one denotes V2 ./ D H 1Cj ./d \ H01 ./d \ L2 ./, j D 0; 1. Moreover, for s 2 .0; 1/ one defines V21Cs ./ D .V21 ./; V22 .//s;2 , where .:; :/s;q denotes the real interpolation functor.

1302

H. Abels and H. Garcke

Theorem 7 (Regularity of Weak Solutions, [5, Theorem 2]). Let m > 0 be independent of c and let c0 2 HN2 ./ such that Ef ree .c0 / < 1 and  c0 C 0 .c0 / 2 H 1 ./. (i) Let d D 2 and let v0 2 V21Cs ./ with s 2 .0; 1. Then every weak solution .v; c/ of (69), (70), (71), (72), (73), (74), and (75) on .0; 1/ satisfies 0

0

v 2 L2 .0; 1I H 2Cs .// \ H 1 .0; 1I H s .// \ BUC .Œ0; 1/I H 1Cs" .// for all s 0 2 Œ0; 12 / \ Œ0; s and " > 0 as well as r 2 c; 0 .c/ 2 L1 .0; 1I Lr .// for every 1 < r < 1. In particular, the weak solution is unique. (ii) Let d D 2; 3. Then for every weak solution .v; c; / of (69), (70), (71), (72), (73), (74), and (75) on .0; 1/ there is some T > 0 such that v 2 L2 .T; 1I H 2Cs .// \ H 1 .T; 1I H s .// \ BUC .ŒT; 1/I H 2" .// for all s 2 Œ0; 12 / and all " > 0 as well as r 2 c; 0 .c/ 2 L1 .T; 1I Lr .// with r D 6 if d D 3 and 1 < r < 1 if d D 2. (iii) If d D 3 and v0 2 V2sC1 ./, s 2 . 21 ; 1, then there is some T0 > 0 such that every weak solution .v; c/ of (69), (70), (71), (72), (73), (74), and (75) on .0; T0 / satisfies 0

0

v 2 L2 .0; T0 I H 2Cs .// \ H 1 .0; T0 I H s .// \ BUC .Œ0; T0 I H 1Cs" .// for all s 0 2 Œ0; 12 / and all " > 0 as well as r 2 c; 0 .c/ 2 L1 .0; T0 I L6 .//. In particular, the weak solution is unique on .0; T0 /. The proof of the latter theorem is essentially based on the fact that c 2 BUC .Œ0; 1/I Wq1 .// for some q > d , which implies that cW   Œ0; 1/ ! R is uniformly continuous. This makes it possible to use regularity results for the Stokes system with variable viscosity .c/, which is the linearization of the right-hand side of (69), together with regularity results for the Cahn-Hilliard equation with convection term (71) and (72). Similar results on existence of weak solutions can also be obtained for the socalled double obstacle potential for , i.e., ( .c/ D

 2c c 2

if c 2 Œ1; 1;

C1

else:

But in this case (72) has to be replaced by the differential inclusion

C c C c c 2 @IŒ1;1 .c/;

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1303

Fig. 3 Logarithmic free energy in (79) for ! 0

where IŒ1;1 is the indicator function of Œ1; 1, i.e., IŒ1;1 .c/ D 0 if c 2 Œ1; 1 and IŒ1;1 .c/ D C1 else. This double obstacle potential is the pointwise limit of in (79), when ! 0, cf. Fig. 3. It can also be shown that the corresponding solutions of (69), (70), (71), (72), (73), (74), and (75) converge as ! 0 to solutions of the system (69), (70), (71), (72), (73), (74), and (75), cf. [1, Section 6.5] or [6]. Because of the regularity of any weak solution for large times, it is possible to prove convergence to stationary solutions as t ! 1. Theorem 8 (Convergence to Stationary Solution, [5, Theorem 3]). Assume that W .1; 1/ ! R is analytic and let .v; c; / be a weak solution of (69), (70), (71), (72), (73), (74), and (75). Then .v.t /; c.t // *t!1 .0; c1 / in H 2" ./d H 2 ./ for all " > 0 and for some c1 2 H 2 ./ with 0 .c1 / 2 L2 ./ solving the stationary Cahn-Hilliard equation  c1 C

0

.c1 / D const:

@  rc1 j@ D 0 Z Z c1 .x/ dx D c0 .x/ dx: 

in ;

(87)

on @;

(88) (89)



The proof is based on the so-called Lojasiewicz-Simon inequality, cf. [5] for details. To prove this inequality, it is important that W .1; 1/ ! R is analytic, which is the case for the canonical example (79). The system (69), (70), (71), and (72) was also considered in the case of nonNewtonian fluids of power-law type. In this case 2.c/Dv in (69) is replaced by general viscous stress tensor S.c; Dv/, which satisfies suitable growth conditions with respect to an exponent p > 1. First analytic results in this case were obtained by Kim, Consiglieri, and Rodrigues [48]. They proved existence of weak solutions

1304

H. Abels and H. Garcke

C2 in the case p  3d , d D 2; 3, using monotone operator techniques. In [44] d C2 Grasselli and Pražak discussed the longtime behavior of solutions of the system in C2 the case p  3d , d D 2; 3, in the case of periodic boundary conditions and d C2 a regular free energy density. For the same p existence of weak solutions with a singular free energy density f was proved by Bosia [26] in the case of a bounded domain in R3 . Moreover, the longtime behavior was studied. Finally, existence of weak solutions was shown by Abels, Diening, and Terasawa [11] in the case that p > d2d using the parabolic Lipschitz truncation method for divergence-free vector C2 fields developed by Breit, Diening, and Schwarzacher [30].

3.4

Analysis for the Model with General Densities Based on the Volume Averaged Velocity

In this subsection analytic results of the system (52), (53), (54), and (55) are discussed, i.e., J/  rv  div.2.'/Dv/ C rp D " div.r' ˝ r'/; @t v C .v C e div v D 0;

(91)

@t ' C v  r' D div.m.'/r /;

D " ' C

(90)

1 "

0

.'/

(92) (93)

in .0; T /, where   Rd , d D 2; 3, is a bounded domain with smooth boundary and 1C' 1' C Q ; 2 2 QC  Q QC  Q e JD J' D  m.'/r : 2 2  D .'/ D QC

(94)

The system is closed with the boundary and initial conditions (73), (74), and (75). Here one sets O D 1 for simplicity. Smooth solutions of (90), (91), (92), and (93) together with (73), (74), and (75) satisfy the same energy dissipation identity as in the case of same densities, i.e., (76), where c is replaced by ' and  D .'/ in (77). In particular, this yields a priori bounds for v 2 L1 0; 1I L2 .// \ L2 .0; 1I H01 ./d /; ' 2 L1 .0; 1I H 1 .//; r 2 L2 .0; 1I L2 .//d if m.'/  m0 > 0: as in the case of same densities.

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1305

So far there are only few results on existence of solutions to the system above. The system was discussed by Abels, Depner, and Garcke in [10] and [9], where existence of weak solutions in the case of singular free energies with nondegenerate and degenerate mobility, respectively, was shown. More precisely, in the nondegenerate case the following result was shown: Theorem 9 (Existence of Weak Solutions, [10, Theorem 3.4]). Let m 2 C 1 .R/ be bounded such that infs2R m.s/ > 0, let Assumption 1 hold true and assume that 00 .s/ additionally lims!˙1 0 .s/ D C1. Then for every v0 2 L2 ./ and '0 2 H 1 ./ R 1 with j'0 j  1 almost everywhere and jj  '0 dx 2 .1; 1/ there exists a weak solution .v; '; / of (90), (91), (92), and (93) together with (73), (74), and (75) such that v 2 BCw .Œ0; 1/I L2 .// \ L2 .0; 1I H01 ./d /; ' 2 BCw .Œ0; 1/I H 1 .// \ L2loc .Œ0; 1/I H 2 .//;

0

.'/ 2 L2loc .Œ0; 1/I L2 .//;

2 L2loc .Œ0; 1/I H 1 .// with r 2 L2 .0; 1I L2 .//d : Here the definition of weak solutions is similar to Definition 3; see [10, Definition 3.3] for the details. The structure of the proof of Theorem 9 is as follows: System (90), (91), (92), and (93) is first approximated with the aid of a semi-implicit time discretization, which satisfies the same kind of energy identity as the continuous system. Hence one obtains a priori bounds for vN 2 L1 .0; 1I L2 .// \ L2 .0; 1I H01 ./d /; ' N 2 L1 .0; 1I H 1 .//; r N 2 L2 .0; 1I L2 ./d / if m.'/  m0 > 0; where .vN; ' N; N / are suitable interpolations of the time-discretized system with discretization parameter h D N1 . In order to pass the limit N ! 1, it is essential to obtain a bound for ' N 2 L2loc .Œ0; 1/; H 2 .//;

0

.' N / 2 L2loc .Œ0; 1/; L2 .//;

which follows from Theorem 4. The latter theorem is also used to obtain existence of solutions for the discrete-time system with the aid of the Leray-Schauder principle and the theory of monotone operators. In the case of degenerate mobility, it is assumed that ‰ 2 C 1 .R/, ( m.s/ D

1  s2

if s 2 Œ1; 1;

0

else

1306

H. Abels and H. Garcke

and  and  are as in Assumption 1. In this case one does not obtain an a priori J WD bound for r in L2 ..0; T /  /. Instead one obtains an a priori bound for b p m.'/r and J WD m.'/r . There one has to avoid r in the weak formulation and has to formulate the equations in terms of J. More precisely, weak solutions are defined as follows, cf. [9, Definition 3.3]. Definition 4. Let T 2 .0; 1/, v0 2 L2 ./ and '0 2 H 1 ./ with j'0 j  1 almost everywhere in . Then the triple .v; '; J/ with the properties v 2 BCw .Œ0; T I L2 .// \ L2 .0; T I H01 ./d /; ' 2 BCw .Œ0; T I H 1 .// \ L2 .0; T I H 2 .// with j'j  1 a.e. in QT ; J 2 L2 .0; T I L2 ./d / and .v; '/ jtD0 D .v0 ; '0 / is called a weak solution of (90), (91), (92), and (93) together with (73), (74), and (75) if the following conditions are satisfied: Z

Z

T

Z



v  @t Z

Z

T

2.'/Dv W D 0

Z

T

Z

T

Z

0

.v ˝ 

QC Q J/ 2

Wr

dx dt

(95)

dx dt

Z

T



D 0,

Z

Z

' @t  dx dt C



Z

dx dt



2 C01 .  .0; T //d with div

for all

T 0

' r'  0

Z

dx dt 



D

Z

div.v ˝ v/  

0

C

Z

dx dt C



0

T

T

Z

.v  r'/  dx dt D 

0

J  r dx dt 0

(96)



for all  2 C01 ..0; T /I C 1 .// and Z

T

Z J   dx dt

0



Z

T

(97)

Z



D 0

0

 .'//  ' div.m.'// dx dt



for all  2 L2 .0; T I H 1 ./d / \ L1 .  .0; T //d which fulfill @  j@ D 0 on @  .0; T /.

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1307

Here C01 .  .0; T //; C01 ..0; T /I X / is the set of all smooth functions 'W   .0; T / ! R, 'W .0; T / ! X with compact support. It is noted that (97) is a weak formulation of J D m.'/ r



0

 .'/  ' :

Theorem 10 (Existence of Weak Solutions, [9, Theorem 3.5]). Let the previous assumptions hold, v0 2 L2 ./ and '0 2 H 1 ./ with j'0 j  1 almost everywhere in . Then there exists a weak solution .v; '; J/ of (90), (91), (92), and (93) together 2 b with (73), (74), and (75) in pthe sense of Definition 4. Moreover for some J 2 L . J and .0; T // it holds that J D m.'/ b Z tZ

2.'/ jDvj2 dx d C

E.'.t /; v.t // C s



Z tZ s

jb Jj2 dx d



(98)

 Etot .'.s/; v.s// for all t 2 Œs; T / and almost all s 2 Œ0; T / including s D 0, where E.'.t /; v.t // is defined as in (76) with c.t / replaced by '.t/. In particular, J D 0 a.e. on the set fj'j D 1g. The theorem is proved by approximating m by a sequence of strictly positive mobilities m" and by " .s/

WD

.s/ C ".1 C s/ ln.1 C s/ C ".1  s/ ln.1  s/;

s 2 Œ1; 1;

where " > 0. Then existence of weak solutions .v" ; '" ; " / for " > 0 follows from Theorem 9. In order to pass to the limit, one uses the energy inequality (83). But this does not give a bound for '" 2 L2 .0; T I H 2 .//, which is essential to pass to the limit in the weak formulation of (90). In order to obtain this bound, one tests the weak formulation of (92) with G"0 .'" /, where G 00 .s/ D m"1.s/ for s 2 .1; 1/ and G"0 .0/ D G" .0/ D 0; see [9, Proof of Lemma 3.7] for the details. Existence of weak solutions of (90), (91), (92), and (93) together with (69), (70), (71), and (72) was proven in the case of power-law-type fluids of exponent p > 2d C2 , d D 2; 3, in [8]. More precisely, 2.'/Dv in (90) is replaced by S.'; Dv/, d C2 d d where SW R  Rdsyd m ! Rsy m satisfies jS.s; M/j  C .jMjp1 C 1/; jS.s1 ; M/  S.s2 ; M/j  C js1  s2 j.jMjp1 C 1/; S.s; M/ W M  !jMjp  C1 for all M 2 Rdsyd m , s; s1 ; s2 2 R and some C; C1 ; ! > 0. Furthermore, the case of constant, positive mobility together with a suitable smooth free energy density is considered. Unfortunately, in this case there is no mechanism, which enables to

1308

H. Abels and H. Garcke

show that 2 Œ1; 1. Hence one has to modify , defined as in (94) for ' 2 Œ1; 1, outside of Œ1; 1 suitably such that it stays positive. But then (48) is no longer valid, and one obtains instead @t  C div.v C b J/ D R;

where R D r

@  r : @'

(99)

Here R is an additional source term, which vanishes in the interior of f' 2 Œ1; 1g. In order to obtain a local dissipation inequality and global energy estimate, the equation of linear momentum (90) has to be modified to   v %@t v C .%v C b J/  rv C R  div S.'; Dv/ C rp D " div r' ˝ r' : 2 Under these assumptions existence of weak solutions is shown with the aid of the so-called L1 -truncation method, cf. [8] for the details.

3.5

Analysis for the Model with General Densities Based on the Mass Averaged Velocity

In the following the known results on existence of weak and strong solutions for the model by Lowengrub and Truskinovsky [52] are discussed, i.e., one considers @t v C v  rv  div S.c; Dv/ C rp D " div.jrcjq2 rc ˝ rc/;

(100)

@t  C div.v/ D 0;

(101)

@t c C v  rc D div.m.c/r /;  D 1

@ p  div..c/jrcjq2 rc/ C  @c

0

.c/;

(102) (103)

cf. (63), (64), (65), and (66), in   .0; T /, where  D .c/ O with 1 1 1c 1 1Cc D C ; .c/ Q 2 QC 2

S.c; Dv/ D 2.c/Dv C .c/ div v Id;

 Rd , d D 2; 3, is a bounded domain with C 3 -boundary and T 2 .0; 1. Moreover, one chooses q

jrcj fO .c; rc/ D "q1 C q

.c/ "

for some q  2. Usually one chooses q D 2 for these kinds of diffuse interface models. But for proving existence of weak solutions, it is necessary so far to choose q > d . The reasons will be explained below.

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1309

The system is closed by adding the boundary and initial conditions @  vj@ D @  S.c; Dv/ C  v j@ D 0

on @  .0; T /;

(104)

@  rcj@ D @  r j@ D 0

on @  .0; T /;

(105)

in ;

(106)

.v; c/jtD0 D .v0 ; c0 /

where 0 <  < 1 is a friction coefficient and denotes the tangential part of a vector field. For the analysis it is important to choose Navier boundary conditions for v (104) instead of no-slip boundary conditions vj@ D 0 as before since this makes it possible to estimate the pressure suitably. In the following it is assumed that Q ¤ QC , that ; m; ; W R ! R are sufficiently smooth and that ; ; m are strictly positive and bounded; see [4, 7] for the precise assumptions. Similar as for the other models, smooth solutions of (100), (101), (102), (103), (104), (105), and (106) satisfy the energy dissipation identity d E.c.t /; v.t // dt Z Z   D 2.c/jDvj2 C .c/j div vj2 dx  m.c/jr j2 dx 

(107)



for all t 2 .0; T /, where E.c; v/ D Ef ree .c/ C Eki n .v/ and Z

"q1

Ef ree .c.t // D Z



jrc.x; t/jq dx C q

Z 

.c.x; t // dx; "

2

Eki n .v.t // D

.c.x; t // 

jv.x; t /j dx: 2

In order to get a priori estimates for the construction of weak solutions, it is essential that  D .c/ O stays positive. Note that .c/ O D

1 ; ˛ C ˇc

where ˇ D

1 1 1 1  ;˛ D C 2QC 2Q 2QC 2Q

and O 2; O0 .c/ D ˇ 2 .c/ as seen in Sect. 3.2. Hence one needs a mechanism, which guarantees that c stays in Œ1; 1 or at least in Œ1  ı; 1 C ı for some sufficiently small ı > 0. Unfortunately, so far it was not possible to work with a singular free energy because of the pressure appearing in (102). But an alternative is to choose q > d , which yields an a priori d bound for c 2 L1 .0; T I Wq1 .// ,! L1 .0; T I C 1 q .//. In this case c can be

1310

H. Abels and H. Garcke

trapped in Œ1  ı; 1 C ı if is chosen “steep enough” outside of the physical interval Œ1; 1. More precisely one has Lemma 1 (Choice of Free Energy, [4, Lemma 2.3]). Let R; ı > 0, q > d , and let 2 C 2 .Œ1; 1/ with .c/ > 0, c 2 Œ1; 1, be given. Then there is an extension 2 C 2 .R/, .c/  0; 00 .c/  M > 1 such that for all c 2 Wq1 ./ Z 

q .c/jrcj O dxC q

Z .c/ O .c/ dx  R

)

c.x/ 2 .1ı; 1Cı/:

(108)



In order to construct weak or strong solutions, it is essential to reformulate (100), (101), (102), and (103) first. To this end one defines gD

.c/ C

p jrcjq C  c; N q 

R 1 and D 0 C , N N D jj N  dx. Moreover, one decomposes g D g0 C g, R 1 gN D jj  g dx. Then (100), (101), (102), and (103) are equivalent to @t v C v  rv  div S.c; Dv/ C rg0 D  0 rc; @t  C div.v/ D 0;

(110)

@t c C v  rc D div.m.c/r 0 /;  0 C 2 gN D ˇ2 g0  div..c/jrcjq2 rc/ C

0

(109)

.c/

(111) (112)

together with Z

Z

0 .t / dx D 

g0 .t / dx D 0

for all t 2 .0; T /;

(113)



cf. [4, Section 3] for the details. Here the specific form of O and the corresponding relations above are essentially used. For the mathematical analysis, it is essential to use a suitable decomposition of g0 , namely, g0 D g1  @t G.v/;

(114)

where G.v/ D div v @  rG.v/j@ D 0

in ;

(115)

on @;

(116)

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

and

R 

1311

G.v/ dx D 0. This implies rG.v/ D .I  P /v;

(117)

where P W L2 ./d ! L2 ./ is the Helmholtz projection. Hence (109) is equivalent to @t P v C v  rv  div S.c; Dv/ C rg1 D  0 rc:

(118)

Here the part g1 has relatively good regularity, e.g., g1 2 L2 .0; 1I Lp .// with 1 < p < d d1 , cf. Theorem 11 below. It is the part @t G.v/, which makes the analysis difficult and which does not allow to use a singular free energy as in (79). It is also noted that for the estimates of g1 , it is important to consider Navier boundary conditions for v and not no-slip boundary conditions. Because of (118) one defines: Definition 5. Let v0 2 L2 ./d , c0 2 Wq1 ./, q > d , and let N with twice continuously differentiable. Then .v; g1 ; c; 0 ; p/

W R ! Œ0; 1/ be

v 2 BCw .Œ0; 1/I L2 ./d / \ L2 .0; 1I H1 .//; g1 2 L2 .0; 1I L1.0/ .//;

c 2 BCw .Œ0; 1/I Wq1 .//;

0 2 L2 .0; 1I H 1 .//;

pN 2 L1loc .Œ0; 1//;

where H1 ./ D fv 2 H 1 ./d W @  vj@ D 0g, and such that 0 <  D .c/ O 2 L1 .Q/ is called a weak solution of (109), (110), (111), (112) and (104), (105), (106) if the following conditions are satisfied: (i) For every ' 2 C01 .0; 1I H1 ./ \ L1 ./d / Z

Z

1

Z



1

Z

P v  @t ' dx dt C 0

.v  r/v  ' dx dt 0



Z

1

Z



Z

1

1

Z

 S.c; Dv/ W D' dx dt C 

C 

0

Z

1

Z

D

Z

1

0

Z



g1 div ' dx dt C 0



0

@

1 v  ' d  dt



0 rc C r1  S.c; Dv/  ' dx dt:



(ii) For every  2 C01 .0; 1I C 1 ./// Z

1

Z

Z

1

Z

@t  dx dt C  0 Z 1Z

v  r dx dt D 0; 0

Z 1Z



c@t  dx dt C 0



Z

1Z

cv  r dx dt D 0



m.c/r 0  r dx dt; 0



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H. Abels and H. Garcke

and Z

1

Z 

0

 q1

0

2

. 0 C  pN 

Z



0



1

G.v/@t .c/ q 2  dx dt C

ˇ

Z

1

 q 2 g1  dx dt

.c// dx dt D ˇ



Z 1Z

1

Z 1Z 0



0

 1

1 q jrcjq2 rc  r dx dt:



(iii) .v; c/jtD0 D .v0 ; c0 /. (iv) .v; c; 0 / satisfy the energy inequality Z tZ E.c.t/; v.t // C s

C

 kv k2L2 .@.s;t//



 S.c; Dv/ W Dv C m.c/jr 0 j2 dx d



 E.c.s/; v.s//

for all t 2 Œs; 1/ and almost all 0  s < 1 including s D 0. Theorem 11 (Existence of Weak Solution, [4, Theorem 2.4]). Let q > d , ı; R > 0. Moreover, let 2 C 2 .R/, .c/  0; 00 .c/  M , be given such that (108) holds. Then for every v0 2 L2 ./d ; c0 2 Wq1 ./ with E.c0 ; v0 /  R there exists a N of (109), (109), (109), (112) and (104), (104), (106) weak solution .v; g1 ; c; 0 ; g/ with the property that c.t; x/ 2 Œ1  ı; 1 C ı

for all x 2 ; t 2 .0; 1/;

g1 2 L2 .0; 1I Lp .//;

pN 2 L2loc .Œ0; 1//:

The proof of Theorem 11 is based on a two-level approximation. First (109), (110), (111), and (112) is regularized by adding the terms ıg0 v2 and ıg0 to R the left-hand sides of (109) and (111), respectively. This gives an extra-term ı  jg0 j2 dx on the right-hand side of (107). Existence of weak solutions for the regularized system is proved by a semi-implicit time discretization. Afterward, one reformulates (109) as (118) together with the extra-term ıg0 2v , derives suitable a priori estimate for g0 , g1 and gN and passes to the limit ı ! 0. Finally, a comment on short-time existence of strong solutions in [7] the following result was shown: Theorem 12 (Existence of Strong Solutions, [7, Theorem 1.2]). Let v0 2 H1 ./; c0 2 H 2 ./ with jc0 .x/j  1 for all x 2  and @ rc0 j@ D 0, d D 2; 3, and let the assumption throughout this subsection hold. Then there is some T > 0 such that there is a unique solution v 2 H 1 .0; T I L2 .//\L2 .0; T I H 2 ./d //; c 2 1 H 2 .0; T I H.0/ .// \ L2 .0; T I H 3 .// solving (109), (110), (111), (112) and (104), (105), (106).

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1313

1 1 Here H.0/ ./ is the dual of H.0/ ./ WD H 1 ./ \ L2.0/ ./. The prove is based on a fixed-point argument and the unique solvability of the linearized system

" @t v  div e S.c0 ; Dv/ C r div.4 rc 0 / D f1 ˇ˛ @t c 0  ˇ 1 div v D f2 ˇ .@  e S.c0 ; D.P v/// C .P v/ ˇ@ D a @  vj@ D @  rcj@ D 0 0

.v; c /jtD0 D

.v0 ; c00 /

in   .0; T /;

(119)

in   .0; T /; on @  .0; T /; on @  .0; T /; in ;

where c 0 corresponds to c. To solve the latter system, one uses the Helmholtz projection P to decompose v D P v C rG.div v/, where .I  P /v D rG.div v/. Moreover, P and I  P are applied to (119). Throughout the analysis one has to solve a kind of damped plate equation of the form @2t c 0  .a.c0 /@t c 0 / C

" div.04 rc 0 / D f ˛ˇ 2

up to lower order terms for some a.c0 / > 0. In order to solve this equation, an abstract result by Chen and Triggiani [33] is applied. The same kind of linearized system arises in the analysis of a Korteweg-type model for compressible fluids with capillary stresses, cf. Kotschote [49]. Furthermore the linearized system differs very much from the linearized system of the model with same densities and the model with volume averaged densities.

4

Sharp Interface Limits

In this section it is shown in a formal way that the diffuse interface model of Abels, Garcke, Grün [12] (52), (53), (54), and (55) and the diffuse interface model (63), (64), (65), and (66) of Lowengrub and Truskinovsky both converge to the classical sharp interface model (2), (3), (4), (5), and (6) if the parameter " tends to zero. It was already noted in the introduction that the energy Z  O 

 " 1 2 jr'j C .'/ dx 2 "

converges to a multiple of the surface energy Hd 1 ./

1314

H. Abels and H. Garcke

where  denotes the sharp interface; see [53, 54]. One would hence expect that all terms involving O will converge to terms involving interfacial energy and curvature, which is the first variation of interfacial energy. This will in fact be the case as one will see in the following analysis. The method of formally matched asymptotic expansions, which is used in the following, is based on the assumption that for small " the domain  can at each time be separated into open subdomains ˙ .t; "/ which are separated by a hypersurface .t; "/. In addition, it is assumed that the solutions have an asymptotic expansion in " in the bulk regions away from .t; "/ and another suitable scaled expansion close to .t; "/. The scaling will be needed in the x–variable as the values of the phase field ' will change its value sharply but smoothly in a region of thickness ". That leads to the formation of internal layers. These expansions then have to be matched in a region where both expansions overlap. A detailed description of the method can be found in [9, 40, 41]. For some phase field models this approach can be justified rigorously, cf. [14, 19, 31, 35].

4.1

Models Based on a Volume Averaged Velocity

In this section the system @t ..'/v/ C div.v ˝ .v C e J//  div.2.'/Dv/ C rp D r'; div v D 0;

O "

(120) (121)

@t ' C v  r' D "m0 ;

(122)

0

(123)

.'/  O " ' D ;

with QC  Q QC  Q e J' D  "m0 r

JD 2 2 and .'/ D QC

1C' 1' C Q 2 2

is studied. This is basically the model (52), (53), (54), and (55) with the reformulation (58) and for simplicity m D "m0 is taken to be constant; see [9] for a more general case. The solution of (120), (121), (122), and (123) is always denoted by v" ; e J" ; '" ; " . In addition, it is always assumed that is of double well form with two global minima at ˙1.

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1315

4.1.1 Outer Expansions It is assumed that v" ; e J" ; '" ; " have in .t; "/ an expansion of the form u" .x; t / D u0 .x; t / C "u1 .x; t / C O."2 /: Substituting these expansions into (120), (121), (122), and (123) leads to equations which have to be solved order by order. The equation (123) gives to leading order "1 0

.'0 / D 0:

The stable solutions of this equation are ˙1, and ˙ are defined to be the sets where '0 D ˙1. The expansions to order "0 of the fluid equations yield ˙ @t v0 C ˙ v0  rv0  ˙ v0 C rp0 D 0; div v0 D 0 with the scaling chosen the equation (122) is fulfilled to leading order "0 . The paper [9] discusses the case when the mobility is scaled to be of order one.

4.1.2 Inner Expansions and Matching Conditions It is now assumed that the zero level sets of '" .; t / converge for " ! 0 to a smooth hypersurface .t / which moves with a normal velocity V. As .t / is smooth one can define the signed distance function d .x; t / of a point x 2  to .t / which is defined such that d .x; t / > 0 if x 2 C .t / and negative if x 2  .t /. Close to  the function d is smooth, and each function u.x; t / close to  is expressed in new coordinates U .s; z; t /, where s is a tangential spatial coordinate on  and z.x; t / D d .x; t /=". In the new coordinates, the relevant differential operators transform as follows: 1 @t u D  V@z U C h:o:t: " 1 rx u D @z U  C r U C h:o:t: " 1 1 x u D 2 @zz U  H @z U  zjSj2 @z U C  U C h:o:t:; " " where  D rx d is the unit normal pointing into C .t /, r is the spatial surface gradient on , jSj is the spectral norm of the Weingarten map S,  is the LaplaceBeltrami operator on .t /, and h.o.t. denotes terms of higher order in " (see the Appendix of [9] for a proof).

1316

H. Abels and H. Garcke

Furthermore, it is assumed that the functions v" ; p" ; '" ; " as functions .v" ; p" ; ˆ" ; M" / in the inner variables have an expansion of the form u" .x; t / D U" .s; z; t / D U0 .t; s; z/ C "U1 .t; s; z/ C : : : : In an "–dependent overlapping domain, the outer and inner expansions have to coincide in a suitable sense when " tends to zero. This leads to the following matching conditions which are derived in [40] and [41]. At a point x 2 .t / with coordinate s, it holds lim U0 .s; z; t / D u˙ 0 .x; t /;

z!˙1

lim @z U0 .s; z; t / D 0;

z!˙1

lim @z U1 .s; z; t / D ru˙ 0 .x; t /  ;

z!˙1

where u˙ 0 denotes the limit lim u0 .x ˙ ı/ at a point x 2 . ı!0

4.1.3 Leading Order Equations In the interfacial region, the equation (123) gives to leading order 1" : 0

.ˆ0 /  @zz ˆ0 D 0

(124)

and matching with the outer solutions gives the following boundary condition at ˙1: lim ˆ0 .z/ D ˙1:

z!˙1

(125)

The problem (124), (125) has a unique solution with the property ˆ0 .0/ D 0; see, e.g., [63, Section 2.6]. This solution is chosen in what follows. The equation div v D 0 gives to the order 1" @z V0   D @z .V0  / D 0 and together with the matching conditions, one obtains that V0   needs to be constant. One hence obtains .vC 0  /.x/ D lim .V0  /.z/ D lim .V0  /.z/ D .v0  /.z/ z!1

z!1

and this gives Œv0  C  D 0:

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

At order

1 "

1317

the diffusion-type equation (122) leads to  V@z ˆ0 C .v0  /@z ˆ0 D m0 @zz M0 :

(126)

The matching conditions lead to @z M0 ! 0 and ˆ0 .z/ ! ˙1 for z ! ˙1. Hence (126) implies V D v0   and M0 D M0 .s; t /: C  In addition, one obtains Œ C  D 0 and hence D D M0 . One now considers the momentum equation to leading order "12 . Expressing rx v and Dx v in the new coordinates gives

1 @z v ˝  C r v C h:o:t:; "

1 1 1 Dx v D .@z v ˝  C  ˝ @z v/ C .r v C .r v/T / C h:o:t:: 2 " 2 rx v D

With the notation E.A/ D 12 .A C AT /, one obtains divx ..'/Dx v/ D

1 1 @z ..ˆ/E.@z V ˝ // C @z ..ˆ/E.r V// "2 " 1 C div ..ˆ/E.@z V ˝ // C div ..ˆ/E.r V// C h:o:t: "

where div is the surface divergence. Using @z V0   D 0 one obtains . ˝ @z V0 / D .@z V0  / D 0 and hence the momentum equation gives to leading order @z ..ˆ0 /@z V0 / D 0: The matching conditions imply that V0 is bounded, and hence the above ODE only has constant solutions. The matching property lim v0 .z/ D v˙ .x/ for x 2  z!˙1

hence implies Œv0 C  D 0:

1318

H. Abels and H. Garcke

4.1.4 Next-Order Equations The equation (123) which defines the chemical potential gives to the order "0 O

00

.ˆ0 /ˆ1  O @zz ˆ1 D M0  O @z ˆ0 H:

(127)

As @z ˆ0 is in the kernel of the differential operator u 7! 00 .ˆ0 /u  @zz u the righthand side of (127) needs to be L2 -orthogonal to @z ˆ0 ; see [9] for details on this Fredholm alternative type of argument. One hence obtains Z 1 0D @z ˆ0 .M0  O @z ˆ0 H /d z 1

Z

1

j@z ˆ0 j2 d z

D 2M0  O H 1

D 2 0   H where  D O c0 with Z

1

j@z ˆ0 j2 :

c0 D 1

It remains to derive the force balance (5) at the interface. One first observes that the term   QC  Q e "m0 div.v ˝ r / div.v ˝ J/ D  2 in the interfacial region gives no contribution to the order 1" . Here one uses the facts that @z M0 D 0 and @z v0 D 0. One hence obtains that (120) to order 1" gives the identity @z ..ˆ0 /V0 /V C @z ..ˆ0 /.V0 ˝ V0 //  2@z ..ˆ0 /E.@z V1 ˝ // 2@z ..ˆ0 /E.r V0 // C @z P0  D @z ˆ0 :

(128)

The matching conditions require lim @z V1 .z/ D rv˙ 0 .x/ and hence z!˙1

@z V1 ˝  C r V0 ! rx v0 for z ! ˙1: Integrating (128) with respect to z now gives C C O Œ0 v0 C  V C Œ0 v0  v0    2Œ".rx v0 /  D 

The identity V D v0  hence gives

Z

1 1

 .@z ˆ0 /2 d z H  C Œp0 C  :

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1319

C 2ŒDv0 C   C Œp0   D  H :

Hence, all equations which appeared in the sharp interface problem (2), (3), (4), (5), and (6) are derived.

4.1.5

The Navier-Stokes/Mullins-Sekerka System as Sharp Interface Limit It is also possible to obtain the Navier-Stokes/Mullins-Sekerka system (24), (25), (26), (27), (28), (29), and (30) as sharp interface limit. To achieve this, one has to use a different scaling in (122). In fact (122) is replaced by @t ' C v  r' D 2m :

(129)

Expansions in ˙ immediately give 0 D 0: At order

1 "

one obtains from (129) that .V C v0  /@z ˆ0 D 2m@zz M1 :

Matching requires @z M1 ! r 0  , and integration of the above equation gives .V C v0  / D mŒr 0  C ; which is precisely equation (30).

4.2

Sharp Interface Expansions for the Lowengrub-Truskinovsky Model

Now the sharp interface limit of the Lowengrub-Truskinovsky model @t .v/ C div.v ˝ v/  div S.c; Dv/ C rp D O " div.rc ˝ rc/; 2

.c/.@t c C rc  v/ D m" O ; 2

div v D ˇ m" O ; O "

0

.c/  O " c  ˇ.c/p D .c/ ;

(130) (131) (132) (133)

is considered. Here, S.c; Dv/ D 2.c/D.v/ C .c/ div v Id, m D "2 m, O and it is assumed that the functional relation between  and c is of simple mixture type; see (61).

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H. Abels and H. Garcke

4.2.1 Outer Expansions In the phases ˙ one obtains as in the preceding section c0 D ˙1 ; 0 D ˙ and hence div v0 D 0: This then implies ˙ @t v0 C ˙ v0  rv0  ˙ v0 C rp0 D 0:

4.2.2 Inner Expansion to Leading Order The expansions in the interface are as in the case of the volume averaged velocity with the two exceptions c" D C" .s; z; t / D C0 .t; s; z/ C "C1 .t; s; z/ C : : : ; p" D P" .s; z; t / D "1 P1 .t; s; z/ C P0 .t; s; z/ C : : : : In the interface the term "rc˝rc will give a contribution to the order "1 which has to be balanced by the pressure. This is due to the fact that in contrast to the volumeaveraged case, one does not work with rc as a capillarity term. Therefore, the inner expansion of the pressure has P1 as the leading order term. For the capillarity-type term " div.rc ˝ rc/, one obtains " div.rc ˝ rc/ D

1 1 @z .j@z C j2 / C .@z C r C / "2 " 1 C div .j@z C j2  ˝ /Cdiv .@z C . ˝ r c Cr c ˝ //; " Ch:o:t: (134)

where an "12 contribution of this term in the momentum balance can be observed. Similar as in the previous section, one obtains from (132) that @z V0   D 0 which leads to Œv0    D 0: The equation (131) gives to leading order .C0 /@z C0 .V C V0  / D 0: Matching implies lim C0 .z/ D ˙1

z!˙1

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

1321

which implies @z C0 6 0 and hence V D v0  : The momentum balance (130) gives to leading order "2  @z ..C0 /@z V0 / C @z P1  D O @z j@z C0 j2 :

(135)

Since @z V0   D 0 one obtains from the normal part of the above equation, P1 .t; s; z/ D PO .t; s/  O j@z C0 j2 .t; s; z/: Matching requires P1 ! 0 and @z C0 ! 0 for z ! ˙1 and hence PO  0 which gives P1 D O j@z C0 j2 : Hence (135) boils down to @z ..C0 /@z V0 / D 0 which implies after matching Œv0  D 0: The equation (133) gives to leading order "1 O

0

.C0 /  O @zz C0  ˇ.C0 /P1 D 0:

Using ˇ D 0 =2 and P1 D O j@z C0 j2 gives O

0

.C0 / 

O .@z .@z C0 // D 0: 

This ODE has a unique solution fulfilling C0 .˙1/ D ˙1 and C0 .0/ D 0. In particular C0 is independent of s and t .

4.2.3 Inner Expansions to Next-Leading Order Using (134) and r C0  0, one obtains from the momentum balance (130) to order "1  2@z ..C0 /E.@z V1 ˝ //  2@z ..C0 /E.r V0 // C @z P0 O  C r P1 C O @z .2@z C0 @z C1 / C O div .j@z C0 j2  ˝ / D 0

1322

H. Abels and H. Garcke

where as above V D v0   is used which yields that the kinetic term gives no contribution. Since C0 is independent of s and t , one obtains that r P1  0. One computes div .j@z C0 j2  ˝ / D j@z C0 j2 div . ˝ / D H j@z C0 j2 : Hence, it follows  2@z ..C0 /E.@z V1 ˝ //  2@z ..C0 /E.r V0 // C @z P0  C O @z .2@z C0 @z C1 /  O H j@z C0 j2  D 0: Integrating and using the matching conditions give similar as in Sect. 4.1.4 C 2ŒDv0 C   C ŒP0   D  H ;

R1 where one uses that 1 @z .@z C0 @z C1 /d z D 0 which follows from matching. One hence obtains that also the Lowengrub-Truskinovsky model yields the sharp interface model (2), (3), (4), (5) and (6) in the asymptotic limit " ! 0.

4.3

Known Results on Sharp Interface Limits

First results on the sharp interface limits of Cahn-Hilliard/Navier-Stokes systems are for a simplified situation due to Lowengrub and Truskinovsky [52]. They used the method of formally matched asymptotic expansions. In the general case the sharp interface limit has been analyzed with formally matched asymptotic expansions by Abels, Garcke, and Grün [12], where also different scalings have been analyzed which lead to quite different asymptotic limits. So far only very few rigorous results for the sharp interface limit exist. Abels and Röger [16] and Abels and Lengeler [14] showed convergence in the sense of varifold solutions, cf. Chen [32], for the case in which m is constant. Abels and Röger [16] studied the case of matched densities and m independent of ". Abels and Lengeler [14] considered the case of a volume averaged velocity and " m independent of " as well as m D m."/ !"!0 0 sublinearly, i.e., m."/ !"!0 0. Moreover, it is shown that certain radially symmetric solutions of (90), (91), (92), and (93) tend to functions which will not satisfy the Young-Laplace law (6) in the limit " ! 0 if the mobility tends to zero faster than "3 . A result on a sharp interface limit to solutions which fulfill the limit equations in a stronger sense is still open. Abels and Schaubeck [17] showed that for mobilities m tending to zero faster than "3 in the convective Cahn-Hilliard equation, i.e., (93), (94) with a given velocity smooth and solenoidal field v, the surface tension term " div.r'" ˝r'" / in general does not converge to a multiple of the mean curvature vector as " tends to zero. For

26 Weak Solutions and Diffuse Interface Models for Incompressible Two-. . .

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a related Allen-Cahn/Stokes system Abels and Liu [15] are able to show converge to solutions which fulfill the sharp interface problem in a strong sense for small times.

5

Conclusions

Because of possible singularities in the interface, the mathematical description of a two-phase of macroscopically immiscible fluids remains a mathematical challenge with many open problems and questions. Weak formulations of the classical sharp interface model for two viscous incompressible, immiscible Newtonian fluids have been discussed. In the absence of surface tension, existence of weak solutions is known, but there is little control of the regularity of the interface known. In particular, it cannot be excluded that it is dense in the domain in general. In the case with surface tension, the energy estimates provide a control of the total surface measure of the interface. But existence of weak solutions is unknown since possible oscillation and concentration effects of the interface prevent from passing to the limit in the weak formulation of the mean curvature vector, which arises due to the Young-Laplace law. Moreover, a nonclassical sharp interface model is discussed, where the classical kinematic condition that the interface is transported by the fluid velocity is replaced by a convective Mullins-Sekerka equation. This model arises as the sharp interface limit of a diffuse interface model if the mobility coefficient in the diffuse interface model does not tend to zero. For this model existence of weak solutions can be shown with similar techniques as for the Mullins-Sekerka system since an additional term in the energy inequality gives rise to a suitable a priori bound of the mean curvature of the interface. In order to describe two-phase flows beyond the occurrence of topological singularities, diffuse interface models, where the macroscopically immiscible fluids are considered as partly miscible, are an important alternative. In these models the sharp interface and the characteristic function of one phase is replaced by an order parameter, which varies smoothly, but with a steep gradient in a thin interfacial region. In the case of different densities, there are different models in dependence of choice of the mean velocity for the fluid mixture. The choice of a volume averaged velocity leads to a divergence-free velocity field and a system, which is very similar to the case of same densities. Results on existence of weak solutions for different choices of the free energy and mobility are discussed. For a barycentric/mass-averaged velocity, the velocity field is no longer divergence free, and the pressure enters the equation for the chemical potential. This leads to significant new difficulties in the mathematical analysis of this model. Moreover, the linearized system is rather different from the case of same densities. In this case existence of weak solutions is only known in the case of a free energy, which is non-quadratic in the gradient of the concentration. Finally, the sharp interface limit of the diffuse interface models to the classical sharp interface model has been derived. This convergence can be discussed using

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the method of formally matched asymptotic expansions. But there are only few mathematical rigorous convergence results. In particular a proof of convergence to strong solutions of the limit equations remains an open problem, even for small times.

6

Cross-References

 Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible

Fluid Mechanics  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Local and Global Solvability of Free Boundary Problems for the Compressible

Navier-Stokes Equations Near Equilibria  Multi-Fluid Models Including Compressible Fluids  Stability of Equilibrium Shapes in Some Free Boundary Problems Involving

Fluids  Variational Modeling and Complex Fluids  Water Waves With or Without Surface Tension

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Water Waves with or Without Surface Tension

27

Diego Córdoba and Charles Fefferman

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Equations of Water Surface Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Global Existence and the Formation of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Splash Singularities for a Viscous Fluid: Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . 6 On the Absence of Splash Singularities with Two Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1329 1331 1333 1337 1340 1343 1345 1346 1346

Abstract

In this survey article, we present results from the last several decades on several systems of PDE that model water waves. Some of those results provide shorttime existence of smooth solutions; other results establish existence of smooth solutions for all time; still other results assert that singularities form in finite time.

1

Introduction

The aim of this paper is to describe the recent analytical results in the Cauchy problem for the dynamics of the water surface: given the initial velocity of the

D. Córdoba () Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, Spain e-mail: [email protected] C. Fefferman Department of Mathematics, Princeton University, Princeton, NJ, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_30

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water u0 and the surface S0 , solve the equations of the motion of the waves. In the case of an incompressible fluid in a fixed domain, whether periodic or in the whole space (with finite energy), the local existence of solutions is well known since in the work conducted by Leray [44] and in dimension 2, the solution exists for all time (see the summary [20] and the references found therein). However, for the case of an incompressible viscid fluid with free boundaries, it was not until 1977 that Solonnikov [53] proved the local existence in time of solutions using the regularizing property of viscosity. In the case of inviscid incompressible fluids, in 1997 S. Wu [59] proved that the problem is well posed for short time intervals; i.e., solutions exist for the problem of waves that preserve the regularity of the initial data u0 and S0 locally in time. In what follows we deduce the equations that describe this phenomenon; we explain the nature of Wu´s result and the most recent results in the formation of singularities as well as the global existence for small initial data. In order to deduce the system of equations for the motion of water, we assume the following properties: • The water is not very viscid, and thus the velocity u.x; t / and the pressure p.x; t/ satisfy the Euler equations: ut C uru D rp  .0; g /; where the vector .0; g / is the exterior forced exerted by gravity. • The water is incompressible, and thus the vector velocity has zero divergence. • Boundary conditions at the free surface: the interface between the air and the water moves with the fluid, and the pressure at the interface is equal to atmospheric pressure (constant along all the surface). With the aim of simplifying this exposition and emphasizing the most relevant ideas, we impose the following conditions in the sections that follow: • We consider that the water is in two dimensions (2D) and that the interface is a curve. • We consider waves in a very deep ocean; we assume an infinite depth. • We consider that the fluid is irrotational. • We consider that the interface is periodic or defined along all the real line that becomes asymptotically flat at infinity. The surface tension at the interface may also be taken into account in the model, since it arises from the forces exerted by the air and the water on the surface of the wave, which prevent the interface from becoming deformed. These forces cause the surface to behave like an elastic membrane, although their effect is small when dealing with large masses of water. The surface tension describes a discontinuity in the pressure along the interface: pressure at the interface = atmospheric pressure –  

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where  is the mean curvature of the surface and  is the surface tension coefficient. For further details about the influence of the capillary forces in the theory of local existence, global existence and the formation of the singularities of waves, see [2, 6, 14, 24].

2

Equations of Water Surface Motion

The velocity u.x; t / D .u1 .x; t /; u2 .x; t //; x D .x1 ; x2 / 2 R2 and t 2 RC , of an irrotational incompressible fluid satisfies r  u D 0;

r  u D 0;

which enables us to define a stream function ' D '.x; t/ and a potential  D .x; t/ with the property u D r D r ? ': The fluid occupies the domain .t/, and the free surface (surface of the water) is parametrized in the following manner: @.t / D fz.˛; t / D .z1 .˛; t /; z2 .˛; t // W ˛ 2 Rg: We consider the interface along all the real line that approaches zero as it tends to infinity (Fig. 1). lim .z.˛; t /  .˛; 0// D 0;

˛!1

or periodic z.˛ C 2k; t / D z.˛; t / C 2k.1; 0/: The potential  satisfies the Laplace equation

a

b VACUUM

FLUID

VACUUM

FLUID

Fig. 1 Waves on the real line and waves in a periodic domain

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 D 0 in the domain .t/; this is a harmonic function and the particular solution  depends on the dynamics of the surface. Our aim now is to write the system of equations in terms of quantities that act on the surface of the wave. We extend the velocity field (irrotational and incompressible velocities) to the whole plane in the following way: we define the vorticity r ?  u in all R2 as r ?  u.x; t / D !.˛; t /ı.x  z.˛; t //; i.e., the vorticity is a Dirac measure with amplitude ! and support at the interface z defined by < r ?  u;  >D

Z !.˛; t /.z.˛; t //d ˛; R

where .x/ is a test function. We are now in a position to employ the tools of the potential theory to solve the Laplace equation in terms of the curve z and the amplitude of the vorticity ! in the domain .t/ (x ¤ z.˛; t /): .x; t/ D

1 PV 2



Z arctan R

 x2  z2 .ˇ; t / !.ˇ; t /d ˇ x1  z1 .ˇ; t /

which yields the velocity 1 u.x; t / D PV 2

Z R

.x  z.ˇ; t //? !.ˇ; t /d ˇ jx  z.ˇ; t /j2

(1)

within the water. Furthermore, by integrating the Euler equations ut C uru D rp  .0; g/; we obtain the pressure in the interior of .t/ 1 p.x; t/ D t .x; t /  ju.x; t /j2  gx2 : 2

(2)

This formula is known as Bernoulli’s law. The next step is to obtain the closed equations of z and !. Taking the limit .x; y/ ! .z1 .˛; t /; z2 .˛; t // 2 @.t / of the expression (1) of the velocity from the interior of the fluid, we obtain the velocity of the interface

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1 @˛ z.˛; t / u.z.˛; t /; t / D BR.z; !/.˛; t / C !.˛; t / ; 2 j@˛ z.˛; t /j2 where BR denotes the Birkhoff-Rott integral 1 BR.z; !/.˛; t / D P:V: 2

Z R

.z1 .˛; t /  z1 .ˇ; t /; z2 .˛; t /  z2 .ˇ; t //? !.ˇ; t /d ˇ: j.z1 .˛; t /  z1 .ˇ; t /; z2 .˛; t /  z2 .ˇ; t //j2

The second term of the velocity is tangent to the curve, which does not modify the geometry of the curve itself. Indeed, the evolution of the interface does not change when the tangential component of the velocity is modified; thus, we may consider the equation @t z.˛; t / D BR.z; !/.˛; t / C c.˛; t /@˛ z.˛; t /

(3)

where c.˛; t / is a parameter we may conveniently choose and which involves a reparametrization of the curve. In order to close the system, we take the limit .x; y/ ! .z1 .˛; t /; z2 .˛; t // 2 @.t / of the expression (2) with the condition that the pressure at the boundary is equal to atmospheric pressure (constant along all the interface) and thereby obtain

@t !.˛; t / D  2@˛ z.˛; t /  @t BR.z; !/.˛; t /   j!j2 .˛; t / C @˛ .c.˛; t /!.˛; t //  @˛ 4j@˛ zj2 C 2c.˛; t/@˛ z.˛; t /  @˛ BR.z; !/.˛; t /  2g@˛ z2 .˛; t /:

(4)

We conclude that the water surface (wave) equations are given by (3) and (4) with initial data z0 D z.˛; 0/ and !0 D !.˛; 0/. The total energy of the water consists of the kinetic component and the potential 1 ET D Ec C Ep D 2

Z

1 juj dx1 dx2 C 2 .t/ 2

Z

g@˛ z1 .˛; t /.z2 .˛; t //2 d ˛ @.t/

which has the property of being conserved over time. It is important to take into account solutions with finite energy, a necessary condition being that our initial data !0 should have zero mean.

3

Existence of Solutions

The first results on the existence of solutions to the Cauchy problem for an inviscid fluid (without a regularizing effect) that is irrotational and incompressible with a free surface are by Nalimov [49], Yosihara [64], and Craig [28]. They showed local

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existence in time in 2D for initial data close to the equilibrium, while Beale-HouLowengrub [11] proved that the linearized problem is well posed. Finally, Wu [59, 60] in her work, [59] and [60], proved that the system of equations, in 2D and 3D, of the water surface admits solutions to the Cauchy problem (in Sobolev spaces H k ) for short time intervals (depending on the initial data) where the solutions conserve the regularity of the initial data without restrictions of size. Lannes obtained the same result [42], but on the assumption that the water has a bottom ( the depth of the ocean is not infinite). In the work by Ambrose-Masmoudi [6] and CórdobaCórdoba-Gancedo [22], the same result obtained by Wu was proved from a different point of view, with the observation of a new magic cancellation that enables the estimates to be closed a priori. In a recent work by Alazard-Burq-Zuily (see [2] and [3]), these authors reduced the regularity required in the initial data in order to obtain local existence, taking advantage of the dispersion of the system. The condition that the fluid should be irrotational is not necessary for the existence of solutions. In the Works by Lindblad [46] (see also the prior work by Cristodoulou and Lindblad [19]), Coutand and Shkoller [24] and Zhang-Zhang [65], it is shown by means of different techniques the existence of rotational solutions to the Euler equation with a free surface. In this section, we explain the reason why the system (3) and (1) with initial data z0 D z.˛; 0/ and !0 D !.˛; 0/ has solutions when assuming a certain regularity of the initial data. On linearizing the system around a flat interface z D .˛; f .˛; t // and ! D w.˛; t /, we obtain from (3) and (4) the following equations: ft .˛; t / D

1 H .w/.˛; t /; 2

wt .˛; t / D 2g@˛ f .˛; t/; where the operator H is the Hilbert transform. Deriving the first equation in regard to time, and substituting in the second equation, it is deduced that ft t .˛; t / D gƒ.f /.˛; t /

2

b. The equation can be resolved where the operator ƒ is defined by ƒ˛ .f / D j j˛ f in Fourier, and the solutions depend on the sign of g 1

g < 0 ) e j j 2 t ; 1

g > 0 ) cos.j j 2 t /;

1

e j j 2 t 1

sin.j j 2 t /;

which distinguishes when the system is either stable or unstable. The energy associated with the system is Z 1 1 EL .t / D .gj@˛ f j2 C jƒ 2 wj2 /d ˛ 2

27 Water Waves with or Without Surface Tension

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d which is conserved over time, since it satisfies dt EL .t / D 0. Thus, in the case g > 0, it ensures a control for all time of the derivatives of f and w. If the constant g < 0, then we have no control over the solutions and the problem is ill posed. In the nonlinear case (the system (3) and (4)), the sign of

 .˛; t / D .rp.z.˛; t /; t //  @? ˛ z.˛; t/ > 0; has the same role as the sign of gravity in the linear case and is known as the Rayleigh-Taylor condition. In her paper [59], Wu proved that this condition is fulfilled. Subsequently, Hou-Caflish (see the citation in [60]) proposed the following demonstration: By applying the divergence operator in the Euler equations p D jrvj2  0 which, together with the conditions that the pressure is constant at the boundary and that gravity has the correct sign, enable us thanks to the Hopf lemma to arrive at the following conclusion  .˛; t /  jz? ˛ .˛; t /j@n p.z.˛; t /; t / > 0; where @n is the normal derivative. A further condition necessary for the problem to be well posed is that the interface z does not self-intersect. This is measured with the chord-arc condition F.z/.˛; ˇ; t /, defined as F.z/.˛; ˇ; t / D

jˇj jz.˛; t /  z.˛  ˇ; t /j

8 ˛; ˇ 2 .; /;

and F.z/.˛; 0; t / D

1 : j@˛ z.˛; t /j

We are now able to state the theorem of local existence. • Theorem [Local Existence] Local existence in time of solutions to the system (3) and (4) for initial data that satisfy z0 .˛/ 2 H k , !0 .˛/ 2 H k1 (k  4), F.z0 /.˛; ˇ/ < 1 y  .˛; 0/ > 0:

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Given that in the linear case we can associate the following energy to the system (in the case z 2 H 4 .T/) Z



E.t / D

Π

 .˛; t / 1 j@4 z.˛; t /j2 C jƒ 2 @3˛ '.˛; t/j2 d ˛ j@˛ z.˛; t /j2 ˛

C kzk2H 3 .t / C kF.z/k2L1 .t / C k!k2H 2 .t / C

jz˛ j2 ; m. /.t /

although in the nonlinear case, it is necessary to take into account the evolution of the chord-arc and the Rayleigh-Taylor condition. The last term measures the possible decay of the Rayleigh-Taylor condition, where m. /.t / D min˛2T  .˛; t /, and the function '.˛; t/ is defined by '.˛; t/ D

!.˛; t /  c.˛; t /j@˛ z.˛; t /j 2j@˛ z.˛; t /j

which was introduced by Ambrose-Masmoudi [6] (in the linear case by Beale-HouLowengrub [11]). The function ' and the choice of c Z zˇ .ˇ; t / ˛C  .BR.z; !//ˇ .ˇ; t /  dˇ c.˛; t / D 2 2 jz ˇ .ˇ; t /j  Z ˛ zˇ .ˇ; t / .BR.z; !//ˇ .ˇ; t /  dˇ  2 jz ˇ .ˇ; t /j  enable us to obtain a cancellation equivalent to the linear case d dt d dt

Z Z

T

 j@4 zj2 d ˛ D Term controlled by E.t / C S jz˛ j2 ˛ 1

T

jƒ 2 @3˛ '.˛; t/j2 d ˛ D Term controlled by E.t/  S

where Z



2

SD 

@4˛ z  z? ˛ ƒ.@3˛ '/d ˛ jz˛ j3

is a higher-order term that cannot be absorbed by E.t /. All the other terms are bounded by powers of E.t /, and the bound is obtained a priori ˇ ˇ ˇd ˇ ˇ E.t/ˇ  Const C .E.t //100 ˇ dt ˇ

27 Water Waves with or Without Surface Tension

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In order to obtain these estimates, it is crucial in Sobolev spaces H k to get explicit upper bounds of the operator .I C T /1

where

T .v/.˛; t / D 2BR.z; v/.˛; t /  @˛ z:

In the paper [22], the techniques of harmonic analysis are used – conformal mapping, Hopfs lemma, and Harnack inequalities in order to obtain upper bounds on the inversibility of the operator in terms of E.t/. To conclude the proof, the system of Eqs. (3) and (4) is regularized so that the Picard theorem guarantees the existence of solutions independently of the sign of the Rayleigh-Taylor condition as well as satisfying the previous a priori estimate. When we consider  .˛; 0/ > 0, the new system yields a uniform time of existence with respect to the regularization that allows us to take the limit. For the details of this proof, see [6] and [22].

4

Global Existence and the Formation of Singularities

Once the initial data problem of the dynamics of the water surface (3) and (4) is shown to be well posed for short time intervals, the following questions arise: • Is it possible to extend these solutions for all time? • Can singularities be developed in finite time? In the case of traveling waves, which are those waves that move with a constant velocity while maintaining their geometry, are solutions that exist for all time, and the interface and potential are solutions of the following type: z.˛; t / D .˛; f .˛  ct // y

.x; t/ D .x1  ct; x2 /

where the profile of the wave is propagated with velocity c. The interface may be regular or may have a singularity. G. Stokes [56] predicted the existence of waves with corner angles and concave at every point except at the crests. But it was not until 1920, with the publication of Nekrasov [50] and Levi-Civitas [45] work, when analytical results on the existence of traveling waves were obtained. These results concern the case in which the velocity of the water is irrotational (i.e., the water does not rotate locally), so that by means of a conformal map, they are reduced to an elliptical problem in a fixed domain. In 1982 Amick-Fraenkel-Toland [7] and Plotnikov [51] independently proved the Stokes conjecture. Recently in [58], it was analyzed whether the Stokes conjecture holds at all points where the vertical velocity is zero, and whether these points are isolated. In [63], S. Wu has recently shown other examples of singularities that are propagated over time and have the property of being self-similar solutions. However, none of these examples provide answers to the questions above.

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Notwithstanding, it is natural to think that the answer to the first question should be in the affirmative if we consider a family of initial data with a small norm; that is, a wave that initially is almost flat and with almost zero velocity should generate large waves when the only external force applied to the water is that of gravity. In this sense, S. Wu [61] demonstrated that in dimension 2, there exists a 0 > 0 such that if the initial data in a certain Sobolev norm (which involves the interface and the initial velocity) is of the order < 0 , then the time of existence can be extended to 0  t  exp 1 . The proof uses the dispersive character of the system and obtains estimates of the analysis of the spatial and temporal resonances. Some years later, S. Wu [62] and Germain-Masmoudi-Shatah [32] benefited from the improvement obtained in larger dimensions in the estimates coming from dispersion and thus proved existence for all time, in dimension 3, when the initial data is small. Global existence in 3D, for small initial data, with surface tension and no gravity, was proved by Germain-Masmoudi-Shatah in [33] and with both, surface tension and gravity, was obtained by Deng-Ionescu-Pausader-Pusateri in [30]. The global existence in dimension 2 was finally solved by Ionescu-Pusateri in [41], by Alazard-Delort in [4, 5] and by Ifrim-Tataru in [39]. And with surface tension by Pusateri-Ionescu in [41] and by Ifrim-Tataru in [40]. For an extensive survey about analytical results on water waves, see the monograph [43]. With regard to the second question, one has only to look at the sea to realize that waves exist that begin by being a regular graph, then curve and finally collapse (Fig. 2).

a

b VACUUM

VACUUM

FLUID

FLUID

c VACUUM

FLUID

Fig. 2 A wave that curves and collapses

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The first simulations showing that the interface curled (the water surface ceased to be a graph) are due to Beale-Hou-Lowengrub [12]. By using the same numerical method (adapting it to a more singular scenario) in [13], we observe that regular interfaces exist which, initially being graphs, do curl and then collapse. This collapse has the particularity that the regularity of the interface is not lost, but the chord-arc ceases to be well defined at one point. We call these types of singularities splash singularities. In two recent works [13, 15, 16] (in collaboration with Ángel Castro, Francisco Gancedo, Javier Gómez-Serrano and María López-Fernández), we prove that the phenomena occur independently: • Theorem [Turning and Splash] 1. Initial data exist in H k , whose interface is a graph, such that the solution to the system (3) and (4) generates interfaces in finite time that cease to be graphs. 2. Initial data exist in H k such that the solution to the system (3) and (4) produces a splash-type singularity. The property that the Eqs. (3) and (4) are reversible over time is used in the proof of these two results. The numerical simulations serve to guide us as well as to create an objective scenario with the desired characteristics (curl and splash). On the basis of this scenario, the aim is to prove a theorem of local existence showing that solutions exist with the characteristics that are described by the simulations (Fig. 3). In the case of splash, it is necessary to choose a conformal map P which separates the point of collapse such that its singular points (where P cannot be inverted) are located outside the water. This map transforms the splash into a closed curved whose chord-arc is well defined. We select an initial velocity that immediately separates the point of collapse. In the new domain, the existence of solutions is proven by using energy estimates and checking that the magic cancellation is also satisfied. In order to return to the original domain and obtain solutions to (3) and (4), it is necessary to invert the map P . In the proof of this type of singularity, it is very important that no fluid exists between the two curves that

Fig. 3 Splat-type singularity

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collapse; in other words, this result cannot be extended to the breakup of a drop of water. The strategy of the proof enables us to extend the scenario to splat-type singularities, that is, for those in which collapse may occur along a curve. • Theorem [Splat Singularities] Initial data exist in H k such that the solution to the system (3) and (4) produces a splat-type singularity. However, unlike in the case of splash, this scenario is unstable, and it is impossible to observe it in numerical simulations. In later work it is shown that these types of singularities, splash and splat, also occur in the rotational case [25] and that surface tension does not prevent their formation [14]. Moreover, by taking advantage of the trick of opening up the splash with the conformal map P it is shown in [23] the existence of stationary solutions with a splash type singularity: • Theorem [Stationary Splash Singularities] For any small enough g, there is some surface tension coefficient  such that there is a periodic solution to the water wave system (3), (4) and surface tension for which the interface has a splash-type singularity. We denote by a periodic solution those in which the pressure, the velocity of the fluid and the geometry of the interface remain invariant under the translation .x; y/ ! .x C 2; y/: These solutions are a perturbation of a particular curve of the family of Crapper waves [29].

5

Splash Singularities for a Viscous Fluid: Navier-Stokes

Here in this section, we consider the motion of a viscous and incompressible fluid in a uniform gravitational field in dimension two. We assume that this fluid occupies a bounded domain .t/ and that is separated from the vacuum by a smooth closed curve @.t /. The velocity of the fluid satisfies the Navier-Stokes equations, and the curve @.t / is transported by the restriction of the velocity to the interface. @t u C .u  r/u D rp C u C .0; g /

in .t/

r uD0

in .t/

@t X .x; t / D u.X .x; t /; t / x 2 0 .t/ D X .0 ; t /

(5)

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with initial conditions u.x; 0/ D u0 .x/

.r  u0 D 0/

(6)

.0/ D 0 The boundary conditions for the viscous case on the interface differ from the ideal case: we impose the continuity of the stress tensor, i.e., 

ˇ  pfluid j  .ru C ruT /ˇ n D patm n;

(7)

where pfluid is the pressure of the fluid at the interface, is the viscosity, u the velocity of the fluid, the atmospheric pressure patm is constant, and n is the unitary normal vector pointing out the fluid. This implies p  patm D n  .ru C ruT /n 0 D  .ru C ruT /n Here  is the unit tangential vector to @.t /. Moreover, this implies a compatibility condition on the initial data u0 : r  u0 D 0 0  .ru0 C ruT0 /n0 D 0

in 0 on @0

In light of previous achievements, we study the existence of solutions that develop splash singularities. A breakdown of the solutions has never been shown for incompressible viscous fluids. However, there exists a huge literature dealing with the problem of local existence and global existence for small initial data. Here we review some of these works (Fig. 4). Fig. 4 Splash singularity for a closed interface

VACUUM

ρ>0 u>0 FLUID

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Some of the earliest papers in this topic were written by Solonnikov. He studied the problem of a viscous fluid bounded by a free boundary in the vacuum (the fluid domain is bounded). He showed local existence of solutions using Holder spaces with [53] and without surface tension [54, 55]. For the case without surface tension, Beale, in [8] (See also [8, 10]), proved the local well-posedness in Sobolev spaces (see also [1] for an extension to Lp spaces). Sylvester, in [57], showed global well-posedness for small initial data. Hataya, in [37, 38], showed the existence of solutions which decay algebraically in time for a periodic in the horizontal variable surface. Guo and Tice, in [34, 35] and [36], have proved algebraic decay rate in time for asymptotically flat surfaces and almost exponential decay rate in time for periodic in the horizontal variable surfaces. For a more extensive list of references, see [17]. Despite the vast work performed in these problems, there are no results studying the formation of singularities although this phenomenon is observed very often in the nature: the wave breaking in a beach, for example. Our purpose is to study the type of singularities we found in [16] for an ideal fluid, in the case of a viscous fluid. Once we know that the free boundary for the Euler equation can develop a splash singularity, it is difficult to find a reason why the viscosity can avoid it. In the viscous case, new technical difficulties arise. The strategy of [16]: we start in a splash curve and solve the equation backwards in time by choosing a suitable initial velocity, cannot work for the present case of nonzero viscosity, since the equations cannot be solved backward in time. In order to get around this difficulty in [17], we use again the conformal map P from [16] that separates the self-intersecting points of the splash curve. But instead of showing local existence backward in time in the tilde domain, we prove local existence in the tilde domain adapting the analysis of Beale [8]. Moreover, we show that the solutions depend stably on the initial conditions. Now we are in position to apply a perturbative argument to prove a splash singularity for Navier-Stokes (for a different proof, see [26]) : • Theorem [Splash for Navier-Stokes] Initial data exist in H k such that the solution to the system (5), (6), and (7) produces a splash-type singularity. In order for this argument to work, it is crucial to be able to choose properly the initial data u0 as for Euler equations. This indeed is also the case in the presence of viscosity, although the compatibility conditions give the following restriction: r  u0 D 0

0  .ru0 C ru0 /n0 D 0

But we can define, for every x in a small neighborhood of @.0/, .x/ D

0 .s/

C

2 .s/

2

27 Water Waves with or Without Surface Tension

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where s is the tangential variable and  is distance of x to the boundary. Then we extend to 0 in some smooth way and define u0 D r ? . The continuity of the stress tensor implies that @2s

0 .s/

D

2 .s/

and the normal velocity is given by u0  nj@0 D @s

0 .s/

Therefore we can pick first 0 in order to have a suitable normal velocity and then 2 .s/ to satisfy the boundary condition. The above proof does not show the existence of splat singularities for NavierStokes. This singularity is a variation of the former in which the fluid interface self-intersects along an arc which can develop for Euler equations. The presence of viscosity may prevent the existence of splat singularities. This is the case, see [21], of an incompressible viscous fluid in a porous media (or in a vertical Hele-shaw cell [52]) which is modeled by Darcy’s law [48]. The proof is shown by contradiction: suppose that there exists a time T where there is a splat singularity, i.e., the smooth interface collapses along an arc at time T. We apply again the conformal P and transform the system to the tilde domain. In the tilde domain we show that due to Darcy’s law there is instant analyticity of the regularity of the interface. Moreover, the strip of analyticity is nonzero as long as the regularity of the curve and the arcchord condition do not fail. So therefore at time T the splat curve in the tilde domain is real-analytic. Thus, applying P 1 , it follows that the analytic curve self-intersects along an arc, hence we get a contradiction.

6

On the Absence of Splash Singularities with Two Fluids

In this section we discuss the possible formation of singularities of an internal wave. If the vacuum is replaced by an incompressible fluid with low density, then physical intuition suggests that the water wave gets close to a splash singularity, but the low-density fluid resists collapse, so the water wave recoils. The approach from our previous analysis for the splash does not work for the case of surface-internal waves, and it demands new ideas (Fig. 5). In a recent paper [31] (see also [27]) it is shown that the presence of a second fluid prevents splash singularities to form in finite time. The crucial observation is that for a splash to develop the amplitude of the vorticity blows-up and the presence of a smooth fluid gives a L1 bound on the measure of the vorticity in the boundary: Let z W R=Z ! R2 smooth and satisfies the chord-arc condition jz.˛; t /  z.˛ 0 ; t /j  CA.t /k˛  ˛ 0 k for ˛; ˛ 0 2 R=Z:

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Fig. 5 Two fluids: internal waves

FLUID 1

ρ ρ

FLUID 2

Here, CA.t / > 0 is the “chord-arc constant,” and k˛  ˛ 0 k denotes the distance from ˛ to ˛ 0 in R=Z. Consider the amplitude of the vorticity !.˛; t /. Then • if j!.˛; t /j remains bounded as t ! T , then, because the interface moves with 1 the fluid, the function F .t/ D CCA.t satisfies / j

dF j  Const:jF jln.jF j C 2/ dt

Hence, F(t) remains bounded as t ! T , so the splash cannot form. • if j@k˛ z.˛; t /j .k D 0; 1; 2; : : : ; 10/ and maxfj@ˇx uI .x; t /j W x 2 I .t /; jˇj  10g remain bounded as t ! T , then j!.˛; t /j remains bounded as t ! T , because ! satisfies a variant of Burger’s equation. Another recent result concerning splash singularities and two fluids is obtained in [23] where it is shown that there are stationary solutions of Euler equations with two fluids and small gravity whose interfaces are arbitrarily close to a splash. A stationary solution of the two fluid system reduces to finding 2-periodic functions !.˛/ and z.˛/  ˛ satisfying 2j@˛ zj2 F .z/ C !.!  2/ D 2; 2BR.z; !/  @˛ z C ! D 2; BR.z; !/  @? ˛ z D 0;

27 Water Waves with or Without Surface Tension

1345

where F is given by F .z/ D 

22 qK.z/  2gz2 C 1 C 2;  2  1

q WD 2 , WD 221 1 ; K.z/ is the curvature of the interface and  is a perturbation of the constant arising in the pure capillary wave problem. The idea of the proof is to perturb a family of exact stationary water waves introduced by Crapper [29]. And use the implicit function theorem to perturb with respect to the density of the upper fluid 1 and the gravity a Crapper wave that is arbitrarily close to the splash.

7

Conclusions

In general it is a challenge to prove the existence of finite time singularities in the dynamics of an incompressible fluid. In particular for water waves, we have shown the existence of a particular kind of singularities, splash and splat, by taking advantage of the absence of a second fluid. Here, in this section, we compile a list of problems that in the opinion of the authors would be interesting, not only from the point of view of the model but also in terms of the analytical challenge they present: • Can singularities (with finite energy) develop when the interface is a graph? Can the curvature diverge? • Once the interface has curled, can singularities develop in finite time that are different from splash and splat singularities? • Can the existence of a solution to the system (3) and (4) be proved that is initially a graph, then curls, and after collapses in a splash? • What influence does surface tension have on the formation of singularities? • If we assume that there is a sea bed, that is, the water does not have infinite depth, can other types of singularity be formed? • Can rotation prevent or give rise to singularities that are different from the irrotational solutions? We are aware that all these questions are currently being addressed by different research groups and feel sure we will have answers to all of them (either partially or completely) in the coming years. In particular, in collaboration with Angel Castro, Francisco Gancedo, and Javier Gómez-Serrano, we would like to exhibit a water wave solution that captures both mechanisms: turnover and splash, i.e., the interface starts as an H 4 -smooth graph at time zero and ends in a splash at time T . We believe that we have the tools that may lead to a rigorous computer-assisted proof of the existence of such a solution. Let us describe briefly the strategy: We have a numerical simulation that starts as a graph and ends in a splash, and our goal is to show that nearby this approximate solution, there is an actual solution of

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water waves that captures both phenomena (turning and splash). There are two main ingredients involved in the plausible proof, and both of them make extensive use of the help of computers. First of all, the structural stability theorem, which we proved in [18], shows that an approximate solution .x;  / of the water wave system remains close to an exact solution .z; $ / of the system if their initial conditions are close. The question to address here is to quantify how fast the two solutions separate from each other and in which topology. To do so, we should give bounds on the constants of the a priori bounds. First of all, one of the difficulties one has to overcome is to write the system in terms only of the difference between the numerical solution and the exact solution .D; d / D .z  x; $   / or the approximate solution itself .x;  / in the tilde domain where the chord-arc and the smoothness of the interface do not change dramatically. The energy of the system E.t / which involves H k d norms of .D; d / satisfies dt E.t / D F .E/.t / with F nonlinear. The second part of the proof is devoted to rigorously bind the norms of the errors that a numerical solution obtained by nonrigorous simulation makes. In order to do so, we will make use of the framework of interval arithmetic (see the seminal work of Moore [47] for the underlying basics of this technique), in which we will work with intervals instead of real numbers. Since a computer can’t produce a perfect result because of rounding errors, or the errors done by performing arithmetic operations, we will give an enclosure of the result, i.e., an interval in which we know that our solution will lie. The width of this interval will be wide enough so that the aforementioned errors are taken into account. The same questions may be posed when the scenario or the model is modified: for example, by studying the dynamics of the surface of a viscid fluid or by considering the interface between two immiscible fluids (in this case, the pressure at the interface is not constant and depends on the dynamics of both fluids), such as oil and water.

8

Cross-References

 Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible

Fluid Mechanics  Weak Solutions and Diffuse Interface Models for Incompressible Two-Phase

Flows Acknowledgements We thank Tania Pernas for providing the figures of the paper. DC was partially supported by the grants MTM2014-59488-P (Spain) and SEV-2015-556. CF was partially supported by NSF grant DMS-1265524, AFOSR grant FA9550-12-1-0425, and Grant No 2014055 from the United States-Israel Binational Science Foundation (BSF).

References 1. H. Abels, The initial-value problem for the Navier-Stokes equations with a free surface in L q-Sobolev spaces. Adv. Differ. Equ. 10(1), 4564 (2005) 2. T. Alazard, N. Burq, C. Zuily, On the water-wave equations with surface tension. Duke Math. J. 158(3), 413499 (2011)

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55. V.A. Solonnikov, Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval. St. Petersburg Math. J. 3(1), 189–220 (1992) 56. G.G. Stokes, Considerations relative to the greatest height of oscillatory irrota- tional waves which can be propagated without change of form, in Mathematical and Physical Papers, ed. by G.G. Stokes, J. Larmor, J.W. Strutt Rayleigh (Cambridge University Press, Cambridge, 1880) 57. D.L.G. Sylvester, Large time existence of small viscous surface waves without surface tension. Commun. Partial Differ. Equ. 15(6), 823–903 (1990) 58. E. Varvaruca, G.S. Weiss, A geometric approach to generalized Stokes conjectures. Acta Math. 206(2), 363–403 (2011) 59. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130(1), 39–72 (1997) 60. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12(2), 445–495 (1999) 61. S. Wu, Almost global wellposedness of the 2-D full water wave problem. Invent. Math. 177(1), 45–135 (2009) 62. S. Wu, Global wellposedness of the 3-D full water wave problem. Invent. Math. 184(1), 125–220 (2011) 63. S. Wu, On a class of self-similar 2d surface water waves (2012). Arxiv preprint arXiv:1206.2208 64. H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci. 18(1), 49–96 (1982) 65. P. Zhang, Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations. Commun. Pure Appl. Math. 61(7), 877–940 (2008)

Part III Compressible Viscous Fluids

Concepts of Solutions in the Thermodynamics of Compressible Fluids

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Conservation/Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Constitutive Relations for Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Equations of Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Euler System, Ideal Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Navier-Stokes-Fourier System, Viscous and Heat-Conducting Fluids . . . . . . . . . . 3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Slip vs. Stick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Boundary Behavior of the Temperature, Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . 4 Well Posedness, Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Classical Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Local-in-Time Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Classical Solvability: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Euler System: Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Navier-Stokes-Fourier System: Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 A Disturbing Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Mathematical Theory of Compressible, Viscous, and Heat-Conducting Fluids . . . . . . . 6.1 Finite Energy Weak Solutions to the Navier-Stokes-Fourier System . . . . . . . . . . . 6.2 Global-in-Time Existence of Finite Energy Weak Solutions . . . . . . . . . . . . . . . . . . 7 Dissipative Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Ballistic Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Relative Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Relative Energy Inequality, Dissipative Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Weak-Strong Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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E. Feireisl () Evolution Differential Equations (EDE), Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, Czech Republic e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_31

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8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Conditional Regularity via the Relative Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The objective of this chapter is to highlight the recent development of the mathematical theory of complete fluids. The word complete means the governing system of equations is rich enough to incorporate the basic physical principles, in particular the first, second, and third laws of thermodynamics, in a correct and integral way into the mathematical model. In the whole text, the platform of classical continuum mechanics is adopted, where the fluid motion is described in terms of observable macroscopic quantities: the mass density, the (absolute) temperature, and the (bulk) velocity.

1

Introduction

This chapter reviews the recent development of the mathematical theory of complete fluid systems, where the word complete refers to the governing system of equations that is rich enough to incorporate the basic underlying physical principles: the first, second, and third laws of thermodynamics in the mathematical model. The standard platform of classical continuum mechanics is systematically used, where the fluid motion is described in terms of observable macroscopic quantities: the mass density, the (absolute) temperature, and the (bulk) velocity. Continuum mechanics describes a fluid in motion in terms of numerical values of macroscopic quantities – fields – depending on the time t and the spatial position x. Throughout the whole text, the Eulerian description is used, where the coordinate frame is attached to the physical domain  occupied by the fluid. The fields are interrelated through a system of field equations – balance laws – reflecting the underlying physical principles of conservation of mass, momentum, energy, as well as other quantities as the case may be. The material properties of a specific fluid are characterized by constitutive relations. The interaction of the fluid with the outer world is specified through boundary conditions.

1.1

State Variables

In accordance with the general principles delineated above, the state of a fluid at any instant t is characterized by its mass density % D %.t; x/ and the absolute temperature # D #.t; x/. The motion is described by means of the macroscopic velocity field u D u.t; x/. Accordingly, the fluid moves along streamlines – the spatial curves X D X.t / solving d X D u.t; X/: dt

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Such a choice of state variables gives rise to a rather limited but still sufficiently rich class of fluids considered in this chapter. Obviously, more complex models are necessary and can be developed to handle more complicated real-world applications.

1.2

Conservation/Balance Laws

The conservation/balance laws in continuum mechanics are usually written in a general differential form @t d .t; x/ C divx F.t; x/ D s.t; x/: A conservation/balance law reflects the underlying physical principle relating the changes of a volume density of a physical quantity d to its flux F and a possible source term s as the case may be. In the Eulerian coordinate system, the flux F consists of a convective (conservative) component d u and, at least for certain physical quantities, a diffusive (dissipative) part proportional to spatial derivatives of d .

1.2.1 Equation of Continuity, Mass Conservation A mathematical formulation of the physical principle of mass conservation reads @t %.t; x/ C divx .%.t; x/u.t; x// D 0:

(1)

The mass flux is purely convective and the source term is absent in (1).

1.2.2 Momentum Equation, Newton’s Second Law The time evolution of the momentum %u is governed by the system of equations @t .%.t; x/u.t; x// C divx .%.t; x/u.t; x/ ˝ u.t; x// D divx T.t; x/ C %.t; x/f.t; x/; (2) where T denotes the Cauchy stress specified below, and f is the volume density of the external forces acting on the fluid.

1.2.3 Energy Balance, First Law of Thermodynamics The volume density of the total energy of the fluid ED

1 %juj2 C %e 2

consists of the kinetic energy component 12 %juj2 and the internal energy %e. In accordance with the specific choice of state variables, the (specific) internal energy e D e.%; #/ is a function of the density % and the temperature #.

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A mathematical formulation of the first law of thermodynamics reads:  @t

    1 1 2 2 %juj C %e .t; x/ C divx %juj C %e .t; x/u.t; x/ 2 2

Cdivx q.t; x/  divx .T.t; x/  u.t; x// D %.t; x/f.t; x/  u.t; x/ C%.t; x/Q.t; x/;

(3)

where q denotes the diffusive part of the internal energy flux and Q the volume density of the external heat sources.

1.3

Constitutive Relations for Fluids

Fluids are characterized by Stokes’ law: T D S  pI;

(4)

where S is the viscous stress tensor and p is a scalar quantity termed pressure. Similarly to the specific energy e, the pressure p D p.%; #/ is a function of the state variables %, #.

1.3.1 Entropy, Second Law of Thermodynamics The second law of thermodynamics postulates the existence of another thermodynamic function – entropy. Here the specific entropy s D s.%; #/ is supposed to be interrelated with the internal energy e and the pressure p by means of Gibbs’ equation: #Ds.%; #/ D De.%; #/ C p.%; #/D

  1 ; %

(5)

where D stands for the differential with respect to % and # (see [28]). Internal Energy Equation In view of Stokes’ relation (4), the total energy balance may be rewritten as the internal energy equation: @t .%e.%; #// C divx .%e.%; #/u/ C divx q D S W rx u  p.%; #/divx u C %Q;

(6)

or, equivalently, in the form of thermal energy balance   @p.%; #/ divx u C %Q; %cv .%; #/ @t # C u  rx # C divx q D S W rx u  % @#

(7)

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where we have introduced the specific heat at constant volume: cv .%; #/ D

@e.%; #/ : @#

Note that the passage from (6) to (7) uses the equation of continuity (1). In addition, the thermodynamic stability hypothesis imposes further restrictions on p D .%; #/ and e.%; #/, specifically, @e.%; #/ @p.%; #/ > 0; cv .%; #/ D >0 @% @#

(8)

for all % > 0, # > 0 (see [2]). Entropy Production The internal energy balance (6) divided by #, together with the equation of continuity (1) and Gibbs’ relation (5), gives rise to the entropy equation: @t .%s.%; #// C divx .%s.%; #/u/ C divx

q #

1 D #

  % q  rx # S W rx u  C Q: # # (9)

Here, the quantity 1 D #



q  rx # S W rx u  #

 0

(10)

is termed entropy production rate and, in accordance with the second law of thermodynamics, is always nonnegative. This may (and will) imply some structural restrictions to be satisfied by the constitutive equations for S and q discussed below.

2

Basic Equations of Fluid Dynamics

In order to close the system of fluid dynamic equations, certain constitutive equations relating the viscous stress S and the internal energy flux q to the basic phase variables are needed.

2.1

Euler System, Ideal Fluids

Ideal fluids are those for which S D 0, q D 0. The associated system of equations is usually called Euler system (see, e.g., [14]): @t % C divx .%u/ D 0;

(11)

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@t .%u/ C divx .%u ˝ u/ C rx p.%; #/ D 0;  @t

(12)

     1 1 %juj2 C %e.%; #/ C divx % %juj2 C %e.%; #/ u C divx .p.%; #/u/ D 0; 2 2 (13)

where, for the sake of simplicity, the effect of external sources is omitted in (12), (13).

2.2

Navier-Stokes-Fourier System, Viscous and Heat-Conducting Fluids

Ideal fluids introduced in the previous section may and should be seen as a hypothetical limit state of real fluids that are both viscous and heat conducting. In such a case, the viscous stress S and the internal energy flux q depend effectively on the velocity gradient rx u and the temperature gradient rx #, respectively.

2.2.1 Newtonian Fluids For Newtonian or linearly viscous fluids, the viscous stress tensor is a linear function of the velocity gradient. Newton’s Law The viscous stress tensor S for a Newtonian fluid is given by Newton’s rheological law (see [11]):   2 S.rx u/ D  rx u C rxt u  divx uI C divx uI; 3

(14)

where the scalar quantities  and  are termed the shear and bulk viscosity coefficient, respectively. In accordance with the second law of thermodynamics enforced through (10),  and  are nonnegative and may depend on the state variables %, # as the case may be. Fourier’s Law Similarly to (14), the internal energy flux q of a linearly viscous fluid is a linear function of rx # determined by Fourier’s law: q D rx #;

(15)

with the heat conductivity coefficient   0 that may depend on % and #.

2.2.2 Navier-Stokes-Fourier System In accordance with the previous discussion, the time evolution of a Newtonian heat-conducting fluid is determined by the Navier-Stokes-Fourier system (see, e.g., [29, 47]):

28 Concepts of Solutions in the Thermodynamics of Compressible Fluids

@t % C divx .%u/ D 0; @t .%u/ C divx .%u ˝ u/ C rx p.%; #/     2 t D divx  rx u C rx u  divx uI C divx uI ; 3   %cv .%; #/ @t # C u  rx #  divx .rx #/     2 @p.%; #/ D  rx u C rxt u  divx uI C divx uI W rx u  % divx u; 3 @#

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(16)

(17)

(18)

where, similarly to the Euler system (11)–(13), the effect of the external sources has been omitted. It is worth noting that equation (18) is formally equivalent to the total energy balance (3), the internal energy balance (6), and even to the entropy balance (9).

3

Boundary Conditions

Fluids are usually confined to a bounded spatial domain ; the unbounded domains considered in certain mathematical models should be seen as an idealization of large fluid domains in the real world. There is a large variety of boundary behavior of both Eulerian and Navier-Stokes fluid determined by its interaction with the outer world. For definiteness, only a very simple situation will be considered in this chapter, where the kinematic boundary @ is at rest and impermeable, meaning u  nj@ D 0

(19)

where the symbol n denotes the outer normal vector to @.

3.1

Slip vs. Stick

While the impermeability condition (19) is sufficient for the description on an inviscid fluid governed by the Euler system (11)–(13), an extra piece of information is needed if the fluid is viscous.

3.1.1 No-Slip Boundary Conditions A commonly accepted hypothesis asserts that a viscous fluid adheres completely to the boundary, meaning, in addition to (19), also the tangential component of the velocity vanishes on @. This can be written concisely in the form of no-slip boundary condition: uj@ D 0:

(20)

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3.1.2 No-Stick, Complete Slip Boundary Conditions Under certain circumstances, e.g., for nanofluids (see [38]), it was observed that the no-slip condition (20) is no longer a relevant description of the fluid behavior. Instead, one may postulate the no-stick or complete slip condition: .S  n/  nj@ D 0:

(21)

In other words, the tangential component of the normal (viscous) stress vanishes on @.

3.1.3 Navier’s Slip A compromise between (20) and (21) is Navier’s slip condition: ŒS  ntan C ˇ%uj@ D 0;

(22)

where ˇ plays a role of a friction coefficient (see [9]).

3.2

Boundary Behavior of the Temperature, Heat Flux

For heat-conducting fluids, the boundary behavior of the temperature must be specified. In energetically insulated systems, the heat flux vanishes in the normal direction to @: q  nj@ D rx #  nj@ D 0:

(23)

Alternatively, the distribution of the temperature on the boundary can be prescribed yielding Dirichlet-type boundary conditions: #j@ D #b :

(24)

Of course, there are many other possibilities of the boundary behavior of # including a combination of (23), (24) imposed on disjoint parts of @.

4

Well Posedness, Classical Solutions

A system of evolutionary partial differential equations, supplemented with suitable boundary conditions, is well posed provided it admits a unique solution for any admissible initial state. The initial state for the Navier-Stokes-Fourier or complete Euler system is given by specifying the initial distribution of the density, velocity, and temperature: %.0; / D %0 ; #.0; / D #0 ; u.0; / D u0 in :

(25)

Alternatively, the initial momentum .%u/0 , the initial internal energy e0 , and/or the entropy s0 may be given. In view of the physical background, the initial data

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should obey certain admissibility conditions, in particular, the density and (absolute) temperature should be strictly positive, the total initial energy finite, among others.

4.1

Classical Solvability

Given smooth and physically admissible initial data, the problems in fluid dynamics are supposed to admit unique classical (smooth) solutions. This is (known to be) true, however, only on a possibly short time interval Œ0; Tmax /. Whether or not Tmax D 1 is in general an open question. Solutions of the (inviscid) Euler system (11)–(13) may develop discontinuities (shock waves) in a finite time no matter how smooth and even small the initial data are (see [41, Chapter 15]). Regularity of solutions to the Navier-Stokes-Fourier system (16)–(18) in the long run is a famous open problem (see [20, 43] for a thorough discussion in the context of incompressible fluids).

4.2

Local-in-Time Existence

There are many results concerning local-in-time existence of smooth solutions for both the Euler and the Navier-Stokes-Fourier system, for different choices of spatial geometries, boundary conditions, classes of initial data, etc.

4.2.1 Euler System: Classical Solutions To avoid technicalities connected with the boundary behavior of solutions, the existence result for the Euler system will be stated in the physically relevant domain R3 (see [5, Chapter 13, Theorem 13.1.]). Theorem 1. Let % > 0, # > 0 be given. Suppose that the pressure p D p.%; #/, e.%; #/ are twice continuously differentiable functions satisfying Gibbs’ relation (5) and the thermodynamic stability condition (8) in an open set U  .0; 1/2 containing Œ%; #. Let the initial data %0 , #0 , u0 be given such that Œ%0 .x/; #0 .x/ belong to a compact subset of U for all x 2 R3 ; %0  %; #0  # 2 W k;2 .R3 /; u0 2 W k;2 .R3 I R3 / for some k >

5 : 2

Then there exists a positive time T > 0 such that the Euler system (11)–(13) admits a solution %, #, u unique in the class %  %; #  # 2 C .Œ0; T I W k;2 .R3 // \ C 1 .Œ0; T I W k1;2 .R3 //; u 2 C .Œ0; T I W k;2 .R3 I R3 // \ C 1 .Œ0; T I W k1;2 .R3 ; R3 //: Remark 1. The symbol W k;2 .R3 / denotes the Sobolev space of functions having (generalized) derivatives up to order k square integrable in R3 .

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4.2.2 Navier-Stokes-Fourier System: Classical Solutions A short-time existence result for the Navier-Stokes-Fourier system (11)–(13), endowed, for definiteness, with the boundary conditions (20), (23) may be stated as follows (see [44, Theorem A and Remark 3.3]). Theorem 2. Let   R3 be a bounded domain of class C 2C ,  > 0. Let the initial data %0 ; #0 2 W 3;2 ./; u0 2 W 3;2 .I R3 / be given such that Œ%0 .x/; u0 .x/ belong to a compact subset of an open set U  .0; 1/2 , and satisfying the compatibility conditions ˇ ˇ u0 j@ D 0; rx #0  nj@ D 0; rx p.%0 ; #0 /ˇ @    ˇ 2 ˇ t D divx .%0 ; #0 / rx u0 C rx u0  divx u0 I C .%0 ; #0 /divx u0 I ˇ : @ 3 Suppose that the pressure p D p.%; #/, the specific heat at constant volume cv D cv .%; #/, as well as the transport coefficients  D .%; #/,  D .%; #/, and  D .%; #/ are three times continuously differentiable in U and satisfy @p.%; #/ > 0; cv .%; #/ > 0; .%; #/ > 0; .%; #/  0; .%; #/ > 0 @% for all Œ%; # 2 U . Then there exists T > 0 such that the Navier-Stokes-Fourier system (11)–(13), supplemented with the boundary conditions (20), (23), admits a unique solution in the class: %; # 2 C .Œ0; T I W 3;2 .// \ C 1 .Œ0; T I W 2;2 .//; u 2 C .Œ0; T I W 3;2 .I R3 // \ C 1 .Œ0; T I W 2;2 .I R3 //: Remark 2. It can be shown that any solution belonging to the class specified in Theorem 2 possesses all the necessary derivatives and is therefore a classical solution in the open set .0; T /  .

4.3

Classical Solvability: Conclusion

The systems of equations considered in mathematical fluid dynamics are nonlinear and as such susceptible to develop singularities, either in the form of steep gradients (shock waves) or concentrations (mass collapse). Such phenomena have been rigorously verified for the inviscid Euler system. What is more, a mathematical theory based on global-in-time classical solutions is beyond the reach of the available mathematical methods and up-to-date knowledge, even for the Navier-Stokes-Fourier system. On the other hand, these problems are being solved numerically with

28 Concepts of Solutions in the Thermodynamics of Compressible Fluids

1363

continuously improving capacity of modern computers. Some concept of solution is therefore needed to perform a rigorous analysis of convergence of the numerical methods. The weak solutions discussed in the next part offer such alternative.

5

Weak Solutions

The idea of weak solutions is based on the concept of generalized derivatives or distributions. Classical functions are replaced by their integral averages or, more precisely Z f W Q 7! R  Q

f '; ' 2 Cc1 .Q/

where the symbol Cc1 .Q/ denotes the set of infinitely differentiable functions with compact support in Q. Differential operators D can be conveniently expressed by means of a formal by-parts integration: Z Df   Q

fD'; ' 2 Cc1 .Q/:

Accordingly, any (locally) integrable function possesses derivatives of arbitrary order! The Sobolev spaces W k;2 used in the previous part are based on distributional derivatives.

5.1

Euler System: Weak Solutions

A trio of functions Œ%; #; u is a weak solution of the Euler system (11)–(13) in the set .0; T /   if: Z

T

Z .%@t ' C %u  rx '/ dx dt D 0

0

(26)



for any ' 2 Cc1 ..0; T /  /; Z

T

Z .%u  @t ' C %u ˝ u W rx ' C p.%; #/divx '/ dx dt D 0

0



for any ' 2 Cc1 ..0; T /  I R3 /; Z

T 0

Z  

 1 %juj2 C %e.%; #/ @t ' 2

(27)

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 C

  1 2 %juj C %e.%; #/ C p.%; #/ u  rx ' dx dt D 0 2

(28)

for any ' 2 Cc1 ..0; T /  /. Note that the integral identities (26-28) are well defined as soon as all the compositions of %, #, u with all nonlinearities are at least locally integrable.

5.1.1 Weak Continuity, Initial and/or Boundary Conditions Functions that are merely (locally) integrable do not posses traces on lowerdimensional structures in , in particular, it is not clear how to define the initial and/or boundary conditions in the class of weak solutions. Fortunately, the necessary piece of information is already encoded in the weak formulation. For example, if % is a weak solution of (26), the choice of a special test function '.t; x/ D .t / .x/, 2 Cc1 .0; T /, 2 Cc1 ./ gives rise to Z

T

0

Z

Z

.t /

%.t; / dx dt D  

0

Z

T

.t / 0

%u.t; /  rx dx dt; 

from which it follows that the function Z t 7! %.t; / dx admits an integrable generalized derivate in .0; T / 

and as such can be represented, upon modification on a set of times of zero measure, by an absolutely continuous function. Thus the initial conditions can be interpreted in the sense of integral averages: Z

Z

%.0; / D %0 

%.t; / dx ! 



%0 dx as t ! 0 C for any 2 Cc1 ./:

The anticipated weak continuity in time makes possible to incorporate the initial conditions into the weak formulation, replacing (26-27) by Z

Z

T

Z .%@t ' C %u  rx '/ dx dt D  

0

%0 '.0; / dx

(29)



for any ' 2 Cc1 .Œ0; T /  /; Z

T

Z .%u  @t ' C %u ˝ u W rx ' C p.%; #/divx '/ dx dt

0



Z

D

%0 u0  '.0; / dx 

for any ' 2 Cc1 .Œ0; T /  I R3 /;

(30)

28 Concepts of Solutions in the Thermodynamics of Compressible Fluids

Z

T

Z  

0

 1 2 %juj C %e.%; #/ @t ' 2

  1 2 C %juj C %e.%; #/ C p.%; #/ u  rx ' dx dt 2  Z  1 D %0 ju0 j2 C %0 e.%0 ; #0 / '.0; / dx  2 

1365

(31)

for any ' 2 Cc1 .Œ0; T /  /. Boundary conditions, or at least the normal traces of the fluxes, can be interpreted in a similar way. This issue will be discussed in the context of the Navier-StokesFourier system. Remark 3. As a matter of fact, the weak formulation can be derived directly (without passing from classical to generalized derivatives) from the underlying physical principles written in their natural integral form (see [25, Chapter 1]).

5.2

Navier-Stokes-Fourier System: Weak Solutions

In order to introduce a weak formulation of the Navier-Stokes-Fourier system, rewrite first the energy equation (18) in the conservative form @t .%e.%; #// C divx .%e.%; #/u/  divx .rx #/     2 D  rx u C rxt u  divx uI C divx uI W rx u  p.%; #/divx u: 3 Note that this is possible as long as p, e, and cv D @# e are interrelated through (5). Accordingly, the weak formulation of the Navier-Stokes-Fourier system (16)– (18) reads as follows: Z

T

Z

Z .%@t ' C %u  rx '/ dx dt D 

0



%0 '.0; / dx

(32)



for any ' 2 Cc1 .Œ0; T /  /; Z

T

Z .%u  @t ' C %u ˝ u W rx ' C p.%; #/divx '/ dx dt

Z

0 T

Z



  rx u C

0



Z

rxt u

 Z TZ 2  divx uI W rx ' dx dt C divx udivx ' dx dt 3  0

%0 u0  '.0; / dx

D 

(33)

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E. Feireisl

for any ' 2 Cc1 .Œ0; T /  I R3 /; Z

T

Z

Z

T

Z

.%e.%; #/@t ' C %e.%; #/u  rx '/ dx dt  0



rx #  rx ' dx dt 0



  Z   2  rx u C rxt u  divx uI C divx uI W rx u' dx dt 3  0 Z Z TZ p.%; #/divx u' dx dt  %0 e.%0 ; #0 /'.0; / dx  Z

T

D

0



(34)



for any ' 2 Cc1 .Œ0; T /  /. Similarly to the previous part, the weak formulation already includes the satisfaction of the initial conditions.

5.2.1 Boundary Conditions The reader will have noticed that, in contrast with the Euler system, the weak formulation of the Navier-Stokes-Fourier system includes first derivatives of the velocity u as well as the temperature #. Anticipating that the first derivatives are integrable functions, the fields u and # have well-defined traces on the boundary @ (see, e.g., [48, Chapter 3]). Thus the Dirichlet-type boundary conditions (20), (24) may be incorporated in the definition of the function spaces the solution belongs to. In particular, the no1;p slip condition (20) corresponds to the Sobolev space W0 ./ of functions with integrable first-order derivatives in power p and vanishing on the boundary. The boundary conditions of Neumann type like (22), (23) can be accommodated in the weak formulation by extending the class of admissible test functions. Thus, for instance, the no-flux condition (23) is enforced by postulating (34) for any ' 2 Cc1 .Œ0; T /  /. The complete slip (19), (21) requires u  nj@ D 0 and (33) to be satisfied for any ' 2 Cc1 .Œ0; T /  I R3 /, '  nj@ D 0, etc.

5.3

A Disturbing Example

The class of weak solutions to a given problem is apparently much larger than required by the classical theory. In other words, it might be easier to establish existence but definitely more delicate to show uniqueness among all possible weak solutions emanating from the same initial data. Indeed there exist weak solutions to the (incompressible) variant of the Euler system that can be obtained in a completely non-constructive way by the method of convex integration recently developed in the context of fluid mechanics in [15]. A further adaptation of this technique provides a rather illustrative but at the same time quite disturbing example of non-uniqueness in the context of fluid thermodynamics. To this end, consider the so-called EulerFourier system: @t % C divx .%u/ D 0;

(35)

28 Concepts of Solutions in the Thermodynamics of Compressible Fluids

1367

@t .%u/ C divx .%u ˝ u/ C rx .%#/ D 0;

(36)

3 Œ@t .%#/ C divx .%#u/  # D %#divx u: 2

(37)

The system (35-37) is a special case of the Navier-Stokes-Fourier system with p D %#, cv D 32 ,  D  D 0,  D 1. Although a correct physical justification of an inviscid heat-conducting fluid may be dubious, the system has been used as a suitable approximation in certain models (see [46]). For the sake of simplicity, the problem will be endowed with the spatially periodic boundary conditions, meaning the underlying spatial domain 3   D T 3 D Œ1; 1 jf1I1g is the “flat” torus. The following result holds true (see [10, Theorem 3.1]). Theorem 3. Let T > 0 be given. Let the initial data satisfy %0 ; #0 2 C 3 .T 3 /; u0 2 C 3 .T 3 I R3 /; %0 > 0; #0 > 0 in T 3 : Then the initial-value problem for the Euler-Fourier system (35-37) admits infinitely many weak solutions in .0; T /   belonging to the class % 2 C 2 .Œ0; T   /; @t # 2 Lp .0; T I Lp .//; rx2 # 2 Lp .0; T I Lp .I R33 // for any 1  p < 1, u 2 Cweak .Œ0; T I L2 .I R3 // \ L1 ..0; T /  I R3 /; divx u 2 C 2 .Œ0; T   /: The conclusion of Theorem 3 reveals the main drawback of the mathematical theory based on the concept of weak solutions, namely, the restrictions imposed by the weak formulation upon the class of possible solutions are too weak to ensure uniqueness. Apparently, the weak formulation must be augmented by certain admissibility conditions dictated by physics to pick up the relevant solution. On the other hand, the extra conditions should not be too strong to prevent global-in-time existence. This as well as other related issues will be addressed in the remaining part of this chapter devoted to the mathematical theory of the complete Navier-StokesFourier system.

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6

E. Feireisl

Mathematical Theory of Compressible, Viscous, and Heat-Conducting Fluids

There is an alternative weak formulation of the Navier-Stokes-Fourier system based on the second law of thermodynamics. The theory accommodates, in particular, the energetically closed systems, mechanically and thermally insulated from the outer world. Accordingly, the conservative boundary conditions uj@ D 0; rx #  nj@ D 0;

(38)

are imposed, in particular the total energy E is a constant of motion: d dt

Z  

 1 %juj2 C %e.%; #/ dx D 0: 2

(39)

The total energy balance (39), together with @t % C divx .%u/ D 0;

(40)

@t .%u/ C divx .%u ˝ u/ C rx p.%; #/ D divx S.rx u/

(41)

and the entropy inequality @t .%s.%; #// C divx .%s.%; #/u/ C divx

q #

1  #



q  rx # S.rx u/ W rx u  #

 (42)

will be used as a basis of a new weak formulation of the Navier-Stokes-Fourier system. Similarly to the above, the viscous stress and the heat flux are taken in the form   2 t S.rx u/ D  rx u C rx u  divx uI C divx uI; q D rx #: (43) 3

6.1

Finite Energy Weak Solutions to the Navier-Stokes-Fourier System

A trio of functions %, #, u is termed a finite energy weak solution to the Navier-Stokes-Fourier system (39-43), supplemented with the boundary conditions (38) if:

28 Concepts of Solutions in the Thermodynamics of Compressible Fluids

1369

• 1 % 2 L1 .0; T I L .//; # 2 L1 .0; T I Lq .// \ L2 .0; T I W 1;2 .// for certain  > 1; q > 1, %  0; # > 0 a.a. in .0; T /  ; u 2 L2 .0; T I W01;r .I R3 // for a certain r > 1I • 1

Z

Z

T

Z .%@t ' C %u  rx '/ dx dt D 



0

%0 '.0; / dx

(44)



for any ' 2 Cc1 .Œ0; T /  /; • 1 Z TZ .%u  @t ' C %u ˝ u W rx ' C p.%; #/divx '/ dx dt 

0

Z

T

Z

D 0



S.rx u/ W rx ' dx dt 

Z %0 u0  '.0; / dx

(45)



for any ' 2 Cc1 .Œ0; T /  I R3 /; • 1  Z TZ  q  rx ' dx dt %s.%; #/@t ' C %s.%; #/u  rx ' C #  0 Z

T

Z

C 0



1 #

  Z q  rx # S.rx u/ W rx u  ' dx dt   %0 s.%0 ; #0 /'.0; / dx #  (46)

for any ' 2 Cc1 .Œ0; T /  /, '  0. • 1   Z  Z  1 1 2 2 %juj C %e.%; #/ . ; / dx D %0 ju0 j C %0 e.%0 ; #0 / dx  2  2

(47)

for a.a. 2 .0; T /. The weak solutions satisfying (44-47) enjoy the important compatibility property, namely, any weak solution that is smooth satisfies the classical formulation of the Navier-Stokes-Fourier system (16-18) (see [25, Chapter 2]).

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6.2

E. Feireisl

Global-in-Time Existence of Finite Energy Weak Solutions

The weak formulation of the Navier-Stokes-Fourier system based on the integral identities (inequalities) (44)–(47) is mathematically tractable. Under certain technical but still physically grounded restrictions imposed on the constitutive relations, the problem admits global-in-time solution for any finite energy initial data.

6.2.1 Hypotheses: Constitutive Relations The thermodynamic functions p D p.%; #/, e D e.%; #/, and s D s.%; #/ are interrelated through Gibbs’ equation (5) and comply with the hypothesis of thermodynamics stability (8). In addition, it is required that the internal energy e D e.%; #/ and the pressure take the form e.%; #/ D em .%; #/ C

a 4 a # ; p.%; #/ D pm .%; #/ C # 4 ; a > 0; % 4

(48)

where em , pm represent molecular components augmented in (48) by radiation (see [25, Chapter 1]). Moreover, pm and em satisfy the monoatomic gas equation of state:

pm .%; #/ D

2 %em .%; #/: 3

(49)

It is easy to see that relation (49) is compatible with Gibbs’ equation (5) provided

pm .%; #/ D # 5=2 P

 %  3 # 3=2  %  # P I whence e : .%; #/ D m # 3=2 2 % # 3=2

(50)

In this setting, the hypothesis of thermodynamics stability (8) gives rise to

P .0/ D 0; P 0 .Z/ > 0; 0
0; Z

(51)

where, in addition, the specific heat at constant volume is required to be uniformly bounded. .Z/ is nonincreasing; whence it is Finally, by virtue of (51), the function Z 7! P Z possible to suppose that

lim

Z!1

P .Z/ D p1 > 0: Z 5=3

(52)

28 Concepts of Solutions in the Thermodynamics of Compressible Fluids

1371

6.2.2 Hypotheses: Transport Coefficients Transport coefficients  D .#/,  D .#/, and  D .#/ appearing in (43) are effective functions of the absolute temperature, specifically: .1 C # ˛ /  .#/  .1 C # ˛ /; j0 .#/j  c for all # > 0; 0  .#/  .1 C # ˛ / for all # > 0;

2 < ˛  1;  > 0; 5 (53) (54)

and .1 C # 3 /  .#/  .1 C # 3 / for all # > 0;  > 0:

(55)

6.2.3 Existence of Finite Energy Weak Solutions The following existence result was proved in [25, Chapter 3, Theorem 3.1]: Theorem 4. Let   R3 be a bounded domain of class C 2C . Suppose that the pressure p and the internal energy e are interrelated through (48)–(50), where P 2 C Œ0; 1/ \ C 3 .0; 1/ satisfies the structural hypotheses (51), (52). Let the transport coefficients , ,  be continuously differentiable functions of the temperature # satisfying (53)–(55). Finally, let the initial data %0 , #0 , u0 be given such that %0 ; #0 2 L1 ./; %0 > 0; #0 > 0 a.a. in ; u0 2 L2 .I R3 /:

(56)

Then the Navier-Stokes-Fourier system (39)–(43), supplemented with the boundary conditions (38), possesses a finite energy weak solution %; #; u in .0; T /   in the sense specified in (44)–(47). The weak solution belongs to the class: %  0 a.a. in .0; T /  ; (57) 1

1

% 2 C .Œ0; T I L .// \ L .0; T I L

5=3

ˇ

.// \ L ..0; T /  /

for a certain ˇ > 53 ; # > 0 a.a. in .0; T /  ; # 2 L1 .0; T I L4 .// \ L2 .0; T I W 1;2 .//; # 3 ; log.#/ 2 L2 .0; T I W 1;2 .//I u 2 L2 .0; T I W01;ƒ .I R3 //; ƒ D

8 ; %u 2 Cweak .0; T I L5=4 .I R3 //: 5˛

(58) (59)

(60)

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E. Feireisl

In the remaining part of this text, various properties of the finite energy weak solutions, the existence of which is guaranteed by Theorem 4, will be discussed. At this point, it is worth noting that an alternative approach based on the internal energy formulation (32)–(34) was proposed in [21]. Although mathematically less sophisticated and physically limited by more restrictive constitutive relations than in Theorem 4, the approach [21] proved to be convenient when studying stability and convergence properties of certain numerical methods [23]. The weak formulation of the Navier-Stokes-Fourier system based on the complete energy balance has also been studied in the framework of weak solutions. In [31], the authors established global existence for radially symmetric data in R3 . They also identified one of the main stumbling blocks in the analysis of the NavierStokes-Fourier system, namely, the (hypothetical) appearance of vacuum zones, where the density vanishes and the classical understanding of the equations breaks down. More recently, a new a priori bound on the density gradient was discovered in [6, 7] leading to global-in-time existence in the truly 3Dsetting conditioned, unfortunately, by a very specific relation imposed on the density-dependent viscosity coefficients and a rather unrealistic formula for the pressure that has to be infinite (negative) for % ! 0. The constraint represented by (44)–(47) may seem too weak to ensure, at least formally, the well posedness of the problem, meaning uniqueness and possibly stability of solutions with respect to the initial data. Note, however, that this issue remains largely open even for the seemingly simpler incompressible Navier-Stokes system despite a concerted effort of generations of excellent mathematicians (see [20]). In the text below, a less ambitious but still interesting question will be discussed, namely, the weak-strong uniqueness principle. This principle asserts that weak and strong solutions emanating from the same initial data coincide as long as the latter exists. To attack the problem, more thermodynamics is needed encoded in the so-called relative energy inequality; the resulting concept of dissipative solution is discussed in the next section.

7

Dissipative Solutions

Motivated by the work [13], a relative energy functional associated to the Navier-Stokes-Fourier system will be introduced. Here again, the second law of thermodynamics, enforced through Gibbs’ equation (7) and the hypothesis of thermodynamics stability (8), will play a crucial role.

7.1

Ballistic Free Energy

In accordance with the terminology of [18], the so-called ballistic free energy functionally takes the form H‚ .%; #/ D %e.%; #/  ‚%s.%; #/:

28 Concepts of Solutions in the Thermodynamics of Compressible Fluids

1373

The thermodynamic stability relation (8) gives rise to the following two properties of the function H‚ : % 7! H‚ .%; ‚/ is strictly convex,

(61)

Q # 7! H‚ .%; #/ attains its global minimum at # D #:

(62)

and

As observed by [2], the above properties are intimately related to stability of the equilibrium solutions to the Navier-Stokes-Fourier system. It will become clear that (61), (62) contain the necessary piece of information used later in the proof of weak-strong uniqueness.

7.2

Relative Energy

The relative energy is defined as ˇ   ˇ E %; #; uˇr; ‚; U  Z  1 @H‚ .r; ‚/ 2 D % ju  Uj C H‚ .%; #/  .%  r/  H‚ .r; ‚/ dx; @%  2 where %; #; u is a weak solution to the Navier-Stokes-Fourier system and r; ‚; U is an arbitrary trio of functions satisfying the relevant compatibility conditions. More precisely, imposing the no-slip conditions (38) requires the test functions to satisfy r > 0; ‚ > 0 and Uj@ D 0:

(63)

Given the coercivity properties free energy stated in (61), (62), ˇof the ballistic   ˇ it is easy to see that E %; #; uˇr; ‚; U plays a role of “distance” between ˇ   ˇ Œ%; #; u and Œr; ‚; U, meaning E %; #; uˇr; ‚; U  0 vanishing only if Œ%; #; u D Œr; ‚; U.

7.3

Relative Energy Inequality, Dissipative Solutions

The strength of the mathematical theory based on the weak solutions in the setting(44)–(47) ˇ consists  in the fact that it is possible to derive a functional relation ˇ for E %; #; uˇr; ‚; U reminiscent of the Gronwall inequality. Specifically, the following result holds:

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E. Feireisl

ˇ h  itD ˇ E %; #; uˇr; ‚; U tD0   Z Z ‚ q.#; rx #/  rx # S.#; rx u/ W rx u  dx dt C # 0  # Z Z  %.u  U/  rx U  .U  u/ dx dt Z

0

Z



Z



Z



C Z



0

C 0

Z

   % s.%; #/  s.r; ‚/ U  u  rx ‚ dx dt   % @t U C U  rx U  .U  u/ dx dt



.S.#; rx u/ W rx U  p.%; #/divx U/ dx dt

C 0 Z

 Z



Z       % s.%; #/  s.r; ‚/ @t ‚ C % s.%; #/  s.r; ‚/ U  rx ‚ dx dt 

0

Z

q.#; rx #/  rx ‚ dx dt #  0 Z Z   % % C 1 @t p.r; ‚/  u  rx p.r; ‚/ dx dt r r 0  

(64)

for any finite energy weak solution of the Navier-Stokes-Fourier system (38)–(43) and any trio of (smooth) test functions satisfying the compatibility conditions (63) (see [26, Section 3]). Motivated by [35], where a similar definition is proposed for the incompressible Euler system, a trio of functions %; #; u is called a dissipative solution to the Navier-Stokes-Fourier system (38)–(43) if • %; #; u belong to the regularity class specified in Theorem 4; • %; #; u satisfy the relative energy inequality (64) for any trio r; ‚; U of sufficiently smooth (for all integrals in (64) to be well defined) test functions satisfying the compatibility conditions (63). As observed in [26, Section 3], any finite energy weak solution of the NavierStokes-Fourier system is a dissipative solution. The reverse implication is an interesting open problem. The concept as well as a relevant existence theory in the framework of dissipative solutions can be extended to a vast class of physical spaces, including unbounded domains in R3 (see [34]).

7.4

Weak-Strong Uniqueness

The important feature of the dissipative solutions is that they comply with the weakstrong uniqueness principle (see [22, Theorem 6.2] and [26, Theorem 2.1]).

28 Concepts of Solutions in the Thermodynamics of Compressible Fluids

1375

Theorem 5. In addition to the hypotheses of Theorem 4, suppose that

s.%; #/ D S

 %  4a # 3 ; with S .Z/ ! 0 as Z ! 1: C # 3=2 3 %

(65)

Let %; #; u be a dissipative (weak) solution to the Navier-Stokes-Fourier system in the set .0; T /  . Suppose that the Navier-Stokes-Fourier system admits a strong Q uQ in the time interval .0; T /, emanating from the same initial data and solution %; Q #; belonging to the class: Q @t u; Q m Q 2 L1 ..0; T /  /; m D 0; 1; 2: Q @m @t %; Q @t #; Q @m x %; x #; @x u Then Q u uQ in .0; T /  : % %; Q # #; Remark 4. The extra hypothesis (65) reflects the third law of thermodynamics and can be possibly relaxed (cf. [3, 4]). As stated in Theorem 2, the Navier-Stokes-Fourier system admits a local-in-time regular solution as soon as the initial data are regular. In view of Theorem 5, any weak solution coincides with this strong solution as long as the latter exists. On the other hand, by virtue of Theorem 4, the weak solutions exist globally in time and as such provide a possible alternative of extending the local smooth solution beyond its existence interval. Whether or not strong solutions exist globally in time is an interesting open question, for small data results in this direction see [36, 37].

7.4.1 Back to the Euler-Fourier System At this moment, it seems interesting and useful to go back to Theorem 3, where an example of a system (Euler-Fourier) possessing infinitely many weak solutions was produced. One can introduce the relative energy and define the dissipative solutions for the Euler-Fourier system (35)–(37), exactly as for the Navier-Stokes-Fourier system. It can be shown, see [10, Theorem 4.1], that the dissipative solutions of the Euler-Fourier system enjoy the property of weak-strong uniqueness similarly to the solutions of the Navier-Stokes-Fourier system. Still these restrictions do not lead to a well-posed problem as the following result shows (see [10, Theorem 4.2]): Theorem 6. Under the hypotheses of Theorem 3, let T > 0 be given, together with the initial data: %0 ; #0 2 C 3 .T 3 /; %0 > 0; #0 > 0 in T 3 :

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Then there exists an initial velocity u0 2 L1 .T 3 ; R3 / such that the corresponding initial-value problem for the Euler-Fourier system (35-37) admits infinitely many dissipative weak solutions in .0; T /   belonging to the class: % 2 C 2 .Œ0; T   /; @t # 2 Lp .0; T I Lp .//; rx2 # 2 Lp .0; T I Lp .I R33 // for any 1  p < 1, u 2 Cweak .Œ0; T I L2 .I R3 // \ L1 ..0; T /  I R3 /; divx u 2 C 2 .Œ0; T   /: It is worth noting that the conclusion of Theorem 6 does not contradict the principle of weak-strong uniqueness as u0 is not smooth. The problem of “maximal” smoothness of such data is closely related to the so-called Onsager’s conjecture that has been intensively studied in the context of the incompressible Euler system, see [16, 17], and, more recently, [8]. It is worth noting that application of the relative energy inequality may provide positive well-posed results even in the context of the Euler system (see [24]).

8

Conclusion

A conditional regularity criterion is a condition which, if satisfied by a weak solution to a given system, implies that the latter is regular. Similarly, such a condition may be applied to guarantee that a local (strong) solution can be extended to a given time interval. The most celebrated conditional regularity criteria are due to [1,39,40], and [12] in the context of the incompressible Navier-Stokes and Euler systems. Recently, similar conditions were obtained also for compressible barotropic fluids and the full Navier-Stokes-Fourier system, the reader may consult [19,33], [42,45], and also the references cited therein. In view of the results of [30, 32], certain discontinuities imposed through the initial data in the compressible Navier-Stokes system propagate in time. In other words, unlike its incompressible counterpart, the hyperbolic-parabolic compressible Navier-Stokes system does not enjoy the smoothing property typical for purely parabolic equations. Analogously, a solution of the full Navier-Stokes-Fourier system can be regular only if regularity is enforced by a proper choice of the initial data.

8.1

Conditional Regularity via the Relative Energy

The chapter is concluded by a short discussion of conditional regularity of weak solutions to problems in fluid dynamics. These are additional restriction that, if satisfied by a weak solution, imply its regularity (smoothness). A possible approach to conditional regularity of weak solutions is to show that:

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• the problem admits local-in-time strong solution; • the problem enjoys the weak-strong uniqueness property; • show conditional regularity for the strong solution. This procedure applied in the context of the finite energy weak solutions to the Navier-Stokes-Fourier system gives rise to the following result (see [27, Theorem 2.1]). Theorem 7. Under the hypotheses of Theorem 5, let %; #; u be a finite energy weak solution of the Navier-Stokes-Fourier system on the time interval .0; T / belonging to the regularity class specified in Theorem 4, emanating from (regular) initial data satisfying the hypotheses of Theorem 2. Suppose, in addition, that ess sup krx u.t; /kL1 .IR33 / < 1: t2.0;T /

Then %; #; u is a classical solution of the Navier-Stokes-Fourier system in .0; T /  .

9

Cross-References

 Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak

Solutions for the Compressible Navier-Stokes Equations  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stabil-

ity, and Longtime Behavior  Weak Solutions for the Compressible Navier-Stokes Equations with Density

Dependent Viscosities  Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical

Cases Acknowledgements The research of E.F. leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.

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29

Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stability, and Longtime Behavior Antonín Novotný and Hana Petzeltová

Contents 1

2

3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Relative Energy and Robustness of the Class of Weak Solutions . . . . . . . . . . . . Thermodynamics of Viscous Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Navier-Stokes-Fourier System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Domain, Conservative Boundary Conditions and Initial Data . . . . . . . . . . . . . . . 2.3 Thermodynamic Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Constraints Imposed by Thermodynamic Stability Conditions . . . . . . . . . . . . . . 2.6 Third Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Barotropic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Mathematical Tools for Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Instantaneous Values of Functions in L1 .0,T I L1 .// . . . . . . . . . . . . . . . . . . . 3.2 Instantaneous Values of Solutions of Conservation Laws . . . . . . . . . . . . . . . . . . 3.3 Weakly Convergent Sequences in L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Convexity, Monotonicity, and Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Inverse of the Div Operator (Bogovskii’s Formula) . . . . . . . . . . . . . . . . . . . 3.6 Poincaré- and Korn-Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Time Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Operator r1 and Riesz-Type Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Some Results of Compensated Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Parametrized (Young) Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Some Elements of the DiPerna-Lions Transport Theory . . . . . . . . . . . . . . . . . . . 3.12 The Gronwall Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1383 1383 1389 1391 1393 1394 1395 1396 1400 1401 1401 1402 1403 1404 1410 1410 1412 1413 1415 1416 1417 1418 1419 1420

A. Novotný () Université de Toulon, IMATH, Toulon, France e-mail: [email protected] H. Petzeltová Department EDE, Mathematical Institute of the Academy of Sciences of the Czech Republic, Praha 1, Czech Republic e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_76

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4

Existence of Weak Solutions to the Compressible Navier-Stokes Equations for Barotropic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Weak Formulation and Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Dissipative Solutions, Relative Energy Inequality, and Weak-Strong Uniqueness Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Relative Energy and Relative Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dissipative Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Relative Energy Inequality with a Strong Solution as a Test Function . . . . . . . . 5.4 Stability and Weak-Strong Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Longtime Behavior of Barotropic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Uniqueness of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Convergence to Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Bounded Absorbing Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Existence of Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Navier-Stokes-Fourier System in the Internal Energy Formulation . . . . . . . . . . . . . . . . 7.1 Definition of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Existence of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Main Ideas of the Proof of Theorem 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Equations Verified by the Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Weak Limits in the Momentum and Renormalized Continuity Equations . . . . . 8.4 Effective Viscous Flux Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Oscillations Defect Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Renormalized Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Strong Convergence of the Density Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Limit in the Thermal Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Navier-Stokes-Fourier System in the Entropy Formulation . . . . . . . . . . . . . . . . . . . . . . 9.1 Definition of Finite Energy Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Relative Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Bounded Energy Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Dissipative Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Constitutive Relations and Transport Coefficients for the Existence Theory . . . 9.6 Existence of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Construction of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Weak Compactness of the Set of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Estimates and Weak Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Strong Convergence of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Strong Convergence of Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Stability Results and Weak-Strong Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Sketch of the Proof of Theorems 43 and 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Longtime Behavior of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Longtime Behavior of Conservative System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Longtime Behavior for Time-Dependent Forcing: Blow Up of Energy . . . . . . . 12.4 Longtime Behavior: Stabilization to Equilibria for Rapidly Oscillating Driving Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1421 1421 1431 1431 1433 1434 1435 1441 1442 1447 1450 1453 1456 1456 1460 1462 1464 1465 1470 1472 1476 1478 1479 1482 1486 1488 1492 1493 1495 1499 1500 1503 1505 1506 1510 1513 1516 1518 1523 1524 1527 1531 1534 1540 1541 1541

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

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Abstract

This double-sized chapter contains two related themes that were supposed to be covered by two independent chapters of the handbook in the original project: (1) weak solutions of the Navier-Stokes equations in the barotropic regime and (2) weak solutions of the Navier-Stokes-Fourier system. We shall discuss for both systems: (1) Various notions of weak solutions, their relevance, and their mutual relations. (2) Global existence of weak solutions. (3) Notions of relative energy functional, dissipative solutions and relative energy inequality and its impact on the investigation of the stability analysis of compressible flows. (4) Weak strong uniqueness principle. (5) Longtime behavior of weak solutions. For physical reasons, we shall limit ourselves to the three-dimensional physical space.

1

Introduction

1.1

Weak Solutions

The Navier-Stokes-Fourier system is a system of partial differential equations describing evolution of density % D %.t; x/, absolute temperature # D #.t; x/, and velocity u D u.t; x/ of a viscous compressible and heat-conducting fluid filling domain  (x 2 ) within the time interval t 2 Œ0; T /. There are several ways to define weak solutions for the complete Navier-Stokes-Fourier system. Here, we shall mention three of them: the convenience of each definition depends on the mathematical assumptions that one imposes on the constitutive laws for pressure (internal energy) on one hand and on the transport coefficients on the other hand. Indeed, the weak formulation of the momentum and continuity equations is standard, while for the weak formulation of the energy conservation, one has at least three reasonable options that are not equivalent within the class of irregular solutions: (1) formulation in terms of the internal energy, (2) formulation in terms of the specific entropy, and (3) formulation in terms of the total energy. The first and second one are continuations of the theories based on the so-called effective viscous flux identity started by P.L. Lions [77], and the third one, due to Bresch and Desjardins [7], can be considered as a continuation of theories based on new a priori estimates in the line started by Kazhikov [72].

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The first approach due to Feireisl [30] based on a weak formulation of the continuity, momentum, and internal energy equations is convenient for the pressure p and internal energy e of type p.%; #/ D pc .%/ C #pth .%/; pc .%/ 1 % ;  >3=2 Z

%

e.%; #/ D ec .%/ C eth .#/; ec .%/ D 1

(1)

p.z/ dz; eth .#/  # !C1 ; !  0; z2

where pth must be monotone and dominated by a certain power of pc .%/ for large 1=3 %’s (more precisely by pc .%/). Here, # denotes the temperature, % denotes the density, and  is the adiabatic coefficient of the fluid. The heat conductivity in this approach has to be temperature dependent (with a convenient power growth), and the viscosity coefficients have to be constant. The second approach was developed in [33] preceded by a compactness result in [32]. It exploits the observation of Ducomet and Feireisl [20,21] on the regularizing effect of the radiative pressure on the weak solutions of the magnetohydrodynamic equations. It involves, besides the standard weak formulation of continuity and momentum equations, the weak formulation of the conservation of energy in terms of the specific entropy that includes explicitly the second law of thermodynamics via the entropy production rate being a nonnegative measure. This approach is applicable for the pressure and internal energy laws p.%; #/, e.%; #/, exhibiting the coercivity of types % and # 4 for large densities and temperatures; a prototype example is p.%; #/  % C #pth .%/ C # 4 ; e.%; #/  % C # !C1 C

(2)

#4 ; !  0; %

where pth is the same as in (1). The viscosity coefficients in this theory are in general temperature dependent and have to behave like .1 C #/ˇ ; the heat conductivity has to behave like .1 C #/˛ , where loosely speaking ˛ > 0 has to be larger when ˇ  0 becomes smaller. For example, for the pressure law of monoatomic gas with radiation that behaves like %5=3 (for large %’s and # fixed) and like # 4 (for large #’s and % fixed) – see Sect. 2.4, in particular (47)–(49) – the theory gives ˇ 2 Œ2=5; 1 within physically reasonable value ˛ D 3 (see [33]), while for the pressure law of type (2) obeying the above asymptotic condition for pth , one can achieve values  ˇ (see [32]). If  > 3 one can achieve values ˇ 2 Œ0; 4=3 provided ˛ D 16 3 C jˇj. The latter situation corresponds rather to 4  ˇ  0 provided ˛  16 3 compressible fluids than to gasses (see [58]). Both above formulations are sufficiently weak to allow existence of variational solutions for large data and reasonable in the sense that any sufficiently regular weak solution is a classical solution.

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The second formulation including balance of entropy as the pointwise conservation of energy in the weak formulation has an important advantage in comparison with the first formulation based on the pointwise energy conservation in terms of the internal energy balance. Indeed, in the second approach, the thermodynamic stability conditions can be reformulated in terms of an evolution variational inequality (called relative energy inequality) governing a specific functional called relative energy functional, which is able to measure a “distance” between a weak solution .%; #; u/ and any other (sufficiently regular) state of the fluid .r; ‚; U/. This inequality is automatically satisfied by any weak solution based on the balance of entropy (see [34] and [49, 50] for the barotropic case). It appears that the relative energy inequality encodes most of the stability properties of compressible fluids and is, in fact, responsible for the robustness of this type of weak solutions with respect to perturbations of initial conditions and external forces as well as with respect to singular limits involving various physically reasonable small parameters appearing in the nondimensional formulation of the Navier-Stokes-Fourier system. The third approach due to Bresch and Desjardins [7, 8] (see also Mellet, Vasseur [83]) is convenient in the case when the shear viscosity  and the bulk viscosity  depend on the density and satisfy the differential identity 

0

2   3

.%/ D 2%0 .%/  2.%/;

and pressure is in the form (1), where however pc .%/ is singular at % ! 0. The main ingredient in the proof in this situation is the fact that the particular relation between viscosities stated above makes possible to establish a new mathematical entropy identity, which provides estimates for the gradient of density. This estimate implies compactness of the sequence of approximating densities. In spite of the compactness, the construction of the solutions in this situation is a tough problem. It was so far possible under the additional nonphysical assumption that pc explodes at the vacua. Only recently, two preprints [75, 105] appeared suggesting an explicit construction of the global solutions in the “simple” barotropic case in the physically reasonable situation when the cold pressure pc is not singular at zero. In this chapter we shall concentrate to the first two formulations; the third formulation is investigated in a separate chapter of the handbook. In the mathematical literature, there is another notion of weak solutions to the compressible Navier-Stokes equations due to D. Hoff [64–68] and references quoted there. Hoff’s solutions must have essentially bounded density, but discontinuities are allowed. Solutions in Hoff’s class are almost unique (see [68]). A drawback is that their existence is guaranteed only for small initial data. They will be treated in a separate chapter of the handbook.

1.1.1 Lions’ Approach and Feireisl’s Approach The concept of weak solutions in fluid dynamics was introduced in 1934 by Leray [74] in the context of incompressible Newtonian fluids. It has been extended more

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than 60 years later to the Newtonian compressible fluids in barotropic regime (meaning that p D p.%/  % ) by Lions [77]. The Lions theory relies on two crucial observations: (1) A discovery of a certain weak continuity property of the quantity  p.%/ C

 4  C  divu 3

called effective viscous flux. This part is essential for the existence proof; it employs certain cancelation properties that are available due to the structure of the equations that are mathematically expressed through a commutator involving density, momentum, and the Riesz operator. The main ideas related to the effective viscous flux identity will be explained in Sect. 8 (namely, in Sect. 8.4). (2) Theory of renormalized solutions to the transport equation that P.L. Lions introduced together with DiPerna in [18]. In the context of compressible Navier-Stokes equations, the DiPerna-Lions transport theory applies to the continuity equation. The theory asserts among others that the limiting density is a renormalized solution to the continuity equation provided it is square integrable. This hypothesis is satisfied only provided   9=5. The condition on the squared integrability of the density is the principal obstacle to the improvement of the Lions result. Notice that some indications on the particular importance of the effective viscous flux were known at about the same time to several authors and used in different problems dealing with small data (see Hoff [64] and Padula [86]) and that the suggestion to use the continuity equations to evaluate the oscillations in the sequence of approximating densities has been formulated and performed in the onedimensional case by D. Serre [97]. All physically reasonable adiabatic coefficients  for gases belong to the interval .1; 5=3/, the value  D 5=3 being reserved for the monoatomic gas. This is the reason why it is interesting and important to relax the condition on the adiabatic coefficient in the Lions theory. This has been done by Feireisl et al. in [47]. The new additional aspects of this extension are based on the previous observations by Feireisl in [27] and are the following: (1) As suggested in [27], the authors have used the oscillations defect measure to evaluate the oscillations in the sequence of approximating densities and proved that it is bounded provided  > 3=2. This part of the proof will be discussed in detail in Sect. 8 (namely, Sect. 8.5). (2) The boundedness of the oscillations defect measure is a criterion that replaces the condition of the squared integrability of the density in the DiPerna-Lions transport theory. Consequently if any term of the sequence of approximating densities satisfies the renormalized continuity equation, and if the oscillations

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defect measure of this sequence is bounded, then the weak limit of the sequence is again a renormalized solution of the continuity equation. This property is discussed in detail in Sect. 8.6. Recently the authors of so far unpublished paper [9] suggested an alternative way to the Lions’ approach of measuring of oscillations in the density sequence, which promises to be slightly more robust than the Lions’ approach.

1.1.2 Weak Solutions for the Complete Navier-Stokes-Fourier System The existence theory for the complete Navier-Stokes-Fourier system (with possibly temperature-dependent viscosities) employs both Lions’ and Feireisl’s techniques. Most of its additional difficulties dwell in the possible existence of vacuum regions in relation to the temperature approximations. In what follows, we describe general ideas on how these problems can be solved: First approach. (1) The procedure to prove strong convergence of the approximated density sequence %n via the Lions-Feireisl approach involves solely continuity and momentum equations. The weak limit of the sequence #n pth .%n / can be identified with expression #pth .%/, where # is a weak limit of the approximated temperature sequence #n and pth .%/ is the weak limit of the sequence pth .%n /. After this observation, the Lions-Feireisl method including effective viscous flux identity bound for the oscillations defect measure and renormalized continuity equation can be performed exactly as in the barotropic case, leading to the almost everywhere convergence of %n to a function %  0. After this observation, the problem is reduced to the limit passage in the internal energy balance. The details are described in Sections 8.2, 8.3, 8.4, 8.5, and 8.6. (2) In this case the internal energy balance provides an estimate of @t .%# !C1 / (and not of @t # !C1 ). Loosely speaking, this estimate eliminates possible oscillations outside vacua in the set f.t; x/ 2 QT j%.t; x/ > 0g (QT D .0; T /  ), but, unfortunately, does not discard oscillations on the vacuum set f.t; x/ 2 QT j%.t; x/ D 0g which can be of nonzero measure. Consequently we can reasonably hope to obtain almost everywhere convergence of the approximated temperature sequence #n to #Q on the set f.t; x/j%.t; x/ > 0g  QT . This observation in combination with the almost everywhere convergence of density established in item (1) allows to pass to the limit in all terms of the weak formulation of the internal energy balance containing multiples of %. (3) The term corresponding to the heat flux divq.#; rx #/ can be written in the form K.#/ with convenient strictly monotone function K provided the heat flux q is given by the Fourier law with the coefficient of heat conductivity dependent only on temperature. The available estimates provide a weak limit K.#/ of the sequence K.#n / in L1 .QT /. One can

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now define a new temperature # D K1 .K.#// that is equal to the almost everywhere limit #Q of the approximated temperature sequence on the set f.t; x/j%.t; x/ > 0g established in item (2). (In the real proof, the sequence K.#n / is bounded only in L1 .QT / which does not prevent concentrations. One can however perform the proof by using convenient truncations of K using a procedure reminiscent to Chacon’s biting limit see [11]). (4) Fortunately, the above term is the only term in the internal energy balance (except the term involving S.rx u/ W rx u, whose limit passage can be treated by the lower weak semi-continuity provided the stress tensor obeys, e.g., the Navier-Stokes law for the Newtonian fluids) which is not a multiple of %. Therefore, we can replace in all remaining terms the temperature #Q by the new temperature #. A detailed development of ideas described in items (2), (3), and (4) is available in Sect. 8.8. Second approach. (1) In order to reduce the investigation to a situation similar to the barotropic case, one has to prove first the almost everywhere convergence of the approximated temperature sequence. In contrast with the previous case, this seems to be possible, thanks to the presence of radiation energy. Indeed the energy conservation allows to estimate @t .# 4 / and thus to discard the possible time oscillations in the approximated temperature sequence. Since in this setting we are dealing with entropy balance rather than with the energy balance, this point involves the treatment of the entropy production rate as a Radon measure and a convenient use of the compensated compactness, namely, of the Div-Curl lemma in combination with the theory of parametrized Young measures. The crucial condition allowing to conclude is the monotonicity of the entropy with respect to temperature. (2) Even after the strong convergence of temperature is known, the weak continuity of the effective viscous flux is not an obvious issue. It requires to use another cancelation property that is mathematically expressed through another commutator including shear viscosity, symmetric velocity gradient, and the Riesz operator. The ideas described in items (1) and (2) are treated in Sects. 10.1 and 10.2. (3) Once the weak continuity property of the effective viscous flux is known, the proof follows the lines of Lions’ and Feireisl’s approaches: (a) one proves first the boundedness of the oscillations defect measure for the sequence of densities; (b) the boundedness of oscillations defect measure implies that the limiting density is a renormalized solution to the continuity equation; and (c) the renormalized continuity equation is used to show that the oscillations in the density sequence do not increase in time. This means the strong convergence of density. The details to this part of the proof are available in Sect. 10.3.

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Weak solutions for the compressible barotropic equations are introduced in Sect. 4 along with the main existence results and their qualitative properties, while those for the complete Navier-Stokes-Fourier system are introduced in Sects. 7 and 9. We provide the detailed description of the main ideas of the existence proofs of weak solutions in Sects. 8 and 10. Weak solutions in the theory of compressible Navier-Stokes equations are usually constructed via several levels of approximations including small parameters via suitable approximating system of PDEs. Construction of weak solutions through numerical schemes is a very recent topic which goes out of scope of this chapter. The reader can consult [55, 71], or monograph [56] for the recent development in this subject.

1.2

Relative Energy and Robustness of the Class of Weak Solutions

Weak solutions are not known to be uniquely determined (cf., e.g., exposition of Fefferman [26] dealing with three-dimensional incompressible Navier-Stokes equations) and may exhibit rather pathological properties (see, e.g., Hoff and Serre [69]). So far, the best property that one may expect in the direction of a unique result is the weak-strong uniqueness, meaning that any weak solution coincides with the strong solution emanating from the same initial data as long as the latter exists. The weak-strong uniqueness principle is known for the incompressible NavierStokes equations since the works of Prodi [95] and Serrin [98] (see [25] for the later development). About 50 years later, the weak-strong uniqueness problem has been revisited by Desjardins [17] and Germain [61] for the compressible NavierStokes equations. They obtained some partial and conditional results. Finally, the unconditional weak-strong uniqueness principle has been proved in [50] (see also related paper [49]). Only very recently the weak-strong uniqueness property has been proved in [34] for weak solutions of the complete Navier-Stokes-Fourier system in the entropy formulation introduced in [33]. In all cases cited above, the weak-strong uniqueness principle has been achieved by the method of relative energy that is reminiscent to the relative entropy method. Relative entropy method was brought to the mathematical fluid mechanics by C. Dafermeos [16] and has been used later in various contexts by different authors (see [77], Saint-Raymond [96], Grenier [63], Masmoudi [80], Ukai [103], Wang and Jiang [107], among others). The notion of dissipative solutions introduced in Lions [76] for the incompressible Euler equations is very much related to the concept of relative entropies. Regardless the fact that [16] is about conservation laws (disregarding the dissipation) while [33] includes dissipative effects, the main difference between the relative energy and relative entropy methods is the following: the starting point of [16] (in the case of complete Euler system) is the balance of internal energy, and the output is the relative entropy inequality, while the starting point in [33] is the balance

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of entropy and the output is the relative energy inequality. The procedure suggested in [16] cannot be repeated in the context of weak solutions to the Navier-StokesFourier system unless one supposes additionally that the density and temperature are bounded from below by positive constants. It is however not known whether the latter condition is satisfied globally in time for any weak solution. The relative energy method is introduced in Sects. 5 and 11. We have already mentioned that the relative energy inequality encodes most of the robustness properties of the weak solutions to the compressible NavierStokes-equations and to the Navier-Stokes-Fourier system. Let us mention a few applications: (1) If one takes for the test state .r; ‚; U/ a strong solution in the relative energy inequality, one obtains a stability estimate of a strong solution (emanating from initial data .r0 ; ‚0 ; U0 / and external force g) within the class of weak solutions (emanating from initial data .%0 ; #0 ; u0 / and external force f), in terms of difference of the external forces and relative energy of the initial data. This statement yields, in particular, the weak-strong uniqueness principle saying that the weak solution coincides with the strong solution as long as the strong solution exists, provided both solutions emanate from the same initial data and external forces (see again [34,49,50] for the barotropic case). These applications will be investigated in Sects. 5 and 11. (2) The large time behavior of weak solutions, namely, convergence to the equilibrium states in the case of conservative forces, energy blow up in the case of nonconservative forces, and questions related to the bounded absorbing sets and attractors can be treated on the basis of the relative energy inequality (see [44] and references quoted there). These applications are investigated in Sects. 6 and 12. (3) There is a bunch of applications of the relative energy inequality related to the investigation of singular limits in the nondimensional version of the compressible Navier-Stokes equations and the Navier-Stokes-Fourier system involving various combinations of low Mach, Froude, Rossby, Péclet numbers, and large Reynolds number toward reduced target systems as long as we know that the target system admits a regular solution (at least locally in time). Practically all so far rigorously obtained singular limits within the complete Navier-StokesFourier system have been obtained by the relative energy method. Another family of problems, where the relative energy inequality appeared to be a crucial tool, are limits connected to dimension reduction. We refer to [3, 5, 35– 38, 52, 79, 100] for a few examples to some of these applications. The problem of the singular limits in the compressible Navier-Stokes equations will be discussed in another two independent chapters of the handbook. (4) The numerical version of the relative energy inequality is employed in [60] to investigate the error estimates of numerical schemes solving the compressible Navier-Stokes equations. The reader can consult also, e.g., [54, 57] among others, for the recent developments of this subject. These applications go far beyond the scope of this handbook.

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The chapter is organized as follows. We start with a short introduction to the thermodynamics of viscous fluids (Sect. 2) followed by a review section collecting the most important specific mathematical tools for the treatment of compressible Navier-Stokes equations (Sect. 3). Sections 4, 4, and 6 are devoted to the compressible Navier-Stokes equations in barotropic regime (treating the notions of weak solutions, finite and bounded energy weak solutions, renormalized weak solutions, dissipative solutions, relative energy inequality, weak-strong uniqueness, and longtime behavior). The same issue is then revisited for the full Navier-StokesFourier system through Sects. 7, 8, 9, 10, 11, and 12.

2

Thermodynamics of Viscous Compressible Fluids

We shall describe the motion of a compressible, viscous and heat-conducting fluid sometimes called also a viscous gas. For simplicity, we suppose that the fluid fills a fixed domain   R3 , and we shall investigate its evolution through an (arbitrary) large time interval .0; T /. We denote by QT D .0; T /   the space-time cylinder. The motion will be described by means of three basic state variables: the mass density % D %.t; x/, the velocity field u D u.t; x/, and the absolute temperature # D #.t; x/, where t 2 .0; T / is the time variable and x 2   R3 is the space variable in the Eulerian coordinate system. The physical nature of density and temperature requires that the density is nonnegative function on QT , and the absolute temperature is positive function on QT . We shall investigate the time evolution of these quantities. It is described by the balance laws of physics expressed through the following partial differential equations: (i) Conservation of mass @t % C divx .%u/ D 0:

(3)

(ii) Conservation of linear momentum @t .%u/ C divx .%u ˝ u/ C rx p.%; #/ D divx S.%; #; rx u/ C %f:

(4)

(iii) Conservation of internal energy – first law of thermodynamics @t .%e.%; #// C divx .%e.%; #/u/ C divx q.%; #; rx #/ C p.%; #/divx u D S.%; #; rx u/ W rx u:

(5)

In these equations p D p.%; #/ is the pressure, e D e.%; #/ is the (specific) internal energy, S D S.%; #; rx u/ is the viscous stress tensor, and q.%; #; rx #/ is the heat flux. They are given functions characterizing the gas. The quantity f D f.t; x/ is a

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given function expressing the specific external forces. For the sake of simplicity, we do not consider the external heat sources. In physics, there are at least two another ways of writing the conservation of energy (5): in terms of the specific total energy and in terms of the specific entropy. Formulation of the first law in terms of the kinetic energy. The specific total energy is the sum of specific kinetic energy ekin D 12 u2 and the specific internal energy e.%; #/ etot .%; u/ D

1 2 u C e.%; #/: 2

(6)

Due to (3)–(5), it must obey equation @t .%etot .%; #// C divx

   %etot .%; #/ C p.%; #/ u C divx q.%; #; rx #/

(7)

  D divx S.%; #; rx u/  u C %f  u: Formulation of the first law in terms of the specific entropy. The second law of thermodynamics postulates existence of the specific entropy s D s.%; #/ defined by the Gibbs relation #ds.%; #/ D de.%; #/ 

p.%; #/ d% %2

(8)

that must obey the balance of entropy equation  @t .%s.%; #// C divx .%s.%; #/u/ C divx

q.%; #; rx #/ #

 D ;

(9)

where the quantity  must be nonnegative. It is called the entropy production rate. In the present situation, D

1 #



S.%; #; rx u/ W rx u 

q.%; #; rx #/  rx # #

 :

(10)

If p, e, S, q are differentiable functions of their respective arguments, if density % and temperature # are positive and sufficiently smooth on QT , and if the velocity field u is sufficiently smooth on QT , then equations (5), (7), and (9)–(10) are equivalent. This equivalence does not need to be necessarily true if the functions above do not possess enough regularity. Therefore, in spite of the fact that weak formulation of the balance of energy based on each of equations (5), (7), and (9), respectively, is equally physically justifiable, it may lead to weak solutions with different properties. It may happen that some of the possible definitions of weak solutions may be more advantageous

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

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in some situations and may even lead to global in time existence results, while other definition will fail to have this property, depending on the flow regimes and constitutive laws characterizing the gas. If % > 0 on QT and %, u belong to C 1 .QT /, then the continuity equation is equivalent to the family of so-called renormalized continuity equations:     @t b.%/ C divx b.%/u C %b 0 .%/  b.%/ divx u D 0 for all b 2 C 1 .0; 1/:

(11)

Again, if the couple .%; u/ does not possess enough regularity, this property does not need to be true, in general.

2.1

Navier-Stokes-Fourier System

We suppose that the viscous stress S is described by Newton’s law 2 S.%;#; rx u/ D .%; #/T.rx u/C.%; #/divx uI; T.rx u/ D rx uC.rx u/T divx uI; 3 (12) where I is the identity tensor, while q is the heat flux satisfying Fourier’s law q D  .%; #/rx #:

(13)

The quantities , , and are called transport coefficients, more specifically, shear and bulk viscosities, and heat conductivity, respectively. According to the second thermodynamical law, they have to be all nonnegative. We are however dealing with viscous and heat conducting fluids; we shall therefore always suppose that the transport coefficients satisfy at least .%; #/ > 0; .%; #/  0; .%; #/ > 0;

(14)

and we shall assume the following minimal regularity, .; ; / 2 C 1 .Œ0; 1/2 /:

(15)

The system of equations (3)–(5) (where (5) may be replaced by (7) or by (9)–(10)) with the constitutive relations (12) and (13) is called Navier-Stokes-Fourier system. Physical considerations suggest that the heat conductivity behaves .#/  # ˛ ; ˛  3 for large values of #

(16)

due to the radiation effects. The approximation of viscosity coefficients by constants  > 0;   0

(17)

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is considered in many situations as satisfactory. The kinetic theory predicts .#/ 

p #; for large values of #

(18)

(see [108]).

2.2

Domain, Conservative Boundary Conditions and Initial Data

2.2.1 Initial Data Equations (3)–(5) are supplemented with initial conditions %.0; / D %0 ; %u.0; / D %0 u0 ; %e.%; #/.0; / D %0 e.%0 ; #0 /; %0  0; #0 > 0; (19) where %0 , #0 , and u0 are given functions.

2.2.2 Boundary Conditions We shall always assume that  has globally uniformly Lipschitz boundary. If  is bounded, we will deal with no-slip boundary conditions for velocity uj@ D 0;

(20)

and zero heat transfer conditions through the boundary q  nj@ D 0;

(21)

where n denotes the external normal to the boundary @ of . The no-slip boundary conditions can be replaced in many cases by the complete slip boundary conditions u  nj@ D 0;

Sn  nj@ D 0;

(22)

or with Navier’s slip boundary conditions  SnnCƒ.un/ j@ D 0; where ƒ  0 is the friction coefficient: (23) If  is an unbounded domain, one has to prescribe in addition to boundary conditions (20), resp. (22), resp. (23), and (21) also the behavior at infinity,

unj@ D 0;



%.t; x/ ! %1  0; in some sense, as jxj ! 1.

u.t; x/ ! u1 2 R3 ;

#.t; x/ ! #1 > 0

(24)

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2.2.3 Global Conservation Properties Suppose now that the domain  is bounded (and sufficiently smooth). Integrating equation for the conservation of global energy under conditions (23), we get d dt

Z  

1 %juj2 C %e.%; #/ 2



Z

juj2 dSx D

dx C ƒ

Z %f  u dx;

@

(25)



provided the trio .%; #; u/ is sufficiently smooth in QT ; in particular, in the case of boundary conditions (20) and (22), the total energy of the system in the volume  is conserved, namely, d dt

Z  

1 %juj2 C %e.%; #/ 2



Z dx D

%f  u dx:

(26)



Under the same smoothness requirement, multiplying equation (9) by a positive constant ‚, integrating over , and subtracting the result from equations (25) and (26), we get the dissipation identity d dt

Z  

1 %juj2 C H‚ .%; #/ 2



Z

Z

dx C ‚

juj2 dSx D

dx C ƒ 

Z

@

%f  u dx; 

(27) respectively, d dt

Z  

1 %juj2 C H‚ .%; #/ 2



Z dx C ‚

Z dx D



%f  u dx;

(28)



where the quantity   H‚ .%; #/ D % e.%; #/  ‚e.%; #/

(29)

is called Helmholtz function or ballistic free energy. It plays an essential role in the stability analysis of weak solutions.

2.3

Thermodynamic Stability Conditions

The fluid characterized by the pressure p.%; #/ and internal energy e.%; #/ verifies the thermodynamic stability conditions if @p.%; #/ @e.%; #/ > 0; > 0 for all %; # > 0: @% @# We easily verify by using Gibbs’ relation (8) that

(30)

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@H# @2 H# #  # @e 1 @p .%; #/ D % .%; #/ and .%; #/: .%; #/ D 2 @# # @# @% % @%

(31)

Thus, the thermodynamic stability in terms of the function H# can be reformulated as follows: % 7! H# .%; #/ is strictly convex,

(32)

# 7! H# .%; #/ attains its global minimum at # D #:

(33)

while

We notice that if the second thermodynamical condition is satisfied, then the map # 7! s.%; #/ is for any % a (strictly) increasing function of temperature; therefore it admits a limit as # ! 0C that is 0 or 1 (after choosing adequately the constant of integration).

2.4

Constitutive Relations

We shall primarily assume a certain minimal regularity of constitutive laws for pressure and internal energy, p 2 C 1 .Œ0; 1/  Œ0; 1//;

e 2 C 1 ..0; 1/  Œ0; 1//

(34)

We shall always assume that the gas obeys the second law of thermodynamics expressed through the Gibbs relation (8) postulating existence of the specific entropy; in particular, it must obey relation @% e.%; #/ D

 1 p.%; #/  #@ p.%; #/ ; # %2

(35)

called Maxwell’s relation. There are several families of constitutive laws enjoying physical justification and allowing for the satisfactory theory of weak solutions. They can be written down in the following framework p.%; #/ D pra .#/ C pmo .%; #/ C pel .%/;

(36)

where the indexes “ra,” “mo,” and “el” refer to “radiative,” “molecular,” and “elastic” (pressure), respectively. Correspondingly, the internal energy reads e.%; #/ D where we have to take

1 era .#/ C emo .%; #/ C eel .%/; %

(37)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

era .#/ D

0 #pra .#/

Z  pra .#/;

%

eel .%/ D 1

1397

pel .z/ dz z2

in order to comply with Maxwell’s relation (35). Under these assumptions, the specific entropy reads s.%; #/ D

1 sra .#/ C smo .%; #/ C sel .%/; %

(38)

and the Helmholtz function is H‚ D Hra;‚ C Hmo;‚ C Hel ; According to (35), the radiative entropy reads Z # 0 era .z/ dz: sra .#/  sra .1/ D z 1

(39)

(40)

Consequently the radiative Helmholtz function is given by Z

#

Hra;‚ .%; #/ D Hra;‚ .#/ D 1

0 era .z/ .z  ‚/dz C Hra;‚ .1/: z

(41)

The contribution of the elastic components of pressure and internal energy to the specific entropy and to the Helmoholtz function is Z sel .%; #/ D 0;

Hel .%; #/ D Hel .%/ D % 1

%

pel .z/ dz D %eel .%/; z2

(42)

respectively, again by virtue of relation (35). In particular, %Hel0 .%/  Hel .%/ D pel .%/;

(43)

and in view of (11) function .t; x/ 7! Hel .%.t; x// verifies @t Hel .%/ C divx .Hel .%/u/ C pel .%/divx u D 0:

(44)

We shall consider two families of molecular pressure constitutive laws: 1. Real gas phenomenological constitutive laws The molecular pressure and internal energy in many real gases enter into the following general framework pmo .%; #/ D #pth .%/;

emo .%; #/ D eth .#/:

In this situation, the specific entropy reads

(45)

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A. Novotný and H. Petzeltová

Z smo .%; #/ D smo;# .#/ C smo;% .%/;

#

smo;# .#/ D 1

Z

%

smo;% .%/ D  1

0 eth .z/ dz; z

pth .z/ dz z2

(46)

and the Helmholtz function is Z  Hmo;‚ D % emo .#/  ‚

# 1

0 emo .z/  dz C % z

Z 1

%

pth .z/ dz: z2

2. Constitutive laws derived in the statistical mechanics They take the general form pmo .%; #/ D # =.1/ P





% # 1=.1/

;  > 1;

(47)

where P 2 C 1 Œ0; 1/:

(48)

In agreement with Gibbs’ relation (8), the (specific) internal energy must be taken as emo .%; #/ D

1 # =.1/  %  P :  1 % # 1=.1/

(49)

In this case, the specific entropy reads smo .%; #/ D S





% # 1=.1/

; where S 0 .Z/ D 

1  P .Z/  P 0 .Z/Z :  1 Z2 (50)

The reader may consult Eliezer, Ghatak, and Hora [23] and [33, Chapter 3] for the physical background and further discussion concerning the structural hypotheses (47), (48), and (49). We shall proceed to several concrete examples. Radiative pressure The radiative pressure and energy are given by the Stefan-Boltzmann law: pra .%; #/ D pra .#/ D era .%; #/ D

a 4 # ; 3

a 4 # where a > 0 is the Stefan-Boltzmann constantI %

consequently we deduce from (40) and (41),

(51)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

sra .%; #/ D

1399

4a 3 4 # ; Hra;‚ .#/ D a.# 4  ‚# 3 /: 3% 3

(52)

Examples of real gas phenomenological molecular pressure constitutive laws Perfect gas – Boyle’s law. For the perfect gas, pmo .%; #/ D R%#;

emo .%; #/ D cv # p ;

p1

(53)

where R > 0 is universal gas constant and cv > 0; we have for the specific entropy

smo .%; #/ D

8 ˆ
=

if p D 1;

 1/  R ln %

> if p > 1 ;

;

(54)

and

Hmo;‚ .%; #/ D

8 ˆ ˆ
> =

: > > ; if p > 1 (55)

Real gases – virial series. According to Becker [2, Chapter 10], the pressure in the real gas can be expressed through the so-called virial series that takes the form p.%; #/ D R#% C

n X

Bi .#/%i ;

n 2 N:

iD1

One of the best approximations of this form is the so-called BeattieBridgeman state equation (see [106, Sections 3.4, 10.10, Chapter 10] for more details). Mie-Gruneisen equations of state are of the form p.%; #/ D pc .%/ C %#G.%; #/; where pc .%/ refers to the “cold” pressure (see [13, 99] for more details). Examples of molecular pressure constitutive laws from statistical mechanics In formulas (47), (48), and (49), at least two values of  are considered to be physically reasonable.

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Monoatomic gas. For monoatomic gases,  D 5=3. Relativistic gas. For the so-called relativistic gas,  D 4=3. See [23, Chapter 3] for more details. Examples of elastic pressure Nuclear fluids. In a simplified model of nuclear fluids, the molecular pressure is given by the Boyle’s law while there is an elastic pressure being composed of two terms:   pel .%/ D c1 %5=3 C c2 %3  c3 %2 ;

c1 ; c2 ; c3 > 0;

where the first term is the so-called Thomas-Fermi-Weizsacker approximation while the second term comes from the so-called Skyrme interaction (see [19]). Perfect gas in isentropic regime. Supposing that the gas evolves in the regime with the constant entropy s, we may deduce from (54)pD1 and (53) pmo .%; #/ pel .%/ D b% ; b D Re s > 0;  D

R C cv : cv

This is the pressure law for isentropic gas. The values of  (that is called adiabatic constant) ranges in the interval .1; 53 /. The value  D 5=3 corresponds to the isentropic flows of monoatomic gas.

2.5

Constraints Imposed by Thermodynamic Stability Conditions

The elastic pressure satisfies thermodynamic stability conditions if and only if pel0 .%/ > 0 for all % > 0:

(56)

The molecular pressure and internal energy given by formula (45) satisfy thermodynamic stability conditions if and only if 0 pth .%/>0 for all %>0;

0 eth .#/>0 for all #>0:

(57)

Likewise, the pressure and internal energy given by formulas (47), (48), and (49) satisfy thermodynamic stability conditions if and only if P 0 .Z/ > 0;

 P .Z/  P 0 .Z/Z >0 Z

for all Z > 0:

(58)

First point to be noticed at this moment is that by virtue of (58), the function Z 7! P .Z/=Z  must be decreasing on .0; 1/ and therefore lim P .Z/=Z  D p1 2 Œ0; 1/:

Z!1

(59)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1401

Second point is that under the thermodynamic stability conditions, function Z 7! S .Z/ is decreasing on interval .0; 1/ in view of (50); it may be chosen by means of a convenient additive constant in such a way that lim S .Z/ D S1 ; where S1 D 0 or S1 D 1:

Z!1

2.6

(60)

Third Law of Thermodynamics

The third thermodynamical law postulates that lim s.%; #/ D 0 for all % > 0:

#!0C

(61)

We notice that the perfect gas whose state equation is given by the Boyle’s law does not obey the third law (see formula (54)). The gases of mechanical statistics whose pressure and internal energy are given by formulas (47)–(49) obey the third law provided S can be taken (by choosing the integration constant in (50)) in such a way that lim S .Z/ D 0:

Z!1

(62)

The third law imposes further constraints on the constitutive laws in extreme regimes close to values # D 0. It is usually not necessary for building up the existence theory (at least on bounded domains). It may however play an important role when one investigates the stability issues.

2.7

Barotropic Flows

A fluid flow is said to be in barotropic regime or the fluid is said to be barotropic if the pressure p depends solely on the density. This can be achieved if we take in (36), pra D 0, and molecular pressure/internal energy given by (45) with pth .%/ D 0. We thus get p.%/ D pel .%/; e.%; #/ D eel .%/ C eth .#/: Supposing moreover that the viscous stress S is independent on the absolute temperature, system (3)–(9) in this situation reads @t % C divx .%u/ D 0;

(63)

@t .%u/ C divx .%u ˝ u/ C rx p.%/ D divx S.%; rx u/ D %f;

(64)

@t .%eth .#// C divx .%eth .#/u/ C divx q.%; #; rx #/ D S.%; rx u/ W rx u;

(65)

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where we have used identity (44) in order to transform (5) to (65). We observe that equation (65) and system (63)–(64) are decoupled in the sense that once the couple .%; u/ is determined from equations (63)–(64), temperature # can be obtained by solving (65) with boundary conditions (21). Moreover, taking a scalar product of equation (64) with u and integrating over  (under the assumption of enough smoothness of %; u and positivity of %) yields @t

Z  Z Z  1 S.%; rx u/ W rx u dx D %f  u dx; %juj2 C H .%/ dx C  2  

(66)

where Z

%

H .%/ D Hel .%/ D % 1

p.z/ dz; z2

(67)

provided the boundary conditions for velocity are conservative as those exposed in (20) or (22). This equation replaces for the barotropic flows the global dissipation identity (28) valid for the (regular) heat-conducting flows. System of partial differential equations (63) and (64) is called compressible Navier-Stokes equations in barotropic regime. It does not describe fully satisfactorily physically realistic situations. However, it is consistent with thermodynamics, and it already contains pretty much of the mathematical difficulties encountered when dealing with the full Navier-Stokes-Fourier system. Its investigation is not only of independent interest, but it can be used as a preliminary toy problem before attacking the full system. The most usual examples of barotropic flows are isothermal flows where p.%/ D R#% describing the flows of the perfect gas with the constant temperature # > 0 and the isentropic flows p.%/ D Re s % ;  D

R C cv cv

describing the flows of the perfect gas with the constant entropy s 2 R. Notice, however, that the requirements of constant temperature or constant entropy violate conservation of energy (65) unless specific external heat sources are not added to (65).

3

Specific Mathematical Tools for Compressible Fluids

We shall gather in this section most of mathematical tools needed to investigate weak solutions to the compressible Navier-Stokes equations or to the Navier-StokesFourier system. As far as the notations are concerned, we employ standard notation

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1403

commonly used in the mathematical analysis and in the theory of partial differential equations, as in the books [30, 33, 59, 88, 102].

3.1

Instantaneous Values of Functions in L1 .0,TI L1 .//

Theorem on Lebesgue points (see, e.g., [10, Appendix]) says that for any v 2 L1 .0; T I X /, X a Banach space, there exists vQ ˙ 2 L1 .0; T I X / such that: (i) 1

1 For a. a. 2 .0; T /; lim C h h!0

Z B ˙ . Ih/

kv.t /  vQ ˙ . /kX dt D 0;

where B C . I h/ D . ; C h/, B  . ; h/ D .  hI /. (ii) 1 For a. a. 2 .0; T /; vQ C . / D vQ  . /: (iii) If v 2 Cweak .Œ0; T I L1 .//, then vQ C . / D vQ  . / D v. / for all 2 Œ0; T . After this reminder, we are ready to define the instantaneous values of functions in L1 .0; T I L1 .//. We define right instantaneous value of v 2 L1 .0; T I L1 .// at

2 Œ0; T / as a continuous linear functional (a measure) v. C/ 2 .C .// < v. C/;  >C ./ D lim inf h!0C

1 h

Z v.t; x/.x/dx for all  2 C ./;

(68)

B C . ;h/

and left instantaneous value of v 2 L1 .0; T I L1 .// at 2 .0; T  as a continuous linear functional (a measure) v. / 2 .C .// < v. /;  >C ./ D lim sup h!0C

1 h

Z v.t; x/.x/dx for all  2 C .//:

(69)

B  . ;h/

The instantaneous values of function v will be defined as follows: 1 instŒv.0/ D v.0C /; instŒv. / D .v. C /Cv.  //; if 2.0; T /; instŒv.T / D v.T  /: 2 (70) If v belongs only to L1 .0; T I L1 .//, then v. / D instŒv. / for a.a. 2 .0; T /. If v 2 L1 .0; T I Lp .//, 1 < p < 1, then instŒv. / 2 Lp ./: Theorem on Lebesgue points described above implies that for any v 2 Cweak .Œ0; T I L1 .//, instŒv. / D v. / for all 2 Œ0; T :

(71)

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A. Novotný and H. Petzeltová

Here and in the sequel, Cweak .Œ0; T I Lp .// is the space of functions in L1 .0; T I Lp .// which are continuous for the weak topology of the space Lp ./, 1  p < 1.

3.2

Instantaneous Values of Solutions of Conservation Laws

3.2.1 The Case of Variational Identity Suppose that d 2 L1 .0; T I L1 .// verifies identity Z

T

Z

Z



0

Z

T

Z F.t; x/  rx '.t; x/ dxdt

Z G.t; x/'.t; x/ dxdt C 



(72)



0

Z

D 0

T

d .t; x/@t '.t; x/ dxdt 



d0 .x/'.0; x/ dx; with any ' 2 Cc1 .Œ0; T //;

where .F; G/ 2 L1 .QT I R4 / and d0 2 L1 ./. We may take in (72) the test functions ˙ 1

;h .t /'.t; x/, ' 2 Cc .Œ0; T //, where 2 .0; T / and h > 0 is sufficiently small, and 

;h

C

;h

8
C .Œ0;T / 

 T

G.t; x/'.t; x/ dxdt C 0



Z

Z

D

F.t; x/rx '.t; x/ dxdt 0



d0 .x/'.0; x/ dx; with any ' 2 

(91) Cc1 .Œ0; T //; '

 0;

where Z TZ < †; ' >C .Œ0;T // 

Z.t; x/'.t; x/ dxdt with any ' 2 C .Œ0; T /; '  0: 0



29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1409

Due to the nonnegativity of †,  T;h

k†kC .Œ0;T / D lim < †; h!0C

> kd kL1 .0;T IL1 .// :

(92)

Moreover there exists a unique nonnegative measure † on the  -algebra of Borel sets of Œ0; T    such that Z < †; ' >C .Œ0;T / D

Choosing in (91) test functions '.t; x/ Z

(93)

we get

Z Z  1 'd† C 'd† 2 Œ0; /  Œ0;  (94) Z Z Z Z D d .t; x/@t '.t; x/ dxt C F.t; x/  rx '.t; x/ dxdt Z

d . ; x/'. ; x/ dx  

˙

;h ,

'd† : Œ0;T 

0

Z



0



Z

C 0

d0 .x/'.0; x/ dx C





G.t; x/'.t; x/ dxdt; with any ' 2 Cc1 .Œ0; T /  /; '  0:

Identity (94) holds for a.a. 2 .0; T / and it is equivalent to (90). In particular, one deduces from the choice .t; x/ D  .t; x/ D z;h .t /1.x/ in (90) Z

Z d . ; x/ dx 



d0 .x/ dx C 

Z



Z

D

1 2

Z



;h .t /1.x/,

Z d† C

Œ0; /

d†



Œ0; 

G.t; x/'.t; x/ dxdt 0

resp.,

(95)



for a.a. 2 .0; T /, and Z Z  1 d . ; x/ dx  d .z; x/ dx C d† C d† 2 Œz; /   .z;  Z Z G.t; x/'.t; x/ dxdt D

Z

Z

z

(96)



for a.a. 0 < z < h < T or foriall values of and z in Œ0; T  if we replace R R  d .; x/ dx by inst  d .; x/ dx ./.

1410

3.3

A. Novotný and H. Petzeltová

Weakly Convergent Sequences in L1

Theorem 1. Let O  RN be a bounded open set and vn W O 7! R be a sequence of measurable functions such that sup kˆ.vn /kL1 .O/ < 1; for a certain application ˆ 2 C Œ0; 1/: n1

Suppose that lim

jzj!1

jzj D 0: ˆ.jzj/

Then there is a subsequence of vn (not relabeled) such that vn * v in L1 .O/:

3.4

Convexity, Monotonicity, and Weak Convergence

It is well known that convex lower semicontinuous functions give rise to L1 – sequentially weakly lower semicontinuous functionals – and give rise to a useful criterion of the a.e. convergence. We present here a convenient formulation of these results taken over from [30, Theorem 2.11 and Corollary 2.2] or [33, Theorem 10.20]. (More general formulation can be found in Brezis [10] or in Ekeland, Temam [22].) The corresponding theorems read: Theorem 2. Let O  RN be a measurable set and fvn g1 nD1 a sequence of functions in L1 .OI RM / such that vn ! v weakly in L1 .OI RM /: Let ˆ W RM ! .1; 1 be a lower semicontinuous convex function. Then ˆ.v/ W O 7! R is integrable and Z

Z ˆ.v/dx  lim inf

O

n!1

ˆ.vn /dx: O

Strictly convex lower semicontinuous functions are involved in a useful criterion of the a.e. convergence. Theorem 3. Let O  RN be a measurable set and fvn g1 nD1 a sequence of functions in L1 .OI RM / such that vn ! v weakly in L1 .OI RM /:

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1411

Let ˆ W RM ! .1; 1 be a lower semicontinuous convex function such that ˆ.vn / 2 L1 .O/ for any n, and ˆ.vn / ! ˆ.v/ weakly in L1 .O/: Then ˆ.v/  ˆ.v/ a.e. on O:

(97)

If, moreover, ˆ is strictly convex on an open convex set U  RM and ˆ.v/ D ˆ.v/ a.e. on O; then vn .y/ ! v.y/ for a.a. y 2 fy 2 O j v.y/ 2 U g

(98)

extracting a subsequence as the case may be. Similar properties are true also for monotone functions as a consequence of the so-called Minti’s trick. The following result is taken from [33, Theorem 10.19]: Theorem 4. Let I  R be an interval, Q  RN a domain, and .P; G/ 2 C .I /  C .I / a couple of nondecreasing functions.

(99)

Assume that %n 2 L1 .QI I / is a sequence such that 8 ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ :

9 > > > > > =

P .%n / ! P .%/; G.%n / ! G.%/; P .%n /G.%n / ! P .%/G.%/

> > > > > ;

weakly in L1 .Q/:

(100)

(i) Then P .%/ G.%/  P .%/G.%/:

(101)

(ii) If, in addition, G 2 C .R/;

G.R/ D R;

G is strictly increasing; (102)

P 2 C .R/;

P is nondecreasing;

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and P .%/G.%/ D P .%/ G%/;

(103)

P .%/ D P ı G 1 .G.%//:

(104)

then

(iii) In particular, if G.z/ D z, then P .%/ D P .%/:

3.5

(105)

The Inverse of the Div Operator (Bogovskii’s Formula)

Theorem 5. Let   RN be a bounded Lipschitz domain. (i) Then there exists a linear mapping B, B W ff j f 2

Cc1 ./;

Z 

f dx D 0g ! Cc1 .I RN /;

such that divx .BŒf / D f with the following properties: (ii) We have kBŒf kW kC1;p .IRN /  ckf kW k;p ./ for any 1 < p < 1; k D 0; 1; : : : ; (106) In particular, B can be extended in a unique way to a bounded linear operator B W ff j f 2 Lp ./;

Z

1;p



f dx D 0g ! W0 .I RN /:

R

(iii) If f 2 Lp ./,  f dx D 0, and, in addition, f D divx g, where g 2 .Lq .//N , 1 < q < 1, and g  nj@ D 0 (in the weak sense of normal traces), then kBŒf kLq .IRN /  ckgkLq .IRN / :

(107)

Operator B has been constructed for the first time by Bogovskii. The reader can consult Galdi [59] or [88, Section 3.3] for more details about this problem.

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

3.6

1413

Poincaré- and Korn-Type Inequalities

Applications in compressible thermodynamics often require refined versions of Poincaré and Korn inequalities that are not directly covered by the standard theory. We shall list some of them and refer the reader to [33, Appendix, Sections 10.8, 10.9] for more systematic treatment. Theorem 6. Let 1  p  1, 0 < < 1, and let   RN be a bounded Lipschitz domain. Let V   be a measurable set such that jV j  V0 > 0: Then there exists a positive constant c D c.p; ; V0 / such that h Z  1 i k v kW 1;p ./  c krx vkW 1;p .;RN / C jvj dx V

for any v 2 W 1;p ./. Theorem 7. Let   RN , N > 2 be a bounded Lipschitz domain, and let 1 < p < 1, M0 > 0, K > 0,  > 1. Then there exists a positive constant c D c.p; M0 ; K;  / such that inequality Z      kvkW 1;p .IRN /  c rx v p C rjvj dx N L .IR /



holds for any v 2 W 1;p .I RN / and any nonnegative function r such that Z Z r dx; r  dx  K: 0 < M0  

(108)



The following lemma is often useful in combination with Theorem 6 to investigate positivity of the temperature (see [33, Lemma 2.1]). Lemma 1. Let  be a bounded Lipschitz domain and p;  > 1. Let S 2 C .0; 1/ be a strictly decreasing function such that limZ!1 S .Z/ D S1 2 f1; 0g and Z  %  n %n S 1=.1/ dx  0 lim sup 1=. 1/ n!1 f%n #n g #n whenever %n  0 is bounded in L ./ and 0 < #n ! 0 in Lp ./. Then for any M0 > 0, 0 > 0, and S 2 R, there exist ˛ D ˛.M0 ; 0 ; S/ > 0, # D #.M0 ; 0 ; S/ > 0 such that for any nonnegative functions %, # satisfying Z Z %dx  M0 ; .% C # p /dx  0 ; 



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and Z %S





% # 1=.1/



dx  S > M0 S1 ;

we have ˇ ˇ ˇ ˇ ˇf#  #gˇ  ˛:

(109)

The classical Korn’s inequality deals with the symmetrized gradients of the vector fields. It reads: Theorem 8. Assume that 1 < p < 1. (i) There exists a positive constant c D c.p; N / such that krvkLp .RN IRN N /  ckrv C r T vkLp .RN IRN N / for any v 2 W 1;p .RN I RN /. (ii) Let   RN be a bounded Lipschitz domain. Then there exists a positive constant c D c.p; N; / > 0 such that Z   jvj dx kvkW 1;p .IRN /  c krv C r T vkLp .;RN N / C 

for any v 2 W 1;p .I RN /. In the fluid dynamics of compressible fluids, we often need a version of Korn’s inequality involving the symmetrized and traceless gradient. It reads: Theorem 9. Let 1 < p < 1 and N > 2. (i) There exists a positive constant c D c.p; N / such that krvkLp .IRN N /  ckrv C r T v 

2 divv IkLp .IRN N / N

for any v 2 W 1;p .RN I RN /, where I D .ıi;j /N i;j D1 is the identity matrix. (ii) Let   RN be a bounded Lipschitz domain. Then there exists a positive constant c D c.p; N; / > 0 such that Z   2 jvj dx kvkW 1;p .IRN /  c krv C r T v  divv IkLp .;RN N / C N  for any v 2 W 1;p .I RN /.

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1415

Finally the generalized version of the above inequality reads: Theorem 10. Let   RN , N > 2 be a bounded Lipschitz domain, and let 1 < p < 1, M0 > 0, K > 0,  > 1. Then there exists c D c.p; K; M0 ;  / > 0 such that kvkW 1;p .IRN / Z    2    c rx v C rxT v  divv I p C rjvj dx L .IRN / N  for any v 2 W 1;p .I RN / and for any nonnegative function r such that Z 0 < M0  

3.7

Z

r  dx  K:

r dx;

(110)



Time Compactness

We report the classical theorem known as Aubin-Lions-Simon lemma [6, Theorem II.5.16]. Theorem 11. Let X  B  Y be Banach spaces, where the symbols  denotes compact and  continuous imbeddings, respectively, and let 1  p; q  1. Let v n be a sequence of functions such that v n is bounded in Lp .0; T I X /; @t v n bounded in Lq .0; T I Y /: Then there exists a subsequence (denoted again by v n ) such that if p < 1; v n ! v (strongly) in Lp .0; T I B/I if p D 1 and q > 1; v n ! v (strongly) in C .Œ0; T I B/: The classical Aubin-Lions lemma is convenient for applications involving time evolution of the quantity v expressed through an equation. It usually cannot be applied to investigate time compactness of quantities evaluating according to differential inequalities. In the latter situation, one may use a weaker variant of the above theorem (see [30, Lemma 6.3]). Theorem 12. Let   RN be a bounded domain and 1 < p < 1. Let v n be a sequence of functions such that v n is bounded in Lp .0; T I Lq ./// \ L1 .0; T I L1 .//; q > @t v n D gn C †n ;

2N ; 2CN

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where †n is a nonnegative distribution and gn is bounded in L1 .0; T I W m;r .// with some m  1, r > 1. Then v n contains a subsequence such that vn ! v (strongly) in Lp .0; T I W 1;2 .//:

3.8

Operator r 1 and Riesz-Type Operators

We introduce operators A D rx 1 and R D rx ˝ rx 1 , .r1 /j .v/ D F 1

h i

j F.v/. / j j2

i

.r ˝ r1 /ij .v/ D F 1

;

h i i j F.v/. / ; j j2 (111)

where F denotes the Fourier transform Z 1 v.x/exp.i  x/dx: ŒF.v/. / D 2 3 R3 We recall the basic properties of these operators (see e.g. Feireisl [30], [33, Sections 0.5 and 10.16] or [88] for more details). Theorem 13. (i) A is a continuous linear operator from L1 \ L2 .R3 / to L2 C L1 .R3 I R3 / and from Lp .R3 / to L3p=.3p/ .R3 I R3 / for any 1 < p < 3. (ii) R is a continuous linear operator from from Lp .R3 / to Lp .R3 I R33 / for any 1 < p < 1. (iii) The following formulas hold R.v/ D RT .v/;

3 X

Rjj .v/ D v; v 2 Lp .R3 /;

j D1

@k Rij .v/ D Rij .@k v/; Rij .@k v/ D Rik .@j v/; v 2 W 1;p .R3 /; where 1 < p < 1; rx A.v/ D R.v/;

divA.v/ D v; v 2 Lp .R3 /;

where 1 < p < 3; Z R3

A.v/wdx D 

Z vA.w/dx; R3

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1417

with 0

0

v 2 Lp .R3 /; w 2 Lq .R3 /; A.w/ 2 Lp .R3 /; A.v/ 2 Lq .R3 /; where 1 < q; p < 3; Z R3

R.v/wdx D

Z

0

vR.w/dx; v 2 Lp .R3 /; w 2 Lp .R3 /;

R3

where 1 < p < 1.

3.9

Some Results of Compensated Compactness

We shall start by the celebrated Div-Curl lemma of Murat and Tartar [84] formulated in the form [33, Lemma 10.1]. Theorem 14. Let Q  RN be an open set. Assume Un ! U weakly in Lp .QI RN /; (112) Vn ! V weakly in Lq .Q; RN /; where 1 1 1 C D < 1: p q r In addition, let 9 div Un r  Un ; = T

curl Vn .rVn  r Vn /

;

 be precompact in

W 1;s .Q/; W 1;s .Q; RN N /;

(113)

for a certain s > 1. Then Un  Vn ! U  V weakly in Lr .Q/: The next theorem involving commutator of Riesz operators may be seen as a consequence of the Div-Curl lemma stated above (see Feireisl [30, Section 6] or [33, Theorem 10.27]).

1418

A. Novotný and H. Petzeltová

Theorem 15. Let V" ! V weakly in Lp .RN I RN /; U" ! U weakly in Lq .RN I RN /; where

1 p

C

1 q

D

1 s

< 1. Then

U"  RŒV"   RŒU"   V" ! U  RŒV  RŒU  V weakly in Ls .RN /: The next theorem is a compensated compactness result in the spirit of Coifman and Meyer [15] (see [33, Theorem 10.28]). Theorem 16. Let w 2 W 1;r .RN / and V 2 Lp .RN I RN / be given, where 1 < r < N;

1 1 1  C < 1: r N p

Then there exists ˛ > 0 and q D q.r; p/ > 1 such that     RŒwV  wRŒV

W ˛;q .RN IRN /

 c.r; p/kwkW 1;r .RN / kVkLp .RN IRN / :

Here W ˛;q .RN / denotes the Sobolev-Slobodeckii space.

3.10

Parametrized (Young) Measures

Let Q  RN be a domain; we say that Q  RM if 8 < for a. a. x 2 Q; the function  7! :

for all  2 RM ; the function x 7!

W Q  RM is a Carathéodory function on 9 .x; / is continuous on RM I = .x; / is measurable on Q:

;

(114)

We recall that  is called a probability measure on RM if it is a nonnegative Borel measure, such that .RM / D 1. In the sequel, we shall deal with families fx gx2Q of probability measures x . We say that the family of measures fx gx2Q is a family of parametrized measures depending measurably on x if for almost all x 2 Q, x is a probability measure and if 8 < :

8 W RM ! R;  2 C .RM / \ L1 .RM /; the function x !

R

RM

./ dx ./ WD< x ;  > is measurable on Q:

9 = ; (115)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1419

Families of parametrized measures are connected to the weak convergence as described in the following theorem (see Pedregal [91, Chapter 6, Theorem 6.2]): N M Theorem 17. Let fvn g1 be a weakly convergent sequence nD1 , vn W Q  R ! R 1 M of functions in L .QI R /, where Q is a domain in RN . Then there exist a subsequence (not relabeled) fvn g1 nD1 and a parameterized family fy gy2Q of probability measures on RM depending measurably on y 2 Q with the following property: For any Carathéodory function ˆ D ˆ.y; z/, y 2 Q, z 2 RM such that

ˆ.; vn / ! ˆ weakly in L1 .Q/; we have Z ˆ.y/ D

.y; z/ dy .z/ for a.a. y 2 Q:

RM

The family of measures fy gy2Q associated to a sequence fvn g1 nD1 , vn * v in L1 .QI RM /; is termed Young measure. Suppose that vn is only a bounded sequence in L1 .Q/. Then there still exists an associated parametrized family fy gy2Q of nonnegative Borel measures with the properties stated in Theorem 17, which, however, do not need to be necessarily probability measures.

3.11

Some Elements of the DiPerna-Lions Transport Theory

In the following theorems, we present some consequences of the DiPerna-Lions transport theory applied to the continuity equation (see [33, Section 10.16]. ) Theorem 18. Let N  2, ˇ; q 2 .1; 1/, q1 C ˇ1 2 .0; 1. Suppose that the functions ˇ

q

1;q

.%; u/ 2 Lloc ..0; T /  RN /  Lloc .0; T I Wloc .RN I RN //, where %  0 a. e. in .0; T /  RN , satisfy the transport equation @t % C divx .%u/ D f in D0 ..0; T /  RN /, where f 2 L1loc ..0; T /  RN /. Then     @t b.%/ C divx .b.%/u C %b 0 .%/  b.%/ divx u D f b 0 .%/

(116)

(117)

in D0 ..0; T /  RN / for any b 2 C 1 .Œ0; 1//;

b 0 2 Cc .Œ0; 1//:

(118)

1420

A. Novotný and H. Petzeltová

Theorem 19. Let   RN , N D 2; 3 be a bounded Lipschitz domain. (i) Suppose that .ˇ; q/ satisfy assumptions of Theorem 18 and that .%; u/ 2 1;q Lˇ ..0; T /  /  Lq .0; T I W0 .//. Then there holds: If the couple .%; u/ 0 satisfies equation (116) in D ..0; T /  /, then it verifies the same equation also in D0 ..0; T /  RN / provided .%; u/ is extended to .0; T /  RN as follows:  .%; u/.t; x/ D

.%; u/.t; x/ for .t; x/ 2 .0; T /  ; .%1  0; 0/ for .t; x/ 2 .0; T /  .RN n /:

(119)

1;q

(ii) Suppose that .%; u/ 2 L1 ..0; T /  /  Lq .0; T I W0 .//, q > 1 verifies renormalized continuity equation (117) in D0 ..0; T /  / with any b belonging to class (118). Then the extension (119) verifies the same equation in D0 ..0; T /  RN / for the same functions b. Theorem 20. Let   RN , N  2 be a bounded domain and let .%; u/, % 2 L1 .0; T I Lˇ .//, u 2 Lq .0; T I W 1;q .//, f 2 Lq ..0; T /  /, %u 2 L1 ..0; T /  / satisfies continuity equation (116) in D0 ..0; T // and renormalized continuity equation (117) with any b in class (118). Then % 2 C .Œ0; T I L1 .//: Lemma 2. Let N  2, ˇ; q 2 Œ1; 1/,

1 q

C

1 ˇ

2 .0; 1. Suppose that the functions

ˇ q 1;q Lloc ..0; T /RN /Lloc .0; T I Wloc .RN I RN //, where %

.%; u/ 2  0 a.e. in .0; T / RN , satisfy the renormalized continuity equation (117) for any b belonging to the class (118). Then we have: p

(i) If f 2 Lloc ..0; T /  RN / for some p > 1, p 0 . qˇ0  1/  ˇ, then equation (117) holds for any 0

b 2 C 1 .Œ0; 1//; jb 0 .s/j  cs ˇ=q 1 ; for s > 1:

(120)

(ii) If f D 0, then equation (117) holds for any 0

b 2 C .Œ0; 1// \ C 1 ..0; 1//; sb 0  b 2 C Œ0; 1/; jb 0 .s/j  cs ˇ=q 1 if s 2 .1; 1/: (121)

3.12

The Gronwall Lemma

We recall a variant of the Gronwall-Bellman lemma. The reader can consult the monograph [89] for the details on this variant and other differential and integral inequalities.

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1421

Theorem 21. Let ˛ 2 L1 .0; T /, ˇ 2 L1 .0; T /, ˇ  0 be given functions. Suppose that a function u 2 L1 .0; T / satisfies inequality Z



ˇ.t /u.t /dt for a.a. 2 .0; T /:

u. /  ˛. / C 0

Then Z

t

u.t /  ˛.t/ C

˛.s/ˇ.s/e

Rt s

ˇ.z/dz

ds for a.a. t 2 .0; T /:

0

4

Existence of Weak Solutions to the Compressible Navier-Stokes Equations for Barotropic Flows

In this section we shall define and investigate weak solutions to the system (63)–(64) in a time cylinder QT D .0; T /  , where  is a bounded domain, with pressure p D p.%/;

p 2 C Œ0; 1/ \ C 1 .0; 1/; p.0/ D 0:

(122)

and stress tensor (12), where  D const: > 0;

 D const:  0:

(123)

The system is completed with initial conditions %.0; / D %0 ./;

%u.0; / D %0 u0 ;

(124)

and no-slip boundary conditions (20), i.e., u.t; /j@ D 0:

4.1

(125)

Weak Formulation and Weak Solutions

We begin with the definition of the Leray-type weak solutions to problem (63)– (64), (122)–(125). It consists of the standard weak formulation of equations (63)– (64). Dissipation identity (66) will be replaced by the dissipation inequality “” in the integral R R form. In fact identity (66) integrated over time contains the functional Z 7! 0  S.Z/ W Z dxdt, Z D ru that is not continuous but only sequentially lower weakly semicontinuous with respect to the weak topology of L2 .Q I R9 /. Consequently, when passing from approximations to a solution, the limit processes will conserve solely the inequality “.”

1422

A. Novotný and H. Petzeltová

Definition 1. Let  be a bounded domain, and let %0 W  ! Œ0; C1/; u0 W  ! R3 ; %0 u20 D 0 a.e. in the set fx 2 j%0 .x/ D 0g (126)

%0 u0 D 0;

with finite energy E0 D

R

1 2  . 2 %0 u0 CH .%0 //dx

and finite mass 0 < M0 D

R 

%0 dx.

We shall say that a pair .%; u/ is a finite energy weak solution to the problem (63)– (64), (122)–(125) emanating from the initial data .%0 ; u0 / if: (a) 1

% 2 L1 .0; T I L1 .//; %  0 a.e. in .0; T /  ; p.%/ 2 L1 .QT /;

(127)

1 u 2 L2 .0; T I W01;2 .//; %u; %u2 ; H .%/ 2 L1 .0; T I L1 .//: 2 (b) % 2 Cweak .Œ0; T I L1 .//, and the continuity equation (63) is satisfied in the following weak sense Z 

Z ˇ

ˇ %'dx ˇ D 0

0

Z 

 %@t ' C %u  rx ' dxdt;

(128)



for all 2 Œ0; T  and for all ' 2 Cc1 .Œ0; T   /: (c) %u 2 Cweak .Œ0; T I L1 .//, and the momentum equation (64) is satisfied in the weak sense, Z 

ˇ Z Z   ˇ %u@t 'C%u˝u W r'Cp.%/div'S.rx u/ W rx 'C%f' dxdt %u'dx ˇ D 0

0



(129)

for all 2 Œ0; T  and for all ' 2 Cc1 .Œ0; T   I R3 /: (d) The dissipation identity (66) is satisfied as inequality in the weak sense: Z

T

0

 0

Z  0

T

Z  Z T Z  1 %juj2 C H .%/ dx C .t / S.rx u/ W rx u dxdt  2  0 (130) Z 1 .t / %f  u dxdt C E0 .0/ for all 2 C Œ0; T /;  0:

.t /



29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

Z

gdx j 0 is meant for

Here and hereafter the symbol

Z



1423

Z g. ; x/dx  

g0 .x/dx. We recall that the Helmholtz function H is defined in (67). Space 

Cweak .Œ0; T I L1 .// is defined in (71). Definition 2. A couple .%; u/ satisfying all requirements of Definition 1 with exception of the energy inequality (130) which is replaced by Z  Z Z  ˇ Z Z 1 ˇ S.rx u/ W rx u dxdt  %f  u dxdt; %juj2 C H .%/ dx ˇ C 0  2   0 0 (131) for almost all 2 .0; T / will be called bounded energy weak solution of problem (63)–(64), (122)–(125). Definition 3. We say that the couple .%; u/ 2 L1 .0; T I L1 .//  L2 .0; T I W 1;p .//; %  0; %u 2 L1 .QT /; p > 1 (132) satisfies continuity equation in the renormalized sense iff it satisfies continuity equation (116) in D0 ..0; T /  / and renormalized continuity equation (117) in D0 ..0; T // with any test function b belonging to the class (118) and with f D 0. Weak solution to problem (63)–(64), (122)–(125) satisfying the continuity equation in the renormalized sense will be called renormalized weak solution. Remark 1. 1. Suppose that .%; u/ is a renormalized weak solution of the continuity equation such that % 2 L1 .0; T I L .//; u 2 L2 .0; T I W 1;2 .//;  > 1; where  is a bounded domain. Then % 2 C .Œ0; T I L1 .//: If, moreover, uj.0;T /@ D 0 and  is a Lipschitz domain, then the renormalized continuity equation is satisfied up to the boundary, namely, Z

Z b.%. ; x//'. ; x/ dx 



Z



D 0

b.%.0; x//'.0; x/ dx 

Z  

 b.%/@t ' C b.%/u  rx '  B.%/divx u' dxdt D 0

(133)

1424

A. Novotný and H. Petzeltová

for all 2 Œ0; T , for all ' 2 Cc1 .QT /, and for all b; B belonging to 5

b 2 C Œ0; 1/ \ C 1 .0; 1/; jb.z/j  c.1 C z 6  /; 

B 2 C Œ0; 1/; jB.%/j  c.1 C % 2 /;

(134)

where b; B are related by the formula B.z/ D zb 0 .z/  b.z/. Moreover, b.%/ 2 C .Œ0; T I L1 .//. If, in addition   2, then the continuity equation is satisfied up to the boundary, namely, Z

Z

Z

%. ; x/'. ; x/ dx  



%.0; x/'.0; x/ dx D

Z 

0



 %@t ' C %u  rx ' dxdt



for all 2 Œ0; T  and for all ' 2 Cc1 .QT /. The above statements follow from the DiPerna-Lions transport theory [18] evoked through Theorems 18, 19, 20, and Lemma 2. The reader can consult [30, Chapter 4, Section 4.1.5], [88, Chapter 6, Section 6.2], [33, Appendix, Section 10.18] for more details and proofs. 2. For any t 2 Œ0; T , the momentum %u.t; / vanishes almost everywhere on the vacuum set of function %.t; /. More precisely, properties % 2 C .Œ0; T I L1 .//, %  0, %u 2 Cweak .Œ0; T I L1 .//, and %u2 2 L1 .0; T I L1 .//, where u 2 L1 .QT /, are enough to conclude that %u.t; / D 0 a.e. on the set fx 2 j%.t; x/ D 0g:

(135)

Similarly, if in addition to previous hypotheses, % 2 L1 .0; T I L3=2 .//, then   %u2 D 0 a.e. on the set [t2Œ0;T  ftg  fx 2 j%.t; x/ D 0g :

(136)

3. We introduce global kinetic energy Ekin W Œ0; T  7! Œ0; 1/ and global elastic energy Eel W Œ0; T  7! Œ0; 1/ hZ 1 i Ekin D inst %u2 .t; x/ dx ;  2

hZ i Eel D inst H .%/.t; x/ dx ;

(137)



where the instantaneous values were introduced in (70). We define global mechanical energy as Emech E D Ekin C Eel :

(138)

With this notation, in agreement with (86)–(88), inequality (130) can be rewritten as identity

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

Z



0

E. / . /  0

Z

Z

Z

.t /

Œ0; /

(139)

Œ0; 

2 C 1 Œ0; T ;

%f  u dxdt C E0 .0/ for all

0

 .t /d

.t /d C

Z



D

1 .t /E.t /dt C 2

1425

 0;



for all 2 Œ0; T , where  is a nonnegative measure on the algebra of Borel sets of interval Œ0; T  satisfying, in particular, 1 2

Z .t /dC

Z Œ0; /

 Z .t /d  Œ0; 

Z .t / S.rx u/ W rx u dxdt; for all



2 C Œ0; T ;

 0:



0

With this definition at hand, we may deduce from inequality (139) in agreement with (86)–(88) that E. /  E0 C

1 2

Z

Z

 Z d D

d C Œ0; /



%u  f dxdt for all 2 Œ0; T ;

0

Œ0; 

Z 

(140) and E. /  E.z/ C

1 2

Z

Z d C

Œz; /

 Z d D

.z; 

z



Z %u  f dxdt 

for all 0 < z < < T . In  Rparticular, Rfunction 7! E. / is a sum of a nonincreasing function 7!  12 Œ0; / d C Œ0;  d (that must have at most a countable number of jumps) R R and an absolutely continuous function 7! 0  %uf dxdt . This representation of E is convenient to use for studying of the longtime behavior of weak solutions. 4. Relation (139) implies that any finite energy weak solution is a bounded energy weak solution. Existence of weak solutions to problems (63)–(64),and (122)–(125) is known provided the pressure verifies in addition to (122) conditions p 0 .%/  a1 %1  b; p.0/ D 0;

% > 0;

p.%/  a2 % C b;

(141)

%  0;

with some  > 3=2, a1 > 0, a2 ; b 2 R. The exact statement of the existence result is announced in the following theorem: Theorem 22 (See [77] for p.%/  % ,   9=5, [47, Theorem 1.1] with p.%/  % ,  > 3=2, [28, Theorem 1.1] for nonmonotone pressure (141) and  > 3=2.). Let  be a bounded domain of class C 2; , T > 0 and f 2 L1 .QT /,

1426

A. Novotný and H. Petzeltová

where QT D .0; T /  . Suppose that the initial data satisfy (126) and that the pressure p belongs to the regularity class (122) and satisfies condition (141) with  > 3=2. Then the problem (63)–(64), (124), (125) admits a renormalized finite energy weak solution with the following additional properties

% 2 C .Œ0; T I L1 .// \ L1 .0; T I L .// \ Lp0 .QT /; p0 D min p.%/ 2 Lp1 .QT /; p1 D p0 = > 1; %u 2 L1 .0; T I L

2  C1

.// \ Cweak .Œ0; T I L

2  C1

.//:

n 5  3 3 o ;  ; 3 2 (142) (143) (144)

The main ideas of the proof of Theorem 22 will be explained in the more general context of heat-conducting gases in Sect. 7. The detailed existence proof can be found in paper [47] for the monotone pressure and  > 3=2 and in [28] for nonmonotone pressure. More details about this problem are available in monographs [30, 77, 88]. On unbounded domains, Definitions 1, 2, and 3 of finite (bounded/renormalized) weak solutions for the problem (63)–(64), (123) (124), (125) must be slightly modified in order to be able to accommodate conditions at infinity (24). We shall first consider the case

%1 D 0; u1 D 0; cf. (24):

(145)

Definition 4. Let  be an unbounded domain. We say that couple .%; u/ is (i) finite energy weak solution, (ii) bounded energy weak solution, (iii) renormalized weak solution of problem (63)–(64), (122)–(125) with zero conditions at infinity (145) iff: it belongs to class (127) with u belonging to L2 .0; T I D01;2 .I R3 // (in place of u 2 L2 .0; T I W01;2 .I R3 //), % 2 Cweak .Œ0; T I L1 .K//, %u 2 Cweak .Œ0; T I L1 .KI R3 // with any compact K   and (i) it satisfies all requirements of Definition 1 (for finite energy weak solution); (ii) it satisfies all requirements of Definition 2 (for bounded energy weak solution); (iii) it satisfies all requirements of Definition 3 (for renormalized weak solution). In the above, we have denoted by D01;2 ./ the homogenous Sobolev space given by   closurekrx kL2 .IR3 / Cc1 ./ : The weak solutions designed in Definitions 1, 2, 3, and 4 enjoy the following stability condition with respect to the variations of the domain:

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1427

Theorem 23 (See [48, Theorem 1.1]). Let n be a sequence of domains in R3 and  be a domain, such that: (i) For any compact set K  , there is a natural number n0 such that for all n  n0 , K  n ; (ii) Sets n n  enjoy the property cap2 .n n / ! 0, where Z cap2 .M / D inff

R3

jrx jdx j  2 Cc1 .R3 /;   1 on M g:

Let .%n ; un / be a sequence of bounded energy renormalized weak solutions to problem (63)–(64), (123), (124), (125) (and (24), (145) if  is unbounded) with pressure p satisfying (122), (141) with  > 3=2 with initial conditions .%n;0  0; un;0 / and external forces fn such that    %n;0 ; %n;0 un;0 ! %0 ; %0 u0 in L1 .R3 I R4 / .when extended by .0; 0/ to R3 /; Z   En;0 D %n;0 jun;0 j2 C H .%n;0 / dx ! E0 ; fn ! f in L1 \ L1 ..0; T /  R3 I R3 /: 

n

Then, extending .%n ; un / by .0; 0/ in .0; T /  .R3 n / and passing to a subsequence as the case may be, we have %n ! % in C .Œ0; T I L1 .R3 //; un * u in L2 .0; T I W 1;2 .R3 //; where .%; u/ is a bounded energy renormalized weak solution of the same problem on .0; T /   for initial conditions .%0 ; u0 /. Remark 2. 1. A sufficient condition guaranteeing (ii) is, for example, n n  bounded, j@n j D j@j D 0, and jn n j ! 0. 2. Existence of weak solutions on nonsmooth domains. Theorem 23 asserts existence of bounded energy weak solutions on a large class of nonsmooth bounded domains. These weak solutions are however not finite energy weak solutions. Finite energy weak solutions do not exist in general on nonsmooth domains, but they are known to exist on domains that are Lipschitz (or even slightly less regular than Lipschitz; see [73]). More exactly, the conclusion of Theorem 22 is valid under the same assumptions for bounded Lipschitz domains. To see this fact, one may approximate domain  by “larger” smooth domains n and construct the finite energy weak solutions .%n ; un / on domains n according to Theorem 22. Since  is Lipschitz, we obtain the crucial estimate H .%n / up to the boundary (in Lp ..0; T /  /, p > 1), thanks to the properties of the Bogovskii operator on Lipschitz domains (see Theorem 5 and its application exposed in item 6 of

1428

A. Novotný and H. Petzeltová

Sect. 7.1). This estimate suffices to pass to the limit in the differential form of the dissipation inequality (130). This is in sharp contrast with the case of a nonsmooth bounded domain when the Bogovskii operator provides only local estimates out of the boundary for the sequence %n . Under this circumstance one does not have almost everywhere convergence of sequence H .%n / up to the boundary, and one must use the lower weak semi-continuity and the weaker integral form (131) of the dissipation inequality for the limit passage. The reader can consult Kukucka [73], Poul [48, 94], and comments in [88, Section 7.12] for related material. 3. Existence of weak solutions on unbounded domains (case %1 ; u1 / D .0; 0/). Large class of unbounded domains (in particular, exterior domains, but many others) can be approximated by C 2; domains in the sense of convergence postulated in Theorem 23. Theorem 23 in combination with the existence Theorem 22 thus guarantees existence of bounded energy weak solutions to problem (63)– (64), (123) (124), (125) endowed with conditions at infinity (145) on an unbounded domain  in the class described in the above alinea, provided hypotheses of Theorem 22 are satisfied on , and f belongs additionally to L1 ..0; T /  /. Existence of finite energy weak solutions in this situation is not known. 4. Existence of (bounded) energy weak solutions on unbounded domains (case %1 > 0; u1 2 R3 ). If u1 D 0, the definition of the bounded energy weak solutions has to be changed as follows: (1) as far as the functional spaces, we must take % 2 L1 .0; T I L1loc .// \ Cweak .Œ0; T I L1 .K// (K any compact subset of /, u 2 L2 .0; T I D01;2 .I R3 //, %u 2 L1 .0; T I L1loc .I R3 // \ Cweak .Œ0; T I L1 .KI R3 //, and p.%/ 2 L1 .0; T I L1loc .//; (2) weak formulations to the continuity and momentum equations remain without changes (see (128), (129)); and (3) the dissipation inequality (131) must be replaced by Z  

 ˇ

ˇ %u2 C H .%/  H 0 .%1 /.%  %1 /  H .%1 / dx ˇ

0

Z



Z

C 0



S.rx u/ W rx u dxdt 

Z



(146)

Z %f  u dxdt:

0



Bounded energy weak solutions are known to exist on a large class of uniformly bounded Lipschitz domains, provided f 2 L1 \ L1 ..0; T /  I R3 / for the R  initial data with finite energy  %0 u20 C H .%0 /  H 0 .%1 /.%0  %1 /   H .%1 / dx. Existence of finite energy weak solutions (where the dissipation inequality (146) is replaced by its differential counterpart) is not known in this situation. The reader can consult [88] and [70] for more details and related material on unbounded domains in this situation.

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1429

The treatment when u1 ¤ 0 is slightly more involved. It is investigated in [88, Definition 7.78, Theorem 7.79] in the case of an exterior domain. 5. One can consider the same problem (63)–(64), (124) with the complete slip (22) or with the Navier slip (23) boundary conditions for the velocity (instead of uj@ D 0) on a bounded domain provided one modifies appropriately the definition of weak solutions. For example, in the case of Navier’s boundary conditions, the necessary modifications in the definition of finite energy weak solutions are the following: (1) In functional spaces (see formula (127)), one has to require u 2 L2 .0; T I W 1;2 .I R3 // and u  nj.0;T /@ D 0 in the sense of traces instead of u 2 L2 .0; T I W01;2 .I R3 //. (2) In the weak formulation of the momentum equation (129), one has to add to the right-hand side term RT R ƒ 0 @ u  'dSx dt and to consider test function ' 2 Cc1 .Œ0; T   /, RT R '  njŒ0;T @ D 0. 3) One has to add term ƒ 0 .t / @ juj2 dSx dt to the left-hand side of the dissipation inequality (130). Once these modifications are done, one can prove existence of finite energy weak solutions under the same assumptions on the regularity of the domain, initial data, external force, constitutive relations, and transport coefficients as in Theorem 22. The solutions constructed in this way enjoy all additional properties mentioned in Theorem 22. Also in this situation, any finite energy weak solution is also a bounded energy weak solution. The reader can consult [77], [88, Section 7.12.2], [33, Chapter 3] for related considerations. 6. Likewise one can consider finite (and bounded) energy weak solutions to the problem (63)–(64), (124) with periodic boundary conditions (i.e.,  is replaced by the periodic cell .Œ0; 1jf0;1g /3 (1 periodic torus)- with period 1 for simplicity. In this case, all function spaces entering into the definition of weak solutions are replaced by the functional spaces of (periodic) functions on the torus with the same regularity and integrability properties. Theorem 22 holds also in this situation. 7. The case of non-homogenous boundary conditions. The reasonable (and natural) definition of weak solutions of problem (63)–(64), (124) with nonzero inflow-outflow boundary conditions (

 )  %.t; x/ D %1 .t; x/ on [t2.0;T / ftg  in .t /g ; u.t; x/ D u1 .t; x/ on .0; T /  @

;

(147)

where in is the inflow part of the boundary, in .t / D fx 2 @ j u1 .t; x/  n.x/ < 0g; has been suggested in [88, Section 7.12.5]. Existence of this weak solution has been proved in Novo [85] (for  a ball and %1 , u1 constant) and in Girinon [62] (where the domain and boundary data can be more general, but the inflow boundary must be convex and contained in the cone and the inflow velocity

1430

A. Novotný and H. Petzeltová

must verify the so-called no-reflux condition). The general result without these limitations has been obtained recently in [12]. 8. Theorem 22 is true also for bounded two-dimensional domains provided  > 1. In the borderline cases ( D 3=2 for the three-dimensional domains and  D 1 for the two-dimensional domains), the main difficulty in proving the existence of weak solutions comes from the limit passage in the convective term (at least in two dimensions). The two-dimensional case has been solved only recently (see [92]); the three-dimensional case still resists. These problems are subject of a separate chapter of the handbook. 9. The progress within the framework of the Lions’ theory (with limitation   9=5) has been made also in another directions. It concerns the relaxation of certain hypotheses on the pressure (allowing more general nonmonotonicity than stipulated in (141)) and the relaxation in the conditions in the form of the viscous stress tensor (allowing small anisotropic perturbations of the stress tensor (12) in the case of constant viscosities (123)) (see D. Bresch and P.E. Jabin [9]). 10. Existence of time periodic solutions is subject of papers [46, 47]. Remark 3. 1. Sometimes, it may be convenient to use another representation of mechanical energy than the representation (137). To this end we introduce lower continuous convex function 8 1 q2 ˆ if r > 0; < 2 r e W R  R3 7! .1; 1; e.r; q/ D (148) 0 if .r; q/ D .0; 0/; ˆ : C1 if r  0; .r; q/ ¤ .0; 0/: We realize that under hypothesis (141), Z

%

p.z/  p.1/ C bz dz; B.%/ D % z2

H .%/ D A.%/ C B.%/; A.%/ D % 1

Z 1

%

p.1/  bz dz; z2

where A is convex continuous function on Œ0; 1/, jA.z/j  c.1 C z / and B is continuous on Œ0; 1/, B.z/  c.1 C zj ln zj/ with some c > 0, for all z 2 .0; 1/. We introduce mechanical energy Z

Z   e %.t; x/; %u.t; x/ dxC H .%.t; x// dx:

Œ0; T  3 t 7! Emech .t / D E.t / D 

Since have

R

1 2  2 %.t; x/u .t; x/

dx D

R 







(149)

e %.t; x/; %u.t; x/ dx for a.a. t 2 .0; T /, we

E.t / D E.t/ for almost all t 2 .0; T /:

(150)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1431

Moreover, according to theorem of lower weak semi-continuity of convex functionals in form formulated in Theorem 2, function Œ0; T  3 t 7! E.t / is lower semicontinuous function;

(151)

in particular E.0/ D E0  lim inf E.t /: t!0C

5

Dissipative Solutions, Relative Energy Inequality, and Weak-Strong Uniqueness Principle

5.1

Relative Energy and Relative Energy Functional

Let us now introduce the notion of the relative energy. We first introduce the relative energy function E W Œ0; 1/  .0; 1/ ! R; .%; r/ 7! E.%jr/ D H .%/  H 0 .r/.%  r/  H .r/;

(152)

where H is defined by (67). If the pressure verifies the monotonicity hypothesis p 0 .%/ > 0 for all % > 0;

(153)

the Helmoholtz function H is strictly convex on Œ0; 1/, and therefore E.%jr/  0 and

E.%jr/ D 0 , % D r:

In fact function E.j/ possesses better coercivity properties than stated above. This is subject of the following lemma whose proof is an easy application of the real analysis of functions of two variables. Lemma 3. Let 0 < a < b < 1 and let p 2 C Œ0; 1/ \ C 1 .0; 1/; p.0/  0;

p 0 .%/ > 0:

Then there exists a number c D c.a; b/ > 0 such that for all % 2 Œ0; 1/ and r 2 Œa; b,   E.%jr/  c.a; b/ 1Ores .%/ C %1Ores .%/ C .%  r/2 1Oess .%/ ; where E is defined in (152) and

(154)

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A. Novotný and H. Petzeltová

Oess D Œa=2; 2b; Ores D Œ0; 1/ n Oess :

(155)

In order to measure the “distance” between a weak solution .%; u/ of the compressible Navier-Stokes system and any other state .r; U/ of the fluid, we introduce the relative energy functional, defined by Z  ˇ  1 ˇ E.%; uˇr; U/ D %ju  Uj2 C E.% j r/ dx:  2

(156)

It appears that any (bounded energy) weak solution satisfies an inequality involving the relative energy functional called relative energy inequality regardless whether the pressure satisfies the thermodynamic stability condition. It is however to be noticed that the relative energy functional measures “a distance” between weak solution and any other state of the fluid only provided thermodynamic stability condition (153) is satisfied. This fact is formulated in the following theorem: Theorem 24. If .%; u/ is a weak solution to problem (63)–(64), (122)–(125) emanating from the finite energy initial data .%0 ; u0 / specified in (126) and external force f 2 L1 .QT I R3 /, then ˇ   ˇ Z ˇ ˇ E %; uˇr; U . /ˇ C 0

Z Z 0

0



Z

  S rx .u  U/ W rx .u  U/ dxdt 

(157)



Z Z Z Z   S rx U W rx .Uu/ dxdtC %@t U.Uu/ dxdt C %urU.Uu/ dxdt 0



Z



0



Z

Z





Z

r % @t p.r/ dxdt r   0 0 Z Z Z Z %  %f  .U  u/ dxdt rx p.r/  u dxdt   r  0 0



p.%/divU dxdt C

for a.a. 2 .0; T /, and for any pair of test functions r 2 C 1 .Œ0; T   /; r > 0; U 2 Cc1 .Œ0; T   I R3 /; Uj.0;T /@ D 0:

(158)

Remark 4. 1. Theorem 24 remains true if one replaces the Dirchlet boundary conditions (125) with the slip (22) or Navier’s conditions (23). In the latter case, weR have to add to the left-hand side of the relative energy inequality

R term ƒ 0 @ ju  Uj2 dSx dt, and the test functions .r; U/ must be taken in the class (158), where however condition Uj.0;T /@ D 0 must be replaced by U  nj.0;T /@ D 0 (see [50, Section 3.2.1]).

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1433

2. Theorem 24 remains valid if one replaces bounded domain with an unbounded domain and considers in addition conditions .%1  0; u1 D 0/ at infinity (cf. (24) and items 3, 4 in Remark 2). In this case the test functions .r; U/ must be taken in class (158), where r %1 2 Cc1 ..Œ0; T / (see [50, Theorem 2.4]). Theorem 24 has been formulated in [50] (see also [49]) under assumptions that p additionally complies with the assumptions (141) of the existence theory and satisfies the thermodynamic stability conditions. The proof from [50] can be repeated line by line without those additional assumptions. The reader can consult similar and more involved proof of Theorem 39 (dealing with the full Navier-Stokes-Fourier system). Under thermodynamic stability conditions, relative energy inequality becomes a powerful tool with many applications, in singular limit investigation [35, 36, 38, 52, 80, 100] and in numerical analysis [60], to name only a few. In what follows, we shall concentrate to the applications closely related to the problem of well posedness of weak solutions: weak-strong uniqueness principle and longtime behavior of weak solutions.

5.2

Dissipative Solutions

Inspired by Theorem 24, and following the philosophy of P.L. Lions [76] for the Euler equations (that can be traced back to Prodi [95] and Serin [98] in the case of incompressible Navier-Stokes equations), we define for the compressible Navier-Stokes equation the notion of dissipative solutions that is weaker than weak solutions. Definition 5. The couple .%; u/ is a dissipative solution of problem (63)– (64), (122)–(125) iff: (a) It belongs to class (127). (b) It satisfies relative energy inequality (157). Remark 5. 1. According to Theorem 24, under assumptions of the existence Theorem 22, problem (63)–(64), (122)–(125), admits at least one dissipative solution. 2. Any bounded energy weak solution .%; u/ to problem (63)–(64), (122)–(125) is a dissipative solution (regardless the thermodynamic stability condition and the asymptotic behavior of % 7! p.%/ for large values of %). The validity of the opposite statement is an open problem; it is not known whether any dissipative solution is a weak solution (even if condition (153) holds). 3. Under the hypotheses (141) of the existence theory (invoked in Theorem 22) and under the thermodynamic stability conditions (153), finite energy weak solutions satisfying relative energy inequality to system (63)–(64), (122)–(125) have been for the first time constructed in [49].

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5.3

Relative Energy Inequality with a Strong Solution as a Test Function

If the test functions .r; U/ in the relative energy inequality (157) obey equations (63)–(64) almost everywhere in QT , the right-hand side of the relative energy becomes quadratic in differences .%  r; u  U/. This observation is subject of the following lemma: Lemma 4. Let  be a bounded Lipschitz domain and f 2 L1 .QT /. Let .%; u/ be a weak solution to the Navier-Stokes equations with initial and boundary conditions (124)–(125). Let .r; U/ that belongs to the class U 2 L1 .0; T I L1 .//;

0 < r  r  r < 1I

(159)

@t r; @t U; rx r; rx U 2 L2 .0; T I L1 .//; be another (weak) solution of the same equations with initial data .r.0/; U.0// D .r0 ; U0 /. Then, under assumptions of Theorem 24, ˇ  ˇ Z ˇ ˇ E %; uˇr; U ˇ C 0

Z

0



Z

  S rx .u"  U/ W rx .u  U/ dxdt

Z



Z





Z

.r/.@t UCUrx U/.Uu/ dxdt C 

0

Z Z C 

0

(160)



.uU/rx U.Uu/ dxdt 0

rx p.r/ .r /.uU/ dxdt  r

Z Z  0



 p./p 0 .r/.r/p.r/ divU dxdt



for a.a. 2 .0; T /. Sketch of the proof. We deduce from regularity (159) and weak formulation of the momentum equation (129) that rx2 U 2 L2 .0; T I L1 .I R27 //. The couple .r; U/ is in fact a strong solution and satisfies momentum and continuity equations a.e. in QT : @t r C div.rU/ D 0 a.e. in .0; T /  ; r@t U C rU  rU C rp.r/ D divS.rU/ C rf a.e. in .0; T /  :

(161) (162)

The scalar product of (162) and u  U integrated over  yields Z 



Z

S.rU/ W r.u  U/ dx D 0;

r@t U C rU  rU  rf C rp.r/  .u  U/ dx C





where we have used the integration by parts in the last integral.

(163)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1435

Now we put together identity (163) and relative energy inequality (157). Formula (160) appears after a straightforward calculation. This finishes proof of Lemma 4.

5.4

Stability and Weak-Strong Uniqueness

We shall show here three versions of theorems on stability of strong solutions in the class of weak solutions and of weak-strong uniqueness theorems. In the first theorem, we shall require for the pressure solely the thermodynamic stability condition, while we shall suppose that the weak solution has density bounded from below and from above by positive constants. Theorem 25. Let   R3 be a bounded Lipschitz domain. Assume that the pressure p is twice continuously differentiable on .0; 1/ and verifies thermodynamic stability condition (153). Let .%; u/ be a weak solution to the Navier-Stokes equations (63)–(64), (124)– (125) emanating from initial data .%0 ; u0 / specified in (126) in the time interval Œ0; T /, T > 0 such that 0 < % < %.t; x/ < % < 1:

(164)

Let .r; U/ be a strong solution of the same equations in the regularity class (159), with initial data .r0 ; U0 / satisfying (126). Then Z  Z    1 1 %ju  Uj2 C j%  rj2 . / dx  c %0 ju0  U0 j2 C j%0  r0 j2 dx:  2  2 Theorem 25 has a drawback: it is a conditional result in the sense that it is not known whether one can construct global in time weak solutions satisfying the additional condition (164). In the second and third theorems, we require for pressure slightly more than the thermodynamic stability conditions. As a counterpart we can deal with bounded energy weak solutions without any additional assumptions. This allows us to get unconditional results. Theorem 26. Let  be a bounded Lipschitz domain. Suppose that pressure satisfies in addition to the thermodynamic stability condition (153) c1 C c2 % C H .%/  p.%/ for all %  R;

(165)

where R, c1 , c2 are some positive constants. Assume further that pressure belongs to the regularity class (122) and is twice continuously differentiable on .0; 1/ and that viscosities ,  verify (123). Assume that the external force f 2 L1 .QT ; R3 /. Let .%; u/ be a weak solution to the Navier-Stokes equations (63)–(64), (124)– (125) emanating from initial data .%0 ; u0 / specified in (126). Let .r; U/ be a strong

1436

A. Novotný and H. Petzeltová

solution of the same equations with initial data .r0 ; U0 / as in (126) that belongs to the class (159). Then there exists a positive number c (dependent on ; T; jj; diam; r; r; kpkC 2 .Œr=2;2r/ , kf; UkL1 .QT IR6 / , k@t U; rU; rrkL2 .0;T IL1 .IR15 // but independent of the weak solution itself) such that E.%; ujr; U/. /  cE.%0 ; u0 jr0 ; U0 /

(166)

for a.a. 2 .0; T /. In particular, if .%0 ; u0 / D .r0 ; U0 /, then % D r; u D U in Œ0; T   :

(167)

The third variant of the weak-strong uniqueness theorem is the following: Theorem 27. Conclusions (166)–(167) of Theorem 26 remain true if we replace the class of strong solutions (159) with the larger class 0 < r  r  r < 1;

U 2 L1 ..0; T /  /;

rx r 2 L2 .0; T I Lq .I R3 //; rx2 U 2 L2 .0; T I Lq .//; q > maxf3;

(168) 6 g: 5  6

and the hypothesis (165) by the stronger hypothesis (141) with  > 6=5. Remark 6. 1. One may verify by using the definition of Helmholtz function H that if pressure satisfies assumptions of Lemma 3 and condition 0
0; %!1 p1 % % %!1

then it satisfies condition (165). In particular, any pressure satisfying the thermodynamic stability condition (153) and assumption (141)>1 verifies condition (165). Consequently, weak solutions constructed in Theorem 22 verify the weak-strong uniqueness principle, provided the pressure is, in addition to the hypotheses in Theorem 22, twice continuously differentiable on .0; 1/ and verifies thermodynamic stability condition (153). 2. Under assumptions that  is a bounded domain of class C 4 , p 2 C 3 .0; 1/, f 2 L2loc .Œ0; 1/I W 2;2 .I R3 //, @t f 2 L2loc .Œ0; 1/I L2 .I R3 // and %0 2 W 3;2 ./, inf %0 > 0; u0 2 W 3;2 .I R2 / satisfying the compatibility condition at the boundary %10  rx p.%0 / C divS.rx u0 / C ˇ ˇ D 0; Valli [104, Theorem A] constructed a %0 f  %0 u0  rx u0 ˇ @ unique strong solution to problem (63)–(64), (123)–(125) in the regularity

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1437

class % 2 C .Œ0; TM /I W 3;2 .//; u 2 L2 .0; TM I W 4;2 .I R3 //; @t % 2 L2 .0; TM I W 2;2 .//; @t u 2 L2 .Œ0; TM /I W 2;2 .I R3 //; 0 < r inf.t;x/2.0;TM / %.t; x/ on a short time interval Œ0; TM / (dependent on the size of the initial data). This class is contained in class (159). This means that any weak solution emanating from Valli initial data coincides with the Valli strong solution at least on a (short) time interval Œ0; TM / provided pressure satisfies assumptions of Theorem 26 (and , f satisfy the Valli regularity hypotheses). 3. Under assumption p 2 C 1 Œ0; 1/,  bounded C 3 domain, %0 2 W 1;q ./, infx2 %0 > 0, u0 2 W01;2 \ W 2;2 ./, f 2 C .Œ0; 1/; L2 .I R3 // \ L2loc .Œ0; 1/I Lq .I R3 //, @t f 2 L2loc .Œ0; 1/I W 1;2 .I R3 //, q 2 .3; 6, Cho, Choe, Kim [14, Proposition 5] constructed a unique strong solution to problem (63)–(64), (123)–(125) in the regularity class % 2 C .Œ0; TM /I W 1;q .//, u 2 C .Œ0; TM /I W 2;2 .I R3 // \ L2 .0; TM I W 2;q .I R3 //, @t % 2 L2 .0; TM I Lq .//, p @t u 2 L2 .0; T I W01;2 .I R3 //, %@t u 2 L1 .0; T I L2 .I R3 // on a (short) maximal existence time interval Œ0; TM / (dependent on the size of initial data). Theorem 27 implies, in particular, that any weak solution emanating from the Cho, Choe, and Kim initial data coincides with the strong solution at least on the maximal existence time interval Œ0; TM / of the Cho, Choe, and Kim strong solution provided pressure satisfies hypotheses of Theorem 27 (and , f satisfy the Cho, Choe, Kim regularity hypotheses). 4. Under additional assumptions (141) with  > 1 and p 0 .%/ > 0, and if 
3=2 and (169) are satisfied, as long as the density component % of the weak solution remains bounded (see [49, Theorem 4.6]).

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A. Novotný and H. Petzeltová

Another consequence of the weak-strong uniqueness principle is the fact that the density in the weak solution must exhibit blowup before developing vacuum. More precisely, if all assumptions of the previous alinea are satisfied, and if density of the weak solution (that exists on the large interval .0; T /) verifies ess inf %. ; x/ D 0 for a certain 2 .0; T /; x2

then h i lim sup ess sup %.t; x/ D 1 t! 

x2

(see [49, Corollary 4.7]). 5. Theorems 26 and 27 hold with obvious modifications with slip boundary conditions (22) or with Navier’s boundary conditions (23) (see [50, Section 4.1.2]). It can be easily extended to a large class of unbounded domains with boundary conditions at infinity .%1  0; u1 D 0/ (see [50, Section 4.2.2 and Theorem 4.6]). Sketch of the proof of Theorems 25, 26, and 27. We shall outline here the main ideas of proof of Theorems 25, 26, and 27. The reader can find all complementary details in [50, Theorem 4.1].

5.4.1 Main Idea: The Gronwall Inequality The main idea is to use the relative energy inequality (157) with the strong solution .r; U/ of system (63)–(64), (124)–(125) in the form derived in Lemma 4. The goal is to find an estimate of the left-hand side of (160) from below by Z



c 0

ku  Uk2W 1;2 .IR3 / dt  c 0

Z

0

ˇ ˇ ˇ

ˇ ˇ ˇ E.%; uˇr; U/dt C E.%; uˇr; U/ˇ ; 0

(171)

and the right-hand side from above by Z



ku 

ı 0

Uk2W 1;2 .IR3 / dt

0

Z



C c .ı/ 0

ˇ ˇ a.t /E.%; uˇr; U/dt

(172)

with any ı > 0, where c > 0 is independent of ı, c 0  0, c 0 D c 0 .ı/ > 0, and a 2 L1 .0; T /. This process leads to the estimate Z ˇ ˇ ˇ ˇ E.%; uˇr; U/. /  E.%0 ; u0 ˇr.0/; U.0// C c

0

ˇ ˇ a.t /E.%; uˇr; U/dt;

(173)

which implies estimate (166) by the Gronwall inequality invoked in Theorem 21. In the rest of this section, we shall perform this program.

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1439

5.4.2 Bound from Below of the Dissipation By virtue of the Korn inequality invoked in Theorem 9 and the standard Poincaré inequality, Z

Z



ku 

c 0

Uk2W 1;2 .IR3 / dt



Z

S.r.u  U/ W r.u  U/ dxdt:

 0

(174)



5.4.3 Essential and Residual Sets We introduce essential and residual sets in . To this end we take in (155) a D r, b D r and define for a.e. t 2 .0; T / the residual and essential subsets of  as follows: ˇ ˇ (175) Ness .t / D fx 2  ˇ%.t; x/ 2 Oess g; Nres .t / D  n Ness .t /: With this definition at hand and having assumption (165) in mind, we deduce from Lemma 3 Z h i Z ˇ h i h i h i2  ˇ c E.%; uˇr; U/ dx (176) 1 C % C p.%/ C %  r dx  res



res

ess

res



with some c D c.r; r/ > 0, where we have set Œhess D h1Ness ; Œhres D h1Nres : for a function h defined a.e. in .0; T /  .

5.4.4

Estimates of the Right-Hand Side of Inequality (160) for Theorem 25 We observe that on essential set, Ness expressions E.rj%/ and .%r/2 are uniformly equivalent, meaning that there are c D c.r; r/ > 0 and c D c.r; r/ > 0, c.%  r/2  E.%jr/  c.%  r/2 whenever % 2 Ness ; r  r  r;

(177)

provided p 2 C 2 .0; 1/, regardless the structural properties of p near zero and infinity. Now, we split all integrals over  at the right-hand side of inequality (160) to the integrals over R R R R the essential R R sets Ness and residual sets Nres ; more precisely, we write D C 0  0 Ness .t/ 0 Nres .t/ . In the case of Theorem 25, all integrals over the residual sets are zero. (Indeed, we may suppose that r  %, r  %.) By virtue of the Cauchy-Schwarz inequality and Taylor’s formula, the upper bound of the integrals over the essential set is Z



Z

c 0

Ness .t/

1 2

2

2



%ju  Uj C j%  rj dx  c

0

Z

0

E.%; ujr; U/dt;

(178)

1440

A. Novotný and H. Petzeltová

where the last estimate holds due to (176). Implementing these observations into (160), we arrive at inequality (173) and conclude the proof of Theorem 25 by the Gronwall lemma (see Theorem 21) applied to (173).

5.4.5

Estimates of the Right-Hand Side of Inequality (160) for Theorems 26 and 27 The essential part of the right-hand side will be treated exactly as in the previous case. The structural assumptions of the pressure will play a role only for the estimates of the residual part of integrals at the right-hand side of inequality (160). Let us show a typical reasoning on the example of the first term of the right-hand side of (160) in the situation of Theorem 26. Recall that Nres D f%  r=2g [ f%  2rg. We shall estimate the integrals over the sets f%  r=2g and f%  2rg separately. Z



Z 1f%r=2g .  r/.@t U C U  rx U/  .U  u/ dxdt

0



Z



Z

 2r 0

Z



 2r Z

0

ı 0





ˇ ˇˇ ˇ ˇ ˇˇ ˇ 1res ˇ@t U C U  rx Uˇ ˇU  uˇ dxdt

    @t U C U  rx U

L1 .IR3 /

    1res 

L2 ./

Z

 2   u  U 2

L .IR3 /



dt C c.ı; r; r/ 0

    u  U

L2 .IR3 /

dt

ˇ   ˇ a.t /E %; uˇr; U dt;

where a D k@t U C U  rx Uk2L1 .IR3 / 2 L1 .0; T /, and Z



Z 1f%2rg .%/.%  r/.@t U C U  rx U/  .U  u/ dxdt

0



Z



Z

2 0

Z



 0



ˇp ˇ ˇ p ˇˇ ˇ ˇ ˇ Œ1res %ˇ@t U C U  rx Uˇ %ˇU  uˇ dxdt

    @t U C U  rx U

L1 .IR3 /

Z



 c.r; r/ 0

h i 1=2   2 1=2      %  1 % u  U  1 dt res L ./

L ./

ˇ  ˇ a.t /E %; uˇr; U dt 

with the same a as before. In all the above three formulas, we have employed (176) in the passage to their last lines. The remaining terms at the right-hand side of the relative energy inequality (160) may be estimated in a similar way. Finally, one gets estimate (173) and applies the Gronwall lemma invoked in Theorem 21. This finishes the sketch of the proof of Theorems 25, 26, and 27.

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

6

1441

Longtime Behavior of Barotropic Flows

In this section, results on the longtime behavior of weak solutions to the barotropic system (63)–(64) with homogenous Dirichlet boundary conditions (125) with viscosities (123) and pressure (122) are discussed. Further restrictions on the pressure, typically p.%/ D a% ; a > 0;

(179)

where   1 will be required later in most of statements starting from Sect. 6.2, mostly for the sake of simplicity. We do not restrict ourself to bounded domains. The barotropic model may be viewed as a special case of the Navier-StokesFourier system with constant temperature or with constant entropy, as described in Sect. 2.7. In this model the mechanical motion is completely separated from thermal effects. The simplified system (63), (64), when considered independently of the thermal energy equation (65), may feature rather different properties than the complete system. For instance, in contrast to the full system, it admits bounded absorbing sets for nonconservative forcing term, f 6 rF , or even nontrivial periodic solutions provided the driving force is time periodic, which is impossible in the full system in domains with thermally insulated boundary (see Sect. 12, Corollary 3, Remark 26 and compare with [51]). In Sect. 6.2, the large-time dynamics of weak solutions to the problem (63)– (64), (125) where the external force is a gradient of a scalar potential F , bounded and Lipschitz continuous on  will be discussed. Formally, the problem (63)–(64), (125) represents a gradient flow which admits a Lyapunov function – the total energy Z   1 EF .t / D %juj2 C H .%/  F % dx;  2 satisfying the energy inequality dEF C dt

Z   4 .  C /jruj2 C jdiv uj2 dx  0 3 

(180)

(see item 3 in Remark 1). Consequently, it is plausible to anticipate that, at least for some sequences tn ! 1, %.tn / ! %s ; %u.tn / ! 0; where %s is a solution to the corresponding stationary problem. Uniqueness of stationary solutions is discussed in Sect. 6.1.

1442

6.1

A. Novotný and H. Petzeltová

Uniqueness of Equilibria

In this section, static (equilibrium) solutions to the problem (63)–(64), (125) are examined in the case that the external force f is a gradient of a potential F which is assumed to be locally Lipschitz continuous on . The system reads Z % dx D m; (181) rp.%/ D %rF; %  0; 

where the parameter m represents the total mass conserved by the flow. Beirao Q da Veiga [1] obtained a necessary and sufficient condition for the existence of a strictly positive solutions of (181) expressed in terms of F and structural properties of p. It is easy to show that such a solution is necessarily unique. On the other hand, this restriction excludes an important class of solutions with vacuum states. The following theorem applies to any domain   Rn and a broad class of nonlinearities p. The uniqueness condition is expressed in terms of the upper level sets of the potential F , ŒF > k fx 2 I F .x/ > kg: Theorem 28 ([39, Theorem 2.1]). Let   Rn be an arbitrary domain. Suppose that pressure p satisfies condition (122) and thermodynamic stability condition (153). Let F be a locally Lipschitz continuous function on , and, in addition, suppose that the the upper level sets ŒF > k are connected in  for any k:

(182)

Then, given m > 0, there is at most one function % 2 L1 loc ./ satisfying (181). Moreover, if such a function exists, it is given by the formula %.x/ D Q1 .F .x/  k /

(183)

for a certain constant k , where

Q.z/ D

8R z ˆ < 0 ˆ :Rz 1

dp.s/ s

if P0 D

dp.s/ s

if

R1 0

R1 0

dp.s/ s

dp.s/ s

is finite

D C1:

Theorem 28 provides the following corollary: Corollary 1. Let p satisfy assumptions of Theorem 28; let P0 be finite, jj D 1, F  0, and Z Q1 .F .x// dx D m0 > 0 finite: 

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1443

Then there are no solutions of (181) with the mass m > m0 . Proof of Corollary 1. If there is such a solution %, then, by virtue of Theorem 28, it would hold %.x/ D Q1 .F .x/ C c/ with c > 0 and, consequently, Z

%.x/ dx  Q1 .c/jj D 1:



t u Remark 7. Two examples involving pressure (179) and different potential forces are given. 1. In the case that p.%/ D a% , the solution formula reads %.x/ D

1   1   1 ŒF .x/ C cC for a certain constant c 2 R: a

2. Let F be the gravity potential of a solid ball surrounded by a viscous gas, i.e., F .x/ D 

! ; x 2  D fx 2 R3 j jxj  rg jxj

for certain positive constants !; r. Consider 4 : 3

p.z/ D z ; 1 <  < A straightforward computation gives Q1 .z/ D

1    1   1 1 z  1 for z  0; 

and, consequently, Z

Q1 .F .x// dx D c. ; !/ 

Z

1

jxj  1 dx; 

where the last integral is finite provided 1 <  < 43 . Applying Corollary 1, it is possible to deduce the existence of a finite critical mass m0 for %, such that the problem (181) does not possess any solution for m > m0 . In such a situation, one

1444

A. Novotný and H. Petzeltová

can anticipate that any solution of the evolution problem (63), (64), (125) with the initial mass m > m0 should divide into two parts, one of which will converge to a stationary state and the other tending locally to zero. The importance of the assumption (182) is illustrated by the following statement: Theorem R 1 dp.s/ 29. Let p satisfy the hypotheses of Theorem 28. Assume P0 is finite, D 1, and there exists k such that the set ŒF > k has two disjoint bounded 1 s open components. Then there is m > 0 and a nonempty interval I such that the problem (181) admits a one-parameter family of solutions % ;  2 I satisfying Z % .x/ dx D m for all  2 I: 

Proof of Theorem 29. Consider the two disjoint components O1 ; O2 from the hypotheses of the theorem. As F is continuous, there exists 0 > k such that the function %0 .x/ D 1O1 Q1 .F .x/  0 / C 1O2 Q1 .F .x/  0 / is a solution of (181) with Z %0 .x/ dx D m > 0 

for a certain finite m. Using continuous dependence of the integral on parameters and monotonicity of Q1 , one can find a small interval I containing 0 and a nonincreasing function q W I 7! I such that % .x/ D 1O1 Q1 .F .x/  / C 1O2 Q1 .F .x/  q.//;  2 I are solutions of (181) satisfying Z % .x/ dx D m: 

t u The next result applies to the pressure p.%/ D a% . In Theorem 28, the solution is uniquely determined by its mass m. One can expect that, prescribing in addition the potential energy e Z 

a %  %F dx D e;  1

(184)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1445

the geometrical condition on the upper level set ŒF > k could be relaxed. This is really the case as stated in the following: Theorem 30 ([42, Theorem 1.2]). Let   RN be an arbitrary domain. Assume F is locally Lipschitz continuous function on , p.%/ D a% ;  > 1. Moreover, suppose  can be decomposed as  D 1 [ 2 ; 1 \ 2 D ;;

(185)

where 1 ; 2  RN are domains (one of them possibly empty) and that ŒF > k \ i is connected in i for i D 1; 2 and for any k 2 R:

(186)

Then, given m, e, the problem (181), (184) admits at most two solutions. The proof, where some elements of convex analysis are used, can be found in [42]. Remark 8. 1. Saying that % is a solution of (181), we require, in particular, all the integrals being convergent, i.e., % 2 L1 \ L ./; %F 2 L1 ./. 2. The previous results were generalized by Erban [24] for F locally Lipschitz continuous and bounded, p.%/ D a% ;  > 1. He showed that there exists critical mass m Q such that: • The system (181) has at most one solution for the mass m 2 Œm; Q 1/. • There is continuum of solutions of the system (181) for the mass m 2 .0; m/. Q Moreover, he defined a critical mass mc such that: • If m 2 Œmc ; 1/, then the stationary problem (181), (187) admits at most two solutions for each energy e 2 R. • If m 2 .0; mc /, then there exists an energy e 2 R such that the system (181), (187) has continuum of solutions. Some consequences of Theorem 28 with p.%/ D a% finish this section. Since the upper level sets ŒF > k are connected in , any solution of the stationary problem (181) with finite mass may be written in the form: %s .x/ D

1   1   1 ŒF .x/  kC ; a

where k is uniquely determined by the mass Z mŒ%s  D

%s dx: 

(187)

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A. Novotný and H. Petzeltová

The mass mŒ%s  considered as a function of the parameter k, mŒ%s  W R 7! Œ0; 1; is continuous nonincreasing. Moreover, clearly, mŒ%s .k/ D 0 for all k  sup F .x/; x2

and mŒ%s  is strictly decreasing on any open interval on which it is finite and strictly positive. We have the following assertion: Lemma 5. Let F be as in Theorem 28. Given m0  0, there exists a stationary solution %s such that Z 

%s dx  m0 ;

and % s  %s for any stationary solution %s such that Z %s dx  m0 :

(188)



Proof of Lemma 5. All stationary solutions are given by the formula (187). Take k D inffkj %s given by (187) satisfies (188)g and set %s .x/ D

1   1   1 : ŒF .x/  kC a

t u To conclude, consider the energy Z eŒ%s  D 

as a function of the parameter k.

a %  F %s dx:  1 s

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1447

Lemma 6. Let F satisfy the hypotheses of Theorem 28. Then the energy eŒ%s  is a nondecreasing function of k with values in Œ1; 0. Moreover, e is strictly increasing on any open interval on which mŒ%s  is finite and strictly positive. Proof of Lemma 6. Expressing %s by means of the formula (187), one has to observe that 1   1   1   1 a   1 F ŒF  kC ŒF  kC   1 a a 

k 7!

is a nondecreasing function of k which may be verified by a direct computation.

t u

Corollary 2. For F satisfying the hypotheses of Theorem 28 and E1 a given number, there is at most one stationary solution %s with finite mass and such that eŒ%s  D E1 :

6.2

Convergence to Equilibria

The aim of this section is to show that any weak solution converges to a fixed stationary state as time goes to infinity, more precisely, %.t / ! %s strongly in L ./;

p

%u.t / ! 0 strongly in ŒL2 ./3 as t ! 1;

under the two basic hypotheses: @ is Lipschitz and compact and the upper level sets satisfy (182): ŒF > k D fx 2 j F .x/ > kg are connected in  for all k: The above assumptions hold in many physically interesting cases, in particular in the situation when  is an exterior domain with spherical boundary and F is the gravitational potential, specifically,  D fx 2 R3 j jxj  Rg; F .x/ D 

! ; jxj

! > 0, modeling the motion of a viscous barotropic gas surrounding a star, considered in [82]. For the sake of simplicity, assume (179)>1 , i.e., p.%/ D a% . Further restrictions on values of  will be required later according to the investigated cases.

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Given a positive number m, the condition (182) is both necessary and sufficient for the stationary problem (181) to admit at most one weak solution %s uniquely determined by its mass Z mŒ%s  D

%s dx;

(189)



(cf. Theorem 28). On the other hand, the mass mŒ%.t / is a conserved quantity even for the weak solutions of the problem (63)–(64), (125) so one is tempted to believe the condition (189) picks up the right candidate to describe the large-time behavior of the density %. This is certainly true for bounded domains, while, if  is unbounded, such a conjecture is false, in general, due to possible “loss of mass at infinity” (cf. Remark 7). It seems interesting that for  bounded and a nonconstant potential F , there always exists m > 0 large in comparison with F such that the unique solution of (181) with the given mass m contain vacuum zones (cf. formula (187)). Thus for any nonconstant F , global solutions approach rest states with vacuum regions as time goes to infinity. We should remark in this context that there are many formal results on convergence of isentropic flows to a stationary state under various hypotheses including uniform (in time) boundedness away from zero of the density (see, e.g., [90]). As just observed, this could be rigorously verified only for solutions representing perturbations of strictly positive rest states (cf. [66, 81]). In particular, it is never true when the driving force rF is large in comparison with the total mass of the data. The main result of this section reads as follows: Theorem 31. Let   R3 be a domain with compact and Lipschitz boundary. Let the potential F is bounded and Lipschitz continuous on , and let the upper level sets ŒF > k be connected in  for any k < supx2 F .x/. Moreover, if  is unbounded, assume  lim ess supx2;jxjR jF .x/j C jrF .x/j/ D 0:

R!1

(190)

Finally, let p verify (179) with  > 3=2, namely, p.%/ D a% ; a > 0;  >

3 : 2

(191)

Then for any finite energy weak solution %; u of the problem (63), (64), (125) there exists a stationary state %s such that %.t / ! %s strongly in L ./;

p

%juj.t / ! 0 strongly in L2 ./ as t ! 1: (192)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1449

The proof consists of several steps: energy estimates, local and boundary estimates, compactness result, and, in the case of unbounded , also estimates at infinity. See [40] for details. p Remark 9. 1. Observe that the quantities % and %juj are continuous as functions  2 of t in the space L ./ and L ./, respectively, endowed with the weak topology, and, consequently, (192) makes sense. 2. The condition (191) seems restrictive from the physical point of view but natural for the mathematical treatment of the problem ensuring local integrability of the product terms appearing in the equations. In fact, such a condition is not necessary provided we know that % is bounded in Lq ./ uniformly in t for a certain 32 < q  1, in particular when the density is uniformly bounded as it is the case for radially symmetric data (cf. [82]). 3. As already mentioned, the mass mŒ%s  of the limiting solution may be strictly less than mŒ%.t / D m0 . Probably the simplest example is F D 0,  unbounded, when, according to Theorem 31, the density %.t / converges to zero in L ./. 4. Another example is furnished by item 2 in Remark 7, where 1 <  < 43 . As shown in the previous section, there is a critical mass m such that there is no solution of the stationary problem with a finite mass greater than m. Taking radially symmetric data, it can be shown that the density %.t / remains bounded uniformly in t ! 1 (see [82, Proposition 1]). In accordance with the Remark 8, Theorem 31 applies even though (191) is not satisfied, yielding convergence for any radially symmetric data. It is clear that the limit mass can never exceed m. Remark 10. The proof of Theorem 31 can be carried out without essential modifications in the following situations (see [40]): 1. If  is a bounded regular domain in R2 , the conclusion of Theorem 31 holds with the same condition (191) with  > 1: However, the case of an exterior domain exhibits some additional difficulties because of the lack of the Sobolev inequality for functions in W 1;2 .R2 /. 2. p is a general strictly increasing function of the density, p.z/  z for large z and p 0 bounded in a neighborhood of zero. Moreover, if  is unbounded, we need Z 0

1

p 0 .z/ dz finite: z

3. The viscosity coefficients ;  may depend on %; u, and a nonpotential and even time-dependent external force f may be added to rF provided it vanishes in a certain sense for large t.

1450

6.3

A. Novotný and H. Petzeltová

Bounded Absorbing Sets

In this part, globally defined finite energy weak solutions of the problem (63)– (64), (125) on a bounded Lipschitz domain , will be dealt with. More exactly, assume that %, u belong to the classes     1;2 3 C  2 C I L ./ ; u 2 L I .W / ./ ; % 2 L1 R R loc loc 0

(193)

the equations (63), (64) hold in D0 .RC  /, and the energy inequality d 4 EŒ%; u.t/C. C/ dt 3

Z

jru.t /j2 dxC

Z



jdivx u.t /j2 dx  

Z %.t /f.t /:u.t / dx 

(194)

is satisfied in D0 .RC /, where the energy EŒ%; u is given by the formula 1 EŒ%; u.t/ D 2

Z

a %.t /ju.t /j dx C   1  2

Z

% .t / dx:



The following result establishes the existence of an absorbing ball for any finite energy weak solution. Theorem 32 ([43, Theorem 1.1]). Let p satisfies (179) with >

5 ; 3

(195)

and let f be a bounded measurable function, n

ess

sup

o jf.t; x/j  K:

(196)

t2RC ; x2

Then there exists a constant E1 , depending solely on  , K and on the total mass m, having the following property: Given E0 , there exists a time T D T .E0 / such that EŒ%; u.t/  E1 for a.e. t > T

(197)

ess lim sup EŒ%; u.t/  E0 ;

(198)

provided

t!0C

and %, u is a (finite energy) weak solution of the problem (63)–(64), (123), (125), satisfying the hypotheses (193)–(194).

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1451

Remark 11. 1. Theorem 32 was proved in [43] under the additional assumption   % 2 L2loc RC I L2 ./ ;

(199)

which is satisfied provided   9=5 (see [41]). In fact, the condition (199) is not necessary in the proof of Theorem 32; it is sufficient to have estimates of the form (262), which are valid for the pressure satisfying (195). Note that estimate of pressure in Lp up to the boundary (whose main ideas are presented in item 6 of Sect. 8.2; see also “pressure estimates” in [44, Section 4.2]) is one of the prerequisites to obtain energy inequality in the differential form, and due to this reason, it constitutes one of the building blocks of the proof of Theorem 32. 2. In agreement with item 3 of Remark 1, the instantaneous values E D instŒEŒ%; u (defined in Sect. 3.2) satisfy inequality (131) everywhere in RC , and consequently inequality (197) is valid for any t > T , provided one replaces EŒ%; u by its instantaneous value E. The proof of Theorem 32 is based on the following Lemma and Proposition. Lemma 7. Assume f satisfies (196). Let %, u belong to the classes (193), (199) and comply with the energy inequality (194). Then, being redefined on a set of measure zero if necessary, the (instantaneous value of) energy E has locally bounded variation on RC , and E.t C/ D lim E.s/  lim E.s/ D E.t / for any t 2 RC : s!t

s!tC

(200)

Moreover,   p E.t2 /  1 C E.t1 C/ e 2mK.t2 t1 /  1 for all 0 < t1 < t2 :

(201)

Sketch of the proof of Lemma 7. It follows from the energy inequality (194) – see item 3 in Sect. 3.2 and item 3 in Remark 1 – that E can be written as a sum of a nonincreasing function and an absolutely continuous one, and, consequently, E is continuous except a countable set of points in which (200) holds. By virtue of (196), the right-hand side of (194) may be estimated as follows: Z %f:u dx  K 

Z 

% dx

 12  Z

%juj2 dx

 12



p

2mK.1 C E/;



whence (201) is a straightforward consequence of the Gronwall lemma. The following assertion plays a crucial role in the proof of Theorem 32.

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Proposition 1. Under the hypotheses of Theorem 32, there exists a constant L, depending solely on  , K, and m, enjoying the following property: If E..T C 1// > E.T C/  1 for a certain T 2 RC ;

(202)

then sup

E.tC/  L:

t2.T;T C1/

The proof of this proposition is carried over by a series of auxiliary results (see [43, Proposition 3.1]). Sketch of proof of Theorem 32. With Lemma 7 and Proposition 1 at hand, Theorem 32 can be proved. To begin, observe there exists T D T .E0 / such that E.t0 C/  L for a certain t0 < T; where L is the constant from Proposition 1. Indeed, if it was not the case then, by virtue of Proposition 1, the energy would become negative. Next claim is that for any integer n  0 E..t0 C n/C/  L: By induction, assume E..t0 C n/C/  L: By Proposition 1, either sup

E.tC/  L;

t2.t0 Cn;t0 CnC1/

and, consequently, E..t0 C n C 1//  L; or E..t0 C n C 1/C/  E..t0 C n C 1//  E..t0 C n/C/  1  L  1: Finally, by virtue of Lemma 7 and (203), take p

E1 D .1 C L/e This completes the proof of Theorem 32.

2mK

 1:

(203)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

6.4

1453

Existence of Attractors

In this part, results from the publication [29, Sections 3–5] are presented. Throughout this section, assume 8 9 5 < p.%/ D a% ;  > 3 ;  is a bounded Lipschitz domain, = :

f 2 F; where F denotes a bounded subset of L .R  /: 1

(204)

;

First, observe that the finite energy weak solution satisfies % 2 Cweek .Œ0; T I L .//; q %u 2 Cweek .Œ0; T I Lp .// with p D

2 ;  C1

and, moreover, the fact that the continuity equation holds in D0 .Œ0; T   R3 / makes it possible to employ the regularizing machinery in the spirit of DiPerna and Lions [18] to deduce % 2 C .Œ0; T /I L˛ .// for any 1  ˛ <  ; cf. Theorem 20 in Sect. 3.11. These relations enable to justify the observation that .%u/.t; x/ D 0 for a.e. x 2 V .t/ D fxI %.t; x/ D 0g for any t 2 Œ0; T ; (cf. item 2 in Remark 1). Now, redefining the total energy on a set of measure zero if necessary, set 1 EŒ%; %u.t/ E.t / D 2

Z %.t/>0

j.%u/j2 a .t / dx C %  1

Z

% dx;

(205)



where t 7! E.t / is lower semicontinuous function on RC (cf. Remark 3). The first result deals with complete bounded trajectories, i.e., the finite energy weak solutions defined on the whole line R whose energy is uniformly bounded on R. Their importance is shown in Proposition 2. Denote  F C D fI f D lim hn .: C n / weak star in L1 .R  / for a certain hn 2 F and n ! 1 :

n !1

We introduce an analogue of the so-called short trajectory in the spirit of [78]. n U s ŒE0 ; F.t0 ; t / D Œ%. /; q. /; 2 Œ0; 1I %. / D %.t C /; q. / D .%u/.t C /; where %; u is a finite energy weak solution of the problem (63)–(64), (125) on an open interval I; o .t0 ; t C 1  I; with f 2 F; and such that lim sup E.t /  E0 E.0/ : t!t0

1454

A. Novotný and H. Petzeltová

Proposition 2. Assume Œ%n ; qn  2 U s ŒE0 ; F.t0 ; tn / for a certain sequence tn ! 1. Then there is a subsequence (not relabeled) such that %n ! % in L ..0; 1/  / and in C .Œ0; 1I L˛ .// for 1  ˛ <  ; 2

qn ! .% u/ in Lp ..0; 1/ / and in Cweek .Œ0; 1I L  C1 .// for any 1  p
k are connected for any k 2 R. Moreover, let .f.: C /  rF / ! 0 weakly in L1 ..0; 1/  / as ! 1:

(210)

Then any finite energy weak solution %; u of the problem (63)–(64), (123), (125), (179) on I D RC satisfies %.t / ! %s in L ./; and the kinetic energy

1 2

Z %>0

jqj2 dx ! 0 as t ! 1; %

where %s is the unique solution of the stationary problem (181). The convergence in (210) is very weak. It requires only that integral means taken with respect to space and time approach a potential driving force. In other words, both the density and the momenta are robust with respect to possible random fluctuations of the driving force both in space and time. Finally, we discuss the dependence of the attractor on the driving force f. The result, in the case of a perturbation of a potential force rF satisfying (182), may be formulated as follows: Theorem 37. Let the assumptions of Theorem 36 be satisfied. Fix ˛ 2 Œ1;  /. Then given any " > 0, there exists ı > 0 such that lim sup k%.t /  %s kL˛ ./ < " t!1

whenever lim sup kf.t /  rF kL1 ./ < ı t!1

for any density component % of a finite energy weak solution of the problem (63)– (64), (123), (125), (179) with the driving force f measurable and bounded on RC . Here %s is the unique solution of the stationary problem (181). The proof, similarly as the proof of Theorem 33, follows from the compactness property stated in Proposition 2.

7

Navier-Stokes-Fourier System in the Internal Energy Formulation

7.1

Definition of Weak Solutions

In this section we shall deal with the Navier-Stokes-Fourier system (3)–(5) with the stress tensor and heat flux given by (12)–(13) and with the pressure and internal

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1457

energy obeying (34)–(37), where the molecular pressure pmo satisfies (45). The material of this section is mostly taken from [30]. In this situation, one can use identity (44) in order to rewrite the internal energy conservation in the simplified form     @t % eth .#/ C era .%; #/ C divx %u eth .#/ C era .%; #/

(211)

  C divx q.%; #; rx #/ C #pth .%/ C pra .#/ divx u D S.%; #; rx u/ W rx u: The right hand of the above identity contains the positive term S.%; #; rx u/ W rx u which will give rise in the weak formulation to the functional of type rx u 7! RT R  S.%; #; rx u/ W rx u dxdt. This functional cannot certainly be continuous, 0 but can be solely lower weakly semicontinuous with respect to the weak topology of the space L2 .QT I R9 /. Therefore, we must replace in the weak formulation of equation (211) the equality sign by the inequality sign “.” In order to compensate the lack of information caused by this operation we add to the weak formulation of the system the total energy balance (26) with sign “.” This motivates the following definition of weak solutions that we shall formulate for the heat flux of a specific form

q D  .#/rx # D rx K.#/; where K.#/ D

Z

#

.z/dz:

(212)

0

Definition 6. Let  be a bounded domain, and let the initial conditions .%0 ; u0 ; #0 / satisfy %0 W  ! Œ0; C1/; u0 W  ! R3 ; #0 W  ! .0; 1/;

(213)

where %0 u0 D 0 and %0 u20 D 0 a.e. in the set fx 2 j%0 .x/ D 0g R 1 2 with finite energy E0 D R  . 2 %0 u0 C Hel .%0 / C %0 eth .#0 / C %0 era .%0 ; #0 //dx and finite mass 0 < M0 D  %0 dx. We shall say that a trio .%; #; u/ is a weak solution to the Navier-Stokes-Fourier system (3)–(5) with boundary conditions (20)–(21), with viscous stress and heat flux (12)–(15), (212), and with pressure and internal energy (34)–(37), where pmo obeys (45), emanating from the initial data .%0 ; #0 ; u0 ; / if:

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% 2 L1 .0; T I L1 .//; # 2 L1 .QT /; %  0; # > 0 a.e. in .0; T /  ; (214) 1 2 1;2 2 1 1 u2L .0; T IW0 .//I %u; %u ; Hel .%/; %.eth .#/Cera .%; #//2L .0; T IL .//; 2

(a) 1

#.pth .%/ C pra .#//; S.%; #; rx u/ W rx u; K.#/ 2 L1 .QT /: (b) % 2 Cweak .Œ0; T I L1 .//, and the continuity equation (3) is satisfied in the following weak sense Z 

Z ˇ

ˇ %'dx ˇ D 0

0

Z 

 %@t ' C %u  rx ' dxdt;

(215)



for all 2 Œ0; T  and for all ' 2 Cc1 .Œ0; T   /: (c) %u 2 Cweak .Œ0; T I L1 .//, and the momentum equation (4) is satisfied in the weak sense, Z 

Z Z  ˇ

 ˇ %u@t 'C%u˝uWr'Cp.%; #/div'S.rx u/ W rx 'C%f' dxdt D 0 %u'dx ˇ D 0

0



(216)

for all 2 Œ0; T  and for all ' 2 Cc1 .Œ0; T   I R3 /: (d) Balance of thermal energy (211) is satisfied as an inequality Z h  Z  i  % eth .#/ C era .%; #/ . / dx  %0 eth .#0 / C era .%0 ; #0 / dx 



Z Z      % eth .#/ C era .%; #/ @t ' C % eth .#/ C era .%; #/ u  rx ' C K.#/'  0



    #pth .%/ C pra .#/ divx u' C S.%; #; rx u/ W rx u' dxdt

(217)

for a.a. 2 .0; T / and for all ' 2 Cc1 .Œ0; T I C 2 .//, rx '  nj.0;T /@ D 0, '  0. (e) The balance of the total energy (26) is satisfied in the weak sense as inequality Z

T

 0

for all

0

Z  Z T Z   1 %juj2 C Hel .%/ C % eth .#/ C era .%; #/ dxdt  .t / .t / %f  u dxdt  2  0 Z    1 (218) %0 ju0 j2 C Hel .%0 / C %0 eth .#0 / C era .%0 ; #0 / dx C .0/  2 2 Cc1 Œ0; T /,

Z We recall that 

 0.

gdx j 0 means

Z

Z g. ; x/dx 



g0 .x/dx. The Helmholtz function 

Hel is defined in (67), and the space Cweak .Œ0; T I L1 .// is defined in (71).

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1459

Definition 7. Weak solution whose density-velocity component .%; u/ satisfies the continuity equation in the renormalized sense (116)–(117) in D0 .QT / with f D 0, with any test function b belonging to (118) is called renormalized weak solution. Remark 12. 1. According to (88) the total energy balance formulation (218) implies Z    ˇ Z Z 1 ˇ %juj2 C Hel .%/ C % eth .#/ C era .%; #/ dx ˇ  %f  udxdt; 0 2 0   (219) for almost all 2 .0; T /. 2. According to (85) applied to the thermal energy conservation (217), the right and left instantaneous values Œ%eth .#/ C %era .%; #/. C/ and Œ%eth .#/ C %era .%; #/. / defined in (68)–(69) are continuous linear functionals on C ./ satisfying Œ%eth .#/ C %era .%; #/. C/  Œ%eth .#/ C %era .%; #/. /:

(220)

3. We deduce from (85) (with  D 1) applied to the thermal energy balance (217) that the function of instantaneous values of thermal energy hZ   i Œ0; T  3 ! 7 Eth . / inst % eth .#.; x// C era .%.; x/; #.; x// dx . / 

(221)

is a sum of an absolutely continuous function and a nondecreasing function (with at most countable number of jumps). 4. Likewise, according to (86)–(89) applied to (218), the function of the instantaneous values of total energy of the weak solution h Z 1  i Œ0; T  3 7! E. / inst %u2 .; x/C%e.%; #/.; x/CHel .%.; x// dx . /  2 (222) is a sum of an absolutely continuous function and a nonincreasing function (with a countable number of jumps). It seems that a significant piece of information is lost when replacing the internal energy equation (5) by the variational inequality (217). However, to compensate this loss, we require that the weak solution obeys the total energy inequality (218). This makes from Definitions 6 and 7 “good” definitions. Indeed, any sufficiently regular weak solution is a classical solution as stated in the following lemma whose proof can be found in Feireisl [30, Section 6].

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Lemma 8. Let the trio .%; #; u/ be a weak solution to problem (3)–(9) with the same constitutive laws for pressure, internal energy, stress tensor, and heat flux as in Definition 6, with boundary conditions (20)–(21) and initial conditions .%0 ; #0 ; u0 / verifying (213) on a Lipschitz bounded domain  in the regularity class .%; #; u/ 2 C 1 .QT /  C 1 .QT / \ C .Œ0; T I C 2 .//  C 1 .QT I R3 / \ C .Œ0; T I C 2 .I R3 // % > 0; # > 0: (223) Then .%; #; u/ is a classical solution to the Navier-Stokes-Fourier system. In particular, it satisfies all energy balance laws (5), (211), (7), (9)–(10) as identities on QT .

7.2

Existence of Weak Solutions

We start by specifying the assumptions under which the existence theorem on weak solutions will be investigated. We shall consider the flow without radiation (i.e., pra D 0, era ) for which the present weak formulation is more appropriate. The reader is invited to confront these assumptions with the physically motivated requirements (34)–(42), (45)–(46), (56)–(57), (16)–(18): (1) Pressure and internal energy. p.%; #/ D pel .%/ C #pth .%/; e.%; #/ D eel .%/ C eth .#/;

(224)

where pel is the same as in the barotropic case, namely, 

1

pel 2 C Œ0; 1/ \ C .0; 1/; pel .0/ D 0;

pel .%/  a1 % C b; pel0 .%/  a2 %1  b;

(225)

for some   1; a1 ; a2 ; b > 0, 0 pth 2 C Œ0; 1/ \ C 1 .0; 1/; pth .0/ D 0; pth .%/  0; pth .%/  c.1 C % / for some 0  :

(226)

In agreement with (45), the thermal energy is given by Z eth .#/ D

#

cv .z/dz; cv 2 C 1 Œ0; 1/;

0 ˛

inf cv .z/ cv > 0;

z2Œ0;1/

cv .#/  c.1 C # 2 1 / where c > 0; ˛  0:

(227)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1461

In agreement with (42), elastic energy is calculated from the elastic pressure pel through the formula Z % pel .z/ %eel .%/ Hel .%/ D % dz: (228) z2 1 (2) Viscous stress and heat flux. The fluid is Newtonian with the viscous stress given by (12) with the constant viscosity coefficients  > 0;   0:

(229)

Heat flux is given by the Fourier law (212), where 2C 2 Œ0; 1/; c1 .1C# ˛ / .#/  c2 .1C# ˛ /; with constants c1 ; c2 > 0, and ˛  0: (230) Under the above assumptions, the Navier-Stokes-Fourier systems admits a weak solution provided the constants  , , and ˛ verify some further restrictions. This statement is subject of the following theorem reported from [30, Theorem 7.1]. Theorem 38. Let   R3 be a bounded domain with boundary of class C 2; ,  > 0. Suppose that pressure, internal energy, viscous stress tensor, and heat flux satisfy assumptions (224)–(230) with  > 3=2; 0   =3; ˛  2: Then the Navier-Stokes-Fourier system (3)–(5) with boundary conditions (20)– (21) and initial conditions (213) with ess inf #0 .x/ > 0 x2

admits a renormalized weak solution with the following additional properties: % 2 C .Œ0; T I L1 .// \ L1 .0; T I L .// \ Lp0 .QT /; p0 D minf

5  3 3 ;  g; 3 2 (231)

pel .%/ 2 Lp1 .QT /; #pth .%/ 2 L2 .QT /; p1 D p0 = > 1; 2

2

%u 2 L1 .0; T I L  C1 .// \ Cweak .Œ0; T I L  C1 .//; # 2 L˛C1 .QT /; 2

(232)

# ˛ ; Œeth .#/ ˛ ˛ 2 L2 .QT / for all ˛ 2 Œ0;

(233) (234)

˛C1 ; ln # 2 L2 .QT /; 2

(235)

6

%eth .#/ 2 L1 .0; T I L1 .// \ L2 .0; T I L  C6 .//;

(236)

1462

A. Novotný and H. Petzeltová

Z ess lim

t!0C 

Z %eth .#/.t; x/.x/ dx D 

%0 eth .#0 / dx;  2 Cc1 ./:

(237)

There exists #Q 2 L2 .0; T I W 1;2 .// such that # D #Q a.e. in f.t; x/j%.t; x/ > 0g: (238)

8

Main Ideas of the Proof of Theorem 38

As in the case of the “simple” barotropic situation, the main issue in the proof of the existence theorem is the understanding of the propagation of the density oscillations. This phenomenon is coupled with the the thermal energy balance and gives rise to further difficulties linked especially to the vanishing density. In fact in the context of weak solutions, we cannot avoid the formation of vacuum regions of nonzero Lebesgue measure. Rather than existence, we shall prove the weak stability of the set of (sufficiently smooth) weak solutions. We shall formulate this property in the subsequent Lemma 9. The proof of this lemma will contain already all main ingredients of the proof of the existence theorem. The reader should however be aware that even after Lemma 9 is established, the construction of solutions remains a hard and tricky job with great amount of difficulties. The construction of weak solutions to this problem goes far beyond the scope of the handbook. There are so far two methods available in the mathematical literature: (1) a functional analytic method based on several levels of approximations by partial differential equations involving several (small) parameters similar to the one reported through (402)–(415), whose details can be found in [30, Chapter 7], and (2) numerical method based on the finite volumes/finite element approximations whose details can be found in [55]. This method needs a further restriction on the adiabatic coefficient  , namely,  > 3. Lemma 9. Let   R3 be a bounded domain with boundary of class C 2; ,  > 0. Suppose that pressure, internal energy, viscous stress tensor, and heat flux satisfy assumptions (224)–(230) with  > 3=2, 0   =3, and ˛  2. Let .%n ; #n ; un / in the regularity class (223) be a sequence of finite energy renormalized weak solutions to problem (3)–(5) with boundary conditions (20)–(21) and initial conditions .%n;0 ; #n;0 ; un;0 / satisfying %n;0 * %0 in L1 ./; %n;0 un;0 * %0 u0 in L1 .I R3 /; %n;0 eth .#n;0 / * %0 eth .#0 / in L1 ./; Z   1 %n;0 jun;0 j2 C %n;0 eth .#n;0 / C Hel .%n;0 / dx  2 Z   1 ! %0 ju0 j2 C %0 eth .#0 / C Hel .%0 / dx;  2

(239)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1463

with bounded from below entropy Z

%n;0 s.%n;0 ; #n;0 / dx  S 2 R;



where .%n;0 ; #n;0 ; un;0 / and .%0 ; #0 ; u0 / verify (213) with Mn;0 > 0; En;0 2 R, and M0 > 0, E0 2 R, respectively. Then there exists a subsequence (denoted again .%n ; #n ; un /) such that %n ! % weakly-* in L1 .0; T I L .//;

(240)

3 5 where % 2 C .Œ0; T I L1 .// \ Lr .QT /; 0 < r  minf  ;   1g; 2 3 un ! u weakly in L2 .0; T I W01;2 .I R3 //; 2

2

%n un !%u weakly-* in L1 .0; T I L  C1 .I R3 //; where %u2Cweak .0; T I L  C1 .//; K! .#n / * ‚! as n ! 1 (weakly) in L1 .QT /; ‚! ! ‚ a.e. in QT as ! ! 0C; with K! .#/ D

Z

#

h! .z/ .z/dz; h! .z/ D 0

1 ; .1 C z/!

(241)

where the trio .%; # D K1 .‚/; u/ is a renormalized weak solution of (3)–(5) with boundary conditions (20)–(21) and initial conditions .%0 ; #0 ; u0 /. Remark 13. 1. It should be noticed that the gradient of the temperature component # of the weak solution is not square integrable, as one would expect from the presence of dissipation in the thermal energy balance. One can show that there is #Q 2 L2 .0; T I W 1;2 .// such that #n * #Q in L2 .0; T I W 1;2 .// coinciding with the temperature component # of the weak solution almost Q D K.#/ almost everywhere outside the vacuum set. Consequently, K.#/ Q ¤ K.#/ in a subset everywhere outside vacua; however it may happen that K.#/ of the vacuum set with the nonzero measure.

1464

8.1

A. Novotný and H. Petzeltová

Equations Verified by the Sequence

By virtue of Lemma 8, the trio .%n ; #n ; un / satisfies equations (3), (4), (5), (211)era D0 , (7), (9)–(10) together with boundary conditions (20)–(21). In particular, @t %nCdivx .%n un /D0 in Œ0;T R3 provided .%n ;un / is extended by .0; 0/ outside ; (242) @t .%n un /Cdivx .%n un ˝un /Crx p.%n ; #n / D divx S.rx un / D %n f in QT ; @t .%n eth .#n // C divx .%n un eth .#n //

(243) (244)

Cdivx q.rx #n / C #n pth .%n /divx un D S.rx un / W rx un in QT ;  @t .%n s.%n ; #n // C divx .%n s.%n ; #n /un / C divx

1 D #n d dt



q.#n ; rx #n / #n

q.#n ; rx #n /  rx #n S.rx un / W rx un  #n

 (245)

 in QT ;

Z Z   1 2 %n f  un dx for all t 2 Œ0; T ; %n jun j C %n eth .#n / C Hel .%n / dx D  2  (246)     @t b.%n / C divx b.%n /un C %n b 0 .%n /  b.%n / divx un D 0; b as in (134); (247) in Œ0; T   R3 provided .%n ; un / is extended by .0; 0/ outside :

This implies     @t Tk .%n / C divx Tk .%n /un C %n Tk0 .%n /  Tk .%n / divx un D 0;

(248)

  @t %n Lk .%n / C divx %n Lk .%n /un C Tk .%n /divx un D 0

(249)

and

in Œ0; T   R3 provided .%n ; un / is extended by .0; 0/ outside ;

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1465

where Z

z

Tk .z/ D kT .z=k/; Lk .z/ D 1

T 2 C 1 Œ0; 1/;

8.2

T .z/ D

8 ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ :

Tk .w/ dw; w2

(250)

z if z 2 Œ0; 1; concave on Œ0; 1/; 2 if z  3:

A Priori Estimates

1. Bounds due to the mass conservation. Integrating equation (242) yields Z

Z %n . / dx D



%n;0 dx; 

in particular k%n kL1 .0;T IL1 .//  c.M0 /:

(251)

2. Bounds due to the global energy conservation. Balance of total energy in the volume  (246) (that is equation (7) integrated over ) yields Z   1 %n jun j2 C %n eth .#n / C Hel .%n / . / dx  2 Z  Z Z  1 D %n f  un dxdt: %n;0 jun;0 j2 C %n;0 eth .#n;0 / C Hel .%n;0 / dx C 0  2  Recalling definition of Hel (42) and (225), we verify that cR1 %R  Hel .%/Cc2 .1C

% ln %/ with c1 ; c2 > 0 (dependent on a1 ; a2 ; b). Further j 0  %n f  un dxdt j  qR R qR 2 kfkL1 .Q IR3 / 0  %n dx  %n .un / dx by virtue of the Cauchy-Schwarz inequality. Employing these facts, the Gronwall lemma (see Theorem 21) and assumptions (227) on the form of eth , we derive from the last center-lined identity the bounds k%n kL1 .0;T IL .//  c.M0 ; E0 ; F0 ; T /;

(252)

1466

A. Novotný and H. Petzeltová

k%n eth .#n /kL1 .0;T IL1 .// ; k%n #n kL1 .0;T IL1 .//  c.M0 ; E0 ; F0 ; T /; k%n jun j2 kL1 .0;T IL1 .//  c.M0 ; E0 ; F0 ; T /;

(253) (254)

where here and hereafter, we denote F0 kfkL1 .QT IR3 / : 3. Bounds due to the entropy balance. Entropy balance (245) integrated over the space time cylinder Q , while taking into account the boundary conditions (20), (21), yields Z

0

Z  1 jrx #n j2  dxdt S.rx un / W rx un C .#n / .#n /2  #n Z Z %n s.%n ; #n /. / dx  %n;0 s.%n;0 ; #n;0 / dx; D 



where the specific entropy s.%; #/ D smo .%; #/ D smo;# .#/ C smo;% .%/ is given by formula (46). Employing (46) and assumptions (226), (227) we find pointwise estimates smo;# .#/1f#1g .#/ c.1 C eth .#//1f#1g .#/; smo;# .#/1f# 6=5. Employing moreover the compact imbedding Lq ./ ,!,! W 1;2 ./, we get the convergence of these quantities in L2 .0; T I W 1;2 .//. Summarizing the above, we have %n ! % in Cweak .Œ0; T I L .// and in L2 .0; T I W 1;2 .//; b.%n / ! b.%/ in Cweak .Œ0; T I Lq .//; and in L2 .0; T I W 1;2 .//; provided b 2 C Œ0; 1/ \ C 1 .0; 1/; b.%n / bounded in L1 .0; T I Lq .//; %n b 0 .%n /  b.%n / bounded in L2 .QT //; %n un ! %u in Cweak .Œ0; T I L2=.C1/ .I R3 // and in L2 .0; T I W 1;2 .; R3 //; %n un ˝ un ! %u ˝ u in L2 .0; T I L6=.4C3/ .; R9 //: (269) The second relation in (269) employed with b D pth in combination with the second relation in (267) yields, in particular, #n pth .%n / * #Q pth .%/ in L2 .QT /; #n pth .%n /Tk .%n / * #Q pth .%/Tk .%/ in L2 .QT /:

(270)

1472

A. Novotný and H. Petzeltová

Now, we are ready to let n ! 1 in equations (242), (243), and (247). We get, in particular, Z

Z

Z

%. ; x/'. ; x/ dx 

T

%0 '.0; / dx D 

Z 

 %@t 'C%urx ' dxdt

(271)



0

for all 2 Œ0; T  and any ' 2 Cc1 .Œ0; T   /; Z

Z %. ; x/'. ; x/ dx  

Z

T

D

%0 u0  '.0; / dx

(272)



Z 

 %u  @t ' C %u ˝ u W rx ' C p.%; #/divx '  S.rx u/ W rx ' C %f  ' dxdt



0

for all 2 Œ0; T  and for any ' 2 Cc1 .Œ0; T   I R3 /, 'j@ D 0; Z Z %Lk .%/. ; x/'. ; x/dx  %0 Lk .%0 /'.0; / dx 

Z



Z

 0

(273)



Z   %Lk .%/ @t ' C u  rx ' dxdt D 



T

Z Tk .%/divu' dxdt; 

0

and Z

Z

Z

Tk .%/. ; x/'. ; x/dx  



Z



0

Z

D 0



Z

Tk .%0 /'.0; /dx 



.%Tk0 .%/

  Tk .%/ @t ' Curx ' dxdt



(274)

 Tk .%//divu ' dxdt;

where 2 Œ0; T  and ' 2 Cc1 .Œ0; T   / and functions Tk , Lk are defined in (250).

8.4

Effective Viscous Flux Identity

The quantity p.%; #/ 

4 3

  C  divx u

called effective viscous flux or effective pressure satisfies a certain weak continuity property discovered by P.L. Lions [77] in the context of barotropic model. This property, in our situation, is formulated in the following lemma. Lemma 10 (See [30, Proposition 6.1]). Let .%n ; #n ; un / be the trio investigated in Lemma 9. Then for any k > 1, there holds

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

4

1473



   Tk .%/divx u  Tk .%/divx u D p.%; #/Tk .%/  p.%; #/ Tk .%/ ; 3 (275) with functions Tk defined in (250). C

In order to get the statement of Lemma 10, we proceed in several steps. Step 1. First, we multiply the momentum equation (243) by test function Q k .%n /; where .t /.x/rx 1 ŒT

2 Cc1 .0; T /; ; Q 2 Cc1 ./

and integrate over the space-time cylinder QT . The (pseudodifferential) operator rx 1 is defined in (111). Employing notation introduced in (111) and in Theorem 13, we get identity Z

T

Z

Z

T

p.%n ; #n /Tk .%n / dxdt  0

0



Z

Q k .%n / dxdt D S.rx un /WRŒT 

7 X

Ini CJn ;

i D1

(276) where Z

T

Z

  Q k .%n /RŒ%n un   %n un  RŒT Q k .%n / dxdt; un  T

Jn D 0



and In1 D In2

Z 0

  Q k .%n / dxdt; S.rx un / W rx  ˝ AŒT



Z

Z

T

Q k .%n / dxdt; p.%n ; #n /rx   AŒT

D 0

In3 In4

Z

T

Z

Z

Z

T

D Z

T

0

  Q k .%n / dxdt; %n un ˝ un W rx  ˝ AŒT

In5 D 



Z

In6 D  Z

T 0

Z

T 0

Q k .%n / dxdt; %n f  AŒT



0

D

In7 D



%n un  AŒTk .%n /rx Q  un  dxdt; 

Z

T 0

0

Z

Q k .%n / dxdt; %n un  AŒT



Z 

Q n T 0 .%n /  %n / dxdt: %n un  AŒ.% k

1474

A. Novotný and H. Petzeltová

When deriving identity (276), we have used several times integration by parts, renormalized continuity equation (248), and item (iii) in Theorem 13. Step 2. Employing in the limiting momentum equation (272) test function .t /.x/rx 1 ŒQ Tk .%/; where

2 Cc1 .0; T /; ; Q 2 Cc1 ./;

we get identity Z

T

Z

Z

Z

T

S.rx u/ W RŒQ Tk .%/ dxdt D

p.%; #/ Tk .%/ dxdt  0





0

7 X

I i CJ;

i D1

(277) where Z

T

Z

  un  Q Tk .%/RŒ%u  %u  RŒQ Tk .%/ dxdt;

J D 0



and I1 D

Z

Z

T 0



Z

2

  S.rx u/ W rx  ˝ AŒQ Tk .%/ dxdt; Z

T

p.%; #/rx   AŒQ Tk .%/ dxdt;

I D 0



Z

3

Z

T

%f  AŒQ Tk .%/ dxdt;

I D 4

Z

Z

T

0

  %u ˝ u W rx  ˝ AŒQ Tk .%/ dxdt;

I D 0

I5 D 



Z

I6 D  T 0

%u  AŒTk .%/rx Q  u dxdt;



Z

T 0

Z

Z

T 0

I7 D



0

Z 

Z

%u  AŒQ Tk .%/ dxdt;



%u  AŒQ .%Tk0 .%/  %/ dxdt:

When deriving identity (277), we have used several times integration by parts, renormalized continuity equation (274), and item (iii) in Theorem 13. Step 3. In view of estimates and induced convergence relations established in previous two sections together with the continuity properties of operators A and R reported in first two items of Theorem 13, it is a relatively easy task to verify that

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

Ini ! I i .as n ! 1/:

1475

(278)

Step 4. Now, we shall establish relation J n ! J .as n ! 1/:

(279)

This relation is the key point in the proof of Lemma 10. In fact, this property does not follow by employing the “standard” compactness argument. Instead, we must use the compensated compactness. Indeed, combining the commutator lemma reported in Theorem 15 with the convergence established in the first two lines of (269), we get 

   Q k .%n / .t / * Q Tk .%/RŒ%u  %u  RŒQ Tk .%/ .t / Q Tk .%n /RŒ%n un   %n un  RŒT

2 (weakly) in Lr .I R3 / with some r > 6=5 (in fact r D C1 ) for all t 2 Œ0; T . The r 1;2 compact imbedding L ./ ,!,! W ./ now yields that the above convergence is strong in W 1;2 ./ for every t 2 Œ0; T . We use this fact together with the weak convergence of the sequence un established in (267) and the Lebsegue dominated convergence theorem used over .0; T / in order to conclude

RT

R

0

!



  Q k .%n /RŒ%n un   %n un  RŒT Q k .%n / dx un  T

RT

R

0

  Q Tk .%/RŒ%u  %u  RŒQ Tk .%/ dx: u   

This is exactly statement (279). Step 5. Integrating twice by parts and employing the property of the Riesz operator listed in item (iii) of Theorem 13, we get identities 4 Z T Z Q S.rx u /WRŒTk .%n / dxdtD  C   Q Tk .%/divx u dxdt; lim n!1 0 3   0 (280) Z T Z 4 Z T Z S.rx u/ W RŒQ Tk .%/ dxdt D  Q Tk .%/divx u dxdt: C 3 0 0   Z

T

Z

n

Step 6. At the point of conclusion, we perform limn!1 in the identity (276) and subtract from its identity (277). We obtain the statement of Lemma 10 in view of (278)–(280).

1476

A. Novotný and H. Petzeltová

8.5

Oscillations Defect Measure

Let %n be a sequence and % its weak limit in L1 .QT /. We introduce the oscillations defect measure of the sequence %n , Z  oscp Œ%n * %.QT / sup lim sup n!1

k1

QT

ˇ ˇp  ˇ ˇ ˇTk .%n /  Tk .%/ˇ dxdt ;

p  1; (281)

where function Tk is defined in (250). The main achievement of the present section is the following lemma. Lemma 11 (see [Proposition 6.2][30]). Let .%n ; #n ; un / be the trio investigated in Lemma 9. Then oscC1 Œ%n * %.QT / < 1:

(282)

Step 1. In view of (225), a2  % C pm .%/ C pb .%/; 

pel .%/ D

(283)

a2  % Cb minfr; %g is nondecreasing function on Œ0; 1/ with % 7! pm .%/ D pel .%/ 2 and pb .%/ D  minfr; %g, where a2 r1 D 2b. With this decomposition and with relation (270) at hand, effective viscous flux identity (275) implies

a2 

Z TZ  Z TZ    % Tk .%/% Tk .%/ dxdt C pm .%/Tk .%/pm .%/ Tk .%/ dxdt C 0



0

Z

T

Z

C 0





3   X Ini ; #Q pth .%/Tk .%/  pth .%/ Tk .%/ dxdt D lim sup n!1

(284)

i D1

where In1

D

In2 D

4

Z

T

C

3 4

0

Z C

3 Z In3 D 

T 0

Z 

 Tk .%n /  Tk .%/ divx un dxdt;



T 0

Z 

Z 

 Tk .%/  Tk .%/ divx un dxdt;



 pb .%n /Tk .%n /  pb .%n /Tk .%/ dxdt:



Step 2. By Hölder’s inequality, lower weak semi-continuity of Lebesgue norms and interpolation

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .  1

1477  C1

jIn1 jCjIn2 j  2 lim sup kdivx un kL2 .QT / kTk .%n /  Tk .%/kL21 .Q / kTk .%n /  Tk .%/kL2  C1 .Q

T/

T

n!1

h i  C1 2  c.M0 ; E0 ; S; F0 ; T / oscC1 Œ%n * %.QT / ;

where we have used bounds (251), (260). Similarly, jIn3 j  c.M0 ; E0 ; S ; F0 ; T /: Step 3. We write Z TZ  Z TZ       %n % Tk .%n /Tk .%/ dxdt % Tk .%/% Tk .%/ dxdt D lim sup n!1



0

Z

T

C 0

Z 



0

  %  % Tk .%/  Tk .%/ dxdt:

(285)





Since % 7! % is convex on Œ0; 1/ and % 7! Tk .%/ is concave, the second integral at the right-hand side of (285) is nonnegative by virtue of Theorem 3. Next, by using the definition (250) of functions Tk and elementary properties of function % 7! % , we easily verify algebraic relations ja  bj  ja  b  j and ja  bj  jTk .a/  Tk .b/j; .a; b/ 2 Œ0; 1/2 : Consequently, formula (285) yields Z

T

0

Z  Z  % Tk .%/  % Tk .%/ dxdt  lim sup n!1



0

T

Z ˇ ˇC1 ˇ ˇ dxdt: ˇTk .%n /  Tk .%/ˇ 

(286) Step 4. Finally, according to Theorem 4, the second and third terms at the left-hand side of relation (284) are nonnegative. Coming back with this information, with relation (286) and with all estimates established in Step 3 to relations (284), we deduce inequality Z

T

lim sup n!1

0

Z ˇ ˇC1 ˇ ˇ dxdt ˇTk .%n /  Tk .%/ˇ 

C1   h i 2  c.M0 ; E0 ; S ; F0 ; T / 1 C oscC1 Œ%n * %.QT / :

The latter formula yields the statement of Lemma 11.

1478

8.6

A. Novotný and H. Petzeltová

Renormalized Continuity Equation

Relation (282) implies that the limit quantities %, u satisfy the renormalized continuity equation. The exact statement reads: Lemma 12 (see [30, Proposition 6.3] and [33, Lemma 3.8]). Let %n * % un * u run * ru

in Lp ..0; T /  R3 /; p > 1; in Lr ..0; T /  R3 I R3 /; in Lr ..0; T /  R3 I R9 /; r > 1:

Let oscq Œ%n * %..0; T /  R3 / < 1

(287)

for q1 < 1  1r , where .%n ; un / solve the renormalized continuity equation (247) (with any b belonging to (118)). Then the limit functions %, u solve the renormalized continuity equation   @t b.%/ C divx .b.%/u/ C %b 0 .%/  b.%/ divx u D 0 in D0 ..0; T /  R3 /

(288)

for any b belonging to the same class (118). We shall outline the proof of this lemma in several steps. Step 1. Passing to the limit in (248), we get   @t Tk .%/ C divx Tk .%/u D .%Tk0 .%/  %/divx u in D0 ..0; T /  R3 /: Since for fixed k > 0, Tk .%/ 2 L1 ..0; T /  R3 /, we can employ Theorem 18 in order to infer that     0 @t bM .Tk .%// C divx bM .Tk .%//u C Tk .%/bM .Tk .%//  bM .Tk .%// divx u (289) 0 .Tk .%// in D0 ..0; T /  R3 / D .%Tk0 .%/  %/divx u bM holds with any bM in class (118) with compact support in Œ0; M /. Step 2. Seeing that by lower weak semi-continuity of L1 norms, Tk .%/ ! % in L1 ..0; T /  R3 / as k ! 1; we obtain from equation (289) by using the Lebesgue dominated convergence theorem

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1479

    0 @t bM .%/ C divx bM .%/u C %bM .%/  b.%/ divx u D 0 in D0 ..0; T /  R3 /; (290) provided we show that     0 .Tk .%// .%Tk0 .%/  %/divx u/bM

L1 ..0;T /R3 /

! 0 as k ! 1:

(291)

To show the latter relation, we use lower weak semi-continuity of L1 norm, Hölder’s 0 inequality, uniform bound of un in Lr .0; T I W 1;r .R3 //, and interpolation of Lr between Lebesgue spaces L1 and Lq to get     0 .Tk .%// 1 .%Tk0 .%/  %/divx u/bM L ..0;T /R3 / Z 0  max jbM .z/j j.%Tk0 .%/  %/divx u/jdxdt z2Œ0;M 

fTk .%/M g

q.r1/r r.q1/ L1 ..0;T /R3 /

 c supn>0 k%n Tk0 .%n /  %n /k

q

lim inf k%n Tk0 .%n /  %n /kLr.q1/ q .fT

k .%/M g/

n!1

:

We have k%n Tk0 .%n /  %n /kL1 ..0;T /R3 /  2supn>0 k%n kL1 .f%n kg/ ! 0 as k ! 1 by virtue of the uniform bound of %n in Lp ..0; T /  R3 / (in the above we have also used algebraic relation zTk0 .z/  Tk .z/  2z1fzkg ), while k%n Tk0 .%n /  %n /kLq .fTk .%/M g/  2kTk .%n /kL1 .fTk .%/M g/

   2 kTk .%n /  Tk .%/kLq ..0;T /R3 / CkTk .%/  Tk .%/kLq ..0;T /R3 / C kTk .%/kLq .fTk .%/M g/ ;

where we have used algebraic relation zTk0 .z/  2Tk .z/ and the Minkowski inequality. Since the latter expression remains bounded, relation (291) is proved. We have thus shown (290). Equation (290) with b D bM however implies (288) with any b in class (118) by virtue of the Lebesgue dominated convergence theorem. Lemma 12 is proved.

8.7

Strong Convergence of the Density Sequence

We deduce from (247) using Lemma 12 with r D 2, p D  , q D  C 1 that Z

Z

Z

%Lk .%/. ; x/'. ; / dx  



Z

%0 Lk .%0 /'.0; / dx  

0

Z





Z

D

Tk .%/divu' dxdt; 0



  %Lk .%/ @t ' C u  rx ' dxdt (292)

1480

A. Novotný and H. Petzeltová

with any 2 Œ0; T  and ' 2 Cc1 .Œ0; T   /, where Lk .%/ is defined in (250). Next, we write (273) and (292) with test function ' D 1 and deduce Z 

Z  %Lk .%/  %Lk .%/ . / dx D 





Z gk dxdt; where gk D Tk .%/divu  Tk .%/divu:

0



(293)

We evaluate function gk by using the effective viscous flux identity (275) with the decomposition of elastic pressure pel D pm .%/  pb .%/; pb 2 Cc2 Œ0; 1/; pb  0;

where pb .z/ D 0 whenever z > r with some r > 0, and pm is an increasing function on Œ0; 1/: gk D gk1 C gk2 C gk3 ; gk2 D

4  3

  gk1 D Tk .%/divu  Tk .%/divu ;

  i 1 h pm .%/Tk .%/  pm .%/ Tk .%/ C #Q pth .%/Tk .%/  pth .%/ Tk .%/ ; C  1  gk3 D 4 pb .%/Tk .%/  pb .%/ Tk .%/ : C 3

Writing ˇZ ˇ ˇ

T 0

Z  

ˇ  ˇ Tk .%/divu  Tk .%/divu dxdt ˇ  kTk .%/  Tk .%/kL2 .QT / kdivukL2 .QT / ;

(294)

and realizing that kTk .%/Tk .%/kL1 .QT /  kTk .%/%kL1 .QT / Ck%Tk .%/kL1 .QT / ! 0 as k ! 1;

(295)

we may use interpolation of L2 between L1 and L C1 together with the boundedness of the oscillations defect measure established in Lemma 282 to show Z QT

gk1 dxdt ! 0 as k ! 1:

(296)

On the other hand, by virtue of Theorem 4, gk2  0:

(297)

Finally, we observe that there is ƒ D ƒ.pb / > 0 such that % 7! ƒ% log %  %pb .%/ and % 7! ƒ% log % C pb .%/

(298)

are convex functions on Œ0; 1/. We have, by employing several times Theorem 2,

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

4 Z

3



D lim

k!1 0



C

Z Z lim

k!1 0



gk3 dxdt

Z  Z  pb .%/Tk .%/  pb .%/ Tk .%/ dxdt D lim 

h Z  lim ƒ k!1

0

h Z  lim ƒ k!1



0

Z 



k!1 0

Z  % log %  % log % dxdt C



0

Z



 .1 C r/ƒ 0

Z 

Z 

 pb .%/%  pb .%/ % dxdt



Z   pb .%/  pb .%/ % dxdt

0

Z Z   % log %  % log % dxdt C 

1481

Z



 i pb .%/  pb .%/ % dxdt



% 0. Now the Gronwall lemma (cf. Theorem 21) says that necessarily Z   % log %  % log % . / dx  0: 

Finally, since the function z ! z log z is strictly convex on .0; 1/, we have % log %  % log % D 0 a. e. in .0; T /  

(300)

%n ! % a.e. in .0; T /  

(301)

and

according to Theorem 3. With relation (301) at hand, we easily establish that pel .%/ D pel .%/; pth .%/ D pth .%/; b.%/ D b.%/; B.%/ D B.%/;

(302)

where b; B are defined in (134). The reader will notice that in the case of elastic pressure, one can deduce (301) immediately after (297). The analysis between formulas (298) and (299) is needed in order to accommodate the locally compactly non monotone elastic pressure. At this place the analysis hits the limits of the Lions-Feireisl method. The reader can consult [30, Section 6.6] for more details and proofs.

1482

8.8

A. Novotný and H. Petzeltová

Limit in the Thermal Energy Equation

Step 1: Strong convergence of the temperature outside vacua By virtue of (259), eth .#n / * eth .#/ in L2 .0; T I W 1;2 .//;

(303)

consequently, 6

%n eth .#n / * %eth .#/ in L2 .0; T I L  C6 .//;

(304)

where we have used the strong convergence of %n in L2 .0; T I W 1;2 ./) established in (269). Next, we evaluate the time derivative @t .%n eth .#n // from the thermal energy equation (244) in order to be able to employ Feiresl’s version of Lions-Aubin theorem (see Theorem 12), and we establish %n eth .#n / ! %eth .#/ (strongly) in L2 .0; T I W 1;2 .//:

(305)

The latter convergence in combination with (303) yields h i2 %n .eth .#n //2 * % eth .#/ in L1 .QT /:

(306)

Writing Z

%.eth .#n //2 dx D

Z



.%  %n /.eth .#n //2 dx C 

Z

%n .eth .#n //2 dx; 

and employing estimate (259) together with (301) and (306), we deduce Z

T 0

Z

 2 % eth .#n /  eth .%/ dxdt ! 0;



which implies eth .#n / ! eth .#/ a.e. in f.t; x/j%.t; x/ > 0g:

(307)

As function eth admits an inverse function eth1 (since it is increasing according to assumption (227)), we get Q D eth .#/ a.e. in f.t; x/j%.t; x/ > 0g; #n ! #Q a.e. in f.t; x/j%.t; x/ > 0g; eth .#/ (308) where #Q is the weak limit of the sequence #n established in (267). Step 2: Renormalized thermal energy equation The goal is now to pass to the limit in the thermal energy equation (244) and get the thermal energy inequality (217)era D0 . The standard argument to achieve this goal would be to multiply equation (244) by test function ' in class (217) and

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1483

integrate conveniently by parts before passing to the limit by using the already established convergence. This procedure allows to pass to the limit n ! 1 (letting appear eventually an inequality due to the lower weak semi-continuity of term RT R RT R S.rx un / W rx un ' dxdt , '  0) in all terms except term 0  divx q.#n /' dxdt D  0 RT R 1 0  K.#n /' dxdt with K.#n / being bounded solely in L .QT /. This bound is not enough to guarantee the limit K.#/ to be a function but merely a measure. To get around this difficulty, we shall first investigate the renormalized version of the thermal energy equation: we multiply equation (244) by functions h! .#n /, ! 2 .0; 1/ introduced in (241). Denoting Z

#

eth;! .#/ D 0

h! .z/cv .z/dz; K! .#/ D

Z

#

h! .z/ .z/dz; 0

and testing by ' vanishing at t D T in class (217), i.e., ' 2 Cc1 .Œ0; T /I C 2 .//; rx '  nj.0;T /@ D 0; '  0;

(309)

we obtain, Z

T

Z 

Z  %n eth;! .#n /@t ' C %n eth;! .#n /un  rx ' dxdt C



0

Z

T

0

Z

Z

T

Z

C 0



h0! .#n / .#n /jrx #n j2 '

(310)



0 T

Kh .#n /' dxdt 

h! .#n /S.rx un / W rx un ' dxdt



Z

Z

Z

h! .#n /#n pth .%n /divx un ' dxdt 

D 0

T

Z dxdt 

%n;0 eth;! .#n;0 /'.0; / dx: 

We now pass to the limit n ! 1 for fixed !. Before starting, we observe that the family of functions h! , ! 2 .0; 1/ verifies h! 2 C 2 Œ0; 1/; h! .0/ D 1; h! nonincreasing;

(311)

lim h! .z/ D 0; h00! .z/ h! .z/  2.h0! .z//2 for all z  0;

z!1

h! % 1 as ! ! 0 C : Writing Z TZ h! .#n /pth .%n /#n divx un ' dxdt D 0

Z TZ 

 pth .%n /  pth .%/ h! .#n /#n divx un ' dxdt

0



Z

T



Z

C

pth .%/h! .#n /#n divx un ' dxdt; 0



1484

A. Novotný and H. Petzeltová

where ' 2 L1 .QT /, we deduce from (301), (308), (267), assumptions (226) =3 , (311), and estimates (252), (259), (266) that Q #p Q th .%/divx u in L1 .QT /: h! .#n /#n pth .%n /divx un * h! .#/

(312)

We proceed in a similar way, to get Q Q %n eth;! .#n / * %eth;! .#/; %n eth;! .#n /un * %eth;! .#/u in L1 .QT /:

(313)

Since h! verifies (311), the function ( R2 3 .s; z/ 7!

h! .s/z2 if s  0; 1 if s < 0;

is convex lower semicontinuous; consequently, we deduce from Theorem 2 that Z



Z

Z



Z

h! .#/S.rx u/ W rx u' dxdt  lim inf n!1



0

h! .#n /S.rx un / W rx un ' dxdt 0



(314)

for any ' 2 L1 .QT /, '  0. The last term to be treated is the term containing K! .#n /: First, according to (266), (212), (230)˛2 , kK! .#n /kL1 .QT /  kK.#n /kL1 .QT /  c.M0 ; E0 ; S ; F0 ; T /

(315)

uniformly in n and ! . Second, since lim

z!1

K! .z/ D 0; K.z/

we can use Theorem 1 to deduce that K! .#n / * K! .#/ in L1 .QT /;

(316)

where by virtue of the almost everywhere convergence established in (308) Q x// for a.a. .t; x/ 2 f.t; x/j%.t; x/ > 0g: K! .#/.t; x/ D K! .#.t;

(317)

With relations (312)–(316), we are ready to pass to the limit n ! 1 in equation (310) and get Z

T



0

Z TZ

Q h! .#/#p th .%/divx u' dxdt 

 0

Z  Q t ' C %eth;! .#/u Q  rx ' dxdt C %eth;! .#/@

Z 



T 0

Z TZ 0



Z 

K! .#/' dxdt

Q h! .#/S.r x u/ W rx u' dxdt 

(318)

Z %0 eth;! .#0 /'.0; / dx 

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

with any ' in class (309), where we have used also the fact that jrx #n j2 ' dxdt has negative sign.

1485

R R 0

0  h! .#n / .#n /

Step 3: Thermal energy inequality

The goal now is to pass to the limit ! ! 0 in (318). As h! % 1, we have Q * %eth .#/; Q Q * %eth .#/u; Q %eth;! .#/ %eth;! .#/u Q #p Q th .%/divx u * #p Q th .%/divx u; h! .#/S.r Q h! .#/ x u/ W rx u * S.rx u/ W rx u

weakly in L1 .QT / as ! ! 0 by the Lebesgue dominated convergence theorem. R R The most difficult term in this limit passage is term 0T  K! .#/' dxdt . We observe that sequence fK! .#.t; x//g!!0 is increasing as ! ! 0 and uniformly bounded in L1 .QT / by virtue of (266). Consequently, by virtue of the monotone convergence theorem and by (318), there is ‚ 2 L1 .QT / such that K! .#.t; x// % ‚.t; x/ for a.a. .t; x/ 2 QT as ! ! 0:

(319)

On the other hand, due to (317) and definition of K! , the value of ‚ is directly calculable outside vacua, namely, Q x// for a.a. .t; x/ 2 f.t; x/j%.t; x/ > 0g: ‚.t; x/ D K.#.t;

(320)

At this stage we set   #.t; x/ D K1 ‚.t; x/ :

(321)

By virtue of (320) Q x/ for a.a. .t; x/ 2 f.t; x/j%.t; x/ > 0g: #.t; x/ D #.t;

Passing to ! ! 0 with these observations at hand in (318), we get the required weak formulation (217) of the thermal energy balance. Step 4: Positivity of temperature and total energy balance

By virtue of Theorem 2,  ln K! .#/  ln K! .#/:

(322)

Moreover according to (230) and (241), K! .#n / is equivalent to #n near 0, and the same is true for ln K! .#n / and ln #n . Therefore, relation (322) together with (257) and (266) implies k ln K! .#/kL2 .QT /  c uniformly with respect to ! 2 .0; 1/:

1486

A. Novotný and H. Petzeltová

Moreover, ‚.t; x/  K! .#/.t; x/ for a.a. .t; x/ 2 QT and at the same time ‚ 2 L1 .QT /. Consequently, ln ‚ 2 L2 .QT /:

After the analysis of behavior of K1 near 0 and near 1 obtained with the help of (230)˛2 , we get from (321) ln # 2 L2 .QT /; # 2 L˛C1 .QT /:

Finally, we obtain from the total energy balance (246) in the limit its weak formulation (218) by virtue of (313), (301), and (263) and the last line in (269). The procedure described above is very much related to the notion of biting limits of bounded sequences in L1 . The reader may consult [30, Sections 6.7.2–6.8.2] for more details of the proofs and on these problems.

9

Navier-Stokes-Fourier System in the Entropy Formulation

In this section we shall deal with the Navier-Stokes-Fourier system in the entropy formulation, where the internal energy balance is replaced by the entropy balance: @t % C divx .%u/ D 0; @t .%u/ C divx .%u ˝ u/ C rx p.%; #/ D divx S.%; #; rx u/ C %f;   q.%; #; rx #/ D : @t .%s.%; #// C divx .%s.%; #/u/ C divx #

(323) (324) (325)

We recall that the specific entropy s is related to the internal energy e and pressure p by the Gibbs relation ds D

p  1 de  2 d% ; # %

(326)

where the pressure and internal energy obey (34). Entropy production rate  is given by formula (10); recall 1 D #

  q.%; #; rx #/  rx # S.%; #; rx u/ W rx u  : #

(327)

We consider Newtonian fluids (12) with the heat flux given by Fourier’s law (13), specifically, S.%; #; rx u/ D .%; #/T.rx u/ C .%; #/divx uI;

2 T.rx u/ D rx u C .rx u/T  divx uI; 3 (328)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

q D  .%; #/rx #;

1487

(329)

where ; ; obey (14)–(15). Equations (323)–(327) are supplemented with initial conditions %.0; / D %0 ; %u.0; / D %0 u0 ; %s.%; #/.0; / D %0 s.%0 ; #0 /;

%0  0; #0 > 0;

(330)

and no-slip boundary conditions for velocity (20) and zero heat transfer conditions (21) on the boundary, recall q  nj.0;T /@ D 0;

(331)

uj@ D 0:

(332)

In [33], the authors have introduced a concept of weak solution to the NavierStokes-Fourier system (323)–(332). This concept postulates, in agreement with the second law of thermodynamics, that the entropy production rate  is a nonnegative measure, 

1 #

  q.#; rx #/  rx # S.#; rx u/ W rx u  : #

(333)

With this postulate, equation (325) becomes inequality. In order to compensate the loss of information, we may postulate that the total energy of the system in the volume  is conserved, namely, d dt

Z  

1 %juj2 C %e.%; #/ 2



Z dx D

%f  u dx:

(334)



Putting together equations (325) and (334), we obtain the so-called dissipation balance d dt Z C 

Z  

# #

 1 2 %juj C H# .%; #/  @% H# .%; #/.%  %/  H# .%; #/ dx 2

(335)

  Z TZ q.%; #; rx #/  rx # S.%; #; rx u/ W rx u  dxdt  %f  u dxdt; # 0 

where we have taken into account inequality (333) and conservation of mass (323). In this inequality % and # are positive constants and H# D %e  #s is the Helmoholtz function introduced in (29). On the other hand, if .%; #; u/ % > 0, # > 0 is a trio of smooth functions satisfying (323)–(332), one may derive, at least formally, the so-called relative energy identity,

1488

A. Novotný and H. Petzeltová

Z  

Z

Z



C

S.%; #; rx u/ W rx u dxdt  #

‚ 0



Z  D 

Z



Z

C 0



 1 %ju  Uj2 C E.%; #jr; ‚/ . ; / dx 2

Z



Z

0

Z

0

0

Z



‚ 0



Z



Z

0



 dx

q.%; #; rx #/  rx ‚ dxdt #





Z h % rx p.r; ‚/ i @t p.r; ‚/  %u  dxdt 1 r r  Z

Z





Z

p.%; #/divx U dxdt C 0

q.%; #; rx #/ W rx # dxdt #2

   % s.r; ‚/  s.%; #/ @t ‚ C u  rx ‚ dxdt

C

C

Z

  % @t U C u  rx U  .U  u/ dxdt

C

Z



1 %0 ju0  U.0; /j2 C E.%0 ; #0 jr.0; /; ‚.0; // 2

S.%; #; rx u/ W rx U dxdt 

Z

Z

(336)



%f  .u  U/ dxdt; 0



where we have denoted E.%; # jr; ‚/ D H‚ .%; #/  @% H‚ .r; ‚/.%  r/  H‚ .r; ‚/; H‚ .%; #/ D %e.%; #/  ‚%s.%; #/:

In (336), .r; ‚/ is a couple of positive sufficiently‘ smooth functions on Œ0; T   , and U is a sufficiently smooth vector field with compact support in Œ0; T   . Conformably to (333), for a weak solution .%; #; u/, the identity (336) has to be replaced by an inequality with the inequality sign . This inequality is usually called the relative energy inequality. Notice that the dissipation balance (335) is a particular case of the relative energy inequality, where r D %, ‚ D # , and U D 0. The material of this section is based on the monograph [33, Chapters 1–3] for the notion of (finite energy) weak solutions and on papers [34] for the notion of relative energy functional and dissipative solutions [70] for the notion of bounded energy weak solutions.

9.1

Definition of Finite Energy Weak Solutions

Definition 8. Let  be a bounded domain, and let the initial functions .%0 ; u0 ; #0 / satisfy condition %0 W  ! Œ0; C1/; u0 W  ! R3 ; #0 W  ! .0; 1/;

(337)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1489

where %0 u0 D 0 and %0 u20 D 0 a.e. in the set fx 2 j%0 .x/ D 0g; R R withRfinite total energy E0 D  . 12 %0 u20 C%0 e.%0 ; #0 // dx, finite mass 0 < M0 D  %0 dx, and  %0 js.%0 ; #0 /j dx S0 < 1. We shall say that the trio .%; #; u/ is a finite energy weak solution to the Navier-StokesFourier system (323)–(332) emanating from the initial data .%0 ; #0 ; u0 ; / if: (a) 1

%; # 2 L1 .0; T I L1 .//; %  0; # > 0 a.e. in .0; T /  ; p.%; #/ 2 L1 .QT /; (338) 1 2 1;q 2 1 1 u 2 L .0; T I W0 .//I %u; %u ; %e.%; #/; %s.%; #/ 2 L .0; T I L .//; q > 1; 2 %s.%; #/u; S.%; #; rx u/;

1 q.%; #; rx #/  rx #  S.%; #; rx u/ W rx u  2 L1 .QT /I # #

(b) % 2 Cweak .Œ0; T I L1 .// and equation (323) is replaced by a family of integral identities Z 

Z ˇ

ˇ %' dx ˇ D 0

0

Z   %@t ' C %u  rx ' dxdt

(339)



for all 2 Œ0; T  and for any ' 2 Cc1 .Œ0; T   /; (c) %u 2 Cweak .Œ0; T I L1 .I R3 // and momentum equation (324) is satisfied in the sense of distributions, specifically Z 

Z



ˇ

ˇ %u  ' dx ˇ D 0

(340)

Z 

0



 %u  @t ' C %u ˝ u W rx ' C p.%; #/divx '  S.%; #; rx u/ W rx ' C %f  ' dx dt

for all 2 Œ0; T  and for any ' 2 Cc1 .Œ0; T   I R3 /; (d) the entropy balance (325), (333) is replaced by a family of integral inequalities Z 

  q.%; #; rx #/  rx # S.%; #; rx u/ W rx u  dxdt 0 # 0   (341)  Z Z  q.%; #; rx #/  rx '  dxdt %s.%; #/@t ' C %s.%; #/u  rx ' C #  0 ˇ Z ˇ %s.%; #/' dx ˇ C



Z

' #

for a.a. 2 .0; T / and for any ' 2 C 1 .Œ0; T   /, '  0;

1490

A. Novotný and H. Petzeltová

(e) the balance of total energy (334) in the volume  is verified in the weak sense Z

T

Z 

 0



1 %juj2 C %e.%; #/ 2 Z



0



Z .t / dxdt D 

Z

T

C

.t / 0

%f  u dxdt for all 



1 %0 ju0 j2 C %0 e.%0 ; #0 / .0/ 2

dx (342)

2 Cc1 Œ0; T /:

Definition 9. Weak solution whose density-velocity component .%; u/ satisfies the continuity equation in the renormalized sense (116)–(117) with f D 0, with any test function b belonging to (118), is called renormalized weak solution. Remark 14. 1. We deduce from (89) and (88) that the total energy balance (342) is equivalent with the formulation Z  

1 %juj2 C %e.%; #/ 2



Z ˇ

ˇ dx ˇ D 0



Z %f  u dxdt for a.a. 2 .0; T /:

0

(343)



2. We deduce from (341) and (77), (85) that Z

Z



Œ%s.%; #/. ; x/ dx C 

Z



D 0

h i %0 s.%0 ; #0 / dx C  Œ0; 

(344)



 Z  q.%; #; rx #/  rx  dxdt for a.a. 2 .0; T /; %s.%; #/u  rx  C # 

where  2 C 1 ./,   0, and  is a nonnegative Radon measure on Borel sets of Œ0; T . Likewise, we deduce from (90), (95), and (341) that Z

Z



h i %0 s.%0 ; #0 / dx C  Œ0;    D 0 for a.a. 2 .0; T /;

Œ%s.%; #/. ; x/ dx C 



(345)

where  is a nonnegative Radon measure on Borel sets of Œ0; T    satisfying h

i

Z



Z

 Œ0;     0



q.%; #; rx #/  rx #  1 S.%; #; rx u/ W rx u  dxdt: # #

3. Putting together (343) and (345) we get the so-called dissipation identity in the form Z  

 ˇ

h i Z ˇ %juj2 C H# .%; #/ dx ˇ C  Œ0;    D 0



Z %f  u dxdt

(346)



0

for a.a. 2 .0; T / and # D const > 0. Similarly, by the same token involving (96), Z  

 ˇ

h i Z ˇ %juj2 C H# .%; #/ dx ˇ C  Œz;    D z



Z %f  u dxdt

z



(347)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1491

for a.a. 0 < z < 2 .0; T / and # D const > 0, where i Z h  Œz;     z



Z 

1 q.%; #; rx #/  rx #  S.%; #; rx u/ W rx u  dxdt: # #

4. According to (94) applied to the entropy balance (341), the right and left instantaneous values Œ%s.%; #/. C/ and Œ%s.%; #/. / defined in (68)–(69) are continuous linear functionals on C ./ satisfying Œ%s.%; #/. C/  Œ%s.%; #/. /:

(348)

5. We deduce from (94) (with ' D 1) applied to the entropy balance (341) that the function of instantaneous values of global entropy i hZ %.; x/s.#.; x/; %.; x// dx . / Œ0; T  3 7! inst

(349)



is a nondecreasing function (with a countable number of jumps). Likewise we deduce from (72)–(73) that the instantaneous values of the total energy  i hZ 1 %juj2 .; x/ C Œ%e.%; #/.; x/ dx . /

3 Œ0; T  D E. / D inst  2

(350)

yield an absolutely continuous function. 6. In the important case of the potential forces f.t; x/ D rx F .t; x/, it is convenient to replace in the definition of finite energy weak solutions the total energy balance (342) with 

 Z TZ  1 %juj2 C%e.%; #/%F 0  2 for all

0

Z .t / dxdt D

.0/ 

  1 %0 ju0 j2 C%0 e.%0 ; #0 /%0 F dx 2 (351)

2 Cc1 Œ0; T / which is equivalent to Z  

1 %juj2 C %e.%; #/  %F 2



ˇ

ˇ dx ˇ D 0 for a.a. 2 .0; T /: 0

(352)

If F 2 L1 .0; T I W 1;1 .// and %u 2 Cweak .Œ0; T I Lq .I R3 // with some q > 1, all formulations (342), (351), (343), and (352) are equivalent. 7. If one considers problem (323)–(331) with slip boundary condition (22), one must modify adequately the Definition 8 of finite energy weak solutions at two points: (1) In the function spaces (338), velocity u must belong to the space L2 .0; T I W 1;q .I R3 //, q > 1 1;q with the normal trace u  nj.0;T /@ D 0 (and not to L2 .0; T I W0 .I R3 //). (2) Test functions in the weak formulation of the momentum equation must belong to class ' 2 Cc1 .Œ0; T   I R3 /; '  nj.0;T /@ D 0: Other items in the definition remain without changes.

(353)

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Considering the entropy production rate as a nonnegative measure satisfying (333) transforms the balance of entropy identity (325) into the variational inequality (341). It may considerably extend the number of weak solution. To compensate this loss of information, we require that the weak solution obeys the global energy conservation (342). This makes from Definition 8 admissible definition. Indeed, any sufficiently regular weak solution is a classical solution as stated in the following lemma whose proof can be found in [33, Section 2]. Lemma 13. Let the trio .%; #; u/ be a finite energy weak solution to problem (323)–(332) in class .%; #; u/ 2 C 1 .QT /  C 1 .QT / \ C .Œ0; T I C 2 .//  C 1 .QT I R3 / \ C .Œ0; T I C 2 .I R3 //;

% > 0; # > 0:

Then .%; #; u/ is a classical solution to the Navier-Stokes-Fourier system. In particular, it satisfies all variants of energy balance laws (5), (7), (9)–(10) as identities on QT .

Lemma 13 remains valid if we replace the homogenous Dirichlet boundary conditions (332) with the slip or Navier’s slip boundary conditions (22) or (23).

9.2

Relative Energy Functional

We shall now define dissipative solutions. This definition is inspired by identity (336). We introduce relative energy as a function of four variables as follows: Œ0; 1/  .0; 1/3 3 .%; #; r; ‚/ 7! E.%; # jr; ‚/ 2 R;

(354)

E.%; # jr; ‚/ D H‚ .%; #/  @% H‚ .r; ‚/.%  r/  H‚ .r; ‚/;

where H‚ .%; #/ D %e.%; #/  ‚%s.%; #/:

If the thermodynamic stability conditions (30) are satisfied, then the function E.j/ has a remarkable property of a “quasi-distance” E.%; #jr; ‚/  0 and E.%; #jr; ‚/ D 0 , .%; #/ D .r; ‚/:

(355)

Indeed, we deduce this property from the splitting E.%; #jr; ‚/ D ŒH‚ .%; #/  H‚ .%; ‚/ C ŒH‚ .%; ‚/  @% H‚ .r; ‚/.%  r/  H‚ .r; ‚/; (356)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1493

by virtue of relations (32)–(33). We may introduce functional E.%; #; u j r; ‚; U/ D

Z   1 .%ju  Uj2 C E.%; #jr; ‚/ dx;  2

(357)

where .%; #; u/, %  0, # > 0 are integrable functions on QT representing the state of the gas and .r; ‚; U/ are arbitrary integrable functions with positive r and ‚ a.e. in QT . According to property (355), if the thermodynamic stability conditions (378) are satisfied, then E.%; #; u j r; ‚; U/  0 and E.%; #; u j r; ‚; U/ D 0 , .%; #; u/ D .r; ‚; U/:

(358)

Consequently the functional E is able to measure the “distance” between a state .%; #; u/ of the gas and arbitrary trio .r; ‚; U/ with positive r and ‚. In fact, under the hypotheses of the thermodynamic stability, the relative energy function E.j/ obeys stronger coercivity properties than (355). They are described in the following lemma: Lemma 14 (see [33, Proposition 3.2 and Lemma 5.1], [44, Lemma 4.1]). Let the constitutive relations for e; p; s obey regularity (34), Gibbs relation (326), and thermodynamic stability conditions (30). Let 0 < r < r; 0 < ‚ < ‚ be given constants. Then there exists c D c.r; r; ‚; ‚/ > 0 such that for all .%; #/ 2 Œ0; 1/  .0; 1/ and all .r; ‚/ 2 Œr; r  Œ‚; ‚ 8 2 2 ˆ < j%  rj C j#  ‚j if .%; #/ 2 Oess E.%; #jr; ‚/  c (359) ˆ : %e.%; #/ C ‚js.%; #/j C 1 if .%; #/ 2 Ores ; where Oess , Ores are essential and residual subsets in the density-temperature twodimensional phase space defined by Oess D Œr=2; 2r  Œ‚=2; 2‚;

Ores D Œ0; 1/  .0; 1/ n Oess :

Proof of Lemma 14 is based on the thermodynamic stability conditions expressed in the form (32), (33) and on the definition of function H‚ (see 354).

9.3

Bounded Energy Weak Solutions

The concept of finite energy weak solutions is not convenient for investigation of weak solutions on unbounded domains. In fact, the finite energy weak solutions are not able to track the conditions at infinity (24). If the thermodynamic stability conditions are satisfied, then E.%; #j%1 ; #1 / D 0 if and only if .%; #/ D .%1 ; #1 /

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according to the property (355) (at least provided %1 > 0, #1 > 0). The conditions (24) will be then verified in the sense that ŒE.%; #; uj%1 ; #1 ; u1 /. / is bounded for a.a. 2 .0; T /. We shall incorporate this property into the definition of weak solutions. Such weak solution will be called bounded energy weak solution. Definition 10. Let  be a bounded or an unbounded domain, and let conditions at infinity .%1 ; #1 ; u1 / specified in (24) be given in the case of unbounded . Suppose that initial data verify %0 W  ! Œ0; C1/; u0 W  ! R3 ; #0 W  ! .0; 1/; with %0 u0 2 L1loc ./; %0 u0 D 0 and %0 u20 D 0 a.e. in the set fx 2 j%0 .x/ D 0g; %0 2 L1loc ./; %0 s.%0 ; #0 / 2 L1loc .//; Z     %0 ju0  uj2 C H# .%0 ; #0 /  @% H# .%; #/ %0  %  H# .%; #/ dx < 1; 

where we have set % D %1 , # D #1 , u D u1 if  is unbounded, and %, # positive numbers, u D 0 in the case of a bounded domain. The trio .%; #; u/ is a bounded energy weak solution to problem (323)–(332) – with conditions at infinity (24), if  is unbounded – provided: (a) %; # 2 L1 .0; T I L1loc .//; %  0; # > 0 a.e. in .0; T /; p.%; #/ 2 L1 .0; T I L1loc .//; (360) 1 2 1;q 2 1 1 u 2 L .0; T I W0;loc .//I %u; %u ; %e.%; #/; %s.%; #/ 2 L .0; T I Lloc .//; q > 1; 2 %s.%; #/u; S.%; #; rx u/ 2 L1 .0; T I L1loc .//; q.%; #; rx #/  rx #  1 S.%; #; rx u/ W rx u  2 L1 .QT /; # #   H# .%; #/@% H# .%; #/ %% H# .%; #/2L1 .0; T I L1 .//; %juuj2 2L1 .0; T I L1 .//I (b) % 2 Cweak .Œ0; T I L1 .K// for any compact subset K  , and weak formulation (339) of the continuity equation holds; (c) %u 2 Cweak .Œ0; T I L1 .KI R3 // for any compact subset K  , and weak formulation (340) of the momentum equation is verified; (d) the weak formulation (341) of the entropy balance is satisfied; (e) the balance of total energy is replaced by the weak formulation of the dissipation inequality (335) in the integral form,

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1495

Z     ˇ

1 ˇ %ju  uj2 C H# .%; #/  @% H# .%; #/ %  %  H# .%; #/ dx ˇ 0  2 Z



Z

C 0



q.%; #; rx #/  rx #  1 S.%; #; rx u/ W rx u  dxdt  # #

Z



(361)

Z %f  u dxdt

0



for a.a. 2 .0; T /. Remark 15. 1. In view of the dissipation balance (346) and continuity equation (339), any finite energy weak solution is a bounded energy weak solution for bounded domains. It is not known whether the opposite statement is true. 2. If one considers the slip boundary conditions (22), one has to modify accordingly 1;q the definition: Condition u 2 L2 .0; T I W0;loc .// in (360) must be replaced by u 2 1;q

L2 .0; T I Wloc .//, u  nj.0;T /@ D 0, and the test function ' in the weak formulation of the momentum equation must be taken in class (353).

9.4

Dissipative Solutions

Definition 11. We say that the triplet .%; #; u/ is a dissipative solution to the Navier-StokesFourier system (323)–(332) if it belongs to class (338) and if it satisfies relative energy inequality Z Z ˇ Z Z S.%; #; rx u/ q.%; #; rx #/ ˇ W rx u dxdt E.%; #; u j r; ‚; U/ˇ C ‚ ‚ W rx # dxdt 0 # #2 0  0  (362) Z Z Z Z q.%; #; rx #/  rx ‚ dxdt  S.%; #; rx u/ W rx Udxdt  #   0 0 Z



Z

C 0

Z

Z



Z





C 0

Z





   % s.r; ‚/  s.%; #/ @t ‚ C u  rx ‚ dxdt

C 0

  % @t U C u  rx U  .U  u/ dxdt

Z h rx p.r; ‚/ i % @t p.r; ‚/  %u  dxdt 1 r r  Z

Z



Z

p.%; #/divx U dxdt C

 0



%f  .u  U/ dxdt 0



for a.a. 2 .0; T / with any .r; ‚; U/ 2 Cc1 .Œ0; T   I R5 /; r > 0; ‚ > 0; Uj.0;T /@ D 0:

(363)

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Remark 16. 1. If one considers the slip boundary conditions (22) in place of the homogenous Dirichlet boundary conditions (20), the definition must be modified: we must replace condition Uj.0;T /@ D 0 in (363) with the condition U  nj.0;T /@ D 0. 2. If one considers unbounded domains with conditions (24) at infinity with u1 D 0 (for simplicity) and with homogenous boundary conditions (332), it is necessary to modify the definition as follows: Inequality (362) remains as it stays, but one must replace (363) by r  %1 ; ‚  #1 2 Cc1 .Œ0; T   /; r > 0; ‚ > 0; Uj@ D 0: The reader can consult [70] to find more details about the dissipative solutions and relative energy inequality in the situations described in items 1. and 2. above.

Bounded energy weak solutions in the sense of Definition 8 are dissipative solutions under mild assumptions on constitutive laws and transport coefficients. This is subject of the following theorem: Theorem 39. Let  be a bounded domain, and let .%; #; u/ be a bounded energy weak solution to the Navier-Stokes-Fourier system (323)–(332) in the sense of Definition 10. Then .%; #; u/ is a dissipative solution; in particular it satisfies relative energy inequality (362). Remark 17. 1. The reader has noticed that relations (323)–(332) include implicitly certain regularity assumptions and sign assumptions on the constitutive laws for p, e (namely, (34)) and transport coefficients , , (namely, (14)–(15)). 2. Theorem 39 holds true regardless whether thermodynamic stability conditions are satisfied. However, it becomes a useful and powerful tool of analysis especially in the case when the thermodynamic stability conditions are satisfied. Indeed, relative energy inequality (362) governs the evolution of the relative energy functional E.%; #; ujr; ‚; U/. If the thermodynamic stability conditions are satisfied, the functional E.%; #; ujr; ‚; U/ measures the “distance” between the weak solution .%; #; u/ and other state .r; ‚; U/ of the fluid by means of Lemma 14. Due to this fact, Theorem 39 has many potential applications. In this chapter of the handbook, we will mention two of them that are directly related to the existence theory: (1) stability and weak-strong uniqueness and (2) longtime behavior. There are other applications, e.g., investigation of various singular limits to the complete Navier-Stokes-Fourier system that goes far beyond the scope of this chapter (see, e.g., [35, 38] and monograph [33]). 3. According to Theorem 39, if  is a bounded domain, then any bounded energy weak solution is a dissipative solution. This statement is not known to be true for the bounded energy weak solution on unbounded domains. However, under certain additional structural assumptions on the constitutive laws, one can construct bounded energy weak solutions that are dissipative. This questions will be discussed later in more details. Proof of Theorem 39. If we take in the continuity equation (339) as test function ' D we obtain the identity Z % 

Z Z   jUj2 ˇˇ

dx ˇ D %U  @t U C u  rU dxdt: 0 2 0 

jUj2 2 ,

(364)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1497

Momentum equation (340) with the test function ' D U reads Z  

Z Z h ˇ

i ˇ %u.@t UCurU/Cp.%; #/divx US.%; #; ru/ W rUC%Uf dxdt: %uUdx ˇ D 0

0 

(365)

Taking in the entropy inequality (341) ' D ‚ as test function, we obtain Z  

ˇ Z ˇ %s.%; #/‚dx ˇ C

Z

‚ q.%; #; rx #/  rx #  S.%; #; rx u/ W rx u  dxdt 0 #  # 0 (366) Z Z h q.%; #; r#/  r‚ i  dxdt: %s.%; #/.@t ‚ C u  r‚/ C #  0

Summing up energy identity (343) with identities (364), (365) and with the inequality (366), we arrive at the inequality Z h % 2



Z



Z

C 0



Z

Z

0



S.%; #; ru/ W rU 

Z



Z

q.%; #; r#/  r‚ dxdt #

  % @t U C u  rU  .U  u/ dxdt

C 

0

Z



Z %s.%; #/.@t ‚ C u  r‚/ dxdt

 0

0

q.%; #; rx #/  rx #  ‚ S.%; #; rx u/ W rx u  dxdt # #



Z

i ˇ

ˇ ju  Uj2 C H‚ .%; #/ dx ˇ



Z

Z



Z

p.%; #/divx Udxdt C

 0

%f  .u  U/ dxdt: 0





Due to the Gibbs relation (326), a@% Hb .a; b/  Hb .a; b/ D p.a; b/: Consequently, Z  

for a.a. 2 .0; T /.

Z  ˇ

ˇ r@% H‚ .r; ‚/  H‚ .r; ‚/ dx ˇ D 0



Z @t p.r; ‚/ dxdt

0



(367)

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Further, continuity equation (339) with test function @% H‚ .r; ‚/ yields Z  Z



Z



ˇ

ˇ %@% H‚ .r; ‚/dx ˇ D 0



 % @t @% H‚ .r; ‚/ C u  rx @% H‚ .r; ‚/ dxdt;

 

0

where, by the Gibbs relation (326),

@y @% Hb .a; b/ D

1 @y p.a; b/  s.a; b/@y b: a

Whence adding to the left-hand side of (367) the term Z h

i ˇ

ˇ  %@% H‚ .r; ‚/ C r@% H‚ .r; ‚/  H‚ .r; ‚/ dx ˇ ; 0



we arrive at the inequality Z h i ˇ

% ˇ ju  Uj2 C H‚ .%; #/  @% H‚ .r; ‚/.%  r/  H‚ .r; ‚/ dx ˇ 0  2 Z



Z

q.%; #; rx #/  rx #  ‚ S.%; #; rx u/ W rx u  dxdt # #

C 0



Z



Z

 0



S.%; #; ru/ W rU 

Z



Z

C 0

Z

Z



Z



C





Z 

1

0



Z



Z



%u  0

Theorem 39 is proved.



  % @t U C u  rU  .U  u/ dxdt

  % s.r; ‚/  s.%; #/ .@t ‚ C u  r‚/ dxdt

C 0

q.%; #; r#/  r‚ dxdt #

% @t p.r; ‚/dxdt  r rx p.r; ‚/ dxdt C r

Z

Z



p.%; #/divU dxdt 0

Z





Z %f  .u  U/ dxdt:

0



(368)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

9.5

1499

Constitutive Relations and Transport Coefficients for the Existence Theory

In the above setting, we will be able to build up existence theory under certain assumptions on constitutive laws on pressure, internal energy, and transport coefficients that are listed in the sequel. The reader is advised to confront these conditions with the physically motivated constraints due to statistical mechanics exposed in (47)–(50), (51), due to thermodynamic stability conditions exposed in (58)–(59), and due to the physical transport properties of the fluid exposed in (16)–(18). (i) Pressure, internal energy, and specific entropy p.%; #/ D # =.1/ P





%

C

# 1=.1/

a 4 # ; a > 0;  > 1; 3

(369)

where P 2 C 1 Œ0; 1/; P .0/ D 0; P 0 .Z/ > 0 for all Z  0 0
0; Z P .Z/ lim D P1 > 0: Z!1 Z 

(370) (371) (372)

The internal energy must write e.%; #/ D

1 # =.1/ P  1 %





% # 1=.1/

Ca

#4 ; %

(373)

and the formula for (specific) entropy reads  s.%; #/ D S

% # 1=.1/

 C

4a # 3 ; 3 %

(374)

where S 0 .Z/ D 

1  P .Z/  P 0 .Z/Z < 0:  1 Z2

(375)

(ii) Transport coefficients ;  2 C 1 Œ0; 1/ \ L1 .0; 1/; 0 2 L1 .0; 1/;

(376)

.1 C # ˇ /  .1 C # ˇ /; 0  .#/  .1 C # ˇ /; 2 C 1 Œ0; 1/;

.1 C # 3 /  .#/  .1 C # 3 /;

where ; ; ; ; are positive constants.

(377)

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A. Novotný and H. Petzeltová

It should be underlined that pressure and internal energy defined through formulas (369)–(375) verify the thermodynamic stability conditions, namely, @e.%; #/ @p.%; #/ > 0; > 0 for all %; # > 0: @% @#

(378)

Recall that these conditions can be rewritten in terms of the Helmoltz function via formulas @2% H# .%; #/ D

1 @p.%; #/ ; % @%

@# H# .%; #/ D %

#  # @e.%; #/ # @#

(379)

with any # > 0, meaning that

9.6

% 7! H# .%; #/ is strictly convex,

(380)

# 7! H# .%; #/ attains its global minimum at # D #:

(381)

Existence of Weak Solutions

We shall present two existence theorems for weak solutions. The first one deals with  D 5=3 (this case corresponds to the monoatomic gas) and ˇ is allowed to vary in a certain range: Theorem 40 (see [33, Theorems 3.1 and 3.2] reproved in Feireisl, Pražák [44, Theorem 4.3]). Let   R3 be a bounded domain of class C 2; ,  2 .0; 1/ and let f 2 L1 .QT I R3 /. Suppose that the thermodynamic functions p, e, and s satisfy hypotheses (369)–(375) and that the transport coefficients , , and obey (376), (377), where  D 5=3; ˇ 2 .2=5; 1: Finally, assume that the initial data (330) verify (337). Then the complete Navier-StokesFourier system (323)–(332) admits at least one renormalized finite energy weak solution with the following additional properties: u 2 Lq .0; T I W 1;p .I R3 // with q D

6 18 ;p D ; 4ˇ 10  ˇ

% 2 C .Œ0; T I L1 .// \ L1 .0; T I L .// \ Lq .QT / with some q >  ; %u 2 L1 .0; T I L

2  C1

.// \ Cweak .Œ0; T I L

2  C1

.//;

ln #; # ˇ 2 L2 .0; T I W 1;2 .//; # 2 L1 .0; T I L4 .//; ˇ 2 Œ0; 3=2; S.#; rx u/ 2 Lq .QT I R9 / with some q > 1:

(382) (383) (384) (385) (386)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1501

There holds Z

Z ess lim

t!0C 

%s.%; #/.x/ dx  

%0 s.%0 ; #0 / dx;  2 Cc1 ./;   0:

(387)

If, moreover, #0 2 W 1;1 ./, then Z

Z ess lim

t!0C 

%s.%; #/.x/ dx D 

%0 s.%0 ; #0 / dx;  2 Cc1 ./:

In the second variant of the existence theorem, we allow  > 3=2 and fix ˇ D 1. Theorem 41 (see [70, Theorems 2.1 and 2.2]). Let   R3 be a bounded domain of class C 2; ,  2 .0; 1/ and let f 2 L1 .QT I R3 /. Suppose that the thermodynamic functions p, e, s satisfy hypotheses (369)–(375) and that the transport coefficients , , and obey (376), (377), where  > 3=2; ˇ D 1: Finally, assume that the initial data (330) verify (337). Then the complete Navier-StokesFourier system (323)–(332) admits at least one renormalized bounded and finite entropy weak solution with further properties (383)–(387) and with u 2 L2 .0; T I W 1;2 .//:

(388)

Remark 18. 1. The conclusion of Theorems 40 and 41 is valid under the same assumptions also for bounded Lipschitz domains as one can verify by using the techniques introduced for this purpose by Kukucka [73] and Poul [94]. 2. One can consider the same problem (323)–(332) with the complete slip (22) boundary conditions for the velocity (instead of uj.0;T /@ D 0) on a bounded domain. After the necessary appropriate modifications in the definition of weak solutions exposed in item 7 of Remark 14, one can prove their existence under the same assumptions on the regularity of the domain, initial data, external force, constitutive relations, and transport coefficients as in Theorems 40 and 41. The solutions constructed in this way enjoy all additional properties mentioned in Theorems 40, resp., 41, according to the case. The reader can consult [33, Chapter 3] for the details. 3. Definition of weak solutions introduced through (337)–(342) and investigated in Theorems 40 and 41 relies essentially on the fact that the fluid system must be mechanically and thermally isolated (meaning that u  nj@ , q  nj@ D 0). If in the boundary conditions one of these assumptions is violated, the theory cannot be applied. 4. The system (323)–(332) on an unbounded domain with condition (24) admits under certain circumstances a bounded energy weak solution. For example, if in (24), %1 > 0; #1 > 0; and u1 D 0, it is known that the system (323)–(332) admits on any unbounded uniformly Lipschitz domain a bounded energy weak and dissipative solution under the same assumptions on constitutive laws p, e and transport coefficients ,  , , as in Theorems 40, resp.,41 provided f D rx F , F 2 L1 .0; T I W 1;1 \ W 1;1 .I R3 //

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(see [50, Theorem 2.5 and Remarks 2.5, 2.6]). The same problem on unbounded domains with the complete slip conditions is investigated in the same paper in Sect. 6. Remark 19. 1. Existence theorems of type Theorem 40 and Theorem 41 are known also to be true for the phenomenological constitutive laws of real gasses (compressible fluids) of general form (36), (37). Indeed, the compactness result established in [32, Theorem 3.1] in combination with the construction of weak solutions suggested in [33, Chapter 3] and existence theorem proved in [58, Theorem 3.1] can be summarized in the following way. Assumptions on the constitutive equations and the transport coefficients are the following: (1) Pressure and internal anergy take form] p.%; #/ D pF .%; #/ C

a 4 # ; a > 0; 3

(389)

#4 ; %

(390)

e.%; #/ D eF .%; #/ C a

where pF , eF satisfy Gibbs’ relation (8) for a certain entropy sF . Moreover, we impose the hypothesis of thermodynamic stability @eF .%; #/ @pF .%; #/ > 0; > 0 for all %; # > 0: @% @#

(391)

Further, we suppose pF 2 C 2 ..0; 1/2 / \ C 1 .Œ0; 1/2 /, lim pF .%; #/ D 0 for any # > 0;

%!0C

lim pF .%; #/ D pc .%/ for any % > 0;

#!0C

(392) with the “cold pressure” pc satisfying p%  pc .%/  p.1 C %/ ; p > 0:

(393)

In addition, we suppose ˇ ˇ   ˇ @pF .%; #/ ˇ ˇ  c 1 C %=3 C # 3 for all 0 < # < ‚c .%/; ˇ ˇ ˇ @#

(394)

where % 7! ‚c .%/ is a continuous curve satisfying ‚c .%/  c%=4  1 for a certain c > 0:

(395)

As for the internal energy eF , we assume eF .%; #/  0; cv .%; #/

lim

Œ%;#!Œ0;0

eF .%; #/ D 0;

@eF .%; #/ 2 C .Œ0; 1/2 /; @#

(396) (397)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

0 < c.1 C #/!  cv .%; #/  c.1 C #/! for all %; # > 0:

1503

(398)

(2) Transport coefficients The viscous stress S.#; rx u/ is given by Newton’s rheological law (12), where  D .#/;  D .#/ 2 W 1;1 Œ0; 1/, 0 < .1 C #/ˇ  .#/  .1 C #/ˇ ;

(399)

0 < .1 C #/ˇ  .#/  .1 C #/ˇ ;

(400)

j0 .#/j  .1 C #/ˇ1 ; j0 .#/j  .1 C #/ˇ1 ; for all # 2 Œ0; 1/. The heat flux q.#; rx #/ is given by Fourrier’s law (13) where 2 C 1 Œ0; 1/ verifies .1 C #/˛  .#/  .1 C #/˛ :

(401)

In the above, ; ; ; ; ; are positive constants. Under assumptions (389)–(401) with  > 3=2; 0  ˇ  4=3; ˛ 

16  ˇ; 0  !  1=2; 3

or  > 3; 4  ˇ  0; ˛ 

16  ˇ; 0  !  1=2; 3

there is a finite energy weak solution in the sense of Definition 8 on a bounded sufficiently smooth domain. 2. We notice that the bulk viscosity coefficient  is supposed to be strictly positive. This assumption can be relaxed in the case ˇ  0, for which the lower bound in (400) can be replaced by   0.

9.7

Construction of Weak Solutions

Proof of Theorems 40 and 41 can be done via several levels of approximations: (i) Continuity equation The equation of continuity (323) is regularized by means of an artificial viscosity term: @t % C divx .%u/ D "% in .0; T /  ;

and supplemented with the homogeneous Neumann boundary condition

(402)

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A. Novotný and H. Petzeltová

rx %  nj@ D 0;

(403)

%.0; / D %0;ı ;

(404)

%0;ı 2 C 2; ./; inf %0;ı .x/ > 0; rx %0;ı  nj@ D 0

(405)

and the initial condition

where x2

is a convenient approximation of the initial density %0 . (ii) Momentum equation The momentum balance (324) expressed through the integral identity (340) is replaced by a Faedo-Galerkin approximation: Z

T 0

Z 

   %u  @t ' C %Œu ˝ u W rx ' C p.%; #/ C ı.% C %2 / divx ' dxdt

(406)



Z

T

D 0

Z  ".rx %rx u/  ' C Sı W rx '  %fı  ' dx dt  .%u/0  ' dx;

Z  



to be satisfied for any test function ' 2 Cc1 .Œ0; T /I Xn /, where Xn  C 2; .I R3 /  L2 .I R3 /

(407)

is a finite-dimensional (n-dimensional) vector space of functions satisfying 'j@ D 0 in the case of the no-slip boundary conditions.

(408)

The spaces Xn  XnC1 are endowed with the Hilbert structure induced by the scalar product of the Lebesgue space L2 .I R3 /, and the linear hull of [n2N Xn is dense in L2 .I R3 /. Furthermore, we set   2 Sı D Sı .#; rx u/ D ..#/ C ı#/ rx u C rxT u  divx u I C .#/divx u I; 3

(409)

while the function fı 2 C 1 .Œ0; T   I R3 /

is a suitable approximation of the driving force f.

(410)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1505

(iii) Entropy balance Instead of the entropy balance (325), we consider a modified internal energy equation in the form @t .%eı .%; #// C divx .%eı .%; #/u/  divx rx Kı .#/ D Sı .#; rx u/ W rx u  p.%; #/divx u C "ı. % 2 C 2/jrx %j2 C ı

(411) 1  "# 5 ; #2

supplemented with the Neumann boundary condition rx #  nj@ D 0;

(412)

#.0; / D #0;ı ;

(413)

and the initial condition

#0;ı 2 W

1;2

1

./ \ L ./; ess inf #0;ı .x/ > 0; x2

(414)

where #0;ı is a convenient approximation of #0 . Here eı .%; #/ D emo;ı .%; #/ C a# 4 ; Kı .#/ D

Z

# 1

emo;ı .%; #/ D emo .%; #/ C ı#;

(415)

 1 : ı .z/ dz; ı .#/ D .#/ C ı # C #

In problem (402)–(415), the quantities ", ı are small positive parameters, while > 0 is a sufficiently large fixed number. Loosely speaking, the "dependent quantities provide more regularity of the approximate solutions modifying the type of the field equations, while the ıdependent quantities prevent concentrations yielding better estimates on the amplitude of the approximate solutions. For technical reasons, the limit passage must be split up in two steps letting first " ! 0 and then ı ! 0. The complete existence proof goes far behind the scope of the handbook. The reader can find all details in [33, Chapter 3]. In the handbook, we shall show solely the weak compactness property of the set of weak solutions. This property already contains the main ingredients of the existence proof. Note, however, that the compressible models are very much “approximation sensitive,” and the way from the weak compactness to the real existence is always a delicate task.

10

Weak Compactness of the Set of Weak Solutions

In this section we show weak compactness of the (hypothetical) set of weak solutions emanating from initial data .%0 ; #0 ; u0 / in the situation corresponding to assumptions of Theorem 41. This exercise follows main ideas exposed in

1506

A. Novotný and H. Petzeltová

[33, Chapter 3] and illustrates all essential difficulties that one faces during the existence proof. Theorem 42. Let   R3 be a bounded Lipschitz domain and f 2 L1 .QT ; R3 /. Suppose that the thermodynamic functions p, e, s satisfy hypotheses (369)–(375) and that the transport coefficients , , and obey (376), (377) with  > 3=2, ˇ D 1. Finally assume that the initial data .%n;0 ; #n;0 ; un;0 / satisfy %n;0 * %0 in L1 ./; %n;0 un;0 * %0 u0 in L1 .I R3 /;

(416)

%n;0 s.#n;0 / * %0 s.#0 / in L1 ./; Z  Z    1 1 2 %n;0 jun;0 j C %n;0 e.%n;0 ; #n;0 / dx ! %0 ju0 j2 C %0 e.%0 ; #0 / dx;  2  2 where %n;0 ; #n;0 ; un;0 and %0 ; #0 ; u0 verify (337) with Mn;0 > 0; En;0 > 0; Sn;0 > 0, and M0 > 0, E0 > 0, S0 > 0, respectively. Let .%n ; #n ; un / be a sequence of renormalized finite energy weak solutions to the complete Navier-Stokes-Fourier system (323)–(332) with initial data .%n;0 ; #n;0 ; un;0 /. Then there exists a subsequence (denoted again .%n ; #n ; un /) such that %n * % in L1 .0; T I L .//; #n * # in L2 .0; T I W 1;2 .//; un * u in L2 .0; T I W01;2 .I R3 //; and the trio .%; #; u/ is a weak solution of the complete Navier-Stokes-Fourier system (323)– (332) with initial data .%0 ; #0 ; u0 /. Remark 20. 1. It is to be noticed that Theorem 42 can be proved with less restrictive conditions on the heat conductivity : One can admit heat conductivity is dependent on both density and temperature, namely, C 1 .Œ0; 1/  Œ0; 1// 3 D .%; #/ enjoying bounds (377) (see [32]). However, in spite of the available compactness result in this situation, and in contrast with the case D .#/, the construction of weak solutions under condition D .%; #/ remains an open problem.

10.1

Estimates and Weak Limits

10.1.1 Estimates Let .%n ; #n ; un / be a sequence of weak solutions of the problem (323)–(332) on .0; T /  . Any trio of this sequence satisfies, in particular, the dissipation inequality (361)uD0 . The dissipation inequality will produce most of a priori estimates that are available in this problem. It will be convenient to split H# .%; #/  @% H# .%; #/.% 

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1507

%/  H# .%; #/ according to (356). Employing (31), (369), (373), and (419), we obtain Z H# .%; #/  H# .%; #/ D

Z

#

@# H# .%; z/dz  4a

#

#

z2 .z  #/dx;

(417)

#

and % Z z

Z H# .%; #/  @% H# .%; #/.%  rQ /  H# .%; #/ D

%

 @2% H .w; #/dw dz

rQ

(418)

 i h h i

% log.%=%/  %  % C %   %1 .%  %/  % ;

where we have used the equivalence P 0 .Z/ 1 C Z 1 ;

Z > 0;

(419)

that can be derived from (371)–(372). With observations (417)–(418) at hand, and using the conservation of mass Z %n dx D M0 ;

(420)



we deduce from the dissipation balance (361) the following estimates: Z esssup.0;T /



%n u2n dx  c.M0 ; E0 ; S0 ; T /;

Z esssup.0;T / Z esssup.0;T /







(421)

%n dx  c.M0 ; E0 ; S0 ; T /;

(422)

#n4 dx  c.M0 ; E0 ; S0 ; T /:

(423)

By virtue of (421)–(422), we deduce for the momentum, k%n un kL1 .0;T IL2=. C1/ .IR3 //  c.M0 ; E0 ; S0 ; T /:

(424)

The “velocity part” of the entropy production yields bounds kT.rx un /k2L2 .0;T IL2 .IR33 // C

RT R 0

1 2  #n jT.rx un /j dxdt

 c.M0 ; E0 ; S0 ; T /I

(425)

whence employing first the Korn type theorem (see Theorem 9) and then the standard Poincaré inequality, we get kun kL2 .0;T IW 1;2 .IR3 //  c.M0 ; E0 ; S0 ; T /:

The “temperature part” of the entropy production rate gives

(426)

1508

A. Novotný and H. Petzeltová ˇ

krx #n kL2 .0;T IL2 .IR3 //  c.M0 ; E0 ; S0 ; T /;

ˇ 2 Œ1; 3=2; (427)

krx log #n kL2 .0;T IL2 .IR3 //  c.M0 ; E0 ; S0 ; T /:

In agreement with (374)–(375), j%s.%; #/j  c.% C %j log %j C %j log #j C # 3 /:

(428)

With this observation at hand, we verify that assumptions of Lemma 1 are satisfied with some 3 < p < 4. Therefore, we deduce from (427) and the Poincaré-type inequality from Theorem 6, ˇ

ˇ

k log #n log #kL2 .0;T IW 1;2 ./ Ck#n # kL2 .0;T IW 1;2 ./  c.M0 ; E0 ; S0 ; T /; ˇ 2 Œ1; 3=2: (429) We get by the Sobolev imbedding and by interpolation from (421)–(429) using eventually (369)–(377) k#n kL3 .0;T IL9 .//  c.M0 ; E0 ; S0 ; T /;

k#n kL17=3 ..0;T //  c.M0 ; E0 ; S0 ; T /; (430)

kS.#n ; rx un /kL2 .0;T IL4=3 .IR33 //  c.M0 ; E0 ; S0 ; T /;

(431)

kq.#n ; rx #n /=#n kL2 .0;T IL8=7 .IR3 //  c.M0 ; E0 ; S0 ; T /;

(432)

k%n s.%n ; #n /kL1 .0;T ILq .//  c.M0 ; E0 ; S0 ; T / with some q > 1;

(433)

k%n s.%n ; #n /un kLq ..0;T /IR3 /  c.M0 ; E0 ; S0 ; T / with some q > 1:

(434)

Under assumptions (369)–(372) jp.%; #/j  c.%# C % C # 4 /:

(435)

Consequently, we can deduce from (422)–(423) only kp.%n ; #n kL1 .0;T IL1 .//  c.M0 ; E0 ; S0 ; T /. We however need for the pressure better estimate than an estimate in L1 ./. To improve this estimate, we use in the momentum equation (340) (written R ! 1 % dx, where with .%n ; #n ; un / on ) the test function ' D .t/B Œ%!n  jj n  ! > 0,  2 Cc1 .0; T / and B is the Bogovskii operator introduced in Theorem 5. A straightforward but laborious calculation (the same as exposed in (262)) leads to the conclusion that RT R p.%n ; #n /%! n dxdt  c.M0 ; E0 ; S0 ; T; !/ with some ! > 0; R0T R q 0  jp.%n ; #n /j dxdt  c.M0 ; E0 ; S0 ; T; q/ with some q > 1:

(436)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1509

10.1.2 Weak Limits Estimates derived in the previous sections together with equations (339), (340), (133)– (134) written with .%n ; #n ; un / give rise to the following convergence relations for a chosen subsequence denoted again .%n ; #n ; un /: %n * % in L1 .0; T I L .//; #n * # in L1 .0; T I L4 .//; un * u in L2 .0; T I W 1;2 .I R3 //; #n * # in L2 .0; T I W 1;2 .//;

(437)

and sequences %n ; b.%n /; %n un ; %n un ˝ un verify convergence relations (269)

(438)

(see relations (267)–(269) for the similar reasoning). Moreover, if we denote by g.%; #; u/ weak limit of the sequence g.%n ; #n ; un / in L1 ..0; T /  //, we have for the nonlinear quantities log #n * log # in L2 .0; T I W 1;2 .//; p.%n ; #n / * p.%; #/ in Lq ..0; T /  / with some q > 1;

(439)

S.#n ; rx un / * S.#; rx u/ in L4=3 ..0; T /  I R33 /; %n s.%n ; #n / * %s.%; #/ in Lq ..0; T /  / with some q > 1; q.#n ; rx #n /=#n * q.#; rx #/=# in L8=7 ..0; T /  I R3 /:

The main goal in what follows is to “remove” bars over all nonlinear quantities in the weak limits (439). This will be done if we show convergence almost everywhere in QT of the sequences %n and #n .

10.1.3 Limiting Momentum, Continuity, and Renormalized Continuity Equations Now, we are ready to let n ! 1 in the weak formulation of the momentum equation, continuity equation, and the renormalized continuity equation. We have, similarly as in (271)–(274), in particular: (1) Limiting momentum equation Z

Z %. ; x/'. ; x/ dx  

Z

T

D 0

%0 u0  '.0; / dx

Z  

(440)



 %u  @t ' C %u ˝ u W rx ' C p.%; #/divx '  S.#; rx u/ W rx ' dxdt

for all 2 Œ0; T  and for any ' 2 Cc1 .Œ0; T   I R3 /, 'j@ D 0;

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A. Novotný and H. Petzeltová

(2) Limiting continuity and renormalized continuity equations in form equations (271), (273), (274) hold;

(441)

where functions Tk , Lk are defined in (250), cf. formulas (248)–(249).

10.2

Strong Convergence of Temperature

10.2.1 Entropy Production Rate as a Nonnegative Radon Measure Entropy balance (341) can be rewritten as identity Z %0 s.%0 ; #0 /'.0; /dxC < n ; ' >

(442)



Z D 0

T

 Z  q.#n ; rx #n /  rx ' %n s.%n ; #n /@t ' C %n s.%n ; #n /un  rx ' C dxdt; #n 

where n is a nonnegative linear functional on the space Cc1 .Œ0; T /  / defined by the above equation. According to (90), (92), there is a sequence of continuous linear functionals †n 2 .C .Œ0; T   // , k†n k.C Œ0;T //  c.M0 ; E0 ; S0 ; T /;

(443)

such that < †n ; ' >C .Œ0;T / D< n ; ' > for all ' 2 Cc1 .Œ0; T /  /:

10.2.2 A Consequence of Div-Curl Lemma We may now apply the Div-Curl lemma (see Theorem 14) to the four-dimensional vector fields Vn D .%n s.%n ; #n /; %n s.%n ; #n /un C q.#n ; rx #n /=#n /;

Wn D .Tk .#n /; 0; 0; 0/:

Since divVn D †n and since the imbedding .C .Œ0; T   // ,! W 1;q ..0; T /  / is compact for any q 2 .1; 4=3/, the assumptions of the lemma on .0; T /   are satisfied. Therefore, 4 4 Tk .#/%smo .%; #/ C aTk .#/# 3 D Tk .#/ %smo .%; #/ C aTk .#/ # 3 ; 3 3

(444)

1

where smo .%; #/ D S .%=#  1 /. We shall first prove that Tk .#/%smo .%; #/  Tk .#/ %smo .%; #/;

(445)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1511

where

Tk .z/ D kT .z=k/;

C Œ0; 1/ 3 T D

8 ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ :

z if z 2 Œ0; 1; T strictly increasing on Œ0; 1/; limz!1 T .z/ D 2:

9 > > > > > > = > > > > > > ;

To this end we write   %n smo .%n ; #n / Tk .#n /  Tk .#/ D h      i  %n smo %n ; Tk1 Tk .#n /  smo %n ; Tk1 Tk .#/ Tk .#n /  Tk .#/     Tk .#n /  Tk .#/ : C%n smo %n ; Tk1 Tk .#/

Therefore, inequality (445) will be shown if we prove that     %n smo %n ; Tk1 Tk .#/ Tk .#n /  Tk .#/ * 0 weakly in L1 ..0; T /  / as n ! 1: (446)

10.2.3 Application of Theorem on Parametrized Young Measures The quantity    %n smo %n ; Tk1 Tk .#/ .t; x// D

.t; x; %n /

(447)

can be regarded as a composition of a Carathéodory function with a weakly convergent sequence %n . Since according to (437), (438) b.%n / ! b.%/ in L2 .0; T I W 1;2 .//; G.#n / ! G.#/ in L2 .0; T I W 1;2 .//;

we have b.%/G.#/ D b.%/ G.#/

(448)

for any b and G 2 W 1;1 ..0; 1//, zb 0  b 2 L1 .0; 1/. This implies (446) by virtue of the fundamental theorem on parametrized Young measures (see Theorem 17). %;# % # Indeed, denote .t;x/ , .t;x/ , and .t;x/ the parametrized Young measures corresponding, in accordance with Theorem 17, to the sequences .%n ; #n /, %n , and #n , respectively. Then we have, due to (448) and in agreement with Theorem 17, Z R2

h./G./d .%;#/ .; / D

Z R

h./d % ./ 

Z R

G./d # ./:

1512

A. Novotný and H. Petzeltová

Consequently, Z .t; x; %/G.#/.t; x/ D

%

# .t; x; / G./ d.t;x/ ./ d.t;x/ ./ D

R2



 .t; x; %/ G.#/ .t; x/:

10.2.4 Monotone Functions Versus Weak Convergence Now we shall use the properties of monotone operators with respect to the weak convergence reported in Theorem 4. Theorem 4 implies, in particular, Tk .#/# 3  Tk .#/ # 3 ;

that in turn with (444)–(445) yields Tk .#/# 3 D Tk .#/ # 3 ;

and finally, by monotone convergence, as k ! 1, # 4 D ## 3 :

(449)

#n ! # a.e. in .0; T /  

(450)

The last identity implies

by virtue of (105) in Theorem 4 and Theorem 3.

10.2.5 Weak Limits of the Momentum Equation and Entropy Balance Coming back with (450) to the momentum equation (440), we obtain Z 

%0 u0  '.0; / dx

(451)



Z

T

D 0

Z  

 %u  @t ' C %u ˝ u W rx ' C p.%; #/divx '  S.#; rx u/ W rx ' C %f  ' dxdt

for any ' 2 Cc1 .Œ0; T /  I R3 /, 'j@ D 0. Moreover, estimates (425), (427) yield boundedness of the sequences s

 2 .#n /  rx un C .rx un /T  divun ; #n 3

s

.#n / divun ; #n

p .#n / rx #n #n

(452)

in L2 ..0; T /  //; whence by the lower weak continuity combined with (450) and (437), one gets Z

T

Z 

0

Z

T

' #

Z

 lim inf n!1

0



  q.#; rx #/  rx # S.#; rx u/ W rx u  dxdt #

' #

  q.#n ; rx #n /  rx #n S.#n ; rx un / W rx un  dxdt; #n

(453)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1513

for any 0  ' 2 Cc .Œ0; T   /. Thus effectuating the limit n ! 1 in (341) (with %n ; #n ; un in place of %; #; u), we get   q.#; rx #/  rx # S.#; rx u/ W rx u  dxdt #  0 (454)  Z  q.#; rx #/  rx ' dxdt %s.%; #/@t ' C %s.%; #/u  rx ' C # 

Z

Z

T

Z

' #

%0 s.%0 ; #0 /'.0; / dx C 

Z

T

 0

for any ' 2 Cc1 .Œ0; T /  /, '  0.

10.3

Strong Convergence of Density

10.3.1 Effective Viscous Flux Identity The main result of this section is the following lemma. Lemma 15 (see [33, formula (3.324) and its proof in Section 3.7.4]). Let .%n ; #n ; un / be the sequence investigated in Theorem 42. Then for any k > 1, there holds p.%; #/Tk .%/  p.%; #/ Tk .%/ D

4 3

.#/ C .#/





Tk .%/divx u  Tk .%/divx u

(455)

with functions Tk defined in (250). Proof. Repeating step-by-step proof of Lemma 10, we arrive at identity Z

T

Z

0

Z

T

Z

D 0





   Q p.%; #/Tk .%/  p.%; #/ Tk .%/ dxdt

  Q k .%/  S.#; u/ W RŒT Q k .%/ dxdt; .t; x/ S.#; u/ W RŒT

(456)

where ; Q 2 Cc1 ..0; T /  /. In order to write the right-hand side of formula (456) in the form of the right-hand side of formula (455), we use properties listed in item (iii) of Theorem 13 yielding identity Z

Z

Z

Z

 2   Q  .#/ C .#/ Tk .%/ divx u dxdt 3 0 0   Z TZ i h io n h  Q k .%/ R W .#/ rx u C .rx u/T  .#/R W rx u C .rx u/T dxdt C T T

0

Q k .%/ dxdt D S.#; u/ W RŒT

T



Z

T

Z

C 0



i h Q k .%/.#/R W rx u C .rx u/T dxdt;  T

1514

A. Novotný and H. Petzeltová

where R W .Z/ D Z TZ 0



P3

i;j D1 Rij .Zij /

and R W Œrx u C .rx u/T  D 2divx u. Consequently,

Q k .%/ dxdt D lim S.#; u/ W RŒT

Z TZ

n!1 0

Z

T

Z

C lim

n!1 0



 Q



4 3

 .#n / C .#n / Tk .%n / divx un dxdt

Q k .%n /!.#n ; un / dxdt; T (457)

and Z

T 0

where

Z TZ  4 Q k .%/ dxdt D S.#; u/ W RŒT  Q .#/ C .#/ Tk .%/ divx u dxdt 3   0 RT R C 0  Q Tk .%/!.#; u/ dxdt; (458)

Z

 h i  

!.#n ; un / D R .t; x/.#n / run C .run /T  .t; x/.#n /R W run C .run /T : In order to treat the difference between the last terms in formulas (457) and (458), we will need two compensated compactness results: Div-Curl lemma reported in Theorem 14 and a specific commutator lemma reported in Theorem 16. Thanks to Theorem 16, the sequence !.#n ; un / is bounded in L1 .0; T I W ˇ;q .I R3 // with some ˇ 2 .0; 1/; q > 1:

(459)

Now we consider the four-dimensional vector fields Vn ŒTk .%n /; Tk .%n /un ;

Un Œ!.#n ; un /; 0; 0; 0:

Seeing that curlUn is compact in W 1;r ..0; T /  I R33 / with some r > 1 by virtue of (459), (423), (430) (and of course divVn is compact in W 1;r ..0; T /  I R33 / because of the fact that .%n ; un / satisfies renormalized continuity equation (133)–(134), with b D Tk ) we may employ Div-Curl lemma reported in Theorem 14 to get !.#n ; un / Tk .%n / * !.#; u/ Tk .%/; in L1 ..0; T /  /; where, due to (450), !.#; u/ D !.#; u/: This result in combination with (456) and (457)–(458) yields the effective viscous flux identity (455).

10.3.2 Oscillations Defect Measure Going back to (419), we deduce employing the hypotheses (369)–(372) that p.%; #/ D d % C pm .%; #/; for some d > 0; where @% pm .%; #/  0. Reasoning as in (285), we get

(460)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

Z

T

Z

 jTk .%n /  Tk .%/jC1 dxdt 1C#

d lim sup n!0

Z

T

0



Z

    .Tk .%n /  Tk .%/ .%n  % / dxdt 1C#

 d lim sup n!1

1515



0

(461)

Z TZ d 0

Z TZ        % Tk .%/% Tk .%/ dxdt  p.%; #/ Tk .%/p.%; #/ Tk .%/ dxdt; 1 C # 0 1 C #

with any  2 Cc1 ..0; T /  /,   0. To derive the last inequality in formula (461), we have employed decomposition (460), the fact that weak  lim p.%n ; #n /g.%n / p.%; #/g.%/ D p.; #/g./ weak  lim p.%n ; #/g.%n / n!1

n!1

for any bounded function g (that holds, thanks to the almost everywhere convergence of the sequence #n ; see (450)), and the relation between the weak limits of monotone functions pm .; #/Tk ./  pm .; #/ Tk .%/  0;

(462)

reported in Theorem 4. Next, we verify that Z

T 0

Z 

jTk .%n /  Tk .%/jq dx D c

hZ

T

Z

0

Z 0

1 1 C #n



Z

 ! 1 q 1 C # jT .% /  T .%/j dxdt n n k k !  .1 C #n / iq=. C1/ jTk .%n /  Tk .%/jC1 dxdt ;

T

where q > 2, provided !. C 1/ D q and !. C 1/=. C 1  q/  17=3, cf. (430). According to (461), expression Z 0

T

Z 

1 jTk .%n /  Tk .%/jC1 dxdt; 1 C #n

which stays at the right-hand side of the last inequality, can be estimated by calculating the right-hand side of (461) from the effective viscous flux identity (455). Now, reasoning as in Step 2 of the proof of Lemma 11, we show the following lemma: Lemma 16. Let .%n ; un / be the density-velocity component of the sequence investigated in Theorem 42. Then oscq Œ%n ! %.QT /  c.M0 ; E0 ; F0 ; T / with some q > 2; where oscq Œ%n ! %.QT / is defined in (281).

(463)

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10.3.3 Renormalized Continuity Equation and Strong Convergence of Density Lemma 16 guarantees satisfaction of all hypotheses of Lemma 12. Using the latter lemma, we deduce that the weak limit .%; u/ constructed in (437) verifies the renormalized continuity equation (133)–(134), in particular equation (292) holds. Recall that also (273) holds in our setting according to (441). We deduce from (273) and (292) with the help of the effective viscous flux identity (455), Z  

Z  %Lk .%/  %Lk .%/ . / dx D 

0



Z 

gk dxdt; where gk D Tk .%/divu  Tk .%/divu; (464)

in particular, gk D gk1 C gk2 ; gk1 D Tk .%/divx u  Tk .%/divx u;   1 p.%; #/Tk .%/  p.%; #/ Tk .%/ : gk2 D 4 3 .#/ C .#/ R R R R Reasoning as in (294), we find that limk!1 0  gk1 dxdt D 0, while 0  gk2 dxdt  0 by virtue of (460) and (462). Now we get from (464) exactly by the same argument leading to (300), completed and modified by decomposition (460), and relation (462) that this formula holds also in the present case. Formula (300) implies %n ! % a.e. in QT by virtue of Theorem 3. This is the last convergence relation needed to conclude the proof of Theorem 42.

11

Stability Results and Weak-Strong Uniqueness

The results presented in this section will rely on the relative energy inequality. They are based on paper [34], where the relative energy method for the Navier-Stokes-Fourier system has been introduced. We have observed in Theorem 39 under very mild assumptions on constitutive laws and transport coefficients that any bounded energy weak solution verifies relative energy inequality (362). If the thermodynamic stability conditions are satisfied, some of various consequences of the relative energy inequality are theorems on the stability of weak solutions with respect to strong solutions and on the weak-strong uniqueness principle. We shall formulate these results in several settings. The least requirement on the constitutive relations is contained in the following result: Theorem 43 (see [31]). Let   R3 be a bounded Lipschitz domain. Assume that the thermodynamic functions p, e are twice continuously differentiable on .0; 1/2 and verify the thermodynamic stability conditions (378). Let .%; #; u/ be a bounded energy weak solution to the Navier-Stokes-Fourier system (323)–(332) in space time cylinder QT , T > 0 in the sense specified in Definition 8, emanating from the initial data (337), verifying in addition

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

0 < % < %.t; x/ < % < 1;

0 < # < #.t; x/ < # < 1 for a.a. .t; x/ 2 QT :

1517

(465)

Finally, suppose that the Navier-Stokes-Fourier system admits a strong solution .r > 0; ‚ > 0; U/ in class X f@t r; @t ‚; @t U; rxm r; rxm ‚; rxm U 2 L1 .QT /;

m D 0; 1; 2g

(466)

emanating from the same initial data. Then .%; #; u/ D .r; ‚; U/: Remark 21. 1. It is to be noticed that conditions (323)–(332) include implicitly requirements (8), (14)–(15), in particular, that e; p; s; ; ; are continuously differentiable functions of density and temperature and that e; p verify the Gibbs relation. 2. The drawback of this theorem dwells in the fact that it is not known whether weak solutions satisfying (465) do exist on (arbitrary large) time interval .0; T /. There are however situations (characterized by the constitutive laws and transport coefficients) when the weak-strong uniqueness principle holds unconditionally in the class of weak solutions whose existence is guaranteed by Theorems 40 and 41. We report the following result: Theorem 44 (see [34] for the original version of the result with  D 5=3; ˇ D 1; see [31] for the case  D 5=3, ˇ 2 .2=5; 1; see [70] for the case  > 3=2, ˇ D 1). Let  be a bounded Lipschitz domain. Let the constitutive laws for e; p and transport coefficients ; ; satisfy all assumptions of existence Theorem 40 or of existence Theorem 41. Assume further that the Third law lim S .Z/ D 0

Z!1

(467)

is verified and that the function P is twice continuously differentiable on .0; 1/. Let .%; #; u/ be a bounded energy weak solution to the Navier-Stokes-Fourier system (323)–(332) in space time cylinder QT , T > 0 emanating from initial data .%0 ; #0 ; u0 / in the class (337) and external force f 2 L1 .QT I R3 / constructed in Theorem 40 or 41 according to the case. Let .r > 0; ‚ > 0; U/ be a strong solution to the Navier-Stokes-Fourier system (323)– (332) in class (466) emanating from the .r0 ; ‚0 ; U0 / 2 (337) and external force g 2 L1 .QT I R3 /. Then there exists a positive constant c depending on the parameters of constitutive laws, transport coefficients, QT , lower bounds of r and ‚, and the norms of the strong solution in class (466) (but independent on the weak solution, initial data, and external forces) such that   E.%; #; u j r; ‚; U/  c E.%0 ; #0 ; u0 j r0 ; ‚0 ; U0 / C kf  gk2L1 .QT / ;

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where E is the relative energy functional introduced in (357). In particular, if .%0 ; #0 ; u0 / D .r0 ; ‚0 ; U0 / and f D g, then .%; #; u/ D .r; ‚; U/: Remark 22. 1. Theorems 43 and 44 remain true for the bounded energy weak solutions with the complete slip (22) boundary conditions (see [70, Section 6] for more details). 2. Since on bounded domains the class of finite energy weak solutions is contained in the class of bounded energy weak solutions, Theorems 43 and 44 are true also for the finite energy weak solutions. 3. Theorems 43 and 44 are formulated in the class of bounded energy weak solutions. They are however true also in the seemingly larger class of dissipative solutions since the proof relies basically on the relative energy inequality. 4. On unbounded domains with boundary conditions (24) one cannot, in general, construct finite energy weak solutions. In some situations satisfied by the conditions at infinity, one can construct bounded energy weak solutions on unbounded domains with uniformly Lipschitz boundary, provided e; p; s; ; ; verify assumptions of Theorem 40 or 41. These bounded energy weak solutions are not necessarily dissipative solutions, and they do not verify the weak-strong uniqueness principle. In the class of bounded energy weak solutions, there are however solutions that are dissipative. Then the weak-strong uniqueness principle holds in the class of bounded energy dissipative solutions. The reader is advised to consult [70], Theorem 2.5 (for the no-slip boundary conditions) and Theorem 6.5 (for the complete slip boundary conditions) to learn more about these problems. 5. Under certain additional hypotheses, a strong solution .r; ‚; U/ exists at least locally in time. For example: If  is a bounded domain of class C 4 , f 2 C 1 .Œ0; T I W 2;2 .//, cv ; ; ; 2 C 3 .0; 1/, cv  c v > 0,    > 0,  > 0, if the initial data verify 0 < r0 2 W 3;2 ./, 0 < ‚0 2 W 3;2 ./, U0 2 W 3;2 .I R3 / and satisfy the natural and classical compatibility conditions at the boundary, then there exists TM > 0 such that the Navier-Stokes-Fourier system (323)–(332) admits a unique strong solution .r > 0; ‚ > 0; U/ on the interval Œ0; TM / in a subclass of (466) (see [104, Theorem A and Remark 3.3]). 6. The weak-strong uniqueness principle turns some of the blow up criteria for strong solutions of the Navier-Stokes-Fourier system to the criteria of regularity of weak solutions (see [53] and [45] for more details about this issue). 7. In some situations, the assumption that the constitutive equations must verify the third law (see assumption (467) in Theorem 44) can be disregarded. This is the case of constitutive laws and weak solutions mentioned in Remark 18 as shown in [58, Theorem 4.1].

11.1

Sketch of the Proof of Theorems 43 and 44

11.1.1 Relative Energy Inequality with a Strong Solution as a Test Function We denote   A D r.@t U C U  rU/ C rp.r; ‚/  rg  .u  U/ C S.‚; rU/ W r.u  U/;

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1519

and  S.‚; rU/ W rU q.‚; r‚/  r‚  q.‚; r‚/  r.‚  #/ C C : BD.#‚/ r.@t s.r; ‚/CUrs.r; ‚// ‚ ‚2 ‚

Since the trio .r; ‚; U/ verifies equations (324)–(325) with (327) and boundary conditions (331)–(332) in the classical sense, there holds Z



Z .A C B/ dxdt D 0:

0



We now add this identity to the right-hand side of the relative energy inequality (362). Employing several times conveniently the Gibbs relation (326) in the form 1r @# p.r; ‚/ D r@ s.r; ‚/ and the continuity equation (323) satisfied by .r; U/, we transform after a long and tedious computation relative energy inequality (362) to the form stated in the following lemma: Lemma 17. Let  be a bounded Lipschitz domain and f; g 2 L1 .QT ; R3 /. Let .%; #; u/ be a bounded energy weak solution to the Navier-Stokes-Fourier system emanating from initial data .%0 ; #0 ; u0 / specified in (337) and external force f. Let .r > 0; ‚ > 0; U/ be a strong solution of the same system emanating from initial data .r0 ; ‚0 ; U0 / in (337) and external force g, in the class (466). Then,

Z



E.%; #; u j r; ‚; U/. /  E.%0 ; #0 ; u0 j r0 ; ‚0 ; U0 / Z Z Z Z Dmech .t; x/ dxdt C Dth .t; x/ dxdt  R.t; x/ dxdt

Z

C 0



0



0

(468)



for a.a. 2 .0; T /, where ‚ # ‚ S.#; ru/WruS.#; ru/ W rU C S.‚; rU/Wr.U  u/ C S.‚; rU/WrU; # ‚  q.#; r#/ ‚ q.#; rx #/ Dth D   rx #   r‚ # # #  #  ‚ q.‚; r‚/ q.‚; r‚/  r.‚  #/ C  r‚ ; C ‚ ‚ ‚   R D .%  r/@t U C .%u  rU/  rx U  .U  u/ C .%f  rg/  .u  U/     S.%; #/  .%  r/@% S.r; ‚/  .#  ‚/@# S.r; ‚/  S.r; ‚/ @t ‚ C U  r‚    p.%; #/  .%  r/@% p.r; ‚/  .#  ‚/@# p.r; ‚/  p.r; ‚/ divU   C% s.r; ‚/  s.%; #/ .u  U/  rx ‚ C .f  g/  .u  U/;

Dmech D

and S.%; #/ D %s.%; #/:

(469)

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11.1.2 Relative Energy Inequality Rewritten We shall investigate separately the cases 0 < # < ‚ and #  ‚. In the first case, we have  1f0 0 for any fixed # > 0: @%

(482)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1525

1;1 Let us fix constant #Q > 0, F 2 W 1;1 ./ and suppose that %Q 2 Wloc ./ verifies (480) a.e. in . Then necessarily %.x/ Q > 0 for all x 2 . Indeed, by virtue of (379)1 , %Q satisfies equation

Q D F C c#Q ; P#Q .%/

(483)

on any positivity component fx 2 j%.x/ Q > 0g  , where c#Q is a constant that may depend on the specific positivity component and where Q 2 P Q .Œ0; 1//  R Œ0; 1/ 3 z 7! P#Q .z/ @% H#Q .z; #/ # is an invertible (increasing) application such that limz!0 P#Q .z/ D 1 by virtue of the thermodynamic stability condition (378)1 extended by (482). Therefore, the right-hand side of (483) is bounded in contrast with the left-hand side, which tends to minus infinity for %Q approaching zero. Consequently, %Q must remain bounded away from zero on . Q maximize the total entropy functional Finally, equilibrium solutions .%; Q #/ Z %s.%; #/ dx .%; #/ ! 

in the class of all measurable functions %  0, # > 0 verifying constraints Z Z  Z Z    Q  %F %e.%; #/  %F dx D %e. Q %; Q #/ Q dx: % dx D %Q dx; 





(484)



In order to see this property, use the definition of Helmholtz function and (484), where F is replaced by using formula (483) to get #Q

Z  

Z  Z   Q  %s.%; #/ dx D Q dx C .%Q  %/F dx %s. Q %; Q #/ H#Q .%; #/  H#Q .%; Q #/ 

D



Z 

Q @H#Q .%; Q #/



@%

H#Q .%; #/  .%  %/ Q

 Q dx: Q #/  H#Q .%;

(485)

Q The most right integral is however nonnegative and equal to zero if and only if .%; #/ D .%; Q #/ by virtue of (355) or alternatively by virtue of Lemma 14. The above discussion leads to the following theorem: Theorem 45 (see [44, Theorem 4.1]). Let   R3 be a bounded domain. Assume that the thermodynamic functions p, e, and s are continuously differentiable in .0; 1/2 and that they satisfy relations (326), (378) together with condition (482). Let F 2 W 1;1 ./: Then for given constants M0 > 0; E0;F , there exists at most one solution %; Q #Q of static problem (480) in the class of locally Lipschitz functions subject to the constraints (481). In addition, %Q is strictly positive in , and, moreover, Z Z Q dx  %s. Q %; Q #/ %s.%; #/ dx 



for any couple %  0; # > 0 of measurable functions satisfying (484).

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A. Novotný and H. Petzeltová

Remark 23. 1. If the solution of problem (480)–(481) with F 2 W 1;1 ./ exists, then %Q 2 W 1;1 ./, and it is given by the formula   %.x/ Q D P 1 F .x/ C c Q #

where c 2 R and #Q > 0 are determined through implicit relations Z Z h 



P 1 Q #

  F .x/ C c dx D M0 ; P 1 Q #



       i F .x/ C c e P 1 F .x/ C c ; #Q  P 1 F .x/ C c F .x/ dx D E0;F Q Q #

#

Q with P#Q ./ D @% H#Q .; #/. 2. The result is based on strict positivity of the equilibrium density, which follows from the assumptions (378)1 and (482). A simple example p.%; #/ D a% with a > 0;  > 1 shows that the solution of (480) may not be strictly positive in  at least for small values of the total mass M0 . Indeed, the function

%.x/ Q D

 1 .F .x/ C c/ a

1 C !  1

; c2R

represents a classical solution of (480). In addition, it can be shown that, in general, the solutions of (480) are not uniquely determined by the total mass M0 (see [24] and Remark after Theorem 30). 3. Existence theory of finite energy solutions with specific constitutive laws for p; e satisfying assumptions (369)–(375) was built in Theorems 40and 41. It is to be noticed that these assumptions obey Gibbs relations (326), thermodynamic stability conditions (341), as well as the additional condition (482). Theorem 45 therefore applies to this situation. The following lemma concludes this section by the observation that boundedness of the entropy and the total mass of a static state imply bounds of its norm, at least when the pressure and internal energy satisfy assumptions (369)–(375) (needed for the existence theory in Theorems 40, 41). This result will be useful in the sequel. Lemma 18 (see [44, Lemma 5.5]). Let the thermodynamic functions p, e, and s be given through (369)–(375). Let %; Q #Q be a solution of the problem (480) such that Z

Z %Q dx D M0 ; 

Q dx SQ 0 %s. Q %; Q #/



for certain constants M0 > 0; SQ 0 2 R: Then there exist constants %; #; %; # depending only on M0 ; SQ 0 ; and kF kL1 ./ such that Q < % for all x 2 : 0 < # < #Q < #; 0 < % < %.x/

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

12.2

1527

Longtime Behavior of Conservative System

Until the end of Sect. 12, hypotheses (369)–(377) with  D 5=3, 1=2  ˇ  1 are assumed. (These values were considered in [44].) In this situation, existence of finite energy weak solutions on time interval .0; T / is guaranteed by Theorem 40. Moreover, pressure and internal energy obey all assumptions of Theorem 45 dealing with the static states. First observation is that the weak solutions constructed in Theorem 40 on time interval .0; T / can be extended to the time interval .0; 1/. This is subject of the following theorem: Theorem 46 (see [44, Theorems 4.4 and 4.5]). Let the hypotheses of Theorem 40 be satisfied and, in addition, 1 f 2 L1 ..0; 1/  /I R3 /; ˇ 2 Œ ; 1: 2 Then there holds: If 0 < T1 < T2 and if .%1 ; #1 ; u1 / is a finite energy weak solution constructed in Theorem 40 on time interval Œ0; T1 /, then there exists a weak solution .%2 ; #2 ; u2 / with the same properties as stated in Theorem 40 on the time interval Œ0; T2 / such that .%2 ; #2 ; u2 /jŒ0;T1 / D .%1 ; #1 ; u1 /: Theorem 47 (see [44, Theorem 4.5]). Let the hypotheses of Theorem 46 be satisfied. Let f%; u; #g be a weak solution of the system (323)–(332) on time interval Œ0; 1/ constructed in Theorem 46, where f D rx F; F D F .x/; F 2 W 1;1 ./: Then there exist %Q D %.x/; #Q D const > 0 solving the static problem (480)–(481) such that 5

%.t; :/ ! %Q in L 3 ./; 5

(486)

.%u/.t; :/ ! 0 in L 4 .I R3 /;

(487)

#.t; :/ ! #Q in L4 ./

(488)

as t ! 1. Sketch of the proof. The main idea of the proof is to show that a norm implying convergence (486)–(488) is dominated by the “distance” of the trajectory f%; u; #g from Q by means of the relative energy functional. In view of the equilibrium state f%; Q 0; #g inequality (471) (or alternatively in view of Lemma 14), the theorem will be proved once we achieve Q 0/ ! 0 as t ! 1: E.%.t; /; #.t; /; u.t; / j %; Q #;

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12.2.1 Weak Compactness of the Set of Weak Solutions The following stability result can be shown in the same way as the similar stability result proved in Theorem 42. Lemma 19 (see [44, Theorem 4.2]). Let the assumptions of Theorem 46 be satisfied. Let fn 2 L1 ..0; T /  /, kfn kL1 ..0;T //  c:

(489)

Let .%n ; #n ; un / be a sequence of finite energy weak solutions to the system (323)–(332), with f D fn , such that  1 2 %n jun j C %n e.%n ; #n / . ; :/ dx  E; ess sup

!0  2 Z %n s.%n ; #n /. ; :/ dx  S; ess inf Z 

!0 

(490) (491)

and Z %n dx  M ;

(492)



uniformly in n, where S > M S1 ; S1 D lim S .Z/  1; Z!1

(493)

and M > 0, E > 0. Finally, suppose that one of the following alternatives holds: either %n .0/ %0;n ! %0 in L1 ./;

(494)

divx un ! 0 in L1 ..0; T /  /:

(495)

or

Then, passing to a subsequence if necessary, we have fn * f in L1 ..0; T /  I R3 /; 5

%n ! % in L1 ..0; T /  / \ Cweek .Œ0; T I L 3 .//; #n * # in L1 .0; T I L4 .// and strongly in L1 ..0; T /  /; rx #n * r# in L2 .0; T I L2 .I R3 /; un * u in L2 .0; T I W01;2 .I R3 //;

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1529

where the trio .%; #; u/ is again a weak solution of the system driven by the force f. Moreover, Z  

  Z  1 1 %n jun j2 C %n e.%n ; #n / dx ! %juj2 C %e.%; #/ dx in L1 .0; T /; 2  2 Z Z %n s.%n ; #n /. ; :/ dx ! %s.%; #/. ; :/ dx for a.a. 2 .0; T /: 



Remark 24. 1. Hypotheses (494), (495) are of rather different character. Assumption (494) prevents possible spatial oscillations of the density field that may be imposed through the initial data. The conclusion of the theorem is then important for existence theory for the initial-boundary value problems. Hypothesis (495) is satisfied when f%n ; n ; un g represent suitable time shifts of a single trajectory, which provides useful information of the longtime behavior of the corresponding single solution. 2. The meaning of (493) is to avoid degenerate states with zero temperature. Since the entropy can be always normalized so that S1 2 f0; 1g, condition (493) reduces to strict positivity of S in the former case and to finiteness of S in the latter.

12.2.2 Time Shifts of the Weak Solution Let .%; #; u/ be a finite energy weak solution determined by Theorem 46. We introduce sequences %n .t; x/ D %.t C n; x/; #n .t; x/ D #.t C n; x/; un .t; x/ D u.t C n; x/; t 2 .0; T /; x 2 : It is a routine matter to show that .%n ; #n ; un / verifies hypotheses of Lemma 19. In particular, it Rfollows from the dissipation balance in the form (347) and the fact that the total entropy 7!  Œ%s.%; #/. ; x/ dx is non decreasing (see (349)) that Z

T 0

kun k2W 1;q .IR9 / ! 0;

Z

T 0

krx # n k2L2 .IR3 / ! 0; q D

8 : 5ˇ

With this information at hand, application of Lemma 19 yields existence of functions Q / such that %Q D %.x/ Q and #Q D #.t %n ! %Q in L1 ..0; T /  / \ Cweak .0; T I L5=3 .//; #n ! #Q in L2 .0; T I W 1;2 .//; Q // is an equilibrium state (480), (481). In accordance with Theorem 45, where .%.t; Q x/; #.t the equilibrium solution is uniquely determined by the constants of motion (481), whence Q / D #Q D const . Moreover, according to Lemma 18, there are numbers %.t; Q x/ D %.x/, Q #.t 0 < # < # < 1, 0 < % < % < 1 (determined by M0 , E0 , SQ 0 and kF kW 1;1 ./ ) such that %  %Q  %;

#  #Q  #:

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12.2.3 Relative Energy Function The above convergence relations yield Z  Z    1 Q  %F %n u2n C H#Q .%n ; #n /  %n F dx ! H#Q .%; Q #/ Q dxI  2  whence, recalling (483), Q 0/ ! 0: Q #; ŒE.%n ; un ; #n j%; Moreover, according to Lemma 14, Z Z

Q 0/. /  c E .%n ; #n ; un j %; Q #;





 4 Q 2ess dx %n u2n dxCŒ1res CŒ%n 5=3 Q 2ess CŒ#n # res CŒ#n res CŒ%n %

in terms of notation (471)–(472), where c D c.%; %; #; #/ > 0. This finishes the proof of Theorem 47. The following theorem asserts that the set of equilibria is a kind of attractor for all trajectories emanating from a set of bounded total mass and energy. It means that all trajectories approach the set of equilibria uniformly with growing time. As the total mass and energy are constants of motion, we cannot expect the attractor to be bounded or even compact in the associated energy norm. It is basically the only situation when the energetically insulated Navier-Sokes-Fourier system possesses an attractor. Theorem 48 (see [44, Theorem 5.1]). Let the assumptions of Theorem 47 be satisfied. Let M > 0; E F ; S be given, with S > M S1 ; S1 D limZ!1 S .Z/  1. Then for any " > 0, there exists a time T D T ."/ such that k%.t; :/  %k Q

5

L 3 ./

k.%u/.t; :/k

5

 ";

L 4 .IR3 /

 ";

Q 4 k#.t; :/  #k L ./  " for a:a: t  T ."/, for any weak solution f%; u; #g of the Navier-Stokes-Fourier system defined on .0; 1/ constructed in Theorem 46 and satisfying Z %.t; :/ dx > M ; t 2 .0; 1/; Z 

(496)



1 . %juj2 C %e.%; #/  %F /.t; :/ dx < E F ; t 2 .0; 1/; 2 Z ess inf %s.%; #/.t; :/ dx > S ; t>0 

(497) (498)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1531

Q is a solution of the static problem (480) determined uniquely by the condition where .%; Q #/ Z

Z %Q dx D 

Z  

% dx; 

Z    1 Q  %F %juj2 C %e.%; #/  %F dx: %e. Q %; Q #/ Q dx D  2

Remark 25. 1. The total mass and the total energy are constant in time, so the specific choice of the initial time does not play any role; the interval .0; 1/ may be replaced by .T; 1/. In general, the case S1 D 1, is possible, so the meaning of the condition (498) is to avoid degenerate states with vanishing absolute temperature. 2. The rate of decay to the set of static solutions characterized by the mapping " ! T ."/ depends on M ; E F , and the structural properties of the constitutive functions. 3. Condition (498) is automatically satisfied if the fluid obeys the third thermodynamical law limZ!1 S .Z/ D 0.

12.3

Longtime Behavior for Time-Dependent Forcing: Blow Up of Energy

The choice of time-independent nonconservative driving force f D f.x/; f 2 L1 .I R3 / such that f 6 rx F reflects a constant supply of the mechanical energy into the system that is, in accordance with second law of thermodynamics, irreversibly converted to heat. As the boundary of  is thermally insulated, the system accumulates the energy, therefore, inevitably E.t/ D

Z   1 %juj2 C %e.%; #/ .t; :/ dx ! 1 as t ! 1:  2

To avoid blow up of E.t/ in the general situation of time-dependent forcing term, the function f must behave like gradient of a scalar potential when time tends to infinity, or f must rapidly oscillate as time tends to infinity. The former situation is described in Theorem 49 and the latter in Theorem 50 in the next section. The main theorem of this section reads: Theorem 49 (see [44, Theorem 5.2]). Let the assumptions of Theorem 46 be satisfied. Then for any finite energy weak solution of the Navier-Stokes-Fourier system defined on the interval .0; 1/ constructed in Theorem 46, one of the following alternatives holds: • Either Z E.t / D 

1 2

 %juj2 C %e.%; #/ .t; :/ dx ! 1 for t ! 1;

(499)

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A. Novotný and H. Petzeltová

• or there is a constant E1 such that E.t/  E1 for a:a: t > 0:

(500)

Moreover, in the latter case, each sequence of times n ! 1 contains a subsequence such that fn .t; x/ D f.t C n ; x/ satisfies fn ! rx F weakly-(*) in L1 ..0; T  /I R3 / for any fixed T > 0;

(501)

where the limit F D F .x/; F 2 W 1;1 ./ may depend on the choice of f n g1 nD1 . Sketch of the proof of Theorem 49. The proof of this Theorem follows from the compactness Lemma 19 and Lemma 18. Assume that there is a solution f%; u; #g such that E. n /  E < 1 for n ! 1: Then, due to the structural properties of e; s, the total entropy is bounded Z %s.%; #/. n ; :/ dx  S; 

and, as the total entropy is nondecreasing in time, assume Z lim

t!1 

%s.%; #/.t; :/ dx D S:

(502)

For time shifts %n .t; x/ D %. n C t; x/; un .t; x/ D u. n C t; x/; #n .t; x/ D #. n C t; x/ it implies, together with the entropy balance (341), that un ; rx un ; r#n ! 0 in, say L1 ..0; T /  I R3 /. Then application of the compactness Lemma 19 gives (weak) Q fg satisfying, in the sense of distributions, convergence of f%n ; #n ; fn g to a limit f%; Q #; Q D %f; rx p.%; Q #/ Q

Z

Z

Q %s. Q %; Q #/.t; :/ dx D S :

%Q dx D M0 ; 



29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1533

The entropy is a strictly increasing function of temperature, so the last equality implies that #Q is independent of t. Then, in accordance with Theorem 45, f is a gradient of a scalar function F .x/. The last point is to show that the energy cannot oscillate, i.e., lim sup E.t/ D 1; lim inf E.t/ < 1 t!1

t!1

is excluded. If this is valid, then the continuity of the energy implies that for any K > 0, there exists a sequence of times n ! 1 such that E. n / D K: Now, define again time shifts of solutions and deduce, as above, that they converge to a static solution satisfying Q D %r Q #/ Q x F; #Q D const > 0; rx p.%; Z

Z %Q dx D M0 ; 

Q %s. Q %; Q #/.t; :/ dx D S :

(503)



and Z 

 Q  %F %e. Q %; Q #/ Q dx D K:

(504)



However, by virtue of Lemma 18, relations (503) and (504) are not compatible for arbitrary (large) K, which concludes the proof of Theorem 49. Examples of external forces which drive the energy to infinity are given in the following corollary. These examples are direct consequences of Theorem 46. R The fact that the blowup E.t/ ! 1 implies the blowup of the thermal energy Eth . / D  %. ; x/e.%. ; x/; #. ; x// dx is formulated in Corollary 4. Corollary 3. Let the assumptions of Theorem 46 be satisfied. Let f 2 L1 ..0; T /  I R3 / satisfies one of the following conditions: (i) (ii) (iii) (iv)

f D f.x/; f 6 rx F ; f is time periodic, nonconstant in time, f.t C T; x/ D f.t; x/ for all t; xI f is almost periodic, nonconstant in timeI f is asymptotic periodic (almost periodic) nonconstant in time, meaning

sup jf.t; x/g.t; x/j!0 as t !1; where g is periodic (almost periodic) nonconstant in time: x2

Then E.t/ D

Z   1 %juj2 C %e.%; #/ .t; :/ dx ! 1 as t ! 1  2

for any finite energy weak solution f%; u; #g of the Navier-Stokes-Fourier system defined on .0; 1/   constructed in Theorem 46.

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Remark 26. 1. The first condition together with Theorem 45 gives a complete description of the longtime behavior of the energetically insulated Navier-Stokes-Fourier system driven by a time-independent external force. 2. In contrast with the static case, the function f.t; x/ D rx F .t; x/ with F periodic and nonconstant in time satisfies condition (ii), which leads to the explosion of the total energy. With the help of Corollary 3, it is possible to construct forces that tend to zero when time goes to infinity and vanish on a large set, but still drive the energy of the system to infinity. See [44, Example 5.1]. The following result shows that boundedness of the internal energy implies boundedness of the total energy. Corollary 4. Let the assumption of Theorem 46 be satisfied. Let f%; u; #g be a global finite energy weak solution on Œ0; 1/ to the Navier-Stokes-Fourier system constructed in Theorem 46 such that E.t/ D

Z   1 %juj2 C %e.%; #/ .t; :/ dx ! 1 as t ! 1:  2

(505)

Then Z %e.%; #/.t; :/ dx D 1:

ess lim sup t!1

In fact, if ess lim supt!1

R



 %e.%; #/

dx < 1; then also

sup k%.t; :/k

5

L 3 ./

 c;

and the total entropy is bounded, which in turn yields a sequence of times n ! 1 such that Z

n C1

n

ku.t; :/kL6 .IR3 / ! 0:

These two relations imply ess lim inft!1 E.t/ < 1, in contrast with (505).

12.4

Longtime Behavior: Stabilization to Equilibria for Rapidly Oscillating Driving Forces

An example of nontrivial driving forces that, in contrast with the examples in the last section, stabilize the system, is given in this section. The previous discussion may suggest that almost all time-dependent driving forces imposed on the energetically insulated NavierStokes-Fourier system result in the blowup of the energy for time tending to infinity. Instead of forces that converge to a conservative form or simply vanish, also rapidly oscillating forces may stabilize the system. This means that the condition (501) allows for some interesting exceptions and that thanks to rapid oscillations the solutions may converge to

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

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the homogeneous static state as time goes to infinity. The specific choice of the driving force was studied in [44], where the following result was proved: Theorem 50 (see [44, Theorem 5.3]). Let the assumptions of Theorem 46 be satisfied. Let the driving force take the form f.t; x/ D !.t ˇ /w.x/;

t > 0; x 2 ;

where w 2 W 1;1 ./; w ¤ 0, and ˇZ ˇ ! 2 L1 .R/; ! ¤ 0; sup ˇˇ

>0

0

ˇ ˇ !.t /dt ˇˇ < 1;

(506)

are given functions, with ˇ > 2. Then any global-in-time finite energy weak solution of the Navier-Stokes-Fourier system constructed in Theorem 46 satisfies %u.t; / ! 0 in L5=4 .I R3 / as t ! 1;

(507)

#.t; / ! #Q in L4 ./ as t ! 1;

(508)

%.t; / ! %Q in L5=3 ./ as t ! 1;

(509)

where %; Q #Q are positive constants, %Q D

1 jj

Z %dx: 

Proof of Theorem 50. The proof is based on the energy estimates obtained by means of the total dissipation balance and on the analysis of possible oscillations of the driving force f. The idea is to apply Lemma 19 on the sequence of time shifts %n .t; x/ D %.t C n; x/; #n .t; x/ D #.t C n; x/; un .t; x/ D u.t C n; x/; fn .t; x/ D !..t C n/ˇ /w.x/; t 2 .0; T /; x 2 : To this end, it should be shown that

Z  E. / D 

fn * 0 in L1 ..0; T /  I R3 /;  1 2 %juj C %e.%; #/ . ; /dx ! E1 for ! 1; 2

(510) (511)

and, exactly as (502), observed that Z %s.%; #/. ; /dx ! S1 as ! 1: 

(512)

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With (510)–(512) at hand, application of Lemma 19 yields %. n C ; / ! %Q in Cweek .Œ0; 1I L5=3 .//; u. n C ; / ! 0 in, say, L1 ..0; 1/  /; #. n C ; / ! #Q in, say, L1 ..0; 1/  / for any n ! 1, where %; Q #Q is the (constant) solution to the stationary problem (480), uniquely determined by Q %e. Q %; Q #/jj D E1 :

%jj Q D M0 ;

To finish the proof, it remains to show convergence (507)–(509). This follows from (511), (512), and the coercivity of Helmholtz function established in Lemma 14 (see (354) and (359)). Hence, it is sufficient to show (510) and (511). Proof of (510). It is enough to see that Z

1

!..t C n/ˇ / .t / dt ! 0 as n ! 1 for any

0

2 Cc1 .0; 1/:

This is a consequence of hypothesis (506), and ˇ > 2: Z

1

!..t C n/ˇ / .t / dtD 

0

1 ˇ

Z

h

1 0

0

O..t C n/ˇ /

i .t C n/1ˇ C.ˇ1/.t C n/ˇ .t / dt!0; (513)

where O. / D

Z



!.t / dt 0

is bounded according to (506). The convergence (511) follows from the total energy balance (343) and the following relation Z

0

!.t ˇ /

Z 

w.x/.%u/.t; x/ dxdt ! I1 2 R for ! 1:

(514)

Proof of (514). First, deduce energy estimates and then uniform bounds via an iteration process. Denoting Z w  .%u/.t; :/ dx;

U .t / D 

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1537

proceed as in (513) to get Z

T

!.t ˇ /

Z

T 1

Z

T

!.t ˇ /U .t / dt 

w.x/.%u/.t; x/ dxdt D T 1



ˇZ  ˇˇ

ˇT 1 ˇˇ 1 ˇˇ T ˇ ˇ ˇ 1ˇ ˇ ˇ 1ˇ d U .t / dt ˇ : U .t /ˇ C ˇ O.t / .1  ˇ/t U .t / C t ˇO.t /t ˇ T 1 ˇ ˇ ˇ T 1 dt Hence, (514) follows provided that U; dtd U are proved to be bounded functions. jU .t /j 

p

p M0 jjwjjL1 .IR3 / jj %u.t; /jjL2 .IR3 / :

Take a test function ' D .t /w.x/ in the momentum equation (324) to get Z  Z  d U .t /D .%Œu ˝ u W rx wCp.%; #/divx wS W rx w/ .t; / dx C %jwj2 .t; /!.t ˇ / dx dt   (515) for a.a. t 2 .0; 1/. To get uniform bounds for dtd U , the total dissipation balance (346) R M0 is used. Now, fix #Q > 0; %Q D jj ; M0 D  % dx and rewrite equation (346) in Q 0/. Denoting D.t / D E.%; #; u j %; Q 0/ and terms of relative energy E.%; Q #; Q #;  #; u j %; R #Q  qr# dx, the equation (335) can be rewritten as follows: Q.t / D  # S W ru  # Z



D. / C

Q M0 ; E0 ; SQ 0 / C Q.t / dt  C .#;

0

Z



!.t ˇ /U .t / dt:

(516)

0

The next goal is to establish uniform bounds for D, which then imply bounds for Q and the time derivative of U . The coercivity of Helmholtz function H#Q gives Z j.%Œu ˝ u W rx w C p.%; #/divx w/ .t; /j dx  c1 E.t/  c2 .1 C D.t // 

for a.a. t 2 .0; 1/. Also, writing r p 2 2 .#/ Œrx u C rxt u  divx uI W Œrx w C rxt w  divx wI .#/# S W rx w D # 3 3 p p C .#/=#divx u divx w .#/#; yields Z 

S W rx w dx

2 

Z 2 .#/ jrx u C rxt u  divuIj2 dx .#/# dx 3  #  Z Z .#/ jdivx uj2 dx .#/# dx: Cckwk2W 1;1 .IR3 /  # 

ckwk2W 1;1 .IR3 /

Z

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Assumptions (376) on  and  give Z

Z

.1 C # 2 /.t; / dx  c2 .1 C

..#/ C .#//#.t; / dx  c1 

p

D.t //:



Altogether, the previous estimates imply Z D. / C

Z





Q.t / dt  c1 C 0

!.t ˇ /U .t / dt;

(517)

0

p D. /  0; Q. /  0; jU .t /j  c2 D.t /;

(518)

ˇ ˇ q p p ˇ d ˇ ˇ U . /ˇ  c3 D. / C c4 Q. / 1 C D. / C c5 ˇ d

ˇ

(519)

and

for a.a. 2 .0; 1/, where constants ci , i D 1; : : : ; 5 depend only on M0 ; E0 ; SQ 0 and on the norms kwkW 1;1 .IR3 / and k!kL1 .R/ . Moreover, the entropy balance equation (345) gives Z

0

Q.t / dt  jSQ 0 j C

Z

%js.%; #/j. / dx  jSQ 0 j C c.%; #/ C D. /:

(520)



Uniform Bounds. Next, estimates (517)–(520) are used to obtain a uniform bound on D. The first step in the proof is to obtain a bound D. /  c 2 . Then an iteration procedure, where by repeating the same argument many times gives better and better bounds in each step, and after finitely many steps the uniform bound is obtained. The initial bound on D, follows from (517), (518):   Z

p .1 C D.t // dt I D. /  c 1 C 0

whence D. /  c 2 for a.a. 2 .1; 1/: This estimate is a starting point for the iteration procedure described in what follows. Assume that the following estimate has been already proved: D. /  c m for a.a. 2 .1; 1/ for a certain m 2 Œ0; 2. Using (521) in (519) gives ˇ ˇ2   ˇd ˇ ˇ U . /ˇ  c 2m C Q. / m=2 for a.a. 2 .1; 1/I ˇ d

ˇ

(521)

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

1539

whence, thanks to (520) and (521) Z

T T 1

ˇ2 ˇ Z ˇ ˇ d ˇ U . /ˇ dt  c1 T 2m C T m=2 ˇ ˇ d

!

T

Q.t / dt

 c2 .T 2m C T 3m=2 /  c3 T 2m

0

(522) provided T > 2. On the other hand, with the bounded (see (506)) primitive function O O. / D

Z



!.t / dt; 0

the following estimate holds Z

T

ˇ

!.t ˇ /U .t / dt 

T 1

ˇZ ˇ ˇ T h i ˇ ˇ ˇ ˇ ˇ 1ˇ 0 Cˇ O.t / .1  ˇ/t U .t / C t U .t / dt ˇ : ˇ T 1 ˇ T 1

ˇT ˇ ˇ ˇ ˇO.t ˇ /t 1ˇ U .t /ˇ Therefore,

ˇ ˇZ ˇ ˇ T   ˇ ˇ ˇ !.t /U .t / dt ˇ  c1 T 1ˇCm=2 C T ˇCm=2 C T mC1ˇ  c2 T 1ˇCm ˇ ˇ ˇ T 1

(523)

using jU .t /j  ct m=2 , and (522). Finally it follows that D. /  c 2ˇCm I

(524)

in particular, (521) implies (524). Hence, using the assumption ˇ > 2, it holds, after finitely many steps esssup 2.0;1/ D. / < 1: Now, it follows from (523) Z



I . / D 0

!. ˇ /U .t / dt ! I1 2 R for ! 1;

and, using the total energy balance (343) also Z  E. / D 

 1 %juj2 C %e.%; #/ . ; / dx ! E1 for ! 1: 2 t u

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Remark 27. 1. Even if the restriction ˇ > 2 is probably not optimal, some uniform growth of frequency is necessary. Indeed, consider f.t; x/ D !.nt /w for t 2 .Tn1 ; Tn /; T0 D 0; where ! is a time-periodic function with zero mean, and the sequence of times Tn is chosen in such a way that E.t/  n for a:a: t  Tn : Such sequence of times is possible to find applying repeatedly the existence Theorem and Corollary 3 to the problem on the intervals .Tn ; 1/, with initial data 5

5

%.Tn ; :/ 2 L 3 ./; %u.Tn ; :/ 2 L 4 .I R3 /; #.Tn ; :/ 2 L4 ./; where # is uniquely determined by the equation  %.Tn ; :/s %.Tn ; :/; #.Tn ; :/ D %s.Tn ; :/: 2. Similar stability result was proved in [4] for unbounded driving forces, when the oscillations are so rapid that they in some sense prevail the growth in time, or the decay in time allows for slower oscillations, specifically, f D t ı !.t ˇ /w; where ! and w satisfy the assumptions of Theorem 50, and ı > 0; ˇ  2ı > 2 or ı  0; ˇ  ı > 2: The assertions (507)–(510) hold true for this kind of forcing terms. The proof of this result follows the same lines as that of Theorem 50; it is based on precise energy estimates together with careful analysis of possible oscillations of the driving force.

13

Conclusion

In spite of the fact that the theory of weak solutions to the compressible Navier-Stokes equations is a young topic, it already benefits of quite large comprehensive literature including monographs. The first results appeared in the pioneering seminal work of P.L. Lions [77] dealing with the equations in barotropic regime. The Lions’ breakthrough was made possible due to the discovery of the so-called effective viscous flux identity and the renormalized transport theory developed previously by DiPerna and Lions in [18]. Another milestone in the understanding of weak solutions for these equations is Feiresl’s monograph [30] containing a comprehensive treatment of the heat-conducting compressible fluids with weak formulation of the energy conservation in terms of the thermal energy balance. In the light of this work (that employs in addition to the techniques introduced by P.L. Lions new

29 Weak Solutions for the Compressible Navier-Stokes Equations:. . .

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ideas related to the notion of oscillations defect measure), the Lions theory is a particular case of Feireisl’s results. Monograph [88] contains an extensive material on weak solutions to the compressible Navier-Stokes equations in barotropic regime ranging from stationary to evolution problems and from bounded to unbounded domains with different boundary conditions containing comprehensive detailed proofs. The theory of weak solutions has been revisited in [93] in view of applications in the control theory. Monograph [33] introduces in Chapter 3 a theory of weak solutions to the complete Navier-Stoke-Fourier system with the energy conservation in terms of the entropy balance and the entropy production rate as a Radon measure (entropy weak solutions). Among others, this work reveals importance of the Helmholtz function (called sometimes ballistic free energy). This quantity plays an essential role in the book [44] devoted to the investigation of the longtime behavior of weak solutions. Thermodynamic stability conditions for entropy weak solutions to the Navier-Stokes-Fourier system as well as for the barotropic equations can be reformulated as a variational inequality called relative energy inequality (see [34, 49, 50]) that becomes a basic tool to prove the weak-strong uniqueness principle for these equations (see again [34, 49, 50]) and has many other applications, e.g., the investigation of various singular limits or deriving error estimates for various numerical schemes.

14

Cross-References

 Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak Solutions for

the Compressible Navier-Stokes Equations  Concepts of Solutions in the Thermodynamics of Compressible Fluids  Symmetric Solutions to the Viscous Gas Equations  Weak and Strong Solutions of Equations of Compressible Magnetohydrodynamics  Weak Solutions for the Compressible Navier-Stokes Equations in the Intermediate

Regularity Class  Weak Solutions for the Compressible Navier-Stokes Equations with Density Dependent

Viscosities  Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical Cases Acknowledgements The work of A.N. was supported by the MODTERCOM project within the APEX programme of the Provence-Alpes-Côte d’Azur region, H.P. was supported in the framework of RVO:67985840.

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53. E. Feireisl, A. Novotný, Y. Sun, A regularity criterion for the weak solutions to the compressible Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 212(1), 219–239 (2014) 54. E. Feireisl, R. Hošek, D. Maltese, A. Novotný, Error estimates for a numerical method for the compressible Navier-Stokes system on sufficiently smooth domains (2015). arXiv preprint arXiv:1508.06432 55. E. Feireisl, T. Karper, A. Novotny, A convergent mixed numerical method for the NavierStokes-Fourier system. IMA J. Numer. Anal. 36, 1477–1535 (2016) 56. E. Feireisl, T. Karper, M. Pokorny, Mathematical Theory of Compressible Viscous Fluids – Analysis and Numerics (Birkhauser, Basel, 2016) 57. E. Feireisl, M. Lukáˇcová-Medvidóvá, S. Neˇcasová, A. Novotný, B. She, Asymptotic preserving error estimates for numerical solutions of compressible Navier-Stokes equations in the low Mach number regime. Inst. Math. Cz. Acad. Sci. (2016). Preprint. http://www. math.cas.cz/recherche/preprints/preprints.php?mode_affichage=3&id_membre=4018& unique=1&lang=0 58. E. Feireisl, A. Novotny, Y. Sun, On the motion of viscous, compressible and heat-conducting liquids. J. Math. Phys. 57(08) (2016). https://doi.org/10.1063/1.4959772 59. G.P. Galdi, An Introduction to the Mathematical Theory of Navier-Stokes Equations, vol. I (Springer, Berlin, 1994) 60. T. Gallouet, R. Herbin, D. Maltese, A. Novotny, Error estimates for a numerical approximation to the compressible barotropic Navier-Stokes equations. IMA J. Numer. Anal. 36, 543–592 (2016) 61. P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J. Math. Fluid Mech. 13(1), 137–146 (2011) 62. V. Girinon, Navier-Stokes equations with non homogenous boundary conditions in a bounded three dimensional domain. J. Math. Fluid. Mech. 13, 309–339 (2011) 63. E. Grenier, Oscillatory perturbations of the Navier-Stokes equations. J. Math. Pures Appl. (9), 76(6), 477–498 (1997) 64. D. Hoff, Global well-posedness of the Cauchy problem for nonisentropic gas dynamics with discontinuous initial data. J. Differ. Equ. 95, 33–37 (1992) 65. D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120, 215–254 (1995) 66. D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Ration. Mech. Anal. 132, 1–14 (1995) 67. D. Hoff, Compressible flow in a half-space with Navier boundary conditions. J. Math. Fluid Mech. 7(3), 315–338 (2005) 68. D. Hoff, Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow. SIAM J. Math. Anal. 37(6), 1742–1760 (electronic) (2006) 69. D. Hoff, D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51, 887–898 (1991) 70. D. Jessle, B.J. Jin, A. Novotny, Navier-Stokes-Fourier system on unbounded domains: weak solutions, relative entropies, weak-strong uniqueness. SIAM J. Math. Anal. 45, 1907–1951 (2013) 71. T.K. Karper, A convergent FEM-DG method for the compressible Navier-Stokes equations. Numer. Math. 125(3), 441–510 (2013) 72. A. Kazhikov, A. Veigant, On the existence of global solution to a two-dimensional Navier-Stokes equations for a compressible viscous flow. Sib. Math. J. 36, 1108–1141 (1995) 73. P. Kukucka, On the existence of finite energy weak solutions to the Navier-Stokes equations in irregular domains. Math. Methods Appl. Sci. 32, 1428–1451 (2009) 74. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934) 75. J. Li, Z. Xin, Global Existence of Weak Solutions to the Barotropic Compressible NavierStokes Flows with Degenerate Viscosities. Preprint. http://arxiv.org/pdf/1504.06826.pdf

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76. P.-L. Lions, Mathematical Topics in Fluid Dynamics: Incompressible Models, vol. 1 (Oxford Science Publication, Oxford, 1996) 77. P.-L. Lions, Mathematical Topics in Fluid Dynamics: Compressible Models, vol. 2 (Oxford Science Publication, Oxford, 1998) 78. J. Málek, J. Neˇcas, A finite-dimensional attractor for the three dimensional flow of incompressible fluid. J. Differ. Equ. 127, 498–518 (1996) 79. D. Maltese, A. Novotny, Compressible Navier-Stokes equations on thin domains. J. Math. Fluid Mech. 16, 571–594 (2014) 80. N. Masmoudi, Incompressible inviscid limit of the compressible Navier-Stokes system. Ann. Inst. H. Poincaré, Anal. non linéaire 18, 199–224 (2001) 81. A. Matsumura, M. Padula, Stability of stationary flow of compressible fluids subject to large external potential forces. SAACM 2, 183–202 (1992) 82. Š. Matušu-Neˇcasová, M. Okada, T. Makino, Free boundary problem for the equation of spherically symmetric motion of viscous gas (III). Jpn. J.Ind. Appl. Math. 14(2), 199–213 (1997) 83. A. Mellet, A. Vasseur, On the barotropic Navier-Stokes equations. Commun. Partial Differ. Equ. 32, 431–452 (2007) 84. F. Murat, Compacité par compensation Ann. Sc. Norm. Super. Pisa Cl. Sci. Ser. 5 IV, 489–507 (1978) 85. S. Novo, Compressible Navier-Stokes model with inflow-outflow boundary conditions. J. Math. Fliud. Mech. 7, 485–514 (2005) 86. A. Novotny, M. Padula, Lp approach to steady flows of viscous comprtessible fluids in exterior domains. Arch. Ration. Mech. Anal. 126, 243–297 (1994) 87. A. Novotný, I. Straškraba, Convergence to equilibria for compressible Navier-Stokes equations with large data. Ann. Mat. Pura ed Appl. CLXXIX(IV), 263–287 (2001) 88. A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow (Oxford University Press, Oxford, 2004) 89. B.G. Pachpatte, Inequalities for Differential and Integral Equations (Academic Press, San Diego, 1998) 90. M. Padula, Stability properties of regular flows of heat-conducting compressible fluids. J. Math. Kyoto Univ. 32(2), 401–442 (1992) 91. P. Pedregal, Parametrized Measures and Variational Principles (Birkhauser, Basel, 1997) 92. P.I. Plotnikov, W. Weigant, Isothermal Navier-Stokes equations and Radon transform. SIAM J. Math. Anal. 47(1), 626–653 (2015) 93. P.I. Plotnikov, J. Sokolowski, Compressible Navier-Stokes Equations. Theory and Shape Optimization. Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series), vol. 73 (Birkhäuser/Springer Basel AG, Basel, 2012) 94. L. Poul, Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains, in Proceedings of the 6th AIMS International Conference. Discrete and Continuous Dynamical Systems, 2007, pp. 834–843 95. G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959) 96. L. Saint-Raymond, Hydrodynamic limits: some improvements of the relative entropy method. Annal. I.H.Poincaré- AN 26, 705–744 (2009) 97. D. Serre, Variation de grande amplitude pour la densité d’un fluid viscueux compressible. Phys. D 48, 113–128 (1991) 98. J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962) 99. K.M. Shyue, A fluid mixture type algorithm for compressbile multicomponent flow with Mie-Gruneisen equation of state. J. Comput. Phys. 171(2), 678–707 (2001) 100. F. Sueur, On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain. J. Math. Fluid Mech. 16, 163–178 (2014) 101. Y. Sun, C. Wang, Z. Zhang, A Beale – Kato – Majda blow-up criterion for the 3-D compressible Navier-Stokes equations. Journal de Mathématiques Pures et Appliquées 95, 36–47 (2011)

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102. R. Temam, Navier-Stokes Equations (North-Holland, Amsterdam, 1977) 103. S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation. J. Math. Kyoto Univ. 26(2), 323–331 (1986) 104. A. Valli, An existence theorem for compressible viscous fluids. Ann. Mat. Pura Appl. 130(IV), 197–213 (1982) 105. A. Vasseur, C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations. Invent. Math. 206, 935–974 (2016) 106. J.G. Van Wyllen, R.E. Sonntag, Fundamentals of Classical Thermodynamics (John Wiley, New-York, 1985) 107. S. Wang, S. Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Commun. Partial Differ. Equ. 31(4–6), 571–591 (2006) 108. Y.B. Zeldowich, Y.P. Raizer, Physics of Schock Waves and High Temperature Hydrodynamics (Academic Press, New York, 1966)

30

Weak Solutions for the Compressible Navier-Stokes Equations with Density Dependent Viscosities Didier Bresch and Benoît Desjardins

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Simple Fluid Flow Models and Historical Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Why Degenerate Viscosities May Help and Why It Seems to be Necessary to Consider Such Situation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Model Behavior When the Density Vanishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Density and Velocity Fluctuations Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Basic a Priori Estimates, Identities, and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Stability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Construction of Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The -entropy and Two-Velocities Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Barotropic Compressible System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Heat-Conducting Compressible System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Relative Entropy and Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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D. Bresch is partially supported by the ANR- 13-BS01-0003-01 project DYFICOLTI. D. Bresch () LAMA UMR 5127 CNRS Batiment le Chablais, Université de Savoie Mont-Blanc, Le Bourget du Lac, France e-mail: [email protected] B. Desjardins Fondation Mathématique Jacques Hadamard, CMLA, ENS Cachan, CNRS and Modélisation Mesures et Applications S.A., Paris, France e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_44

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Abstract

In this chapter, we focus on compressible Navier-Stokes equations with densitydependent viscosities in the multidimensional space case. The main objective of these notes is to present at the level of beginners an introduction to such systems showing the difference with the constant viscosity case. The guideline is to show a nonlinear hypercoercivity property due to the density dependency of the viscosities, to explain how it may be used to provide global existence of weak solutions to the barotropic compressible Navier-Stokes equations, and to the heat-conducting Navier-Stokes equations with a total energy formulation. We will also focus on the relative entropy method for such systems showing the difficulty coming from the density dependency. We hope to motivate by this chapter young researchers to work on such difficult topic trying to fill the gap between the constant viscosities case and the density-dependent viscosities satisfying the BD relation, trying to relax some modeling hypotheses and to extend the results. AMS Classifications. 35Q30, 35D30, 54D30, 42B37, 35Q86, 92B05.

1

Introduction

The aim of this chapter is to describe some recent progress related to the mathematical analysis of the Navier-Stokes equations for compressible viscous and heat-conducting fluids in the case of density-dependent viscosity coefficients. Some basic notations of fluid dynamics variables are introduced together with arguments motivating the analysis of density-dependent viscosities. Next, properties of the so-called BD entropy are detailed in the framework of barotropic flows, i.e., in the case when pressure does not depend on temperature. Such results are then extended to the full system including heat-conducting equations, for which the construction of suitable approximate solutions is described. More detailed introduction on fluid dynamics models and related mathematical results of NavierStokes equation can be found in monographs [26,44,45,51] and papers [8,9, 12, 13] to name a few. In order to avoid technicalities and additional mathematical difficulties (some of them being completely unsolved so far) associated to boundaries or infinite spaces, the space domain will be  D Td where d D 2 or 3, i.e., a d-dimensional box with periodic boundary conditions. A compressible heat-conducting fluid in  governed by the Navier-Stokes equations satisfies the following system 8 ˆ < @t  C div .u/ D 0; @t .u/ C div .u ˝ u/ C rp D div  C f; ˆ : @t .E/ C div .uH / D div .  u/ C div .r / C f  u;

(1)

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where u 2 Rd denotes the velocity field of the fluid,  the density,  the temperature,  the thermal conductivity coefficient,  the viscous stress tensor, p the pressure field, e the specific internal energy, and h D e C p= the specific enthalpy. The specific total energy and the associated specific total enthalpy are denoted, respectively, by E DeC

juj2 2

and

H DhC

juj2 : 2

Equations (1) express the conservation of mass, momentum, and total energy, respectively. In order to close this system, two additional ingredients are necessary. First, the fluid is assumed to be Newtonian, i.e., there exist two viscosity coefficients .;  / and .;  / such that the stress tensor expresses as D   pIRn D 2 D.u/ C . div u  p/IRn where D.u/ D .ru C rut /=2 denotes the strain rate tensor defined as the symmetric part of the velocity gradient ru and IRn the identity tensor. As a second condition, a thermodynamical closure law defines the pressure p and the internal energy e as functions of the density  and the temperature  : p D P.;  /; e D E.;  /:

(2)

For perfect gases, the pressure law and the internal energy read p D R and e D Cv , R and Cv being positive constants. It is important to recall to the readers that the difficult problem of global existence of weak solutions to the heat-conducting case with constant viscosities or viscosities depending on the temperature has been firstly established by E. Feireisl [26] and then by E. Feireisl and A. Novotný in [28]. Here we will focus on the case where  and  depend on the density. This will crudely change the situation and will ask for new mathematical tools initiated by the two authors of this chapter and C.K. Lin for the Korteweg compressible system, see [13].

2

Simple Fluid Flow Models and Historical Comments

Below we discuss mathematical properties of some particular cases of system (1). I) Incompressible Navier-Stokes equations without temperature (homogeneous case). First, if we assume that  is a constant, the continuity equation in (1) reduces to the incompressibility constraint div u D 0 with an associated Lagrangian multiplier. Forgetting the total energy equation, we obtain (

div u D 0; .@t u C div .u ˝ u// D div C f;

(3)

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where D 2D.u/  p IRn with p the Lagrangian multiplier associated to the incompressibility constraint. Notice that if the viscosity  is constant, the viscous dissipation term rewrites div .2D.u// D  u, so that the momentum equation can be rewritten as p   @t u C div .u ˝ u/ D u  r C f:   Defining the kinematic viscosity  D = and P D p=; one can rewrite the incompressible equations (3) as (

div u D 0; @t u C div .u ˝ u/ C rP D  u C f;

(4)

which is the classical incompressible Navier-Stokes equations for homogeneous fluids. The first global existence of weak solutions result for such system is due to [42]. Such system may be obtained from the barotropic compressible NavierStokes equations by letting the Mach number go to zero. II) Incompressible Navier-Stokes equations without temperature (nonhomogeneous case ). If we just assume that the fluid is incompressible and forget the total energy equation, we end up with the nonhomogeneous incompressible NavierStokes equations 8 ˆ < div u D 0; @t  C div .u/ D 0; ˆ : @t .u/ C div .u ˝ u/ D div C f

(5)

where D 2./D.u/  pId. Existence results of global weak solutions for such system are due to A. Kazhikhov [40] (initial density far from vacuum and constant viscosity), J. Simon [56] (initial density with possible vacuum and constant viscosity) and E. Fernández-Cara and F. Guillén-Gonzalez [31], P.L. Lions [44] (initial density with possible vacuum and density-dependent viscosity but strictly positive). Note that such a system may be obtained from the compressible Navier-Stokes equations with heat conductivity letting the Mach number go to zero. The interested reader is referred to [1] and reference cited therein for such derivation. More general models may be encountered, for instance, addressing large heat-release properties: see, for instance, [15] and [1]. In that case, the divergence-free constraint may be replaced, for instance, by divu D g./ with g a given function which encodes the heat-conducting pressure law. III) Barotropic compressible Navier-Stokes equations. If we assume that the temperature  is constant, the pressure p only depends on the density , which corresponds to the so-called “barotropic” case (also sometimes improperly called

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“isentropic”: entropy production rate is eventually proportional to the work of viscous stresses, and must therefore be positive). In the framework of barotropic flows, the most commonly found equation of state is the so-called law, i.e., p./ D a , where  1; a > 0 are constants. Thus, System (1) becomes 8 ˆ < @t  C div .u/ D 0; @t .u/ C div .u ˝ u/ C rp D div .2D.u// C r. div u/ C f; ˆ : p D p./

(6)

supplemented with the initial conditions jtD0 D 0 ;

.u/jtD0 D m0 :

(7)

Several remarks should be given on the barotropic Navier-Stokes equations (6): Remark 1. The first existence result on global weak solutions in the multidimensional in space case was obtained by P.L. Lions [45] in 1998 for large enough exponents  3d =.d C 2/, d D 2; 3. This result has been later extended to the somehow optimal case > d =2 by E. Feireisl et al. [29] in 2001. Note the recent paper by P. Plotnikov and W. Weigant [53] where the case D 1 is covered in the two-dimensional space case. Remark 2. Note that the pressure law can be generalized, a typical example being p 2 C 1 .Œ0; C1//; a 1  b  p 0 ./ 

p.0/ D 0 with

1 1 C b with > d =2  a

for some constants a > 0I b  0: See E. Feireisl [25], B. Ducomet, E. Feireisl, H. Petzeltova, I. Straskraba [24] for slightly more general assumptions. However it is always required that p./ be increasing after a certain critical value of . Note that a new global existence result without monotonicity assumptions on the pressure law has been recently proved by D. Bresch and P.E. Jabin in [17]: It requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation. Remark 3. In the result of P.L. Lions and E. Feireisl, the viscosity coefficients  and  need to be constants. The case when  and  depend on the density, i.e.,  D ./; and  D ./ is an open problem in its full generality. Only partial results have been obtained, such as D. Bresch, B. Desjardins et al. [8, 12, 13], A. Mellet and A. Vasseur [46], J. Li and Z. Xin [43], and A. Vasseur and C. Yu [59] based on a new mathematical entropy discovered by D. Bresch and B. Desjardins if the relation ./ D 2.0 ./.// is satisfied. See also the work by A. Kazhikhov and W. Weigant [41] where they consider the case ./ D cste and ./ D ˇ with

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ˇ > 3 in a periodic box  D f.x; y/ 2 R2 W 0 < x < 1; 0 < y < 1g: They get global existence of strong solution in 2D using this density dependency of . More precisely, they prove the following theorem Theorem 1. Let ./ D 1;

./ D ˇ with ˇ > 3

and P ./ D R with R > 0 and  0. Assuming 0 < m0  0 .x; y/  M0 < C1;

0 2 W 1;q ./ with q>2;

u0 2 H 2 ./

then there exists a unique strong solution to the compressible Navier-Stokes system (6) satisfying the initial conditions with moreover u 2 L2 .0; T I H 2 .// \ H 1 .0; T I L2 .//;  2 L1 .0; T I W 1;q .// \ W 1;1 .0; T I Lq .//: The solution exists for all time and the density is an L1 function and is bounded away from vacuum. Note that recently several authors succeeded to improve the hypothesis on ˇ (see B. Haspot, Z.P. Xin et al. and others) and to consider bounded domains (see, for instance, B. Ducomet and S. Necasova). See also the work by M. Perepetlisa (see [52]) regarding the existence of global weak solution with interesting feature regarding non-vacuum and boundedness for the density. Remark 4. The essential difficulties of the incompressible Navier-Stokes equations (5) are also present in the framework of barotropic flows for constant viscosities. More precisely, in (5), the main difficulty is how to pass to the limit in n un and more precisely in the term div .u ˝ u/. We will explain later on quickly how to handle this difficulty. For barotropic flow, compactness of the pressure law is also necessary. Such issues will be discussed in detail in the next section. This is the main difficulty for compressible barotropic Navier-Stokes equations with constant viscosities and power law pressure state.

3

Why Degenerate Viscosities May Help and Why It Seems to be Necessary to Consider Such Situation?

Let us explain why density-dependent viscosities may be important and why new velocity field may be encountered for strongly heterogeneous flows. We focus on model behavior when the density vanishes, dimensional analysis, and density/velocity fluctuations modeling.

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

3.1

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Model Behavior When the Density Vanishes

When the density  tends to zero, compressible fluid models are no longer valid from a physical viewpoint, rarefied gas models being in that case more relevant. Still, Navier-Stokes numerical codes need to be robust near regions where the density  ! 0, which occurs in many industrial simulations of complex flows. As a matter of fact, the compressible Navier-Stokes equations when the density tends to vacuum behaves like the degenerate model div .2D.u//  r.div u/ D 0 where  D . D 0; / and  D . D 0; /. Such an equation has no physical meaning. Using density-dependent viscosity coefficient .; / vanishing when  ! 0 has the advantage to remove the above “ghost elliptic equation” in vacuum regions. The reader is referred to [37] and [54] for discussion on the problem of continuity with respect to the initial data when the viscosities are assumed to be constant.

3.2

Dimensional Analysis

Once the idea of using viscosity coefficients vanishing as a function of the density when  tends to zero has been introduced, dimensional analysis may help find relevant velocities linked to density or viscosity variations. First, the following gradient length associated with viscosity variations can be defined L D

 krk

A (dimensionless) Reynolds number can then be introduced Re D

VL ; 

which defines a velocity scale V associated with viscosity variations, V 

kr./k  D : L 

Hence velocities such as r= seem to play a role in the dynamics of heterogeneities. We will see that this is exactly this quantity which will play a crucial role in the BD entropy and more generally in the global existence of weak solutions for compressible Navier-Stokes equations with degenerate viscosities.

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Density and Velocity Fluctuations Modeling

Turbulence models can be derived by systematic use of averaging operators, leading to correlations of products of variable fluctuations. Such statistical derivation of turbulence models can be found, for instance, in Ishii [38] and Drew and Passman [23] in the context of multiphase flows. It leads to averaged models involving velocities associated with density or viscosity small-scale fluctuations.

3.3.1 Reynolds Average Reynolds average of a given variable is defined as the ensemble average of this quantity (mean over a large number of realizations of the same experiment) satisfying the following properties of any observable variables a and b a D a;

aCb DaCb ab D ab

@t a D @t a

@xi a D @xi a

8i D 1 : : : d

Reynolds fluctuations are then defined as the difference of observable a with its average value a: a D a C a0 Application to homogeneous flows   const ant leads to the derivation of the socalled RANS equations (Reynolds Averaged Navier-Stokes equations). Note that time or combined space-times averages may also be used instead of set average.

3.3.2 Favre Average Favre average is adapted to heterogeneous flows (compressibility/large density variations). Indeed, the density is used as a weight in Reynolds statistical average for a given variable a: aQ D

a 

Favre fluctuations are similarly defined by a D aQ C a00 Example 1. Taking the average of mass conservation equation leads to @t  C div .u/ D 0;

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which simplifies using the properties of the Reynolds average @t  C div .u/ D 0: Since by definition u D uQ , one has @t  C div .uQ / D 0; so that the averaged mass conservation equations is exactly preserved by the nonhomogeneous averaging process. Averaging compressible fluid mechanics models leads to correlations of fluctuations of physical quantities a with velocity fluctuations u00 u00 a00 : Boussinesq-Reynolds closure consists in assuming that this correlation is proportional to the gradient of the Favre average of a u00 a00 D 

t r aQ a

where t is the turbulent viscosity, a is a constant (Prandtl-Schmidt number). This closure can be rigorously mathematically justified under some restrictive hypotheses on the fluctuations (see [2], for instance). Example 2. Difference between Favre and Reynolds velocity u D uv

v D 1= t D uQ C u00 v 00 D u   r vQ a where

since vQ D v= D 1=, one ends up with u D uQ   D uQ C

t 1 r a 

t r : a 

Consequently, velocities based on density gradients seem to play a role when modeling turbulent compressible flows with significant density heterogeneities. Note the importance of the quantity t r log  which is a part in the velocity dedicated to counteract osmotic effect as mentioned by A. EINSTEIN: See the book by NELSON [49].

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Basic a Priori Estimates, Identities, and Comments

This section is devoted to basic estimates on solutions of fluid equations introduced in Sect. 1. Let us recall that the domain is assumed to be a torus  D Td , i.e., a box with periodic boundary conditions. This part is for people who are not familiar with partial differential equations arising in fluid mechanics modeling. I) Incompressible Navier-Stokes equations without temperature (homogeneous case). Taking the scalar product of u with the momentum equation in (4) and integrating over the space domain , one obtains 1 d 2 dt

Z

Z

2

juj C 

Z

2

jruj D





f  u:

(8)



R Here the fact that  rp  u D 0 has been used. In the case of vanishing bulk forces f  0, one deduces from (8) that for all positive time T u 2 L1 .0; T I L2 .// \ L2 .0; T I H 1 .//: Similar bounds can be obtained for nonzero f if suitable regularity is assumed such as f 2 L2 .0; T I H 1 .// for all T > 0. II) Incompressible Navier-Stokes equations without temperature (nonhomogeneous case). In this case, multiplying by u the momentum equation in (5) and the mass equation by juj2 =2, adding the results, and integrating with respect to x 2  leads to Z Z Z 1 d 2 2 juj C  jruj D f  u: (9) 2 dt    Similarly, if f  0, one deduces from (9) that p

u 2 L1 .0; T I L2 .//; ru 2 L2 .0; T I L2 .//:

Similar bounds can be obtained for nonzero f if f 2 L1 .0; T I L2 .//. This uses the fact that  2 L1 ..0; T /  / if the initially density 0 is bounded (note that u is divergence-free). We just have to bound the term coming from the external force as follows Z p p u  f  k kL1 ./ k ukL2 ./ kf kL2 ./ : 

III) Barotropic compressible Navier-Stokes equations. The same integration procedure as in the incompressible case of (5) gives the total energy balance 1 d 2 dt

Z

2

Z

2

juj C 2 

Z

jD.u/j C 

2

Z

Z

jdiv uj C 

u  rp D 

f  u: 

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

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From the mass conservation equation @t  C div .u/ D 0 in (6), noRinformation on div u can be extracted as in the incompressible case, since the term  u  rp has to be controlled. Let us define such that  0 ./  ./ D p./, then the following equation reads at least formally @t ./ C div. ./u/ C p./divu D 0: Integrating in space and time and adding to the previous relation, we get using the periodic boundary condition 1 d 2 dt

Z

juj2 dx C 

d dt

Z

Z

jD.u/j2 dx C 

./dx C 2 

Z



Z D

jdiv uj2 dx



f  udx

(10)



where is defined by  0 ./  ./ D p./: Choosing ./ D e./, we find that e is defined as Z  e./ D p.s/=s 2 ds ref

where ref is a constant reference density. Assuming that f  0 and ;  are constants, one deduces that p u 2 L1 .0; T I L2 .//I

ru 2 L2 .0; T I L2 .//I

./ 2 L1 .0; T I L1 .//: Assuming the pressure satisfies C 1   C  p./  C  C C for some constant C > 0, similar bounds can be obtained for nonzero f if f 2 L1 .0; T I L2 =. 1/ .// noticing that Z 

p p u  f  kukL2 =. C1/ kf kL2 =. 1/  k kL2 ./ k ukL2 ./ kf kL2 =. 1/ :

Remark 5. If we assume a -type law p./ D a ; > 1 and ref D 0, we find that e./ D a 1 =.  1/: Thus (10) implies that  2 L1 .0; T I L .//: In order to get the global weak solutions of (6), three steps are necessary: • Derive A priori estimates • Obtain stability properties of approximate solutions sequences • Construct approximate solutions

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D. Bresch and B. Desjardins

From the basic energy estimates, using the equations and Aubin-Lions-Simon compactness lemma, it is possible to obtain the following results: 1. For homogeneous incompressible equations (4): Stability in u ˝ u can be proved (See [42]). 2. For nonhomogeneous incompressible equations (5): Stability in u, u ˝ u can be obtained if ./  C > 0 (See [44]). 3. For compressible barotropic models (6): – Constant viscosities case: Stability in u, u ˝ u can be proved and stability in  requires more work (See [45, 51]). – Density-dependent case: stability on  can be proved with an extra estimate and stability in u ˝ u asks for more work. Assuming that .n ; un / is a sequence of approximate solutions satisfying energy estimates for the barotropic system and the equation in a weak sense. (I) Barotropic case: (I-1) The case  and  constants (nondegenerate case). In that case, the main difficulty is to pass to the limit in the pressure term proportional to n (the p compactness of n un will be discussed later on). Additional information is required on density oscillations: how to get compactness of n in Lebesgue spaces. How to pass to the limit in ./n is the main difficulty in P.L. Lions’ and E. Feireisl’s results. Their methodology uses strongly that viscosities are constants. Recall that the equations of (6) reads (

@t  C div .u/ D 0;

(11)

@t .u/ C div .u ˝ u/   u  . C /rdiv u C rp D f supplemented with the initial conditions jtD0 D 0 ;

.u/jtD0 D m0 :

(12)

Applying formally the divergence operator div to the momentum equation leads to div .@t .u/ C div .u ˝ u//   div u  . C / div u C p D div .f /: Remarking    div u  . C / div u C p D a  .2 C / div u where a  .2 C / div u is called the effective flux. The equation above shows that in some sense there exists some compactness on the effective flux: With one derivative of the equation, we find the Laplacian of this quantity. We use strongly the fact that  and  are constants to commute the divergence operator with the

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

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diffusion operator. Note that there exists several steps to prove global existence of weak solutions for the barotropic compressible Navier-Stokes equations with constant viscosities: • Energy estimates: See calculation before • Extra integrability on :  2 L C ..0; T /   with   2d =  1): Use Bogovski operator. • Effective flux property:   divu  divu D p./  p./ =.2 C / where  denotes the weak limit • Compactness on the density:  @t . ln    ln / C div. ln u   ln u/ D divu  divu D p./   p./ =.2 C / Use monotonicity of p to have a sign of the right-hand side and the strict convexity of s 7! s ln s to provide the propagation of compactness on n if initially compact. p The compactness on n un in L2 ..0; T /  / follows in some sense what is done for the incompressible nonhomogeneous Navier-Stokes equations: we will discuss that later on. For reader’s convenience, let us give the existence result in the case without external force, namely, f D 0 and in a periodic box  D Td : Theorem 2. Let p.s/ D as with > d =2 and a > 0. Assume the initial data satisfy Z 0 2 L ./; 0  0; 0 D M0 > 0 

and m0 2 L1 ./;

m0 D 0 if 0 D 0;

jm0 j2 =0 2 L1 ./:

Then there exists a global weak solutions of the compressible Navier-Stokes system (11) and (12) that means a solution satisfying the energy inequality and the system in a weak sense. With some modifications, the existence result may be extended to the case mentioned in Remark 2 as proved in [25]. Note also the recent result by A. Plotnikov and W. Weigant related to D 1 in the two dimensional in space case. For details, interested readers are referred to the other contributions in this handbook. Remark 6. Note the recent work by D. Bresch and P.E. Jabin [17] where they relax the hypothesis on the pressure law in the constant viscosities case; more precisely they assume p to be locally lipschitz on Œ0; C1/, with p.0/ D 0 C 1   C  p./  C  C C

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D. Bresch and B. Desjardins

and for all s  0, they only assume Q jp 0 .s/j  s 1

for some Q > 1. Then they prove the following result without external force, namely, f D 0 and in a periodic box  D Td : Theorem 3. Let .0 ; u0 / such that Z E.0 ; u0 / D where e.s/ D

Rs 0

Td

jm0 j2 C 0 e.0 / < C1 20

p./= 2 d . Let p satisfying the previous hypothesis with   > max.2; Q / C 1 d =.d C 2/

then there exists a global weak solution to the compressible Navier-Stokes equations (11) and (12). Note that the authors write in [18] an introduction of the memoir [17]. This allows to explain, on a simpler but still relevant and important system, the tools recently introduced by the authors and to discuss the important results that have been obtained on the compressible Navier-Stokes equations. To get such global existence of weak solutions result, the two authors have revisited the classical compactness theory on the density by obtaining precise quantitative regularity estimates: This requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation (by introducing appropriate weights). (I-2) The case  and  not constants (degenerate case: density dependency). Here the difficulty will be to pass to the limit u ˝ u. The main difficulty is that less information on the velocity compared to the constant viscosities case is known: no L2 .0; T I H 1 .// bound on u. What kind of additional estimate can be derived to help to conclude? Some results about nonconstant viscosities have been obtained: W. Vaˇıgant and A. Kazhikov [58], D. Bresch and B Desjardins [8,12], A. Mellet and A. Vasseur [46], A. Vasseur and C. Yu [59], and J. Li and Z. Xin [43]. In particular, under the relation ./ D 2.0 ./  .//, a mathematical entropy has been discovered by Bresch and Desjardins [8] (so-called BD entropy): a kind of nonlinear hypocoercivity property which is the cornerstone of all the previous works except the one by W. Vaˇıgant and A. Kazhikov who considered the case ./ D cste and ./ D ˇ with ˇ  3. Using the BD entropy gives control of gradient of  but asks to work with degenerate parabolic behavior for the velocity. Extra work is therefore necessary to ensure extra integrability on u if it is the case initially. Let us present the BD entropy, explain why it asks for degeneracy, and discuss the relation assumed between the two viscosities ./ D 2.0 ./  .//.

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

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For the reader’s convenience, let us give the very nice existence result which is actually known: stability and construction of approximate solutions. The novelty is the construction of approximate solutions (which is really more complicated than for the constant viscosities case) and is due independently to A. Vasseur, C. Yu in [59] and J. Li, Z.P. Xin [43]. We consider no external force, namely, f D 0 and a periodic box  D Td : Theorem 4. Let ./ D 0 and ./ D  with P ./ D a with > 1 for d D 2 and 1 < < 3 for d D 3. Assume 0 2 L ./; m0 2 L1 ./;

0  0;

p r 0 2 L2 ./; jm0 j2 =0 2 L1 ./:

m0 D 0 if 0 D 0;

Assume moreover that Z

0 .1 C ju0 j2 / ln.1 C ju0 j2 / dx < C1:



Then there exists a global weak solution .; u/ of the degenerate compressible Navier-Stokes equations that means satisfying the energy inequality, the BD entropy, the Mellet-Vasseur estimate, and the compressible Navier-Stokes equation in a weak sense. Note that the more general case is still in progress in [21] with the Bresch-Desjardins relation ./ D 2.0 ./  .//. Partial results were obtained in [43] with a nonphysical stress tensor S D ./ru C ./divuIRn instead of the symmetric one S D 2./D.u/ C ./divuIRn . Anyway this paper is really interesting because it helps to understand more the difficult problem of constructing approximate solutions in the density-dependent viscosities case for compressible Navier-Stokes equations. (I-2-a) BD entropy: Barotropic case/nonlinear hypocoercivity – An extra estimate. In this subsection, we will introduce the BD entropy to the equations (6) in detail. Lemma 1. Assume that ./ D 2.0 ./  .//. The formal equality associated with BD entropy can be written as 1 d 2 dt

Z

ju C 2r'j2 C 2 

d C dt

./ C 2 

./jA.u/j2 

Z

Z

Z



p 0 ./0 ./jrj2 D 

Z f  .u C 2r'/; 

(13)

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D. Bresch and B. Desjardins

where A.u/ D .ru  rut /=2 denotes the skew-symmetric part of the velocity gradient. Note that in the one dimensional in space case, such property has been noticed by Y. Kanel [39]. Proof. Multiplying the mass conservation equation by 0 ./, one deduces that @t ./ C div .u.// C .0 ./  .// div u D 0: Applying the operator r to the above equation leads to @t r./ C div ../ru/ C div .u ˝ r.// C r.0 ./  .//div u D 0: Defining 0 .s/ D ' 0 .s/=s; then the above equation becomes @t .r'.// C div .u'.// C div ../D.u// C div ../A.u// C r.0 ./  .//div u D 0:

(14)

The momentum equation reads @t .u/ C div .u ˝ u/  2div ../D.u//  r../div u/ C rp D f: Multiplying Eq. (14) by 2 and adding it to the momentum equation, it gives @t ..u C 2r'// C div .u ˝ .u C 2r'// C 2div ../A.u// C rp D f: Then one can multiply the above equation by u C 2r' and the mass equation by ju C 2r'j2 =2. Summing up the results and integrating over ; one obtains that 1 d 2 dt

Z

Z

2

ju C 2r'j C 2 

./jA.u/j C 

Z D

Z

2

rp  .u C 2r'/ 

f  .u C 2r'/: 

Notice that Z

Z rp  .u C 2r'/ D 

rp  u C 2 

D then we obtain the equality (13).

Z

d dt

Z

rp  r' 

Z

./ C 2 



p 0 ./0 ./jrj2 ; 

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

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Assume that f  0 and p 0 ./ 0 ./ > 0; the new entropy estimates give the following additional regularities: p  .u C 2r'/ 2 L1 .0; T I L2 .//I ./ A.u/ 2 L2 .0; T I L2 .//I s p 0 ./0 ./ r 2 L2 .0; T I L2 .//I ./ 2 L1 .0; T I L1 .//: 

p

In view of the energy estimates, we already know that p

 u 2 L1 .0; T I L2 .//I

p

./ D.u/ 2 L2 .0; T I L2 .//I

./ 2 L1 .0; T I L1 .//: Combining with the above results, one has p

 r' 2 L1 .0; T I L2 .//I

p

./ ru 2 L2 .0; T I L2 .//:

Obviously, additional information has been obtained on  (depending on the behavior of ). Remark 7. Assume ./ D ; then './ D log : The new estimate is on 2r log /:

p

.u C

(I-2-b) The problem of degeneracy occurring with the BD entropy. The property ./ D 2.0 ./  .//; ./  c > 0; and ./  0 are impossible to be satisfied simultaneously. Then ./ C 2./=d  0; ./  0 is the only way that has been done. Recall that   ˇ div u ˇˇ2 2 jdiv uj2 : ./jD.u/j2 C ./jdiv uj2 D ./ˇD.u/  I Rn C  C d d Assume ./ D ˛ ; then d ./ C 2./ D 2.d ˛  d C 1/˛  0 if ˛  .d  1/=d: We will see in the case of heat-conducting fluids by Bresch and Desjardins [9] that ( ./

2 3

n

with

n

m

with

m1

close to vacuum, far to vacuum.

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D. Bresch and B. Desjardins

It means that the viscosities are degenerate. Thus more information is needed on u since a H 1 bound in space is no longer available. Remark 8. Hoff and Serre [37] obtained that it is a failure of continuous dependence on initial data for the compressible Navier-Stokes equations if the viscosities are constants in one dimension in space. Thus it is necessary to consider the degenerate viscosities in the compressible Navier-Stokes equations. It is a really interesting and challenging work to perform the same study in the multidimensional case, see, for instance, the paper by D. Serre [54]. (I-2-c) Comments on the relation between ./ and ./. It is really strange to ask such relation between the two viscosities (namely, ./ D 2.0 ./  .//) and the authors do not know if there is some physical meaning for such relation. Anyway let us give an example where similar relation occurs directly. This example is linked to Korteweg compressible systems. Recall that the compressible EulerKorteweg system reads @t % C div.%u/ D 0; @t .%u/ C div.%u ˝ u/ C rp.%/ D div.K/;

(15) (16)

where % denotes the fluid density, u the fluid velocity, p.%/ the fluid pressure, and K the Korteweg stress tensor defined as  1 0 2 K D %div.K.%/r%/ C .K.%/  %K .%//jr%j IRn  K.%/r% ˝ r%: 2 

(17)

with K.%/ the capillary coefficient. In [7], it has been observed that divK may be written   divK D 2div 1 .%/rv1 C r 1 ./divv1 p p with 1 ./ D 2Œ01 .%/%1 .%/ and where v1 D r'1 ./ with %'10 .%/ D K.%/, p 01 .%/ D K.%/ %=2. Remark that we find an interesting second-order operator on v1 with coefficients satisfying the same structure than the viscosities ./ and ./. This is perhaps an algebraic coincidence, but this is interesting enough to be noticed in this chapter. (II) The full heat-conducting system case (stability results) The main difficulties associated with weak solutions for heat-conducting fluids are described now. Note that there are fundamental differences between the framework considered by E. Feireisl with constant viscosities (or temperaturedependent viscosities) and that of D. Bresch and B. Desjardins with densitydependent viscosities. More precisely, E. Feireisl studies pressure laws expressed as

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

1565

p.; / D pe ./ C  p ./ where p ./ pe ./. The assumptions considered by D. Bresch and B. Desjardins are different, since pressure is the form of p.; / D pc ./ C r ; with a so-called cold pressure component pc ./ (pressure at zero temperature) defined close to vacuum such that it vanishes away from vacuum. In particular, perfect gas pressure laws can be considered far from vacuum. Remark that interesting other pressure law states have been considered by E. Feireisl and A. Novotný [28] such that those with radiative part helping to get compactness on temperature. One mathematical difficulty is that a priori estimates are not sufficient to define weak solution ; u, and  . From the total energy estimates and BD entropy estimates, the following bounds hold  log  2 L1 .0; T I L1 .//; jr log  j2 ;

jD.u/j2 2 L1 .0; T I L1 .//: 

But fluxes such as u.juj2 =2 C e/ C pu in the total energy conservation equation are not necessarily integrable. The second difficulty appears in the compactness of the temperature. The first challenges are to prove the following properties: (i) The control juju2 2 Lp .0; T I Lp .// with p > 1 (ii) The control r 2 Lp .0; T I Lp .// with p > 1 (iii) The compactness on  : We need more information on some negative power of  as in [9] or radiative term as proposed in [47]. We will prove that the generalization of the BD entropy for the heat-conducting Navier-Stokes equations and the presence of a cold pressure degenerating close to vacuum will help to conclude of global existence of weak solutions via the total energy formulation. This cold pressure part is crucial to deduce some compactness on the temperature and pass to the limit in the total energy formulation.

4.1

Stability Properties

4.1.1 The Barotropic Case In this subsection, we want to prove the weak stability of the approximate solutions .n ; un / of the compressible barotropic Navier-Stokes equations. The main difficulty for the density-dependent viscosity case is to pass to the limit in the term n un ˝ un , p which requires the strong convergence of n un : We will see that passing to the limit in the pressure quantity an is simple due to the BD entropy information. Let us explain, by hand, the difference with the constant viscosities case to pass to the limit in n un and n un ˝ un . In the constant viscosities case, we can write the following identities: Z TZ n un D< n ; un >H 1 ./H 1 ./ 0



1566

D. Bresch and B. Desjardins

and Z

T

Z

0

n jun j2 D< n un I un >H 1 ./H 1 ./



using that un 2 L2 .0; T W H 1 .//. Then use the strong convergence of n un in L2 .0; T I H 1 .//, n in L2 .0; T I H 1 .// [using, respectively, the momentum equation, the mass equation, and Aubin-Lions-Simon compactness lemma] and the weak convergence of un in L2 .0; T I H 1 .// to pass to the limit in the relation p written above. This gives the convergence in norm of n un in L2 .0; T I L2 .// p and thus the announced strong convergence using weak convergence of n un to p 2 u in L ..0; T /  /. The first strong convergence in negative Sobolev spaces is obtained using the bounds given by the energy estimates, the extra integrability on n and the bound on @t .n un / through the momentum equation and the energy estimates (Aubin-Lions-Simon compactness). With density-dependent viscosities which is degenerated close to vacuum, we understand that we loose this last estimate un 2 L2 .0; T I H 1 .// and therefore loose a priori the convergence p of n un in L2 .0; T I L2 .//. The key ingredient to achieve is an additional estimate which bounds n u2n in a space better than L1 .0; T I L1 .//; namely, L1 .0; T I L log L.//: We will give the sketch proof here, see [46] for more details. Remark that compared to the constant viscosities case, pass to the limit in the pressure term is quite easy in the density-dependent viscosities case using the BD entropy. Let us explain the various convergence for the degenerate densitydependent viscosity case. This is a summary of what may be found in [46] or in [11]. Let us show the following result Theorem 5. Let us consider a sequence .n ; un /n2N satisfying uniformly the energy estimates, the BD entropy, and the compressible Navier-Stokes equations in a weak sense. Let us also assume that it satisfies the bound Z

T

Z n f .jun j/ < C1

0



uniformly with respect to n with f an increasing function such that s 2 =f .s/ ! 0 when s ! C1: Then there exists a subsequence such that the weak limit satisfies the energy and BD entropy estimates and the compressible Navier-Stokes equations in a weak sense. Step 1: Convergence of

p

n

Using the mass conservation equation, it follows that for all t  0 kn .t; /kL1 ./ D kn;0 kL1 ./ :

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

1567

Hence up to the extraction of a subsequence, one may assume that .n /n1 tends in the sense of distributions to some nonnegative density . The BD entropy introduced p in (I-2-a) leads to the fact that n is bounded in L1 .0; T I H 1 .//. Next, we notice that @t

1p p p n D  n div un  un  r n 2 1p p D n div un  div .un n /: 2

p which allows to conclude that @t n is bounded in L2 ..0; T /  / C 1 1 L .0; T I H .//. Then using Aubin-Lions lemma leads to the strong p p convergence of n in L2loc ..0; T /  / to . Step 2: Convergence of pressure

Due to the BD entropy and the energy estimates, it follows that n 2 L .0; T I L3 .//: Since n is bounded in L1 .0; T I L1 .//; Hölder inequality gives 1

kn k

2

5

L 3 ..0;T //



3

 kn kL5 1 .0;T IL1 .// kn kL5 1 .0;T IL3 .//  C:

5



Hence n is bounded in L 3 ..0; T /  /. Since we already know that n converges almost everywhere to  , those bounds yield the strong convergence of n in 1 Lloc ..0; T /  /: Step 3: Convergence of the momentum In this step, we will prove that n un ! m in L2 .0; T I Lp / with p 2 Œ1; 32 /: In particular, n un ! m almost everywhere .x; t / 2   .0; T /: p p Proof. We just consider the case of d D 3. First, n un D n  n un . From the BD p entropy as in the previous subsection, we know that  r'./ 2 L1 .0; T I L2 .//. p If  satisfies that 0  c > 0; then we get r  2 L1 .0; T I L2 .//: Similar to p p 1 6 [46], one deduces that  2 L .0; T I L .//: Besides, we know that n un 2 3 L1 .0; T I L2 .// from the energy estimates. Then n un 2 L1 .0; T I L 2 .// by HRolder inequality. Next we consider @i .n un /; i D 1; 2; 3; @i .n unj / D n @i unj C @i n unj D

p p p p n n @i unj C 2 n unj @i n :

Using the BD entropy estimates, the property of 0  c > 0; and ./  c, we obtain that @i .n un / 2 L2 .0; T I L1 .//:

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D. Bresch and B. Desjardins

Finally, we want to derive the regularity of @t .n un /: The momentum equation is @t .n un / C div .n un ˝ un /  2 div ..n /D.un //  r..n /div un / C arn D 0: From the facts that n un ˝ un D

p

p n un 2 L1 .0; T I L1 .//;

n un ˝





and n 2 L1 .0; T I L1 .//; it follows that rn 2 L1 .0; T I W 1;1 .// and div .n un ˝ un / 2 L1 .0; T I W 1;1 .//. .n / 2 Using the assumptions on  with respect to  in [46], it follows that p n p L1 .0; T I L6 .// and n r'.n / 2 L1 .0; T I L2 .//: One can refer to [46] for details. Then .n /run D r..n /un /  un  r.n /   .n / p p p  n un  n un  n r'.n /: Dr p n It deduces that div ..n /run / 2 L1 .0; T I W 1;1 .//. It is the same procedure for r..n / div un / . Applying Aubin-Lions lemma, the proof is complete. p Step 4: Convergence of n un p p Next, we will prove the following facts n un ! u in L2loc .0; T  /: p First of all, since mn = n is bounded in L1 .0; T I L2 .//; and Fatou’s lemma yields that Z lim inf

m2n dx < 1: n

We define u such that 8m < uD  : 0

when

 ¤ 0;

when

 D 0:

Then Z

m2 dx D 

Z

juj2 dx < 1:

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

1569

Moreover, from Step 3 and Fatou’s lemma, it yields that Z

Z f .juj/dx 

lim inf n f .jun j/dx Z

 lim inf

n f .jun j/dx;

and thus f .u/ is in L1 .0; T I L1 .//: Next, since mn and n converge almost p p everywhere, it is readily seen that in f.x; t / ¤ 0g; n un D mn = n converges p p almost everywhere to u D m= : Moreover p

n un 1jun jM !

p

u1jujM almost everywhere:

As a matter of fact, the convergence holds almost everywhere in f.x; t / ¤ 0g; and p p in f.x; t / D 0g; thus n un 1jun jM  M n ! 0: For M > 0; we cut the L2 norm as follows: Z

p p j n un  uj2 dxdt  Z C2

Z

p p j n un 1jun jM j  u1jujM j2 dxdt

p j n un 1jun jM dxdt j C 2

Z

p j u1jujM dxdt j:

p n un 1jun jM is bounded uniformly in L1 .0; T I It is obvious that 3 L .//: Then the convergence of the above first holds, i.e., Z

p p j n un 1jun jM  u1jujM j2 dxdt ! 0:

Finally, we write Z

M2 p j n un 1jun jM j2 dxdt  f .M /

Z n f .jun j/dxdt;

and Z

M2 p j u1jujM jdxdt  f .M /

Z f .juj/dxdt:

Summing the above results, we deduce that Z lim sup n!1

CM 2 p p j n un  uj2 dxdt  f .M /

1570

D. Bresch and B. Desjardins

for all M > 0: Then the strong convergence of using the required property on f .

p u follows by taking M ! 1

Step 5: Convergence of the diffusion terms. In this step the following facts hold ./run ! ./ru in D0 ; ./div un ! ./div u in D0 ; as n ! 1: The proof is based on the energy estimates and compactness arguments, see [46] for more details. Remark 9. It is important to note that we don’t need Mellet-Vasseur estimate if we add extra terms in the momentum equations which will help to get, for instance, the bound juj2Cı 2 L1 ..0; T /  / for some ı > 0. This is the case, if we add drag terms such as jujı u in the momentum equations or a singular pressure law close to vacuum. This has been discussed by D. Bresch, B. Desjardins, and coauthors several times.

An interesting propagation property: Mellet-Vasseur property. It is interesting to note that it is not necessary to have extra terms such as drag terms or singular pressure laws to get extra integrability on un . Let us show that it is included in the compressible Navier-Stokes equations with degenerate viscosities using energy and BD entropy. We consider the case ./ D   and ./ D 0 for simplicity. For more general cases, the reader is referred to [46]. Multiplying the momentum equation by .1 C ln.1 C juj2 //u, multiplying the mass equation by .1 C juj2 / ln .1 C juj2 /=2, integrating in space, and adding the result, we get d dt

Z

Z 1 C juj2 2  ln.1 C juj / C   Œ1 C ln.1 C juj2 //jD.u/j2 2   Z Z   Œ1 C ln.1 C ju2 //u  r C C jruj2 : 



It remains to bound the right-hand side. The second term is controlled using the energy and BD entropy. Concerning the first term, it suffices to integrate by parts to rewrite terms and put space derivative on u instead of  to get the bound Z ˇZ ˇ Z  ˇ ˇ jruj2 C  Œ1 C ln.1 C juj2 //jD.u/j2 ˇ Œ1 C ln.1 C ju2 //u  r ˇ  2    Z CC Œ2 C ln.1 C juj2 /j2 1 : 

The first term is controlled using the energy and BD entropy estimates; the second term is absorbed in the left-hand side of the inequality written before. Let us remark that the last term is controlled through the estimate

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

Z

Œ2C ln.1Cjuj2 /j2 1 

Z



2 ı=21

 Z 2=.2ı/ .2ı/=2

1571

ı=2 Œ2C ln.1Cjuj2 /2=ı



with ı 2 .0; 2/. The right-hand side is bounded in time using the energy and BD entropy controls. All the calculations show that juj2 ln.1 C juj2 / 2 L1 .0; T I L1 .// if initially 0 ju0 j2 ln.1 C ju0 j2 / 2 L1 ./. This plays a crucial role in [59] to get existence of global weak solutions for barotropic compressible Navier-Stokes equations with the degenerate viscosity ./ D , ./ D 0. This propagation of integrability is really due to the control obtained using the BD entropy and the energy estimates. Remark 10. It is important to note that A. Mellet and A. Vasseur have proved in [46] the bound n jun j2 ln.1Cjun j2 / 2 L1 .0; T I L1 .// if this quantity is integrable initially using the energy estimate and the BD entropy. Remark that we can get also such bound if an additional term is included in the momentum equations such as drag term or singular close to vacuum pressure laws, see [14], and thus get the p compactness on n un .

4.1.2 Heat-Conducting Fluids In this subsection we consider the full Navier-Stokes system. We will focus on the various energy estimates and compare the difference between the barotropic case and the full system case. The full Navier-Stokes system consists of three equations: • Mass equation!  • Momentum equation ! P.;  / • Total energy equality! E D e C

juj2 ;e 2

D E.;  /

Energy estimate for heat-conducting flows: Integrating the total energy conservation equation, we get Z



juj2  eC 2 

d dt



Z D

f  u: 

Note that in this estimate, we loose any information on the gradient of the velocity. This is one of the difficulty for heat-conducting Navier-Stokes equations. Let’s try to have some extra control: Multiplying the momentum equation by u and the mass equation by juj2 =2, summing the result and integration by part, we get that 1 d 2 dt

Z 

juj2 C

Z 



 2jD.u/j2 C jdiv uj2 C

Z

Z rp  u D 

f  u: 

We need to control the last term in the left-hand side. The pressure law is assumed to express as p.; / D pc ./ C ph .;  /, where pc ./ (also called cold pressure,

1572

D. Bresch and B. Desjardins

associated with pressure at zero temperature) will generate a positive term at the left-hand side of the equality, ph .;  / is a hot pressure. In order to be consistent with the second principle of thermodynamics, the following compatibility condition, so-called “Maxwell equation” between P and E has to be satisfied: P.;  / D 2

@P ˇˇ @E ˇˇ ˇ C ˇ : @  @ 

(18)

The specific entropy s D S.; e/ is defined up to an additive constant by: 1 @S ˇˇ p @S ˇˇ ˇ D ; ˇ D 2 : @e   @    Another important assumption on the entropy function is made: the entropy S is a concave function of .1 ;  /; which ensures the nonnegativity of the so-called Cv coefficient given by Cv D

@E ˇˇ 1 @2 S ˇˇ1 ˇ D 2 2ˇ : @   @e 

Finally, we assume that the equations of state (2) are of ideal polytropic gas type: p D r C pc ./; e D Cv  C ec ./;

(19)

(I) Assumptions. This part deals with assumptions regarding physical coefficients, such as viscosity, thermal conductivity, and equation of state and the assumptions on the initial data. Viscosities, thermal conductivity, and equation of state. First of all, the viscosity coefficients  and  are assumed to be, respectively, C 0 .RC / and C 0 .RC \ C 1 .RC // functions of the density only, such that .0/ D 0, and the following constraints are satisfied: there exists positive constants c0 , c1 , A, m  1, and .d  1/=d < n < 1 such that for all s > 0; .s/ D 2.s0 .s/  .s//; for all s < A; .s/  c0 s n and d .s/ C 2.s/  c0 s n ; for all s  A; c1 s m  .s/ 

sm c0

and ca s m  d .s/ C 2.s/ 

(20) sm : c1

We require that ec is a C 2 nonnegative function on RC and the following constraint is satisfied in order to satisfy assumption (18)

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

pc ./ D 2

1573

dec ./: d

We also require that there exists  > 0;  > 0; k > 1; ` > 1; C > 0; C0 > 0, and C > 0 such that for all  2 .0;  /; `1 `1  pc0 ./  C `1 ;  ec0 ./  C0 `1 ; C C0 2n.3m  2/  1; where `  m1

(21)

and for all  >  ; 1 0  pc0 ./  C k1 ; 0  ec0 ./  C k1 ;    1 5.` C 1/  6n where k  m  : 2 `C1n



(22)

Initial data. Concerning the initial data, we assume that the functions 0 , m0 , and G0 satisfy 0  0 a.e. on ;

and

jm0 j2 D0 0

a.e. on

fx 2  = 0 .x/ D 0g;

(23)

and G0 has to be taken in such a way that G0 .x/ 2 0 .x/E.0 .x/; RC / for a.e. x 2 ;

(24)

which allows to define the initial temperature 0 on fx 2  = 0 .x/ ¤ 0g, which is assumed to be non negative n o 0 .x/ D E.0 .x/; /1 G0 .x/=0 .x/  0 a.e. on fx 2  = 0 .x/ ¤ 0g:

(25)

(II) Main result. Let us give the result Theorem 6. Let us assume that the viscosity, thermal conduction, and equation of state satisfy the assumptions mentioned before. The initial data .0 ; m0 ; G0 / are taken in such a way that the assumptions mentioned before are satisfied and that H.0/ D

Z  

jm0 j2 G0 C 20

 dx < C1;

(26)

1574

D. Bresch and B. Desjardins

that the initial density 0 satisfies 0 2 L1 ./;

r.0 / 2 L2 ./d ; p 0

and

(27)

and that the initial entropy density s0 D Cv log.0 =0 / satisfies 0 s0 2 L1 ./:

(28)

Then, there exists a global in time weak solution to heat-conducting Navier-Stokes equations (mass equation, momentum equation, and total energy satisfied + initial data in the distribution sense) and weak regularity r./ 2 L1 .0; T I L2 ./d /; p 

e and juj2 2 L1 .0; T I L1 .//;

(29)

p ./ru 2 L2 .0; T I L2 ./d d /; .1 C

p /r a=2

and

.1 C

p

/

(30)

r 2 L2 .0; T I L2 ./d /; 

(31)

for a  2. Finally, one has for some large enough positive number  and E 2 C .RC I H  .//;

u 2 C .RC I H  ./d /:

(32)

(III) Physical energy. As it is pointed out above, the physical energy equality d dt

  Z juj2 dx D  eC f  u 2  

Z

(33)

can be obtained in a classical way by multiplying the momentum equation by u and using the energy equation. Remark 11. For the barotropic case, we obtain the estimate 1 d 2 dt

Z 

juj2 C 2

Z 

./jD.u/j2 C

d dt

Z

Z

./jdiv uj2 D

./ C 



Z f  u; 

which provides a H 1 bound in space on the velocity u. But for the full system with heat conductive equation, the following facts hold:

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

1 d 2 dt

Z

2

Z

juj C 2 



Z

Z

c ./ C 

./jdiv uj2

./jD.u/j C 

Z

d C dt

Z

2

1575

rph .;  /  u D 

f  u; 

where c ./ D ec ./; ph .;  / D p.; /pc ./; and eh .;  / D e.;  /ec ./. If we want to control on ru; we have to control Z

Z rph .;  /  u D 



ph .;  /  div u: 

In [26], E. Feireisl takes ph .;  / D  p ./; pc ./  ; p ./  3 . Then it holds that ˇZ ˇ ˇ ˇ ˇ  p ./div uˇ  kkL6 ./ kp ./kL3 ./ kdiv ukL2 ./ 



 C k kH 1 ./ kkL3 ./ kukH 1 ./ ; where we used the regularity of  ; , and u. For simplicity, we omit the details. (IV) Thermal equations. Two different reformations of the internal energy equations will lead to useful bounds on the temperature. The first one is written as   Cv @t . / C div . / C r  div u D 2D.u/ W D.u/ C jdiv uj2 C div .  r /   2 jdiv uj2 C div .r /; D 2S .u/ W S .u/ C  C d

(34)

where the derivatotic part S .u/ of the strain rate tensor D.u/ is defined as the zero trace component: S .u/ D D.u/  .div u/I=d: Using the assumption that 0  0 a.e. on  and the fact that the first two terms of the right-hand side of (34) are nonnegative, the minimum principle formally applies to the temperature. The second form of the internal energy equation is the most physically relevant, since it involves the specific entropy s: Indeed, we deduce from the definition of s that it satisfies formally: .@t .s/ C div .us// D 2D.u/ W D.u/ C jdiv uj2 C div .r /:

(35)

1576

D. Bresch and B. Desjardins

Hence dividing (35) by  and integrating over , we end up with Z  

2   jD.u/j2 C jdiv uj2 C 2 jr j2   



Z

d D dt

s: 

Using the assumption that 0 .s0 / 2 L1 ./; for t  0; one has Z tZ 0

Z

 1 2D.u/ W D.u/ C jdiv uj2 C 

Z tZ 0

 jr j2 2



(36)

.s.t; / C 0 .s0 / / ;

 

where s0 D max.s0 ; 0/: Obviously, the property on s will give the control of Z

T 0

Z 

 jr j2 < C1: 2

(37)

When the equation of state takes the form of (19), the physical entropy can be  written in terms of  and  as s D Cv log.  /; where  D  1: Introducing  1 > 0 and recalling that s is defined up to an additive constant, we may write 1 s D Cv  log  C Cv  log. /  1 /:  Cv  C Cv  log.  Then it follows that Z

Z



Z

1 s  Cv  C Cv  log    

 :

It remains to control the above term, which is done by using the mass conservation equation in a renormalized way with ˇ1 ./ D  log.1 =/; @t ˇ1 ./ C div .ˇ1 ./u/  div u D 0: The right-hand side of (36) is estimated by Z

Z

Z 0 .s0 / C 

Z

Cv .t; / C Cv 

T

Z

ˇ1 .0 / C 

 jdiv uj :

0



The last term above can be estimated by the left-hand side of (36) as follows: Z tZ

Z tZ

1

jdiv uj  0



0

1

2 

.d  C 2/

1 2



.d  C 2/ 2 

1 2

p  jdiv uj  ;

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

1577

where we use the Young inequality with the bound of  in L1 .0; T I L1 .// and the assumption that s 7! .d .s/ C 2.s// belongs to L1 .RC / since n  1 and m  1: p  From above, it concludes that jr j 2 L2 .0; T I L2 .//: From the properties  of ; i.e., .;  / D 0 .;  /.1 C /.1 C  ˛ /; ˛  2;

(38)

where 0 is a C 0 .R2C / function satisfying for all positive  and , c3  0 .;  / 

1 ; c3

p 1 for some p positive constant c3 ; the following four quantities .d  C 2/ 2 jdiv uj=  ; 1 a p p  2 S .u/= ; . C1/r 2 , and .1C /r log  are a priori bounds in L2 ..0; T / / as soon as 0 .s0 / and 0 log.1 =0 / are bounded in L1 ./: (V) A priori estimates: Kinds of Energy and BD entropy. The additional estimates are deduced as Z Z Z Z 1 d juj2 C 2./D.u/ W D.u/ C ./jdiv uj2 D p.; /div u; 2 dt     and 1 d 2 dt

Z ju C 2r'./j2 C 2./A.u/ W A.u/   Z D  rp.; /.u C 2r'.//;

Z



where A.u/ D .ru  rut /=2 denotes the skew symmetric part of ru and ' is defined up to a constant by ' 0 . / D 0 . /= : Adding the above two equalities, the following terms have to be controlled : Z

Z p div u; and



rp  r'./:

(39)



Remark 12. For the barotropic case, the two terms in (39) are easily controlled by Z rp  u D 

d dt

Z

./; 

and Z

Z rp  r' D





p 0 ./0 ./ jrj2 : 

1578

D. Bresch and B. Desjardins

Z rp  r'. In order to bound such an integral, the thermodynam-

(V-1) Control of 

ical properties of the fluid have to be used. Defining the “hot” pressure and energy components as ph .;  / D p.; /  pc ./; and eh .;  / D e.;  /  ec ./; i.e., the pressure and energy associated with nonzero temperature effects. Notice that in the case of an equation of state satisfying (19) with coefficients > 1 and Cv ; where the preceding five coefficients are constants and given by  D  1: Then tedious but straightforward computations lead to the differential identity valid for any equation of state, rp D rpc C r.r C r /: Similar to the barotropic case, we obtain the following facts: Z

Z rp  r' D 



pc0 ./0 ./ jrj2 C 

Z rph  r': 

Taking ph .;  / D r ; it follows that Z

Z

jrj2 C rph  r' D r ./    0

Z

r0 ./r  r;



where ' 0 ./ D 0 ./=: Hence, the cold component of the pressure yields the following integral for some positive constant c0 W Z 

pc0 ./0 ./

jrj2  c0 

Z

jr./.`C1n/=2 j2 



1 

Z 

jr./j2 ; 

the lower bound being deduced from assumption (22) and  being taken such that ./ D  for    =2 and ./ D 0 for    : On the other hand, it follows that by using Cauchy-Schwarz-type inequalities jr0 ./r  rj  .;  /  .;  /

jrj2 r 2  2 jr./j2 C 2  c jr j2 C  jr'./j2 ; 2 0

where (38) is used in the last inequality Zand c > 0 is a constant. Due to the rp  r'. Note that integrability of

inequality (37) one can control the term 

negative power of the density will be really important to get the compactness on the temperature and Z to pass to the limit in the total energy formulation. p div u. It remains to bound the term

(V-2) Control of 

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

Z

Z

Z

p  div u D 

pc ./div u C Z

1579



ph ./  div u 



2 dec

Z

r div u div u C d  Z Z d ec ./ C r div u: D dt  

D



In the case of a perfect gas equation of state, we split the integrated expression into bounded and unbounded densities ˇZ ˇ ˇ ˇ ˇ rdiv uˇ ˇ ˇ 

1

crk.d  C 2/ 2 div ukL2 ./  2 1  k 5 kL2 ./ k.65n/=10 1A kL3 ./ k kL6 ./ c 1 k.d  C 2/ 2 div uk2L2 ./ C kk2L1 ./  c r./ C .1 C k k2L6 ./ /.k p k2L2 ./ C kk2L1 ./ /;   for all positive  and constant c: Using together with Sobolev embedding, we deduce that t 7! k.t; /k2L6 ./ is a priori bounded in L1loc .RC /: Taking  small enough in 1

order to absorb k.d  C 2/ 2 div uk2L2 ./ by the left-hand side coming from viscous 1 dissipation, observing that  is already known to belong to L1 loc .RC I L .//; the third term will be estimated by a Gronwall-type lemma. With those estimates and some local integrability analysis of various energy fluxes such as ujuj2 , eu, pu, and r, we can study the compactness of sequences of approximate solutions .n ; un ; n / and pass to the limit in nonlinear terms to establish the global existence of weak solutions, we omit it here and the reader can refer [9] for the details. We will make only several comments. Compared to the barotropic case and after getting the a priori estimates we have presented before, the difficulty is of course the temperature dependency. Remark that using the cold pressure term, we can get some bounds on the velocity with weights depending on the density, for instance, on 1=3 u in Lq with q > 3, we can also get some bounds on the temperature and the heat flux. To get strong compactness for the internal energy and the temperature, the first step is to derive uniform bounds on @t .n En / using the bounds described before and then to get

1580

D. Bresch and B. Desjardins

p enough space compactness on n n taking advantage of the strong convergence p 1=2 of n un and that En D Cv n C jun j2 =2. Using the strong compactness of n (obtained using the singular pressure close to vacuum) will give compactness on n . Remark just that to pass to the limit in the total energy conservation equation, the only difficulty is to pass to the limit in the energy flux n un .en C jun j2 =2/, the heat flux .n ; n /rn , and the stress flux 2.n /D.un /  un , .n /un divun . For p the energy flux, n un n converges strongly to u in L1 as a product of n un p 1=3 and n n which converges in L2 . Moreover, n un converges to 1=3 u in L3 . For 1 the heat flux, it converges in L using the compactness on the temperature obtained through the control of negative power of the density. Finally the stress flux converges weakly in L1 using strong convergence on un , and on n and weak convergence on .n /1=2 run . We refer to [9] for details and also mentioned [47] where heatconducting compressible mixture with a degenerate viscosity with multicomponent diffusion is considered.

4.2

Construction of Approximate Solutions

Compactness of the set of weak solutions has been proved. It remains to eventually construct solutions of approximate systems for which existence and uniqueness can be easily obtained. The challenge is to add smoothing terms to the partial differential equations that preserve the mathematical structure of the original equations, in particular energy estimates and BD entropy estimates obtained, respectively, by multiplying the momentum conservation equation either by u or by r'./, where ' 0 .s/ D 0 .s/=s. This issue is not straightforward for compressible Navier-Stokes equation with density dependent viscosities. Let us explain how it can be done if a cold pressure is added for the heat-conducting Navier-Stokes equations: The stability was proved in [10]. We will also explain the construction of approximate solutions for the barotropic compressible Navier-Stokes equation with degenerate viscosities. It has been, for instance, fully described in [60] and [34] based on proposition made in [10]. (I) Barotropic case. Let us present the approximate procedure used in [60] and based on propositions written in [10]: @t  C div .u/  "  D 0 @t .u/ C div .u ˝ u/ C rp C rpc ./  2div.D.u// C "r  ru

(40)

1 p C 2 u C r0 u C r1 juj˛ u  r. p /  ır 2sC1  D f 

(41)

with s large enough and ˛ > 0. The interested reader is also referred to [48] where they design appropriate approximate solutions to a model of two component compressible reactive flows. Note that all terms , pc ./, 2 u, and r 2sC1 

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

1581

are important to build approximate solutions (the first one is also present in the constant viscosities case, see [51]). Then it is possible to pass to the limit letting ", ı, and  tend to zero. It is possible to build approximate solutions satisfying uniformly the energy estimates and the BD entropy but which does not satisfy the Mellet-Vasseur estimate uniformly (Galerkin method and fixed-point procedure are the usual steps). Therefore, at this stage, we are just able to obtain the following existence results: • If pc ./ maintained, we can let the other quantities go to zero, namely, r0 ; r1 ; : Global existence of compressible Navier-Stokes equations with singular pressure (cold pressure). • If r1 is maintained, we can let the other quantities go to zero. Global existence of compressible Navier-Stokes equations with turbulent drag term. • We can prove the existence with r0 ; r1 > 0 and  > 0. Global existence of compressible quantum Navier-Stokes equations with drag terms. It is important to note the last papers written recently by A. Vasseur and C. Yu [59] and by J. Li and Z. Xin [43] where they get global existence of weak solutions of compressible Navier-Stokes equations with degenerate viscosities without any additional terms. In [59], they use the existence of global weak solutions for compressible Navier-Stokes quantize equations with drag terms (choosing ˛ D 2) p p and quantum term r. = / to perform a kind of renormalization technic on juj to show the possibility to pass to the limit with respect to the capillarity coefficient using truncation close to vacuum and for large density (a link between the truncated parameter and the capillarity one is necessary). Then they pass to the limit with respect to the drag coefficients. In [43], they propose an appropriate regularized system and show the limit passage. The interested reader is referred to these works. (II) Heat-conducting case. Let us present the different quantities step by step in order to explain things for beginners. Step 1: Density control. The first step is to add to the right-hand side of the momentum conservation equation a high-order Korteweg-like term and an additional pressure term in order to control the density: @t  C div .u/  "  D 0 @t .u/ C div .u ˝ u/ C rp C rpc ./ C "r  ru  2div.D.u// D f   Cr 2sC1  @t .E/ C div .uH C pa u/ D div .  u/ C div .r / C f  u   Cu  r 2sC1  :

(42)

1582

D. Bresch and B. Desjardins

The additional cold pressure term is designed to get high integrability properties for 1=: one can choose, for instance, pc ./ D N 1N where N D 6n. The corresponding cold internal energy therefore satisfies ec ./ D N . Let us observe that (42) has an additional term proportional to " associated with the work of high regularity Korteweg force in the momentum conservation equation. As a consequence, the entropy evolution equation remains identical to (35)  .@t s C div .us// D 2D.u/ W D.u/ C jdiv uj2 C div .r / ; so that the estimates resulting from integration of the entropy equation remain valid. In particular, the estimates involving Z p div u 

remain unchanged. In particular, one obtains L1 .0; T I L1 .// bounds on "N . It remains to identify the contribution of the high regularity Korteweg force in the two estimates obtained by multiplying the momentum conservation by u and 1 r./. On the one hand, multiplying the high regularity Korteweg force by the velocity u and integrating by parts using the mass conservation equation leads to Z



2sC1

u  r





Z

@t  2sC1 

 D 

D

1 d 2 dt

Z

jjDj2sC1 j2 : 

As a consequence of this bound, one may write



2 1

D  C

 2 L ./

2

D 



1



L1 ./ 2

C

L2 ./

kDk2L1 ./



1



3

! :

L2 ./

Using the fact that .s/  c0 s n , one concludes that for " > 0 given 1 belongs to L1 ..0; T /  /, so that the density is bounded from below by a positive constant. On the other hand, multiplication of Korteweg-like force by 1 r leads to Z  

 2sC1  D 

Z

jjDj2sC2 j2 : 

Step 2: Velocity smoothing. In order to smooth out the velocity field u, we add a smoothing term depending on parameter  > 0

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

1583

@t  C div .u/ D "  @t .u/ C div .u ˝ u/ C rp C rpa ./ D div  C f C 2sC1 .u/ C r 2sC1   "r  ru @t .E/ C div .uH / D div .  u/ C div .r / C f  u Cu  2sC1 .u/ C u  r 2sC1 :

(43)

The contribution of this additional term  2sC1 .u/ to the estimates is the following: scalar multiplication by u and integration over  of the momentum conservation equation leads to an additional dissipative term Z

jjDj2sC1 .u/j2 ; 

whereas its contribution to the entropy equation is zero because the work of such forces has been added to the total energy conservation equation. We therefore obtain an addition bound in L2 .0; T I H 2m .// on u. On the other hand, the BD entropy estimate obtained by multiplying the momentum conservation equation by './ leads to Z Z  2sC1 .u/  r log  D r 2sC1 .u/  r 2sC1 : 



A Cauchy-Schwarz argument for the right-hand side then allows to conclude that this term is estimated by the L2 .0; T I H 2sC1 .// bound on u, using the fact that high derivatives of functions of  are L2 integrable for given " > 0. Step 3: Temperature smoothing and control. In order to smooth out the temperature while preserving the structure of the entropy evolution equation, we add the following perturbation to the temperature equation for some small parameter !>0   ˇ Cv @t . C ˇ 4 / C div .. C ˇ 4 /u/ C .r  C  4 / div u  div. r / 3  5 D 2D.u/ W D.u/C 2    (44) where  is the regularized conductivity coefficient. Note that the two terms    5 are added as in [25] or [47] to ensure the temperature stays away from 2 zero and is bounded from above. The added radiative quantity in the pressure ˇ 4 is to ensure compactness of density without playing with the cold pressure. The corresponding entropy evolution equation integrated by parts leads to Z  

2  jD.u/j2 C 2 jr j2  

 D

d dt

Z s: 

1584

D. Bresch and B. Desjardins

Step 4: Construction of smooth solutions. Given fixed positive parameters, the a priori estimates induced allow to build solutions by using fixed-point procedure via Galerkin type methods: continuity equation and temperature equation (regularizing the equation in such a way that the classical theory for quasilinear parabolic equations could be applied). Thus uniform estimates provide global in time solvability. A fundamental estimate is obtained on the entropy dividing the internal energy equation by  and integrating. Limit passage in the Galerkin approximation. The strong convergence of the temperature is obtained by stability process. It remains now to let the parameters go to zero using the stability process described before due to uniform bounds. We will not detail things and refer to [47] for details in the case of heat-conducting, compressible mixtures with multicomponent diffusion.

5

The Ä-entropy and Two-Velocities Hydrodynamics

In this section, we will discuss a new concept of solution introduced in [14]. This shows interesting phenomena with two-velocities hydrodynamic in the barotropic compressible system and an arbitrary mixing parameter  2 .0; 1/: This presence of two velocities is due to the density dependency of the viscosities. This will give a nice  nonlinear hypocoercivity property which will be helpful for asymptotic analysis through relative entropy techniques. It is strange that some properties artificially introduced in the constant viscosities case by H. Brenner appear in this situation, see [4–6]. For reader’s convenience, let us first present H. Brenner’s system studied mathematically in [30]: 8 < @t  C div.um / D 0; @ .%u/ C div.u˝ um /C rp D divS;  : t 1 2 @t . 2 jvj C e/ C div . 21 jvj2 C e/vm C div.pv/ C divq D div.S v/

(45)

with pressure laws p D pe ./ C  pt ./ where  is the temperature. The internal energy splits into two parts e.;  / D ee ./ C cv  with cv > 0 the two-velocity fields are related through the following constitutive relation v  vm D Kr log  and the Fourier’s law is given by q C KPe ./r log  D r

30 Weak Solutions for the Compressible Navier-Stokes Equations with. . .

1585

where  is the heat conductivity coefficient and S is given through the relation  2 S D  2D.u/  divu IRn C divu IRn 3 with D.u/ D .ru C r t u/=2 is the strain tensor. Remark the two-velocities hydrodynamic introduced by H. BRENNER artificially. The new model provides a relatively simple and rather transparent modification of the classical system replacing the Eulerian mass velocity um by its volume counterpart u in the viscous stress tensor and the specific momentum, where um and u are interrelated through a kind of Fick formula. The two velocities coincide in the “incompressible” regime when  D const. We will see that such property (two-velocities framework) is in fact included in the compressible Navier-Stokes equations when the two viscosities depend on the density, see [14] for discussions around heat-conducting flows.

5.1

The Barotropic Compressible System

Let us recall the compressible Navier-Stokes equations with the following viscosities .%/ D  , ./ D 0: 

@t  C div.u/ D 0; @t .%u/ C div.u ˝ u/ C rp.%/  2div.D.u// D 0

(46)

with D.u/ D .ru C .ru/T /=2. Recently in [14], it has been observed that such compressible Navier-Stokes system may be reformulated through an augmented system. The main idea is to use a parameter  2 .0; 1/ to mix energy and BD entropy, namely, we consider the following quantity in the kinetic energy juj2 C .1  /ju C 2r log j2 : This may be seen as a two-velocity hydrodynamic with 1 D , u1 D u and 2 D .1  /, u2 D u C 2r log , see [32] and [55]. This motivates the introduction p of an intermediate velocity v D u C 2r log , a drift velocity w D 2 .1  /r log , and a mixture coefficient , then the system reads as a two-velocity hydrodynamic system: 8 ˆ ˆ @t % C div.% .v  2r log // D 0; ˆ ˆ ˆ ˆ @t .%v/ C div.%v ˝ .v  2r log / C rp.%/ ˆ ˆ < D div.2.1  /D.v//  p Cdiv.2%A.v//  div 2 .1  /%rw ; ˆ ˆ ˆ ˆ ˆ ˆ @t .w/ C div.w ˝ .v  2r log // p ˆ ˆ : D div.2%rw/  div.2 .1  /%.rv/T /:

(47)

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with A.v/ D .rv  .rv/T /=2. The associated -entropy reads for all t 2 Œ0; T :

sup

Z h

.

 2Œ0;t 

i jwj2 jvj2 C / C ./ . /dx 2 2 C2

 p p 2 2  jA.v/j C jD. 1  v/  r. w/j dx ds 

Rt R 0

Z tZ C2 0



p 0 ./ jrj2 dx ds  

Z h

%.



i jwj2 jvj2 C / C .%/ .0/ 2 2 (48)

with s 0 .s/  .s/ D p.s/. This -entropy is obtained by taking the scalar product of equation satisfied by v and w, respectively, with v and w, adding the results and using the mass equation. Note that the more general case with .%/ arbitrary and .%/ D 2.0 .%/%  .%// for the compressible Navier-Stokes equations is covered in [14] (but with extra drag or cold pressure terms for the existence). This allows to define appropriate global solutions named -entropy solutions to the compressible Navier-Stokes equation with degenerate viscosities. It works well also for general viscosities ./ and ./ such that ./ D 2.0 ./  .//: See [14] for details. It is then possible to prove the following mathematical result. Theorem 7. Let ./ D 2.0 ./  .// with P ./ D a with > 1. Assume 0 2 L ./; m0 2 L1 ./;

0  0; m0 D 0 if 0 D 0;

p r.0 /= 0 2 L2 ./; jm0 j2 =0 2 L1 ./:

Then there exists a global -entropy solution .; u/ of the degenerate compressible Navier-Stokes equations with extra drag terms or cold pressure that means satisfying a -entropy similar to (48), and the compressible Navier-Stokes equation with its boundary conditions in a weak sense. Remark 13. It is important to note that global existence of -entropy solution has been proved in [14] for the compressible Navier-Stokes with a drag term or singular cold pressure. Note that it is not difficult to prove that a -entropy solution satisfies also the augmented formulation in a weak sense. Let us just precise here the regularized system in the case ./ D  and ./ D 0 which allows to build approximate solutions which will give after passing to the limit a global -entropy solution:

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8 ˆ @t % C div.%vı /  2  D 0; ˆ ˆ ˆ ˆ @t .%v/ C div.v ˝ .%vı  2r/ C rp.%/ C " 2s v  div..1 C jrvj2 /rv/ ˆ ˆ ˆ < D div.2.1  /D.v//  p Cdiv.2%A.v//  div 2 ˆ .1  /%rw ; ˆ ˆ ˆ ˆ ˆ @t .w/ C div.w ˝ .vı  2r// ˆ p ˆ : D div.2%rw/  div.2 .1  /%.rv/T / (49) where the smoothing parameter ı designs standard mollification with respect to time and space. The smoothing high-order derivative term 2s with s  2, depending on small parameter  > 0, has to be introduced to control large spatial variations of v, because div v is not a priori bounded in L1 .0I T I .L1 .//. Such bound will be required to be able to have bounds on the density. We will also need to show that w D 2r log  at some point of the construction process and the second term in the regularization process will be helpful. Remark 14. This is interesting to note that, by changing the definition of the velocity, a diffusive term appears in the mass equation. No need to add it as previously: compressible Navier-Stokes equations with density-dependent viscosities ensure in some sense parabolicity property of the density. This property has been also observed by B. Haspot, see [36]. Note also the cross-diffusion effect, we get, between the velocity v and the drift velocity w. Remark 15. Note that this formulation will be very helpful to obtain global existence of weak solutions for compressible Navier-Stokes equations with degenerate viscosities for quite large range of viscosities ./, ./ such that ./ D 2.0 ./  .//. This is the purpose of the forthcoming paper [21]. It requires to consider a third-order regularized term compatible with the viscous term.

5.2

The Heat-Conducting Compressible System

In this section, we present the equations of motions for the heat-conducting fluid written in terms of the two velocities u and u C 2r'./ with corresponding densities .1  / and : This presentation follows [14]. We do not aim at proving the existence result for such system but on showing that the two-velocity hydrodynamics in the spirit of the work by S.M. SHUGRIN [55] is consistent with the study performed for the low Mach number system in the first part of this diptych in [15]. More precisely, we will show that the formal low Mach number limit for the two-velocities system gives the augmented system used in [15] to construct the approximate solution. An important observation is that the system presented

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below does not coincide with the usual heat-conducting compressible Navier-Stokes equations. Indeed, the two-velocities description of the dynamics of the fluid lead to different energy equations with a generalized temperature, called the -temperature. However, this is not a priori the usual temperature, unless the system reduces to the angle velocity one (i.e., the density  is equal to 0). This property was also explained in the works [55] and [32] where the authors discuss the capillary temperature. As mentioned these calculations may be found in [14]. We assume that  is a periodic box in R3 , i.e.,  D T3 , and we consider the following two-velocity system 8 ˆ @t  C div.u/ D 0 ˆ ˆ ˆ ˆ @ ˆ t u C div.u ˝ u/  div.2./D.u//  r../divu/ C rp.; e / D 0 ˆ ˆ < @t ..u C 2r'.// C div.u ˝ .u C 2r'.///  div.2./A.u// Crp.; e / D 0 ˆ ˆ ˆ ˆ @ E C div.E u/ C div.p Œ.1  /u C .u C 2r'.// C div Q ˆ t   ˆ ˆ ˆ : div .1  /S1 .u/ C S2 .u C 2r'.// D 0; (50) where we denoted D.u/ D 12 .ru C r t u/ and A.u/ D 12 .ru  r t u/. The viscosity coefficients ./, ./ satisfy the BRESCH-DESJARDINS relation ./ D 20 ./  2./: The total -energy E is defined as follows  .1  / 2  juj C ju C 2r'./j2 : E D  e C 2 2

Remark 16. Note that E u is expressed as a sum of two energies



juj2 ju C 2r'./j2 C  e C .1  / e C 2 2 similarly to energy from [55]. Integrating the total -energy equation with respect to space, we obtain d dt

Z E D 0: 

Thus (5.2) and the identity .1  / 2  ju C 2r'./j2 j2r'./j2 juj C ju C 2r'./j2 D C .1  / 2 2 2 2

(51)

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yields the following conservation property Z  1 .1  / d  e C ju C r'./j2 C jr'./j2 D 0: dt  2 2 This quantity may be treated as a generalization of the -entropy, found for the barotropic case, to the heat-conducting case. The viscous tensors S1 and S2 are given by S1 D 2./D.u/ C ./div u Id and S2 D 2./A.u/: The heat flux Q is given by standard Fourier’s law, i.e., Q D Kr ; where K is the positive heat-conductivity coefficient and  denotes the generalized temperature (the -temperature). Let us consider an ideal polytropic gas, namely, p.; e / D r C pc ./;

e D Cv  C ec ./;

where r and Cv are two positive constant coefficients, see, for instance, [9]. For convenience, we denote D 1 C r=Cv . Moreover, the additional pressure and internal energy, pc and ec , respectively, are associated to the “zero Kelvin isothermal,” which roughly speaking means that lim pc ./ D 1:

!0C

Further, we require that ec is a C 2 .0; 1/ nonnegative function and that the following constraint is satisfied pc ./ D 2

dec ./: d

Below we present two different formulations of the internal energy equation which lead to useful bounds on -temperature similarly as in [9] for the usual temperature. The first formulation reads  Cv @t . / C div.u / C  div w D 2.1  /./jD.u/j2 C 2./jA.u/j2 C2.1  /.0 ./  .//jdiv uj2 C div.Kr /;

(52)

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with  the Gruneisen parameter and where the mixing temperature  becomes the usual temperature if  D 0. Note that for 0    1, the -temperature remains nonnegative in view of the maximum principle. The second formulation is based on the notion of generalized -entropy s . It is the usual entropy in which the standard temperature has been replaced by the -temperature, i.e., s D Cv log   r log ; thus, when ;  is sufficiently regular we can derive the following equation @t .s / C div.us /  div.Kr log  / D 2.1  /

(53)

./jD.u/j2 ./jA.u/j2 C 2  

C2.1  /

.0 ./  .//jdiv uj2 jr j2  2 './ C K :  2

Note that, recalling the relation ˇ ˇ2 ˇ ˇ 1 1 ˇ jD.u/j D ˇD.u/  divu Idˇˇ C jdivuj2 ; 3 3 2

the terms on the right-hand side, when integrated over space, give nonnegative contribution using the assumption on 3./ C 2./ and ./ D 2.0 ./  .//. Indeed, it suffices to check that for the penultimate term reads Z 

 './ D 

Z

 ' 0 ./jrj2  0: 

Using all this information, it could be possible to prove global existence of -entropy solution for the heat-conducting compressible Navier-Stokes system under analogous assumptions as in [9], replacing the usual temperature by the temperature. The existence of the approximate solution could be then proven by using the augmented system written in terms of w D u C 2r'./ and v D 2r'./ as it was done in [55] or in [14] addressing barotropic flows @t  C div.Œwı /  2div.Œ0 ./˛ r/ D 0;

(54)

0

@t .w/ C div..Œwı  2Œ ./˛ r/ ˝ w/  r.../ 2.0 ./  .///div.w  v//  2.1  /div../D.w// 2div../A.w//C" 2s w  "div..1 C jrwj2 /rw/Crp.; e / D 2.1  /div../rv/;

(55)

0

@t .v/ C div..Œwı  2Œ ./˛ r/ ˝ v/  2div../rv/ C2r..0 ./  .//div.w  v// D 2div../r t w/

(56)

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with the -total energy supplemented by the  correction corresponding to the  regularization of the momentum   (57) @t .E / C div. Œwı  2Œ0 ./˛ r E / C div.p w/ C divQ  div S1 w C .1  / S2 v C " s w  w  "div..1 C jrwj2 /jwj2 D 0 (58) and the set of initial conditions. Above, the total -energy E is defined as 1 .1  / 2 E D e C jwj2 C jvj : 2 2 Note, however, this construction would not lead to the usual heat-conducting compressible Navier-Stokes system in the limit  ! 0. Indeed, the difference is again due to the -temperature that is not the usual one. But, performing a formal low Mach number limit for this system, we would get p D 1, div w D 0 (comparing terms of the same order). In the equation on w, being now incompressible, the pressure gradient rp would be replaced by Lagrangian multiplier r . As a result, we would get the augmented system defined in [15] in the part devoted to construction of solution.

6

Relative Entropy and Some Applications

(I) Relative entropy for barotropic Navier-Stokes equations with density-dependent viscosities. In this section, let us provide recent results obtained by D. Bresch, P. Noble, and J.P. Vila in [19] with details in [20]. Since the pioneering work by C. Dafermos and H.T. Yau, relative entropy methods have become a crucial and widely used tool in the study of asymptotic analysis (singular limits, long-time behavior). The very rough idea is work with an energy and make a Taylor expansion of this energy around a state written in terms of conservative quantities. In our case, linked to the -entropy ˇ estimate, let us consider the relative energy functional, denoted E.; v; wˇr; V; W /, defined by Z ˇ 1 ˇ E.; v; w r; V; W / D %.j w  W j2 C jv  V j2 /dx (59) 2  Z C .F .%/  F .r/  F 0 .r/.%  r//dx 

which measures the distance between a -entropic weak solution .%; v; w/ to any smooth enough test function .r; V; W /. This is similar to the one that we can find in [27] for compressible Navier-Stokes with constant viscosities but here with twovelocity fields because we will play with the augmented system presented in the

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previous section. In the following theorem, we consider the case where ./ D   with  a strictly positive constant (for the general case, the reader is referred to [16]). We will explain later on for readers who don’t know how to find the appropriate relative entropy quantity in the compressible Navier-Stokes equations setting. Consider the case where ./ D , ./ D 0, then we can prove: Theorem 8. Any weak solution .; v; w/ of the compressible Navier-Stokes equations satisfies the following so-called relative entropy inequality: For all  2 Œ0; T  and for any pair of test functions such that r 2 C 1 .Œ0; T   / with r > 0, and V; W 2 C 1 .Œ0; T   / W ˇ ˇ E.; v; wˇr; V; W /. /  E.; v; wˇr; V; W /.0/ Z Z Z Z p p C2 %ŒA.v  V /j2 C 2 %jD. .1  /.v  V /  .w  W //j2 Z

0

Z

C2 Z



Z

0



 0





h

i h i % p 0 .%/r log %  p 0 .r/r log r  r log %  r log r





r

 w/  rW /  .W  w/ .1  / Z Z  % @t W  .W w/C@t V  .V v/ w/  rV /  .V  v/ C

% ..v  0



r

C..v Z



Z

 .1  /

Z



Z

0



r h rF 0 .r/  %.v 

 w/ .1  /   0 0 r r i Z Z   r.V  .p.r/  p.%// div.V  W/ C W/ .1  / .1  /  0 Z Z Z Z  p  1 rr 0 p p .%/r%  Œ2 % D .1  /V /  W  C 2 r .1  / 0 0     p  p p r. W / W D .1  /.V  v//  r. .W  w/ Z Z Z Z rr r% % 0 C2 %A.V / W A.V  v/ C 2 p .r/rr  .  / r %   r 0 0 Z Z h i p % A.W / W A.v  V /  A.w  W / W A.V / : (60) C2 .1  / C

@t F 0 .r/.r  %/ 

0



These are simple calculations and interested readers are referred to [19] (and [20] for details). By using a density argument, we can of course relax the regularity on the test functions using the regularity of the -entropy solutions as it was done in [27] for the constant viscosities. Remark that here, we do not assume W to be a gradient. The third line is also original compared to [35] and allows to relax the strong constraint imposed in [35] that the viscosity is proportional to the pressure

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law and covering now the physically founded case of the shallow-water equations. Note the recent work in [16], where they prove that the augmented system is really interesting to define relative entropy estimates for Euler-Korteweg systems in a simplest way compared to recent works [22] and [33] with less hypothesis on the capillarity coefficient. They also provide relative entropy tools for compressible Navier-Stokes-Korteweg with density-dependent viscosities satisfying the BreschDesjardins relation ./ D 2.0 ./  ./. This algebraic relation plays again a crucial role to get the results. (I-a) How to get relative entropy terms? Let us present for reader’s convenience how to get modulated quantity in the framework of the compressible Navier-Stokes equation. In the -entropy, the kinetic quantity reads 1 2

Z

.jvj2 C jwj2 / C



Z F ./: 

These quantities correspond to , v, and w in the time derivatives of mass and momentum equations. We therefore linearized the kinetic energy with respect to this variables. The quantity coming from the pressure F ./ is modulated as follows (linearization around r) I1 D F ./  F .r/  F 0 .r/.  v/ and the quantity related to jvj2 or jwj2 as follows (linearization around .r; rV / or .r; rW /) I2 D jvj2 =  jrV j2 =r C

jrV j2 rV  .v  rV / .  r/  2 2 r r

D jvj2 C jV j2  2V  v D jv  V j2 : The same conclusion occurs for .r; W /. We therefore find the relative kinetic quantity already present for compressible Navier-Stokes equations with constant viscosities: see [28]. Let us now modulate the quantities coming from the viscous quantities and the pressure term in the  entropy. Remark that in the viscous quantity the conserved quantity is D.u/ or A.u/ and remark that for the pressure term the quantity is  and v. Concerning the pressure term, it reads p 0 ./jvj2 . Then after calculation, we get the following modulated quantity I3 D

 p 0 .r/ 0 p 0 ./ p 0 .r/ p 0 .r/ jvj2  jrV j2  .rV /  .v  rV /: jrV j2 .  r/  2  r r r

Concerning the viscosity term jD.u/j2 =, the calculation is the same than for the kinetic quantity and the modulated quantity reads

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I4 D

 1 0 1 1 jD.u/j2  jrD.U /j2  jrD.U /j2 .  r/  r r 1 2 rD.U /  .D.u/  rD.U // D jD.u/  D.U /j2 (61) r

The expression is similar for ./A.u/. Remark that we can write I3 D %Œp 0 .%/r log %  p 0 .r/r log r  Œr log %  r log r CrŒp./  p.r/  p 0 .r/.  r/  r log r: This explains the chosen modulated quantity for the term coming from the pressure noticing that the last quantity is related to the relative kinetic entropy. For more general viscosities, the reader is referred to [16]. The interesting feature is that relative entropy for compressible Navier-Stokes equation work using the BD relation between  and , namely,  D 2.0 ./  .//. (I–b) Some applications. Let us now present several applications that we can consider using this relative entropy tool: see [28] for the compressible Navier-Stokes equations with constant viscosities. Using the relative entropy (60) and the identity %Œp 0 .%/r log %  p 0 .r/r log r  Œr log %  r log r 0

D %p .%/jr log %  r log rj

(62)

2

CrŒp.%/  p.r/  p 0 .r/.%  r/  r log r ˇ   %.p 0 .%/  p 0 .r//  p 00 .r/.%  r/r ˇr log rj2 ; we can justify several mathematical results. The interested reader is referred to [20] for details of the proof. Let us mention two of them which extend to the densitydependent viscosities the well-known results for constant viscosities. (I) Weak-strong uniqueness.

Let us consider a -entropy solution .%; u/ and recall v D u C 2r log % and p w D 2 .1  /r log %. Assume that .r; W; V / satisfies pthe augmented system with the regularity written before and assume that W D 2 .1  /r log r. Then we prove that .%; v; w/ D .r; V; W / that means uniqueness property: p p weak-strong this gives .%; u/ D .r; U / with U D V  W = .1  /. More precisely the following result holds. Theorem 9. Let  be a periodic box. Suppose that p.%/ D a% with > 1. Let .%; u/ be a -entropy solution to the compressible Navier-Stokes system (46). Assume that there exists a strong solution .r; U / of the compressible NavierStokes equations (46) such that the terms in (60) are defined with r > 0 and

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r 2 L2 .0; T I W 1;1 .// \ L1 .0; T I W 2;1 .//. Then the following weak-strong uniqueness result holds: .%; u/ D .r; U /. (I-c) Inviscid limit: convergence to dissipative solution. Let us recall the definition of a dissipative solution of compressible Euler equations. Such concept has been introduced by P.L. LIONS in the incompressible setting: see, for instance, [45]. The reader is referred to [27] and [3] for the extension to the compressible framework with constant viscosities. Here we deal with an example of density-dependent viscosities with a dissipative solution target. Of course the target could be the local strong solution of the compressible Euler equations similarly to [57]. Definition. The pair .%; u/ is a dissipative solution of the compressible Euler equations if and only if .%; u/ satisfies the relative energy inequality Z t ˇ ˇ   kdivU ./kL1 ./ d  E.%; u; 0ˇr; U; 0/.t /  E.%; u; 0ˇr; U; 0/.0/ exp c0 .r/ Z

t

C

h

Z

t

exp c0 .r/ 0

kdivU ./kL1 ./ d  s

iZ

0

%E.r; U /  .U  u/ dxds



for all smooth test functions .r; U / defined on Œ0; T / so that r is bounded above and below away from zero and .r; U / satisfies @t r C div.rU / D 0;

@t U C U  rU C rF 0 .r/ D E.r; U /

for some residual E.r; U /. We prove the following result in [20]: Theorem 10. Let .%" ; u" / be any finite -entropy solution to the compressible Navier-Stokes equations (46) in the periodic setting replacing  by ". Then, any weak limit .%; u/ of .%" ; u" / in the sense %" ! % weakly in L1 .0; T I L .//; %" jv" j2 ! % juj2 weakly in L1 .0; T I L1 .// %" jw" j2 ! 0 weakly in L1 .0; T I L1 .// p with v" D u" C 2"r log %" and w" D 2" .1  / log %" as " tends to zero is a dissipative solution to the compressible Euler equations. As a by-product this justifies the limit between a viscous shallow-water system to the inviscid shallow-water system. Using the relative entropy, it is possible to prove the convergence of the viscous shallow-water system to the incompressible Euler equations: low Froude and inviscid limit using the mean velocity plus the oscillating part as target functions (see [25] for the constant viscosities case and [20] for the density-dependent viscosities case).

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(II) The heat-conducting Navier-Stokes equations. Concerning relative entropy and heat-conducting Navier-Stokes equations, nothing has been done for the density-dependent viscosities case. The only results known by the authors are the nice works by E. Feireisl and A. Novotný et al. concerning the heat-conducting Navier-Stokes equations with constant viscosities: The interested reader is referred to [50] and references cited therein.

7

Conclusion

In this chapter, we discussed the compressible Navier-Stokes equations with density-dependent viscosities. We hope to have shown that it is not straightforward to extend to this framework the important properties proved in the constant viscosties case by P.L. Lions, E. FEIREISL, A. NOVOTNÝ et al., and more recently D. Bresch and P.E. Jabin. The main challenging question is to understand how to make a link between these two frameworks which seem to be so orthogonal perhaps playing with the  parameter. Concerning the relative entropy, it would be interesting to extend what has been done for the barotropic case to the heat-conducting Navier-Stokes equations when density dependent viscosities are taken into account. It would be also interesting to understand the two-velocities hydrodynamic problem for the heatconducting Navier-Stokes equations and this new PDEs that we discussed with a perturbed temperature definition due to the heterogeneity.

8

Cross-References

 Concepts of Solutions in the Thermodynamics of Compressible Fluids  Singular Limits for Models of Compressible, Viscous, Heat Conducting, and/or

Rotating Fluids  Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stabil-

ity, and Longtime Behavior  Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical

Cases

References 1. Th. Alazard, Low Mach number limit of the full Navier-Stokes equations. Arch. Ration. Mech. Anal. 180(1), 1–73 (2006) 2. M. Avellaneda, A.J. Majda, Mathematical models with exact renormalization for turbulent transport II: fractal interfaces, non-Gaussian statistics and the sweeping effect. Commun. Math. Phys. 146, 139–204 (1992) 3. C. Bardos, T. Nguyen, Remarks on the inviscid limit for the compressible flows. Recent Advances in Partial Differential Equations and Applications. In honor of H. Beirao da

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Veiga’s 70th birthday. Contemporary mathematics, vol. 666 (American Mathematical Society, Providence, 2016), pp. 55–67 4. H. Brenner, Kinematics of volume transport. Phys. A 349, 11–59 (2005) 5. H. Brenner, Navier-Stokes revisited. Phys. A 349(1–2), 60–132 (2005) 6. H. Brenner, Fluid mechanics revisited. Phys. A 370, 190–224 (2006) 7. D. Bresch, F. Couderc, P. Noble, J.-P. Vila, A generalization of the quantum Bohm identity: hyperbolic CFL condition for Euler-Korteweg equations. C.R. Acad. Sci. Paris 354(1), 39–43 (2016) 8. D. Bresch, B. Desjardins, Some diffusive capillary models of korteweg type. C.R. Acad. Sci. Paris, Section Mécanique 332(11), 881–886 (2004) 9. D. Bresch, B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pures Appl. (9), 87(1), 57–90 (2007) 10. D. Bresch, B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models. J. Math. Pures Appl. 86(4), 362–368 (2006) 11. D. Bresch, B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys. 238(1–2), 211–223 (2003) 12. D. Bresch, B. Desjardins, D. Gérard-Varet, On compressible Navier-Stokes equations with density dependent viscosities in bounded domains. J. Math. Pures Appl. (9), 87(2), 227–235 (2007) 13. D. Bresch, B. Desjardins, C.-K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Commun. Partial Differ. Equ. 28(3–4), 843–868 (2003) 14. D. Bresch, B. Desjardins, E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: part II existence of global –entropy solutions to the compressible Navier-Stokes systems with degenerate viscosities. J. Math. Pure Appl. 104(4), 801–836 (2015) 15. D. Bresch, V. Giovangigli, E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: part I well posedness for zero Mach number systems. J. Math. Pures Appl. 104(4), 762–800 (2015) 16. D. Bresch, M. Gisclon, V. Ingrid, On Navier-Stokes-Korteweg and Euler-Korteweg: application to the quantum hydrodynamic. Submitted (2017) 17. D. Bresch, P.-E. Jabin, Global existence of weak solutions for compresssible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor. Submitted (2015) (See arXiv:1507.04629) 18. D. Bresch, P.-E. Jabin, Global weak solutions of PDEs for compressible media: a compactness criterion to cover new physical situations. To appear in Springer INdAM-Series, special issue dedicated to G. Métivier, Eds F. Colombini, D. Del Santo, D. Lannes (2017) 19. D. Bresch, P. Noble, J.-P. Vila, Relative entropy for compressible Navier-Stokes equations with density-dependent viscosities and applications. C.R. Acad. Sci. 354(1), 45–49 (2016) 20. D. Bresch, P. Noble, J.-P. Vila, Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and various applications. Workshop on low Velocity flows – Application to low Mach and low Froude regimes. S. Dellacherie, G. Faccanoni, B. Grec, F. Lagoutière, Y. Penel Eds. To appear in ESAIM Proc. (2017) 21. D. Bresch, A. Vasseur, C. Yu, A remark on the existence of compressible Navier-Stokes equations. In Preparation (2016) 22. D. Donatelli, E. Feireisl, P. Marcati, Well/ill posedness for the Euler-Korteweg-Poisson system and related problems. Commun. Partial Differ. Equ. 40, 1314–1335 (2015) 23. D.A. Drew, S.L. Passman, Theory of Multicomponent Fluids. Applied Mathematical Sciences, vol 135 (Springer, New York, 1999) 24. B. Ducomet, E. Feireisl, H. Petzeltova, I. Straskraba, Global in time weak solutions for compressible barotropic self-gravitating fluids. Disc. Contin. Dyn. Syst. 11(2), 113–130 (2004) 25. E. Feireisl, Compressible Navier-Stokes equations with a non-monotone pressure law. J. Differ. Equ. 184(1), 97–108 (2002)

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26. E. Feireisl, Dynamics of Viscous Compressible Fluids. Oxford Lecture series in mathematics and its applications, vol 26 (Oxford University Press, Oxford, 2004) 27. E. Feireisl, B.J. Jin, A. Novotný, Relative entropies, suitable weak solutions and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech. 14(4), 717–730 (2012) 28. E. Feireisl, A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids. Advances in mathematical fluid mechanics (Basel, Birkhauser-Springer, 2009) 29. E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001) 30. E. Feireisl, A. Vasseur, New Perspective in Fluid Dynamics: Mathematical Analysis of a Model Proposed by Howard Brenner. Advances in fluid mechanics (Birkhauser, 2009), pp. 153–179 31. E. Fernández-Cara, F. Guillén-Gonzalez. Some new existence results for the variable density Navier-Stokes. Annales de la Faculté des sciences de Toulouse: Mathématiques, Série 6: Tome 2(2), 185–204 (1993) 32. S.L. Gavrilyuk, S.M. Shugrin, Media with equations of state that depend on derivatives. J. Appl. Mech. Tech. Phys. 37(2), 177–189 (1996) 33. J. Giesselmann, C. Lattanzio, A.-E. Tzavaras, Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics. Arch. Ration. Mech. Anal. 223(3), 1427–1484 (2017) 34. M. Gisclon, I. Violet, About the barotropic compressible quantum Navier-Stokes equations. Nonlinear Anal. 128, 106–121 (2015) 35. B. Haspot, Weak-Strong Uniqueness for Compressible Navier-Stokes System With Degenerate Viscosity Coefficient and Vacuum in One Dimension. Commun. Math. Sci. 15(3), 587–591 (2017) 36. B. Haspot, New formulation of the compressible Navier-Stokes equations and parabolicity of the density. (2014) HAL Id: hal-01081580 37. D. Hoff, D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51(4), 887–898 (1991) 38. M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow (Eyrolles, Paris, 1975) 39. Y. Kanel, On a model system of equations of one-dimensional gas motion. Differ. Equ. 4, 374–380 (1968) 40. A. Kazhikhov, Resolution of boundary value problems for non homogeneous viscous fluids. Dokl. Akad. Nauk. 216, 1008–1010 (1974) 41. A. Kazhikhov, W. Weigant, On existence of global solutions to the two dimensional NavierStokes equations for a compressible viscous fluid. Sib. Math. J. 36(6), 1108–1141 (1995) 42. J. Leray, Sur le mouvement dun fluide visqueux remplissant lespace. Acta Math. 63, 193–248 (1934) 43. J. Li, Z. Xin, Global existence of weak solutions to the barotropic compressible Navier-Stokes flows with degenerate viscosities. Submitted (2015) (see arXiv:1504.06826) 44. P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1. Volume 3 of Oxford lecture series in mathematics and its applications (The Clarendon Press Oxford University Press, New York, 1996). Incompressible models, Oxford Science Publications 45. P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2. Volume 10 of Oxford lecture series in mathematics and its applications. (The Clarendon Press Oxford University Press, New York, 1998). Compressible models, Oxford Science Publications 46. A. Mellet, A. Vasseur, On the barotropic compressible Navier-Stokes equations. Commun. Partial Differ. Equ. 32(1–3), 431–452 (2007) 47. P. Mucha, M. Pokorny, E. Zatorska, Heat-conducting, compressible mixtures with multicomponent diffusion: construction of a weak solution. SIAM J. Math. Anal. 47(5), 3747–3797 (2015) 48. P. Mucha, M. Pokorny, E. Zatorska, Approximate solutions to a model of two-component reactive flow. Discrete Contin. Dyn. Syst. Ser. S 7(5), 1079–1099 (2014) 49. E. Nelson, Dynamical Theories of Brownian Motion. (Princeton University press, Princeton, 1967)

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31

Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical Cases P. I. Plotnikov and W. Weigant

Contents 1 2 3 4 5 6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extension: Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Integral Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isothermal Flow: Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Step 1: First Integral Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Step 2: Positivity – Properties of W",M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Step 3: Positivity – Representation of CM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Step 4: The Second Integral Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Step 5: Monotonicity – The Function W",M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Step 6: Proof of the Main Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1602 1606 1614 1621 1624 1637 1639 1644 1647 1651 1661 1663 1669 1670 1670

Abstract

In this chapter the compressible Navier-Stokes equations with the critical adiabatic exponents are considered. The crucial point in this situation are new estimates of the Radon measure of solutions. These estimates are applied

P.I. Plotnikov () Mathematical Department, Novosibirsk State University, Novosibirsk, Russia Siberian Division of Russian Academy of Sciences, Lavryentyev Institute of Hydrodynamics, Novosibirsk, Russia e-mail: [email protected]; [email protected] W. Weigant Institute für Angewandte Mathematik, Universität Bonn, Bonn, Germany e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_75

1601

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P.I. Plotnikov and W. Weigant

to the boundary value problem for the compressible Navier-Stokes equations with the critical adiabatic exponents. The existence of weak solutions to 2D isothermal problem is proved. The cancelation of concentrations for 3D nonstationary initial-boundary value problem with the critical adiabatic exponent 3/2 is established.

1

Introduction

Suppose a viscous compressible fluid occupies a bounded domain ˝  Rd , d D 2, 3. The state of the fluid is characterized by the macroscopic quantities: the density %.x; t / and the velocity u.x; t /. The problem is to find u.x; t / and %.x; t / satisfying the following equations and boundary conditions in the cylinder QT D ˝  .0; T /. @t .%u/ C div .%u ˝ u/ C r% D div S.u/ C %f @t % C div .%u/ D 0;

in QT ;

%  0 in QT ;

u D 0 on @˝  .0; T /; u.x; 0/ D u0 .x/;

%.x; 0/ D %0 .x/ > 0 in ˝:

(1a) (1b) (1c) (1d)

Here, the vector field f 2 L1 .QT / denotes the density of external mass forces; the viscous stress tensor S.u/ has the form   S.u/ D 1 ru C ru> C 2 div u I; (1e) where the viscosity coefficients satisfy the inequalities 1 > 0 and 21 C 2 > 0. The first nonlocal results concerning the mathematical theory of compressible Navier-Stokes equations are due to P.-L. Lions. In monograph [8] he established the existence of a renormalized solution to nonstationary boundary value problem for the Navier-Stokes equations with the pressure function p  % for all  > 5=3. More recently, Feireisl, Novotný, and Petzeltová, see [5], proved the existence result for all  > d =2, see also monographs [4, 10], and [13] for references and details. For d D 3 and   3=2, the question on solvability of problem (1) is still open. The main difficulty is the so-called concentration problem, see [8] ch.6.6. It is easy to see that in the three-dimensional case, the energy estimates and embedding theorems guarantee the inclusion %juj2 2 Ls .˝/ with s > 1 if and only if  > d =2. Hence, for   d =2, only L1 estimate for the density of the kinetic energy is available. The question is under what conditions will a weak limit of approximate solutions to equations (1) be a solution. Let a sequence of approximate solutions .%" ; u" /, " > 0 to problem (1) satisfies the energy inequality k%d" =2 kL1 .0;T IL1 .˝// C k%" ju" j2 kL1 .0;T IL1 .˝// C ku" kL2 .0;T IW 1;2 .˝//  c; where c is independent of ". For d D 2 the density admits the extra estimate k%" ln.%" /kL1 .0;T IL1 .˝//  c:

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1603

Therefore, there is a subsequence of the sequence .%" ; u" /, still denoted by .%" ; u" / such that   u" ! u weakly in L2 0; T I W01;2 .˝/ ; %" ! % star weakly in L1 .0; T I L .˝//; %" u" ˝ u" ! M star weakly in the space of Radon measures as " ! 0; where M D .Mi;j /d d denotes a d  d matrix-valued Radon measure in ˝. In the general case, the weak star defect measure M  %u ˝ u ¤ 0. This leads to the so-called concentration problem, which was widely discussed in the mathematical literature in connection with vortex sheets dynamics. Hence, the question is to describe the structure of the defect measure and to find conditions under which it is equal to 0. The goal of this paper is to investigate the concentration problem in the critical case  D d =2. In order to make the presentation clear and to avoid unnecessary technical difficulties, it is assumed that the flow domain and the given data satisfy the hypotheses: Condition 1 • The flow domain ˝  Rd is a bounded domain with C 2 boundary. • The data %0 ; u0 2 L1 .˝/, and f 2 L1 .QT / satisfy the estimate ku0 kW 1;2 .˝/ C k%0 kL1 .˝/ C kfkL1 .QT /  ce ;

%0 > c > 0;

0

(2)

where ce and c are positive constants. Further, c denotes generic constants depending only on ˝; T; k%0 kL1 .˝/ , ku0 kL2 .˝/ ; kfkL1 .QT / , and i . In order to regularize problem (1) the artificial pressure method is used, and equations (1) are replaced by the regularized equations @t .%u/ C div .%u ˝ u/ C rp.%/ D div S.u/ C %f @t % C div .%u/ D 0

in QT ;

in QT ;

u D 0 on @˝  .0; T /; u.x; 0/ D u0 .x/;

(3) (4) (5)

%.x; 0/ D %0 .x/ in ˝:

(6)

Here, the artificial pressure function is given by p.%/ D %d =2 C "%m ;

" 2 .0; 1;

m  6:

(7)

The existence of weak renormalized solutions to problem (3), (4), (5), (6), and (7) was established in monographs [4] and [8]. The following proposition is a consequence of these results.

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Proposition 1. Let domain ˝ and functions u0 , %0 , f satisfy Condition 1. Then problem (3), (4), (5), (6), and (7) has a weak solution .%" ; u" / with the following properties: (i) The functions %"  0 and u" satisfy the energy inequality Z Z ˚  2 3=2 m ess sup %" ju" j C %" C "%" .x; t / dx C jru" j2 dxdt  c ˝

t2.0;T /

(8)

QT

for d D 3, and Z

˚

2

%" ju" j C %" ln.1 C %" / C

ess sup ˝

t2.0;T /

"%m "



Z .x; t / dx C

jruj2 dxdt  c

QT

(9) for d D 2. Moreover, there are r > 1 and a constant c."/, depending on ", such that k%" ju" j2 kLr .QT / C k%" C "%m " kLr .QT /  c."/:

(10)

(ii) The integral identity Z





Z



 %" u" ˝ u" C p.%" / I  S.u" / W r dxdt C

%" u"  @t  dxdt C

QT

QT

Z

Z

%" f   dxdt C QT

%0 .x/u0 .x/  .x; 0/ dx D 0

(11)

˝

holds for all vector fields  2 C 1 .QT / satisfying .x; T / D 0

in ˝;

.x; t / D 0

on @˝  .0; T /:

(12)

(iii) The integral identity Z



%" @t

Z    dxdt C . %0 /.x; 0/ dx D 0 C % " u"  r

QT

holds for all smooth functions ˝  ft D T g.

(13)

˝

, vanishing in a neighborhood of the top

The following consequences of Proposition 1 will be used throughout the paper. Corollary 1. Let d D 3 and  D 3=2. Assume that the pair .%" ; u" / meets all requirements of Proposition 1. Then the estimates

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

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k%" u" kL1 .0;T IL6=5 .˝//  c hold true for all " 2 .0; 1. Proof. The desired estimate obviously follows from estimate (8) and the Hölder inequality. If d D 3 and  D 3=2, then there is a subsequence of the sequence .%" ; u" /, still denoted by .%" ; u" /, and there are % 2 L1 .0; T I L3=2 .˝//, u 2 L2 .0; T I W01;2 .˝// with the properties %" .t / ! %.t / weakly in L3=2 .˝/ for a.e. t 2 .0; T /; %" u" .t / ! %.t /u.t / weakly in L6=5 .˝/ for a.e. t 2 .0; T /;   u" ! u weakly in L2 0; T I W01;2 .˝/ :

(14)

If d D 2 and  D 1, then there is a subsequence of the sequence .%" ; u" /, still denoted by .%" ; u" /, and there are % 2 L1 .0; T I L1 .˝//, u 2 L2 .0; T I W01;2 .˝// with the properties   % ln.1 C %/ 2 L1 0; T I L1 .˝/ ; %" .t / ! %.t / weak in L1 .˝/ for a.e. t 2 .0; T /;   u" ! u weak in L2 0; T I W01;2 .˝/ : (15) In 1986 Padula, [11] and [12], announced the result on existence of a weak solution to problem (1) in the isothermal case with d D 2 and  D 1. The complete proof was given recently in [14]. In this chapter the following theorem on solvability of the 2D isothermal problem is proved. Theorem 2. Let d D 2 and  D 1. Assume that Condition 1 is fulfilled. Then problem (1) has a weak solution which satisfies the estimate kukL2 .0;T IW 1;2 .˝// C k%juj2 kL1 .0;T IL1 .˝// C k% log.1 C %/kL1 .0;T IL1 .˝//  c: 0

(16) Moreover, for every cylinder Q0 D ˝ 0  Œ˛; ˇ b QT and  2 .0; 1=8/, there is c.Q0 ; / such that k%kL1C .Q0 /  c.Q0 ; /:

(17)

Rotationally symmetric flows may be considered as intermediate case between 2D and 3D flows. In the case of spherically symmetric flows, the existence of weak global solutions was proved for all  > 1 in [6] (see also [2]). Axisymmetric flows were studied in [7], where the existence of weak solutions was also proved for all  > 1. However, in contrast to the spherically symmetric case, the resulting

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solutions do not satisfy the equations on the axis of symmetry, where concentrations may appear. In paper [15] the existence of rotationally symmetric weak solutions  p  was established for all   7 C 73 =12. The optimal lower estimate for admissible  is unknown. In general 3D case with  D 3=2, the question on the solvability of problem (1) remains open. The goal of these notes is to prove the cancelation of concentrations in 3D case. The result is given by the following: Theorem 3. Let d D 3 and  D 3=2. Assume that Condition 1 is fulfilled. There is a subsequence of the sequence .%" ; u" /, still denoted by .%" ; u" /, such that Z

T

Z

Z

T

Z

h%" u";i u";j dxdt ! 0

˝

h%ui uj dxdt as " ! 0 0

(18)

˝

for every function h 2 L1 .˝  .0; T // compactly supported in ˝  .0; T /. The remaining part of the paper is devoted to the proof of Theorems 2 and 3.

2

Preliminaries

Function spaces. For every s 2 Œ1; 1 and integer l, W l;s .Rd / denotes the Sobolev space of all measurable functions u in Rd with the finite norm kukW l;s .Rd / D

X

kD ˛ ukLs .Rd / :

(19)

j˛jl 0

For s > 1, the dual space W l;s .Rd /0 s 0 D s=.s  1/ is denoted by W l;s .Rd /. Further the notation C0 .Rd / stands for the Banach space of all continuous functions u W Rd ! R such that u.x/ ! 0 as jxj ! 1. The space C0 .Rd / is supplemented with the norm kukC0 .Rd / D sup ju.x/j:

(20)

Rd

The dual space M.Rd / D C0 .Rd /0 consists of all finite signed Radon measures  on Rd . The Banach space M.Rd / is supplemented with the norm kkM.Rd / D jj.Rd /:

(21)

Here the nonnegative measure jj D C C  is a variation of the measure ,  D C   is the Jordan decomposition of .

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

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Let 1 < r  1. The Banach space Lrw .0; T I M.Rd // consists of all tempered measures .t /, t 2 .0; T /, such that the mapping Z t 7!

g.x/d .t / R3

is measurable for every g 2 C0 .Rd / and   kkLrw .0;T IM.Rd // WD k.t /kM.R2 / Lr .0;T / < 1: Integral operators Let BR  Rd be the ball ˚  BR D x 2 Rd W jxj  R :

(22)

Let f 2 Lr .Rd / be an arbitrary function such that spt f  BR . Then, see [3], the Poisson equation u D f

in Rd ;

(23)

has a solution with the following properties: It is analytic outside BR and satisfies lim sup .log jxj/1 ju.x/j < 1 for d D 2;

lim sup jxjju.x/j < 1 for d D 3; jxj!1

jxj!1

kukW 2;r .BN /  c.N /kf kLr .Rd / : The relation f ! u determines a linear operator 1 . For d D 3 it admits the integral representation uD

1 4

Z

jx  yj1 f .y/ dy:

(24)

R3

Introduce the integral operators Z Kij f .x/ D

Rd

Kij .x  y/f .y/ dy;

1  i; j  d;

(25)

with the kernels   Kij .x/ @2xi xj jxj D jxj1 ıij  jxj3 xi xj :

(26)

The following properties of Kij are obvious jKij .x/j  jxj1 ; Kij .x/yi yj  0 for y 2 Rd :

(27)

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P.I. Plotnikov and W. Weigant

It follows from the estimates of the Newton and Riesz potentials, [1], that for every f supported in the ball BR the operators Kij admit the estimates kKij f kL6 .R3 /  c.R/kf kL6=5 .R3 / ;

(28)

kKij f kL6 .R2 /  c.R/kf kL3=2 .R2 / : Introduce also the integral operator Qj given by Z Qj f .x/ D Qj .x  y/f .y/ dy; Qj .x/ D jxj1 xj ;

1  j  d:

(29)

R3

Radon transform. For every function f 2 L1 .Rd /, the Radon transform Rf W R  Sd 1 ! R is defined as follows Z Rf . ; !/ D f .x/ dx; 2 R1 ; ! 2 Sd 1 ; (30) ˘. ;!/

where ˚  ˘. ; !/ D x 2 Rd W !  x D ;

˚  Sd 1 D ! 2 Rd W j!j D 1 :

(31)

The Radon transform admits the obvious estimate kRf kL1 .Sd 1 IL1 .R//  kf kL1 .Rd / for all f 2 L1 .R3 /:

(32)

The following relation between the Fourier transform F and the Radon transform Z (33) Ff . / D .2/d =2 e ij j Rf . ; !/ d for D j j! R

is a straightforward consequence of (30). It is convenient to introduce the notation. For every function g 2 L2 .R/ denote by F1 the one-dimensional Fourier transform Z 1=2 F1 g./ D .2/ e is g.s/ ds; g 2 L2 .R/: (34) R

The following identity is known as the Bessel formula for the Radon transform. Lemma 1. The identity Z

Rf . ; !/ Rg. ; !/ d D 2.2/d 1 Re R

Z

1

.F f /.j j!/ .F g/.j j!/ d j j 0

holds for all real-valued functions f; g 2 C01 .Rd /.

(35)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1609

Proof. It follows from the Bessel identity that Z

Z R

   F1 .Rf .; !// ./ F1 .Rg.; !// ./d 



Rf . ; !/ Rg. ; !/ d D R

(36)

for every ! 2 Sd 1 . Notice that the equality F1 .Rf .; !//./ D F1 .R.; !//./ holds for every real-valued function f . It follows from this and (36) that Z Z 1     F1 .Rf .; !// ./ F1 .Rg.; !// ./d : Rf . ; !/ Rg. ; !/ d D 2Re R

0

(37)

Next, identity (33) implies   Ff .j j!/ D .2/1=2d =2 F1 .Rf .; !// .j j/:

(38)

This leads to the relation .2/d 1

Z

1

.F f /.j j!// .F g/.j j; !//d j j 0

Z

1

D



   F1 .Rf .; !// .j j/ F1 .Rg.; !// .j j/ d j j;

0

which along with (37) yields (35). Corollary 2. The identity Z

Z Sd 1

Z R

Rf . ; !/ Rg. ; !/ d d ! D cd

jx  yj1 f .x/g.y/ dxdy

R2d

holds for all f; g 2 C01 .Rd /. Here the constant cd depends only on d . Proof. Relation (35) implies Z

Z Rf .!/ Rg.!/ d d ! Sd 1

R d 1

D 2.2/

Z

Z

D 2.2/d 1 Re

1

.F f /.j j!/ .F g/.j j!/ d j jd !

Re Z

Sd 1

Rd

0

Ff . / Fg. / j j1d d :

(39)

1610

P.I. Plotnikov and W. Weigant

It remains to note that Z Z Ff . / Fg. / j j1d d D ./.1d /=2 f .x/ g.x/ dx Rd

Rd

and to recall the definition of the Riesz potential, [1], Z .1d /=2 ./ f .x/ D cd jx  yj1 f .y/ dy: Rd

Corollary 3. Let f 2 L2 .R3 / be a real-valued function. Then k@ Rf kL2 .S2 IL2 .R// D 21=2 kf kL2 .R3 / :

(40)

If, in addition, f is supported in the ball BR , then kRf kL2 .S2 IW 1;2 .R//  c.R/kf kL2 .R3 / ;

(41)

kRf kL2 .S2 IC .R//  c.R/kf kL2 .R3 / :

(42)

Proof. The Bessel identity implies Z

Z

1 2

2

1

j j2 jF1 .Rf .; !//.j j/j2 d j j:

jj jF1 .Rf .; !//./j d  D 2 1

(43)

0

Notice that Z 1

2

Z

2

1

j@ Rf . ; !/j2 d :

jj jF1 .Rf .; !//./j d  D 1

(44)

1

Substituting (44) into (43) gives the equality Z

1

j@ Rf . ; !/j2 d D 2 1

Z

1

j j2 jF f .j j!/j2 d j j: 0

Integrating both sides with respect to ! and noting that Z Z S2

1

j j2 jFf .j j!/j2 d j jd ! D

0

Z R3

jF f . /j2 d D

Z

jf .x/j2 dx

R3

leads to (40). If f is supported in BR , then Rf .; !/ is supported in the interval j j  R. In this case the norms k@ Rf .; !/kL2 .R/ and kRf .; !/kW 1;2 .R/ are equivalent which yields (41). This leads to the following. Lemma 2. The estimates

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

kRf kL2 .S2 IL2 .R//  c.R/kf kL6=5 .R3 / ; kRf kL2 .S1 IL2 .R//  c.R/kf kL3=2 .R2 /

1611

(45)

hold true for every f 2 L6=5 .R3 / (f 2 L3=2 .R2 /) supported in the ball BR . Proof. It suffices to show (45) for d D 3 and f 2 C01 .R3 /. It follows from (28) that ˇZ ˇ ˇ

R6

ˇ ˇ jx  yj1 f .x/f .y/ dxdy ˇ D kf Kf kL1 .R3 /  ckKf kL6 .R3 / kf kL6=5 .R3 /  c.R/kf k2L6=5 .R3 / :

(46)

On the other hand, relation (39) yields Z Z S2

jRf . ; !/j2 d d ! D c R

Z

jx  yj1 f .x/f .y/ dxdy:

R3

Combining this equality with (46) gives (45). The following proposition will be used throughout the paper. Proposition 2. Let functions fij ; fi ; gj ; g 2 L6=5 .R3 / (fij ; fi ; gj ; g 2 L3=2 .R2 /) be supported in the ball BR . Then Z

Z Sd 1

R

Z

cd

R2d

!i !j Rfij .!/ Rg.!/ d d ! D Kij .x  y/fij .y/g.x/ dydx;

(47a)

Z

Z Sd 1

Z

cd

R

R2d

!i !j Rfi .!/ Rgj .!/ d d ! D Kij .x  y/fi .y/gj .x/ dydx;

(47b)

where cd , d D 2; 3, is independent of f and g. Proof. It suffices to prove (47a). The proof of (47b) is similar. Assume that fij ; g 2 C01 .Rd /. It follows from (35) that Z Sd 1

Z R

!i !j Rf Rg d d !

1612

P.I. Plotnikov and W. Weigant

d 1

D 2.2/

D 2.2/d 1

Z

Z Re Z Rd

Sd 1

1

!i !j Ff .j j!/ Fg.j j!/ d j jd !

0

j jd 1 i j .F f /. / .F g/. / d :

Notice that  F @xi xj ./.1Cd /=2 fij /. / D j jd 1 i j .F f /. /; which implies Z

Z Sd 1

R

d 1

Z

!i !j Rf Rg d d ! D 2.2/

Rd

Z D cd

j j1d i j .F f /. / .F g/. / d

g@2xi xj ./.d C1/=2 fij dx:

Rd

(48)

The integral representation for the Riesz potential yields @2xi xj ./.d C1/=2 fij .x/ D cd

Z Rd

Kij .x  y/fij .y/ dy:

Combining this identity with (48) leads to desired identity (47a), which can be rewritten in the equivalent form Z

Z Sd 1

R

Z !i !j Rfij . ; !/ Rg. ; !/ d d ! D cd

Rd

Kij fij .x/ g.x/ dx:

(49)

This proves relation (47a) for functions fij ; g 2 C01 .˝/. Set r D 6=5 for d D 3 and r D 3=2 for d D 2. Fix R > 0 and denote by LrR .Rd / the totality of all functions f 2 Lr .Rd / vanishing outside of the ball BR . Denote by CR1 .Rd / the totality of all C 1 -functions compactly supported in BR . It is clear that LrR .Rd / is a closed subset of Lr .Rd / and CR1 .Rd / is dense in LrR .Rd /. It follows from (45) and (28) that the operators R W LrR .Rd / ! L2 .Sd 1 I L2 .R//;

Kij W LrR .Rd / ! L6 .Rd /

(50)

are continuous and their norms a bounded by constants depending only on R. Since (49) holds true for fij ; g 2 CR1 .Rd / and CR1 .Rd / is dense in LrR .Rd /, the identity (49) is fulfilled for all fij ; g 2 LrR .Rd /. Proposition 3. Let fij 2 L1 .Rd / and g 2 L2 .Rd / be supported in the ball BR . Then

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z

Z Sd 1

!i !j

1613

Z

R

Rfij . ; !/ Rg. ; !/ d d ! D cd

Rd

Kij g.x/ fij dx;

(51)

and ˇZ ˇ ˇ

Rd

ˇ ˇ Kij g.x/ fij dx ˇ  c.R/kfij kL1 .Rd / kgkL2 .Rd / :

(52)

Proof. There are sequences gn ; fn;ij 2 C01 .Rd / such that the functions gn ; fn;ij are supported in the ball BR and kfn;ij  fij kL1 .Rd / C kgn  gkL2 .Rd / ! 0 as n ! 1: It follows from this and (32), (42) that kRgn  RgkL2 .S d 1 IC .R// C kRfn;ij  Rfij kL1 .S d 1 IL1 .R// ! 0 as n ! 1 which yields Z

Z

Sd 1

R

Z

Z !i !j Rgn Rfn;ij d d ! !

Sd 1

R

!i !j RgRfij d d ! as n ! 1: (53)

Next, notice that the estimate kKij gkC .Rd /  c.R/kgkL2 .Rd /

(54)

holds for every function g 2 L2 .Rd / supported in the ball BR . It follows that Kij gn ! Kij g in C .Rd / as n ! 1 which yields Z

Z Rd

Kij gn fn;ij dx !

Rd

Kij fij g dx as n ! 1:

(55)

Since gn ; fn;ij 2 C01 .Rd /, Proposition 2 implies Z Sd 1

Z

Z R

!i !j Rgn .!/Rfn;ij .!/ d d ! D cd

Rd

Kij gn fn;ij dx:

Letting n ! 1 and using (53), (54), and (55) gives (51). It remains to note that estimate (52) is a straightforward consequence of (54).

1614

3

P.I. Plotnikov and W. Weigant

Extension: Compactness

Due to technical reasons, it is convenient to extend the system of compressible Navier-Stokes equations to the whole space using the cutoff functions. Introduce nonnegative functions ; & 2 C01 .R3 / with the following properties spt &  ˝;

spt  ˝;

D 1 on

spt & :

(56)

Next set

" D &%" in ˝  Œ0; T ;

" D 0 in .Rd n ˝/  Œ0; T ;

v" D u" in ˝  Œ0; T ;

v" D 0 in .Rd n ˝/  Œ0; T ;

(57)

where .%" ; u" / is a solution to problem (3), (4), (5), (6), and (7) given by Proposition 1. Along with the extended densities, velocities, and momentum, we will consider the sequence of the energy tensors E" with the entries E";ij D " v";i v";j C p" ıij ;

1  i; j  d;

(58)

where p" D &%" C &"%m " 0

(59)

is supported in the cylinder BR  Œ0; T . The energy density is defined by E" tr E" D " jv" j2 C dp" :

(60)

It follows from (56) and Corollary 1 that the functions . " ; v" / satisfy the following conditions: H.1 There is R > 0 such that the functions . " ; v" / are supported in the cylinder BR  Œ0; T . The functions " are nonnegative. H.2 There is a constant c > 0 such that kv" kL2 .0;T IW 1;2 .Rd // C k " jv" j2 kL1 .0;T IL1 .Rd //  c; k " kL1 .0;T IL3=2 .R3 // C k " v" kL1 .0;T IL6=5 .R3 //  c for d D 3; k " ln.1 C %" /kL1 .0;T IL1 .R2 // C k " v" kL1 .0;T IL1 .R2 //  c for d D 2: (61) H.3 There are functions 2 L1 .0; T I Ld =2 .Rd //, L2 .0; T I W 1;2 .Rd //, with the following properties



v" ! v weakly in L2 .0; T I W 1;2 .Rd // as " ! 0;

0, and v

2

(62)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1615

" .t / ! .t /weakly in Ld =2 .R3 /;

" .t /v" .t / ! .t /v.t /weakly in L6=5 .R3 / for d D 3;

(63)

" .t /v" .t / ! .t /v.t /weakly in L1 .R2 / for d D 2 as " ! 0 for a.e. t 2 .0; T /. H.4 For every function ' 2 C Œ0; 1 with '.0/ D 0 and a.e. t 2 .0; T /, there is ' 2 L1 .Rd  .0; T // such that '. " .t // ! '.t / star weakly in L1 .Rd //

(64)

as " ! 0 for a.e. t 2 .0; T /. replaced by &  and &

Moreover, relations (11) and (13) with  and integral identities Z

T

Z Rd

0

. " v" @t   & S.v" / W r/ dxdt C Z

T

Z

C Z 0

T

0

Z Rd

. " @t

Rd

Z   F" dxdt C

C " v"  r

Rd

Z

T

imply the

Z Rd

0

@xj i E";ij dxdt

(65)

0 v0  .0/dx D 0;

C g" /dxdt C

Z Rd

0 .0/dx D 0:

(66)

Here  and are arbitrary functions of the class C 1 .Rd  .0; T // vanishing at the top ft D T g. The vector field F" and the function g" are defined by the equalities   F";i D @xj & %" u";i u";j C p.%" /ıij  Sij .u" / C &%" fi ; g" D r&  u" %" in ˝  .0; T /; F" D 0;

(67)

g" D 0 in .Rd n ˝/  .0; T /:

It follows from estimates (61) that kF" kL2 .0;T IL1 .Rd //  c;

kg" kL1 .0;T IL6=5 .R3 //  c for d D 3;

(68)

kg" kL1 .0;T IL1 .R2 //  c for d D 2: Hence the task is to investigate the compactness properties of the sequence . " ; v" /. This section is devoted to the proof of preliminary results. The first goal is to show that the artificial velocities v" converge strongly outside the vacuum zone f D 0g. This result is given by the following proposition which is the first main result of this section. Introduce the function

1616

P.I. Plotnikov and W. Weigant

.x; t / D 1 when .x; t / > 0 and .x; t / D 0 otherwise :

(69)

Proposition 4. Let r 2 Œ1; 2/ and q 2 Œ1; 6/. Then there is a subsequence of the sequence v" still denoted by v" such that   v" ! v in Lr 0; T I Lq .Rd / as " ! 0:

(70)

The second result of this section concerns the weak convergence of a sequence '. " /. Proposition 5. There is a set E  .0; T / of full measure in .0; T / with the following properties: For every function ' W RC ! R satisfying the condition j'.s/j  c;

j' 0 .s/j  c.1 C jsj/2 ;

'.0/ D 0;

(71)

there is a function ' 2 L1 .0; T I L1 .˝// such that '. " .t // ! '.t / star weakly in L1 .Rd / as " ! 0 for all t 2 E:

(72)

For every M > 0 set

";M .x; t / D minf " .x; t /; M g:

(73)

In view of Proposition 5, there is a function M such that  

";M .t / ! M .t / star weakly in L1 Rd as " ! 0

(74)

for a.e. t 2 .0; T /. The function M does not equal minf ; M g. However, it satisfies the conditions 0  .x; t /  M;

M .x; t / % .x; t / as M % 1 for a.e. .x; t /:

(75)

The following proposition which is the second main result of this section is a consequence of Propositions 4 and 5. Proposition 6. There is a subsequence of the sequence . " ; v" / still denoted by . " ; v" / such that  

";M .t /v" ! M .t /v.t / weakly in Lq Rd ; as " ! 0 for all M > 0 and a.e. t 2 .0; T /. Introduce the measures

1  q < 2;

(76)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

d ";ij .t / D E";ij .t / dx;

d " .t / D E" .t / dx; d ";ij .t / D F";i .t / dx:

1617

(77)

d 2 d Proposition 7. There are measures ij 2L1 w .0; T I M.R //, i 2 Lw .0; T I M.R //, and a subsequence of the sequence ";ij still denoted by ";ij such that

";ij ! ij ;

";i ! i star weakly in L1 .0; T I M.Rd //:

(78)

This means that Z

Z

T

hg.t /; ";ij .t /i dt ! 0

T

hg.t /; ij .t /i dt as " ! 0

(79)

0

for all g 2 L1 .0; T I C0 .Rd //. The similar relations hold for ";i . Proof. It suffices to prove relation (78) for the measure ij . The proof of this 3 relation for the measure ij is similar. Recall that L1 w .0; T I M.R // consists of all weakly measurable families of measures .t / such that kkM.R3 / 2 L1 .0; T /. The mapping t ! .t / is weakly measurable if the function t ! hg; .t /i is measurable for every g 2 C0 .R3 /. Notice that the L1 .0; T I C0 .Rd / is the dual space d 1 d to L1 w .0; T I M.R //. Since L .0; T I C0 .R // is separable Banach space and the 1 sequence ";ij is bounded in Lw .0; T I M.Rd //, there is a subsequence still denoted by ";ij satisfying (78). The rest of the section is devoted to the proof of Propositions 4, 5, and 6. Proof of Proposition 4. It suffices to prove the proposition in the case d D 3. The proof for d D 2 is similar. The sequence v" is bounded in L2 .0; T I L6 .R3 //. Hence it suffices to show that v" .t / ! v.t / in Lq .R3 / a.e. in .0; T /. Choose an arbitrary N > 0 and denote by E the set of all t such that kv.t /kW 1;2 .Rd /  N: Consider the functions w";N .t / D v" .t / when kv" kW 1;2 .R3 /  N and w";N .t / D v.t / otherwise:

(80)

Let us prove that .t /w" .t / ! .t /v.t / in Lq .R3 /

(81)

as " ! 0 for a.e. t 2 .0; E/. Notice that kw" .t /kW 1;2 .R3 /  N for all t 2 E. Notice also that the functions w" are uniformly compactly supported in R3 . Therefore, the sequences w" .t / and .t /w" .t / are relatively compact in Lq .R3 / for all t 2 E. Hence it suffices to show that for t 2 E, the sequence .t /w" .t / has the only limiting point .t /v.t / in Lq .R3 /. Choose an arbitrary t 2 E. Let w0 and w00 be

1618

P.I. Plotnikov and W. Weigant

limiting points of the sequence w" .t / in Lq .R3 /. This means that there are the sequences w"0 .t / ! w0 ;

  w"00 .t / ! w00 in Lq R3 :

Assume that 1=q C 2=3 < 1: The set f"0 g consists of two disjoint subsets f"0i g and f"0j g such that w"0i .t / D v"0i .t /;

w"0j .t / D v.t /:

(82)

Assume that the set f"0i g is infinite. Since "0 .t / ! .t / weakly in L3=2 .R3 / and w"0 .t / ! w0 strongly in Lq .R3 /, it follows that

"0i .t /w"0i .t / ! .t /w0 weakly in L˛ .R3 /;

1=˛ D 1  1=q  2=3:

(83)

Since w"0i .t / D v"0i .t /, relation (63) in Condition H.3 implies

"0i .t /w"0i .t / ! .t /v.t / weakly in L6=5 .R3 /: Combining this equality with (83) gives

.t /w0 D .t /v.t /:

(84)

Now assume that the set f"0j g is infinite. It follows that

"0j .t /w"0j .t / ! .t /w0 weakly in L˛ .R3 /: Since w"0j .t / D v.t /, it obviously follows from this and (82) that w0 and v satisfy (84). Repeating these arguments gives the equality .t /w00 .t / D .t /v.t /. Hence .t /w0 D .t /w00 . Hence the sequence w" .t / has the only limiting point .t /v.t / in Lq .R3 /, which yields (81). Since the sequence w" is bounded in L2 .EI W 1;2 .R3 //, it is bounded in Lr .EI Lq .R3 // for all r 2 .1; 2/ and q 2 .1; 6/. Notice that the set of admissible .r; q/ is open. It follows from this and (81) that kw"  vkLr .EILq .R3 // ! 0 as " ! 0:

(85)

Denote by E" the set of all t 2 E such that kv" .t /kW 1;2 .R3 /  N . It is easily seen that v" .t /  v.t / D w" .t /  v.t / for t 2 E n E" ;

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1619

which leads to the equality kv" .t /  v.t /kLq .R3 / D kw" .t /  v.t /kLq .R3 / for t 2 E n E" : It follows that kv"  vkrLr .0;T ILq .R3 //  kw" .t /  v.t /krLr .EILq .R3 // C I" ;

(86)

where Z I" D .0;T /nE/[E"

k.v"  v/krLq .R3 / dt 

Z

T

c 0

kv"  vk2Lq .R3 / dt

r=2

.2r/=2  meas ..0; T / n E/ [ E" :

Next, the estimate Z

Z

T

0

kv"  vk2Lq .R3 / dt  c

0

T

kv"  vk2W 1;2 .R3 / dt  c

implies the inequality .2r/=2  : I"  c meas ..0; T / n E/ [ E"

(87)

Since kv.t /kW 1;2 .R3 /  N for t 2 .0; T / n E, the Chebyshev inequality yields the estimate meas ..0; T / n E/  N 2

Z 0

T

kv.t /k2W 1;2 .R3 / dt  cN 2 :

Since kv" .t /kW 1;2 .R3 /  N for t 2 E" , the Chebyshev inequality implies meas E"  N 2

Z 0

T

kv" .t /k2W 1;2 .R3 / dt  cN 2

Combining these results with (87) leads to the inequality I"  cN r2 ! 0 as N ! 1: It follows from this, estimate (86), and relation (85) that lim sup kv"  vkrLr .0;T ILq .R3 //  cN r2 ! 0 as N ! 1; "!0

and the proposition follows.

1620

P.I. Plotnikov and W. Weigant

Proof of Proposition 5. The proof is based on the following renormalization lemma. Lemma 3. Assume that functions " 2 L1 .0; T I L6 .R3 //, 0 2 L6 .Rd / vanish for jxj  R, and a function g 2 L1 .0; T I L1 .Rd // also vanishes for jxj  R. Furthermore, assume that a vector field v" 2 L2 .0; T I W 1;2 .Rd // and the functions

" , g satisfy integral identity (66) for all 2 C 1 .Rd  .0; T // vanishing for all sufficiently large jxj and t D T . Then the integral identity Z

T 0

Z

T

Z

C 0

Rd

Z Rd

.'. " /@t

C '. " /v"  r

0

.'. " /  ' . " / " /div v" dxdt C

C ' 0 . " /g" /dxdt (88)

Z Rd

0 .0/dx D 0

holds for every C 1 function ' W R ! R satisfying condition (71). It follows from (88) that the function '. " / satisfies the equation @t '. " / D  div .'. " /v" / C .'. " /  ' 0 . " / " / div v" C ' 0 . " /g" :

(89)

It follows from expression (67) for g and estimates (61) for " and v" that the sequence @t '. " / is bounded in L2 .0; T I W 1;2 .Rd //; hence the sequence '. " / W Œ0; T  ! W 1;2 .Rd / is equi-continuous. Moreover, since the functions '. " /.t / are uniformly bounded and supported in the ball BR the set the functions '. " /.t /, " 2 Œ0; 1/, t 2 .0; T / is relatively compact in W 1;2 .Rd /. By the Ascoli theorem, the sequence '. " / is relatively compact in C .0; T I W 1;2 .Rd //. Hence, there are a subsequence of the sequence '. " /, still denoted by '. " /, and a function ' 2 C .0; T I W 1;2 .Rd // such that '. " /.t / ! '.t / in W 1;2 .Rd / as " ! 0:

(90)

On the other hand, there is a set E 2 .0; T / with the properties

" .t / ! .t / weakly in L1 .Rd / for t 2 E;

meas .0; T / n E D 0:

In particular, the sequence " .t / is bounded in L1 .Rd / for every t 2 E. Hence the sequence '. " /.t / is bounded in L1 .Rd / for every t 2 E. It follows from this and (90) that '.t / 2 L1 .Rd / and '. " /.t / ! '.t / star weakly in L1 .Rd / for t 2 E. This completes the proof of Proposition 5 2 Proof of Proposition 6. Since ";M jv" j2  M %" jv" j2 and the sequence %" jv" j2 1 1 d is bounded in L .0; T I L .R //, the sequence ";M v" is bounded in L1 .0; T I L2 .R3 //. In view of Proposition 5, the sequence ";M .t / converges to M .t / star weakly in L1 .R3 / for a.e. t 2 .0; T /. On the other hand, Proposition 4 implies that

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1621

the sequence .t /v" .t / converges strongly in Lq .Rd / for every q 2 .1; 2/ and a.e. t 2 .0; T /. It follows that .t / ";M .t /v" .t / ! .t / M .t /v.t / weakly in Lq .Rd / as " ! 0

(91)

for a.e. t 2 .0; T /. Next, the relations Z

Z 0 Rd

.1.t // ";M .t / dx !

Z

Rd

.1.t // M .t / dx 

.1.t // .t / dx D 0: Rd

imply that .1  .t // ";M .t / ! 0 in measure in Rd as " ! 0. Since the sequence

";M .t / is bounded in L1 .Rd / and uniformly compactly supported in R3 , it follows that .1  .t // ";M .t /v" .t / ! 0 in Lq .Rd / as " ! 0: Combining this relation with (91) gives (76), and the proposition follows.

4

Basic Integral Identity

This section is devoted to the proof of the main integral identity for the function " and the vector field v" given by (57). The result is given by the following: Proposition 8. Let . " ; v" / satisfy conditions H.1–H.5 and integral identities (65)– (59). Then there are a subsequence of the sequence . " ; v" / still denoted by . " ; v" / and a set T of full measure in .0; T / with the following properties. The integral identity Z

T

Z

Z

Rd

0

Z

T

Z

T



Rd

Z

0

Rd

Rd

0

D 

Z

Kij . " /E";ij dxdt 

Z

0

T



Kij & S.v" /ij " dxdt 

Z

Kij . " v";i / " v";j dxdt

T

Z

0

Z

Qj . " /F";j dxdt C Z  Rd

Rd

Rd

" v";j Qj g" dxdt

Qj . " /.T  / " .T  /v";j .T  / dx

Qj . 0 / 0 v0;j dx

(92)

holds true for a.e. T  2 T and for all " from the selected subsequence. Proof of Proposition 8. Choose an arbitrary T  2 .0; T ,  2 C 1 .0; T W C 2 .Rd // and set

1622

P.I. Plotnikov and W. Weigant

 h .x; t / D .x; t / for t 2 .0; T   h/;

 h .x; t / D 0 for t  T  ;

 h .x; t / D h1 .T   t /.x; t / for t 2 .T   h; T  /: Substitute  h into integral identity (65) and let h ! 0. There is a set T of full measure in .0; T / with the property: The integral identity Z

T 0

Z

T

Z

Z

Rd

C 0

Rd

. " v" @t   & S.v" / W r/ dxdt C

T

0

Z

  F" dxdt 

Z

Rd

Z Rd

@xj i E";ij dxdt

.T  /v" .T  /  .T  /dx C

(93)

Z

0 v0  .0/dx D 0

Rd

holds for every T  2 T and all  2 C 1 .0; T I C 2 .Rd //. Let # 2 C01 .Rd / be a standard mollifying kernel satisfying the conditions Z #  0;

#.x/ dx D 1:

#.x/ D #.x/; Rd

Introduce the mollifier Sh , h > 0, defined by Sh u.x/ D h

d

Z

#.h1 .x  y//u.y/ dy:

(94)

R3

Choose an arbitrary  2 C01 .0; T /. Substituting D hd .t / #.h1 .x0  x// into integral identity (59) leads to the equality Z

T 0

Z

T

0 .t /Sh " .x0 ; t / dt

Z .t / div Sh . " v" /.x0 ; t / dtC

0

T

.t /.Sh g" /.x0 ; t /; dt D 0

0

which yields the differential equation @t Sh " .x; t / D  div Sh . " v" /.x; t / C .Sh g" /.x; t /

(95)

in Rd  .0; T /. It follows that Sh " and @t Sh " are infinitely differentiable with respect to x. Consider the vector fields  i D Qi Sh " ;

1  i  d;

where the integral operator Qi is given by (29). Identity @xj Qi D Kij implies @xi j D Kij Sh " :

(96)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1623

In its turn relation (95) yields @t i D Kij Sh . " v";j / C Qi Sh g" :

(97)

Substituting  into (93) leads to the integral identity Z

T

Z

Z Rd

0

Z

T

Z

T

Z

Kij .Sh . " v";i // " v";j dxdt D

Z

Rd

T

Kij .&S.v" /ij / Sh . " / dxdt  0

Z

 0

Rd

0

Rd

0

Z

T

Kij .Sh " /E";ij dxdt 

Z Qj Sh . " /F";j dxdt C Z  Rd

Z Rd

" v";j Qj Sh g" dxdt

Qj Sh . " /.T  / " .T  /v";j .T  / dx

Rd

Qj Sh . 0 / 0 v0;j dx:

(98)

Now the task is to pass to the limit in (98) as h ! 0. Notice that for a fixed ", the sequences Sh " and E";ij are bounded in L1 .0; T I Lm .R3 // and L1 .0; T I L1 .R3 //. Moreover, they are supported in BR  .0; T /. Since m  6, Sh " .t / ! " .t / in L6 .Rd / as h ! 0 for a.e. t 2 .0; T /. It follows from the standard properties of the Newton potential that the sequence Kij Sh " is bounded in L1 .0; T I C .Rd // and Kij .Sh " /.t / ! Kij " .t / in C .Rd / as h ! 0 for a.e. t 2 .0; T /, which yields Z

T

Z

Z Rd

0

Z

T

Kij .Sh " /E";ij dxdt !

Kij . " /E";ij dxdt as h ! 0:

Rd

0

(99)

The similar arguments lead to the relation Z

T Z 0

Rd

  Kij & S.v" /ij .Sh " / dxdt !

T Z

Z 0

Rd

  Kij & S.v" /ij . " / dxdt as h ! 0:

(100) Next, the sequence " is bounded in L1 .0; T I Lm .Rd //, and the sequence v" is bounded in L2 .0; T I L6 .Rd //. Hence, the sequence " v" is bounded in L2 .0; T I L2 .Rd /. Moreover, " and v" are supported in BR  .0; T /. It follows that Sh . " v" / ! " v" in L2 .0; T I L2 .Rd // as h ! 0;

1624

P.I. Plotnikov and W. Weigant

and hence Kij Sh . " v" / ! " v" in L2 .0; T I L2 .BR // as h ! 0: It follows that Z

T

Z

Z Rd

0

T

Z

Kij .Sh . " v";i // " v";j dxdt !

Kij . " v";i / " v";j dxdt

Rd

0

(101)

as h ! 0. Since kQj f kC .Rd /  ckf kL1 .Rd / for every function f supported in the ball BR , we obviously have Z

T

Z

Z

 Rd

0

Z C Rd

0

T

Z

Rd

Rd

Rd

Z

T

0 



Qj Sh . " /F";j dxdt

Qj Sh . 0 / 0 v0;j dx !

" v";j Qj g" dxdt  

C

R3

Z

Z

 0

Z

" v";j Qj Sh g" dxdt 

Qj Sh . " /.T  / " .T  /v";j .T  / dx  Z

T

Qj . " /.T / " .T /v";j .T / dx 

Z

(102)

Z Rd

Rd

Qj " F";j dxdt

Qj . 0 / 0 v0;j dx

as h ! 0. Letting h ! 0 in identity (98) and using relations (99), (100), (101), and (102) we arrive at desired identity (92).

5

Isothermal Flow: Proof of Theorem 2

This section is devoted to the proof of Theorem 2. Only the main ideas of the proof will be given, the reader is referred to the paper [14] for details. It follows from general compactness results for compressible Navier-Stokes equations, see [5, 8] ch.5, [13] Sec. 4.4, that the theorem will be proved if it will be established that there exist exponents  > 0 and  > 1 such that the inequalities k%" kL1C .Q0 / C k%" ju" j2 kL .Q0 /  c hold for every cylinder Q0 D ˝ 0  Œ˛; ˇ b QT . Here c is independent of ". Choose a function & satisfying conditions (56) and a function 2 C01 .QT / such that 0  & ;  1;

& D 1;

D 1 in Q0 :

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1625

It suffices to prove that k& %" kL1C .QT / C k& %" ju" j2 kL .QT /  c:

(103)

The proof of estimate (103) falls into a sequence of lemmas. The first establishes the positivity of the left-hand side of identity (92). Lemma 4. Under the assumptions of Theorem 2, Z

Z

T

Z R2

0

T

Z

" v";i v";j Kij " dxdt 

Kij . " v";i / " v";j dxdt  0:

R2

0

(104)

Proof. Notice that " .t / 2 L6 .R2 / and v" 2 W 1;2 .R2 /. Since " and v" are supported in BR , it follows from the continuity of the embedding W 1;2 .BR / ,! L6 .BR / that the functions " .t /jv" .t /j2 and " .t /v" .t / belong to the class L2 .R2 /. Moreover, they are supported in BR . It follows from this and Proposition 2 that Z Z Z

" v";i v";j Kij . " / dxdt D cd !i !j R. " v";i v";j /R. " / d d !; R2

S1

R

S1

R

Z Z

Z

R2

" v";i Kij . " v";j / dxdt D cd

!i !j R. " v";i /R. " v";j / d d !

for a.e. t 2 .0; T /. On the other hand, the definition of the Radon transform R yields the equalities Z

" .!  v" /2 d ˘

!i !j R. " v";i v";j /R. " / D ˘. ;!/

Z

" d ˘ ; ˘. ;!/

Z

2

!i !j R. " v";i /R. " v";j / D

" .!  v" /d ˘

:

˘. ;!/

It follows that TZ

Z 0

Z Z (Z S2

Z R2

2

Z R2

0

" d ˘ 

˘. ;!/

Kij . " v";i / " v";j dxdt D 2 )

Z

Z

" .!  v" / d ˘

R

T

" v";i v";j Kij . " / dxdt 

˘. ;!/

d d !:

" .!  v" /d ˘ ˘. ;!/

It remains to note that the Cauchy inequality implies Z

2

Z

Z

" .!  v" / d ˘ ˘. ;!/

and the lemma follows.

" d ˘  ˘. ;!/

2

" .!  v" /d ˘

˘. ;!/

;

1626

P.I. Plotnikov and W. Weigant

The second lemma gives the estimates for the density in the dual space. Lemma 5. There is a positive constant c independent of " and T  such that Z

T

k " .t /k2W 1;2 .R2 / dt  c:

0

(105)

Proof. Consider integral identity (92). Since E";ij D " v";i v";i C . " C " "m /ıij ; Lemma 4 implies the inequality Z

T

Z

Z

R2

0

T

Z

Kij . " /E";ij dxdt 

R2

0

Z

T

Z

R2

0

Kij . " v";i / " v";j dxdt  Ki i . " /. " C " "m / dxdt:

Notice that Ki i is the integral operator with the nonnegative kernel jx  yj1 , which yields the estimate Z

T

Z

Z

R2

0

T

Z

Kij . " /E";ij dxdt 

R2

0

Z 0

T

Z

Kij . " v";i / " v";j dxdt 

" .x/ " .y/jx  yj1 dxdydt:

R2 R2

It follows from the theory of the Riesz potential that the inequalities Z R2 R2

Z

" .x/ " .y/jx  yj1 dxdy D C

R2

" .1=2 / " dx  C k " k2W 1;2 .R2 /

hold for every compactly supported function " . Here the constant C depends only on the diameter of spt " . Combining the obtained results leads to the inequality Z

T

0

Z 0

T

Z R2

Z Kij . " /E";ij dxdt  0

T

k " k2W 1;2 .R2 / dt 

Z R2

Kij . " v";i / " v";j dxdt:

Substituting this inequality into (92) gives the estimate

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z

T

k " k2W 1;2 .R2 / dt  c

0

Z

T

Z R2

0

R2

Z

R2

jKij S .v" /ij /j " dxdt C

T

Z

" v";j jQj g" j dxdt C c 0 

C

Z

T

Z

c 0

Z



1627

Z



jQj . " /.T / " .T /v";j .T / jdx C

jQj . " /F";j j dxdt

R2

R2

(106)

jQj . 0 / 0 v0;j j dx:

Notice that the functions g" , F" , " v" are supported in the ball BR and admit the estimate kg" kL1 .0;T IL1 .R2 // C kF" kL1 .0;T IL1 .R2 // C k " v" kL1 .0;T IL1 .R2 //  c:

(107)

It follows from this and representation (29) for the operator Qj that kQj g" kL1 .0;T IC .BR // C kQj F" kL1 .0;T IC .BR // C kQj . " v" /kL1 .0;T IC .BR //  c: (108) Substituting (107) and (108) into (106) gives the estimate Z

Z

T

k " k2W 1;2 .R2 / dt

0

T

Z

c 0

R2

jKij S .v" /ij /j " dxdt C c:

(109)

It remains to estimate the integral in the right-hand side of this inequality. Recall that the functions Sij .v" / supported in the ball BR and admit the estimates kSij .v" /kL2 .0;T IL2 .R2 //  c: It follows that kKij Sij .v" /kL2 .0;T IW 1;2 .BR //  c:

(110)

Represent Kij Sij .v" / in the form Kij Sij .v" / D M .t/w; where M .t/ D kKij Sij .v" /.t /kW 1;2 .BR // ;

w D M 1 Kij Sij .v" /:

(111)

It follows from (110) that kM kL2 .0;T /  c and kw.t /kW 1;2 .BR // D 1. The Trudinger inequality implies the estimate k exp.jw.t /j/kL1 .BR /  c: It follows from this, energy estimate (9), and the Young inequality that Z 0

T

Z R2

Z jKij S .v" /ij /j " dxdt D

Z

T

M .t/ 0

jw.t /j " .t / dxdt  BR

1628

P.I. Plotnikov and W. Weigant

Z

Z

T

Z

M .t/



exp.jw.t /j/ dx

0

.1 C " .t // ln.1 C " .t // dx

BR

dt

BR

Z

T

c

M .t/ dt  c: 0

Substituting this inequality into (109) gives desired estimate (105). Denote by rot the differential operators rot w D @x2 w1  @x1 w2 ; and consider the function F D .F1 ; F2 / defined by the equalities F1 D div 1 . " v" /;

F2 D rot 1 . " v" /:

(112)

The following lemma gives Lp - estimates for F. Lemma 6. For every positive ı < 1, there is a constant c.ı/ such that kFk

8

L4 .0;T IL 3Cı .BR //

 c.ı/:

(113)

Proof. Fix an arbitrary positive ı < 1. Integral representation for the operator @x 1 implies the inequalities Z

jx  yj1 " jv" j.y; t / dy 

c c b.x; t/L.x; t / C b 1 .x; t /Q.x; t /; 2 2

jx  yj1 " jv" j2 .y; t / dy;

Q.x; t / D

jF.x; t /j  c ˝

where L.x; t / D

Z ˝

Z

jx  yj1 " .y; t / dy;

˝

b is an arbitrary positive function. Notice that if L.x; t / or Q.x; t / vanishes at least at one point .x; t /, then F.; t / vanishes in R2 . In opposite case we can take b D p Q=L. Thus we get p p jF.x; t /j  c L.x; t / Q.x; t / a.e. in R2  .0; T /: Now the task is to estimate L and Q. We have Z   1Cı   1ı

" jv" j2 2 jx  yj1ıC˛ " jv" j2 2 jx  yj˛ dy; L.x; t / D BR

where ˛ D ı=2 > 0. It follows that

(114)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

L.x; t /  c.R/

Z



" jv" j2

 1Cı 2

1629

  1ı jx  yj1ıC˛ " jv" j2 2 dy for x 2 BR :

BR

The Hölder inequality implies the estimate L.x; t /  c

Z

2

" jv" j jx  yj

2˛ 2C 1Cı

Z 1Cı 2 dy

2

1ı 2

" jv" j dy

BR

:

BR

This leads to the inequality Z

L.x; t /

2 1Cı

Z

1ı Z 1Cı

2

dx

BR

BR

Z c

Z

2˛

" jv" j2 .y; t /jx  yj2C 1Cı dxdy

" jv" j dy BR

" jv" j2 .y; t / dy

BR

2 1Cı

:

BR

It follows from this and energy estimate (9) that kLk

2

L1 .0;T IL 1Cı .BR //

 ck " jv" j2 kL1 .0;T IL1 .R2 // (115)

 ck%" ju" j2 kL1 .0;T IL1 .˝//  c: Now the task is to estimate Q. It is easily seen that Q.x; t /  c.R/

Z R2

jx  yj1 " .y; t / dy D c./1=2 " .x; t / WD G.x; t /:

On the other hand, the estimate of the Riesz potential yields kGkL2 .0;T IW 1=2;2 .BR /  k " kL2 .0;T IW 1=2;2 .BR //  c: Since the embedding W 1=2;2 .BR / ,! L4 .BR / is bounded, see [1], thm. 7.57, it follows that kQkL2 .0;T IL4 .BR //  c.R/kGkL2 .0;T IL4 .BR //  ckGkL2 .0;T IW 1=2;2 .BR //  c: Combining these inequalities with (115) leads to the estimates p k Lk

4 L1 .0;T IL 1Cı

.BR //

 c;

p k QkL4 .0;T IL8 .BR //  c:

Next, the Hölder inequality implies that p p p p k L QkL .0;T ILr .BR //  k LkL 1 .0;T ILr1 .BR // k QkL 2 .0;T ILr2 .BR //

(116)

1630

P.I. Plotnikov and W. Weigant

for all ; r; i ; ri 2 Œ1; 1 satisfying the condition 1 D 11 C 21 ;

r 1 D r11 C r21 :

Set D 2 D 4, 1 D 1, r1 D 4=.1 C ı/, r2 D 8, and r D 8=.3 C 2ı/, and recall inequalities (116). It follows that p p k L QkL4 .0;T IL8=.3C2ı/ .BR //  c.; ı; R/E: Combining this result with (114) gives (113), and the lemma follows. Now the task is to derive Lp estimate for %" . The derivation is based on the special integral identity, which is a heart of the theory of compressible Navier-Stokes equations. This identity was introduced in [8] and was used in [5] for the proof of global solvability of initial boundary value problem. Following [13] (Theorem 4.5.2 and formula (6.0.9)) this identity can be formulated as follows. Choose an arbitrary function ' 2 C 1 Œ0; 1 satisfying condition (71). It follows from Proposition 1 that %" , u" , and '" D '.%" / serve as a weak solutions to equations @t .%" u" / C div .%" u" ˝ u" / D div T C %" f in QT ; @t %" C div .%" u" / D 0

in QT ;

@t '" C div .'" u" / C $ D 0

in QT ;

where T D S.u" /  .%" C "%m " /I;

$ D .'.%" /  %" ' 0 .%" // div u" :

(117)

It follows from (9) and (10) that T 2 Lr .QT /; r > 1;

'" 2 L1 .QT /;

$ 2 L2 .QT /:

Hence the functions u" , %" , '" , $ and the tensor T meet all requirements of Theorem 4.5.2 in [13] with g D g' D 0 and G D 0. In view of this theorem, the integral identity Z

'" B W Œ& T dxdt D QT

Z

.H  u" C &%" .P C Q/  u" C S/ dxdt

(118)

QT

holds for every 2 C01 .QT / and & 2 C01 .˝/. Here, we denote H D BŒ '" .&%" u" /  . '" /BŒ&%" u" ; S D .r& ˝ AŒ '" / W .%" u" ˝ u"  T/ C &AŒ '"   %" f; P D AŒ'" @t C '" u"  r ;

Q D AŒ$:

(119)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1631

The Riesz operator B D .Bij / and the potential operator A D .Aj / are given by the relations Bij D @xi @xj 1 ;

Aj D @xj 1 ;

(120)

where the potential 1 is defined as a solution to equation (23). Let us rewrite identity (118) in more suitable form. The identity B W Œ&pI D & pI implies Z

'" B W Œ& T dxdt D QT

Z  QT

&'" .%" C "%m " / dxdt C

Z

'" B W Œ& S.u" / dxdt:

(121)

QT

Notice that the functions .t/ and & are supported in ˝ b BR . The vector field u" .t / 2 W01;2 .˝/ for a.e. t 2 .0; T /. Hence all functions in (118) are extended by zero to R2  .0; T /. Next, the sequence " is bounded in L1 .0; T I Lm .˝//, m  6, and the sequence u" is bounded in L2 .0; T I L6 .˝//. Hence, the extended sequence " u" is bounded in L2 .0; T I L3 .R2 /. Notice that BŒ " u"  2 L2 .0; T I L3 .R2 /, since the Riesz operator B W Lr .R2 / ! Lr .R2 / is continuous for every r 2 .1; 1/. It follows from this that ju" j.j " u" j C jBŒ " u" j/ 2 L1 .0; T I L6=5 .R2 /:

(122)

Denote by Sh the mollifier given by (94). Since '" 2 L1 .R2  .0; T /, the relations BSh . '" /.t / ! B. '" /.t /;

Sh . '" /.t / ! . '" /.t / as h ! 0

hold true for a.e. t 2 .0; T /. Moreover, the sequences BSh . '" / and Sh . '" / are bounded in L1 .0; T I L6 .˝//. It follows from this and (122) and (119) that Z H  u" dxdt D Z lim

h!0 QT

QT





(123)

BŒSh . '" /.&%" u" /  Sh . '" /BŒ&%" u"   u" dxdt

Since &%" u" D " v" , we have &%" u" D rF1 C r ? F2 ; where F D .F1 ; F2 / is a solution to problem (112). The identity Bij @xi Aj ŒSh . '" / implies the equality Z u"  BŒSh . '" /.&%" u" / dxdt D Z

R2 .0;T /

R2 .0;T /



(124) ?

u";j @xj ˚  .rF1 C r F2 / dxdt;

1632

P.I. Plotnikov and W. Weigant

where ˚ D AŒSh . '" /:

(125)

Integrating by parts in the right-hand side of (124) leads to the equality Z R2 .0;T /

u"  BŒSh . '" /.&%" u" / dxdt D

Z

 R2 .0;T /

Z



 R2 .0;T /

F1 r.Sh . '" //  u" dxdt

(126)

 F1 ru";j  @xj ˚ C F2 r ? u";j  @xj ˚ dxdt:

Since   BŒ& " u"  D r 1 div .& " u" / D rF1 ; it follows that Z



 Sh . '" /BŒ&%" u"   u" dxdt D

QT

Z R2 .0;T /

Sh . '" /rF1  u" dxdt:

(127)

Combining (126) with (127) and using the relation Z

Z R2 .0;T /

F1 r.Sh . '" //  u" dxdt C

R2 .0;T /

Sh . '" /rF1  u" dxdt D

Z

 R2 .0;T /

Sh . '" /F1 div u" dxdt;

leads to the identity Z



 BŒSh . '" /.&%" u" /  Sh . '" /BŒ&%" u"   u" dxdt D

QT

Z R2 .0;T /

Z

Sh . '" /F1 div u" dxdt

 F1 ru";j  @xj ˚ C F2 r ? u";j  @xj ˚ dxdt:



 R2 .0;T /

Finally, let h ! 0 in (123) and (128) to obtain

(128)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z

Z H  u" dxdt D

QT

Z

R2 .0;T /

. '" /F1 div u" dxdt

 F1 ru";j  @xj AΠ'"  C F2 r ? u";j  @xj AΠ'"  dxdt:



 R2 .0;T /

1633

(129)

Next, fix  2 .0; 1=8/ and an arbitrary N > 0, and choose a function ' 2 C 1 Œ0; 1 such that '.%/ D .1 C %/ for 2 Œ0; N ;

'.%/ D .1 C N / for 2 ŒN C 1; 1/; j'.%/j C j'.%/  %' 0 .%/j  c.1 C %/ ; (130)

where c is independent of N . Now the task is to estimate all terms in the right-hand side of (118). Lemma 7. The estimate ˇZ ˇ ˇ

QT

ˇ ˇ H  u" dxdt ˇ  c

(131)

holds true for every function ', satisfying condition (130). Here the constant c is independent on N . Proof. Notice that energy estimate (9) implies the inequality k '" kL1 .0;T IL1= .R2 // C k@xj AΠ'" kL1 .0;T IL1= .R2 //  c:

(132)

Next, identity (129) yields the estimate ˇZ ˇ ˇ

QT

Z ˇ ˇ H  u" dxdt ˇ  c

R2 .0;T /

.j '" j C j@xj AΠ'" j/jFj j& ru" j dxdt:

(133)

Since  < 1=8, there is ı > 0 such that  C .3 C ı/=8  1=2. It follows from this and the Hölder inequality that ˇZ ˇ ˇ

QT

ˇ ˇ H  u" dxdt ˇ  c.k '" kL1 .0;T IL1= .R2 // C k@xj AŒ '" kL1 .0;T IL1= .R2 // /  kFkL4 .0;T IL.8=.3Cı/.BR // kru" kL2 .0;T IL2 .BR // : (134)

It remains to note that desired estimate (131) obviously follows from (134), estimate (132), estimate (113) in Lemma 6, and energy estimate (9).

1634

P.I. Plotnikov and W. Weigant

Lemma 8. The inequality ˇZ ˇ ˇ

QT

ˇ ˇZ ˇ ˇ &%" P  u" dxdt ˇ C ˇ

QT

ˇ ˇZ ˇ ˇ &%" Q  u" dxdt ˇ C ˇ

QT

ˇ ˇ Sdxdt ˇ  c

(135)

holds true for every function ', satisfying condition (130). Here c is independent on N. Proof. The function '" is supported in BR  Œ0; T  and admits estimate (132), which yields kAŒ '" kL1 .0;T IC .BR //  ckAŒ '" kL1 .0;T IW 1;1= .BR //  c:

(136)

It follows from this, expression (119), and energy estimate (9) that ˇZ ˇ ˇ

QT

Z ˇ ˇ S dxdt ˇ  c

.jr& jj.%" u" ˝ u"  Tj C & j%" j/ dx 

QT

ck " u" j2 kL1 .BR .0;T // C ckS.u" /kL1 .BR .0;T // C

(137)

ck " C "%m " kL1 .BR .0;T //  c: Next, the embedding theorem implies ku" .t /kL1= .R2 /  cku" .t /kW 1;2 .BR / . It follows from this and energy estimate (9) that ku" .t /kL2 .0;T IL1= .R2 //  c; which along with estimate (132) yields the estimate kr  '" u" kL2 .0;T IL1=2 .R2 //  c: The function r  '" u" is supported in BR and 1=2 > 2. It follows that kAŒr  '" u" kL2 .0;T IC .R2 //  c: Combining this estimates with (136) and recalling expression (119) for P leads to the estimate ˇZ ˇ ˇ

QT

ˇ ˇ &%" P  u" dxdt ˇ 

ckAŒ'" @t C r  '" u" kL2 .0;T IC .R2 // k& " u" kL2 .0;T IL1 .R2 //  c: Next, the identities

(138)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z

&%" Q  u" dxdt D 

QT

Z $ A  .&%" u" / dxdt D QT

Z

$1 div .&%" u" / dxdt D 



1635

QT

Z $ F1 dxdt QT

and the expression for $ lead to the estimate ˇZ ˇ ˇ

QT

Z ˇ ˇ &%" Qu" dxdt ˇ  c

.1 C %" / jdiv u" jjF1 j dxdt 

QT

ck1 C %" kL1 .0;T IL1 .BR // kru" kL2 .0;T IL2 .BR // kF1 kL4 .0;T IL8=.3Cı/ .BR // :  c: Combining this result with (137) and (138) gives desired estimate (135). Lemma 9. There exists a constant c independent of " such that Z QT

  & "1C C " "mC dxdt  c:

(139)

Proof. Substituting equality (121) into the left-hand side of (118) and using estimates (131) and (135) lead to the inequality Z & . " C QT

" "m /'"

ˇZ ˇ dxdt  ˇ

QT

ˇ ˇ

'" B W Œ& S.u" / dxdt ˇ C c:

(140)

Since the operator B W L2 .R2 / ! L2 .R2 / is bounded, estimate (132) and energy estimate (9) imply ˇZ ˇ ˇ

QT

ˇ ˇ

'" B W Œ& S.u" / dxdt ˇ  ck '" kL1 .0;T IL1= .BR // kru" kL2 .0;T IL2 .BR //  c:

It follows from this and (140) that Z & . " C " "m /'" dxdt  c:

(141)

QT

Recall that the nonnegative function ' depends on N and '" D .1 C %" / for %"   N . Letting N ! 1 in (141) gives desired estimate (139). Estimate (139) and energy estimate (9) imply one-parametric family of estimates of %" in anisotropic Lebesgue spaces. In order to derive these estimates, choose an arbitrary r 2 .2; 1/ and set s D r=.r  / > 1 and ˛ D .1 C /=r 2 .0; 1/. Obviously, .1  ˛/=1 C ˛=.1 C / D 1=r;

1  ˛ C ˛=.1 C / D 1=s:

1636

P.I. Plotnikov and W. Weigant

It follows from this, estimates (139), (9), and the interpolation inequality that ˛ k& %" kLr .0;T ILs .˝//  k& %" k1˛ L1 .0;T IL1 .˝// k& %" kL1C .0;T IL1C .˝// < c:

(142)

Inequality (139) leads to estimate for the momentum %" u" . The result is given by the following: Lemma 10. There are exponents p 2 .2; 1/ and q 2 .1; 1/ such that k& %" u" kLp .0;T ILq .˝//  c:

(143)

Proof. Represent & %" ju" j it in the form & %" ju" j D .& %" / .& %" ju" j2 /1 ˇju" j21 :

(144)

Let us show that there exist  2 .1=2; 1/, and p; q; 2 .1; 1/ such that =r C .2  1/=2 D 1=p;

=s C 1   C = D 1=q;

(145)

where r and s are exponents in estimate (142). To this end notice that these relations imply the equality  D .1=2 C 1=p/.1 C 1=r/1 . The inequalities 1=2 <  < 1 are fulfilled if and only if 2r=.r C 2/ < p < 2r. Since r > 2, there exists p > 2 satisfying these inequalities. On the other hand, it follows from s > 1 that 0 < 1 C .1=s  1/ < 1. Hence there is 2 .1; 1/ such that q 2 .1; 1/. This completes the proof of (145) It follows from (144), the Hölder inequality, and estimates (142), (9) that k& %" u" kLp .0;T ILq .˝//  k& %" kLr= .0;T ILs= .˝//  k& %1 ju" j2.1/ kL1 .0;T IL1=.1/ .˝// kju" j.21/ kL2=.21/ .0;T IL =.21/ .˝// " 

1

21

D k& %" kLr .0;T ILs .˝// k& %" ju" j2 kL1 .0;T IL1 .˝// ku" kL2 .0;T IL .˝//  c: This completes the proof of the lemma. The last lemma gives the estimate for the kinetic energy density. Lemma 11. There exists an exponent  > 1, depending only on  such that Z QT

where c is independent on ".

.& j " ju" j2 / dxdt  c;

(146)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1637

Proof. Since p > 2 and q > 1, there are 1 ; 2 ; ! 2 .1; 1/ such that 1=p C 1=2 D 1=1 and 1=q C 1=! D 1=2 . Set  D minfi g. Next, the Hölder inequality and estimate (143) for %" u" imply k& %" ju" j2 kL .˝.0;T //  ck& %" ju" j2 kL1 .0;T IL2 .˝ 0 //  ck& %" u" jkLp .0;T ILq .˝// ku" kL2 .0;T IL! .˝//  cku" kL2 .0;T IW 1;2 .˝//  c: 0

Estimate (139) and (146) yield desired estimate (103). This completes the proof of Theorem 2.

6

Proof of Theorem 3

This section is devoted to the proof of Theorem 3. Let us explain the main ideas of the proof. It is necessary to prove that there exists a subsequence of the sequence . " ; v" /, still denoted by . " ; v" /, such that the equality Z

T

Z

lim

"!0 0

R3

h. " v";i v";j  vi vj / dxdt D 0

(147)

holds for all h 2 L1 .R3  .0; T //. Recall that

" * star weakly in L1 .0; T I L3=2 .R3 //; ";M * M star weakly in L1 .R3 .0; T //; and M % as M ! 1. Moreover, ";M v";i v";j ! M vi vj weakly in L1 .R3  .0; T //. Therefore, it suffices to show that Z

T

Z

lim lim

M !1 "!0 0

R3

. "  ";M /jv"; j2 dxdt D 0:

(148)

The proof of this equality is based on the fact that the integral in the left-hand side of (148) can be estimated via the defect Z W";M D

˚

R3

 Kij . "  ";M / E";ij  Kij .. "  ";M /v";i / . "  ";M /v";j dx:

In particular, Lemma 21 constitutes the estimate Z 0

T

Z

2

R3

Z

. "  ";M /jv" j dxdt  c

!4=15

T

W";M dt

;

0

which holds true for every T  2 .0; T . Therefore, in order to proof equality (148) it suffices to show that

1638

P.I. Plotnikov and W. Weigant

Z

T

W";M dt D 0:

lim lim

M !1 "!0 0

(149)

The proof of (149) is based on the monotonicity arguments and occupies Sects. 6.1, 6.2, 6.3, 6.4, and 6.5. The main idea is to derive and next to compare two alternative integral identities. The first can be obtained as the limit of main integral identity (92) as " ! 0. It reads Z

Z

T 0

Z

T

CM .t / dt  0

Z lim

"!0 0

T

AM .t / dt  2

BM .t / dt C 0

Z

T

T

W";M .t / dt D

(150)

.V .t /  D.t // dt C G: 0

Here the function CM .t / is defined by the equality Z CM D

R3

Kij . M / d ij :

AM , BM , V , and D are unessential low-order terms given by (152), (153), (154), (155), (156), and (157). Recall that the matrix-valued Radon measure fij g is a weak limit as " ! 0 of the energy tensors fE";ij g. The derivation of integral identity (150) is given in Sect. 6.1. In order to obtain the alternative integral identity, it is necessary to pass to the limit as " ! 0 in integral identities (65)–(66) and next to obtain the new integral identity for the functions .%; v/ arguing as in the proof of (92). The corresponding result is given by Theorem 5 in Sect. 6.4. It follows from this theorem that Z

T

Z

Z

0

R3

Z

T

˚.x; t / .x; t /dt D

A dt C 0

T

.V .t /  D.t // dt C G;

(151)

0

where ˚ D Kij ? ij and Z A.t/ D R3

Kij . vi / . vj / dx:

The properties of the function ˚ are described by Proposition 10 in Sect. 6.3. It follows from this proposition that ˚ is nonnegative and all integrals in (151) are well defined. The last step is the passage to the limit in integral identity (150) as M ! 1. The justification of this procedure is given in the proof of Proposition 12 in Sect. 6.5. It follows from this proposition that the passage to the limit as M ! 1 in (150) leads to the integral identity

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z

T

Z

Z

T

˚ dxdt  Z

0

Z

T

T

W";M .t / dt D

lim lim

M !1 "!0 0

A.t/ dt C

R3

0

1639

.V .t /  D.t // dt C G; 0

which, being combined with (151), implies desired equality (149), and the theorem follows. The rest of the chapter is devoted to the formal proof of Theorem 3. The proof falls into six steps.

6.1

Step 1: First Integral Identity

In this section the first basic integral identity for the weak limit . ; v/ of the sequence . " ; v" / is derived. Recall that M is the weak limit of the sequence

";M D minf " ; M g as " ! 0. The properties of ";M and M are described by relations (72), (73), (74), (75), and (76). Introduce the auxiliary functions which play the crucial role in the further analysis. For every M > 0 set Z A";M .t / D

R3

Kij . ";M v";i / . ";M v";j / dx;

R3

Z B";M .t / D

R3

Z R3

Kij . M vi / . M vj / dx;

Kij .. "  ";M /v";i / . ";M v";j / dx; (153)

Z

BM .t / D C";M D

(152)

Z

AM .t / D

Kij ..  M /vi / . M vj / dx;

R3

Z

Kij . ";M / E";ij dx;

CM D

R3

Z W";M D Z

 R3

R3

Kij . M / d ij ;

(154)

Kij . "  ";M / E";ij dx (155)

Kij .. "  ";M /v";i / . "  ";M /v";j dx:

We also introduce the mappings D; V W .0; T / ! R and a quantity G defined by the equalities Z D" .t / D Z V" .t / D

R3

R3

Z Qj d ";j ;

Kij " & S.v" /ij dx

D.t / D Z

V .t/ D R3

R3

Qj d j ;

Kij & S.v/ij dx;

(156)

1640

P.I. Plotnikov and W. Weigant

Z G" D

R3

Qj . " /.T  / " .T  /v";j .T  / dx 

Z



GD R3



Z R3

Qj . 0 / 0 v0;j dx; (157)

Z



Qj . /.T / .T /vj .T / dx 

R3

Qj . 0 / 0 v0;j dx:

The following theorem is the main result of this section. Theorem 4. The integral identity Z

Z

T

Z

T

CM .t / dt 

AM .t / dt  2

0

0

Z

Z

(158)

T

W";M .t / dt D

lim

BM .t / dt C 0

T

"!0 0

T

.V .t /  D.t // dt C G 0

holds for all T  and for all M > 0. Moreover, the functions AM and BM admit the estimate ˇZ ˇ ˇ

T

0

ˇ ˇZ ˇ ˇ AM .t / dt ˇ C ˇ

T 0

ˇ ˇ BM .t / dt ˇ  c;

(159)

where c is independent of M . Proof of Theorem 4. Rewrite identity (92) in the form Z

Z

T

0

Z

T

C";M .t /dt  0

Z

T

A";M .t /dt  2 0

Z

Z

T

V" dt  0

T

B";M .t /dt C

W";M .t / dt D 0

T

D" dt C G" :

(160)

0

The rest of the proof is based on the following lemmas. Lemma 12. There are a set E  .0; T / of the full measure in .0; T / and subsequences of the sequences f"g and fM g still denoted by f"g and fM g with the following properties. The inequalities jA";M .t /j C jAM .t /j  c; jB";M .t /j C jBM .t /j  c

(161)

hold for every t 2 E and for every ", M . Here the constant c is independent of " and M . Moreover, we have A";M .t / ! AM .t /;

B";M .t / ! BM .t / as " ! 0 for all t 2 E:

(162)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1641

Proof. In view of estimate (61) in Condition H.2, for every " there is a set E" of full measure in .0; T / such that k " .t /v" .t /kL6=5 .R3 /  c for all t 2 E" :

(163)

Choose an arbitrary countable sequence " ! 0 and set E D \" E" . Obviously estimate k ";M .t /v" .t /kL6=5 .R3 /  c

(164)

holds true for all such ", for all t 2 E, and for all M > 0. Since the functions " .t / are supported in the ball BR , it follows that kKij . ";M .t /v" .t //kL6 .R3 /  c

(165)

for all t 2 E and for all M > 0. Combining (164) and (165) leads to desired estimate (161) for A";M . Repeating these arguments gives estimate (161) for AM , B";M , and BM . It remains to prove relations (162). Choose q 2 .1; 2/ such that q > 3=2. It follows from relation (76) in Proposition 6 that there is a set E of the full measure in .0; T / such that

";M .t /v";i .t / ! M .t /vi .t / weakly in Lq .R3 / as " ! 0

(166)

for all t 2 E. Since the functions ";M .t /v" are uniformly compactly supported in R3 , it follows from the standard estimates of the Newton potential that Kij . ";M v";i /.t / ! Kij . M vi /.t / in C0 .R3 / as " ! 0

(167)

for every t 2 E. It follows from (166)–(167) that A";M .t / ! AM .t / for all t 2 E. Next, it follows from relation (63) in Condition H.3 and relation (76) in Proposition 6 that . " .t /  ";M .t //v";j .t / ! . .t /  M .t /vj .t / weakly in L6=5 .R3 / as " ! 0 (168) for all t 2 E. It remains to note that relation B";M .t / ! BM .t / for all t 2 E obviously follows from (167)–(168). Lemma 13. There are a set E  .0; T / of the full measure in .0; T / and subsequences of the sequences f"g and fM g still denoted by f"g and fM g such that jC";M .t /j C jCM .t /j  cM for all t 2 E;

(169)

1642

P.I. Plotnikov and W. Weigant

and C";M ! CM star weakly in L1 .0; T / as " ! 0:

(170)

Proof. There exist subsequence of the sequence ."; M / still denoted by ."; M / and a set E of full measure in .0; T / such that estimate k";ij .t /kM.R3 / C kij .t /kM.R3 /  c

(171)

holds for all t 2 E. Notice that ";M and M are bounded by the constant M and compactly supported in the ball BR , which leads to the estimate kKij ";M .t /kC0 .R3 / C kKij M .t /kC0 .R3 /  cM:

(172)

Combining identities (154) with inequalities (171) and (172) gives desired estimate (169). It remains to prove (170). Choose an arbitrary g 2 L1 .0; T /. It follows from (74) that ";M .t / ! M .t / star weakly in L1 .R3 / for almost every t 2 .0; T /. Moreover, ";M .t / and M .t / are supported in the ball BR . Therefore, the functions Kij ";M .t / are uniformly bounded and Kij ";M .t / ! Kij M .t / in C0 .R3 /: It follows that g Kij ";M ! g Kij M in L1 .0; T I C0 .R3 //:

(173)

On the other hand, relation (78) in Proposition 7 implies that ";ij ! ij in 3 L1 w .0; T I M.R //. It follows from this and (173) that

Z

Z g .0;T /

R3

Kij ";M

 Z d ";ij .t / dt !

Z g .0;T /

R3

 Kij M d ij .t / dt; (174)

which yields Z

Z gC";M dt !

.0;T /

gCM .t / dt:

(175)

.0;T /

In other words, the sequence C";M converges to CM star weakly in L1 .0; T / as " ! 0. Lemma 14. Under the above assumptions,

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z

Z

T

D" dt ! 0

Z

T

0

Z

T

D dt;

T

V" ! 0

1643

V:

(176)

0

G" ! G as " ! 0:

(177)

Proof. First observe that

" .t / ! .t / weakly in L3=2 .R3 /;

" v" .t / ! v.t / weakly in L6=5 .R3 /

(178)

for a.e. t 2 .0; T / as " ! 0. Moreover, these functions are supported in the ball BR . It follows that

" .T  / ! .T  / weakly in L3=2 .R3 /;

" v" .T  / ! v.T  / weakly in L6=5 .R3 /

(179)

for a.e. T  , which yields Qj " .t / ! Qj .t / in C .R3 / for a.e. t 2 .0; T /; Qj " .T  / ! Qj .T  / in C .R3 /:

(180)

Next, notice that kQj " kL1 .0;T IC .R3 //  c:

(181)

Qj " ! Qj in Lp .0; T I C .R3 //

(182)

Hence

for every p 2 Œ1; 1/. On the other hand, Proposition 7 implies that ";j ! ij star 3 weakly in L1 w .0; T I M.R /. From this and (182) it follows that Z

Z

T

T

D" dt !

D dt as " ! 0:

0

(183)

0

Relation (62) implies that S.v" /ij ! S.v/ weakly in L2 .0; T I L2 .R3 /. Recall that these functions are supported in cylinder BR  .0; T /. This result along with relation (182) yields Z

Z

T

V" dt ! 0

T

V dt as " ! 0: 0

(184)

1644

P.I. Plotnikov and W. Weigant

Finally noting that (179) and (180) imply the relation G" ! G. This completes the proof of the lemma. It remains to note that Theorem 4 obviously follows from Lemmas 12, 13, and 14.

6.2

Step 2: Positivity – Properties of W",M

The properties of the functions W";M and the functions C";M are described by the following: Proposition 9. The inequalities C";M .t /  0; k. " .t /  ";M .t //5=4 kW 1;2 .R3 /

W";M  0; p  c W";M

(185)

hold for a.e. t 2 .0; T / and M > 0. Here the constant c is independent of " and M . Proposition 9 along with integral identity (158) leads to Corollary 4. Under the assumptions of Theorem 4 the function CM satisfies the inequalities Z

T

CM  0;

CM .t / dt  c;

(186)

0

where c is independent of M . Proof of Proposition 9. The proof is based on the following lemma Lemma 15. The inequalities C";M .t /  0;

W";M  0; (187)  Z . "  ";M /3=2 d ˘ . "  ";M /d ˘ d !d

Z Z Z W";M .t /  c

R

S2

˘.!; /

˘.!; /

(188) hold true for a.e. t 2 .0; T /. Here a positive constant c is independent of " and M . Proof. Recall that " 2 L1 .0; T I L6 .R3 //. It follows from this and Proposition 3 that

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z Z

Z C";M D

1645

R3

Kij ";M E";ij dx D c3

S2

R

!i !j .R ";M /.RE";ij /d d !

(189)

and Z

Z Z

R3

Kij . "  ";M /E";ij dx D c3

S2

!i !j R. "  ";M /.RE";ij /d d !:

R

(190)

On the other hand, Proposition 2 implies Z R3

Z Z c3

S2

  Kij . "  ";M /v";i . "  ";M /v";j / dx D (191)

   !i !j R . "  ";M /v";i R . "  ";M /v";j d d !: 

R

Combining (190) and (191), (155) gives the representation Z Z W";M D c3 Z Z  S2

R

S2

R

!i !j .R. "  ";M /.RE";ij /d d ! (192)

    !i !j R . "  ";M /v";i R . "  ";M /v";j d d !:

Expression (58) for E";ij implies Z RE";ij .!; ; t / D

Z

" v";i v";j d ˘ C

p" ıij d ˘ :

˘.!; /

˘.!; /

Notice that Z R ";M D

";M d ˘ ; ˘.!; /

  R . "  ";M /v";j D

Z

. "  ";M /v";j d ˘ : ˘.!; /

It follows from this and (190) and (192) that Z Z Z C";M D c3

W";M D c3

S2

R





" .!  v" /2 C p" d ˘

˘.!; /

S2

";M d ˘

d d !;

˘.!; /

(193)

Z Z nZ R



Z

˘.!; /

Z . "  ";M / d ˘ ˘.!; /

" .!  v" /2 d ˘

1646

P.I. Plotnikov and W. Weigant



Z

. "  ";M /.!  v" / d ˘ ˘.!; /

Z Z Z S2

d !d 

Z . "  ";M / d ˘

C R

2 o

p" d ˘

˘.!; /

d !d :

(194)

˘.!; /

Since p"  0, inequality (193) yields inequality (187) for C";M . Next, the Cauchy inequality implies Z

Z

. "  ";M /.!  v" /2 d ˘

. "  ";M / d ˘ ˘.!; /

˘.!; /

2

Z . "  ";M /.!  v" / d ˘



:

˘.!; /

It follows from this and (194) that Z Z Z W";M  c3

R



Z . "  ";M / d ˘

S2

˘.!; /

p" d ˘

d !d  0;

˘.!; /

which yields estimate (187) for W";M . It remains to prove estimate (188). It suffices to note that (59) the artificial pressure satisfies the inequalities p"  c&%3=2  "3=2  c. "  ";M /3=2 which leads to desired estimate (188) for W";M . Let us turn to the proof of Proposition 9. The Hölder inequality implies Z

. "  ";M /3=2 d ˘

˘.!; /

Z . "  ";M /d ˘  ˘.!; /

Z

5=4

. "  ";M /

2 d˘

2  D R . "  ";M /5=4 :

˘.!; /

It follows from this and (188) that Z Z R

S2

2  R . "  ";M /5=4 d !d  cW";M .t /:

Since " .t / 2 L6 .R3 / for a.e. t 2 .0; T /, the function . " .t / ";M .t //5=4 belongs to the class L6=5 .R3 /. Moreover this function is supported in BR . Hence this function meets all requirements of Proposition 2 which yields the identity

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z  R3

1647

  . " .t /  ";M .t //5=4 1 " .t /  ";M .t //5=4 dx D Z Z c3

R

S2

2  R . "  ";M /5=4 d !d  cW";M .t /:

It remains to note that k. " .t /  ";M .t //5=4 kW 1;2 .R3 / 

Z  R3

  . " .t /  ";M .t //5=4 1 " .t /  ";M .t //5=4 dx;

and the proposition follows. Proof of Corollary 4. Relation (170) in Lemma 13 and inequality (185) implies that CM is nonnegative. Since W";M  0, integral identity (158) and inequality (159) yield the desired estimate Z

Z

T

0

Z

T

CM .t / dt 

AM .t / dt C 2 0

Z

T

BM .t / dt C 0

T

.V .t /  D.t // dt C G 0

Z cC

T

.V .t /  D.t // dt C G  c: 0

6.3

Step 3: Positivity – Representation of CM

This section is devoted to the proof of the important technical result on the representation of the potential Kij ? ij and the function CM . In order to formulate it introduce the auxiliary notation. In view of (26) the kernel Kij admits the representation Kij .x/ D .8jxj/3 aij .x/;

aij .x/ D jxj2 ıij  xi xj :

(195)

For a.e. .x; t / 2 R3  Œ0; T  define a Borel measure x;t by the relation Z R3

u.y/d x;t D

1 8

Z R3

u.y/aij .x  y/d ij .y; t / for all u 2 C0 .R3 /:

(196)

Since ij is supported in the ball BR , the measure x;t belongs to the class M.R3 / and is supported in BR . Proposition 10. There is a nonnegative function ˚ W R3  .0; T / ! R with the following properties. The function ˚ belongs to the class L1 .0; T I Lr .BN // for every r 2 Œ1; 3/ and N > 0. For a.e. t, the function ˚.t / is continuous in R3 n B2R and admits the estimate

1648

P.I. Plotnikov and W. Weigant

j˚.x; t /j  cjxj1 for jxj  2R;

(197)

where c is independent of t . The equality Z

T 0

Z R3

Z

T

Z

.Kij u.x; t // d ij .t / dt D

˚.x; t /u.x; t /dxdt 0

(198)

R3

holds for all u 2 L1 .0; T I Lq .R//, q 2 .3=2; 1/, supported in BR  Œ0; T . There is a constant c < 1 such that Z

T

Z ˚.x; t / .x; t /dxdt  c:

0

(199)

R3

The next proposition gives the expression for ˚ in terms of the measure x;t . Proposition 11. The measure x;t is nonnegative and Z ˚.x; t / D R3

jx  yj3 d x;t

(200)

for a.e. .x; t / 2 R3  .0; T /. The rest of the section is devoted to the proof of Propositions 10 and 11. Proof of Proposition 10. Introduce the functions a0;ij .x/ D maxf0; aij .x/g;

aij D a0;ij  a1;ij : (201) Denote by ij D 0;ij  1;ij the Jordan decomposition of the measure ij . The measures ˛;ij are nonnegative and are supported in the ball BR . Set K˛;ij .x/ D

a1;ij .x/ D  minf0; aij .x/g;

1 jxj3 a˛;ij .x/; 8

Z K˛;ij u.x/ D

R3

K˛;ij .x  y/ u.y/ dy:

(202)

K˛;ij u d ˇ;ij :

(203)

It is easy to see that Z R3

Kij u d ij D

Z X .1/˛Cˇ ˛;ˇ

R3

Choose an arbitrary N > R and a function u 2 L1 .0; T I Lq .R3 //, q 2 .3=2; 1/, compactly supported in BN Œ0; T . Since K˛;ij .x/  cjxj1 , the function K˛;ij u.t / is continuous and kK˛;ij u.t /kC0 .R3 /  c.N /ku.t /kLq .BN / :

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1649

Notice that the function Z ˚˛;ˇ .x; t / D K˛;ij ˇ;ij .x; t / D

R3

K˛;ij .x  y/ d ˇ;ij .y; t /

(204)

is locally integrable. Choose an arbitrary function u 2 L1 .0; T I C .R3 // which is supported in the cylinder BN  .0; T /. The estimates k0;ij .t /kM.R3 / C k1;ij .t /kM.R3 / D 0;ij .t /.R3 / C 1;ij .t /.R3 / D jij .t /j.R3 / D kij .t /kM.R3 /  c and the identities Z Z K˛;ij u d ˇ;ij .t /D R3

R6

Z K˛;ij .xy/u.y/ d ˇ;ij .t / dyD

R3

u.x; t / ˚˛;ˇ .x; t /dx

imply the inequalities ˇZ ˇ ˇ

T

Z R3

0

ˇ ˇ u.x; t / ˚˛;ˇ .x; t /dxdt ˇ  c.N /kukL1 .0;T ILq .BN // kˇ;ij kL1 3 w .0;T IM.R //  c.N /kukL1 .0;T ILq .BN // :

Since u 2 L1 .0; T I Cc .BN // is an arbitrary function, it follows from this that k˚˛;ˇ kL1 .0;T ILr .BN //  c.N / for r D q=.q  1/ 2 Œ1; 3/:

(205)

Obviously ˚˛;ˇ is nonnegative. It is clear that the function X

˚D

.1/˛Cˇ ˚˛;ˇ

(206)

˛;ˇD0;1

satisfies equality (198) and admits the estimate k˚ kL1 .0;T ILr .BN //  c.N / for r D q=.q  1/ 2 Œ1; 3/:

(207)

It follows that ˚ belongs to the class L1 .0; T I Lr .BN // for every r 2 Œ1; 3/ and N > 0. Let us prove that ˚ is nonnegative. Choose an arbitrary N > 0 and a nonnegative function u 2 L1 .0; T I Cc .R3 // supported in BN  .0; T /. Recall the identity Z

T 0

Z

1 Kij u d ";ij .t /dt D 8 R3

Z

T 0

Z R3

jx  yj3 aij .x  y/E";ij .y; t / dxdt; (208)

1650

P.I. Plotnikov and W. Weigant

where E";ij is given by (58). It follows from (58) and expression (195) for aij that   aij .xy/E";ij .y; t / D " .jxyj2 jv" j2 ..xy/v" /2 Cp" jxyj2  0:

(209)

Combining this equality with (208) gives Z

T

Z R3

0

Kij u d ";ij .t /  0:

(210)

Next, relation (78) yields Z

T

Z

Z

T

Z

u.x; t / ˚.x; t /dxdt D 0

R3

Z

T

Z

D lim

"!0 0

R3

0

R3

Kij u d ij .t / dt

Kij u d ";ij .t /dt  0:

Hence ˚ is nonnegative in BN  .0; T / for all N > 0. It remains to prove estimate (199). Since M 2 L1 .0; T I L1 .R3 // is supported in BR  Œ0; T , relation (198) implies Z

T

Z

Z R3

0

T

Z

˚ M dxdt D

Z

R3

0

Kij M d ij .t / dt D

T

CM .t /dt:

(211)

0

It follows from this and estimate (186) in Corollary 4 that Z

T 0

Z R3

˚ M dxdt  c;

where c is independent of M . It remains to note that desired estimate (199) obviously follows from the relation M % as M ! 1. Proof of Proposition 11. First prove that the measure x;t is nonnegative. Choose an arbitrary N > 0 and a nonnegative function u 2 L1 .0; T I Cc .R3 // supported in BN  .0; T /. It follows from the definition of the measure x;t that Z

T 0

Z

Z R3

T

Z

u.y; t / d x;t .y; t / dt D lim

"!0 0

u.y; t /aij .x  y/ d ";ij .y; t /:

R3

(212)

Next, inequality (209) yields Z

T 0

Z

Z R3

u.y; t /aij .xy/ d ";ij .y; t /dt D 0

T

Z R3

uaij .xy/E";ij .y; t / dxdt  0:

3 Recall that ";ij ! ij star weakly in L1 w .0; T I M.R //. It follows that we have

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z

1651

TZ

0

R3

u.y; t /aij .x  y/ d ij .y; t /dt Z

T

Z

D lim

"!0 0

R3

u.y; t /aij .x  y/ d ";ij .y; t /dt  0:

Combining this result with (212) gives the inequality Z

T

Z R3

0

u.y; t / d x;t .y; t / dt  0:

Hence x;t is nonnegative. Next, introduce the functions ˚  Ph .x/ D min jxj3 ; h3 ;

h > 0;

and set .h/

Z

˚˛;ˇ D

R3

Ph .x  y/a˛;ij .x  y/ d ˇ;ij .y; t /; X

˚ .h/ D

.h/

.1/˛Cˇ ˚˛;ˇ :

˛;ˇD0;1

The definition of the measure x;t implies the representation ˚ .h/ .x; t / D

Z R3

Ph .x  y/ d x;t .y/:

(213)

It follows from the Fatou theorem that .h/

0  ˚˛;ˇ  ˚˛;ˇ ;

.h/

˚˛;ˇ % ˚˛;ˇ as h & 0;

and ˚ .h/ .x; t / ! ˚.x; t / as h ! 0 for all .x; t / 2 R3  .0; T /:

(214)

Notice that Ph .x/ % jxj3 as h & 0 for x ¤ 0. Letting h ! 0 in (213) and applying relation (214) and the Fatou theorem we arrive at the desired representation (200).

6.4

Step 4: The Second Integral Identity

This section is devoted to the proof of the integral identity, which is complementary to integral identity (158) given by Theorem 4. The result is given by the following

1652

P.I. Plotnikov and W. Weigant

Theorem 5. The integral identity Z

T

Z

Z

Z

T

˚.x; t / .x; t /dt D R3

0

T

A dt C

.V .t /  D.t // dt C G

0

(215)

0

holds for a.e. T  2 .0; T /. Here ˚ is given by Proposition 10, the function A is given by Z A.t/ D R3

Kij . vi / . vj / dx;

(216)

the functions D, V , and a constant G are given by (156) and (157). The rest of the section is devoted to the proof of this theorem. The proof falls into a sequence of lemmas. The first lemma constitutes the weak formulation of the limiting equations for and v. Lemma 16. The integral identity Z

Z

T 0

Z

R3 T

. v@t   & S.v/ W r/ dxdt C

T

Z R3

0

Z

C 0

Z

R3

Z

i d i .t /dt D Z

T

Z R3

0

Z

. @t



v.T /  .T /dx 

R3

R3

C v  r

(217)

Z

R3

.T  / .T  /dx C





@xj i d ij .t /dt

0 v0  .0/dx;

C %r&  u/dxdt

(218)

Z R3

0 .0/dx D 0

2 C 1 .R3  .0; T /, and for a.e. holds for all  2 C 1 .R3  .0; T //, for all  T 2 Œ0; T . Here the measures ij ; j are given by (77). Proof. It follows from relations (62) and (63) that v" ! v weakly in L2 .0; T I W 1;2 .R3 //;

" .t / ! .t / weakly in L3=2 .R3 /;

" .t /v" .t / ! .t /v.t / weakly in L6=5 .R3 / as " ! 0 for a.e. t 2 .0; T /. Since the sequence " is bounded in L1 .0; T I L3=2 .R3 // and " v" is bounded in L1 .0; T I L6=5 .R3 //, it follows that

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1653

3=2

" ! star weakly in L1 .R3 //; w .0; T I L 6=5 .R3 //;

" v" ! v star weakly in L1 w .0; T I L

(219)

S.v" /ij ! S.v/ij weakly in L2 .0; T I L2 .R3 //;

" v" .T  / ! v.T  / weakly in L6=5 .R3 / as " ! 0:

(220)

Now Proposition 7 implies that ";i ! i star weakly in L2 .0; T I M.R3 //;

";ij ! ij ;

(221)

where the measures ";ij and ";i are defined by the relations d ";ij D E";ij dx;

d ";j D F";j dx:

Letting " ! 0 in integral identities (59) and (93) and using (219), (220), and (221) gives desired integral identity (217). The next technical lemma concerns the properties of the mollifiers. Let # 2 C01 .R3 / be a standard mollifying kernel satisfying the conditions Z #  0;

#.x/ D #.x/;

#.x/ dx D 1:

(222)

R3

The mollifier Sh , h > 0, is defined by the equality 3

Sh u.x/ D h

Z

#.h1 .x  y//f .y/ dy for all f 2 L1 .R3 /:

(223)

R3

For every locally integrable function f denote by f h the special mollifier f h .x/ D

3 4h3

Z f .y/ dy:

(224)

jxyjh

Now consider a sequence of mollifying kernels #n , n  1, with the following properties. The kernel #n satisfies conditions (222), it is supported in the ball fjxj  1 C 1=ng and #n .x/ & 3.4/1 for jxj  1;

#n .x/ & 0 for jxj > 1:

(225)

Denote by Sn;h the mollifier given by expression (223) with # replaced by #n . Lemma 17. The relation kSn;h f  f h kC .R3 / ! 0 as n ! 1:

(226)

1654

P.I. Plotnikov and W. Weigant

holds true for every f 2 L1 .R3 / and h > 0. Proof. Since #n ! 3.4/1 uniformly in B1 as n ! 1, the desired relation obviously follows from the inequalities ˇZ ˇ jSn;h f  f h j  h3 ˇ ˇZ 3 ˇ Ch ˇ

ˇ  ˇ #n .h1 .x  y/  3=.4/ f .y/ dy ˇ



jxyjh

hjxyjhC1=n

ˇ ˇ #n .h1 .x  y/f .y/ dy ˇ Z

 h3 sup j#n .x/  3.4/1 j

jf .y/j dy jxyjh

jxj1 3

Z

Cch

jf .y/j dy ! 0 as n ! 1: hjxyjhC1=n

Now the task is to derive the approximation of integral identity (217). The result is given by the following lemma. Lemma 18. The integral identity Z

T

Z R3

0

Z

T 0

Z

Z

  Kij Πh d ij  Z R3

T

Z

0

R3

Z

Kij S .v/ij / h dxdt 

Qj h .T  / .T  /vj .T  / dx 

C



 Kij Πvi h vj dxdt D

R3

Z

T

0

R3

Z R3

Qj h d j

Qj 0 h 0 v0;j dx

(227)

holds true for a.e. T  2 .0; T /. Proof. The proof imitates the proof of identity (98) in Proposition 8. Let mollifier Sh with a kernel  be defined Choose an arbitrary  2 C01 .0; T /. Substituting D h3 .t / #.h1 .x0  x// into integral identity (59) gives the equality Z 0

T

0 .t /Sh .x0 ; t / dt 

Z

T

.t / div Sh . v/.x0 ; t / dt D 0; 0

which is equivalent to the differential equation @t Sh .x; t / D  div Sh . v/.x; t / in R3  .0; T /:

(228)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1655

It follows that Sh " and @t Sh " are infinitely differentiable with respect to x. Introduce the vector fields  i D Qi Sh ;

i D 1; 2; 3;

where the integral operator Qi is given by (29). The identity @xj Qi D Kij implies @xi j D Kij Sh : In its turn, relation (228) yields @t i D Kij Sh . vj /;

i D 1; 2; 3:

Substituting  j into (93) gives the integral identity Z

Z

T

Z

R3

0

Z

T

R3

Kij .Sh . vi // vj dxdt D

R3

0

Z

Z

R3

Z

Z

Kij .Sh / d ij .t /dt 

0

C

T

Z

T

Kij S .v/ij / Sh . / dxdt  0

Qj Sh . /.T  / .T  /vj .T  / dx 

R3

Z

Qj Sh . / d j dt

Qj Sh . 0 / 0 v0;j dx:

R3

Next, replace Sh by Sn;h to obtain Z

Z

T

Z R3

0

Z

T

Z C R3

Z

Kij .Sn;h / d ij .t /dt 

R3

0

Z R3

0

T

Z

Kij .Sn;h . vi // vj dxdt D T

Z

Kij S .v/ij / Sn;h . / dxdt 

Qj Sn;h . /.T  / .T  /vj .T  / dx 

R3

0

Z

R3

Qj Sn;h . /d j dt

Qj Sn;h . 0 / 0 v0;j dx:

(229)

In view of Lemma 17 the sequences Sn;h . v/ and Sn;h . / are bounded in 1 3 3 //. Moreover, Sn;h . v/.t / and Sn;h . /.t / converge in C .R / to L .0; T I C .R

v h .t / and h .t / as n ! 1. It follows that Z

T

Z

0

Z 0

T

Z

Z

R3

R3

T

Z

Kij . vi /Sn;h . vj / dxdt 

R3

0

Z

T

Kij .Sij .v//.t / Sn;h .t / dxdt ! 0

Kij .Sn;h .t // d ij .t /dt C Z R3



Kij . vi / vj h dxdt 

1656

P.I. Plotnikov and W. Weigant

Z

T

Z R3

0

Z

  Kij Πh .t / d ij .t /dt C

T

Z R3

0

Kij .Sij .v//.t / h .t / dxdt (230)

and Z

T

Z

0

Z

T

Z

Z R3

R3

0

Qj Sh . /d j dt  Qj h d j dt 

Z

R3

R3

Qj Sn;h . /.T  / .T  /vj .T  / dx !

Qj h .T  / .T  /vj .T  / dx

(231)

as n ! 1. Letting n ! 1 in (229) leads to desired identity (227). Now the task is to pass to the limit as h ! 0 in integral identity (227). The first result in this direction is given by the following lemma, which is the key point in the proof of Theorem 5. Lemma 19. Let ˚ be given by Proposition 10. Then the equality Z lim

h!0 0

T

Kij h d ij .t / dt D

Z

T

˚.x; t / .x; t / dxdt;

(232)

0

holds true for a.e. T  2 .0; T /. Proof. Notice that Kij u D @2xi xj 2 u;

(233)

for every integrable compactly supported function u.x/. Here the inverse to the biharmonic operator 2 is defined by 1  uD 8 2

Z jx  yju.y/ dy:

(234)

R3

Definition (224) of the mollifier  h implies the identity

h D h ; h .x/ D

3 for jxj  h; 4h3

h .x/ D 0 for jxj > h:

(235)

It follows that Kij h .x; t / D

Z R3

Nh;j i .x  y/ .y; t/ dy;

Nh .x/ D @2xi xj 'h .x/

(236)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1657

where 'h is a unique solution to the equation 2 'h D h in R3 ; 'h .x/ 

1 jxj ! 0 as jxj ! 1: 8

(237)

Calculations show that 'h .x/ D

1 2 3h 1 C r  r 4 for r  h; 32 16h 160h3

'h .x/ D

1 h2 rC for r  h; where r D jxj: 8 40r

which yields the representation  Nh;ij D

 1 1 1 3 aij .x/ C xi xj for r  h;   8hr 2 40h3 8hr 2 40h3  1 h2 h2 a .x/ D  .x/ C xi xj for r  h: Nh;jj ij 8r 3 40r 5 20r 5 (238)

Next, split the kernel Nh;ij into two parts Nh;ij .x/ D Uh;ij .x/ C Th;ij .x/;

(239)

where Uh;ij D

1 aj i .x/ for r  h; 8r 3

1 aj i .x/ for r  h; 8h3 1 Th;ij .x/ D Nh;ij .x/  for r  h; 8h3 1 aij .x/ for r  h: Th;jj .x/ D Nh;ij .x/  8r 3 Uh;ij D

(240)

(241)

For every h > 0 the kernels Uh;ij , Tij are continuous and admit the estimates jTh;ij .x/j  jUh;ij .x/j 

c for jxj  h; h

cjxj2 for jxj  h; h3

This leads to the representation

jTh;ij .x/j 

ch2 for jxj  h; jxj3

jUh;ij .x/j 

c for jxj  h: jxj

(242) (243)

1658

P.I. Plotnikov and W. Weigant

Z

T 0

Kij h d ij .t / dt D Z

Z

T

Z

Z

T

R3

0

0

R3

Z

Z R3

R3

 Uh;ij .x  y/ d ij .t / dxdt C 

Th;ij .x  y/ .y; t /dy

d ij .t /dt: (244)

Expression (196) for the measure x;t and formula (240) for Uh;ij implies the equality Z R3

Z

Uh;ij .x  y/ d ij .t / ˚ .h/ D

R3

Ph .x  y/d x;t ;

where Ph .x/ D minfjxj3 ; h3 g. It follows that Z

T

Z

Z

R3

0

R3

 Z Uh;ij .x  y/ d ij .t / dxdt D

T

Z

.x; t /˚ .h/ .x; t / dxdt:

R3

0

Notice that and ˚ .h/ are nonnegative. Moreover, we have ˚ .h/ % ˚ as h & 0 a.e. in R3  .0; T /. It also follows from (199) that ˚ belongs to L1 .R3  .0; T //. It follows from this and the Fatou theorem that Z

T

h!0 0

Z

Z

lim

R3

R3

 Z Uh;ij .x  y/ d ij .t / dxdt D

T

Z

.x; t /˚.x; t / dxdt:

0

R3

(245) Next, introduce the mappings Jh;ij .t / defined by ˇZ ˇ Jh;ij .t / D sup ˇ jxjR

R3

ˇ ˇ Th;ij .x  y/ .y; t /dy ˇ:

(246)

Estimate (242) implies the inequality .1/

.2/

.3/

Jh;ij .t /  Jh .t / C Jh .t / C Jh .t /; where Z p .1/ Jh .t / D c h sup jxjR

.2/

Jh .t / D ch2 sup

jxjR

Z p

p jxyj h

.y; t / dy;

jx  yj3 .y; t / dy;

hjxyj h

.3/

Jh .t / D ch1 sup

jxjR

Z

.y; t / dy: jxyjh

(247)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1659

Since 2 L1 .0; T I L3=2 .R3 // is supported in BR  .0; T /, the first term in the right-hand side of (247) admits the estimate p .1/ Jh .t /  c h:

(248)

Next, the Hölder inequality implies .2/

Z

ch2

p hjxyj h

2

jx  yj9 dy

Z

1=3 sup

p jxyj h

jxjR

!1=3

p h

Z

 ch

r

7

Z

dr

sup

h

p jxyj h

jxjR

Z  c sup

p jxyj h

jxjR

Jh .t /  2=3

.y; t /3=2 dy

.y; t /3=2 dy

.y; t /3=2 dy

2=3

(249)

2=3

and .3/

Jh .t /  c

1 h

Z

Z

1=3 dy

.y; t /3=2 dy

sup

jxyjh

2=3

jxyjh

jxjR

Z

.y; t /3=2 dy

 c sup

(250)

2=3 :

jxyjh

jxjR

Substituting (248), (249), and (250) into (247) gives the estimate Z p Jh;ij .t /  c h C c sup jxjR

p jxyj h

.y; t /3=2 dy

2=3  c:

(251)

Since 2 L1 .0; T I L3=2 .R3 //, it follows that Jh;ij .t / ! 0 as h ! 0 for a.e. t 2 .0; T /:

(252)

On the other hand, expression (246) for Jh;ij and (251)–(252) imply the inequalities ˇZ ˇ ˇ

0

Z c 0

T

Z

Z

R3

 R3

Z

T

Jh;ij .t / kij .t /kM.R3 / dt  c

Th;ij .x  y/ .y; t /dy

ˇ ˇ d ij .t /ˇ 

T

Jh;ij .t /dt  c.h/ ! 0 as h ! 0: 0

(253)

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P.I. Plotnikov and W. Weigant

Letting h ! 0 in decomposition (244) leads to desired equality (232). In order to complete the proof of Theorem 5, notice that vi 2 L1 .0; T I L6=5 .R3 //, 1 6 3 which yields i / 2 L .0; T I L .R //. On the other hand, the the

inclusion Kij . v 1 6=5 3 sequence vj h is bounded in L .0; T I L .R // and

.t /vj .t / h ! .t /vj .t / in L6=5 .R3 / as h ! 0

for a.e. t 2 .0; T /. Hence

vj

h

! vj in Lm .0; T I L6=5 .R3 // as h ! 0

for every m 2 Œ1; 1/. It follows from this that Z

T

Z R3

0

Kij . vi / vj h dxdt !

Z

T

Z R3

0

Z

Kij . vi / vj dxdt (254)

T

A.t/ dt as h ! 0:

D 0

2 3 Next, notice ij Sij belongs to the class L .0; T I C .R //. The that the function K 1 3=2 3 sequence h is bounded in L .0; T I L .R //. It is supported in the cylinder BR  .0; T / and



.t / h ! .t / in L3=2 .R3 / as h ! 0: Hence

h ! in Lm .0; T I L3=2 .R3 // as h ! 0 for every m 2 Œ1; 1/. It follows from this that Z 0

T

Z R3

Kij .Sij / h dxdt !

Z

T

Z R3

0

Z

Kij .Sij / dxdt

T

D

V .t / dt as h ! 0: 0

Obviously, the sequence Qj h is bounded in L1 .0; T I C .R3 // and

Qj .t / h ! Qj .t / in C .R3 / as h ! 0 for a.e. t 2 .0; T /, which leads to the relations

(255)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z

T

Z R3

0

Qj h d j .t /dt ! Z

Z

T

1661

Z R3

0

Qj d j dt (256)

T

D.t / dt as h ! 0:

D 0

Finally notice that Z R3

Z

Qj h .T  / vj .T  / dxdt !

T

Z R3

0

Qj .T  / vj .T  / dxdt as h ! 0: (257)

It remains to note that desired identity (215) obviously follows from integral identity (227) and relations (232), (254), (255), (256), and (257).

6.5

Step 5: Monotonicity – The Function W",M

This section is the heart of the work. Here integral identities (191) and (215) given by the equation 236 and Theorem 5 are employed in order to investigate the limits of the function W";M . The following proposition plays a crucial role in the proof of the main Theorem 3. Proposition 12. Let W";M be given by (155). Then the relation Z

T

W";M .t / dt D 0

lim sup lim

M !1 "!0

(258)

0

holds for a.e. T  2 .0; T /. The rest of the section is devoted to the proof of this fact. Proof of Proposition 12. Notice that identity (158) in Theorem 4 and identity (215) in Theorem 5 read Z

Z

T

CM .t / dt C lim

"!0 0

0

Z

Z

T

T

W";M .t / dt D

.AM C 2BM / dt 0

T

C

.V .t /  D.t // dt C G; 0

Z

T

Z

Z ˚.x; t / .x; t /dt D

0

R3

It follows that

Z

T

A dt C 0

T

.V .t /  D.t // dt C G: 0

1662

P.I. Plotnikov and W. Weigant

Z

Z

T

lim

"!0 0

Z

T

W";M .t / dt C

T

Z

CM .t / dt 

˚.x; t / .x; t /dt

0

R3

0

Z

Z

T

.AM C 2BM / 

D 0

(259)

T

A dt 0

Now the task is to pass to the limit in this equality as M ! 1. Notice that the function M .t / is bounded by a constant M . Expression (154) for CM and relation (198) with u replaced by M imply Z

Z

T

T

Z

CM .t / dt D 0

0

R3

M ˚ dxdt;

where ˚ is defined by Proposition 10. Since M % as M ! 1, the Fatou theorem implies the relation Z

Z

T

CM .t / dt D 0

0

T

Z R3

Z

M ˚ dxdt !

T

Z

˚ dxdt

0

(260)

R3

as M ! 1. Notice that the integral in the right-hand side of this relation converges in view of Proposition 10. Next, recall that v belongs to the class L1 .0; T I L6=5 .R3 //. Since M % as M ! 1, it follows that .  M /6=5 jvj6=5  6=5 jvj6=5 2 L1 .0; T I L1 .R3 //; .  M /6=5 jvj6=5 & 0 a.e. in R3  .0; T / as M % 1:

(261)

It follows from this and the Fatou theorem that k.  M /6=5 jvj6=5 kLr .0;T IL1 .R3 // ! 0 as M ! 1 for every r < 1. This leads to the relation k v  M vkL2 .0;T IL6=5 .R3 // ! 0 as M ! 1:

(262)

Recall that the operator Kij W L6=5 .R3 / ! L6 .R3 / is continuous, which yields kKij . vi  M vi /kL2 .0;T IL6 .R3 // ! 0 as M ! 1: It follows that

(263)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

Z

Z

T

T

AM .t / dt D 0

Z

T

Z

! 0

R3

Z R3

0

1663

Kij . M .t /vi .t //. M .t /vj .t // dxdt Z

(264)

T

Kij . .t /vi .t //. .t /vj .t // dx D

A.t/ dt 0

as M ! 1. Finally notice that expression (153) for BM yields the estimate Z

T

jBM .t /j dt  k M vkL2 .0;T IL6=5 .R3 // kKij . vi  M vi /kL2 .0;T IL6 .R3 // ; 0

which along with (262)–(263) implies the relation Z

T

(265)

BM .t / dt ! 0 as M ! 1: 0

It remains to note that identity (259) and relations (260), (264), and (265) lead to desired equality (258).

6.6

Step 6: Proof of the Main Theorem 3

In order to prove Theorem 3, it suffices to show that Z

T

Z

Z R3

0

T

Z

h " v";i v";j dxdt !

R3

0

h vi vj dxdt as " ! 0

(266)

for every h 2 L1 .R3  .0; T // and for a.e. T  2 .0; T /. The proof of (266) falls into a sequence of lemmas. The first two lemmas give the estimate of the kinetic energy density in terms of the function W";M . Lemma 20. There is a constant c independent of "; M , and T  such that the inequality Z 0

T

Z

2

R3

Z

. "  ";M /juj dxdt  c

!2=5

T

W";M dt

kukL1 .0;T IW 1;2 .R3 //

0

(267) holds for every u 2 L1 .0; T I W 1;2 .R3 // supported in BR  .0; T / and for every T  2 .0; T . Proof. It follows from the Hölder inequality and the embedding W 1;2 .R3 / ,! L6 .R3 / that

1664

P.I. Plotnikov and W. Weigant

Z Z R3

. "  ";M /

6=5

Z c Z c R3

Z c R3

. "  ";M /juj2 dx 

R3

R3

juj

6=5

. "  ";M /

5=6 Z

6=5

juj

dx

juj dx 5=6

6=5

dx

f. "  ";M /5=4 jujg24=25 juj6=25 dx 4=5 Z

. "  ";M /5=4 juj dx

juj6 dx 5=4

c R3

. "  ";M /

R3

kukW 1;2 .R3 / D 5=6 kukW 1;2 .R3 / 

1=30

R3

Z

1=6

6

kukW 1;2 .R3 /  4=5

juj dx

6=5

kukW 1;2 .R3 / 

4=5

ck. "  ";M /5=4 kW 1;2 .R3 / kuk2W 1;2 .R3 / : This result and estimate (185) in Proposition 9 imply the estimate Z

T

R3

0

Z

!2=5 Z

T

c

k. "  0

";M /5=4 k2W 1;2 .R3 / Z

Z

. "  ";M /juj2 dxdt  T

dt 0

!3=5 10=3 kukW 1;2 .R3 /

dt



!2=5

T

W";M .t / dt 0

kuk2L1 .0;T IW 1;2 .R3 // ;

and the lemma follows. Lemma 21. There is a constant c independent of "; M , and T  such that the inequality Z 0

T

Z

2

R3

Z

. "  ";M /jv" j dxdt  c

!4=15

T

W";M dt

(268)

0

holds for every T  2 .0; T . Proof. Choose an arbitrary N > 0 and set u" .t / D v" when kv" kW 1;2 .R3 /  N;

u" .t / D 0 when kv" kW 1;2 .R3 /  N:

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1665

It follows that Z

T

Z R3

0

Z

T

Z R3

0

. "  ";M /ju" j2 dxdt C

. "  ";M /jv" j2 dxdt D (269)

Z

Z EN

. "  ";M /jv" j2 dxdt;

R3

where EN D ft 2 .0; T / W kv" kW 1;2 .R3 /  N g: Next, inequality (267) in Lemma 20 yields Z

T

Z R3

0

. "  ";M /ju" j2 dxdt  c

Z

T

W";M dt

2=5

N:

(270)

0

On the other hand, the Chebyshev inequality meas EN  N 2 kv" k2L2 .0;T IW 1;2 .R3 //  cN 2 implies the estimates Z

Z EN

R3

Z

Z

2

. "  ";M /jv" j dxdt 

EN

R3

" jv" j2 dxdt 

c meas EN k " jv" j2 kL1 .0;T IL1 .R3 //  c meas EN  cN 2 : Combining this result with (270) and (269) leads to the inequality Z

T

Z R3

0

2

. "  ";M /ju" j dxdt  c

Z

T

W";M dt

2=5

N C cN 2 :

0

Substituting N D

Z

T

W";M dt

2=15

0

in this inequality gives desired estimate (268). Lemma 22. The relation Z

T

Z

lim lim sup

M !1

"!0

0

R3

jhj. "  ";M / jv" j2 dxdt D 0

(271)

1666

P.I. Plotnikov and W. Weigant

holds true for every h 2 L1 .R3  .0; T //. Proof. Inequality (268) in Lemma 21 yields Z

T

Z R3

0

jhj. "  ";M /jv" j2 dxdt c

Z

T

W";M dt

4=15

:

0

Letting " ! 0, M ! 1 and recalling relation (258) in Proposition 12 we arrive at (271). Lemma 23. The equality Z

T

Z

lim

"!0

R3

0

  h ";M v";i v";j  M vi vj dxdt D 0

(272)

holds true for every h 2 L1 .R3  .0; T // and M > 0. Proof. Introduce the function  defined by .x; t / D 1 when .x; t / > 0 and  D 0 when D 0: The equality .1  / M D 0 yields the representation Z

T

R3

0

Z

Z

T

Z

R3

0

Z

T

  h ";M v";i v";j  M vi vj dxdt D   h ";M v";i v";j  M vi vj dxdt

Z

C 0

(273)

R3

h .1  / ";M v";i v";j dxdt:

On the other hand, Proposition 4 implies the relation v" ! v in L2 .0; T I L2 .R3 //. It follows that v";i v";j  vi vj ! 0 in L1 .R3  .0; T // as " ! 0: Recall that ";M ! M star weakly in L1 .R3  .0; T //, which gives Z

T 0

Z R3

  h ";M v";i v";j  M vi vj dxdt ! 0 as " ! 0:

(274)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1667

Next, notice that the functions v" and v are supported in the cylinder BR  .0; T /. In view of estimate (61) in Condition (H.2) the sequence v" is bounded in L2 .0; T I W 1;2 .R3 //. Since the embedding W 1;2 .R3 / ,! L6 .R3 / is continuous, it follows from this that kjv" j2 kL1 .0;T WL3 .R3 //  c:

(275)

Now set TN D ft 2 .0; T  / W kjv" .t /j2 kL3 .R3 /  N g: Obviously, the representation Z nZ TN

T

Z R3

0

Z

.1  / ";M jv" j2 dxdt D (276)

oZ

C

2

R3

.0;T  /nTN

.1  / ";M jv" j dxdt

holds for every N > 0. Notice that estimate (275) and the Chebyshev inequality implies meas .0; T  / n TN  cN 1 : It follows from this and energy estimate (61) that Z

Z

R3

.0;T  /nTN

.1  / ";M jv" j2 dxdt (277) 

 k " jv" j kL1 .0;T IL1 .R3 // meas .0; T / n TN  cN 2

1

:

On the other hand, the Hölder inequality implies the estimate Z

Z Z TN

TN

R3

.1  / ";M jv" j2 dxdt 

kjv" .t /j2 kL3 .R3 / k.1  / ";M .t /kL3=2 .R3 / dt  Z

(278)

T

N

k.1  / ";M .t /kL3=2 .R3 / dt: 0

Combining (276) with (278) and (277) leads to the inequality Z 0

T

Z R3

.1  / ";M jv" j2 dxdt  N

Z

T 0

k.1  / ";M .t /kL3=2 .R3 / dt C cN 1 ;

1668

P.I. Plotnikov and W. Weigant

which holds true for every N > 0. It follows that Z

T 0

Z R3

.1  / ";M jv" j2 dxdt  c

Z

T

k.1  / ";M .t /kL3=2 .R3 / dt

1=2

:

0

(279)

Since the nonnegative functions ";M converge star weakly in L1 .R3  .0; T // to

M and .1  / M D 0, we have Z

T

.1  / ";M dxdt ! 0 as " ! 0: 0

Hence .1  / ";M ! 0 in measure as " ! 0. On the other hand, the functions ";M are bounded by the constant M , which gives Z

T

k.1  / ";M .t /kL3=2 .R3 / dt ! 0 as " ! 0:

(280)

0

Letting " ! 0 in (279) and using (280) gives the relation Z

T

Z R3

0

.1  / ";M jv" j2 dxdt ! 0 as " ! 0:

(281)

It remains to note that desired equality (272) obviously follows from (273), (274), and (281). Lemma 24. The equality Z

T

Z

lim

M !1

R3

0

 h  M / jvj2 dxdt D 0

(282)

holds true for every h 2 L1 .R3  .0; T //. Proof. The energy estimate (61) implies Z

T 0

Z

h jvj2 dxdt < 1:

R3

On the other hand, M % as M % 1 in R3  .0; T /. It follows from this and the Fatou theorem that Z 0

T

Z R3

jhj M jvj2 dxdt !

Z 0

T

Z R3

jhj jvj2 dxdt:

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

1669

This completes the proof of the lemma. Lemma 25. The equality Z

T

Z

lim

"!0

R3

0

 h " v";i v";j  vi vj / dxdt D 0:

(283)

holds true for every h 2 L1 .R3  .0; T //. Proof. Notice that Z

T

0

Z 0

Z

Z



R3 T

T

Z

h " v";i v";j  vi vj / dxdt D 0

Z R3



R3

Z



  h "  ";M v";i v";j dxdt C T

h ";M v";i v";j  M vi vj dxdt C 0

Z R3

 h M  / vi vj dxdt

Letting " ! 0 and applying Lemma 23 leads to the inequality ˇZ ˇ lim sup ˇ "!0

Z

T

R3

0

Z

T

ˇ  ˇ h " v";i v";j  vi vj / dxdt ˇ  Z

lim sup "!0

0

R3

Z 0

  jhj "  ";M jv" j2 dxdt C

T

Z R3

 h  M / jvj2 dxdt:

Letting M ! 1 and applying Lemmas 22 and 24 give the desired relation (283). It remains to note that the assertion of Theorem 3 is a straightforward consequence of Lemma 25.

7

Conclusion

In recent years significant progress has been made in the theory of the compressible Navier-Stokes equations with subcritical and critical adiabatic exponents. Among the obtained results are the proof of existence of weak solutions to stationary boundary value problems for all adiabatic exponents greater than one, and the proof of the existence results for symmetric solutions to nonstationary initial-boundary value problems. The reader is referred to chapters “ Existence of Stationary Weak Solutions for Isentropic and Isothermal Compressible Flows” and “ Symmetric Solutions to the Viscous Gas Equations” for the state of art on the domain.

1670

P.I. Plotnikov and W. Weigant

At the same time, many important questions remain unsolved. The Lions problem, [9], on existence of global weak solutions to 3D nonstationary compressible Navier-Stokes equations with the adiabatic exponent   3=2 still is not solved. The main difficulty is that the pressure can be concentrated on sets of small measure. This leads to the so-called concentration problem, which is well known in the theory of the Euler and the Navier-Stokes equations. In this chapter it was shown that there is no concentrations of the kinetic energy density in the case of the critical adiabatic exponent. However, the problem on cancelation of concentrations of the pressure function remains unsolved. Rotationally symmetric flows may be considered as intermediate case between 2D and 3D flows. Therefore, it is of interest to consider such flows in more details. Finally notice that it would also be interesting to apply the method developed in this chapter to 2D problem for heat conducting flows with the classic Mendeleev Clapeyron relation p D c% between the pressure p, the density %, and the temperature  .

8

Cross-References

 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows  Existence of Stationary Weak Solutions for Isentropic and Isothermal Compress-

ible Flows  Symmetric Solutions to the Viscous Gas Equations  Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stabil-

ity, and Longtime Behavior  Weak Solutions for the Compressible Navier-Stokes Equations in the Intermediate

Regularity Class Acknowledgements This work was supported by Russian Science Foundation, project 15-1120019.

References 1. R.A. Adams, Sobolev Spaces (Academic press, New-York, 1975) 2. B. Ducomet, S. Neˇcasová, A. Vasseur, On spherically symmetric motions of a viscous compressible barotropic and selfgravitating gas. J. Math. Fluid Mech. 99, 1–24 (2009) 3. L. C. Evans, Partial Differential Equations (Am. Math. Soc., Providence, RI, 1998) 4. E. Feireisl, Dynamics of Viscous Compressible Fluids (Oxford University Press, Oxford, 2004) 5. E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3(3), 358–392 (2001) 6. S. Jiang, P. Zhang, On spherically symmetric solutions of the compressible isentropic NavierStokes equations. Commun. Math. Phys. 215, 559–581 (2001) 7. S. Jiang, P. Zhang, Axisymmetric solutions to the 3D Navier-Stokes equations for compressible isentropic fluids. J. Math. Pures Appl. 82, 949–973 (2003) 8. P.L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models (Clarendon Press, Oxford, 1998)

31 Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in. . .

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9. P.L. Lions, On some chalenging problems in nonlinear partial differential equations, in Mathematics: Frontiers and Perspectives, ed. by V. Arnold, M. Atyah, P. Lax, D. Mazur (AMS, Providence, 2000) 10. A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and Its Applications, vol. 27 (Oxford University Press, Oxford, 2004) 11. M. Padula, Existence of global solutions foe two-dimensional viscous compressible flows. J. Funct. Anal. 69(1), 1–20 (1986) 12. M. Padula, Correction. J. Funct. Anal. 76(1), 70–76 (1988) 13. P.I. Plotnikov, J. Sokolowski, Compressible Navier-Stokes Equations. Theory and Shape Optimization (Birkhauser, Basel, 2012) 14. P.I. Plotnikov, W. Weigant, Isothermal Navier-Stokes equations and radon transform. SIAM J. Math. Anal. 47, 626–652 (2015) 15. P.I. Plotnikov, W. Weigant, Rotationally Symmetric Viscous Gas Flows. Computational mathematics and mathematical physics 57, 387–400 (2017)

Weak Solutions for the Compressible Navier-Stokes Equations in the Intermediate Regularity Class

32

Misha Perepelitsa

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 On the Consistency of the Model of the Navier-Stokes Equations . . . . . . . . . . . . . 1.4 Weak Solutions in the Intermediate Regularity Class . . . . . . . . . . . . . . . . . . . . . . . 1.5 On the Regularity of the Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Conditions at the Interface of Density Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notation and Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Estimates for Solutions of Poisson’s and Lamé’s Equations . . . . . . . . . . . . . . . . . . 2.3 Bogovski’s Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Embedding Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Local Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Formulation of the Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Uniqueness and Continuous Dependence on Data . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Estimates Based on the Equation of Mass Conservation . . . . . . . . . . . . . . . . . . . . . 4.3 Compactness Properties of Approximating Solutions . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1674 1675 1675 1677 1677 1677 1679 1680 1680 1681 1688 1688 1689 1690 1691 1694 1695 1701 1706 1707 1708 1709

Abstract

In this chapter we discuss an initial-boundary value problem for the NavierStokes equations for compressible flows in bounded domains with the no-slip

M. Perepelitsa () Department of Mathematics, University of Houston, Houston, TX, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_45

1673

1674

M. Perepelitsa

boundary conditions for the velocity. We demonstrate the existence of weak solutions that belong to the intermediate regularity class, with strictly positive and uniformly bounded density and Hölder continuous velocity. The result is proved under the assumption that the initial data are close to a static equilibrium.

1

Introduction

This chapter discusses the problem of existence of solutions to the system of the Navier-Stokes equations for compressible fluid flows in the so-called intermediate regularity class. For a typical solution .; u/ in this class, density  is a bounded, discontinuous function, and velocity u is Hölder continuous in space and time. Among other properties of solutions of the intermediate regularity is the existence of uniquely defined flow trajectories and non-formation of vacuum. The interest in studying solutions of this type was initiated by works [9, 10, 28] which describe how discontinuities in the initial data are resolved in the onedimensional model of the Navier-Stokes equations and compare the weak solutions to those of the Euler equations of gas dynamics (system (1) with  D  D 0), with the same initial data. In particular, the authors consider the Riemann problem, i.e., the problem in which the initial data consist of two states . ; u / and .C ; uC /; to the left and to the right of a point x0 ; respectively. The corresponding solution of the isentropic Euler equations is represented by a superposition of two selfsimilar waves, where each wave is either a shock or a rarefaction wave, see for example [3]. In a shock wave both the density and the velocity experience nonzero, constant jumps in values across the shock interface. The wave moves with the constant velocity, different from the fluid velocities on the left and on the right of the shock. As shown in [9, 10, 28], the weak solutions of the Navier-Stokes equations with the Riemann initial data also contain interfaces of discontinuity of the flow parameters. The qualitative properties of the interface are however very different from those on a shock interface, even for the diminishingly small values of the viscosity. Due to the presence of diffusion in the momentum equation, the velocity instantaneously gains regularity and becomes continuous. The density remains discontinuous across an interface convected by the flow velocity, and the size of the discontinuity of the density decays with time t at the rate inversely proportional to the viscosity. In the limit of vanishing viscosity, the difference of values of the density on both sides of the interface diminishes, and the interface becomes a part of smooth flow, away from the shock. These findings lead to further investigation of properties of the viscous, compressible flows with bounded, discontinuous densities and resulted in the development of new techniques for studying equations (1), which was successfully applied to multidimensional problems in [11–18, 20, 27].

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1.1

1675

Equations

The homogeneous Navier-Stokes equations express the principles of conservation of mass and momentum in a flow of gas in the absence of exterior forces: @t  C div .u/ D 0; @t .uj / C div .uj u/  uj  . C /@xj div u C @xj P ./ D 0; j D 1::3; (1) where .x; t / 2 QT D ˝  .0; T /; ˝ is an open, bounded set in R3 ;  and u D .u1 ; u2 ; u3 / are the unknown functions of x and t representing the density and the velocity; P D  ;   1;  > 0; is the isentropic pressure;  and  are viscosity coefficients, verifying conditions  > 0;

2  C   0: 3

(2)

The shear viscosity coefficient  describes the intensity of friction, and  D C 23 ; the bulk viscosity coefficient, describes the intensity of energy dissipation due to expansion/contraction of a fluid element. The system (1) is solved subject to the initial conditions ..x; 0/; u.x; 0// D .0 .x/; u0 .x//;

x 2 ˝;

(3)

and the boundary conditions, which we assume to be no-slip conditions u.x; t / D 0;

.x; t/ 2 @˝  Œ0; T /:

(4)

Finally, in to be in agreement with the model, we require density to be a positive function.

1.2

Weak Solutions

A notion of a weak solution extends the notion of a classical solution by allowing the description of a state of the fluid by functions without well-defined point-wise behavior. In such description the integral averages are used instead. Accordingly, equations in (1) are interpreted in the weak sense; for any test function 2 C01 .˝  Œ0; T //; “

Z .@t C u  r / dxdt C

0 .x/ .x; 0/ dx D 0;

(5)

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M. Perepelitsa

and for all

2 C01 .˝  Œ0; T //3 ;

“ u  .@t

C u  r /  @xk u  @xk “ C

 . C / div u div

dxdt

Z P div

dxdt C

0 u0 .x/ 

.x; 0/; dxdt D 0:

(6)

An additional restriction, the balance of total energy, is imposed on a weak solution: Z ˝

Z t2 Z 1  .x; t2 / 2 juj .x; t2 / C dx C ru W ru C . C /j div uj2 dxdt 2  1 t1 ˝ Z  .x; t1 / 1 (7) juj2 .x; t1 / C dx; a.a. t2 > t1 > 0:   1 ˝ 2

For classical solutions the balance of total energy holds with equality sign, and it follows from the equations in (1). Weak solutions are used to model fluid flows in which some parameters of the flow are discontinuous (singular) functions. Such discontinuities might be present in the problem by design, when the data of the problem are discontinuous or, potentially, may result from complex nonlinear dynamics. A notion of a weak, energy solution was introduced [23, 24]. It is defined as a pair of functions .; u/ with the properties  2 L1 .0; T I L .˝//I juj2 2 L1 .0; T I L1 .˝//I ru 2 L2 .˝  .0; T //33 ; with .x; t /  0; a.e. .x; t /; that verify integral relations (5), (6) and (7), and the boundary conditions (4), in a sense of traces of weakly differentiable on ˝ functions. As can be seen from the balance of the total energy, the above requirements are minimal requirements one has to impose on functions for the weak formulation to make sense. Remarkably, a result of [23] shows that, if the polytropy constant   9=5; and given initial data with the finite total energy, one can construct a weak, energy solution .; u/ for any T > 0: The result was further strengthened in [8] for the range  > 3=2 that includes the physically relevant case of monoatomic gas. Analytic tools introduced in [23,24] proved to be crucial for the subsequent development of the theory of weak solutions. A thorough exposition of this theory is contained in a monograph [7].

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1.3

1677

On the Consistency of the Model of the Navier-Stokes Equations

It should be noted carefully that the system of equations (1) is derived under the condition that fluid occupies all domain ˝: If a part of ˝ is initially void of mass (vacuum) or if, hypothetically, the vacuum forms as a result of flow dynamics, the Navier-Stokes equations cease to be a correct model of fluid flow. Instead, the flow must be modeled as a free boundary problem, with the Navier-Stokes equations inside the domain where f > 0g; and an appropriate boundary conditions on the interface between gas and vacuum. The result [19], for one-dimensional flows, illustrates the difference between two models. Almost everywhere positivity of the density is a local property, which is hard to verify for weak solutions. Up to the moment, this chapter has been written it is not known if weak, energy solutions of (1) preserve this property for t > 0; if initially there is no vacuum.

1.4

Weak Solutions in the Intermediate Regularity Class

The weak solutions in this class are described by the properties of the material derivative uP D @t u C u  ru: A weak, energy solution of (1) will be called a weak solution in the intermediate regularity class, or Hoff’s regularity class, on ˝  .0; T /; if the distributional derivative uP can be represented by a locally integrable function, for any 2 .0; T / uP 2 L1 . ; T I L2 .˝//3 ;

(8)

r uP 2 L2 .˝  . ; T //33 ;

(9)

and ess sup  < C1; ess inf  > 0: ˝.0;T /

˝.0;T /

As we shall see later in this chapter, the norms of uP in spaces in (8), (9) appear in higher-order energy balance inequalities.

1.5

On the Regularity of the Velocity

The information on the material derivative uP (acceleration), and ; in the definition of a weak solution can be readily used to observe various integrability/ differentia-

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M. Perepelitsa

bility properties of the velocity u; as a weak solution to a uniformly elliptic system of the Lamé equations: u C . C /r div u D uP C rP;

(10)

for each fixed t > 0: For example, elliptic regularity and Sobolev’s embedding lemmas, described in the following sections, imply that for t > 0; u.x; t / is in C ˛ .˝/; for all ˛ 2 .0; 1/: Moreover, applying curl and div operators to the above system of equations, we obtain to Poisson’s equations:  curl u D curl .uP /;

(11)

.. C 2/ div u  P / D div .uP /;

(12)

for curl u and function F D . C 2/ div u  P; called viscous flux. The regularity of the weak solutions of Poisson’s equation implies that for any t > 0; r curl u.; t / 2 L6loc .˝/33 ; rF .; t / 2 L6loc .˝/3 :

(13)

Furthermore, it can be shown that 3 1 1 curl u 2 L1 . ; T I L1 loc .˝// ; F 2 L . ; T I Lloc .˝//;

for any 2 .0; T /: The validity of the above properties globally in ˝ depends on the choice of the boundary conditions for the velocity. With the no-slip conditions (4), these properties do not extend to all of ˝; unless  has better regularity at the boundary @˝: For ˝ D R3 or for an open domain with the Navier (slip) boundary conditions, the properties hold globally: curl u 2 L1 . ; T I L1 .˝//3 ; F 2 L1 . ; T I L1 .˝//; for any 2 .0; T /: The reader may consult [11, 13] for the details. The viscous flux, F; plays a central role in the analysis of weak solutions of the Navier-Stokes equations in both P.L. Lions and Hoff’s approaches, because it allows a suitable control of the density. When the divergence, div u; is expressed using the viscous flux, the equation of conservation of mass can be written as lnP C

F P D :  C 2  C 2

In this equation coefficient,  C 2 > 0: The pressure works against the growth of the density, while the viscous flux is effectively controlled by, for example, property (13).

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1.6

1679

Conditions at the Interface of Density Discontinuity

In the results presented in this chapter, we assume that the density of the flow is an element of L1 space and thus is generically discontinuous. To get some insight of the properties of viscous fluid flows with discontinuous density, it is instructive to start with a prototypical example, in which the density is piecewise smooth with a jump discontinuity across a regular, two-dimensional surface, and write down the Rankine-Hugoniot conditions on the surface of discontinuity. The following presentation can be found in [11, 29]. We will proceed informally, assuming that there is a regular surface S  ˝  .0; T / of dimension 3, such that each section S .t / D S \ ftg  ˝ is a regular two dimensional surface. Denote by .n; O n/ N 2 R4 ; 3 nN 2 R ; the normal vectors to S and to S .t /; respectively. Assume that .; u; ru/ have continuous first derivatives in .x; t / 2 ˝ .0; T /nS and have well-defined limits on both sides of S .t /: By Œ; Œu; and Œru, we denote the increments of the values of the corresponding functions across interface S .t /: Finally, we assume that .; u/ is a weak solution of (1) in ˝  .0; T /: As a consequence of viscous stresses acting in fluid flow, we expect velocity to be continuous across the interface: Œu D 0;

x 2 S .t /:

It follows, that for any tangent vector to S .t / at point x; Œru D 0: Since the tangent space has dimension 2; we can express the jump in the rate of strain as Œru D aN ˝ n; N

(14)

for some aN D a.x; N t / 2 R3 : The Rankine-Hugoniot condition for the equation of mass conservation takes a form of the orthogonality nO C u  nN D 0; which implies that S consists of trajectories of the flow. It follows that the acceleration is continuous across S .t / W ŒPu D 0;

x 2 S .t /:

(15)

The Rankine-Hugoniot condition for the equation of conservation of momentum reads ŒrunN C . C /Œ div unN C ŒP nN D 0: Using formula (14) we find from here that Œru D

ŒP  . C / div u nN ˝ nN :  jnj N 2

Thus, Œru is a symmetric matrix, and Œ curl u D 0;

x 2 S .t /:

(16)

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M. Perepelitsa

Computing the trace of matrices in the expression for Œru; we arrive at Œ. C 2/ div u  P  D 0;

x 2 S .t /:

(17)

The condition states that viscous flux F is continuous across the interface, while div u is not. Turning our attention to Œ; we use the equation of conservation of mass to obtain an ODE for the evolution of Œ; expressing Œ div u through ŒP  from condition (17). In this we way, we obtain equation ŒlnP  C

1 ŒP  D 0:  C 2

If the density is lower bounded away from vacuum, infx;t  > 0and then Πdecays exponentially fast along a trajectory of the flow, with the rate inversely proportional to viscosities: . C 2/1 : Summarizing our findings, we observe that conditions (15), (16), (17) indicate that functions uP ; curl u; and F have better regularity than div u and : In the framework of weak solutions in the intermediate regularity class, this improved regularity is expressed by conditions (9) and (13). Accordingly, the theory of weak solutions focuses on the properties of uP ; curl u; and F: For example, the higherorder energy inequalities, that are used in the proof of the main result, Theorem 5, are derived for L2 norms of uP and its derivatives. Full information on u is retrieved from an elliptic system of PDEs (10). As we mentioned earlier, the class of weak solutions in the intermediate regularity class was introduced with an idea to model fluid flows with piecewise smooth initial data and a regular interface separating the domains where data are smooth. A refined analysis of weak solutions in the intermediate regularity class can be performed to show that there exists global in time solution in which the density has a piecewise smooth structure for all times t > 0; with an interface transported by the fluid flow, [12].

2

Preliminaries

2.1

Notation and Functional Spaces

By ˝ we denote an open, bounded, connected domain in R3 with the boundary of class C 3C! ; for some ! 2 .0; 1/: By QT we denote an open cylinder ˝  .0; T /: The conventional notation is used for Hölder spaces C ˛ .˝/ and C ˛;ˇ .QT /; ˛; ˇ > 0: W k;p .˝/; k – positive integer, p 2 Œ1; C1; is a Sobolev’s space of Lp .˝/ integrable, together with all their weak derivatives up to order k; functions, P 1=p p m with the norm ; where m 2 R3 is a multi-index of m W jmjk kD f kLp .˝/

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1681

nonnegative integers. When p D 2; W k;2 .˝/ is abbreviated to H k .˝/: H01 .˝/ is a subspace of H 1 .˝/ of functions with zero trace on @˝: To measure the density, we use norm ˛ .˝/ D kkL1 .˝/ C kkC@˝ ess sup

.x;y/2˝˝

j.x/  .y/j ; maxfdx ; jx  yjg˛

(18)

˛ where dx D dist.x; @˝/; ˛ 2 .0; 1/: By C@˝ .˝/ we denote a subspace of L1 .˝/ ˛ .˝/ norm. Functions in C ˛ .˝/ have C ˛ .@˝/ traces on functions with finite k  kC@˝ @˝ the boundary of ˝: Norm (18) is equivalent to

kkL1 .˝/ C sup ess sup x2@˝

y2˝

j.x/  .y/j ; jx  yj˛

for which we use the same notation. When X is a Banach space, we denote by Lp .0; T I X /; p 2 Œ1; C1; a space of Bochner measurable functions from .0; T / ! X with finite norm kkf .; t /kX kLp .0;T / :

2.2

Estimates for Solutions of Poisson’s and Lamé’s Equations

This section contains classical estimates for solutions of Poisson’s and Lamé’s equations. It is assumed, without further mentioned it, that ˝  R3 is a bounded connected domain of class C 3C˛ ; even though some results hold under weaker conditions on the regularity of the boundary of the domain. Consider a homogeneous Dirichlet problem for Poisson’s equation:  u D f; x 2 ˝;

u.x/ D 0; x 2 @˝:

(19)

The following results are classical and can be found in [6]. Theorem 1. i. Let f D @xi g; for some i D 1::3; with g 2 Lp .˝/; for some 1;p p 2 .1; C1/. Then, problem (19) has a unique weak solution u 2 W0 .˝/ and kukW 1;p .˝/  C kgkLp .˝/ ;

(20)

with C independent of g: ii. Let f D @xi g; for some i D 1::3; with g 2 C ˛ .˝/; for some ˛ 2 .0; 1/. Then, problem (19) has a unique weak solution and kukC 1C˛ .˝/  C kgkC ˛ .˝/ ; with C independent of g:

(21)

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M. Perepelitsa

iii. Let f 2 L2 .˝/: Then, there is a unique strong solution u 2 H01 .˝/ \ H 2 .˝/; and kukH 2 .˝/  C kf kL2 .˝/ ;

(22)

with C independent of f: iv. Let f 2 C ˛ .˝/; for some ˛ 2 .0; 1/: Then, there is a unique classical solution u 2 C 2C˛ .˝/; and kukC 2C˛ .˝/  C kf kC ˛ .˝/ ;

(23)

with C independent of f: Consider a homogeneous Dirichlet problem for Lamé’s equations:  u  . C /r div u D f; x 2 ˝;

u.x/ D 0; x 2 @˝;

(24)

where  > 0 and  C 2=3  0: The Lamé equations can be written in a matrix form, with u D .u1 ; u2 ; u3 /; as Lu D

X

Lj k uk D f j ;

(25)

k

where Lj k D  ıj k   . C /

@2 : @xk @xj

The symbol of L; P ./ D fPj k ./g; equals   Pj k ./ D .2 /2 jj2 ıj k C . C /k J : An operator L from (25) is called strongly elliptic if 8. ; / 2 R6 ;

T P ./  c0 jj2 j j2 ;

(26)

for some c0 independent of .; /: It follows that Lamé’s operator L is strongly elliptic if and only if  > 0;  C 2 > 0: This property is verified under condition (2) on the viscosity coefficients. The theory of linear, constant coefficient, strongly elliptic operators parallels that of Poisson’s equation. The next theorem contains various properties of solutions of (24). The estimates follow from the theory of elliptic operators [1] and from the potential theory for Lamé’s equations [26].

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1683

Theorem 2. i. Let f D rg; with g 2 Lp .˝/ for some p 2 .1; C1/: There is a unique weak solution of (24) and kukW 1;p .˝/3  C kgkLp .˝/ : ii. Let f D rg; with g 2 C ˛ .˝/ for some ˛ 2 .0; 1/: There is a unique weak solution of (24) and kukC 1C˛ .˝/3  C kgkC ˛ .˝/ : iii. Let f 2 Lp .˝/3 ; p 2 .1; C1/: Then, there is a unique strong solution of (24) and kukW 2;p .˝/3  C kf kLp .˝/3 : iv. Let f 2 C ˛ .˝/3 ; ˛ 2 .0; 1/: Then, there is a unique classical solution of (24) and kukC 2C˛ .˝/3  C kf kC ˛ .˝/3 : In the above estimates, C is independent of f or g: The following estimates hold for solutions of vector Poisson’s equations with a gradient right-hand side. ˛ Lemma 1. Consider vector Poisson’s equations (19) with f D r;  2 C@˝ .˝/: The weak solution u verifies the following estimates:

i. For some C > 0; independent of ; ˛ k div u  kC ˛ .˝/  C kkC@˝ .˝/ :

(27)

ii. For some C > 0; independent of ; and 8.x; z/ 2 @˝  ˝; ˛ ju.x/  u.z/j  C kkC@˝ .˝/ jx  zj:

(28)

Proof. Let H .x/ be the fundamental solution of the Laplace equation in R3 ; G.x; y/ be Green’s function for the Laplace equation on ˝; and H  .x; y/ D H .x  y/  G.x; y/: For u.x/ as in the conditions of the lemma, we obtain representations Z div u.x/ D .x/ C ˝

@2xi ;yi H  .x; y/.y/ dy

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M. Perepelitsa

and Z ry G.x; y/.y/ dy:

u.x/ D  ˝

The estimates of parts i. and ii. follow from the standard potential estimates for the singular integral operators. Consider now the homogeneous Dirichlet problem (24) with f D  r./; ˛  2 C@˝ .˝/: Lemma 2. i. The viscous flux F D div u 

   C 2

is Hölder continuous in ˝ and ˛ kF kC ˛ .˝/  C kkC@˝ .˝/ ;

(29)

where C is independent of : ii. For all .x; y/ 2 @˝  ˝; ˛ ju.x/  u.y/j  C kkC@˝ .˝/ jx  yj;

where C is independent of : Proof. Split the solution of Lamé’s equations u D us C ur into a singular part, us ; the gradient of which might be discontinuous, and a regular part ur ; with a Hölder continuous gradient in the following way. Define us to be the weak solution of the problem  u D r

   ; x 2 ˝;  C 2

u.x/ D 0; x 2 @˝:

By part i. of Lemma 1 k div us 

 C ˛ kC ˛ .˝/  kkC@˝ .˝/ :  C 2  C 2

On the other hand, ur D u  us solves the equations  Lur D . C /r

div us 

   ; . C 2/

(30)

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1685

with the no-slip boundary condition. The elliptic regularity estimate of Theorem 2 results in krur kC ˛ .˝/33  C k div us 

 kC ˛ .˝/ : . C 2/

Thus, us matches all discontinuities of u: Using (30), one arrives at k div u 

 ˛ kC ˛ .˝/  C kkC@˝ .˝/ :  C 2

On the other hand part ii. of Lemma 1 states that ˛ jus .x/  us .y/j  C kkC@˝ .˝/ jx  yj;

for .x; y/ 2 @˝  ˝: Combining this with the above estimate on the difference ru  rus concludes the second statement of the lemma. To establish long-time stability of solutions of the Navier-Stokes equations, the estimate on the viscous flux from part i. of the previous lemma needs to be refined. ˛ .˝/ and u be a weak solution of (24) with f D r./: Lemma 3. Let  2 C@˝ There is C0 independent of .; ; / and ; such that

kF kC ˛ .˝/  C0

 ˛ kkC@˝ .˝/ ; . C 2/

for all ;  > 0: Proof. We use the method of [25] for solving Lamé’s equations. If the values of F on @˝ are given, then the solution u.x/ of the problem (24), with f D  r./; can be obtained from formulas Z C  u.x/ C ry G.x; y/.y/ dy xF .x/ D  2  C 2 ˝ Z C C @ny G.yF .y// d  ; 2 @˝ Z 3 C 5 @2ny ;xi G.x; y/.yi  xi /F .y/ d  F .x/ D C @˝ Z 2 C @2 H  .x; y/.y/ dy; . C 2/. C / ˝ xi ;yi where as before G.x; y/ D H .x  y/  H  .x; y/ is the Green’s function for the Laplace operator on ˝ and H .x  y/ is the fundamental solution of the

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M. Perepelitsa

Laplace equation on R3 : The integrals in the formula for F are discontinuous when x ! x0 2 @˝ W Z lim

1 @2ny ;xi G.x; y/.yi  xi /F .y/ d  D  F .x0 / 2 @˝ Z C @2ny ;xi G.x0 ; y/.yi  x0;i /F .y/ d  ; @˝

Z lim ˝

@2xi ;yi H  .x; y/.y/ dy

1 D  .x0 / C 2

Z ˝

@2xi ;yi H  .x0 ; y/.y/ dy:

This leads to an integral equation for F on @˝ W 5 C 7 F .x/  C D

Z @˝

2 . C 2/. C /

@2ny ;xi G.x; y/.yi  xi /F .y/ d 

Z ˝

@2xi ;yi H  .x; y/.y/ dy

 .x/; . C 2/. C /

x 2 @˝: (31)

This equation can also be expressed as Q C KŒF  D F0 ; F

5 C 7 Q D C

(32)

where Z KŒF .x/ D  @˝

@2ny ;xi G.x; y/.yi  xi /F .y/ d 

and F0 denotes the right-hand side of the equation (31). Using standard potential estimates, it is observed that kF0 kC ˛ .˝/ 

C  ˛ kkC@˝ .˝/ ; . C 2/. C /

(33)

with some C independent of  and ; : The integral kernel in KŒF  is weakly singular, and operator KŒF  is compact from C ˛ .@˝/ to C ˛ .@˝/: For any Q in the resolvent set of K and any F0 2 C ˛ .˝/; there is a unique solution of (32), and for some C; independent of F0 ; kF kC ˛ .@˝/  C kF0 kC ˛ .@˝/ :

(34)

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1687

It needs to be shown that for all ;  > 0; the number C from (34) remains bounded. Since Q 2 Œ5; 7; it suffices to show this interval is in the resolvent set. This fact is established in the lemma below. Lemma 4. The resolvent set of K contains interval .7=2; C1/: Proof. Let FQ be a C ˛ .@˝/ solution of the equation Q FQ C KŒFQ  D 0: Define function Z

@ny G.x; y/.y  x/FQ .y/ d  ;

v.x/ D

x 2 ˝:

@˝

Direct computations show that Z v D  2r Z div v D @˝

@ny G.x; y/FQ .y/ d ;

@˝

@2ny ;xi G.x; y/.yi  xi /FQ .y/ d   3

Z

(35)

@ny G.x; y/FQ .y/ d  ; @˝

(36) and v.x/ D 0 on @˝: Consider functions Z F1 .x/ D @˝

@2ny ;xi G.x; y/.yi  xi /FQ .y/ d ; F2 .x/ D

Z

@ny G.x; y/FQ .y/ d : @˝

Both functions are harmonic in ˝: Moreover, as x converges to a boundary point, F2 converges to FQ .x/; and F1 converges to 1  FQ .x/  KŒFQ .x/ D .Q  1=2/FQ .x/: 2 By the uniqueness of the Dirichlet problem for the Laplace equation, F1 D .Q  1=2/F2 : Then, from (36), div v D .Q  7=2/F2 : Substituting this in (35), one obtains an equation for v W v C

2 r div v D 0; Q  7=2

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M. Perepelitsa

which, for Q > 7=2, has only zero solution, because v D 0 on @˝: This implies that FQ D 0: The proof of the Lemma 3 is concluded by noticing that since F is harmonic in ˝; kF kC ˛ .˝/  C kF kC ˛ .@˝/ : This estimate together with (33) and (34) provides the statement of the lemma.

2.3

Bogovski’s Operator

Let ˝ be a bounded, Lipschitz domain in R3 : For a function g 2 Lp .˝/; consider the problem div v D g; x 2 ˝;

v D 0; x 2 @˝:

(37)

The proof of the following lemma can be found in [5], theorem 3.3. p

Lemma 5. Let p 2 .1; C1/: There is a linear bounded operator B W Lm .˝/ ! 1;p p W0 .˝/3 ; where Lm .˝/ is the space of Lp integrable functions with zero mean, such that v D BŒg solves problem (37). Moreover, if g D div h; for some h 2 Lr .˝/3 ; with h  n D 0; on @˝; and r 2 .1; C1/; then kBŒgkLr .˝/3  C khkLr .˝/3 ; for some C > 0 independent of h:

2.4

Embedding Lemmas

The following embedding estimates are frequently used in this chapter. Their proofs are well known and can be found, for example, in [31]. Lemma 6. Let u 2 W 1;2 .˝/: Then, u 2 L6 .˝/ and 6p

3p6

kukLp .˝/  C kukL2p2 .˝/ krukL22p.˝/3 ;

8p 2 Œ2; 6;

for some C independent of u: Lemma 7. If u 2 W 1;p .˝/; with p > 3; then u has a Hölder continuous representation (still denoted by u), and kukC ˛ .˝/  C .kukL2 .˝/ C krukLp .˝/3 /; for some C independent of u:

˛ D 1  3=p;

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

2.5

1689

Local Classical Solutions

In this section we state the result of [30] in the form that applies to the initialboundary value problem (1), (2), (3) and (4). Let ˝ be an open, bounded, connected set in R3 with the boundary @˝ of class 2C˛ C ; for some ˛ 2 .0; 1/: Let .0 ; u0 / 2 C 1C˛ .˝/  C 2C˛ .˝/3 ; inf 0 > 0 ˝

and such that for any x 2 @˝; j

j

div .0 u0 u0 /  u0  . C /@xj div u0 C @xj P .0 / D 0; j D 1; 2; 3:

(38)

The result is stated using parabolic Hölder spaces H nC˛ .QT /; for the velocity u; where n is a nonnegative integer and ˛ 2 .0; 1/; with the norm kukH nC˛ .QT / D

n X 2rCjsjD0

sup j@rt Dxs uj C QT

X

h@rt Dxs ui˛QT ;

2rCjsjDn

where hui˛QT D

sup .x;t/6D.y; /2QT

ju.x; t /  u.y; /j : .jx  yj C jt  j1=2 /˛

The density  is an element of B 1C˛ .QT / which is defined by the norm kkB 1C˛ .QT / D

1 X rCjsjD0

sup j@rt Dxs j C QT

X

h@rt Dxs i˛QT :

rCjsjD1

Theorem 3. If the initial data .0 ; u0 / verify compatibility conditions (38), there is T > 0; such that the problem (1), (2), (3) and (4) has a unique classical solution .; u/ on QT ; and .; u/ 2 B 1C˛ .QT /  H 2C˛ .QT /3 ; inf  > 0: QT

More regularity of the data and the boundary @˝ translates into the improved regularity of the classical solution. Suppose that @˝ of class C 3C˛ ; for some ˛ 2 .0; 1/: Let .0 ; u0 / 2 C 2C˛ .˝/  C 3C˛ .˝/3 ; inf 0 > 0: ˝

Theorem 4. If the initial data .0 ; u0 / verify necessary compatibility conditions on @˝; there is T > 0; such that the problem (1), (2), (3) and (4) has a unique classical solution .; u/ on QT ; which, in addition to properties stated in Theorem 3, verifies

1690

M. Perepelitsa

D 2 B 1C˛ .QT /;

Du 2 H 2C˛ .QT /3 ;

where D; Du stand for any of the first .x; t /; derivatives of  and u:

3

Formulation of the Result

Our main result, Theorem 5, states sufficient conditions on the parameters and the initial data of problem (1), (2), (3) and (4) for existence of a weak solution in the intermediate regularity class. The result can also be described as a stability of an equilibrium solution .; N 0/ 2 R4 ; with N > 0; of (1), (2), (3) and (4), when ˛ deviations from .; N 0/ are measured by the norm of C@˝ .˝/  H01 .˝/3 : Let ˝ be an open bounded domain with the boundary of class C 3C! ; ! 2 .0; 1/: Let N > 0 and let ˛ 2 .0; minf!; 1=4g/: We assume that coefficients ;  are positive constants. Select the initial data for (1) ˛ .0 ; u0 / 2 C@˝ .˝/  H01 .˝/3 ;

such that ess inf 0 > 0;

0 dx D : N

˝

˝

Theorem 5. There is N depending on .˛; ˝/ and M depending on . ; ; ; ; ˛; ; N ˝/ such that, if  > 0;

 N 

(39)

and ˛ .˝/ C ku0 k 1 k0  k N C@˝ H .˝/3  M;

(40)

then the problem (1), (2), (3) and (4) has a weak solution in the intermediate regularity class, defined on ˝  .0; C1/: The solution, .; u/; verifies the following properties: ˛  2 L1 .0; C1I C@˝ .˝//;

1 1

ess inf  > 0; ˝.0;C1/

u 2 L1 .0; C1I H01 .˝//3 \ C 2 ; 8 .˝  . ; C1//3 ;

8 > 0:

There are two types of conditions imposed by the theorem. Conditions (39) restrict the range of the parameters of the model (1). They are postulated in order to control the deviation of the density from the ground state : N Under these

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1691

conditions, the viscous damping of sound waves dominates the growth of amplitude of oscillations due to the reflection of waves from the boundary, providing for the global stability. With  and  selected according to (39), condition (40) restricts the initial data .0 ; u0 / to be close to a constant equilibrium state .; N 0/: This is another condition for the global stability, under which the viscous dissipation controls growth due to the nonlinear terms in (1). Let us remark that weak solutions of the intermediate regularity can be obtained under weaker assumptions on the initial data. The result [11], for flows in Rn ; builds the solutions with the initial data .0 ; u0 / 2 L1 .Rn /  .Lp .Rn //n ; with sufficiently large p > 1: In particular, it allows for Riemann problem-like initial data, with jump discontinuities of the density and the velocity across surfaces of co-dimension 1. The weak solutions with such data contain initial layers, in which the velocity instantaneously gains H 1 .Rn /n regularity. To control the solution in the initial layer, the ratio of the viscosity coefficients needs to be restricted so that  < bn ; where bn is a positive number determined by the dimension of the space  n: This condition should not be confused with conditions (39), which accommodate for the boundary effects. The proof of this theorem is given in the sections below.

3.1

Uniqueness and Continuous Dependence on Data

Uniqueness and continuous dependence on data of weak solutions in the intermediate regularity class is an outstanding open question. The following result of Hoff [14] describes a class of weak solutions within which solutions depend continuously on the data. This class in many respects is close to the class of weak solutions of the intermediate regularity; however it requires, in addition, the velocity to be Lipschitz continuous in space with the Lipschitz constant integrable in time, see condition (44) below. This property is not known to hold for solutions from Theorem 5. Examples of genuinely discontinuous solutions that fit into the framework described in Theorem 6 are solutions with piecewise smooth structure constructed in [12]. We compare solutions of the nonhomogeneous Navier-Stokes equations on Rn  .0; T /; n D 2; 3; given by equations @t  C div .u/ D 0; @t .uj / C div .uj u/  uj  . C /@xj div u C @xj P ./ D f; j D 1::n: (41) We refer to a solution of (41) as a triple .; u; f /; f being a part of the data of the problem. Let Q > 0 be a fixed positive number. The following conditions on the data and solutions are assumed.

1692

M. Perepelitsa

  Q is a bounded map from Œ0; T  into L1loc .Rn / \ H 1 .Rn / and   0 a.e.; (42) 0 u0 2 L2 .Rn / u; P  P ./; Q ru; f 2 L2 .Rn  .0; T // juj2 2 L1 .Rn  .0; T //

9 > > = > > ;

(43)

Two weak solutions .; u; f / and .; N u; f /; that we compare in the next theorem, assumed to satisfy 1 n u; u 2 C .Rn  .0; T / \ L1 .0; T I W 1;1 .Rn // \ L1 loc .0; T I L .R //

(44)

and   ; Q N  ; Q u; u; f; f 2 L2 .Rn  .0; T //:

(45)

Additionally, one of the solutions .; u; f / is required to satisfy ; 1 2 L1 .Rn  .0; T //

(46)

and Z

T 0

Z

jujr dxdt < 1

(47)

Rn

for some r > n; and the other solution .; N u; f / is required to satisfy Z

T 0

.ku.; t /k2L1 C t kru.; t /k2L1 C tkrF ; r curl uk2L2 C .tkrF ; r curl uk2L4 /a / dt < 1;

(48)

where F D . C 2/ div u  P ./; N a D 2=3 for n D 2 and a D 4=5 for n D 3; and f 2 L1 .0; T I L2q .Rn //;

(49)

for some q 2 Œ1; 1: Finally, it is assumed that 0  N0 2 .L2 \ L2p /.Rn /;

(50)

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1693

where p is the Hölder conjugate of q: By X .y; t; s/ we denote the flow trajectories of u W 8 @X ˆ ˆ .y; t; s/ D u.X .y; t; s/; t / < @t ˆ ˆ : X .y; s; s/ D y

(51)

and by X trajectories of u: Due to (44) trajectories are defined uniquly. Denote the map S .x; t / D X .X .x; 0; t /; t; 0/ that maps the position of a fluid particle under flow of u at time t to a corresponding position of the fluid particle under flow of u; at time t: Theorem 6. Given P D ; M; T; and r > n; where n D 2 or 3; there is a constant C depending on K; M; T; and r such that if .; u; f / and .; N u; f / are weak solutions of (41) satisfying (42), (43), (44) and (45), with .; u; f / satisfying (46) and (47) and .; N u; f / satisfying (48) and (49); if (50) holds, and if all norm occurring in these conditions are bounded by M; then Z

T

Z

1=2

2

ju  uj dxdt/ Rn

0

C sup k.  /.; N t /kH 1 .Rn / t2Œ0;T 

"

Z

 C k0  N0 kL2 \L2p C k0 u0  N0 u0 kL2 C

T

Z

2

1=2 #

jf  f ı S j dxdt 0

:

Rn

(52) RT If 0 t krf .; t /kL1 dt  M , then f ı S in the above inequality can replaced by f : The same result holds with no restriction on the pressure function P provided that     P ..; t //  P ..; N t //   sup  r   .; t /  .; N t/

t2Œ0;T 

< 1;

L˛

where ˛ > 2 when n D 2 and ˛ D 3 when n D 3I in this case the constant C depends additionally on ˛ and on the above sup. We refer an interested reader to [14] for the proof of this result.

1694

4

M. Perepelitsa

Proof of Theorem 5

The proof is based on the method of a priori estimates. First, we select smooth initial data .0 ; u0 / that approximate .0 ; u0 / in L2 .˝/  H 1 .˝/3 norm. Solving problem (1), (2), (3) and (4) with the new initial data, we obtain a classical solution .; u / that exists on some time interval Œ0; T  /: We estimate this local solution in ˛ energy norms for the velocity and the acceleration and in C@˝ norm for the density. Most of the work in the proof is done on obtaining the estimates that are independent of the length of the interval of existence of the local solution, T  ; and : Such estimates are possible due to the fact that we restrict the data of the problem to flow regimes in which diffusion dominates inertia. Since the amount of the dissipated kinetic energy is controlled by the total initial energy, the initial data can be selected to be so close to the equilibrium, as expressed by condition (40), that the energy norms of the velocity and the acceleration remain uniformly bounded on any time interval Œ0; T  /: The diffusion also has a stabilizing effect on the amplitude of the density oscillations (sound waves). Ratio (39) controls the magnitude of the rate of growth of amplitude of sound waves due to reflection from @˝; relative to the magnitude of the rate of viscous damping, see equation (72) and estimate (78). The ratio is assumed to be small so that the viscous damping is dominant. With a priori estimates at hand, the local solution can be extended to all times t > 0: At the final step of the proof, we use the compactness results of P.L.Lions [24], to extract strongly compact sequence f.n ; un /g of solutions that converge to a weak solution in the intermediate regularity class, with the properties described in the theorem. If pressure P depends linearly on ; the weak compactness of f g and strong compactness of fu g; that follow from a priori estimates, are sufficient to pass to the limit in the equations and obtain a weak solution of (1), (2), (3) and (4). The latter argument was implemented in [11]. Let .0 ; u0 / be the initial data verifying the conditions of the theorem. There is family of smooth functions f.0 ; u0 /g  C 2C˛ .˝/  C 3C˛ .˝/3 ; such that densities 0 are positive with the mean over ˝ equal to I N velocities verify the no-slip boundary conditions; as  ! 0; .0 ; u0 / ! .0 ; u0 / in the norm of L2 .˝/  H 1 .˝/3 I each .0 ; u0 / verifies conditions of the local existence Theorem 4 from the preliminaries. The latter establishes the existence of the classical solution .; u / of (1), (2), (3) and (4) with the initial data .0e ; u0 / and defined on cylinders QT : Throughout the remaining of this section ; .; u/ represents an element of the family f. ; u /g defined for t 2 Œ0; T ; where T may depend on : Denote by

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

A.T / D sup k.; t /  k N 2C ˛

@˝ .˝/

t2Œ0;T 

B.T / D sup t2Œ0;T 

; A0 D A.0/;



ku.; t /k2H 1 .˝/3 C  .t /kPu.; t /k2L2 .˝/3 Z

1695



T

kru.; t /k2L2 .˝/33 C  .t /kr uP .; t /k2L2 .˝/33 dt;

C 0

and B0 D B.0/; with  .t / D minf1; tg: One of the principle observations is that A.T /; B.T / can be estimated only in term of the initial values A0 ; B0 ; as stated in the next lemmas. Lemma 8. Suppose that A.T /  minf1; =2g: N There are positive numbers N ˝/; such that .D1 ; C1 /; with D1 < 1; depending only on . ; ; ; ; ˛; ; B.T /  C1 .A0 C B0 /1=2 ;

8.A0 ; B0 / W A0 C B0 2 .0; D1 /:

(53)

N Lemma 9. There are positive numbers .A ; B  ; D2 ; C2 / with A  minf1; =2g; B   1; depending only on . ; ; ; ; ˛; ; N ˝/ such that whenever A.T /  A ; B.T /  B  ; A0 C B0  D2 ; then A.T /  C2 .A0 C B0 /1=2 : The proofs of the lemmas will be given below.

4.1

Energy Estimates

In order to simplify the presentation, the estimates are established only for the pressure law P D : The general case requires only minimal modifications. The estimates of the following lemma were established in [11]. In this section we assume that conditions of Lemma 8 hold. In particular, .x; t /  =2: N Lemma 10. There is C depending on ; N ; ; ; and ˝; such that .; u/ verifies the following estimates. Z

2

2

Z

T

Z

ju.; t /j C j.; t /  j N dx C

sup t2Œ0;T  ˝

0

jruj2 dxdt  C .A0 C B0 /I

˝

(54) Z sup t2Œ0;T  ˝

jru.; t /j2 dx C

Z TZ 0

˝

jPuj2 dxdt  C .A0 C B0 / C C

Z 0

TZ

jruj3 dxdt I

˝

(55)

1696

M. Perepelitsa

Z

 jPu.; t /j2 dxC

sup

Z TZ

t2Œ0;T  ˝

0

jr uP j2 dxdt C .A0 CB0 /CC

˝

Z TZ 0

 jruj4 dxdt: ˝

(56) Proof. Throughout this section uP D @t u C u  ru and @i stands for @xi : The first estimate of the lemma follows from the balance of total energy: for all t 2 Œ0; T ; Z

.; t /ju.; t /j2 =2 C ˚..; t /; / N dx C

˝

Z tZ Z

jruj2 C . C /j div uj2 dxdt

˝

0

0 ./ju0 ./j2 =2 C ˚.0 ./; / N dx;

D ˝

R

N 2 ds is the potential energy of the gas. where ˚.; / N D N N .s  /=s To obtain the second estimate, the momentum equations are expressed as uP i  @2l;l ui  . C /@2i;l ul C @i ./ D 0; i D 1::3;

(57)

where the summation over a repeated index is always assumed. The equations are multiplied by uP i; summed over i, and integrated over the domain ˝: Consider each term of the resulting equation separately. Z

d @i ./Pu dx D dt ˝ i

Z

@i ./u  ˝

d dt Z

˝

Z

D  

˝

D 

Z

i

d dt

@2i;t ./ui  @i ./uk @k ui dx Z

.  / N div u dx C

.@t ./ C div .u// div u dx

˝

˝

@i uk @k ui dx Z

Z

@i uk @k ui dx:

.  / N div u dx  ˝

˝

It follows that ˇZ ˇ ˇ ˇ

ˇ ˇZ ˇ ˇ ˇ ˇ @i ./Pui dxdt ˇˇ  ˇˇ ..; T /  / N div u.; T / dx ˇˇ C C .A0 C B0 /: ˝ ˝ 0 (58) Consider now the term T

Z

Z ˝

@2l;l ui uP i

Z d dx D  jru.; t /j2 dx dt ˝ Z @l ui @l uk @k ui C @l ui @2k;l ui uk dx:  ˝

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1697

Notice that due to zero boundary condition for the velocity, Z ˝

@l ui @2k;l ui uk dx D 

1 2

Z

jru.; t /j2 div u.; t / dx:

˝

Using this in the previous computation and integrating the later over Œ0; T ; the following equation is obtained. Z

T

Z ˝

0

@2l;l ui uP i dxdt D 

Z

jru.; T /j2 dx C ˝

Z

T

 0

Z

jru0 ./j2 dx

˝

Z

1 @l ui @l uk @k ui  jruj2 div u dxdt: 2 ˝

(59)

Similar computations lead to Z

T

Z ˝

0

@2l;i ul uP i dxdt D 

Z

j div u.; T /j2 dx C ˝

Z

 0

T

Z

j div u0 ./j2 dx ˝

Z

1 . div u/@i ul @l ui  . div u/3 dxdt: 2 ˝

(60)

Combining estimates (58), (59) and (60) results in (55). To derive the last energy estimate, operator @t ./ C div .u/ is applied to each equation in (57): .@t uP i C uk @k uP i /  @t @2l;l ui  @k .uk @2l;l ui /  . C /@t @2l;i ul  . C /@k .uk @l;i ul / C @2t;i ./ C @k .uk @i .// D 0: The equations are multiplied by  uP i ; summed over i and integrated over the domain ˝: Consider each term separately. Z

d .@t uP C u @k uP /Pu dx D dt ˝ i

k

i

i

Z ˝

 jPuj2 dx  2

Z ˝

0 jPuj2 dx: 2

(61)

Next, the term containing pressure is expressed as Z ˝

 .@2t;i ./ C @k .uk @i .///Pui dx D 

Z

 .@t ./ C @k .uk //@i uP i dx ˝

Z 

 @k .@i uk /Pui dx

˝

Z

 @i uk @k uP i dx:

D ˝

1698

M. Perepelitsa

It follows that ˇ ˇZ T Z Z ˇ ˇ 2 k i ˇ ˇ  .@t;i ./ C @k .u @i .///Pu dxdt ˇ  C ˇ ˝

0

T

Z  jrujjr uP j dxdt: ˝

0

Consider the viscosity term Z Z  .@t @2l;i ul C @k .uk @2l;i ul //Pui dx D   .@2t;l ul C @k .uk @l ul //@i uP i dx ˝

(62)

˝

Z

 @k .@i uk @l ul /Pui dx

 Z

˝

Z

2

D

 @l uk @k ul @i uP i dx

j div uP j dx C ˝

˝

Z

j div uj2 div uP dx

 ˝

Z

 @i uk @l ul @k uP i dx:

C ˝

It follows that Z tZ ˝

0

 .@t @2l;i ul C @k .uk @2l;i ul //Pui dx D  Z tZ

 @l uk @k ul @i uP i dxdt 

C 0

˝

Z tZ

 j div uP j2 dxdt

˝

0

Z tZ

 j div uj2 div uP dxdt

0

˝

0

˝

Z tZ C

 @i uk @l ul @k uP i dxdt:

(63)

The next equation is obtained by similar arguments. Z tZ ˝

0

 .@t @2l;l ui C @k .uk @2l;l ui //Pui dx D 

Z tZ

k

C

i

k

i

i

Z tZ 0

 jr uP j2 dxdt

˝

Z tZ

 .@l u @l u  @k u @l u /@l uP dxdt C 0

˝

0

 @l uk @l ui @k uP i dxdt:

˝

(64) Combining estimates (61), (62), (63) and (64) one can obtain (56). For any t 2 Œ0; T ; define v.; t / D BŒ.; t /  ; N where BŒ is Bogovski’s operator from Lemma 5. Since .; u/ verifies the equation of conservation of mass, @t v D BŒ@t  D  BŒ div .u/: By Lemma 5 the following estimates are valid: N L2 .˝/ ; k@t vkL2 .˝/3  C kukL2 .˝/3 : krvkL2 .˝/33  C k  k

(65)

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1699

Multiplying the momentum equations by v and integrating in .t; x/; after some simplifications and making use of the first energy estimate (54) and (65), one obtains estimate Z tZ

j  j N 2 dxdt  C .A0 C B0 / C C

˝

0

0

By Poincare’s inequality Z tZ 0

Z tZ

R ˝

R

juj2 dx  C

juj2 dxdt:

˝

jruj2 dx; it is concluded that

˝

j  j N 2 dxdt  C .A0 C B0 /;

8t 2 Œ0; T :

(66)

˝

In the next lemmas, the theory of Lamé’s equations is used to estimate terms on the right-hand side of the energy inequalities. Lemma 11. For all t 2 Œ0; T ; sup

2Œ0;t

kru.; /k2L2 .˝/33 Z

Z tZ

jPuj2 dxd  C .A0 C B0 /

C 0

˝

t

kru.; t /k6L2 .˝/33 d :

CC 0

There are positive numbers .1 ; C /; with 1 < 1; such that sup kru.; t /k2L2 .˝/33 C

Z

T 0

t2Œ0;T 

Z

jPuj2 dxdt  C .A0 C B0 /;

˝

whenever A0 C B0  1 : Lemma 12. For all t 2 Œ0; T ; sup

2Œ0;t

 . /kPu.; /k2L2 .˝/3 1=2

Z

 C .A0 CB0 /

0

Z tZ

jr uP j2 dxd

C 0

t

kPuk2L2 .˝/3

˝

  1=2 d  sup  . /kPu.; /k2L2 .˝/3 CC .A0 CB0 /:

2Œ0;t

There are positive numbers .2 ; C /; with 2 < 1; such that sup  .t /kPu.; t /k2L2 .˝/3 C

t2Œ0;T 

whenever A0 C B0  2 :

Z

T 0

Z

jr uP j2 dxdt  C .A0 C B0 /1=2 ; ˝

1700

M. Perepelitsa

Proof (Lemma 11). In the remaining proofs, C stands for a certain function of .; ; ; ˛; ; N ˝/: Consider the momentum equations as a system of Lamé’s equations. That is, for each t 2 Œ0; T  consider the problem Lu D  uP  r..  //; N x 2 ˝;

u D 0; x 2 @˝:

The velocity can be split into u D v C w; where v; w solve problems Lv D  uP ; x 2 ˝;

v D 0; x 2 @˝;

Lw D  r..  //; N x 2 ˝;

(67)

w D 0; x 2 @˝:

(68)

The elliptic regularity, stated in Theorem 2, provides estimates kvkH 2 .˝/3  C kuP kL2 .˝/3  C kPukL2 .˝/3 ;

(69)

N Lp .˝/ ; krwkLp .˝/33  Cp k  k

(70)

8p 2 .1; C1/:

Using these estimates we obtain the following inequalities. krukL3 .˝/33  krvkL3 .˝/33 C krwkL3 .˝/33 1=2

1=2

 C krvkL2 .˝/33 kPukL2 .˝/33 C C k  k N L3 .˝/ 1=2

1=2

1=2

 C .krukL2 .˝/33 C krwkL2 .˝/33 /kPukL2 .˝/33 C C k  k N L3 .˝/ 1=2

1=2

1=2

 C .krukL2 .˝/33 C k  k N L2 .˝/ /kPukL2 .˝/33 C C k  k N L3 .˝/ : In the second inequality above, we used Sobolev’s embedding Lemma 6 and (69): 1=2

1=2

1=2

1=2

krvkL3  C krvkL2 kD 2 vkL2  C krvkL2 kPukL2 : It follows then that for any  > 0; there C such that Z

t 0

Z t Z t  kruk3L3 .˝/33 d   kPuk2L2 .˝/3 d CC N 6L2 .˝/ d

kruk6L2 .˝/33 Ckk 0

0

Z

t

CC 0

k  k N 3L3 .˝/ d :

(71)

Upon using this expression in the energy estimate (55) with the properly selected  > 0 and noticing that Z 0

t

k  k N 3L3 .˝/ d  C

Z 0

t

k  k N 2L2 .˝/ d  C .A0 C B0 /;

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1701

which holds true by (66), the first statement of Lemma R t 11 is concluded. The second statement follows from the first, using the estimate 0 kruk2L2 dxdt  C .A0 CB0 /; which is contained in (54). Proof (Lemma 12). Following the line of arguments in the previous lemma, one obtains estimate krukL4 .˝/33  krvkL4 .˝/33 C krwkL4 .˝/33 1=4

3=4

 C krvkL2 .˝/33 kPukL2 .˝/33 C C k  k N L4 .˝/ 1=4

1=4

3=4

 C .krukL2 .˝/33 C krwkL2 .˝/33 /kPukL2 .˝/33 C C k  k N L4 .˝/ 1=4

1=4

3=4

 C .krukL2 .˝/33 C k  k N L2 .˝/ /kPukL2 .˝/33 C C k  k N L4 .˝/ : In the second inequality above, we used Sobolev’s embedding Lemma 6 and (69): 1=4

3=4

1=4

3=4

krvkL4  C krvkL2 kD 2 vkL2  C krvkL2 kPukL2 : Then, using the first energy estimate (54) and (66), Z

t 0

 kruk4L4 .˝/33 d  C

Z

t 0

 .krukL2 .˝/33 C k  k N L2 .˝/33 /kPuk3L2 .˝/33 d

Z

CC 0

t

k  k N 4L4 .˝/ d

 C .A0 C B0 /1=2

Z

t 0

kPuk2L2 .˝/3 d

 sup

2Œ0;t



 kPuk2L2 .˝/3

1=2

C C .A0 C B0 /: Combining this with (56), the statement of the lemma follows. Proof (Lemma 8). The statement of this lemma follows from Lemmas 11 and 12, with D1 D minf1 ; 2 g and an appropriate C1 :

4.2

Estimates Based on the Equation of Mass Conservation

The first equation in (1) can be expressed as d  ln  C .  / N D F1  div v; dt  C 2

F1 D div w

 .  /; N  C 2

(72)

where v; w come from splitting of the velocity into regular and singular parts as defined in (67) and (68).

1702

M. Perepelitsa

Proof (Lemma 9). Choose first A D minf1; =2g; N B  D 1; and assume that   A.T /  A ; B.T /  B : Let .D1 ; C1 / be from Lemma 8. Assume that A0 C B0 < D1 ; so that (53) holds. Since ˛  1=4; from the embedding Lemmas 6 and 7 and elliptic estimates, it follows that 1=4

3=4

krvkC ˛ .˝/33  C kvkW 2;4 .˝/3  C krvkL2 .˝/33 C C kPukL2 .˝/3 kr uP kL2 .˝/33 : (73) Estimating krvkL2 as in (11) and using the definition of B.T /; one arrives at krv.; t /kC ˛ .˝/33  C .A0 C B0 /1=2 C  .t /kr uP .; t /k2L2 .˝/33 C CB.T / 4=5 : (74) Let .x; x0 / 2 ˝  @˝: Lemma 2 shows that jw.x; t /  w.x0 ; t /j ˛ .˝/ :  C k.; t /  k N C@˝ jx  x0 j

(75)

Thus, if a.t / D

sup x2˝; x0 2@˝

ju.x; t /  u.x0 ; t /j ; jx  x0 j

(76)

then a.t /  C .A0 C B0 /1=2 C  .t /kr uP k2L2 .˝/33 C CB.T / 4=5 .t / C CA.T /; and Z

t

a.s/ ds  C .A0 C B0 /1=2 t C CB.T / maxft 1=5 ; tg C CA.T /t:

0

In particular, it is possible to choose .A ; B  ; D2 / with D2 < D1 ; so small that the inequality  exp 

 N .t  / C ˛ 2. C 2/

Z



t

a.s/ ds

  C exp 



  N .t  / ; 4. C 2/ (77)

for all < t 2 Œ0; T ; holds as long as A.T / < A ; B.T / < B  ; A0 C B0 < D2 : The last conditions are assumed to hold hereafter. The estimate of F1 is obtained from Lemma 3:

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1703

C0 ˛ k.; t /  k N C@˝ .˝/ ; . C 2/

(78)

kF1 .; t /kC ˛ .˝/ 

where C0 D C0 .˝; ˛/: Next, the equation of mass conservation is integrated along the trajectories of the flow with velocity u.x; t /: Denote by Xt ./ the trajectory that starts at point x D  at t D 0: Let .; x0 / 2 ˝  @˝: Because of the no-slip boundary conditions, Xt .x0 / D x0 ; for all t ’s. Notice also, that any < t; jX ./  x0 j  exp jXt ./  x0 j

Z

t

 a.s/ ds ;

(79)



where a.t / was introduced above. For a generic function, f .x; t/ defines an operator S .f /.t / D f .Xt ./; t /  f .x0 ; t /; to denote the difference of the values of the function on two (fixed) trajectories. In this way, d  S .ln / C S ./ D  S .F1 /  S . div v/: dt  C 2 Since, S ./ D S Q .ln /; for some Q 2 Œ=2; N 3=2; N Z t  N  N e  2.C2/ .t / jS .F1 /j. / d

jS .ln /.t /j  jS .ln /.0/je  2.C2/ t C Z C

0 t

e

 N  2.C2/ .t /

jS . div v/j. / d :

0

Dividing the inequality by jXt ./  x0 j˛ ; taking the supremum over .; x0 / 2 ˝  @˝; and using (74), (77), we obtain from (79)

˛ N C@˝ sup Œ.; /˛;@˝  C k0  k .˝/ C C0

2Œ0;t

 C 2 sup kF1 .; /kC ˛ .˝/ 

2Œ0;t C C .A0 C B0 /1=2 C CB.T /:

Combining this with (53), and (78) justifies estimate

˛ ˛ sup k.; /  k N C@˝ N C@˝ .˝/  C k0  k .˝/ C C0

2Œ0;t

 ˛ N C@˝ sup k.; /  k .˝/  2Œ0;t C C .A0 C B0 /1=2 ;

1704

M. Perepelitsa

where C0 depends only on .˝; ˛/: Assuming that ratio = is sufficiently small, it follows that 1=2 ˛ .˝/  C .A0 C B0 / N C@˝ ; sup k.; /  k

2Œ0;t

which concludes the proof of Lemma 8. With Lemmas 8 and 9 at hand, define (   2   2 ) A B 1 ; ; M D min D1 ; D2 ; A ; B  ; 2 2C2 2C2 and assume that the initial data are so small that A0 CB0 < M: Functions A.t /; B.t /; defined at the beginning of the section, are continuous on Œ0; T : There is T1 2 .0; T  such that A.t/  A ; B.t /  B  ;

t 2 Œ0; T1 :

By Lemmas 8 and 9 and the choice of M; A.t/  A =2; B.t /  B  =2;

t 2 Œ0; T1 :

It follows by continuity that A.t/  C2 .A0 C B0 /1=2 ; B.t /  C1 .A0 C B0 /1=2 ;

t 2 Œ0; T :

(80)

With independent of t bounds (80), the local solution .; u/ can be extended to interval Œ0; C1/; on which (80) are also valid. The next lemma shows that estimates (80) control the Hölder norm of the velocity u: We follow here the arguments of [11]. Lemma 13. For any > 0, there is C . /; such that kukC 1=2;1=8 .˝. ;C1//3  C; and for all T > ; kF kL8=3 . ;T IC 1=4 .˝// ; k div ukL8=3 . ;T IC 1=4 .˝//  C T: Proof. Consider v and w components of the velocity from (67) to (68). For any t 2 Œ0; T ; and p 2 .1; C1/; krw.; t /kLp  C k.; t /  k N Lp : By the embedding Lemma 7, and a priori estimates for A.T /; for any ˛ 2 .0; 1/;

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1705

sup kw.; t /kC ˛ .˝/3  C; t2Œ0;T 

for some C; independent of T: On the other hand, a priori estimates for B.T / imply that for any > 0 and t 2 Œ ; T ; kv.; t /kC 1=2 .˝/3  C krv.; t /kL6 .˝/33  kr 2 v.; t /kL2 .˝/333  C . /; for some C . /; independent of T: In this manner, sup ku.; t /kC 1=2 .˝/3  C . /: t2Œ ;T 

To estimate the modulus of continuity in t; let x 2 ˝; t2 > t1 > ; and consider the average u.y; t / dy; Br .x/

where it is assumed that u is extended by zero outside of ˝: It follows then that u.y; t / dyj  C . /r 1=2 :

ju.x; t /  Br .x/

The difference between velocities at different times is estimated as Z

t2

j@t u.y; t /j dydt C C . /r 1=2

ju.x; t1 /  u.x; t2 /j  t1

Br .x/

 C r 3=2 jt1  t2 j1=2

Z

t2

t1

j@t u.y; t /j2 dydt

1=2

C C . /r 1=2 :

Br .x/

Writing @t u D uP  u  ru and using estimates of Lemmas 8 and 9, one concludes that sup ju.x; t1 /  u.x; t2 /j  C . /.r 3=2 jt1  t2 j1=2 C r 1=2 /: x

The first estimate in the statement of the lemma follows by choosing r D jt1 t2 j1=4 : The second follows from the estimates (73), and (75) for v and w and the energy estimates for A.T /; B.T /:

1706

4.3

M. Perepelitsa

Compactness Properties of Approximating Solutions

Consider a family f. ; u /g of classical solutions defined globally on ˝ .0; C1/; and such that for any T > 0; ˛ f g bounded in L1 .0; T I C@˝ .˝//; infx;t   =2I N  1 f@t  g bounded in L .0; T I H 1 .˝//I fu g bounded in L1 .0; T I H 1 .˝//3 I 8 > 0; fu g bounded in C 1=2;1=8 .˝  . ; T //3 ; 8 > 0; fF  g bounded in L8=3 . ; T I C 1=4 .˝//; with F  D div u  8 > 0; f div u g bounded in L8=3 . ; T I L1 .˝//:

  I C2

Passing to a limit on a suitable sequence (still labeled by ) shows that there is a pair of functions .; u/ 2 L1 .QT /  L1 .0; T I H01 .˝//3 ; with N u 2 C 1=2;1=8 .˝  . ; T //3 ; div u 2 L8=3 . ; T I L1 .˝//; ess infx;t   =2; for any ; T > 0; and such that  !  *–weakly in L1 .QT /I u ! u *–weakly in L1 .0; T I H 1 .˝//3 I u ! u in C .˝  . ; T //3 : If pressure is a linear function of ; the above compactness properties suffice to pass to the limit in the equations. The pointwise convergence is needed to show ˛ that  2 L1 .0; T I C@˝ .˝// and to handle the case of P D  ;  > 1: This convergence follows by the argument of [24], which is outlined below. Lemma 14. There is a subsequence of f g; still labeled by ; such that  !  in L2 .QT / and a.e. .x; t /: Proof. Consider the pair .; u/ constructed above.  is a weak solution of @t  C div .u/ D 0; and, by corollary II.2 of [4], it is a renormalized solution. In particular, function ˚ D .x; t / ln .x; t / is a weak solution of @t ˚ C div .˚u/ C  div u D 0:

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1707

On the other hand, for any  > 0; ˚  D  .x; t / ln  .x; t / solves @t ˚  C div .˚  u / C

. /C1 C  F  D 0:  C 2

Compactness lemma 5.1 of [24] applies to the product  F  ! F D  div u 

   C 2

in D0 .QT /;

where  denotes L2 .QT /; weak limit of . / : Passing to the limit in the equation for ˚  and comparing the result with the equation for ˚; it is observed that R D ˚  ˚ solves the equation    C1   D 0; @t R C div .Ru/ C  C 2 and R.x; 0/ D 0; since the initial data 0 converges to 0 in L2 .˝/: By convexity, R; C1    0

a.e. QT :

After integration over QT W “

C1   dxdt  0; QT

and so, C1   D 0 a.e. QT : This implies that  !  in L2 .QT /; and the lemma follows. The lemma concludes the proof of Theorem 5.

5

Conclusion

The following is a short summary of results in the theory of weak solutions of intermediate regularity. In dimension 1 the Cauchy problem for (1) in the class of weak solutions was considered in [9,10,28]. The initial data are assumed to be integrable functions with the density in BV .R/ space. The results establish global existence of weak solutions for initial data with arbitrary large norms. For the weak solutions of this type, the density is function of bounded variation for all times. Let us also mention that the global existence of weak and classical solutions for the Navier-Stokes equations in dimension 1 was first obtained in [21, 22]. For the Cauchy problem in multidimensions, work [11] develops a theory of weak solutions for flows in all of the space Rn ; n D 2; 3: The initial data are assumed to be .0 ; u0 / 2 L1 .Rn /  Lp .Rn /3 , where p is sufficiently large (depending on the space dimension), and the energy of the initial data is assumed to be small. The

1708

M. Perepelitsa

uniqueness and continuous dependence on data, under some additional regularity properties of weak solutions, was demonstrated in [14]. A similar result for the initial-boundary value problem for a half-space in R3 with the Navier (slip) boundary conditions was established in [13]. The boundary conditions are expressed as u  n D 0;

..ru C rut /n C Ku/tan D 0; K  0; .x; t/ 2 @˝  Œ0; T /;

where n is the external normal vector and .u/tan is the projection of vector u to the tangent plane to @˝ at x: The existence of weak solutions is shown under the conditions similar to the conditions in [13]. The result is further generalized by considering system (1) with an external force in the equation of the balance of momentum. Flows with the no-slip boundary conditions in a half-space of R3 are ˛ ; as a suitable measure the density, is introduced considered in [27]. The space C@˝ in that paper. An interesting class of solutions was constructed in [12]. The density of the solution in this class has a piecewise C ˛ structure with a regular C 1C˛ interface separating different “phases” of the flow. The solution retains this structure for all times t > 0: The propagation of the density discontinuities along trajectories of a fluid flow in R3 for a class of weak solutions similar to weak solutions of the intermediate regularity was considered in [18]. The analysis validates the qualitative picture of the dynamics of the density discontinuity described in the introduction. The problem for motion of an interface of discontinuity becomes singular when the interface is in contact with the boundary of the flow domain, on which the no-slip boundary conditions are postulated. A detailed analysis around a contact point was performed in [16, 17], first for a model problem and then for the the Navier-Stokes equations. It is shown that the singularity in the velocity leads to an instantaneous change in the geometry of the interface. There are results on weak solutions of the intermediate regularity that go beyond the Navier-Stokes equations. In [15], the authors consider the Navier-Stokes-Fourier model within the class of data with radial symmetry. The theory of weak solutions to exothermically reacting flows in dimension 1, in Lagrangian coordinates, is developed in [2]. Weak solutions for 3-D compressible magnetohydrodynamics are constructed in [20].

6

Cross-References

 Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Finite Time Blow-Up of Regular Solutions for Compressible Flows  Global Existence of Regular Solutions with Large Oscillations and Vacuum for

Compressible Flows

32 Weak Solutions for the Compressible Navier-Stokes Equations in the. . .

1709

 Local and Global Solvability of Free Boundary Problems for the Compressible

Navier-Stokes Equations Near Equilibria  Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves  Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stabil-

ity, and Longtime Behavior  Weak Solutions for the Compressible Navier-Stokes Equations with Density

Dependent Viscosities  Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical

Cases

References 1. S. Agmon, A. Dougllis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II. Commun. Pure Appl. Math. 17, 35–92 (1964) 2. G.-Q. Chen, D. Hoff, K. Trivisa, Global solutions to a model for exothermically reacting, compressible flows with large discontinuous initial data. Arch. Ration. Mech. Anal. 166(4), 321–358 (2003) 3. R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves (Wiley Interscience, NewYork, 1948) 4. R.J. DiPerna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989) 5. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, I (Springer, New-York, 1994) 6. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, New-York, 1998) 7. E. Feireisl, Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and Its Applications, vol. 26 (Oxford University Press, Oxford, 2004) 8. E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids. J. Math. Fluid Dyn. 3, 358–392 (2001) 9. D. Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with non-smooth initial data. Proc. R. Soc. Edinb. Sect. A 103(3–4), 301–315 (1986) 10. D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Am. Math. Soc. 303(1), 169–181 (1987) 11. D. Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible flow. Arch. Ration. Mech. Anal. 114(1), 15–46 (1991) 12. D. Hoff, Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Commun. Pure Appl. Math. 55(11), 1365–1407 (2002) 13. D. Hoff, Compressible flow in half-space with Navier boundary conditions. J. Math. Fluid Mech. 7(3), 315–338 (2005) 14. D. Hoff, Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow. SIAM J. Math. Anal. 37(6), 1742–1760 (2006) 15. D. Hoff, K.Jenssen, Symmetric nonbarotropic flows with large data and forces. Arch. Ration. Mech. Anal. 173(3), 297–343 (2004) 16. D. Hoff, M. Perepelitsa, Instantaneous boundary tangency and cusp formation in twodimensional fluid flow. SIAM J. Math. Anal. SIAM J. Math. Anal. 41, 753–780 (2009) 17. D. Hoff, M. Perepelitsa, Boundary tangency for density interfaces in compressible viscous flows. J. Differ. Equ. 253(12), 3543–3567 (2012)

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18. D. Hoff, M. Santos, Lagrangean structure and propagation of singularities in multidimensional compressible flow. Arch. Ration. Mech. Anal. 188(3), 509–543 (2008) 19. D. Hoff, D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51(4), 887–898 (1991) 20. D. Hoff, A. Suen, Global low-energy weak solutions of the equations of 3D compressible magnetohydrodynamics. Arch. Ration. Mech. Anal. 205(1), 27–58 (2012) 21. J.I. Kanel, Ob odnoi modelnoi sisteme uravnenii odnomernogo dvizheniya gaza: A model system of equations for the one-dimensional motion. Diff. Uravneniya 4(4), 721–734 (1968) 22. A.V. Kazhikhov, V.V. Shelukhin, Unique global solution with respect to time of initial boundary value problem for one-dimensional equations of a viscous gas. Prikl. Mat. Mekh. 41, 282–291 (1977) 23. P.L. Lions, Existence globale de solutions pour les equations de Navier-Stokes compressibles isentropiques. C. R. Acad. Sci. Paris Ser. I Math. 316, 1335–1340 (1993) 24. P.-L. Lions, Mathematical Topics in Fluid Dynamics. Compressible Models, vol. 2 (Oxford Science Publication, Oxford, 1998) 25. L. Lichtenstein, Uber die erste Randwertaufgabe der Elastizitatstheorie. Math. Z. 20, 21–28 (1924) 26. S.G. Mikhlin, N.E. Morozov, M.E. Paukshto, The Integral Equations of the Theory of Elasticity (Teubner-Texte zur Mathematik, Stuttgart, 1995) 27. M. Perepelitsa, Weak solutions of the Navier-Stokes equations for compressible flows with no-slip boundary conditions. Arch. Ration. Mech. Anal. 212(3), 709–726 (2014) 28. D. Serre, Evolution d’une masse finie de fluide en dimension 1. Nonlinear Partial Differential Equations and their Applications. Lect. Coll. de France Semin., Paris/Fr. 1987–88, Vol. X, Pitman Res. Notes Math. Ser. 220, 304–319 (1991). 29. D. Serre, Variations de grande amplitude pour la densite d’un fluide visqueux compressible. Phys. D 48(1), 113–128 (1991) 30. A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion. Publ. RIMS Kyoto Univ. 13, 193–253 (1977) 31. W. Zimmer, Weakly Differentiable Functions (Springer, New York, 1989)

Symmetric Solutions to the Viscous Gas Equations

33

Song Jiang and Qiangchang Ju

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Classical Compressible Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Isentropic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nonisentropic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Density-/Temperature-Dependent Viscosity and/or Heat Conductivity . . . . . . . . . . . . . . 3.1 Isentropic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Nonisentropic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1712 1715 1716 1728 1737 1738 1742 1743 1744 1744 1745

Abstract

In the last decades, significant progress has been made on the existence and uniqueness of symmetric solutions to the compressible Navier-Stokes equations. In this chapter a brief review of some existence and large-time behavior results of symmetric (spherically symmetric, axisymmetric, etc) solutions with large data will be presented. The different cases: isentropic or nonisentropic flows, constant or the density-/temperature-dependent viscosity and heat conductivity, weak or strong (smooth) solutions, etc., will be discussed. The ideas and developed techniques used in analysis will be presented and analyzed, and some open questions will be addressed.

S. Jiang () • Q. Ju Institute of Applied Physics and Computational Mathematics, Beijing, China e-mail: [email protected]; [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_35

1711

1712

1

S. Jiang and Q. Ju

Introduction

The Navier-Stokes equations for a compressible heat-conducting fluid in Rn (n D 2; 3) can be written in Eulerian coordinates as follows: @t  C div .v/ D 0;

(1)

@t .v/ C div .v ˝ v/ C rp D div T; h i Et C div .E C p/v D div .Tv/  div q:

(2) (3)

Here  and v D .v1 ; : : : ; vn / denote the density and velocity of the fluid, respectively, E WD jvj2 =2 C e denotes the total energy, e is the specific internal energy, q is the heat flux, p is the pressure, T denotes the viscous stress tensor of the form for a Newtonian fluid: Tij D div vıij C 2Dij ;

(4)

 0, and D.v/ is the  and  are the viscosity coefficients satisfying  > 0,  C 2 n deformation tensor 1 D.v/ D .dij /ni;j D1 ; dij D .@i vj C @j vi /: (5) 2 The system (1), (2), and (3) describes the motion of a viscous heat-conducting fluid and is derived from the conservation laws of mass, momentum, and energy (e.g., cf. [2, 74]). This chapter is mainly concerned with the physical situation of ideal gases for which e and p have the following form: e D cV  ;

p D R

.polytropic ideal gas/;

(6)

where cV and R are positive constants and the heat flux obeys the Fourier’s law q D r ;

(7)

with heat conductivity   0. Substituting (6) and (7) into (1), (2), and (3), one obtains the complete system for the unknowns ; v;  in Eulerian coordinates .t; x/: @t  C div .v/ D 0;

(8)

@t .v/ C div .v ˝ v/ C Rr. / D r. div v/ C div .2D.v//; i h  jvj2  i h  jvj2 C cV  C div  C cV  C R v @t  2 2 D div .r / C div .vdiv v/ C div .2D.v/  v/:

(9)

(10)

33 Symmetric Solutions to the Viscous Gas Equations

1713

In the case of isentropic flows (recalling p D a with a depending on the entropy, more precisely,  > 1 corresponds to the isentropic case, whereas  D 1 corresponds to the isothermal case), the system (8), (9), and (10) reduces to @t  C div .v/ D 0;

(11)

@t .v/ C div .v ˝ v/ C ar D r. div v/ C div .2D.v//;

(12)

where   1 is the specific heat ratio and a is a positive constant. The main purpose in this chapter is to study the global well-posedness theory of symmetric solutions to the isentropic Navier-Stokes equations (11)–(12) and the full Navier-Stokes equations (8), (9), and (10), respectively. First the case that , , and  are constants in (11)–(12) or (8), (9), and (10) is discussed. This corresponds to the classical Navier-Stokes equations. Then one turns to the case that , , and  may depend on the density or temperature in (11)–(12) or (8), (9), and (10). Now the symmetric forms of (8), (9), and (10) is derived when , , and  are constants, while the corresponding symmetric forms of (8), (9), and (10) with density-/temperature-dependent viscosities and heat conductivity can be written in the same manner and will be discussed in Sect. 3. The spherically symmetric and axisymmetric forms of the system (8), (9), and (10) are mainly given in the following. The spherically symmetric solutions to (8), (9), and (10) have the form vi .x; t / D

xi v.r; t /; i D 1; : : : ; n; .x; t / D .r; t /; .x; t / D .r; t /; (13) r

where x D .x1 ; : : : ; xn / 2 Rn , r WD jxj. Denoting ˇ D C2, one thus reduces the system (8), (9), and (10) to the following equations for .r; t /; v.r; t /, and .r; t /: t C .v/r C .v/t C Œv 2 C p.; /r C

n1 v r

n1 v 2 r

D 0;

D ˇvrr C ˇ.n  1/

h i

v ; r r

. /t C .v /r C n1 v C p.;  /.vr C n1 v/ r r    2 D  rr C n1 r C  vr C n1 v C 2.vr /2 C 2 n1 v2: r r r2 For an axisymmetric flow in R3 , there is no flow in the angular direction, and all angular derivatives are identically zero. So only two variables are considered, q

r D x12 C x22 the radial direction and z D x3 the axial direction. The axisymmetric solution to (8), (9), and (10) has the form vi .x; t / D

xi r

v.r; z; t /; i D 1; 2;

v3 .x; t / D w.r; z; t /; .x; t / D .r; z; t /; .x; t / D .r; z; t /:

1714

S. Jiang and Q. Ju

Then .r; t /; v.r; t /; w.r; t /, and .r; t / satisfy the following equations: t C 1r .rv/r C .w/z D 0;

h i .v/t C 1r .rv 2 /r C .vw/z C p.; /r D  1r .rvr /r C vzz ; i h C. C / 1r .rv/r C wz   rv2 ; r i h .w/t C 1r .rvw/r C .w2 /z C p.; /z D  1r .rwr /r C wzz ; h i C. C / 1r .rv/r C wz ; z

. /t C . v/r C

v r

C .w/z C p.;  /.vr C 1r v C wz /;

.rr C 1r r C zz / D Q with 1 1 2 2 Q D .vr C v C wz /2 C 2.vr C v/2  .v /r C 2.wz /2 C .wr C vz /2 : r r r In the case of isentropic flows, the corresponding spherically symmetric and axisymmetric equations take the following forms, respectively: t C .v/r C

m v r

D 0;

(14)

h i .v/t C Œv 2 C a r C mr v 2 D ˇurr C ˇm vr ; r

(15)

with m D n  1, and t C 1r .rv/r C .w/z D 0; i h .v/t C 1r .rv 2 /r C .vw/z C a. /r D  1r .rvr /r C vzz h i C. C / 1r .rv/r C wz   rv2 ; r h i .w/t C 1r .rvw/r C .w2 /z C a. /z D  1r .rwr /r C wzz h i C. C / 1r .rv/r C wz : z

(16)

(17)

(18)

In the following sections, a short survey on recent mathematical results on global symmetric solutions will be given. The rest of this chapter is organized as follows. Section 2 is devoted to the classical compressible Navier-Stokes equations, and the global existence of symmetric weak and strong solutions to the isentropic Navier-Stokes equations is investigated. In this section, the spherically symmetric,

33 Symmetric Solutions to the Viscous Gas Equations

1715

axisymmetric, and other symmetric solutions are discussed, respectively. Then one turns to the nonisentropic case. In Sect. 3 the spherically symmetric solutions to the compressible isentropic Navier-Stokes equations with density-dependent viscosity coefficients are considered first. Then, the results on the symmetric solutions in the nonisentropic case are reviewed. In Sect. 4, some related problems are shortly discussed. A list of references has been included here, although it is not intended to be exhaustive. Of course the selection of the material follows personal interests and experiences. Notations (used throughout this chapter): Let be a domain in Rn (n D 1; 2; 3). m;p Let m be an integer and 1  p  1. By W m;p . / (W0 . /), one denotes the m;2 m usual Sobolev space defined over . W . /  H . / (W0m;2  H0m . /) and W 0;p . /  Lp . / with norm k  kLp . / . For simplicity, one also uses the following abbreviations: RC WD .0; 1/; RC 0 WD Œ0; 1/: Lp .I; B/ and k  kLp .I;B/ denote the space of all strongly measurable, pth-power integrable (essentially bounded if p D 1) functions from I to B and its norm, respectively, I  R an interval, B a Banach space. C 0 .I; B  w/ is the space of all functions which are in L1 .I; B/ and continuous in t with values in B endowed with the weak topology. B  is the dual space of B. By B ˛ Œ1; 1/, ˛ 2 .0; 1/, one denotes the space of bounded functions over Œ1; 1/ which are uniformly Hölder continuous with exponent ˛. For an integer k, BkC˛ Œ1; 1/ denotes the space of functions satisfying @ix v 2 B ˛ Œ1; 1/ for all integers i 2 Œ0; k. For a domain QT  Œ0; 1/ Œ0; T , B ˛;˛=2 .QT / denotes the space of uniformly Hölder continuous functions with exponent ˛ in x and ˛=2 in t. For an integer k, B kC˛;.kC˛/=2 .QT / j denotes the space of the functions satisfying @ix u; @t u 2 B ˛;˛=2 .QT / for all integers i 2 Œ0; k and j 2 Œ0; Œ k2 , where Œ k2  is the integer part of k=2.

2

Classical Compressible Navier-Stokes Equations

For the classical Navier-Stokes equations of compressible (heat-conducting) fluids, significant progress has been achieved since the work by Nash [68] in the last 1960s. In one dimension, it is well known that global (smooth and weak) solutions exist for large initial data and are time-asymptotically stable. In more than one dimension, the global smooth and weak solutions exist and converge to the corresponding stationary solutions as time tends to infinity for sufficiently small initial data (see, e.g., [63,64, 77]), and there are a vast amount of literature contributing to this issue; the reader can refer to the extensive bibliography in [58], for example. For large initial data, the global existence and large-time behavior of strong or smooth solutions to the Navier-Stokes equations for compressible heat-conducting

1716

S. Jiang and Q. Ju

fluids have been obtained in the spherically symmetric case (see, e.g., [22, 42, 43, 66, 69]). However, the case of general large initial data in general domains becomes more complex, and a number of important questions still remain open, for example, the existence of global solutions in the case of heat-conducting gases and the uniqueness of global weak solutions. For the classical isentropic Navier-Stokes equations with general large data, Lions [58] first established the existence of global weak solutions by using the weak convergence method, provided that the specific heat ratio  is appropriately large, for example,   3n=.n C 2/; n D 2; 3. Then by combining Lions’ techniques and convex analysis to reduce the integrability requirement of the density around the origin, Jiang and Zhang [46] showed the global existence of weak solutions for any  > 1 to the Cauchy problem when the initial data are spherically symmetric. Adapting Lions’ arguments and exploiting the Div-Curl Lemma and an idea from [46], Feireisl, Novotný, and Petzeltová [20, 21] extended Lions’ global existence to the case  > n=2 (n D 2; 3). The global existence of axisymmetric weak solutions for any  > 1 was shown in [48]. When  D 1, which corresponds to isothermal flows, the global existence of weak solutions was obtained first by Hoff [29] for spherically symmetric BV-initial data, and the two-dimensional result in [29] was extended to Lp -initial data by Jiang and Zhang in [47]. Later, Sun, Jiang, and Guo [75] proved the global existence of weak solutions for any  > 1 when the Cauchy data are helically symmetric. For general initial data and  2 .1; 3=2, however, it still remains open to establish the global existence of weak solutions to the classical isentropic Navier-Stokes equations. For compressible heat-conducting flows with positive initial density, the global existence of spherically and cylindrically symmetric weak solutions was established by Hoff and Jenssen in [30], where the momentum equation only holds weakly in the entire space-time domain with a nonstandard interpretation of the viscosity terms as distributions. When the heat conductivity  satisfies certain growth conditions, the result of [30] was extended to the standard sense by Zhang, Jiang, and Xie in [89]. Furthermore, the recent work by Jiang, Jiang, and Yin [45] improved the twodimensional result in [30]. The reader can refer to the extensive bibliography in [58, 70] and the survey article [33] for more references on mathematical theory of the multidimensional compressible Navier-Stokes equations. The main issue in this section is the global well-posedness theory for the classical multidimensional compressible Navier-Stokes equations with large symmetric data. The isentropic flow case will be discussed first and, then, the nonisentropic flow case.

2.1

Isentropic Flows

In this part, the spherically symmetric, axisymmetric, and other symmetric solutions to the classical compressible isentropic Navier-Stokes equations are studied, respectively.

33 Symmetric Solutions to the Viscous Gas Equations

1717

2.1.1 Spherically Symmetric Solutions First, the global existence of spherically symmetric weak solutions to the Cauchy problem is considered for the isentropic Navier-Stokes equations (11)–(12) in Rn (n D 2; 3) for which the corresponding spherically symmetric form is (14)– (15). The initial data and the (boundary) condition at the origin for (14)–(15) are imposed: .0; r/ D 0 .r/;

.v/.0; r/ D m0 .r/;

x  0;

(19)

v.t; 0/ D 0; t  0:

(20)

Before stating the existence result, one recalls the definition of weak solutions to (14), (15), (19), and (20). Definition 1. ..t; r/; v.t; r// is called a global weak solution of (14), (15), (19), and (20) if (1) .t; r/  0 a.e. and for any T > 0,  2 L1 .0; T I L .RC //; v 2 2 L1 .0; T I L1 .RC //; vr ; v=r 2 L2 .0; T I L2 .RC //; 2



 C1 C 0  2 C 0 .Œ0; T ; Lloc .RC 0 /  w/; v 2 C .Œ0; T ; Lloc .R0 /  w/;



2

 C1 C .; v/.0; r/ D .0 ; m0 /.r/ weakly in Lloc .RC 0 / Lloc .R0 /:

Here and throughout this subsection, one denotes Z n o jf .r/jp r m dr < 1 Lp . / WD f 2 L1loc . / W

R p 1 with norm k  kLp . / WD . j  jp r m dr/1=p ; Lloc . / and Hloc . / are defined p 1 similarly to Lloc . / and Hloc . /, respectively. (2) For any t2  t1  0 and any 2 C01 .R RC 0 /, there holds Z 0

1

ˇt2 Z ˇ  r dr ˇ  m

t1

t2

Z

1

. t C v r /r m drdt D 0:

0

t1

(3) For any t2  t1  0 and any 2 C01 .R RC 0 / satisfying .t; 0/ D 0, there holds Z 0

1

ˇt2 Z ˇ v r m dr ˇ  t1

t2

t1

Z 0

1

 m o m r drdt: v t C v 2 r C a r C r

n

1718

S. Jiang and Q. Ju

Z

t2

Z

1

D ˇ



vr r C

t1

0

mv  m r drdt: r2

For the problem (14), (15), (19), and (20), it is shown by Jiang and Zhang [46] that 2

Theorem 1. Let  > 1, 0  0 2 L1 .RC / \ L .RC /, m0 2 L  C1 .RC /, m20 =0 2 L1 .RC /. Then there exists a global weak solution .; v/ of (14), (15), (19), and (20), such that Z

1

 v 2

C

2

0

Z tZ  m ./ .t; r/r dr C ˇ 0

Z

1

 m2 0



20

0

C

1 0



vr2 C

mv 2  m r drd  r2



.0 / .r/r m dr 8t  0;

a  . Moreover, for any T > 0 and any 1 2 C01 .RC where ./ D 1 0 / with n

1 .r/ D O.r / as r ! 0, there holds

Z 0

T

Z

1 0

2 .t; r/ 14 .r/ dxdt  C:

If, in addition,  > n=2, then for any T > 0 and any 2 2 C01 .RC 0 / with 2 .r/ D O.r m / as r ! 0, there holds Z

T 0

Z 0

1

C .t; r/ 2 .r/ drdt  C

8
0, where  is the weak limit of the  th power of the approximate density sequence. The L2= -integrability of    , though not the L2 -integrability of  itself, is sufficient to conclude that the concentration of mass cannot develop at the origin. Moreover, it seems that this L2= -integrability is a multidimensional property (cf. [20, 46]). Next, the main steps of the proof in [46] are given.

33 Symmetric Solutions to the Viscous Gas Equations

1719

Step 1. The first step is to construct the approximate solutions similarly to that in [29]. First, mollify the initial data as follows: n m  o m 0 .r/ WD r   r  0 j =2 .r/ C ; n m  o n o mp

m m 0 .r/ WD r  2 0 r 2 pm00 j =2 .r/; v0 .r/ WD  0 .r/; 0

where j =2 is the standard Friedrichs mollifier and is a smooth function satisfying .r/ D 0 for r  and .r/ D 1 for r  2 . Then one considers the following approximate problem of (14), (15), (19), and (20) in the domain . ; 1/: m v r

Πt C Πv r C Πv t C Π.v /2 C a. / r C

m  .v /2 r

D 0;

(21)

D ˇŒu rr C ˇ.n  1/

hv i

r

r

; (22)

together with initial and boundary conditions  .0; r/ D 0 .r/;

v .0; r/ D v0 .r/;

r  ;

(23)

v .t; / D 0; t  0:

(24)

By Theorem 4.1 of [29], the problem (21), (22), (23), and (24) has a global weak solutions . .t; r/; v .t; r// on Œ0; 1/ Œ ; 1/ with positive  , such that 

 2 C 0 .Œ0; 1/; Lloc .Œ ; 1///; v 2 C 0 .Œ0; 1/; L2 . ; 1// \ C 0 ..0; 1/; H01 . ; 1//;  .t; r/ is pointwise bounded from above, and Z

1



1 2

 v 2 C Z 

1



. /

.t; r/r m dr C ˇ

Z tZ 0

1 2



0 .v0 /2 C

.0 /





1



Œv 2r C m

r m dr  C;

v 2  m r drdt r2

8 t  0;

(25)

where ./

WD

a a 1    a 1  C a  :  1  1

Step 2. In order to prove Theorem 1, one has to show the precompactness of the approximate solution sequence . ; v /. For this purpose, besides (25), an additional higher space-time integration estimate for the density  is needed.

1720

S. Jiang and Q. Ju

Let 2 C01 .Œ0; 1// and .r/ D O.r n / as r ! 0. One multiplies (22) with

and integrates over .r; 1/ (r 2 Π; 1/) to obtain Z

Z

1

Z

1

1

m v 2

dy y r r r Z 1h v mv i 

y dy; .v /y C D .a C  v 2 /  ˇ.v /r  ˇm  ˇ r y r @t

. v 2

 v dy 

a  / y

C

dy C

which yields that Z 1  v  2 

  1C v 2 C   @t

 v dy a D ˇ .v /r C m r r Z 1 Z 1 m v 2  2   . v C a / y dy C  

dy y r r Z 1h mv i 

y dy: .v /y C Cˇ y r

(26)

Now, mollifying the equation (21) by introducing  ı .t; r/ D  .t; / jı , multiplying the resulting equation by . ı /1 , and taking the limit ı ! 0, one gets that 

@t   C @r .v   / C

m v D .1   /.v /r   r

(27)

in D0 ..0; T / . ; 1//. Thus, 

Z

1

 @t r

h Z

 v dy D @t  

1

r

Z h i

 v dy C @r v  

1

 v dy

i

r

Z  h m v i 1 

 v dy C v 2  1C : C .  1/.v /r  C r r

(28)

Inserting (28) into (26), one finds that  h Z 1 i v  2 

C @t   a D ˇ .v /r C m

 v dy r r Z 1 h i h  C C@r v 

 v dy C .  1/.v /r   C

Z  m v i 1



r Z 1

C

r

r

r

 v dy   

m v 2

dy C ˇ y

Z

Z

1

. v 2 C a  / y dy

r 1

h

.v /y C

r

mv i

y dy: y

(29)

33 Symmetric Solutions to the Viscous Gas Equations

1721

Multiplying (29) by 3 , and integrating over .0; T / . ; 1/, one gets Z

T

Z

0



1

 2 4 dxdt D

Z

Z

T 0

1



1 fR.H.S. of (29)g 3 drdt: a

(30)

Then, carefully estimating each term on the right-hand side of (30), the following lemma can be established. Lemma 1. For any T > 0 and any 2 C01 .Œ0; 1//, .r/ D O.r n / as r ! 0, there exists a positive constant C , independent of , such that Z

T

Z

0

1

 2 4 .r/ drdt  C:



Step 3. One shall extract a limiting solution .; v/ from the approximate solution sequence . ; v / of (21), (22), (23), and (24), thus getting a global weak solution to (14), (15), (19), and (20). First, both v .t; x/ and  .t; x/ are extended to be zero for 0  x  . For simplicity, this extension is still denoted by .v .t; x/;  .t; x//. In view of (25) and Lemma 1, one can extract a subsequence of . ; v /, still denoted by . ; v /, such that 

2

 *  weak  in L1 .0; T I Lloc .RC // and weakly in L2 .0; T I Lloc .RC //; 1 v * v weakly in L2 .0; T I Hloc .RC //:

For any 0 <  <  , a subsequence of   @x v is extracted such that it 2

2

C

 C .RC //. Similarly,  is weakly convergent in L  C .0; T I Lloc 2

2  C



;  ;   are

weakly convergent in L  C .0; T I Lloc .RC //, in L2 .0; T I L2loc .RC //, and in 2

2

 .RC //, respectively. For the sake of convenience, denote by f ./ L  .0; T I Lloc the weak limit of f . / (in the sense of distributions) as ! 0. Using Lemma 1 and the weak convergence arguments, one can prove that

Lemma 2. For any 0 <  <  , we have aC  ˇQ D a   ˇ vx ; where Q is the weak limit of   @x v . Furthermore, Lemma 2 implies the following L2= -integrability of   , which suffices to exclude the concentration of mass at the origin.

1722

S. Jiang and Q. Ju

Lemma 3. Let 0 <  < 1 satisfy 12 .1   C

p 1 C 6 C  2 /   . Then

   2 L2= .0; T I L2= .0; 1//: Proof. By Lemma 2, one has a.C    / D ˇ.Q   vx /:

(31)

By virtue of convexity, it holds that   = C           .   /  0:

(32)

By (25), (31), and (32), 2

2

 .   / 2 L 2C .0; T I L 2C .0; 1//: 2

2 2 Recalling 2  C  C2 when  satisfies 12 .1   C can use (33) and Young’s inequality to infer that



  

 2

p

(33)

1 C 6 C  2 /   , one

2  2   C2  2   C2     D    2     C2 2 2   C    1 C   C2 2 n   C2  2 2 o  C 1 C  C      C2 2 L1 .0; T I L1 .0; 1//;

which proves the lemma.  For   1, one easily shows by the same arguments as in the proof of (27) that @t   C @r .v   / C

m  v D .1   /.v /r   : r

(34)

Now, taking ! 0 in (21), (22), and (34), one obtains t C .v/r C .v/t C Œv 2 C a r C

.n1/ v r

.n1/ v 2 r

D 0;

(35)

D ˇurr C ˇ.n  1/

h i

v ; r r

(36)

and . /t C . v/r C

.n  1/   v D .1   /Q r

in D0 ..0; T / RC /:

(37)

33 Symmetric Solutions to the Viscous Gas Equations

1723

From (35), (37), and Lemma 2, one gets that .   /t C ..   /v/r C

.n  1/  .   /v  .1   /.   /vr : (38) r

Let f  f .t; r/ WD r m .   /. Then, a multiplication of (38) by r m gives @t f C @r .vf /  .1   /f vr : By a careful analysis, one can conclude that @t f 1= C @r .vf 1= /  0 in D0 ..0; T / RC /:

(39)

Now, let 2 C01 .RC / satisfy 

.r/ D

0; r  =2 or r  1 C 1; 1;  r  1 ;

and j@r .r/j  C 1 for r  and j@r .r/j  C for r  1 . One multiplies (39) with and integrates it over .0; t/ .0; 1/ to find that Z

Z Z C t f .t; r/ .r/ dr  jvjf 1= drds 0 0 0 Z tZ 1 CC jvjf 1= drds: 1

1=

0

(40)

1=

By letting ! 0 in (40), it holds that  .t; r/ D  .t; r/;

a:e: .t; r/ 2 RC RC :

Hence by virtue of convexity, the Young measure associated with f .t; r/g is the Dirac measure, and in terms of Proposition 3.1.7 in [53] and Lemma 1, it holds that p

C  !  strongly in Lloc .RC 0 R /; 8 p < 2 :

Once the strong convergence of the density is proved, the proof of Theorem 1 can be completed by the standard weak convergence arguments. Remark 1. (i) For the case limr!1 0 .r/ D 1 > 0, limr!1 v0 .r/ D v1 , one could use cutoff function arguments similar to those used in [58] to obtain a similar existence theorem. (ii) If .t; x/ WD .t; jxj/; U .t; x/ WD v.t; jxj/x=jxj is defined for t  0 and x 2 Rn , then it is easy to see that ..t; x/; U .t; x// is a weak solution of the Cauchy problem for the compressible isentropic Navier-Stokes equations in Rn .

1724

S. Jiang and Q. Ju

Furthermore, the properties of vacuum states in weak solutions to the problem (14), (15), (19), and (20) have been studied by Xin and Yuan [85]. It is shown that vacuum states cannot be developed later on in time in a region far away from the center of symmetry, provided that there is no vacuum state initially, and two initially non-interacting vacuum regions never meet each other in the future. When the system (14), (15) is considered in an annular domain, Itaya [39] establishes the existence of the spherically symmetric solutions globally in time for large initial data. Then, Matsumura in [62] studied the global existence and largetime behavior of the spherically symmetric solutions to an isothermal model with external forces. This result was extended then by Higuchi [28] to an isentropic model later, while Nakamura, Nishibata, and Yanagi [67] extended the results in [28, 62] to the case of an exterior domain. As for the strong solutions with vacuum, Salvi and Straškraba in [73] and Cho, Choe, and Kim in [8, 10, 11] established some local and global existence results for the isentropic Navier-Stokes equations in bounded or unbounded domains, provided that the initial data satisfy certain regularity and the compatibility conditions. In particular, Choe and Kim in [11] proved that the radially symmetric strong solution exists globally in time for any   2 in an annular domain. Moreover, Fan, Jiang, and Ni [18] extended this result to the case   1. The local classical solution was obtained by Cho and Kim in [9]. When the initial energy is assumed to be small in R3 , Huang, Li, and Xin [37] recently proved the existence and uniqueness of global classical solutions to the isentropic compressible Navier-Stokes equations. The global spherically symmetric classical solution to the compressible isentropic Navier-Stokes equations with vacuum in an annular or exterior domain was obtained by Ding, Wen, Yao, and Zhu in [14] provided that the initial data satisfy certain compatibility condition.

2.1.2 Axisymmetric Solutions For an axisymmetric flow, the system (16), (17), and (18) is supplemented with initial and boundary conditions: .0; r; z/ D 0 ; v.0; r; z/ D m0 ; .r; z/ 2 RC R;

(41)

v.t; 0; z/ D 0; @r w.t; 0; z/ D 0; t  0; z 2 R;

(42)

where v D .v; w/ and m0 D .m10 ; m20 /. Comparing with the 2D or the spherically symmetric case, the difficulties here lie in the singularity at r D 0, the fact that the singularity set here is the plane 3 RC 0 f0g R in R but not a line as in the spherically symmetric case, and the Neumann boundary condition for w which could induce concentration of singularity involving w at r D 0 in passing to the limit r ! 0. For the simplicity of presentation, it is assumed that  C   0. It is easy to see that the case  C  > 0 will not arouse any new difficulties. Now the definition of the so-called finite energy solutions to the system (16), (17), (18), (41), and (42) is modified in the following way:

33 Symmetric Solutions to the Viscous Gas Equations

1725

Definition 2. ..t; r; z/; v.t; r; z// is called a finite energy weak solution of (16), (17), (18), (41), and (42) if (1)   0 a.e., and for any T > 0,  2 L1 .0; T I L .RC R//; jvj2 2 L1 .0; T I L1 .RC R//; rv; v=r 2 L2 .0; T I L2 .RC R//; 

2=.C1/

0  2 C 0 .Œ0; T ; Lloc .RC 0 R/  w/; v in C .Œ0; T ; Lloc

.RC 0 R/  w/; 2



 C1 C .; v/.0; x/ D .0 ; m0 /.x/ weakly in Lloc .RC 0 R/ Lloc .R0 R/;

where Z n o p Lp . / WD f 2 Lloc . / W jf .r; z//jp r drd z < 1

(43)



R p with norm k  kLp . / WD . j  jp r m drd z/1=p ; Lloc . / is defined similarly to p Lloc . /. (2) For any b 2 C 1 .R/ such that jb.s/j C jb 0 .s/sj  C for all s 2 R, there holds: v  1 C div v D 0 @t b./ C @r Œrb./v C @z Œb./w C Œb 0 ./  b./ r r in D0 ..0; T / RC R/, i.e., .; v/ is a renormalized solution of (16). 2 C01 .R3 /, ' 2 C01 .R3 / with '.t; 0; z/ D 0, (3) For any t2  t1  0 and any 1 3

2 C0 .R / with r .t; 0; z/ D 0, the following equations hold: Z RC R

ˇt2 Z ˇ  rdrd zˇ 

Z

RC R

D Z RC R

t1

t2

.

Z

RC R

t1

ˇt2 Z ˇ v'rdrd zˇ  Z

Z t2

t1

t1

n

h'

t1

t1

Z

RC R

t

C v

r

C w

z /rdrd zdt

D 0I

.v't C v 2 'r C vw'z /rdrd zdt

i v o C 'r  @r v'r   2 ' rdrd zdt I r r t1 RC R ˇt2 Z t2 Z ˇ w rdrd zˇ  .w t C vw r C w2 z /rdrd zdt t2

Z

a

t2

Z

D t1

RC R

RC R

fa z  @r w r  @z w z grdrd zdt:

1726

S. Jiang and Q. Ju

Z tZ ˇ  jvj2 v2  a  ˇ .4/ C .t; r; z/rdrd z C   ˇrvj2 C 2 rdrd zdt 2  1 r RC R RC R 0 Z    jm j2 a0 0 rdrd z 8t  0: C  C 2  1 0 R R Z

The existence result reads as follows, see [48]. p Theorem 2. Let  > 1; 0  0 2 L .RC R/ \ L1 .RC R/, and m0 = 0 2 L2 .RC R/. Then there exists a global weak solution of (16),(17), (18), (41), and (42), such that for any T; L > 0, and ˛ 2 .0; 1/, Z

T 0

Z 0

1

Z

L

. C v 2 /r ˛ drd zdt  C:

(44)

L

Remark 2. If one defines .t; x/ WD .t; r; z/; U .t; x/ WD ..x 0 =r/v.t; r; z/; w.t; r; z//, where x D .x1 ; x2 ; x3 / 2 R3 , x 0 D .x1 ; x2 / 2 R2 , and r D jx 0 j, then it is easy to see that ..t; x/; U .t; x// is a weak (finite energy) solution to the Cauchy problem for the compressible isentropic Navier-Stokes equations in R3 (cf. the proof of Theorem 5.7 in [29]). Roughly speaking, Theorem 2 is shown by adapting the techniques from the proof of Theorem 1 and the arguments from [20, 58] and the concentration compactness principle. In order to exclude the possible concentration of singularity at the symmetric axis induced by the term w2 , namely, the possible concentration in the weak limit  .w /2 rdrd zdt * w2 rdrd zdt C

X

ci ı.ti ; 0; zi /;

i2J

X

ci < 1;

i 2J

in the sense of measure, where . ; w /, J , and ci are the approximate solution, an at most countable set (possibly empty) and constants, respectively. To show ci D 0, besides modifying the arguments in the proof of Theorem 1, one has to adapt Lions’ concentration compactness method for stationary isothermal flows. As a by-product, a new integrability (44) of the density near r D 0 is obtained.

2.1.3 Other Symmetric Solutions First, other symmetric solutions to the system (14), (15), helically symmetric solutions, for example, will be introduced. Consider the system (14), (15) with initial data .0; x/ D 0 .x/;

.v/.0; x/ D m0 .x/;

x 2 R3 ;

33 Symmetric Solutions to the Viscous Gas Equations

1727

that are helically symmetric, i.e., 0 and m0 are periodic in x3 of period 2=˛ (0 < ˛ 2 R). For helically symmetric flows, in cylindrical coordinates .r;  ; z/ (0 < r < 1; 0    2; 1 < z < 1), the velocity vector v and the pressure a do not depend on  and z independently, but only on the linear combination  D N  C ˛z where N is a given even integer. Namely, for helically symmetric flows,

v.t; x/ D



.t; x/ D .t; r; /; x1 u .t; r; / r 1



x2 u .t; r; /; xr2 u1 .t; r; / r 2

C

x1 u .t; r; /; u3 .t; r; / r 2



for some .t; r; / and u.t; r; / D .u1 .t; r; /; u2 .t; r; /; u3 .t; r; q//, where  and u

are periodic in  of period 2, x D .x1 ; x2 ; x3 / 2 R3 , and r D helical symmetry of the initial data .0 ; m0 / means that

m0 .x/ 



x12 C x22 . Then the

0 .x/  0 .r; /; x1 0 m1 .r; / r



x2 0 m2 .r; /; xr2 m01 .r; / r

C

x1 0 m2 .r; /; m03 .r; / r



for some 0 .r; / and .m01 ; m02 ; m03 / that are periodic in  of period 2. The helically symmetric form of the compressible Navier-Stokes equations (14), (15) for the unknowns .t; r; / and u.t; r; / reads as @t  C 1r @r .ru1 / C

N r

@ .u2 / C ˛@ .u3 / D 0;

(45)

u22

@t .u1 / C 1r @r .ru21 / C Nr @ .u1 u2 / C ˛@ .u1 u3 /  r C @r p  h  i 2 @ u D  1r @r .r@r u1 / C ˛ 2 C Nr 2 @2 u1  ur 21  2N ;  2 2 r @t .u2 / C 1r @r .ru1 u2 / C Nr @ .u22 / C ˛@ .u2 u3 / C ur1 u2 C h  i  2 D  1r @r .r@r u2 / C ˛ 2 C Nr 2 @2 u2  ur 22 C 2N ; @ u  1 2 r

N r

(46)

@ p

@t .u3 / C 1r @r .ru1 u3 / C Nr @ .u2 u3 / C ˛@ .u23 / C ˛@ p  h  i 2 D  1r @r .r@r u3 / C ˛ 2 C Nr 2 @2 u3 ;

(47)

(48)

together with initial values .0; r; / D 0 .r; /; .u/.0; r; / D m0 .0; r; /; .r; / 2 RC R

(49)

and boundary conditions (at the symmetric axis) u1 .t; 0; / D u2 .t; 0; / D @r u3 .t; 0; / D 0; t > 0;  2 R;

(50)

 and u are periodic in  of period 2:

(51)

1728

S. Jiang and Q. Ju

Here p D a and m0 D .m01 ; m02 ; m03 /, and for simplicity, it is assumed that N is an even integer. The space Lp .RC .0; 2// is also defined as in (43). For the problem (45), (46), (47), (48), (49), (50), and (51), there is the following existence result [75]: Theorem 3. Let  > 1. Assume that 0 ; m0 are helically symmetric and that 0  p 0 2 L1 .RC .0; 2// \ L .RC .0; 2//; m0 = 0 2 L2 .RC .0; 2//, and 0 ; m0 are periodic in  of 2 period. Then there exists a global finite energy weak solution .; v/ of 2 -period to (11) and (12), which is helically symmetric for some ˛ .; u/ that is a weak solution to (45), (46), (47), (48), (49), (50), and (51). Moreover, for any T > 0 and ˇ 2 .0; 1/, we have Z 0

T

Z 0

1

Z 0

2

. C u21 C u22 /r ˇ drd dt  C:

The authors of [75] combined the techniques from the proof of Theorem 2 and the arguments from [20, 58] to prove Theorem 3. Compared with the axisymmetric case, the difficulties lie in the following: First, for helically symmetric flows, there are three components in the velocity field, and some swirls are allowed in the flows, and hence the equations in the symmetric form become much more complex and contain the new cross terms, which induce new difficulties and have to be dealt with carefully in weak convergence; second, at the first glance, when using the effective viscous pressure to derive higher-order estimates for the density, one should use three equations for the velocities (46), (47), and (48). Unfortunately, this could bring difficulties in defining properly the inverse of a (degenerate) elliptic operator needed in the derivation of the higher estimates. Instead, the authors in [75] use only two equations (46) and (48) to obtain the higher estimates for . Remark 3. The choice of boundary conditions (50) follows from the fact that when N is even, a smooth helically symmetric solution to (14), (15) satisfies (50) automatically. When N is odd, one has to impose the following boundary conditions, instead of (50), .u1 C N @ u2 /.t; 0; / D .u2  N @ u1 /.t; 0; / D @ u3 .t; 0; / D 0: A similar theorem can be obtained without essential changes in the arguments for N being even.

2.2

Nonisentropic Flows

In this section, the Navier-Stokes equations (8), (9), and (10) are considered for compressible heat-conducting fluids with symmetric initial data.

33 Symmetric Solutions to the Viscous Gas Equations

1729

The global existence of weak solutions to (8), (9), and (10) with spherically or cylindrically symmetric initial data and external forces in the whole space was first analyzed by Hoff and Jenssen in [30]. Here for simplicity, only the spherical symmetric case is considered. With the external force F, the Navier-Stokes equations (8), (9), and (10) are rewritten as follows: @t  C div .v/ D 0;

(52)

@t .v/ C div .v ˝ v/ C rp.; / D r.div v/ C div .2D.v// C F; (53) h  jvj2 i h  jvj2   i C cV  C div  C cV  C p v @t  2 2 D div .r / C div .vdiv v/ C div .2D.v/  v/ C v  F: (54) Consider the system (52), (53), and (54) in a ball of radius b centered at the origin in R3 with boundary conditions v D 0;

@ D 0 on @ @n

(55)

and initial conditions .; v;  /jtD0 D .0 ; v0 ; 0 /;

(56)

where n is the outer normal vector to @ . In the spherically symmetric case, namely, x x v.x; t / D v.r; t / ; .x; t/ D .r; t /; .x; t / D .r; t /; F D f .r; t/ ; (57) r r where r WD jxj, the system (52), (53), and (54) takes the following form: t C .v/r C 2r v D 0; .v/t C Œv 2 C p.; /r C 2r v 2 D ˇurr C 2ˇ

h i

v r r

C f;

. /t C .v /r C 2r v C p.;  /.vr C 2r v/ 2    v2 D  rr C 2r r C  vr C 2r v C 2.vr /2 C 4 2 ; r while the boundary and initial conditions become v D r D 0 at r D b; .; v;  /jtD0 D .0 ; v0 ; 0 /;

0 < r < b:

1730

S. Jiang and Q. Ju

Next we shall describe the existence result from [30]. The external force F is assumed to satisfy F 2 L1 .0; T I L1 . // \ L1 .0; T I Lq . //;

(58)

for each T > 0 and some q > 2. The initial data .0 ; v0 ; 0 / are assumed to satisfy C01  0  C0 ; C01  0 a:e:

Z

0 S .0 ; v0 ; 0 /dx  C0

for some positive constant C0 ;

(59) (60)



where S is the entropy density 1 S .; v;  / D K‰.1 / C ‰. / C jvj2 2 with ‰.s/ D s  log s  1. It should be noted that here S is mathematical rather than physical entropy. The global weak solutions are obtained as the limit as j ! 1 of the approximate smooth solutions .j ; vj ;  j / in an annular regions j D fx j aj < r.x/ < bg, where aj is a sequence of positive inner radii tending to zero, with modified j j j initial data .0 ; v0 ; 0 /. In describing the limit of these approximate solutions, an 1 increasing C function W Œ0; 1/ ! Œ0; 1 is fixed with  0 on Œ0; 1 and  1 on Œ2; 1/, and for R > 0, R .r/ D .r=R/ is defined. The main existence result in [30] reads as follows. Theorem 4. Assume that the parameters ; , and  in (52), (53), and (54) are positive; that p.; / D K, where K is a constant; and that the external force F and the initial data .0 ; v0 ; 0 / are spherically symmetric and satisfy the conditions (58), (59), and (60). Then the initial-boundary problem (52), (53), (54), (55), and (56) has a global weak solution .; v;  / in the form of (57) satisfying (a) supp is bounded on the left by a semicontinuous curve r W Œ0; 1/ ! Œ0; 1/. Moreover, if F is the “fluid region,” defined by F WD f.t; x/j t  0 and r.t / < r.x/  bg; then F \ ft > 0g \ fr < bg is open. (b) The density  2 L1 loc .F /, v and  are locally Hölder continuous in F \ ft > 0g, and the Navier-Stokes equations (52), (53), and (54) hold in D0 .F \ ft > 0g \ fr < bg/. N and if v (c) The density  2 C .Œ0; 1/I W 1;1 . / /. Also, .; t /  0 in n F, N is taken to be zero in n F, the weak form of the mass equation (52) holds for the N Œt1 ; t2 /: test functions 2 C 1 .

33 Symmetric Solutions to the Viscous Gas Equations

Z

ˇ t2 Z ˇ  dx ˇ D t1



1731

Z

t2

. t1

t

C v  r / dxdt:



N Œt1 ; t2 / such that (d) For t1 < t2 fixed, there are distributions Ui 2 C 2 . the weak form of the momentum equation (53) holds in the following sense: if 2 N Œt1 ; t2 / vanishes on @ , then for i D 1; 2; 3, C 2 . ˇt2 Z ˇ dx ˇ 

Z vi

t1



Z

t2

.vi t1

t

C vi v  r

C p.; /

xi / dxdt



Z

t2

Z .Fi

D t1

dxdt C Ui . /:



The distribution Ui is given by Z

t2

Z

Œ. C /vj  r

Ui . / D lim lim

R!0 j !1 t1



R xi

C .vj /i 

R

dxdt;

where R .x; t / WD R .jxj/ .x; t /. (e) The gradients rv; r 2 L1loc .F / and the weak form of the energy N Œt1 ; t2 / for which there is an equation (54) holds for test functions 2 C 2 .  > 0 such that supp .; t /  fx W r.t / C   r.x/g for each t 2 Œt1 ; t2 : Z

ˇt2 Z ˇ E dx ˇ  t1

Z

t2

t2

.E

t1

Z

Z

Z

t2

F  v dxdt 

D t1



t1



t

C .E C p.; //v  r /dxdt

Z 

1 r C rjvj2 2

C .div v/v C .rv/v  r dxdt: (f) The total energy, minus the mechanical work done by the external force, is weakly nonincreasing in time. That is, if E.t / WD

Z

h i 1 .x; t / .x; t / C jvj2 dx; 2

then E.t / D E.0/ C

Z

T

Z

Z v  F dxdt  lim lim

0



R!0 j !1 aj r.x/R

as a function of T in D0 .0; 1/, where E j D

1 j 2 jv j C  j . 2

.j E j /.x; T /dx

1732

S. Jiang and Q. Ju

Next, a brief description of the proof of Theorem 4 in [30] will be given. The solution .; v;  / is obtained as the limit as j ! 1 of the approximate solutions .j ; vj ;  j / in the annular faj < r.x/ < bg. To prove the limit, one of the key points is to establish pointwise bounds for j , independent of aj . Specifically, one j defines the particle position rh .t / by Z hD

j

rh .t/

j .r; t /r m dr

aj

for any given h  0 and shows that, for h > 0 and T > 0, there is a constant C D C .h; T / such that C .h; T /1  j .x; t /  C .h; T /

(61)

j

for t  T and r.x/  rh .t /. Then using these pointwise bounds, one can obtain the higher-order regularity estimates for vj and  j away from t D 0. Note that the bounding constants in these estimates are independent of aj . On the other hand, j

@rh @t

j

D uj .rh ; t /, so that uniform estimates for vj imply the convergence as aj ! 0 j (ignoring subsequences) of rh to a Hölder continuous curve rh ./ for h > 0. Since rh .t / is increasing in h, the limit r.t / D lim rh .t / h!0

exists, and one can define the “fluid region” F. Since r is semicontinuous, F \ ft > 0g \ fr < bg is an open set. The sequences fvj g and f j g are then uniformly Hölder continuous on compact subsets of F \ ft > 0g. The pointwise bounds (61) yield a weak limit j .; t / * .; t / on fx W r.t / C   r.x/  bg for  > 0, and it can be shown that j .; t / ! 0 in L1 .fx W r.x/  r.t /g. These limiting arguments, together with the facts that the approximations .j ; vj ;  j / are weak solutions, then enable us to conclude that .; v;  / is a weak solution as described in (b)–(f) of Theorem 4. It should be remarked that the momentum equations in Theorem 4 only hold weakly with a nonstandard interpretation of the viscous terms as distributions. When the heat conductivity  WD .;  / satisfies the following growth conditions 1 .1 C /q  .;  /; j .;  /j  2 .1 C  /q ; with q  2;

(62)

for some positive constants 1 < 2 , Zhang, Jiang, and Xie [89] proved the global existence of weak solutions for screw pinches arisen from plasma physics, which satisfy the mass, momentum, and magnetic field equations in the entire space-time domain in the standard sense of distributions. From the mathematical point of view, the condition (62) provides more integrability information on the temperature from

33 Symmetric Solutions to the Viscous Gas Equations

1733

the basic energy-entropy estimates, which helps us to derive a number of higherorder estimates of the approximate quantities (see [89]). On the other hand, it also makes the derivation of an upper bound of the temperature and thus the higher-order energy estimate on the temperature more complicated. Recently, Jiang, Jiang, and Yin [45] improved the result in [30]. They proved the global existence of weak solutions to the Navier-Stokes equations of compressible heat-conducting fluids in two dimensions with large and spherically symmetric initial data and external forces. In [45], the weak solution obtained satisfies the mass and momentum equations in the entire space-time domain, and the energy equation in any compact subset of the “fluid region,” in the sense of distributions. The crucial point of the proof in [45] is to derive some new uniform global integrability of the approximate solutions by exploiting the theory of the Orlicz spaces in two dimensions. Next, the global existence and large time behavior of strong solutions to the compressible heat-conductive Navier-Stokes equations in an exterior domain will be investigated. The spherically symmetric form of (8), (9), and (10) in Lagrangian coordinates in the exterior domain fx 2 Rn I jxj > 1g can be written as (63) ut D .r n1 v/y ; h .r n1 v/ i y R ; y 2 .0; 1/; t > 0; (64) vt D r n1 Q u u y h r 2n2  i 1 y cV t D  Q n1 v/y R .r n1 v/y 2.n  1/.r n2 v 2 /y ; (65) C Œ.r y u u where u  u.t; y/ is the specific volume, v  v.t; y/ is the velocity component in the radial direction, .t; y/ and .t; r/ denote the Lagrangian and Eulerian coordinates, respectively, Q D  C 2, and r  r.t; y/ is given by Z r.t; y/ D r0 .y/ C

t

v.y; / d ; y; t  0; 0

Z n r0 .y/ WD 1 C n

y

u0 ./ d 

o1=n

:

0

Consider nonslip and thermally insulated boundary, i.e., v.t; 0/ D 0;

y .t; 0/ D 0; t  0I

(66)

and initial conditions .u.0; y/; v.0; y/; .0; y// D .u0 .y/; v0 .y/; 0 .y//; y  0:

(67)

1734

S. Jiang and Q. Ju

For the problem (63), (64), (65), (66), and (67), one has the following existence theorem [42]: Theorem 5. Assume that u0  1; v0 ; 0  1; r0n1 @y u0 ; r0n1 @y v0 ; r0n1 @y 0 2 L2 .0; 1/I u0 .y/; 0 .y/ > 0 for all y 2 Œ0; 1/; and that the initial data are compatible with the boundary conditions (66). Then the problem (63), (64), (65), (66), and (67) has a unique solution fu; v;  g with u;  > 0, such that for any T > 0, u  1; v;   1 2 L1 .0; T I H 1 .0; 1//;

ut 2 L1 .0; T I L2 .0; 1//;

vt ; t ; uyt ; vyy ; yy ; r n1 y ; r 2n2 yy 2 L2 ..0; T / .0; 1//; and .u; v;  / satisfies (63), (64), (65), (66), and (67) almost everywhere in .0; T / .0; 1/ and takes on the given boundary and initial conditions in the sense of traces. Moreover, when n D 3, there are positive constants u; uN independent of t and y, such that u  u.t; y/  uN for all t; y  0, and for an arbitrary but fixed integer j  2, kv.t /kL2j .0;1/ ! 0 as t ! 1. The proof of Theorem 5 is essentially based on a careful examination of a priori estimates and a limit procedure. Since the domain is unbounded and the coefficients tend to infinity as y ! 1, some difficulties arise; for example, from the a priori estimates, one could get only v.y; t/ D o.y 1=2C1=.2n/ / as y ! 1; but this is not sufficient to guarantee integration by parts in the proof where v D o.y 1C1=n / is required. To overcome such difficulties, one first studies an approximate problem in the bounded interval .0; k/ and shows the a priori estimates independent of k by utilizing some cutoff function and modifying a technique of Kazhikhov [1, 55] for the one-dimensional case, and then letting k tend to infinity and using the obtained a priori estimates, one obtains a global spherically symmetric solution as the limit. The large-time behavior is obtained by adapting an idea of Kazhikhov [55] to derive a representation of u from which the uniform boundedness of u from below and above follows and utilizing tricky energy estimates to derive L2j -boundedness of v uniformly in t . Remark 4. The same techniques work and an analogous theorem is obtained when (66) is replaced by the following boundary conditions: v.t; 0/ D 0; .t; 0/ D 1; t  0: The results in Theorem 5 have been extended to the case that the potential forces are present in the system (8), (9), and (10) by Nakamura and Nishibata [66].

33 Symmetric Solutions to the Viscous Gas Equations

1735

In this case the system (8), (9), and (10) is changed into the following system with an external force f: @t  C div .v/ D 0;

(68)

f@t v C .v  r/vg C Rr. / D r.div v/ C div .2D.v// C f;

(69)

cV ft C .v  r/g C Rr  v D  C .div v/ C 2D.v/  D.v/: (70) 2

As in [42], the flow is over an exterior domain WD fx 2 Rn W jxj > 1g with n  3. The initial and the boundary conditions are prescribed by ..x; 0/; v.x; 0/; .x; 0// D .0 .x/; v0 .x/; 0 .x//; @ ˇˇ vj@ D 0; ˇ D 0; @ @

(71) (72)

where  denotes the unit outward normal vector. Assume that the external force f is given by the spherically symmetric potential force x r D jxj; (73) f WD rU D  Ur .r/; r then the spherically symmetric form of (68), (69), and (70) reads as t C .v/r C

n1 v r

D 0;

.vt C vvr / C R. /r D ˇvrr C ˇ.n  1/

(74) h i

v r r

 Ur ;

cV .t C v/r C R.vr C n1 v/ r    2 D  rr C n1 r C  vr C n1 v C 2.vr /2 C 2 n1 v2; r r r2

(75)

(76)

where ˇ D 2 C  > 0. The initial and boundary conditions (71) and (72) reduce to ..r; 0/; v.r; 0/; .r; 0// D .0 .r/; v0 .r/; 0 .r//;

(77)

v.1; t / D 0 D 0; r .1; t/ D 0:

(78)

The initial data are assumed to satisfy

inf

r2Œ1;1/

0 .r/ > 0;

inf

r2Œ1;1/

0 .r/ > 0;

lim .0 .r/; v0 .r/; 0 .r// D .C ; vC ; C /; C > 0; C > 0;

r!1

(79) (80)

1736

S. Jiang and Q. Ju

where C ; vC ; C are constants. Moreover, the initial data .0 ; v0 ; 0 / are supposed to be compatible with boundary data (78): v0 .1/ D 0; 0r .1/ D 0; n  .r n1 v /  o 0 r ˇ jrD1 D 0:  R. v /  U 0 0 r r r r n1

(81) (82)

N The stationary solution ..r/; N v.r/; N .r// is a solution to equations (74), (75), and (76) independent of time t and satisfies the same boundary and spatial N asymptotic conditions (78) and (80). The stationary solution ..r/; N v.r/; N .r// is obtained explicitly as follows:  .r/ N D C exp 

 1 U .r/ ; RC

v.r/ N D 0;

N .r/ D C ;

(83)

where we have assumed that Z lim U .r/ D lim

r!1

r!1 1

r

Ur ./ d  C U .1/ D 0;

without loss of generality. Thus, one obtains the following large-time behavior [66]. Theorem 6. Suppose that the initial data .0 ; v0 ; 0 / satisfy r .n1/=2 .0  ; N v0 ; 0  C ; .0  / N r ; v0r ; 0r / 2 L2 .1; 1/; 0 2 B 1C Œ1; 1/; .v0 ; 0 / 2 B 2C Œ1; 1/ for a certain constant  2 .0; 1/. Then there exists a constant ı > 0, such that if the external force Ur 2 C 1 Œ1; 1/ \ B  Œ1; 1/ satisfies ı  Ur .r/ for any r  1, then the initial boundary value problem (74), (75), (76), (77), and (78) has a unique solution .; v;  / satisfying r .n1/=2 .  ; N v;   C ; .  / N r ; vr ; r / 2 C 0 .Œ0; T I L2 .1; 1//;  2 B 1C;1C=2 .Œ1; 1/ Œ0; T /; .v; / 2 B 2C;1C=2 .Œ1; 1/ Œ0; T / for any T > 0. Moreover, the solution .; v;  / converges to the corresponding stationary solution (83) as time tends to infinity: lim

sup j..r; t /  .r/; N v.r; t /; .r; t /  C /j D 0:

t!1 r2Œ1;1/

The idea of the proof of Theorem 6 is as follows. To show Theorem 6, one first transforms the equations in Eulerian coordinates into that in Lagrangian coordinates

33 Symmetric Solutions to the Viscous Gas Equations

1737

and considers an approximate problem restricted in the bounded interval .0; m/ with a positive integer m. Then, the a priori estimates uniformly in m are established. By letting m ! 1, the local-in-time existence in the unbounded interval .0; 1/ is proved. To obtain a global solution and the convergence to the stationary solution, one derives the pointwise boundedness of the density by using the representation formula of the density with basic energy estimates. Finally, the Hölder estimates of the solution are derived to translate the stability theorem in Lagrangian coordinates into that in Eulerian coordinates. As remarked in [66], for the Dirichlet boundary condition on the temperature, the same results as in Theorem 6 hold if the temperature boundary condition .1; t /DC is imposed, in place of r .1; t/ D 0 in (78). The case of .1; t / ¤ C still remains open. There are other works contributing to the existence and large-time behavior of spherically symmetric solutions in annular domains. Nikolaev [69] first studied the initial boundary value problem (63), (64), (65), (67), and (68) in an annular domain and proved the global existence of smooth spherically symmetric solutions for strictly positive initial density and temperature, while the exponential decay as t ! 1 of the corresponding spherically symmetric solutions was shown in [43]. Yashima and Benabidallah [22, 23] dealt with the case of nonnegative initial density and temperature and established the global existence of weak spherically symmetric solutions. Using a difference scheme and a limit procedure, Chen and Kratka obtained the global spherically symmetric solutions for a free boundary problem in an annulus [6]. In [91], Zheng and Qin studied the existence of maximal attractors for the spherically symmetric smooth solutions in an annulus. It should also be mentioned that for viscous polytropic ideal gases, the local existence of unique strong solutions with vacuum was shown by Cho and Kim in [9] when initial data satisfy some compatibility condition, and the global existence of classical solutions to the Cauchy problem with small initial energy and non-vacuum state at infinity was established recently by Huang and Li [34].

3

Density-/Temperature-Dependent Viscosity and/or Heat Conductivity

One turns to the case when , , and  are not constants in (11)–(12) or (8), (9), and (10), but may depend on the density or temperature. The compressible Navier-Stokes equations with constant viscosity behave singularly in the presence of vacuum, see, e.g., [31, 60, 84]. By certain considerations from both mathematical and physical points of view, Liu, Xin, and Yang [60] introduced a modified NavierStokes equations in which the viscosity coefficients depend on the density. From the physical point of view, one can find that viscosity and heat conductivity are not constants but depend on the temperature in the derivation of the compressible Navier-Stokes equations from the Boltzmann equation through the ChapmanEnskog expansion to the second order (cf. [24]) as discussed in [60]. For isentropic

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fluids, this dependence is translated into the dependence on the density by the laws of Boyle and Gay-Lussac for ideal gases . In this section, the symmetric solutions to the isentropic compressible NavierStokes equations with density-dependent viscosity coefficients will be considered first, and then the symmetric solutions to the nonisentropic compressible NavierStokes equations will be discussed.

3.1

Isentropic Flows

The compressible Navier-Stokes equations with density-dependent viscosity coefficients can be written as @t  C div .v/ D 0;

(84)

@t .v/ C div .v ˝ v/ C ar D r../div v/ C div ../D.v//;

(85)

where t 2 .0; 1/ and x 2 Rn (n D 2; 3), .x; t / and v.x; t / stand for the fluid density and velocity, respectively, and D.v/ is the deformation tensor as in (5), ./, and ./ are the Lamé viscosity coefficients satisfying ./ > 0;

./ C n./  0:

In particular, the viscous Saint-Venant system for shallow water is expressed exactly as (84) and (85) with n D 2, ./ D , ./ D 0, and  D 2. Now, consider the case ./ D  and ./ D 0, and investigate the spherically symmetric solutions to (84), (85) in the ball of radius R centered at the origin in R3 . The initial and boundary conditions for (84), (85) are imposed as .; v/jtD0 D .0 ; m0 /;

(86)

m D v D 0 on @ :

(87)

For spherically symmetric solutions, one sets .x; t / D .r; t /;

x v.x; t / D v.r; t / ; r

r D jxj:

Then the system (84), (85) reduces to the following equations in spherical symmetry for r > 0: t C .v/r C

2v r

D 0;

.v/t C Œv 2 C a r C 2r v 2  .ur /r  

 

2v r r

D0

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with initial data .; v/jtD0 D .0 ; m0 / and boundary conditions v.0; t / D 0; v.R; t/ D 0: The existence of global weak solutions to (84), (85), (86), and (87) was established by Guo, Jiu, and Xin [25] when the initial data are large and spherically symmetric. Before stating this existence result, a definition of weak solutions to (84), (85), (86), and (87) from [25] is given. For simplicity, one takes that D.v/  rv, which holds for the spherically symmetric case, i.e., for v being spherically symmetric; also see Remark 5. Definition 3. A pair .; v/ is said to be a weak solution to (84), (85) provided that (1)   0 a.e. and  2 L1 .0; T I L1 . / \ L . // \ C .Œ0; 1/; W 1;1 . / /; p p  2 L1 .0; T I H 1 . //; v 2 L1 .0; T I L2 . //; where W 1;1 . / is the dual space of W 1;1 . /; N Œt1 ; t2 /, the mass equation (84) holds (2) For any t2  t1  0 and any  2 C 1 . in the following sense: Z

ˇt2 Z ˇ  dx ˇ D t1

t2

Z .t C v  r/ dxdtI

t1



N Œ0; T / satisfying (3) for any D . 1 ; 2 ; 3 / 2 C 2 . .x; T / D 0, it holds that Z

Z m0 

T

Z

.0; / dx C



0

Z

T

Z

C 0

p p Π. v/  @t

C

p

.x; t / D 0 on @ and

v ˝

p

v W r  dxdt



 div

dxdt C hrv; r i D 0;



where the diffusion term makes sense when written as TZ

Z

p p . v/ dxdt 2

hrv; r i D  0



TZ

Z 0



p p . v/.r r/ dxdt:

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The initial data are assumed to satisfy 0  0 a.e. in I m0 D 0 a.e. on fx 2 j0 .x/ D 0gI

(88)

2C

0 2 W 1;2 . /I

m20 m0 2 L1 . /I 2 L1 . /; 1C 0 0

(89)

where  2 .0; 1/ is some small constant. Then one has the following existence result [25]. Theorem 7. For n D 3 and 1 <  < 3, if the initial data have the form 0 D 0 .jxj/;

v0 D v0 .jxj/

x r

satisfying (88), (89), then the initial-boundary value problem (84), (85), (86), and (87) possesses a global spherically symmetric weak solution in the form  D .jxj; t /;

v D v.jxj; t /

x r

satisfying, for all T > 0, p v 2 L1 .0; T I L2 . //; .x; t / 2 C .Œ0; T I L3=2 . //; Z Z .x; t / dx D 0 .x/ dx:



Moreover, it holds that Z   1 1 p sup jvj2 C  C jr j2 C jvj2C dx  C;  1 t2Œ0;T  2 where C is a positive constant. The proof of Theorem 7 is based on the construction of suitable approximate solutions and a priori estimates for the approximate solutions. Motivated by the approach of Jiang, Xin, and Zhang [49], in which the one-dimensional case was dealt with, the authors of [25] construct the approximate solutions by solving an approximate system of (84), (85) with  ./ D ./ C ˇ and  ./ D ./ C .ˇ  1/ˇ for some fixed 0 < ˇ < 1 instead of ./ and ./ in (84) and (85). Furthermore, the corresponding radially symmetric equations are used in the annular domain D nBN .0/, where B .0/ is the ball with radius and center 0 and the corresponding equations in Lagrangian coordinates are introduced to obtain lower bounds of the approximate solutions. By this approach, a class of

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approximate solutions are constructed with desired a priori estimates, such as the energy estimates and entropy estimates. Then, to take to the limit of the approximate solutions in order to obtain a weak solution defined on the entire domain , an p appropriate extension is performed to keep the uniform boundedness of  in L1 .0; T I H 1 . //. Finally, the global existence of a weak solution to the original system is obtained by taking to the limit of the approximate solutions based on the uniform in estimates. Remark 5. As aforementioned, Theorem 7 was proved in [25] under the assumption on the deformation tensor D.v/  rv which is actually not a restriction for spherically symmetric flows since D.v/ D rv for any spherically symmetric v. On the other hand, however, for non-spherically symmetric flows, the situation is quite different. When the general initial data are considered, some partial results were obtained independently by Vasseur and Yu [78] and Li and Xin [57] on the existence of global weak solutions to the three-dimensional compressible Navier-Stokes equations with degenerate viscosities. In particular, two cases of the deformation tensor (either D.v/  rv or (5)) were discussed in [57]. It is remarked in [25] that the analysis can be applied to slightly more general viscosity coefficients ./ D ˛ and ./ D .˛  1/˛ for some ˛ > n1 .n D n 2; 3/. More general, ./ and ./ satisfying ./ D 0 ./  ./ and other restrictions given in [65] can be handled in a similar way. In addition, the shallow water equations corresponding to the case of n D 2, ˛ D 1, and  D 2 in (84) and (85) are covered. The free boundary value problem for (84), (85) with stress free across a free surface has been studied by Guo, Li, and Xin [26] for the case ./ D  and ./ D 0. In this case, the domain is changed into T D f.r; t / j 0  r  a.t /; 0  t  T g: At the center of symmetry, the following Dirichlet boundary condition is imposed: v.0; t / D 0; and the free surface @ t moves in the radial direction along the “particle path” r D a.t / with stress-free boundary condition 2 .  .vr C v//.a.t /; t / D 0; r

t > 0;

where a0 .t / D v.a.t /; t /, t > 0, and a.0/ D a0 . Now, the main results given in [26] are as follows. First, it is shown that a global spherically symmetric entropy weak solution to the free boundary problem for (84) and (85) exists for general initial data, and the free surface moves along particle paths in radial direction. Then the Lagrangian structure of the global weak solution

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is investigated, and it is shown that starting from any point in initial non-vacuum regions, a particle path is uniquely defined globally in time, along which the flow density is strictly positive and bounded from below and above in finite time. Any two particle paths starting from two initially separated points in non-vacuum regions shall be separated uniformly from each other for all time. Between the two particle paths, vacuum states shall not form in any finite time so long as there is no vacuum state initially, and the initial regularities of solutions are maintained. In addition, the free surface is shown to expand outward in the radial direction at an algebraic rate in time, and the density decays to zero time asymptotically almost everywhere away from the symmetry center; this leads to the formation of vacuum states as time goes to infinity. Finally, the dynamics of vacuum states for the global spherically symmetric entropy weak solution are studied, and the large-time behavior of any global entropy weak solutions is shown in [26]. For one-dimensional compressible isentropic flows, there is an extensive literature on the density-dependent viscosity system, see, e.g., [27,44,49,56,60,61,71,79, 86–88], and the reference cited therein. The case of the multidimensional spherically symmetric domain between a solid core and a free boundary was investigated in [6, 7, 81]. For general domains in higher dimensions, assuming that  is a constant and ./ D aˇ with ˇ > 3, Vaigant and Kazhikhov [76] first proved the global existence of strong and classical solutions to the two-dimensional system of (84) and (85) with slip boundary condition, while the condition ˇ > 3 is relaxed to ˇ > 4=3 for the Cauchy problem and the problem of periodic boundary conditions in [35, 36, 50–52]. For the case of  D ./ and  D ./, the first multidimensional result is due to Bresch, Desjardins, and Lin [5], who established the L1 -stability of weak solutions to the Korteweg system (with the Korteweg stress tensor kr). Bresch and Desjardins [3] improved the result in [5] to the case of vanishing capillarity k D 0 but with an additional quadratic friction term. Mellet and Vasseur [65] proved the L1 -stability of weak solutions to the system (11) and (12) based on the new entropy estimates, extending the corresponding L1 -stability in [3, 5] to the case of vanishing capillarity and friction term. Recently, the existence of global weak solutions to the three-dimensional compressible Navier-Stokes equations with degenerate viscosities was obtained independently by Vasseur and Yu [78] and Li and Xin [57].

3.2

Nonisentropic Flows

As mentioned at the beginning of this section, both viscosity and heat conductivity depend on the temperature in the derivation of the compressible Navier-Stokes equations from the Boltzmann equation through the Chapman-Enskog expansion. Global existence theory was established for the equations of one-dimensional viscous heat-conductive real gases in [41, 54] and for the equations of nonlinear thermoviscoelasticity in [12, 13] when viscosity depends on  , and heat conductivity is allowed to depend on the temperature in a special way with a positive lower bound

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and balanced with corresponding constitution relations. We refer the readers to the book by Hsiao [32] and the survey article by Hsiao and Jiang [33] for more references. Jenssen and Karper [40] proved the global existence of a weak solution for ideal polytropic gas flow, when viscosity is a constant and heat conductivity depends on the temperature in power law. And the corresponding existence of global strong solutions was obtained by Pan and Zhang in [72] recently. It should be remarked that the case of temperature-dependent viscosity still remains to be a difficult problem, and some progress on the global existence has been made recently. When the adiabatic exponent is close to 1, Liu, Yang, Zhao, and Zou in [59] established the global existence of large solutions to the one-dimensional compressible Navier-Stokes equations with temperature-dependent viscosity, while in [80] Wang and Zhao investigated the case that the viscosity coefficient behaves like h./ ˛ with ˛ sufficiently small and h satisfying certain growth conditions. For the multidimensional full compressible Navier-Stokes equations with temperature-dependent viscosity and/or heat conductivity, to the best knowledge of the authors of this chapter, mathematical results on global solutions are mainly limited to the case of special pressure, viscosity, and heat conductivity. Under certain growth conditions upon the pressure, viscosity, and heat conductivity (i.e., radiative gases), Feireisl established the global existence of the so-called variational solutions in the sense that the energy equation is replaced by an energy inequality in the distributional sense; see [19], for example. Bresch and Desjardins [4] obtained the global existence of weak solutions in the standard sense of distributions to the Cauchy problem or spatially periodic problem when viscosity, heat conductivity, pressure, and internal energy satisfy some assumptions. It should be pointed out that the models in [4, 19] do not cover the case of viscous polytropic ideal gases. Recently, the global existence of classical and strong solutions in an annulus, with spherically or cylindrically symmetric initial data, was given by Wen and Zhu [83] when the viscosity coefficients are constants, and the pressure and heat conductivity satisfy some assumptions. The corresponding one-dimensional case was dealt with by the same authors in [82].

4

Related Problems

In this subsection some related results concerning the symmetric solutions to the Navier-Stokes-Poisson system for a self-gravitating barotropic gas are presented. In the case of constant viscosity, Jang [38] established the local-in-time well-posedness of strong solutions to a vacuum free boundary problem for spherically symmetric isentropic flows. When the viscosity coefficients are density dependent, Zlotnik and Ducomet [17, 92] proved the existence, uniqueness, and global behavior of the spherically symmetric solutions to the compressible Navier-Stokes equations with a general mass force and a solid core, while a free boundary problem without a solid core was studied by Zhang and Fang [90]. The global existence of spherically symmetric weak solutions to the Cauchy problem was shown by Ducomet, Necasova, and Vasseur in [15, 16].

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Conclusion

The global existence of weak solutions to the compressible isentropic Navier-Stokes equations in Rn (n D 2; 3) with large and symmetric (spherically symmetric, axisymmetric, or helically symmetric) initial data has been established for any specific heat ratio  > 1. When  D 1, which corresponds to isothermal flows, the global existence of weak solutions has been obtained for spherically symmetric BVinitial data in three dimensions and Lp -initial data in two dimensions. On the other hand, for compressible heat-conducting flows, the global existence of spherically symmetric weak solutions still remains open in general, even though some partial progress has been achieved for positive initial density. The global existence and large-time behavior of spherically symmetric strong solutions to the compressible heat-conducting Navier-Stokes equations have been obtained only in symmetric domains without the origin. Recently, there has been much progress on the existence of global symmetric solutions to the compressible Navier-Stokes equations with density-/temperaturedependent viscosity and heat conductivity in multidimensions. The global existence of spherically symmetric weak solutions for isentropic flows with density-dependent viscosity has been shown for large initial data in a ball, while the corresponding free boundary value problem has been studied as well. Concerning the multidimensional nonisentropic compressible Navier-Stokes equations with temperature-dependent viscosity and/or heat conductivity, the present mathematical results on global solutions are mainly restricted to the cases of special pressure, viscosity, and heat conductivity.

6

Cross-References

 Concepts of Solutions in the Thermodynamics of Compressible Fluids  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows  Existence of Stationary Weak Solutions for Isentropic and Isothermal Compress-

ible Flows  Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical

Cases  Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stabil-

ity, and Longtime Behavior Acknowledgements Jiang is supported in part by the National Basic Research Program (Grant Nos. 2014CB745000, 2011CB309705), NSFC (Grant Nos. 11229101, 11371065), and Beijing Center for Mathematics and Information Interdisciplinary Sciences. Ju is supported by NSFC (Grant Nos. 11171035, 11571046, 11471028), the National Basic Research Program (Grant No. 2014CB745000) and BJNSF (Grant No. 1142001).

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44. S. Jiang, Global smooth solutions of the equations of a viscous, heat-conducting, onedimensional gas with density-dependent viscosity. Math. Nachr. 190, 169–183 (1998) 45. F. Jiang, S. Jiang, J.P. Yin, Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and force. Discrete Contin. Dyn. Syst. 34, 567–587 (2014) 46. S. Jiang, P. Zhang, Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Commun. Math. Phys. 215, 559–581 (2001) 47. S. Jiang, P. Zhang, Remarks on weak solutions to the Navier-Stokes equations for 2-D compressible isothermal fluids with spherically symmetric initial data. Indiana Univ. Math. J. 51, 345–355 (2002) 48. S. Jiang, P. Zhang, Axisymmetric solutions of the 3-D Navier-Stokes equations for compressible isentropic fluids. J. Pures Appl. Math. 82, 949–973 (2003) 49. S. Jiang, Z.P. Xin, P. Zhang, Global weak solutions to 1D compressible isentropic NavierStokes equations with density-dependent viscosity. Methods Appl. Anal. 12, 239–252 (2005) 50. Q. Jiu, Y. Wang, Z. Xin, Global well-posedness of the Cauchy problem of two dimensional compressible Navier-Stokes equations in weighted spaces. J. Differ. Equ. 255, 351–404 (2013) 51. Q. Jiu, Y. Wang, Z. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum. J. Math. Fluid Mech. 16, 483–521 (2014) 52. Q. Jiu, Y. Wang, Z. Xin, Global classical solutions to the two-dimensional compressible NavierStokes equations in R2 . arXiv:1209.0157 53. J.L. Joly, G. Métivier, J. Rauch, Focusing at a point and absorption of nonlinear oscillations. Trans. Am. Math. Soc. 347, 3921–3970 (1995) 54. B. Kawohl, Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas. J. Differ. Equ. 58, 76–103 (1985) 55. A.V. Kazhikhov, On the theory of initial-boundary value problems for the equations of onedimensional nonstationary motion of a viscous heat-conducting gas. Din. Sploshnoj Sredy 50, 37–62 (1981) (Russian) 56. H.L. Li, J. Li, Z. Xin, Vanishing of vacuum states and blow up phenomena of the compressible Navier-Stokes equations. Commun. Math. Phys. 281, 401–444 (2008) 57. J. Li, Z. Xin, Global existence of weak solutions to the barotropic compressible Navier-Stokes flows with degenerate viscosities. arXiv:1504.0682v1. 58. P.L. Lions, Mathematical Topics in Fluid Mechanics, vol. II, Compressible Models (Oxford Science Publications/Clarendon Press, Oxford, 1998) 59. H. Liu, T. Yang, H. Zhao, Q. Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data. SIAM J. Math. Anal. 46, 2185–2228 (2014) 60. T.P. Liu, Z. Xin, T. Yang, Vacuum states of compressible flow. Discrete Contin. Dyn. Syst. 4, 1–32 (1998) 61. T. Makino, On a local existence theorem for the evolution equations of gaseous stars, in Patterns and Wave Qualitative Analysis of Nonlinear Differential Equations (North-Holland, Amsterdam/New York, 1986), pp. 459–479 62. A. Matsumura, Large time behavior of the spherically symmetric solutions of an isothermal model of compressible viscous gas. Trans. Theorem Stat. Phys. 21, 579–592 (1992) 63. A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980) 64. A. Matsumura, T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89, 445–464 (1983) 65. A. Mellet, A. Vasseur, On the barotropic compressible Navier-Stokes equation. Commun. Partial Differ. Equ. 32, 431–452 (2007) 66. T. Nakamura, S. Nishibata, Large-time behavior of spherical flow of heat-conducting gas in a field of potential forces. Indiana Univ. Math. J. 57, 1019–1054 (2008)

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67. T. Nakamura, S. Nishibata, S. Yanagi, Large-time behavior of spherically symmetric solutions to an isentropic model of compressible viscous fluid in a field of external forces. Math. Models Method Appl. Sci. 14, 1849–1879 (2004) 68. J. Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962) 69. V.B. Nikolaev, On the solvability of mixed problem for one-dimensional axisymmetrical viscous gas flow. Dinamicheskie zadachi Mekhaniki sploshnoj sredy 63, Sibirsk. Otd. Acad. Nauk SSSR, Inst. Gidrodinamiki (1983) (Russian) 70. A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow (Oxford University Press, Oxford, 2004) 71. M. Okada, Š. Matuš˙u-Neˇcasová, T. Makino, Free boundary problem for the equation of onedimensional motion of compressible gas with density-dependent viscosity. Ann. Univ. Ferrara Sez. VII (N.S.) 48, 1–20 (2002) 72. R.H. Pan, W. Zhang, Compressible Navier-Stokes equations with temperature dependent heat conductivities. Commun. Math. Sci. 13, 401–425 (2015) 73. R. Salvi, I. Straškraba, Global existence for viscous compressible fluids and their behavior as t ! 1. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40, 17–51 (1993) 74. J. Serrin, Mathematical principles of classical fluid mechanics. Handbuch der Physik VIII/1, (Springer, Berlin/Heidelberg/New York, 1972), pp. 125–262 75. W.J. Sun, S. Jiang, Z.H. Guo, Helically symmetric solutions to the 3-D Navier-Stokes equations for compressible isentropic fluids. J. Differ. Equ. 222, 263–296 (2006) 76. V.A. Vaigant, A.V. Kazhikhov, On the existence of global solutions of two dimensional NavierStokes equations of a compressible viscous fluid. Sibirsk. Mat. Zh. 36, 1283–1316 (1995) (Russian) 77. A. Valli, W.M. Zaja¸czkowski, Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103, 259–296 (1986) 78. A. Vasseur, C. Yu, Existence of global weak solutions for 3D degenerate compressible NavierStokes equations. arXiv:1501.06803v4. 79. S.W. Vong, T. Yang, C.J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum (II). J. Differ. Equ. 192, 475–501 (2003) 80. T. Wang, H. Zhao, Global large solutions to a viscous-heat conducting one dimensional gas with temperature-dependent viscosity. arXiv:1505.05252 81. M. Wei, T. Zhang, D. Fang, Global behavior of spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients. SIAM J. Math. Anal. 40, 869–904 (2008) 82. H.Y. Wen, C.J. Zhu, Global classical large solutions to Navier-Stokes equations for viscous compressible and heat conducting fluids with vacuum. SIAM J. Math. Anal. 45, 431–468 (2013) 83. H.Y. Wen, C.J. Zhu, Global symmetric classical solutions of the full compressible NavierStokes equations with vacuum and large initial data. J. Math. Pures Appl. 102, 498–545 (2014) 84. Z. Xin, Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Commun. Pure Appl. Math. 51, 229–240 (1998) 85. Z. Xin, H. Yuan, Vacuum state for spherically symmetric solutions of the compressible NavierStokes equations. J. Hyper. Differ. Equ. 3, 403–442 (2006) 86. T. Yang, Z.A. Yao, C.J. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Commun. Partial Differ. Equ. 26, 965–981 (2001) 87. T. Yang, C.J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Commun. Math. Phys. 230, 329–363 (2002) 88. T. Yang, H.J. Zhao, A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. J. Differ. Equ. 184, 163–184 (2002) 89. J. Zhang, S. Jiang, F. Xie, Global weak solutions of an initial boundary value problem for screw pinches in plasma physics. Math. Models Methods Appl. Sci. 19, 833–875 (2009)

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90. T. Zhang, D. Fang, Global behavior of spherically symmetric Navier-Stokes-Poisson system with degenerate viscosity coefficients. Arch. Ration. Mech. Anal. 191, 195–243 (2009) 91. S. Zheng, Y. Qin, Universal attractors for the Navier-Stokes equations of compressible and heat-conductive fluid in bounded annular domains in Rn . Arch. Ration. Mech. Anal. 160, 153–179 (2001) 92. A.A. Zlotnik, B. Ducomet, The stabilization rate and stability of viscous compressible barotropic symmetric flows with a free boundary for a general mass force. Sb. Math. 196, 1745–1799 (2005)

Local and Global Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Energy Method

34

Jan Burczak, Yoshihiro Shibata, and Wojciech M. Zaja¸czkowski

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Short-Time Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Global-in-time Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Energy Methods in Free Boundary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Notation and Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Local Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Estimates for Approximative Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Estimates for Differences of Approximative Sequences . . . . . . . . . . . . . . . . . . . . . 4.3 Proof of Theorem 4 (Local-in-time Existence) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Global Existence for Small Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Lower-Order Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1752 1753 1754 1756 1761 1766 1766 1766 1768 1772 1774 1776 1808 1815 1818 1820

J. Burczak () Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland OxPDE, Mathematical Institute, University of Oxford, Oxford, UK e-mail: [email protected] Y. Shibata Department of Mathematics and Research Institute for Science and Engineering, Waseda University, Shinjuku-ku, Tokyo, Japan e-mail: [email protected] W.M. Zaja¸czkowski Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, Warsaw, Poland e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_47

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5.2 Higher-Order Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Differential Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Proof of Theorem 5 (Large-Time Existence for Small Data) . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1821 1831 1835 1837 1837 1839

Abstract

In this chapter we present the classical energy approach for existence of regular solutions to the equations of compressible, heat-conducting fluids in a bounded three-dimensional domain. Firstly, we provide a state of the art and recall representative results in this field. Next, we give a proof of one of them, concerning Dirichlet boundary conditions for velocity and temperature. The result and thus the proof is divided into two main parts. A local-in-time existence result in high-regularity norms, via a method of successive approximations, occupies the former one. In the latter part, a differential inequality is derived, which allows us to extend the local-in-time solution to the global-in-time solution, provided a certain smallness condition is satisfied. This smallness condition is in fact an equilibrium proximity condition, since it involves differences between data and constants, whereas the data for temperature and density may be large themselves. All our considerations are performed within the L2 -approach. The proved result is close to that of Valli and Zaja¸czkowski (Commun Math Phys 103:259–296, 1986), but the techniques used here: the method of successive approximations (instead of a Leray-Schauder fixed-point argument there) as well as a clear continuation argument renders our exposition more traceable. Moreover, one may easily derive now an explicit smallness condition via our approach. Besides, the thermodynamic restriction on viscosities is relaxed, certain technicalities are improved and a possibly useful approach to deal with certain difficulties at the boundary in similar problems is provided.

1

Introduction

In this chapter we discuss the energy method for existence of regular solutions to the evolutionary equations of compressible, heat-conducting fluids in a threedimensional, bounded domain. Despite being probably the second most classical approach for solving partial differential equations, the energy estimates are still worth studying, since in many cases they provide the sharpest available results. Moreover, however sometimes technically repellent, they are generally less troublesome than the use of potentials (possibly, the “first most classical approach”) and are much better suited to nonlinear problems. In fact, the general idea behind them is immediate to grasp, and they are in principle more arduous than difficult. This is especially the case of involved systems of PDEs, where there is no hope for their well-posedness and regularity “in the large”, due to fundamental obstacles present already in much simpler systems. Our situation here is precisely of this type: the

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difficulties imminent to the classical Navier-Stokes system indicate, that with any of the currently known methods, the best possible results in three-dimensional case involving certain extra assumptions. The most typical ones are that of smallness, either of the time of evolution or of data. This chapter is divided into two main parts. The first one (Sect. 2), after introducing the considered equations, recalls a range of results concerning it, including citing representative theorems, with the emphasis on those obtained by the energy method. The second part (remaining sections) announces our results (Theorems 4 and 5) and provides their proofs. Both results and proofs can be seen as another representative of an existence result for a heat-conducting compressible flow, obtained within the energy approach. In fact, Theorems 4 and 5 are a slight modification of those presented in [39], with new, much more traceable proofs that are free from certain previous flaws. In addition, we expect that presenting details of proofs may be advantageous for readers that intend to make themselves acquainted with details of the energy method.

2

Overview

In a three-dimensional domain, a viscous, heat-conducting fluid obeys the following system of equations %.vt C v  rv/  .v C rdiv v/ C rp D %f; %t C div .%v/ D 0 %cv .t C v  r / C  p div v   D

(1)  jD.v/j2 C .  /.div v/2 C %b: 2

The unknowns are the fluid velocity field v, its scalar density %  0 and temperature   0. The pressure p D p.%; / and the specific heat at a constant volume cv D cv .%;  / are known functions of unknowns %,  . Finally, f and b are, respectively, given external force field per unit mass and heat sources per unit mass. By Du we Tu . denote the symmetric part of gradient, i.e., Du D ruCr 2 The simplification, where p and % completely determine each other (no temperature dependence of p), is referred to as the barotropic case. The isothermal case neglects temperature dependence entirely. On the other hand, in full generality, the coefficients ;  may become non-constants. Naturally, to complete the system (1), one needs to prescribe the considered domain and related data. For the sake of flexibility, in the following presentation of known results, we denote them separately for each presented problem. Some of the known results concerning the existence of regular solutions to the evolutionary, compressible Navier-Stokes equations are recalled in what follows. In particular, these focused on weak solutions are not mentioned. For the competent presentations of other results related to compressible Navier-Stokes system, we refer to other chapters in this handbook, including  “Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows”,  “Existence of Stationary Weak Solutions for Isentropic and Isothermal Compressible Flows”,

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 “Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows” (stationary problems), “ Weak Solutions for the Compressible NavierStokes Equations with Density Dependent Viscosities,” “ Weak Solutions for

the Compressible Navier-Stokes Equations: Existence, Stability, and Longtime Behavior,” “ Weak Solutions for the Compressible Navier-Stokes Equations in the Intermediate Regularity Class,” “ Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical Cases” (weak solutions), “ Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak Solutions for the Compressible Navier-Stokes Equations,” “ Finite Time Blow-Up of Regular Solutions for Compressible Flows” (blowup-related issues), “ Multi-Fluid Models Including Compressible Fluids,” “ Solutions for Models of Chemically Reacting Compressible Mixtures” (compressible multifluids), “ Global Existence of Classical Solutions and Optimal Decay Rate for Compressible Flows via the Theory of Semigroups” (space/time asymptotics), and “ Singular Limits for Models of Compressible, Viscous, Heat Conducting, and/or Rotating Fluids” (singular limits). The closest to our results are those related to existence of smooth solutions for 3D evolutionary problems, including “ Global Existence of Regular Solutions with Large Oscillations and Vacuum for Compressible Flows,” “ Local and Global Existence of Strong Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Maximal Regularity,” “ Local and Global Solvability of Free Boundary Problems for the Compressible Navier-Stokes Equations Near Equilibria.”. Since we recall in more detail certain results (including theorem formulations) related to energy methods, it is necessary to introduce certain notation before we proceed. Derivatives with respect to t, x, z,  (tangent directions), and n (normal directions) will be denoted by @t u D u;t , @x u D u;x , @z u D u;z , @ u D u; , @n u D u;3 , or even without the commas. The norms of the usual Sobolev W2l .˝/ D H l .˝/ spaces and Lebesgue Lp .˝/ spaces are denoted by k  kl;˝ , l  0, and j  jp;˝ , 1  p  1, respectively. For time-dependent functions, we write typically Lp .0; T I H k .˝//, Lq .0; T I Lp .˝//, Wpk;l .˝T /, etc. Let us also introduce the space kl .˝/ of functions u.t; x/ with its norm X k@it u.t /kli;˝ ku.t /k l .˝/ .D ju.t /jl;k;˝ / D k

ilk

(observe that t is fixed) where l > 0 and 0  k  l as well as the related spaces Lp .0; T I kl .˝//, C .0; T I kl .˝//. We will use also Sobolev trace spaces related k 1

1

to the boundary @˝ D S , i.e., W2 2 .S / D H k 2 .S /. The related evolutionary spaces and kl .S / spaces are derived analogously.

2.1

Short-Time Regularity

Supposedly, the first result on existence of local-in-time classical solutions for compressible Navier-Stokes system was given by Nash [29] in 1962. He provided short-time existence in Hölder spaces via a method of successive approximations

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for the problem in R3 . Remarkably, he already allowed for heat conduction and dependence of pressure on both density and temperature. For a similar results, see also Itaya [14] and Volpert and Hudjaev [13] (problem in an infinite strip). In 1976, Solonnikov showed in [31] local-in-time well-posedness in Lp ; p > 3 framework, restricting himself to the isothermal case and bounded domains. In 1977, Tani [35] proved short-time regularity in Hölder spaces for both bounded (with homogenous Dirichlet conditions) and unbounded domains as well as non-constant coefficients ; . In 1980s, further generalizations were obtained: over inhomogenous Dirichlet boundary conditions, see Valli [36, 37]; allowing for inflow, see Fiszdon and Zaja¸czkowski [12]; and for both inflow and outflow, see Łukaszewicz [24]. Compare also Ströhmer [32] for the semigroup approach. As a representative result here, one may regard the result [39]. It considers (1) with  > 13  in a domain ˝ T D ˝  .0; T /, where ˝  R3 is bounded (we will write @˝ D S ), supplemented with the following: • Initial conditions vjtD0 D v0 2 H 1 .˝/; jtD0 D 0 2 H 1 .˝/; 0   > 0;

(2)

%jtD0 D %0 2 H 2 .˝/; %0  % > 0 • Boundary conditions: vjS D vb ;

jS D b

on S T ;

3 12

vb ; b 2 L2 .0; T I 1

.S //:

(3)

For well-posedness, one needs also 1

• Initial-boundary data compatibility conditions in H 2 .S /: vb jtD0 D v0jS ; vb ;t jtD0 D Œv0  rv0 C %1 0 .v0 C rdiv v0  rp.%0 ; 0 // C f .0; / jS ; (4) b jtD0 D 0jS ; h  b ;t jtD0 D v0  r0 C .%0 cv .%0 ; 0 //1  0 p .%0 ; 0 /div v0 C 0 i  C jD.v0 /j2 C .  /.div v0 /2 C %0 b.0; / : 2 jS Moreover, in the case when the inflow boundary S1 .t / D fx 2 S j vb  n < 0g (which by assumption is a time-independent, closed set) has a positive measure, we require also • Inflow-boundary condition for density

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%jS1 D %1

on S1T : 3

(5) 1

• Inflow compatibility conditions in, respectively, H 2 .S / and H 2 .S / %1 jtD0 D %0 jS1 ;

%1;t jtD0 D Œdiv .%0 v0 / jS1 :

(6)

Then one has Theorem 1 (Valli and Zaja¸czkowski [39]). Let S 2 C 3 , f 2 L2 .0; T I 01 .˝//, b 2 L2 .0; T I 01 .˝//, p D p.%; / 2 C 2 , 0 < cv D cv .%;  / 2 C 1 . Then, there exists T  > 0 small enough such that there exists solution .v;  ; %/: .v;  / 2 L2 .0; T  I 13 .˝//;

% 2 C .Œ0; T  I 12 .˝//

to problem (1) supplemented with (2), (3) and (4), provided S1 D ;. In the case S1 ¤ ;, the same result holds, provided additionally conditions (5) and (6) are satisfied and %1 2 L2 .0; T I 02 .˝//. The above result can be found as Theorem 2.5 in [39] for the case S1 D ;. The case S1 ¤ ; is announced in the last paragraph of Sect. 2 of [39], at p. 274. Please observe that from considerations in [39], it follows the Dirichlet boundary condition for  can be replaced by homogenous Neumann condition. Moreover, that an analogous result to Theorem 1 is also available for sufficiently smooth nonconstant  D . ;  /;  D . ; /;  D . ;  /. More recently, the local-in-time well-posedness of compressible Navier-Stokes equations has drawn again some attention, in the context of Fourier analysis methods. Here, one refers to a survey [8] by Danchin and its references as well as to Danchin and Chikami [9] (concerning density-dependent coefficients), see also “ Fourier Analysis Methods for the Compressible Navier-Stokes Equations.” Also a short-time well-posedness result for a coupling of compressible Navier-Stokes and Allen-Cahn equations should be mentioned here; see Kotschote [20].

2.2

Global-in-time Regularity

The existence results of regular solutions to compressible Navier-Stokes system in the large are very scarce. Firstly, one recalls Kazikhov and Weigant [18]. The authors consider the isothermal, barotropic case p. /   , whose main dissipative part equals .v C r..%/div v//, in two-dimensional periodic torus as well as under the slip boundary conditions. They show global-in-time regularity, under the assumption .%/  c%ˇ with ˇ > 3. This result was recently extended by Ducomet and Neˇcasova [10] to the problem in a more general two-dimensional domain. On the other hand, in R3 (barotropic case with heat conduction), Z. Xin in [40] indicated a mechanism for blowup of solutions. However, it relies crucially on compactness

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of the initial density (far-field vacuum); see also “ Finite Time Blow-Up of Regular Solutions for Compressible Flows.” The above results indicate that the solvability in the large, even for a generic isothermal problem, is generally open in any physically relevant dimension.

2.2.1 Solutions Near Equilibria The global-in-time regularity results of the compressible Navier-Stokes equations are naturally much easier available under certain data-smallness assumptions. In this direction, the fundamental results were obtained in early 1980s by Matsumura and Nishida. In [25,26] they consider the initial-value problem in the whole space (the Cauchy problem) for equations (1) describing motion of viscous and heat-conductive gases, under certain additional assumptions. For instance, in [25] the authors assume that the gas is ideal and polytropic, hence p  %2 . There, a global existence result in the class .%; u;  / 2 L1 .RC I H 3 .R3 //; r% 2 L2 .RC I H 2 .R3 //; ru; r 2 L2 .RC I H 3 .R3 // for small data .%; u; /.0/ 2 H 3 .R3 / is provided. Furthermore, the solution decays as follows: lim kr.%; u;  /.t /kH 1 .R3 / D 0:

t!1

The result follows from a combination of energy method and use of potentials. In [27] initial-boundary problem for more general gases is studied. The authors consider problem (1) in a half-space or in an external domain. The main Theorem 1.1 of [27] reads Theorem 2 (Matsumura and Nishida [26]). In a half-space ˝ D R3C or in an external domain with smooth boundary, consider problem (1), (2), (3) and (4) with vb D 0 and a constant b D N . Assume that k%.0/  %; N u; .0/  N k3 is sufficiently small, where .%; N 0; N / is the constant state with %N > 0, N > 0. Let p; cv ; ; ;  be smooth and such that p; cv ; ; ; p% ; p are strictly positive as well as 23  C   0. Let the force f D r , where 2 H 5 and b D 0. Then, there exists a global solution such that %  %Q 2 C .RC I H 3 .˝// \ C 1 .RC I H 2 .˝//; u;   N 2 C .RC I H 3 .˝// \ C 1 .RC I H 1 .˝//; that decays lim k.%  %; Q   N /.t /kC .˝/ D 0:

t!1

Above, %Q comes from the stationary solution .%; N 0; N /, with %Q given by R %.x/ N p% . ;/ Q d D  .x/. %N

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The used method is again a combination of energy methods and use of potentials. D 0. Moreover, the authors consider also the Neumann problem for  , i.e., @ @n j@˝ T For bounded domains, one may refer also to Valli. In [38], he considered compressible viscous barotropic motions. In order to present the result, let us introduce .t/ D kv.t /k21 C k .t /k22 C k ;t .t /k21 C kv;t .t /k20 C Œv.t / 22 ; ˚.t / D kv.t /k23 C k .t /k22 C k ;t .t /k21 C kv;t .t /k20 ; where Œv.t / k denotes a sum of L2 norms of only interior and tangential derivatives of order k. Then, via the energy method, the following differential inequality is shown d  C ˚  c˚ 2 C cF; dt where F D kf k21 C kf;t k21 . This differential inequality allows us to obtain globalin-time solvability for small data, in classes related to ; ˚ . Let us remark that the result does not contain estimate for the full second derivative of v (compare definition of Œv.t / k ). To generalize the result of [38], Valli and Zaja¸czkowski considered in [39] the full problem (1) under the thermodynamical restriction  > 13 , in a bounded domain, supplemented with initial-boundary and compatibility conditions (2), (3), (4), (5) and (6). The setting is analogous to that of local-in-time smoothness result, i.e., the quoted Theorem 1. Additionally, more restrictions on inflow/outflow are needed than for the local-in-time smoothness result. Since in the case of inflow and outflow, i.e., v  nN jS ¤ 0, the equation of continuity implies that necessarily d dt

Z

Z %dx D 

˝

%v  ndS N S

in order to prove global existence, a sufficiently fast decreasing in time of either inflow or outflow is necessary. The most interesting case is a compensation between inflow and outflow such that the difference is small in time. This problem is still open. In the paper [39], one has N /j22;1 C j.  N /.t /j22;1 ; .t/ D jv.t /j22;1 C j.%  %/.t ˚.t / D jv.t /j23;2 C j.%  %/.t N /j22;1 C j.  N /.t /j23;2 ; R where 0 < %N D –˝ %0 .x/dx, N D jS ; uN D ujS ; v D u  uN and P1 D jN  1 j22;1 ;

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1759

P2 D jNuj23;2 C jN  1 j23;2 C jNu;t t j22 C jN ;t t j22 C jbj21;0 C jrj21;0 ; where b is the external force, r is the heat source per unit mass and by definition N Then, the following differential inequality holds 1 D infS T . d  C ˚  c. C  3 /˚ C c.P1 C P2 / dt via the energy method. This inequality for small .0/ and P1 ; P2 implies the existence of a global solution that is close to the equilibrium solution .Nu; %; N N /. Moreover (in relation to smallness of ), this inequality indicates that the solution space is determined by Z sup .t/ C t

t

˚.t 0 /dt 0  C:

0

Next, similar results were obtained by Ströhmer [33] via semigroup approach for the linearized problem (in Lagrangian coordinates). For the results via linear estimates, compare also Mucha and Zaja¸czkowski [28]. In [19] global existence of solutions to compressible barotropic viscous fluid in a bounded domain with the slip boundary conditions is proved via the energy method in L2 -based Besov spaces. For results based on linear estimates obtained via semigroup approach, see Kagei and Kobayashi [17] and Kagei [16] (unbounded domain) and more recent Enomoto and Shibata [11] in the Lp -Lq setting, bounded domain. For the global-in-time result based on linearization in Eulerian coordinates, compare Kotschote [21].

2.2.2 General Stability Results Let us recall also here another type of results concerning the existence of regular solutions, namely, stability-type results in a vicinity of an arbitrary, sufficiently smooth solution (a “benchmark” solution). On one hand, such results are more general than the aforementioned ones (since they concern an arbitrary smooth solution, as opposed to the special smooth solution used before: the equilibrium one). On the other hand, for complex fluids systems, not much is generally known about the existence of smooth solutions (“benchmarks”), beyond the simplest ones: an equilibrium solution or, for certain cases, a solution with the spherical symmetry. A representative of such stability result can be found in Bella, Feireisl, Jin, and Novotný [2]. The authors there consider the compressible viscous Navier-Stokes system (barotropic case with no forcing) (1) in a bounded three-dimensional domain and supplement it with the initial conditions and the Navier slip conditions: v  nN D 0;

˘  nN  nN D 0

on S T ;

where ˘ D 2.Du  13 div uI / C div uI . The result of [2] reads as follows: if there exists a smooth solution to the considered system, emanating from the initial data %0 ; v0 , then for any perturbed data %0; ; v0; , there exists a strong solution to the

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considered problem emanating from %0; ; v0; , as long as the perturbations %0;  %0 ; v0;  v0 are small in, respectively, W41 .˝/ and H 2 .˝/. Let us also note our recent result [5] on incompressible Navier-Stokes system, that provides a similar stability-type result and suggest its use to search for possible singularities. Another example of a stability result around an arbitrary regular solution can be found in [1]. There, the problem (1) supplemented with homogenous Dirchlet boundary conditions was considered. Assuming that there exists a special solution .vs ; %s / to (1) such that for a certain k 2 N0 it holds .%s ; vs / 2 C .kT; .k C 1/T I 12 .˝//, vs;t t 2 C .kT; .k C 1/T I L2 .˝//, 0 < %  %.x; t /  % < 1, one introduces the perturbed quantities u D vvs , D %%s , g D f fs as well as .t/ D ku.t /; .t /k2 2 .˝/ ;

N.˝  .kT; .k C 1/T // D f.u; / W sup ./ < 1g;

1

˚.t /

D ku.t /k2 3 .˝/ 2

C

 t

k .t /k2 2 .˝/ ; 1

  Z t ˚. /d  < 1 M.˝  .kT; .k C 1/T // D .u; / W sup './ C  t

0

and B.t / D kvs .t /; %s .t /k2 2 .˝/ C kvs .t /; %s .t /k3 2 .˝/ C kvs .t /; %s .t /k2H 3 .˝/ 1

C

kvs;t t .t /k2L2 .˝/

1

C

kfs .t /k2 1 .˝/ 0

G.t / D kvs .t /; %s .t /k2 2 .˝/ kg.t /k2 1 .˝/ : 1

0

Then it holds Theorem 3 (Bae and Zajaczkowski [1]). Assume that fs ; g 2 L1 .kT; .k C 1/T I 01 .˝//. Next, assume there exists a sufficiently small constant  that Z .0/   ;

sup kgkL1 .kT;.kC1/T I 1 .˝//  c ;

k2N0

0

.kC1/T

G.t /dt  ˛ : kT

R .kC1/T B.t /dt  C T , where the relation between ˛ and C is Assume also that kT given by ˛e C T C e C T  1. Then .u; / 2 M.˝  .kT; .k C 1/T //

for any k 2 N0

The example of such a special global regular solutions .vs ; %s / may be a spherically symmetric one. Let us

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

2.3

1761

Energy Methods in Free Boundary Problems

The energy method developed in [39] was next extended to the case of free boundary problems for viscous compressible barotropic fluids. In this paragraph we recall certain results related to analysis of free boundary problems via the energy approach. For a complementary presentation, we refer to other chapters of this handbook, including “ Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible Fluid Mechanics” and “ Local and Global Solvability of Free Boundary Problems for the Compressible Navier-Stokes Equations Near Equilibria.” We denote by ˝.t / the domain at time t and by S .t / the boundary at time t. In [42, 47, 53] the following free boundary problem has been considered in ˝.t /;

t 2 RC ;

in ˝.t /;

t 2 RC ;

%.vt C v  rv/  div ˘.v; %/ D %f %t C div .%v/ D 0

(7)

with the following initial conditions %jtD0 D %0 ;

vjtD0 D v0 ;

˝.t /jtD0 D ˝0 ;

S .t /jtD0 D S0

(8)

and boundary conditions v  nN D 

t

jr j

˘ nN D p0 nN

;

on S .t /;

on S .t /;

t 2 RC ;

t 2 RC ;

(9) (10)

r where f .x; t / D 0g describes the boundary S .t /, nN D jr is the unit outward j normal vector to S .t /, and, by definition, tensor ˘ D 2Dv C .  /div vI  pI: Let

.t/ D jvj23;0;˝.t/ C jp  p0 j23;0;˝.t/ ; ˚.t / D jvj24;1;˝.t/ C jp  p0 j23;0;˝.t/ ; N and %N const that is close to %0 . where p0 D p.%/ Then, by the energy method the following differential inequality is proved   d  C ˚  C .1 C  3 /˚ C  2 .0/ C sup t var t ; (11) dt R A   where t D 1 ˝.t/ % dx for p D A% ,  > 1. It is proved independently that supt var t is small for all times. Hence, from (11), the global existence for small data follows. Moreover, the obtained solutions satisfy for any t 2 ŒkT; .k C 1/T

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Z

.kC1/T

.t/ C

k 2 N0 :

˚. /d   C;

(12)

kT

In order to describe further results, let us introduce the equilibrium state (equilibrium solution) as follows v D 0; % D %e const; f D 0; ˝.t / ˝e :

(13)

M , where M is the total fluid mass and p D p  p0 ; % D %  %e . Then %e D ˝ e In [48], the differential inequality (11) for (7), (8), (9), and (10) with f D 0 is improved to the following one

d N  C c0 ˚  C .1 C  2 /˚ C c2 N ; dt

(14)

where N D



Z

% ˝.t/

 D N C

X

jDx˛ @it vj2 C

j˛jC12

Z

t 0

p% p1 2 % C %  %

X

jDx˛ @i g j2 dx;

1j˛jCi 2

kv.t 0 /k23;˝.t 0 / dt 0 ;

˚ D jvj23;1;˝.t/ C k% k22;˝.t/ C k%t k22;˝.t/ C k%t t k21;˝.t/ ; N D kp k20;˝.t/ ; and the quantity p1 is calculated from the relation Z

1

p D .%  %e /

p 0 .%e C s.%  %e //ds p1 % :

0

Inequality (14) in conjunction with a local existence theorem implies existence of global solutions to (7), (8), (9), and (10) such that v 2 C .RC I 02 .˝.t /// \ L2 ..kT; .k C 1/T I 13 .˝.t ///; % 2 L2 .kT; .k C 1/T I H 2 .˝.t /// \ H 2 .kT; .k C 1/T I H 1 .˝.t ///; k 2 N0 . Furthermore, in [54] the case of the viscous compressible barotropic fluid in a bounded domain ˝.t / with free boundary S .t /, t 2 RC governed by the surface tension. In this case the boundary condition (10) is replaced by the following one ˘ nN   H nN D p0 nN

on S .t /;

t 2 RC ;

(15)

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1763

where  is the surface tension coefficient and H is the double mean curvature of S .t /. It is negative for convex domains and can be expressed via the LaplaceBeltrami operator S.t/ on S .t / H nN D S.t/ .t /x;

x D .x1 ; x2 ; x3 /

(16)

In [54] global existence of solutions to (7), (8), (9), (10), and (15) that are close to an equilibrium solution is proved. By an equilibrium solution, we mean here (16) with ˝e D BRe (ball with radius Re /. Denoting by M the total mass of the fluid, we have from the mass conservation %e D

M 4 3 Re 3

;

and Re is a solution to the equation p

M 4 3 Re 3

! D

2 C p0 ; Re

where p0 is the external pressure. Let p D p  p0  R2e , D p.%pe /% and S .t / be determined by x D x.s t /, where s t D .s 1t ; s 2t / are local coordinates. Next, let g D .g˛;ˇ / be the fundamental tensor of S .t / and let G D detg. Then 1

1

S.t/ .t / D G  2 @s ˛ G 2 g˛;ˇ @s ˇ : Let us define .t/ D jvj23;0;˝ t C jp  p0 j23;0;˝ t  Z t  Z t Z X  k 0 k 0 nN  C g˛;ˇ nN  @s vs 1 ;s ˛ dt @s vs 2 ;s ˇ dt dS .t/ 2 S.t/ 0 0 jkj2

C

C

C

 2  2  2

Z S.t/

Z S.t/

Z S.t/

ˇ2 X ˇˇ Z t ˇ k 0ˇ ˇnN  @ v dt 1 2 s s ;s ˇ ˇ dS .t/

jkj2

0

 2 Z t 2  XX 1 2 nN  @ks vs i ;s i dt 0 C 2@ks H .; 0/ C dS .t/ 2 Re 0 iD1

jkj2

X

k k g˛;ˇ nN  Dt;s vs ˛ nN  Dt;s vs ˇ dS .t /:

jkj2

Let jxj D R.!; t /; ! 2 S 2 (the unit sphere) describes S .t /. Next, one introduces

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˚.t / D jvj24;1;˝.t/ C jp j23;0;˝.t/ ; .t/ D kvk20;˝.t/ C kp k20;˝.t/ C kR.; t /  R.; 0/k20;S 2 ; F D jf j22;0;˝.t/ C kft t t k20;˝.t/ and Q .t/ D .t/ C

Z

t

kvk23;˝./ d  ;

0

Q / D ˚.t / C ˚.t

Z

0

t

kvk24;˝./ d  :

By the energy method, the following inequality is proved d Q .1 Q C Q 3 /˚ C cF C c˚ C ckH .; 0/ C 2 k4 2  C ˚  cP ./ dt R0 2;S  Z t C c kR.; t /  R.; 0/k24:5;S 2 k vd  k23;S.t/ CkR.; t / 

R.; 0/k23;S 2 k

0

Z

t

vd  k24;S.t/

0

(17)

;

where P denotes a polynomial which has to be considered in tandem with the Laplace-Beltrami operator on S .t /. Together with a local existence theorem, inequality (17) implies existence of global solutions v 2 C .RC ; 03 .˝.t /// \ W24;2 .˝.t /t2fkT;.kC1/T g / 3;3=2

p 2 C .RC ; 03 .˝.t /// \ W2

.˝.t /t2fkT;.kC1/T g /

k 2 N0 . Furthermore, in [43, 44, 51, 52, 55] the following free boundary problem is considered %.vt C v  rv/  div ˘.v; %/ D %f %t C div .%v/ D 0 %cv .t C v  r / C  p div v   D

in ˝.t /;

t 2 RC ;

 jD.v/j2 C .  /.div v/2 2 C %b

with the following initial conditions

t 2 RC ;

in ˝.t /;

in ˝.t /;

t 2 RC ;

(18)

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

%jtD0 D %0 ; jtD0 D 0 ;

vjtD0 D v0 ;

˝.t /jtD0 D ˝0 ;

1765

S .t /jtD0 D S0 (19)

and boundary conditions v  nN D 

t

jr j

@ D 1 @n

;

on S .t /;

(20)

on S .t /;

(21)

˘ nN D p0 nN on S .t /:

(22)

The notation is analogous to that of system (7), (8), (9) and (10). The new unknown  is the temperature, b is the heat source per unit mass, and 1 is the given heat flux through S .t /. The pressure p is now additionally  -dependent. Next, cv D cv .%;  / is the specific heat at a constant volume. Finally, thermodynamic restrictions imply cv > 0,  > 0,  > 13  > 0. Papers [43, 44, 51] contain a global existence result for problem (18), (19), (20), (21) and (22). The obtained solutions are sufficiently close to the equilibrium state, defined as follows M : %e (23) The method of the proof, similarly as in the barotropic case, consists in deriving by the energy method a differential inequality, which implies a small-data global existence, provided a local existence holds. In fact, a range of differential inequalities is given, each leading to a slightly different kind of solution. Since their formulations are complicated, we do not recall them here in detail and recommend the interested reader to refer directly to the aforementioned papers. In [45, 46, 49, 50] the authors investigate the problem (18), (19), (20), and (21) with the surface-tension boundary condition (15) in place of (22). In this case the involved R equilibrium solution differs from (23): ˝e is a ball of radius Re , 1 e D j˝j ˝ 0 dx, and Re is a solution to the equation f D b D 1 D 0;

v D 0;

% D %e D const;

p

M ; e 4 3 Re 3

! D

˝.t / D ˝e D const;

2 C p0 ; Re

j˝e j D

(24)

(it is assumed that (24) is solvable for Re ). In these papers, as before, certain differential inequalities for quantities close to the equilibrium state are derived by the energy method. Then, using local existence, the global existence in vicinity of the equilibrium solution follows. A deep summary of a wide range of results for compressible free boundary problems via the energy method can be found in Zadrzy´nska [41].

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Further Remarks

Recently, there is a growing literature on regularity conditions for the compressible Navier-Stokes equations; see Sun, Wang, and Zhang [34] and its references as well as “ Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak Solutions for the Compressible Navier-Stokes Equations.” Let us consider the case of an (initial) density bounded from below by a positive constant. (In fact, this assumption can be relaxed in certain cases, so that vacuum (zero-density regions) is allowed.) Then, the short-time strong solution well-posedness in the case of compressible Navier-Stokes system without heat conduction (with p.%/ being a C 1 function, satisfying an additional compatibility condition) can be found in Cho, Choe and Kim [7]. The generalization to the heatconducting case (under certain additional assumptions) was provided in [6] by Cho and Kim.

3

Results

In the second part of this chapter, we present a new version of a representative result obtained for viscous, compressible, heat-conducting fluids by means of the energy method, namely, local-in-time existence of regular solutions and global-intime existence of regular solutions that are in a vicinity of the equilibrium solution. For clarity, we formulate below precisely the problem that we deal with. The equations are identical to those presented in the first part, but now for clarity we immediately supplement them with the boundary and initial conditions considered here. Next, in Sect. 3.2 we present used notation and certain auxiliary notions (a small portion of this section repeats the notation used in the first overview part, for reader’s convenience). In Sect. 3.3 we state our main results: Theorem 4 presents the local-in-time solvability, whereas Theorem 5 is the global-in-time result (with data restrictions). The subsequent sections contain proofs.

3.1

Problem

The motion of a compressible, viscous heat-conducting fluid in a bounded domain ˝  R3 is described by the following system of equations in ˝ T ;

%.vt C v  rv/  .v C rdiv v/ C rp D %f %t C div .%v/ D 0 %cv .t C v  r / C  p div v   D

in ˝ T ;

 jD.v/j2 C .  /.div v/2 2 C %b

in ˝ T ;

(25)

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1767

where x D .x1 ; x2 ; x3 / are the Cartesian coordinates, ˝ T D ˝  .0; T /, % D %.x; t / 2 RC is the density of the fluid, v D .v 1 .x; t /; v 2 .x; t /; v 3 .x; t // 2 R3 denotes the velocity,  D .x; t / 2 RC – the temperature, f D .f 1 .x; t /; f 2 .x; t /; f 3 .x; t // 2 R3 is the external force field per unit mass and b D b.x; t/ 2 R are the heat sources per unit mass. Moreover, p D p.%; / is the pressure and cv D cv .%;  / – the specific heat at a constant volume. These last two quantities are assumed to be known functions of % and  . We will sometimes use the following operator notation for the main part of the system for velocity: A D  C rdiv . It is worth emphasizing that no explicit growth assumptions on p.%; / or cv .%;  / are needed. It suffices that p 2 C 2 .RC  RC /, cv 2 C 1 .RC  RC /, and their derivatives are bounded for bounded arguments and that cv does not vanish for positive arguments ,i.e., 8"1 >0;"2 >0 9ı>0 j1 j  "1 ; j2 j  "2 H) cv .1 ; 2 /  ı."1 ; "2 /:

(26)

The viscosity coefficients  and  and the coefficient of the heat conductivity  are assumed to be constant and to satisfy the following thermodynamic restrictions  > 0;

 > 0;

 > 0:

(27)

Remark 1. Observe that our restriction  > 0 is more general than the usual  > 13 , present in [39]. Let S be the boundary of ˝. One supplements (25) with the following boundary conditions vjS D vb ;

jS D b ;

vb  nj N S D 0;

(28)

where nN is the unit outward normal vector to S and  denotes the scalar product in R3 . The boundary condition (28) guarantees no inflow or outflow. One could also allow for a strictly controlled and very small inflow and outflow, but at a cost of heavy technical complications. Finally, one imposes the initial conditions vjtD0 D v0 ;

jtD0 D 0 ;

%jtD0 D %0 ;

(29)

for which there exist such  > 0 and % > 0 that 0   and %0  % . In view of (28) and (29) and in relation to the function spaces appearing in our main results, the following initial-boundary data compatibility conditions are needed (compare, for instance, (36) in what follows) vb jtD0 D v0jS

3

in H 2 .S /;

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vb ;t jtD0 D Œv0  rv0 C %1 0 .v0 C rdiv v0  rp.%0 ; 0 // C f .0; / jS 1

in H 2 .S /; 3

b jtD0 D 0jS in H 2 .S /;  h b ;t jtD0 D v0  r0 C .%0 cv .%0 ; 0 //1  0 p .%0 ; 0 /div v0 C0 C

i  jD.v0 /j2 C .  /.div v0 /2 C %0 b.0; / 2 jS

(30)

1

in H 2 .S /:

Since the considered problem is highly nonlinear, the results need to be shown in a class of regular functions. As already mentioned, contrary to Valli and Zaja¸czkowski [39], we do not use here the Leray-Schauder fixed-point argument. Consequently, the exposition is much more traceable and, moreover, one may easier derive an explicit smallness condition related to Theorem 5.

3.2

Notation and Auxiliary Results

By C one denotes a generic constant that may change its value between formulas. Similarly, ' denotes a generic increasing positive function that may also change its precise form between lines. The conjunction a ^ b denotes the minimum of a and b. Derivatives with respect to t , x, z,  (tangent directions), and n (normal directions) are denoted by @t u D u;t , @x u D u;x , @z u D u;z , @ u D u; , @n u D u;3 or even without the commas, when there is no risk of confusion with an element of a sequence.

3.2.1 Function Spaces Let k  kl;˝ , l  0 and j  jp;˝ , 1  p  1 denote, respectively, the norms of the usual Sobolev W2l .˝/ D H l .˝/ spaces and Lebesgue Lp .˝/ spaces. For timedependent functions we write kukLp .0;T IH k .˝// D kukk;p;˝ T ;

p 2 Œ1; 1 ; k 2 N0

kukLq .0;T ILp .˝// D huip;q;˝ T ;

p; q 2 Œ1; 1 :

Next, let us introduce the space kl .˝/ of functions u.t; x/ with the finite norm ku.t /k l .˝/ D

X

k

k@it u.t /kli;˝ ju.t /jl;k;˝ ;

ilk

where l > 0 and 0  k  l. Furthermore, let us write kukLp .0;T I l .˝// D jujl;k;p;˝ T ; k

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1769

where p 2 .1; 1/ and kukC .0;T I l .˝// D jujl;k;1;˝ T : k

For space-only norms, the subscript ˝ will be dropped. In order to avoid confusion, we will retain the space-time subscript ˝ T . Since the introduced notation may seem confusing at the first sight, let us put together all of it in the following formula. It may serve as a reference while reading the remainder of the paper. jujp D kukp;˝ D kukLp .˝/ ; kukl D kukl;˝ D kukH l .˝/ ; X k@it u.t /kH li .˝/ ; ju.t /jl;k;˝ D ku.t /k l .˝/ D k

ilk

jujp;˝ T D kukLp .˝ T / ; huip;q;˝ T D kukLq .0;T ILp .˝// ; kukk;p;˝ T D kukLp .0;T IH k .˝// ; jujl;k;p;˝ T D kukLp .0;T I l .˝// : k

For brevity, for a given u1 ; u2 ; : : : ; un , we will write ku1 ; u2 ; : : : ; un k2L D

n X

kui k2L ;

iD1

where L is any space used in this paper. The following quantities will be also needed .t/ D ku.t /; .t /; #.t /k2 2 .˝/ .u; ; #/; 1

˚.t / D

ku.t /; #.t /k2 3 .˝/ 2

C k .t /k2 2 .˝/ ˚.u; ; #/: 1

Now let us define the spaces needed for the formulation of our main result N.˝ t / D f.u; ; #/ W sup ./ < 1g  t

and t



Z

M.˝ / D .u; ; #/ W sup './ C  t

0

t

 ˚. /d  < 1 :

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The letters X; X1 ; Y; Z are reserved for denoting certain norms as follows X .f1 ; : : : ; fn ; t / D jf1 ; : : : ; fn j2;1;1;˝ t Y .f1 ; : : : ; fn ; t / D jf1 ; : : : ; fn j3;2;2;˝ t ; Z DX CY and X1 .f1 ; : : : ; fn ; t / D kf1 ; : : : ; fn k2;1;˝ t ; for instance, X .un ; #n ; n ; t / D jun ; #n ; n j2;1;1;˝ t , Y .un ; t / D jun j3;2;2;˝ t . Finally, one will need the following interpolation inequality (see Besov, Il’in and Nikolskii [3, Sect. 15]) X

˛

kD ukLp .˝/  c

j˛jDr

X



˛

kD ukLp2 .˝/

kuk1 Lp .˝/ C ckukL1 .˝/ ; 1

(31)

j˛jDl

which holds for ˝  R3 and p1 ; p2 2 Œ1; 1 , 0  r < l,  3 3 3 l ;  r D .1   / C  p p1 p2

r    1; l

l; r 2 N [ f0g.

3.2.2 Zero-Trace Reformulations and Data Extensions We will seek a global, regular solutions to (25), (26), (27), (28), (29), and (30) in a vicinity of a certain constant (equilibrium) solutions. Therefore, let us introduce the following quantities Z Z %N D  %0 dx; N D  0 dx; (32) ˝

R

where ˝ means

1 j˝j

R

˝.

˝

Moreover, let us extend the boundary data to !; : !jS D vb ;

jS D b  N

(33)

and reformulate the problem (25), (26), (27), (28), (29), and (30) by considering new unknowns

D %  %; N

u D v  !;

# D   N 

(34)

The solutions to the reformulated problems begin their evolution at 0 D %0  %; N u0 D v0  w0 , and #0 D 0  N  0 and have zero-Dirichlet boundary data. Since these auxiliary problems are presented in a two slightly different ways in

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1771

Sect. 3 (local-in-time solvability) and Sect. 4 (global-in-time result), one should refer directly there for their formulations.

3.2.3 Partition of Unity S Let us introduce a partition of unity: .f!Q i g; f˝Q i g; fi g/, where ˝D i ˝Q i , !Q i  ˝Q i , suppi  ˝Q i and ij!Q i D 1 is a cutoff function. The more precise definition follows. For clarity of the following explanation, let !, Q ˝Q be one of the !Q i0 s, ˝Q i0 s and .x/ – the corresponding cutoff function. If ˝Q is an interior subdomain, then let the closure Q Otherwise one assumes that ˝NQ \ S 6D ; and !NQ \ S 6D ;. Now, certain !NQ  ˝. details on the procedure of the boundary straightening will be provided. Let  2 Q Next, a new (“local”) coordinate system y D Y .x/ with its !NQ \ S  ˝NQ \ S S. origin at  will be introduced. The mapping Y is a composition of a translation and a rotation. Assume that its local portion of boundary SQ is described by y3 D G.y1 ; y2 / in the coordinates y, where G is sufficiently regular. Then, by definition ˝Q D fy W jyi j < 2; i D 1; 2; !Q D fy W jyi j < ; i D 1; 2; supp  ˝Q i ;

j!Q i D 1;

G.y 0 / < y3 < G.y 0 / C 2; y 0 D .y1 ; y2 /g; G.y 0 / < y3 < G.y 0 / C ; y 0 D .y1 ; y2 /g

:

jry.k/ j  ck k

Further, one introduces the new (“straightened”) Cartesian variables z by Q 1 ; y2 /; y 2 ˝; Q zi D yi ; i D 1; 2; z3 D y3  G.y Q is an extension of G to ˝. Q The which will be denoted by z D ˚.y/, where G respective associated straightened sets read Q D fz W jzi j < 2; i D 1; 2; 0 < z3 < 2g ˝O D ˚.˝/

and

SO D ˚.SQ /

Hence SO D fz W z3 D 0; jzi j < 2; i D 1; 2g. Let D ˚ ı Y (the change of variables x ! z). In relation to the chain rule @zi @ @ D @x , one introduces the notation @xk k @zi @zi rO k jxD 1 .z/ D rz @xk jxD 1 .z/ i and uQ .x/ D u.x/.x/; O uQ .z/ D uO .z/.z/;

Q ˝Q \ S D ;; x 2 ˝; O z 2 ˝;

where uO .z/ D u. 1 .z//. Due to the assumed boundary regularity, the estimates for derivatives  hold also for derivatives O (with another constant).

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Whereas A D  C rdiv .D Ax D x C rx div x / denotes our dissipative operator in the original variables x, by Az we mean A with respect to the O straightened variables z (i.e., Az D z C rz div z ). Finally AO D O C  rO div 1 O is the operator Ax at x D .z/ expressed through the chain rule (compare rk above). It will be important that for any , one can construct our partition of unity so that Q 1 C jr Gj Q 21 /  c: .jr Gj

(35)

This is possible thanks to the regularity of our boundary. For more details on the partition of unity, see Chapter IV of [22]. In particular, the property (35) is given as (7.16) there.

3.3

Main Results

In Sect. 4, local-in-time existence of solutions to problem (25) is proved by the method of successive approximations. More precisely, let us define M ./ D j!j3;2;2;˝  C j!j2;1;1;˝  C j j3;2;2;˝  C j j2;0;1;˝  C jf j1;0;1;˝  C jbj1;0;1;˝  C ju0 j2;1 C j 0 j2;1 C j0 j2;1 : where !; are extensions of boundary data vb D vjS and b  N , respectively. Since it is more rigorous to use quantities involving exclusively the data, let us immediately observe that there is a bounded extension-restriction correspondence 5 3 between a (boundary) function g 2 L2 .0; T I H 2 .S // \ H 1 .0; T I H 2 .S // and its extension gQ 2 L2 .0; T I H 3 .˝// \ H 1 .0; T I H 2 .˝//, whereas the equivalent norm for the latter space is our jgj Q 3;2;2;˝ t . Similarly, for jgj Q 2;1;1;˝ t < 1, one has the pair (recall that in our notation for norms the time-related ‘1’ indicates continuous functions) gQ 2 C .0; T I H 2 .˝// \ C 1 .0; T I H 1 .˝//; 3

1

g 2 C .0; T I H 2 .S // \ C 1 .0; T I H 2 .S //: Hence instead of M ./, one may write M 0 . / D jvb j

5

3

L2 .0;IH 2 .S //\H 1 .0;IH 2 .S //

jb  N j

5

3

C jvb j

L2 .0;IH 2 .S //\H 1 .0;IH 2 .S //

3

1

C .0; IH 2 .S //\C 1 .0;IH 2 .S //

C jb  N j

3

C 1

C .0; IH 2 .S //\C 1 .0;IH 2 .S //

(36) C jbj1;0;1;˝  C jf j1;0;1;˝  C ju0 j2;1 C j 0 j2;1 C j0 j2;1

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1773

and M 0 differs from M by a numerical constant. It holds Theorem 4. [Local-in-time solvability] Let @˝ D S 2 C 3 and functions p.%; /; cv .%;  / (pressure and specific heat) be, respectively, twice and once differentiable, with their highest derivatives bounded for bounded arguments and cv satisfying (26). Assume that for a  > 0 our data enjoy the following regularity: M ./ < 1 or equivalently M 0 . / < 1. Moreover, let the initial density and temperature be bounded away from 0 and sufficiently flat; more precisely, let 1 j0  N j1  N ; 2

N1 j%0  %j

1 %: N 2

(37)

N 2%N < % and any Take any numbers 0 < % < % such that 0 < % < 14 %, 1 N   N 0 <  <  such that 0 <  < 4  , 2 <  . Then, there exists increasing  1  function C1 .%1  ; % ;  ;  ; M / such that for any t   satisfying  1  N 2 ^ N 4 ; t C1 .%1  ; % ;  ;  ; M .t //  1 ^ %

(38)

there exists .v; %;  / 2 N.˝ t / \ M.˝ t /;

% 2 Œ% ; % ;

 2 Π;   :

solving the problem (25), (26), (27), (28), (29), and (30). In Sect. 5, a differential inequality that guarantees prolongation of the regular localin-time solution given by Theorem 4, provided it begins its evolution sufficiently close to the equilibrium solution (v D 0, % D const,  D const), will be derived. This prolongated solution does not necessarily converge to the equilibrium (no decay of external forces is assumed), but remains close to it for all times. Hence, our second main result can be seen as a stability-type one. More precisely, we have for N 2 C j%.t /  %j .t/ WD ju.t /j22;1 C j.t /  j N 22;1 ; 2;1 P .t / D j j23;2 C j j22;0 C j!j23;2 C j!t j21;0 C jf j21;0 C jbj21;0 Theorem 5. [Global-in-time solvability near equilibria] Let @˝ D S 2 C 3 and functions p.%; /; cv .%;  / (pressure and specific heat) be, respectively, twice and once differentiable, with their highest derivatives bounded for bounded arguments and cv satisfying (26). Assume that data enjoy the following regularity: M .1/ < 1 or equivalently M 0 .1/ < 1. Moreover, let the initial density and temperature be bounded away from 0 and sufficiently flat according to (37). Then, for sufficiently small .0/ and P .t / with t 2 Œ0; 1/, there exists C .M .1// and a global solution

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.u;  ; / to the problem (25), (26), (27), (28), (29), and (30) such that .u;  ; / 2 N.˝ 1 / \ M.˝ 1 /; .t/ C

1 4

Z

t2

P .t /dt C .M /.t2  t1 / C .t1 /

for any 0  t1  t  t2 < 1

t1

and 1 j.t /  N j1  N ; 2

j%.t /  %j N1

1 %: N 2

Remark 2. The smallness conditions as well as the form of C .M .1// are given in Lemma 20. Remark 3. Observe that the assumed pointwise smallness on P is not compatible with ingredients of M . One can relax this assumption to become compatible with RT M , i.e., to a smallness assumption on 0 P .t /dt and a pointwise smallness on lower-order terms, at the cost of a balance between (any) prefixed time of existence TQ and a smallness condition. For an additional explanation, compare Remark 5, page 87. Remark 4. Theorems 4 and 5 should be compared with results of [39]. Our results here do not involve inflow or outflow conditions. On the other hand, they allow for more relaxed thermodynamic conditions; the obtained smoothness is richer and the conditions are more explicit. Moreover, the proofs appear to as as being easier to follow.

4

Local Existence

In this section Theorem 4 on local-in-time existence of solutions to problem (25), (26), (27), (28), (29) and (30) will be proved. To this end, one first expresses it in the following form %ut  Au D %.u C !/  r.u C !/ C A!  p% .%;  /r  p .%;  /r C %f F .%;  ; u; f /;

(39)

ujS D 0; ujtD0 D u.0/;

N u D %div N !;

t C .u C !/  r C div u C div ! C %div

jtD0 D .0/;

(40)

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1775

%cv .%;  /#t  # D %cv .u C !/  r#  %cv .u C !/  r   p div .u C !/  %cv .%;  / t C  C

 jD.u C !/j2 2

C .  /jdiv .u C !/j2 C %b

(41)

G.%;  ; u; b/; #jS D 0; #jtD0 D #.0/; where % D %N C ,  D N C C #, v D u C ! and A! D ! C rdiv !. Next, we will provide existence of solutions to problems (39), (40), and (41) by the method of successive approximations. More precisely, the steps are as follows 1. For given functions un , n , where n D N C C#n , one solves the linear problems N corresponding to (39), (40) and (41), i.e., for unC1 , #nC1 and n .D %n  %/ %n unC1;t  AunC1 D F .%n ; n ; un ; f /; unC1 jS D 0;

(42)

unC1 jtD0 D u.0/;

N un  %div N ! D 0;

n;t C.un C !/  r n C n div un C n div ! C %div

n jtD0 D .0/;

(43)

%n cv .%n ; n /#nC1;t  #nC1 D G.%n ; n ; un ; b/ #nC1 jS D 0;

(44)

#nC1 jtD0 D #.0/: As the zero-approximation u0 and #0 , one chooses an extension of initial data u.0/ and #.0/. Then 0 is a solution to (43). Since dealing with the linear problems (42), (43), and (44) is the most classical one, it suffices to remark here that for a given un , existence of n solving the problem (43) follows from the method of characteristics. Next, for given un , %n , n , existence of solutions to problems (42), (44) follows from the technique of “regularizer” see, for instance, Ladyzhenskaya, Solonnikov, and Uraltseva [22, Ch. 4]. For more details, compare also Solonnikov [30]. In the following proofs, one may need more regularity of the involved linear systems than available for the data from respective classes. Rigorously, one mollifies data so that the solutions are smooth, perform the proofs, obtain thesis that does

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not involve this extra smoothness, and takes limit with respect to mollification parameter. For clarity of our nevertheless technical proofs, let us proceed without such mollification. 2. Recall that the letters X; X1 ; Y; Z are reserved for denoting certain norms and that Z D X C Y D j : : : j2;1;1;˝ t C j : : : j3;2;2;˝ t To prove existence of solutions to (39), (40), and (41) by the method of successive approximations, having already solutions to the linear problems (42), (43), and (44), it suffices to show uniformly in n (i) Estimates for approximative sequences in Z, see Sect. 4.1. (ii) Estimates of the difference of the approximative sequence at the step nC1 by the difference of the approximative sequence at the step n, with a coefficient that is small for small time (implying that they are Cauchy sequences for sufficiently small time in certain spaces, possibly of lower regularity than that of Z), see Sect. 4.2. The second information allows to take the limit in weak formulations of the approximative problems and the first one – to conclude that these limits have regularity of Z. Observe that merely (i) would be insufficient, since it provides convergences up to a subsequence and the approximative solutions use the entire approximative sequence, which is a technical drawback of the method of successive approximations.

4.1

Estimates for Approximative Sequences

Some estimates presented in what follows, particularly the first ones, are at some points clearly rough. They are nevertheless sufficient, since our reasoning will be closed on a high-regularity level related to the quantity Z.

4.1.1 Estimate for n Providing its Boundedness Lemma 1. Assume that un ; ! 2 L2 .0; t I 23 .˝// and that .0/ 2 H 2 .˝/. For solutions to problem (43) it holds   1 k n .t /k2  exp C t 2 jjun C !jj3;2;˝ t .1 C jj .0/jj2 /; (45)   1 1 k n .t /  .0/k2  C t 2 jjun C !jj3;2;˝ t exp C t 2 jjun C !jj3;2;˝ t .1 C jj .0/jj2 /: (46) Moreover assume that j .0/j1  12 %N and take numbers % ; % such that 0 < % < 1 %, N 2%N < % . Then for any t  tn , where tn satisfies 4

34 Local and Global Solutions for the Compressible Navier-Stokes. . .



1 2



1 2

C tn jjun C !jj3;2;˝ t exp C tn jjun C !jj3;2;˝ t .1 C jj .0/jj2 / 

1777

1 %; N 4

(47)

with constants as in (46), one has %  %n  % :

(48)

Proof. Applying @ix to (43), multiplying the result by @ix n , integrating over ˝ and summing with respect to i from 0 to 2, one obtains Z ˝

@ix n;t @ix n dx C Z ˝

Z

@ix . n div .un

˝

C

@ix Œ.un C !/  r n @ix n dxC !//@ix n dx

Z C

(49) %@ N ix div .un

C

!/@ix n dx

D 0:

Let us focus on the highest-order case i D 2 of (49). The first term is 1 d 2 dt

Z

j@2x n j2 dx:

˝

The second term equals Z X 1 ˝ kD0

k 2 @2k x .un C !/r@x n @x n dx C

Z ˝

.un C !/r@2x n @2x n I C II;

(50)

where I can be bounded via the Hölder and the Sobolev inequalities as follows I  .j@2x .un C !/j4 j@x n j4 C j@x .un C !/j1 j@2x n j2 /j@2x n j2  C jjun C !jj3 jj n jj2 j@2x n j2 :

(51)

In II one integrates by parts and uses the boundary conditions unjS D 0, !  njS D 0 to write Z Z 1 1 2 2 .un C !/rj@x n j D  r  .un C !/j@2x n j2 ; 2 ˝ 2 ˝ which by the Hölder inequality and the Sobolev embedding is controlled by C j@x .un C !/j1 j@2x n j22  C jjun C !jj3 j@2x n j22 : All in all, I C II  C jjun C !jj3 jj n jj22 :

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Similarly, the third term of the case i D 2 of (49) is bounded by C jjun C !jj3 jj n jj2 j@2x n j2 and the last one by C jjun C !jj3 j@2x n j2 : Summing up, the case i D 2 of (49) is controlled with C jjun C !jj3 jj n jj22 C C jjun C !jj3 jj n jj2 : This, together with easier considerations for the case i D 0; 1 of (49) (where no integration by parts is necessary), implies 1 d jj n jj22  C jjun C !jj3 jj n jj2 .1 C jj n jj2 /: 2 dt Simplification yields d jj n jj2  C jjun C !jj3 .1 C jj n jj2 /; dt

(52)

d .1 C jj n jj2 /  C jjun C !jj3 .1 C jj n jj2 /: dt

(53)

hence

This gives  Z t 1 C jj n .t /jj2  exp C jj.un C !/.s/jj3 ds .1 C jj n .0/jj2 /;

(54)

0

which implies (45) due to n .0/ D .0/ and the Hölder inequality. To show (46), one proceeds analogously. More precisely, one applies @ix to (43), multiplies the result by @ix . n .t /  .0// and, along the lines leading to (45), obtains the following analogue of (52) d .jj n .t /  .0/jj2 / C jjun C !jj3 .1 C jj n jj2 /  dt  Z t jj.un C!/.s/jj3 ds .1Cjj .0/jj2 /; C jjun C!jj3 exp C 0

where for the second estimate we have used (54). Integration in time of the above inequality yields

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

Z

t

jj n .t /  .0/jj2  C

1779

 Z t jj.un C !/.s/jj3 ds exp C jj.un C !/.s/jj3 ds

0

0

.1 C jj .0/jj2 /  1  1  C t 2 jjun C !jj3;2;˝ t exp C t 2 jjun C !jj3;2;˝ t .1 C jj .0/jj2 /; where the second inequality follows from the Hölder inequality. This yields (46). From (46) one obtains   1 1 j n j1  C t 2 jjun C !jj3;2;˝ t exp C t 2 jjun C !jj3;2;˝ t .1 C jj .0/jj2 / C j .0/j1 : Using that j .0/j1  12 %N and the assumption that t is so small that   1 1 1 N C t 2 jjun C !jj3;2;˝ t exp C t 2 jjun C !jj3;2;˝ t .1 C jj .0/jj2 /  %; 4 N Since by definition n D %n  %, N the proof is concluded. one has j n j  34 %.

t u

4.1.2 First-Order Estimates for unC1 and #nC1 Recall that X1 . n ; un ; #n ; ; t / D k n ; un ; #n ; k2;1;˝ t and that ' is a general increasing positive function of its arguments. Moreover, by assumptions, functions p.%; /; cv .%;  / (pressure and specific heat) are, respectively, twice and once differentiable, with their highest derivatives bounded for bounded arguments. Hence, from this point, one uses that jp.%; /j  '.j%j1;˝ ; j j1;˝ /, jp% .%;  /j  '.j%j1;˝ ; j j1;˝ /, jp .%;  /j  '.j%j1;˝ ; j j1;˝ / and similar facts for second derivatives of p as well as for first derivatives of cv , without referring to these assumptions explicitly at each such occasion. First-Order Estimates for unC1 Lemma 2. Let X1 . n ; un ; #n ; ; t / < 1. For f 2 L2 .0; t I L2 .˝// and u.0/ 2 L2 .˝/, the following estimate holds junC1 .t /j22 C kunC1 k21;2;˝ t  t '.X1 . n ; un ; #n ; ; t /; k!k1;1;˝ t ; % ; % /.1 C ju.0/j22 C jf j22;2;˝ t / C C j% f j22;2;˝ t

% C ju.0/j22 %

(55)

for % ; % and any t  tn given by Lemma 1. Proof. Here and in proofs of next lemmas, let us suppress t-dependence of quantity X1 for brevity. Multiplying .42/1 by unC1 , integrating over ˝ and applying the Poincaré, Hölder and Young inequalities yield

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Z

1 2

˝

C

%n @t u2nC1 dx C kunC1 k21  c.% /.k!k41 C kun k41 / '.X1 . n ; #n //.jr#n j22

C

jr n j22

C

jr j22 /

(56) 

C C j% f

j22 :

Continuing, one has d dt

Z ˝



%n u2nC1 dx C kunC1 k21

k n;t k0 junC1 j24

(57) 

C '.X1 . n ; #n ; un ; /; k!k1;1;˝ t / C C j% f

j22 :

To estimate the first term on the r.h.s., the following estimate given by (43) is needed k n;t k0  c.kun k1 k n k2 C k!k1 k n k2 C kun k1 C k!k1 /  '.X1 . n ; un /; k!k1;1;˝ t /;

(58)

where ' is the second-order polynomial with respect to the first argument. Employing the interpolation junC1 j4  "1=4 jrunC1 j2 C c"3=4 junC1 j2 , " 2 .0; 1/, and (58) in (57) implies d dt

Z ˝

%n u2nC1 dx C kunC1 k21  '.X1 . n ; un /; k!k1;1;˝ t /kunC1 k20 

C '.X1 . n ; un ; #n ; /; k!k1;1;˝ t / C C j% f

(59)

j22 :

Dropping the second term on l.h.s. and employing the Gronwall lemma and Lemma 1 that implies %n  % , one derives p k %n unC1 .t /k20  '.X1 . n ; un ; #n ; /; k!k1;1;˝ t ; % ; % /.1Cju.0/j22 Cj% f j22;˝ t /: This estimate used in time-integrated (59) yields Z ˝

.%n u2nC1 /.t /dx C kunC1 k21;2;˝ t  t '.X1 . n ; un ; #n ; /; k!k1;1;˝ t ; % ; % /.1C ju.0/j22 C jf j22;2;˝ t / C C j% f j22;2;˝ t C j% u.0/j22

that gives (55).

t u

First-Order Estimates for #nC1 The central issue connected with estimates involving #n is a possible degeneracy of the evolutionary term, due to conceivable vanishing of %n cv .%n ; n /. One excludes it as follows: Lemma 1 implies that %n  % , so it suffices to show that cv .%n ; n / is bounded away from 0. Recall that by the assumption (26) on cv , it holds

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1781

8"1 >0;"2 >0 9ı>0 j1 j  "1 ; j2 j  "2 H) cv .1 ; 2 /  ı."1 ; "2 /; hence, if there exists  > 0 such that n   , one obtains %n cv .%n ; n /  % ı.% ;  /:

(60)

Hence non-degeneracy of evolutionary term hinges on n   . In the estimates of this section, one simply assumes this for every estimate involving nC1 . Finally, in the iterative procedure of controlling the approximative sequences, one will obtain that in fact n   implies nC1   for t small enough. This and assumption 0   will justify i   ; i D 1; 2; : : : , used in the estimates. Lemma 3. Let the assumptions of Lemma 2 hold. Moreover, let b 2 L2 .˝ t /, %.0/; #.0/ 2 H 2 .˝/, j j2;1;2;˝ t C j!j2;0;1;˝ t C hbi2;2;˝ t < 1 and n   . Then for solutions to problem (44) one has

sup jj#nC1 .t /jj20 C jj#nC1 jj21;2;˝ t  t



Z ˝

 '.X1 . n ; #n ; t // t exp t % ı.% ;  / % ı.% ;  /

%.0/cv .%.0/; .0//# 2 .0/dx C '.X1 . n ; un ; #n //.1 C t C j j22;1;2;˝ t (61) 

C jj!jj42;1;˝ t C jbj22;2;˝ t / 1 C % ı.% ;  /

Z

%.0/cv .%.0/; .0//# 2 .0/dx; ˝

for % ; % and any t  tn given by Lemma 1. Proof. Multiply .44/1 with #nC1 and integrate the result over ˝ to get 1 2

Z %n cv .%n ; n / ˝

@ 2 dx C jr#nC1 j22 D # @t nC1;t

Z G.%n ; n ; un ; b/#nC1 dx; ˝

(62)

where the first integral in (62) equals 1 d 2 dt

Z ˝

2 %n cv .%n ; n /#nC1 dx

1  2

Z

The second integral in (63) is bounded by

˝

@ 2 .%n cv .%n ; n //#nC1 dx: @t

(63)

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2 Œj%n;t jjcv .%n ; n /j C j%n j.jcv;%n .%n ; n /jj%n;t j C jcv;n .%n ; n /jjn;t j/ #nC1 dx 

Z

C ˝

2 .j%n;t j C j#n;t j C j ;t j/'.jj%n jj2 ; jjn jj2 /#nC1 dx  '.jj%n jj2 ; jjn jj2 /.j%n;t j3

C j#n;t j3 C j ;t j3 /j#nC1 j23 DW I; (64) where for the middle inequality one used that jcv .%n ; n /jCjcv;%n .%n ; n /jCjcv;n .%n ; n /j  '.j%n j1 ; jn j1 /  '.jj%n jj2 ; jjn jj2 /: Let us continue estimating I , using the interpolation j#nC1 j23  j#nC1 j2 j#nC1 j6 and the Sobolev embedding, to end up with I  '.jj%n jj2 ; jjn jj2 /.jj%n;t jj1 C jj#n;t jj1 /jj#nC1 j2 jj#nC1 jj1  jj#nC1 jj21 C

1 2 ' .jj%n jj2 ; jjn jj2 /.jj%n;t jj21 C jj#n;t jj21 C jj ;t jj21 /j#nC1 j22 : 2

(65)

Altogether, (62), after considering (63) with I gives 1 d 2 dt

Z ˝

2 %n cv .%n ; n /#nC1 dx C 

Z ˝

jr#nC1 j22 

ˇZ ˇ ˇ ˇ ˇ G.%n ; n ; un ; b/#nC1 dx ˇ C jj#nC1 jj2 1 ˇ ˇ

(66)

˝

C

1 2 ' .jj%n jj2 ; jjn jj2 /.jj%n;t jj21 C jj#n;t jj21 C jj ;t jj21 /j#nC1 j22 : 2

The first term on r.h.s. of (66) will be now estimated. In view of the form of G (compare (41)), one has ˇ ˇZ Z h ˇ ˇ ˇ G.%n ; n ; un ; b/#nC1 dx ˇ  '.jj%n jj2 ; jjn jj2 / j.un C w/  r#n #nC1 j ˇ ˇ ˝

˝

C j.un C w/  r #nC1 j C jdiv .un C w/#nC1 j C j ;t #nC1 j C j #nC1 j i C jr.un C w/j2 j#nC1 j C jbjj#nC1 j dx DW J: The Sobolev embedding with Hölder’s and Young’s inequalities imply  J  j#nC1 j22 C C  1 ' 2 .jj%n jj2 ; jjn jj2 / jjun C !jj22 .jj#n jj21 C jj jj21 C 1/ C j j22;1  Cjjun C !jj42 C jbj22 :

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

This in (66), together with the Poincaré inequality for #nC1 , yields for  D d dt

Z

1783  2

2 %n cv .%n ; n /#nC1 dx C k#nC1 k21

˝

   '.jj%n jj2 ; jjn jj2 / jj%n;t jj21 C jj#n;t jj21 C jj ;t jj21 j#nC1 j22  C '.jj%n jj2 ; jjn jj2 / jjun C !jj22 .jj#n jj21 C jj jj21 C 1/ C j j22;1  Cjjun C !jj42 C jbj22 :

(67)

Now, recall (60). It implies in (67) d dt

Z ˝

2 %n cv .%n ; n /#nC1 dx C k#nC1 k21 

Z  '.jj%n jj2 ; jjn jj2 / jj%n;t jj21 C jj#n;t jj21 C jj ;t jj21 2 %n cv .%n ; n /#nC1 dxC % ı.% ;  / ˝ (68)   2 2 2 2 4 '.jj%n jj2 ; jjn jj2 / jjun C wjj2 .jj#n jj1 C jj jj1 C 1/ C j j2;1 C jjun C !jj2 C jbj22 : Dropping the second summand of l.h.s., one arrives via the Gronwall inequality and the definition of X1 ./ D k  k2;1;˝ t at Z sup t

˝

Z

 ˝

2 %n cv .%n ; n /#nC1 dx



'.X1 . n ; #n /; t /  exp t % ı.% ;  /



%.0/cv .%.0/; .0//# 2 .0/dx C '.X1 . n ; un ; #n /; t /.1 C t C j j22;1;2;˝ t

C jj!jj42;1;˝ t C jbj22;˝ t / : Using the above inequality in the time-integrated (68) implies thesis.

t u

4.1.3 First-Order Estimates for Time Derivatives of unC1 and #nC1 Recall that by definition X .: : : / D j : : : j2;1;1;˝ t ; where the norm concerns space C .12 /. First-Order Estimate for unC1,t Lemma 4. Assume that X1 . n ; un ; !; t / < 1, X .un ; #n ; !; ; t / < 1, f 2 L1 .0; t I H 1 .˝//, f;t 2 L1 .0; t I L2 .˝//, u;t .0/ 2 L2 .˝/. Then the following inequality holds

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J. Burczak et al.

junC1;t .t /j22 C kunC1;t k21;2;˝ t  h  1 C kft k20;1;˝ t C

t '.X1 . n ; un ; #n ; t /; X .un ; #n ; !; ; t // % i C kf k21;1;˝ t C ju;t .0/j22 (69)

1 j!j2 t % 3;2;2;˝

for %n  % . Proof. Differentiation .42/1 with respect to t, multiplication the result by unC1;t and integration over ˝ yields Z .%n unC1;t t C %n;t unC1;t /unC1;t dx C c.; /kunC1;t k21 ˝

D

Z h

%n;t .un C !/  r.un C !/ C %n .un;t C !;t /  r.un C !/

˝

i

Z

A!;t  unC1;t dx

C %n .un C !/  r.un;t C !;t /  unC1;t dx C C

(70)

˝

Z h

p%% .%n ; n / n;t r n C p% .%n ; n /n;t r n C p% .%n ; n /r n;t v

˝

i C p% .%n ; n / n;t rn C p .%n ; n /n;t rn C p .%n ; n /rn;t  unC1;t dx Z C .%n;t f C %n f;t /  unC1;t dx: ˝

The first term on the l.h.s. of (70) equals Z Z Z Z 1 1 d %n @t u2nC1;t dx C

n;t u2nC1;t dx D %n u2nC1;t dx C

n;t u2nC1;t dx; 2 ˝ 2 dt ˝ ˝ ˝ where, using (58), the second integral is bounded by k n;t k0 junC1;t j24  "kunC1;t k21 C c.1="/'.X1 . n ; un /; k!k1;1;˝ t /  junC1;t j22 : The first integral on the r.h.s. of (70) is bounded by "kunC1;t k21 C C .1="/.'.X1 . n // C k n;t k21 /jun C !j42;1 : One controls the third integral on the r.h.s. of (70) by X1 . n ; n /junC1;t j6 .k n;t k1 C kn;t k1 C .j n j2;1 C jn j2;1 /2 /: Finally, the last term on the r.h.s. of (70) is estimated by "junC1;t j26 C c.1="/.j n;t j22 jf j23 C '.X1 . n //jf;t j22 /:

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1785

Employing the above estimates in (70) and assuming that " is sufficiently small, one has Z d %n u2nC1;t dx C kunC1;t k21  '.X1 . n ; un /; k!k1;1 /junC1;t j22 dt ˝  C .'.X1 . n // C k n;t k21 /jun C !j42;1 C cX1 . n ; #n ; / k n;t k21 C k#n;t C ;t k21  C .j n j2;1 C j#n C j2;1 /4 C '.X. n //jf;t j22 C '.X1 . n ; un //kf k21 C j!j23;2 ; (71) where the last term is a consequence of estimating From (43) it holds (pointwisely in time)

R ˝

A!;t  unC1;t dx of (70).

k n;t k1  c.kun C !k2 k n k2 C kun C !k2 /  '.X1 . n ; un ; !//:

(72)

In view of (72) and the definition of X one derives from (71) the following inequality d dt

Z ˝

%n u2nC1;t dx C kunC1;t k21  '.X1 . n /; X. n ; #n ; un ; !; //

(73)

C '.X1 . n ; un ; !//jf;t j22 C '.X1 . n ; un //kf k21 C j!j23;2 : Integrating (73) with respect to time, we obtain (69). t u Observe that to write above unC1;t .0/ D u;t .0/ one uses the fact that our linear problems are solved in a high-regularity class. This matter reappears also in further lemmas.

First-Order Estimate for #nC1,t Lemma 5. Assume that X .un ; %n ; n ; !; t / < 1, b;t 2 L1 .0; t I L2 .˝//, b 2 L1 .0; t I H 1 .˝//, ;t t 2 L2 .0; t I L2 .˝//, 2 L1 .0; t I H 2 .˝//, #;t .0/ 2 L2 .˝/. Then for solutions to (44) the following inequality is valid

j#nC1;t .t /j22 C k#nC1;t k21;2;˝ t  t ' % ;  ; X .un ; %n ; n ; !; t /; k ;t t k20;2;˝ t Z  2 2 2 Ck k2;1;˝ t C kbk1;1;˝ t C kb;t k0;1;˝ t C %.0/cv .jj%.0/jj2 ; jj.0/jj2 /j#;t .0/j2 ˝

(74) as long as n   and %n  % . Proof. Differentiate (44)1 with respect to t and test with #nC1;t to get via the Poincaré inequality Z Z 2 .%n cv .%n ; n //;t #nC1;t %n cv .%n ; n /#nC1;t t #nC1;t dx C dx C jj#nC1;t jj21 ˝

˝

Z D

.G.%n ; n ; un ; b//;t #nC1;t dx: ˝

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J. Burczak et al.

Hence via the Hölder inequality one obtains 1 2

Z %n cv .%n ; n / ˝

@ j#nC1;t j2 dx C jj#nC1;t jj21  "j#nC1;t j26 @t

C C "1 j .%n cv .%n ; n // ;t j23 j#nC1;t j23 C C "1 jG 0 .%n ; n ; un ; b//;t j26 5 ˇZ ˇ ˇ ˇ C ˇˇ  ;t #nC1;t dx ˇˇ ; ˝

where G 0 denotes G without  . The Sobolev embedding and an appropriate choice of " implies Z

@ j#nC1;t j2 dx C jj#nC1;t jj21  C j .%n cv .%n ; n // ;t j23 j#nC1;t j23 @t ˇZ ˇ ˇ ˇ C C jG 0 .%n ; n ; un ; b//;t j26 C ˇˇ  ;t #nC1;t dx ˇˇ DW CI1 j#nC1;t j23 C CI2 : 5

%n cv .%n ; n / ˝

˝

(75) Next, let us examine particular terms of r.h.s. of (75) I1  '.j%n ; n j1 /.j%n;t j23 C jn;t j23 /  '.jj%n ; n jj2 /.jj%n;t ; #n;t ; ;t jj21 /; I2  j.%n cv .%n ; n /.un C !/r#n /;t j26 C j.%n cv .%n ; n /.un C !/r /;t j26 5 5 ˇZ ˇ ˇ ˇ C j.n pn .%n ; n /div .un C !//;t j26 C j.%n cv .%n ; n / t /;t j26 C ˇˇ  ;t #nC1;t dx ˇˇ 5

5

˝

C jr.un;t C !;t /r.un C !/j26 C j.%n b/;t j26 DW 5

5

7 X

I2i ;

i D1

where  I21  '.j%n ; n j1 / jun C wj21 jj%n;t ; n;t jj21 jj#n jj22 C jjun;t C !;t jj21 jj#n jj22  Cjjun C !jj22 jj#n;t jj21 ;  I22  '.j%n ; n j1 / jun C !j21 jj%n;t ; n;t jj21 jj jj22 C jjun;t C !;t jj21 jj jj22  Cjjun C !jj22 jj ;t jj21 ;   I23  '.j%n ; n j1 / jj%n;t ; n;t jj21 jjun C !jj22 C jjun;t C !;t jj21 ;   I24  '.j%n ; n j1 / jj%n;t ; n;t jj21 jj t jj21 C jj t t jj20 ;

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

I25

1787

ˇZ ˇ ˇ ˇ ˇ D ˇ r ;t r#nC1;t dx ˇˇ  "jjr#nC1;t jj20 C c"1 jjr ;t jj20 ; ˝

I26  C jjun;t C !;t jj21 jjun C !jj22 ; I27  C j%n j22;1 jbj21;0 : The estimates for I1 and I2 used in (75) give Z

@ j#nC1;t j2 dx C jj#nC1;t jj21 @t ˝ h  '.jj%n ; n jj2 / jj%n;t ; n;t jj21 .1 C jjun C !jj22 .1 C jjn jj22 C jj jj22 // i Cjun C!j22;1 .j#n j22;1 Cj j22;1 /Cjj t jj21 Cjj t t jj20 Cjun C!j42;1 Cj%n j22;1 jbj21;0 DW A: %n cv .%n ; n /

Hence Z

d dt

˝

%n cv .%n ; n /j#nC1;t j2 dx C jj#nC1;t jj21 Z

 '.jj%n ; n jj2 / ˝

2 .j%n;t j C jn;t j/#nC1;t dx C A

(76)

 '.jj%n ; n jj2 /.j%n;t j3 C jn;t j3 /j#nC1;t j2 j#nC1;t j6 C A 

 jj#nC1;t jj21 C '.jj%n ; n jj2 /.jj%n;t jj21 C jjn;t jj21 /j#nC1;t j22 C A; 2

where the Hölder, the Young and the Sobolev inequalities were also invoked. Now, one proceeds similarly as in the last part of Lemma 3. Assumptions n   , %n  % imply %n cv .%n ; n /  % ı.% ;  /: Consequently (76) takes the form Z

d dt

˝

%n cv .%n ; n /j#nC1;t j2 dx C jj#nC1;t jj21;2;˝ t 

'.jj%n ; n jj2 /.jj%n;t jj21 C jjn;t jj21 / % ı.% ;  /

(77)

Z

2

%n cv .%n ; n /j#nC1;t j dx C A: ˝

Skipping the second term on l.h.s. of (77) yields Z sup t

˝

%n cv .%n ; n /j#nC1;t j2 dx

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J. Burczak et al.

Z  exp 0

t

'.X .%n ; n // % ı.% ;  /

Z

2

Z

t

%.0/cv .%.0/; .0//j#;t .0/j dx C ˝

A :

0

Next, the use of the above estimate to time-integrated (77) gives Z sup t

t

˝

%n cv .%n ; n /j#nC1;t j2 dx C jj#nC1;t jj21;2;˝ t 

'.X .%n ; n // exp % ı.% ;  /

Z 0

t

Z Z t '.X .%n ; n // %.0/cv .%.0/; .0//j#;t .0/j2 dx C A % ı.% ;  / ˝ 0 Z Z t AC %.0/cv .jj%.0/jj2 ; jj.0/jj2 /j#;t .0/j2 dx: C ˝

0

Using the form of A, the above inequality implies

j#nC1;t .t /j22 C k#nC1;t k21;2;˝ t 

 t ' % ;  ; X .un ; %n ; n ; !/; k ;t tk20;2;˝ t C k k22;1;˝ t C kbk21;1;˝ t C kb;tk20;1;˝ t Z %.0/cv .jj%.0/jj2 ; jj.0/jj2 /j#;t .0/j2 dx; (78) C ˝

which is our thesis.

t u

4.1.4 Higher-Order Estimates for unC1 In order to prove the following third-order estimates for unC1 and second-order estimates for unC1;t , one resorts to local considerations. Providing second-order estimates for unC1 itself will be omitted, since third-order estimates for unC1 are necessary to close our reasoning anyway and the full H 3 norm can be obtained by an interpolation between the first-order norm of a previous paragraph and obtained estimates for the third-order seminorm in what follows. Turning to local considerations is a standard way to single out the sole direction (normal to the boundary) that one cannot differentiate along at the boundary; consequently integration by parts in energy estimates works fine only for tangential directions to the boundary. One deals with the derivatives along the normal direction (now canonical thanks to localization and boundary straightening) via the equation itself. We present in what follows only estimates for the neighborhoods near the boundary, because estimates in interior subdomains are similar and simpler (they involve only localization, with no need for a variable change). Let us localize in space problem (42) to a neighborhood of a portion of boundary S . Consequently, we have

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1789

%O n uQ nC1;t  Az uQ nC1 D O O unC1  .2rO uO nC1 rO O C uO nC1 O /  .Az  A/Q

(79)

O C FQ .%O n ; On ; uO n ; fO /; O uO nC1 rO O C rO uO nC1 rO O C uO nC1 rO rO /  .div where the notation of Sect. 2 is implemented. In particular, recall that  is a function that cuts off to a generic local set ˝Q having its length comparable to  and that O denotes an object (function, set) related to the original x-variables at x D 1 .z/ expressed in the “straight” variables z. We write Az D z C rz div z and O . AO D O C  rO div For some more details, see Sect. 3.2.3. Observe that there are no lower-order terms related to FQ , since it depends on non-localized functions. In the following @@ will denote derivatives along tangent directions to the @ straightened boundary (i.e.,  is either z1 or z2 ) and @n – normal derivatives (i.e.,  n is z3 ). The dependence of constants on % ; % will be also suppressed, because it is controlled by considering only times t  tn given by Lemma 1. Third-Order Estimate for unC1 Lemma 6. Assume that X1 . n ; un ; #; !/ < 1, f 2 L1 .0; T I H 1 .˝//, f;t 2 L1 .0; t I L2 .˝//, u.0/ 2 H 2 .˝/, and ! 2 L2 .0; T I H 3 .˝//. Then for any t  tn given by Lemma 1, as long as kunC1;t k22;2;˝ t is finite, one has for any " > 0 kunC1 .t /k22;˝ C kunC1 k23;2;˝ t  "kunC1;t k22;2;˝ t C cku.0/k22;˝ C ck!k23;2;˝ t C t '.X1 . n ; un ; #n ; ; !/; "/.1 C ju.0/j22 C kft k20;1;˝ t C kf k21;1;˝ t /: (80) Proof. Differentiate (79) twice with respect to tangential directions , multiply the result by uQ nC1; , integrate over a generic local “straightened” set ˝O and by parts to obtain Z Z Z 2 .%O n uQ nC1;t /;  uQ nC1; d z C jr uQ nC1; j d z C jdiv uQ nC1; j2 d zC ˝O



˝O

ckQunC1; k21;˝O

˝O

 C c kFQ ; k20;˝ C c4 kOunC1 k22;˝O ;

(81)

O The recall that   1 is proportional to the side length of a local domain ˝. important  in front of the first r.h.s. term of (81) follows from the smoothness O Let us explain this of the variable change map and the size of a local domain ˝. in some more details. Recall that by definition Q 1 ; y2 /; y 2 ˝Q zi D yi ; i D 1; 2; z3 D y3  G.y and

(82)

1790

J. Burczak et al.

@zi rz : rO k D @xk i Let us assume for a moment that the map Y changing the variables x ! y is identity. Then @zi rO k D rz : @yk i We will explain how to perform the estimate in relation to variable x by the end of this reasoning. We have constructed our partition of unity so that for any number , it holds Q 1 C jr Gj Q 21 /  c; .jr Gj

(83)

recall (35). Consider now the main part O unC1 D .z  /Q O unC1  .rz div z  rO div O /QunC1 .Az  A/Q of (79). The differentiation @  and testing with uQ nC1; yields Z  Z

˝O

 ˝O

O uknC1;  /QuknC1;  d z C .z uQ knC1;˛ ˇ  Q ˛ ˇ ˛ ˇ

j j .rzk rzj uQ nC1;˛ ˇ  rO k rO j uQ nC1;˛ ˇ /QuknC1;˛ ˇ d z C III DW I C II C III;

(84) where ˛; ˇ D 1; 2 and i; j; l; k D 1; 2; 3 and the lower-order term Z III  C ./ ˝O

junC1;z  jjunC1;  jd z:

Let us consider I Z  I D  ˝O

Z  dz D   ˝O



@2zj zj

@zj  @z @yk j

ıjj 0 

@zj @zj 0 @yk @yk



@zj 0 @z 0 @yk j



.QuknC1;˛ ˇ / uQ knC1;˛ ˇ

@2zj zj 0 uQ knC1;˛ ˇ uQ knC1;˛ ˇ

@zj @yl @2 zj 0 @z 0 uQ k uQ k d z D I 1 C I 2: @yk @zj @yk yl j nC1;˛ ˇ nC1;˛ ˇ

The lower-order term (with respect to derivatives of u) I 2 can be estimated simply as follows

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

I 2  C ./

1791

Z ˝O

junC1;z  jjunC1; jd z:

Integrating by parts once in I 1 and using that uQ   j@˝O D 0, we get Z  I D ˝O

ıjj 0 

@zj @zj 0 @yk @yk



@zj uQ knC1;˛ ˇ @zj 0 uQ knC1;˛ ˇ d z C I 4 D I 3 C I 4 ;

where I 4 is the lower-order term involving derivatives of cutoff functions; hence jI 4 j  C ./

Z ˝O

junC1;z  jjunC1;  jd z:

Finally, I 3 in view of (82) reads I3 D 

Z

˝O

 ıjj 0  ıjj 0 fj;j 0 D1;2g @zj uQ knC1;˛ ˇ @zj 0 uQ knC1;˛ ˇ d z

Z 

@zj @z3 C ıj 3  @zj uQ knC1;˛ ˇ @z3 uQ knC1;˛ ˇ @yk @yk ˝O Z  @z3 @z3 ıj 3  dz D 0 C  @z1 uQ knC1;˛ ˇ  @z2 uQ knC1;˛ ˇ @y @y O 1 2 ˝ @z3 @z3 @z3 uQ knC1;˛ ˇ d z  @z uQ k @yk @yk 3 nC1;˛ ˇ In the last formula ı33 and

@2 z3 @y32

cancel out each other and consequently

Q 1 C jr Gj Q 21 / jI 3 j  C  .jr Gj

Z ˝O

junC1;z  j2 d z  c

Z ˝O

junC1;z  j2 d z;

with the latter estimate coming from (83). Hence we see that the contribution of I agrees with (81). We deal similarly with the part II of (84). Namely, its highest-order part reads Z  II D  ˝O

@zi @zi 0 j @zk @zj  @zi @zi 0 .QunC1;˛ ˇ / uQ knC1;˛ ˇ d z: @yk @yj

Again, integrating by parts and using that uQ   j@˝O D 0, we get Z  II D 

@zi 0 @zi j j @zj uQ nC1;˛ ˇ @zk uQ knC1;˛ ˇ  @zi 0 uQ nC1;˛ ˇ @zi uQ knC1;˛ ˇ d zC @y @yk ˝O Z j 0 @z @z i i j @zi 0 d z D II 1 C II 2  uQ nC1;˛ ˇ @z uQ k @yk i nC1;˛ ˇ @yj ˝O

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The lower-order term satisfies jII 2 j  C ./

Z ˝O

junC1;z  jjunC1; jd z;

whereas for the high-order term we have analogously as before, thanks to (82) Q 21 jII j  C  jr Gj 1

Z ˝O

junC1;z  j2 d z:

We see that the contribution of II also agrees with (81). Altogether, we see that (81) holds in relation to the change of variables y ! z. Finally, we need to justify that the above reasoning works also for the change of variables x ! z, which means we need now to take into account the map x ! y. This is however a composition of a translation and a rotation. Hence all the estimates justifying (81) hold also in this case. Having (81), let us observe that the first term on the l.h.s. of (81) is proportional to Z Z %O n uQ nC1;  t uQ nC1; d  C .%O n; uQ nC1;t C %O n; uQ nC1;t /  uQ nC1; d  ˝O

D D

1 2

˝O

Z ˝O

1 d 2 dt

%O n @t jQunC1; j2 d  C Z

Z

˝O

C ˝O



1 d 2 dt

Z

%O n jQunC1; j2 d  

˝O

.%O n; uQ nC1;t C %O n; uQ nC1;t /  uQ nC1; d 

Z

˝O

%O n;t uQ 2nC1; d 

.%O n; uQ nC1;t C %O n; uQ nC1;t /  uQ nC1; d 

Z

˝O

%O n jQunC1; j2 d   I1 C I2 ;

with c jI1 j  "jQunC1; j26;˝O C j%O n;t j23;˝O jQunC1; j22;˝O  "kQunC1; k21;˝O " c c 2 C k On;t k1;˝O jQunC1; j22;˝O  "kQunC1; k21;˝O C '.X1 . n ; un ; !//jQunC1; j22;˝O ; " " where (72) was used. Next c jI2 j  "jQunC1; j26;˝O C k%O n k22;˝O kQunC1;t k21;˝O  " c 2 "kQunC1; k1;˝O C '.X1 . n //kQunC1;t k21;˝O : " Hence (81) gives

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1793

Z Z 1 d 2 %O n jQunC1; j C jr uQ nC1; j2 d z  c. C "/kQunC1; k21;˝O 2 dt ˝O ˝O c c C '.X1 . n ; un ; !//kQunC1 k22;˝O C '.X1 . n //kQunC1;t k21;˝O " "  C c kFQ ; k20;˝O C c4 kOunC1 k22;˝O :

(85)

Finally, writing vn WD un C !, one estimates kFQ k2

1;˝O

 ck!k Q 2

3;˝O

Cck%O n k2 O kfQ k2 2;˝

1;˝O

Cck On;z vO n  rO vO n C %O n vO n;z  rO vO n C %O n vO n  rO vO n;z k2

0;˝O

2 C ckp%% On;z On;z C p% On;z On;z C p% On;z On;z C p On;z

 ck!k Q 2

3;˝O

Cck%O n k2 O kfQ k2 2;˝

1;˝O

C p% On;zz C p On;zz k0;˝O

C'.k On k2;˝O ; kOun C!k2;˝O /C'.k On k2;˝O ; kOn k2;˝O /:

Employing the above estimates in (85) yields Z Z 1 d %O n jQunC1; j2 d z C jr uQ nC1; j2 d z  c. C "/kQunC1; k21;˝O 2 dt ˝O ˝O c c C '.X1 . n ; un ; !//kQunC1 k22;˝O C '.X1 . n //kQunC1;t k21;˝O C (86) " " ck!k Q 2 Cck%O n k2 kfQ k2 C'.k On k O ; kOn k O ; kOun C!k O /Cc4 kOunC1 k2 : 3;˝O

2;˝O

1;˝O

2;˝

2;˝

2;˝

2;˝O

Integrating (86) with respect to time, using (55) to gain control of L2 norms of u; on r.h.s., which, via interpolation, allows to control also H 1 norms of u; , so that consequently one has full norms on r.h.s., implies jQunC1; .t /j22;˝O C kQunC1; k21;2;˝O t  c. C "/kQunC1; k21;2˝O t C c '.X1 . n ; un ; !//kOunC1 k2;2;˝O t C '.X1 . n //kQunC1;t k21;2;˝O t C ck!k Q 23;2;˝O t C 4 C t '.X1 .un ; n ; #n ; ; !// C ckQu.0/k2 C '.X1 .%n //kfQ k2 1;2;˝O t

2;˝O

t '.X1 . n ; un ; #n ; /; k!k1;1;˝ t /.1 C ju.0/j22 C jf j22;2;˝ t /; (87) where the last summand on r.h.s. follows from (55). Above, a smallness of " was used as well as the fact that assumed tn  t (of Lemma 39) implies %Q n  % (dependence of c on % was suppressed in notation). In order to obtain full third-order estimate, one needs to supplement (86) with estimates containing second- and third-order normal derivatives. As already mentioned, since differentiation by parts would produce here unwanted non-zero boundary terms, one resorts directly to our equation. A control on full kQunC1 k3;2;˝O t will be obtained in two steps.

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(i) Firstly, differentiate (79) once with respect to  , and, using Az uQ D z uQ C rz div z uQ , rewrite it as O unC1;  z uQ nC1; D rz div z uQ   .%O n uQ nC1;t /;  .Az  A/Q O ;  .2rO uO nC1 rO O C uO nC1 O /

(88)

O ; C FQ ; .%O n ; On ; uO n ; fO /: O uO nC1 rO O C rO uO nC1 rO O C uO nC1 rO rO /  .div Hence, forming on both sides of (88) L2 norms, using Calderón-Zygmund estimates (to write all second-order norms out of z ) and interpolation (to write full H 3 norm using only the highest- and lowest-order terms), one obtains along the computation from the previous page kQunC1; k22  ckdiv uQ nC1; k21 C '.X1 . n //kQunC1;t k21;˝O C ckQunC1; k22;˝O C ck!k Q 23;˝O C ck%O n k22;˝O kfQ k21;˝O C '.k On k2;˝O ; k#O n C k O 2;˝O ; kOun C !k2;˝O / C c6 kOunC1 k22;˝O ; (89) i.e., it suffices to control kdiv uQ nC1; k21 . Actually, thanks to (87) that controls jdiv uQ nC1; j22;2 , it suffices to control jdiv uQ nC1;3 j22;2 (derivation with respect to normal direction is denoted by ;3 ). To this end, one uses the classical and crucial observation on cancellation in the dissipative term. Namely, let us write Az uQ nC1 D z uQ nC1 Crz div z uQ nC1 D .C/rz div z uQ nC1 C.z uQ nC1 rz div z uQ nC1 /: (90) Its third (normal) component reads z uQ 3nC1 C r3 div z uQ nC1 D . C /div z uQ nC1;3 C .Qu1nC1;11 C uQ 2nC1;22  uQ 1nC1;13  uQ 2nC1;23 /; where a cancellation of uQ 333 can be observed. Formula (90) allows to write yet another form of (79) differentiated once with respect to . Its third (normal) component reads in view of the above cancellation  . C /div z uQ nC1;3 D .Qu1nC1;11 C uQ 2nC1;22  uQ 1nC1;13  uQ 2nC1;23 /  3 O 3  .%O n uQ nC1;t /3  .Az  AO k /QunC1;  .2rO k uO nC1 rO k O C uO nC1 O k / 

(91)

O 3 C FQ 3 .%O n ; On ; uO n ; fO /: O k uO nC1 rO k O C rO k uO nC1 rO k O C uO nC1 rO k rO k /  .div   Since r.h.s. of (91) contains only first-order normal derivatives (except for the -small term involving .Az  AO k /), it is controlled via (87) by its r.h.s. Now (91) governs jdiv uQ nC1;n j22;2 ; hence the entire r.h.s. of time-integrated (89) is via (91)

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1795

under control by means of r.h.s of (87). Consequently jQunC1; .t /j22;˝O C kQunC1; k22;2;˝O t  ckQunC1; k22;2;˝O t C c '.X1 . n ; un ; !//kOunC1 k2;2;˝O t C '.X1 . n //kQunC1;t k21;2;˝O t C ck!k Q 23;2;˝O t C 4 C t '.X1 .un ; n ; #n ; // C ckQu.0/k2 C '.X1 .%n //kfQ k2 1;2;˝O t

t '.X1 . n ; un ; #n ; /; k!k1;1;˝ t /.1 C

2;˝O

ju.0/j22

C jf

j22;2;˝ t /:

(92) After choosing small  and using (55) to estimate term involving kQunC1;t k2 O t , one 1;2;˝ simplifies (92) to jQunC1; .t /j22;˝O C kQunC1; k22;2;˝O t  '.X1 . n ; un ; !//kOunC1 k22;2;˝O t C ck!k Q 23;2;˝O t C '.X1 .%n //kfQ k21;2;˝O t C ckQu.0/k22;˝O

(93)

C t '.X1 . n ; un ; #n ; /; k!k1;1;˝ t /.1 C ju.0/j22 C jf j22;2;˝ t /: (ii) In the second step, one repeats the first step for the normal derivative. Namely, differentiating (79) once with respect to the normal derivative, we get a counterpart of (88) that reads Q 23;˝O kQunC1;3 k22  ckdiv uQ nC1;3 k21 C '.X1 . n //kQunC1;t k21;˝O C ckQunC1;3 k22;˝O C ck!k Cck%O n k22;˝O kfQ k21;˝O C'.k On k2;˝O ; k#O n k2;˝O ; kOun C!k2;˝O /Cc6 kOunC1 k22;˝O :

(94)

Again, it suffices to control kdiv uQ nC1;3 k21 . Moreover, thanks to (93) resulting from the first step, in fact one needs only to control the pure normal derivative jdiv uQ nC1;33 j22 . Here, as in (i), thanks to cancellation of type (91), one controls jdiv uQ nC1;33 j22 using only second-order normal derivatives (C-small normal derivatives) that are governed by (93). All in all, one arrives at jQunC1; .t /j22;˝O C kQunC1 k23;2;˝O t  '.X1 . n ; un ; !//kOunC1 k22;2;˝O t C ck!k Q 23;2;˝O t C '.X1 .%n //kfQ k21;2;˝O t C ckQu.0/k22;˝O

(95)

C t '.X1 . n ; un ; #n ; /; k!k1;1;˝ t /.1 C ju.0/j22 C jf j22;2;˝ t /: This ends the second step. Next one passes in neighborhoods near the boundary to variables x, derives a similar estimate to (95) in an interior subdomain, and sums up over all neighborhoods of the partition of unity to arrive at global estimate

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junC1; .t /j22;˝ C kunC1 k23;2;˝ t  '.X1 . n ; un ; !//kunC1 k22;2;˝ t C ck!k23;2;˝ t C '.X1 .%n //kf k21;2;˝ t C cku.0/k22;˝

(96)

C t '.X1 . n ; un ; #n ; /; k!k1;1;˝ t /.1 C ju.0/j22 C jf j22;2;˝ t /: (The right-hand side after summation takes the above form, because the partition of unity can be constructed so that every point belongs to a uniformly bounded number of neighbourhoods.) To conclude the proof, the presence of full second derivatives in the evolutionary term is needed. To this end one uses brute force, writing d dt

Z ˝

u2nC1;xx dx  "kunC1;t k22;˝ C .c="/junC1;xx j22;˝ :

This, integrated over time, supplemented with (69) and interpolation to control lower-order norms, together with (96) implies kunC1 .t /k22;˝ C kunC1 k23;2;˝ t  "kunC1;t k22;2;˝ t C '.X1 . n ; un ; !/; "/kunC1 k22;2;˝ t C '.X1 .%n //kf k21;2;˝ t C cku.0/k22;˝ C ck!k23;2;˝ t

(97)

t '.X1 . n ; un ; #n ; /; k!k1;1;˝ t /.1 C ju.0/j22 C kft k20;1;˝ t C kf k21;1;˝ t /: This, in tandem with interpolation '.X1 . n ; un ; !/; "/kunC1 k22;2;˝ t 

1 kunC1 k23;2;˝ t C '.X1 . n ; un ; !/; "/kunC1 k21;2;˝ t 2

and first-order estimate (55) implies thesis, where norm of f is estimated by the supremum norm in time to gain multiplication with t . u t Adding Second-Order Estimates for Time Derivative of unC1 The first term on the r.h.s. of (80), i.e., "kunC1;t k22;2;˝ t is of the highest order. In the following result, one gets rid of it and provides the final high-order estimate. Lemma 7. Let X .un ; n ; #n ; ; !; t / < 1 as well as ! 2 L2 .0; T I 23 .˝//. Moreover, assume that f 2 L1 .0; t I H 1 .˝//, f;t 2 L1 .0; t I L2 .˝// and u.0/ 2 H 2 .˝/, u;t .0/ 2 H 1 .˝/. Then for any t  tn given by Lemma 1, it holds kunC1;t .t /k21;˝ C kunC1;t k22;2;˝ t C kunC1 .t /k22;˝ C kunC1 k23;2;˝ t  ku;t .0/k21;˝ C ku.0/k22;˝ C t '.X .un ; n ; #n ; ; !; t //Œ1 C

(98) kft k20;1;˝ t

C kf

k21;1;˝ t

C

ju.0/j22

C

cj!j23;2;2;˝ t :

Proof. This proof follows the lines of the previous one, except for the final part. Formally, up to counterpart of estimate (96), one substitutes one tangent derivative

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1797

in space by the time derivative. Namely, let us differentiate (79) once with respect to time and once with respect to a tangential direction  , multiply the result by uQ nC1;t , integrating over a generic local “straightened” set ˝O and by parts to obtain a counterpart of (81) Z ˝O



Z .%O n uQ nC1;t /;t uQ nC1;t d z C ckQunC1;t k22;˝O

˝O

jr uQ nC1;t j2 d z C

Z ˝O

jdiv uQ nC1;t j2 d zC

 2 4 2 Q C c kF;t k0;˝ C c kOunC1;t k1;˝O :

(99)

Recalling that X differs from X1 by involving time derivative and that X .%n /  X1 .%n ; un ; !/ by (72), (99) leads to the following counterpart of (87) jQunC1;t .t /j22;˝O C kQunC1;t k21;2;˝O t  ckQunC1;t k22;2;˝O t C c '.X . n ; un ; !//jOunC1 j1;0;2;˝O t C '.X1 . n ; un ; !//kQunC1;t k21;2;˝O t C ck!; Q t k22;2;˝O t C 4 C ckQu;t .0/k2 C '.X1 .%n ; un ; !//jfQ j2 1;0;2;˝O t



t '.X .un ; n ; #n ; ; !// 1 C

1;˝O

kft k20;1;˝ t

 C kf k21;1;˝ t C ju;t .0/j22;˝ C cj!j23;2;2;˝ t ; (100)

where the last line follows from r.h.s. of (69), employed to control lower-order terms of norms. Analogously, one obtains the following estimate related to (96) junC1;t .t /j22;˝ C kunC1;t k22;2;˝ t  '.X1 .%n ; un ; !//jf j21;0;2;˝ t C cku;t .0/k21;˝ C   t '.X .un ; n ; #n ; ; !// 1 C kft k20;1;˝ t C kf k21;1;˝ t C ju;t .0/j22;˝ C cj!j23;2;2;˝ t : (101) Now we diverge from reasoning of the proof of Lemma 6. First, since (101) controls the " part of r.h.s. of (80), one obtains by adding (101) and (80) and choosing small " junC1;t .t /j22;˝ C kunC1;t k22;2;˝ t C kunC1 .t /k22;˝ C kunC1 k23;2;˝ t  

'.X1 .%n ; un ; !// jf j21;0;2;˝ t C t ju.0/j22 C cku;t .0/k21;˝ C cku.0/k22;˝ t '.X .un ; n ; #n ; ; !//Œ1 C kft k20;1;˝ t C kf k21;1;˝ t C ju.0/j22

(102)

C ju;t .0/j22;˝ C cj!j23;2;2;˝ t : This closes the estimate for the highest-order terms that was inconsistent in formulation of Lemma 6. To conclude our proof, an estimate of the term junC1;t3 .t /j22;˝ is neccessary. Here, using brute force as in Lemma 6 would produce an uncontrolled term with two time derivatives. Therefore let us proceed differently. Namely, via Eq. (42), one

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estimates the full evolutionary term, whereas the dissipative part will be controlled by (102). More precisely, differentiating (42) with respect to t , multiplying the result by unC1;t and integrating over ˝ yields Z

Z .%n unC1;t /;t unC1;t C

 ˝

AunC1;t unC1;t

˝

Z

Z

D

.%n .un C !/  r.un C !//;t  unC1;t dx  ˝

A!t unC1;t dx

(103)

˝

Z

Z

C

Œp% .%n ; n /r n C p .%n ; n /rn ;t  unC1;t dx  ˝

.%n f /;t  unC1;t dx: ˝

In the first term on the l.h.s. of (103), an integration by parts produces no unwanted boundary terms, since time derivative (as tangent) vanishes on the boundary. Hence Z

Z .%n unC1;t /;t unC1;t D

 ˝

.r n;t unC1;t ˝

C n;t runC1;t C r n unC1;t t C %n runC1;t t /  runC1;t dx DW

4 X

Ii :

i D1

Now, let us examine the particular terms from

P4

iD1 Ii .

c jI1 j  "jrunC1;t j26 C jr n;t j22 junC1;t j23 : " In view of (72) and interpolation, the second term of r.h.s. is bounded by '.X1 . n ; un ; !//kunC1;t k0 kunC1;t k2 ; hence c jI1 j  "kunC1;t k22 C '.X1 . n ; un ; !//kunC1;t k20 : " Next 3

1

jI2 j  j%n;t j3 jrunC1;t j23  k%n;t k1 kunC1;t k22 kunC1;t k02 c  "kunC1;t k22 C '.X1 . n ; un ; !//kunC1;t k20 ; " where one used interpolation and (72). To examine I3 , calculate unC1;t t from .42/1 . Hence it holds

34 Local and Global Solutions for the Compressible Navier-Stokes. . .



Z I3 D

r n runC1;t  ˝

1 AunC1;t %n



 

1

n;t AunC1 %2n

1799



 C .un;t C !;t /  r.un C !/ C .un C !/  r.un;t C !t /  C

 1 1 A!;t  2 n;t A! C p%% .%n ; n / n;t r n C p% .%n ; n /#n;t r n %n %n

C p% .%n ; n /  r n;t C p% .%n ; n / n;t r.#n C / C p .%n ; n /.#n C /;t r.#n C / C p .%n ; n /r.#n C /;t /  .p% .%n ; n //r n Cp .%n ; n /r.#n C /

1 %n

 6  X 1 C f; dx DW I3i :

n;t t %2n i D1

Let us examine separately each of the six I3i terms, according to the division above, given by large parentheses. In view of Lemma 1 and by interpolation c I31  "jAunC1;t j22 C jr n j26 jrunC1;t j23  "jAunC1;t j22 " 3 1 c C k n k22 kunC1;t k22 kunC1;t k02  2"kunC1;t k22 C c" k n k82 kunC1;t k20 : " The second summand in I3 is bounded via the Hölder, Sobolev, and Young inequalities as follows I32 

Z jr n j j n;t j jrunC1;t j jAunC1 jdx  jr n j6 j n;t j6 jrunC1;t j6 jAunC1 j2 ˝

c  "kunC1;t k22;˝ C k n k22 k n;t k21 kunC1 k22 " 4 2 c  "kunC1;t k22;˝ C k n k22 k n;t k21 kunC1 k33 kunC1 k03 " c c k n k62 k n;t k61 kunC1 k20  "kunC1;t k22;˝ C "1 kunC1 k23 C " "1 " c c  "kunC1;t k22;˝ C "1 kunC1 k23 C '.X1 . n //'.X1 . n ; un ; !//kunC1 k20 ; " "1 " where for the last estimate, (72) was used. Next c 1 I33  "jrunC1;t j26 C jr n j26 jun C !j42;1  "kunC1;t k22 C '.X1 . n /; X .un ; !//: " " Recalling that for t  tn %n 2 Œ% ; % , one estimates the fourth term I34 by

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J. Burczak et al.

c.% /.jrunC1;t j3 jr n j6 k!;tk2 C jrunC1;t j6 jr n j6 j n;t j6 jA!j2 / 1

1

 c.% /.jrunC1;t j62 jrunC1;t j22 jr n j6 k!;tk2 C jrunC1;t j6 jr n j6 j n;t j6 jA!j2 /

 c.% / '.X1 . n // jrunC1;t j22 C k!k22 "

 c.% /  "kunC1;t k22 C k!;tk22 C '.X1 . n // kunC1;t k2 kunC1;t k0 C k!k22 "

 c.% /  2"kunC1;t k22 C k!;tk22 C '.X1 . n // kunC1;t k20 C k!k22 : "  "jrunC1;t j26 C k!;tk22 C

Similarly, for the last two terms in I3 holds I35 C I36  "jrunC1;t j26 C

 c.% /

'.X1 . n /; X .un ; #n ; ; !// C X12 . n /jf;t j22 : "

Concerning I4 , it can be expressed in the form 1 I4 D 2

Z

1 d %n @t jrunC1;t j D 2 dt ˝ 2

Z

1 %n jrunC1;t j  2 ˝ 2

Z

n;t jrunC1;t j2 ;

˝

where, by interpolation, the second integral of r.h.s. is not greater than c "krunC1;t k21 C k n;t k41 junC1;t j22  "krunC1;t k21 C '.X1 . n ; un ; !//junC1;t j22 " with the second inequality given by (72). Summing up, for the first term on the l.h.s. of (103), it holds Z

Z 4 X 1 d  .%n unC1;t /;t  unC1;t D Ii  %n jrunC1;t j2 dx  "kunC1;t k22 2 dt ˝ ˝ iD1  c.% / '.X1 . n /; X .un ; #n ; ; !//C  "1 kunC1 k23  k!;t k22  ""1 '.X1 . n ; un ; !//.junC1;t j20 C junC1 j20 / C X12 . n /.jf;t j22 C k!k22 / : (104) Next, one estimates the r.h.s. of (103). The first term is bounded by "junC1;t j22

c C "

Z

Œj n;t j2 jun C !j2 jr.un C !/j2 ˝

C j%n j2 jun;t C !;t j2 jr.un C !/j2 C j%n j2 jun C !j2 jr.un;t C !;t /j2 dx; where the second integral is bounded by

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1801

'.X1 . n ; un ; !/; X .un ; !//  '.X1 . n /; X .un ; !//: The second integral on the r.h.s. of (103) is not greater than c "junC1;t j22 C k!;t k22 : " Finally, the last two terms on the r.h.s. of (103) can be estimated, by the same token as before, by "junC1;t j22 C

 c

'.X . n ; un ; #n ; ; !// C X12 . n ; un ; !/kf k21 C % jf;t j22 : "

Putting together all the estimates for (103): (104) for its l.h.s., the above three ones for its r.h.s, we get 1 d 2 dt C

Z

2

Z

%n jrunC1;t j C ˝

˝

AunC1;t unC1;t  "kunC1;t k22 C "1 kunC1 k23 C k!;tk22

c.% /  '.X . n ; un ; #n ; ; !// C '.X1 . n ; un ; !//.junC1;t j20 C junC1 j20 / ""1 (105) 

C X12 . n /.k!k22 C kunC1;t k20 C jf;t j22 / C X12 . n ; un ; !/kf k21 : As already mentioned, the second term on l.h.s. is not used as a dissipative term now, but transferred it to r.h.s. and estimated by C kunC1;t k22;˝ . Integrating the resulting inequality in time and employing (69) for the lower-order terms gives kunC1;t .t /k21;˝  C kunC1;t k22;2;˝ t C "1 kunC1 k23;2;˝ t C cj!j23;2;2;˝ t C kunC1;t .0/k21;˝ h i C t '.X .un ; n ; #n ; ; !// 1 C kft k20;1;˝ t C kf k21;1;˝ t :

(106)

Finally, one utilizes (102) to deal with C kunC1;t k22;2;˝ t , and "1 kunC1 k23;2;˝ t on the r.h.s. above and arrives at our thesis. u t As already mentioned, we suppose that the “brute force” approach of Lemma 6 and estimating the lacking normal derivatives via equation differentiated with respect to time (Lemma 8) may prove useful to deal with difficulties at the boundary in similar problems.

4.1.5 Closing Estimates for n and %n Observe that in assumptions of Lemma 7, more regularity on n and #n is needed, than already provided. In this final section concerning estimates for approximative sequences, this gap is closed.

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J. Burczak et al.

Estimate of X.n / Recall that Y .un ; #n ; t / D jun ; #n j3;2;2;˝ t . Lemma 8. For any t  T , Y .un C !; t/ < C1 and jj n;t .0/jj21 C jj .0/jj2 < C1 holds  1  1  X . n /  jj n;t .0/jj21 exp t 2 .1 C jj .0/jj2 /Y .un C !; t/ exp C t 2 Y .un C !; t/ C   1 (107) exp C t 2 Y .un C !; t/ .1 C jj .0/jj2 /: Proof. Lemma 1 gives   1 k n .t /k2  exp C t 2 jjun C !jj3;2;˝ t .1 C jj .0/jj2 /;

(108)

so it suffices to obtain the estimate of k n;t .t /k1 (the bound given by (72) does not involve time t that will provide smallness in what follows; therefore it is not sufficient). Testing @xt .43/ with n;xt yields d dt

Z

2

Z

j n;xt j  C ˝

j n;xt j

2

Œjjun C !jj3 C jj.un C !/;t jj2 .C C jj n jj2 /

˝

Z C jjun C !jj3

j n;t j2 ;

(109)

˝

where integration by parts was used to remove the third-order derivative from n . Adding to (109) the lower-order estimate (resulting from testing @t .43/ with n;t ) implies d jj n;t jj21  C jj n;t jj21 Œjjun C !jj3 C jj.un C !/;t jj2 .C C jj n jj2 / : dt Hence the Gronwall inequality yields jj n;t .t /jj21



jj n;t .0/jj21

Z

t

Œjjun C !jj3 C jj.un C !/;t jj2 jj n jj2 dt

exp

0



0

  1  jj n;t .0/jj21 exp jj n jj2;1 t 2 jun C !j3;2;2;˝ t : Using estimate (108) to control jj n jj2;1 and to control the entire X . n / provides thesis. t u

4.1.6 Estimate of Y.#nC1 / Observe that in the following lemma n   must be ensured to provide estimate for nC1 . See also our remarks directly after formula (60).

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1803

Lemma 9. Let n   and t  tn of Lemma 1. Let the assumptions of Lemma 5 be satisfied. Assume that X .%n ; n ; un / < 1 and that un 2 L2 .0; t I 23 .˝//. Concerning data, let Z.!; / < 1, additionally 2 L1 .0; T I 02 .˝//, b 2 L1 .0; T I 01 .˝// and #;t .0/ 2 H 1 .˝/, #.0/ 2 H 2 .˝/, %.0/ 2 H 2 .˝/. Then for any "1 > 0 k#nC1;t .t /k21 C

 k#nC1;t k22;2;˝ t C jj#nC1 jj23;2;˝ t  "1 kun k23;2;˝ t C % ı.% ;  /

(110) C .jj jj23;2;˝ t C k t k22;2;˝ t C jj!jj23;2;˝ t / C .t C t 2 /'   X .%n ; n ; un C !/; j j2;0;1;˝ t ; jbj1;0;1;˝ t ; k%.0/k2 ; k#.0/k2 ; k#;t .0/k1 ; "1 : Proof. Assumptions n   and t  tn of Lemma 1 allow to use (60), i.e., %n cv .%n ; n /  % ı.% ;  /: Divide (44)1 by %n cv .%n ; n / to get #nC1;t 

 G.%n ; n ; un ; b/ #nC1 D : %n cv .%n ; n / %n cv .%n ; n /

Differentiating the above with respect to t, multiplying the result with #nC1;t gives Z Z  1 d jr#nC1 j2 C j#nC1;t j2 D 2 dt ˝ ˝ %n cv .%n ; n / Z  Z   G.%n ; n ; un ; b/ ;t #nC1 #nC1;t  ;t #nC1;t : %n cv .%n ; n / %n cv .%n ; n / ˝ ˝ By %n cv .%n ; n /  % ı.% ;  /, the Hölder and the Young inequality, it yields Z  j#nC1;t j2  % ı.% ;  /    ˝ ˝ ˇ2 ˇ2 ! Z ˇ Z ˇ ˇ ˇ ˇ G.%n ; n ; un ; b/ ˇ % ı.% ;  /  2 ˇ ˇ ˇ ;t ˇ j#nC1 j C ;t ˇˇ : ˇ ˇ 2 %n cv .%n ; n / %n cv .%n ; n / ˝ ˝ d dt

Z

jr#nC1 j2 C

The first term on r.h.s. is bounded by c "jj#nC1 jj23 C '.j%n ; n j2;1 /j#nC1 j22 " and the last by

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"1 kun C !k23 C C k t k22 C '.j%n ; n ; un C !j2;1 ; j j2;0 ; jbj1;0 ; "1 /; where "1 kun C !k23 stems from the following estimate of quadratic terms of G Z C ˝

j.un C !/;x .un C !/;xt j2  C jr.un C !/j21 jun C !j22;1

 C k.un C !/k3 k.un C !/k2 jun C !j22;1 and from the Young inequality. Hence, adding the lower-order estimate (74) and using (61) to control the term c" '.j%n ; n j2;1 /j#nC1 j22 , one arrives at k#nC1;t .t /k21 C

 k#nC1;t k22;2;˝ t  "jj#nC1 jj23;2;˝ t % ı.% ;  /

c C t '.j%n ; n j2;1;1;˝ t /j#nC1 j22;1;˝ t C "1 kun C !k23;2;˝ t C C k t k22;2;˝ t " Z t C '.j%n ; n ; un C !j2;1 ; j j2;0 ; jbj1;0 ; "1 /dt 0 C k#;t .0/k21 C r.h.s. of (74) 0

c   "jj#nC1 jj23;2;˝ t C "1 kun k23;2;˝ t C .t C t 2 / ' X .%n ; n ; un ; !/; j j2;0;1;˝ t ; " Z  1 jbj1;0;1;˝ t ; j%.0/j2 ; j.0/j2 C %.0/cv .%.0/; .0//# 2 .0/dx % ı.% ;  / ˝ C k#;t .0/k21 C C .k t k22;2;˝ t C k!k23;2;˝ t /:

(111)

In order to deal with the "jj#nC1 jj23;2;˝ t on the r.h.s. of (111), one differentiates (44) with respect to x and puts the evolutionary part on the r.h.s. After multiplication with #nC1;x and integration over space, it results in jj#nC1 jj23  '.jj%n ; n jj2 /jj#nC1;t jj21 C C jj%n .cv .%n ; n //x #nC1;t jj20 C C jjrG.%n ; n ; un ; b/jj20  "1 kun k23 C '.X1 .%n ; n ; un C !//.jj#nC1;t jj21 C jjbjj21 C jj t jj21 C jj jj22 C 1/ C C .jj jj23 C jj!jj23 /; where standard by now estimates were invoked, including, similarly as before, interpolation to deal with quadratic terms of G that produces "1 part. Time integration implies jj#nC1 jj23;2;˝ t  "1 kun k23;2;˝ t C C .jj jj23;2;˝ t C jj!jj23;2;˝ t / C '.X1 .%n ; n ; un C !//jj#nC1;t jj21;2;˝ t

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1805



 C t ' X1 .%n ; n ; un C !/ jjbjj21;1;˝ t C jj t jj21;1;˝ t C jj jj21;1;˝ t C 1 : The term jj#nC1;t jj21;2;˝ t above is estimated by means of Lemma 5 to get jj#nC1 jj23;2;˝ t  "1 kun k23;2;˝ t C C .jj jj23;2;˝ t C jj!jj23;2;˝ t /  C .t C t 2 /' X .%n ; n ; un C !/; j j2;0;1;˝ t ; jbj1;0;1;˝ t ;  k%.0/k2 ; k#.0/k2 ; j#;t .0/j2 ; "1 :

(112)

It allows to control the " part on the r.h.s. of (111). Altogether, (111) and (112) yield k#nC1;t .t /k21 C

 k#nC1;t k22;2;˝ t C jj#nC1 jj23;2;˝ t  2"1 kun k23;2;˝ t C % ı.% ;  / C .jj jj23;2;˝ t C k t k22;2;˝ t C jj!jj23;2;˝ t /C

  .tCt 2 /' X .%n ; n ; un C!/; j j2;0;1;˝ t ; jbj1;0;1;˝ t ; k%.0/k2 ; k#.0/k2 ; k#;t .0/k1 ; "1 ; t u

i.e., the thesis.

4.1.7 Boundedness of Z Finally, the preparations to show boundedness of Z. n ; un ; #n ; t / are complete. Recall that Z D X C Y , where X .un ; #n ; n ; t / D jun ; #n ; n j2;1;1;˝ t , Y .un ; #n ; t / D jun ; #n j3;2;2;˝ t . It is important to pay attention to times tn needed in Lemmas 2, 3, 4, 5, 6, 7, 8, and 9 to have n 2 Œ% ; % and n   ; in fact we will need below to show n 2 Œ ;   for finite numbers   >  > 0. For n 2 Œ% ; % , the appropriate tn ’s are given simply by Lemma 1. For n the situation is more complex due to the nature of the preceding estimates. In what follows, a justification that n 2 Œ ;   will be given at every iteration step. Define M by M D jf j1;0;1;˝ t C j!j3;2;2;˝ t C j!j2;1;1;˝ t C jbj1;0;1;˝ t C j j3;2;2;˝ t C j j2;0;1;˝ t C ju0 j2;1 C j 0 j2;1 C j#0 j2;1 :

(113)

For a fixed n, as long as n 2 Π;   , the following facts are valid. Lemma 7 gives Z.unC1 ; t /  t '.X .un ; #n ; !; t //.1CC .M //CC .M /  t '.X .un ; #n ; t /; M /CC .M / (114) for any t  tn D

c %N 2 . 1/ .1 C jj 0 jj2 /2 '.Z.u Q n ; t / C kwk3;2;˝ t /

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given by Lemma 1. Next, Lemmas 3 and 9 imply Z.#nC1;t /  "1 kun k23;2;˝ t C .t C t 2 /'.X .un ; #n ; %n ; !; t /; M; "1 / C C .M /  1 Z.un ; t / C t '.X .un ; #n ; %n ; !; t /; M / C C .M /; 2

(115)

where for the second inequality, one has chosen "1 and used t 2  C t , since here small times only are considered. Finally, Lemma 8 implies  1  1  X . i /  C .M / exp t 2 C .M /Y .ui C !/ exp C t 2 Y .ui C !/  1  C C .M / exp C t 2 Y .ui C !/  1   1  C .M / exp t 2 C .M /Y .ui C !/ exp C t 2 Y .ui C !/ :

(116)

For i D n C 1 and via (114), one obtains consequently  1 X . nC1 /  C .M / exp t 2 C .M /.t '.X .un ; #n /; M /   1 CC .M // exp C t 2 .t '.X .un ; #n /; M / C C .M // :

(117)

Finally, (114), (115), and (117) give together 1 Z.un / C t '.X .un ; #n ; %n ; !/; M / 2  1 C C .M / C C .M / exp t 2 C .M /.t '.X .un ; #n /; M /   1 CC .M // exp C t 2 .t '.X .un ; #n /; M / C C .M // ;

Z.unC1 ; #nC1 / C X . nC1 / 

(118)

as long as n 2 Π;   and t  tn . Next, one shows Lemma 10. Assume that 0 is bounded away from 0 and that it is flat enough: N 1  1 . N For any 0 <  <   such that 0 <  < 1 N , 2N <   , there exist j0  j 2 4 C3 .M / and CQ .M /, possibly large, such that if 1

C3 .M /.t  / 4  then for any t  t  , it holds uniformly in n

1N ; 4

(119)

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

Z.un ; #n ; t / C X . n ; t /  CQ .M /

1807

(120)

and n 2 Π;   : Proof. We proceed by induction for n. For the first step i D 0, (120) concerning u0 ; #0 with, say, 12 CQ .M / on its r.h.s., follows from definitions of u0 , #0 as extension of boundary data. For clarity, let us fix immediately CQ .M /, such that it satisfies the already stated (120) for u0 ; #0 and such that CQ .M /  100C .M /;

(121)

where C .M / is taken from (118). The form of nonlinearity ' is fixed from (118) as well. To get estimate (120) for X . 0 /, where 0 solves (43), one chooses t  and CQ .M / in relation with (116) so that    1 1 1 C .M / exp .t  / 2 C .M /Y .u0 C !/ exp C .t  / 2 Y .u0 C !/  CQ .M /: 2 (122) This and (116) imply estimate on X . 0 /. Hence Z. 0 ; u0 ; #0 /  CQ .M /. Observe that at the first step, one did not use bounds for n . They will be needed in next steps. At the step from n to n C 1, the inductive assumption is (120). Now (118) gives 1Q C .M / C t '.CQ .M /; M / 2  1 (123) C C .M / C C .M / exp t 2 C .M /.t '.CQ .M /; M /   1 CC .M // exp t 2 C .M /.t '.CQ .M /; M / C C .M // :

Z.unC1 ; #nC1 / C X . nC1 / 

Since one can freely choose small t  , (123) implies (120) for n C 1. It remains to show that nC1 2 Œ ;   . But now (123) gives an uniform bound Z.#nC1 /  CQ .M /. This, via embedding, controls constant of Hölder continuity of #nC1 via K CQ .M /, where K is a numerical constant related to embedding. Hence, smallness of time t  in relation to K CQ .M / implies nC1 2 Œ ;   via its Hölder continuity. More precisely, since nC1 2 L2 .0; t I 23 .˝//, one has by embedding nC1 2 1 C 0; 4 .˝ t /, with its Hölder norm controlled by K CQ .M /. Consequently N  jnC1 .x; t /  .x; 0/j C j.x; 0/  N j jnC1 .x; t /  j 1 1  K CQ .M /t 4 C N ; 2 where the last term follows from assumptions. Hence for

(124)

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J. Burczak et al. 1

K CQ .M /t 4 

1N  4

(125)

the inductive thesis holds, because  <   was chosen so that 0 <  < 14 N , 2N <   . Finally, (119) holds with a constant C3 .M / that combines smallness needed for (123) and for (125). t u The obtained (120) is the first needed element (boundedness) for the method of successive approximations.

4.2

Estimates for Differences of Approximative Sequences

Now estimates for differences of approximative sequences will be shown that imply that they are Cauchy sequences. Let us emphasize that, thanks to having already boundedness (120) in high-regularity norms, it suffices to close the convergence estimates on a lower regularity level. To this end, let us introduce equations for differences, denoting fNnC1 D fnC1  fn . They read (all with null initial-boundary values) %n uN nC1;t  ANunC1 D %N n un;t C %N n .un C !/r.un C !/ C %n1 uN n r.un C !/ C %n1 un1 r uN n C p% .%n ; n /r Nn C .p% .%n ; n /  p% .%n1 ; n1 //r n1

(126)

C p .%n ; n /r #N n C .p .%n ; n /  p .%n1 ; n1 //rn1 C %N n f;

NnC1;t C .unC1 C !/r NnC1 C uN nC1 r n C NnC1 div unC1 C . n C %/div N uN nC1 C NnC1 div ! D 0:

(127)

Finally, from (44) one has %n cv .%n ; n /#N nC1;t C Œ%n cv .%n ; n /  %n1 cv .%n1 ; n1 / #n;t  #N nC1 D N G.%n ; #n ; un ; b/  G.%n1 ; #n1 ; un1 ; b/ D G; consequently %n cv .%n ; n /#N nC1;t C Œ%N n cv .%n ; n / C %n1 .cv .%n ; n / N  cv .%n1 ; n1 // #n;t  #N nC1 D G; where N .%n cv .%n ; n /r#n  %n1 cv .%n1 ; n1 /r#n1 / G

(128)

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1809

 .%n cv .%n ; n /.un C !/r  %n1 cv .%n1 ; n1 /.un1 C !/r /  .n p .%n ; n /div .un C !/  n1 p .%n1 ; n1 /div .un1 C !//

(129)

 .%n cv .%n ; n / t  %n1 cv .%n1 ; n1 / t / C

 .jD.un C !/j2  jD.un1 C !/j2 / 2

C .  /.jdiv .un C !/j2  jdiv .un1 C !/j2 / C %N n b

6 X

Ii C %N n b:

i D1

The ingredients of

P6

iD1 Ii

read

I1 D %N n cv .%n ; n /r#n  %n1 .cv .%n ; n /  cv .%n1 ; n1 //r#n %n1 cv .%n1 ; n1 /r #N n ; I2 D %N n cv .%n ; n /.un C w/r  %n1 .cv .%n ; n /  cv .%n1 ; n1 //.un C w/r %n1 cv .%n1 ; n1 /r uN n r ; I3 D #N n p .%n ; n /div .un C w/  n1 .p .%n ; n /  p .%n1 ; n1 //div .un C w/ n1 p .%n1 ; n1 /div uN n ; I4 D %N n cv .%n ; n / t  %n1 .cv .%n ; n /  cv .%n1 ; n1 // t ; I5 D

 .D.un C !/ C D.un1 C !//  D.Nun /; 2

I6 D .  /.div .un C !/ C div .un1 C !//div .Nun /: First observe that, having already boundedness of sequences of approximate solutions un , %n , n by (120) (in high – regularity norms), one can easily obtain from .126/, .127/, and .128/ first- and second-order estimate for differences. The details follow. Recall that by definition M D jf j1;0;1;˝ t C j!j3;2;2;˝ t C j!j2;1;1;˝ t C jbj1;0;1;˝ t C j j3;2;2;˝ t C j j2;0;1;˝ t C ju0 j2;1 C j 0 j2;1 C j#0 j2;1 and that by Lemma 10, it holds Z. n ; un ; #n /  CQ .M /: In what follows C .M / denotes the common bound of M and Z. n ; un ; #n /.

4.2.1 Estimates for nC1 Lemma 11. For finite data quantity M , one has

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jj NnC1 jj1;1;˝ t  C .M /t 2 jjNunC1 jj2;2;˝ t and jj NnC1;t jj20;2;˝ t  C .M /tjjNunC1 jj21;1;˝ t C C .M /t 2 jjNunC1 jj22;2;˝ t : Proof. Test @x .127/ with NnC1;x and integrate in space. Integrating by parts in the term containing r NnC1;x NnC1;x , one arrives after straightforward computations at d jjr NnC1 jj20  jjr NnC1 jj20 jjunC1 C wjj3 C jjr NnC1 jj0 jjNunC1 jj2 jj n jj2 : dt Together with the lower-order estimate (or directly via the Poincaré inequality), it yields d jj NnC1 jj21  jr NnC1 j22 jjunC1 C wjj3 C jr NnC1 jj0 jjNunC1 jj2 jj n jj2 : dt This is differential inequality of type p XP  CX C D X that implies for X .0/ D 0 Z X .T /  4

2

T

D.t /dt

exp

RT 0

C .t/dt

;

0

i.e., jj NnC1 jj21

Z

2

T

4

jjNunC1 jj2 jj n jj2 0

exp

C .M /

 4jj n jj22;1

Z

T

2 jjNunC1 jj2 expC .M / ;

0

which gives the first estimate of the thesis. In order to get the second one, one tests .127/ with NnC1;t and integrates in space and time. This implies Z tZ 0

j NnC1;t j2 

˝

Z 0

t

j.unC1 C !/j21 jr NnC1 j22 C jNunC1 j23 jr n j26

C j NnC1 j26 jdiv .unC1 C !/j23 C j. n C %/j N 21 jdiv uN nC1 j22 (130) Z t  C .M / jj NnC1 jj21 C jjNunC1 jj21  C .M /t .jj NnC1 jj21;1 C jjNunC1 jj21;1 / 0

 C .M /tjjNunC1 jj21;1;˝ t C C .M /t 2 jjNunC1 jj2;2;˝ t : t u

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1811

4.2.2 Estimates for unC1 Lemma 12. For any t  t  of Lemma 10 and finite data quantity M , it holds jjNunC1;t jj20;2;˝ t C jjNunC1 jj21;1;˝ t C jjNunC1 jj22;2;˝ t    C .M / jjNun jj21;2;˝ t C jj%N n jj21;2;˝ t C jj#N n jj21;2;˝ t : Proof. Denote r.h.s. of .126/ by K.%N n ; uN n ; #N n /, hence %n uN nC1;t  ANunC1 D K.%N n ; uN n ; #N n /:

(131)

In order to derive second-order estimates, one may test (131) with NunC1 or with ANunC1 . The latter choice leads to a shorter proof in the case of secondorder estimates, since testing with NunC1 needs integration by parts in relation to r  div part of A. This, in turn, calls for localization in order to deal with boundary. This localization was thoroughly presented in Lemmas 6 and 7 and cannot be avoided when one derives third-order estimates. Here, since second-order estimate suffices for differences, computations related to testing with ANunC1 will be presented. Observe that A is Legendre-Hadamard elliptic for any  > 0;  > 0 (in [39] the authors use the strong ellipticity valid only for the case  > 13 , but differential symbol Aij .@/.uj / D .ıij @2kk C @2ij /.uj / of A implies Aij ./ i N j D j j2 jj2 C i i Nj N j  j j2 jj2 ). Testing (131) with ANunC1 and integrating in space implies Z

%n uN nC1;t ANunC1 C ˝

1 2

Z

jANunC1 j2 

Z

˝

jK.%N n ; uN n ; #N n /j2 :

(132)

˝

The evolutionary part above can be written as Z

1 2

1 2

A %n uN nC1;t A uN nC1 C

˝

Z

Z

1

˝

1 d D A %n uN nC1;t A uN nC1 C 2 dt ˝ 1 2

1

%n A 2 uN nC1;t A 2 uN nC1

1 2

Z

1 2

2

Z

%n jA uN nC1 j  ˝

(133) 1 2

2

%n;t jA uN nC1 j : ˝

Since A is a Legendre-Hadamard elliptic operator, from (132) to (133) the following inequality can be derived Z Z d %n jr uN nC1 j2 C jr 2 uN nC1 j22  dt ˝ ˝ Z Z Z jr%n jjNunC1;t jjr uN nC1 j C j%n;t jjr uN nC1 j2 C jK.%N n ; uN n ; #N n /j2 : C ˝

˝

Three terms on r.h.s. are estimated, respectively, as

˝

(134)

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"jr uN nC1;t j26 C C" jr%n j23 jNunC1;t j22 ; "jr uN nC1;t j26 C C" j%n;t j23 jr uN nC1 j22 and finally Z ˝

jK.%N n ; uN n ; #N n /j2  C .M /jj%N n ; uN n ; #N n jj21 :

Hence, choosing sufficiently small " implies d dt

Z

Z

2

%n jr uN nC1 j C ˝

˝

jr 2 uN nC1 j22  C .M /.jj%N n ; uN n ; #N n jj21 C jNunC1;t j22 C jjNunC1 jj21 /:

(135) In order to estimate jNunC1;t j22 and jjNunC1 jj21 on r.h.s. of (135), one tests .126/ with uN nC1;t and obtains Z

d dt

%n jNunC1;t j2 C

˝

Z

jr uN nC1 j2  "

˝

d jNunC1;t j C dt ˝ 2

jNunC1;t j2 C ˝

Use of %n  % and choice " D Z

Z

Z ˝

% 2

C jK.%N n ; uN n ; #N n /j22 : "

yield

jr uN nC1 j2  C .M /jj%N n ; uN n ; #N n jj21 :

(136)

Integration of the above inequality in time and the Poincaré inequality imply that jNunC1;t j22 C jjNunC1 jj21;1;˝ t  C .M /jj%N n ; uN n ; #N n jj21;2;˝ t :

(137)

Hence, (135) and (137) give d dt

Z ˝

%n jr uN nC1 j2 C jjNunC1 jj22 C jNunC1;t j22 

C .M /jj%N n ; uN n ; #N n jj21 C C .M /jj%N n ; uN n ; #N n jj21;2 ; which, after integration in time, provides thesis.

t u

4.2.3 Estimate for # nC1 First estimate of Lemma 11 (for n) used at the r.h.s. of Lemma 12 gives jjNunC1;t jj20;2;˝ t C jjNunC1 jj21;1;˝ t C jjNunC1 jj22;2;˝ t    C .M / jjNun jj21;2;˝ t C t 2 jjNun jj22;2;˝ t C jj#N n jj21;2;˝ t    C .M /t jjNun jj21;1;˝ t C jjNun jj22;2;˝ t C jj#N n jj21;1;˝ t ;

(138)

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1813

where one used that t n  C .n/t , since we are interested in finite times. Hence the second estimate of Lemma 11 takes the form   jj NnC1;t jj20;2;˝ t  C .M /t jjNun jj21;1;˝ t C jjNun jj22;2;˝ t C jj#N n jj21;1;˝ t :

(139)

Using (138) in the first estimate of Lemma 11 implies together with (139) jjNunC1;t jj20;2;˝ t C jjNunC1 jj21;1;˝ t C jjNunC1 jj22;2;˝ t C jj NnC1 jj21;1;˝ t C jj NnC1;t jj20;2;˝ t (140)  C .M /t ŒjjNun jj21;1;˝ t C jj#N n jj21;1;˝ t C jjNun jj22;2;˝ t : In (140), control over jj#N n jj21;1 is lacking. Hence one needs Lemma 13. For any t  t  of Lemma 10 and finite data quantity M holds jj#N nC1;t jj20;2;˝ t C jj#N nC1 jj21;1;˝ t C jj#N nC1 jj22;2;˝ t  C .M /t ŒjjNun jj21;1;˝ t C jjNun jj22;2;˝ t C jj#N n jj21;1;˝ t : Proof. Test .128/ with #N nC1 . After space-time integration and straightforward computations, one obtains d dt

Z

%n cv .%n ; n /jr #N nC1 j2 C

˝

Z

N 2CC jGj

C ˝

Z

CC

Z

j#N nC1 j2

Z h

˝

i %N n cv .%n ; n / C %n1 .cv .%n ; n /  cv .%n1 ; n1 // #n;t j2

˝

j.%n cv .%n ; n //;x jjr #N nC1 jj#N nC1;t j

(141)

˝

Z CC

j.%n cv .%n ; n //;t jjr #N nC1 j2 DW

˝

4 X

Ii :

iD1

N a standard by Concerning estimate of the terms Ii , recalling splitting (129) of G, now computation yields I1  X .%n ; #n ; un ; %n1 ; #n1 ; un1 ; ; !; b/x.i:e; / .k%N n k21 C k#N n k21 C kNun k1 kNun k2 C kNun k21 /  C .M /.k%N n k21 C k#N n k21 C kNun k1 kNun k2 C kNun k21 /: Similarly

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J. Burczak et al.

I2  X .%n ; #n /.k%N n k21 C k#N n k21 /  C .M /.k%N n k21 C k#N n k21 /: Next one has I3  C jr #N nC1 j6 j.%n C #n /x j6 j%n C #n j1 j#N nC1;t j2  "k#N nC1 k22 C C .M /j#N nC1;t j22 and I4  C jr #N nC1 j6 j.%n C #n /t j6 j%n C #n j1 j#N nC1;t j2  "k#N nC1 k22 C C .M /j#N nC1;t j22 : All these estimates in (141) imply, after time integration and use of the lower bound on %n cv .%n ; n / kr #N nC1 k20;1;˝ t C kr 2 #N nC1 k20;2;˝ t  "k#N nC1 k22;2;˝ t C C .M /k#N nC1;t k20;2;˝ t C C .M /t .k%N n k21;1;˝ t C k#N n k21;1;˝ t C kNun k21;1;˝ t / p C C .M / tkNun k1;1;˝ t kNun k2;2;˝ t 

(142)

"k#N nC1 k22;2;˝ t C C .M /k#N nC1;t k20;2;˝ t C C .M /t .k%N n k21;1;˝ t C k#N n k21;1;˝ t C kNun k21;1;˝ t C kNun k22;2;˝ t /: In order to control term k#N nC1;t k20;2;˝ t on r.h.s. of (142), we test .128/ with #N nC1;t and obtain Z

d j#N nC1;t j2 C dt ˝

Z

jr #N nC1 j2  C

˝

Z

N 2; jGj

(143)

˝

where already the lower bound on %n cv .%n ; n / was used. Time integration of (143) and use of estimate for I1 gives, together with the Poincaré inequality jj#N nC1;t jj20;2;˝ t C jj#N nC1 jj21;1;˝ t  C .M /t Œk%N n k21;1;˝ t C jjNun jj21;1;˝ t C jjNun jj22;2;˝ t C jj#N n jj21;1;˝ t : This and (142) implies thesis, where also Lemma 11 was invoked in order to remove %N n from r.h.s. t u

4.2.4 Cauchy Sequence Altogether, Lemma 13 with (140) give for t small enough the Cauchy sequence. More precisely, it holds Proposition 1. For any t  t  of Lemma 10 and finite data quantity M there holds

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1815

jjNunC1;t jj20;2;˝ t C jjNunC1 jj21;1;˝ t C jjNunC1 jj22;2;˝ t C jj NnC1 jj21;1;˝ t C jj NnC1;t jj20;2;˝ t C jj#N nC1;t jj20;2;˝ t C jj#N nC1 jj21;1;˝ t C jj#N nC1 jj22;2;˝ t    C .M /t jjNun jj21;1;˝ t C jjNun jj22;2;˝ t C jj#N n jj21;1;˝ t :

4.3

Proof of Theorem 4 (Local-in-time Existence)

Proposition 1 implies that for t sufficiently small fun ; n ; #n g are Cauchy sequences in respective spaces. Let us denote their limits by fu; ; #g. We intend to perform the limit passage in the weak formulations of the approximative problems, where boundedness of strong norms of these sequences, provided by (120), will be utilized. As already mentioned, neither use of weak convergences (in strong norms) nor strong convergences (via compact embeddings) implied by (120) is sufficient, due to the character of the method of successive approximations. Namely, the convergences implied by (120) involve merely a subsequence, whereas approximative problems need the entire sequence (every problem is related to the step n ! n C 1). Let us illustrate how to perform the limit passage in relation to the most complex case, namely, the equation for temperature (44). Writing it in its space-time weak formulation with a smooth test function,  reads Z tZ Z tZ %n cv .%n ; n /#nC1;t  C r#nC1 r D G.%n ; n ; un ; b/; 0

˝

0

˝

(144)

#nC1 jS D 0; #nC1 jtD0 D #.0/; In order to justify that the limit equation holds, i.e., Z tZ Z tZ %cv .%;  /#;t  C r#r D G.%;  ; u; b/; 0

0

˝

˝

(145)

#jS D 0; #jtD0 D #.0/; consider the differences Z tZ .%n cv .%n ; n /#nC1;t  %cv .%;  /#;t / DW D1 ; ˝

0

Z tZ .r#nC1  r#/r DW D2 ;

Z tZ

0

˝

.G.%n ; n ; un ; b/  G.%;  ; u; b// DW D3 ; 0

˝

They can be written, compare (128) and (129) as

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Z tZ D1 D

%N n cv .%n ; n /#nC1;t  C %Œcv .%n ; n /  cv .%;  / #nC1;t  ˝

0

C %cv .%;  /#N nC1;t ; Z tZ D2 D r #N nC1 r; ˝ 0 Z tZ N D3 D G;

(146)

˝

0

where N denotes now the difference from the limit, i.e., fNi D fi  f . First, let us focus on D1 . Using (120), one has ˇ ˇZ t Z ˇ ˇ ˇ %N n cv .%n ; n /#nC1;t  ˇˇ  CQ .M /jj%N n jj0;2;˝ t jj1;˝ t ; ˇ ˝ 0 ˇ ˇZ t Z ˇ ˇ ˇ %Œcv .%n ; n /  cv .%;  / #nC1;t  ˇˇ  ˇ ˝

0

CQ .M; .rcv /.M //jj%N n ; #N n jj0;2;˝ t jj#nC1;t jj0;2;˝ t jj1;˝ t  C .M /jj%N n ; #N n jj0;2;˝ t jj1;˝ t ; and ˇZ t Z ˇ ˇ ˇ N N ˇ ˇ %c .%;  / # v nC1;t ˇ  C .M /jj#nC1;t jj0;2;˝ t jj1;˝ t : ˇ 0

˝

For D2 one gets ˇ ˇZ t Z ˇ ˇ N ˇ r #nC1 r ˇˇ  C .M /jjr #N nC1 jj0;2;˝ t jrj1;˝ t : ˇ 0

˝

Proposition 1 implies that the r.h.s. of the four estimates above vanish as n ! 1, hence D1 and D2 do. Similarly, D3 of (146) is dealt with. Recalling Ii of (129), one obtains ˇ ˇZ t Z ˇ ˇˇZ t Z 6 ˇ X ˇ ˇ ˇ ˇ ˇ ˇ N ˇ I C % N b G jD3 j D ˇ ˇ jj1;˝ t i n ˇ ˇ ˇ 0 0 ˝ ˝ iD1 (147) Z tZ 6 X DW jJi jjj1;˝ t C j%N n bjjj1;˝ t iD1

0

˝

with Z tZ J1 D 0

˝

%N n cv .%n ; n /r#n  %.cv .%n ; n /  cv .%;  //r#n  %cv .%;  /r #N n ;

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1817

Z tZ J2 D

%N n cv .%n ; n /.un C w/r  %.cv .%n ; n /  cv .%;  //.un C w/r 0

˝

 %cv .%;  /r uN n r ; Z tZ J3 D #N n p .%n ; n /div .un C w/  .p .%n ; n /  p .%;  //div .un C w/ 0

˝

 p .%;  /div uN n ; Z tZ J4 D %N n cv .%n ; n / t  %.cv .%n ; n /  cv .%;  // t ; 0

˝

Z tZ J5 D 0

˝

0

˝

 .D.un C !/ C D.u C !//  D.Nun /; 2

Z tZ J6 D

.  /.div .un C !/ C div .u C !//div .Nun /:

Thanks to (120), analogously to the estimate for I1 of proof of Lemma 13, it holds 6 X

jJi j  C .M /.jj%N n jj1;2;˝ t C jj#N n jj1;2;˝ t C jjNun jj2;2;˝ t /:

iD1

This in (147) implies jD3 j  C .M /.jj%N n jj1;2;˝ t C jj#N n jj1;2;˝ t C jjNun jj2;2;˝ t /jj1;˝ t :

(148)

Therefore, Proposition 1 implies that D3 of (146) vanishes as n ! 1. Summing up, a justification of the limit passage of equation for temperature (44) was provided. One deals similarly with the rest of equations to obtain the weak formulations for fu; ; #g, i.e., weak formulations of nonlinear problems (39), (40), and (41), consequently of the original equations (25), (26), (27), (28), (29), and (30). Now, Lemma 10 with sequential weak- pre-compactness of respective bounded sets and uniqueness of limits fu; ; #g implies regularity of these limits, postulated in Theorem 4. In fact, two more matters need to be mentioned. Firstly, the presented method results in L1 spaces, where spaces of continuous functions should appear – recall that, by definition, the norm j  jl;p;1 concerns space C .0; T I kl .˝//. In order to obtain continuity in time (which will be also needed for the proof of the next section), one needs in fact to refer again to the equation and use time-increments (compare Ladyzhenskaya, Solonnikov, and Uraltseva [22], III.4) or duality (see, for instance, Zeidler [56], Chapter 23). Secondly, having this continuity in time, one recovers also the initial and boundary values in respective spaces. Due to spacelimitation as well as standard methods involved, these matters are merely signalled here. Observe, however, that our ‘L1 procedure’ implies limits in spaces that are strong enough to justify keeping the initial and boundary values, via traces, in certain lower-regularity norms.

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J. Burczak et al.

Let us recheck and gather the assumptions needed for validity of Theorem 4: these are the assumptions of Proposition 1 and Lemma 10, i.e., (i) the data quantity M of (113) must be finite, (ii) the initial temperature and density are to be bounded away from 0, compare (29), (iii) the initial temperature and density must be flat N 1  1 N (see Lemma 10) and j%0  %j enough, i.e., j0  j N 1  12 %N (see Lemma 1) 2 (iv) the involved known functions p; cv and domain must be smooth enough with cv bounded away from 0 as in (26). Finally recall our local time of existence t depends on. For a fixed data quantity M , Lemma 10 gives condition (119). Lemma 1 needs a similar t C .M /  %N 2 ; see (47). Finally, Proposition 1 needs t C .M / < 1. All in all, it is sufficient that t C .M /  1 ^ %N 2 ^ N 4 . This is the condition of Theorem 4.

5

Global Existence for Small Data

In this section, Theorem 5 on small data global-in-time existence of solutions to problem (25), (26), (27), (28), (29), and (30) is proved. The estimates of this section are very similar to these from the previous one. The new ingredient is differential inequality that implies large-time existence for small data. Consequently, less details on proofs of the estimates in what follows will be provided. First, let us reformulate the problem (25), (26), (27), (28), (29), and (30) via the following procedure, which groups the new unknowns u; ; # in a slightly different order than for proof of Theorem 4. Recall that according to (33), ! is an extension of the boundary data vb and of N so b  , !jS D vb ;

jS D b  N :

(149)

Then one considers the differences

D %  %; N

u D v  w;

# D   N  ;

(150)

which solve the following problems N C p1 r C p2 r# D f0 ut  Au

in ˝ T ;

uD0

on S T ;

ujtD0 D u0 v0  !jtD0

(151)

in ˝;

N D u where Au D u C rdiv u, Au N C rdiv N u and N D

 , %N

N D %N ;

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

in ˝ T ;

t C .u C !/  r C %div N uDh

jtD0 D 0 %0  %N

p2 p3

 N , %c N v .%; N /

#t  # N C N p3 div u D g

in ˝ T ;

# D0

on S T ;

p1 D

N p% .%; N / %N

(152)

in ˝I

#jtD0 D #0 0  jtD0 where N D

1819

> 0, p2 D

N p .%; N / , %N

(153)

in ˝; p3 D

N > 0. Let D cv .%; N /

p2 N cv .%; N /

and consequently

f0 f1 C f2 C f3 ;

(154)

where 

p% .%N C ; N C C #/ r ; f 1 D p1  %N C

 p .%N C ; N C C #/ r#; f2 D p2  %N C

f3 D .u C !/  r.u C !/ C

N 1 Au C f  !t  A!; %N C

%N C

and h D  div u  .%N C /div !:

(155)

g g1 C g2 C g3 ;

(156)

Finally,

where

N N C C # p .%N C ; N C C #/ N / p .%; N  div u g1 D  N %N C cv .%N C ; N C C #/ %c N v .%; N /

    #; N %c N v .%; N / .%N C /cv .%N C ; N C C #/

g2 D

 .%N C /cv .%N C ; N C C #/

g3 D  t  .u  !/  r.# C /

 

N C C # p .%N C ; N C C #/ div !; %N C cv .%N C ; N C C #/

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J. Burczak et al.



1

 C jD.u C !/j2 C .  /jdiv .u C !/j2 N .%N C /cv .%N C ;  C C #/ 2 C



b : cv .%N C ; N C # C /

In order to prove the wanted differential inequality, the existence of sufficiently regular local solutions is needed. Hence one uses the local existence from the previous section. Moreover, one needs to ensure that there exist positive constants % , % ,  ,   such that %  %  % ;

     

(157)

for all times. The relations (157) will be justified by a differential inequality. In this section any equivalent norm in H s .˝/ is denoted with the same symbol k  ks . Constants may depend on constants of (157), but we do not mark it explicitly, since they are fixed for all times eventually.

5.1

Lower-Order Estimates

Lemma 14. For a sufficiently regular solution to problems (151), (152), (153), (154), (155), and (156), it holds d ku; ut ; #; #t ; ; t k20 C ku; #; ut ; #t k21 C k ; t k20  cY12 ; dt

(158)

where Y12 D kf0 ; g; ft ; gt ; h; ht k20 C ku; !k23 k t k20 C kut ; !t k21 k k22 : Proof. Multiplying .151/1 by u, .152/1 by over ˝ and by parts yields

p1

, %N

.153/1 by

p2 N 3 #, p

adding, integrating

d .kuk20 C k k20 C k#k20 / C kuk21 C k#k21  "1 k k20 C cX12 ; dt

(159)

where 1 X12 D jf0 j26=5 C jgj26=5 C jhj22 C .kuk21 C k!k21 /k k21 : " Differentiating .151/1 , .152/1 , and .153/1 with respect to time, multiplying by ut , p1

, Np2 #t , respectively, adding and integrating over ˝, one obtains %N t p 3

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

d .kut k20 C k t k20 C k#t k20 / C kut k21 C k#t k21  "2 k t k20 C cX22 ; dt

1821

(160)

where X22 D jf0t j26=5 C jgt j26=5 C C

1 1 kht k20 C .kut k21 C k!t k21 /k k22 "2 "2

1 .kuk23 C k!k23 /k t k20 : "2

Rearrange now (151) as an equation for r

N u C u N C f0 p1 r D ut  p2 r# C rdiv

in ˝:

(161)

Multiplying the above identity by  that solves the problem div RD with j@˝ D 0 and using the fact that % has a constant mean over time (hence –˝ D 0) in view of (25) yields   k k20  c kuk21 C k#k21 C kut k20 C kf0 k20 ;

(162)

where we used the estimate kk1  C k k0 (see Bogovskii [4] or Kapitanskii and Pileckas [15]). Moreover, .152/1 gives k t k20  c.kuk21 C khk20 C .kuk21 C k!k21 /k k22 /:

(163)

From (159), (160), (162), and (163), one derives (158). This concludes the proof. t u

5.2

Higher-Order Estimates

As before, in order to derive estimates for the second and the third space derivatives, one needs the partition of unity and the local coordinates. As before, the considerations will be restricted to boundary neighbourhoods only, because estimates in interior neighbourhoods can be derived much easier. In the local coordinates introduced in Sect. 3.2, Eqs. (151) and (152) take the following form uQ t  AN z uQ C p1 rz Q C p2 rz #Q D fQ C k1 ;

(164)

O  rO Q C %div N z uQ D hQ C k2 ;

Qt C .Ou C !/

(165)

where ki D ki1 C ki2 , i D 1; 2 and O Q C p2 .rz  r/ O #Q  . O u  .r O /Qu; N z  /Q N z div z  rO div k11 p1 .rz  r/ (166)

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J. Burczak et al.

O k12 p1 OrO O C p2 #O rO O  .2 N rO k uO rO k O C uO O / O O uO rO O C rO uO k rO k O C uO k rO rO k /;  .div

Applying the operator

O u; k21 %.div N z  div/Q

(168)

O O u C !/ O rO O C %O O urO : k22 .O

(169)

CN N  rz %N

to (165) and adding the result to (164), one has

N C N rz Qt C p1 rz Q D . N z uQ  rz div z uQ /  uQ t  p2 rz #Q %N 

(167)

N C N N C N Q rz ..Ou C !/ rz h C k3 ; O  rO / Q C fQ C %N %N

(170)

N  rz k2 C k1 and k1 , k2 are defined above. Denote  D zi , i D 1; 2, where k3 CN %N and n D z3 . One expresses (164) in the form

. C /rz div z uQ D .z uQ  rz div z uQ / C uQ t C p1 rz Q C p2 rz #Q  fQ  k1 :

(171)

Next, a result on the second-order space derivatives for u and the first for will be provided. Lemma 15. For a sufficiently regular solution to (151), (152) and (153), it holds d .kux k20 C k x k20 / C kuk22 C k k21  c.k k20 C k#k21 C kuk21 / dt

(172)

C c.kuk22 C k!k22 /k k22 C c.kf k20 C khk21 /: Proof. Differentiate (164) and (165) with respect to , multiply the results by uQ  O Then and p%N1 Q , respectively, add and integrate over ˝. d dt

Z 

p1 2

Q d z C kQuz k20 C kdiv uQ  k20 %N ˝O Z Z Z p1 p1 Q O D ..Ou C !/ O  r / Q ; Q d z  f  uQ   d z C hQ  Q d z %N ˝O %N ˝O ˝O Z Z p1  k1  uQ   d z C k2; Q d z: %N ˝O ˝O uQ 2 C

(173)

Let us now examine the particular terms from the r.h.s. of (173). The first term equals

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

I1 

p1 %N

Z ˝O

.Ou C !/ O ;  rO Q Q d z 

p1 %N

Z ˝O

1823

.Ou C !/ O  rO Q Q dx;

where integration by parts in the second term yields the integral p1 2%N

Z ˝O

O u C !/ r.O O Q2 dx:

Hence O 22;˝O k k Q 22 : jI1 j  "k Q k20 C .c="/kOu C !k Employing (166) and (167), the last but one term on the r.h.s. of (173) is bounded by ˇ ˇZ ˇ ˇ Q 2/ ˇ k11  uQ   d zˇˇ  "kQu  k20 C .c="/.k k Q 21 C kQuk22 C k#k 1 ˇ ˝O

and ˇZ ˇ ˇ ˇ 2 O 2 C kOuk2 /: ˇ ˇ k  u Q d z O 20;˝O C k#k ˇ O 12   ˇ  "kQu  k0 C .c="/.k k 0;˝O 1;˝O ˝

Similarly, employing (168) and (169), one estimates the last term on the r.h.s. of (173) by ˇZ ˇ ˇ ˇ 2 2 2 ˇ ˇ k

Q d z ˇ O 21;  ˇ  "k Q k0 C .c="/kQuk2 C .c="/kQuk1 ˝

and

ˇZ ˇ ˇ ˇ 2 2 ˇ ˇ O 22;˝O /k k O 22;˝O C .c="/kOuk21;˝O : ˇ O k22; Q d zˇ  "k Q k0 C .c="/.kOuk2;˝O C k!k ˝

Using the Hölder and the Young inequalities to the second and the third term on the r.h.s. of (173) and employing the above estimates, one obtains for a sufficiently small " the inequality Z  p1 2 d 2 uQ  C Q d z C kQuz k20 C kdiv uQ  k20  "k Q k20 dt ˝O %N c Q 2 / C c .k k O 2 C kOuk2 / (174) Q 21 C kQuk22 C k#k O 20;˝O C k#k C .k k 1 0;˝O 1;˝O " " c Q 2 c O 22;˝O /k k O 22;˝O C ckfQ k20 C khk C .kOuk22;˝O C k!k 1: " " Multiplying the third component of (170) by Qn and integrating the result over ˝O yield

1824

J. Burczak et al.

N C N d k Qn k20 C p1 k Qn k20 2%N dt Q 2 C kfQ k2 C khk Q 2/  c.kQuz k20 C kQut k20 C k#k 1 0 1 Z N C N  @n Œ.Ou C !/ O  rO Q Qn d z C ckk3 k20 : %N ˝O

(175)

The last but one term equals

1 @n .Ou C !/ O  rO Qn2 d z O  rO Q Qn C .Ou C !/ 2 ˝O

Z N C N 1O 2 O D @n .Ou C !/ O  r Q Qn  r  .Ou C !/ O Qn d z: %N 2 ˝O 

N C N %N

Z

Hence it is estimated by "k Qn k20 C .c="/.kOuk22;˝O C k!k O 22;˝O /k k Q 22 : Finally Q 2 C kQuk2 / C c.k k O 2 C kOuk2 / Q 21 C k#k O 20;˝O C k#k kk3 k20  c.k k 1 2 0;˝O 1;˝O O 22;˝O /k k O 22;˝O : C c.kOuk22;˝O C k!k Employing the above estimates in (175) and using that " is sufficiently small, one obtains d Q 2 C kfQ0 k2 C khk Q 2/ k Qn k20 C k Qn k20  c.kQuz k20 C kQut k20 C k#k 1 0 1 dt Q 2 C kQuk2 / C c.k k O 2 C kOuk2 / O 2 C k#k C c.k k Q 2 C k#k 1

1

2

0;˝O

0;˝O

1;˝O

(176)

O 22;˝O /k k O 22;˝O : C c.kOuk22;˝O C k!k Taking the L2 -norm of the third component of (171) yields Q 2 C kfQ0 k2 / kdiv uQ n k20  c.kQuz k20 C kQut k20 / C c.k Qn k20 C k#k 1 0 Q 2 C kQuk2 / C c.k k O 2 C kOuk2 /: C c.k k Q 21 C k#k O 20;˝O C k#k 1 2 0;˝O 1;˝O Adding appropriately (174), (176), and (177), one has

(177)

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1825

d .kQu k20 C k Qz k20 / C kQuz k20 C kdiv uQ k21 C k Qn k20 dt Q 2 C kQuk2 / C c.k k O 2 C kQu k2  "k Q k20 C c.k k Q 21 C k#k O 20;˝O C k#k 1 2 0 1;˝O

(178)

Q 2 /: C kOuk21;˝O / C c.kOuk22;˝O C k!k O 22;˝O /k k O 22;˝O C c.kfQ0 k20 C khk 1 Deriving an inequality similar to (178) in an interior subdomain, summing it and (178) over all subdomains of the partition of unity and using the notation that uN  does not contain the normal derivative to the boundary in any neighborhood near the boundary yield d .kNu k20 C k x k20 / C kux k20 C kdiv uk21 C k Nn k20 dt  "k  k20 C c.k k21 C k#k21 C kuk22 / C c.k k20 C k#k21 C kuk21

(179)

C kut k20 / C c.kuk22 C k!k22 /k k22 C c.kf0 k20 C khk21 /; where Nn denotes that near the boundary, there is only normal derivative. Now let us rearrange (164) so that it resembles the stationary Stokes problem, as follows N uQ C fQ0 C k1  Q N u C p1 r Q D Qut  p2 r #Q C rdiv

O in ˝;

div uQ D div uQ

O in ˝;

(180)

uQ jSO D 0: For the problem (180), one has the estimate kuk22 C k k21  c.kut k20 C kk21 C kdiv uk21 C kf0 k20 C kuk21 C k k21 /:

(181)

This is estimate of type 77 [23] for div uQ D F . Adding (179) and (181) and using that " is sufficiently small, one derives d .kNu k20 C k x k20 / C kuk22 C k k21  ck#k21 C c.k k20 C k#k20 C kuk21 / dt C c.kuk22 C k!k22 /k k22 C c.kf k20 C khk21 /: (182) N integrating over ˝, and employing boundary condition Multiplying (151) by Au, give 1 d 2 dt 

Z ˝

N 20 .jruj2 C jdiv uj2 /dx C kAuk

c.k x k20

C

k#x k20

C kf

k20 /:

(183)

1826

J. Burczak et al.

t u

Adding (182) and (183) implies (172). This concludes the proof. Lemma 16. For a sufficiently regular solution to (151), (152) and (153), it holds

d .kut ; #t ; u; #; k21 C k t k20 /Cku; #; ut ; #t k22 C k ; t k21  cY22 C cku; !k22 j j22;1 ; dt (184) where Y22 D Y12 C khk21 .

Proof. Inequalities (158) and (172) imply d ku; ux ; #; #t ; ; x ; t k20 C kuk22 C k ; #; ut ; #t k21 C k t k20 dt

(185)

 c.Y12 C khk21 / C cku; !k22 j j22;1 : From .152/1 , it follows that k t k21  c.kdiv uk21 C khk21 C ku; !k22 k k22 /:

(186)

Multiplying .153/1 by #, integrating over ˝ and using boundary condition .153/2 give d k#x k20 C k#k22  c.kuk21 C kgk20 /: dt

(187)

N t , integrating over Differentiating .151/1 with respect to time, multiplying by Au ˝ and using boundary condition .151/2 give d dt

Z ˝

.jrut j2 C jdiv ut j2 /dx C kut k22  c.k t k21 C k#t k21 C kf0t k20 /:

(188)

Differentiating .153/1 with respect to time, multiplying the result by #t , integrating over ˝ and using the boundary condition .153/2 yield d k#xt k20 C k#t k22  c.kut k21 C kgt k20 /: dt

(189)

Inequalities (188) and (189) imply d kuxt ; #xt k20 C kut ; #t k22  c.k t k21 C kut k21 C kf0t ; gt k20 /: dt

(190)

From (185), (186), (187) and (190), one obtains (184). This concludes the proof. t u

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1827

Next, an estimate for the third spatial derivatives of u and # as well as second spatial derivatives of will be derived. Lemma 17. For a sufficiently regular solution to (151), (152) and (153), one has d ju; #; j22;1 C ju; #j23;2 C j j22;1  c.jf0 ; gj21;0 C jhj22;1 / C cju; !j23;2 j j22;1 : dt

(191)

Proof. Differentiating (164) and (165) twice with respect to  , multiplying the result by uQ  , p%N1 Q , respectively, integrating over ˝O and adding, one gets d .kQu k20 C k Q  k20 / C kQuz  k20 C kdiv uQ   k20 dt Z Z Z p1 fQ0 uQ   d z C hQ   Q  d z D  Œ.Ou C !/ O  rO Q ;  Q  d z C %N ˝O ˝O ˝O Z Z p1 Q 2: C k1; uQ   d z C k2; Q  d z C ck#k 2 %N ˝O ˝O

(192)

The first term on the r.h.s. is bounded by c O 23;˝O k k Q 22 : "k Q  k20 C kOu C !k " The second via the integration by parts by c "kQu   k20 C kfQ k20 " and the third by c "k Q  k20 C khQ   k20 : " To estimate the fourth term, let us utilise expressions (166) and (167). Hence, examine ˇ ˇZ ˇ ˇ c 2 2 2 Q 2 ˇ ˇ ˇ O k11;  uQ   d zˇ  "kQu   k0 C " .k Qz k0 C k#z k0 C kQuzz k0 / ˝

c Q 2 C kQuk2 /; C .k k Q 21 C k#k 1 2 " ˇ ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ 2 ˇ ˇ ˇ ˇ ˇ O k12;  uQ   d zˇ D ˇ O k12;  uQ    d zˇ  "kQu   k0 ˝

˝

c O 2 C kOuk2 /: C .k k O 21;˝O C k#k 1;˝O 2;˝O "

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J. Burczak et al.

To estimate the last term on the r.h.s. of (192), one obtains, in view of (168) and (169), the estimates ˇ ˇZ 2 ˇ ˇ ˇ  "k Q  k2 C c  kQuz  k2 C c kQuk2 ˇ k

Q d z 21;   0 0 2 ˇ ˇ O " " ˝ and ˇ ˇZ ˇ ˇ c c 2 2 ˇ ˇ O 22;˝O /k k O 22;˝O C kOuk22;˝O : ˇ O k22; Q d zˇ  "k Q  k0 C " .kOuk2;˝O C k!k " ˝ Choosing " and  sufficiently small and using the above estimates in (192), one derives d .kQu k20 C k Q  k20 / C kQuz  k20 C kdiv uQ   k20 dt  "k Q  k2 C c.k Qz k2 C k#Q z k2 C kQuzz k2 /C 0

0

0

0

(193)

c O 2 C kOuk2 / C c .kfQ0 k2 C khQ   k2 / .k k O 21;˝O C k#k 0 0 2;˝O 2;˝O " " c C .kOuk23;˝O C k!k O 23;˝O /k k O 22;˝O : "

Differentiation of the third component of (170) with respect to  , multiplication by

Qn , and integration over ˝O yield d k Qn k20 C k Qn k20  ckQuz  k20 C c.k Qz k20 C kQuzz k20 C k#Q z k20 / dt O 2 C kOuk2 / C kOu C !k O 2 C k#k O 2 k k Q 2 C ckQut k2 C c.k k 1

1;˝O

2;˝O

2;˝O

3;˝O

2

(194)

C kfQ0 k20 C khQ z k20 : Differentiating the third component of (171) with respect to  , one obtains kdiv z uQ n k20  c.kQuz  k20 C k Qn k20 / O 2 C kOuk2 CkfQ0 k2 /: C c.k Qz k20 C k#Q z k20 C kQuzz k20 / C c.kQut k21 C k ; O #k 0 1;˝O 2;˝O (195) Let us recall the reformulation (180). Differentiating (180) with respect to  and using the estimate for the stationary Stokes problem provide kQuzz k20 C k Qz k20  c.kdiv uQ  k21 C k#Q z k20 C kQu t k20 C kfQ k20 /C O 2 C kOuk2 /: c.k Qz k20 C k#Q z k20 C kQuzz k20 / C c.k ; O #k 1;˝O 2;˝O

(196)

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1829

Inequalities (193), (194), (195), (196), and assumption that " and  are sufficiently small imply d O .kQu k20 C k Q  k20 C k Qn k20 / C kQuzz k20 C k Qz k20  ckQut k21 C cX12 .˝/; dt (197) where O uO k2 C kOu C !k Q 2: O D k k X12 .˝/ O 21;˝O C k#; O 23;˝O k k O 22;˝O C kfQ0 k21 C khk 2 2;˝O

(198)

Multiplying the normal derivative of the third component of (170) by Qnn and integrating over ˝O yield d Q 2 k Qnn k20 C k Qnn k20  ckQuzz k20 C ckQut k21 C ck#k dt C c.k Qzn k20 C k#Q zn k20 C kQuzzn k20 / O 2 C kOuk2 C kOu C !k Q 2 /: O 23;˝O k k O 22;˝O / C c.kfQ0 k21 C khk C c.k k O 21;˝O C k#k 2 1;˝O 2;˝O (199) Differentiating (171) with respect to n and taking the L2 -norm yield kdiv uQ zn k20  c.kQuzz k20 C k Qzn k20 C k#Q zn k20 C kfQ0 k21 / Q 2 C kQuk2 / C c.k k O 2 C kOuk2 /: Q 22 C k#k O 21;˝O C k#k C ckQut k21 C c.k k 2 3 2;˝O 2;˝O (200) Inequalities (197), (199), and (200) imply d .kQu k20 C k Qzz k20 / C kQuzz k20 C kdiv uQ zz k20 C k Qzz k20 dt Q 2/  c.kQut k21 C kfQ0 k21 C khk 2

(201)

O 2 C kOuk2 C kOu C !k O 23;˝O k k O 22;˝O /; C c.k k O 21;˝O C k#k 2;˝O 2;˝O where one used that  is sufficiently small. From estimates related to problem (180), it follows Q 2 C kfQ0 k2 / C c.k Qzz k2 kQuk23 C k k Q 22  c.kdiv uQ k22 C kQut k21 C k#k 2 1 0 O 2 C kOuk2 /: C k#Q zz k20 C kQuzzz k20 / C c.k k O 21;˝O C k#k 1;˝O 2;˝O For sufficiently small , inequality (202) takes the form

(202)

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J. Burczak et al.

Q 2 C kfQ0 k2 / kQuk23 C k k Q 22  c.kdiv uQ k22 C kQut k21 C k#k 2 1 O 2 C kOuk2 /: C c.k k O 21;˝O C k#k 1;˝O 2;˝O

(203)

Inequalities (201) and (203) imply the inequality d Q 2/ .kQu k20 C k Qzz k20 / C kQuk23 C k k Q 22  c.kQut k21 C kfQ0 k21 C khk 2 dt O 2 C kOuk2 C kOu C !k O 2 k k O 2 /: C c.k k O 2 C k#k 1;˝O

2;˝O

2;˝O

3;˝O

(204)

2;˝O

Since under the time derivative in (204) solely the norm kQu  k20 appears, it suffices to consider the expression d kQuzz k20 D 2 dt

Z ˝O

uQ zz uQ zzt d z  "1 kQuzzt k20 C

c kQuzz k20 : "1

(205)

Then, (204) and (205) imply the inequality d Q 2/ .kQuzz k20 C k Qzz k20 / C kQuk23 C k k Q 22  "1 kQuzzt k20 C c.kQut k21 C kfQ k21 C khk 2 dt O 2 C .c="1 /kOuk2 C kOu C !k O 2 k k O 2 /: C c.k k O 2 C k#k 1;˝O

2;˝O

2;˝O

3;˝O

2;˝O

(206) Finally, let us examine problem (153) in the form N D g   p3 div u g ; #t  # #jS D 0; #jtD0 D #0 :

(207)

Multiplying (207) by #, integrating over ˝, and employing H 2 -estimate for the Poisson equation yield d kr#k20 C k#k22  c.kgk20 C kdiv uk20 /: dt

(208)

From (207), it follows k#k23  c.k#t k21 C kgk21 C kdiv uk21 /:

(209)

d k#xx k20  "2 k#xxt k20 C .c="2 /k#xx k20 : dt

(210)

Moreover, one has

From (208), (209) and (210), one derives the inequality

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1831

d k#k22 C k#k23  "2 k#t k22 C c.k#t k21 C kgk21 C kdiv uk21 C .c="2 /k#k22 /: dt (211) Passing in (206) to old variables x and summing over all neighborhoods of the partition of unity yield d .kuxx k20 C k xx k20 / C kuk23 C k k22  ".kuxxt k20 C k#xxt k20 / dt C c.kut k21 C k#k22 C kf0 k21 C khk22 / C .c="/.kuk22 C k#k22;˝ /

(212)

C c.k k21 C k#k21 C .c="/kuk22;˝ / C cku C !k23 k k22 : From the equation of continuity (152), one has d k xt k20  "k xt k20 C .c="/.kut k22 C khxt k20 /Cc.ku; !k23 k t k21 C kut ; !t k22 k k22 /: dt (213) From (211), (212), (213), (184), and (158) follows (191). This concludes the proof. t u

5.3

Differential Inequality

Let us define the quantities .t/ D ju.t /j22;1 C j#.t/j22;1 C j .t /j22;1 ;

(214)

˚.t / D ju.t /j23;2 C j#.t/j23;2 C j .t /j22;1 :

(215)

.t/  ˚.t /:

(216)

p.t/ D j j22;1 C j!j22;1 ;

(217)

Then

Moreover, let

P .t / D j j23;2 C j j22;0 C j!j23;2 C j!t j21;0 C jf j21;0 C jbj21;0

(218)

and recall that f0 D f1 C f2 C f3 ; g D g1 C g2 C g3 ; h are r.h.s. of problems (151), (152), and (153). The data related quantity from the previous section will be also used, namely, M D jf j1;0;1;˝ t C j!j3;2;2;˝ t C j!j2;1;1;˝ t C jbj1;0;1;˝ t C j j3;2;2;˝ t ju0 j2;1 C j j2;0;1;˝ t C j 0 j2;1 C j#0 j2;1 :

1832

J. Burczak et al.

Recall that for the local-in-time solution u; #;

Z.u; #/ C X . /  C .M /; see (120). This bound will be needed to prove the following Lemma 18. Let the assumptions of Theorem 4 hold. As long as the solution given by Theorem 4 exists, there is such C .M / that jf0 j21;0 C jhj22;1 C jgj21;0  C .M /Œ 2 .t / C .t /˚.t / C P .t / C p.t /˚.t / ; (219) Proof. Using the estimate for the local solution we obtain estimates for f0 (compare (154)), h (compare (155)), and g (compare (156)). Focusing now on estimates for f0 , one has jjf1 jj20  '.jj ; ; #jj2 /jj ; ; #jj21 jj jj22 "  p .%N C ; N C C #/ %  2 jjf1;x jj0  

x  %N C

 p% .%N C ; N C C #/ C #x %N C

# #   2 p% .%N C ; N C C #/  C

x r   %N C

0

C

'.jj ; ; #jj2 /j ; ; #j21 jj jj22

 '.jj ; ; #jj2 /jj ; ; #jj22 jj jj22 :

Next jjf1;t jj20

"  p .%N C ; N C C #/ %  

t  %N C

p% .%N C ; N C C #/ %N C



p% .%N C ; N C C #/ C %N C



 C 

C Hence

#t #

# 2   r t  

'.jj ; ; #jj2 /j ; ; #j21 jj t jj21

0

 '.jj ; ; #jj2 /j ; ; #j22;1 j j22;1 :

34 Local and Global Solutions for the Compressible Navier-Stokes. . .

1833

jf1 j21;0  '.M /j ; ; #j22;1 j j2;1 ; where one used the estimate for the local-in-time solution. Similarly, jf2 j21;0  '.M /j ; ; #j22;1 j#j2;1 : N one The remaining estimate for f3 is needed. Recalling our assumption j j  12 %, has jjf3;x jj20  jj.ux C !x /  r.u C !/jj20 C jj.u C !/  r.ux C !x /jj20 N x jj20 / N 20 C jj Au C C%N .jj x Aujj C jjfx jj20 C jj!xt jj20 C C%N .jj x A!jj20 C jj A!x jj20 /  jju C !jj42 C C%N jj jj22 jjujj23 C jjf jj21 C jj!t jj21 C C%N jj jj22 jj!jj23 : Finally, we calculate jjf3;t jj20  jj.ut C !t /  r.u C !/jj20 C jj.u C !/  r.ut C !t /jj20 N t jj20 / N 20 C jj Au C C%N .jj t Aujj C jjft jj20 C jj!t t jj20 C C%N .jj t A!jj20 C jj A!t jj20 /  ju C !j42;1 C C%N j j22;1 juj23;2 C jjft j20 C jj!t t jj20 C C%N j j22;1 j!j23;2 : Summarizing, we have for f1 ; f2 and f3 jf1 j21;0  C .M /.j j22;1 C j#j22;1 C j j22;1 /j j22;1  C .M /Œ 2 .t / C P .t / ; jf2 j21;0  C .M /.j j22;1 C j#j22;1 C j j22;1 /j#j22;1  C .M /Œ 2 .t / C P .t / ;   jf3 j21;0  C .juj22;1 C j!j22;1 /2 Cj j22;1 juj23;2 C jf j21;0 C j!t j21;0 C j!j23;2 Cj j22;1 j!j23;2  C .M /Œ 2 .t / C .t /˚.t / C P .t / : Analogously, one obtains jhj22;1  C Œj j22;1 .juj23;2 C j!j23;2 / C j!j23;2  C .M /Œ.t /˚.t / C P .t / ; jg1 j21;0  C .M /.j j22;1 C j#j22;1 C j j22;1 /.juj22;1 C j#j23;2 /  C .M /Œ 2 .t / C .t /˚.t / C P .t / C p.t /˚.t / ; jg2 j21;0  C .M /Œj j23;2 C .j j22;1 C j#j22;1 C j j22;1 /.j j23;2 C j#j23;2 C j!j22;1 / ;  C .M /Œ 2 .t / C .t /˚.t / C P .t / C p.t /˚.t / ; jg3 j21;0  C .M /Œj t j21;0 C .juj22;1 C j!j22;1 /.j j22;1 C j#j22;1 /

1834

J. Burczak et al.

C .kuk22 C k!k22 /.juj23;2 C j!j23;2 / C .j j22;1 C j#j22;1 C j j22;1 / .juj42;1 C j!j42;1 / C jbj21;0 C j j22;1 jbj21;0  C .M /Œ 2 .t / C .t /˚.t / C P .t / :

From the above estimates (219) follows. This concludes the proof.

t u

Employing (219) in (191) yields Proposition 2. Let the assumptions of Theorem 4 hold. As long as the solution given by Theorem 4 exists, there is such C2 .M /  2 that t .t / C ˚.t /  C2 .M /Œ 2 .t / C .t /˚.t / C P .t / C p.t /˚.t / :

(220)

All the ingredients for proving our large-time-regularity Theorem 5 are now prepared. Let us begin with Lemma 19. Let the assumptions of Theorem 4 be satisfied, and let T be its local time of existence. Take C2 .M / of Proposition 2, and for a  2 .0; 1 , assume that .0/ 

 4C2 .M /

(221)

and P .t /
Then .t / D in (224) yield

Cı 4C2 .M /

and .t/
0, we have D k u 2 Lq and 1

1

kD k ukLq  C kC1 kCd . p  q / kukLp :

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1847

• There exists a constant C so that for any k 2 N, p 2 Œ1; 1, and function u of Lp with Supp uO  f 2 Rd = r  jj  Rg for some  > 0, we have k kukLp  C kC1 kD k ukLp : As general solutions to nonlinear PDEs need not be spectrally localized, we want a device for splitting any function or distribution into a sum of spectrally localized functions. To this end, fix some smooth radial nonincreasing function  with Supp   B.0; 43 / and   1 on B.0; 34 /, and then set './ D .=2/  ./. We thus have X '.2j / D 1 in Rd n f0g j 2Z

The homogeneous dyadic blocks P j are defined by def def P j u D '.2j D/u D F 1 .'.2j /F u/ D 2jd h.2j / ? u

def

with h D F 1 ':

We also introduce the low-frequency cutoff operator SP j : def def L j / ? u SP j u D .2j D/u D F 1 ..2j /F u/ D 2jd h.2

def with hL D F 1 :

Let us emphasize that operators P j and SP j are continuous on Lp , with norm independent of j , a property that would fail if taking a rough cutoff function  (unless p D 2 of course). The price to pay for smooth cutoff is that P j is not an L2 orthogonal projector. However, the following important quasi-orthogonality property is fulfilled: P j P k D 0 if jj  kj > 1:

(6)

The homogeneous Littlewood-Paley decomposition for u reads uD

X

P j u:

(7)

j

This equality holds modulo polynomials only. In order to have equality in the distributional sense, one may consider the set Sh0 of tempered distributions u such that lim kSP j ukL1 D 0:

j !1

As distributions of Sh0 tend to 0 at infinity, one can easily see that (7) holds true in S 0 whenever u is in Sh0 .

1848

2.2

R. Danchin

Besov Spaces

It is obvious that for all s 2 R, we have X C 1 kuk2HP s  22js kP j uk2L2  C kuk2HP s ;

(8)

j 2Z

and it is not difficult to prove the following inequalities involving homogeneous Hölder semi-norms: C 1 kukCP 0;s  sup 2js kP j ukL1  C kukCP 0;s ;

s 2 .0; 1/:

(9)

j 2Z

In (8) and (9), we observe that three parameters come into play: the regularity parameter s, the Lebesgue exponent that is used for P j u, and the type of summation that is done over Z. This motivates the following definition. Definition 1. For s 2 R and 1  p; r  1, we set def

kukBP p;r D s

X

2 kP j ukrLp rjs

 1r

def if r < 1 and kukBP p;1 D sup 2js kP j ukLp : s j 2Z

j 2Z

s is the subset of distributions u 2 Sh0 such that The homogeneous Besov space BP p;r kukBP p;r < 1. s

We shall often use the following classical properties: • Scaling invariance: For any s 2 R and .p; r/ 2 Œ1; C12 , there exists a constant s C such that for all  > 0 and u 2 BP p;r , we have d

d

 ku./kBP p;r  C s p kukBP p;r C 1 s p kukBP p;r s s s :

(10)

s is a Banach space whenever s < d =p or s  d =p and r D 1. • Completeness: BP p;r • Density: The space S0 of functions in S whose Fourier transforms are supported s away from 0 is dense in Bp;r if both p and r are finite. s • Fatou property: If .un /n2N is a bounded sequence of functions of BP p;r that 0 0 s P converges in S to some u 2 Sh , then u 2 Bp;r and kukBP p;r  C lim inf kun kBP p;r s s . • Duality: If u is in Sh0 , then

kukBP p;r  C suphu; i s 

where the supremum is taken over those  in S \ BP ps0 ;r 0 such that kkBP s0 0  1. p ;r

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1849

• Interpolation: The following inequalities are satisfied for all 1  p; r1 ; r2 ; r  1, s1 6D s2 , and 2 .0; 1/: kukBP s2 C.1 /s1  C kuk1 kuk BP s2 : s BP 1 p;r

p;r1

p;r2

• Action of Fourier multipliers: If F is a smooth homogeneous of degree m function on Rd n f0g and F .D/ maps Sh0 to itself, then s sm ! BP p;r : F .D/ W BP p;r

(11)

s s1 In particular, the gradient operator maps BP p;r in BP p;r .

Proposition 2 (Embedding for Besov spaces on Rd ). 0 0 ,! Lp ,! BP p;1 . 1. For any p 2 Œ1; 1, we have the continuous embedding BP p;1 sd .

1



1

/

p p 2. If s 2 R, 1  p1  p2  1, and 1  r1  r2  1, then BP ps 1 ;r1 ,! BP p2 ;r2 1 2 . 0 2 0 0 3. For all .s; s / 2 R and 1  p; r1 ; r2  1 with d =p  s < s  d =p, the s s0 in BP p;r is locally compact. More precisely, for any ' 2 S, embedding of BP p;r 1 2 s s0 the map u 7! 'u is compact from BP p;r to BP p;r . 1 2 d

p is continuously embedded in the set of bounded continuous 4. The space BP p;1 functions (going to 0 at infinity if, additionally, p < 1).

2.3

Paraproduct and Nonlinear Estimates

Formally, the product of two tempered distributions u and v may be decomposed into uv D Tu v C R.u; v/ C Tv u

(12)

with def X P def X Sj 1 u P j v and R.u; v/ D

Tu v D

j

j

X

P j u P j 0 v:

jj 0 j j1

The above operator T is called paraproduct, whereas R is called remainder. The decomposition (12) has been first introduced by J.-M. Bony in [3]. We observe that in Fourier variables the sum in Tu v is locally finite; hence, Tu v is always defined. We shall see however that it cannot be smoother than what is given by high frequencies, namely, v. As for the remainder, it may be not defined, but if it is, then the regularity exponents add up. All that is detailed below (where we agree from now on that A . B means that A  CB for some “harmless” positive constant C ).

1850

R. Danchin

Proposition 3. Let .s; r/ 2 R  Œ1; 1 and 1  p; p1 ; p2  1 with 1=p D 1=p1 C 1=p2 : • For the paraproduct, the following continuity properties hold true: . kukLp1 kvkBP ps kTu vkBP p;r s

2 ;r

and

kTu vkBP p;r sCt . kukB P pt

1 ;1

kvkBP ps ;r ;

if t < 0:

2

• If s1 C s2 > 0 and 1=r D 1=r1 C 1=r2  1, then kR.u; v/kBP s1 Cs2 . kukBP ps1 ;r kvkBP ps2 ;r : 1 1

p;r

2 2

• If s1 C s2 D 0 and 1=r1 C 1=r2  1, then . kukBP ps1 ;r kvkBP ps2 ;r : kR.u; v/kBP p;1 0 1 1

2 2

Putting together decomposition (12) and the above results, one may get the following product estimate that depends only linearly on the highest norm of u and v. s Corollary 1. Let u and v be in L1 \ BP p;r for some s > 0 and .p; r/ 2 Œ1; 12 . Then there exists a constant C depending only on d , p, and s and such that

   C kukL1 kvkBP p;r C kvkL1 kukBP p;r kuvkBP p;r : s s s d

p is embedded in L1 , we deduce that whenever p < C1, Remark 1. Because BP p;1 d

d

p p the product of two functions in BP p;1 is also in BP p;1 and that for some constant C D C .p; d /,

kuvk

d

p BP p;1

 C kuk

d

p BP p;1

kvk

d

p BP p;1

:

Let us finally state a composition result. Proposition 4. Let F W R ! R be smooth with F .0/ D 0. For all 1  p; r  1, s s and s > 0, we have F .u/ 2 BP p;r \ L1 for u 2 BP p;r \ L1 , and kF .u/kBP p;r  C kukBP p;r s s with C depending only on kukL1 , F 0 (and higher derivatives), s, p, and d .

(13)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

2.4

1851

Endpoint Maximal Regularity for the Linear Heat Equation

This paragraph is dedicated to maximal regularity issues for the basic heat equation: @t u  u D f;

ujtD0 D u0 :

(14)

In the case u0  0, we say that the functional space X endowed with norm k  kX has the maximal regularity property if k@t u; Dx2 ukX  C kf kX :

(15)

Fourier-Plancherel theorem implies that (15) holds true for X D L2 .RC  Rd /. Using more elaborate tools based on singular integrals and heat kernel estimates, one may gather that (15) is true for X D Lr .RC I Lp .Rd // whenever 1 < p; r < C1. However, the endpoint cases where one of the exponents p or r is 1 or C1 are false. One of the keys to the approach presented in these notes is that (15) is true for s X D L1 .RC I BP p;1 .Rd //, a consequence of the following lemma: Lemma 1. There exist two positive constants c0 and C such that for any j 2 Z, p 2 Œ1; 1, and  2 RC , we have for all u 2 S 0 with P j u in Lp 2j

ke  P j ukLp  C e c0 2 kP j ukLp : Proof. A suitable change of variable reduces the proof to the case j D 0. Then consider a function  in D.Rd n f0g/ with value 1 on a neighborhood of Supp '. We have   2 e  P 0 u D F 1 e jj P 0 u Z 2 def D g ? P 0 u with g .x/ D .2 /d e i.xj/ ./e jj d :

b

Integrating by parts yields 2 d

g .x/ D .1 C jxj /

Z Rd

  2 e i.xj/ .Id   /d ./e jj d :

Combining Leibniz and Faá-di-Bruno’s formulae to bound the integrant, we get jg .x/j  C .1 C jxj2 /d e c0  ; and thus kg kL1  C e c0  : Now the desired inequality (with j D 0) just follows from L1 ? Lp ! Lp .

(16)

1852

R. Danchin

Theorem 1. Let u satisfy (14). Then for any p 2 Œ1; 1 and s 2 R, the following inequality holds true for all t > 0 W Z ku.t /kBP s C p;1

t 2

kr ukBP s

p;1

0

  Z t d   C ku0 kBP s C kf kBP s d   p;1

p;1

0

(17)

Proof. If u satisfies (14), then we have, for any j 2 Z, P j u.t / D e t P j u0 C

Z

t

e .t / P j f ./ d :

0

Taking advantage of Lemma 1, we thus have 2j

kP j u.t /kLp . e c0 2 t kP j u0 kLp C

Z

t

e c0 2

2j .t /

kP j f ./kLp d  :

(18)

0

Multiplying by 2js and summing up over j 2 Z yield X

2 kP j u.t /kLp . js

X

j

e

Z t X 2j P 2 kj u0 kLp C e c0 2 .t / kP j f ./kLp d 

c0 22j t js

0

j

j

whence kukL1 .0;tIBP s

p;1 /

. ku0 kBP s C kf kL1 .0;tIBP s / : p;1

p;1

Note that integrating (18) with respect to time also yields    2j 22j kP j ukL1 .0;tILp / . 1  e c0 2 t kP j u0 kLp C kP j f kL1 .0;tILp / : Therefore, multiplying by 2js , using Bernstein inequality, and summing up over j yield kr 2 ukL1 .0;tIBP s

p;1 /

.

X

1  e c0 2

2j t



  2js kP j u0 kLp C kP j f kL1 .0;tILp / ;

(19)

j

which is even slightly better than what we wanted to prove Remark 2. Starting from (18) and using general convolution inequalities in RC give a whole family of estimates for the heat equation. However, as time integration has been performed before summation over j , the norms that naturally appear are



def j D 2 k  kLat .Lb / c b;c / `

k  ke Lat .BP

where

def

k  kLat .Lb / D k  kLa ..0;t/ILb .Rd // :

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1853

With this notation, (18) implies that C kf k s2C 2 kuk sC 2 . ku0 kBP p;r s 2 / e e Lt 1 .BP p;r 1 / Lt 2 .BP p;r

for 1  2  1  1:

The relevancy of the above norms in the maximal regularity estimates has been first noticed (in a particular case) in the pioneering work by J.-Y. Chemin and N. Lerner [7] and then extended to general Besov spaces in [6]. They will play a fundamental role in the proof of decay estimates (see the last section). Let us point out that results in the spirit of Propositions 3 and 4 may be easily

proved for e Lt .BP p;r / spaces, the general rule being just that the time exponents behave according to Hölder inequality.

2.5

The Linear Transport Equation

Here we give estimates in Besov spaces for the following transport equation: 

@t a C v  ra C a D f ajtD0 D a0

in R  Rd in Rd ;

(20)

where the initial data a0 D a0 .x/, the source term f D f .t; x/, the damping coefficient   0, and the time-dependent transport field v D v.t; x/ are given. Assuming that a0 2 X and f 2 L1loc .RC I X /, the relevant assumptions on v for (20) to be uniquely solvable depend on the nature of the Banach space X . Broadly speaking, in the classical theory based on Cauchy-Lipschitz theorem, v has to be at least integrable in time with values in the set of Lipschitz functions, so that it has a flow . This enables us to get the following explicit solution for (20): a.t; x/ D e t a0 .

1 t .x//

Z C

t

e .t / f .;

.

0

1 t .x/// d  :

(21)

Theorem 2. Let 1  p; p1  1, 1  r  1, and s 2 R satisfy d

 min

p1

;

  d d d ;  < s < 1 C min p0 p p1

Then any smooth enough solution to (20) fulfills   C V .t/ ka k C kf k kake 1 .B 1 .B s s / C kake s /  e s / 0 P P P P e B L1 . B L L t t t p;r p;r p;r p;r Z t def with V .t/ D krv./k 0

d

p BP p11;1 \L1

d:

1854

R. Danchin

d

If s D 1 C min krv.t /k pd .

p

 ; pd1 and r D 1, then the above inequality holds with V 0 .t / D

P 1 B p ;1 1

Proof. Applying P j to (20) gives P j def with R D Œv  r; P j a:

Pj @t P j a C v  r P j a C P j a D P j f C R

(22)

Therefore, from classical Lp estimates for the transport equation, we get kP j a.t /kLp C kP j akL1t .Lp /  kP j a0 kLp Z t  1 P j kLp C kdiv vkL kP j akLp d  : C kP j f kLp C kR p 0

(23)

P j satisfies We claim that the remainder term R P j .t /kLp  C cj .t /2js krv.t /k kR

d

p BP p11;1 \L1

ka.t /kBP p;r s

with k.cj .t //k`r D 1:

(24) Indeed, from Bony’s decomposition, we infer that (with the summation convention over repeated indices) P j D ŒTvk ; P j @k a C T P v k  P j T@k a v k C R.v k ; @k P j a/  P j R.v k ; @k a/: R @k j a (25) The first term is the only one where having a commutator improves the estimate. Indeed, owing to the properties of spectral localization of the Littlewood-Paley decomposition, we have X ŒTvk ; P j @k a D ŒSP j 0 1 v k ; P j @k P j 0 a: jj j 0 j4

Now, remark that ŒSP j 0 1 v k ; P j @k P j 0 a.x/ Z   jd h.2j .x  y// SP j 0 1 v k .x/  SP j 0 1 v k .y/ @k P j 0 a.y/ dy: D2 Rd

Hence, using the mean value formula and Bernstein inequalities yields kŒTvk ; P j @k akLp . 2j kSP j 0 1 rvkL1

X

k@k P j 0 akLp

jj 0 j j4

. krvkL1

X jj 0 j j4

kP j 0 akLp :

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1855

Bounding the third and last term in (25) follows from Proposition 3. Regarding the second term, we may write X

T@k P j a v k D

SP j 0 1 @k P j a P j 0 v k I

j 0 j 3

hence, using Bernstein inequality, kT@k P j a v k kLp .

X

0

2j j kP j akLp kr P j 0 vkL1 ;

j 0 j 3

and convolution inequality for series thus ensures (24) for that term. Finally, we have P j R.v k ; @k a/ D P j

 X

 k P P P P 0 0 0 0 @k j a .j 1 C j C j C1 /v ;

j 0 j 3

whence, by virtue of Bernstein inequality, kR.v k ; @k a/kLp .

X

kP j 0 akLp krvkL1 ;

jj 0 j j1

and that term is thus also bounded by the r.h.s. of (24). Let us resume to (22). Using (23) and (24), multiplying by 2js , then summing up over j , and using the notations of Remark 2 yield Z kake C kake C kf ke s s /  ka0 kB s / C C P p;r Ps L1 L1t .BP p;r L1t .BP p;r t .Bp;r /

t 0

V 0 kakBP p;r s d:

Then applying Gronwall’s lemma gives the desired inequality for a.

3

The Local Existence in Critical Spaces

This section is dedicated to solving (4) locally in time, in critical spaces. For def

simplicity, we focus on the case where the density goes to 1 at 1. Setting a D %1 and looking for reasonably smooth solutions with positive density, System (4) is equivalent to 8 ˆ < @t a C u  ra D .1 C a/div u;   Au 1 ˆ : @t u C u  ru  C rG.a/ D div 2e .a/D.u/ C e .a/div u Id ; 1Ca 1Ca (26)

1856

R. Danchin def P 0 .1Ca/

where G 0 .a/ D

1Ca

def

def

def

, A D  C . C /rdiv with  D .1/ and  D .1/,

def def .z/ D .1 C z/  .1/: e .z/ D .1 C z/  .1/ , and e

Let us emphasize that with our approach, the exact value of functions G, e , and e  do not matter. We shall only need enough smoothness and vanishing at 0 for e  and e . The scaling invariance properties for .a; u/ are those pointed out for (4). Critical norms for the initial data are thus invariant for all ` > 0 by a0 .x/

a0 .`x/ and

u0 .x/

`u0 .`x/:

In all that follows, we shall only consider homogeneous Besov spaces having the above scaling invariance and last index 1. There are good reasons for that choice, which will be explained throughout. Remembering (10), we thus take d

d

p a0 2 BP p;1

and

1

p u0 2 BP p;1 :

(27)

In order to guess what is the relevant solution space, we just use the fact that a is governed by a transport equation and that u may be seen as the solution to the following Lamé equation: @t u  Au D f;

ujtD0 D u0 :

(28)

In the whole space case, the solutions to (28) also satisfy the estimates of Theorem 1 and Remark 2 whenever the following ellipticity condition is fulfilled: >0

and

def

 D  C 2 > 0:

(29)

Indeed, if we denote by P and Q the orthogonal projectors over divergence-free and potential vector fields, then we have @t Pu  Pu D Pf

and

@t Qu  Qu D Qf:

In particular, applying Theorem 1 yields for all t  0 Z kPu.t /kBP s C  p;1

t

kQu.t /kBP s C  p;1

p;1

0

Z 0

  Z t kr 2 PukBP s d   C kPu0 kBP s C kPf kBP s d  ;

t

kr QukBP s 2

p;1

p;1

0

p;1

  Z t d   C kQu0 kBP s C kQf kBP s d   p;1

0

p;1

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1857

s As P and Q are continuous on BP p;1 (being zero-order multipliers), we conclude that

Z ku.t /kBP s C min.; / p;1

t 0

  Z t kr 2 ukBP s d   C ku0 kBP s C kf kBP s d   p;1

p;1

0

p;1

(30)

d p 1

So, in short, starting from u0 2 BP p;1 , we expect u in (26) to be in d d

def ˚ p 1 p 1 /; @t u; r 2 u 2 L1 .0; T I BP p;1 /  Ep .T / D u 2 C.Œ0; T I BP p;1

In particular ru has exactly the regularity needed in Theorem 2 to ensure the d

p conservation of the initial regularity for a, and we thus expect a 2 C.Œ0; T I BP p;1 /. The rest of this section is devoted to the proof of the following statement.

Theorem 3. Let the viscosity coefficients  and  depend smoothly on % and satisfy (29). Assume that .a0 ; u0 / fulfills (27) for some 1  p < 2d and that d  2. If in addition 1 C a0 is bounded away from 0, then (26) has a unique local-ind time solution .a; u/ with a in C.Œ0; T I BP p / and u in Ep .T /. Furthermore, u is in p;1

d 1 Pp e L1 T .Bp;1 /

d

and a is in

e Pp L1 T .Bp;1 /.

We propose two different proofs for Theorem 3. The first one uses an iterative scheme for approximating (26) which consists in solving a linear transport equation and the Lamé equation with appropriate right-hand sides. Taking advantage of Theorem 2 and of (30), it is easy to bound the sequence in the expected solution space on some fixed time interval Œ0; T  with T > 0. However, because the whole system is not fully parabolic, the strong convergence of the sequence is shown for a weaker norm corresponding to “a loss of one derivative.” For that reason, that approach works only if 1  p < d and d  3 (it is in fact possible to modify the iterative scheme for constructing solutions and combine compactness arguments to get the existence for the full ranges 1  p < 2d and d  2). The same restriction occurs as regards the uniqueness issue, although the limit case p D d , or d D 2 and p  2, is tractable by taking advantage of a logarithmic interpolation inequality combined with Osgood lemma (see the end of this section). The second proof consists in rewriting (26) in Lagrangian coordinates. Then the density becomes essentially time independent, and one just has to concentrate on the velocity equation that is of parabolic type for small enough time and can thus be solved by the contracting mapping argument.

3.1

The Classical Proof in Eulerian Coordinates

We here present the direct approach for solving (26). Our proof covers only the case d  3 and 1  p < d as regards existence and 1  p  d with d  2 for uniqueness.

1858

R. Danchin

To simplify the presentation, we assume that  and  are density independent. def

a 1Ca

Denoting I .a/ D



def P 0 .1Ca/

and G 0 .a/ D

1Ca

, System (26) thus rewrites

@t a C u  ra D .1 C a/div u; @t u  Au D u  ru  I .a/Au  r.G.a//:

Finally, we suppose that for a small enough constant c D c.p; d; G/, we have ka0 k

 c:

d

p BP p;1

(31)

Step 1: An Iterative Scheme def def We set a0n D SP n a0 and un0 D SP n u0 and define the first term .a0 ; u0 / of the sequence of approximate solutions to be def

a0 D a00

def

u0 D e tA u00 ;

and

where .e tA /t0 stands for the semigroup of operators associated with (28). Next, once .an ; un / has been constructed, we define anC1 and unC1 to be the solutions to the following linear transport and Lamé equations: 

@t anC1 C un  ranC1 D .1 C an /div un ; @t unC1  AunC1 D un  run  I .an /Aun  r.G.an //;

(32)

supplemented with initial data a0nC1 and unC1 . 0

Step 2: Uniform Estimates in the Case 1  p < 2d and d  2 As the data are smooth, it is not difficult to check (by induction) that an and un are smooth and globally defined. We claim that there exists some T > 0 such that d p .an /n2N is bounded in C.Œ0; T I BP p;1 / and .un /n2N is bounded in the space Ep .T /. d

p is stable by product imply that for some Indeed, Theorem 2 and the fact that BP p;1 C  1,

kanC1 .T /k

d p BP p;1

 ka0nC1 k

d p BP p;1

Z

T

.1 C kan k

CC 0

Z CC 0

def

Let U n .T / D krun k

d p

P / L1T .B p;1

of a0nC1 , we thus get

d

p BP p;1

/kdiv un k

d

p BP p;1

dt

T

krun k

d

p BP p;1

kanC1 k

d

p BP p;1

dt:

. Applying Gronwall’s lemma and using the definition

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

kanC1 k

d Pp L1 T .Bp;1 /

 C eC U

n .T /

ka0 k

d p BP p;1

C .1 C kan k

d Pp L1 T .Bp;1 /

1859

  n / e C U .T /  1 

(33)

Therefore, assuming that a0 fulfills (31) with some small enough c that kan k

d p

P L1 T .Bp;1 /

 4C c

(34)

and that C U n .T /  log.1 C c/;

(35)

we conclude from (33) that anC1 also satisfies (34) for the same T . At this point, let d

p us observe that as BP p;1 is continuously embedded in L1 , one may take c so small as

kan k

d Pp L1 T .Bp;1 /

 4C c

implies

kan kL1 .Œ0;T Rd /  1=2:

(36)

Let us now prove estimates for the velocity. From (30), we get for some constant C depending only on  and :   Z T kunC1 kEp .T /  C ku0 k dp 1 C kun  run C I .an /Aun C r.G.an //k dp 1 dt  BP p;1

BP p;1

0

The terms in the r.h.s. may be bounded by means of Propositions 3 and 4 (remembering (36)) if d  2 and 1  p < 2d . We get for some C 0 D C 0 .p; d; G/    n C kun k kunC1 kEp .T /  C 0 ku0 k dp 1 C kan k d d 1 kru k d p p p P L1 T .Bp;1 /

BP p;1

P L1 T .Bp;1

P / L1T .B p;1

/

C T kan k





d

Pp L1 T .Bp;1 /

Using (34) and the definition of U n , this implies that   kunC1 kEp .T /  C 0 ku0 k dp 1 C .4C c C U n .T //kun kEp .T / C 4C cT : BP p;1

Therefore, assuming that (35) is fulfilled and taking smaller c if needed, we get kunC1 kEp .T / 

1 n ku kEp .T / C C 0 .ku0 k dp 1 C 4C cT /; 2 BP p;1

and thus, if  kun kEp .T /  2C 0 ku0 k

d 1

p BP p;1

then unC1 also satisfies (37).

C 4cC T



(37)

1860

R. Danchin

To complete the proof, we still have to justify that (35) is fulfilled at rank n C 1. From the definition of k  kEp .T / , embedding, and (37) (at rank n C 1), we know that there exists some constant C 00 so that  U nC1 .T /  C 00 ku0 k

d 1 p BP p;1

 C cT :

Hence, there exists a constant c 0 > 0 such that if T and u0 satisfy ku0 k

d 1

p BP p;1

C cT  c 0

(38)

then both (35) and (37) are fulfilled at rank n C 1. If ku0 k

def

 c 0 , then we split un into unL C uQ n with unL .t / D e tA un0 . Denoting

d 1

Pp B p;1

def

uL D e tA u0 and observing that unL D SP n uL , we discover that U n .T /  krunL k

d

p L1T .BP p;1 /

C kr uQ n k

d

p L1T .BP p;1 /

 C kruL k

d

p L1T .BP p;1 /

C kr uQ n k

d

p L1T .BP p;1 /

:

The term with uL goes to 0 for T tending to 0 with a speed of convergence that may be described according to (19). To handle the second term, we observe that uQ nC1 satisfies @t uQ nC1  AQunC1 D Qun  run  unL  r uQ n  unL  runL  r.G.an //  I .an /Aun : Because uQ nC1 .0/ D 0, combining (30), product laws in Besov spaces, and (31), we get

kQunC1 kEp .T /  C

Z

kQun k

d 1 p BP p;1

0

C kuL k

d C1

Pp L1T .B p;1

kuL k /

Z

T

d 1

Pp L1 T .Bp;1

krun k

d p BP p;1

n

C ka k

kuL k

d

d

p BP p;1

0 n

Pp L1 T .Bp;1 /

/

T

dt C ku k

d 1

p L1T .BP p;1

kQun k

d

p BP p;1

n

C T ka k

dt  d

Pp L1 T .Bp;1 /

/

 (39)

Arguing by interpolation yields, for any ˇ > 0, Z

T

kuL k 0

d

p BP p;1

kQun k

d

p BP p;1

CC ˇ 1 kuL k

dt  ˇkuL k

d C1 p L1T .BP p;1 /

d 1

Pp L1 T .Bp;1

kQun k

d 1

Pp L1 T .Bp;1

: /

kQun k /

d C1

p L1T .BP p;1 /

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

Besides, as kuL k

d 1 p

P L1 T .Bp;1

nC1

kQu

 C ku0 k /

d 1

p BP p;1

  kEp .T /  C U n .T /Cˇku0 k

d 1 p BP p;1

C ku0 k

d 1 p BP p;1

d C1 Pp L1T .B p;1 /

d C1 p L1T .BP p;1 /

Choosing ˇ D 1=.4C ku0 k

. Inequality (39) implies that

Cˇ 1 kuL k

kuL k

1861

Ckan k

d

Pp L1 T .B /

C .T C kuL k

d C1 p L1T .BP p;1 /

 n kQu kEp .T /

/kan k

d Pp L1 T .Bp;1 /

 

/, remembering (34) and (35) (taking c smaller if

d 1

p BP p;1

needed), and assuming that .1 C ku0 k

d 1

p BP p;1

/kuL k

d C1 p

L1T .BP p;1

 c;

(40)

/

we conclude that there exists C 000 so that kQunC1 kEp .T / 

1 n kQu kEp .T / C C 000 c: 2

Hence, kQun kEp .T /  2C 000 c implies that kQunC1 kEp .T /  2C 000 c, and thus (37) is fulfilled at rank n C 1 if c has been chosen small enough. Finally, let us notice that there exists some T > 0 so that X

2j T

1  e c0 2



d

2j . p 1/ kP j u0 kLp 

c 1 C ku0 k



d 1 p BP p;1

j

Hence, (19) ensures that (40) is satisfied for this choice of T .

Step 3: Convergence in the Case 1  p < d and d  3 def

def

Let ıan D anC1  an and ıun D unC1  un . The couple .ıanC1 ; ıunC1 / satisfies 

P @t ıanC1 C unC1  rıanC1 D 3iD1 ıFin ; P @t ıunC1  AıunC1 D 5iD1 ıGin ;

def

with ıF1n D  ıun  ranC1 ;

def

ıF2n D  ıan div unC1 ;

(41)

def

ıF3n D  .1 C an /div ıun ;

 def def def ıG1n D I .an /I .anC1 / AunC1 ; ıG2n D I .an /Aıun; ıG3n D r.G.an /G.anC1 //; def

ıG4n D  unC1  rıun ; and

def

ıG5n D  ıun  run :

1862

R. Danchin

Owing to the first equation, one can perform estimates for .ıan ; ıun / only in a space d

1

p with one less derivative, namely, in C.Œ0; T I BP p;1 /  Fp .T / with

d

d

2

def p p / \ L1 .0; T I BP p;1 /: Fp .T / D C.Œ0; T I BP p;1

Now, using the same type of computations as in step 3, we get for all t 2 Œ0; T  kıanC1 .t /k

d 1

Pp B p;1

 kıanC1 .0/k Z

CC

d 1

p BP p;1

t

krunC1 k

d p BP p;1

0

Z



kıan k

d 1 p BP p;1

t

1 C C kan k

C 0

C kıanC1 k

d p BP p;1

d 1 p BP p;1

C C kanC1 k



d

d p BP p;1



kıun k

d

p BP p;1

d:

Using the bounds provided by the previous step, we thus get, taking c smaller if needed, kıanC1 k

d 1 Pp L1 t .Bp;1 /

 kıanC1 .0/k

d 1 p BP p;1

1 n C kıan k d 1 C 2kıu kFp .t/ : Pp 8 L1 t .Bp;1 /

(42)

As in the previous step, bounding ıunC1 in Fp .T / follows from (30) and product laws. However, as less regularity is available, one has to make the stronger assumption d 3

1  p < d:

and

(43)

Taking c smaller if needed, we eventually get thanks to (34), (35), and (37) kıunC1 kFp .T /  C kıunC1 .0/k

d 2 p BP p;1

C

 1 n n kıa k d 1 C kıu kFp .T / : p P 8 L1 T .Bp;1 /

Combining with (42) yields kıanC1 k

d 1 Pp L1 T .Bp;1 /

 C 4kıunC1 kFp .T /  C kıanC1 .0/k

d 1 p BP p;1

C

C 4kıunC1 .0/k

d 2 p BP p;1

 5 n n kıa k d 1 C 4kıu kFp .T / : p P 8 L1 T .Bp;1 /



35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1863

Summing up over n 2 N, we conclude that .an  a0 /n2N and .un  u0 /n2N converge d

1

p in C.Œ0; T I BP p;1 / and in Fp .T /, respectively.

Step 4: Checking that the Limit Is a Solution and Upgrading Regularity From step 3, we know that there exist a and u so that d

1

p in L1 .0; T I BP p;1 / and

an  a0 ! a  a0

un  u0 ! u  u0

in Fp .T /:

The bounds of step 2 combined with Banach-Alaoglu theorem imply that in addition d

d

p / weak an * a in L1 .0; T I BP p;1

1

p un * u in L1 .0; T I BP p;1 / weak :

and

Routine verifications thus allow to pass to the limit in (32). d

2

p The previous step tells us that r 2 u is in L1 .0; T I BP p;1 /. To upgrade the regularity exponent by 1, let us write that for all J 2 N,

X Z jj jJ

T

j . dp 1/

2

kP j r 2 ukLp dt 

0

X Z

X Z jj jJ

d

2j . p 1/ kP j r 2 un kLp dt

0

jj jJ

C 2J

T

T

d

2j . p 2/ kP j r 2 u  P j r 2 un kLp dt:

0

The first term is bounded by the r.h.s. of (37), while, by virtue of step 3, the second one tends to 0 when n goes 0. Hence, letting J tend to C1 ensures that kr 2 uk d 1 is finite. Next, as .a; u/ satisfies (26), Theorem 2 and Inequality (30) p P L1T .B p;1

/

d

d

1

Pp e1 P p imply that a 2 e L1 T .Bp;1 / and that u 2 LT .Bp;1 /, which, combined with the d

d

1

2

p p fact that a  a0 2 C.Œ0; T I BP p;1 / and u  u0 2 C.Œ0; T I BP p;1 /, implies that d

d

1

p p a 2 C.Œ0; T I BP p;1 / and u 2 C.Œ0; T I BP p;1 /. Indeed, one may write, for any J 2 Z 0 and 0  t  t  T ,

ka.t 0 /  a.t /k

d Pp B p;1

 2J

X

d

2j . p 1/ kP j a.t 0 /  P j a.t /kLp

j J

C2

X j >J

d

2j p sup kP j a./kLp :  2Œ0;T 

1864

R. Danchin d

Pp Because a 2 e L1 T .Bp;1 /, the last term in the r.h.s. converges to 0 when J goes to d

1

p / ensures that for any fixed J 2 Z, C1, and the fact that a  a0 2 C.Œ0; T I BP p;1 the first term tends to 0 when t 0 ! t. Proving the continuity of u is totally similar.

Step 5: Uniqueness Consider two solutions .a1 ; u1 / and .a2 ; u2 / of (26) with the above regularity. The def

difference .ıa; ıu/ D .a2 a1 ; u2 u1 / satisfies 

P @t ıa C u2  rıa D 3iD1 ıFi ; P5 @t ıu  Aıu D iD1 ıGi ;

def

with ıF1 D  ıu  ra1 ;

def

ıF3 D  .1 C a1 /div ıu;

def

def

ıG3 D r.G.a1 /  G.a2 //;

ıF2 D  ıa div u2 ;

 def ıG1 D I .a1 /  I .a2 / Au2 ;

(44)

ıG2 D  I .a1 / Aıu;

def

ıG4 D  u2  rıu; and

def

def

ıG5 D  ıu  ru1 :

Mimicking the computations of step 3, it is easy to see that if (43) is fulfilled, then d

1

p .ıa; ıu/  0 in C.Œ0; T I BP p;1 /  Fp .T /. It turns out that the limit case d D 2 or p D d is tractable even though



0 0 ıa 2 BP d;1 and Au2 2 BP d;1



implies

1 ıaAu2 2 BP d;1

only:

(45)

Indeed, applying Theorem 2 and product laws (see Proposition 3) gives kıakL1 .BP 0

d;1 /

T

  kıa.0/kBP 0

d;1

C .1 C ka1 kL1 .BP 1 / /kıukL1 .BP 1 T

d;1

T

d;1 /



e

ku2 kL1 .BP 2 T

d;1 /

:

Regarding ıu, owing to (45), one has to apply Remark 2 rather than Theorem 1, which enables us to control the following quantity: def j P kıuke 1 L1T .BP d;1 / D sup 2 kj ıukL1T .Ld / ; j

which is slightly weaker than kıukL1 .BP 1 / . T d;1 Inserting the following logarithmic interpolation inequality (see [16]):

kıukL1 .BP 1 T

d;1 /

  kıuke 0 2 L1T .BP d;1 / C kıuke L1T .BP d;1 / log e C d;1 / kıuke 1 L1T .BP d;1 /

. kıuke L1 .BP 1 T

(46)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1865

in the estimate for ıa and using Osgood lemma (see, e.g., [1], Chap. 3), we end up with kıakL1 P0 t .B

d;1

P 1 / CkıukL1 t .B

e

1 P1 d;1 /\Lt .Bd;1 /

 . kıa.0/kBP 0

d;1

Ckıu.0/kBP 1

exp. R0t ˛ d /

d;1

where ˛ is in L1 .0; T / and depends only on the high norms of the two solutions. This yields uniqueness on Œ0; T .

3.2

The Lagrangian Approach

We now propose another proof of the local well-posedness of (26), which will provide us with the statement of Theorem 3 in its full generality. It is based on the Lagrangian formulation of the system under consideration. To make it more precise, we need to introduce some notation. First, we agree that for a C 1 function F W Rd ! Rd  Rm , then div F W Rd ! Rm is defined by d

def X

.div F /j D

@i Fij

for 1  j  m;

iD1

and that for A D .Aij /1i;j d and B D .Bij /1i;j d two d  d matrices, A W B D Tr AB D

X

Aij Bj i :

i;j

The notation adj .A/ designates the adjugate matrix that is the transposed cofactor matrix of A. Of course if A is invertible, then we have adj .A/ D .det A/ A1 . Finally, given some matrix A, we define the “twisted” deformation tensor and divergence operator (acting on vector fields z) by the formulae def 1 

DA .z/ D

2



Dz  A C TA  rz

and

def

divA z D TA W rz D Dz W A:

We recall the following classical result (see the proof in, e.g., [17]). Lemma 2. Let K be a C 1 scalar function over Rd and H , a C 1 vector field. Let X def def be a C 1 diffeomorphism such that J D det.Dy X / > 0 and denote FN D F ı X for any function F W Rd ! R. Then we have N rx K D J 1 divy .adj .Dy X /K/;

(47)

N /: divx H D J 1 divy .adj .Dy X /H

(48)

1866

R. Danchin

Let X be the flow associated with the vector field u that is the solution to Z t u. ; X . ; y// d  : X .t; y/ D y C

(49)

0

def

Let %.t; N y/ D %.t; X .t; y// and uN .t; y/ D u.t; X .t; y//. Formally, we see from the chain rule and Lemma 2 above that .%; u/ satisfies (4) if and only if .%; N uN / fulfills (

N D 0 @t .J %/   N A .Nu/ C .%/ N divA uN Id C P .%/Id N D0 %0 @t uN  div adj .DX / 2.%/D (50) def

def

1

with J D det DX , A D .Dy X /

and Z

t

uN . ; y/ d  :

X .t; y/ D y C

(51)

0

The first equation means that %N D J 1 %0 , and the velocity equation thus recasts in:   L%0 .Nu/ D %1 u; uN / C I2 .Nu; uN / C I3 .Nu; uN / C I4 .Nu/ 0 div I1 .N with   def L%0 .u/ D @t u  %1 0 div 2.%0 /D.u/ C .%0 /div u Id

(52)

and  def I1 .v; w/ D .adj .DXv /  Id / .Jv1 %0 /.Dw Av C TAv rw/ C.Jv1 %0 /.TAv W rw/Id def

I2 .v; w/ D ..Jv1 %0 /  .%0 //.Dw Av C TAv rw/ C..Jv1 %0 /  .%0 //.TAv W rw/Id   def I3 .v; w/ D .%0 / Dw.Av  Id / C T.Av  Id /rw C .%0 /.T.Av  Id / W rw/Id I4 .v/

def

D adj .DXv /P .%0 Jv1 /; def

def

where Xv is given by (49) with v instead of u, Av D .DXv /1 and Jv D det DXv . So finally, in order to solve (50) locally, it suffices to show that the map ˚ W v 7! u with u the solution to    L%0 .u/ D %1 0 div I1 .v; v/ C I2 .v; v/ C I3 .v; v/ C I4 .v/ ; ujtD0 D u0 has a fixed point in Ep .T / for small enough T .

(53)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1867

In order to treat the case where % is just bounded away from zero, we need to generalize (30) to the following Lamé system with nonconstant coefficients: @t u  2adiv .D.u//  br.div u/ D f;

(54)

where a, b, , and  satisfy the following uniform ellipticity condition:  def ˛ D min

inf

.t;x/2Œ0;T Rd

.a/.t; x/;

inf

.t;x/2Œ0;T Rd

 .2a C b/.t; x/ > 0:

(55)

In [17], the following statement has been proved. Proposition 5. Let a, b, , and  be bounded functions satisfying (55). Assume that d

1

p ar, br, ra, and rb are in L1 .0; T I BP p;1 / for some 1 < p < 1. There exist  > 0 and ˛ > 0 such that if for some m 2 Z, we have



min

inf

.t;x/2Œ0;T Rd

SP m .2a C b/.t; x/;

k.Id  SP m /.ra; ar; rb; br/k

 ˛ SP m .a/.t; x/  ; 2 .t;x/2Œ0;T Rd inf

d 1 p

P L1 T .Bp;1

 ˛

(56) (57)

/

then the solutions to (54) satisfy for all t 2 Œ0; T  kukL1 Ps t .B

p;1 /

C ˛kukL1 .BP sC2 / t

p;1

 Z t    C 2 P  C ku0 kBP s Ckf kL1t .BP s / exp kSm .ra; ar; rb; br/k d d  p;1 p;1 p ˛ 0 BP p;1 whenever  min.d =p; d =p 0 / < s  d =p  1. Other important ingredients in the proof of existence of a fixed point for ˚ are the “flow estimates” that we recall below. More details may be found in [17] or [21]. Recall that any measurable time-dependent vector field v W Œ0; T /  Rd ! Rd such that t 7! v.t; x/ is in L1 .0; T / for all x 2 Rd and in addition rv 2 L1 .0; T I L1 / has, by virtue of the Cauchy-Lipschitz theorem, a unique C 1 flow Xv satisfying Z Xv .t; y/ D y C

t

v.; Xv . ; y// d 

for all t 2 Œ0; T ;

0

and that, for all t 2 Œ0; T , the map Xv .t; / is a C 1 diffeomorphism over Rd .

1868

R. Danchin def

Lemma 3. Let p 2 Œ1; 1/ and v.t; N y/ D v.t; Xv .t; y//. Under Assumption (68), we have, for all t 2 Œ0; T , kId  adj .DXv .t //k

. kD vk N

d

p BP p;1

kId  Av .t /k

. kD vk N

d

p BP p;1

d p

L1t .BP p;1 /

kadj .DXv .t //TAv .t /  Id k

d

p BP p;1

kJv˙1 .t /  1k

d

p BP p;1

;

d

p L1t .BP p;1 /

. kD vk N

(58)

;

(59)

. kD vk N

d

p L1t .BP p;1 /

d p

L1t .BP p;1 /

;

(60)

:

(61)

Proof. As an example, let us prove the last item. We have thanks to the chain rule: Jv .t; y/ D 1 C

Rt

0 div v.; Xv . ; y// Jv . ; y/ d  Rt D 1 C 0 .D vN W adj .DXv //. ; y/ d  :

(62)

Hence, if Condition (68) holds, then we have (61) for Jv , a consequence of the fact d

p that BP p;1 is a Banach algebra and of (58). In order to get the inequality for Jv1 , it suffices to notice that

Jv1 .t; y/

 1 D .1 C .Jv .t; y/  1//

1

Z t X k 1D .1/ D vN W adj .DXv / d  : 0

k1

def

Lemma 4. Let vN 1 and vN 2 be two vector fields satisfying (68) and ıv D vN 2  vN 1 . Then we have for all p 2 Œ1; 1/ and t 2 Œ0; T  kAv2  Av1 k

d

Pp L1 t .Bp;1 /

. kDıvk

kadj .DXv2 /  adj .DXv1 /k

d p

kJv2  Jv1 k

. kDıvk

P L1 t .Bp;1 /

d p

P L1 t .Bp;1 /

d

p L1t .BP p;1 /

;

. kDıvk

d p

L1t .BP p;1 /

(63)

d p

L1t .BP p;1 /

;

:

(64) (65)

Proof. In order to prove the first inequality, we use the fact that, for i D 1; 2, we have 1

Avi D .Id C Ci /

X D .1/k Cik k0

Z with Ci .t / D

t

D vN i d  : 0

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1869

Hence, Av2  Av1 D

X

XX k1  Z t j k1j Dıv d  C1 C2 : C2k  C1k D 0

k1

k1 j D0

d

p So using the fact that BP p;1 is a Banach algebra, it is easy to conclude to (63). Proving the second inequality is similar. One can now tackle the proof of existence of a fixed point in Ep .T / for the function ˚ that has been defined in (53). To this end, we introduce as in the previous subsection the solution uL in Ep .T / to

L1 uL D 0;

ujtD0 D u0 :

We want to apply Banach fixed point theorem to ˚ in some suitable closed ball def def BN Ep .T / .uL ; R/. Let v be in BN Ep .T / .uL ; R/ and u D ˚.v/. Denoting uQ D u  uL , we see that    L%0 uQ D %1 0 div I1 .v; v/ C I2 .v; v/ C I3 .v; v/ C I4 .v/ C .L1  L%0 /uL ; uQ jtD0 D 0: (66) The existence of some m 2 Z so that   .% /  .% /  ˛ .%0 /  0 0 > min inf SP m 2 ; inf SP m C %0 %0 %0 2 Rd Rd   .% / 0 0  .%0 / .%0 /  .%0 / 0 r% ; r% ; r% ; r% and .Id  SP m / dp 1  ˛ 0 0 0 0 %0 %0 %20 %20 BP p;1 is ensured by the fact that all the coefficients (minus some constant) belong to the d

p space BP p;1 which is defined in terms of a convergent series and embeds continuously in the set of bounded continuous functions. Hence, if one can show that the rightd

1

p hand side of (66) is in L1 .0; T I BP p;1 / (which will be carried out in the next step), then we will be allowed to apply Proposition 5 to bound uQ in Ep .T /.

First Step: Stability of BEp .T/ .uL , R) def Let v 2 BN Ep .T / .uL ; R/ and uQ be given by (66). Let a0 D %0  1. Proposition 5, product laws in Besov spaces, and Proposition 4 imply that  kQukEp .T /  C e C%0 T k.L1  L%0 /uL k

d 1 p

L1T .BP p;1

 C 1 C ka0 k

d p BP p;1



kI4 .v/k

d p L1T .BP p;1 /

for some constant C%0 depending only on %0 .

/

C

3 X i D1

kIi .v; v/k

d p L1T .BP p;1 /



(67)

1870

R. Danchin

In what follows, we assume that T and R have been chosen so that, for a small enough positive constant c, Z T krvk dp dt  c: (68) BP p;1

0

Now, using the decomposition   .L1  L%0 /uL D .%1 0  1/div 2.%0 /D.uL / C .%0 /div uL Id   C div 2..%0 /  .1//D.u/ C ..%0 /  .1//div u Id ; d

1

p / and and Proposition 4, we see that .L1  L%0 /uL 2 L1 .0; T I BP p;1

k.L1  L%0 /uL k

d 1 p

L1T .BP p;1

. ka0 k

d

p BP p;1

/

.1 C ka0 k

d

p BP p;1

/kDuL k

d p

L1T .BP p;1 /

:

(69)

Likewise, flow and composition estimates (see in particular Lemma 3) ensure that kIi .v; w/k

d 1 p

P L1T .B p;1

. .1 C ka0 k

d

p BP p;1

/

/kDvk

d p

P / L1T .B p;1

kDwk

for i D 1; 2; 3,

d p

L1T .BP p;1 /

(70) kI4 .v/k

d p

P / L1T .B p;1

. T .1 C ka0 k

d

p BP p;1

/.1 C kDvk

d p

L1T .BP p;1 /

/:

(71)

So plugging the above inequalities in (67) and keeping in mind that v satisfies (68), we get after decomposing v into vQ C uL : kQukEp .T /  C e C%0 T .1 C ka0 k C kDuL k2

d Pp / L1T .B p;1

d p BP p;1

 /2 .T C ka0 k

d

p BP p;1

 C kDuL k

d p L1T .BP p;1 /

kDuL k

d p

L1T .BP p;1 /

C kD vk Q

d p L1T .BP p;1 /

/



kD vk Q



d p L1T .BP p;1 /

:

Now, because vQ 2 BN Ep .T / .0; R/, kQukEp .T /  C e C%0 T .1 C ka0 k

d p BP p;1

 /2 .T C ka0 k

d

p BP p;1

C .R C kDuL k

kDuL k

d Pp / L1T .B p;1

d

p L1T .BP p;1 /

/

/kDuL k

d Pp / L1T .B p;1

 C R2 :

Therefore, if we first choose R so that for a small enough constant , .1 C ka0 k

d

p BP p;1

/2 R  

(72)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1871

and then take T so that C%0 T  log 2;

T  R2 ;

ka0 k

d

p BP p;1

kDuL k

d p L1T .BP p;1 /

 R2 ;

kDuL k

d

p L1T .BP p;1 /

 R; (73)

then we may conclude that ˚ maps BN Ep .T / .uL ; R/ into itself.

Second Step: Contraction Estimates Let us now establish that, under Condition (73), the map ˚ is contractive. We def consider two vector fields v 1 and v 2 in BN Ep .T / .uL ; R/ and set u1 D ˚.v 1 / and def

def

def

u2 D ˚.v 2 /. Let ıu D u2  u1 and ıv D v 2  v 1 . We have  2 2 1 1 L%0 ıu D %1 0 div .I1 .v ; v /  I1 .v ; v //  C .I2 .v 2 ; v 2 /  I2 .v 1 ; v 1 // C .I3 .v 2 ; v 2 /  I3 .v 1 ; v 1 // C .I4 .v 2 /  I4 .v 1 //  So applying Proposition 5 (recall that C%0 T  log 2), we get

kıukEp .T /  C .1 C ka0 k

d p BP p;1

3 X

/

kIi .v 2 ; v 2 /  Ii .v 1 ; v 1 /k

d p

L1T .BP p;1 /

iD1

C kI4 .v 2 /  I4 .v 1 /k

 d Pp / L1T .B p;1



(74)

In order to deal with the first term of the right-hand side, we use the decomposition  T  2 adj .DXv2 /  adj .DXv1 / I1 .v 2 ; v 2 /  I1 .v 1 ; v 1 / D .Jv1 2 %0 / Av 2 W rv   T  1 Av2 W rv 2 C adj .DXv1 /  Id .Jv1 2 %0 /  .Jv 1 %0 /     T T 1 T C adj .DXv1 /  Id .Jv1 1 %0 / . Av 2  Av 1 / W rv C Av 2 W rıv C terms pertaining to : Taking advantage of product laws in Besov spaces, of Proposition 4, and of the flow estimates of Lemma 3, we deduce that for some constant C%0 depending only on %0 : kI1 .v 2 ; v 2 /  I1 .v 1 ; v 1 /k

d p L1T .BP p;1 /

 C%0 k.Dv 1 ; Dv 2 /k

d

p L1T .BP p;1 /

kDıvk

d

p L1T .BP p;1 /

:

Similar estimates may be proved for the next two terms of the right-hand side of (74). Concerning the last one, we use the decomposition   I4 .v 2 /  I4 .v 1 / D adj .DXv1 /  adj .DXv2 / P .Jv1 2 %0 /   1  adj .DXv1 / P .Jv1 2 %0 /  P .Jv 1 %0 / :

1872

R. Danchin

Hence, kI4 .v 2 /  I4 .v 1 /k

d p

L1T .BP p;1 /

 C .1 C ka0 k

d

p BP p;1

/T kDıvk

d p

P / L1T .B p;1

:

We end up with kıukEp .T /  C .1 C ka0 k

d p BP p;1

 /2 T C k.Dv 1 ; Dv 2 /k

d p L1T .BP p;1 /



kDıvk

d p

L1T .BP p;1 /

:

Given that v 1 and v 2 are in BN Ep .T / .uL ; R/, our hypotheses over T and R (with smaller  in (72) if need be) thus ensure that kıukEp .T / 

1 kıvkEp .T / : 2

Hence, ˚ admits a unique fixed point in BN Ep .T / .uL ; R/.

Third Step: Regularity of the Density def

def

Set % D Ju1 %0 . By construction .%; u/ satisfies (50), and a D %  1 is given by a D .Ju1  1/a0 C a0 : d

p /, we have that Ju1  1 belongs to From Lemma 3, as Du 2 L1 .0; T I BP p;1 d

d

d

p p p /. Hence, a is in C.Œ0; T I BP p;1 /, too. Because BP p;1 is continuously C.Œ0; T I BP p;1 embedded in L1 , the density remains bounded away from 0 on Œ0; T  (taking T smaller if needed).

Last Step: Uniqueness and Continuity of the Flow Map Let the data .%10 ; u10 / and .%20 ; u20 / fulfill the assumptions of Theorem 3, and let def

.%1 ; u1 / and .%2 ; u2 / be the corresponding solutions. Setting ıu D u2  u1 , we see that L%1 .ıu/ D .L%1  L%2 /.u2 / 0

0

0

C .%10 /1 div

3 X 

  .Ij2 .u2 ; u2 /  Ij2 .u1 ; u1 / C .I42 .u2 /  I42 .u1 //

j D1

C .%10 /1 div

3 X

 ..Ij2  Ij1 /.u1 ; u1 / C .I42  I41 /.u1 / ;

j D1

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1873

where I1i , I2i , I3i , and I4i correspond to the quantities that have been defined just above (53), with density %i0 . Compared to the second step, the only definitely new terms are .L%1  L%2 /.u2 / and the last line. As regards .L%1  L%2 /.u2 /, we have, 0 0 0 0 for t  T , k.L%1  L%2 /.u2 /k 0

0

d 1 p L1t .BP p;1 /

 C%1 ;%2 kı 0 k 0

kDu2 k

d

p BP p;1

0

d

Pp / L1t .B p;1

:

The other new terms satisfy analogous estimates. Hence, applying Proposition 5 yields, if ı 0 is small enough:  kıukEp .t/  C%1 .t C kDu1 k 0

d p

L1t .BP p;1 /

C kıukEp .t/ /kıukEp .t/

C kıu0 k

d p BP p;1

C kı 0 k

d p BP p;1

.t C kDu1 k

d p L1t .BP p;1 /

 / :

An obvious bootstrap argument thus shows that if t, ıu0 , and ı 0 are small enough, then  kıukEp .t/  2C%0 kıu0 k

d p BP p;1

C kı 0 k

d p BP p;1



:

1 1 2 As regards the density, we have ıa D Ju1 1 ıa0 C .Ju2  Ju1 /a0 . Hence, for all t 2 Œ0; T ,

kıa.t /k

d

p BP p;1

 C .1 C kDu1 k

d

p L1t .BP p;1 /

/kıa0 k

d

p BP p;1

kDıuk

d

p L1t .BP p;1 /

:

So we get uniqueness and continuity of the flow map on a small time interval. Then iterating the proof yields uniqueness on the initial time interval Œ0; T , as well as Lipschitz continuity of the flow map. It is now easy to conclude to Theorem 3 in its full generality, as a mere corollary of the following proposition which states the equivalence of Systems (26) and (50) in our functional setting (see the proof in [17]). Proposition 6. Let 1  p < 2d . Assume that the couple .%; u/ with .%  1/ 2 d

p C.Œ0; T I BP p;1 / and u 2 Ep .T / is a solution to (26) such that

Z

T

kruk 0

d

p BP p;1

dt  c:

(75) def

Let X be the flow of u defined in (49). Then the couple .%; N uN / D .%ıX; uıX / belongs to the same functional space as .%; u/ and satisfies (50).

1874

R. Danchin d

p Conversely, if .%N  1; uN / belongs to C.Œ0; T I BP p;1 /  Ep .T / and .%; N uN / satisfies (50) and, for a small enough constant c,

Z

T

kr uN k 0

d

p BP p;1

dt  c

(76)

def

then the map Xt D X .t; / defined in (51) is a C 1 diffeomorphism on Rd , and the def

couple .%; u/.t / D .%.t N / ı Xt1 ; uN .t / ı Xt1 / satisfies (26) and has the same regularity as .%; N uN /.

4

The Global Existence Issue

This section is devoted to the proof of global existence of strong solutions for small perturbations of the constant state .%; u/ D .1; 0/, under the stability assumption P 0 .1/ > 0. For simplicity, we assume that the viscosity functions  and  are constant. Let us emphasize that the approach we used so far to solve (26) cannot provide us with global-in-time estimates (even if both a0 and u0 are small) because we completely ignored the coupling between the mass and momentum equation through the pressure term and looked at it as a low-order source term, just writing @t u  Au D u  ru  I .a/Au  r.G.a//: Then, applying Inequality (30) and product laws in Besov spaces led to  kukEp .t/  C ku0 k

d 1 p BP p;1

 C kukEp .t/ kak

d Pp L1 t .Bp;1 /

 C kukEp .t/ C

Z

t

kak 0

d p BP p;1

 d 

(77) At the same time, as a is a solution to a transport equation, we can only get bounds on kak d , and the last term of (77) is thus out of control for t ! C1. p P L1 t .Bp;1 /

4.1

The Linearized Compressible Navier-Stokes System and Main Result

The key to proving global results is a refined analysis of the linearized system (26) about .a; u/ D .0; 0/ taking the coupling between the mass and momentum equation through the pressure term into account. The system in question reads: 

@t a C div u D f; @t u  u  . C /rdiv u C P 0 .1/ra D g:

(78)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1875

Applying the orthogonal projectors P and Q over divergence-free and potential def

vector fields, respectively, to the second equation and setting ˛ D P 0 .1/ and def

 D  C 2, System (78) translates into 8 < @t a C div Qu D f; @ Qu  Qu C ˛ra D Qg; : t @t Pu  Pu D Pg:

(79)

We see that Pu satisfies an ordinary heat equation, which is uncoupled from a and Qu. For studying the coupling between a and Qu, it is convenient to set def

def

v D jDj1 div u (with F.jDjs u/./ D jjs uO ./), keeping in mind that, according to (11), bounding v or Qu is equivalent, as one can go from v to Qu or from Qu to v by means of a 0-order homogeneous Fourier multiplier. For notational simplicity, we assume from now on that ˛ D  D 1. This is not restrictive as the rescaling p p    p  a.t; x/ D aQ ˛ t; ˛ x and u.t; x/ D ˛ uQ ˛ t; ˛ x (80) ensures that .a; Q uQ / satisfies (79) with ˛ D  D 1. This implies that .a; v/ satisfies the following 2  2 system: ( @t a C jDjv D f;

(81)

def

@t v  v  jDja D h D jDj1 div g: def

Taking the Fourier transform with respect to x and denoting D jj with  2 Rd the Fourier variable, System (81) translates into !       b d b f a b a def 0  D M C with M D  (82) b v b v

 2 dt b h • In the low-frequency regime < 2, M has two complex conjugated eigenvalues: s

2 4 def ˙ . / D  .1 ˙ iS . // with S . / D 1 2

2 def

with real part  2 =2, exactly as for the heat operator with diffusion 1=2. • In the high-frequency regime > 2, there are two distinct real eigenvalues:

2 ˙ . / D  .1 ˙ R. // 2 def

s def

with R. / D

1

4 

2

As 1  R. / 2= 2 for ! C1, we have C . /  2 and  . / 1 In other words, a parabolic and a damped mode coexist.

1876

R. Danchin

Optimal a priori estimates may be derived by computing explicitly the solution of (81) in the Fourier space. Below, we present another method which is generalizable to more complicated systems where explicit computations are no longer possible (see, e.g., [18]). Fix some  0 and consider the corresponding solution .a; O v/ O of (82) in the case fO D hO D 0. Omitting the dependency with respect to , we get 1 d NO D 0; jaj O 2 C Re .aO v/ 2 dt 1 d NO D 0; O 2  Re .aO v/ jvj O 2 C 2 jvj 2 dt d NO D 0; NO C jvj O 2 C 2 Re .aO v/ Re .aO v/ O 2  jaj dt

(83) (84) (85)

from which we deduce 1 d 2 L C 2 j.a; O v/j O 2D0 2 dt

def NO with L2 D 2j.a; O v/j O 2 C j aj O 2  2 Re .aO v/:

(86)

Using Young inequality, we discover that there exists some constant C0 > 0 independent of so that C01 L2  j.a; O a; O v/j O 2  C0 L2 :

(87)

Combining with (86), we conclude that there exists a universal constant c0 > 0 so that 2

L2 .t /  e c0 min.1; /t L2 .0/

for all t  0:

(88)

For general source terms fO and hO in (82), using Duhamel’s formula thus leads to Rt j.a; O a; O v/.t O /j C min.1; 2 / 0 j.a; O a; O v/j O d   Rt O O O  C j.aO 0 ; aO 0 ; vO 0 /j C 0 j.f; f; h/j d   Note that as O @t vO C 2 vO D hO C a; we also have

2

Z

Z

t

t

jv./j O d   jvO 0 j C 0

0

O jh./j d C

Z

t

j a./j O d: 0

(89)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1877

And, thus, bounding the last term according to (89), we get the following inequality which provides the full parabolic smoothing for v: 2

Z

t

j.a; O a; O v/.t O /j C min. ; /

jaj O d C 0

2

Z

t

jvj O d 0



Z

t

 C j.aO 0 ; aO 0 ; vO 0 /j C

 O O O j.f; f; h/j d  

(90)

0

From Inequality (88) and Fourier-Plancherel theorem, it is easy to obtain estimates of L2 type for the solutions to (78). Clearly, optimal informations will be obtained if splitting the unknowns into packets corresponding to frequencies of comparable sizes. This prompts us to apply P k to (79). We get 8 < @t P k a C div P k Qu D P k f; @ P Qu  P k Qu C r P k a D P k Qg; : t Pk @t k Pu  P k Pu D P k Pg:

(91)

In the case with no source term, then using (88) combined with Fourier-Plancherel theorem (and Lemma 1 to handle Pu) readily yields for some universal constant C0 , and c0 depending only on , k.P k a; P k ra; P k u/.t /kL2  C0 e c0 min.1;2

2k /t

k.P k a; P k ra; P k u/.0/kL2 :

Then, for general source terms, using Duhamel’s formula and repeating the computations leading to (90), we end up with k.P k a; P k ra; P k u/.t /kL2 Cmin.1; 2k / 

 C k.P k a0 ; P k ra0 ; P k u0 /kL2 C

Z

Z

t

kP k rakL2 d  C22k

0

t

Z

t

kP k ukL2 d 

0

 P P P k.k f; k rf; k g/kL2 d  

(92)

0

Multiplying both sides by 2ks , taking the supremum on Œ0; t, and then summing up on k  k0 or k  k0 , we conclude to the following: Proposition 7. Let s 2 R and .a; u/ satisfy (78) with P 0 .1/ D  D 1. Let k0 2 Z. Then we have for some constant C depending only on k0 and , and all t  0, ` k.a; u/ke Ps L1 .B t

2;1 /

C k.a; u/k` 1

sC2 Lt .BP 2;1 /

   C k.a0 ; u0 /k`BP s C k.f; g/k`L1 .BP s / ; 2;1

h h kakh 1 P sC1 C kakh 1 P sC1 C kuke P s C kukL1 .BP sC2 / L1 e Lt .B2;1 / Lt .B2;1 / t .B2;1 / t 2;1  h h  ka0 k P sC1 C ku0 kBP s C kf kh 1 B2;1

2;1

sC1 Lt .BP 2;1 /

t

2;1

C kgkhL1 .BP s t

2;1 /



;

1878

R. Danchin

where we used the notation X 2k kP k zkLp kzk`BP D p;1

and

kzkhBP D p;1

kk0

X

2k kP k zkLp :

(93)

kk0

The high-frequency inequality means that in order to get optimal estimates, it is suitable to work with the same regularity for ra and u. In contrast, for low frequencies, one has to work in the same space for a and u, a fact which does not follow from our scaling considerations (5) but is fundamental to keep the pressure term under control in (26). Granted with the above proposition, it is now natural to look at (26) as System (78) with right-hand side f D div .au/

and

def

g D uruI .a/Auk.a/ra where k.a/ D G 0 .a/G 0 .0/:

The problem is that f will cause a loss of one derivative as there is no smoothing effect for a in high frequency. A second limitation of Proposition 7 is that it concerns Besov spaces related to L2 even though we know the system to be locally well-posed in more general Besov spaces (see Theorem 3). To overcome the first problem, let us include the convection terms in our linear analysis, thus considering 

@t a C v  ra C div u D f; @t u C v  ru  u  . C /rdiv u C ˛ra D g;

(94)

where v stands for a given time-dependent vector field. Proposition 8. Let d =2 < s  d =2 and .a; u/ satisfy (94) with ˛ D  D 1. Let k0 2 Z. Then we have for some constant C depending only on k0 and , and all t  0, k.a; ra; u/ke Ps L1 t .B

2;1 /

C kak` 1

sC2 / Lt .BP 2;1



C krakhL1 .BP s

2;1 /

t

Z

 C k.a0 ; ra0 ; u0 /kBP s Ck.f; rf; g/kL1t .BP s / C 2;1

2;1

C kukL1 .BP sC2 / t

t

krvk 0

d 2 BP 2;1

2;1

 k.a; ra; u/kBP s d  

Proof. Applying P k to (94) yields  @t ak C P k .v  ra/ C div uk D fk ; @t uk C P k .v  ru/  uk  . C /rdiv uk C rak D gk ; def def def def with ak D P k a, uk D P k u, fk D P k f , and gk D P k g. Keeping in mind the proof of Proposition 7, we introduce def

L2k D 2k.ak ; uk /k2L2 C krak k2L2 C .uk j rak /L2 :

2;1

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1879

Now, remembering that  C 2 D 1, we get   1 d 2 Lk CkrPuk k2L2 Ck.rQuk ; rak /k2L2 D gk j .2ukCrak / L2 C2.fk j ak /L2 2 dt       C rfk j rak L2  2 P k .v  ra/ j ak L2  2 P k .v  ru/juk L2       P k r.v  ra/ j rak L2  P k .v  ru/ j rak L2  P k r.v  ra/ j uk L2 : (95) def

Let us explain how to bound the convection terms. Set Rk .v; b/ D P k .v  rb/  v  r P k b for b 2 fa; u1 ;    ; ud g. To handle the second and third terms of the second line of (95), we integrate by parts and get Z Z   P k .v  rb/ j bk L2 D .v  rbk /  bk dx C Rk .v; b/ bk dx 

1 2

Z

jbk j2 div v dx C kRk .v; b/kL2 kbk kL2 :

Bounding the last term according to (24), we thus get ˇ  ˇ ˇ P k .v  rb/ j bk 2 ˇ  C ck 2ks krvk L

d

2 BP 2;1

kbkBP s kbk kL2 2;1

with .ck /k2Z in the unit sphere of `1 .Z/. Next, we use the fact that for i 2 f1;    ; d g, eik .v; a/ @i P k .v  ra/ D v  r@i ak C R

eik .v; a/def with R D Œ@i P k ; v  ra:

By adapting the proof of (24), it is easy to prove that ek .v; a/kL2  C ck 2ks krvk kR

d

2 BP 2;1

krakBP s : 2;1

Then using an integration by parts, exactly as above, we conclude that ˇ  ˇˇ ˇ P ˇ k r.v  ra/jrak L2 ˇ  C ck 2ks krvk

d

2 BP 2;1

krakBP s krak kL2 : 2;1

Finally, to handle the last two convection terms, we use the fact that 

   P k .v  ru/ j rak L2 C P k r.v  ra/ j uk L2         ek .v; a/ j uk 2 : D vruk jrak L2 C .vr/rak /juk L2 C Rk .v; u/ j rak L2 C R L

1880

R. Danchin

Integrating by parts in the first two terms of the second line and using again (24) to bound the last two terms eventually lead to ˇ    ˇˇ ˇ P ˇ k .v  ru/ j rak L2 C P k r.v  ra/ j uk L2 ˇ    C ck 2ks krvk d2 krakBP s kuk kL2 C kukBP s krak kL2 : BP 2;1

2;1

2;1

Because Lk  k.ak ; rak ; uk /kL2 , we thus conclude that 1 d 2 L C krPuk k2L2 C k.rQuk ; rak /k2L2 2 dt k   k.fk ; rfk ; gk /kL2 C C ck 2ks krvk

d

2 BP 2;1

 k.a; ra; u/kBP s Lk ;

(96)

2;1

which after time integration and multiplication by 2ks yields

2ks Lk .t / C c0 2ks min.1; 22k /

Z

t

k.ak ; rak ; uk /kL2 d   2ks Lk .0/

0

Z

Z

t ks

t

2 kgk kL2 d  C

C 0

ck krvk 0

d

2 BP 2;1

k.a; ra; u/kBP s d  : 2;1

Taking the supremum on Œ0; t and then summing up over k, we thus get Z k.a; ra; u/ke Ps L1 t .B

2;1

/C

t

k.a; u/k`P sC2 d  C B2;1

0

Z . k.a; ra; u/.0/kBP s C 2;1

Z

t 0

k.ra; u/khBP s d 

t 0

k.f; rf; g/kBP s d  C 2;1

2;1

Z

t

krvk 0

d

2 BP 2;1

k.a; ra; u/kBP s d  : 2;1

(97) Finally, using the fact that @t u C v  ru  u  . C /rdiv u D g  ra;

(98)

localizing (98) according to P k , and arguing as above, we find out Z t Z t Z t kuk . ku.0/k C kgrak d C krvk kuke s s 1 Ps C sC2 BP BP Lt .B / BP 2;1

0

2;1

2;1

0

2;1

0

d

2 BP 2;1

kukBP s d  :

Then bounding ra according to (97) completes the proof of the proposition.

2;1

t u

It turns out to be possible to extend the above proposition to more general Besov spaces related to the Lp spaces with p 6D 2. The proof relies on a paralinearized

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1881

version of System (94) combined with a Lagrangian change of variables (see [5,8]). Here, in order to solve (26) globally, we shall follow a more elementary approach based on the paper by B. Haspot [26]: we use Proposition 7 only for bounding low frequencies and perform a suitable quasi-diagonalization of the system to handle high frequencies. This eventually leads to the following statement that will be proved in the rest of this section. The reader may refer to [19] for a slightly more general result. Theorem 4. Let d  2. Let p 2 Œ2; min.4; 2d =.d  2/ with, additionally, p 6D 4 if d D 2. Assume with no loss of generality that P 0 .1/ D 1 and  D 1. There exist a universal integer k0 2 N and a small constant c D c.p; d; ; G/ such d

d

d

1

p p 2 1 and u0 2 BP p;1 with besides .a0` ; u`0 / in BP 2;1 (with the notation that if a0 2 BP p;1 z` D SP k0 C1 z and zh D z  z` ) satisfy

def

Xp;0 D k.a0 ; u0 /k`

d 1

2 BP 2;1

C ka0 kh d C ku0 kh d 1  c p BP p;1

(99)

p BP p;1

then (26) has a unique global-in-time solution .a; u/ in the space Xp defined by d

1

d

C1

d

1

d

2 2 C b .RC I BP 2;1 / \ L1 .RC I BP 2;1 /; .a; u/` 2 e

d

p p ah 2 e C b .RC I BP p;1 / \ L1 .RC I BP p;1 /; d

C1

p p uh 2 e C b .RC I BP p;1 / \ L1 .RC I BP p;1 /

s def s s where e C b .RC I BP q;1 / D C.RC I BP q;1 /\e L1 .RC I BP q;1 /, s 2 R, and 1  q  1. Furthermore, for some constant C D C .p; d; ; G/, there holds

k.a; u/kXp  CXp;0 :

(100)

Remark 3. Condition (99) is satisfied for small a0 and large highly oscillating velocities: take u"0 W x 7! .x/ sin."1 x  !/ n with ! and n in Sd 1 and  2 S.Rd /. Then d

ku"0 k

d 1 p BP p;1

and ku"0 k`

d 1

 C "1 p

if p > d;

has fast decay with respect to ". Hence, such data with small enough

P 2 B 2;1

" generate global unique solutions in dimension d D 2; 3. Remark 4. One may extend the above global result to 2d =.d C2/  p < 2 provided the following smallness condition is fulfilled: ka0 k

d 1

2 BP 2;1

d

2 \BP 2;1

C ku0 k

d 1

2 BP 2;1

 :

1882

R. Danchin

Indeed, Theorem 5 provides a global small solution in X2 . Therefore, it is only a matter of checking that the constructed solution has additional regularity Xp . This may be achieved by following steps 3 and 4 of the proof below, knowing already that the solution is in X2 . The condition that 2d =.d C 2/  p comes from the part d u`  ra of the convection term in the mass equation, as ru` is only in L1 .RC I BP 2 /, 2;1

d

p . Hence, we need to have d =p  d =2 C 1 and the regularity to be transported is BP p;1 (see Theorem 2). The same condition appears when handling k.a/ra.

e b .RC I BP s / rather than just Cb .RC I BP s / is not essential Remark 5. Using space C 2;1 2;1 in the proof of Theorem 4. We chose to present that slightly more accurate result, as it will be needed when investigating time decay estimates, at the end of the survey.

4.2

Global A Priori Estimates

Consider a smooth solution .a; u/ to (26) satisfying, say, kakL1 .RC Rd /  1=2:

(101)

We want to find conditions under which the following quantity: def

` Xp .t / D k.a; u/k` d 1 C k.a; u/k d C1 2 P 2 e L1t .BP 2;1 L1 / t .B2;1 / h C kakh C kakh d C kukh d d 1 C kuk d C1 p p p p 1 .B 1 .B P P P e e L1 . B / / L / L L1t .BP p;1 / t t t p;1 p;1 p;1

satisfies (100) for all t 2 RC . Rewriting System (26) as follows: ( def @t a C div u D f D  div .au/; def

@t u  Au C ra D g D  u  ru  I .a/Au  k.a/ra; we shall take advantage of Proposition 7 with s 0 D d =2  1 to bound the lowfrequency part of .a; u/. To handle high frequencies, following [26], we shall use that, up to low-order terms: • Pu satisfies a heat equation (hence, parabolic smoothing in any Besov space); • The effective velocity def

w D r./1 .a  div u/ satisfies a heat equation; • The high frequencies of a have exponential decay.

(102)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1883

First Step: Low Frequencies From Proposition 7, we readily infer that ` k.a; u/k` . k.a0 ; u0 /k` d 1 C k.f; g/k` d 1 : d 1 C k.a; u/k d 2 1 P 2 C1 2 2 P e L1 . B / . B / / L L1t .BP 2;1 BP 2;1 t t 2;1 2;1 (103)

Second Step: High Frequencies, the Incompressible Part of the Velocity To handle Pu, we just use the fact that @t Pu  Pu D Pg: Hence, according to Remark 2 (restricted to high frequencies),   h h h kPukh d 1 C kPuk d C1  C kPu0 k d 1 C kPgk d 1 : p p p p P e L1 L1t .BP p;1 / BP p;1 L1t .BP p;1 / t .Bp;1 /

(104)

Third Step: High Frequencies of the Effective Velocity and of the Density On the one hand, the effective velocity w defined in (102) fulfills @t w  w D r./1 .f  div g/ C w  ./1 ra: Therefore, Theorem 1 and the fact that r./1 is a homogeneous Fourier multiplier of degree 1 imply that  def h h h kwke D kwkh d 1 C kwk d C1  C kw0 k d 1 E p .t/ p p p 1 P P e L BP p;1 L1 . B / . B / t t p;1 p;1 C kf  div gkh

d 2 p L1t .BP p;1 /

C kw  ./1 rakh

d 1 p L1t .BP p;1 /



:

(105)

On the other hand, we have @t a C div .au/ C a D div w:

(106)

We claim that  h kakh C kak  C ka0 kh d C kdiv wkh d d d p p p 1 1 Pp / e Lt .BP p;1 / Lt .BP p;1 / BP p;1 L1t .B p;1 Z t C kruk dp kak 0

BP p;1

d p BP p;1

 d 

Indeed, as in the proof of Theorem 2, let us apply P k to (106). We get P k; @t P k a C u  r P k a C P k a D P k .adiv u/  P k div w C R

(107)

1884

R. Danchin

P k satisfies where, according to (24), the remainder term R d

P k kLp  C ck 2k p kruk 8k 2 Z; kR

d p BP p;1

kak

X

with

d p BP p;1

ck D 1

k2Z

and where ka div uk

 C kdiv uk

d

p BP p;1

d

p BP p;1

kak

d

p BP p;1

:

Therefore, evaluating the Lp norm of P k a seen as the solution to a transport d equation, multiplying by 2k p , and summing up over k  k0 yield (107). Now, owing to the high-frequency cutoff, we have, for some universal constant C , kwkh d 1  C 22k0 kwkh d C1 p BP p;1

p BP p;1

k./1 rakh d 1  C 22k0 kakh d :

and

p BP p;1

p BP p;1

(108) In consequence, combining (105) and (107) and choosing k0 large enough yield  h h h kwke C ka.t /k d C kak  kw0 kh d 1 C ka0 kh d d E .t/ p

p BP p;1

p

p BP p;1

L1t .BP p;1 /

C kf  div gkh

Z d 2

p L1t .BP p;1

C /

p BP p;1

t

kruk 0

d p BP p;1

kak

d p BP p;1

 d 

(109)

Fourth Step: End of the Proof of the Linear Estimate Putting Inequality (109) together with (103) and (104) and observing that   h h h h kuke ;  kPuke C kwke C C kakh d 2 C kak d E p .t/ E p .t/ E p .t/ p p P e L1t .BP p;1 / L1 t .Bp;1 / we come to the conclusion (if k0 has been taken large enough) that  Z t  k.f; g/k` Xp .t /  Ck0 ; Xp .0/ C 0

d 1

2 BP 2;1

C kf kh d 2 C kgkh d 1 p BP p;1

p BP p;1

C kruk

d

p BP p;1

kak

d

p BP p;1



 d 

Fifth Step: Nonlinear Estimates It is only a matter of proving that under hypothesis (101), we have Z t   k.f; g/k` d 1 C kf kh d 2 C kgkh d 1 C kruk dp kak dp d   CXp2 .t /: 0

P 2 B 2;1

p BP p;1

p BP p;1

BP p;1

BP p;1

(110)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1885

As p  2, it is clear that the last term of the r.h.s. of (110) is bounded by CXp2 .t /. Next, arguing exactly as in the proof of the local existence, we easily get, for 1  p < 2d , kf k

d 1 p

L1t .BP p;1

kgk

d 1 Pp / L1t .B p;1

  C kuk

 C kak

d 1 p

P L1 t .Bp;1

C kak

d Pp L1 t .Bp;1 /

d p

L2t .BP p;1 /

/

kruk

kuk

d p

L2t .BP p;1 /

;

d p

P / L1t .B p;1

/

kr 2 uk

d 1 p L1t .BP p;1 /

C kak

d Pp / L2t .B p;1

krak

d 1 p L2t .BP p;1 /





Let us observe that by interpolation and embedding (recall that p  2), we have kak

d p

P / L2t .B p;1

 kak`

C

d p

C kakh

d p

L2t .BP p;1 /

L2t .BP p;1 /

kak`

kak`



d 1

P 2 L1 t .B2;1

1=2 d C1

2 L1t .BP 2;1

/

 C kak`

d

Pp L1 t .Bp;1 /

/

kak`

1=2 



d

Pp / L1t .B p;1

Hence, kak

d

p L2t .BP p;1 /

 CXp .t /:

As a similar inequality may be proved for kuk

d

p L2t .BP p;1 /

k.f; g/kh

d 1 p L1t .BP p;1 /

(111)

, we eventually get

 CXp2 .t /:

So we are left with the proof of k.f; g/k`

d 1 2 L1t .BP 2;1 /

 CXp2 .t /:

(112)

Let us admit the following two inequalities (the first one being proved in [19] and the second one being a particular case of Proposition 3 followed by suitable embedding, owing to 1  p=2  2): kTa bk

s1C d2  d p P B 2;1

kR.a; b/k

s1C d2  d p P B 2;1

 C kak

d 1 p BP p;1

 C kak

kbkBP s

d 1 p BP p;1

p;1

kbkBP s

p;1

  ; if d  2 and 1  p  min 4; d2d 2 if s > 1min

d

; d p p0



(113) and 1  p  4: (114)

1886

R. Danchin d

2 In order to prove (112) for f , it suffices to bound .au/` in L1 .0; t I BP 2;1 /. Now, ` h using Bony’s decomposition and the fact that a D a C a , we see that

  `  `  ` .au/` D Ta u/` C R.a; u/ C Tu a` C Tu ah :

(115)

The first three terms may be bounded, thanks to Prop. 3 and Inequalities (113) and (114) with s D dp  1. Observing that kzk`BP  C kzkBP for any Besov norm, we r;1 r;1 get k.Ta u/` k

d

2 L1t .BP 2;1 / `

 C kak

k.R.a; u// k

d

2 L1t .BP 2;1 /

k.Tu a` /` k

d

2 L1t .BP 2;1 /

d

d 1 p

P L1 t .Bp;1

 C kak

kuk d 1 p

P L1 t .Bp;1

d C1 p

L1t .BP p;1

/

; /

kuk

d C1 p

L1t .BP p;1

/

 C kukL1 ka` k P 1 t .B1;1 /

d C1

2 L1t .BP 2;1

; /

: /

1

p 1 is embedded in BP 1;1 , the above right-hand sides may be bounded by As BP p;1 2 CXp .t /. To handle the last term of (115), we just have to observe that owing to the definition of the low- and high-frequency cutoff, there exists a universal integer N0 so that  X  `  SP k1 u P k ah  Tu ah D SP k0 C1

jkk0 jN0

Hence, k.Tu ah /` k

d

d

2 BP 2;1

. 2k0 2

P

jkk0 jN0

kSP k1 u P k ah kL2 . Now, if 2  p 

min.d; 2d =.d  2//, then we may use for jk  k0 j  N0  d  d 2k0 2 kSP k1 u P k ah kL2  C 2k0 kSP k1 ukLd 2k p kP k ah kLp ; and if d  p  4, then  d  d  d 2k0 2 kSP k1 u P k ah kL2  C 2k0 2k. p 1/ kSP k1 ukLp 2k p kP k ah kLp : Hence, one may conclude that f satisfies (112). Bounding g is similar (see [19]).

Last Step: Global Estimate Putting all the previous estimates together, we get  Xp .t /  C Xp .0/ C Xp2 .t //:

(116)

Now it is clear that as long as 2CXp .t /  1;

(117)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1887

Inequality (116) ensures that Xp .t /  2CXp .0/:

(118)

Using a bootstrap argument, one may conclude that if Xp .0/ is small enough, then (101) and (117) are satisfied as long as the solution exists. Hence, (118) holds globally in time.

The Proof of Theorem 4 We just give the important steps. We fix some initial data so that X0 is small enough. First, Theorem 3 implies that there exists a unique maximal solution .a; u/ to (26) d p 1 /, kakL1 .0;T  Rd /  1=2, on some time interval Œ0; T  /, with a 2 C.Œ0; T  /I BP p;1 d

d

1

C1

p p and u 2 C.Œ0; T  /I BP p;1 / \ L1loc .0; T  I BP p;1 /. From (26) and Proposition 7, one may check that the additional low-frequency information is preserved on Œ0; T  /: we have

d

d

2 1 2 C1 a` 2 C.Œ0; T  /I BP 2;1 / \ L1 .0; T  I BP 2;1 / and d

d

2 1 2 C1 u` 2 C.Œ0; T  /I BP 2;1 / \ L1loc .0; T  I BP 2;1 /:

Let us assume (by contradiction) that T  < 1. Then applying (118) for all t < T  yields kak C kuk d d 1  CX0 : p p e e L1 .BP p;1 / L1 .BP p;1 / T T If X0 is so small that (118) implies that both (31) and (75) are fulfilled on Œ0; T  /, then, for all t0 2 Œ0; T  /, one can solve (26) starting with data .a.t0 /; u.t0 // at time t D t0 and get a solution according to Theorem 3 on the interval Œt0 ; T C t0  with T independent of t0 . Choosing t0 > T   T thus shows that the solution can be t u continued beyond T  , a contradiction.

5

Asymptotic Results

In this section, we focus on two types of asymptotic issues for small global solutions to (4) that received a lot of attention since the 1980s: the low Mach number asymptotic and the long-time behavior. We shall see that essentially optimal results may be obtained by very simple arguments from the global result we established in the previous section.

1888

R. Danchin

5.1

The Low Mach Number Limit

This subsection is devoted to the rigorous justification of the convergence of (4) to the incompressible Navier-Stokes equations:  @t u C u  ru  u C r˘ D 0; (119) div u D 0; in the so-called

def

ill-prepared data case, where we only assume that a0" D "1 def

. 0"  1/ and u"0 are suitably bounded. Therefore, if we set a" D "1 . "  1/, then .@t a" ; @t u" /jtD0 is of order 1=", so that one cannot exclude highly oscillating acoustic waves. More concretely, we have to pass to the limit " ! 0 in 8 div u" ˆ " ˆ a C @ D div .a" u" /; t ˆ ˆ " ˆ ˆ ˆ < k."a" / " Au" ra" D ra C @t u" C u"  ru"  " ˆ 1 C "a " " ˆ ˆ ˆ ˆ   ˆ 1 ˆ : C div 2e ."a" /D.u" / C e ."a" /div u" Id : 1 C "a" (120) Before stating our main results, let us introduce some notation. In this section, we agree that for z 2 S 0 .Rd /, def X

P j z

z`;ˇ D

and

def X

zh;ˇ D

2j ˇ2j0

P j z;

(121)

2j ˇ>2j0

for some large enough nonnegative integer j0 depending only on p and d and on def

the functions k, =, and = with  D  C 2. We also need the notation `;ˇ

def X

kzkBP D p;1

2j kP j zkLp

and

h;ˇ

p;1

2j ˇ2j0

def X

kzkBP D

2j kP j zkLp :

(122)

2j ˇ2j0

Keeping in mind the linear analysis we performed for (78) in the case  D 1 and " D 1 and combining with the change of variable def

.a; u/.t; x/ D ".a" ; u" /."2 t; "x/;

(123) def

we expect the threshold between low and high frequencies to be at 1=Q" with "Q D ", and it is thus natural to consider families of data .a0" ; u"0 / such that " " " " h;e " h;e k.a0" ; u"0 /k`;e d 1 C ka0 k d C ku0 k d 1 P 2 B 2;1

p BP p;1

p BP p;1

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1889

is bounded independently of ". We expect the corresponding solutions of (120) to p be uniformly in the space X"; defined by: d

d

2 1 2 C1 " `;e • .a`;e ; u "/ 2 e C b .RC I BP 2;1 / \ L1 .RC I BP 2;1 /, d

d

p p " • ah;e 2e C b .RC I BP p;1 / \ L1 .RC I BP p;1 /, d

d

1

C1

p p " 2e C b .RC I BP p;1 / \ L1 .RC I BP p;1 /, • uh;e

and endowed with the norm: def `;e " h;e " " p D k.a; u/k "kakh;e k.a; u/kX"; d 1 C kuk d 1 Ce d p p 2 1 1 P e L1 .BP 2;1 / e e L .Bp;1 / L .BP p;1 / " C k.a; u/k`;e

d C1

2 L1 .BP 2;1

" C kukh;e /

d C1 p

L1 .BP p;1

" C "1 kakh;e /

d p

:

L1 .BP p;1 /

Let us now state a low Mach number limit result in the small data case, the reader being referred to [14,15] for the large data case and stronger results of convergence. Theorem 5. Assume that the fluid domain is either Rd or Td , that the initial data .a0" ; u"0 / are as above, and that p is as in Theorem 4. There exists a constant  independent of " and of  such that if def

" " " " h;e C0"; D k.a0" ; u"0 /k`;e "ka0" kh;e d 1 C ku0 k d 1 Ce d  ; P 2 B 2;1

p BP p;1

(124)

p BP p;1

then System (120) with initial data .a0" ; u"0 / has a unique global solution .a" ; u" / in p the space X"; with, for some constant C independent of " and , "; p  CC k.a" ; u" /kX"; 0 :

(125)

In addition, Qu" converges weakly to 0 when " goes to 0, and, if Pu"0 * v0 , then Pu" converges in the sense of distributions to the unique solution of (119) supplemented with initial data v0 . Proof. Performing the change of unknowns given in (123) and the change of data def

.a0 ; u0 /.x/ D ".a0" ; u"0 /."x/

(126)

reduces the proof of the global existence to the case  D 1 and " D 1, which was done in Theorem 4. Back to the original variables yields the desired uniform estimate (125) under Condition (124). Indeed, up to some harmless constant,

1890

R. Danchin

 " " " `;1 h;1 h;1  " h;e k.a0" ; u"0 /k`;e "ka0" kh;e d 1 Cku0 k d 1 Ce d D  k.a0 ; u0 /k d 1 Cku0 k d 1 Cka0 k d 2 BP 2;1

p BP p;1

p BP p;1

p BP p;1

2 BP 2;1

p BP p;1

and p D k.a; u/k p : k.a" ; u" /kX"; X 1;1

Granted with the uniform estimates established in the previous section, it is now easy to pass to the limit in the system in the sense of distributions, by adapting the compactness arguments of P.-L. Lions and N. Masmoudi in [30]. More precisely, consider a family .a0" ; u"0 / of data satisfying (124) and Pu"0 * v0 when " goes to 0. Let .a" ; u" / be the corresponding solution of (120) given by the first part of Theorem 5. Because " " ka0" kh;e "ka0" kh;e d 1 . e d ; p BP p;1

(127)

p BP p;1

d

d

1

p p the data .a0" ; u"0 / are uniformly bounded in BP p;1  BP p;1 . Likewise, (125) ensures that d

1

p /. Therefore, there exists a sequence .a" ; u" / is bounded in the space Cb .RC I BP p;1 d

1

p (with Pu0 D v0 ) and ."n /n2N decaying to 0 so that .a0"n ; u"0n / * .a0 ; u0 / in BP p;1

d

1

p .a"n ; u"n / * .a; u/ in L1 .RC I BP p;1 /

weak :

(128)

The strong convergence of the density to 1 is obvious: we have %"n D 1 C "n a"n , d

p and .a"n /n2N is bounded in L2 .RC I BP p;1 / (argue as in (111)). In order to justify that div u D 0, we rewrite the mass equation as follows:

div u"n D "n div .a"n u"n /  "n @t a"n : d

p p Given that a"n and u"n are bounded in L2 .RC I BP p;1 / (use the definition of X"; and d

1

p /. As interpolation), the first term in the right-hand side is O."n / in L1 .RC I BP p;1 for the last term, it tends to 0 in the sense of distributions, by virtue of (128). We thus have div u"n * 0, whence div u D 0. To establish that u is a solution to (119), let us project the velocity equation onto divergence-free vector fields:

@t Pu"n  Pu"n D P.u"n  ru"n /  P



 1 "n  Au 1 C "n a"n

(129)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1891

Because Qu D 0, the left-hand side weakly converges to @t u  u. To prove that " " the last term tends to 0, we use the fact that having e ".a" /h;e and .a" /`;e bounded in d

d

1

p p / and L1 .BP p;1 /, respectively, implies that, for all ˛ 2 Œ0; 1, L1 .BP p;1

d

1C˛

p is bounded in L1 .BP p;1

e "˛ a "

d

/:

(130)

1

p Now, Au" is bounded in L1 .BP p;1 / and p < 2d . Hence, according to product laws in Besov spaces, composition inequality, and (130), we get .1 C "a" /1 Au" D d

2C˛

p O.e "1˛ / in L1 .BP p;1 /, whenever 2 max.0; 1  dp / < ˛  1. Hence, the last term of (129) goes strongly to 0 for some appropriate norm. In order to prove that P.u"n  ru"n / * P.u  ru/, we note that

u"n  ru"n D

1 rjQu"n j2 C Pu"n  ru"n C Qu"n  rPu"n : 2

Projecting the first term onto divergence-free vector fields gives 0, and we also know that Pu D u. Hence, we just have to prove that P.Pu"n  ru"n / * P.Pu  ru/

and

P.Qu"n  rPu"n / * 0:

(131)

This requires our proving results of strong convergence for Pu"n . To this end, one may exhibit uniform bounds for @t Pu"n in a suitable space. First, arguing by d

C 2 3

p m interpolation, we see that .r 2 u"n / is bounded in Lm .BP p;1 / for any m  1. 2 2 Choosing m > 1 so that m  3 > d min. p ; 1/ (this is possible as p < 2d ) and d

p /, we thus get ..1C"n a"n /1 Au"n / remembering that ."n a"n / is bounded in L1 .BP p;1 d

2 Cm 3

p bounded in Lm .BP p;1

/. Similarly, combining the facts that .u"n / and .ru"n / are

d p 1

d

2 Cm 2

p bounded in L1 .BP p;1 / and Lm .BP p;1 d 2 p C m 3

is bounded in Lm .BP p;1

/, too. Computing @t Pu"n from (129), it is now clear d

2 Cm 3

p that .@t Pu"n / is bounded in Lm .BP p;1 d 2 p C m 3

1 in C 1 m .RC I BP p;1 d

1

/, respectively, we see that .u"n  ru"n /

/. Hence, .Pu"n  Pu"0n / is bounded d

1

p /. As Pu"n is also bounded in Cb .RC I BP p;1 / and as the d

C 2 3

p p m in BP p;1 is locally compact (see, e.g., [1], page 108), we embedding of BP p;1 conclude by means of Ascoli theorem that, up to a new extraction, for all  2 S.Rd / and T > 0,

Pu"n ! Pu

d

2 Cm 3

p in C.Œ0; T I BP p;1

/:

(132)

1892

R. Danchin d

1

p Interpolating with the bounds in Cb .RC I BP p;1 /, we can upgrade the strong converd

1˛

p / for all small enough ˛ > 0 and all gence in (132) to the space C.Œ0; T I BP p;1 T > 0. Combining with the properties of weak convergence for ru"n to ru and Qu"n to 0 that may be deduced from the bounds of u"n , it is now easy to conclude to (131). One can use, for instance, the fact that for all m > 1, we have

d

2 Cm 2

p ru"n * ru in Lm .BP p;1

5.2

d

2 Cm 1

p / weak and Qu"n * 0 in Lm .BP p;1

/ weak :

Time Decay Estimates

In the present subsection, we show that under a mild additional decay assumption 0 that is satisfied if the data are in L1 .Rd /, for instance, the L2 norm (the BP 2;1 d

norm in fact) of the global solutions constructed in Theorem 4 decays like t  4 for t ! C1, exactly as for the linearized equations. This fact has been first observed by A. Matsumura and T. Nishida in [31] in the case of solutions with high Sobolev regularity. The adaptation to the L2 -type critical regularity framework has been carried out recently by M. Okita in [33], in dimension d  3. Below, we give a more accurate description of the time decay, emphasizing a better decay for high frequencies. This is the key to handling any dimension d  2. For simplicity, we concentrate on the L2 -type framework, even though similar results are true in the more general Lp framework of Theorem 4 (see [22]). Theorem 6. Let the data .a0 ; u0 / satisfy the assumptions of Theorem 4 with p D 2, and assume with no loss of generality that P 0 .1/ D 1 and that  D 1. Denote defp def h i D 1 C  2 . Let " > 0 be small enough and denote ˛ D min. d4 C 2; d2 C 12  "/. There exists a positive constant c so that if in addition   def D0 D sup kF.P k a0 /kL1 C kF.P k u0 /kL1  c

(133)

kk0

then the global solution .a; u/ given by Theorem 4 satisfies, for all t  0,  D.t /  C D0 C k.a0 ; ra0 ; u0 /k

def

with D.t / D

sup s2Πd2

C";2

d

d 1 2 BP 2;1



(134)

s

kh i 4 C 2 .a; u/k`L1 .BP s t

2;1 /

h C khi˛ .ra; u/kh d 1 C k ruk d : 2 P P 2 e e L1 . B / L1 t t .B2;1 / 2;1

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1893

Proof. Throughout the proof, we shall use repeatedly that for 0 < 1  2 , we have Z t ht  i 1 h i 2 d  . ht i 1 if in addition 2 > 1: (135) 0

Step 1: Bounds for the Low Frequencies Denoting by .S .t //t0 the semigroup associated with (78), we have, for all k 2 Z, 

P k a.t / P k u.t /



 D S .t /

def

P k a0 P k u0



Z



t



S .t  / 0

def

 P k f1 . / d P k .f2 Cf3 Cf4 /. /

def

(136)

def

with f1 D div .au/, f2 D u  ru, f3 D k.a/ra, and f4 D I .a/Au. From an explicit computation of S .t / in Fourier variables (see, e.g., [5]), we discover that there exist positive constants c and C depending only on k0 and such that 2

jF .S .t /U /./j  C e c0 tjj jF U ./j

for all jj  2k0 :

Therefore, for all k  k0 , kS .t /P k U k2L2 .

Z

2 e 2c0 jj t jF P k U ./j2 d  2k

. kF P k U k2L1 2kd e c0 2 t : We thus get up to a change of c0 : d

s

t 4 C2

X

 X p d 2k 2ks kS .t /P k U kL2 . sup kF P k U kL1 . t 2k / 2 Cs e c0 2 t : kk0

kk0

kk0

(137) As for any > 0, there exists a constant C so that sup t0

X



t 2 2k e c0 2

2k t

 C ;

k2Z

we get from (137) that for s > d =2, d

s

sup t 4 C 2 kS .t /U k`BP s  Cs sup kF P k U kL1 : 2;1

t0

kk0

It is also obvious that for s > d =2, kS .t /U k`BP s . kU k`BP s . sup kF P k U kL1 : 2;1

2;1

kk0

(138)

1894

R. Danchin

Hence, we conclude that d

s

suphti 4 C 2 kS .t /U k`BP s . sup kF P k U kL1 : 2;1

t0

(139)

kk0

Next, we claim that for all s 2 .d =2; 2 and i 2 f1;    ; 4g, we have Z t  d s d s ht   i 4  2 sup kF P k fi . /kL1 d  . ht i 4  2 D 2 .t / C X 2 .t / 0

(140)

kk0

Z

def

with X .t / D k.a; ra; u/k 1 d2 1 C e Lt .BP 2;1 /

t

kak`

d C1 2 BP 2;1

0

C kakh d C kuk 2 BP 2;1

d C1



2 BP 2;1

d .

Of course, as the Fourier transform maps L1 to L1 , it suffices to prove (140) with kfi kL1 instead of supkk0 kF P k fi kL1 . To bound the term with f1 , we use the following decomposition: f1 D u  ra C a div u` C a div uh : Now, from Cauchy-Schwarz inequality and the definition of D.t /, one may write Z t   d s d ht   i 4  2 k.u  ra/. /kL1 d   sup hi 4 ku. /kL2 0t

0



d

1

sup hi 4 C 2 kra./kL2



0t

Z

t



d

s

d

1

ht   i 4  2 hi 2  2 d 

0

. hti

 d4  2s

D 2 .t /;

where we used (135) and the fact that 0 < d4 C 2s  d2 C 12 for s  2. Bounding the term with a div u` is totally similar. Regarding the term with a div uh , we use that if t  2, Z

t

ht i

 d4  2s

Z

h

1

k.adiv u /. /kL1 d  .

0

Z C

d

s

ht i 4  2 ka./kL2 kdiv uh . /kL2 d 

0 t

  d s d  d ht  i 4  2 h i1 4 hi 4 ka./kL2 kdiv uh . /kL2 d  :

1

Therefore, as d =2 < s  2, we get ht i

s d 2C 4

Z

t

ht  i

 d4  2s

h

k.adiv u /. /kL1 d  .

0

C



d



sup ka./kL2

 2Œ0;t

sup h i 4 ka./kL2

 2Œ0;t

Z

t

kdiv uh . /kL2 d  0



 sup  kdiv uh . /kL2 ;

 2Œ0;t

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1895

and (140) is thus satisfied by the term with a div uh if t  2, the case t  2 being obvious as ht i  1 and ht   i  1 for 0    t  2, and one may write Z

t

0

h 2 ka div uh kL1 d   kakL1 2 kdiv u kL1 .L2 / . X .t /: t .L / t

Handling the terms with f2 and f3 is totally similar: k.a/ra and u  ru` may be treated as u  ra, and u  ruh , as a div uh . For f4 , we write that f4 D I .a/Au` C I .a/Auh : Now, we have Z

t

d

s

ht   i 4  2 kI .a/Au` kL1 d 

0

.



d

sup h i 4 ka./kL2



d

sup h i 4 C1 kr 2 u` . /kL2

Z

 2Œ0;t

2Œ0;t

t

d

s

d

ht i 4  2 hi1 2 d  :

0

Hence, thanks to (135), the term with I .a/Au` fulfills (140). Finally, for t  2, Z

t

d

s

d

s

ht   i 4  2 kI .a/Auh kL1 d  . ht i 4  2

0

Z

1

kakL2 kr 2 uh kL2 d 

0

Z

t

C

  d s d  d ht  i 4  2 h i1 4 hi 4 ka./kL2 kr 2 uh . /kL2 d I

1 h hence, because d =2 < s  2 and k r 2 uh kL1 2 . k ruk d , t .L / P 2 e L1 t .B2;1 /

Z

t

 d s d s ht   i 4  2 kI .a/Auh kL1 d  . ht i 4  2 D 2 .t / C X 2 .t /

for t  2:

0

Obviously, as ht i  1 and ht   i  1 for 0    t  2, we have the following inequality: Z

t

d

s

d

s

ht   i 4  2 kI .a/Auh kL1 d  . ht i 4  2 X 2 .t / for t  2;

0

which completes the proof of (140). Combining with (139) and using Duhamel’s formula, we conclude that for all t  0 and s 2 .d =2; 2, d

s

ht i 4 C 2 k.a; u/k`BP s . D0 C D 2 .t / C X 2 .t /: 2;1

(141)

1896

R. Danchin

Step 2: Decay Estimates for the High Frequencies of .r a, u/ We here want to bound the second term of D.t /. Recall that Theorem 4 ensures that k.ra; u/kh d 1  CX .0/ P 2 e L1 T .B2;1 /

for all T  0:

Therefore, it suffices to bound def X k. d2 1/ 2 sup t ˛ k.P k ra; P k u/.t /kL2 : kt ˛ .ra; u/kh d 1 D P 2 / e t2Œ2;T  L1 . B 2;T 2;1 kk0

The starting point is Inequality (95) which implies that for k  k0 and for some c0 D c.k0 ; / > 0, we have  1 d 2 Lk C c0 L2k  k.rfk ; gk /kL2 2 dt

 ek .u; a/kL2 C krukL1 Lk Lk C kRk .u; a/kL2 C kRk .u; u/kL2 C kR

def def with f D  adiv u, g D k.a/ra  I .a/Au, Rk .u; b/ D P k .u  rb/  u  r P k b ei .u; a/def D @i P k .u  ra/  u  r@i P k a. for b 2 fa; ug, and R k

After time integration, we discover that

e

c0 t

Lk .t /  Lk .0/ C

Z 0

t

 e c0  k.rfk ; gk /kL2 C kRk .u; a/kL2  ek .u; a/kL2 C krukL1 Lk d  ; C kRk .u; u/kL2 C kR

whence, remembering that Lk  k.P k ra; P k u/kL2 for k  k0 , t ˛ k.P k ra; P k u/.t /kL2 .t ˛ e c0 t k.P k ra; P k u/.0/kL2 Ct ˛

Z

t c0 . t/ 0e



k.rfk ; gk /kL2

 ek .u; a/kL2 CkrukL1 k.P k ra; P k u/kL2 d  CkRk .u; a/kL2 CkRk .u; u/kL2 CkR d

and, thus, multiplying both sides by 2k. 2 1/ , taking the supremum on Œ2; T , and summing up over k  k0 ,  Z t  X h ˛ c0 . t/ k. d2 1/ t . k.ra ; u /k C sup e 2 S d  kt ˛ .ra; u/kh 0 0 k d 1 d 1 2 P 2 2tT e 0 L1 BP 2;1 2;T .B2;1 / kk0 (142)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations def P5

with Sk D

1897

Ski and

iD1

def

Sk1 D k.rfk ; gk /kL2 ;

def

def

Sk2 D kRk .u; a/kL2 ;

Sk3 D kRk .u; u/kL2 ;

def

def

ek .u; a/kL2 ; Sk4 D kR

Sk5 D krukL1 k.P k ra; P k u/kL2 :

To bound the supremum on Œ2; T , we split the integral on Œ0; t into integrals on Œ0; 1 and Œ1; t, respectively. The Œ0; 1 part of the integral is easy to handle: we have X kk0

sup t ˛

Z

2tT

1

d

e c0 . t/ 2k. 2 1/ Sk . / d  

0

X kk0

Z .

1

c0

sup t ˛ e  2 t

Z

2tT

X

1

d

2k. 2 1/ Sk d 

0 d

2k. 2 1/ Sk d  :

0 kk 0

Hence, arguing as in the proof of Theorem 4, X kk0

 Z 1  d t ˛ e c0 . t/ 2k. 2 1/ Sk . / d  . X 2 .1/:

sup 2tT

(143)

0

Let us finally consider the Œ1; t part of the integral for 2  t  T . We shall use repeatedly the following inequality: k ruk 1 d2 . D.t /; P / e Lt .B 2;1

(144)

which is straightforward as regards the high frequencies of u and stems from d

1

d

1

k ruk` . kh i 4 C 2 ruk` . kh i 4 C 2 uk`L1 .BP 1 /  D.t / d d t 2 1 P 2 2;1 P e e L1 . B / L . B / t t 2;1 2;1 for the low frequencies of u. Regarding the contribution of Sk1 , we first notice that, by virtue of (135), X kk0

sup t 2tT

˛

Z

t 1

d

e c0 . t/ 2k. 2 1/ Sk1 . / d  . k ˛ .rf; g/kh d 1 : P 2 e L1 T .B2;1 /

Now, product laws in tilde spaces ensure that ˛1 ak 1 d2 k div uk 1 d2 : k ˛ rf kh d 1 . k 1 2 e e LT .BP 2;1 / LT .BP 2;1 / P e LT .B2;1 /

(145)

1898

R. Danchin

The high frequencies of the first term of the r.h.s. are obviously bounded by D.T /. As for the low frequencies, we notice that if d  4, then for all small enough " > 0, d

d

. k 2 " ak` k 2 " ak` d d 2"  D.T / P 2/ P 2 e L1 . B / L1 T 2;1 T .B2;1

(146)

and if d  5, taking s D 2 in the first term of D.T /, d

d

k 4 C1 ak` . k 4 C1 ak`L1 .BP 2 /  D.T /: d T 2;1 P 2/ e L1 . B T 2;1

(147)

Therefore, using (144) and remembering the definition of ˛, we get 2 k ˛ rf kh d 1 . D .T /: P 2 / e L1 . B T 2;1

Next, we have  X .T /D.T / k ˛ .k.a/rah /k 1 d2 1 . kak 1 d2 k ˛ akh d P / e e LT .BP 2;1 / LT .B P 2 e L1 2;1 T .B2;1 / and, according to (146) and (147), . D 2 .T /: k ˛ .k.a/ra` /k 1 d2 1 . k 1" ak 1 d2 k ˛1C" ak` d P / e e LT .BP 2;1 / LT .B P 2 e L1 2;1 T .B2;1 / We also see that   ˛1 akh k ˛ I .a/Auk 1 d2 1 . k r 2 uk 1 d2 1 k ˛1 ak` : d C k d P e e LT .BP 2;1 / LT .B P 2 P 2 e e L1 L1 2;1 / T .B2;1 / T .B2;1 / The first term of the r.h.s. may be bounded by virtue of (144), and it is also clear that the last term is bounded by D.T /. As for the second one, we use again (146) and (147). Resuming to (145), we end up with X kk0

sup t 2tT

˛

Z

t 1

d

e c0 . t/ 2k. 2 1/ Sk1 . / d  . D 2 .T /:

To bound the term with Sk2 , we use the fact that Z 1

t

e c0 .t/ kRk .u; a/kL2 d   kRk .u;  ˛1 a/kL1 2 t .L /

Z

Hence, thanks to (135) and to (24) (adapted to tilde spaces),

1

t

e c0 . t/  ˛ d  :

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

X kk0

1899

 Z t  X d ˛ c0 . t/ k. d2 1/ 2 sup t e 2 Sk . / d  . 2k. 2 1/ kRk .u;  ˛1 a/kL1 2 T .L /

2tT

1

kk0

. k ruk 1 d2 k ˛1 ak 1 d2 1 : P / e e LT .B LT .BP 2;1 / 2;1 The first term of the r.h.s. may be bounded thanks to (144), and the high frequencies of the last one are obviously bounded by D.T /. To bound k ˛1 ak` d 1 , we P 2 e L1 T .B2;1 / use the following two inequalities: ˛1 k ˛1 ak` ak` d 1 . k d 12" 1 2 P P 2 e L1 LT .B2;1 / / T .B2;1

if d  6;

˛1 ak`L1 .BP 2 / k ˛1 ak` d 1 . k T 2;1 P 2 e L1 T .B2;1 /

if d  7:

Because ˛  1 D d2  12  " if d  6 and ˛  1 D are bounded by D.T /. We eventually get X kk0

sup t ˛ 2tT

Z

t 1

d 4

C 1 if d  7, the r.h.s. above

d

e c0 . t/ 2k. 2 1/ Sk2 . / d  . D 2 .T /:

The terms Sk3 and Sk4 may be treated along the same lines. Finally, using product laws and (135), we get X

sup t ˛

kk0

Z

2tT

t

1

d

e c0 . t/ 2k. 2 1/ Sk5 . / d 

. k ruk 1 d2 k ˛1 .ra; u/kh sup t ˛ d 1 1 2 P / e LT .B P e LT .B2;1 / 2tT 2;1

Z

t

e c0 . t/  ˛ d  . D 2 .T /:

1

Putting all the above inequalities together, we conclude that X kk0

sup 2tT

 Z t  d t ˛ e c0 . t/ 2k. 2 1/ Sk . / d  . D.T /X .T / C D 2 .T /: 1

Then plugging this latter inequality and (143) in (142) yields h 2 2 kh i˛ .ra; u/kh d 1 . k.ra0 ; u0 /k d 1 C X .T / C D .T /: 1 2 2 P P e LT .B2;1 / B2;1

(148)

1900

R. Danchin

Step 3: Decay Estimates with Gain of Regularity for the High Frequencies of r u In order to bound the last term of D.t /, we shall use the fact that the velocity u satisfies def

@t u  Au D F D  .1 C k.a//ra  u  ru  I .a/Au; whence @t .tAu/  A.tAu/ D Au C t AF: Because the maximal regularity estimates for the Lamé semigroup are the same as for the heat semigroup (see (30)), we deduce from Remark 2 below Theorem 1 that ktAukh . kAukh d 1 C ktAF kh d d 3 ; 1 P 2 1 2 P 2 e e Lt .B2;1 / / L1 L1t .BP 2;1 t .B2;1 / whence, using the bounds given by Theorem 5, kt rukh . X .0/ C kF kh d d 1 : P 2 P 2 e e L1 L1 t .B2;1 / t .B2;1 / In order to bound the last term, we notice that, because ˛  1, we have ˛ h k rakh d 1 . kh i ak d : P 2 P 2 e e L1 L1 t .B2;1 / t .B2;1 /

Next, product and composition estimates adapted to tilde spaces give 1

2 2  D 2 .t /; k k.a/rakh d 1 . k ak d P 2 P 2 e e L1 L1 t .B2;1 / t .B2;1 /

as well as k u  rukh d 1 k ruk d d 1 . kuk P 2 P 2 e e L1 L1 P 2 e L1 t .B2;1 / t .B2;1 / t .B2;1 / and 2 k I .a/Aukh d k r uk d 1 : d 1 . kak 1 P 2 2 P 2 e e L . B / L1 P e L1 . B / t t .B2;1 / 2;1 t 2;1

Therefore, resuming to (149) and remembering (144), we get . X .0/ C D.t /X .t / C D 2 .t / C kh i˛ akh kt rukh d d : 1 2 P P 2 e e Lt .B2;1 / L1 t .B2;1 /

(149)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1901

Finally, bounding the last term according to (148) and adding up the obtained inequality to (141) and (148) yield D.t / . D0 C k.ra0 ; u0 /kh d 1 C X 2 .t / C D 2 .t /: 2 BP 2;1

As Theorem 5 ensures that X .t / . X .0/ 1, one can now conclude that (134) is fulfilled for all time if D0 and k.ra0 ; u0 /kh d 1 are small enough. 2 BP 2;1

6

Conclusion

We here presented several results where Fourier analysis techniques and in particular Littlewood-Paley decomposition combined with scaling arguments play a fundamental role in the study of the compressible Navier-Stokes equations. It goes without saying that our approach goes far beyond this particular system. In fact, as regards the well-posedness issue, combining Fourier analysis and scaling invariant framework proves to give essentially optimal results for a number of nonlinear evolutionary systems of parabolic, hyperbolic-parabolic, hyperbolic, or even dispersive type that arise from, e.g., mathematical physics. The reader will find more details and examples in [1]. The main limitation of our approach is that it does not allow to handle boundaries. We mean that for a viscous compressible fluid in a domain ˝ of Rd , Fourier analysis is supposed to give good insights of the evolution inside ˝, far from the boundary, but completely different techniques (that are beyond the scope of that chapter) have to be used to investigate the behavior in the vicinity of the boundary. If ˝ is the half-space, however, we expect Fourier analysis to be still an appropriate tool (see the two recent papers [20] and [23] dedicated to the inhomogeneous incompressible Navier-Stokes equations). Let us finally mention a few questions concerning the qualitative behavior of solutions to (4) that could be treated efficiently by means of Fourier analysis tools. The first one is a more accurate description of long-time asymptotics of global solutions, in the small data case. For example, it should be possible to extend the fine description of Hoff-Zumbrun in [28] (dedicated to the linearized compressible Navier-Stokes equations) to (4) within a suitable critical framework. It would be also interesting to study whether the time decay rates of Theorem 6 are optimal. Indeed, it is well known that for the incompressible Navier-Stokes equations, better time decay may be achieved for initial data satisfying suitable cancelations (see, e.g., [4]). Another interesting issue is the propagation of discontinuous densities along interfaces within a functional framework that ensures uniqueness. In the 2D case, D. Hoff in [27] constructed solutions with density discontinuous along smooth enough interfaces. However, the uniqueness within a class of solutions, where this special structure is not imposed, is an open question.

1902

7

R. Danchin

Cross-References

 Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak

Solutions for the Compressible Navier-Stokes Equations  Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value

Problem  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Global Existence of Classical Solutions and Optimal Decay Rate for Compress-

ible Flows via the Theory of Semigroups  Global Existence of Regular Solutions with Large Oscillations and Vacuum for

Compressible Flows  Large Time Behavior of the Navier-Stokes Flow  Low Mach Number Limits and Acoustic Waves  Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical

Cases

References 1. H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 343 (Springer, Heidelberg/New York, 2011) 2. G.K. Batchelor, An Introduction to Fluids Dynamics (Cambridge University Press, Cambridge, 1967) 3. J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Annales Scientifiques de l’École Normale Supérieure 14(4), 209–246 (1981) 4. L. Brandolese, Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier-Stokes equations. Rev. Mat. Iberoam. 20(1), 223–256 (2004) 5. F. Charve, R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical Lp framework. Arch. Rat. Mech. Anal. 198(1), 233–271 (2010) 6. J.-Y. Chemin, Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel. J. d’Anal. Math. 77, 27–50 (1999) 7. J.-Y. Chemin, N. Lerner, Flot de champs de vecteurs non-lipschitziens et équations de NavierStokes. J. Differ. Equ. 121, 314–328 (1995) 8. Q. Chen, C. Miao, Z. Zhang, Global well-posedness for the compressible Navier-Stokes equations with the highly oscillating initial velocity. Comm. Pure Appl. Math. 63(9), 1173– 1224 (2010) 9. Q. Chen, C. Miao, Z. Zhang, On the ill-posedness of the compressible Navier-Stokes equations in the critical Besov spaces. Rev. Mat. Iberoam. 31(4), 1375–1402 (2015) 10. N. Chikami, R. Danchin, On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces. J. Differ. Equ. 258(10), 3435–3467 (2015) 11. R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141(3), 579–614 (2000) 12. R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases. Commun. Partial Differ. Equ. 26, 1183–1233 (2001) 13. R. Danchin, Global existence in critical spaces for flows of compressible viscous and heatconductive gases. Archiv. Ration. Mech. Anal. 160(1), 1–39 (2001)

35 Fourier Analysis Methods for the Compressible Navier-Stokes Equations

1903

14. R. Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations. Ann. Sci. de l’École Normale Supérieure 35(1), 27–75 (2002) 15. R. Danchin, Zero Mach number limit for compressible flows with periodic boundary conditions. Am. J. Math. 124(6), 1153–1219 (2002) 16. R. Danchin, On the uniqueness in critical spaces for compressible Navier-Stokes equations. NoDEA Nonlinear Differ. Equ. Appl. 12(1), 111–128 (2005) 17. R. Danchin, A Lagrangian approach for the compressible Navier-Stokes equations. Ann. de l’Institut Fourier 64, 753–791 (2014) 18. R. Danchin, B. Ducomet, On a simplified model for radiating flows. J. Evol. Equ. 14, 155–195 (2013) 19. R. Danchin, L. He, The incompressible limit in Lp type critical spaces. Math. Ann. (in press) 20. R. Danchin, P.B. Mucha, A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space. J. Funct. Anal. 256(3), 881–927 (2009) 21. R. Danchin, P.B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density. Commun. Pure. Appl. Math. 65, 1458–1480 (2012) 22. R. Danchin, J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical Lp framework. arXiv:1605.00893 (2016) 23. R. Danchin, P. Zhang, Inhomogeneous Navier-Stokes equations in the half-space, with only bounded density. J. Funct. Anal. 267(7), 2371–2436 (2014) 24. H. Fujita, T. Kato, On the Navier-Stokes initial value problem I. Archiv. Ration. Mech. Anal. 16, 269–315 (1964) 25. B. Haspot, Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces. J. Differ. Equ. 251, 2262–2295 (2011) 26. B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids. Archiv. Ration. Mech. Anal. 202(2), 427–460 (2011) 27. D. Hoff, Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Commun. Pure Appl. Math. 55(11), 1365–1407 (2002) 28. D. Hoff, K. Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves. Z. Angew. Math. Phys. 48(4), 597–614 (1997) 29. P.-L. Lions, Compressible Models. Mathematical Topics in Fluid Dynamics, vol. 2 (Oxford University Press, Oxford, 1998) 30. P.-L. Lions, N. Masmoudi, Une approche locale de la limite incompressible. C. R. Acad. Sci. Paris Sér. I Math. 329(5), 387–392 (1999) 31. A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Uni. 20, 67–104 (1980) 32. J. Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. de la Soc. Math. de France 90, 487–497 (1962) 33. M. Okita, Optimal decay rate for strong solutions in critical spaces to the compressible NavierStokes equations. J. Differ. Equ. 257, 3850–3867 (2014) 34. J. Serrin, On the uniqueness of compressible fluid motions. Archiv. Ration. Mech. Anal. 3, 271–288 (1959) 35. A. Valli, W. Zaja¸czkowski, Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103(2), 259–296 (1986)

Local and Global Existence of Strong Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Maximal Regularity

36

Matthias Kotschote

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notations and Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Stationary Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Main Result: Exponential Stability of Stationary Solutions . . . . . . . . . . . . . . . . . . 3 The Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Euler Coordinates vs. Lagrange Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The “Correct” Fixed Point Mapping for Euler Coordinates . . . . . . . . . . . . . . . . . . 3.3 The Linear Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Nonlinear Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1906 1909 1909 1910 1914 1918 1921 1923 1928 1940 1943 1944 1944

Abstract

The purpose of this contribution is to show how the maximal regularity method can serve to prove existence of strong solutions to the Navier-Stokes equations. In order to illustrate the method, existence and uniqueness of global solutions to the Navier-Stokes equations for compressible fluids with or without heat conductivity in bounded domains shall be proved. The initial data have to be near equilibria that may be nonconstant due to considering large external potential forces. The exponential stability of equilibria in the phase space is shown and, above all, the problem is studied in Eulerian coordinates. The latter seems to be a novelty, since in works by other authors, global strong Lp -solutions have been investigated only in Lagrangian coordinates; Eulerian coordinates are even

M. Kotschote () Department of Mathematics and Statistics, University of Konstanz, Konstanz, Germany e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_50

1905

1906

M. Kotschote

declared as impossible to deal with, cf. on page 418 in Mucha, Zaja¸czkowski (ZAMM 84(6):417–424, 2004). The proof is based on a careful derivation and study of the associated linear problem.

1

Introduction

The purpose of this contribution is to prove existence and uniqueness of global solutions to the Navier-Stokes equations for compressible fluids with or without heat conductivity in bounded domains. The type of solutions that are of interest is strong Lp -solutions, meaning that all terms in the differential equations belong to Lp -spaces regarding time and space. The initial data have to be sufficiently close to the equilibria, and in this case, exponential decay to equilibria can be established. As large external potential forces are allowed, equilibria of the form .0; # 1 ; 1 .x// are possible solutions of the stationary problem, that is, the stationary velocity u1 is zero, the stationary temperature # 1 is a positive constant, while the stationary density 1 may depend on x; the pair .# 1 ; 1 / solves a functional equation that involves pressure, internal energy, and external force. In what follows,   Rn , n  1, denotes a bounded domain and J  R a compact time interval Œ0; T , T > 0. Mathematically, the fluid flows are mappings .u; #; / W J   ! Rn  .0; 1/  .0; 1/: The state variables u, #, and  correspond to velocity, temperature, and density. Apart from effects of viscosity and heat conduction, the physics of the fluid is captured, e.g., in its Helmholtz energy F or internal energy E that are mappings from .0; 1/  .0; 1/ ! R, . ; #/ 7! FL .; #/ D FL .1=; #/ DW F .; #/; L L . ; s/ 7! E.; s/ D E.1=; s/ DW E.; s/: Here s and  D 1= denote the entropy density and the specific volume. The laws of thermodynamics relate these energy potentials through Legendre transforms, L D FL C s#; E

L s D @# F;

L # D @s E:

Due to these relations, one can also think of E as a function of .; #/ according to E D F .; #/  #F# .; #/: The Cauchy stress tensor T is supposed to be given as the sum of the Newtonian viscous stress and the hydrostatic stress, i.e., T D S  I

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1907

with  D @ FL D 2 @ F denoting the thermodynamic pressure and I the unit tensor. Viscous stress S.u/ D 2 D0 .u/ C ru I; D0 .u/ D D.u/  n1 r  u I;

D.u/ D 12 .ru C .ru/T /

and heat flux, ˇr#, are quantified by means of the coefficients , , and ˇ of shear viscosity, bulk viscosity, and thermal conductivity, all three given functions of  and # that satisfy positivity and regularity conditions, ; ; ˇ 2 C2 ..0; 1/2 /;

. /; . /; ˇ. / > 0;

8 2 .0; 1/2 :

(1)

The fluid motion is described by the balance laws for mass, momentum, and energy, @t  C r.u/ D 0;

.t; x/ 2 J  ;

@t .u/ C r.u ˝ u/  rT D fext ./;

.t; x/ 2 J  ;

@t .E/ C r.uE/  rŒˇr# C T  u D fext ./  u;

(2)

.t; x/ 2 J  ;

where E denotes the total energy density 1 E WD E C juj2 : 2 The external force fext is assumed to be a smooth potential, fext .; x/ D r'.; x/; '.; x/  0;

' 2 C3 .RC I H2p ./ \ C1 .//;

8.; x/ 2 RC  ;

'.0; x/ D 0;

(3)

8x 2 :

System (2) is supplemented by the initial conditions u.0; x/ D u0 .x/;

#.0; x/ D #0 .x/;

.0; x/ D 0 .x/;

8x 2 ;

(4)

and boundary conditions. Regarding the latter, two cases are of interest for the velocity u, namely, the no-slip boundary condition, u.t; x/ D 0;

.t; x/ 2 J  0 ;

(5)

that is, the fluid adheres completely to the boundary and the Navier boundary condition .u.t; x/ j .x// D 0;

.t; x/ 2 J  s ;

Q..x//S.u.t; x//  .x/ C ˛Q..x//u.t; x/ D 0;

.t; x/ 2 J  s ;

(6)

1908

M. Kotschote

which admits the fluid to slip along the boundary. In case of ˛ D 0, this boundary P condition reduces to the pure slip boundary condition. Here .a j b/ WD i ai bi denotes the Euclidean inner product and .x/ the outer normal of WD @ at position x 2 , while Q./ WD I  ˝ projects a vector field on its tangential part. The nonnegative constant ˛  0 is an empirical quantity that expresses resistance on the boundary. The compact boundary WD @ of class C 2 decomposes disjointly as P s D d [ P n

D 0 [

(7)

meaning that is the disjoint union of 0 and s and of d and n , where each set is open and closed in and may be empty. Observe that these assumptions imply dist . 0 ; s / > 0 and dist . d ; n / > 0. As for boundary conditions for #, it is considered #.t; x/ D gd .t; x/;

.t; x/ 2 J  d ;

.r#.t; x/ j .x// D gn .t; x/;

.t; x/ 2 J  n :

(8)

Due to these boundary conditions, two different situations are of interest, namely, (i) d 6D ¿ with general data gd , gn ; (ii) d D ¿ and gn D 0 to have energy conserved, i.e., Z

Z

Œ0 E0 C ˆ.0 /.x/ dx DW e0 > 0;

ŒE C ˆ./.t; x/ dx D 

8t 2 RC ;



Z

s

ˆ.s; x/ WD

'.; x/ d ; 0

E0 WD EjtD0 D E.0 ; #0 /.x/ C 12 ju0 .x/j2 : (9)

In view of .u j / D 0 on , the total mass is conserved as well, Z

Z .t; x/ dx D 

0 .x/ dx DW m0 > 0;

8t 2 RC :

(10)



Since in the analysis of (2) the energy F will play an important role, it is introduced ‰.; #; x/ WD F .; #/ C ˆ.; x/ for which strict convexity w.r.t.  and strict concavity w.r.t. # are required, F 2 C3 .R2C /;

@2 ‰.; #; x/ > 0;

8 .; #; x/ 2 .0; 1/2  ;

@# .E.; #//  # @2# F .; #/ > 0;

8 .; #/ 2 .0; 1/2 :

(11)

The latter assumption is needed in order that the heat equation does not degenerate. Observe that in case of '  0, strict convexity of ‰ w.r.t.  is equivalent to strict

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1909

convexity of FL w.r.t.  being a natural assumption in physics. Finally, for the proof of existence and uniqueness of the stationary solution, the following assumptions are needed: F .; 0/ D 0;

8 2 RC ;

F .0; #/ D @ F .0; #/ D 0;

2

The Main Result

2.1

Notations and Function Spaces

8# 2 RC :

(12)

As strong solutions in the Lp -sense are object of interest, the subsequent regularity classes seem to be natural on the basis of the differential equations (2). In fact, .u; #; / is sought in E.J / WD E1 .J /  E2 .J /  E3 .J /, where E1 .J / WD E.J I Rn /;

E2 .J / WD E.J I R/;

E3 .J / WD Z.J / \ Z.J /

with E.J I K/ WD H1p .J I Lp .I K// \ Lp .J I H2p .I K//; K 2 fRn ; Rg; Z.J / WD H1p .J I Lp .// \ Lp .J I H1p .//; Z.J /WD C1 .J I Lp .// \ C.J I H1p .//: If J is a compact time interval, then Z.J / ,! Z.J /. As usual, Hsp stand for the s s , Bpq denote Bessel potential spaces and Wps for the Slobodeckij spaces (Wps  Bpp n Besov spaces); see [48]. Let BUC.R / be the collection of all bounded uniformly continuous functions on Rn equipped with norm kf k1 WD supx2Rn jf .x/j. Let k 2 N, it is then set BUCk .Rn / WD ff 2 C.Rn / W D ˛ f 2 BUC.Rn /; j˛j  kg; similarly the space BUC.RC / is defined. If G is a bounded Lipschitz domain in Rn , then C.G/ is the collection of all continuous functions on G equipped with norm kf k1 WD kf kC.G/ D supx2G jf .x/j. Finally, if 0 2 J and F.J / denotes any function space on J , it is set 0 F.J / WD f 2 F.J / W .0/ D 0g whenever the trace at t D 0 exists, e.g., 0 H1p .J I Lp .//. Furthermore, the following function spaces are needed: X1 .J / WD Lp .J I Lp .I Rn //; X2 .J / WD Lp .J I Lp .//; X3 .J / WD X2 .J / \ C.J I Lp .//; 1=p/ 1=p .J I Lp . I K// \ Lp .J I W2j . I K//; j D 0; 1; Yj .J  I K/ WD W1=2.2j p p

Y1 .J / WD Y0 .J  0 I Rn /; Y2 .J / WD Y0 .J  s I R/  Y1 .J  s I Rn /; Y3 .J / WD Y0 .J  d I R/; Y4 .J / WD Y1 .J  n I R/; .I Rn /  W22=p ./  H1p ./; V WD W22=p p p

1910

M. Kotschote

which are related to base spaces of the partial differential equations, boundary equations, and initial condition. Here p is restricted to p 2 .p; O 1/ with pO D maxfn; 2g to assure the crucial embeddings H1p .I K/ ,! C.I K/;

Z.J / ,! C.J  /;

E.J I K/ ,! Lp .J I C1 .I K// \ C.J I H1p .I K// \ C1=2C 0 .J I Lp .I K//; Y0 .J  I K/ ,! C1=4 .J I Lp . I K// \ Lp .J I C1 . I K// with 0 < 0 < 1=2  1=p. These follow from Sobolev embeddings and .I K//; E.J I K/ ,! Hp .J I H2.1/ p

 2 .0; 1/;

(13) (14)

a consequence of the mixed derivative theorem [37]. The regularity classes are briefly discussed. Due to the interested in strong solutions, i.e., all derivatives are supposed to be in Lp , the choices of E1 .J /, E2 .J /, and Z.J / are obvious. To understand the regularity  2 Z.J /, let u 2 E1 .J / with .u j /j D 0 and  2 Z.J / solve the continuity of (2). Applying r to the continuity equation and testing the resulting equation with rjrjp2 , p  2, lead to the identity Z d 1p kr.t /kLp .IRn / D .1  1=p/ ru.t /jr.t /jp dxkr.t /kLp .IRn / dt  Z 1p .ru.t /  r.t /C.t /rru.t / j r.t // jr.t /jp2 dxkr.t /kLp .IRn / :  

(15) The right-hand side can be estimated by a constant times kru.t /kC.IRnn / kr.t /kLp .IRn / C k.t /kC./ krr  u.t /kLp .IRn / : This estimate, the embeddings (13), and Gronwall’s lemma show r 2 L1 .J I Lp .I Rn //. Moreover, this regularity combined with @t  D r  u  ru shows @t  2 L1 .J I Lp .//. Thus, one can expect to obtain  2 Z.J /. This simple observation will be important for later investigations.

2.2

The Stationary Problem

The stationary problem of (2) is studied. To obtain solutions of the form .u; #; /.x/ D .0; # 1 ; 1 .x//; that is, the velocity field is zero, the temperature is constant, and the density may depend on x, the limit boundary data related to (6) are specified according to

36 Local and Global Existence of Strong Solutions for the Compressible. . .

gd1 2 RC nf0g

and

1911

gn1 D 0:

The stationary problem of (2) then reads as r.u/ D 0;   r  u ˝ u  S.u/ C I C r' D 0;   r Eu  ˇr#  S.u/  u C u C r'  u D 0;

x 2 ; x 2 ;

(16)

x2

with the boundary conditions u D 0; # D gd1 ;

x 2 0 ; x 2 d ;

.u j / D 0; @ # D 0;

Q./S.u/   C ˛u D 0;

x 2 s ;

x 2 n :

(17)

Due to the boundary conditions (5), (6), and (8) of the dynamic problem and conservation of mass (10) and energy (9), the following conditions Z

Z .x/ dx D m0 > 0;

and if d D ¿ W



.E C ˆ/.x/ dx D e0 > 0 

(18)

have to be added. The constants e0 , m0 are given and the same as in (9) and (10). If #.x/  0 for a.a. x 2  is assumed, then it is possible to prove that u.x/ D 0 and #.x/ D const hold for a.a. x 2 . To see this, one first derives the following two identities: Z Z ˇ@ # d  D ˛juj2 d  ;

d

Z

s

ˇ ˇ Z Z ˇ r# ˇ2 1 1 ˇ ˇ ˇˇ dx C ˛juj2 d  D 0: S.u/ W D.u/ dx C 1 # ˇ g d s   #

The first one results from integrating the energy equation over  and using the boundary conditions and the continuity equation, while the second follows from testing the energy equation with 1=#, using the momentum equation, the boundary conditions, the continuity equation, and the first identity. It is noted that S.u/ W D.u/  0 holds for a.a. x 2  in view of the assumption (1), S.u/ W D.u/ D 2.; #/D0 .u/ W D0 .u/ C .; #/jruj2 : Hence, Eqs. (16) and (17) reduce to ‰ ..x/; #; x/ D k; # D gd1 ;

x 2 ; x 2 d

(19)

with some constant k that has to be determined too. Two cases are discussed now.

1912

M. Kotschote

(i) d 6D ¿. In this case, the temperature is uniquely given by gd1 , i.e., # 1 D gd1 is constant and positive. The density can be obtained from (19) and (18). Due the assumption ‰ > 0 (see (11)), a unique function P W RC  R   ! R;

.#; k; x/ 7! P .#; k; x/

exists such that ‰ .P .#; k; x/; #; x/ D k;

8.#; k; x/ 2 RC  R  :

The constant k can be determined by using the first condition of (18), Z m0 D m.k/;

P .# 1 ; k; x/ dx:

m.k/ WD 

The function m W R ! R is smooth due to the regularity assumption on ‰, and, since m0 .k/ D

Z

Pk .# 1 ; k; x/ dx D

Z



‰ .P .# 1 ; k; x/; # 1 ; x/1 dx > 0; 8k 2 R;



the unique solution of the equation m0 D m.k/ is given by k 1 D m1 .m0 /. Hence, the density 1 D P .# 1 ; m1 .m0 /; / is unique as well. As m.0/ D 0 following from (3) and (12) and m0 .k/ > 0, the constant k 1 has to be positive which induces positivity of 1 . The smoothness assumptions (3) and (11) finally imply 1 2 H2p ./ \ C./. (ii) d D ¿. This case is considerably more involved, since the temperature is not given by the Dirichlet boundary data, but has to be determined as well. As before one finds the relation  D P .#; k; /, such that the unknowns k, # satisfy the nonlinear system Z m0 D m.#; k/;

m.#; k/ WD

P .#; k; x/ dx; 

Z e0 D e.#; k/;

e.#; k/ WD 

 Œ.x/E..x/; #/ C ˆ..x/; x/ j.x/DP .#;k;x/ dx (20)

with given constants m0 ; e0 > 0. The functions m; e W RC  R ! R are smooth and satisfy m.#; 0/ D 0

and

e.0; k/  0;

8.#; k/ 2 RC  R:

The first equation of (20) is solved for k. The strict monotonicity

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1913

8.#; k/ 2 .0; 1/  R

mk .#; k/ > 0;

provides the existence of a unique function … W .0; 1/  R ! R, .#; m0 / 7! ….#; m0 /, such that m0 D m.#; ….#; m0 //;

8# 2 RC :

It remains to solve the equation e0 D eO .#/ for #, where eO W RC ! R is defined by eO .#/ WD e.#; ….#; m0 //. The function eO is strictly increasing, since eO 0 .#/ > 0 for all # > 0. To see this, first keep in mind that eO 0 .#/ D e# C ek  …# with Z 

 Z ‰# E#  .E C ˆ/ dx D  #F# # dx e# D ‰   Z Z 2 ‰# ‰ ‰# dx C # dx;  ‰  ‰  Z 1 Z Z ‰  #‰# ‰# 1 ek D dx; …# D dx  dx : ‰   ‰  ‰ The equation ‰ .P .#; k; x/; #; x/ D k (D const) and the inequality Z 

1 ‰# ‰

2 dx

Z  

2 1 ‰# ‰

Z dx 

1 ‰ dx;

which follows from Hölder’s inequality, lead to eO 0 .#/ 

Z 



 #F# # .; #/

 jDP .#;k;x/

dx > 0;

8# 2 .0; 1/:

This yields a unique temperature # 1 D eO 1 .e0 / and constant k 1 D ….Oe 1 .e0 /; m0 /. Positivity of the density 1 D P .# 1 ; k 1 ; / is obtained as before, while # 1 > 0 is a consequence of the normalization F .; 0/ D 0 (see (12)) and (3). In conclusion, the following lemma has been proved. Lemma 1. Let   Rn be a bounded domain. Supposing that (3), (11), (12), and gd1 2 .0; 1/ and m0 ; e0 > 0. Let u  0 in the stationary problem (16), (17), and (18). Then, for all m0 ; e0 > 0, there exists a unique solution w1 D .0; # 1 ; 1 / of (16), (17), and (18) with the following properties: 1. .# 1 ; 1 / satisfies (19); 2. 1 2 H2p ./ \ C./ with 1 .x/ > 0 for all x 2  and # 1 > 0 is constant.

1914

2.3

M. Kotschote

Main Result: Exponential Stability of Stationary Solutions

The main result can be stated now. Theorem 1. Let  be a bounded domain in Rn , n  1, with compact C 2 -boundary

decomposing disjointly according to (7), p 2 .p; O 1/ with pO D maxfn; 2g, ˛  0, and the conditions (1), (3), and (11) be satisfied. Supposing that 1. (a) if d 6D ¿: gd1 2 .0; 1/, gd  gd1 2 Y0 .RC  d /, and gn 2 Y1 .RC  n /; (b) if d D ¿: gn D 0; 2. .u0 ; #0 ; 0 / 2 VC WD f.u; #; / 2 V W #;  > 0 on g; 3. compatibility conditions: u0j 0 D 0;

.u0 j /j s D 0; #0j d D gd jtD0 ;

Q./S.u0 /  j s C ˛u0j s D 0; @ #0 D gnjtD0 :

Then, there exist constants r > 0 and ı0 > 0 such that, if k.u0 ; #0  # 1 ; 0  1 /kV C ke ı .gd  gd1 /kY3 .RC / C ke ı gn kY4 .RC /  r

(21)

with ı 2 Œ0; ı0 / and .0; # 1 ; 1 / denoting the stationary solution of (16), (17), and (18), the system (2), (4), (5), (6), and (8) possesses a unique global solution, i.e., .u; #; / 2 E.Œ0; T / for every T > 0 and e ı .u; #  # 1 ;   1 / 2 E.RC / for all ı < ı0 . In particular, the equilibrium .0; # 1 ; 1 / is exponentially stable in the phase space VC , i.e., lim e ıt k.u.t /; #.t /  # 1 ; .t /  1 /kV D 0:

(22)

t!1

Moreover, the total mass is conserved and if d D ¿ the total energy as well. Remark 1. (i) It should be noted that this main result could also be formulated for different time and space Lebesgue exponents. Such a situation is more flexible to reduce the regularity of the initial data. More precisely, the condition on p is only needed to treat the nonlinear problem, that is, to prove boundedness of nonlinear terms. In case of different Lebesgue exponents, this condition changes to 2=p C n=q < 1, where p 2 .2; 1/ and q 2 .n; 1/ denote the Lebesgue exponents regarding time and space, respectively. This generalization is easily accessible, since the associated linear problem (see proof of Theorem 2) is tackled by methods allowing for different Lebesgue exponents, cf. the results of [7]. More precisely, the phase space V and trace spaces Yj .J  I K/ (see (2.1)) change to n 22=p 1 V D B22=p qp .I R /  Bqp ./  Hq ./; 1=p/ 1=q Yj .J  I K/ D F1=2.2j .J I Lq . I K// \ Lp .J I B2j . I K//; pq qq

j D 0; 1;

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1915

where Fspq stands for the vector-valued Triebel-Lizorkin spaces; see [48] for definition and properties. (ii) The first notable result for local-in-time classical solutions of the compressible Navier-Stokes equations was proved by Serrin [41] for bounded domains and by Nash [34] considering the whole space case  D Rn . In [47] existence and uniqueness of local-in-time Hölder continuous solutions are established for both bounded and unbounded domains. Existence and uniqueness for strong solutions were then proved in case of sufficiently regular initial data and positive initial density as well as global solutions provided that the data are small in some sense, e.g., [5, 6, 15, 18, 20, 30, 31, 43, 49] and references therein. In [50], Valli and Zaja¸czkowski study global existence, stability, and existence of periodic and stationary solutions; they are also able to consider the case of inflow and outflow through the boundary, cf. also [27, 28]. Another important issue concerns nonnegative initial densities 0 , i.e., 0 may vanish in an open subset of the domain or in case of unbounded domains decay at infinity. In [3], cf. also [4, 46], this situation has been studied, and existence and uniqueness of strong solutions have been shown for a large class of initial data. It should be mentioned that their linear problem is strongly related to the linear problem (46), (47); the linear continuity equation in (46) differs slightly and leads to a strong coupling. Moreover, the existence and uniqueness result for positive initial data is very similar to Theorem 2 as well as its proof. Nevertheless, the focus of their work lies on considering initial density with vacuum. As for studying the associated linear problem @t U C AU D F , resulting from linearizing in Lagrangian coordinates, Mucha and Zaja¸czkowski [32, 33] and Ströhmer [44, 45] have studied this problem. Lp -estimates of the linearized system are established which are used to obtain global existence for the nonlinear problem. Ströhmer [45] also proves that A generates an analytic semigroup decaying exponentially as t ! 1 which is strongly related to Theorem 3. Analyticity of the semigroup is also verified by Shibata and Tanaka [42]. There are, of course, many results to the existence of weak solutions as well as existence results for discontinuous initial data, cf. the survey article [8] and the books [35] and [12]. It is therefore only mentioned [14] and [2]. In [14] Feireisl and Pražák use the method of “relative energies” to prove a very general result on stability of steady states in the weak sense, i.e., for every initial data 0 2 L1 ./;

#0 2 L1 ./;

0 u0 2 L1 ./;

0 .x/ > 0; #0 .x/ > 0 a.a. x 2 

there exists a global weak solution .u; #; / that converges to .0; # 1 ; 1 / in the following way: .u; #; /.t / ! .0; # 1 ; 1 / in L5=4 .I R3 /  L4 ./  L5=3 ./ as t ! 1: In contrast to the assumptions on energy, viscosities, and heat conductivity (see (1) and (11)), the authors have to assume more structural hypotheses on these given functions that are needed to prove the existence of global week solutions. Stability

1916

M. Kotschote

regarding of any sufficiently regular solution as well as robustness with respect to variations of the domain have been investigated in [2]. Finally, in [13] it is proved that weak solutions coincide with strong (classical) solutions, emanating from the same initial data, as long as the latter exists. This result is strongly related to the Theorem 1, since it provides the existence and uniqueness of global strong solutions to the Navier-Stokes system (2), of course, provided that the initial data are sufficiently close to the stationary solution. Moreover, applying this principle to the linearized problem and its local existence and uniqueness result Theorem 2 seems to be interesting as well, since the latter result holds for any smooth (large) initial data. (iii) Results to exponential stability of steady states to the Navier-Stokes equations for compressible fluids there are only few, at least in the multidimensional case. In fact, most of them concern the 1D case or start with considering a 3D problem, but then reduce it to a one-dimensional problem by assuming symmetry, cf. [17], [38–40] and references therein. More precisely, considering the 1D case the transition from Euler (x 2 .0; 1/) to Lagrangian coordinates (y) becomes very simple by using the density as transformation, 1 y D y.t; x/ D m0

Z

x

.t; / d : 0

R1 This relation implies y 2 .0; 1/ as y.t; 0/ D 0 and y.t; 1/ D 1=m0 0 .t; / d D 1 due to conservation of mass (10) and the continuity equation transforms to a linear equation, @t   @y u D 0. The multidimensional case with the restriction to barotropic/isothermal fluids can be found in the monograph [36] on the stability of compressible flows, where energy methods have been used to prove exponential decay. This result coincides with the Theorem 1 in the barotropic case and constant coefficients. On the other hand, the strength of Theorem 1 is its generality, i.e., nonconstant coefficients, general pressure laws, different boundary conditions, and heat-conductive fluids can be covered. Finally, in [11] exponential decay to the equilibrium for weak solutions of isentropic fluids is proved, provided that the density is upper bounded. (iv) At last, it is shortly discussed the time decay of solutions in case of unbounded domains. There are a lot of papers regarding existence, stability of constant states, and convergence rates of solutions for both isentropic and heatconductive fluids. Another focus lies on the choice of the domain  itself, such as the whole space Rn and half space RnC , e.g., investigated by Kagei and Kobayashi [19], Li et al. [26] and Kobayashi and Shibata [22], and exterior domains, e.g., in [21]. In all these contexts the time decay is persuaded, e.g., the optimal L2 time-decay rate in three dimension is established as k..t /  1 ; u.t //kL2 .R3 /  C .1 C t /3=4 , while the optimal Lp convergence rate in R3 is established by Duan et al. [10] for the non-isentropic compressible flow as k..t /  1 ; u.t /; #.t /  # 1 /kLp .R3 /  C .1 C t /3=2.11=p/ ;

2  p  6:

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1917

Finally, the same issue has been questioned for compressible fluids with exterior forces, cf. [9]. The proof of the Theorem 1 is based on the contraction mapping principle , i.e., an iteration mapping ƒ associated to (2) has to be devised in such a way that ƒ is a contractive self-mapping. The proof will therefore run as follows: • In Sect. 3 the nonlinear equations (2) are linearized around those steady states that are constructed in Lemma 1, i.e., the deviation .v;  ; %/ D .u  0; #  # 1 ;   1 / from equilibrium .0; # 1 ; 1 / is studied. This approach usually causes a loss of regularity which is due to the linearization of the (hyperbolic) continuity equation that reads as @t % C r.1 v/ D r.%v/ D r%  v  %rv: Considering the right-hand side now as given input, i.e., replacing r .%v/ by r  .%Q v/ Q with given .v; Q %/ Q 2 E1 .RC /  E3 .RC /, the regularity r %Q  vQ 2 Lp .RC I Lp .// is insufficient to establish estimates for the deviation in E.RC /. Many authors therefore evade this problem by considering Lagrangian coordinates to remove this critical term. This issue is deliberately discussed in Sect. 3.1 and related problems are displayed. • The observations of Sect. 3.1 are very fertile, since they bring on an alternative linearization of the continuity equation and thereby the iteration mapping ƒ W E.J / ! E.J /, .vn ; n ; %n / 7! .vnC1 ; nC1 ; %nC1 /, which basically means to solve a certain linear problem (see (37)), will work properly. Still, the globalin-time solvability of the linear problem, in particular wherefrom the exponential decay arises, presents the main challenge for this mapping. The first important step consists in splitting up this linear problem into two sub-problems. While the solution theory and the global-in-time solvability of the first (auxiliary) linear equation is straightforward by using the concept of maximal Lp -regularity, the global-in-time solvability of second linear problem is more involved and represents the key issue. In fact, it directly connects the spectral properties of a certain linear operator with the global solvability of the starting linear problem (37). How to set about this linear equation is described in great detail in Sect. 3.2. Then in Sect. 3.3 the corresponding Theorems 2 and 3 regarding solvability of the linear problems are stated and proved. These results are crucial for both the correctness of the mapping ƒ and the property of self-mapping. • As the hyperbolic continuity equation is involved, a well-known predicament occurring in symmetric quasilinear hyperbolic systems is present, namely, the fact that contraction cannot be proved in the usual setting; here this means to show contraction for ƒ in the regularity class E.J /. One possibility to resolve

1918

M. Kotschote

this difficulty consists in considering contraction in a rougher topology, i.e a suitable larger function space F.J / is needed for which contraction of ƒ can be shown. However, this concept only works whenever closedness of E.J / regarding the topology of F.J / is guaranteed. According to Majda [29], this idea goes back to Kato and Lax. This approach has been already used in [23] to prove local well-posedness of (2) and will be applied here again. • Finally, in Sect. 4 the contraction mapping principle is applied to ƒ as described above providing a unique fix point which is the unique solution of the nonlinear problem (2).

3

The Linearization

This section addresses the linearization of the nonlinear system (2). The deviation w.t; x/ WD .v.t; x/; .t; x/; %.t; x// from the equilibrium w1 .x/ D .0; # 1 ; 1 .x// is given by v.t; x/ WD u.t; x/  0;

.t; x/ WD #.t; x/  # 1 ;

%.t; x/ WD .t; x/  1 .x/:

Here and in the following, the notation H 1 .x/ WD H .# 1 ; 1 .x//, H denoting any coefficient below, is used again. The quasilinear differential operators and nonlinearities are linearized around .0; # 1 ; 1 .x// as usual, H .#; /  H 1 C H # . ; %/;

H 2 f; ; ˇ; Eg;

H .#; /  H 1 C H1 % C H#1  C H # . ; %/; S.u/  S 1 .v/ C S # .v/;

H 2 f; E; 'g;

ˇr#  ˇr  ˇ 1 r C ˇ # . ; %/r ;

S 1 .v/ WD 21 D0 .v/ C 1 rv I;

S # .w/ WD 2# . ; %/D0 .v/ C # . ; %/rv I:

Observe that the remainders, which are defined by the expansions, suffice the conditions H 2 f; ; ˇ; Eg W

H # 2 C1 .R2 /;

H 2 f; Eg W #

1

H # 2 C2 .R2 /;

2

' 2 C .RI Hp .//;

#

H # .0; 0/ D 0;

H # .0; 0/ D 0;

' .0; x/ D 0;

#

@.%;/ H # .0; 0/ D 0;

@% ' .0; x/ D 0;

(23)

8x 2 :

Moreover, keeping in mind that .# 1 ; 1 / solves (19) and # 1 D const, the identity 1 %/ C r.#1  / C r # r.#; /  fext .; x/ D 1 r.‰

C %r.'1 % C ' # / C 1 r' #

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1919

is easily verified. The linearized system for w D .v;  ; %/ then reads 1 1 @t v  rS 1 .v/ C 1 r.‰ %/ C r.#1  / D F1 .w/;

.t; x/ 2 J  ;

1 E#1 @t  C .1 E 1 C ˆ1 / @t %  r.ˇ 1 r /C r.Œ1 E 1 C  1 C 1 ' 1 v/ D F2 .w/; @t % C r.1 v/ D F3 .w/;

.t; x/ 2 J  ; .t; x/ 2 J  ; (24)

which is supplemented by the boundary and initial conditions v D 0; .v j / D 0;

.t; x/ 2 J  0 ;

1

Q./S .v/   C ˛v D gs .w/;

.t; x/ 2 J  s ;

 D gd  # 1 ;

.t; x/ 2 J  d ;

.r j / D gn ;

(25)

.t; x/ 2 J  n ;

w D w0 ;

.t; x/ 2 f0g  :

The right-hand sides F1 , F2 , F3 , gs , and w0 are given by F1 .w/ WD F10 .w/  rF11 .w/  rF12 .w/  %@t v;

F11 .w/ WD S # .v/;

F10 .w/ WD 12 %2 r'1  ' # r1 C %r' #  r 'Q # C .1 C %/rv  v; Z % s@s ' # .s; x/ ds; F12 .w/ WD  # C 12 '1 %2 C 1 ' #  'Q # ; 'Q # D 0

F2 .w/ WD rF21 .w/  @t F22 .w/;

F3 .w/ WD r.%v/;

F21 .w/ WD .E/# v  ˇ # r  ŒS 1 .v/ C S # .v/ W D.v/

(26)

C Œ1 % C #1  C  # v C ' 1 %v C Œ'1 % C ' # .1 C %/v; Z % 1 ' # .s; x/ ds; F22 .w/ WD  2C% jvj2 C .E/# C '1 21 j%j2 C 0

#

w0 WD .v0 ; 0 ; %0 / WD .u0 ; #0  # 1 ; 0  1 /:

gs .w/ WD ˛ 1 C# v;

The linearization requires some remarks. First, the equations have been linearized in such a way that the divergence structure is maintained. A consequence of this approach is that conservation of mass and energy can be recovered for the fixed point mapping. More precisely, conservation of mass (10) and energy (9) along with the assumption (18) yield Z

Z %.t; x/ dx D 0;





Œ.t/E.t /1 E 1 .x/CŒˆ.t /ˆ1 .x/ dx D 0;

t  0: (27)

1920

M. Kotschote

Linearizing the second integrand gives rise to the equivalent relation Z Lw.t / WD 

1 E#1  C .1 E 1 C ˆ1 / % dx D

Z F22 .w/ dx DW „.w/.t /; 

(28)

which can be also obtained by integrating the second equation in (24) and using the condition Lw0 D „.w0 /. Note that this condition corresponds to taking t D 0 in (27), which is exactly satisfied by the stationary solution w1 . This is the reason for maintaining the structure of the energy equation, i.e., F2 D r  F21  @t F22 . Finally, observe that the linear term .1 E 1 C ˆ1 / @t % is of lower order due to the continuity equation in (24). The nonlinear problem (24) and (25) is associated with the abstract equation Lw D .F .w/; G.w/; w0 /;

(29)

where L reflects the linear operator on the left-hand side of (24) and (25), i.e.,   Lw WD M@t w C A.D/w; B.D/w; j0 w ; Mw WD .1 v; 1 E#1  C .1 E 1 C ˆ1 / %; %/T ; 1 %/ C r.#1  /; r.ˇ 1 r / A.D/w WD .rS 1 .v/ C 1 r.‰

C r.Œ1 E 1 C  1 C 1 ' 1 v/; r.1 v//T ; B.D/w WD .B1 .D/v; B2 .D/ /;

j0 w WD wjtD0 D w.0/;

B1 .D/v WD .vj 0 ; .v j /j s ; Q./S 1 .v/  j s C ˛vj s /; B2 .D/ D .j d ; @ j n /; (30) and according to (26) and (25) the nonlinearities F.w/ and G.w/ are given by F.w/ D .F1 .w/; F2 .w/; F3 .w//;

G.w/ WD .0; 0; gs .w/; gd  # 1 ; gn /:

Since exponential decay to equilibria is to be established, the above problem has to be rewritten for wı .t / WD e ıt w.t /, where 0  ı is chosen later. The point is that the spectral bound of the linear operator Ap (see (35)) is negative and ı is chosen smaller than this value. Rewriting (29) for wı leads to Lı wı D .Fı .wı /; Gı .wı /; w0 / where   Lb w WD MŒ@t w C bw C A.D/w; B.D/w; j0 w ; Fb .w/ WD .F1b .w/; F2b .w/; F3b .w//; Fib .w/ WD e bt Fi .e bt w/;

b 2 R;

Gb .w/ WD .0; 0; gsb .w/; gd  # 1 ; gn /;

i D 1; 2; 3;

gsb .w/ WD e bt gs .e bt w/:

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1921

As Fi and gs are genuinely nonlinear functions, the factor e ıt is always absorbed in Fiı and gsı , e.g., F3ı .wı / WD e ıt r .e 2ıt %ı v ı / D r .e ıt %ı v ı /. Moreover, commuting e ıt with the time derivative, the right-hand side of the energy equation can be rewritten as F2ı .w/ D e ıt F2 .e ıt w/ D r  e ıt F21 .e ıt w/  Œ@t  ıI e ıt F22 .e ıt w/: Using these notations, the first natural attempt for an iteration mapping would be ƒ W E.J / ! E.J /; wnC1 solves:

wn D .vn ; n ; %n / 7! ƒ.wn / WD wnC1 ; Lı wnC1 D .Fı .wn /; Gı .wn /; w0 /:

Studying the solvability of this linear problem, it turns out that the set of equations with the resulting regularity of .Fı .wn /; Gı .wn /; w0 / with wn 2 E.J / cannot provide the regularity wnC1 2 E.J /. In fact, there always occurs a regularity loss due to the continuity equation, since the right-hand side F3ı .wn / belongs to X3 .J / only, which is insufficient to find wnC1 2 E.J /. Looking at the starting problem (2) and, above all, the continuity equation involved therein @t  C .r j u/ C ru D 0; .0/ D 0 ;

.t; x/ 2 J  ; x 2 ;

(31)

it is well known that space regularity of  is governed by the operator A.u/ WD .r j u/. More precisely, given u 2 E1 .J / with .u j / D 0 on RC  , one can show by using Kato’s theory, cf. the precise result in [23], that there is a solution operator RŒu depending on u nonlinearly such that the unique solution of (31) is given by  D RŒu.0; 0 / 2 Z.J /: Thus, spatial regularity of  can be gained only from the hyperbolic problem (31) and not from the momentum equation. To cope with this difficulty and above all to get rid of the crucial term .r% j v/ in the continuity equation, many authors use Lagrangian coordinates. In that approach however things become really tedious, if not impossible. For instance, coefficients occurring in the operator L that depend on 1 become time dependent as Lagrangian coordinates involve the time-dependent velocity field. This is the main reason to consider only constant coefficients or constant stationary solutions.

3.1

Euler Coordinates vs. Lagrange Coordinates

In this section the linearization of the continuity equation (31) is considered regarding Euler and Lagrangian coordinates. In what follows, the multidimensional

1922

M. Kotschote

case is studied. The relation between the Eulerian .t; x/ and Lagrangian .t; / coordinates is given by Z t xD C vL .s; / ds D X .t; /; 0

where vL and %L denote velocity and density, respectively, in Lagrangian coordinates, i.e., vL .t; / WD v.t; X .t; //;

%L .t; / WD %.t; X .t; //;

%.t; x/ D .t; x/  1 .x/:

The crucial point of this approach is that the nonconstant equilibria 1 transforms to L1 .t; / WD 1 .X .t; //, thus getting time dependent. Both situations are summarized Euler:

@t %.t; x/ C rx  .1 .x/v.t; x// D FE .t; x/ 2 C.J I Lp .//;

Lagrange:

@t %L .t; / C r  .L1 .t; /vL .t; // D FL .t; / 2 C.J I H1p .//;

where FE WD  .rx % j v/  %rx  v;

FL WD Q.vL ; D/.L1 vL /  %L rvL  vL

and rvL WD Œ@x i @ i  r  CQ.vL ; D/, Q.vL ; D/ denotes a quasilinear firstorder operator. The point is that there must be some hidden regularity contained in the nonlinear problem that has not yet been taken into account. In fact, applying r to the continuity equation and testing it with r%ı jr%ı jp2 and dividing by 2p kr%ı .t /kLp .IRn / result in d 1 kr%ı .t /k2Lp .IRn /  ıkr%ı k2Lp .IRn / C dt 2 2p

hrr.1 v ı .t // ; r%ı .t /jr%ı .t /jp2 ikr%ı .t /kLp .IRn / 2p

D hrF3ı .wı .t // ; r%ı .t /jr%ı .t /jp2 ikr%ı .t /kLp .IRn / ; where Z .f .x/ j g.x// dx;

hf ; gi WD

f; g 2 L2 .I Rn /



denotes the inner product of L2 .I Rn /. The right-hand side belongs now to L1 .RC / \ Lp=2 .RC / for every wı D .v ı ;  ı ; %ı / 2 E.RC /. This fact follows by the same computations as carried out in (15), namely, there holds the estimate ı jF03 .wı /.t /j  e ıt kv ı .t /kH2p .IRn / k%ı .t /k2H1 ./ : p

How to use this property in the linear problem is shown in the next section.

36 Local and Global Existence of Strong Solutions for the Compressible. . .

3.2

1923

The “Correct” Fixed Point Mapping for Euler Coordinates

The fixed point mapping has to be constructed in such a way that the property of the right-hand side F3ı can be used. The continuity equation is therefore linearized as follows: Q DW C.t; v; Q D/% @t %  ı% C r.1 v/ D r.e ıt v%/ where vQ 2 E1 .RC / is given; note that the right-hand side is still linear containing the unknown Q D/% has the structure to get the above estimate R %. Moreover, C.t; v; and fulfills  C.t; v; Q D/% dx D 0 whenever .vQ j / D 0 on RC  . Fı is therefore re-defined according to Q D .F1ı .w/; Q F2ı .w/; Q 0/; Fı .w/

F1ı .w/ Q D e ıt F1 .e ıt w/; Q

Q D r  e ıt F21 .e ıt w/ Q  Œ@t  ıI e ıt F22 .e ıt w/; Q F2ı .w/ Q D .0; 0; gsı .w/; Q gd  # 1 ; gn /; Gı .w/

(32)

gsı .w/ Q D e ıt gs .e ıt w/ Q

with F1 , F21 , F22 , and gs as introduced in (26). If wQ D .v; Q Q ; %/ Q 2 E.RC / with w.0/ Q D w0 and .vQ j / D 0 on RC  , another important property is obtained by integrating F2ı over  and applying the divergence theorem, Z d F2ı .w/.t; Q x/ dx D .t/  ı .t /; 8t  0 (33) dt  with Z

e ıt F22 .e ıt w.t; Q x// dx;

.t/ WD

.0/ D Lw0 ;



2 H1p .RC /:

The condition .0/ D Lw0 means conservation of energy (see (28)), which is needed for the case d D ¿. In conclusion, the mapping ƒ W †R  E.RC / ! E.RC / is re-defined as follows: w Q WD .v; Q Q ; %/ Q 7! ƒ.w/ Q WD w D .v;  ; %/; w solves: Lı w D K.v/%C.F Q Q Gı .w/; Q w0 /; ı .w/;

(34)

K.v/% Q WD .0; 0; C.t; v; Q D/%; 0; 0/ with †R WDfw D .v;  ; %/ 2 E.RC / W wjtD0 D .u0 ; #0  # 1 ; 0  1 /; kwkE.RC /  R; Z .v j / D 0 on RC  ; % dx D 0 8t  0; 

if d D ¿ W .r j / D 0 on RC  g:

1924

M. Kotschote

The constant 0 < R  1 is specified later and will, of course, be chosen sufficiently small. The reason for writing the linear problem as above, i.e., not to include the linear term C.t; v; Q D/% into Lı w, is that the long-time behavior of the linear problem is essentially determined by Lı . Letting (see (30)) Ap WD M1 A.D/ W D.Ap /  Xp ! Xp

(35)

with domain and base space O 1p ./ W B.D/w D 0; if d D ¿ W Lw D 0g; D.Ap / D fw 2 H2p .I Rn / H2p ./ H Z 1 1 O dx D 0g; k%kHO 1 ./ D kr%kLp .IRn / ; Hp ./ WD f 2 Hp ./ W p



Xp WD Xp1  Xp2  Xp3 ;

O 1p ./; Xp1 WD Lp .I Rn /; Xp3 WD H Z dx D 0g; Xp2 WD f 2 Lp ./ W if d D ¿ W 

(36)

where Xp is equipped with norm k.v;  ; %/k2Xp WD kvk2Lp .IRn / C kk2Lp ./ C kr%k2Lp .IRn / ; the spectrum of the linear operator Ap plays a decisive role for the asymptotics of solutions of (34). More precisely, the strategy for solving the linear problem Lı w D K.v/% Q C .f1 ; f2 ; 0; h; w0 /

(37)

with inhomogeneities .f1 ; f2 ; 0/ and h D .h1 ; h21 ; h22 ; h3 ; h4 /, which correspond to Fı .w/ Q and Gı .w/, Q is the following: O solve the linear problems 1. Splitting. First, let wb and w Lbı wb D K.v/% Q b C f b;

f b WD .f1 ; f2b ; 0; h; w0 /;

f2b WD f2 C b ; (38)

O D K.v/ Q %O C .bwb ; 0; 0/  .0; b ; 0; 0; 0/ Lı w

(39)

b O solves (37). Here,  2 Lp ./ denotes some function with Rthen w D w C w  dx D 1 and, since f2 will be assumed to have the property (see (33) and  below)

9 2 H1p .RC /; .0/ D Lw0 W

Z f2 dx D 

d  ı ; dt

(40)

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1925

R d the data f2b satisfies  f2b dx D dt C .b  ı/ . The point is now that wb fulfills d d the identity dt Lwb .t /C.b ı/Lwb .t / D dt .t/C.b ı/ .t/, which is obtained by integrating the energy equation of (38). As .0/ D Lw0 holds, one concludes d Lwb .t / D .t/ for all t  0. Likewise, w O fulfills dt LwıL O w O D b.Lwb  / D 0 and thus Lw.t O / D 0, t  0, in view of Lw.0/ O D 0 resulting from the initial condition w.0/ O D 0. Back to the solution theory of (38). This linear problem can be solved locally in time for every b 2 R and globally for sufficiently large b > 0, i.e., wb 2 E.RC / (see Theorem 2). The second linear problem (39) that can also be written as the abstract Cauchy problem O  ıw O C Ap w O D bwb C .0; b ; C.t; v; Q D/%/; O @t w

t > 0;

w.0/ O D 0;

(41)

possesses at least a unique local solution by Theorem 2, i.e., w O 2 E.Œ0; T / for every T > 0. However the spectral properties of Ap will provide even global bounded solutions. 2. Analyticity of Ap . The linear operator Ap W D.Ap /  Xp ! Xp is the generator of an analytic semigroup TAp .t / and has negative spectral bound, s.Ap / < ı0 < 0 with some ı0 > 0 (see Theorem 3). The same applies to the perturbed operator Aıp WD Ap  ıI wtih D.Aıp / D D.Ap / and s.Aıp / < .ı0 ı/ < 0 as long as ı < ı0 . Since for analytic semigroups the spectral bound and the growth bound 1 coincide, the following relation jTAıp .t /j  M! e !t ;

8t  0

(42)

holds for all ! > s.Aıp / with some M!  1. As s.Aıp / is negative, one can choose ! D "0 with some "0 2 .0; ı0  ı/ in (42), i.e., TAıp .t / is exponentially stable. Moreover, these results also imply that Aıp is "0 -accretive, i.e., 0. Dividing (44) with right-hand side as estimated above by kw.t O /kXp yields d O /kXp  b.kwb .t /kXp C k .t/kXp2 / kw.t O /kXp C "0 kw.t dt C C e ıt kv.t Q /kH2p .IRn / kw.t O /kXp : Gronwall’s lemma gives rise to Z kw.t O /kXp 

0

e "0 .ts/ bkwb .s/kXp C bk .s/kXp2

t

 ds CC e ıs kv.s/k Q k w.s/k O 2 n X Hp .IR / p

and using Young’s inequality and that kvk Q Lp .RC IH2p .IRn //  kwk Q E.RC /  R entail

kwk O L1 .RC IXp /  c."0 ; p  / b.kwb kLp .RC IXp / C k kLp .RC IXp2 / /  C CRkwk O L1 .RC IXp / ;

kwk O Lp .RC IXp /  c."0 ; 1/ b.kwb kLp .RC IXp / C k kLp .RC IXp2 / /  C CRkwk O L1 .RC IXp / with c."0 ; q/ WD ke "0  kLq .RC / . Choosing R smaller than 1=.C c."0 ; p  // boundedness of w O follows,  kwk O L1 .RC IXp /  Œ1  c."0 ; p  /CR1 c."0 ; p  /b kwb kLp .RC IXp /  C kkLp ./ k kLp .RC /   kwk O Lp .RC IXp /  C1 kwb kE.RC / C k kLp .RC / C C2 kwk O L1 .RC IXp /  b   C3 kw kE.RC / C k kLp .RC / with some constants C1 ; C2 ; C3 > 0. Once boundedness of wO in Lp .RC I Xp / has been proved, boundedness of wO in 0 H1p .RC I Xp /\Lp .RC I D.Ap //  E.RC / follows (see page 1939). Moreover, using the embeddings (13), (14) in the estimate k@t %k O C.RC ILp .//  kr.1 v/k O C.RC ILp .// C kb%b kC.RC ILp .// C kr.e ı vQ %/k O C.RC ILp .//  C4 kvk O H1=2 .R p

1 n C IHp .IR //

C C6 kvk Q H1=2 .R p

C C5 k%b kE3 .RC /

1 n C IHp .IR //

k%k O C.RC IHO 1 .// ; p

1928

M. Kotschote

one finally obtains %O 2 Z.RC / and thus %O 2 E3 .RC / and w O 2 E.RC /. 4. Boundedness of w. Since w D wb C w O solves (37) and by the Theorems 2 and 3 (see below) the solutions wb and wO are globally bounded with bounds (49) and (51), the following estimate for w is obtained: kwkE.RC /  kwb kE.RC / C kwk O E.RC /  M0 k.f1 ; f2b ; h; w0 /kX.RC /Y.RC /V C kwk O E.RC /   M0 k.f1 ; f2b ; h; w0 /kX.RC /Y.RC /V CM1 kwb kE.RC / Ck kLp .RC / /    M2 k.f1 ; f2 ; h; w0 /kX.RC /Y.RC /V C k kLp .RC / (45) with some M2 > 0 and function spaces X.RC /, Y.RC / as defined in Theorem 2, in particular there exists a unique solution w 2 E.RC / of (37).

3.3

The Linear Problem

The linear problem La w D K.v/% Q C f b , a 2 R, which is (38) by taking a D b  ı, is now studied. Written out this system in full detail reads 1 1 Œ@t v C av  rS 1 .v/ C 1 r.‰ %/ C r.#1  / D f1 .t; x/; .t; x/ 2 J  ;

1 E#1 Œ@t  C a C .1 E 1 C ˆ1 / Œ@t % C a% r.ˇ 1 r / C r.Œ1 E 1 C  1 C 1 ' 1 v/ D f2b .t; x/; .t; x/ 2 J  ; Q D 0; .t; x/ 2 J   Œ@t % C a% C r.1 v/ C r.e ıt %v/ (46) and v D h1 ; .v j / D h21 ;

Q./S 1 .v/   C ˛v D h22 .t; x/;

.t; x/ 2 J  0 ; .t; x/ 2 J  s ;

 D h3 .t; x/;

.t; x/ 2 J  d ;

@  D h4 .t; x/;

.t; x/ 2 J  n ;

.v;  ; %/ D w0 .x/;

(47)

.t; x/ 2 f0g  :

The idea of proving existence and uniqueness for (46) and (47) is the same as in [23]. First, the continuity equation is solved, i.e., the density is given by 1 % D Ra Œe ıt v.r. Q v/; %0 / with Ra denoting the solution operator of Q D r.1 v/; @t % C a% C r.e ıt v%/ %.0/ D %0 ;

x 2 ;

.t; x/ 2 RC  ;

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1929

which is exactly the third equation of (46) plus initial condition. More precisely, let Q E1 .RC /  R  1 and .vQ j / D 0 on RC  , and %0 2 H1p ./, vQ 2 E1 .RC / with kvk the “inhomogeneity” r.1 v/ in Lp .RC I H1p .// \ C.RC I Lp .//, which is the case for v 2 E1 .RC / and p > 2. Then there is a unique local solution % 2 E3 .J / which is global as long as a > 0 is chosen sufficiently large, i.e., % 2 E3 .RC /. Furthermore, the following estimate k%kE3 .RC /  ma .k%0 kH1p ./ C kvkE1 .RC / /

(48)

holds and the constant ma > 0 tends to zero as a ! 1. Then consider % as Q .1 v/; %0 / in that sub-problem of (46) and (47) the nonlocal term Ra Œe ıt v.r which is obtained by dropping the continuity equation as well as the related initial condition. This system can now be solved by using maximal Lp -regularity. Theorem 2. Let  be a bounded domain in Rn , n  1, with compact C 2 -boundary

decomposing according to (7), J D Œ0; T , p 2 .2; 1/ with p 6D 3, and a 2 R. Assuming that (1), and (a) F 2 C3 .R2C /, ' 2 C2 .RC I H2p ./ \ C1 .//, and ‰.; #/ D F .; #/ C ˆ.; x/, @ ˆ D '; (b) # 1 > 0, 1 2 H2p ./ \ C./ with 1 .x/ > 0 for all x 2 ; (c) vQ 2 E1 .RC / with .vQ j / D 0 on RC  . Then problem (46) and (47) possesses a unique solution wa D .v a ;  a ; %a / 2 E.Œ0; T /, for every T > 0, if and only if the data f1 , f2b , h D .h1 ; h2 ; h3 ; h4 / with h2 D .h21 ; h22 /, and w0 WD .v0 ; 0 ; %0 / satisfy the following conditions: 1. 2. 3. 4.

.f1 ; f2b / 2 X.Œ0; T / WD X1 .Œ0; T /  X2 .Œ0; T /; h 2 Y.Œ0; T / WD Y1 .Œ0; T /  Y2 .Œ0; T /  Y3 .Œ0; T /  Y4 .Œ0; T /; .I Rn /  W22=p ./  H1p ./; w0 2 V WD W22=p p p compatibility conditions: (a) v0j 0 D h1jtD0 in W23=p . 0 I Rn /, .v0 j /j s D h21jtD0 in W23=p . s /, p p 23=p 0j d D h3jtD0 in Wp . d /; (b) if p > 3: Q./S 1 .v0 /  j s C ˛v0j s D h22jtD0 in W13=p . s I Rn /, p 13=p .r0 j /j n D h4jtD0 in Wp . n /.

If additionally .f1 ; f2b / 2 X.RC /, h 2 Y.RC / and a > 0 is chosen sufficiently large, the solution is globally bounded, i.e., wa 2 E.RC / and there exists a constant M0 > 0 such that   kwa kE.RC /  M0 k.f1 ; f2b /kX.RC / C khkY.RC / C kw0 kV :

(49)

R R Moreover, if  %0 .x/ dx D 0 then conservation of mass holds, i.e.,  %a .t; x/ dx D 0 for all t  0. If d D ¿ and h4  0 and f2b D f2 C b fulfilling (40), then Lwa .t / D .t/ holds for all t  0.

1930

M. Kotschote

Proof. The condition p > 2 is needed to ensure the embedding r  v 2 n n H1=2 p .J I Lp .I R // ,! C.J I Lp .I R //, which is a consequence of the embedding (14) with  D 1=2, providing continuity of @t %. In what follows a > 0 is chosen sufficiently large, such that global bounded solutions will be shown. A direct consequence of global well-posedness is the local solvability for every a 2 R. (i) Necessity. Suppose that w D .v;  ; %/ solves (46), and (47) and belongs to E.RC /. This regularity and the differential equations of (46) immediately imply f1 2 X1 .RC / and f2b 2 X2 .RC /. The regularities of the boundary and initial data follow from well-known trace theorems, e.g., cf. [1, 7, 25]. Moreover, as p > 2, the compatibility conditions in 4.(a) hold and, if p > 3, the conditions in 4.(b) as well. The compatibility conditions are obtained by taking trace at t D 0 in the boundary equations. The value p D 3 has been excluded, since the trace theorem does not hold for this critical value. (ii) Sufficiency. As already mentioned above % is considered as Ra Œe ıt v.r Q  1 . v/; %0 / with a > 0 being sufficiently large. Splitting % according to % D % 0 C %1 ;

%0 D Ra Œe ıt v.0; Q %0 /

%1 D Ra Œe ıt v.r Q  .1 v/; 0/;

the system (46), and (47) is associated with M0 Œ@t C a I .v; / C A0 .D/.v;  / D .f1 ; f2b /  B0 %0  B1 v; B.v;  / D h;

.t; x/ 2 J  ; .t; x/ 2 J  ;

.v;  / D .v0 ; 0 /;

.t; x/ 2 f0g  ; (50)

where B1 v WD B.D/%1 ;

B0 %0 WD B.D/%0 ;

1 /; .1 E 1 C ˆ1 / r.e ıt v // Q B.D/ WD .1 r.‰

and A0 .D/.v;  / WD .rS 1 .v/ C r.#1  /; r.ˇ 1 r / C r.Œ1 E 1 C  1 C 1 ' 1 v/  .1 E 1 C ˆ1 / r.1 v//;

M0 .v;  / WD .1 v; 1 E#1  /:

As a direct consequence of [7], cf. also [23, Theorem 2], [24, Theorem 4.2], the linear operator La WD .M0 Œ@t C a I  C A0 .D/; B; jt D0 / with jt D0 denoting the trace operator at time t D 0 is an isomorphism between the regularity class E1 .J /  E2 .J / and the space of data D.J / WD X.J /  Y.J /  W22=p .I Rn /  W22=p ./, i.e., La 2 Li s.E1 .J /  E2 .J /; D.J // with J D Œ0; T  p p

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1931

and every T > 0. Moreover, choosing a > 0 sufficiently large the same holds for J D RC and kL1 a k  Ma with Ma > 0 tending to zero as a ! 1. Hence, the problem (50) can be rewritten as b 1 .v;  / D L1 a .f1 ; f2 ; h; v0 ; 0 /  La .B0 %0 C B1 v; 0; 0; 0/

and by using (48) estimated as follows:

k.v;  /kE1 .RC /E2 .RC /  Ma k..f1 ; f2b /; h; .v0 ; 0 //kD.RC / 

C ma C0 .k%0 kH1p ./ C kvkE1 .RC / /

with some C0 > 0. Setting Ka .v;  / WD L1 a .B1 v; 0; 0/, then Ka W E1 .RC /  E2 .RC / ! E1 .RC /  E2 .RC / is a bounded operator and its norm can be estimated as follows: kKa .v;  /kE1 .RC /E2 .RC /  Ma kB1 vkX1 .RC /X2 .RC /  C0 Ma k%1 kLp .RC IH1p .//  ma C0 Ma kvkE1 .RC /  1=2k.v;  /kE1 .RC /E2 .RC / : The latter inequality can be accomplished by choosing a > 0 large enough, since Ma ma ! 0 as a ! 1. This yields .I C Ka /1 2 B.E1 .RC /  E2 .RC // as well as  .v;  / D .I C Ka /1 L1 .f1 ; f2b ; h; v0 ; 0 /.B0 %0 ; 0; 0; 0/ 2 E1 .RC /E2 .RC /; a t u

which finishes the proof. The next main result concerns global solvability of (41) reading @t w O  ıw O C Ap w O D bwb C .0; b ; C.t; v; Q D/%/; O

t > 0;

w.0/ O D 0; since local solvability is already guaranteed by Theorem 2, i.e., there exists a unique solution w O D .v; O O ; %/ O 2 E.Œ0; T // for every T > 0. The global boundedness of w O will follow from the fact that Ap generates an analytic C0 -semigroup with negative spectral bound and boundedness of w O in Lp .RC I Xp / (see the calculations on pages 1924–1927). In the spectral analysis of Ap conservation of mass and energy will play an important role, which are included in the domain D.Ap / (see (36)). On the Q D/%/ O satisfies other hand, the inhomogeneity bwb C .0; b ; C.t; v; Z 

b%b C C.t; v; Q D/%O dx D 0;

Lwb .t /  .t/ D 0;

t  0;

1932

M. Kotschote

R implying that  %.t; O x/ dx D 0 and Lw.t O / D 0 for all t  0 by integrating the corresponding equations, but this means w.t O / 2 D.Ap / for a.a. t  0. The following crucial theorem can be proved. Theorem 3. Let  be a bounded domain in Rn , n  1, with compact C 2 -boundary

decomposing according to (7), and p 2 Œ2; 1/ with p 6D 3. Assuming that (1), and (11), and the conditions (a), (b) of Theorem 2 are satisfied. Then Ap W D.Ap /  Xp ! Xp generates an analytic C0 -semigroup TAp D fTAp .t /gt0 on Xp and there is ı0 > 0 such that s.Ap / < ı0 , i.e., the spectral bound is negative. Moreover, supposing that (c) wQ 2 †R with 0 < R R 1 being sufficiently small (see proof); (d)  2 Lp ./ satisfies  .x/ dx D 1 and 2 H1p .RC /; R (e) wb 2 E.RC / with  %b .t; x/ dx D 0 for all t  0 and, if d D ¿, Lwb .t / D .t/ for all t  0. O is globally bounded, i.e., w O 2 Then for every ı < ı0 the unique local solution w 1 0 Hp .RC I Xp / \ Lp .RC I D.Ap // and there is M1 > 0 such that   kwk O E.RC /  M1 kwb kLp .RC IXp / C k kLp .RC / :

(51)

Proof. First, it is shown that Ap , p 2 .1; 1/, is sectorial with spectral angle =2C Ap and some Ap 2 .0; =2/ and vertex at some  2 R, i.e., the set †=2C Ap ; D fz 2 Cnfg W j arg.z  /j < =2 C Ap g belongs to the resolvent set .Ap / and the resolvent .I C Ap /1 W Xp ! Xp is bounded, k.I C Ap /1 kB.Xp /  C jj1 ;

8 2 †=2C Ap ; :

(52)

To see this, the following operators are introduced: S v WD 1=1 rS 1 .v/;

S W D.S /  Lp .I Rn / ! Lp .I Rn /;

B WD .1 E#1 /1 r.ˇ 1 r /; 1 H v WD r.‰ r.1 v//

C1  WD .1 /1 r.#1  /;

B W D.B/  Lp ./ ! Lp ./;

H W D.H /  Lp .I Rn / ! Lp .I Rn /; C1 W D.C1 /  Lp ./ ! Lp .I Rn /;

C2 v WD .1 E#1 /1 # 1 #1 r  v;

C2 W D.C2 /  Lp .I Rn / ! Lp ./

with domains D.S / D fv 2 H2p .I Rn / W B1 v D 0g; D.H / D D.S /;

D.B/ D f 2 H22 ./ W B2  D 0g;

D.C1 / D H1p ./;

D.C2 / D H1p .I Rn /:

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1933

The operators S and B are sectorial with †=2C S ;S  .S / and †=2C B ;B  .B/, where S ; B 2 .0; =2/ and S ; B 2 R, and have compact resolvents. Moreover, the operators H and C1 are S -bounded while C2 is B-bounded, i.e., there are positive constants dH , dC1 , dC2 such that kH .I C S /1 kB.Lp .IRn //  dH ;

8 2 †=2C S ;S

kC1 .I C S /1 kB.Lp .IRn //  dC1 jj1=2 ;

8 2 †=2C S ;S ;

kC2 .I C B/1 kB.Lp .IRn //  dC2 jj1=2 ;

8 2 †=2C B ;B :

(53)

Note that C1 and C2 are first-order operators yielding the factor jj1=2 . The energy equation of the resolvent problem .I C Ap /w D f reading as    1 E#1  C .1 E 1 C ˆ1 / % C r.Œ1 E 1 C  1 C 1 ' 1 v  ˇ 1 r / D f2 can be reformulated by using the relations r 1 C 1 r' 1 D 0, .1 /2 rE 1 D . 1  # 1 #1 /r1 , and # 1 D const which results to the equivalent formulation 1 1 E#1   r.ˇ 1 r / C # 1 #1 rv D f2 C # 1 ‰# f3 :

(54)

This simplification and % D  1 .f3  r  .1 v//,  6D 0, lead to the following equivalent system: .I C S /v C H v C C1  D g1 ;  C B C C2 v D g2

(55)

with g D .g1 ; g2 / 2 Yp WD Lp .I Rn /  Lp ./ given by 1 g1 D f1 =1  r.‰ f3 /;

1 g2 D .1 E#1 /1 .f2 C # 1 ‰# f3 /:

(56)

Restricting  2 Cnf0g to the set M D .S / \ .B/ \ .CnBr0 .0//;

r0 WD dH C dC1 dC2 ;

the solution .v;  / of (55) can be represented as  1 v D .I C S /1  1 I C  1 ŒH  C1 .I C B/1 C2 .I C S /1 ˚ g1  C1 .I C B/1 g2 ;  D .I C B/1 g2  .I C B/1 C2 v: Sectoriality of S and B and the estimates (53) yield (52), in particular the resolvent set .Ap / contains the set M. Hence, Ap is the generator of an analytic

1934

M. Kotschote

C0 -semigroup TAp and the same applies to Aıp D Ap C ıI , as ıI is a bounded perturbation. The next purpose concerns a more precise investigation of the resolvent set, which is more involved due to the lack of compactness for the resolvent .I C Ap /1 . Lemma 2. Let Ap W D.Ap /  Xp ! Xp be defined according to (35) and p  2. (a) Every eigenvalue of Ap lies in C D fz 2 C W 0 such that for all  2 p .Ap / with =m 6D 0 there holds m1 < jj < m2 . (c) The resolvent set.Ap / contains the set M [ fz 2 C W ı0 g with some ı0 > 0. Proof of Lemma 2. It will be shown (i) N .I C Ap / D f0g and N .  I C Ap / D f0g for all  2 CC , which already gives (a); (ii) R.Ap / D R.Ap / and, in view of (a), thus Bı1 .0/  .Ap / with some ı1 > 0; (iii) R.I C Ap / D R.I C Ap / for all  2 CC nB"C .0/ and every " > 0, where B"C .0/ WD B" .0/ \ CC . The strategy to prove this statement is the following. As usual let wn D .vn ; n ; %n / 2 D.Ap / be a solution of the resolvent problem .I C Ap /wn D fn , where fn WD .f1n ; f2n ; f3n / 2 Xp converges strongly to some f 2 Xp . Eliminating %n leads to the problem (55) with right-hand side gn ! g in Yp D Lp .I RnC1 /. The left-hand side of (55) defines a closed operator Tp ./ W D.Tp .//  Yp ! Yp with domain D.Tp .// D f.v;  / W H2p .I RnC1 / W .B1 .D/v; B2 .D/ / D 0g. It is possible now to show that for all " > 0 there exists a constant c."/ > 0 such that kTp ./.v;  /kY2  c."/k.v;  /kY2 ;

8.v;  / 2 D.Tp .//; 8 2 CC nB"C .0/: (57)

This inequality and closedness of Tp ./ for all p 2 .1; 1/ imply R.T2 .// D R.T2 .// and then R.Tp .// D R.Tp .// for every p  2. In fact, the inequality (57) for p D 2 implies the convergence xn D .vn ; n / ! x D .v;  / in Y2 with some x 2 Y2 and closedness of T2 ./ shows x 2 D.T2 .// as well as g D T2 ./x. The embedding D.T2 .// ,! Yp , which holds for all p 2 Œ1; 1 as long as n  4, gives rise to x 2 Yp and xn ! x in Yp . Closedness of Tp ./ leads to x 2 D.Tp .// and g D Tp ./x. In case of n > 4 the embedding only 2n holds for p  p 0 D n4 , such that x belongs to D.Tp0 .// only. However, a simple bootstrapping argument can be applied to reach every p  p 0 > 2. Finally, %n given by %n D  1 .f3n r.1 vn // converges to  1 .f3 r.1 v// O p1 ./, since f n ! f3 in H O p1 ./ as well as vn ! v in H2p .I Rn / in H 3 as proved before. Therefore, the inequality (57) has to be established to obtain (iii).

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1935

The claims (i) and (iii) yield CC nB"C .0/  .Ap /. Choosing " < ı1 and using claim (ii) the inclusion CC [ Bı1 .0/  .Ap / follows. Moreover, since also M  .Ap / and the resolvent set is open, there is some ı0 > 0 such that 0, in particular c0 kvk2L2 .IRn /  h1 S v ; vi D

Z

S 1 .v/ W D.v/ dx; 

8v 2 D.S /

(61)

1936

M. Kotschote

and Z Z Z 1 1 1 2 S .v/ W D.v/ dx D 2 D0 .v/ W D0 .v/ C  jrvj dx  1 jrvj2 dx 



 1

 min  x2

.x/krvk2L2 ./

D

c1 krvk2L2 ./ ;

8v 2 D.S /;

since then R jj  j 0 on  and  %.x/ dx D 0 imply % D 0. As for the case d D ¿, conservation of energy Z L.v;  ; %/ WD 

1 E#1  C .1 E 1 C ˆ1 / % dx D 0

has to be taken into account. In fact, v D 0 reduces the momentum equation to 1 %/ C r.#1  / D 0 which is equivalent to 1 r.‰ 1 1 1 ‰ % C ‰#  D c;

x2

with some c 2 R. Here  D const and r‰1 D 0 have been used. Testing this identity by % yields Z 

1



1 .x/‰ .x/j%.x/j2

Z dx C 

1 ‰# .x/%.x/ dx D 0:

(62)

1 1 The identity .1 E 1 C ˆ1 / D ‰1  # 1 ‰# D k 1  # 1 ‰# , k 1 2 R, now implies

Z 0 D  L.v;  ; %/ D  

Z D 

1 .x/E#1 .x/ C .1 .x/E 1 .x/ C ˆ1 .x// %.x/ dx

1 .x/E#1 .x/j j2 dx  # 1

Z 

1 ‰# .x/%.x/ dx;

(63) R where  %.x/ dx D 0 has been used again. Summing up the identities (62) and (63) results in Z Z 1 0D 1 .x/‰ .x/j%.x/j2 dx C # 11 1 .x/E#1 .x/j j2 dx: 



Positivity of # 1 , 1 and assumption (11) lead to % D  D 0 and thus N .Ap / D f0g. In case of  D i r with r 2 Rnf0g, (60) implies again that v D 0 holds. The third equation of (58) implies % D 0 and the second one  D 0. This finishes the proof of (i1).

1938

M. Kotschote

(i2) - N .  I C Ap / D f0g for all  2 CC . One easily verifies that .v;  ; %/ 2 N .I C Ap / means to solve 1 v  rS 1 .v/ C 1 r%  Œ1 .E 1 C ' 1 / C  1 r D 0; 

1

1

E#1 

% C . E

1

1

D 0;

x 2 ;

1 ‰ r.1 v/

D 0;

x2

 r.ˇ r /  1

C ˆ /  

x 2 ;

#1 rv

with .v;  ; %/ 2 D.Ap /  D.Ap / and 1=p C 1=p  D 1. The point is now that one can derive similar relations as for Ap and that solutions are smooth enough. (ii) - closedness of R.Ap /. Let wn D .vn ; n ; %n / 2 D.Ap / be a solution of the resolvent problem with  D 0 and right-hand side fn D .f1n ; f2n ; f3n / 2 Xp , where fn ! f in Xp . By means of the classical Stokes problem the equation O p1 ./ ! r .1 vn / D f3n can be solved, i.e., there is a solution operator R W H 2 n 2 n Hp;B1 .D/ .I R / WD fu 2 Hp .I R / W B1 .D/u D 0g such that vn D 1=1 R.f3n / and therefore vn ! v D 1=1 R.f3 / 2 H2p;B1 .D/ .I Rn /. Solving the second equation leads to n !  2 f 2 H2p ./ W B2 .D/ D 0g, since f2n ! f2 in Lp ./ and vn ! v in H2p;B1 .D/ .I Rn /. The first equation is rewritten to r.‰ 1 %n / D fQ n with right-hand side fQ n D f1 C rS 1 .vn /  r. 1 n / 

1

1

#

O p1 .; d / D f 2 converging to f1 C rS 1 .v/  r.#1  / in Lp .I Rn /. Let H R 1 1 Hp .; d / W  d  D 0g with measure d .x/ D ‰ .x/1 dx. Then r W O p1 .; d / ! Lp .I Rn / is injective and has closed range. Thus, there exists some H 1 1 1 O p1 .I d /. Let % D %Œ‰ O p1 .I d / such that %O n D ‰ %n ! %O in H O   then %O in H 1 O %n converges to % in Hp ./ which finally shows closedness of R.Ap /. (iii) - inequality (57). In case of  6D 0 the resolvent problem is equivalent to (55). This system defines a closed linear operator Tp ./ W D.Tp .//  Yp ! Yp with domain D.Tp .// D f.v;  / 2 H2p .I RnC1 / W .B1 .D/v; B2 .D/ / D 0g, i.e., Tp ./.v;  / D g./ with g D .g1 ; g2 / 2 Yp given by (56). Testing the first equation 1 of (55) with 1 v and the second one with #1  and adding the first result with the complex conjugate of the second one yield p k 1 vk2L2 .IRn / C

Z

S 1 .v/ W D.v/ C



ˇ1 jr j2 #1

dxC

Z q q 2 1 2  1 1 1  E# kL2 ./ C jj2 k ‰ r  . v/kL2 ./ D

 k #1



1 g  1

1  v C #1  g2 dx:

(64) When admitting only  2 C"1 WD f 2 C W 0, the following estimate is obtained by taking the real part of (64), q p 1 k 1 E#1  k2L2 ./ /C "1 .k 1 vk2L2 .IRn / C #1 

Z 

1 S 1 .v/ W D.v/C ˇ#1 jr j2 dx

1 kg1 kL2 .IRn / kvkL2 .IRn / "1

C

1 kg2 kL2 ./ k kL2 ./ : #1

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1939

Young’s inequality, positivity of 1 , E#1 , #1 as well as nonnegativity of the integrals yield the inequality (57) for all  2 C"1 , in particular for "1 > 0 being arbitrarily small. Since also M  .Ap / and the resolvent set is open, the inequality (57) only needs to be shown for  D ˙i r with r 2 ŒR0 ; R1 , 0 < R0 < R1 , to get (iii). Taking again the real part of (64) and applying the inequality (61) gives rise to r 2 c0 kvk2L2 .IRn /  R1 kg1 kL2 .IRn / kvkL2 .IRn / C

R12 kg2 kL2 ./ k kL2 ./ : #1

Considering the imaginary part of (64) and taking into account this estimate, the following inequality can be accomplished: q q 2 1 r.1 v/k2 r 2 c3 kk2L2 ./  #r1 k 1 E#1  k2L2 ./ C k ‰ L2 ./ p 1  r 2 k 1 vk2L2 .IRn / C kg1 kL2 .IRn / kvkL2 .IRn / C r #1 kg2 kL2 ./ k kL2 ./  C .R1 /.kg1 kL2 .IRn / kvkL2 .IRn / C kg2 kL2 ./ k kL2 ./ / with c3 D c2 =# 1 D minx2 1 .x/E#1 .x/=# 1 > 0. These estimates and Young’s inequality finally lead to   R02 Œc0  "1 kvk2L2 .IRn / C Œc3  "1 k k2L2 ./  C .R1 ; "1 /k.g1 ; g2 /k2Y2 ; provided that 0 < "1 < minfc0 ; c3 g. This inequality entails (57), which finishes the proof of the Lemma (2). t u Property (c) of the Lemma (2) implies that s.Aıp /  ı0 < 0 and Theorem 2 guarantees the existence of a unique local solution w O 2 E.Œ0; T / of (41) for every ı 2 R. On the pages 1924–1928 boundedness of w O D .v; O O ; %/ O in Lp .RC I Xp /, i.e., O L .R IL .// ; kr %k O Lp .RC ILp .IRn // < 1; kvk O Lp .RC ILp .IRn // ; kk p C p has been proved, provided that ı < ı0 . To see w O 2 0 H1p .RC I Xp / \ Lp .RC I D.Ap // and the estimate (51), one just studies that sub-problem of (41) for variables .v; O O / which is obtained by reformulating the energy equation as in (54). In fact, @t %O is replaced by means of the continuity equation, all terms containing r %O are put on the right-hand side, the continuity equation is dropped and on both sides the term O is added with b > 0. These series of steps result to the Cauchy O 1 E#1 / b.1 v; problem @t .v; O O / C b.v; O O / C A0p .v; O O / D .q1 ; q2 /; O .v; O /.0/ D0

t > 0;

1940

M. Kotschote

with .q1 ; q2 / 2 Lp .RC I Xp1  Xp2 / and A0p W D.A0p /  Xp1  Xp2 ! Xp1  Xp2 , D.A0p / D f.v;  / 2 H2p .I Rn /  H2p ./ W .B1 .D/v; B2 .D/ / D 0 on g; being generator of an analytic semigroup on Xp1  Xp2 . Choosing b > 0 sufficiently large boundedness of .v; O O / in 0 H1p .RC I Xp1  Xp2 / \ Lp .RC I D.A0p // easily follows, which finishes the proof of Theorem 3. t u

4

The Nonlinear Problem

As already remarked, the existence of a unique fixed point of the mapping ƒ by means of the contraction mapping principle cannot be proved as usual, since contraction of ƒ does not work in the regularity class E.RC /. The strategy is therefore the following: 1. Prove self-mapping of ƒ, i.e., ƒ W †R ! †R with 0 < R < 1=.C c."0 ; p  // specified later (see page 1927 for the first condition on R). The property of selfmapping then induces a bounded sequence fwn g1 nD1  †R which is defined successively via wnC1 D ƒ.wn /, n  1, with some given w1 2 †R . 2. Let J D Œ0; T , T > 0, be a compact time interval. Contraction will be established in F.J / WD F1 .J /  F2 .J /  F3 .J /; for every T > 0, where F1 .J / WD F .J I Rn /;

F2 .J / WD F .J I R/;

F3 .J / WD L2 .J I L2 .//;

1 F .J I K/ WD H1=2 2 .J I L2 .I K// \ L2 .J I H2 .I K//;

K 2 fRn ; Rg:

The restrictions wnjJ of wn 2 †R , n  1, on J belong to †TR WD f jJ W 2 †R g trivially. It is now necessary to know that †TR  F.J / is a closed subset regarding the topology of F.J /. This assertion just follows from the lemma below, which can be found in [23]. Lemma 3. Let X , Y be Banach spaces with Y being reflexive. If the embedding Y ,! X is dense, then the ball Br .0/  Y is closed regarding the topology of X . Observe that E.J / is reflexive, since Ei .J /, i D 1; 2, are reflexive as p 2 .p; O 1/, while reflexivity of E3 .J / WD Z.J / \ Z.J / follows from the fact that E3 .J /  Z.J / is a closed subset of the reflexive Banach space Z.J /. Taking into account the embeddings (13) and (14), the embedding E.J / ,! F.J / is easily verified. 3. These results lead to the convergence wn ! w in †TR and the limit w belongs to †R as well, which is due to the estimate kwkE.J /  lim kwnjJ kE.J /  lim kwn kE.RC /  R; n!1

n!1

8T > 0:

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1941

1. Self-mapping. Let w Q D .v; Q Q ; %/ Q 2 †R be given and assumption (21) be satisfied. Then it is an immediate consequence of the definition of Fı and Gı (see (32) Q Gı .w// Q maps E.RC / to and (26)) and the embeddings (13), that wQ 7! .Fı .w/; X.RC /  Y.RC /, and all compatibility conditions stated in Theorem 2 are fulfilled, in particular F2ı satisfies (40). Moreover, property (23) and the embeddings (13) yield the estimate k.Fı .w/; Q G.w/; Q w0 /kX.RC /Y.RC /V  M3 .R2 C Rr C r/ Q gives rise to with some M3 > 0. Applying the estimate (45) to w D ƒ.w/  kwkE.RC /  M2 k.Fı .w/; Q Gı .w/; Q w0 /kX.RC /Y.RC /V Z  C k e ı F22 .e ı w.; Q x// dxkLp .RC / 

 M2 M3 .R2 C Rr C r/  R; where the last inequality holds by choosing R and r small enough. Thus, selfmapping of ƒ has been established. 2. Contraction. As already mentioned, contraction is shown in the larger space F.J /, and for this approach, the proof of [23, Proposition 4.3] is adopted. First, an (equivalent) exponential weighted L2 -norm w.r.t. the time variable t 2 J is introduced, ˛

L2;˛ .J I K/ WD ff 2 L2 .J I K/ W ke  2  f kL2 .J IK/ < 1g;

˛ WD ˛0 C ı

with ˛0 > 0 chosen later. In what follows, the L2 .J /-spaces occurring in F.J / and W .J / WD L2 .J I L2 .I K//, K 2 fR; Rn ; Rnn g are endowed with this norm. The point is that contraction will follow from smallness of R and 1=˛0 , since nonlinear terms will supply a factor of C  R, while linear terms of lower order will give C =˛0 . Let fwn g1 nD1  †R be the bounded sequence as constructed above, i.e., wnC1 D ƒ.wn /, n  1, with some given w1 2 †R , and let G WD @t with domain D.G/ D 0 H12 .J I L2 .I K//. Then the following inequalities will be crucial for proving contraction: k%j C1  %kC1 kW .J /  kG ˇ G ˇ f kW .J / 

 M4  j C1 kv  v kC1 kF1 .J / C kv j  v k kF1 .J / ; ˛0

C1 C1 kG ˇ f kW .J /  ˇ kf kW .J / ; ˛ˇ ˛0

ˇ 2 Œ0; 1;

G ˇ f 2 W .J /;

kH .wj C1 /  H .wkC1 /kW .J /  C2  Rkwj  wk kF.J / (65)

1942

M. Kotschote

with some constants M4 ; C1 ; C2 > 0 and nonlinear function H satisfying (23). The first inequality can be easily shown by adopting the proof of [23, Lemma 4.3], while the second one is obtained by using complex interpolation (see [48]). The third inequality follows from the mean value theorem and property (23). The contraction inequality results basically from the following theorem. Theorem 4. Let the generic assumptions of Theorem 2 be satisfied, T > 0, and nC1 fwn g1 D ƒ.wn /, n  1, with some given nD1  †R successively defined by w 1 w 2 †R . Then there exist constants M5 ; M6 > 0 independent of T such that 2 X  kv j C1  v kC1 kF1 .J /  M5 kv j C1  v kC1 kW .J / C kF1sı .wj /  F1sı .wk /kW .J / C sD0

kŒ%j %k @t .e ı v j /kW .J / Ck%k e ı Œv j v k kH1=2 .J IL 2

C

2 .//

Cke ı Œv j v k @t %k kW .J /

 1 j C1 1 kv  v kC1 kF1 .J / C kv j  v k kF1 .J / ˛0 ˛0

(66)

and  k j C1  kC1 kF2 .J /  M6 k.v j C1 ;  j C1 /.v kC1 ;  kC1 /kW .J / CRk%j C1 %kC1 kW .J / ı ı C Rkv j  v k kW .J / C kF21 .wj /  F21 .wk /kW .J / C

ıC1 1=2

˛0

ı ı kF22 .wj /  F22 .wk /kW .J /

ı ı C k@% F22 .wj /  @% F22 .wk /kW .J / k@t %j kLp .J ILp .// ı C k@% F22 .wk //kLp .J IC1 .// k1 Œv j  v k  C e ı Œ%j v j 1  %k v k1 kW .J / ı ı C [email protected];/ F22 .wj /  @.v;/ F22 .wk /kW .J / k@t .v j ;  j /kLp .J ILp .IRnC1 // ı C [email protected];/ F22 .wk /Œ.v j ;  j /  .v k ;  k /k

1=2 0 H2 .J IL2 .//

C

 ı kŒ.v j ;  j /  .v k ;  k /@t @.v;/ F22 .wk /kW .J / :

(67)

Proof. The idea of the proof is the same as in [23, Sec. 4.3], where now the stronger assumption fwn g1 t u nD1  †R can be used. Contraction for ƒ is obtained by adding the estimates (66), and (67), and the first one of (65) and then performing the following procedure: • Differences of the form .v j C1 ;  j C1 /  .v kC1 ;  kC1 / are absorbed into the lefthand side by using the estimate

36 Local and Global Existence of Strong Solutions for the Compressible. . .

1943

k.v j C1 ;  j C1 /  .v kC1 ;  kC1 /kW .J / D kG 1=2 G 1=2 Œ.v j C1 ;  j C1 /  .v kC1 ;  kC1 /kW .J / 

C1 1=2 ˛0

kG 1=2 Œ.v j C1 ;  j C1 /  .v kC1 ;  kC1 /kW .J / 

C1 1=2

˛0

k.v j C1 ;  j C1 /  .v kC1 ;  kC1 /kF1 .J /F2 .J /

and choosing ˛0 large enough, while the difference %j C1 %kC1 appearing in (67) is absorbed into the left-hand side by choosing R small enough. • Differences of nonlinear functions only occur regarding the W .J I E/-norm and can be estimated by using (65). • In (66) and (67) there are some H1=2 2 .J I L2 .//-norms which are treated as follows:   C kmk1 k.v j ;  k /.v k ;  k /kH1=2 .J IL .// 2 2 2  k.v j ;  j /  .v k ;  k /kL2 .J IL 2p .//  CRkwj  wk kF.J / ;

kmŒ.v j ;  j /.v k ;  k /kH1=2 .J IL C kmk

1

C 2 C 0 .J ILp .//

2 .//

p2

where 0 < 0 < 1=2  1=p and m D m.wk / denotes some multiplicator having the regularity as needed in the estimate. The first inequality can be found, e.g., in [24, Lemma 2.3], while the second unequal sign results from wk 2 †R . In conclusion, there is some M7 > 0 and sufficiently small 1=˛0 ,R > 0 such that kwj C1  wkC1 kF.J /  M7 .R C 1=˛0 /kwj  wk kF.J /  1=2kwj  wk kF.J / : The contraction mapping principle then gives rise to a unique fixed point e ı w 2 E.RC / of (34). Finally, trace theory yields e ı w 2 E.RC / ,! BUC0 .RC I V/ leading to exponential stability of the equilibrium .0; # 1 ; 1 / in the phase space V (see (22)). The proof of Theorem 1 is complete.

5

Conclusion

The stability issue of stationary solutions for the Navier-Stokes-Fourier equations for compressible fluids was treated by using Lagrangian coordinates yet, at least the problem is considered in the strong Lp -sense. Many existing difficulties can be overcome by employing the Lagrangian approach, but then others are created when nonconstant stationary solutions are considered. Such solutions arise, e.g., by considering large external potential forces.

1944

M. Kotschote

The main contribution of this chapter exactly consists in dealing stability for nonconstant stationary solutions in the framework of Eulerian coordinates; the latter feature seemed to be even an open problem and has been declared as an impossibility to deal with, cf. on p. 418 in [33]. Once one has understood how to correctly linearize the hyperbolic conservation of mass equation and to work with it, the proof of exponential stability for stationary solutions becomes realizable, since the spectral bound of the associated linear operator can be directly related to exponential stability of equilibria for the starting (nonlinear) problem. Finally, the method proposed here opens up a possibility to deal with more complicated situations, e.g., stability of nonconstant stationary solutions for compressible magnetofluiddynamic systems or compressible two-phase fluids as modeled in [16].

6

Cross-References

 Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak

Solutions for the Compressible Navier-Stokes Equations  Concepts of Solutions in the Thermodynamics of Compressible Fluids  Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows  Existence of Stationary Weak Solutions for Isentropic and Isothermal Compress-

ible Flows  Global Existence of Classical Solutions and Optimal Decay Rate for Compress-

ible Flows via the Theory of Semigroups  Well-Posedness and Asymptotic Behavior for Compressible Flows in One Dimen-

sion

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10. R. Duan, S. Ukai, T. Yang, H.-J. Zhao, Optimal convergence rates for the compressible NavierStokes equations with potential forces. Math. Models Methods Appl. Sci. 17, 737–758 (2007) 11. D. Fang, R. Zi, T. Zhang, Decay estimates for isentropic compressible Navier-Stokes equations in bounded domain. J. Math. Anal. Appl. 386(2), 939–947 (2012) 12. E. Feireisl, Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and Its Applications (Oxford University Press, Oxford/New York, 2004) 13. E. Feireisl, A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012) 14. E. Feireisl, D. Pražák, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics. AIMS Series on Applied Mathematics (American Institute of Mathematical Sciences, Springfield, 2010) 15. W. Fiszdon, W.M. Zaja¸czkowski, Existence and uniqueness of solutions of the initial-boundary value problem for the flow of a barotropic viscous fluid, local in time. Arch. Mech. (Arch. Mech. Stos.) 35(4), 497–516 (1984) 16. H. Freistühler, M. Kotschote, Phase-field and Korteweg-type models for the time-dependent flow of compressible two-phase fluids. Arch. Ration. Mech. Anal. 224(1), 1–20 (2017) 17. L. Huang, D. Nie, Exponential stability for a one-dimensional compressible viscous micropolar fluid. Math. Methods Appl. Sci. 38(18), 5197–5206 (2015) 18. N. Itaya, On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluids. Kodai Math. Sem. Rep. 23, 60–120 (1971) 19. Y. Kagei, T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space. Arch. Ration. Mech. Anal. 177, 231–330 (2005) 20. M. Kawashita, On global solutions of Cauchy problems for compressible Navier-Stokes equations. Nonlinear Anal. 48, 1087–1105 (2002) 21. T. Kobayashi, Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3 . Commun. Math. Phys. 200(3), 621–659 (1999) 22. T. Kobayashi, Y. Shibata, Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations. Pac. J. Math. 207(1), 199–234 (2002) 23. M. Kotschote, Strong solutions to the compressible non-isothermal Navier-Stokes equations. AMSA 22, 319–347 (2012) 24. M. Kotschote, R. Zacher, Strong solutions in the dynamical theory of compressible fluid mixtures. Math. Models Methods Appl. Sci. 25(7), 1217–1256 (2015) 25. O.A. Ladyzenskaya, V.A. Solonikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs (American Mathematical Society, Providence, 1968) 26. H.-L. Li, A. Matsumura, G. Zhang, Optimal decay rate of the compressible Navier-StokesPoisson system in R3 . Arch. Ration. Mech. Anal. 196(2), 681–713 (2010) 27. G. Lukaszewicz, An existence theorem for compressible viscous and heat conducting fluids. Math. Methods Appl. Sci. 6, 234–247 (1984) 28. G. Lukaszewicz, On the first initial-boundary value problem for the equations of motion of viscous and heat conducting gas. Arch. Mech. 36, 234–247 (1984) 29. A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (Springer, New York/Heidelberg, 1984) 30. A. Matsumura, T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Jpn. Acad. Ser. A 55, 337–342 (1979) 31. A. Matsumura, T. Nishida, The initial value problems for the equations of motion of compressible viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980) 32. P.B. Mucha, W.M. Zaja¸czkowski, On a Lp -estimate for the linearized compressible NavierStokes equations with the Dirichlet boundary conditions. J. Differ. Equ. 186, 377–393 (2002) 33. P.B. Mucha, W.M. Zaja¸czkowski, Global existence of solutions of the Dirichlet problem for the compressible Navier-Stokes equations. ZAMM 84(6), 417–424 (2004) 34. J. Nash, Le Problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962)

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35. A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and its Applications, vol. 27 (Oxford University Press, Oxford, 2004) 36. M. Padula, Asymptotic Stability of Steady Compressible Fluids (Springer, Heidelberg, 2010) 37. J. Prüss, Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in Lp -spaces. Math. Bohem. 127(2), 311–327 (2002) 38. Y. Qin, Exponential stability for the compressible Navier-Stokes equations with the cylinder symmetry in R3 . Nonlinear Anal. Real World Appl. 11(5), 3590–3607 (2010) 39. Y. Qin, L. Jiang, Global existence and exponential stability of solutions in H 4 for the compressible Navier-Stokes equations with the cylinder symmetry. J. Differ. Equ. 249(6), 1353–1384 (2010) 40. Y. Qin, J. Zhang, X. Su, J. Cao, Global existence and exponential stability of spherically symmetric solutions to a compressible combustion radiative and reactive gas. J. Math. Fluid Mech. 18(3), 415–461 (2016) 41. J. Serrin, On the uniqueness of compressible fluid motions. Arch. Ration. Mech. Anal. 3, 271–288 (1959) 42. Y. Shibata, K. Tanaka, On a resolvent problem for the linearized system from the dynamical system describing the compressible viscous fluid motion. Math. Methods Appl. Sci. 27, 1579–1606 (2004) 43. V.A. Solonnikov, The solvability of the initial-boundary value problem for the equations of motion of a viscous compressible fluid (Russian). Zap. Nau. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 56, 128–142 (1976). Investigations on linear operators and theory of functions, VI 44. G. Ströhmer, About the resolvent of an operator from fluid dynamics. Math. Z. 194, 183–191 (1987) 45. G. Ströhmer, About compressible viscous fluid flow in a bounded region. Pac. J. Math. 143, 359–375 (2002) 46. Z. Tan, Y. Zhang, Strong solutions of the coupled Navier-Stokes-Poisson equations for isentropic compressible fluids. Acta Math. Sci. Ser. B Engl. Ed. 30(4), 1280–290 (2010) 47. A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion. Publ. Res. Inst. Math. Sci. 13, 193–253 (1977) 48. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam, 1978) 49. A. Valli, An existence theorem for compressible viscous fluids. Ann. Mat. Pura Appl. (IV) 13(132), 197–213, 399–400 (1982) 50. A. Valli, W.M. Zaja¸czkowski, Navier-Stokes equations for compressible fluids; Global existence and qualitative properties of solutions in the general case. Commun. Math. Phys. 103, 259–296 (1986)

Local and Global Solvability of Free Boundary Problems for the Compressible Navier-Stokes Equations Near Equilibria

37

Irina Vladimirovna Denisova and Vsevolod Alexeevich Solonnikov

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linear Problem in a Half-Space for a Single Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Homogeneous System (15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Solvability of the Linear Problem in a Bounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . 4 Local-in-Time Solvability of Nonlinear Problem (12) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Problem on the Motion of Two Compressible Fluids Separated by a Closed Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction and Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Local Solvability in the Sobolev-Slobodetskiˇı Spaces . . . . . . . . . . . . . . . . . . . . . . . 5.3 Model Problem with a Plane Interface Between the Liquids . . . . . . . . . . . . . . . . . . 5.4 Local Solvability in Weighted Hölder Spaces for the Problem with Closed Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Evolution of Compressible and Incompressible Fluids Separated by a Closed Free Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Local Existence Theorem, the Case of Positive Surface Tension . . . . . . . . . . . . . . 6.2 Model Problem with a Plane Interface Between the Fluids . . . . . . . . . . . . . . . . . . . 6.3 Problem (105), (106), (4) in a Bounded Domain with  D 0 . . . . . . . . . . . . . . . . . 6.4 Global Solvability of Problem (105), (106), (4) Without Surface Tension . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1948 1956 1957 1963 1965 1976 1986 1986 1988 1990 1996 1998 1998 2002 2006 2009 2031 2032 2032

I.V. Denisova Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg, Russia e-mail: [email protected]; [email protected] V.A. Solonnikov Laboratory of Mathematical Physics, St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia e-mail: [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_51

1947

1948

I.V. Denisova and V.A. Solonnikov

Abstract

The chapter is concerned with free boundary and interface problems for equations governing viscous compressible flow. The main difficulty of such problems is due to the fact that the surface of the fluid is unknown. A proof of the classical solvability is outlined of the problem on the motion of a drop in vacuum in a finite time interval both in the case of the presence of surface tension on the free boundary and without it. The motion of two compressible fluids and fluids of different types, compressible and incompressible, separated by an unknown interface is also studied. For the latter problem, the global-in-time solvability l;l=2 in the Sobolev-Slobodetskiˇı spaces W2 is proved in the case where surface tension is not taken into account and the data are small. The basic tools of analysis of free boundary problems are the passage to Lagrangian coordinates, the Fourier-Laplace transform, and the Plancherel theorem. An exponential energy inequality is also obtained; it is applied to show global existence and exponential decay of a solution in the Sobolev-Slobodetskiˇı spaces. In addition, some results of potential theory are used in studying Hölder continuous solutions.

1

Introduction

This chapter deals with an unsteady motion of a finite volume of a compressible fluid in vacuum or in another liquid medium. The liquid is assumed to be barotropic. Along with the velocity vector field and the density, the boundary of the fluid (or a part of it) is unknown and must be defined in solving process. For this reason the mathematical analysis of these problems started only a few decades ago. One of the first papers on this topic was the article of Secchi and Valli [27], where the local (in time) solvability of a free boundary problem for a single compressible viscous heat-conducting fluid without surface tension was proved in the anisotropic Sobolev space W24;2 . Tani considered similar problems for one and two fluids in [45, 46]. Next, Secchi [25, 26] analyzed the problem with radiation effect and self-gravitation forces, but without external pressure, also in the Sobolev spaces. For the temperature classical boundary conditions (Neumann or of the third type) were stated [25]. The Lagrangian coordinates were employed to transform this free boundary problem into a fixed boundary one. The local-intime existence of a solution is shown by linearization, followed by a (Schauder) fixed-point argument. The case where the capillary forces on the free boundary were taken into account was first studied by Solonnikov and Tani [42,43]. It is essentially more difficult to investigate this case because surface tension generates noncoercive boundary conditions. The authors established the local-in-time well-posedness 2Cl;1Cl=2 for this problem in the Sobolev-Slobodetskiˇı spaces W2 ; l > 1=2; by adapting the technique developed for an incompressible fluid [35, 36] to the case of compressible one [42]. In a similar way, a global existence theorem was proved for the problem near a rest state in [43]. Another result concerning global solvability was obtained by Zajaczkowski [48, 49]. The stability of the equilibrium state of

37 Local and Global Solvability of Free Boundary Problems for the. . .

1949

a compressible barotropic fluid without and with surface tension was established. This required a detailed analysis of the problem on an infinite time interval. In [48], velocity vector field belonged to W24;2 . Zadrzynska and Zajaczkowski [47] obtained also the existence of local and global solutions in the Sobolev spaces with lower regularity. In the series of papers of Shibata and his collaborators [15, 17, 18], solvability theory was developed for problems of hydrodynamics of viscous fluids, both compressible and incompressible, in anisotropic Sobolev spaces with different orders of summability with respect to the spatial and time variables. The proof of solvability of these problems was based on coercive a priori estimates of solutions in the spaces abovementioned; in turn, the estimates were deduced from the Weis theorem on the operator-valued Fourier Lp -multipliers. In particular, the problems governing the motion of two fluids separated by a free interface in a container were studied for the compressible liquids and those of different types: one compressible and another incompressible. For such problems, local existence theorems were established in [17,18], where the extra restrictions for viscosity coefficients imposed in the preceding papers by Denisova [5, 6] were removed. A model problem for a two-phase liquid with phase transition was investigated by Shibata [28]. In addition, local and global well-posedness in the class of maximal Lp  Lq -regularity was stated for the problem on single compressible fluid motion without capillary force action [29, 30]. In the present chapter, the demonstration of a similar local result is given in Hölder classes of functions (Sects. 3 and 4). It was first published in [14]. The proof is based on the solvability of a linear half-space problem on an arbitrary finite time interval (Sect. 2) which was established in [13]. These results on local solvability were extended also to the case of the motion of a two-phase compressible fluid [5, 8] (Sect. 5) and to the case of the motion of two fluids of different types: compressible and incompressible, in the Sobolev-Slobodetskiˇı spaces (Sect. 6) [6]. In Sect. 6.4, global solvability of the latter problem is obtained for the case of zero surface tension. The basic ideas of the proofs are the passage to Lagrangian coordinates from Eulerian ones and the application of fixed-point theorem. In Lagrangian coordinates, the free boundary problem for the Navier-Stokes system becomes a nonlinear problem in a fixed domain. Fixed-point argument provides the existence of a local solution to the nonlinear problem. It is based on the unique solvability of the corresponding linear boundary value problem, which is solved by constructing a regularizer and using maximal regularity estimates of the solution (Sect. 3). Finally, these estimates are obtained by analyzing explicit solution to the half-space problem in the dual Fourier-Laplace space (Sect. 2). There is some analogy with the case of an incompressible fluid. Comparing the cases of compressible and incompressible fluids, one can say that the case of a compressible fluid is simpler to investigate in the following sense. Since the liquid is assumed to be barotropic, the passage to Lagrangian coordinates enables one to exclude the density from the system. The resulting initial–boundary value problem involves a system only for the fluid velocity as an unknown, and this

1950

I.V. Denisova and V.A. Solonnikov

system is parabolic in the sense of Petrovskiˇı. It is easier for studying than the Stokes system. Of course, if the surface tension is taken into account, even the linear problem has noncoercive boundary conditions, but in this case, the operators obtained coincide with those yet investigated in the incompressible case. Thus, the absence of unknown pressure function simplifies the analysis of the problem “in the small.” There is an opposite situation with the global solvability of the problem. The resolvent to boundary problem operator (it is described at the end of the chapter in Remark 11) is not completely continuous, which is one of the main difficulties in the analysis of the compressible fluids. In view of the last remark, it has been necessary to find a new method for proving the global solvability of the nonlinear problem. This is done for the interface problem governing the evolution of two fluids of different types with small initial data when mass forces vanish. The method is based on the global solvability of a linear problem and an exponential decay of the solution at infinity with respect to time. The latter is proved by using exponential estimates of a generalized energy. This idea was proposed by Padula [22]. A similar result for a two-phase compressible fluid was recently published by one of the authors [40]. The problem governing the motion of a single fluid of finite volume can be stated as follows. At the initial instant t D 0, the liquid is assumed to fill a known domain ˝0  R3 . The boundary @˝0 of this domain is denoted by  . For every t > 0; it is required to find the free surface t D @˝t , the velocity vector field v.x; t / D .v1 ; v2 ; v3 /, and the density function .x; t / > 0 of the fluid that solve the following free boundary problem for the Navier-Stokes system for a compressible fluid: .Dt v C .v  r/v/  r  T D f;

Dt  C r  .v/ D 0

in ˝t ; t > 0;

vjtD0 D v0 .x/; jtD0 D 0 .x/; x 2 ˝0 ; Tnjt D  H n  pe n on t :

(1) (2) (3)

Here Dt D @=@t, T is the stress tensor for a compressible fluid given by the formula T D .p./ C 1 r  v/I C S.v/;

.S.v//ij D

@vj @vi C ; @xj @xi

i; j D 1; 2; 3;

I is identity matrix;  is the dynamical viscosity coefficient; 1 is a constant such that 23  C 1 > 0, p./ is liquid pressure which is a known smooth function of the density; f is the vector field of exterior forces; pe D pe .x; t / is the exterior pressure function defined for x 2 R3 ; t > 0; v0 and 0 are the initial values of the fluid velocity and density, respectively; n is the unit outward normal vector to ˝t ;  > 0 is the surface tension coefficient, and H .x; t / is the doubled mean curvature of t ( H < 0 at the points where t is convex); r  T means the vector with @Tij components .r  T/j D , j D 1; 2; 3. Henceforth the summation is implied @xi

37 Local and Global Solvability of Free Boundary Problems for the. . .

1951

from 1 to 3 or from 1 to 2 with respect to repeated indices denoted by the Latin or Greek letters, respectively. A Cartesian coordinate system is assumed to be fixed in R3 ; the centered dot means the Cartesian scalar product. The vectors are denoted by boldface letters. In order to exclude the loss of mass through the free surface, the boundary t is assumed to be formed by the points x.y; t / the radius vector x.y; t / of which is a solution of the Cauchy problem: Dt x D v.x.y; t /; t /; x.y; 0/ D y; y 2 ;

(4)

so that t D fx.y; t /jy 2  g and ˝t D fx.y; t /jy 2 ˝0 g. Condition (4) completes system (1), (2), and (3), allowing one to get rid of the unknown boundary by transforming the Eulerian coordinates into the Lagrangian ones in accordance with the formula: Z t x.y; t / D y C u.y;  / d  Xu .y; t /; (5) 0

where u.y; t / D v.x.y; t /; t / is the velocity vector field in the Lagrangian coordinates. The Jacobian Ju .y; t / D detfaij g3j D1 of transformation (5), where aij .y; t / D ıji C

Z

t

0

@ui d ; @yj

is a solution of the Cauchy problem: Dt Ju .y; t / 

@aij @ui  Ju .y; t /.r  vjxDXu /; Aij D Aij @t @yj

Ju .y; 0/ D 1;

(6)

where ıji are the Kronecker symbols, Aij are the cofactors of the elements aij of the Jacobi matrix, and A D fAij g3i;j D1 . Integrating (6), one obtains Z Ju .y; t / D 1 C

t

Ar  u d

(7)

0

and Z



t

Ju .y; t / D exp

r  vjxDXu d 0

 where ru 

@yl @ @xk @yl

3 kD1

D Ju1 Ar.

Z  exp 0

t

 ru  u d ;

(8)

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I.V. Denisova and V.A. Solonnikov

The transformation (5) puts the second equation in (1) in the form: Dt O C r O u  u D 0; whence, using (8), one can deduce the representation of the density in the Lagrangian coordinates: Z



t

.y; O t / D 0 .y/ exp  0

 ru  u d D 0 .y/Ju1 .y; t /:

(9)

The substitution of this expression into the first equation in (1) eliminates the density. The vector H n can be expressed by a well-known formula: H n D .t /x  .t /Xu ;

(10)

where .t / is the Laplace-Beltrami operator on t . It is convenient to split vectorvalued dynamic boundary condition (3) into the tangential and normal components. This can be done by projecting this condition first to the tangent plane of t and then to that of  via the projection operators ˘ and ˘0 , respectively. The outward normal n0 to  and the normal n to  are related to each other by the formula n.Xu / D

Ju1 An0 An0 .y/ D : jJu1 An0 j jAn0 .y/j

(11)

The result of the above transformations is a system equivalent to (1), (2), (3) and (4) if n  n0 > 0, namely,  Dt u  01 .y/Ar  T0u .u/ D f  01 .y/Arp 0 Ju1

in ˝0 ; t > 0;

ujtD0 D v0 ; ˘0 ˘ Su .u/nj D 0; ˇ ˇ ˚

ˇ n0  T0u .u/nˇ  n0  .t /Xu ˇ D .n0  n/ p.0 Ju1 /  pe .Xu ; t / ˇ ;

(12)

where T0u .w/

D .1 ru  w/I C Su .w/;

.Su .w//ij D

Ju1



@wj @wi Aik C Aj k @yk @yk

 ; (13)

˘0 ! D !  .n0  !/n0 ;

˘ ! D !  .n  !/n:

Although the coefficients in (12) are more complicated than those in (1), (2), and (3), system (12) has an advantage of being an initial–boundary value problem for a single unknown vector-valued function u in the known domain ˝0 . As has been mentioned above, problem (12) will be studied in the Hölder spaces C ˛;˛=2 .

37 Local and Global Solvability of Free Boundary Problems for the. . .

1953

The definition of the Hölder and Sobolev-Slobodetskiˇı spaces is as follows. Let ˝ be a domain in Rn , n 2 N; T > 0; QT D ˝  .0; T /, and let ˛ 2 .0; 1/. The Hölder space C ˛;˛=2 .QT / is the class of functions defined in QT and having finite norm: .˛;˛=2/

jf jQT

.˛;˛=2/

D jf jQT C hf iQT

;

where jf jQT D

.˛;˛=2/

sup jf .x; t/j;

hf iQT

.x;t/2QT .˛/

.˛/

.˛=2/

D hf ix;QT C hf it;QT ;

sup jf .x; t/  f .y; t/jjx  yj˛ ;

hf ix;QT D sup

x;y2˝ t2.0;T /

.ˇ/

hf it;QT D sup sup jf .x; t/  f .x; /jjt   jˇ ; ˇ 2 .0; 1/: x2˝ t;2.0;T /

It is also convenient to define the norms: .˛/

.˛/

jf jx;QT D jf jQT C hf ix;QT I

.ˇ/

.ˇ/

jf jt;QT D jf jQT C hf it;QT :

Let Dxr D @jrj =@x1r1 : : : @xnrn ; r D .r1 ; : : : rn /; ri 2 N [ f0g; jrj D r1 C : : : C kC˛ rn ; Dt s D @ s =@t s ; s 2 N [ f0g, and k 2 N. The space C kC˛; 2 .QT / consists of the functions with finite norm: 

kC˛; kC˛ 2



jf jQT

X

D



jDxr Dt s f jQT

kC˛; kC˛ 2

C hf iQT



;

jrjC2s6k

where 

kC˛; kC˛ 2

hf iQT



D

X

.˛;˛=2/ hDxr Dt s f iQT

jrjC2sDk

C



X

1C˛

2 hDxr Dt s f it;QT



:

jrjC2sDk1

kC˛ kC˛ The symbol CV kC˛; 2 .QT / means the subspace of C kC˛; 2 .QT / formed by  functions f such that Dti f jtD0 D 0, i D 0; : : : ; kC˛ . 2 By C kC˛ .˝/; k 2 N [ f0g, one denotes the space of the functions f .x/; x 2 ˝, with the norm:

.kC˛/

jf j˝

D

X jrj6k

.kC˛/

jDxr f j˝ C hf i˝

;

1954

I.V. Denisova and V.A. Solonnikov

where .kC˛/

hf i˝

D

X

.˛/

hDxr f i˝ D sup

x;y2˝

jrjDk

X

jDxr f .x/  Dyr f .y/jjx  yj˛ :

jrjDk

The Sobolev-Slobodetskiˇı space W2l .˝/ for l > 0 is defined as the space of functions u with finite norm [31, 32]: 0 kukW l .˝/ D @

X

2

11=2 kDxr uk2˝ C kuk2WN l .˝/ A

;

2

jrj 0. The weighted space H .QT / is the space of functions admitting zero continuation to the domain t < 0 without loss of regularity. This space is equipped with the norm:  kukH l;l=2 .Q

T/

D

kuk2 l;0 H .QT /

C

kuk2 0;l=2 H .QT /

1=2 :

37 Local and Global Solvability of Free Boundary Problems for the. . .

1955

Here

kuk2 l;0 H .QT /

ZT D

  e2 t kuk2WN l .˝/ C l kuk2˝ dt; 2

0

kuk2 0;l=2 .QT / H

ZT D 0

l=2 2

@ u

e2 t

@t l=2 dt ˝

for

Œl=2 D l=2;

and

kuk2 0;l=2 H .QT /

ZT D

2 t

e 0

Z1 k

@ u0 .; t / @k u0 .; t   / 2 d

 dt

@t k

k 1Cl2k @t ˝  0

for k D Œl=2 < l=2, where u0 is the extension of u by zero in the domain t < 0. In addition, in the case l > 1, ˇ @i u ˇˇ D 0; @t i ˇtD0

 i D 0; : : : ;

 l 1 : 2

Via local charts and partitions of unity, all the above spaces can be introduced on any smooth manifold, in particular, on  and GT D   .0; T /. A vector is said to belong to a space if each of its components belongs to this space. The norm of a vector is defined as the sum of the norms of its components. The numeration of constants is individual for each section. Let R3T D R3  .0; T /; RC D fx 2 Rjx > 0g. Theorem 1 (Local solvability in Hölder spaces). Let, for some ˛ 2 .0; 1/ and T < 1,  2 C 3C˛ , 0 2 C 1C˛ .˝0 /, 0 .y/ > r0 > 0, the constant  > 0, ˛ the pressure function p 2 C 3 .RC /, and v0 2 C2C˛ .˝0 /, f; Dy f 2 C˛; 2 .R3T /, 1C˛ pe ; Dy pe 2 C 1C˛; 2 .R3T /. Moreover, the following compatibility condition  p.0 /n0 C 1 .r  v0 /n0 C S.v0 /n0 j D  H n0  pe n0 jtD0

(14)

is assumed to hold. Then on a certain time interval .0; T0 / with T0 6 T problem (12) has a unique solution u 2 C2C˛;1C˛=2 .QT0 /, QT0 D ˝0  .0; T0 /. The value of T0 depends on the norms of f; v0 ; p; pe ; 0 and on the curvature of  . Remark 1. Theorem 1 remains true if  is a positive function of class C 1C˛ . /.

1956

I.V. Denisova and V.A. Solonnikov

Remark 2. A similar theorem is valid for  D 0. In this case, it is sufficient to take  2 C 2C˛ . A local existence theorem in the Sobolev spaces for the same problem is stated as follows. 5=2Cl

Theorem 2 (Local solvability in Sobolev-Slobodetskiˇı spaces). Let  2 W2 , l 2 .1=2; 1/ 0 2 W21Cl .˝/, 0 .y/ > R0 > 0,  > 0, f.; t / 2 C2 .R3 / for 8t 2 Œ0; T , f.y; /, rf.y; / 2 Cˇ .0; T / for 8y 2 R3 with some ˇ 2 Œ1=2; 1/, pe 2 C 3 .R3 / for 8t 2 Œ0; T , pe , rpe satisfy the Lipschitz condition with respect to t, v0 2 W1Cl 2 .˝/, and let the compatibility condition (14) hold. 2Cl;1Cl=2 Then problem (12) has a unique solution u 2 W2 .QT0 / on a finite time interval .0; T0 / whose magnitude T0 depends on the data, i.e., on the norms of f, pe , v0 , 0 and on the curvature of the surface  . Only Theorem 1 is proved here in detail. One can find the proof of Theorem 2 in [42]. Remark 3. In fact, Theorem 2 remains valid with  2 W22Cl . If  D 0; then the 3=2Cl theorem is true for any initial boundary  2 W2 , and T0 is independent of the mean curvature of  .

2

Linear Problem in a Half-Space for a Single Fluid

In this section, the proof of classical solvability of the problem (Theorem 1) is started with a study of the regularity of a solution to a linear problem in the halfspace R3C D fx 2 R3 jx3 > 0g. The latter problem is generated by the linearization of problem (12) that was proposed in [42] where an explicit solution of the halfspace problem in the dual Fourier-Laplace space was found and estimated in the Sobolev-Slobodetskiˇı classes of functions. The Hölder estimates of the solution were obtained in [13]. The presentation of the material in this section follows that paper. Here, the method proposed in [37] in order to estimate a solution to the Stokes problem in Hölder spaces with reduced regularity with respect to time variable is developed. In the case of compressible liquid, this technique permits to obtain solution estimates in classical anisotropic Hölder spaces. It is noted that this method is simpler than that related to Fourier multiplier theorem which was applied in [20] in the case of incompressible fluid [41]. The initial–boundary value problem for the Lamé system for an unknown vector field w D .w1 ; w2 ; w3 /

37 Local and Global Solvability of Free Boundary Problems for the. . .

Dt w  . C 1 /r.r  w/  w D f;



@wˇ @w3 C @x3 @xˇ

ˇ ˇ ˇ

x3 D0

Z

t 0

(15)

jxj!1

D dˇ .t; x 0 /;

@w3 C 0 1 r  w C 2 @x3

t > 0;

w ! 0;

wjtD0 D w0 ; 

x3 > 0;

1957

x 0 D .x1 ; x2 / 2 R2 ;

ˇ w3 d ˇx3 D0 D d3 C 

is studied. Here  D r  r; 0 D

@2 @x12

C

@2 , @x22

Z

ˇ D 1; 2; (16)

t

D d

on

R

2

0

; , and 0 are positive constants,

 D 0 , 1 D 1 0 , 1 C 2 =3 > 0, and the constant  > 0; f, w0 , d D fdj g3j D1 , and D are given functions. Let RTC D R3C  .0; T / and R2T D R2  .0; T /. The main result of the Sect. 2 is given in the theorem below. ˛

Theorem 3. Let ˛ 2 .0; 1/ and T < 1. It is assumed that f 2 C˛; 2 .RTC /; 1C˛ ˛ w0 2 C2C˛ .R3C /; d 2 C1C˛; 2 .R2T /, D 2 C ˛; 2 .R2T /. In addition, it is supposed that all given functions together with their derivatives decay at infinity as power functions, and the compatibility conditions  @w



03

@xˇ

C

@w0ˇ  D dˇ .0; x 0 /; ˇ D 1; 2; @x3

1 r  w0 C 2

(17)

@w03 D d3 .0; x 0 /; x 0 2 R2 ; @x3

hold. Then, there exists a unique solution w 2 C 2C˛;1C˛=2 .RTC / of problem (15) and (16), which satisfies the estimate jwj

.2C˛;1C˛=2/ RTC

˚ .˛;˛=2/ .1C˛; 1C˛ .2C˛/ .˛;˛=2/

2 / ; 6 C .T / jfj C Cjw0 jR3 CjdjR2 C jDjR2 RT

C

T

(18)

T

where C .T / is a nondecreasing function of T .

2.1

Homogeneous System (15)

In this subsection, classical unique solvability is proved of system (15) with f D 0, w0 D 0: Dt v  . C 0 /r.r  v/  v D 0; x3 > 0; t > 0; ˇ ! 0; vˇtD0 D 0; v w jxj!1

(19)

1958

I.V. Denisova and V.A. Solonnikov

and with boundary conditions (16) with dˇ .t; x 0 / D bˇ .t; x 0 /; ˇ D 1; 2, Z t Z t d3 C  D d D b3 C  B d  b30 .t; x 0 /: All the given functions are 0

0

assumed to have a power-like decay at infinity and to vanish at t D 0. A solution of problem (19), (16) was constructed in explicit form in [42] by means of the Fourier transform in tangent space variables and the Laplace transform in time variable. The basic idea of regularity analysis consists in reducing (19), (16) to an analogous problem with  D 0, i.e., to system (19) with the boundary conditions:  

@vˇ @v3 C @x3 @xˇ

1 r  v C 2

ˇ ˇ ˇ

x3 D0

D aˇ .t; x 0 /; ˇ D 1; 2;

@v3 ˇˇ D a3 .t; x 0 /; x 0 2 R2 : @x3 x3 D0

(20)

Equation system in (19) is Petrovskiˇı parabolic. For problem (19), (20), the complementing condition (in the sense of Lopatinskiˇı) holds. Hence, according to Theorem 4.9 in [33], this problem is uniquely solvable. Moreover, the solution satisfies the estimate .2C˛;1C˛=2/

jvj

RTC

6c

3 X j D1

.1C˛; 1C˛ 2 /

jaj jR2

;

(21)

T

where c is independent of T . By applying the Fourier-Laplace transform given by the formula Z

1

est

uQ .s; ; x3 / D

Z R2

0

0

eix u.t; x/ dx 0 dt; x 0 D .x1 ; x2 /;

(22)

to problem (19), (16), one obtains the boundary value problem for the system of ordinary differential equations: 2  X d2 vQ ˇ  s dv3 2 C  v Q  C .1 C /i i˛ vQ ˛ C ˇ ˇ 2

dx3 dx3 ˛D1

.2 C / 

! D 0;

 d X d2 vQ 3  s 2  i˛ vQ ˛ D 0; C  vQ 3 C .1 C / 2

dx3 ˛ dx3

 d vQ

ˇ

dx3

ˇ ˇ C iˇ vQ 3 ˇ

x3 D0

D bQ ˇ ;

ˇ D 1; 2;

e v.s; ; x3 /

e d vQ 3   2 ˇˇ B .2 C 1 / C 1 i˛ vQ ˛  D bQ 3 C   bQ 30 ; v3 ˇ x3 D0 dx3 s s

ˇ D 1; 2;

x3 > 0;

(23)

w ! 0; jxj!1

(24)

37 Local and Global Solvability of Free Boundary Problems for the. . .

where  2 D 12 C 22 and D

1959

1

1 D . A similar system can also be derived



from (19), (20).  2  The characteristic determinant of system (23) equals s C  2  2 s C .2 C

1 / 2  .2 C 1 / 2 . It has six roots with respect to , three of which should be dropped in order to obtain a solution decreasing at infinity with respect to x3 . The remaininig roots 1;2 D r, 3 D r1 with negative real part appear as multipliers in the exponents of the exponential functions in the solution formula. By analogy with the case of an incompressible fluid (see [41]), the general solution of (23) is as follows: 0

1 0 1 0 1 r 0 i1 e v D C1 @ 0 A e rx3 C C2 @ r A e rx3 C C3 @ i2 A e r1 x3 ; i1 i2 r1

x3 > 0;

(25)

where r rD

s C  2;

r r1 D

s C  2; 2 C 1

(26)

p it is assumed that j arg zj < =2 for 8z 2 C. The constants Ck are arbitrary. One can find the coefficients Ck by substituting (25) into boundary conditions (24) which leads to .r 2 Cˇ  iˇ H C 2iˇ r1 C3 / D bQ ˇ ;

ˇ D 1; 2;

H 

2 X

i˛ C˛ ;

˛D1

 2rH C .2 C 1 /.r12   2 /C3 C 2 2 C3 

 2  .H  r1 C3 / D bQ 30 : s

(27)

These equations imply

.r 2 C  2 /H  2r1  2 C3 D A 

2 X

iˇ bQ ˇ ;

ˇD1

  2rH C .r 2 C  2 /C3   2 .H  r1 C3 / D bQ 30 : s

(28)

The determinant of system (28) is given by ˇ ˇ .r 2 C  2 / P D ˇˇ 2r  s  2

ˇ ˇ 2r1  2 2 2 2 2 2 2 2 ˇ  2 2 2 ˇ D  .r C / 4 rr1  C0 r1  : .r C  / C s r1  (29)

1960

I.V. Denisova and V.A. Solonnikov

Hence one finds

H D

  .r 2 C  2 / C s r1  2 A C 2r1  2 bQ 30 P

 C3 D

;

2r C s  2 A C .r 2 C  2 /bQ 30 ; P

and by (27)

Cˇ D

bQ ˇ  2iˇ r1 C3 C iˇ H ; r 2

ˇ D 1; 2:

For convenience of the forthcoming estimates, this particular solution is written in the form equivalent to (25): e v D We0 C Ve1 ;

(30)

where  T W D !1 ; !2 ; !3 ;

e0 D erx3 I

T  V D C3 .r  r1 / i1 ; i2 ; r1 ;

e1 D

erx3  er1 x3 I r  r1

the symbol T denotes the transposition, and !k are linear combinations of Ck : !ˇ D rCˇ C iˇ C3 ;

ˇ D 1; 2I

!3 D H  r1 C3 :

ˇ It is easily seen that e vˇx3 D0 D W. Explicit formula for the coefficient !3 will be required in the sequel: !3 D

o 1 n 0 s  2.rr1   2 / A  0 sr1 bQ 30 : P

Here determinant (29) is written in the form: P D 0 sM C 0 r1  2 ;

M D 0 s C

4. C 1 / 2 r ; 2 C 1 r C r1

The solution e v of (23), (24) can be viewed as a solution of the problem with  D 0 and with the boundary data:

37 Local and Global Solvability of Free Boundary Problems for the. . .

1961

aQ ˇ .s; / D bQ ˇ .s; /;

ˇ D 1; 2;   2 ˇˇ  2 D bQ 30 C aQ 3 .s; / D bQ 30 C vQ 3 ˇ !3 x3 D0 s s D bQ 3 C

2 X 0 r1 iˇ ˇD1

C

P

0  2 iˇ bQ 3 C P



 2. C 1 /r 1 A .2 C 1 /.r C r1 /

0 M e B: P

(31)

It is worth noting that this procedure was used in [37] for estimating a solution of a similar model problem in a half-space for an incompressible fluid. The following two auxiliary propositions were proved in [13] to evaluate ai .x 0 ; t /. Lemma 1. Let s 2 C be such that Res C ~jImsj > ıjj2 ; where ~ and ı are small positive numbers. Then c1 .jsj C jj2 /1=2 6 Rer.s; / 6 jr.s; /j 6 c2 .jsj C jj2 /1=2 ; c3 .jsj C jj2 /1=2 6 Rer1 .s; / 6 jr1 .s; /j 6 c4 .jsj C jj2 /1=2 ; r r c5 6 Re 6j j 6 c6 ; r C r1 r C r1 c7 .jsj C jj2 / 6 jM .s; /j 6 c8 .jsj C jj2 /: Moreover, these inequalities (possibly with other constants) hold if ˇ is replaced by ˇ D ˇ C iˇ , ˇ ; ˇ 2 R, ˇ D 1; 2, jj 6 ı1 jj;

(32)

where ı1 is a sufficiently small positive number. Lemma 2. For arbitrary s 2 C such that Res C ~jImsj > a   2 and for arbitrary ˇ D ˇ C iˇ , ˇ ; ˇ 2 R, ˇ D 1; 2, that satisfy (32), the expression P . / .s; / D sM .s; / C   r1 .s; /hi2

1962

I.V. Denisova and V.A. Solonnikov

q q q s s 2 ; r .s; / D 2 , hi D with r.s; / D C hi C hi 12 C 22 , is 1

2 C 1 estimated as follows: c9 .jsj.jsj C  2 / C jr1 j 2 / 6 jP . / .s; /j 6 c10 .jsj.jsj C  2 / C  jr1 j 2 /I (33) an estimate similar to (33) is valid also for jP . / .s  ı2 hi; /j, provided that ~; and

p

a

are small enough. The constants c9 ; c10 are independent of .

ı2 

Now, one performs the Fourier-Laplace transform inverse to (22) Z Z 1 1 0 ix 0  e est uQ .s; ; x3 / ds d; a > 0; .FL/ uQ  u.t; x ; x3 / D .2 /3 i R2 ResDa in (31) and obtains a3 in the form: 0

2 Z t Z X

0

a3 .t; x / D b3 .t; x / C

ˇD1 0

C

2 Z t Z X R2

ˇD1 0

Z tZ C 0

R2

R2

Wˇ .t  ; x 0  y 0 /

W3 .t  ; x 0  y 0 /

@b3 . ; y 0 /dy 0 d @yˇ

@bˇ . ; y 0 / 0 dy d @yˇ

W4 .t  ; x 0  y 0 /B. ; y 0 /dy 0 d

(34)

with 0 iˇ r1 ; ˇ D 1; 2; P  2  2. C  0 / r 1 0  1 ; W3 D .FL/ P 2 C  0 r C r1

Wˇ D .FL/1

W4 D .FL/1

0 M : P

(35)

To estimate the kernels Wj and their derivatives, the following lemma is applied. It is proved on the base of Lemmas 1 and 2 in [13] (see also [37]). Lemma 3. For the functions V1;k .t; x 0 / D .FL/1

r1 .s; /pk1 ./ ; k > 2; P .s; /

V2;k .t; x 0 / D .FL/1 V3;k .t; x 0 / D .FL/1

q.r; r1 /pk ./ ; k > 1; P .s; /

M .s; /pk2 ./ ; k > 2; P .s; /

37 Local and Global Solvability of Free Boundary Problems for the. . .

1963

where pk ./ is a homogeneous polynomial of degree k with respect to ˇ ; jj, and q.r; r1 / is a homogeneous function of degree zero with respect to r and r1 , the inequalities jDjx ; Vˇ;k .t; x 0 /j 6

c11 .t /t =2 ; .jx 0 j C t /kCCjjj

ˇ D 1; 2;

 2 Œ0; 1;

jDjx ; V3;k .t; x 0 /j 6 c12 .t /.jx 0 j C t /.kCjjj/ ; jDt Vˇ;k .t; x 0 /j 6

c13 .t / t

1 2

.jx 0 j C t /kC

;

ˇ D 1; 2;

 2 Œ0; 1;

(36)

hold for t > 0. Here, c11 .t /; c12 .t /; and c13 .t / are increasing functions of t. If t < 0, then each function Vi;k .t; x 0 / D 0. Inequalities (36) enable one to estimate potentials (35), the theorem on equivalent normalization in the Hölder spaces being used [16]. Theorem 4. Let ˛ 2 .0; 1/ and T > 0. It is assumed that the functions bj 2 1C˛; 1C˛

˛;˛=2

2 .R2T /, j D 1; 2; 3, B 2 CV .R2T / decay as power functions, together CV 0 with their derivatives, as jx j ! 1. Then system (19), (16) has a solution v 2 C2C˛;1C˛=2 .RTC / and 8 9 3 '.x 0 /g.

37 Local and Global Solvability of Free Boundary Problems for the. . .

1967

Let b u .t; z/ D u .t; Y .z//, etc. The gradient in the new coordinates is equal to @zi 3 g D fZik g3i;kD1 , r1 D Z T r with Z D f @x k i;kD1 1 1 0 0 A D @0 1 0 0 0 0 'z1 'z2 1C 'z3 0

Z 1

the unit normal is nO 0 D

p

'z01 1 C jr'j2

'z02

!

1

;p ; p , and the 1 C jr'j2 1 C jr'j2

stress tensor is 3  X

b w/ D 1 r1  b w C b S.b w/; b Sij D T0 .b

mD1

@wO j @wO i Zmi C Zmj @zm @zm

 :

Hence, system (48) can be written in the coordinates fzg as follows: b1 .b Dt b u  O01 .0/r  T0 .b u / D b fO C K w/ C O01 .z/Œr1  b T0 .b u /  r  T0 .b u / C ŒO01 .z/  O01 .0/r  T0 .b u /  F.t; z/; b w0  ; u jtD0 D b ˇ  ˇ b 2ˇ .b Sˇ3 .b ab /ˇ C K u /ˇz3 D0 D .˘0b w/   Sˇ3 .b u / C .˘0b n0 /ˇ ˇz3 D0 S.b u /b  bˇ ;

0 T33 .b u / C 0

Z

t

ˇ D 1; 2;

uO 3 djz3 D0 D aO b C 

0

Z 0

t

(49)

b 3 .b b w/ C  AO d C K

0 C T33 .b u /  b n0  b T0 .b u /b n0 C 

Z

ˇ u d ˇz3 D0  b3 C  Cb n0    b

Z

t 0

b 4 .b w/d K

t

0 uO 3

0

Z

t

B d ; 0

 where 0 D @2 =@z21 C @2 =@z22 , ˇ  0 b 3 .b aO C K w/ C T33 .b u /  b n0  T0 .b u /b n0 ˇz3 D0 ; b3 D b ˇ  b 4 .b B Db AO C K u ˇz3 D0 : w/ C 0 uO 3 C b n0   b Since the support of u is contained in ˝ D ˝ \B ; B D fjxj < g, it follows that suppb u  V D Y 1 ˝ ; supp F  V , and supp aˇ ; supp h, supp H  V 0 D u , b F can be V \ fz3 D 0g: Consequently, the functions bj , B and the vector fields b

1968

I.V. Denisova and V.A. Solonnikov

extended by zero into R2 nV 0 and into R3C nV , respectively. Moreover, it is assumed that O0 is extended into R3C in such a way that O0 2 C 1C˛ .R3C / and O0 .z/ > r0 > 0. Then (49) can be viewed as an initial–boundary value problem in RTC . Due to Theorem 3, for b u inequality (18) holds; hence, jb u j

.2C˛;1C ˛2 / RTC

3 n .˛; ˛ / o X .1C˛; 1C˛ .˛; ˛ / .2C˛/ 2 2 / O 6 c2 .T / jFj C / C jb w0  jR3 C jbj jR2 C jBjR2 2 : RT

C

j D1

T

T

(50) Next, the norms on the right-hand side of (50) are estimated. It is obvious that .1/

max jO01 .z/  O01 .0/j 6 c.r0 / jO0 jR3 V

C

and due to (46) sup jZij .z/  ıji j 6 c : V

b1 .b w/ does not involve higher-order derivatives of b w, it is easily Since the operator K seen that jFj

.˛;˛=2/ RTC

 .1C˛; 1C˛ / .2C˛;1C˛=2/ .˛;˛=2/ 6 c.r0 / C c jO0 jR3 jb u j C C c. /jb wjV ;T 2 C jb fO j C ; C

RT

RT

(51)

where V ;T D V  .0; T /. The remaining functions on the right-hand side of (49) are estimated in a similar way. One can use the relations S.b u /b u / C .˘0b n0 /ˇ D Sˇ3 C Sˇ3 .b

3 X

n0  b Sb n0 /; SO ˇj nO 0j  nO 0ˇ .b

j D1 0 0 b0 .b T33 .u /  b n0  T u /b n0 D T33  nO 203 Tb0 33 

X

nO 0i nO 0j Tb0 ij ;

iCj 0g. The transformation Zj takes a neighborhood j of the point j on  to a part of the plane R2j . Let

v0 .x; t / D

N X

vj .x; t /j .x/;

j D1

where vj .x; t / D wj .Zj .x/; t /, and wj are solutions of the following problems with plane boundaries R2j , j D 1; : : : ; N : Dt wj  01 .j /r  T0 .wj / D 0 in Dj  .0; T /  Dj;T ; wj jtD0 D 0; wj

! 0; w jxj!1

˘j T0 .wj /nj jR2j D aj ; Z t Z t wj d D aj C  Aj d : nj  T0 .wj /nj   nj  0j 0

(57)

0

Here ˘j ! D !  .nj  !/nj is the projection to the plane R2j , 0j is the Laplacian on R2j , aj .z; t / D ˘j ˘0 a.x; t /j .x/jxDZj1 .z/ , aj .z; t / D a.x; t /j .x/jxDZj1 .z/ and the Aj are defined in a similar way. In the coordinates x D Zj1 .z/, the vector fields vj .x; t / satisfy the system: Dt vj  01 .j /rj  T.j / .vj / D 0; vj jtD0 D 0;

vj

! 0; w jxj!1

(58)

ˇ ˘j T.j / .vj /nj ˇj D ˘j ˘0 aj ; Z t Z t ˇ e0j vj d D aj C  Aj d ; nj  T.j / .vj /nj ˇj   nj   0

0

where rj D .Jj1 /T r, Jj1 is the Jacobi matrix of the transformation Zj1 with the elements Jjmk ; the tensor T.j / .v/ has the components

37 Local and Global Solvability of Free Boundary Problems for the. . .

.j /

Tik D 1 ıik rj  v C 

3  X

Jjmk

mD1

1973

@vi @vk  ; C Jjmi @xm @xm

e0 D Zj1 .0 /. and j D Zj1 .R2j /,  j j Let the vector function v00 be a solution of the initial–boundary value problem:   Dt v00  01 .x/ . C 1 /r.r  v00 /  r 2 v00 D g v00 jtD0 D 0;

in QT ;

(59)

v00 j D 0

with

gD

N X ˚



01 .x/r  T0 .vj j /  01 .j /j rj  T.j / .vj /

j D1 N N X X  1  D 0 .x/01 .j / r  T0 .vj j /C 01 .j / r  T0 .vj j /j r  T0 .vj / j D1

C

N X

j D1

 01 .j / r  T0 .vj /  rj  T.j / .vj / j :

j D1

By definition, we set R D D w  v0 C v00 : Clearly, R is a linear operator. Its continuity follows from Theorem 4 and from Theorem 4.9 in [33], where unique solvability was stated for problem (58) and problem (59), respectively. Now, the operator M is calculated. To this end, one sums systems (58) over j D 1; : : : ; N , multiplying them beforehand by j and combining two boundary conditions into a single vector condition. It is easily seen that w is a solution of the homogeneous problem (43) with the boundary conditions: ˇ ˇ ˘0 T0 .w/n0 ˇ D ˘0 S.w/n0 ˇ D ˘0 a C ˘0 a1 C b C ˘0 d; Z t ˇ 0 ˇ w d D n0  T .w/n0   n0   0

Z

t

DaC

Z

A d C n0  a1 C n0  d C b C  0

t

.B C B1 C B2 / d : 0

1974

I.V. Denisova and V.A. Solonnikov

Here

a1 D

N X

ˇ .˘j ˘0 a  ˘0 a/j ; b D ˘0 T0 .v00 /n0 ˇ ;

j D1

dD

N n X 

T0 .vj j /  j T0 .vj / n0

j D1

oˇ  Cj T0 .vj /n0  T.j / .vj /nj C j .nj  n0 /nj  T.j / .vj /nj ˇ ; b D n0  T0 .v00 /n0 j ; B1 D n0 

N X

B D n0   v00 ;

. .vj j /  j  vj /;

B2 D

j D1

X

e0j vj /: j .n0   vj  nj  

j

The operator M is defined by the relation:  M D D ˘0 a1 C b C ˘0 d; n0  a1 C b C n0  d; B C B1 C B2 : (b) Estimation of the norm kM Dk. It is needed to prove that M satisfies (55). On the basis of arguments similar to those of 1ı , from (56), it is easy to deduce the estimate: .1C˛; 1C˛ 2 /

jb C djGt

.1C˛; 1C˛ /

.˛; ˛ /

2 C jbjGt C jB C B1 C B2 jGt 2 n .2C˛;1C ˛2 / 6 c1 ."1 C ˛ c2 / max jvj jGt

j 6N

C

.2C˛;1C ˛2 / jv00 jQt

Z C c3 . ; "1 / max

j 6N

0

t

.2C˛;1C ˛2 /

jvj jG

o d :

(60)

The norms of the vector fields vj can be estimated in terms of the norms of the solutions of problems (57). By Theorem 4, .2C˛;1C˛=2/

jwj jDj;t

o n .1C˛; 1C˛ .1C˛; 1C˛ .˛;˛=2/ 2 / 2 / 6 c.T / jaj jR2 C jaj jR2 C jAj jR2 j;t

j;t

j;t

n .1C˛; 1C˛ / o .1C˛; 1C˛ .˛;˛=2/ 2 2 / C jajGt C jAjGt 6 c4 .T / jajGt 6 c4 .T /P .t /; where R2j;t D R2j  .0; t/. This implies that .2C˛;1C˛=2/

max jvj jDt

j 6N

6 c5 .T /P .t /:

(61)

37 Local and Global Solvability of Free Boundary Problems for the. . .

1975

As to the vector v00 , it is a solution of the first initial–boundary value problem for a Petrovskiˇı parabolic system. Since the compatibility conditions for that system are fulfilled, one can apply Theorem 4.9 in [33] to obtain .2C˛;1C˛=2/

jv00 jQt

.˛;˛=2/

6 c.T /jgjQt

o n .1C˛; 1C˛ .2C˛;1C˛=2/ 2 / 6 c.T; r0 / ˛ max jvj jQt C c. / max jvj jQt j 6N

j 6N

Zt n o .2C˛;1C ˛2 / .2C˛;1C ˛2 / ˛ Cc."2 ; / max jvj jQ d : 6 c.T; r0 / ."2 C / maxjvj jQt j 6N

j 6N

(62)

0 .1C˛; 1C˛ /

2 The norm ja1 jGt is estimated now. Since ˘j ˘0 a  ˘0 a D nj .˘0 a .n0  nj //, it is obvious that

Z t ˇX ˇ.1C˛; 1C˛ / 2 .1C˛; 1C˛ .1C˛; 1C˛ ˇ ˇ 2 / 2 / j .˘j ˘0 a  ˘0 a/ˇ 6 c jajGt C c. / jajG d : ˇ Gt

j

0

(63) Thus, for sufficiently small ; "1 and "2 from inequalities (60), (61), (62), and (63), it follows that Z t kM Dk 6 "P .t/ C c."/ P . / d with " < 1; 0

which implies (55). 3ı . The general case is reduced to that treated above by representing a solution of (43) in the form w D w0 C w00 , where w0 is a solution of the Cauchy problem: Dt w0  01 .x/r  T0 .w0 / D f0 in R3 ;

w0 jtD0 D w00 ;

(64)

and w00 is a solution of the homogeneous problem (43) with b D a  T0 .w0 /n0 ; b D a  n0  T0 .w0 /n0   D;

(65)

B D A  n0   w0 C Dt D (D is a smooth function). In (64), the vectors f;0 w00 are extensions of f and w0 to the entire space R3 with class preservation. By Theorem 4.10 in [33], problem (64) is uniquely solvable in ˛ C2C˛;1C 2 .R3T /, R3T D R3  .0; T /, and .2C˛;1C˛=2/

jw0 jR3

T

 .˛;˛=2/  .˛;˛=2/ .2C˛/ .2C˛/ 6 c jf0 jR3 C jw00 jR3 C jw0 j˝ 6 c jfjQ : T

1976

I.V. Denisova and V.A. Solonnikov

The function D in (65) can be chosen so that the boundary functions satisfy the zero initial conditions. This is obvious for the vector b in view of the compatibility conditions (44), but, in general, the function B D A  n0   w0 does not vanish at t D 0. Therefore, B is to be replaced by a sum B C Dt D, where D 2 C 2C˛;1C˛=2 .GT /; DjtD0 D 0, Dt DjtD0 D n0   w0  AjtD0 , and  .˛;˛=2/ .2C˛/ 6 c jAjGT C jw0 j˝ I

.2C˛;1C˛=2/

jDjGt

clearly,  D should be subtracted from b. The uniqueness of the solution follows from the estimate (45) applied to the difference of two smooth solutions of system (43). u t

4

Local-in-Time Solvability of Nonlinear Problem (12)

Before Theorem 1 is proved, some transformations of (12) are made and several auxiliary estimates are obtained. The last boundary condition in (12) is transformed by using the identities: Z

t

n0  .t /Xu j D n0  .t /y C n0  .t / Z

t

D H0 .y/ C n0  Cn0 

0

Z tn

u.y;  / d 0

P / .

0

P /y d C n0  .0/ .

Z

t

u.y;  / d 0

Z



o  u.y;  0 /d 0 C . /  .0/ u.y;  / d ;

0

P / D Dt .t /. where H0 .y/ D n0 .y/.0/y is twice the mean curvature of  and .t Then problem (12) is rewritten as Dt u  01 .y/r  T0 .u/ D f.Xu ; t /  01 .y/Arp.0 Ju1 / C F1 .u/ on ˝; t > 0; ujtD0 D v0 ;

n0  T0 .u/n0  n0  .0/

Z

t 0

˘0 S.u/n0 j D F2 .u/;

(66)

˚

ˇ u dj D  H0 .y/ C .n0  n/ p.0 Ju1 /  pe .Xu ; t / ˇ Z

t

C F3 .u/ C  F4 .u/ C 

F5 .u/ d 0

Here the following notation has been used:  F1 .u/ D 01 .y/ Ar  T0u .u/  r  T0 .u/ ; ˇ  F2 .u/ D ˘0 ˘0 S.u/n0  ˘ Su .u/n ˇ ;

on  ;

t > 0:

37 Local and Global Solvability of Free Boundary Problems for the. . .

1977

 ˇ F3 .u/ D n0  T0 .u/n0  T0u .u/n ˇ ; Z t P /yj d; F4 .u/ D n0  . 0

n

P / F5 .u/ D n0  .t

Z



oˇ  ˇ u.y;  0 /d 0 C .t /  .0/ u.y; t / ˇ : 

0

(67)

Let .2C˛;1C˛=2/

Nt .f /  jf jQt

:

To estimate the functions (67), the following obvious statements are needed. Lemma 4. Let vectors u and u0 satisfy the inequalities: Nt .u/ 6 ı;

Nt .u0 / 6 ı:

(68)

Then for the corresponding cofactor matrices A and A0 of the Jacobi matrices of the transformations (5), the estimates jA 

.1C˛/ A0 jx;Qt . 1C˛ /

hA  A0 it;Q2t .˛=2/

Z

t

6 c.1 C ı/ 0

6 c.1 C ı/t

1˛ 2

.2C˛/

ju  u0 j˝

d 6 ctNt .u  u0 /;

jr.u  u0 /jQt ;

.˛=2/

hA  A0 it;Qt C hr.A  A0 /it;Qt 6 ct 1˛=2 Nt .u  u0 /

(69)

hold. In particular, .1C˛/

jA  Ijx;QT 6 ctNt .u/; . 1C˛ /

hAit;Q2t .˛=2/

6 ct

1˛ 2

jrujQt ;

.˛=2/

hAit;Qt C hrAit;Qt 6 ct 1˛=2 Nt .u/:

(70)

Lemma 5. If u and u0 satisfy (68) and if ı is so small that jAn0 j; jA0 n0 j > c > 0; then the difference n  n0 D An0 =jAn0 j  A0 n0 =jA0 n0 j is subject to the inequalities .1C˛/

jn  n0 jx;Gt 6 c . 1C˛ /

hn  n0 it;G2t

Z

6 ct

t 0

1˛ 2

.2C˛/

ju  u0 j˝

d 6 ctNt .u  u0 /;

jr.u  u0 /jQt :

1978

I.V. Denisova and V.A. Solonnikov

Corollary 1. If u satisfies the assumptions of Lemma 5, then Z

.1C˛/

jn  n0 jGt

6c 0

t

.2C˛/

juj˝

. 1C˛ /

hn  n0 it;G2t

d;

6 ct

1˛ 2

jrujQt :

P /. Similar estimates are also valid for the coefficients of the operators .t / and .t The statement of these results proved in [21] is presented below. Let  be covered by balls of a sufficiently small radius d and with centers at points yk 2  ; k D 1; : : : ; M . It is required that, in local Cartesian coordinates .k/ fi g D fi g, i D 1; 2; 3; the surfaces k D fy 2  ; jy  yk j < d g be defined as .k/

.k/

.k/

.k/

3 D ˚ .k/ .1 ; 2 /; .k/

.k/

j1 j2 C j2 j2 6 d 2 ;

.k/

where 1 , 2 lie on the tangent plane to  at the point yk , and the third basis .k/ vector corresponding to 3 is directed along n0 .yk /. The Laplace-Beltrami operator on k .t / is expressed in the coordinates fg (without the index k) by the formula: 2 2 2 2 X X @f 1 X @  ˇ p @f  ˇ @ f .t /f D p g D g g C h ; g @ˇ @ @ˇ @ @ D1 ˇ; D1

ˇ; D1

where g D detfgˇ g2ˇ; D1 ; gˇ D

@Xu @Xu  , the g ˇ are the elements of the @ˇ @

inverse matrix fgˇ g1 , and 2 1 X @  ˇ p h D p g g ; g @ˇ ˇD1

  Z t Z t Z t Xu .y; t / D 1 C !1 .;  /d; 2 C !2 .;  /d ; ˚ C !3 .;  /d Z

0

0

0

t

!.;  /d:

 yC 0

 Here y D 1 ; 2 ; ˚.1 ; 2 / and ! D u.1 ; 2 ; ˚ ; t /. Hence, the elements of the metric tensor and their derivatives with respect to t are expressed on k .t / by .0/

gˇ D gˇ C

Z

@y  @

t 0

Z

@y @! dC  @ˇ @ˇ

0

t

Z

@! dC @

0

t

Z

@! d  @ˇ

0

t

@! d ; @

(71)

37 Local and Global Solvability of Free Boundary Problems for the. . . .0/

where gˇ D

gP ˇ D

1979

@y @y  , and @ˇ @

@y @! @y @! @!  C  C  @ @ˇ @ˇ @ @ˇ

Z

t

0

@! @! d C  @ @

Z

t 0

@! d : @ˇ

The above formulas show that the following statement is true. Lemma 6. Let vectors u and u0 satisfy inequalities (68) with ı so small that the 0 determinants g D det.gˇ / and g 0 D det.gˇ / of the corresponding metric tensors 0 fgˇ g and fgˇ g calculated by (71) are strictly positive: g 0 > c > 0:

g > c > 0;

Then for the coefficients of the corresponding operators .t / and 0 .t /, the estimates jg

ˇ



.1C˛/ g 0ˇ jx;G k t

C jh 

.˛/ h0 jx;G k

Z

t

.˛=2/

t

6c 0

.2C˛/

ju  u0 jk

d ;

.˛=2/

hg ˇ  g 0ˇ it;G k C hh  h0 it;G k 6 ct 1˛=2 .1 C ı/jr.u  u0 /jGtk ; t

jgP

ˇ

 gP

0ˇ

.1C˛/

jGtk 6 cjr.u  u0 /jGtk ;

.˛/

.2C˛/

jgP ˇ  gP 0ˇ jx;G k C jhP  hP 0 jx;G k 6 cju  u0 jx;G k ; t

hgP

ˇ



. ˛2 / gP 0ˇ it;G k t

t

C hhP 

. ˛2 / hP 0 it;G k t

t

. ˛2 / 1 ˛2 6 c hr.u  u0 /it;G jr.u  u0 /jGtk (72) k C t ˚

t

hold. Here Gtk D k  .0; t/. Corollary 2. If u satisfies the assumptions of Lemma 6, then Z t .2C˛/ ˇ .0/ˇ .1C˛/ .0/ .˛/ jg  g jx;G k C jh  h jx;G k 6 c jujk d ; t

t

.˛=2/ hg ˇ it;G k

C

.˛=2/ hh it;G k t

0

6 ct

1˛=2

jrujGtk ;

.1C˛/ .˛/ .2C˛/ jgP ˇ jx;G k C jhP jx;G k 6 cjujx;G k ; t

.˛=2/ hgP ˇ it;G k

t

t

˚ .˛=2/ .˛=2/ C hhP it;G k 6 c hruit;G k C t 1˛=2 jrujGtk : t

t

On the basis of the statements formulated above, one can estimate the functions occurring on the right-hand side of (66).

1980

I.V. Denisova and V.A. Solonnikov

Lemma 7. It is assumed that u and u0 satisfy estimates (68) and that ujtD0 D u0 jtD0 D v0 : Then .1C˛; 1C˛ 2 /

.˛; ˛ /

jF1 .u/  F1 .u0 /jQt 2 C jF2 .u/  F2 .u0 /jGt .1C˛; 1C˛ 2 /

C jF4 .u/  F4 .u0 /jGt

.1C˛; 1C˛ 2 /

C jF3 .u/  F3 .u0 /jGt .˛; ˛2 /

C jF5 .u/  F5 .u0 /jGt

˛

6 c.t C t 1 2 /Nt .u  u0 /: (73)

Proof. First, ˇ R t an estimation ˇ is made of the Jacobian Ju which is given by (7). Since ˇ 0 Ar  u d ˇ 6 t jAjQt jrujQt 6 c1 ıt; it is evident that 1  c1 ıt 6 jJu j 6 1 C c1 ıt: Taking Lemma 4 into account, it is easy to show that .1C˛/

.1C˛/

Z

jJu jx;Qt 6 jAjx;Qt . 1C˛ /

hJu it;Q2t

.˛;˛=2/

jJu1 jQt

6 ct

1˛ 2

0

t

.1C˛/

jruj˝

d 6 cıt;

jrujQt ;

6 cı.t C t 1˛=2 /; . 1C˛ /

hJu1 it;Q2t

.1C˛/

jJu1 jx;Qt 6 ct ı; . 1C˛ /

6 jJu j2 hJu it;Q2t

6 ct

1˛ 2

jrujQt :

For the difference of the Jacobians corresponding u and u0 , the estimates .1C˛/

.1C˛/

.˛;˛=2/

C jJu1  Ju1 jQt jJu  Ju0 jx;Qt C jJu1  Ju1 0 jx;Q t . 1C˛ /

hJu  Ju0 it;Q2t

. 1C˛ /

2 C hJu1  Ju1 0 it;Q t

6 ct

6 c.t C t 1˛=2 /Nt .u  u0 /;

1˛ 2

jr.u  u0 /jQt

(74)

hold. The increments of the operators F1 , F2 , and F3 are estimated in a similar way, with the help of (69) and (74). For instance, the increment of F1 can be written as n  F1 .u/  F1 .u0 / D 01 .y/ .Ar  A0 r/  T0u .u/  .Ar  r/  T0u0 .u0 /  T0u .u/   o Cr  T0u .u/  T0 .u/  Tu0 .u0 /  T0 .u0 / :

37 Local and Global Solvability of Free Boundary Problems for the. . .

1981

Obviously, ˇ.˛=2/ ˇ ˇ ˇ.˛=2/ .˛=2/ j.Ar  A0 r/  T0u .u/jt;Qt 6 ˇA  A0 ˇt;Qt ˇr  T0u .u/ˇt;Qt 6 c.t C t 1˛=2 /Nt .u/Nt .u  u0 /; ˇ ˇ  ˇ.Ar  r/  T0 0 .u0 /  T0 .u/ ˇ.˛=2/ 6 c.1 C ı/ı.t C t 1˛=2 /Nt .u  u0 /; u u t;Qt ˇ  0 0   ˇ ˇr  T 0 .u /  T0 .u0 /  T0 .u/  T0 .u/ ˙ T0 0 .u/  T0 .u/ ˇ.˛=2/ u u u t;Qt 6 c.t C t 1˛=2 /ıNt .u  u0 /; and similarly, .˛/

jF1 .u/  F1 .u0 /jx;Qt 6 cıtNt .u  u0 /: Thus, estimate (73) has been proved for the increment of the operator F1 . By Lemma 5, the operators (F2 ; F3 ) satisfy the inequalities: .1C˛/

.1C˛/

jF2 .u/  F2 .u0 /jx;Gt C jF3 .u/  F3 .u0 /jx;Gt 6 cıtNt .u  u0 /; . 1C˛ /

. 1C˛ /

hF2 .u/  F2 .u0 /it;G2t C hF3 .u/  F3 .u0 /it;G2t o n 1˛ 6 cı tNt .u  u0 / C t 2 jr.u  u0 /jGt 6 cıtNt .u  u0 /: Since u  u0 jtD0 D 0; the estimate jr.u  u0 /jGt 6 t

1C˛ 2

. 1C˛ /

hr.u  u0 /it;G2t

6t

1C˛ 2

Nt .u  u0 /

(75)

could be used in the last of the above inequalities. Z t

Next, one considers F4 .u/ D  n0  0

ˇ P /yˇ d . By virtue of the equality . 

2 X @y ˇˇ @y h give no contribution to F4 .u/. n0  ˇ D 0, D 1; 2, the lower terms @  @ D1

Therefore, in view of (72) .1C˛/

. 1C˛ /

jF4 .u/  F4 .u0 /jx;Gt C hF4 .u/  F4 .u0 /it;G2t X ˚

1˛ .2C˛/ 6c jk jC 3C˛ t ju  u0 jx;Gt C t 2 jr.u  u0 /jGt 6 ctNt .u  u0 /: k

1982

I.V. Denisova and V.A. Solonnikov

Finally, the operator F5 .u/ is estimated as .˛;˛=2/

jF5 .u/  F5 .u0 /jGt

6 cı.t C t 1˛=2 /Nt .u  u0 /;

by Lemma 6. Thus, estimate (73) is proved completely. Lemma 8. If u 2 C2C˛; .˛;˛=2/

jF1 .u/jQt

.1C˛; 1C˛ 2 /

jF4 .u/jGt

1C˛ 2

t u

.Qt / satisfies Nt .u/ 6 ı and ujtD0 D v0 , then .1C˛; 1C˛ 2 /

C jF2 .u/jGt

.1C˛; 1C˛ 2 /

C jF3 .u/jGt

.˛;˛=2/

C jF5 .u/jGt

˚ 1˛ 6 c ıtNt .u/ C t 2 jrv0 j2˝ ;

(76)

˚ 1˛ 6 c tNt .u/ C t 2 jrv0 j :

(77)

This lemma can be proved by an almost word-for-word repetition of the proof of Lemma 7, but the inequality jrujGt 6 t

1C˛ 2

. 1C˛ /

hruit;G2t

C jrv0 j :

should be used in place of (75). Along with the operators Fi , it is necessary to estimate the increments of the given functions on the right-hand side of (66) and also of the density .y; O t / in the Lagrangian coordinates. Lemma 9. If vectors u and u0 satisfy (68), then .1C˛; 1C˛ 2 /

j.y; O  /  O 0 .y;  /jQt

6 cf.t C t 1˛=2 /Nt .u  u0 / C t

1˛ 2

ˇ jr.u  u0 /ˇtD0 j˝ g; (78)

ˇ ˇ .˛;˛=2/ ˇf.Xu ;  /  f.X 0 ;  /ˇ.˛;˛=2/ C jArp..y; O  //  A0 rp.O0 /jQt u Qt .1C˛; 1C˛ 2 /

Cjp./ O  p.O0 /jGt

.1C˛; 1C˛ 2 /

C jpe .Xu ;  /  pe .Xu0 ;  /jGt

n  o ˇ ˇ 1˛ 6 c tNt .u  u0 / C t 2 jr.u  u0 /ˇtD0 j˝ C j.u  u0 /ˇtD0 j˝ : Proof. Since .y; O t / D 0 .y/Ju1 .y; t /;

(79)

37 Local and Global Solvability of Free Boundary Problems for the. . .

1983

one can use (74) to obtain .1C˛/

.1C˛/

j.y; O  /  O0 .y;  /jy;Qt 6 j0 j˝ . 1C˛ /

hO  O0 it;Q2t

.1C˛/

jJu1  Ju1 6 c.t C t 1˛=2 /Nt .u  u0 /; 0 jy;Q t

. 1C˛ /

2 6 j0 j˝ hJu1  Ju1 0 it;Q t

˚

1˛ 6 c tNt .u  u0 / C t 2 jr.u  u0 /jtD0 j˝ :

Inequality (79) for the increments of f and pe is a direct consequence of the representation:

f.Xu ; t /  f.Xu0 ; t / D

3 Z X kD1

Z

1 0

Dxk f.Xus ; t / ds

0

t

.uk  u0k / d ;

where us D u C s.u0  u/. For estimating the norms of the functions involving the pressure p, the integral representation for the increment and inequalities (69), (70), and (78) are employed. t u Corollary 3. If u 2 C2C˛;1C˛=2 .Qt / and ujtD0 D v0 ; Nt .u/ 6 ı; then .1C˛; 1C˛ 2 /

j.y; O  /  0 .y/jQt

˚ 1˛ 6 c .t C t 1˛=2 /Nt .u/ C t 2 jrv0 j˝ ;

.˛;˛=2/

jf.Xu ;  /  f.y;  /jQt

.˛;˛=2/

C jArp./ O  rp.0 /jQt

.1C˛; 1C˛ /

(80)

.1C˛; 1C˛ 2 /

2 Cjp./ O  p.0 /jGt C jpe .Xu ;  /  pe .y;  /jGt ˚

1˛ 6 c tNt .u/ C t 2 .jrv0 j˝ C jv0 j˝ / :

The preliminary estimation is complete. Proof (of Theorem 1). System (66) is equivalent to u D u0 C w  H.u/:

(81)

Here u0 is a solution of a linear problem with given functions on the right-hand side: Dt u0  01 .y/r  T0 .u0 / D f.y; t /  01 .y/rp.0 .y// in QT ; ˇ u0 jtD0 D v0 ; ˘0 S.u0 /n0 ˇGT D 0; Z t 0 u0 djGT D  H0 .y/ C p.0 .y//  pe .y; t /jGT ; n0  T .u0 /n0  n0  .0/ 0

1984

I.V. Denisova and V.A. Solonnikov

and w is a solution of the same problem (43), but with functions of u in the role of the right-hand sides: ˚

O  rp.0 / C F1 .u/; f D f.Xu ; t /  f.y; t / C 01 .y/ Arp./ w0 D 0;

˘0 a D F2 .u/;

ˇ a D p./.n O 0  n/  p.0 / C pe .y; t /  pe .Xu ; t /ˇGt C F3 .u/ C  F4 .u/;

A D F5 .u/: ˛

By Bd .QT0 /, the subset of the ball of radius d in the space C2C˛;1C 2 .QT0 / such that the vector fields belonging to Bd .QT0 / become equal to v0 at t D 0 is denoted. The operator H.u/ is proved to be a contraction on the closed set Bı .QT0 / defined by NT0 .u/ 6 ı; T0  1; ujtD0 D v0 with ı > 0. Moreover, H.u/ takes this set into itself. Indeed, by virtue of Theorem 5 and inequalities (76), (77), (80), for any element u of Bı .QT0 /, the estimates .2C˛;1C˛=2/

jH.u/jQT

0

6 NT0 .u0 / C NT0 .w/ 1˛   6 NT0 .u0 / C c1 ıT0 1 C NT0 .u/ C c2 T0 2 jrv0 j2˝ C jrv0 j˝ C jv0 j˝

hold. (The compatibility conditions (44) follow from (14).) Let T0 be chosen such that  1 c1 T0 1 C NT0 .u/ 6 : 2 Now, if one takes ı so as to ensure the inequality 1˛ 2

NT0 .u0 / C c2 T0



ı jrv0 j2˝ C jrv0 j˝ C jv0 j˝ 6 ; 2

then H.u/ will belong to the same set Bı .QT0 /: .2C˛;1C˛=2/

jH.u/jQT

0

6 ı; H.u/jtD0 D v0 :

Next, Lemmas 7 and 9 imply that arbitrary u and u0 in Bı .QT0 / satisfy the inequality: .2C˛;1C˛=2/

jH.u/  H.u0 /jQT

0

 1˛=2 6 c3 .1 C ı/ T0 C T0 NT0 .u  u0 /:

37 Local and Global Solvability of Free Boundary Problems for the. . .

1985

If T0 is chosen so small that 1˛=2

c3 .1 C ı/.T0 C T0

/ 6 1;

then H.u/ is seen to be a contraction. Hence, by the Banach theorem, equation (81) (and, consequently, problem (12)) has a unique solution in Bı .QT0 /. It remains to prove that this problem can’t have two solutions of class ˛ C2C˛;1C 2 .QT0 /. Let u and u0 be two such solutions. Their difference u  u0 D v solves the problem: Dt v  01 .y/r  T0 .v/ D f.Xu ; t /  f.Xu0 ; t /  0 0 01 .y/ Arp.0 Ju1 /  A0 rp.0 Ju1 0 / C F1 .u/  F1 .u /  G1 .u; u /; ˇ vjtD0 D 0; ˘0 S.v/n0 ˇ D F2 .u/  F2 .u0 /; 0

n0  T .v/n0   n0  .0/

Zt 0

˚

0

C .n0  n /

ˇ ˚

vˇ d D n0  .n  n0 / p.0 Ju1 /  pe .Xu ; t / C

p.0 Ju1 /  p.0 Ju1 0 /  pe .Xu ; t / C pe .Xu0 ; t /



0

Zt



C  F4 .u/  F4 .u / C 

0

C F3 .u/  F3 .u0 /C Zt

0

F5 .u/  F5 .u / d  G2 .u; u / C  0

F5 .u/ 0

0

 F5 .u / d : To prove that v  0, we estimate the vector field v by Theorem 5 in some interval .0;  /;  < T0 . By (73) and (66), .˛;˛=2/

jG1 .u; u0 /jQ

.1C˛; 1C˛ 2 /

C jF2 .u/  F2 .u0 /j .˛;˛=2/

CjF5 .u/  F5 .u0 /j

6 c4 

1˛ 2

.1C˛; 1C˛ 2 /

C jG2 .u; u0 /j .2C˛;1C˛=2/

jvjQ

:

Inequality (45) shows that .2C˛;1C˛=2/

jvjQ

6 c5 

1˛ 2

.2C˛;1C˛=2/

jvjQ

: 2=.1˛/

This estimate implies that v D 0 in an interval .0; / with  < c5 these arguments shows that v D 0 for t 2 .0; T0 /. This completes the proof of Theorem 1.

. Repeating t u

1986

I.V. Denisova and V.A. Solonnikov

5

Problem on the Motion of Two Compressible Fluids Separated by a Closed Interface

5.1

Introduction and Statement of the Problem

This section is devoted to the problem of the simultaneous evolution for two viscous compressible capillary liquids for which local (with respect to time) unique solvability in the Sobolev-Slobodetskiˇı [5] and Hölder spaces [8] is proved. To analyze this problem, the previous scheme developed for the investigation of a single fluid of finite volume is used. But investigation of the evolution of a bubble in a compressible liquid medium is more difficult than that of the motion of a bubble in vacuum. In particular, an explicit solution of the model problem in Fourier–Laplace coordinates involves more parameters and has the form more complicated than in the case of a single liquid. By estimating the solution, it is not so easy to prove that the Lopatinskiˇı denominator P is bounded away from zero. For these reasons, the estimation of P was carried out first under some additional restrictions on fluid viscosity coefficients (see (87)). This means that stable coexistence was only guaranteed for two compressible fluids with close characteristics. However, it is worth noting that the restrictions mentioned above are not necessary. Now these difficulties are overcome; a way is presented to obtain problem solvability without any unnatural conditions for fluid viscosities. One can employ the idea that the linear problem without including surface tension into consideration is coercive and should have a unique solution because the Lopatinskiˇı complementing condition is satisfied for this problem. It is necessary to consider first the explicit solution of the model problem in the dual Fourier-Laplace space with  D 0. In this case, the denominator of the solution becomes simpler; its strict positiveness follows from  the above arguments. Next, one can extend this property for the general case by Lemma 10 using denominator continuity with respect to the dual time variable. This scheme is also applied for the problem on the motion of two fluids of different types (see Sect. 6.2). In concluding this brief introduction, it should be noted that a similar two-phase problem (but without surface tension taken into account) was studied by Tani [46]. The problem governing the motion of two incompressible capillary liquids is mentioned in [41]; there, its local unique solvability is stated both in the SobolevSlobodetskiˇı and Hölder spaces. Moreover, global solvability for this problem near an equilibrium is proved in the Hölder classes of functions. A general problem on unsteady motion of two compressible barotropic liquids separated by a closed free interface is stated as follows. At the initial instant t D 0, a fluid with dynamical viscosity C > 0, C 1  C C ˇ  , occupies a bounded domain ˝0C and a fluid with dynamical viscosity  > C    3 ˙ 0,  1  ˇ  , occupies the “exterior” domain ˝0  R n ˝0 . The constants ˇ are assumed to belong to the interval .2=3; 1/. For every t > 0, it is necessary to find the free interface t D @˝tC between the liquids, as well as their density

37 Local and Global Solvability of Free Boundary Problems for the. . .

1987

functions .x; t / > 0 and the velocity vector field v.x; t / D .v1 ; v2 ; v3 / satisfying the following initial–boundary value problem for the Navier-Stokes system: .Dt v C .v  r/v/  r  T D f; jtD0 D 0 .x/;

vjtD0 D v0 .x/ in ˝0 [ ˝0C ;

ˇ Œvˇt 

v.x/ 

lim

x!x0 2t ; x2˝tC

lim

in ˝t [ ˝tC ; t > 0;

Dt  C r  .v/ D 0

x!x0 2t ; x2˝t

v.x/ D 0;

! 0; v w jxj!1

ŒTnjt D  H n

on

t ;

(82)

@v @vi C @xji , @xj C equal to C , C 1 in ˝t C vector to ˝t . The fluid

˙ where T D .p./ C ˙ 1 r  v/IC  S.v/ is the stress tensor, .S.v//ij D

I is the identity matrix, and ˙ , ˙ 1 are the step functions  and  ,  in ˝ , respectively; n is the outward normal t 1 pressure p./ is a known smooth function of the density. The other notations are the same as in the case of a single liquid. Kinematic condition (4) on the interface t completes system (82). Then t D fx.y; t /jy 2 0 g, where   0 D @˝0C is the initial interface; and ˝t˙ D fx.y; t /jy 2 ˝0˙ g. As in Sect. 1, the transformation of the Eulerian coordinates fxg into the Lagrangian ones fyg is performed by the formula (5). This leads to a problem with a constant interface instead of (82) and (4). The Jacobian of transformation (5) Ju .y; t / D detfaij g3ij D1 , being a solution of the Cauchy problem (6), is expressed by (8). Problem (82) goes over into the system with unknowns b  and u: b Dt u  ru  Tu .u;b / D b b f; b jtD0 D 0 ; ˇ Œuˇ D 0;

ujtD0 D v0

Dtb  Cb ru  u D 0 in ˝  [ ˝ C ;

ˇ ŒTu .u;b /nˇ D  H n;

in

˝ ˙  ˝0˙ ; t > 0;

! 0; u w jxj!1 (83)

f.y; / C ˙ where b .y; t / D .Xu ; t /, b 1 ru  w/I C  t / D f.Xu ; t /, Tu .w;b  / D .p.b @w

@wi C Aik @ykj : The outward normals n0 and n ˙ Su .w/, .Su .w//ij D Ju1 Aj k @y k to  and t , respectively, are connected by the relation (11). In view of equality (8) and the second equation in (83), expression (9) for the density is deduced in accordance with the Cauchy formula. Substitution of this expression in the first equation of system (83) leads to an initial–boundary value problem for the single unknown function u. Then formula (10) is applied for twice the surface mean curvature H and project the last boundary condition in (83) onto the tangent plane to t and then onto that to  . The result of these transformations is a problem analogous to (12), it being equivalent to (83) provided that n  n0 > 0:

1988

I.V. Denisova and V.A. Solonnikov

Dt u  01 .y/Ar  T0u .u/ D f  01 .y/Arp.0 Ju1 / ˇ uˇtD0 D v0 ; u ! 0;

in ˝ ˙ ;

t > 0;

jyj!1

ˇ Œuˇ D 0;

ˇ Œ˙ ˘0 ˘ Su .u/nˇ D 0;

ˇ ˇ ˇ /ˇ  n0  .t /Xu ˇ D .n0  n/Œp.0 Ju1 /ˇ ; Œn0  T0u .u;b

(84)

/Cp.b /I, and ˘0 ! D !.n0 !/n0 , ˘ ! D !.n!/n. where T0u .w/ D Tu .w;b The solvability of problem (84) will be proved by its linearization and by successive approximations. Solutions to problems of the following type play the role of the approximations: Dt w  1 .y/Ar  T0u .w/ D f in QT˙ D ˝ ˙  .0; T /; ˇ wˇtD0 D w0 in ˝  [ ˝ C ; w ! 0;

(85)

jyj!1

ˇ Œ˙ ˘0 ˘ Su .w/nˇGT D ˘0 a;

ˇ ŒwˇGT D 0; Œn0 

ˇ

T0u .w/nˇ

Zt

.GT    .0; T //;

ˇ w d ˇ D b C 

 n0  .t / 0

Zt B d ; t 2 .0; T /: 0

If u D 0, then system (85) reduces to the problem: Dt w  01 .y/r  T0 .w/ D f in DT  QT [ QTC ; ˇ wˇtD0 D w0 in ˝  [ ˝ C ; w ! 0; jyj!1

ˇ Œwˇ

GT

D 0;

ˇ Œn0  T .w/n0 ˇ   n0  .0/ 0

ˇ Œ ˘0 S.w/n0 ˇGT D ˘0 a; ˙

Zt

ˇ w d ˇ D b C 

0

(86)

Zt B d ; t 2 .0; T /; 0

˙ here T0 .w/ D .˙ 1 r  w/I C  S.w/: Problems (84), (85) and (86) will be studied in the Sobolev-Slobodetskiˇı spaces, both weighted and ordinary. Let kukH l;l=2 .D / D kukH l;l=2 .Q / C kukH l;l=2 .QC / :

5.2

T



T



T

Local Solvability in the Sobolev-Slobodetskiˇı Spaces

Global existence theorem for the linear problem arising from the simultaneous motion of two compressible fluids separated by a known closed interface is as follows:

37 Local and Global Solvability of Free Boundary Problems for the. . .

1989

3=2Cl

Theorem 6. For fixed l > 1=2, let  2 W2 , 0 2 W21Cl .[˝ ˙ /, 0 .x/ > l;l=2 lC1=2;l=2C1=4 lC1=2;l=2C1=4 R0 > 0, f 2 H .DT /, a 2 H .GT /, b 2 H .GT /, B 2 l1=2;l=21=4 .GT / with T 6 1. Moreover, w0 is assumed to be zero. H Then, for sufficiently large > 0 > 0, system (86) has a unique solution 2Cl;1Cl=2 .DT / that satisfies the inequality: w 2 H kwkH2Cl;1Cl=2 .D

T/



 6 c1 kfkHl;l=2 .D

T/



C kakHlC1=2;l=2C1=4 .G / C kbkH lC1=2;l=2C1=4 .G / T T 

C  kBkH l1=2;l=21=4 .G

T/

with a constant c1 independent of T . Remark 4. Although this theorem was stated in [5] under the following inequalities for the main liquid viscosities,  6 C 6 2 ; 2

(87)

it remains valid also without these restrictions, as will be shown below. To formulate existence theorem for the linearized problem, it is necessary to introduce the norms whose squares are defined by the formulas 

.l;l=2/

kukDT



2

D kuk2

l;l=2

W2

.2Cl;1Cl=2/

kukDT

2

D kuk2

.DT /

2Cl;1Cl=2

W2

C T l kuk2DT ; .DT /

C

o X 1 n ˛ 2 2 uk C kD uk kD t DT DT x Tl j˛jD2

C sup ku.; t /k2 t6T .l;l=2/

The norm kukDT with 8T < 1.

W21Cl .[˝ ˙ /

is equivalent to kukH l;l=2 .D 0

:

T/

for l < 1 and to kukW l;l=2 .D 2

T/

3=2Cl

Theorem 7. Let  2 W2 , 0 2 W21Cl .[˝ ˙ / for l 2 .1=2; 1/, 0 .y/ > R0 > 0. In addition, it is assumed that a vector u changes continuously when the boundary  is crossed and that for some T < 1 it satisfies the inequality: .2Cl;1Cl=2/

T 1=2 kukDT

6ı

(88)

with a small number ı. l;l=2 lC1=2;l=2C1=4 ˙ Then for any f 2 W2 .DT /, w0 2 W1Cl .GT /, b 2 2 .[˝ /, a 2 W2 lC1=2;l=2C1=4 l1=2;l=21=4 W2 .GT /, and B 2 W2 .GT / which satisfy the compatibility conditions

1990

I.V. Denisova and V.A. Solonnikov

ˇ Œw0 ˇ D 0;

ˇ ˇ Œ˙ ˘0 S.w0 /n0 ˇ D ˘0 aˇtD0 ;

ˇ ˇ Œn0  T0 .w0 /n0 ˇ D b ˇtD0 ;

2Cl;1Cl=2

problem (85) is uniquely solvable in the space W2 w, the estimate .2Cl;1Cl=2/

kwkDT

.DT /, and for the solution

˚ .l;l=2/ 6 c2 .T / kfkDT C kw0 kW1Cl .[˝ ˙ / C kakW lC1=2;l=2C1=4 .G / T 2 2

C kbkW lC1=2;l=2C1=4 .G / C  kBkW l1=2;l=21=4 .G / 2

T

2

T

holds with a nondecreasing function c2 .T /. Local existence theorem for the nonlinear problem written in the Lagrangian coordinates is now stated. 5=2Cl

Theorem 8. Let  2 W2 , 0 2 W21Cl .[˝ ˙ /, 0 .y/ > R0 > 0, p 2 C 3 .RC / for some l 2 .1=2; 1/. In addition, let f.; t / 2 C2 .R3 /, 8t 2 Œ0; T , f.y; /, ˇ rf.y; / 2 ˙ ˇ Cˇ .0; T / for 8y 2 R3 with some ˇ 2 Œ1=2; 1/, v0 2 W1Cl 2 .[˝ /, Œv0   D 0, and the compatibility condition ˇ ˇ ˇ ˇ Œp.0 /n0 C ˙ S.v0 /n0 C ˙ 1 .r  v0 /n0  y2 D  H n0 tD0 hold. Then there exists a constant T0 2 .0; T  such that problem (84) is uniquely 2Cl;1Cl=2 .DT0 /, where the solvable in the interval .0; T0 /, and its solution u 2 W2 value T0 depends on the norms of f, v0 , 0 and on the curvature of the surface  . Theorems 6, 7 and 8 are analogs of Theorems 1.1–1.3 in [42]. Their proofs are based on the explicit solution of the model problem with a plane fluid interface which will be constructed and estimated in Sect. 5.3. The differences in the study of two fluids occupying the space R3 , in comparison with the case of a single fluid in a bounded domain, are absolutely insignificant for the Sobolev spaces. It is worth noting that the same situation was in the “incompressible” case, where existence theorems for problems with a closed boundary were generalized to the case of two liquids without considerable modifications (see [41]).

5.3

Model Problem with a Plane Interface Between the Liquids

In this subsection, it is assumed that the initial velocity of the fluids w0 D 0 and that their densities form the step function 0˙ . Let DT3 D DT [ DTC ; T 6 1:

DT˙ D R3˙  .0; T /; R3˙ D f˙x3 > 0g; R21 D R2  .0; 1/;

37 Local and Global Solvability of Free Boundary Problems for the. . .

1991

Problem (86) with the plane fx3 D 0g playing the role of  has the form: 3 ; Dt w  . ˙ C 1˙ /r.r  w/  ˙ r 2 w D f in D1 ˇ ˇ ! 0; Œwˇx D0 D 0; wˇtD0 D 0; w w jxj!1 3   ˇ  ˇ @w˛ @w3 ˇ C D d˛ .x 0 ; t /; x 0 D .x1 ; x2 ; 0/;  ˙ ˇ @x3 @x˛ x3 D0

Œ˙ 1 r

ˇ  Zt Zt ˇ ˇ ˙ @w3 ˇ 0 ˇ  w x3 D0 C 2 C  w3 d D d3 C  D d  d30 ; @x3 ˇx3 D0 0

˙ ˙ ˙ ˙ 0 where 1˙ D ˙ 1 =0 , D  =0 ,  D

1 , 2

Theorem 9. Let l > lC 12 ; 2l C 14

H

C

2Cl;1C 2l

l; l

l 12 ; 2l  14

2Cl;1C 2l .DT3 / H

6 c. / kfk

.DT3 /

@2 . @x22

T 6 1. Then for arbitrary f 2 H 2 .DT3 /, dk 2 .R2T / problem (90) has a unique

such that

( H

@2 @x12

0

.R2T /, k D 1; 2; 3, and D 2 H

solution w 2

kwk

˛ D 1; 2; (89)

l; l

C

H 2 .DT3 /

3 X iD1

) kdi k

lC 12 ; 2l C 14

H

.R2T /

C  kDk

l 12 ; 2l  41

H

.R2T /

;

(90) where c. / is independent of T . Theorem 9 was proved in detail in [5]. Here, only some ideas of the proof are given. l;l=2 l;l=2 3 At first, problem (90) is studied in the spaces H .D1 /, H .R21 / under the assumption that f D 0. l;l=2 ˙ For a function f 2 H .D1 /, the Fourier-Laplace transform in x 0 D .x1 ; x2 / and t > 0 calculated by formula (22) defines an analytic function fQ in the half-plane Res > . Transformation (22) reduces problem (90) to the following system of ordinary differential equations:    2  de v˛ dvQ 3 2  ˙ D 0; ˛ D 1; 2; se v ˛  . ˙ C 1˙ /i˛ i1 vQ 1 C i2e v2 C   v Q ˛ dx3 dx32    2  d vQ 3 d dvQ 3 2 s vQ 3  . ˙ C 1˙ / i1e  ˙ (91) v 1 C i2 vQ 2 C   v Q 3 D 0; dx3 dx3 dx32

1992

I.V. Denisova and V.A. Solonnikov

with the boundary conditions ˇ ! 0; e Œe vˇx3 D0 D 0; v w jxj!1    ˇ ˇ dvQ ˛  ˙ C i˛e v 3 ˇˇ D bQ ˛ ; ˛ D 1; 2; dx3 x3 D0   ˇ  ˇ dvQ 3  Q 3 ˇˇ ˙ dv i C 2 ˙ v Q C i e v C   2 vQ 3 ˇx3 D0 D e b 03 : (92) 1 1 2 2 1 ˇ dx3 dx3 x3 D0 s Here the notation on the right-hand side is changed: the functions d1 , d2 , d30 are replaced by b1 , b2 , b30 , and the vector w is done by v. In addition,  2 D 12 C 22 . The ˇgeneral solution ˇof (91) in the two half-spaces is given by the functions e vC D e vˇx3 >0 and e v D e vˇx3 0; i1 i2 r1˙ (93) q q p s C  2 , in doing so j arg zj < =2 for where r ˙ D s˙ C  2 , r1˙ D ˙ ˙ 2 C 1

8z, and Ck˙ are free constants. It is worth noting that a solution of a problem similar to (90) for two incompressible liquids was constructed and estimated in [3] (see also [41]). For the convenience of the following estimates, particular solutions of (91), (92) are sought in a form equivalent to (93): e v D W˙ e r

˙x 3

C V˙

e r

˙x 3

˙

 e r1 r ˙  r1˙

x3

;

˙x3 > 0:

(94)

T T   Here W˙ D !1˙ ; !2˙ ; !3˙ , V˙ D C3˙ .r ˙  r1˙ / i1 ; i2 ; r1˙ , and !i˙ are linear combinations of Ci˙ . Substituting solution (94) in boundary conditions (92) yields the following expressions for C3˙ and !i˙ : C3C D

  r1 r    2 h A  2i  C  C  C 0 .r C r /C 2.   /r C  P s s

bQ 30   C r  r    2 C C2 2 f0 r1 .r C r  / C 1 Π.r C  2 / .r  C  2 /g; P s ( ) C C i 2 h r   r A  0C .r C C r  / 1 2.C   /r    2 C3 D P s s C



bQ 30 C C C r C r C   2 C C2 2 f0 r1 .r C r  / 1 Π.r C  2 / .r  C  2 /g; P s

37 Local and Global Solvability of Free Boundary Problems for the. . .

1993

!kC D!k  !k ;

k D 1; 2; 3; " #  C C C  2 .   /.r r   / 1 C C C C 1 bQ ˛ C i˛ C3  .r  r1 /  !˛ D C C  r C  r  rC C r   .C   /.r1 r    2 /     C i˛ C3  .r  r1 / C ; ˛ D 1; 2; rC C r !3 D

.r1C r C   2 /C3C C .r1 r    2 /C3 ; rC C r

B where A D i1 bQ 1 C i2 bQ 2 , e b 03 D b3 C  e , and P is as follows: s

P D

X

2

0˙ s.r1˙ r ˙   2 / C 0C 0 s.r1C r  C r1 r C  2 2 /

˙

(

C 4C  s 2 1  C 

X ˙

C 4s 2

X

2

˙ r ˙

˙

˙

r  s

r1˙

r ˙  r1˙ r  r1   2 r˙ s s

)

X r r    2 r  r1   2 0˙ r1˙ 1 C  2 : s s

(95)

˙

For estimation of solution (94), it is important to find a lower bound for the denominator P . To this end, one can use the following lemma. Lemma 10. Let  2 R2 , s 2 C, Re s D > 0, and let the expression P have the form P .s; / D sM .s; / C   2 q.s; /;

(96)

where M and q possess the following properties: c1 .jsj C  2 / 6 jM .s; /j 6 c2 .jsj C  2 /; jDs M .s; /j 6 C;

M .0; / D m0  2 ;

(97)

m0 > 0;

c3 .jsj C  2 /1=2 6 jq.s; /j 6 c4 .jsj C  2 /1=2 ; jDs q.s; /j 6 c.jsjC 2 /1=2 ;

(98) q.0; /  q0 jj > 0:

If  1, then ˚

˚

c5 jsj.jsj C  2 / C   2 .jsj C  2 /1=2 6 jP j 6 c6 jsj.jsj C  2 / C   2 .jsj C  2 /1=2 : (99)

1994

I.V. Denisova and V.A. Solonnikov

Proof. 1. At first, it is assumed that jsj 6 ˛ 2 with ˛ 2 .0; 1/. Obviously, P .; s/=s D m0  2 C

 q0 jj3 C N .s; /; s

where N .s; / D M .s; /  M .0; / C

  2  q.s; /  q.0; / : s

By expressing the increments of M and q in terms of derivatives with respect to s, one obtains jN j 6 C jsj C C1

2 C1 jsj 6 C jsj C C1 jj 6 C ˛ 2 C . 2 C  1 /: jjjsj 2

Hence ˇ ˇ ˇ C1 jsj ˇˇ ˇ ˇ ˇ jP j=jsj > ˇRe.P =s/ˇ > ˇm0  2  .C ˛ C C1 =2/ 2  ˇ > c.jsj C  2 /; 2 if ; ˛ are small and is large enough. And   c3  jj3 6   2 jqj 6 c jP j C jsjjM j 6 c jP j C c2 jsj.jsj C  2 / 6 cjP j: 2. Now let jsj > ˛ 2 : In view of the estimates c1 jsj.jsjC 2 / 6 jsjjM j 6 jP jC  2 jqj 6 jP jCc.˛/  2 jsj1=2 6 jP jC

c.˛/ 2  jsj; 1=2

the choice of large enough yields jsj.jsj C  2 / 6 cjP j and  2 jqj 6 c.˛/jsj1=2  2 6

c.˛/ c1 .˛/ c1 .˛/  jsj 2 6 1=2 jsjjM j 6 1=2 jP j C  jqj 2 : 1=2

Hence for large , the inequality jqj 2 6 cjP j holds. The estimate of jP j from above is obvious. This completes the proof of the lemma. t u Lemma 11. For 8 2 R2 , 8s 2 C, Res D > 0 > 0, and for  > 0 the expression P defined by (95) satisfies estimates (99) provided that 0 is sufficiently large.

37 Local and Global Solvability of Free Boundary Problems for the. . .

1995

Proof. It is evident that the denominator P has form (96), where the expression M C r C .r  r   2 /C r  .r C r C  2 /

0 1 1 is “the Lopatinskiˇı determinant” in the case  D0, qD 0 1 1 . s Due to the Lopatinskiˇı condition, M is subject to inequalities (97). One can also demonstrate this by direct calculation taking into account that

r ˙  r1˙ .1 C ˇ ˙ / ; D s .2 ˙ C 1˙ /.r ˙ C r1˙ /

r ˙ r1˙   2 s C .3 C ˇ ˙ / 2 ; D s .2 ˙ C 1˙ /.r ˙ r1˙ C  2 / (100) similarly to the case of compressible and incompressible fluids [6] (see Sect. 6.2). Kubo et al. [18] deduced the same estimates for M (in the case of zero surface tension) from solution uniqueness by contradiction. r ˙ r1˙ s s!0

Moreover, one obtains that jDs M .s; /j 6 C and, in view of lim r ˙ r1˙  2 1Cˇ ˙ , lim s 2.2 ˙ C 1˙ /jj s!0

M .0; / D

D

3Cˇ ˙ , 2.2 ˙ C 1˙ /

D

that

 0  2 .1 C ˇ  /.3 C ˇ C / C 0C  2 .1 C ˇ C /.3 C ˇ  / C .2 C ˇ C /.2  C 1 / .2 C ˇ  /.2 C C 1C / C

20C 0  2 .5 C 2ˇ C C 2ˇ  C ˇ C ˇ  / > m0  2 ; .2 C ˇ C /.2 C ˇ  /

m0 > 0:

(101)

Properties (98) of q are easily verified. In particular, q.0; / D

! 0 .3 C ˇ C / 0C .3 C ˇ  / C jj  q0 jj > 0: 2.2  C 1 / 2.2 C C 1C /

(102) t u

Then Lemma 10 guarantees (99).

Remark 5. Lemma 11 is valid for ˇ ˙ > .1; 1/. This follows from the energy estimate for the problem in the case  D 0 and from (101), (102) for such ˇ ˙ . Next, on the basis of Lemma 11, the following estimates of solution (94) were obtained [5]. Lemma 12. For 8 2 R2 , 8s 2 C, Res D > 0 > 0, the vectors W, V˙ defined by formulas (94) are subject to the inequalities 3 X

e  jBj C jW˙ j 6 c. 0 / p jsj C  2 jsj C  2 iD1 ! 3 X e jBj ˙ Q : jbi j C p jV j 6 c. 0 / jsj C  2 iD1 jbQ i j

! ;

(103)

1996

I.V. Denisova and V.A. Solonnikov

The strict positiveness of jP j guarantees the uniqueness of the solution of problem (90) with f D 0 by the inverse Fourier-Laplace transform. Using the Parseval equality for transformation (22), Z1 Z

e. C i 0 ; ; x3 /j2 d d0 D .2 /3 jf

1 R2

Z1 e

2 t

Z

jf .x; t/j2 dx 0 dt;

(104)

R2

0

one can make sure that estimate (90) is valid for problem (90) with f D 0 on the basis of (103) (see some explanation in Sect. 6.2). In studying the case of general f, the estimates are used of a solution to the Cauchy problem for the equation in (90). In this way, Theorem 9 is proved (see [5]).

5.4

Local Solvability in Weighted Hölder Spaces for the Problem with Closed Interface

This subsection is concerned with problem (84) that is studied in Hölder spaces with power-like weights. These classes of functions were introduced in [12] for the analysis of a similar problem governing the motion of two incompressible fluids. The Hölder spaces with power-like weights are defined as follows. Let ˝ be a domain in Rn , n 2 N, and for T > 0 QT D ˝  .0; T /; in addition, let ˛ 2 .0; 1/, ˛;˛=2 .QT / denotes the set of functions f on QT having norm: ˇ > 0. The symbol Cˇ .˛;˛=2/

jf jˇ;QT

.˛;˛=2/

D jf jˇ;QT C hf iˇ;QT ;

where jf jˇ;QT D sup sup .1 C jxj/ˇ jf .x; t/j; t2.0;T / x2˝

.˛;˛=2/

hf iˇ;QT

.˛/

.˛=2/

D hf ix;ˇ;QT C hf it;ˇ;QT ;

.˛/

hf ix;ˇ;QT D sup sup .1 C jxj/ˇ sup jf .x; t /  f .y; t/jjx  yj˛ ; t2.0;T / x2˝

./

hf it;ˇ;QT D sup .1 C jxj/ˇ x2˝

y2˝

sup jf .x; t/  f .x; /jjt   j ;

 2 .0; 1/:

t;2.0;T /

Here and below, the notation .˛/

.˛/

jf jx;ˇ;QT D jf jˇ;QT C hf ix;ˇ;QT ; is introduced.

./

./

jf jt;ˇ;QT D jf jˇ;QT C hf it;ˇ;QT

37 Local and Global Solvability of Free Boundary Problems for the. . . kC˛;.kC˛/=2

Let k 2 N. By definition, the space Cˇ f with finite norm: .kC˛; kC˛ 2 /

jf jˇ;QT

X

D

1997

.QT / consists of the functions .kC˛; kC˛ 2 /

jDxr Dt s f jˇ;QT C hf iˇ;QT

;

jrjC2s6k

where .kC˛; kC˛ 2 /

hf iˇ;QT

D

X

X

.˛;˛=2/

hDxr Dt s f iˇ;QT C

jrjC2sDk

. 1C˛ /

2 hDxr Dt s f it;ˇ;Q : T

jrjC2sDk1

The symbol CˇkC˛ .˝/, k 2 N [ f0g, means the space of functions f .x/, x 2 ˝, with the norm: X .kC˛/ .kC˛/ jf jˇ;˝ D jDxr f jˇ;˝ ; Chf iˇ;˝ ; jrj6k

where .kC˛/

hf iˇ;˝

D

X

.˛/

hDxr f iˇ;˝ D

jrjDk

X jrjDk

sup .1Cjxj/ˇ sup jDxr f .x/Dyr f .y/jjxyj˛ :

x2˝

y2˝

The subscript ˇ in all these norms is omitted if it equals zero. Theorem 10. Let ˛ 2 .0; 1/, > 0, T < 1, and  2 C 3C˛ , 0  1 2 C 1C˛ .˝0 [ ˝0C /, R1 > 0 .y/ > R0 > 0, 1 > 0, p 2 C 3 .RC /, f; Dx f 2 ˛; ˛

.˝0 [ ˝0C /,  2 C 1C˛ . /,  .x/ > 0 > 0 for x 2  . In C 2 .R3T /, v0 2 C2C˛ addition, the compatibility conditions Œv0 j D 0;

h

iˇ ˇ 01 .y/r  T.v0 ; p.0 // ˇ D 0; 

ŒT.v0 ; p.0 //n0 j D  H0 n0

are assumed to hold; here H0 .y/ D n0  .0/y is the doubled mean curvature of  . Then the problem (84) is uniquely solvable on some finite time interval .0; T0 /, T0 6 T , whose length depends on the norms of f, v0 ; p, 0 and on the curvature of 2C˛;1C˛=2 .DT0 /.  . The solution u 2 C This theorem was proved on the basis of the Banach theorem, the problem (84) being considered as a perturbation of the linear system (86) [8]. The solvability of problem (86) is established by constructing a regularizer, while the uniqueness follows from a priori coercive estimates obtained by the Schauder method. Both these methods use explicit solution (94) of model problem (90). In the proof, arguments similar to those for the case of a single liquid are employed (see Sects. 3 and 4). For Theorem 10, Remark 5 is also valid.

1998

I.V. Denisova and V.A. Solonnikov

6

Evolution of Compressible and Incompressible Fluids Separated by a Closed Free Interface

6.1

Local Existence Theorem, the Case of Positive Surface Tension

Let, for definiteness, at the initial instant t D 0 a compressible fluid fill a bounded region ˝0C  R3 , located inside an incompressible fluid occupying the domain ˝0  R3 n ˝0C . Nothing hinders to change the positions of the fluids with respect to each other. 2 C Let C > 0 be the dynamic viscosity of the compressible liquid, C 1 > 3 . The kinematic viscosity of the incompressible fluid and its density coefficient are denoted by the constants  > 0 and  > 0, respectively. The compressible fluid is assumed to be barotropic. For t > 0, it is necessary to determine the interface t  ˝t \ ˝tC . Moreover, it is required to find the density function C .x; t / > 0 of the compressible fluid, the pressure function p  .x; t / of the incompressible fluid, as well as the velocity vector field of both fluids v.x; t / D .v1 ; v2 ; v3 / which solve the initial–boundary value problem [7]: ˙ .Dt v C .v  r/v/  r  T D ˙ f; Dt ˙ C r  .˙ v/ D 0 in ˝t [ ˝tC ; vjtD0 D v0 C jtD0 D 0C ˇ Œvˇt 

lim

x!x0 2t ; x2˝tC

v.x/ 

in ˝0 [ ˝0C ;

in ˝0C I lim

x!x0 2t ; x2˝t

v ! 0; jxj!1

v.x/ D 0;

(105) p  ! 0I jxj!1

ˇ ŒTnˇt D  H n

on t ;

(106) where the function ˙ is equal to C .x/ in ˝tC and to the constant  in ˝t ; the stress tensor is as follows: TD

( C p C .C / C C 1 r  v I C  S.v/ p  I C  S.v/

in ˝tC ; in ˝t ;

(107)

.S.v//ik D @vi =@xk C @vk =@xi ; i; k D 1; 2; 3I  D   ; p C .C / is the pressure of the compressible fluid given by a smooth positive strictly increasing function of a positive argument; f is the given vector field of mass forces; v0 is the initial value of the velocity vector field; 0C is the initial density distribution of the compressible fluid;  > 0 is the surface tension coefficient; n is the outward normal vector to ˝tC ; H .x; t / is twice the mean curvature of t ;   0 D @˝0C is a given surface.

37 Local and Global Solvability of Free Boundary Problems for the. . .

1999

Since the liquids are supposed to be immiscible, condition (4) should be imposed on system (105) and (106). Then again ˝t˙ D fx D x.y; t /jy 2 ˝0˙ g. This problem was studied in [6] where its local solvability in the SobolevSlobodetskiˇı spaces was obtained. The technique applied there was developed in [35, 36] for the study of drop evolution in vacuum and modified by Solonnikov and Tani [42] for the case of bubble motion in vacuum. Let T 2 .0; 1, t > 0. Henceforth, the following notation is used: ˝  ˝t [ ˝tC ; QT D ˝  .0; T /I [˝t˙  ˝t [ ˝tC ; QT˙ D ˝0˙  .0; T /; DT D QT [ QTC : Let Bd be the ball fx W jxj < d g and BdT  .Bd n ˝0C /  .0; T /. A coordinate system fxg is chosen so that ˝0C is contained in the ball Bd ; d < 1. Theorem 11 (Local solvability of (105), (106) with  > 0). For some l 2 5=2Cl .1=2; 1/, it is assumed that  2 W2 , 0C 2 W21Cl .˝0C /, 0 < R0 6 0C .y/ 6 l;l=2 R1 < 1, y 2 ˝0C , p C 2 C 3 .RC /, f 2 W2 .R3T /, 0 < T < 1, f.; t / 2 3 ˇ 2 C .R / for 8t 2 Œ0; T , f.y; /, rf.y; / 2 C .0; T / for 8y 2 R3 with some ˙ ˇ 2 .1=2; 1/. In addition, let the initial velocity vector field v0 2 W1Cl 2 .[˝0 / satisfy the compatibility conditions: ˇ r  v0 D 0 in ˝0 ; Œv0 ˇ D 0;

ˇ Œ˙ ˘0 S.v0 /n0 ˇ D 0:

Under these hypotheses, there exists a constant T0 2 .0; T  such that problem (105), (106), (4) is uniquely solvable in the interval .0; T0 / and its solution .u; q/ written in the Lagrangian coordinates has the properties: u 2 2Cl;1C l

l; l

l; l

lC 12 ; 2l C 14

2 2 W2 .DT0 /, q 2 W2;loc .QT0 /, rq 2 W2 2 .QT0 /, qjGT0 2 W2 and the following inequality holds:

.2Cl;1Cl=2/

N Œu; q  kukDT

.l;l=2/

C krqkQ

T0

0

.l;l=2/

C kqkB 

d T0

C kqkW lC1=2;l=2C1=4 .G 2

1l  n 6 c1 c2 C c3 T0 2 kv0 kW1Cl .[˝ ˙ / kjfjkR3 C kv0 kW1Cl .[˝ ˙ / T0 0 0 2 2 o C C C kH0 kW lC1=2 . / C kp .0 /kW 1Cl .˝ C / : 2

2

.GT0 /,

T0 /

6

0

The value T0 depends on the norms of f, v0 , 0 , p C and on the doubled curvature H0 of  . Remark 6. It should be noted that this theorem was stated in [6] under some artificial restrictions on the liquid viscosities, namely,

 < C =R1 ;

 > C

2000

I.V. Denisova and V.A. Solonnikov

which arose from the estimates of explicit solution (111) of a model problem with plane interface (see (116)). But now a method is presented to evaluate this solution without any restrictions on the viscosities (see Lemma 10 above). The main ideas of the proof of Theorem 11 are given below. After transformation to the Lagrangian coordinates (5), problem (105), (106), (4) is solved by successive approximations, similarly to the case of a single incompressible fluid [36] or a single compressible one [42]. By using the formulas for twice the mean curvature (10) and ˇ .t /Xu ˇ D .0/y C

Z

t

P /y d C .t / .

0

Z

t

u d ; 0

the normal part of the transformed last boundary condition in (106) can be rewritten as follows: ˇ  n0  T0u .u; q/n ˇGT   n0  .t /

Z

t 0

ˇ uˇ

Z d D H0 .y/ C  GT

t

P /y d n0  .

0

ˇ C .n0  n/p C .0C Ju1 /ˇGT ;

C C 0   where T0u .w; q/  C 1 Iru  w C  Su .w/ in ˝0 and Tu .w; q/  q I C  Su .w/  in ˝0 . One puts u.0/ D 0; q .0/ D 0 and defines u.mC1/ ; q .mC1/ ; m 2 N[0; as a solution of the problem:

0C .y/Dt u.mC1/ Am rT0m .u.mC1/ / D 0C .y/f.Xm ; t /Am rp C .0C Jm1 /  Dt u.mC1/  rm2 u.mC1/ Crm q .mC1/ D f.Xm ; t /;

rm  u.mC1/ D0

ˇ ! 0; u.mC1/ ˇtD0 D v0 in ˝0 [ ˝0C ; u.mC1/  jxj!1 ˇ ˇ Œu.mC1/ ˇGT D 0; Œ˙ ˘0 ˘m Sm .u.mC1/ /nm ˇGT D 0; 

ˇ n0  T0m .u.mC1/ ; q .mC1/ /nm ˇGT  n0  m .t / Z

t

D  H0 .y/ C  0

Z 0

t

in QTC ; in QT ;

! 0; q .mC1/  jxj!1 (108)

ˇ u.mC1/ ˇGT d D

ˇ n0  P m . /y d C .n0  nm /p C .0C Jm1 /ˇGT :

Here the notation rm D ru.m/ etc. is used; m .t / is the Laplace-Beltrami operator on m .t /; nm is an outward normal to the surface m .t / D fx D Xm .y; t /jy 2  g; ˘m is the on its tangential plane; Am is the cofactors matrix for the Jacobi n projector o .m/ matrix aij of (5) corresponding to the vector field u.m/ .

37 Local and Global Solvability of Free Boundary Problems for the. . .

2001

For the linearized problem with a given vector u 0C .y/Dt w  Ar  T0u .w/ D f 0 Dt w   ru2 w C ru s D f; ˇ wˇtD0 D w0

Œn0 

ˇ

ru  w D r

in QT ;

in ˝0 [ ˝0C ;

ˇ ŒwˇGT D 0; T0u .w; s/nˇ

in QTC ;

! 0; s ! 0; w jxj!1 jxj!1 ˇ Œ˙ ˘0 ˘ Su .w/nˇGT D ˘0 a; Zt

  n0  .t /

ˇ wˇ d D b C 

0

(109)

Zt B d ;

t 2 .0; T /;

0

unique solvability theorem is valid for any finite time interval .0; T / and for a vector 2Cl;1Cl=2 u 2 W2 .DT / that changes continuously, when the boundary  is crossed, and satisfies the inequality (88) with a small number ı. l1=2;l=21=4 2Cl;1Cl=2 Since n0  P m . /y 2 W2 .GT / if u.m/ 2 W2 .DT / [36], and lC1=2 since H0 2 W2 . /, by the existence theorem for linearized system (109), problem (108) is solvable in an interval .0; Tm / in which u.m/ ; q .m/ are determined and u.m/ satisfies condition (88) with a sufficiently small ı > 0. It is necessary to show that for 8m Tm ˚> T 0 > 0, N Œu.m/ ; q .m/  are uniformly bounded in .0; T 0 / and that the sequence u.m/ ; q .m/ converges to a solution of the nonlinear problem. The proof of these facts is the same as in the case of a single fluid; it was presented in [36] in detail. A thorough proof of existence theorem for linearized system (109) was given in [6] (Theorem 1.2). It is also based on successive approximation method, and as in the case of (108), the role of the first approximation plays a solution of system (109) with u D 0. Existence theorem for this problem is stated as follows. 3=2Cl

Theorem 12. Let  2 W2

, 0C 2 W21Cl .˝0C / for some l 2 .1=2; 1/, 0 < l; l

R0 6 0C .x/ 6 R1 < 1, x 2 ˝0C . In addition, it is assumed that f 2 W2 2 .DT /, 1Cl; 12 C 2l

r 2 W2

lC 12 ; 2l C 14

0;1C 2l

.QT /, r D r  R; R 2 W2 lC 12 ; 2l C 14

W2 .GT /, b 2 W2 the compatibility conditions ˇ Œw0 ˇ D 0;

˙ .QT /, w0 2 W1Cl 2 .[˝0 /, a 2

l 12 ; 2l  14

.GT /; B 2 W2

ˇ ˇ Œ˙ ˘0 S.w0 /n0 ˇ D ˘0 aˇtD0 ;

.GT / with T < 1 and that

ˇ r  w0 D r ˇtD0

in ˝0

are satisfied. Then problem (109) with u D 0 is uniquely solvable, and its solution .v; p/ ˇ 2Cl;1C 2l l; l l; l has the properties: v 2 W .DT /, p 2 W 2 .Q /, rp 2 W 2 .Q /, p ˇ 2 2

lC 1 ; l C 1 W2 2 2 4 .GT /

and

2;loc

T

2

T

GT

2002

I.V. Denisova and V.A. Solonnikov .2Cl;1Cl=2/

N Œv; p  kvkDT

.l;l=2/

C krpkQ

.l;l=2/

C kpkW lC1=2;l=2C1=4 .G / 6 T dT 2  .l;l=2/ 6 c16 .T / kfkDT C kw0 kW1Cl .[˝ ˙ / C krkW 1Cl;0 .Q / C kRkW0;1Cl=2 .Q / T

C kpkB 

0

2

C T l=2 kDt RkQT C kakWlC1=2;l=2C1=4 .G 2

T

T

2

T

C kbkW lC1=2;l=2C1=4 .G / T 2  C  kBkW l1=2;l=21=4 .G /  c16 .T /F; / 2

C T l=2 kbkW 1=2;0 .G

2

T/

T

2

c16 .T / being a nondecreasing function of T . This theorem can be proved by constructing a regularizer [36, 42] or by analyzing a generalized solution [11]. Both these methods use maximal regularity estimates of a solution to the model problem with plane interface  which will be studied in the next subsection. For two-phase problems governing the flow of one-type liquids, similar results are valid as well. It is worth noting that in the case of two incompressible fluids, they were stated in [4] (see also [41]). As for two compressible fluids, Theorem 11 is an analog to Theorem 8.

6.2

Model Problem with a Plane Interface Between the Fluids

This section concerns the problem: Dt v   r 2 v C

1 rp D 0; 0

r vD0

Dt v  C r 2 v  . C C 1C /r.r  v/ D 0 ˇ vˇtD0 D 0 ˇ Œvˇx3 D0 D 0;

pC

C 1 r

on

R3C [ R3 ;

in

 R1 D R3  .0; 1/;

C in R1 D R3C  .0; 1/;

! 0; v jxj!1

! 0; p jxj!1

(110)

ˇ   @v˛ @v3 ˇˇ  ˙ C D b˛ .x 0 ; t /; ˛ D 1; 2I @x3 @x˛ ˇx3 D0

ˇ  Zt Zt ˇ ˇ ˙ @v3 ˇ 0 ˇ  vC 2 C v3 x3 D0 d D b3 C  B d  b30 @x3 ˇx3 D0 0

on R21 :

0

Here the following notation has been used: R3˙ D f˙x3 > 0g; R21 D R2  C C C D C =0C ; 0 D @2 =@x12 C .0; 1/; 1C D C 1 =0 ; 0 D constant > 0;

2 2 0 @ =@x2 ; x D .x1 ; x2 /. By taking the Fourier-Laplace transform (22), problem (110) is converted into a system of ordinary differential equations for unknown functions e v; p. Q After solving

37 Local and Global Solvability of Free Boundary Problems for the. . .

2003

this system explicitly, the solution is written in the following form convenient for proving the forthcoming estimates: e v D We0˙ C V˙ e1˙ ;

˙x3 > 0;

(111)

pQ D C3 0 sejjx3 D  C3 .r   jj/.r  C jj/ejjx3 ; x3 < 0; q q q p s s 2, r C D 2 ; jj D C  C  12 C 22 ; j arg zj where r ˙ D C C 1 ˙ .2Cˇ /

< =2 for 8z, ˇ C D 1C = C ; e0˙

r ˙ x3

De

;

e1

D

er

x

 ejjx3 ; r   jj 3

e1C

D

er

Cx

C

 er1 r C  r1C 3

x3

;

(112)

1 1 0 1 0 i1 i1 !1 W D @ !2 A ; V D C3 .r   jj/ @ i2 A ; VC D C3C .r C  r1C / @ i2 A ; !3 jj r1C 0

˚

2 A 2 r  C s  2 .r C r1C   2 / C C r  .r C  2r C r1C C  2 / C 0C sr C D .r   jj/P ˚

2 2 bQ 30  .r C r1C  2 /.r  C 2 /CC  2 .r C C 2 2r C r1C /C0C sr  r1C C ; .r   jj/P      A  3 C  C  C C  r .r C jj/ C r .r  jj/ C 2 r jj C jj C (113) C3 D P s     bQ 30  C C 2 C C2 2  r jj.r C jj/ C .r  jj/ C  .r C  /jj ;  P ˚ 

bQ ˛ C i˛  .r  jj/C3 CC .r C  r1C /C3C C .  C /!3 !˛ D ; ˛D1; 2;  r  CC r C C3

!3 D

.r   jj/jjC3  .r C r1C   2 /C3C : r C rC

In formulas (113), the following notation has been applied: A D i1e b 1 C i2 bQ 2 , e Qb 0 D e b 3 C s B; 3 n o 2 P D 0C s 2 jj C  s.r  C jj/ 0C .r C jj C r  r1C C 2 2 / C 0 .r C r1C   2 / C o n C4.C   / 2 C r C .r C  r1C /jj   r  .r C r1C   2 / C C

o  jj3 n C C 0 r1 s C  .r  C jj/.r C r1C   2 / : s

(114)

2004

I.V. Denisova and V.A. Solonnikov

 q Solution (111), (113) can be obtained by the passage to the limit r1  s C 2 ! jj as ˇ  ! 1 from solution (94) of the model problem with a .2Cˇ  /  plane interface between two compressible fluids (90). On the other hand, it goes over as r1C ! jj into a solution of the corresponding problem for two incompressible liquids [3] (see also [41]). The estimates of solution (111), (113) were considered in details in [6]. It should be only remarked that C3 occurs in all of the expressions with multiplier .r   jj/ which is canceled out with the denominator in C3 . Hence the estimates of solution (111), (113) depend only on the lower bound of the denominator P . The uniqueness of solution (111), (113) for Re s > 0 > 0 is guaranteed by the fact that in this case jP j is bounded away from zero.

Lemma 13. Let  > 0. Then for 8 2 R2 , 8s 2 C, Re s D  1, it holds   1   1 c4 jsj2 C jsj 2 C jj3 jsj 2 Cjj 6 jP j 6 c5 jsj2 C jsj 2 C jj3 jsj 2 Cjj : (115) Remark 7. As it has been mentioned above, Lemma 13 was proved earlier under the assumptions:

 < C;

 > C :

(116)

Now the proof of Lemma 13 eliminating these restrictions is presented. Proof. Similarly to Sect. 5.3, in order to apply Lemma 10 to denominator (114), it has to be rewritten in the form:  P .s; /  .r  C jj/P1 .s; / D .r  C jj/ sM .s; / C  jj2 q.s; / ; where 2

M .s; / D 0C

n o sjj C C   C 2  C C 2 C   .r jj C r r  2 / C  .r r   / C 0 0 1 1 r  C jj

 C C 2 r C .r C  r1C / 2 2 r .r r1   / C 4  .r  C jj/s .r  C jj/s n r C r  .r C  r C / jj.r C r1C   2 / o 1 C 4C   2 C .r  C jj/s .r  C jj/s n rC r C r1C   2 o q.s; / D jj 0C  1 C  : r C jj s 2

C 4C jj3

In view of (100), it is not difficult to show that every summand in M has positive real part and then that M satisfies (97). This was considered in [10] in detail where

37 Local and Global Solvability of Free Boundary Problems for the. . .

2005

P with  D 0 was estimated by (120) (see Lemma 15). By direct calculations, one can verify that (

) 2 2  .3 C ˇ C / C .1 C ˇ C / C  C C 20  M .0; / D  > m0  2 ; 2 C C 1C 2 C C 1C ! 0C  .3 C ˇ C / jj  q0 jj; q0 > 0: q.0; / D C 2 2.2 C C 1C / 2

m0 > 0;

as well as estimates (98) are satisfied. Then for the expression P1 , inequalities (99) are valid which imply estimates (115). t u Remark 8. Lemma 13 is true for ˇ C > 1. The following lemma was proved in [6] (Lemma 2.3) on the basis of (115). Lemma 14. For the coefficients W, V˙ defined by formulas (113), the inequalities ! e jbQ 3 jjj C  jBj ; C p jsj C  2 jsj C  2 ˛D1 ! 2 Q 3 jjj C  jBj X e j b jbQ ˛ j C p jsj C  2 ˛D1 2 X

jWj 6 c6

jV˙ j 6 c7

jbQ ˛ j

(117)

hold. Using the estimates [35] Z

Z 0 ˇ j  ˇ dj e  .x / ˇ2 ˇ jr  j2j 1 C jj2j 1 3 ˇ ˇ ˇ d e1 .x3 / ˇ2  2j 1 0 dx 6 cjr j ; dx 6 c ; ˇ ˇ ˇ ˇ 3 3 j j jr  j2 dx3 dx3 1 1 Z 0 Z 0 ˇ j  dj e0 .x3 / ˇˇ2 dx3 dz ˇ d e0 .x3  z/  (118) ˇ ˇ 1C2~ 6 cjr  j2.j C~/1 ; j j jzj dx3 dx3 1 1 Z 0 Z 0 ˇ j  dj e1 .x3 / ˇˇ2 dx3 dz jr  j2.j C~/1 C jj2.j C~/1 ˇ d e1 .x3  z/  6 c ; ˇ ˇ j j jzj1C2~ jr  j2 dx3 dx3 1 1 0

j 2 N [ 0, for e0 , e1 defined by (112), and similar inequalities for e0C ; e1C with jr1C j instead of jj, one can show by means of the Parseval equality (104) that the relation kjvjk22Cl; ;D 3 6 c 1

Z R2

Z

1

1



jWj2 jr ˙ j2lC3 C jV˙ j2 jr ˙ j2lC1 d0 d

2006

I.V. Denisova and V.A. Solonnikov m;m=2

3 holds; here, kjvjkm; ;D1 .D1 /. 3 is a norm equivalent to the norm in H jjx3  As the function e satisfies inequalities (118) with jr j replaced by jj, for the pressure gradient r pQ D  V .r  C jj/ejjx3 ; x3 < 0; the inequality

Z

Z

 6 c kjrpjk2l; ;D1

R2

1

jV j2 jr  j2 jj2l1 d0 d 1

is valid. And finally, as a consequence of Lemma 14, one concludes that kjvjk2Cl; ;D1 3 C kjrpjkl; ;D  6 1 6c

X 2

 kjb˛ jklC1=2; ;R21 C jb3 jl; ;R21 C kjBjkl1=2; ;R21 ;

˛D1

where jujm; ;R21  where r D

6.3

Z

Z

1

jQu. C i0 ; ; 0/j2 jrj2m d0 d

jj R2

1=2

;

1

p s C  2 (see [6] for detail).

Problem (105), (106), (4) in a Bounded Domain with  D 0

Problem (105), (106), (4) is considered in a domain bounded by a closed surface ˙ ; it is assumed that ˇ vˇ˙ D 0:

(119)

(See Fig. 1.) Let  D 0. The mean density of the compressible fluid is denoted by C m

1  j˝tC j

Z ˝tC

C dx:

It is observed R to be independent of t because of the conservation of compressible fluid mass ˝ C C dx and that of its volume: j˝tC j D j˝j  j˝t j D constant, t where ˝ is the domain bounded by the surface ˙ . C Moreover, the new pressure function p1 D p  p C .m / is introduced: p1C D C  C C C C  C C  p . /p .m / in ˝t and p1 D p p .m / in ˝t . Then nothing is changed in system (105), (106), (119). This problem is noted to be coercive.

37 Local and Global Solvability of Free Boundary Problems for the. . .

2007

Fig. 1 Motion of two-phase fluid in a container

Local existence theorem obtained in [10] is stated below. It was proved on the base of the coercive estimates for explicit solution (111), (112) and (113) of the model problem. It was shown by direct calculations that expression jP j (114) with  D 0 is strictly positive for any positive 0˙ , ˙ and C C 1C > 0. Lemma 15. For 8 2 R2 , 8s 2 C, Re s D > 0, the estimates   jP .s; /j > cjsj jsj1=2 C jj jsj C jsj1=2 jj C  2 ;

(120)

jP j > c.0˙ ; ˙ ; 1C / 5=2 hold. Similar estimates without restrictions (116) on the coefficients of fluid viscosities for the determinate P with  D 0 were obtained by Kubo et al. [17] but it was done in a nonconstructive way. 3=2Cl

Theorem 13. For some l 2 .1=2; 1/, it is assumed that  ; ˙ 2 W2 , C 1Cl C C C   C 2 0 2 W2 .˝0 /, 0 < 0 6 0 .y/ 6 1 < 1, y 2 ˝0 , p l;l=2 C 3 .RC /, f 2 W2 .QT /, 0 < T < 1, f.; t / 2 C2 .˝/ for 8t 2 Œ0; T , f.y; /, rf.y; / 2 Cˇ .0; T / for 8y 2 R3 with some ˇ 2 Œ1=2; 1/. In addition, ˙ let the initial velocity vector field v0 2 W1Cl 2 .[˝0 / satisfy the compatibility conditions: r  v0 D 0 in ˝0 ;

ˇ v0 ˇ˙ D 0;

ˇ Œv0 ˇ D 0;

ˇ Œ˙ ˘0 S.v0 /n0 ˇ D 0:

2008

I.V. Denisova and V.A. Solonnikov

Under these hypotheses, there exists a constant T0 2 .0; T  such that problem (105), (106), (4) with  D 0 is uniquely solvable in the interval .0; T0 , and its solution in the Lagrangian coordinates .u; q/ has the properties: 2Cl;1Cl=2 l;l=2 l;l=2 .DT0 /, q 2 W2 .QT0 /, rq 2 W2 .QT0 /, qjGT0 2 u 2 W2 lC1=2;l=2C1=4

W2

.GT0 / and

.2Cl;1Cl=2/

kukDT

.l;l=2/

C krqkQ

.l;l=2/

C kqkW lC1=2;l=2C1=4 .G / 6 T0 2    1l .1;ˇ/ .0;ˇ/ 6 c1 c2 C c3 T0 2 kv0 kW1Cl .[˝ ˙ / kfkWl;l=2 .Q / C jfjQT C jrfjQT C T0

0

C kqkQ

T0

0

2

2

C kv0 kW1Cl .[˝ ˙ / C kp 2

0

C C kC  m kW 1Cl .˝ C / 6 c4 k0C  m kW 1Cl .˝ C / ; 2

t

2

C

T

.0C /

p

C



C .m /kW 1Cl .˝ C / 2 0

;

8t 2 .0; T0 :

0

The value of T0 and constants c1 ; c4 depend on the norms of f, v0 , 0C and p C . Remark 9. Theorem 11, similar to Theorem 13, has been stated in Sect. 6.1 with 5=2Cl  2 W2 since the surface tension has been there taken into account. In that case, one needs the additional smoothness of the interface because it is necessary to calculate the norm of the mean curvature of  . In the present case, Theorem 13 is 3=2Cl that is as regular as a solid boundary. valid for any initial interface  2 W2 It was shown in [10] that the L2 –norms of the velocity and deviation of compressible fluid density from the mean value decay exponentially with respect to time as t ! 1. The proof was based on Theorem 13 and on the idea of constructing a function of generalized energy, proposed by Padula in [22]. Theorem 14. A solution of problem (105), (106), (4) with  D 0 is assumed to l ˙ ˙ be defined on Œ0; T , T 2 .0; C1/ W v 2 W1Cl 2 .[˝t /, Dt v 2 W2 .[˝t /,  2 1Cl C C l W2 .˝t /, Dt  2 W2 .˝t / for almost all t 2 Œ0; T . Let, in addition, f.;  / 2 RT L2 .˝/, t 2 .0; T , and 0 eˇ kf.;  /k2˝ d < 1 with some ˇ > 0. Then for t 2 .0; T  n C 2 C 2 kv.; t /k2˝ C kC .; t /  m k C 6 cebt kv0 k2˝ C k0C  m k C ˝t

˝0

Z

t

C 0

o

eb kf.;  /k2˝ d ;

where b ˇ is a positive constant independent of t.

37 Local and Global Solvability of Free Boundary Problems for the. . .

6.4

2009

Global Solvability of Problem (105), (106), (4) Without Surface Tension

6.4.1 Statement of the Main Result This subsection contains the proof of global-in-time solvability of evolutionary free boundary problem (105), (106), (119), (4) with  D 0 governing the motion of two viscous fluids of different types contained in a bounded vessel and separated by a free interface. The domain ˝ D ˝tC [ t [ ˝t is fixed; the surface ˙ D @˝ is bounded away from t . The viscous part of the stress tensor (107) is given by ( 0

T .v/ D

TC .v/ D C S.v/ C C 1 Ir  v; T .v/ D  S.v/;

x 2 ˝tC ;

x 2 ˝t :

It is clear that j˝t˙ j D mes˝t˙ are independent of t . Let M ˙ denote the total masses of the fluids. The density of the compressible fluid is represented in the form C C .x; t / D m C  C .x; t /, and the pressure of the incompressible fluid is done as   C follows: p .x; pm C   .x; t /, where m D M C =j˝tC j is the mean value R t/ D C C  C C of  .x; t /, ˝ C  .x; t / dx D 0 and pm D p .m /, p C .C /  p.C /. Then t dynamic jump conditions (106) across t reduce to ˇ Œvˇt D 0;



ˇ C  p C .C / C p C .m / C   n C ŒT0 .v/nˇt D 0:

In addition, in view of the transport theorem one has dj˝tC j D dt

Z ˝tC

r  v.x; t / dx D 0:

(121)

The passage is performed to the Lagrangian coordinates y 2 ˝ connected with the Eulerian coordinates x 2 ˝ by (5). Under this transformation, the variable domains ˝t˙ go over into the fixed domains ˝0˙ and problem (105), (106), (119), (4) with  D 0 is converted into  8 C C C / D 0; Dt # C C OC ru  u D 0 in Q1 ; O Dt u ru  T0u .u/Cru p.OC / p.m ˆ ˆ ˆ ˆ  ˆ <  Dt u ru  T0u .u/ C ru #  D 0; ru  u D 0 in Q1 ; ˇ ˇ ˇ C C C  Cˇ ˇ ˇ ˆ ˆ ˆ u tD0 D v0 in ˝0 [ ˝0 ; # tD0 D #0 in ˝0 ; u ˙ D 0; ˆ ˆ ˇ ˇ : ˇ Œu D 0; .p C .OC / C p C .C / C #  /n C ŒT0 .u/nˇ D 0 on G1 ; 0

m

u

0

(122) where OC .y; t / D .X .y; t /; t /, # ˙ .y; t / D  ˙ .X .y; t /; t /, QT˙ D ˝0˙  .0; T /, DT D QT [ QTC , GT D 0  .0; T /, T 1. Here the notation of Sect. 1 is used (see formulas (7), (11), (13), and so on). One notes that Ju  1 in ˝0 . The viscous

2010

I.V. Denisova and V.A. Solonnikov

C part of the transformed stress tensor T0u .w/ is given by TC u .w/ D  Su .w/ C C   1 Iru  w, Tu .w/ D  Su .w/. By virtue of (121), problem (122) can be written in the form:

8 C C m Dt u  r  T0 .u/ C p1 r# C D lC ˆ 1 .u; # /; ˆ ˆ Z ˆ   ˆ ˆ C ˆ D # C C C r  u  1 ˆ r  u.z; t / dz D l2C .u; # C / in Q1 ; t ˆ m C ˆ ˆ j˝0 j ˝0C ˆ ˆ ˆ <    r  u D l2 .u/ in Q1 ;  Dt u  r  T0 .u/ C r#  D l 1 .u; # /; ˇ ˇ ˇ ˆ ˆ ˇ ˆ u D v0 in ˝0 [ ˝0C ; # C ˇtD0 D #0C in ˝0C ; uˇ˙ D 0; ˆ ˆ ˆ tD0 ˆ ˇ ˇ ˆ ˆ ˆ Œuˇ D 0; ˘0 ŒT0 .u/n0 ˇ D l3 .u/; ˆ 0 0 ˆ ˆ ˆ ˇ : C  0  p1 # C # C Œn0  T .u/n0 ˇ0 D l4 .u; # C / on G1 ;

(123)

C C where p1 D p 0 .m / is the derivative of the pressure p./ calculated at the point m , and C C C C C lC 1 .u; # / D # Dt u C ru  Tu  r  T .u/ C p1 r#  C C  ru p.m C # C /  p.m / ;      l 1 .u; # / D ru  Tu .u/  r  T .u/ C r#  ru # ; Z   1 l2C .u; # C / D  # C r  u  r  u.z; t / dz j˝0C j ˝0C Z    1 C C C .m C # / .rru /  u  r  u.z; t / 1J .z; t / dz ; u j˝0C j ˝0C

l2 .u/ Dr  u  ru  u D r  L.u/;

L.u/ D .I  AT /u;

(124)

ˇ l3 .u/ D˘0 Œ˘0 T0 .u/n0  ˘ T0u .u/nˇ0 ;

ˇ C C C # C /  p.m /  p1 # C  Œn0  T0 .u/n0  n  T0u .u/nˇ0 : l4 .u; # C / Dp.m Theorem 15 (Global existence and uniqueness for the nonlinear problem). Let 3

Cl

˙; 0 2 W22 ; l 2 . 21 ; 1/. For arbitrary v0 2 W21Cl .[˝0˙ /, #0C 2 W21Cl .˝0C / satisfying the compatibility and smallness conditions ˇ r  v0 D0 in ˝0; Œv0 ˇ0 D 0; Z #0C .z/ dz D 0;

ˇ Œ˘0 T0 .v0 /n0 ˇ0 D 0;

v0 j˙ D 0;

˝0C

kv0 kW lC1 .[˝ ˙ / C k#0C kW 1Cl .˝ C / 6   1; 2

0

2

0

(125)

37 Local and Global Solvability of Free Boundary Problems for the. . .

2011

problem (123) has a unique solution defined for all t > 0 and satisfying the inequality keˇt ukW 2Cl;1Cl=2 .D /C keˇt # C kW 1Cl;0 .QC /C keˇt Dt # C kW 1Cl;0 .QC / 1

2

1

2

C

1

2





C ke Dt # kW l=2 ..0;1/IW 1 .˝ C // C ke r# kW l;l=2 .Q /C ke # kW lC1=2;l=2C1=4 .G / 1 1 0 2 2 2 2 n o

c kv0 kW 1Cl .[˝ ˙ /C k#0C kW 1Cl .˝ C / ; (126) ˇt

ˇt

2

0

2

ˇt

0

where ˇ > 0.

6.4.2 Linear Problems The proof of Theorem 15 is based on the analysis of the linear problem: 8 C m Dt v  r  T0 .v/ C p1 r C D f; ˆ ˆ ˆ Z ˆ   ˆ ˆ 1 C C ˆ ˆ D  C  r  v.z; t / dz D h in QTC ; r  v  t ˆ m C ˆ C ˆ j˝ j ˝ 0 ˆ 0 ˆ ˆ <  0   Dt v  r  T .v/ C r# D f; r  v D h in QT ; ˇ ˇ ˆ ˆ Cˇ ˇ ˆ v D v ;  D 0C ; ˆ 0 ˆ tD0 tD0 ˆ ˆ ˇ ˇ ˇ ˆ ˆ ˆ vˇ˙ D 0; Œvˇ0 D 0; Œ˘0 T0 .v/n0 ˇ0 D ˘0 b; ˆ ˆ ˆ ˆ ˇ :  p1  C C   C Œn0  T0 .v/n0 ˇ0 D b  n0 on GT ;

(127)

3=2Cl

Theorem 16 (Global solvability for the linear problem). Let ˙, 0 2 W2 , l;l=2 lC1;0 C l 2 .1=2; 1/, T 1. For arbitrary f 2 W2 .DT /, hjQC 2 W2 .QT / \ T

l=2

W2 ..0; T /I W21 .˝0C //; hjQT 2 W21Cl;0 .QT / such that hjQT D r  H, H 2 0;1Cl=2 lC1=2;l=2C1=4 W2 .QT /, b 2 W2 .GT /, v0 2 W2lC1 .[˝0˙ /, 0C 2 W21Cl .˝0C /, satisfying the compatibility conditions: ˇ ˇ r  v0 .y/ D h.y; 0/ in ˝0 ; Œv0 ˇ0 D 0; Œ˘0 T0 .v0 /n0 ˇ0 D ˘0 bjtD0 ; v0 j˙ D 0; problem (127) has a unique solution subjected to the inequality: kvkW 2Cl;1Cl=2 .D / Ck C kW 1Cl;0 .QC / C kDt  C kW 1Cl;0 .QC / CkDt  C kW l=2 ..0;T /IW 1 .˝ C // T T T 2 2 2 2 0 2 n C kr  kW l;l=2 .Q / Ck  kW lC1=2;l=2C1=4 .G / c.T / kfkW l;l=2 .D / C kv0 kW 1Cl .[˝ ˙ / 2

T

T

2

2

C k0C kW 1Cl .˝ C / C khkW 1Cl;0 .DT / C khkW l=2 ..0;T /IW 1 .˝ C // 2 0 2 2 0 2 o C kHkW 0;1Cl=2 .Q / C kbkW lC1=2;l=2C1=4 .G / : 2

T

2

T

T

0

2

(128)

2012

I.V. Denisova and V.A. Solonnikov

Moreover, if hj˝0 D 0,

R

˝0C

h.z; t / dz D 0;

kebt hkW 1Cl;0 .QC / , kebt bkW lC1=2;l=2C1=4 .G T

2

keˇt vkW 2Cl;1Cl=2 .D 2

2

T/

T/

R

˝0C

0C .z/ dz D 0, kebt fkW l;l=2 .D / , 2

T

< 1, b > 0, then

C keˇt  C kW 1Cl;0 .QC / C keˇt Dt  C kW 1Cl;0 .QC / 2

2

T

C



T



C ke Dt  kW l=2 ..0;T /IW 1 .˝ C // C ke r kW l;l=2 .Q / C ke  kW lC1=2;l=2C1=4 .G / T 2 2 2 T 0 2 n

c kebt fkW l;l=2 .D / C kv0 kW 1Cl .[˝ ˙ / C k0C kW 1Cl .˝ C / C kebt hkW l=2 ..0;T /IW 1 .˝ C // T 2 0 0 2 2 2 2 0 o (129) C kebt hkW 1Cl;0 .QC / C kebt bkW lC1=2;l=2C1=4 .G / ˇt

ˇt

T

2

ˇt

T

2

with 0 < ˇ b and c independent of T 6 1. As in [1], problem (127) with zero initial data is firstly considered, as well as the parameter-dependent problem: 8 C m sv  r  T0 .v/ C p1 r C D f; ˆ ˆ ˆ ˆ Z ˆ   ˆ 1 ˆ C ˆ < s C C m r v r  v.z; t / dz D hC in ˝0C ; C j˝0 j ˝0C ˆ ˆ  ˆ  sv  r  T0 .v/ C r  D f; r  v D 0 in ˝0 ; ˆ ˆ ˆ ˇ ˆ ˆ : Œvˇ D 0;   p  C C   n C ŒT0 .v/n ˇˇ D b on  ; 1 0 0 0 0 0

vj˙ D 0; (130) to which the main attention is given. All the functions in (130) are assumed to be complex-valued. 3=2Cl

Theorem 17. Let ˙ , 0 2 W2 lC 12

b 2 W2

; l 2 . 21 ; 1/, f 2 W2l .[˝0˙ /, hC 2 W21Cl .˝0C /,

.0 /. Moreover, it is supposed that Z ˝0C

hC .y/ dy D 0:

Then there exists 0 > 0 such that problem (130) with s 2 C: Res >  0 has a unique solution v 2 W 2Cl .[˝0˙ /,  C 2 W2lC1 .˝0C /;   2 W21Cl .˝0 /, and jjjvjjj2Cl;[˝ ˙ C .1 C jsj/k C kW lC1 .˝ C / C jsj1Cl=2 k C kW 1 .˝ C / C jjjr  jjjl;˝0 C 2 2 0 0 0 ˚

 C l=2 C jjj jjjlC1=2;0 c jjjfjjjl;[˝ ˙ Ckh kW 1Cl .˝ C /Cjsj kh kW 1 .˝ C /CjjjbjjjlC1=2;0 ; 0

2

0

2

0

(131)

37 Local and Global Solvability of Free Boundary Problems for the. . .

2013

where jjjujjjm;˝ D kukW2m .˝/ C jsjm=2 kukL2 .˝/ : Proof (of Theorem 17). Below, the main ideas are outlined of the proof of estimate (131) which is divided into several steps. A Model Problem The model problem is first under consideration: 8 C C r  w D hC m sw  r  T0 .w/ C p1 r# C D f; s# C C m ˆ ˆ ˆ ˆ ˆ ˆ <  sw  r  T0 .w/ C r#  D f; r  w D 0 in Q ; ˇ ˇ 0 ˆ Œwˇy3 D0 D 0; ŒT˛3 .w/ˇy3 D0 D b˛ ; ˛ D 1; 2; ˆ ˆ ˆ ˆ ˆ :  p # C C #  C ŒT 0 .w/ˇˇ D b3 on R2 ; 1 33

in QC ; (132)

y3 D0

where w and # ˙ are supposed to be functions with the supports contained in the closure of [Q˙ D Q0  [I ˙ , I ˙ D f˙y3 2 .0; d0 /g, Q0 D fjy˛ j < d0 ; ˛ D 1; 2g (this assumption is made in view of the forthcoming application of localization procedure). In contrast to the previous sections, for the analysis of problem (132), the decomposition of functions into the Fourier series of the type 1 X 0 0 uQ . 0 /ei y ; u.y / D .2d0 /2 2 0

0



 D

k2Z



k1 ; k2 ; d0 d0

k D .k1 ; k2 /;

is used, where Fu  uQ . 0 / D

Z

0

0

ei y u.y 0 / dy 0 ; Q0

y 0 D .y1 ; y2 /:

(133)

This enables one to reduce problem (132) to the system for the Fourier coefficients (133) of wi and  allowing Res D s1 to take small negative values. This system will be studied separately in the cases jkj > 0 and k D 0. In the first case, the given functions f and hC will be extended by zero with respect to y3 into the domains jy3 j > d0 and y3 > d0 , respectively, and the problem will be considered for y3 2 R˙ , while in the second case, the problem will be analyzed in I ˙ with additional boundary conditions wjjy3 jDd0 D 0. Reduction of Problem (132) with jk1 j C jk2 j > 0 to the Case f D 0, h D 0 Problem (132) is analyzed now with vanishing zero mode. To this end, R it is assumed that the functions on the right-hand side have no zero mode, i.e., Q0 f dy 0 D 0,

2014

R

I.V. Denisova and V.A. Solonnikov

hC dy 0 D 0. The reduction is carried out by construction of some auxiliary functions. Functions f and hC are extended by zero into the domains jy3 j > d0 and y3 > d0 , respectively. The restrictions fjQ˙ , where Q˙  Q0  R˙ , R˙  f˙y3 > 0g, are C ˙ C denoted by f˙ . Moreover, f˙  and h denote extensions of f and h , respectively, 0 with into Q D Q  R carried out in such a way that R ˙ preservation R of class 0 f dy 0 D 0, hC  dy D 0. Q0

Q0

Q0

The auxiliary functions u˙ ;  ˙ are defined as solutions to the systems: 8 <  su  r  T0 .u / C r  D f ; r  u D 0 in Q ; ˇ ˇ   ˇ   ˇ : T˛3 .u / y3 D0 D 0; ˛ D 1; 2;    T33 .u / y3 D0 D 0;

! 0; u ;    jxj!1 (134)

and 8 C < m suC  r  TC .uC / C p1 r C D fC ; : uC jy3 D0 D u jy3 D0 ;

C s C C m r  uC D hC

in QC ;

! 0: uC ;  C  jxj!1 (135)

The detailed discussion of problem (134) for the Stokes equations is omitted. The estimate of .u ;   / has the form: jjju jjj2Cl;Q C jjjr  jjjl;Q C jjj  jjjlC1=2;Q0 6 cjjjf jjjl;Q ;

Res >  0 ;

(136)

The solution .uC ;  C / of (135) is sought in the form uC D u1 C u2 ,  C D 1 C 2 with .ui ; i / defined as solutions of C m su1  r  TC .u1 / C p1 r1 D fC ;

C s1 C m r  u1 D hC 

in Q;

(137)

and (

C su2  r  TC .u2 / C p1 r2 D 0; m

u2 jy3 D0 D .u  u1 /jy3 D0  a;

C s2 C m r  u2 D 0

! 0: u2 ; 2  jxj!1

The Fourier transform Z uO ./ D Q0

dy 0

Z R

0

0

e i y i3 y3 u.y/ dy3 ;

in QC ; (138)

37 Local and Global Solvability of Free Boundary Problems for the. . .

2015

where  0 D . k1 =d0 ; k2 =d0 /, jk1 j C jk2 j > 1; 3 2 R, converts (137) into the algebraic system:  8  C < m .s C C  2 /uO 1 C . C C 1C /.  uO 1 / C p1 i O 1 D OfC ; :

C .i  uO 1 / D hO C s O 1 C m ;

 D . 0 ; 3 /;

 2 D j 0 j2 C 32 ;

where  0 D . k1 =d0 ; k2 =d0 /, jk1 j C jk2 j > 1; 3 2 R. The elimination of O 1 leads to the equation for uO 1 :  1 .s C C  2 /uO 1 C C C 1C .s/ .  uO 1 / D C m

OC OfC  p1 i h  s

!  gO ;

(139)

where 1C .s/ D 1C C p1 =s. The solution of (139) is given by  uO 1 D HC gO = s C C  2 ; where HC is the matrix with the elements  C

C 1C .s/ j k  D ıj k  HjCk D ıj k  s C 2 C C 1C .s/  2

s bsCp1

(140)

j k ; s C C 2 C  2



b D C C 1C > 0. This formula defines the solution of equation (137) for Res > 0. To estimate .u1 ; 1 / for Res >  0 , one can use the estimate of “generalized energy”, according to M.Padula [23]. By repeating the arguments presented below in the proof of (162), one shows that the solution of (138) satisfies the inequality .Res C 0 /.ku1 k2L2 .Q/ C k1 k2L2 .Q/ / C kru1 k2L2 .Q/

c.kf k2L2 .Q/ C kDy3 h kL2 .Q/ /: Moreover, applying this estimate to the finite differences kj .hj /u1 and kj .hj /1 , where j D 1; 2; 3, kj > 2 C l, one easily arrives at ku1 k2

W22Cl .Q/

C k1 k2

W2lC1 .Q/

c.kf k2W l .Q/ C kDy3 h kW lC1 .Q/ /; 2

2

(141)

as in [40]. Equations (137) can be written in the form 8 C m .s  C r 2 /u1˛ D f1˛ C f˛ ; ˛ D 1; 2; ˆ ˆ ˆ ˆ < C m p1 s C 2 .s  C r 2 /u13  m r u13 D f13 C f1  Dy3 hC ; ˆ bs C p1 bs C p 1 ˆ ˆ ˆ : C r  u1 D hC s1 C m ;

(142)

2016

I.V. Denisova and V.A. Solonnikov

where 0

C .Dy3 r 0 u1 r 2 u13 /;r 0D.Dy1 ; Dy2 /: f1˛ D . C C 1C /Dy˛ ru1 p1 Dy˛ 1 ; f13 D m

This implies jjju1 jjj22Cl;Q c.jjjf jjj2l;Q C jjjf1 jjj2l;Q C jjjDy3 h jjj2l;Q /;

(143)

because the expression

P C .; s/ D

 s sbjsj2 C p1 s 2 s C 2 s C C 2 C  2 D C  C 2 bs C p1 jbs C p1 j2 bs C p1

is an analytic function of s if Res > p1 =b. It satisfies the inequality jP C j  c.jsj C jj2 / provided that jj  =d0 with small d0 . The norm jjjf1 jjjl;Q is controlled by "jjju1 jjj2Cl;Q C c."/ku1 kW 2Cl .Q/ ; 2

"  1;

in view of interpolation inequalities for mixed derivatives; hence, estimates (141), (143) and equations (137) yield jjju1 jjj2Cl;Q C .1 C jsj/k1 kW 1Cl .Q/ C jsj1Cl=2 k1 kW 1 .Q/ 2 2 C l=2 C

c.jjjfC  jjjl;Q C kh kjW2lC1 .Q/ C jsj kh kW21 .Q/

c.jjfC jjjl;QC C khC kW lC1 .QC / C jsjl=2 khC kW 1 .QC / : 2

2

(144)

Next, the solution .u2 ; 2 / of (138) is constructed. The construction is preceded by the analysis of a similar problem for the transformed Lamé system: 8 C < m sv  r  TC .v/ D 0 in QC ; ˇ : vˇ D a; v ! 0: y3 D0 jxj!1

The solution v is given by formula (30) with W D e a and C3 D 

. C C 1C /.e A  r aQ 3 / .2 C C 1C /.r  r1 /.r1 C ~r/

;

e A D i˛ aQ ˛ ;

~D

C ;

C C 1C

37 Local and Global Solvability of Free Boundary Problems for the. . .

2017

where r, r1 are defined by (26) with D C , 1 D 1C . Using the inequalities 1=2  jrj; jr1 j; jr1 C ~rj > c jsj C j 0 j2

(145)

C 2

which are valid under the assumption Res >  d 2 , it is easy to show that 0

jjjvjjj2Cl;QC 6 cjjjajjjlC3=2;Q0 :

(146)

The function u2 in (138) is given by the same formula (30) with W D e a and 1 , r1 replaced by 1 .s/ D 1C C p1 =s and r11 , respectively, which yields e ae0  C .i1 ; i2 ; r11 /T e11 ; u2 D e

y3 > 0;

(147)

where .bs C p1 /.e A  r aQ 3 / CD  ; .as C p1 /.r11 C ~1 .s/r/ e11 D

ery3  er11 y3 ; r  r11

Cs ~1 .s/ D ; as C p1

b D C C 1C > 0;

s r11 D

s2 C j 0 j2 ; as C p1

a D 2 C C 1C > 0:

Inequality (145) still holds if r1 is replaced with r11 . Indeed, let s D s1 C is2 : Since 2 r11 D

s ajsj2 Cp .s 2 s22 /

and 1 jasCp1 1 j12 s2 , there hold

s1 ajsj2 C p1 .s12  s22 / ajsj2 C 2p1 s1 C is C j 0 j2 ; 2 jas C p1 j2 jas C p1 j2 ,

ajsj2 C2p1 s1 jasCp1 j2

are uniformly bounded by constants independent of

2 c1 .jsj C j 0 j2 / 6 jr11 j 6 c2 .jsj C j 0 j2 /

and c3 .jsj C j 0 j2 /1=2 6 Rer11 6 jr11 j 6 c4 .jsj C j 0 j2 /1=2 ; provided that d0 is small. 2 The function r11 C ~1 .s/r is analytic with respect to s if Res >  minf pa1 ; d 2 g, 0 and it can be shown that jr11 C ~1 .s/rj > cjrj:

2018

I.V. Denisova and V.A. Solonnikov

For definiteness, s2 is assumed to be nonnegative: s2 > 0 (in the case s2 6 0, the picture will be symmetric with respect to the axis Im s D 0). If s1 > 0, then arg~1 .s/ 2 .0; =2/, and the angle between the vectors r11 and ~1 .s/r on the complex plane is less than 3 =4. Hence, for nonnegative s2 , the inequality jr11 C ~1 .s/rj > c max.jr11 j; j~.s/rj/ > cjrj holds. Now let s1 < 0. Since ~1 .s/  ~1 .is2 / D D



C  s1 C is2 is2  b s1 C is2 C p1 =b is2 C p1 =b s 1 p1

C 2 b .s1 C is2 C p1 =b/.is2 C p1 =b/

and s1 C p1 =b > 0; it is clear that j~1 .s/  ~1 .is2 /j 6 c1 js1 j. Hence jr11 C ~1 .s/rj > jr11 C ~1 .is2 /rj  j~1 .s/  ~1 .is2 /jjrj > c2 jrj; if s1 satisfies the additional requirement js1 j < c=c1 . Thus, (146) is true also for the function u2 defined by (147). As above, 2 is estimated by using (138), which yields jjju2 jjj2Cl;QC C .1 C jsj/k2 kW lC1 .QC / C jsj1Cl=2 k2 kW 1 .QC / cjjjajjjlC3=2;Q0 : 2

2

Together with (144), this inequality leads to jjjuC jjj2Cl;QC C .1 C jsj/k C kW lC1 .QC / C jsj1Cl=2 k C kW 1 .QC / 2 2

˚

c jjjfjjjl;QC C khC kW 1Cl .QC / C jsjl=2 khC kW 1 .QC / C jjjajjjlC3=2;Q0 : 2

2

Finally, in view of (136), one can conclude that X ˙

jjju˙ jjj2Cl;Q˙ C.1Cjsj/k C kW lC1 .QC / Cjsj1Cl=2 k C kW 1 .QC / Ckjjr  jjjl;Q 2

2

˚ C jjj  jjjlC1=2;Q0 c jjjfjjjl;[Q˙ C khC kW 1Cl .QC / C jsjl=2 khC kW 1 .QC / : 2

2

Problem (132) with jk1 j C jk2 j > 0 for the Homogeneous Equations Now, a solution of (132) withe f D 0, e h D 0 is analyzed. The corresponding problem for the Lamé-Stokes system

37 Local and Global Solvability of Free Boundary Problems for the. . .

2019

8 C m sv  r  T0 .v/ D 0; y3 > 0; ˆ ˆ <  sv  r  T0 .v/ C r#  D 0; r  v D 0; y3 < 0; ˆ ˇ ˇ ˇ ˆ : ˇˇ 0 0 Œv y3 D0 D 0; ŒT˛3 .v/ˇy3 D0 Db˛ ; ˛ D 1; 2; #  ˇy3 D0 C ŒT33 .v/ˇy3 D0 D b3 : has been studied in Sect. 6.2. The solution in the dual Fourier space is given by (111) with  D 0, for which Lemma 14 guarantees the inequalities following from (117):

jWj 6 c

3 X

! 1 Q jbi jjrj ;

2 X

˙

jV j 6 c

iD1

! 0 1 Q Q jb˛ j C jb3 jj jjrj ;

(148)

˛D1

where jrj D .jsj C j 0 j2 /1=2 . By employing estimates (118) for e0 , e1 and for e0C ; e1C defined by (112) with  0 instead of , one can show by means of the Parseval equality for the transform (133) X

0

2

jQu.s;  ; x3 /j D .2d0 /

2

Z

k2Z2

0

Q0

2

0

ju.s; x ; x3 /j dx ;



0

 D



k1 ; k2 ; d0 d0

k D .k1 ; k2 /; that in view of (148) jjjv.s; /jjj22Cl;[Q˙ 6c

X

jWj2 jrj2lC3 C jV˙ j2 jrj2lC1



k2Z2

6c

2 X X k2Z2

! jbQ i j2 jrj2lC1 C jbQ 3 jrj2l j 0 j

iD1

and, hence, that jjjvjjj2Cl;[Q˙ C jjjr#  jjjl;Q C jjj#  jjjlC1=2;Q0 6 cjjjbjjjlC1=2;Q0 :

(149)

As above, in the case of problem (138), it can be shown that the replacement of 1 with 1 .s/ in (111) does not influence the validity of (149), if s1 and d0 satisfy the above assumptions. Hence the solution .w; # ˙ / of (132) is subject to jjjwjjjlC2;[Q˙ C .1 C jsj/k# C kW lC1 .QC / C jsj1Cl=2 k# C kW 1 .QC / C jjjr#  jjjl;Q 2

C jjj#  jjjlC1=2;Q0 cjjjbjjjlC1=2;Q0

2

2020

I.V. Denisova and V.A. Solonnikov

in the case f D 0, h D 0 and to jjjwjjjlC2;[Q˙ C .1 C jsj/k# C kW lC1 .QC / C jsj1Cl=2 k# C kW 1 .QC / C jjjr#  jjjl;Q 2 2 ˚  C C jjj# jjjlC1=2;Q0 c jjjfjjjl;[Q˙ C kh kW lC1 .QC / C jsjl=2 khC kW 1 .QC / 2 2

(150) C jjjbjjjlC1=2;Q0 in the general case (but with vanishing zero mode). On Problem (132) with k D 0 If  0 D 0, then (132) is decomposed in two one-dimensional problems: 8 2 Q˛ ˆ C ˙ d w ˆ s w Q   D fQ˛ ; y3 2 I ˙ ;  ˆ < m ˛ dy32 h ˆ ˇ dwQ ˛ iˇˇ ˆ ˆ : Œw D bQ ˛ ; Q ˛ ˇy3 D0 D 0;  ˙ ˇ dy3 y3 D0

(151) w Q ˛ jy3 D˙d0 D 0; ˛ D 1; 2;

and 8 d2 w3 d#Q C Q 3 QC ˆ C C dw ˆ sw Q 3  .2C C C / C p1 D fQ3 ; s #Q C C m D h ; y3 2 I C; ˆ m 1 ˆ 2 ˆ dy dy dy 3 3 ˆ 3 ˆ ˆ ˆ ˆ 2  Q ˆ d# dw Q3 dw Q3 ˆ <  s w Q 3   C D fQ3 ; D 0; y3 2 I  ; 2 dy3 dy3 dy3 ˆ ˇ ˆ ˆ ˆ ŒwQ 3 ˇy3 D0 D 0; w Q 3 jy3 D˙d0 D 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ Q 3 ˇˇ dwQ 3 ˇˇ ˆ  dw :  p1 #Q C .0/ C #Q  .0/ C .2C C C /  2 D bQ 3 : ˇ ˇ 1 dy3 y3 !0C dy3 y3 !0 The latter is equivalent to 8 d2 w Q3 d#Q C ˆ C C ˆ ˆ m sw Q 3  .2C C C / C p D fQ3 ; s #Q C C m 1 ˆ 1 2 ˆ dy dy ˆ 3 3 ˆ < Q 3 jI  D 0; wQ 3 jy3 D0;d0 D 0; w ˆ ˆ ˆ Z 0 ˆ  ˇ ˆ QC ˇ ˆ C dw 3  C C Q Q Q ˆ C b3 ˇ  : # .y3 / D p1 #  .2 C 1 / y3 D0 dy3 y3

dwQ 3 Q C D h ; y3 2 I C ; dy3

fQ3 .z3 / dz3 : (152)

37 Local and Global Solvability of Free Boundary Problems for the. . .

2021

Since (151) is a transformed parabolic problem, it is easily shown for s: Res > ˙ 2  min˙ d 2 that w˛ satisfies the inequality: 0

˚ 

jjjw˛ jjjlC2;[I ˙ 6 c jjjf˛ jjjl;[I ˙ C 1 C jsjl=2C1=4 jb˛ .s/j ;

˛ D 1; 2:

(153)

# C in I C . The System (152) is a problem for the unknown functions wQ 3 and e elimination of e # C leads to the boundary value problem for the single equation: sw Q 3  .2 C C 1C .s//

d2 w 1 Q C p1 dhQ C Q3 D .f3  /  gQ 3 ; C m s dy3 dy32 ˇ D 0; wQ 3 ˇ

y3 2 I C ;

y3 D0;d0

which is equivalent to R.s/w Q3 

d2 w s Q3 D gQ 3 ; as C p1 dy32

w Q 3 jy3 D0;d0 D 0;

(154)

where R.s/ D s 2 =.as C p1 /. The function gQ 3 is expanded in the series in sin k y, d0 3 k D 1; 2; : : :, the eigenfunctions of the Dirichlet problem for the operator

d2 dy32

in the

interval y3 2 Œ0; d0 , and the Fourier transform is applied in (154). For the Fourier coefficients w L 3 of wQ 3 .y3 /, the formula w L3 D

gL 3 s ; as C p1 R.s/ C j3 j2

3 D k=d0 ;

k D ˙1; : : : ;

is obtained; hence jjjw3 jjj2Cl;I C C .1 C jsj/k# C kW lC1 .I C / C jsj1Cl=2 k# C kW 1 .I C / 2 2 o n

c jjjf3 jjjl;I C C khC kW lC1 .I C / C jsjl=2 khC kW 1 .I C / : 2

(155)

2

The function #  is defined by the last line in (152). It is easily seen that jjjr#  jjjl;I  6 cjjjf3 jjjl;I  ; n ˇ   1 C jsjl=2C1=4 j#  ˇy3 D0 j 6 c jjjwjjj2Cl;[I ˙ C 1 C jsjl=2C1=4  o ˇ  j# C ˇy3 D0 j C jb3 .s/j : Estimates (153), (155), and (156) imply the inequality

(156)

2022

I.V. Denisova and V.A. Solonnikov

jjjwjjjlC2;[I ˙ C .1 C jsj/k# C kW lC1 .I C / C jsj1Cl=2 k# C kW 1 .I C / 2 2 ˇ  l=2C1=4  ˇ j# y3 D0 j C jjjr# jjjl;I  C 1 C jsj n

c jjjfjjjl;[I ˙ C khC kW lC1 .I C / C jsjl=2 khC kW 1 .I C / 2 2 o  C 1 C jsjl=2C1=4 jb.s/j : for the solution .w; # C ; #  /. Thus, (150) has been shown both for jkj > 0 and for jkj D 0. Proof of Estimate (131) in a Bounded Domain Estimate (131) is obtained by localization method combined with the estimate of the L2 -norms of v and . Let y0 2 0 . The solution of problem (130) multiplied by the cutoff function .y/ equal to one near y0 and to zero for jy  y0 j > d , d < d0 , satisfies the system 8 C m sw  r  T0 .w/ C p1 r# C D f C FC .v;  C ; /; ˆ ˆ ˆ ˆ ˆ C < s# C C m r  w D hC  C H C .v; / in ˝0C ; ˆ  sw  r  T0 .w/ C r#  D f C F .v;   ; /; r  w D v  r in ˝0 ; ˆ ˆ ˆ ˆ ˇ ˇ : ˇˇ Œw 0 D0; .p1 # C C #  /n0 C ŒT0 .w/n0 ˇ0 Db C B.v; / on 0 ; wˇ˙ D 0; where w D v; # ˙ D  ˙  and F˙ ; H C ; B are differential expressions containing lower order derivatives of v and  . Rectifying the boundary near y0 gives rise to the appearance of higher-order terms with small coefficients (if d is small). As a result, a model problem similar to (132) arises with some differential expression on the right-hand side that can be taken into account with the help of interpolation inequalities. Similar model problems appear also in estimating the solution on compact subsets of ˝0˙ and near the exterior boundary ˙. They are completely analogous to (137) or well studied in the theory of the Stokes equations. Therefore, upon choosing an appropriate “partition of unity” fk g in ˝, estimating vk ,  ˙ k and applying interpolation inequalities, one obtains jjjvjjj2Cl;[˝ ˙ C .1 C jsj/k C kW 1Cl .˝ C / C jsj1Cl=2 k C kW 1 .˝ C / C jjjr  jjjl;˝0 2 2 0 0 0   C l=2 C jjj jjjlC1=2;0 c1 jjjfjjjl;[˝ ˙ C kh kW 1Cl .˝ C / C jsj khC kW 1 .˝ C / (157) 2 2 0 0 0  C jjjbjjjlC1=2;0 C c2 kvk˝ C k k˝ :

37 Local and Global Solvability of Free Boundary Problems for the. . .

2023

The L2 -norms of v and  C can be estimated by the generalized energy method developed by M. Padula [22, 23]. The solution of (130) satisfies the energy relation: Z q 2 ˙ sk m vk˝ C

[˝0˙

0

T .v/ W rv dyp1

Z ˝0C

Z

C

 r  v dyD

Z f  v dy C

˝

b  v dS 0

(158)  with m   . By using the second equation in (130), equality (158) is transformed into q q p 1 2 ˙ vk2 C sN 1 k C k2 C ˙ S.v/k2˝ C C sk m k C ˝ 1 kr  vk˝ C C ˝0 m 2 0 Z Z Z p1 D f  v dy C C  C hN C dy C b  v dS; m ˝0C ˝ 0 where sN and hN are complex conjugates of s and h, a  b D ai bN i . It follows that  q  q 1 p1 C 2 2 2 ˙ Res k m vk˝ C C k k C C k ˙ S.v/k2˝ C C 1 kr  vk˝ C ˝0 m 2 0 ! Z Z Z p1 D Re f  v dy C C  C hN C dy C b  v dS ; C  ˝ 0 m ˝0

(159)

Along with (158), one has C sm

Z ˝0C

Z v  W dy C

˝0C

TC .v/ W rW dy  p1

Z ˝0C

 C r  W dy D

Z ˝0C

f  W dy (160)

with W satisfying y 2 ˝0C ;

r  W D  C ;

Wj0 D 0;

kWkW 1 .˝ C / 6 ck C k˝ C : 2

0

0

Hence Z 

˝0C

 C r  W dy D

Z ˝0C

j C j2 dy:

(161)

2024

I.V. Denisova and V.A. Solonnikov

Summation of (159) and the real part of (160) multiplied by a small > 0 gives q  q p1 C 2  1 C 2 ˙ vk2 C k C p k k k k C ˙ S.v/k2˝ Res k m 1 C C ˝ C ˝0 ˝0 m 2 Z Z  2 C C C C kr  vk C Re T .v/ W rW dy C s v  W dy C m 1 ˝0

D Re

Z

˝0C

f  .v C W/ dy C

˝

p1 C m

Z ˝0C

 C hN C dy C

Z

˝0C

 b  v dS :

0

R C s ˝ C v  W dy. From inequality (161) One considers now the integral J D m 0 and the second equation in (130), it follows that   jJ j c jsjkvk˝ C k C k˝ C 6 c kvk˝ C kvkW 1 .˝ C / C khC k˝ C : 0

0

2

0

0

0

By taking the assumption 23 C C C 1 > 0 into account, one concludes that there is a number ! > 0 such that C >  23 .C  !/. By virtue of kr  vk2 C 6 1 3 kS.v/k2 C ; 4 ˝

˝0

this implies the estimate

0

C 1 2 2 kS.v/k2 C C C 1 kr  vk˝ C > !kS.v/k˝ C ; ˝ 2 2 0 0 0 Finally, using the Korn inequality, which is valid in view of vj˙ D 0 [34], one deduces q 1 1 2 2  2 k ˙ S.v/k2˝ C C 1 kr  vk˝ C > !kS.v/k˝ C C 2 krvk˝0 2 2 0 0 > c1 kvk2

W21 .˝0 [˝0C /

:

Choosing sufficiently small , one arrives at the estimate:    .Res C 0 / kvk2˝ C k C k2 C 6 c kfk2˝ C khC k2 C C kbk20 ˝0

˝0

(162)

with a certain value 0 > 0 depending on ; c1 . Now it is necessary to estimate the function   which satisfies the relations: r 2   D r  f in ˝0 ;

ˇ   j0 D p1  C  Œn0  T0 .v/n0 ˇ0  p0 ;

@  ˇˇ D n˙  f   n˙  rot rot v: @n˙ ˙

37 Local and Global Solvability of Free Boundary Problems for the. . .

2025

It can be represented as the sum   D q1 C q2 , where the functions q1 , q2 are solutions of the problems in ˝0 ;

r 2 q1 D r  f

ˇ q1 ˇ0 D 0;

@q1 ˇˇ ˇ D f  n˙ ; @n˙ ˙

and r 2 q2 D 0

in ˝0 ;

ˇ q2 ˇ0 D p0 ;

@q2 ˇˇ ˇ D  rotn˙ rot v: @n˙ ˙

Let  be a solution of in ˝0 ;

r 2  D q2

j0 D 0;

@ ˇˇ D 0: @n˙ ˙

From well-known energy and elliptic estimates, it follows that kq1 k˝0 6 ckrq1 k˝0 6 ckfk˝0 ; Z Z Z @N 2 2N  q2 r  dx D  p0 dS C  rotn˙ rot vN dS kq2 k˝0 D @n0 ˝0 0 ˙ Z Z @N p0 dS   .r  rot v/  n˙ dS D @n0 0 ˙  6 ckkW 2 .˝  / kp0 k0 C k rot vk˙ ; 2

0

whence  kq2 k˝0 6 c k C k0 C krvk0 [˙ : and n  k  k˝0 6 c kfk˝0 C 1 kvkW 2Cl .[˝ ˙ / C k C kW lC1 .˝ C / 2 0 0 2 o  C c.1 / kvk˝ C k C k˝ C ; 1  1: 0

(163)

Substituting (162), (163) into (157) and choosing 1 sufficiently small, one obtains (131). As for the solvability of problem (130), it can be proved by constructing a regularizer, as in Theorem 13 above (see also [39]). This completes the proof of Theorem 17. t u Proof (of Theorem 16). To avoid technicalities, only the case h D 0 is considered. If v0 D 0 and 0 D 0, then the solution of problem (127) in the infinite time interval t > 0 is obtained

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I.V. Denisova and V.A. Solonnikov

from the solution of problem (130) by the inverse Laplace transform and (128) for T D 1 follows from (131) and the Plancherel theorem. As in the estimates for parabolic problems (see [1]), this implies inequality (129) for arbitrary finite T and ˇ D s1 D Res. The case of nonzero initial data can be treated by constructing auxiliary functions U.y; t /,  C .y; t / such that UjtD0 D v0 ;

 C jtD0 D 0C ;

ŒUj0 D 0;

C

U.y; t / D 0;

 .y; t / D 0 for t > 2; kUkW 2Cl;1Cl=2 [˝ ˙ .0;2/ ckv0 kW 1Cl .[˝ ˙ / ; 2

0

2

0

k kW 1Cl;0 ˝ C .0;2/ C kDt  C kW 1Cl;0 ˝ C .0;2/ C kDt  C kW l=2 .0;2/IW 1 .˝ C // C

0

2



0

2

2

ck0C kW 1Cl .˝ C / : 2 0

2

0

(164)

The existence of U with these properties follows from the trace theorem for the anisotropic Sobolev spaces, and  C can be taken in the form  C .y; t / D 0C .y/.t /; where .t / is a smooth function vanishing for t > 2 and equal to one for small t . The differences u  U;  C   C satisfy the problem with zero initial data and can be treated as above (see Theorem 17). t u R Remark 10. The condition ˝ C  C dz D 0 in the second statement of Theorem 16 0 is important, because otherwise problem (127) for the homogeneous system of equations has a solution v D 0,   D 0,  C D const ¤ 0. Remark 11. Following [2], it is possible to write a homogeneous problem (127) in the form: ˇ Ut C AU D 0; U ˇtD0 D U0 ; where U D .vC ;  C ; v /T and 0

1 1  0 C C  r  T .v / C p r 1 C B m C B C Z B C B C C 1 C C B C:  r  v  r  v dz AU D B m C C j˝0 j ˝0C B C B C @ 1  A    r  T .v / C r   Here

is the solution of the problem: r2

D0

in ˝0 ;

ˇ ˇ

0

ˇ D p1  C  Œn0  T0 .v/n0 ˇ0 :

37 Local and Global Solvability of Free Boundary Problems for the. . .

2027

The domain of A is the subspace H1 of W22 .˝0C /  W21 .˝0C /  W22 .˝0 / defined by ˇ Œvˇ0 D 0;

ˇ Œ˘0 T0 .v/n0 ˇ0 D 0;

rv D 0;

ˇ v ˇ˙ D 0;

Z ˝0C

 C .z/ dz D 0:

and the range of A is the subspace H2 of L2 .˝0C /  W21 .˝0C /  L2 .˝0 / whose elements .fC ; hC ; f / satisfy the relations: r  f D 0;

Z ˝0C

hC .z/ dz D 0:

Inequality (131) that is also valid for l D 0, implies the estimate k.sI C A/ 1 kH2 !H1 6

c ; 1 C jsj

8s 2 C W Res >  0 ;

where I is the unite operator. This means that A generates analytical semigroup. This property is not used in the present chapter, although it is of central importance (see [17, 44]).

6.4.3 Nonlinear Problem This subsection outlines the proof of Theorem 15. It is based on Theorem 16 and on an estimate of nonlinear terms (124). Lemma 16. Let the pressure function p C ./ belong to C 2 in the interval Rt C C =2; 3m =2/, and let U.y; t / D u.y;  / d. If .m 0

sup kU.; t /kW 2Cl .[˝ ˙ / 6 ı1 ; 2

t 0;

C =2; sup j# C .y; t /j 6 m

(165)

QTC

then  Z.T / 6 c ı1 Y .T / C Y 2 .T / ; where T > 1 and

(166)

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I.V. Denisova and V.A. Solonnikov

Z.T /  ZŒu; # D

X

˙ kl˙ 1 .u; # /k

˙

C kl2C kW l=2 .0;T /IW 1 .QC / C kL.u/k 2

2

C kl4 .u; # C /k

l; l W2 2

0;1C 2l

W2

T

1C 2l ; 2l C 14

W2

Y .T /  Y Œu; # D kuk

.QT /

2

C kl3 .u/k

T

lC 21 ; 2l C 14

W2



.GT /

; .GT /

2Cl;1C 2l

W2

.DT /

C k# C kW lC1;0 .QC / C kDt # C kW lC1;0 .QC / l; 2l

W2

2

T

2

C kDt # C kW l=2 ..0;T /IW 1 .QC / C kr#  k 2

.QT˙ /

C kl2˙ .u; # C /kW lC1;0 .Q˙ /

.QT /

T

2

C k#  k

lC 12 ; 2l C 41

W2

; .GT /

(167) l 2 .1=2; 1/: The proof of this lemma is similar to that in [24]. In what follows, a procedure is presented of constructing a solution to nonlinear problem (123) in the time interval t > 0. The solution to this problem satisfies orthogonality condition: Z ˝0C

# C .x; t /Ju .x; t / dx D 0

(168)

(in view of the mass conservation), while in the case of a linear problem, the corresponding condition has the form: Z ˝0C

# C .x; t / dx D 0:

(169)

Therefore the solution is to be constructed step by step, from the time interval Œ.k  1/T; kT  to ŒkT; .k C 1/T  with a certain large enough but finite T , the discrepancy between the conditions (168) and (169) being eliminated at every step. This procedure was proposed in [38]. Let u D u0 C u00 , # D # 0 C # 00 , where .u0 ; # 0 / solves the problem with homogeneous equations and boundary conditions: 8 C 0C 0 0 0 ˆ ˆ m Dt u  r  T .u / C p1 r# D 0; ˆ ˆ ! ˆ Z ˆ ˆ 1 ˆ 0C C 0 0 ˆ ˆ Dt # C m r  u  r  u dz D 0 ˆ ˆ j˝0C j ˝0C ˆ < 0

in QTC ;

 Dt u0  r  T0 .u0 / C r#  D 0; r  u0 D 0 in QT ; ˆ ˆ ˆ ˆ   ˆ ˇ ˇ  ˆ 0C 0 0 ˇ ˆ ˆ Œu n0 C T0 .u0 /n0 ˇ0 D 0 on GT ;  D 0; p # C # 1 ˆ 0 ˆ ˆ ˆ ˆ ˇ 0 ˇ : 0 ˇˇ u ˙ D 0; u0 ˇtD0 D u0 ; # C ˇtD0 D #0C ;

(170)

37 Local and Global Solvability of Free Boundary Problems for the. . .

2029

while .u00 ; # 00 / is a solution to the problem with homogeneous initial data: 8 C 00 C m Dt u00  r  T0 .u00 / C p1 r# C D lC ˆ 1 .u; # /; ˆ ˆ ˆ Z ˆ  ˆ 1 00 C ˆ C 00 ˆ D # C  .r  u  r  u00 dz D l2C .u; # C / in QTC ; ˆ t m ˆ C ˆ j˝0 j ˝0C ˆ ˆ ˆ ˆ <  00  r  u00 D l2 .u/ in QT ;  Dt u00  r  T0 .u00 / C r#  D l 1 .u; # /; ˇ  ˇ ˆ ˆ ˆ Œu00 ˇ0 D 0; ˘0 T0 .u00 /n0 ˇ0 D l3 .u/; ˆ ˆ ˆ ˆ ˆ  ˇ  00 00 ˆ ˆ  p1 # C C #  n0 C n0  T0 .u00 /n0 ˇ0 D l4 .u; # C / on GT ; ˆ ˆ ˆ ˆ ˆ ˇ ˇ 00 ˇ : u00 ˇ˙ D 0; u00 ˇtD0 D 0; # C ˇtD0 D 0: (171) R C Since ˝ C #0 .x/ dx D 0, by Theorem 16 linear problem (170) has a global 0 solution satisfying (129) (with f D 0, h D 0) for arbitrary T > 0, in particular, eˇT N Œu0 .; T /; #

0C

.; T / 6 c1 N Œu0 ; #0C ;

Y Œu0 ; # 0  6 cN Œu0 ; #0C ;

(172)

where N Œu; # C  D kukW lC1 .[˝ ˙ / C k# C kW lC1 .˝ C / : 2

0

2

0

Time interval T is fixed by the condition: c1 eˇT 6

1 : 8

to Lemma 16, p According  the nonlinear terms in (171) satisfy (166) with ı1 D T Y Œu0 ; # 0  C Y Œu00 ; # 00  which together with (172) yields ZŒu0 C u00 ; # 0 C # 00  6 c

p



T Y 2 Œu0 ; # 0  C Y 2 Œu00 ; # 00 

6 c.T /N Œu0 ; #0C  C c.T /Y 2 Œu00 ; # 00  with  from (125). Choosing  small, one can construct a solution of (171) in the time interval .0; T / by successive approximations. Moreover, it can be shown that Y Œu00 ; # 00  6 c2 N Œu0 ; #0C : Hence, by (172), N Œu.; T /; # C .; T / 6 N Œu0 .; T /; #

0C

.; T / C N Œu00 .; T /; #

6 .c1 eˇT C c2 /N Œu0 ; #0C  6

00 C

.; T /

1 N Œu0 ; #0C  4

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I.V. Denisova and V.A. Solonnikov

if  is small. It is assumed that if the solution of (123) is found for t 6 kT , the inequalities Nk 6

1 1 Nk1 6 : : : 6 k N0 4 4

hold with Nk D N Œu.; kT /; #.; kT / and inequality (165) is satisfied in the time interval .0; kT / with ı1  1. Let u D u0 C u00 ; # D # 0 C # 00 in the time interval t 2 kT; .k C 1/T /, where 0 .u ; # 0 / and .u00 ; # 00 / solve problems (170) and (171) with the initial data u0k .y/ D R C 0 00 u.y; kT /; #kC .y/ D # C .y; kT /  1C # .z; kT / dz and u00k D 0; #k C D j˝0 j

1 j˝0C j

R



C



˝0C

# .z; kT / 1  Ju .z; kT / dz, respectively.

˝0C

From the condition (165) with t D kT , it follows that ˇ ˇ k1  Ju ˇtDkT kW lC1 .˝ C / 6 ckUˇtDkT kW 2Cl .˝ C / 6 cı1 ; 2

0

2

0

which implies the inequalities ˇ 00 k#k C kW lC1 .˝ C / 6 ck# C .; kT /kW lC1 .˝ C / k1  Ju ˇtDkT kW lC1 .˝ C / 2

0

2

0

2

0

C

6 cı1 k# .; kT /kW lC1 .˝ C / : 2

0

Hence, by virtue of (172), N Œu0 .; .k C 1/T /; #

0C

.; .k C 1/T / 6 c1 eˇT Nk ;

moreover, for small ı1 the solution of problem (171) can be constructed and the estimates Yk Œu00 ; # 00  6 c2 ı1 Nk ;

Yk Œu; # 6 .c1 eˇT C c2 ı1 /Nk

(173)

can be obtained. Here Yk Œu; # means the sum of norms (??) of u, # on the interval .kT; .k C 1/T /. It follows that NkC1 6 14 Nk : It is clear that (173) is also true for all j < k, hence Yj Œu; # 6 Yj Œu0 ; # 0  C Yj Œu00 ; # 00  6 cNj 6 c4j N0 : If ˇ is so small that eˇT 6 2, then k X j D0

Yj2 Œu; #e2ˇTj 6 c

k X e2ˇTj j D0

42j

N02 6 cN02

(174)

37 Local and Global Solvability of Free Boundary Problems for the. . .

2031

which is equivalent to (126). Verification of the condition (165) for U in the interval .0; kT / reduces to Z sup kU.; t /kW 2Cl .[˝ ˙ / 6 c 2

t 1=2 is embedded in the space of continuous functions; hence, the second inequality in (165) also follows from (174). Finally, the expansion of the interface t can be estimated by evaluating the magnitude of interface displacement: Z

Z

1

1

max ju.; t /jdt 6 c 0

˝0C

0

kukW 1Cl .˝ C / dt 6 c4 : 2

0

Consequently, if one takes  so small that c4  is less than the distance between the initial interface 0 and the solid boundary ˙, these surfaces will never intersect. The proof of Theorem 15 is complete. Remark 12. By the same method, a solution of the Navier-Stokes equations with mass forces f.x; t / can be constructed, provided f decays exponentially as t ! 1 (see [9, 10]).

7

Conclusions

Thus, unsteady motion of compressible and incompressible fluids with free boundaries and interfaces has been considered. The compressible liquid has been assumed to be barotropic with the pressure given by a strictly positive smooth function of the density. For a single compressible fluid, local (in time) existence theorems have been stated for nonnegative surface tension in the Sobolev-Slobodetskiˇı [42] and Hölder spaces of functions, in the latter case the detailed proof of the theorems being cited from [13, 14]. The problem governing the motion of two compressible fluids separated by a closed unknown interface has been also studied in the Sobolev [5] and Hölder spaces [8] where local solvability was obtained in both cases with restrictions on the fluid viscosities. These restrictions have been discussed and a way to eliminate them has been presented. In the last section, the problem on the evolution of a bubble in an incompressible l;l=2 continuum has been analyzed in the spaces W2 . A local existence theorem for the problem has been proved in the case of nonnegative surface tension without restrictions on the viscosities and the densities imposed in [6]. The case where a drop is surrounded by a gas can be studied in the same way. Finally, the global unique solvability has been obtained for the problem without surface tension forces

2032

I.V. Denisova and V.A. Solonnikov

on the interface and with small data, the liquids being located in a container of finite volume. The proof has been based on an exponential global estimate for a generalized energy in a linear problem. Moreover, the maximal displacement of the interface throughout the time has been estimated. By choosing small data, one can guarantee that the bubble will be always contained strictly inside the incompressible liquid. Thus, the conclusion can be made that the stability of a solution to the interface problem takes place in the sense that the solution is small and decays exponentially in time under a small deviation of the initial data from zero.

8

Cross-References

 Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible

Fluid Mechanics  Fourier Analysis Methods for the Compressible Navier-Stokes Equations  Global Existence of Classical Solutions and Optimal Decay Rate for Compress-

ible Flows via the Theory of Semigroups  Local and Global Existence of Strong Solutions for the Compressible Navier-S-

tokes Equations Near Equilibria via the Maximal Regularity  Local and Global Solutions for the Compressible Navier-Stokes Equations Near

Equilibria via the Energy Method  Multi-Fluid Models Including Compressible Fluids  Weak and Strong Solutions of Equations

of

Compressible

Magnetohydrodynamics

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27. P. Secchi, A. Valli, A free boundary problem for compressible viscous fluid. J. Reine Angew. Math. 341, 1–31 (1983) 28. Y. Shibata, On the R-boundedness for the two phase problem with phase transition: compressible-incompressible model problem. Funkcial. Ekvac. 59, 243–287 (2016) 29. Y. Shibata, Local well-posedness of free surface problem for the Navier–Stokes equations in a general domain. Discrete Contin. Dyn. Syst. Ser. S 9(1), 315–342 (2016) 30. Y. Shibata, On the global well-posedness of some free boundary problem for a compressible barotropic viscous fluid flow. Contemp. Math. 666, 341–356 (2016) 31. L.N. Slobodeckiˇı, Generalized Sobolev spaces and their application to boundary problems for partial differential equations. Sci. comm. Leningrad. Gos. Ped. Inst. Herzen. 197, 54–112 (1958, in Russian) (English transl.: in Am. Math. Soc. Transl. (2) 57, 207–275) 32. L.N. Slobodeckiˇı, S.L. Sobolev’s spaces of fractional order and their application to boundary problems for partial differential equations. Dokl. Akad. Nauk SSSR (N.S.) 118, 243–246 (1958, in Russian) 33. V.A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations in general form. Trudy Steklov Mat. Inst. 83, 3–162 (1965) (English transl.: (1967) in Proc. Steklov Inst. Math.) 34. V.A. Solonnikov, On non-stationary motion of an isolated mass of a viscous incompressible fluid. Isvestia Acad Sci USSR 51(5), 1065–1087 (1987) (English transl.: (1988) in Math USSRIzv 31(2), 381–405) 35. V.A. Solonnikov, On an initial-boundary value problem for the Stokes systems arising in the study of a problem with a free boundary. Trudy Mat. Inst. Steklov 188, 150–188 (1990) (English transl.: (1991) in Proc. Steklov Inst. Math. 3, 191–239) 36. V.A. Solonnikov, Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval. Algebra i Analiz 3(1), 222–257 (1991) (English transl.: (1992) in St. Petersburg Math. J. 3(1), 189–220) 37. V.A. Solonnikov, On the justification of the quasi-stationary approximation in the problem of motion of a viscous capillary drop. Interfaces Free Bound 1(2), 125–173 (1999) 38. V.A. Solonnikov, On problem of stability of equilibrium figures of ritating viscous incompressible self-gravitating liquids, in Instability in Models Connected with Fluid Flows II, ed. by C. Bardos, A. Fursikov. International Mathematical Series, vol. 7 (Springer, Dordrecht, 2007), pp. 189–254 39. V.A. Solonnikov, On the linear problem arising in the study of a free boundary problem for the Navier–Stokes equations. Algebra i Analiz 22(6), 235–269 (2010) 40. V.A. Solonnikov, On the solvability of free boundary problem for viscous compressible fluids in an infinite time interval, in Mathematical Fluid Dynamics, Present and Future, ed. by Y. Suzuki, Y. Shibata. Springer Proceedings in Mathematics & Statistics, vol. 183 (Springer, Tokyo, 2016), pp. 287–315 41. V.A. Solonnikov, I.V. Denisova, Classical well-posedness of free boundary problems in fluid mechanics, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids I ed. by Y. Giga, A. Novotny (Springer International Publishing AG, 2017) 42. V.A. Solonnikov, A. Tani, Free boundary problem for a viscous compressible flow with surface tension, in Constantin Carathéodory: An International Tribute, ed. by T.M. Rassias (World Scientific, Singapore, 1991), pp. 1270–1303 43. V.A. Solonnikov, A. Tani, Evolution free boundary problem for equations of motion of viscous compressible barotropic liquid. Lect. Notes Math. 1530, 30–55 (1992) 44. G. Ströhmer, About the resolvent of an operator from fluid dynamics. Math. Z. 194, 183–191 (1987) 45. A. Tani, On the free boundary problem for compressible viscous fluid motion. J. Math. Kyoto Univ. 21(4), 839–859 (1981) 46. A. Tani, Two-phase free boundary problem for compressible viscous fluid motion. J. Math. Kyoto Univ. 24(2), 243–267 (1984)

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47. E. Zadrzynska, W.M. Zajaczkowski, On non-stationary motion of a compressible barotropic viscous capillary fluid bounded by a free surface. Colloq. Math. 79, 283–310 (1999) 48. W.M. Zajaczkowski, On non-stationary motion of a compressible viscous fluid bounded by a free surface. Diss. Math. 324, 1–101 (1993) 49. W.M. Zajaczkowski, On non-stationary motion of a compressible barotropic viscous capillary fluid bounded by a free surface. SIAM J. Math. Anal. 25, 1–84 (1994)

Global Existence of Regular Solutions with Large Oscillations and Vacuum for Compressible Flows

38

Jing Li and Zhou Ping Xin

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Proofs of Theorems 1, 2, and 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2038 2041 2047 2049 2073 2080 2081 2081

Abstract

The global existence of smooth solutions to the compressible Navier-Stokes equations is investigated. In particular, results are reviewed concerning the global existence and uniqueness of classical solutions to the Cauchy problem for the barotropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum. The initial density is allowed to vanish and the spatial measure of the vacuum set can be arbitrarily large, in particular, the initial density can even have compact support. These results generalize previous ones on classical solutions for initial

J. Li () Institute of Applied Mathematics, AMSS and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China e-mail: [email protected] Z.P. Xin The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, Hong Kong, China e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_58

2037

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J. Li and Z.P. Xin

densities being strictly away from vacuum, and are the first for global classical solutions that may have large oscillations and can contain vacuum states.

1

Introduction

The time evolution of the density and velocity of a general viscous barotropic compressible fluid occupying a domain   RN .N D 2; 3/ is governed by the compressible Navier-Stokes equations: (

t C div.u/ D 0; .u/t C div.u ˝ u/  u  . C /rdivu C rP ./ D 0;

(1)

where   0, u D .u1 ;    ; uN / and P ./ D a .a > 0;  > 1/ are the unknown fluid density, velocity, and pressure, respectively. The constant viscosity coefficients  and  satisfy the physical restrictions:  > 0;

C

N   0: 2

(2)

Let  D RN and 1 be a fixed nonnegative constant. One looks for the solutions, ..x; t /; u.x; t //, to the Cauchy problem for (1) with the far field behavior: u.x; t / ! 0;

.x; t / ! 1  0;

as jxj ! 1;

(3)

and initial data, .; u/jtD0 D .0 ; u0 /;

x 2 RN :

(4)

There is a great deal of literature on the large time existence and behavior of solutions to (1). The one-dimensional problem has been studied extensively by many people; see [11,23,37,38] and the references therein. For the multidimensional case, the local existence and uniqueness of classical solutions are discussed in [33, 39] in the absence of vacuum and recently, for strong solutions also in [3, 5, 6, 18, 27, 36] for the case where the initial density need not be positive and may vanish in open sets. The global classical solutions were first obtained by Matsumura-Nishida [32] for initial data close to a nonvacuum equilibrium in some Sobolev space H s . In particular, the theory [32] requires that the solution has small oscillations from a uniform nonvacuum state so that the density is strictly away from vacuum states and the gradient of the density remains bounded uniformly in time. Indeed, for the system (1), it holds that:

38 Global Existence of Regular Solutions with Large Oscillations and. . .

2039

Proposition 1 ([32]). For P D P ./ 2 C 3 .0; 1/, let 1 > 0 be such that P 0 .1 / > 0 and suppose that 0  1 ; u0 2 H 3 .R3 /: Then there exists a positive constant " depending on ; ; 1 ; and P; such that if k.0  1 ; u0 /kH 3 .R3 /  "; the Cauchy problem (1), (2), (3), and (4) has a unique global classical solution .; u/ in R3  .0; 1/ satisfying sup k.  0t 0 in some unbounded domains (including the whole space) and [22] for the general treatment of the full Navier-Stokes-Fourier system (with barotropic equations as a particular case). However, little is known of the structure of such weak solutions. Recently, for the far field density away from vacuum .1 > 0/, under the additional assumptions that the viscosity coefficients  and  satisfy  > maxf4; g;

(5)

Hoff [14,16,17] obtained a new type of global weak solutions with small energy that have extra regularity information compared with those large weak ones constructed by Lions [30] and Feireisl et al. [9]. Note that here the weak solutions may contain vacuum states although the spatial measure of the vacuum set has to be small. Moreover, under some additional conditions that prevent the appearance of vacuum

2040

J. Li and Z.P. Xin

states in the data, Hoff [14, 17] and Zhang [42] also obtained classical solutions. Indeed, for the initial energy defined as:  Z  1 (6) 0 ju0 j2 C G.0 / dx; C0 D  2 where G denotes the potential energy density given by Z  P .s/  P .1 / ds; G./ ,  s2 1 Zhang [42] proved that: Proposition 2 ([42]). Let  D RN .N D 2; 3/ and N > 1 > 0. Assume that  > 0;   0 and that the pressure P 2 C 1 Œ0; 1/ \ C 3 .0; 1/ satisfies P .0/ D 0; P 0 .1 / > 0; .  1 /.P ./  P .1 // > 0; for  6D 1 :

(7)

Then, for given positive numbers M (not necessarily small) and N1 2 .1 ; /, N there is a positive number ", such that the Cauchy problem (1), (2), (3), and (4) with the initial data .0 ; u0 / satisfying 0 ju0 j2 ; G.0 / 2 L1 ./;

(8)

and (

0  1 ; u0 2 H 3 .RN /; kru0 kL2 .RN /  M;

0 < 1  0  N1 ;

C0  ";

(9)

has a unique global classical solution .; u/ satisfying for any T > 0, 1   .x; t /  ; N x 2 RN ; t  0; 2 1 .  1 ; u/ 2 C 1 .RN  Œ0; T / \ C .Œ0; T I H 3 .RN // \ C 1 .Œ0; T I H 2 .RN //; Z lim .j  1 j4 C juj4 C jruj2 /.x; t /dx D 0: t!1

However, it should be noted that in the presence of vacuum states, the global wellposedness of classical solutions and the regularity and uniqueness of those weak solutions [9, 14, 30] remain completely open. In fact, this is a subtle issue because usually one would not expect such general results due to Xin’s blow-up results in [40], where it is shown that in the case where the initial density has compact support, any smooth solution to the Cauchy problem of the full compressible Navier-Stokes system without heat conduction blows up in finite time for any space dimension, and the same holds for the barotropic case (1), at least in one-dimension, and the symmetric two-dimensional case [31]. See also the generalizations to the cases for

38 Global Existence of Regular Solutions with Large Oscillations and. . .

2041

the full compressible Navier-Stokes system with heat conduction [4] and for noncompact but rapidly decreasing at far field initial densities [35]. More recently, the assumptions of [40] that the initial density has compact support and that the smooth solution has finite energy are removed by Xin-Yan [41] for a large class of initial data containing vacuum states. Therefore, it is of great importance to study the global existence and uniqueness of classical solutions to the Cauchy problem for the barotropic compressible NavierStokes equations, (1), when the initial density is allowed to vanish, and even has compact support. Notations: Before stating the main results, the notations and conventions used throughout this chapter are explained. For R > 0 and  D RN .N D 2; 3/, set Z BR , fx 2 j jxj < Rg ;

Z f dx ,

f dx: 

Moreover, for 1  r  1; k  1; and ˇ > 0, the standard homogeneous and inhomogeneous Sobolev spaces are defined as follows. 8 ˆ ˆLr D Lr ./; D k;r D D k;r ./ D fv 2 L1loc ./jr k v 2 Lr ./g; ˆ < k k;2 k;r k k;2 ; W k;r D D DD ˇ W ./;Z H D W ;   ˆ ˇ ˆ 2ˇ O 2 2 ˇ ˆ ˇ P :H D f W  ! R ˇkf kHP ˇ D j j jf . /j d < 1 ; wherefO is the Fourier transform of f .

2

Main Results

In this section, we study the global existence and uniqueness of classical solutions to the Cauchy problem for the barotropic compressible Navier-Stokes equations, (1) in multidimensional spaces with smooth initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum .1 D 0/ or nonvacuum .1 > 0/I in particular, the initial density is allowed to vanish, and even has compact support. In fact, combining [20] and [18] yields Theorem 1. Let  D R3 . For q 2 .3; 6/ and for given numbers M > 0 (not necessarily small), ˇ 2 .1=2; 1, and N  1 C 1, in addition to (8), suppose that the initial data .0 ; u0 / satisfy P ˇ \ D1 \ D2; u0 2 H

.0  1 ; P .0 /  P .1 // 2 H 2 \ W 2;q ;

0  inf 0  sup 0  ; N

ku0 kHP ˇ  M;

(10) (11)

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and the compatibility condition 1=2

 4u0  . C /rdivu0 C rP .0 / D 0 g;

(12)

for some g 2 L2 . If 1 D 0; assume that 0 2 L3=2

(13)

in addition. Then there exists a positive constant " depending on ; ; 1 ; a;  ; ; N ˇ, and M such that for C0 as in (6) if C0  ";

(14)

the Cauchy problem (1), (2), (3), and (4) has a unique global classical solution .; u/ in   .0; 1/ satisfying 0  .x; t /  2; N

x 2 R3 ; t  0;

8 ˆ .  1 ; P  P .1 // 2 C .Œ0; 1/I H 2 \ W 2;q /; ˆ ˆ ˆ 0;

. ; 1/;

for 1 D 0:

(17)

If 1 D 0, one can even have the following decay rates of the classical solutions .; u/ obtained in Theorem 1. Theorem 2 ([29]). Let 1 D 0 and the conditions of Theorem 1 hold. If  > 3=2, assume in addition that 0 2 L1 :

(18)

Then, for r 2 .1; 1/, there exist positive constants C .r/ and C depending on ; ;  ; ; N ˇ, and M such that for the classical solutions .; u/ obtained in Theorem 1, it holds that

38 Global Existence of Regular Solutions with Large Oscillations and. . .

8 1C1=p ˆ ; for p 2 Œ2; 6; ˆ 3=2, C .r/ and C both depend on k0 kL1 .R3 / also. If 1 > 0, similar to the previous studies on the Stokes approximation equations in [28], one can obtain from (16) the following large-time behavior of the gradient of the density when vacuum states appear initially and the far field density is away from vacuum, which is completely in contrast to the classical theory [7, 17, 32]. Theorem 3. In addition to the conditions of Theorem 1, assume further that there exists some point x0 2 R3 such that 0 .x0 / D 0. Then if 1 > 0, the unique global classical solution .; u/ to the Cauchy problem (1), (2), (3), and (4) obtained in Theorem 1 has to blow up as t ! 1, in the sense that for any r > 3, lim kr.; t /kLr D 1:

t!1

Recently, when  D R2 and 1 D 0, for strong and classical solutions, [29] obtained some a priori decay with rates (in large-time) for both the pressure and the spatial gradient of the velocity field provided that the initial total energy was suitably small. Moreover, by using these key decay rates and some analysis of the expansion rates of the essential support of the density, [29] established the global existence and uniqueness of classical solutions (which may possibly be of large oscillations) in two spatial dimensions, provided the smooth initial data have small total energy. In addition, the initial density can even have compact support. The result concerning the global existence of strong and classical solutions is then collected. (Proof can be found in [29]). Theorem 4 ([29]). Let  D R2 and 1 D 0. Suppose that in addition to (8), the initial data .0 ; u0 / satisfy for given numbers M > 0, N  1, a > 1; q > 2; and ˇ 2 .0; 1, 1=2

P ˇ \ D 1 ;  u0 2 L2 ; N xN a 0 2 L1 \ H 1 \ W 1;q ; u0 2 H 0 2 Œ0; ; 0

(20)

and ku0 kHP ˇ C k0 xN a kL1  M;

(21)

xN , .e C jxj2 /1=2 log2 .e C jxj2 /:

(22)

where

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J. Li and Z.P. Xin

Then for positive constant N0 satisfying Z 0 dx  BN0

1 2

Z 0 dx;

(23)

there exists a positive constant " depending on ; ;  ; a; ; N ˇ; N0 , M , and k0 kL1 such that if C0  ";

(24)

the problem (1), (2), (3), and (4) has a unique global strong solution .; u/ satisfying for any 0 < T < 1, .x; t / 2 R2  Œ0; T ;

0  .x; t /  2; N

8 ˆ  2 C .Œ0; T I L1 \ H 1 \ W 1;q /; ˆ ˆ ˆ ˆ ˆ xN a  2 L1 .0; T I L1 \ H 1 \ W 1;q /; ˆ ˆ ˆ 1 and some positive constant N1 depending only on ˛; N0 ; M , and k0 kL1 . Moreover, .; u/ has the following decay rates; that is, for t  1, 8 1C1=p ˆ ; for p 2 Œ2; 1/; ˆ 2/ [31]. The blowup phenomena in both 1D and 2D are based on the fact that the support of the density will not grow in time due to the framework of Sobolev space H 1 . However, for the two- or three-dimensional case, the framework of homogeneous spaces D 1 allows the slow decay of the velocity field for large values of the spatial variable x. In fact, if the smooth solution decays quickly enough for large values of the spatial variable x, it will blow up in finite time (see [35] for details). Remark 5. It should be emphasized that in Theorem 1, the viscosity coefficients are only assumed to satisfy the physical conditions (2). The theory on small energy weak solutions, developed in [14, 17], requires the additional assumption (5), which is crucial in establishing the time-independent upper bound for the density in the arguments in [14, 17]. Remark 6. For the incompressible Navier-Stokes system, a lot of results on the global well-posedness in scaling invariant spaces are available [10, 24, 25]. In particular, Fujita-Kato [10] and Kato [24] proved that the system is globally wellP 1=2 or in L3 . Here, posed for small initial data in the homogeneous Sobolev spaces H because the initial energy is small, one needs the boundedness assumptions on the P ˇ -norm of the initial velocity. It should be noted here that H P ˇ ,! L6=.32ˇ/ and H 6=.3  2ˇ/ > 3 for ˇ > 1=2, implies that, compared with the results in [10, 24], the conditions on the initial velocity may be optimal under the smallness conditions on the initial energy. In this chapter we mainly prove Theorems 1, 2 and 3. Note that for initial data P ˇ , the local existence and in the class satisfying (10), (11), and (12) except u0 2 H uniqueness of classical solutions to the Cauchy problem, (1), (2), (3), and (4), have been established recently in [5,18]. Thus, to extend the classical solution globally in time, we need global a priori estimates on 3 to (1), (2), (3), and (4) in suitable higher norms. Some of the new main difficulties are due to the appearance of vacuum states and that there are no other constraints on the viscosity coefficients beyond the physical conditions (2). It turns out that the key issue in this chapter is to derive both the time-independent upper bound for the density and the time-dependent higher norm estimates of the smooth solution .; u/. We start with the basic energy estimate and the initial layer analysis, and succeed in deriving an estimate on the spatial weighted L3 -norm of the velocity, and the weighted spatial mean estimates on both the gradient and the material derivatives of the velocity. This is achieved by modifying the basic elegant estimates of the material derivatives of the velocity developed by Hoff [12, 14, 15] in the theory of small energy weak solutions with nonvacuum far fields and an interpolation argument. Then we can obtain the desired

38 Global Existence of Regular Solutions with Large Oscillations and. . .

2047

estimates on L1 .0; minf1; T gI L1 .R3 //-norm and the time-independent ones on L8=3 .minf1; T g; T I L1 .R3 //-norm of the effective viscous flux (see (38) for the definition). It follows from these key estimates and Zlotnik’s inequality (see Lemma 4) that the density admits a time-uniform upper bound which is the key for global estimates of classical solutions. This approach to estimating a uniform upper bound for the density is motivated by the previous analysis on the two-dimensional Stokes approximation equations in [28]. The next main step is to bound the gradients of the density and the velocity. Motivated by recent studies [19,21] on the blow-up criteria of classical (or strong) solutions to (1), such bounds can be obtained by solving a logarithm Gronwall inequality based on a Beal-Kato-Majda type inequality (see Lemma 5) and the a priori estimates we have just derived and, moreover, such a derivation simultaneously also yields the bound for L1 .0; T I L1 .R3 //-norm of the gradient of the velocity; see Lemma 11 and its proof. It should be noted here that smallness of the gradient of the initial density which prevents the appearance of a vacuum [17,32] is not required. Finally, with these a priori estimates on the gradients of the density and the velocity at hand, we can estimate the higher-order derivatives by using the same arguments as in [21] to obtain the desired results. The rest of this chapter is organized as follows: In Sect. 3, some elementary facts and inequalities needed in later analysis are collected. Section 4 is devoted to deriving the necessary a priori estimates on classical solutions needed to extend the local solution to all time. Then finally, the main results, Theorems 1, 2, and 3, are proved in Sect. 5.

3

Preliminaries

In this section, some known facts and elementary inequalities that are later used frequently are recalled. We start with the local existence and uniqueness of classical solutions when the initial density may not be positive and may vanish in an open set. Lemma 1 ([5, 18]). Let  D R3 . For 1  0, assume that the initial data .0  P ˇ . Then there exist a small time 0; u0 / satisfy (8), (10), (11), and (12) except u0 2 H T > 0 and a unique classical solution .; u/ to the Cauchy problem (1), (2), (3), and (4) on R3  .0; T  such that 8 ˆ .  1 ; P  P .1 // 2 C .Œ0; T I H 2 \ W 2;q /; ˆ ˆ ˆ ˆ ˆ u 2 C .Œ0; T I D 1 \ D 2 /; ru 2 L2 .0; T I H 2 / \ Lp0 .0; T I W 2;q /; ˆ ˆ ˆ 0 that may depend on q; r such that for f 2 H 1 .R3 / and g 2 Lq .R3 / \ D 1;r .R3 /; one has p

.6p/=2

kf kLp  C kf kL2

q.r3/=.3rCq.r3//

kgkC R3   C kgkLq

.3p6/=2

krf kL2

;

(35)

3r=.3rCq.r3//

krgkLr

:

(36)

Some elementary estimates follow from (35) and the standard Lp -estimate for the following elliptic system derived from the momentum equations in (1): 4F D div.uP /;

4! D r  .uP /;

(37)

where fP , ft C u  rf;

F , .2 C /divu  P ./ C P .1 /;

! , r  u;

(38)

are the material derivatives of f , the effective viscous flux, and the vorticity, respectively. Lemma 3. Let  D R3 and .; u/ be a smooth solution of (1) and (3). Then there exists a generic positive constant C depending only on  and  such that for any p 2 Œ2; 6 krF kLp C kr!kLp  C kuP kLp ; .3p6/=.2p/

kF kLp C k!kLp  C kuP kL2

.krukL2 C kP  P .1 /kL2 /.6p/=.2p/ ; (40)

krukLp  C .kF kLp C k!kLp / C C kP  P .1 /kLp ; .6p/=.2p/

krukLp  C krukL2

(39)

.kuP kL2 C kP  P .1 /kL6 /.3p6/=.2p/ :

(41) (42)

Proof. The standard Lp -estimate for the elliptic system (37) directly yields (39), which, together with (35) and (38), gives (40). Note that u D rdivu C r  !, which implies that ru D r./1 rdivu C r./1 r  !:

38 Global Existence of Regular Solutions with Large Oscillations and. . .

2049

Thus the standard Lp estimate shows that krukLp  C .kdivukLp C k!kLp /; for p 2 Œ2; 6; which, together with (38), gives (41). Now (42) follows from (35), (41), and (39). Next, the following Zlotnik inequality is used to get the uniform (in time) upper bound of the density . Lemma 4 ([43]). Let the function y satisfy y 0 .t / D g.y/ C b 0 .t / on Œ0; T ;

y.0/ D y 0 ;

with g 2 C .R/ and y; b 2 W 1;1 .0; T /. If g.1/ D 1 and b.t2 /  b.t1 /  N0 C N1 .t2  t1 /

(43)

for all 0  t1 < t2  T with some N0  0 and N1  0, then n o y.t/  max y 0 ; C N0 < 1 on Œ0; T ; where is a constant such that g. /  N1

for

 :

(44)

Finally, the following Beale-Kato-Majda type inequality, which can be found in [19] and was first proved in [1] when divu  0, is used later to estimate krukL1 and krkL2 \L6 . Lemma 5 ([19]). For 3 < q < 1, there is a constant C .q/ such that the following estimate holds for all ru 2 L2 .R3 / \ D 1;q .R3 /,  krukL1 .R3 /  C kdivukL1 .R3 / C k!kL1 .R3 / log.e C kr 2 ukLq .R3 / / C C krukL2 .R3 / C C:

4

A Priori Estimates

In this section, some necessary a priori bounds are established for smooth solutions to the Cauchy problem (1), (2), (3), and (4) to extend the local classical solution guaranteed by Lemma 1. Thus, let T > 0 be a fixed time and .; u/ be the smooth solution to (1), (2), (3), and (4) on R3  .0; T  in the class (33) with smooth initial data .0 ; u0 / satisfying (10), (11), and (12). To estimate this solution, for  .t / ,

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J. Li and Z.P. Xin

minf1; tg, define

A1 .T / , sup



t2Œ0;T 

A2 .T / , sup  3

kruk2L2

Z



Z

T

Z

C

jPuj2 dxdt;

(45)

 3 jr uP j2 dxdt;

(46)

0

jPuj2 dx C

Z

T

Z

0

t2Œ0;T 

and Z A3 .T / , sup

juj3 .x; t /dx:

0tT

One has the following key a priori estimates on .; u/. Proposition 3. Under the conditions of Theorem 1, for ı0 , .2ˇ  1/=.4ˇ/ 2 .0; 1=4;

(47)

N ˇ and M such there exists some positive constant " depending on , , 1 , a,  , , that if .; u/ is a smooth solution of (1), (2), (3), and (4) on R3  .0; T  satisfying N sup   2; R3 Œ0;T 

1=2

A1 .T / C A2 .T /  2C0 ;

A3 . .T //  2C0ı0 ;

(48)

A3 . .T //  C0ı0 ;

(49)

the following estimates hold sup   7=4; N R3 Œ0;T 

1=2

A1 .T / C A2 .T /  C0 ;

provided C0  ". Proof. Proposition 3 is an easy consequence of the following Lemmas 8, 9, and 10. The following uses the convention that C denotes a generic positive constant depending on , , 1 , a,  , , N ˇ, and M , and one writes C .˛/ to emphasize that C depends on ˛. We start with the following standard energy estimate for .; u/ and preliminary L2 bounds for ru and uP . Lemma 6. Let .; u/ be a smooth solution of (1), (2), (3), and (4) on R3  .0; T  with 0  .x; t /  2. N Then there is a positive constant C D C ./ N such that

38 Global Existence of Regular Solutions with Large Oscillations and. . .

Z  sup 0tT

2051

 Z TZ  1 2 juj C G./ dx C jruj2 C . C /.divu/2 dxdt  C0 ; 2 0 (50) Z TZ jruj3 dxdt; (51) A1 .T /  C C0 C C 0

and Z

T

Z

A2 .T /  C C0 C CA1 .T / C C

 3 jruj4 dxdt:

(52)

0

Proof. Multiplying the first equation in (1) by G 0 ./ and the second by uj and integrating, then applying the far field condition (3), the energy inequality (50) is easily obtained. The proof of (51) and (52) is due to Hoff [12]. For m  0, multiplying (1)2 by  m uP and then integrating the resulting equality over R3 lead to Z

 m jPuj2 dx D

,

Z

. m uP  rP C  m 4u  uP C . C / m rdivu  uP /dx

3 X

Mi :

iD1

(53) Using (1)1 and integrating by parts give Z M1 D  Z D

. m .divu/t .P  P .1 //   m .u  ru/  rP /dx

Z

m



 divu.P  P .1 //dx

D Z C Z 

 m uP  rP dx

 m m1  0

Z divu.P  P .1 //dx

t

  m P 0 .divu/2  P .divu/2 C P @i uj @j ui dx  m divu.P  P .1 //dx



C m m1  0 kP  P .1 /kL2 krukL2

t 2 C C ./kruk N L2 Z  m 2   divu.P  P .1 //dx C C ./kruk N N 2  2.m1/  0 C0 : L2 C C ./m t

(54)

2052

J. Li and Z.P. Xin

Integration by parts implies Z M2 D

 m 4u  uP dx

Z  m m m1 0  kruk2L2 t C   kruk2L2   m @i uj @i .uk @k uj /dx 2 2 Z  m   kruk2L2 t C C m m1 kruk2L2 C C  m jruj3 dx; 2 (55) and similarly, D

C m m. C / m1  kdivuk2L2 t C  kdivuk2L2 2 2 Z Œ3pt  . C / m divudiv.u  ru/dx M3 D 



C m  kdivuk2L2 t C C m m1 kruk2L2 C C 2

Z

 m jruj3 dx: (56)

Combining (53), (54), (55), and (56) leads to

0

m

Z

. B.t // C  .C m

m1

C

 m jPuj2 dx 2 C .//kruk N L2

2 2.m1/ 0

C C ./m N 

 C0 C C

Z

 m jruj3 dx; (57)

where  . C / B.t / , kruk2L2 C kdivuk2L2  2 2

Z divu.P  P .1 //dx



 . C / 1=2 kruk2L2 C kdivuk2L2  C C0 kdivukL2 2 2



 . C / kruk2L2 C kdivuk2L2  C C0 : 4 2

(58)

Integrating (57) over .0; T /, choosing m D 1, and using (58), we get (51). j Next, for m  0, operating  m uP j Œ@=@t C div.u/ to (1)2 , summing with respect 3 to j , and integrating the resulting equation over R , we obtain, after integration by parts,

38 Global Existence of Regular Solutions with Large Oscillations and. . .



m 2

Z



2

jPuj dx t

Z

m   m1  0 2

Z

jPuj2 dx

 m uP j Œ@j Pt C div.@j P u/dx C 

D

Z

Z

j

 m uP j Œ4ut C div.u4uj /dx

 m uP j Œ@t @j divu C div.u@j divu/dx ,

C . C /

2053

3 X

Ni :

i D1

(59) It follows from integration by parts and using the equation (1)1 that Z

 m uP j Œ@j Pt C div.@j P u/dx

N1 D  Z

0

 m ŒP divu@j uP j C @k .@j uP j uk /P  P @j .@k uP j uk /dx

D

(60)

 C ./ N m krukL2 kr uP kL2  ı m kr uP k2L2 C C .; N ı/ m kruk2L2 : Integration by parts leads to Z

j

 m uP j Œ4ut C div.u4uj /dx

N2 D 

Z D  

3 4

 m Œjr uP j2 C @i uP j @k uk @i uj  @i uP j @i uk @k uj  @i uj @i uk @k uP j dx Z

 m jr uP j2 dx C C

Z

 m jruj4 dx: (61)

Similarly, N3  

C 2

Z

 m .divPu/2 dx C C

Z

 m jruj4 dx:

(62)

Substituting (60), (61), and (62) into (59) shows that for ı suitably small, it holds that  

m

Z

2



Z

jPuj dx

 m m1  0

Z

C

 m jr uP j2 dx C . C /

Z

 m .divPu/2 dx

t

jPuj2 dx C C  m kruk4L4 C C ./ N m kruk2L2 :

(63)

2054

J. Li and Z.P. Xin

Taking m D 3 in (63) and noticing that Z

T

3

 2 0

Z

jPuj2 dxdt  CA1 .T /;

0

we immediately obtain (52) after integrating (63) over .0; T /. The proof of Lemma 6 is completed. Next, the following lemma plays important roles in the estimates on both Ai . .T //.i D 1; 2; 3/ and the uniform upper bound of the density for small time. Lemma 7. Let .; u/ be a smooth solution of (1), (2), (3), and (4) on R3  .0; T  satisfying (48). Then there exist positive constants K and "0 both depending only on , , 1 , a,  , , N ˇ, and M such that t

sup

1ˇ

0t.T /

t 2ˇ

sup

Z

kruk2L2

Z

.T /

C

t

1ˇ

Z

jPuj2 dxdt  K.; N M /;

(64)

0

jPuj2 dx C

Z

.T /

t 2ˇ

Z

jr uP j2 dxdt  K.; N M /;

(65)

0

0t.T /

provided C0  "0 . Proof. As in [15], w1 and w2 are defined to be the solution to: Lw1 D 0;

w1 .x; 0/ D w10 .x/;

(66)

and Lw2 D rP ./;

w2 .x; 0/ D 0;

(67)

respectively, with L being the linear differential operator defined by j

.Lw/j , wt C u  rwj  .wj C . C /divwxj / D w P j  .wj C . C /divwxj /;

j D 1; 2; 3:

Straightforward energy estimates show that: Z sup 0t.T /

2

Z

jw1 j dx C

.T /

Z

2

jrw1 j dxdt  C ./ 0

Z

jw10 j2 dx;

(68)

38 Global Existence of Regular Solutions with Large Oscillations and. . .

2055

and Z sup

jw2 j2 dx C

Z

.T /

Z

jrw2 j2 dxdt  C ./C0 :

(69)

0

0t.T /

It follows from (66) and the standard L2 -estimate for an elliptic system that P 1 kL2 : krw1 kL6  C kr 2 w1 kL2  C kw

(70)

Multiplying (66) by w1t and integrating the resulting equality over R3 , one gets by (70) and (48)3 that Z 1 krw1 k2L2 C . C /kdivw1 k2L2 t C jwP1 j2 dx 2 Z D wP1 .u  rw1 /dx Z  C ./ N 

jwP1 j2 dx

ı =3 C ./C N 00

Z

1=2 Z

juj3 dx

1=3 krw1 kL6

jwP1 j2 dx;

which, together with Gronwall’s inequality and (68), gives Z .T / Z 2 sup krw1 kL2 C jwP1 j2 dxdt  C krw10 k2L2 ;

(71)

0

0t.T /

and sup 0t.T /

tkrw1 k2L2 C

Z

Z

.T /

t 0

jwP1 j2 dxdt  C kw10 k2L2 ;

(72)

provided C0  "01 , .2C .// N 3=ı0 . The solution operator w10 7! w1 .; t / is linear, therefore by the standard SteinWeiss interpolation argument [2], one can deduce from (71) and (72) that for any 2 Œˇ; 1, Z Z .T / sup t 1 krw1 k2L2 C t 1 jwP1 j2 dxdt  C kw10 k2HP ; (73) 0t.T /

0

with a uniform constant C independent of . Next, one estimates w2 . It follows in a similar way to (39) and (41) that (

P 2 kL2 ; kr..2 C /divw2  .P  P .1 ///kL2  C kw krw2 kL6  C .kw P 2 kL2 C kP  P .1 /kL6 /:

(74)

2056

J. Li and Z.P. Xin

Multiplying (67) by w2t , integrating the resulting equation over R3 and using (74), one has   Z Z 1 krw2 k2L2 C. C /kdivw2 k2L2  2 .P P .1 //divw2 dx C jwP2 j2 dx 2 t Z Z D wP2 .u  rw2 /dx  Pt divw2 dx Z  C ./ N

2

1=2 Z

1=3

3

juj dx

jwP2 j dx

Z C

krw2 kL6 Z

divw2 div..P  P .1 //u/dx C ı =3

 C ./C N 00

Z

jwP2 j2 dx

1=2

.P .1 / C .  1/P /divudivw2 dx



k1=2 w P 2 kL2 C kP  P .1 /kL6



  P  P .1 / dx .P  P .1 //u  r divw2  2 C  Z 1 C .P  P .1 //2 divudx C C kruk2L2 C C krw2 k2L2 2.2 C / Z ı =3 1=3 jwP2 j2 dx C C C0 C C kP  P .1 /kL3 kukL6 k1=2 wP 2 kL2  C ./C N 00 Z



C C kP  P .1 /k4L4 C C kruk2L2 C C krw2 k2L2 Z ı =3 1=3 jwP2 j2 dx C C C0 C C kruk2L2 C C krw2 k2L2 ;  C ./C N 00 which, together with (69) and Gronwall’s inequality, gives

sup 0t.T /

krw2 k2L2 C

Z

.T /

0

Z

1=3

jwP2 j2 dxdt  C C0 ;

(75)

N 3=ı0 . Taking w10 D u0 so that w1 C w2 D u, one then provided C0  "02 , .2C .// concludes from (73) and (75) that for any 2 Œˇ; 1,

sup 0t.T /

t 1 kruk2L2 C

Z 0

.T /

t 1

Z

1=3

jPuj2 dxdt  C ku0 k2HP C C C0 ;

(76)

provided C0  "0 , minf"01 ; "02 g. Thus, (64) follows from (76) directly. To prove (65), one takes m D 2  ˇ in (63) to obtain, after integrating (63) over .0;  .T // and using (76) and (42), that

38 Global Existence of Regular Solutions with Large Oscillations and. . .

t 2ˇ

sup

Z

jPuj2 dx C

t 2ˇ

Z

jr uP j2 dxdt

.T /

t 2ˇ kruk4L4 dt C C .; N M/

C 0

Z

.T /

 t 2ˇ krukL2 kuP k3L2 C kP  P .1 /k3L6 dt C C .; N M/

.T /

 1=2 2ˇ 1=2 2 1=2 1ˇ 1=2 2 t .2ˇ1/=2 t 1ˇ kruk2L2 .t k uP kL2 / .t k uP kL2 /dt

C Z

.T / 0

0t.T /

Z

Z

2057

0

C 0

C C .; N M/  C .; N M/

t

sup

2ˇ

!1=2

Z

2

jPuj dx

C C .; N M /;

0t.T /

which implies (65). Thus, we finish the proof of Lemma 7. The following Lemma 8 gives an estimate on A3 . .T //. Lemma 8. If .; u/ is a smooth solution of (1), (2), (3), and (4) on R3  .0; T  N satisfying (48), there exists a positive constant "1 depending on , , 1 , a,  , , ˇ, and M such that the following estimate holds for ı0 defined by (47), Z juj3 .x; t /dx  C0ı0 ; (77) sup 0t.T /

provided C0  "1 . Proof. Multiplying (1)2 by 3juju, and integrating the resulting equation over R3 , one obtains by (42) that Z d juj3 dx dt Z Z  C jujjruj2 dx C C jP  P .1 /jjujjrujdx 3=2

1=2

 C kukL6 krukL2 krukL6 C C kP  P .1 /kL3 kukL6 krukL2 5=2

1=6

 C krukL2 .kuP kL2 C kP  P .1 /kL6 /1=2 C C C0 kruk2L2  1=2 5=2 1=6 1=6  C krukL2 kuP kL2 C C0 C C C0 kruk2L2 0  C t .2ı0 3=2/.1ˇ/ .t 1ˇ kruk2L2 /2ı0 C5=4 .t 1ˇ k1=2 uP k2L2 /1=4 kruk4ı L2

1=12 3.1ˇ/=4

C C C0

t

1=6

.t 1ˇ kruk2L2 /3=4 krukL2 C C C0 kruk2L2 ;

2058

J. Li and Z.P. Xin

which together with (64) and (50) gives Z juj3 dx sup 0t.T /

Z

.T /

 C .; N M/

t



2.34ı0 /.1ˇ/ 38ı0

!.38ı0 /=4 Z

C

1=12 C .; N M /C0

Z C

kruk2L2 dt

dt

0

0

Z

.T /

t

3.1ˇ/=2

!1=2 Z

!1=2

.T /

dt

0

!2ı0

.T /

0

(78)

kruk2L2 dt

0 ju0 j3 dx C C C0

 C .; N M /C02ı0 ; provided C0  "0 , where in the last inequality we have used the following simple facts. Z 3.2ˇ1/=.4ˇ/ Z 3=.2ˇ/ 3 2 0 ju0 j dx ku0 kHP ˇ 0 ju0 j dx  C (79)  C .; N M /C02ı0 ; and ˇ.2ˇ  1/ 2.3  4ı0 /.1  ˇ/ 0. Hence, for any 0 <  < T  T  with T finite, it follows from Lemmas 14 and 15 that 8 T  , such that (48) holds for T D T  , which contradicts (142). Hence, (145) holds. Lemmas 1, 14, 15, and (143) thus show that .; u/ is in fact the unique classical solution defined on R3  .0; T  for any 0 < T < T  D 1. Finally, to finish the proof of Theorem 1, it remains to prove (16). Multiplying (84) by 4.P  P .1 //3 and integrating the resulting equality over 3 R , we have 

kP 

0 P .1 /k4L4

Z

.P  P .1 //4 divudx

.t / D  .4  1/ Z 

P .1 /.P  P .1 //3 divudx;

which yields Z

ˇ 0 ˇˇ ˇ ˇ kP  P .1 /k4L4 .t /ˇ dt

1 1

Z

1

C



kP 

1

P .1 /k4L4

C

kruk4L4



(146) dt  C;

due to (87). Combining (87) with (146) leads to lim kP  P .1 /kL4 D 0;

t!1

which together with (50) implies Z lim

t!1

j  1 jr dx D 0;

for all r satisfying (17). Note that (50) and (35) imply Z

1=2 juj4 dx 

Z

juj2 dx

1=2

kuk3L6  C kruk3L2 :

Thus (16) follows provided that lim krukL2 D 0:

t!1

Setting I .t / ,

C  kruk2L2 C kdivuk2L2 ; 2 2

(147)

2076

J. Li and Z.P. Xin

choosing m D 0 in (53), and using (55) and (56), we have Z

jI 0 .t /j  C

1=2

jPuj2 dx C C kruk3L3 C C C0 kr uP kL2 ;

(148)

where we have used the following simple estimate: ˇZ ˇ ˇ ˇ ˇ jM1 j D ˇ uP  rP dx ˇˇ ˇ ˇZ ˇ ˇ D ˇˇ .P  P .1 //divPudx ˇˇ 1=2

 C C0 kr uP kL2 : We thus deduce from (148), (80), and (87) that Z

1

jI 0 .t /j2 dt  C

1

Z

1



1



k1=2 uP k4L2 C kruk2L2 kruk4L4 C kr uP k2L2 dt

1

Z C

1

k1=2 uP k2L2 C kruk4L4 C kr uP k2L2 dt

 C; which, together with Z

1

jI .t /j2 dt  C 1

Z

1 1

kruk2L2 dt  C;

implies (147). The proof of Theorem 1 is finished. Proof of Theorem 2. The proof of Theorem 2 is due to [29]; we only sketch it here for completeness. In fact, it suffices to prove (19). It follows from Proposition 3 that there exists some " depending only on ; ;  ; ; N ˇ, and M such that 

krukL2 C kkL \L1 C k1=2 uP kL2

sup



1t 0/; r2 C z

Analogously, (47) and (48) can be proved. This completes the proof of Lemma 5. t u Continuation of the proof of Lemma 4. Let 1; 0

 2  2   1 X 1 1 Jk .1; f / Ij jCk C hj jCk ; D kŠ . / . / kD0

2;

0 D

 2  2   1 X 1 Jk .1; f / Ij jCkC2 C h1 ; j jCkC2 kŠ . / . / kD0

3; 0

D

1 N X i  X kD0

kŠ

 Jk .!j ; gj / Ij jCkC1

j D1



2 . /

C

h1 j jCkC1



2 . /



and then @ x ŒP0 . /.f; g/.x/ D

1; ˛ C ˇ 2; 1 3; 0 .x/ C 0 .x/   .x/: . / . / . / 0

;

2118

Y. Shibata and Y. Enomoto

  Let N be an even number. Applying Lemma 5 to I` 2 =. / yields that 1;

0 D N 2 log



2 . /

X   mC N2 1 1 J2mj j .1; f / 1 1   2m .2m  j j/Š 2 . / j j m

C

1 X m

j j1 2

2

J2mC1j j .1; f /  N .1/mC 2 .2m C 1  j j/Š 2



2 . /

mC N 1 2

  2  2  1 X 1 N 1 C Jk .1; f / hj jCk C hj jCk ; kŠ . / . / kD0 2;

0 D N log



2 . /



1 X m

C

1 X m

C

j jC1 2

j j 2 C1

  mC N2 1 J2m2j j .1; f / 1 1 2m2   .2m  2  j j/Š 2 . /

J2m1j j .1; f /  N .1/mC 2 .2m  1  j j/Š 2



2 . /

mC N 1 2

  2  2  1 X 1 1 Jk .1; f / hN C h ; j jCkC2 j jCkC2 kŠ . / . / kD0

3;

0 D N 1 log



2 . /

X 1 m

j jC1 2

  mC N2 1 N X J2m1j j .i!j ; gj / 1 1 2m1   .2m  1  j j/Š 2 . / j D1

 2 mC N 1 1 X N 2 X J2mj j .i!j ; gj /  mC N2 .1/ C .2m  j j/Š 2 . / j j j D1 m

2

  2  2  1 X N X 1 N 1 C Jk .i!j ; gj / hj jCkC1 C hj jCkC1 : kŠ . / . / kD0 j D1

In view of Lemma 2, let 2 > 0 be a positive number chosen in such a way that j 2 =. /j  12 when j j  2 . By (44), ˇ ˇ ˇ ˇ 1   mC N2 1 ˇ ˇX J 1 1 2mj j .1; f / ˇ 2m ˇ   ˇ ˇ ˇ ˇ .2m  j j/Š 2 . / ˇ ˇm j j 2

  C ;p;R

2 2

N2 1C j j2

j j 2

1 X m

1 X

1  C ;p;N;R .2m/Š mD0

j j 2

1 .2m  j j/Š

!2mj j p 2 2R 2 kf kLp .RN / 

!2m p 2 2R 2 kf kLp;R .RN / ; 

(49)

39 Global Existence of Classical Solutions and Optimal Decay Rate for. . .

2119

ˇ ˇ ˇ 1 ˇ mC N 1  ˇ X J ˇ 2 2 2mC1j j .1; f /  ˇ mC N2 2m ˇ  .1/ ˇ ˇ ˇ ˇ .2m C 1  j j/Š 2 . / ˇm j j1 ˇ 2

1 X

1  C ;p;N;R .2m C 1/Š mD0

!2mC1 p 2 2R 2 kf kLp;R .RN / : 

(50)

1 When jzj  1=2, it follows from the definitions of hN ` .z/ and h` .z/ that there exist C > 0 and M > 1 such that 1 ` jhN ` .z/j C jh` .z/j  CM :

(51)

Therefore, by (51) ˇ1  2  2  ˇˇ  ˇX 1 ˇ ˇ N 1 Jk .1; f / hj jCk C hj jCk ˇ ˇ ˇ kŠ . / . / ˇ kD0

 C ;p;R

1 X .2RM /k kD0

kŠ

kf kLp .RN / :

(52)

Thus, by (49), (50), and (52), there exist families of operators G1j . / with G1j . / 2 A.U 2 ; L.Lp;R .RN /; Lp .BR /// .j D 1; 2/ such that 1;

0 D N 2 log G11 . / C G12 . /

for j j  2 :

(53)

Analogously, there exist families of operators G2j . / and G3j . / with G2j . / 2 A.U 2 ; L.Lp;R .RN /; Lp .BR ///; G3j . / 2 A.U 2 ; L.Lp;R .RN /N ; Lp .BR /// .j D 1; 2/ such that 2;

(54)

3;

(55)

0 D N G21 . / C G22 . /; 0 D N 1 log G31 . / C G32 . /

for j j  2 . The formula (38) follows from (32), (54), and (55), when N is an even number.

2120

Y. Shibata and Y. Enomoto

  Let N be an odd number. Applying Lemma 5 to I` 2 =. / yields that

1; 0

 2 mC N2 1 1 X J2mj j .1; f /  mC N 1 2 D .1/ .2m  j j/Š 2 . / j j m

2

C N 2 log



2 . /

X 1 m

C

j j1 2

  mC N 1 2 J2mC1j j .1; f / 1 1 2mC1   .2m C 1  j j/Š 2 . /

 2  2   1 X 1 1 C h ; Jk .1; f / hN j jCk j jCk kŠ . / . / kD0

2; 0

D

1 X m

j j 2 C1

C N log

J2m2j j .1; f /  N 1 .1/mC 2 .2m  2  j j/Š 2



2 . /

X 1 m

C

j jC1 2



2 . /

mC N2 1

  mC N 1 2 J2m1j j .1; f / 1 1 2m1   .2m  1  j j/Š 2 . /

 2  2   1 X 1 1 C h ; Jk .1; f / hN j jCkC2 j jCkC2 kŠ . / . / kD0

3; 0

D

 2 mC N2 1 1 N X X J2m1j j .i !j ; gj /  mC N 1 2 .1/ .2m  1  j j/Š 2 . / j jC1 j D1

m

2

  mC N 1  2 X 1 X N 2 J2mj j .i !j ; gj / 1 1   C N 1 log 2m . / .2m  j j/Š 2 . / j j j D1 m

2

 2  2   1 X N X 1 N 1 C hj jCkC1 : Jk .i !j ; gj / hj jCkC1 C kŠ . / . / j D1 kD0

The same argumentation as that in the proof of (38) yields (41) when N is an odd number. Analogously, (39), (40), (42), and (43) can be proved. This completes the proof of Lemma 4. t u At this point, the assertion (1) of Theorem 21 has been proved. A proof of the assertion (2) in Theorem 21. In view of (32) and (33), for m .f; g/ 2 Xp;R .RN /, let Pj .0/.f; g/, V1j .0/.f; g/, and V2j .0/.f; g/ be defined by the formulas:

39 Global Existence of Classical Solutions and Optimal Decay Rate for. . .

# .˛ C ˇ/jj2 fO ./  i    gO ./ 'j ./ .x/; ŒPj .0/.f; g/.x/ D F  2 jj2

 g./ 'j ./  gO ./ 1 1 'j ./O .x/; ŒVj .0/.f; g/.x/ D F  ˛jj2 ˛jj4 " # O 2 1 'j ./i   f ./ ŒVj .0/.f; g/.x/ D F  2 jj2

2121

"

1

(56)

for j D 0; 1. Lemma 6. Let 1 < p < 1, let L and R be positive numbers, and let m be a nonnegative number. Let 1 and 2 be the same constants as in (36) and Lemma 4, m .RN / respectively. Assume that N  3. Then, for any .f; g/ 2 Xp;R kP1 . /.f; g/  P1 .0/.f; g/kWpmC1 .RN /  C j j.kf kWpmC1 .RN / C kgkWpm .RN / /; kV11 . /.f; g/  V11 .0/.f; g/kWpmC2 .RN /  C j jkgkWpm .RN / ; kV21 . /.f; g/  V21 .0/.f; g/kWpmC2 .RN /  C j j.kf kWpmC1 .RN / C kgkWpm .RN / / when 2 U 1 , and kP0 . /.f; g/  P0 .0/.f; g/kWpmC1 .BL /  C j j.kf kWpmC1 .RN / C kgkWpm .RN / /; kV10 . /.f; g/  V10 .0/.f; g/kWpmC2 .BL /  C j jkgkWpm .RN / ; kV20 . /.f; g/  V20 .0/.f; g/kWpmC2 .BL /  C j j.kf kWpmC1 .RN / C kgkWpm .RN / / when 2 V ;1 . Proof. Let h P1j . /f D F 1 'j ./

i C .˛ C ˇ/jj2 O ./ f 2 C ..˛ C ˇ/ C  2 /jj2

for j D 0 and j D 1. Write P1j . /f  P1j .0/f D

h i 'j ./fO ./ 1 F  2 C .˛ C ˇ/ 2 . 2 C .˛ C ˇ/ /1 C jj2



h i 'j ./fO ./ .˛ C ˇ/ 2 1 F  2 . 2 C .˛ C ˇ/ / 2 . 2 C .˛ C ˇ/ /1 C jj2



h i 'j ./jj2 fO ./ .˛ C ˇ/2 1 F :  2 . 2 C .˛ C ˇ/ / 2 . 2 C .˛ C ˇ/ /1 C jj2

2122

Y. Shibata and Y. Enomoto

By (36), ˇ  ˇ ˇ@

ˇ '1 ./ ˇ ˇ  C jj2j j : 2 . 2 C .˛ C ˇ/ /1 C jj2

for any multi-index 2 NN 0 and 2 U 1 . In view of Lemma 2, it may be assumed that ˇ ˇ ˇ

ˇ ˇ ˇ  C j j 2  C .˛ C ˇ/

(57)

for any 2 U 1 , so that by the Fourier multiplier theorem kP11 . /f  P11 .0/f kWpmC1 .RN /  C j jkf kWpmC1 .RN / when 2 U 1 . To treat the P10 part, in view of Lemma 2, it may be assumed that j.˛ C ˇ/ C 2  j  2 .j j C 1/ with some constant 2 provided that 2 C and j j  2 , so that by Lemma 2 j@ x .ŒP10 . /f .x/  ŒP10 .0/f .x//j  C 01 j j

Z jj2

jfO ./j d jj2

for any multi-index 2 NN 0 with some constant C depending on whenever 2 V ; 2 . Since N  3, Z

jj2 d  D jj2

Z

Z dS! j!jD1

2

r N 3 dr < 1;

0

where dS! is the surface element of the unit surface f! 2 RN j j!j D 1g. Since jfO ./j  kf kL1 .RN /  CR;p kf kLp .RN / as follows from the assumption that f.x/ vanishes when jxj > R, it holds that kP10 . /f  P10 .0/f kWpmC2 .BL /  Cm;p;R;L j jkf kLp .RN / : In this way, the assertions for P0 , V10 , and V20 can be proved, which completes the proof of Lemma 6. t u Let M0 .f; g/ D

X j D0;1

Pj .0/.f; g/;

V0 .f; g/ D

X X j D0;1 i D1;2

Vij .0/.f; g/;

39 Global Existence of Classical Solutions and Optimal Decay Rate for. . .

2123

and then

 ˛Cˇ i 1   gO ./ .x/; f .x/  F M0 .f; g/.x/ D 2  jj2 





 1 1   gO ./ i 1  O 1 1 gO ./ f ./ .x/ F V0 .f; g/.x/ D F .x/ F .x/: ˛ jj2  jj2 ˛ jj4 (58) Since P. /.f; g/ 2 WpmC1 .RN / and V. /.f; g/ 2 WpmC2 .RN /N for 2 V ; 3 , where 3 D min. 1 ; 2 /, it follows from (56) and Lemma 6 that M0 .f; g/ 2 mC1 mC2 N N Wp;loc.R and (31) in Theorem 21. Moreover, applying N / , V0 .f; g/ 2 Wp;loc .R / the Fourier multiplier theorem to the right-hand side of the formulas in (58) yields that X krM0 .f; g/kWpm .RN /C k@ x V0 .f; g/kWpmC2 .RN /  C .kf kWpmC1 .RN /CkgkWpm .RN / /: j jD2

Passing to the limit in (24) as j j ! 0 withj j  =4 and using (31) yield .0 ; v0 / satisfies the equation (24) with D 0. To prove X sup jxjN 1 jŒM0 .f; g/.x/j C sup jxj.N 2Cj j/ j@ x ŒV0 .f; g/.x/j jxj2R jxj2R j j1 (59)  CR;p;n .kf kLp .RN / C kgkLp .RN / /; h i g./ the term, I .x/ D F 1 O .x/ in (58), is only considered, because the other 4 jj terms can be treated in the same manner. Let

   gO ./ .x/ .i D 0; 1/ Ii .x/ D F 1 'i ./ jj4 and then I .x/ D I0 .x/ C I1 .x/. Let m be any integer satisfying the condition, m  N  1 C j j, and then j@ x I1 .x/j  C ;m jxjm .x 6D 0/: h i .i /   In fact, let K1j k .x/ D F 1 '1 ./ jj4j k .x/ and then j th component of

@ x I1 .x/

D

N X

(60)

ŒK1j k  gk .x/

kD1

where g.x/ D .g1 .x/; : : : ; gN .x// and the symbol  stands for the convolution operator. Using the identity N X xj @e ix D e ix 2 @ i jxj j j D1

2124

Y. Shibata and Y. Enomoto

gives F 1

h ' ./.i /   i  @   ' ./.i /    X  ix  1 Z 1 1 j k j k .x/ D d : e ix 4 2 N @ jj jxj .2/ jj4 RN jjDm

Choosing m in such a way that m C 2  j j  N C 1 yields that ˇ ˇ

 Z ˇ ˇ m 1 '1 ./.i / j k d ˇ  Cm ˇjxj F .x/ < 1: ˇ ˇ 4 N C1 jj jj1 jj Thus, if m > N  1 C j j, then jK1j k .x/j 

Cm jxjm

.jxj 6D 0/:

Since gk .x/ D 0 for jxj > R, when jxj  2R, it holds that ˇ ˇ

 Z ˇ ˇ 1 '1 ./.i / j k gOk ./ ˇ ˇF .x/ˇ  Cm ˇ jj2

jyjR

m



4 Cm jxj2m

jgk .y/j dy jx  yj2m

Z

jgk .y/jdy  jyjR

Cm;p;R kgk kLp .RN / ; jxj2m

which furnishes (60). The following theorem (Theorem 22) is used to prove that j@ x I0 .x/j  CR jxj.N 2Cj j/ kgkLp .RN / when jxj 6D 0:

(61)

Theorem 22 (Shibata-Shimizu [49]). Let X be a Banach space with a norm j  jX . Let M bea nonnegative  integer and 0 <   1. Set ı D M C   N . Assume that f 2 C 1 RN nf0gI X satisfies the following conditions: 

@ f ./ 2 L1 .RN W X / for j j  M; Let g.x/ D

R

RN

X

e ix f ./d , and then 

jg.x/jX  CN;ı

ˇ ˇ ˇ  ˇ ˇ@ f ./ˇ  C jjıjj for any  :

Note that @x I0 .x/ D F 1

h



max C jxj.N Cı/

jjM C2

.i / '0 ./Og./ jj4

i

for jxj 6D 0:

.x/. Observe that

39 Global Existence of Classical Solutions and Optimal Decay Rate for. . .

@ 



.i / '0 ./  gO ./ jj4

D

N X

X

C ;ı;; @ı

`D1 ıCCD

2125

.i / `  @ '0 ./ @ gO` ./: jj4

Since ˇ ˇ ˇ ˇ ˇ ˇ ˇ ı .i / ` ˇ ˇ ˇ ˇ  2Cj jjıj ˇ  ˇ ˇ@ ; ˇ@ '0 ./ˇ  jjjj sup ˇjjjj @ '0 ./ˇ ; ˇ  jj4 ˇ  Cı jj N 2R ˇ ˇ ˇ ˇ ˇ ˇ ˇ  ˇ ˇ@ gO` ./ˇ  jjjj sup ˇjjjj @ gO` ./ˇ  C;p;R jjjj kgkLp .RN / ; 2supp '0

it holds that ˇ  ˇ ˇ '0 ./.i /   gO ./ ˇ ˇ@ ˇ  C ;p;R jj2Cj jj j kgkL .RN / : p ˇ  ˇ jj4 Hence by Theorem 22, (61) holds. Combining (60) and (61) yields that j@ x I .x/j  C jxj.N 2Cj j/ kgkLp .RN / when jxj  2R. Analogously, the other terms are estimated, which implies (59). This completes the proof of Theorem 21.

3.2

On the Resolvent Problem in ˝

A purpose of this subsection and next subsection is to analyze the resolvent problem in an exterior domain: 8 ˆ in ˝; ˆ <  C  div v D f (62) v  ˛v  ˇrdiv v C  r D g in ˝; ˆ ˆ :vj D 0; 

mC1;mC2 .˝/// near D 0. By Theorem 20, . I CA/1 2 A.K ; L.WpmC1;m .˝/; Wp;0 for any 0 < < =2. The main result of this subsection is

Theorem 23. Let ˝ be an exterior domain in R3 . Let 1 < p < 1, m 2 N0 , and m .˝/ be let R be a large number such that   BR . Let ˝R D ˝ \ BR and let Xp;R the space defined in (18). Then, there exist a positive number 1 and families of the operators M and V with mC1 m .˝/; Wp;loc .˝///; M 2 A.UP 1 ; L.Xp;R

mC2 m V 2 A.UP 1 ; L.Xp;R .˝/; Wp;loc .˝/3 //

2126

Y. Shibata and Y. Enomoto

m such that for any .f; g/ 2 Xp;R .˝/ and 2 V ; 1

.M .f; g/; V .f; g// D . I C A/1 .f; g/ in ˝:

(63)

mC1;mC2 m .˝/; Wp;loc .˝// .i D 0; 1/ and families of Moreover, there exist Gi 2 L..Xp;R mC1;mC2 m P operators J . / with J . / 2 A.U 1 ; L.Xp;R .˝/; Wp;loc .˝/// such that

.M ; V / D G0 C 1=2 G1 C . log /J . /

(64)

m In this subsection, the operator .M ; V / is defined for .f; g/ 2 Xp;a;R .˝/, where m Xp;a;R .˝/ is the space defined in (18). On the other hand, in the next subsection, m .˝/ by some bootstrap .M ; V / can be extended to the operators defined on Xp;R argument, which also implies the expansion formula (64) in Theorem 23.

Lemma 7. Let ˝ be an exterior domain in R3 with C 3 boundary  . Let 1 2 1 < p < 1. Let .; u/ 2 Wp;loc .˝/  Wp;loc .˝/3 be a solution to the following equation:  div u D 0; ˛u C  r D 0 in ˝; uj D 0: (65)  1   2   2  Assume that u.x/ D O jxj , ru.x/ D O jxj , and .x/ D O jxj as jxj ! 1. Then u.x/ D 0 and .x/ D 0. Proof. It follows from the hypo-ellipticity of the Stokes equation that u 2 2 1 Wq;loc .˝/3 and  2 Wq;loc .˝/ for all q 2 .1; 1/, which furnishes that u 2 2 1 3 W2;loc .˝/ and  2 W2;loc .˝/. Let  be a cutoff function in C01 .R3 / such that .x/ D 1 for jxj < 1 and .x/ D 0 for jxj > 2. Let R1 .x/ D .x=R1 / for any large number R1 . By the second equation of (65), 0 D .˛u C  r; R1 u/ D ˛.ru; R1 ru/ C ˛ .ru; .rR1 /  u/   .; .rR1 /  u/ : Since j.ru; .rR1 /  u/j  CR11 ;

j .; .rR1 /  u/j  CR11 ;

letting R1 ! 1 yields that krukL2 .˝/ D 0, which, combined with the boundary condition, furnishes that u.x/ D 0. Hence,  by the second equation of (65), r D 0, which, combined with .x/ D O jxj2 , furnishes that .x/ D 0. This completes the proof of Lemma 7. t u Let b1 , b2 , and b3 be constants such that R C 1 < b1 < b2 < b3 . Let '.x/ be a cutoff function in C01 .R3 / such that '.x/ D 1 for jxj  b1 and '.x/ D 0 m for jxj  b2 . Given .f; g/ 2 Xp;a;b .˝/, let .f0 ; g0 / be a 4-vector of functions in 3 R m 3 Xp;b3 C1 .R / such that .f0 ; g0 / D .f:g/ in ˝, R3 f0 .x/ dx D 0 and

39 Global Existence of Classical Solutions and Optimal Decay Rate for. . .

2127

kf0 kWpmC1 .R3 / C kg0 kWpm .R3 /  C .kf kWpmC1 .˝/ C kgkWpm .˝/ /: Since such extension is constructed near the boundary of ˝ \ Bb3 , if .f; g/ vanishes outside of Db1 ;b2 D fx 2 R3 j b1  jxj  b2 g, then it may be assumed that ( .f0 ; g0 / D

.f; g/

in ˝

.0; 0/

in R3 n ˝

:

(66)

m Let .f; g/j˝b3 be the restriction of .f; g/ to ˝b3 . Given .f; g/ 2 Xp;a;R .˝/, let u 2 mC2 mC1 Wp .˝b3 / and  2 Wp .˝b3 / be solutions to the Cattabriga-Stokes problem (cf. [18]):

 div u D f j˝b3 ; ˛u  ˇrdiv u C  r D gj˝b3

in ˝b3 ;

uj@˝b3 D 0: (67)

Let J and W be operators defined by relations:  D J ..f; g/j˝b3 /

and

u D W..f; g/j˝b3 /;

and then kJ ..f; g/j˝b3 /kWpmC1 .˝

b3 /

C kW..f; g/j˝b3 /kWpmC2 .˝

b3 /

 C fkf kWpmC1 .˝

C kgkWpm .˝b3 / g:

b3 /

(68)

Let M0 and V0 be the operators given in Theorem 21 (2). If  satisfies (67), then for any constant c,  C c also satisfies (67), and therefore it may be assumed without loss of generality that Z ˝

˚ '.x/ M0 .f0 ; g0 /  J ..f; g/j˝b3 / dx D 0:

(69)

Let M and V be the operators given in Theorem 21, and let ˚ and  be operators defined by the relations: ˚ .f; g/ D .1  '/M .f0 ; g0 / C 'J ..f; g/j˝b3 /;

(70)

 .f; g/ D .1  '/V .f0 ; g0 / C 'W..f; g/j˝b3 /

(71)

m for any .f; g/ 2 Xp;R .˝/. Then

8 ˆ ˚ .f; g/ C  div  .f; g/ D f C S .f; g/ ˆ ˆ ˆ <  .f; g/  ˛ .f; g/  ˇrdiv  .f; g/ C  r˚ .f; g/ ˆ D g C T .f; g/ ˆ ˆ ˆ :  .f; g/j D 0;

in ˝; in ˝;

2128

Y. Shibata and Y. Enomoto

where S .f; g/ D 'J ..f; g/j˝b3 /  .r'/  V .f0 ; g0 / C .r'/  W..f; g/j˝b3 /; T .f; g/ D 2˛.r'/ W rV .f0 ; g0 / C ˛.'/V .f0 ; g0 / C ˇr..r'/  V .f0 ; g0 // C ˇ.r'/div V .f0 ; g0 /  .r'/M .f0 ; g0 / C 'W..f; g/j˝b3 /  2˛.r'/ W rW..f; g/j˝b3 /  ˛.'/W..f; g/j˝b3 /  ˇr..r'/W..f; g/j˝b3 //  ˇ.r'/div W..f; g/j˝b3 / C .r'/J ..f; g/j˝b3 /: For any 3-vector of functions g D .g1 ; g2 ; g3 / and scalar function f , let rf W rg D .rf  rg1 ; rf  rg2 ; rf  rg3 /. By Theorem 23 and (68) mC1 .˝/ S .f; g/ 2 Wp;b 3

lim

j j!0 j arg j=4

and

mC1 T .f; g/ 2 Wp;b .˝/3 I 3

k.S ; T /  .S0 ; T0 /kLW mC1 .˝/W m .˝/3  D 0: p

p

R m .˝/, so that Since S .f; g/dx 6D 0, .S .f; g/; T .f; g// does not belong to Xp;a;b 3 S .f; g/ and T .f; g/ should be modified as follows. Let .x/ be a function in R C01 .R3 / such that supp  Db1 ;b2 and R3 .x/dx D 1. Let SQ .f; g/ D S .f; g/ C

Z ˝

TQ .f; g/ D T .f; g/ C  ˚Q .f; g/ D ˚ .f; g/ C

˚ '.x/ M .f0 ; g0 /  J ..f; g/j˝b3 / dx

Z

˝

Z ˝

.x/;

˚ '.x/ M .f0 ; g0 /  J ..f; g/j˝b3 / dx r .x/;

˚ '.x/ M .f0 ; g0 /  J ..f; g/j˝b3 / dx

.x/;

(72)

and then ˚Q .f; g/,  .f; g/, SQ .f; g/, and TQ .f; g/ satisfy the following equation: 8 ˆ ˚Q .f; g/ C  div  .f; g/ D f C SQ .f; g/ ˆ ˆ ˆ <  .f; g/  ˛ .f; g/  ˇrdiv  .f; g/ C  r ˚Q .f; g/ ˆ D g C TQ .f; g/ ˆ ˆ ˆ :  .f; g/j D 0:

in ˝; in ˝; (73)

Moreover, mC2;mC1 m m .SQ .f; g/; TQ .f; g// 2 Xp;a;b .˝/; .SQ 0 .f; g/; TQ0 .f; g// 2 Xp;a;b .˝/\Wp;b .˝/; 3 3 3

lim

!0; 2K"

k.SQ ; TQ /  .SQ 0 ; TQ0 /kL.WpmC1;m .˝// D 0:

39 Global Existence of Classical Solutions and Optimal Decay Rate for. . .

2129

m In particular, .SQ 0 ; TQ0 / is a compact operator on Xp;a;b .˝/ by the Rellich compact3 ness theorem.

Lemma 8. Let 1 < p < 1. Then, I C .SQ 0 ; TQ0 / has a bounded inverse .I C m .SQ 0 ; TQ0 //1 2 L.Xp;a;b .˝//. 3 m .˝/ into itself, in view of Proof. Since .SQ 0 ; TQ0 / is a compact operator on Xp;a;b 3 Fredholm’s alternative theorem, to prove the lemma, it is sufficient to prove the m injectivity of I C .SQ 0 ; TQ0 /. Thus, pick up an element .f; g/ 2 Xp;a;b .˝/ such that 3 Q Q .I C .S0 ; T0 //.f; g/ D .0; 0/ in ˝. The task is to prove that .f; g/ D .0; 0/. By (73), ˚Q 0 .f; g/ and 0 .f; g/ satisfy the following equation: 8 ˆ in ˝; ˆ 0 small enough yield that I1 .t / C I2 .t /  C .k.0 ; u0 /k2H 1 C I.t /3=2 C I.t /2 /:

(196)

Next, I3 .t / is estimated. Applying (154)–(187) and using (13) yield that

k.r 2 1 ; r 2 u1 /.; t /k2L2 .R3 / C

Z

 C .k.0 ; u0 /k2H 2 Ck.; t /k2H 1 C 2

t

k.r 3 u1 ; r@t u1 ; r 2 1 /.; s/k2L2 .R3 / ds

0

Z

t 0

fk.r 2 .'1 fn CF1 /; r.'1 gn CG1 //k2L2 .R3 /

2

C k.ru  r 1 ; r u  r1 /k2L2 .R3 / C ks k2H 1 C krukH 1 g ds/

(197)

By (13), kr 2 .'1 fn /kL2 .R3 /  C fk.ru  r 2 ; r 2 u  r; r 3 u  /k C k.ru  r; r 2 u  ; ru  /kL2 .˝R / g  C fkr 2 kkruk1 C krk3 k.r 2 u; ru/k6 C kk1 k.r 3 u; r 2 u; ru/k  C krkH 1 krukH 2 I kr 2 F1 kL2 .R3 /  C k.u  ; u/kH 2 .˝R /  C .krukH 1 krkH 1 C krukH 1 /I kr.'1 gn /kL2 .R3 /  C fk.r 3 u  ; r 2 u  r; r 2   ; r  r/k C k.r 2 u  ; r  /kL2 .˝R / g  C krukH 2 krkH 1 I krG1 kL2 .R3 /  C .kukH 2 .˝R / C kkH 1 .˝R / /  C .krukH 1 C krk/I kru  r 2 1 kL2 .R3 /  C kruk1 .kr 2 kH 2 C kkH 1 .˝R / /  C krukH 2 krkH 1 I kr 2 u  r1 kL2 .R3 /  C kr 2 uk6 kkH 2 .˝R /  C krukH 2 krkH 1 :

(198)

Combining (197) and (198) yields that 2

k.r 1 ; r

2

u1 /.; t /k2L2 .R3 /

Z

t

C 0

k.r 3 u1 ; r@s u1 ; r 2 1 /.; s/k2L2 .R3 / ds

 C .k.0 ; u0 /k2H 2 C I1 .t / C I2 .t / C I.t /2 /:

(199) Next, the second-order derivatives of  and u are estimated near the boundary point x0 2  . Applying (179) and using (13) and (138) yield that solutions 0 and u0 of the equation (185) satisfy the estimate:

39 Global Existence of Classical Solutions and Optimal Decay Rate for. . .

2 0

2 0

k.r  ; r u

/.; t /k2L .R3 / 2 C

Z

 C fk.0 ; u0 /k2H 2 C kg0 .; t /k2L 0 2 kH 2 .R3 / C

C kf

t

C 0

k.r 3 u0 ; r@s u0 ; r 2 0 /.; s/k2L

3 2 .RC

C kr.; t /k2 C /

0

C k@s f kL2 .R3 / C krw  C

r0 k2L .R3 / 2 C

Z

t 0

r0 k2H 1 .R3 / C

3 2 .RC /

.krg0 k2L

ds

3 2 .RC /

C k@s g0 k21

C krw  @s 0 k2H 1 .R3

C/

2

(200)

provided that k.0 ; u0 /kH 2  1. Here, the term: k.@t 0 ; @t u0 /.; t /k2L

has been

C k@s w 

C

kruk2H 1

2165

2

C kus k C k@s k / ds

3 2 .RC /

omitted in the left-hand side and the estimate: k.@t 0 ; @t u0 /.; 0/kL2 .R3 /  C .k.0 ; u0 /kH 2 C k.0 ; u0 /k2H 2 / C

has been used. Analogously to (198), by (13), kf 0 kH 2 .R3 /  C'; fkrkH 1 krukH 2 C krukH 1 gI C

0

k@s f kL2 .R3 /  C'; .ks  ru;   rus ;   us ; s  u/kL2 .˝R / C kus k/ C

 C'; .ks k6 kruk3 C kk1 krus k C kk6 kus k3 C kuk6 ks k3 / C kus k/  C'; .krukH 1 ks kH 1 C kus kH 1 krkH 1 C kus k/I krg0 kL2 .R3 /  C'; .krkH 1 krukH 2 C krk2H 1 C krukH 1 C krk/ C

C C  k.r 3 u0 ; r 2 0 /kI kg0 kL2 .R3 /  C'; .kr 2 ukkk1 C krkkk1 C kukH 1 .˝R / C kkH 1 .˝R / / C C  kr 2 u0 kI C

 C' fkrkH 1 .kr 2 uk C krk/ C krukH 1 C krkg C C  kr 2 u0 kI krw  r0 kH 1 .R3 /  C krukH 2 krkH 1 I krw  @s 0 kH 1 .R3 /  C'; k@s kH 1 krukH 2 I C

C

0

k@s w  r kL2 .R3 /  C'; krkH 1 krus k: C

(201)

Moreover, by (13), j.r 2 ut ; /j D j.rut ; r. //j  C krut kkrkH 1 k kH 1 I j.t ru; /j  kr 2 ukkt k6 k k3  C kr 2 ukkrt kk kH 1 I j.rt ; /j  krt kkk6 k k3  C krt kkrkk kH 1 I j.t r; /j  kt k6 krkk k3  C krt kkrkk kH 1 I j. r 2 ut ; /j  j. rut ; r /jCj.ut ; r..r / //j  .k k1 krut kCC kut k/k kH 1 :

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Y. Shibata and Y. Enomoto

2 H01 .˝/. And also, by (13),

for any

j.@t h1 ; /j  fC  kr@t u0 kL2 .R3 / C C'; .kut k C kt k/gk kH 1 .R3 / ; C

C

j.@t h2 ; /j  C'; kut kk kH 1 .R3 / : C

2 H01 .R3C /. Applying these estimations to (172) yields that

for any

k@t g0 k1  C'; .kut kH 1 krkH 1 C kt kH 1 krukH 1 C kt kH 1 krkH 1 C kut k C kt k/ C C  kru0t kL2 .R3 / : C (202) Combining (200), (201), and (202) and choosing  > 0 small enough yield that k.r 2 0 ; r 2 u0 /.; t /k2L 

Z 3 2 .RC /

C fk.0 ; u0 /k2H 2

t

C

k.r 3 u0 ; r@s u0 ; r 2 0 /.; s/k2L

3 2 .RC /

0

ds (203)

2

C I1 .t / C I2 .t / C I.t / g:

provided that k.0 ; u0 /kH 2  1. Since the boundary  is compact, by (199) and (203), k.r 2 ; r 2 u/.; t /k2L 

Z 3 2 .RC /

C fk.0 ; u0 /k2H 2

t

C 0

k.r 3 u; r@s u; r 2 /.; s/k2L

3 2 .RC /

ds (204)

2

C I1 .t / C I2 .t / C I.t / g

provided that k.0 ; u0 /kH 2  1. By the first equation of (1) and (13), krt k  kr 2 . C /uk  C . C krkH 1 /krukH 1 ;

(205)

which, combined with (204), furnishes that I3 .t /  C fk.0 ; u0 /k2H 2 C I1 .t / C I2 .t / C I.t /2 g

(206)

provided that k.0 ; u0 /kH 2  1. Combining (196) and (206) yields (117), which completes the proof of (117), and therefore, the proof of Theorem 1 is completed.

5.6

Decay Estimate: A Proof of Theorem 2

A purpose of this subsection is to prove Theorem 2. For this purpose, it suffices to prove that D.t /  K2 .k.0 ; u0 /k1 C k.0 ; u0 /kH 2 C D.t /2 /

(207)

39 Global Existence of Classical Solutions and Optimal Decay Rate for. . .

2167

with some constant K2 . In what follows, write W21;0 .˝/ by H 1;0 , k  kL2 .˝/ by k  k, k  kLp .˝/ by k  kp (p 6D 2), and k  kH s .˝/ by k  kH s for s D 1; 2 for the sake of simplicity. First, D0 .t / and D1 .t / are estimated with the help of Theorem 3. Let fT .t /gt0 be the analytic semigroup associated with the linearized problem: 8 ˆ ˆ 0. Let D.t / D D0 .t / C D1 .t / C D2 .t /, and then k.fn  u  r/.; s/k1 C k.fn  u  r/.; s/kH 1  C .1 C s/2 D.s/2 ; kgn .s/k1 C kg.s/k  C .1 C s/2 D.s/2 :

(210)

In fact, by Hölder’s inequality and (13), k.fn  u  r/.; s/k1  k.; s/kkru.; s/k C ku.; s/kkr.; s/k  .1 C s/2 D0 .s/D1 .s/I k.fn  u  r/.; s/kH 1  C k.u  r 2 ; ru  r;   r 2 u;   ru; u  r/.; s/k  C .ku.; s/k1 kr.; s/kH 1 C kru.; s/k6 kr.; s/k3 C k.; s/k1 kru.; s/kH 1 /  C .1 C s/5=2 .D1 .t / C D2 .t //2 I kgn .; s/k1  C .kr 2 u.; s/kk.; s/k C kr.; s/kk.; s/k/  C .1 C s/2 D0 .s/.D1 .s/ C D2 .s//I

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Y. Shibata and Y. Enomoto

kgn .; s/k  C .kr 2 u.; s/kk.; s/k1 C kr.; s/kk.; s/k1 /  C kr.; s/kH 1 .kr 2 u.; s/k C kr.; s/k/  C .1 C s/5=2 .D1 .s/ C D2 .s//2 : Thus, applying (209) and the usual analytic semigroup estimate yields that  Z t1  Z t 3 kU.t /k  C .t  s/ 4 .1 C s/2 ds C .1 C s/2 ds D.t /2 ; 0

Z

t1 t1

krU.t /k  C

5

.t  s/ 4 .1 C s/2 ds C

0

Observing Z Z

t1

Z

t

 1 .t  s/ 2 .1 C s/2 ds D.t /2 :

t1

k

k

.t  s/ 4 .1 C s/2 ds  C t  4

(211)

.k D 3; 5/;

0 t

`

.t  s/ 2 .1 C s/2 ds  .2=.2  `//t 2

.` D 0; 1/;

t1

and using (211) give 3

kU.t /k  C t  4 D.t /2 ; (212)

5

krU.t /k  C t  4 D.t /2

for t  1. Since k.; u/.; t /kH 2  C k.0 ; u0 /kH 2 as follows from (8), by (209) and (212), D0 .t / C D1 .t /  C fk.0 ; u0 /k1 C k.0 ; u0 /kH 2 C D.t /2 g:

(213)

Next, k.t ; vt /.; t /k is estimated. Lemma 11. Let f .t/ be a nonnegative C 1 .Œ0; 1// function and let gi .t / .i D 1; 2; 3; 4/ be nonnegative functions such that gi 2 C 0 ..0; 1// .i D 1; 2; 3/ and g4 2 L2 ..0; 1//. Assume that d f .t/ C cf .t /  g1 .t / C g2 .t / C g3 .t /g4 .t / dt

(214)

for any t > T0 with some constant c > 0. Then, for any ˛ > 0, there exists a T1  T0 such that .1 C t /˛ f .t/  .1 C T1 /˛ f .T1 / C .2=c/. sup .1 C s/˛ g1 .s// T1 0 provided the latter exists in the time interval Œ0; T . Note that this theorem is not a standard weak-strong uniqueness resulting in the spirit of [22], in the sense that the weak solution is assumed to satisfy (202). However, since Theorem 14 concerns with the conditional regularity, such an assumption is acceptable. It obviously improves condition (197) in Theorem 13. This is due to rather simple structural assumptions, especially that the viscous coefficients ;  are constants.

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Conclusion

In the first half part of this contribution, possible blow-up mechanism is discussed for the compressible Navier-Stokes equations. Based on regularizing effect of the effective viscous flux, it is shown that the upper bound of the density controls regularity of the solution for the barotropic case. To the complete Navier-StokesFourier system for ideal viscous fluids, with additional bounds on the temperature and specific volume, the same conclusion holds. Physically, these blow-up criteria are reasonable since compressible Navier-Stokes equations are used to describe fluids in normal situation, that is, far away from vacuum, high density, and temperature. Note that constant value of the dissipative coefficients ; , and  plays an important role in the proof, especially in justifying the application of elliptic estimates. Also to the full system of ideal fluids, the special relation between the pressure and internal energy is crucial as pointed out in Remark 10. There are still some unsolved problems, especially the following one about the appearance of vacuum in the barotropic case. Question: Does vacuum appear simultaneously with “blowup” of the density? An affirmative answer to this question will be a consequence of the following blowup criterion in terms of lower bound of the density. Blow-up criterion: If there exists a positive constant % such that

%.t; x/  % in Œ0; T  /  ;

then the strong solution .%; u/ to (18), (19), and (20) will not blow up at the time T  . Another question concerns the complete system. As it is pointed before, up to now all known results on blow-up criteria mostly considered the ideal fluids with constant dissipative coefficients. On the other hand, local existence of strong or classical solution to the complete system has been known under very general structural conditions. Is it possible to give some blow-up criterion in terms of pointwise bound of the solution under more general structural assumption? The answer seems far from satisfactory; see [11] for partial relevant results. In the second part of this chapter, the problem of conditional regularity is investigated for weak solutions to the complete Navier-Stokes-Fourier system. Under different structural conditions on the state equation and dissipative coefficients, regularity conditions are found for two classes of weak solutions which have been constructed in [19] and [21], respectively. The proof is based on weak-strong uniqueness for weak solutions and blow-up criteria for strong solutions. Several comments are given below concerning the hypotheses of Theorems 13 and 14 (and Theorems 9 and 10, respectively). According to the results obtained by Fan et al. [18], the best regularity condition is expected to be

41 Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak. . .

Z

2321

T

kru.t; /kL1 ./ dt < 1: 0

However, this bound seems not enough due to the effective dependence of the dissipative coefficients on the temperature. It is even plausible to expect that (197) could be relaxed to Z

T

0

q

kru.t; /kL1 ./ dt < 1

for some q < 1. In Theorem 14 (and Theorem 10), a relatively simple model is considered in the sense that the viscous coefficients are constants and the constitutive relations are simple. However, as pointed out in Remark 13, one can replace kdivukL1 .0;T IL1 .// by the upper bound of the density in (121). While for the regularity condition (202) to weak solutions in Theorem 14, this seems less obvious since bound on kdivukL1 .0;T IL1 .// is necessary to control the density from below on the level of weak solutions. Certainly, the conclusion of Theorem 14 will still hold true if one replaces the bound on divu by the upper and lower bound of the density. However, this generalization is not considered here in order to comply with Theorem 10. Finally, the methods presented here could be extended to more general classes of domain  as well as other types of boundary conditions such as Navier type for the velocity. It is also possible to extend the results to other models of compressible Navier-Stokes-Fourier system with different constitutive relations and structural conditions, as soon as the existence of weak solutions is known.

10

Cross-References

 Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak

Solutions for the Compressible Navier-Stokes Equations  Concepts of Solutions in the Thermodynamics of Compressible Fluids  Finite Time Blow-Up of Regular Solutions for Compressible Flows  Global Existence of Regular Solutions with Large Oscillations and Vacuum for

Compressible Flows  Local and Global Existence of Strong Solutions for the Compressible Navier-S-

tokes Equations Near Equilibria via the Maximal Regularity  Weak Solutions for the Compressible Navier-Stokes Equations with Density

Dependent Viscosities  Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stabil-

ity, and Longtime Behavior

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Acknowledgements The research of Yongzhong Sun is supported by NSF of China under Grant No. 11571167 and PAPD of Jiangsu Higher Education Institutions, and Zhifei Zhang is partially supported by the NSF of China under Grant No. 11371039 and 11425103.

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Well-Posedness and Asymptotic Behavior for Compressible Flows in One Dimension

42

Yuming Qin

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Nonexhaustive Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Eulerian and Lagrangian Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Polytropic, Ideal, and Viscous Conducting Real Gases . . . . . . . . . . . . . . . . . . . . . . 1.4 Functional Spaces and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A 1D System of Viscous Heat-Conducting Real Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Global Existence and Exponential Stability of Generalized Solutions in H 1 . . . . 2.2 Global Existence and Exponential Stability of Strong Solutions in H 2 . . . . . . . . . 2.3 Global Existence and Exponential Stability of Strong Solutions in H 4 . . . . . . . . . 2.4 Universal Attractors in H 1 , H 2 , and H 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A 1D System of Viscous Polytropic Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 An Initial Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Examples of Results in Other Functional Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Initial Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2326 2326 2327 2329 2331 2333 2333 2367 2376 2383 2393 2394 2398 2406 2406 2411 2415 2416 2416

Abstract

This short survey introduces some results on the global well-posedness and asymptotic behavior of strong (generalized) solutions (including the existence of universal (global) attractors) to the 1D nonstationary equations of viscous polytropic gases, heat-conducting gases, and barotropic gases. In Sect. 3, the existence theory framework of H i .i D 1; 2; 4/global (generalized) solutions

Y. Qin () Department of Applied Mathematics, College of Science, Donghua University, Shanghai, China e-mail: [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_34

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and global attractors of viscous real gases for the initial boundary value problem are introduced. In Sect. 4, such an existence theory framework for the Cauchy problem with some modifications is also given.

1

Introduction

The system of the 1D nonisentropic viscous polytropic and/or real gas is a system of conservations of mass, momentum, and energy. The main source of classical information on the analysis of this system is the pioneering monograph by Antontsev et al. [2]. Since its publication, great progress in the subject has been made. It is impossible to provide an exhaustive overview of the results in this field. In this survey, we deal with the global well-posedness (existence and uniqueness), large-time behavior (including attractors) for the solutions of the boundary value and/or Cauchy problems emanating from strictly positive initial density and temperature, in the Sobolev H i -setting. These results presented in Sects. 2 and 3 are based on our papers [55–58, 60] (Chapter 2 and Section 3.2 of Chapter 3) and [71]. They are presented in a self-consistent way including the essence of proofs. In order to give a flavor of the results available in other functional settings, we have chosen to present several representative theorems without proofs in Sect. 4. We start this treatment with (a necessarily nonexhaustive) list of the bibliography about this subject in the next section.

1.1

Nonexhaustive Bibliography

For the Cauchy problem of the 1D viscous polytropic gas, Itaya [29], Kanel [34], Kazhikhov [39], and Antontsev, Kazhikhov, and Monakhov [2] obtained the global existence and large-time behavior (only for velocity and absolute temperature) of H 1 generalized solutions; Okada and Kawashima [53] proved the global existence and large-time behavior of the classical (or H 1  generalized) solution with small initial data; Jiang [32] considered the large-time behavior of the H 1 generalized strong solution with weighted small initial data; Kawashima and Nishida [35] established the existence and uniqueness of smooth solutions; Qin et al. [71] and Qin [60] (Section 3.2 of Chapter 3) dealt with the existence and asymptotic behavior of global H i global strong solutions .i D 2; 4/. Hoff [21, 22, 24] studied the onedimensional and multidimensional Cauchy problem with discontinuous initial data. Hoff [21–24] discussed the discontinuous solutions for the Cauchy problem (see also references therein). Also see Matsumura and Nishida [44, 45] for the Cauchy problem of the 3D compressible viscous heat-conducting fluids. For the initial boundary value problems of the 1D viscous polytropic gas, Kazhikhov [37, 38] and Kazhikhov and Shelukhin [40] established the global existence and uniqueness of smooth (H 1  generalized) solutions; Qin [54, 60] (Section 3.1 of Chapter 3) proved the global existence and uniqueness of smooth solutions and asymptotic behavior in H 1 ; Nagasawa [48–51] also obtained results

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2327

for outer pressure problems and for stress-free boundary conditions. Duan, Fang, Huang, Liu, Qin, Vong, Xin, Yang, Yao, Zhang, and Zhu [10–15,33,42,43,62,64,65, 76,78–80] proved the well-posedness of solutions to the 1D isentropic compressible Navier-Stokes equations with vacuum, density-dependent viscosity, and degenerate viscosity coefficient. Also see references therein. For the initial boundary value problems of the 1D viscous heat-conducting real gas, Kawohl [36] established the global existence of smooth solutions; Jiang [31] also established the global existence of smooth solutions for some boundary conditions; Jiang [30] additionally established the exponential decay of global smooth large solutions in H 1 ; Qin [55–58, 60] (Chapter 2) established the global existence and exponential stability of (smooth) H i  global strong (generalized) solutions .i D 1; 2; 4/. Ducomet and Zlotnik [8, 9] also established global existence and asymptotic behavior of solutions for the 1D initial boundary value problem of viscous compressible Navier-Stokes equations of heat-conducting real gas with nonmonotone state function (pressure p D p.u;  //. Duan, Fang, Huang, Liu, Qin, Vong, Xin, Yang, Yao, Zhang, Zhu [10–15, 33, 42, 43, 62, 64, 65, 76, 78–80] proved the well-posedness of solutions to the 1D isentropic compressible Navier-Stokes equations with vacuum, density-dependent viscosity, and degenerate viscosity coefficient (see also references therein). Chen, Hoff, and Trivisa [4] discussed the discontinuous solutions for the initial boundary value problem (see also references therein). Stra˘skraba and Zlotnik [73, 74] considered the compressible barotropic fluid equations; also see [84–89] and references therein. See Matsumura and Nishida [46, 47] and Lions [41] for the 3D compressible viscous heat-conducting fluids of Navier-Stokes equations for the initial boundary value problem and Xin [77] for the blow-up of solutions to the Cauchy problem of 3D compressible viscous isentropic Navier-Stokes equations. Concerning the existence of attractors, Hoff and Ziane [25, 26] obtained the existence of a compact global attractor for the 1D isentropic polytropic viscous gas; Zheng and Qin [82] and Qin [60] (Sections 2.5–2.6 of Chapter 2) proved the existence of maximal (universal) attractors in H i .i D 1; 2; 4/ for the 1D nonisentropic polytropic viscous gas. Qin and Muñoz Rivera [59,69] established the existence of universal attractors in H i .i D 1; 2; 4/ for 1D viscous compressible Navier-Stokes equations of real gases (see also [66] for a 1D thermoviscoelastic model). Also refer to Feireisl [17] for the existence of global attractors for the isentropic compressible viscous flow in a bounded domain in R3 , and to Sell [72] for the existence of uniform attractors for the nonautonomous, incompressible NavierStokes equations in a bounded domain in R3 , and to Chepyzhov and Vishik [5] for some results on attractors of the 2D and 3D incompressible Navier-Stokes equations.

1.2

Eulerian and Lagrangian Descriptions

The motion for a compressible viscous, heat-conductive, isotropic Newtonian fluid is described by the system of equations in the Eulerian coordinates .y; t /,

2328

Y. Qin

t C .v/y D 0;

(1)

vt C vvy D y ;

(2)

et C vey D Qy C  vy ;

(3)

with boundary and initial conditions v.0; t / D v.1; t / D 0; Q.0; t / D Q.1; t / D 0; t  0;

(4)

.y; 0/ D 0 .y/; v.y; 0/ D v0 .y/; e.y; 0/ D e0 .y/; y 2 0 ;

(5)

in the domain .y; t / 2 0  Œ0; C1/ with 0 D .0; 1/, where the unknown functions .y; t /, v.y; t/, and e.y; t/ represent the density, velocity, and the internal energy of fluid, respectively. Function  .y; t / denotes the stress and Q.y; t / stands for the heat flux. For convenience, transform the system (1), (2), and (3) into Lagrangian mass coordinates. The Eulerian coordinates .y; t / are connected to the Lagrangian mass coordinates .x; t / by the relation Z

t

y.x; t/ D y0 .x/ C

v.x; Q /d ;

(6)

0

where v.x; Q t / D v.y.x; t /; t / and Z

1

y

y0 .x/ D  .x/; .y/ D

0 .s/ds; y 2 .0; 1/:

(7)

0

It then follows from Eqs. (1) and (4) that Z

Z

y.x;t/

y0 .x/

.s; t /ds D 0

0 .s/ds D x;

(8)

0

and 0 is transformed to  D .0; L/ with Z

Z

1

0 .s/ds D

LD 0

1

.s; t /ds; 0

where L is invariant along the trajectory f.s; t / W 0  s  1; t  0g and assume that without loss of generality L D 1. Moreover, we derive that @x y.x; t/ D Œ.y.x; t /; t /1 :

(9)

In general, for any function '.y; t/, denote '.x; Q t/ WD '.y.x; t/; t/; then it follows from (4), (5), (6), and (7) by the chain rule that

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

@t '.x; Q t/ D @t '.y; t/ C v@y '.y; t/; @x '.x; Q t/ D @y '.y; t/@x y.x; t/ D

2329

(10)

@y '.y; t/ : .y; t /

(11)

The second law of thermodynamics is expressed by the Clausius-Duhem inequality @t ..y; t // C v@y .y; t / C

1 @y 



Q.y; t / .y; t /

 D 0:

(12)

Finally, without danger of confusion, still denote .; Q v; Q eQ / by .; v; e/ and use u WD 1 to represent the specific volume. Thus by virtue of (9), (10), and (11), Eqs. (1), (2), and (3) in the Eulerian coordinates .y; t / can be rewritten in the Lagrangian coordinates in the new variables .x; t /; x 2  D .0; 1/; t  0 with the reference density 0 D 1 as



ut  vx D 0;

(13)

vt  x D 0;

(14)

 . v/x C Qx D 0;

(15)

2

eC

v 2

t

subject to the boundary conditions of the form v.0; t / D v.1; t / D 0;

Q.0; t / D Q.1; t / D 0; t  0;

(16)

and initial conditions u.x; 0/ D u0 .x/; v.x; 0/ D v0 .x/; .x; 0/ D 0 .x/ on

Œ0; 1;

(17)

and (12) now reduces to  t C

Q 

  0;

(18)

x

where subscripts indicate partial differentiations, u D 1 ; v;  ; e; Q; , and  denote the specific volume, velocity, stress, internal energy, heat flux, specific entropy, and temperature, respectively. Note that u; , and e may only take positive values.

1.3

Polytropic, Ideal, and Viscous Conducting Real Gases

A gas is called a one-dimensional homogeneous real gas, if the internal energy e, stress  , specific entropy , and heat flux Q satisfy the constitutive relations

2330

Y. Qin

e D e.u;  /;  D  .u;  ; vx /;  D .u;  /; Q D Q.u;  ; x /

(19)

which, in order to be consistent with (18), must satisfy  .u;  ; 0/ D ‰u .u;  /; .u;  / D ‰ .u;  /;

(20)

. .u;  ; w/   .u;  ; 0//w  0; Q.u;  ; g/g  0

(21)

where ‰ D e   is the Helmholtz free energy function. The gas is called an ideal gas if the pressure p, the specific volume u, temperature  , and internal energy e satisfy pu D const ant;  D .e/; 

(22)

with  0 .e/ > 0. We call a fluid polytropic ideal if the specific volume u, internal energy e, the temperature , and the pressure p satisfy the equation of state: eD

pu D CV  1

where > 1 is the adiabatic exponent and CV > 0 is a constant. When vx x  e D cv  ; p D ;  D p C  ; Q D k ; u u u

(23)

(24)

with suitable positive constants cv ; ; , and k, then it is a polytropic ideal gas. As is known, the constitutive equations of a real gas are well approximated within moderate ranges of u and  by the model of an ideal gas (24). However, under very high temperatures and densities, (24) becomes inadequate. Thus from the description above a more realistic model than (24) would be a linearly viscous real gas (or Newtonian fluid) if its constitutive relations satisfy  .u;  ; vx / D p.u;  / C

.u;  / vx ; u

(25)

satisfying Fourier’s law of heat flux Q.u;  ; x / D 

k.u;  / x ; u

(26)

whose internal energy e and pressure p are coupled by the standard thermodynamical relation eu .u;  / D p.u;  / C  p .u;  /; to be consistent with (18).

(27)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2331

Assume that e; p; , and k are twice continuously differential on 0 < u < C1 and 0   < C1, and there are the exponents q and r satisfying one of the following relations 0  r  1=3; 1=3 < qI 1=3 < r < 4=7; .2r C 1/=5 < qI 4=7  r  1; .5r C 1/=9 < qI

(28)

1 < r  13=3; .9r C 1/=15 < qI 13=3 < r; .11r C 3/=19 < q;

(29)

and concerning the growth of the temperature, also assume that there are positive constants ; p1 ; p2 ; k0 , and for any u > 0 that there are positive constants N .u/; p3 .u/; p4 .u/, and k1 .u/ such that for any u  u and   0 the following conditions hold. 0  e.u; 0/; .1 C  r /  e .u;  /  N .u/.1 C  r /;

(30)

0 < p1  up.u;  /  p2 .1 C  rC1 /;

(31)

p3 .u/Œl C .1  l/ C 

rC1

  pu .u;  /

 p4 .u/Œl C .1  l/ C  rC1 ; l D 0 or 1; r

jp .u;  /j  p4 .u/.1 C  /;

(32) (33)

k0 .1 C  q /  k.u;  /  k1 .u/.1 C  q /; jku .u;  /j C jkuu .u;  /j  k1 .u/.1 C  q /:

(34)

Assume that the viscosity coefficient .u;  / is a positive constant; that is, 0 < 0 D .u;  /:

(35)

Existence, uniqueness, stability, and large-time behavior of solutions to the equations of real gases (13), (14), (15), (16), and (17) under assumptions (25), (26), (27), (28), (29), (30), (31), (32), (33), (34), and (35) are investigated in Sect. 3. Note that the polytropic gas (24) with the exponents r D q D 0 is not included in (28), the existence, uniqueness, stability, and large-time behavior of solutions to a model (13), (14), (15), (16), and (17) with more general constitutive relations (292), (293), (294), (295), (296), and (297) covering (24) are studied in Sect. 3.1. Moreover, the Cauchy problem of the polytropic gas (13), (14), (15), (16), and (17) with (24) is also investigated in Sect. 3.2.

1.4

Functional Spaces and Notations

In this brief survey, we use for the functional spaces the same notation as used in the monograph of Adams [1]): the symbols Lp ; 1  p  C1; and H 1 D W 1;2 ;

2332

Y. Qin

H01 D W01;2 ; H 2 D W 2;2 ; H 4 D W 2;4 denote the usual Lebesgue and Sobolev spaces on .0; 1/; k:kB denotes the norm in the space B; in particular if the subscript B is dropped, we mean the L2  norm, k:kW D k:kL2 : Let QT D   .0; T / with  D .0; 1/, introduce the spaces W .QT / and W22;1 .QT / with the norms kwkW .QT / D kwkL2;1 .QT / C kDx wkL2;1 .QT / C kDt wkL2 .QT / ; kwkW 2;1 .QT / D kwkV2 .QT / C kDx wkV2 .QT / C kDt wkL2 .QT / 2

where kwkV2 .QT / D kwkL2;1 .QT / C kDx wkL2 .QT / d and Dt D @t or dt or a subscript t, and, likewise, Dx D @x or a subscript x, denotes the partial derivatives with respect to t and x in the distribution sense, respectively, Lq;s .Q/ WD Ls .RC ; Lq .// is the anisotropic Lebesgue space with the norm kwkLq;s .Q/ WD kkwkLq ./ kLs .RC / ; Q D   RC and V2 .QT / is the standard space of functions w being finite (parabolic) energy with the norm kwkV2 .QT / D kwkL2;1 .QT / C kwx kL2 .QT / . Note that W22;1 .QT /  W .QT /  C .QT /. The following standard notation for Hölder norms is also used in Sect. 4.2:

< w >a;b A D

jw.x; t /  w.y; s/j : jx  yja C jt  sjb .x;t/;.y;s/2A sup

Furthermore, letters C (sometimes C 0 ; C 00 ) denote universal constants depending only on the initial data, but independent of any length of time t . Constants Ci .i D 1; 2; 3; 4/ denote the universal constant depending only on the H i norm of initial data .u0 .x/; v0 .x/; 0 .x//, min u0 .x/, and min 0 .x/. Without danger x2Œ0;1

x2Œ0;1

of confusion, the same symbol denotes the state functions as well as their values along a thermodynamic process, for example, p.u;  /, and p.u.x; t /; .x; t // and p.x; t/. Let g.x/ W Œa; b ! R be a function. If X D fxk jk D 1; 2;    ; ng is a net over Œa; b with a D x0 < x1 <    < xn D b, then V .X / D †nkD1 jg.xk /  g.xk1 /j is called the variation of g with respect to X . Define T VŒa;b .g/ D supfV .X / W X i s a net over Œa; bgI then T VŒa;b .g/ is called the total variation of g over Œa; b. Other notations, not described above, are explained where they appear.

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2333

2

A 1D System of Viscous Heat-Conducting Real Gas

2.1

Global Existence and Exponential Stability of Generalized Solutions in H1

Before stating the first theorem, define three function classes as follows. n 1 D .u; v;  / 2 H 1 Œ0; 1  H 1 Œ0; 1  H 1 Œ0; 1 W u.x/ > 0; .x/ > 0; x 2 Œ0; 1; HC o vjxD0 D vjxD1 D 0 ; n 2 D .u; v;  / 2 H 2 Œ0; 1  H 2 Œ0; 1  H 2 Œ0; 1 W u.x/ > 0; .x/ > 0; x 2 Œ0; 1; HC o vjxD0 D vjxD1 D 0; x jxD0 D x jxD1 D 0 ; and n 4 HC D .u; v;  / 2 H 4 Œ0; 1  H 4 Œ0; 1  H 4 Œ0; 1 W u.x/ > 0; .x/ > 0; x 2 Œ0; 1; o vjxD0 D vjxD1 D 0; x jxD0 D x jxD1 D 0 : The first result concerns the global existence and exponential stability of solution .u; v;  / in H 1 . Theorem 1. Assume that e; p; , and k are C 2 functions satisfying (19), (20), (21), (25), (26), (27), (30), (31), (32), (33), (34), and (35) on 0 < u < C1 and 0  1  < C1, and q; r satisfy assumptions (28) and (29). If .u0 ; v0 ; 0 / 2 HC , then the problem (13), (14), (15), (16), and (17) admits a unique global generalized solution 1 1 .u.t /; v.t /; .t // in HC which defines a nonlinear C0 semigroup S .t / on HC . 1 Moreover, for any .u0 ; v0 ; 0 / 2 HC , there exists a constant 1 D 1 .C1 / > 0 such that for any fixed 2 .0; 1  and for any t > 0, the following inequality k.u.t /; v.t /; .t //  .Nu; 0; N /k2H 1 D kS .t /.u0 ; v0 ; 0 /  .Nu; 0; N /k2H 1  C1 e  t ; C

C

(36) 1 which means that the semigroup S .t / decays exponentially on HC . Here

Z

1

u0 .x/dx;

uN D

(37)

0

and N > 0 is uniquely determined by e.Nu; N / D

Z

1

 e.u0 ; 0 / C

0

v02 2

 .x/dx:

(38)

2334

Y. Qin

holds, To prove Theorem 1, the following strategy is used. First, some estimates can be obtained easily by some conservations such as mass and energy (see Lemmas 1 and 2) and by the second law of thermodynamics (see Lemma 3). Second, the key estimates are to derive the positive lower and upper bounds of the specific volume of u.x; t /. To achieve this, it suffices to give representations of u.x; t / (see Lemmas 4 and 5) and to use previous estimates in Lemmas 1, 2, and 3 (see Lemma 6). Third, after establishing this estimate, try to use Corollary 1 fully to reduce the higher order of the temperature  so that the growth exponents .r; q/ have as large a range as possible. Fourth, we need to estimate .u; v/ in H 1 in terms of sup k.s/kL1 0st

and to use delicate interpolation techniques by trying to give as many different estimates as one can for the same term (see Lemmas 7, 8, 9, 10, and 11) in which the key estimate is (99) in Lemma 11. This key estimate (99) finally determines the uniform bound of sup k.s/kL1 and the range of exponents .r; q/ (see (43) 0st

and (44)) by the Young inequality. Thus the global existence of .u; v;  / follows. Fifth, to derive the asymptotic behavior of global solutions .u; v;  /, it suffices to employ Lemma 6.2.1 in Zheng [81] (see Lemmas 13, 14, and 15). Sixth, to study the global attractors, it suffices to prove existence of the C0 semigroup S .t / on H 1 in Lemma 16. Seventh, to investigate the exponential stability of the solution in H 1 around the unique equilibrium .Nu; 0; N /, we first need to derive the positive lower bound of  by combining the asymptotic behavior of  in H 1 with the contradiction argument (see Lemma 17), and then construct the functional E.u; v; / in (161) around the equilibrium .Nu; 0; / N and derive that the key estimate (162) holds. Finally we multiply the factor e t for small enough so that (167) holds, and hence this implies the exponential stability of global solutions in H 1 L2 L2 (see Lemma 19) which is the essential estimate for exponential stability in higher regularity space H i .i D 1; 2; 4/ (see Lemmas 18 and 19, and subsequent Lemmas 25 and 26 and Lemmas 32–34). Assume assumptions in Theorem 1 hold throughout this subsection. 1 Lemma 1. If .u0 ; v0 ; 0 / 2 HC , then there exists a unique global strong solution 1 .u.t /; v.t /; .t // in HC to the problem (13), (14), (15), (16), and (17) such that

ut ; vt ; t ; x ; vx ; ux ; vxx ; xx 2 L2 .Œ0; C1/; L2 /;

(39)

0 < .x; t/  C1 ;

(40)

0
0;

(42)

and there exist positive constants C10 ; t0 ; C1 , independent of t , such that for all t  t0 ,

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

k.u.t /  uN ; v.t /; .t /  N /kH 1  C10 e C1 t :

2335

(43)

To prove Lemma 1, the following lemmas are needed. Lemma 2. The following estimates hold.

Z

.x; t / > 0; 8 .x; t / 2 Œ0; 1  Œ0; C1/; (44) Z 1 1 1 2 1 .e.x; t / C v .x; t //dx D .e.x; 0/ C v02 .x//dx D E0 ; 8t > 0; (45) 2 2 0 0 Z 1 . C  1Cr /.x; t /dx  C1 ; 8t > 0; (46) 0

Z

Z

1

1

u.x; t /dx D 0

u0 .x/dx D u0 ; 8t > 0: (47) 0

Proof. Estimates (45), (46), and (47) are based on the laws of conservations of mass and energy; see Jiang [31] and Qin [54–58, 60] for details. The proof is now complete. Lemma 3. There holds that for any t > 0, Z

Z tZ

1

1

Œ. log 1/C 1Cr Cv 2 .x; t/dxC 0

0

0



 vx2 .1C q /x2 C .x; s/dxdsC1 : u u 2 (48)

Proof. Estimate (48) is based on the second law of thermodynamics; see Jiang [31] and Qin [54–58, 60] for details. The proof is complete. Remark 1. It follows from the convexity of the function  ln y that there exist two positive constants r1 ; r2 only depending on the initial data such that 0 < r1  R1 0 dx  r2 : To proceed, it suffices to estimate the positive lower and upper bounds of the specific volume u (see Lemma 6) by giving representations of it in the next two lemmas. Lemma 4. For any t  0, there exists one point x1 D x1 .t / 2 Œ0; 1 such that the solution u.x; t / to the problem (13), (14), (15), (16), and (17) possesses the expression:  Z u.x; t / D D.x; t /Z.t / 1 C 1 0

0

where

t

 u.x; s/p.x; s/ ds ; D.x; s/Z.s/

(49)

2336

Y. Qin

 D.x; t / D u0 .x/ exp 1 C u0

Z

Z

1

Z

Z

x

x

v.y; t/dy  x1 .t/

v0 .y/dy 0



x

u0 .x/ 0

1 0

v0 .y/dydx

;

(50)

0



Z tZ

1 Z.t / D exp  0 u0



1 2

.v C up/.y; s/dyds : 0

(51)

0

Proof. Let Z

Z

x

t

v0 .y/dy C

h.x; t/ D 0

 .x;  /d  :

(52)

0

Then from (14), h.x; t/ satisfies hx D v; ht D  ; and solves the following equation, ht D p C

0 hxx u

(53)

with x D 0; 1 W hx D v D 0:

(54)

.uh/t D hvx  up C 0 hxx :

(55)

Hence

Integrating (55) over Qt , and using the boundary condition (54), one arrives at Z

Z

1

Z tZ

1

uhdx D

u0 h0 dx 

0

0

0

1

.up C v 2 /dxd   .t/:

(56)

0

Then for any t  0, there exists one point x1 D x1 .t / 2 Œ0; 1 such that Z

Z

1

1

uhdx D

.t/  0

udx  h.x1 .t /; t / D u0  h.x1 .t /; t /I 0

that is, Z

Z

t

x1 .t/

p.x1 .t /;  /d  D 0

v0 .y/dy C 0 log 0

u.x1 .t /; t / .t/  u0 .x1 .t // u0

(57)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2337

with boundary conditions Z Z tZ 1 2 .v C up/.x; s/dxds C

.t/ D 

(58)

0

0

Z

1

x

u0 .x/ 0

v0 .y/dydx: 0

On the other hand, (14) can be rewritten as vt  0 .log u/xt D px D px :

(59)

Integrating (59) over Qt for any fixed t > 0, one gets Z x  1 u0 .x/u.x1 .t /; t / exp .v.y; t /  v0 .y//dy u.x; t / D u0 .x1 .t // 0 x1 .t/  Z t C .p.x;  /  p.x1 .t /;  //d  :

(60)

0

Inserting (57) into (60) and noting (50), (51), and (58), one has u1 .x; t / exp



1 0

Z



t

p.x; s/ds

D D 1 .x; t /Z 1 .t /

(61)

0

which implies that  Z t  1 d 1 1 exp p.x; s/ds D D .x; t /Z 1 .t /u.x; t /p.x; t /I dt 0 0 0 that is, 

1 exp 0

Z



t

p.x; s/ds 0

1 D1C 0

Z

t

D 1 .x; s/Z 1 .s/u.x; s/p.x; s/ds:

0

(62) Thus (49) follows from (61) and (62). The proof is now complete. Lemma 5. For any t  0, there exists one point a.t / 2 Œ0; 1 such that the solution u.x; t / to the problem (13), (14), (15), (16), and (17) possesses the expression: u.x; t / D B

1

.x; t /Y

1



1 .t / 1 C 0

Z

t

 u.x; s/p.x; s/Y .s/B.x; s/ds ;

(63)

0

where 

 Z t 1 p.x; s/ds ; (64) 0 0  Z x  1 1 B.x; t / D exp .v0 .y/  v.y; t //dy ; u.a.t /; t / D u0 : (65) u0 .x/u0 0 a.t/

Y .t / D u0 .a.t // exp

2338

Y. Qin

Proof. It follows from (47) that there exists a.t / 2 Œ0; 1 such that the second equality of (65) holds. Substituting x1 .t / by a.t / in (57) yields

u.x; t / D Y

1

.t /B

1



1 .x; t / exp 0

Z



t

p.x; s/ds ; 0

 Z t  1 d 1 exp p.x; s/ds D p.x; t/u.x; t /Y .t /B.x; t /; dt 0 0 0 which yields (63). The proof is complete. Using Lemmas 4 and 5, the next lemma can be obtained. Lemma 6. There holds that 0 < C11  u.x; t /  C1 ; 8.x; t / 2 Œ0; 1  Œ0; C1/:

(66)

Proof. Let Mu .t / D max u.x; t /; mu .t / D min u.x; t /; M .t / D max .x; t /; m .t / x2Œ0;1

x2Œ0;1

x2Œ0;1

D min .x; t /: x2Œ0;1

R1 It follows from (45) and convexity of the function  log y that 0  dx  R1 R1 log 0 dx  1  0 .  log   1/dx  C1 which, along with Remark 1, implies that there exist b.t / 2 Œ0; 1 and two positive constants r1 ; r2 such that R1 0 < r1  0 .x; t /dx D .b.t /; t /  r2 with r1 ; r2 being two positive roots of the equation y  log y  1 D C1 : Thus (29) and (45) yield 1 0 < a1  0 u0

Z

1

.up C v 2 /.x; s/dx  a2

(67)

0

with a1 D p1 =0 u0 , a2 D .p2 C1 C C1 C p2 /=0 u0 . On the other hand, it is easy to obtain j m1 .x; t /   m1 .b.t /; t /j  C j

Z

x

 m1 1 x dyj

b.t/

Z

1

C 0 1=2

x2 .1 C  q / dx u 2

 C V1 .t /Mu1=2 .t /

 12 Z

1 0

u 2m1 dx 1 C q

 12

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2339

where Z V1 .t / D 0

1

x2 .1 C  q /=u 2 dx; 0  m1  m D .q C r C 1/=2

and Z

Z

t

1

 2m1 =.1 C  q /dx  C1

V1 .s/ds  C1 ; 0

Z

0

1

.1 C  1Cr /dx  C1 : 0

Thus C11  C1 V1 .t /Mu .t /   2m1 .x; t /  C1 C C1 V1 .t /Mu .t /:

(68)

Using Lemmas 2, 3, and 4, (50), (68), and noticing that 0 < C11  D.x; t /  C1 and u.x; s/p.x; s/  p2 .1 C  rC1 /  C1 .1 C  2m /  C1 C C1 V1 .s/Mu .s/; one obtains 

Z



t

u.x; t /  C1 1 C

V1 .s/ exp.a1 .t  s//Mu .s/ds 0



Z

 C1 1 C



t

V1 .s/Mu .s/ds ; 0

that is,   Z t Mu .t /  C1 1 C V1 .s/Mu .s/ds :

(69)

0

Thus it follows from Gronwall’s inequality and

Rt 0

V1 .s/ds  C1 that

Mu .t /  C1

(70)

C11  C1 V1 .t /   2m1 .x; t /  C1 .1 C V1 .t //:

(71)

which, along with (68), leads to

Similarly, Lemmas 2, 3, and 4, (51), and (67) yield   Z t exp.a1 t /  Z.t /  exp.a2 t /; u.x; t /  C11 e a2 t C e a2 .ts/ ds 0



C11 .1

e

a2 t

/:

2340

Y. Qin

Thus there exists t0 > 0 such that for all t  t0 , it holds that u.x; t /  C11 :

(72)

Moreover, it follows from Lemma 5 and (71) that Z



1

Z tZ

u.x; t /Y .t /dx  C1 1 C

u0 Y .t / D 0



.1 C  0

Z



t

 C1 1 C



1 2m

/dxY .s/ds

0

.1 C V1 .s//Y .s/ds : 0

By Gronwall’s inequality and noting Z

Rt 0

V1 .s/ds  C1 ; there holds that

t

.1 C V1 .s//ds/  C1 exp.C1 t /:

Y .t /  C1 exp.C1

(73)

0

Thus for all 0  t  t0 ; u.x; t /  B 1 .x; t /Y 1 .t /  C11 Y 1 .t /  C11 :

(74)

By (70), (72), and (74), (66) follows. The proof is complete. Corollary 1. There holds that for any .x; t / 2 Œ0; 1  Œ0; C1/, C11  C1 V2 .t /   2m1 .x; t /  C1 C C1 V2 .t / with 0  m1  m D .q C r C 1/=2 and V2 .t / D R C1 V2 .t /dt < C1. 0

R1 0

.1C q /x2 dx 2

(75) satisfying

From now on, all estimates are bounded in terms of sup k.s/kL1 . 0st

Lemma 7. The following estimates hold for any t > 0, Z

0

kux .t /k2 C

Z tZ 0

0

kv.s/k2L1 ds  C1 ;

(76)

.1 C  /2m v 2 dxds  C1 ;

(77)

0

Z tZ

t

1

0 1

.1 C  rC1 /u2x dxds  C1 .1 C sup k.s/kL1 /ˇ ;

with ˇ D max.r C 1  q; 0/.

0st

(78)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2341

Proof. It follows from (71), Lemma 2, and Corollary 1 that Z 1 Z tZ 1 2 vx2 vx dx dxds  C1 ;  dxds  C1 0 0 0  0 0 0  Z tZ 1 Z tZ 1 Z tZ 1 2m 2 2 .1 C / v dxds  C1 v dxds C C1 V2 .s/v 2 dxds  C1 : Z

0

t

kv.s/k2L1 ds 

0

Z tZ

1

0

0

0

0

Equation (14) can be rewritten as 

v  0

ux C pu .u;  /ux D p .u;  /x : u t

(79)

Multiplying (111) by v  0 uux and then integrating the result over Qt , one has Z tZ

1

0 pu .u;  /u2x dxds u 0 0 Z t Z 1h  1 u0x 2 0 ux i dxds: D kv0  0 k  pu ux v C p x v  2 u0 u 0 0

ux 1 kv  0 k2 C 2 u

By (32) and Corollary 1 (with m1 D r C 1), and noting the following facts, Z tZ 0

1

0

 2 .1 C  r /2 2 v dxds  C1 1C sup k.s/kL1 1 C q 0st

!ı Z Z t 0

1

.1C /2m v 2 dxds

0

!ı  C1 1 C sup k.s/kL1

;

0st

Z tZ 0

1 0

.1 C  r /2 2  dxds  C1 1 C  rC1 x

Z tZ 0

0

1

.1 C  /r1 x2 dxds !ˇ Z

t

 C1 1 C sup k.s/kL1

V2 .s/ds

0st

0

!ˇ  C1 1 C sup k.s/kL1

;

0st

Z tZ 0

0

1

u2x dxds  C1

one arrives at, for all  > 0,

Z

t

V2 .s/kux k2 dsCC1 0

Z tZ 0

0

1

 rC1 u2x dxds; (80)

2342

Y. Qin

Z tZ 1 ux 2 k C Œl C .1  l/ C  rC1 u2x dxds u 0 0 Z t Z 1h  ux i  C1 C C1 j dxds .1 C  rC1 /jux vj C .1 C  r /jx v  0 u 0 0 Z tZ 1 Z t .1 C  1Cr /.u2x C C1 v 2 /dxds C C1 V2 .s/ds  C1 C C1

kv  0

Z tZ

0

0

1

CC1 0

0

Z tZ

1

CC1 0

0

2

 .1 C  / 2 v dxds C C1  1 C q

0

.1 C  r /2 x2 dxds 1 C  1Cr !max.ˇ;ı/

 C1 1 C sup k.s/kL1

1

0

Z tZ C C1 

0st

Z

0

Z tZ

r 2

0

0

.1 C  1Cr /u2x dxds

1

.1 C  1Cr /u2x dxds

t

kvk2L1 ds

CC 0

!ˇ

 C1 1 C sup k.s/kL1

Z tZ

0st

0

!ˇ  C1 1C sup k.s/k

L1

1

C C1  Z

.1 C  1Cr /u2x dxds Z tZ

t 2

C C1 

0st

0

1

V2 .s/kux k dsCC1  0

0

0

 rC1 u2x dxds (81)

with ˇ D max.r C 1  q; 0/  ı. Thus for small  > 0 in (81) and applying the generalized Bellman-Gronwall inequality, one gets Z tZ

2

!ˇ

1

kux k C

Œl C .1  l/ C  0

0

rC1

u2x dxds

 C1 1 C sup k.s/kL1 0st

which for l D 1 and (81) (for l D 0) yield the desired estimate (78). The proof is complete. Lemma 8. There holds that for any t > 0, Z tZ

!ˇ

1

.1 C 0

0

 /2m u2x dxds

 C1 1 C sup k.s/kL1

:

(82)

0st

Proof. The inequality (82) follows from Corollary 1 and Lemma 7. The proof is complete.

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2343

Lemma 9. The following estimates hold that for any t > 0, Z

!ˇ=2

t 2

kvx .s/k ds  C1 1 C sup k.s/kL1

kvx .t /k2 C

Z

kvxx .s/k2 ds  C1 1 C sup k.s/kL1

(84)

;

(85)

!ˇ28 kvx .s/k2L1 ds

Z

2

;

0st

t

0

(83)

!ˇ27

t

0

Z

;

0st

0

 C1 1 C sup k.s/kL1 0st

!ˇ1

t 2

kvx .t /k C

kvt .s/k ds  C1 1 C sup k.s/kL1

(86)

0st

0

with ˇ27 D max.5ˇ=2; ˇ1 /; ˇ28 D ˇ=4 C ˇ27 =2 and ˇ1 D max.2 C 2r  q; 0/: Proof. Multiplying (14) by v; vxx and vt , respectively, and then integrating the resultants over Qt , using (76), (77), (78), and (82) and Lemmas 6, 7, and 8, one gets Z tZ 1 2 vx 2 dxds kv.t /k C 20 0 0 u ˇZ t Z 1 ˇ ˇ ˇ ˇ pu ux C p x /vdxds ˇˇ  C1 C C 1 ˇ 0

0

Z tZ

1

Œ.1 C  1Cr /jux vj C .1 C  r /jx vjdxds

 C1 C C1 0

0

Z t Z

1

.1 C  /rC1 u2x dxds

 C1 CC1 0

Z

0

1=2 Z t Z

t

CC1

V2 .s/ds 0

0

Z t Z 0

0

.1 C  /2m u2x dxds

.1 C  /rC1 v 2 dxds

0

CC1 1 C sup k.s/kL1 0

1=2

0

1=2 Z t Z

!ı=2 Z Z t 0st

1

.1 C  /2r  2 v 2 dxds 1 C q

1

 C1 C C1 0

0

1

1=2 Z t Z

1=2

1

.1 C  /2m v 2 dxds

1=2

0

1

.1 C  /2m v 2 dxds

1=2

0

!ˇ=2  C1 1C sup k.s/kL1 0st

;

(87)

2344

Y. Qin

kvx .t /k2 C

Z

t

kvxx .s/k2 ds 0

Z tZ

1

Œjux vx vxx j C .1 C  1Cr /jux vxx j C .1C r /jx vxx jdxds

 C1 C C 1 0

Z

1  C1 C 4

0

Z tZ

t 2

1

kvxx .s/k ds C C 0

0

0

Œvx2 u2x C .1 C  1Cr /2 u2x

C.1 C  /2r x2 dxds 1  C1 C 4

Z

!ˇ Z

t 2

kvxx .s/k ds C C

t

kvx .s/k2L1 ds

1C sup k.s/kL1 0st

0

!ˇ Z Z t

1

.1 C  /2m u2x dxds

CC1 1 C sup k.s/kL1 0st

0

0

!ˇ1 Z

0

t

CC1 1 C sup k.s/kL1

V2 .s/ds

0st

0

!2ˇ  C1 1 C sup k.s/kL1 0st

1 C 4

!ˇ Z

t

CC1 1C sup k.s/kL1 0st

Z

t

kvxx .s/k2 ds

0

 Z t 1=2 kvx .s/k2 ds 1=2 kvxx .s/k2 ds

0

0

!ˇ1 CC1 1 C sup k.s/kL1 0st

!5ˇ=2  C1 1 C sup k.s/kL1

!ˇ1 C C1 1 C sup k.s/kL1

0st

C

1 2

Z

0st

t

kvxx .s/k2 ds

0

That is, kvx .t /k2 C

Z

!ˇ27

t

kvxx .s/k2 ds  C1 1 C sup k.s/kL1 0st

0

and kvx .t /k2 C

Z 0

t

kvt .s/k2 ds

;

(88)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

Z t

Z

2

1

kpx k C

 C1 C C 1 0

0

Z tZ

 jvx j3 dx ds u2

1

Œ.1 C  /2rC2 u2x C .1 C  /2r x2 C jvx j3 dxds

 C1 C C1 0

0

!2ˇ  C1 1 C sup k.s/kL1

!ˇ1 CC

0st

Z

2345

1 C sup k.s/kL1 0st

t

kvx .s/k5=2 kvxx .s/k1=2 ds

CC1 0

!ˇ1  C1 1 C sup k.s/kL1 0st

Z

t

kvx k2 ds

CC1 sup kvx .s/k 0st

0

3=4 Z

t

kvxx .s/k2 ds

1=4

0

!ˇ1  C1 1 C sup k.s/kL1

!3ˇ=4Cˇ27 =2 C C1 1 C sup k.s/kL1

0st

C

0st

1 sup kvx .s/k2 2 0st !ˇ1

 C1 1 C sup k.s/kL1

C

0st

1 sup kvx .s/k2 ; 2 0st

which, together with (88), yields the estimates (85) and (86) with ˇ1  3ˇ=4 C ˇ27 =2. The proof is complete. Corollary 2. The following estimates hold for any t > 0, Z tZ

!ˇ1

1

.1 C 0

0

Z tZ 0

Z tZ

1

;

(89)

0st

!ˇ29 .1 C  /2mC1 vx2 dxds  C1 1 C sup k.s/kL1

;

(90)

;

(91)

;

(92)

0st

!ˇ30 qC1

.1 C  /

3

jvx j dxds  C1 1 C sup k.s/kL1 0st

0

Z tZ

!ˇ31

1

.1 C 0

 C1 1 C sup k.s/kL1

1

0

0

 /2m vx2 dxds

0

 /qr vx4 dxds

 C1 1 C sup k.s/kL1 0st

2346

Y. Qin

where ˇ29 D min.1 C ˇ1 ; 2m C 1 C ˇ=2/; ˇ30 D minŒq1 C .5ˇ1 C ˇ27 /=4; q C 1 C ˇ1 =2 C 3ˇ=8 C ˇ27 =4; ˇ31 D minŒmax.q  r; 0/ C ˇ1 C ˇ=4 C ˇ27 =2; q2 C 3ˇ1 =2 C ˇ27 =2; q1 D max..q C 1  3r/=4; 0/; q2 D max..q  3r  1/=2; 0/: Proof. By Corollary 1 and Lemmas 7, 8, and 9, using the same method as that in Lemma 9, the proof is complete. Lemma 10. There holds that for any t > 0,

k.t / C  1Cr .t /k2 C

Z tZ 0

!ˇ32

1

.1 C  /qCr x2 dxds  C1 1 C sup k.s/kL1

;

0st

0

(93) where ˇ9 D minŒmax.2r C 1  2q; 0/; max.3r C 3  2q; 0/=2; ˇ32 D maxŒˇ28 ; 3ˇ=2; ˇ9 ; ˇ C 1: Proof. Equation (15) can be rewritten as  et   vx 

kx u

 D 0:

(94)

x

Multiplying (94) by e, integrating the result over Qt and using (16) and Lemmas 2, 3, 4, 5, 6, 7, 8, and 9, one gets k C  rC1 k2 C

Z tZ 0

ˇZ t Z ˇ  C1 C C1 ˇˇ 0

Z

0



0

.1 C  /qCr x2 dxds

0 vx2 e kx eu ux C v.ep/x  u u

t

 C1 C C1 0

kvx .s/k2L1

Z tZ CC1 0

1

1

0

Z

rC1

ˇ ˇ dxds ˇˇ Z tZ

1

.1 C  0



/dxds C C1 0

1

.1 C  /2rC2 jvux jdxds

0

1

Œ.1 C  /2rC1 jvx j C .1 C  /qCrC1 jx ux jdxds;

(95)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

Z tZ

2rC2

.1 C  / 0

Z t Z

1

1

jvux jdxds  C1

.1 C

0

0

Z t Z 0

2347

0

 /2rC2 u2x dxds

1

.1 C  /2rC2 v 2 dxds

 1=2

 1=2

0

 C1 .1 C sup k.s/kL1 /ˇ 0st

Z t Z 0

1

.1 C  /2rC2 u2x dxds

0

Z t Z

1

.1 C  /2m v 2 dxds

 0

 1=2  1=2

0

!3ˇ=2  C1 1 C sup k.s/kL1

:

(96)

0st

Similarly, Z tZ

.1C/ 0

Z

1 2rC1

1=2 Z tZ

t

jvx jdxds 

1

V2 .s/ds

0

0

0

0

 2 .1C /4rC2 v 2 dxds 1C q !max.3rC32q;0/=2

 C1 1 C sup k.s/kL1 0st

Z t Z 0

1

.1 C  /2m v 2 dxds

1=2

0

!max.3rC32q;0/=2  C1 1 C sup k.s/kL1

;

0st

Z tZ

1

.1 C / 0

0

2rC1

1 jvx jdxds  4

Z tZ 0

1

.1 C  /qCr x2 dxds

0

!max.2rC12q;0/ CC1 1 C sup k.s/kL1 0st

Z tZ 0



1 4

Z tZ 0

1

.1 C  /2m v 2 dxds 0

0

1

.1 C  /qCr x2 dxds !max.2rC12q;0/

CC1 1 C sup k.s/kL1 0st

:

1=2

2348

Y. Qin

Thus Z tZ

1

.1 C  / 0

2rC1

0

1 jvx jdxds  4

Z tZ 0

1 0

.1 C  /qCr x2 dxds !ˇ9

C C1 1 C sup k.s/kL1

:

(97)

0st

On the other hand, by the Cauchy inequality and Lemma 6, one obtains Z tZ 0

1

.1 C  /qCrC1 jx ux jdxds 0

1  4

Z tZ 0

0

!ˇC1

1

.1 C  /qCr x2 dxds C C1 1 C sup k.s/kL1

: (98)

0st

Therefore it follows from (95), (96), (97), and (98) and Lemmas 2, 3, 4, 5, 6, 7, 8, and 9 that !ˇ28 Z Z t

1

k C  1Cr k2 C

0

0

.1 C  /qCr x2 dxds  C1 1 C sup k.s/kL1 0st

!3ˇ=2 CC1 1 C sup k.s/kL1 0st

!ˇ9 CC1 1 C sup k.s/kL1 0st

!ˇC1 CC1 1 C sup k.s/kL1 0st

!ˇ32  C1 1 C sup k.s/kL1 0st

which implies (93). The proof is complete. The next lemma plays a crucial role in determining the uniform bound of sup k.s/kL1 ; see Lemma 12 for details. 0st

Lemma 11. There holds that for any t > 0, Z 0

1

.1 C /2q x2 dx C

Z tZ 0

0

!ˇ35

1

.1 C  /qCr t2 dxds C1 1 C sup k.s/kL1

;

0st

8t > 0;

(99)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2349

where ˇ36 D Œmax.3q C 2  r; 0/ C ˇ27 C ˇ32 =2; ˇ33 D minŒˇ36 ; .3q C 4 C ˇ27 /=2; ˚ ˇ37 D max 2 max.q  r; 0/ C 2ˇ C ˇ32 ; max.q  r; 0/ C ˇ C .ˇ32 C ˇ29 /=2;

max.q  r; 0/ C ˇ C .ˇ32 C ˇ31 /=2 ; ˚ ˇ38 D max max.q  r; 0/ C q C 2 C ˇ; 2 max.q  r; 0/ C r C 2 C 2ˇ;

max.q  r; 0/CˇC.ˇ29 C r C 2/=2; max.q  r; 0/CˇC.ˇ31 C r C 2/=2 ; ˇ34 D min.ˇ37 ; ˇ38 /; ˇ35 D maxŒˇ29 ; ˇ30 ; ˇ31 ; ˇ33 ; ˇ34 : Proof. Let Z



H .x; t / D H .u;  / D Z D 0

0

Z tZ

k.u; / d ; X .t / D u

0

1 0

.1 C  /qCr t2 dxds; Y .t /

1

.1 C  /2q x2 dx:

Then it is easy to verify that     kt kx k Ht D Hu vx C ; Hxt D C Hu vxx C Huu vx ux C ux t : u u t u u Multiplying (15) by Ht and integrating the result over Qt , one knows Z tZ 0

1



0

0 vx2 e t C  p vx  u



Z tZ

1

Ht dxds C 0

0

kx Htx dxds D 0: u

(100)

But it follows from (34) and (35) that jHu j C jHuu j  C1 .1 C  /qC1 :

(101)

Now estimate each term in (100) by using (31) and (32), (101), Lemmas 2, 3, 4, 5, 6, 7, 8, 9, and 10, and Corollary 2. It easily follows from (33) and (34) and Lemma 6 that Z tZ

!ˇ29

1

e t Ht dxds  C1 X .t /  C1 1 C sup k.s/kL1 0

Similarly,

0

0st

:

(102)

2350

Y. Qin

ˇZ t Z ˇ ˇ ˇ 0

1 0



0 vx2 p vx  u



!ˇ29 ˇ ˇ C1 ˇ Ht dxds ˇ  X .t /CC1 1C sup k.s/kL1 8 0st !ˇ30 CC1 1 C sup k.s/kL1 0st

!ˇ31 CC1 1C sup k.s/kL1

; (103)

0st

Z tZ 0

ˇZ t Z ˇ ˇ ˇ 0

0

1

kx u



kx u

 dxds  C1 Y .t /  C1 ;

(104)

t

!ˇ36 ˇ ˇ kx ; .Hu vxx C Huu vx ux /dxds ˇˇ  C1 1 C sup k.s/kL1 u 0st

1 0

where ˇ27  ˇ C ˇ28 . On the other hand, it is easy to verify ˇZ t Z ˇ ˇ ˇ 0

!.3qC4Cˇ27 /=2 ˇ ˇ kx ˇ .Hu vxx C Huu vx ux /dxds ˇ  C1 1 C sup k.s/kL1 : u 0st

1 0

Therefore ˇZ t Z ˇ ˇ ˇ 0

1

0

ˇ ˇ kx .Hu vxx C Huu vx ux /dxds ˇˇ  C1 .1 C sup k.s/kL1 /ˇ33 : u 0st

By Lemmas 2, 3, 4, 5, 6, 7, 8, 9, and 10, one gets ˇ   ˇ kx k ux t dxds ˇˇ u u u 0 Z tZ 1 kx ux t jdxds .1 C  /q j  C1 u 0 0  Z t Z 1 C1 kx 2 X .t / C C1  .1 C  /qr u2x dxds 8 u 0 0 !max.qr;0/Cˇ Z  2 t  C1 kx    1  X .t / C C1 1 C sup k.s/kL  u  1 ds 8 0st 0 L

ˇZ t Z ˇ ˇ ˇ 0

1

C1  X .t / C C1 1 C sup k.s/kL1 8 0st

!max.qr;0/Cˇ

(105)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2351

   ˇ # Z t " Z 1ˇ ˇ kx kx ˇ  kx 2 ˇ ˇ dx .s/ds   ˇ u  u  C u xˇ 0 0 C1 X .t / C C1 1 C sup k.s/kL1  8 0st



8
0, Z

t

.kp  k2 C k  k2 /.s/ds  C1 ;

(112)

0

d kp  .t /k2  C1 .kvt .t /k2 C kt .t /k2 C 1/; dt d k  .t /k2  C1 .kvt .t /k2 C kt .t /k2 C 1/ dt where p  D p 

R1 0

pdx and   D  

R1 0

(113) (114)

 dx:

Proof. Equation (14) can be rewritten as vt  0

v x

u

x

D px :

(115)

Using (115) and integrating by parts, one sees that  Z kp  k2 D px ;

  v Z x x D vt  0 ; p  dy u x 0 0   Z x Z x  @ vx  D ;p vt dy; p  dy C 0 @x 0 u 0 Z x  Z x  v @ x   ; p ; vdy; p C vdy; pt C 0 D @t u 0 0 x

p  dy



where .; / is the L2 inner product. Thus Z 0

t

kp  .s/k2 ds  C1 C kv.t /kL1 kp  .t /kL1 C

Z 0

t

kv.s/k2L1 ds

1=2

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

Z

t

kpt .s/k2 ds

0

Z

1=2 C

1 2

Z

t

2355

kp  .s/k2 ds

0

t

kvx .s/k2 ds

CC1 0

Z  C1 C C1

.kt k2 C kvx k2 /.s/ds

1=2 C

0

Z

1  C1 C 2

t

t

1 2

Z

t

kp  .s/k2 ds

0

kp  .s/k2 ds;

0

which implies that Z

t

kp  .s/k2 ds  C1 ;

0

Z

t

k  .s/k2 ds  C1

0

(116)

Z t

(117)

.kp  k2 C kvx k2 /.s/ds  C1 :

(118)

0

Z

t

 C1

 v 

k2 .s/ds

kp  k2 C k

x

u

0

On the other hand, it follows from Lemmas 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 that   Z x d  2     kp .t /k D 2.p ; pt / D 2 px ;  pt dy dt 0   Z x  Z x vx pt dy/  2 0 ; pt dy D 2.vt ; u x 0 0 Z x pt dyk C kvx .t /kkpt .t /k/  C1 .kvt .t /kk 0

2

 C1 .kvt .t /k C kt .t /k2 C 1/: Similarly, noting the equalities t D pt C 0 Z 0

x

vt vxt dy D C u u

Z 0

x



v x

u

t

D pu vx  p t C 0

vt ux dy; u2

one easily gets     Z x Z x d t dy D 2 vt ;  t dy k  .t /k2 D 2 x ;  dt 0 0

vxt v2  x2 u u

 ;

2356

Y. Qin



Z

2

x

 C1 kvt k C k 0

t dyk2



2

 C1 .kvt .t /k Ckvx .t /k2 Ckt .t /k2 C kvx .t /k4 C kvt .t /k2 kux .t /k2 /  C1 .kvt .t /k2 C kt .t /k2 C 1/: The proof is complete. Lemma 14. The following estimates hold for any t > 0, d kux .t /k2  kvxx .t /k2 C kux .t /k2 ; dt

Z 1 d 2 .1 C  /qr xx dx  C1 .kvxx .t /k2 C 1/; kx .t /k2 C C11 dt 0 Z tZ 1 2 kx .t /k2 C .1 C  /qr xx dxds  C1 : 0

(119) (120) (121)

0

Proof. Differentiating (13) with respect to x and multiplying the result by ux yields the estimate (119). Multiplying (15) by e1 xx and integrating the result on Œ0; 1 leads to, for all  > 0, d kx .t /k2 C 2 dt Z D

Z

1

0 1



0

k 2  e dx u xx  p vx vx2 .k=u/x x   e e e

 xx dx

 kxx .t /k2 C C1 .kvx .t /k2 C kvx .t /k4L4 C kx .t /k4L4 Ckux .t /x .t /k2 /  kxx .t /k2 C C1 .kvx .t /k2 C kvx .t /k3 kvxx .t /k C kvx .t /k4 Ckx .t /k3 kxx .t /k C kx .t /k4 C kx .t /k2L1 /  2kxx .t /k2 C C1 .kvx .t /k2 C kvxx .t /k2 C kx .t /k2 /: Hence for small , one can check that d kx .t /k2 C C11 dt

Z

1 0

2 .1 C  /qr xx dx  C1 .kvx .t /k2 C kvxx .t /k2 C kx .t /k2 /

 C1 .kvxx .t /k2 C 1/ which implies

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

kx .t /k2 C

Z tZ 0

0

1 2 .1C /qr xx dxdsC1 C C1

Z

2357

t

.kvx k2 Ckvxx k2 Ckx k2 /.s/ds

0

 C1 : The proof is complete. Lemma 15. As t ! C1, there holds that ku.t /  u0 kH 1 ! 0; kux .t /k ! 0; ku.t /  u0 kL1 ! 0; (122) kv.t /kH 1 ! 0; kvx .t /k ! 0; (123) kx .t /k ! 0; (124) N L1 ! 0; kp  .t /k ! 0; k  .t /k ! 0; (125) k.t /  N kH 1 ! 0; k.t /  k k .u;  ; vx /.t / C p.u0 ; N /k ! 0: (126) Moreover, the decay estimate (43) holds. Proof. By Lemmas 12, 13, and 14 and applying Lemma 6.2.1 in Zheng [81], one concludes, as t ! C1 kux .t /k ! 0; kx .t /k ! 0; kp  .t /k ! 0; k  .t /k ! 0:

(127)

Thus (122) and (123) follow from the embedding theorem and (127). It is obvious from (127) that as t ! C1; k

 v  x

u

k2  C1 .k  k2 C kp  k2 / ! 0;

and kvx .t /k  C1 k

   Z 1  v  vx  vux vx x k  C1 k kCk dxk  C .k k 1 2 u u u u 0

Ckux .t /k/ ! 0:

(128)

Thus kv.t /kH 1  C1 kvx .t /k ! 0: By (45), Lemmas 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14 and the Poincaré inequality, one can get ke.u;  /  e.Nu; N /k  ke.u;  / 

Z 0

1

e.u;  /dxk C kv.t /k2 =2

2358

Y. Qin

 C1 .kex .t /k C kv.t /k/  C1 .kux .t /k C kx .t /k C kvx .t /k/;

(129)

R1 R1 with uN D 0 udx D 0 u0 dx D u0 . By the mean value theorem, there are uQ and Q with uQ D  C .1  /N and uQ D u C .1  /Nu such that e.u;  /  e.Nu; uN / D eu .Qu; Q /.u  uN / C e .Qu; Q /.  N /:

(130)

By Lemmas 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14, one infers that 0 < C11  mi nfu; uN g  uQ  C1 ; 0 < mi nf ; N g  Q  C1 ; which, along with (126), (127), and (129), gives that as t ! C1 k.t /  N k  ke1 .Qu; Q /Œe.u;  /  e.Nu; N /k C ke1 .Qu; Q /eu .Qu; Q /.u.t /  uN /k  1 ke.u;  /  e.Nu; N /k C C1 1 ku  uN k  C1 .kux .t /k C kx .t /k C kvx .t /k/ ! 0:

(131)

Thus (125) follows from (127) and (128). Noting that  .u;  ; vx / C p.Nu; N / D Œp.u;  /  p.Nu; N / C vx =u; one can derive (126) from (127) and (128) and the mean value theorem. By a similar method to that in Okada and Kawashima [53], (45) follows. The proof is complete. Proof of Lemma 1. By Lemmas 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15, the proof of Lemma 1 is complete. 1 Lemma 16. The unique global generalized solution .u.t /; v.t /; .t // in HC defines 1 1 a nonlinear C0 semigroup S .t / on HC . Moreover, for any .u0 ; v0 ; 0 / 2 HC , the global strong solution .u.t /; v.t /; .t // to the problem (13), (14), (15), (16), and (17) satisfies 1 .u.t /; v.t /; .t // D S .t /.u0 ; v0 ; 0 / 2 C .Œ0; C1/; HC /;

(132)

u.t / 2 C 1=2 .Œ0; C1/; H 1 /; v.t /; .t / 2 C 1=2 .Œ0; C1/; L2 /:

(133)

Proof. For any t1  0; t > 0, integrating (13) over .t1 ; t / and using Lemma 1, one obtains Z t ku.t /  u.t1 /kH 1  C1 j .kvx k2 C kvxx k2 /d j1=2 jt  t1 j1=2  C1 jt  t1 j1=2 t1

which implies

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2359

u.t / 2 C 1=2 .Œ0; C1/; H 1 /: In the same manner, one can easily prove v.t /; .t / 2 C 1=2 .Œ0; C1/; L2 /. Thus (133) follows. By Lemma 1, it follows that for any t > 0, the operator S .t / W .u0 ; v0 ; 0 / 2 1 1 HC ! .u.t /; v.t /; .t // 2 HC exists and, by the uniqueness of global generalized 1 that for any t1 ; t2 2 Œ0; C1/, solutions, satisfies on HC S .t1 C t2 / D S .t1 /S .t2 / D S .t2 /S .t1 /:

(134)

1 with respect to t > 0; Moreover, by Lemma 1, S .t / is uniformly bounded on HC that is,

kS .t /kL.H 1 ;H 1 /  C1 : C

C

(135)

We first need to verify the continuity of S .t / with respect to the initial data 1 1 in HC for any fixed t > 0. To this end, assume that .u0j ; v0j ; 0j / 2 HC , .uj ; vj ; j / D S .t /.u0j ; v0j ; 0j / .j D 1; 2/, and .u; v;  / D .u1 ; v1 ; 1 /  .u2 ; v2 ; 2 /. Subtracting the corresponding Eqs. (13), (15), and (15) satisfied by .u1 ; v1 ; 1 / and .u2 ; v2 ; 2 /, one obtains ut D vx ;

(136)

vt D pu .u1 ; 1 /ux  .pu .u1 ; 1 /  pu .u2 ; 2 //u2x  p .u1 ; 1 /x     vxx vx u1x v2x u ; .p .u1 ; 1 /  p .u2 ; 2 //2x C 0   u1 u1 u2 x u21 (137) e .u1 ; 1 /t D .e .u1 ; 1 /  .e .u2 ; 2 //2t  .eu .u1 ; 1 /  eu .u2 ; 2 //v2x eu .u1 ; 1 /vx  p.u1 ; 1 /vx  .p.u1 ; 1 /  p.u2 ; 2 //v2x CŒk.u1 ; 1 /x =u1 C .k.u1 ; 1 /=u1  k.u2 ; 2 =u2 //2x x ;

(138)

t D 0 W u D u0 ; v D v0 ;  D 0 ; x D 0; 1 W v D 0; x D 0:

(139)

By Lemma 1, one can verify that for any t > 0 and j D 1; 2, k.uj .t /; vj .t /; j .t //k2H 1 C 2

Z

C kvjt k /. /d   C1 ;

0

t

.kujx k2 C kvj k2H 2 C kjx k2H 1 C kjt .t /k2 (140)

2360

Y. Qin

Here and hereafter in the proof of this lemma, C1 > 0 denotes the universal constant depending only on the H 1 norm of the initial data .u0j ; v0j ; 0j /, and min u0j .x/ x2Œ0;1

and min 0j .x/ .j D 1; 2/, but independent of t . x2Œ0;1

Multiplying (136), (137), and (138) by u; v, and  respectively, adding them up and integrating the result over Œ0; 1, and using Lemma 1, (139) and (140), the Cauchy inequality, the embedding theorem, and the mean value theorem, one deduces that for any small  > 0, p 1 d .ku.t /k2 C kv.t /k2 C k e .u1 ; 1 /.t /k2 / C 2 dt 2

2

Z

1 0



 0 vx2 2 C k.u1 ; 1 /x dx u1

2

 .kvx .t /k C kx .t /k / C C1 H1 .t /.ku.t /k C k.t /k2H 1 /; which, together with Lemma 1, leads to p d .ku.t /k2 C kv.t /k2 C k e .u1 ; 1 /.t /k2 / C C11 .kvx .t /k2 C kx .t /k2 / dt  C1 H1 .t /.ku.t /k2 C k.t /k2H 1 /;

(141)

where, by (140), H1 .t / D k1t .t /k2 C k2t .t /k2 C kv1xx .t /k2 C kv2xx .t /k2 C k1xx .t /k2 C k2xx .t /k2 C 1 satisfies for any t > 0, Z

t

H1 . /d   C1 .1 C t /:

(142)

0

By Lemma 1, (135), the embedding theorem, and the mean value theorem, one gets ˚

kvxx .t /k2  C1 kvt .t /k2 C kvx .t /k2L1 C k.t /k2H 1 C .1 C kv2xx .t /k2 /ku.t /k2H 1 

1 kvxx .t /k2 C C1 .kvt .t /k2 C k.t /k2H 1 / 2 ˚

CC1 kvx .t /k2 C .1 C kv2xx .t /k2 /ku.t /k2H 1 ;

which gives kvxx .t /k2  C1 kvt .t /k2 C C1 H1 .t /.kvx .t /k2 C ku.t /k2H 1 C k.t /k2H 1 /:

(143)

Differentiating (136) with respect to x, multiplying the result by ux , integrating by parts, and using (143), one derives that for any small ı > 0, d kux .t /k2  C1 ıkvt .t /k2 C C1 .ı/H1 .t /.kvx .t /k2 C ku.t /k2H 1 C k.t /k2H 1 /: dt (144)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2361

Multiplying (137) by vt , integrating the result over Œ0; 1, and using Lemma 1, (139), the embedding theorem, and the mean value theorem, one obtains d vx k p .t /k2 C C11 kvt .t /k2  C1 H1 .t /.kvx .t /k2 C ku.t /k2H 1 C k.t /k2H 1 /: dt u1 (145) Similarly to (143), by (138), one derives that kxx .t /k2  C1 kt .t /k2 C C1 H1 .t /.kvx .t /k2 C ku.t /k2H 1 C k.t /k2H 1 /:

(146)

Similarly to (145), multiplying (138) by t and using (139) and (140), one gets d p k k.u1 ; 1 /x .t /k2 CC11 kt .t /k2 C1 H1 .t /.kvx .t /k2 Cku.t /k2H 1 Ck.t /k2H 1 /: dt (147) Adding up (141), (144), (145), and (147), and taking ı > 0 small enough, one concludes d M1 .t /  C1 H1 .t /.kvx .t /k2 C ku.t /k2H 1 C k.t /k2H 1 /  C1 H1 .t /M1 .t / dt (148) where p vx M1 .t / D ku.t /k2 C kux .t /k2 C kv.t /k2 C k p .t /k2 C k e .u1 ; 1 /.t /k2 u1 p C k k.u1 ; 1 /x .t /k2 satisfies C11 .kv.t /k2H 1 C ku.t /k2H 1 C k.t /k2H 1 / M1 .t /  C1 .kv.t /k2H 1 C ku.t /k2H 1 C k.t /k2H 1 /:

(149)

Thus (148), combined with Gronwall’s inequality, (142), and (149), implies for any fixed t > 0,  Z t  ku.t /k2H 1 C kv.t /k2H 1 C k.t /k2H 1  C1 M1 .0/ exp C1 H1 . /d  0



C1 exp.C1 t /.ku0 k2H 1

C kv0 k2H 1 C k0 k2H 1 /:

That is, kS .t /.u01 ; v01 ; 01 /  S .t /.u02 ; v02 ; 02 /kH 1  C1 exp.C1 t /k.u01 ; v01 ; 01 / C

 .u02 ; v02 ; 02 /kH 1

C

(150)

2362

Y. Qin

1 which leads to the continuity of S .t / with respect to the initial data in HC . By (134) 1 and (135), in order to derive (132), it suffices to show that for any .u0 ; v0 ; 0 / 2 HC ,

kS .t /.u0 ; v0 ; 0 /  .u0 ; v0 ; 0 /kH 1 ! 0; as t ! 0C ;

(151)

C

which also yields S .0/ D I

(152)

1 with I being the unit operator on HC . To derive (151), one may choose a function m m m m m sequence .u0 ; v0 ; 0 / which is smooth enough, for example, .um 0 ; v0 ; 0 / 2 1 1C˛ 2C˛ 2C˛ .0; 1/  C .0; 1/  C .0; 1// \ HC for some ˛ 2 .0; 1/, such that .C m m k.um 0 ; v0 ; 0 /  .u0 ; v0 ; 0 /kH 1 ! 0; as m ! C1:

(153)

C

By the regularity results (see also Remark 2 below), it follows that for arbitrary but fixed T > 0, there exists a unique global smooth solution .um .t /; v m .t /;  m .t // 2 1 .C 1C˛ .QT /  C 2C˛ .QT /  C 2C˛ .QT // \ HC ; QT D .0; 1/  .0; T /. This gives for m D 1; 2; 3; : : : m m C k.um .t /; v m .t /;  m .t //  .um 0 ; v0 ; 0 /kH 1 ! 0; as t ! 0 :

(154)

C

Fixing T D 1, by the continuity of the operator S .t /, (150), and (153), for any t 2 Œ0; 1, as m ! C1; m m k.um .t /; v m .t /;  m .t //  .u.t /; v.t /; .t //kH 1 D kS .t /.um 0 ; v0 ; 0 / C

S .t /.u0 ; v0 ; 0 /kH 1

C

m m  C1 k.um 0 ; v0 ; 0 /

.u0 ; v0 ; 0 /kH 1 !0: (155) C

This, together with (153) and (154), implies kS .t /.u0 ; v0 ; 0 /  .u0 ; v0 ; 0 /kH 1 D k.u.t /; v.t /; .t //  .u0 ; v0 ; 0 /kH 1 C

C

m

m

m

 k.u .t /; v .t /;  .t // .u.t /; v.t /; .t //kH 1

C

m m Ck.um .t /; v m .t /;  m .t //  .um 0 ; v0 ; 0 /kH 1

C

m m Ck.um 0 ; v0 ; 0 /

 .u0 ; v0 ; 0 /kH 1 ! 0; C

C

as m ! C1; t ! 0 ;

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2363

1 which gives (151) and (152). Thus S .t / is a C0 semigroup on HC satisfying (132) and (133). The proof is complete.

The next lemma concerns the uniform global (in time) positive lower boundedness (independent of t ) of the absolute temperature  . 1 Lemma 17. If .u0 ; v0 ; 0 / 2 HC , then the global strong solution .u.t /; v.t /; .t // to the problem (13), (14), (15), (16), and (17) satisfies

0 < C11  .x; t /;

8.x; t / 2 Œ0; 1  Œ0; C1/:

(156)

Proof. Estimate (156) is proved by contradiction. If (156) is not true, that is, inf .x; t / D 0; then there exists a sequence .xn ; tn / 2 Œ0; 1  Œ0; C1/

.x;t/2Œ0;1Œ0;C1/

such that as n ! C1, .xn ; tn / ! 0:

(157)

If the sequence ftn g has a subsequence, denoted also by tn , converging to C1, then by the asymptotic behavior results in Lemma 1, it holds that as n ! C1, .xn ; tn / ! N > 0, which contradicts (157). If the sequence ftn g is bounded i.e., there exists a constant M > 0, independent of n, such that for any n D 1;    ; 0 < tn  M . Thus there exists a point .x  ; t  / 2 Œ0; 1  Œ0; M  such that .xn ; tn / ! .x  ; t  / as n ! C1. On the other hand, by (157) and the continuity of solutions in Lemmas 1 and 16, one concludes that .xn ; tn / ! .x  ; t  / D 0 as n ! C1, which contradicts (40). Thus the proof is complete. In what follows, the exponential stability of the C0 semigroup S .t /, that is, (36), is proved. A modified idea in Okada and Kawashima [53] is used to prove (36). Now introducing the density of the gas,  D 1=u; then from (20), (21) and (25), (26), (27) it follows that the entropy  D .1=; / satisfies @=@ D p =2 ;

@=@ D e =:

(158)

Consider the transform A W .;  / 2 D; D f.;  / W  > 0;  > 0g ! .u; / 2 AD; ;

(159)

where u D 1= and  D .1=; /. Owing to the Jacobian [email protected]; /=@.;  /j D e =2  < 0 on D; , there is a unique inverse function  D .u; / as the smooth function of .u; / 2 AD; . (In fact, D; and AD; are bounded domains, e.g., Lemmas 1 and 16). Thus the functions e; p can also be regarded as the smooth functions of .u; /. Denoting by e D e.u; / W e.u; .u; // D e.1=; /; p D p.u; / W p.u; .u; // D p.1=; /; it then follows from (20), (21), (25), (26), (27), and (158), (159) that e; p satisfy

2364

Y. Qin

eu D  p; e D  ; pu D .2 p C  p2 =e /; p D  p =e ; u D  p =e ;  D=e :

(160)

Define the following energy form, E.u; v; / D

v2 @e @e Ce.u; /e.Nu; / N .Nu; /.u N  uN / .Nu; /. N  /; N 2 @u @

(161)

where N D 1=Nu; N D .1=; N N /: The next two lemmas concern the exponential 1 decay of the global strong solution .u.t /; v.t /; .t // in HC (or equivalently, of the i C0 semigroup S .t / on HC ). Lemma 18. The unique global generalized solution .u.t /; v.t /; .t // to the problem (13), (14), (15), (16), and (17) satisfies the estimates: v2 v2 N 2 /  E.u; v; /  N 2 /: CC11 .ju uN j2 Cj j CC1 .ju uN j2 Cj j 2 2

(162)

Proof. By the mean value theorem, there exists a point .Qu; / Q between .u; / and .Nu; / N such that v2 1 E.u; v; / D C 2 2



@2 e @2 e 2 .Qu; /.u Q  uN /.  / N .Q u ; /.u Q  u N / C 2 @u2 @u@  @2 e 2 (163) C 2 .Qu; /. Q  / N @

where uQ D 0 uN C .1  0 /u;

Q D 0 N C .1  0 /; 0  0  1:

It follows from Lemmas 1, 16, and 17 that 0 < C11  uQ  C1 ; implies j

@2 e @2 e @2 e .Qu; /j Q Cj Q  C1 : .Qu; /j Q C j 2 .Qu; /j 2 @u @u@ @

jj Q  C1 which

(164)

Thus (162) and (163) and the Cauchy inequality give E.u; v; / 

v2 C C1 Œ.u  uN /2 C .  / N 2 : 2

(165)

On the other hand, it follows from (160) that euu D pu D 2 p C  p2 =e ; eu D p D u D  p =e ; e D  D =e ;

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2365

which yields the Hessian of e.u; / is positive definite for any u > 0 and  > 0. Thus from (162) one derives E.u; v; /

v2 v2 C mi n .Qu; /Œ.u Q  uN /2 C.  / N 2   CC11 Œ.u  uN /2 C.  / N 2 2 2 (166)

Q C11 / is the smaller characteristic root of the Hessian of e.Qu; /. Q where mi n .Qu; /. Thus the combination of (165) and (166) gives (162). The proof is complete. Lemma 19. There exists a positive constant 10 D 10 .C1 / > 0 such that for any 1 fixed 2 .0; 10 , the global strong solution .u.t /; v.t /; .t // in HC to the problem (13), (14), (15), (16), and (17) satisfies the following estimate. e t .kv.t /k2 C ku.t /  uN k2 C k.t /  N k2 C kux .t /k2 C kx .t /k2 / Z t e  .kux k2 Ckx k2 Ckx k2 Ckvx k2 /. /d   C1 ; 8 t > 0: (167) C 0

Proof. By Eqs. (13), (14), (15), (16), and (17), it is easy to verify that .; v; / satisfies   v2 eC D Œpv C 0 vvx C kx x ; (168) 2 t t D .kx =/x C k.x =/2 C 0 vx2 =:

(169)

Owing to uN t D 0; Nt D 0, it follows from (168), (169), and (13), (14), (15), (16), (17) that N /Œ0 v 2 C k 2 = D Œ0 vvx C k.1  = N /x Et .1=; v; / C .= x x  .p  p.1=; N N //vx ;

(170)

Œ20 .x =/2 =2 C 0 x v=t C 0 p x2 = D  0 p x x =  0 .vvx /x C 0 vx2 :

(171)

Multiplying (170) and (171) by e t and ˇe t , respectively, and then adding the results up, one gets @ N /.0 v 2 Ck 2 =/= Cˇ.0 p 2 =  0 v 2 C0 p x x =/ G.t /Ce t Œ.= x x x x @t D e t ŒE.1=; v; / C ˇ.20 .x =/2 =2 C 0 x v=/ C e t Œ.1  ˇ/0 vvx N /x  .p  p.; Ck.1  = N N //vx where G.t / D e t ŒE.1=; v; / C ˇ.20 .x =/2 =2 C 0 vx =/.

(172)

2366

Y. Qin

Integrating (172) over Œ0; 1  Œ0; t, by Lemmas 1, 16, and 17, Cauchy’s inequality, and Poincaré’s inequality, one deduces that for small ˇ > 0 and for any > 0, Z t

N 2 C kv.t /k2 C k.t /  k N 2 C kx .t /k2 C e  Œkx k2 C kvx k2 e t k.t /  k Ckx k2 ./d   C1 C C1

0

Z

t

e  .kvk2 C k  k N 2 C k  N k2

0

Ckx k2 /. /d  :

(173)

Using Lemmas 1, 16, and 17, (13), (45), (129), and (131), the mean value theorem, and the Poincaré inequality, one has ku.t /  uN k  C1 kux .t /k;

(174)

k.t /  N k  C1 .ke.u;  /  e.Nu; N /k C ku.t /  uN k/  C1 .ke.u;  /  e.Nu; N /k C kux .t /k/ which, combined with (172), gives k.t /  N k  C1 .kux .t /k C kvx .t /k C kx .t /k/:

(175)

Similarly, it holds that C11 ku.t /  uN k  k.t /  k N  C1 ku.t /  uN k; N C ku.t /  uN k/: k.t /  N k  C1 .k.t /  k

(176) (177)

It follows from (172), (173), (174), (175), and (176) that there exists a constant 10 D 10 .C1 / > 0 such that for any fixed 2 .0; 10 , (167) holds. The proof is complete. Lemma 20. There exists a positive constant 1 D 1 .C1 /  10 such that for any 1 fixed 2 .0; 1 ; the global strong solution .u.t /; v.t /; .t // in HC to the problem (13), (14), (15), (16), and (17) satisfies the estimate: t

2

2

Z

t

e  .kvxx k2 C kxx k2 C kvt k2 C kt k2 /. /d 

e .kvx .t /k C kx .t /k / C 0

 C1 ;

8t > 0:

(178)

Proof. By (14) and (15) and Lemmas 15, 16, 17, and Poincaré’s inequality, one gets kvx .t /k  C1 kvxx .t /k; kvt .t /k  C1 .kux .t /k C kx .t /k C kvxx .t /k/; (179) kx .t /k  C1 kxx .t /k; kt .t /k  C1 .kxx .t /k C kvxx .t /k/:

(180)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2367

Multiplying (14) and (15) by e t vxx ; e t xx , respectively, integrating the results over Œ0; 1  Œ0; t, and adding them up, using Young’s inequality, the embedding theorem, and Lemmas 1, 16, 17, and 19, one concludes Z t p p t 2 2 1 e .kvx .t /k C k e x .t /k / C C1 e  .kvxx k2 C k kxx k2 /. /d  Z  C1 C C1

0

t

˚ e  .kux k C kx k C kux kkvx k1=2 kvxx k1=2 /kvxx k

0

C.kvx k C kvx k3=2 kvxx k C kux kkx k1=2 kxx k1=2 /kxx k d  Z t ˚

e  kvx k2 C kux k2 C kx k2 C .kvx k C kt k/kx k1=2 kxx k1=2 . /d  CC1 0

Z

t

e  .kvxx k2 C kxx k2 /. /d 

 C1 C 1=.2C1 / 0

which, together with Lemmas 1, 16, 17, and 19, Eqs. (13), (14), (15), and (179), (180), gives (178). The proof is complete. We have now completed the proof of Theorem 1.

2.2

Global Existence and Exponential Stability of Strong Solutions in H2

In this subsection, we establish the global existence and exponential stability of strong solutions in H 2 . Theorem 2. Assume that e; p; , and k are C 3 functions satisfying (19), (20), (21), (25), (26), (27), and (30), (31), (32), (33), (34), (35) on 0 < u < C1 and 0   < C1, and q; r satisfy (28)–(29). Then there exists a unique global generalized 2 solution .u.t /; v.t /; .t // in HC to the problem (13), (14), (15), (16), and (17) which defines a nonlinear C0 semigroup S .t / (also denoted by S .t / by the uniqueness of 1 2 2 solution in HC ) on HC . Moreover, for any .u0 ; v0 ; 0 / 2 HC , there exists a constant 2 D 2 .C2 / > 0 such that for any fixed 2 .0; 2  and for any t > 0, the following inequality holds, k.u.t /; v.t /; .t //  .Nu; 0; N /k2H 2 D kS .t /.u0 ; v0 ; 0 /  .Nu; 0; N /k2H 2  C2 e  t ; C

C

(181) 2 which implies that the semigroup S .t / decays exponentially on HC . 2 Remark 2. Recall that the global strong solution .u.t /; v.t /; .t // 2 HC obtained 2 in Theorem 2 is not the classical one. Indeed, if .u0 ; v0 ; 0 / 2 HC , by the 1 embedding theorem, there holds that u0 ; v0 ; 0 2 C 1C 2 .0; 1/. If one imposes

2368

Y. Qin

on the higher regularities of v0 ; 0 2 C 2C˛ .0; 1/; ˛ 2 .0; 1/, the following results on the global existence of classical (smooth) solutions are obtained in Qin [55– 58, 60]. If in addition to the assumptions in Theorem 2, one may further assume that u0 2 C 1C˛ .0; 1/; v0 ; 0 2 C 2C˛ .0; 1/; ˛ 2 .0; 1/ and the compatibility conditions ut jxD0;1 D vt jxD0;1 D t jxD0;1 D 0 hold, then the global strong 2 solution .u.t /; v.t /; .t // 2 HC obtained in Theorem 2 is the classical one ˛ ˛ satisfying u.x; t / 2 C 1C˛;1C 2 .QT /; v.x; t /; .x; t / 2 C 2C˛;2C 2 .QT / for any T > 0; QT D .0; 1/  .0; T /. Moreover, if the initial data possess the higher regularities, then the global strong solutions also possess the higher regularities. 2 Therefore the global strong solution .u.t /; v.t /; .t // in HC can be understood as a global strong solution between the global classical solution and the global 1 generalized solution .u.t /; v.t /; .t // in HC . To prove Theorem 2, it suffices to establish the estimates of solution .u; v;  / in H 2 . First, estimate kvt k; kt k and then by Eqs. (14) and (15) to obtain estimates of kvxx k; kxx k yielding the estimates in H 2 . Second, estimate u in H 2 . To do so, the key estimate (194) is needed and derived from Eqs. (13) and (14), and hence the global existence in H 2 follows. Third, to study the universal (global) attractors in H 2 , the existence of C0 semigroup S .t / on H 2 (see Lemma 24) needs to be shown. Fourth, in order to prove the exponential stability of .v;  / in H 2 around the equilibrium .0; N /, first estimate .vt ; t / in L2 , then derive the exponential stability of .vxx ; xx / in L2 by Eqs. (14) and (15), eventually yielding the exponential stability of .v;   N / in H 2 (see Lemma 25). Finally, to derive the exponential stability of u in H 2 , just multiply (195) by an exponential factor to get the exponential stability of uxx in L2 , which immediately implies the exponential stability of u  uN in H 2 (see Lemma 26). Lemma 21. Under the assumptions in Theorem 2, the problem (13), (14), (15), 2 (16), and (17) admits a unique global strong solution .u.t /; v.t /; .t // in HC , which defines a nonlinear C0 semigroup S .t / (also denoted by S .t / by the uniqueness of 1 2 2 the solution in HC ) on HC such that for any .u0 ; v0 ; 0 / 2 HC ,

kS .t /.u0 ; v0 ; 0 /kH 2 D k.u.t /; v.t /; .t //kH 2  C2 ; 8t > 0: C

C

2 S .t /.u0 ; v0 ; 0 / D .u.t /; v.t /; .t // 2 C .Œ0; C1/; HC /;

(182) (183)

u.t / 2 C 1=2 .Œ0; C1/; H 2 /; v.t /; .t / 2 C 1=2 .Œ0; C1/; H 1 /:

(184)

To prove Lemma 21, it suffices to use the following lemmas. The next two lemmas are concerned with the uniform estimates of .v;  / in H 2 : 2 Lemma 22. For any .u0 ; v0 ; 0 / 2 HC , the following estimates hold.

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2

Z

2

t

.kvxt k2 C kxt k2 /. /d   C2 ; 8t > 0; (185)

kt .t /k C kvt .t /k C kvxx .t /k2 C kxx .t /k2 C

Z

2369

0 t

.kvxxx k2 C kxxx k2 /. /d   C2 ; 8t > 0: (186)

0

Proof. Differentiating (14) with respect to t, multiplying the result by vt , and integrating the result over .0; 1/, one has that d 1 kvt .t /k2 CC11 kvxt .t /k2  kvxt .t /k2 CC1 .kvx .t /k2 Ckvx .t /k4L4 Ckt .t /k2 / dt 2C1 

1 kvxt .t /k2 C C1 .kvxx .t /k2 C kt .t /k2 / 2C1

which, together with Lemma 1, yields kvt .t /k2 C

Z

t

kvxt . /k2 d   C2 C C1 0

Z

t

.kvxx k2 C kt k2 /. /d   C2 :

(187)

0

On the other hand, using Eq. (14), Lemmas 1, 16, and 17, Sobolev’s embedding theorem, and Young’s inequality, one has kvxx .t /k  C1 .kvt .t /k C kx .t /k C kux .t /k C kvx .t /k1=2 kvxx .t /k1=2 /  Z

1 kvxx .t /k C C1 .kvt .t /k C 1/; 2

t

kvxxx . /k2 d   C2 ; 0

which leads to kvxx .t /k  C2 ;

kvx .t /kL1  C2 ;

8t > 0:

(188)

Similarly, kxx .t /k  C1 .kt .t /k C 1/; 8t > 0:

(189)

Similarly to (187), by Eq. (15), one derives that for any ı1 > 0, ˚ d p k e t .t /k2 C C11 kxt .t /k2  ı1 kxt .t /k2 C C1 kx .t /k2 C kvx .t /k2 dt Ckt .t /k3L3 C kt .t /k2 C kvxt .t /k2 C .kt .t /k

Ckt .t /k1=2 ktx .t /k1=2 /ktx .t /k : (190)

2370

Y. Qin

Integration of (190) gives, for all ı1 > 0, kt .t /k2 C

Z

t

kxt . /k2 d   C2 C C1 ı1 0

Z CC1

Z

t

ktx . /k2 d  0

t

.kt . /k5=2 ktx . /k1=2 C kt . /k3 /d 

0

Z

t

ktx . /k2 d  C C1 sup kt . /k4=3

 C 2 C C1 ı1

0 t

0

Z

t

ktx . /k2 d  C

 C2 C C1 ı1 0

1 sup kt . /k2 : 2 0 t

That is, sup kt . /k2 C

Z

0 t

t

kxt . /k2 d   C2 CC1 ı1 0

Z

t

ktx . /k2 d  C

0

1 sup kt . /k2 ; 2 0 t

which, by taking ı1 > 0 small enough, implies 2

Z

t

kxt . /k2 d   C2 ;

sup kt . /k C 0t

8t > 0:

(191)

0

By (15) and (189), one easily gets kxx .t /k2 C

Z

t

kxxx . /k2 d   C2 ;

8t > 0:

(192)

0

Thus (185) and (186) follow from (187), (188), (189), (190), (191), and (192) and Lemma 1. The proof is complete. The following shows the uniform estimate of the specific volume u in H 2 . 2 , the following estimate holds. Lemma 23. For any .u0 ; v0 ; 0 / 2 HC

ku.t /kH 2  C2 ; 8t > 0:

(193)

Proof. Differentiating (14) with respect to x, using (13) (utxx D vxxx ), it follows that 0 with

@  uxx  pu uxx D vtx C E.x; t /; @t u

(194)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2371

E.x; t/ D .puu u2x C 2pu x ux C p x2 / C p xx  20 vx u2x =u3 C 20 ux vxx =u2 : Multiplying (194) by uxx =u, and by Young’s inequality, Lemmas 1, 16, and 17, and (50), one can deduce that d uxx uxx k .t /k2 C C11 k .t /k2 dt u u 1 uxx 2 k CC1 .kx .t /k4L4 Ckux .t /k4L4 Ckvxt .t /k2 Ckxx .t /k2 Ckvx u2x .t /k2 /  k 4C1 u 

1 uxx .t /k2 C C2 .kxx .t /k2 C kux .t /k2 C kvxt .t /k2 /; k 2C1 u

(195)

which, combined with Lemmas 1 and 22, gives 2

Z

t

kuxx . /k2 d   C2 ;

kuxx .t /k C

8t > 0:

(196)

0

Thus (193) follows from Lemma 1 and (196). The proof is complete. The estimate (182) and the global existence of strong solution .u.t /; v.t /; .t // 2 2 HC follow from Lemma 1 and Lemmas 22 and 23. Similarly to (133), it is easy to prove that the relation (184) holds. To complete the proof of Lemma 21, it suffices 2 to prove (182) and the continuity of S .t / with respect to .u0 ; v0 ; 0 / 2 HC , which 2 also leads to the uniqueness of the global strong solutions in HC . This is done in the next lemma. 2 Lemma 24. The global strong solution .u.t /; v.t /; .t // in HC defines a nonlinear 2 C0 semigroup S .t / on HC . 2 1 Proof. The uniqueness of global strong solutions in HC follows from that in HC . 2 Thus S .t / satisfies (134) on HC and by Lemmas 22 and 23, kS .t /kL.H 2 ;H 2 /  C2 . C C In the same manner as in the proof of Lemma 16, assume that .u0j ; v0j ; 0j / 2 2 HC ; .j D 1; 2/, .uj ; vj ; j / D S .t /.u0j ; v0j ; 0j /, and .u; v;  / D .u1 ; v1 ; 1 /  .u2 ; v2 ; 2 /. Denote by e j D e.uj ; j /; p j D p.uj ; j /; k j D k.uj ; j /; .j D 1; 2/. Subtracting the corresponding Eqs. (13), (14), and (15) satisfied by .u1 ; v1 ; 1 /, and .u2 ; v2 ; 2 /, one obtains Eqs. (136), (137), (138), and (139). Similarly to (143), one has

kxx .t /k2  C1 .kt .t /k2 C H1 .t /M1 .t //  C2 .kt .t /k2 C ku.t /k2H 1 C kv.t /k2H 1 C k.t /k2H 1 /: Differentiating (137) with respect to x, it holds that

(197)

2372

Y. Qin

vtx D 0 .vxxx =u1  2vxx u1x =u21 / C R.x; t /;

(198)

where 1 1 1 1 2 R.x; t / D .puu u1x Cpu 1x /ux pu1 uxx .pu1 pu2 /u2xx Œpuu ux C .puu  puu /u2x 1 1 2 1 1 Cpu x C .pu  pu /2x u2x  .pu u1x C p 1x /x  p1 xx 1 1 2 1 1 .p1  p2 /2xx  Œpu ux C .pu  pu /u2x C p x C .p 2 p /2x 2x  0 .vx u1xx =u21 C 2vx u21x =u31 /;

with p j D p.uj ; j / .j D 1; 2/: By Lemmas 1, 16, 17, and 21, 22, 23, the embedding theorem, and the mean value theorem, one can easily obtain kR.t /k2  C2 .kux .t /k2 C kuxx .t /k2 C ku.t /k2L1 C k.t /k2L1 C kx .t /k2 Ckxx .t /k2 C kvx .t /k2L1 /  C2 .ku.t /k2H 2 C k.t /k2H 2 /;

(199)

Here and hereafter in the proof of this lemma, C2 > 0 denotes the universal constant depending only on the H 2 norm of the initial data .u0j ; v0j ; 0j / and min u0j .x/ x2Œ0;1

and min 0j .x/ .j D 1; 2/, but independent of t . x2Œ0;1

By (198) and (199) and the embedding theorem, there holds that kvxxx .t /k2  C1 kvtx .t /k2 C C2 .kvxx .t /k2L1 C kR.t /k2 / 

1 kvxxx .t /k2 CC1 kvtx .t /k2 C C2 .kvxx .t /k2 Cku.t /k2H 2 C k.t /k2H 2 / 2

which gives kvxxx .t /k2  C1 kvtx .t /k2 C C2 .kvxx .t /k2 C ku.t /k2H 2 C k.t /k2H 2 /: (200) Differentiating (136) twice with respect to x, multiplying the result by uxx , integrating the resulting equation over Œ0; 1, using (200) and the Cauchy inequality, one can obtain d kuxx .t /k2  C1 kvtx .t /k2 C C2 .ku.t /k2H 2 C kvxx .t /k2 C k.t /k2H 2 / dt  C1 kvtx .t /k2 C C2 .ku.t /k2H 2 C kv.t /k2H 2 C k.t /k2H 2 /: (201) Differentiating (137) with respect to t , multiplying it by vt , integrating the resulting equation over Œ0; 1, using Lemmas 1, 16, 17, and 21, 22, 23, one deduces that

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2373

d kvt .t /k2 C C11 kvtx .t /k2  C2 .1 C kv2xt .t /k2 /.kvt .t /k2 C kt .t /k2 C ku.t /k2H 1 dt Ckv.t /k2H 1 C k.t /k2H 1 /:

(202)

Multiplying (15) by e1 , differentiating the resulting equation with respect to t , one arrives at t t D I1 .u; v;  / C I2 .u; v;  / C I3 .u; v;  / C I4 .u; v;  / C I5 .u; v;  /

(203)

where I1 .u; v;  / D .et =e2 /.kx =u/x ; I2 .u; v;  / D .kx =u/xt =e ; I3 .u; v;  / D p vx et =e2 ; I4 .u; v;  / D .t p vx C  pt vx C  p vxt /=e ; I5 .u; v;  / D 0 Œ2vx vxt =e u  vx2 .et u C e vx /=e2 u2 : Denote by j

Ii D Ii .uj ; vj ; j /;

j D 1; 2;

i D 1; 2; 3; 4; 5:

By Lemmas 1, 16, 17, and 21, 21, 23, (143), the embedding theorem, and the mean value theorem, one derives that for .u; v;  / D .u1  u2 ; v1  v2 ; 1  2 /, kI11  I12 k2  C2 .1Ck1xxx .t /k2 /.ku.t /k2H 1 Ckv.t /k2H 1 C k.t /k2H 1 C kt .t /k2 /; (204) kI31



I32 k2



C2 .ku.t /k2H 1

C

kv.t /k2H 1

C

k.t /k2H 1

2

2

C kt .t /k C kvxx .t /k /

 C2 .ku.t /k2H 1 C kv.t /k2H 1 C k.t /k2H 1 C kt .t /k2 C kvt .t /k2 /; (205) kI41  I42 k2  C2 kvxt .t /k2 C C2 .1 C kv1xt .t /k2 /.ku.t /k2H 1 C kv.t /k2H 1 C k.t /k2H 1 Ckt .t /k2 C kvt .t /k2 /; kI51



I52 k2

2

(206) 2

 C2 kvxt .t /k C C2 .1 C kv2xt .t /k

/.ku.t /k2H 1

C kv.t /k2H 1

C k.t /k2H 1

Ckt .t /k2 C kvt .t /k2 /:

(207)

Subtracting the corresponding Eqs. (203) satisfied by .u1 ; v1 ; 1 / and .u2 ; v2 ; 2 /, respectively, multiplying the resulting equation by t D .1  2 /t , and using (204), (205), (206), and (207), one easily obtains that d kt .t /k2  dt

Z 0

1

.I21  I22 /t dx C C2 kvxt .t /k2 CC2 .1 C k1xxx .t /k2 C kv1xt .t /k2

Ckv2xt .t /k2 /.ku.t /k2H 1 C kv.t /k2H 1 C k.t /k2H 1 C kt .t /k2 C kvt .t /k2 /:

(208)

2374

Y. Qin

In (208), using (139) and integration by parts, the first term on the right-hand side can be estimated as follows for any small ı2 > 0: Z

1 0

.I21  I22 /t dx D 

Z

Z

1 0

2 k 1 xt =e1 u1 dx C J1 C J2 C J3 ;

(209)

1

J1 D  0

.tx =e1 /Œkt1 x =u1  k 1 x v1x =u21 C ..u2 2x .k 1 k 2 /  k 2 u2x /=u1 u2 /t dx

 ı2 ktx .t /k2 C C2 .1 C k2xt .t /k2 /.ku.t /k2H 1 C kv.t /k2H 1 C k.t /k2H 1 Ckt .t /k2 /; Z 1 1 .ex t =.e1 /2 /Œ.k 1 tx C kt1 x /=u1 C k 1 x v1x =u21 dx J2 D

(210)

 ı2 ktx .t /k2 C C2 .k.t /k2H 1 C kt .t /k2 /; Z 1 1 .ex t =.e1 /2 /Œ.u2 2x .k 1  k 2 /  k 2 u2x /=u1 u2 t dx J3 D

(211)

0

0

 C2 .1 C k2xt .t /k2 /.ku.t /k2H 1 C kv.t /k2H 1 C k.t /k2H 1 C kt .t /k2 /;

(212)

where k j D k.uj ; j /; e j D e.uj ; j /; j D 1; 2: Taking ı2 > 0 small enough in (210) and (211), using Lemmas 1, 16 and 17, and inserting (209), (210), (211), and (212) into (208), one concludes d kt .t /k2 C C11 ktx .t /k2  C1 kvtx .t /k2 C C2 H2 .t /.kvt .t /k2 C kt .t /k2 dt Cku.t /k2H 1 C kv.t /k2H 1 C k.t /k2H 1 /;

(213)

where, by Lemma 22, H2 .t / D 1 C k1xxx .t /k2 C k2xt .t /k2 C kv1xt .t /k2 C kv2xt .t /k2 satisfies Z t H2 . /d   C2 .1 C t /; 8t > 0: (214) 0

Similarly to (143) and (197), one easily obtains from (169) and (170), kvt .t /k2  C2 .kvxx .t /k2 C ku.t /k2H 1 C kv.t /k2H 1 C k.t /k2H 1 /;

(215)

kt .t /k2  C2 .kxx .t /k2 C ku.t /k2H 1 C kv.t /k2H 1 C k.t /k2H 1 /:

(216)

Now multiplying (202) by a large number N2 > 2C12 , then adding up the result, (201), and (213), one concludes d M2 .t /  C2 H2 .t /.kvt .t /k2 C kt .t /k2 C ku.t /k2H 2 C k.t /k2H 1 C kv.t /k2H 1 / dt (217)  C2 H2 .t /.M1 .t / C M2 .t //;

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2375

where M2 .t / D kuxx .t /k2 C N2 kvt .t /k2 C kt .t /k2 : Adding (179) to (217) gives d M3 .t /  C2 H2 .t /M3 .t /; dt

(218)

where, by (143), (146), (149), and (215), (216), M3 .t / D M1 .t / C M2 .t / satisfies C21 .ku.t /k2H 2 C kv.t /k2H 2 C k.t /k2H 2 /  M3 .t /  C2 .ku.t /k2H 2 C kv.t /k2H 2 Ck.t /k2H 2 /:

(219)

Thus it follows from (218), Gronwall’s inequality, (215), and (219) that ku.t /k2H 2

C

kv.t /k2H 2

C

k.t /k2H 2



 C2 M3 .t /  C2 M3 .0/ exp C2

Z

t

H2 . /d 



0

C2 exp.C2 t /.ku0 k2H 2



C

kv0 k2H 2

C k0 k2H 2 /;

8t > 0; 2 . Similarly which implies the continuity of S .t / with respect to the initial data in HC to the proof of (132), one can prove that (182) holds. Thus the proof is complete.

Proof of Lemma 21. From Lemmas 22, 23, 24, the proof of Lemma 21 follows immediately. The next two lemmas concern the exponential decay of global strong solution 2 2 .u.t /; v.t /; .t // in HC (or equivalently, of semigroup S .t / on HC ). Lemma 25. There exists a positive constant 20 D 20 .C2 /  1 such that for any 2 fixed 2 .0; 20 , the global strong solution .u.t /; v.t /; .t // in HC to the problem (13), (14), (15), (16), and (17) satisfies the following estimate e .k.t /  N k2H 2 C kv.t /k2H 2 / C t

Z

t

e  .kvxt k2 C kxt k2 /. /d   C2 ; 0

8t > 0: (220)

Proof. Differentiating equation (15) with respect to t, multiplying the result by vt e t , and integrating the resulting equation over Œ0; 1  Œ0; t, by Lemmas 1 and 16, 17, 18, 19, 20 and Young’s inequality, one easily concludes Z t Z t p 1 t e kvt .t /k2 C 0 e  kvxt = uk2 . /d   C2 C =2 e  kvt k2 . /d  2 0 0 Z t Z t p e  .kvx k2 C kt k2 C kvx k4L4 /. /d  C 0 =2 e  kvxt = uk2 . /d  CC1 0

0

2376

Y. Qin

Z

t

p



 C2 C .C2 C 0 =2/

Z

2

t

e  .kt k2 C kvx k2

e kvxt = uk . /d  C C1 0

0

Ckvxx k2 /. /d  which, combined with Lemma 1, Lemmas 19, 20, and (188), implies that there exists a constant 20 D 20 .C2 /  1 such that for any fixed 2 .0; 20 ; t

2

2

Z

t

e  kvxt . /k2 d   C2 ; 8t > 0:

e .kvt .t /k C kvxx .t /k / C

(221)

0

In the same manner, multiplying (203) by t e t , integrating the result over Œ0; 1  Œ0; t, and using Lemmas 1, 16,17, and (189), it follows that e t .kt .t /k2 C kxx .t /k2 / C

Z

t

e  kxt . /k2 d   C2 ; 0

which, together with (221) and Lemmas 18 and 19, yields (220). The proof is complete. Lemma 26. There exists a positive constant 2 D 2 .C2 /  20 such that for any 2 fixed 2 .0; 2 , the global solution .u.t /; v.t /; .t // in HC to the problem (13), (14), (15), (16), and (17) satisfies ku.t /  uN k2H 2  C2 e  t :

(222)

Proof. Multiplying (195) by e t=2C1 and choosing so small that  min. 20 ; 1=4C1 / D 2 .C2 /, and using Lemmas 19, 20, and 25, one concludes that kuxx .t /k2  C2 e t=2C1 C C2 e  t  C2 e  t ; which, together with Lemmas 19 and 20, gives (222). The proof is complete. We have now completed the proof of Theorem 2.

2.3

Global Existence and Exponential Stability of Strong Solutions in H4

This subsection further introduces the global existence and exponential stability of strong solutions in H 4 to problem (13), (14), (15), (16), and (17). Because the higher regularity arguments in H 4 follow the same lines as those in Sects. 3.1 and 3.2, one only sketches them.

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2377

Theorem 3. Assume that e; p; , and k are C 5 functions satisfying (19), (19), (21), (25), (26), (27), and (30), (31), (32), (33), (34), (35) on 0 < u < C1 and 0   < 4 C1, and q; r satisfy assumptions (28) and (29). Then for any .u0 ; v0 ; 0 / 2 HC , 4 there exists a unique global strong solution .u.t /; v.t /; .t // 2 C .Œ0; C1/I HC / to problem (13), (14), (15), (13), and (17) satisfying that for any .x; t / 2 Œ0; 1  Œ0; C1/; 0 < C11  .x; t /  C1 ; 0 < C11  u.x; t /  C1 ;

(223)

and for any t > 0, ku.t /Nuk2H 4 Cku.t /  uN k2W 3;1 Ckut .t /k2H 3 Ckut t .t /k2H 1 Ckv.t /k2H 4 Ckv.t /k2W 3;1 N 2 4 C k.t /  N k2 3;1 C kt .t /k2 2 Ckvt .t /k2H 2 C kvt t .t /k2 C k.t /  k H W H Ckt t .t /k2  C4 ; (224) Z t .ku  uN k2H 4 C ku  uN k2W 3;1 Ckut k2H 4 Ckut t k2H 2 Ckut t t k2 Ckvk2H 5 Ckv.t /k2W 4;1 0

Ckvt k2H 3 C kvt t k2H 1 C k  N k2H 5 C k.t /  N k2W 4;1 C kt k2H 3 Ckt t k2H 1 /. /d   C4 :

(225)

4 Moreover, the global strong solution .u.t /; v.t /; .t // 2 HC defines a nonlinear 4 4 C0 semigroup S .t / on HC that maps HC into itself and satisfies that for any 4 , .u0 ; v0 ; 0 / 2 HC 4 /; S .t /.u0 ; v0 ; 0 / D .u.t /; v.t /; .t // 2 C .Œ0; C1/I HC

(226)

and S .t / is continuous with respect to initial data; that is, kS .t /.u01 ; v01 ; 01 /  S .t /.u02 ; v02 ; 02 /kH 4  C4 k.u01 ; v01 ; 01 / C

.u02 ; v02 ; 02 /kH 4 ; (227) C

where .uj .t /; vj .t /; j .t // .j D 1; 2/ is the unique global strong solution with 4 4 initial datum .u0j ; v0j ; 0j / 2 HC .j D 1; 2/. Finally, for any .u0 ; v0 ; 0 / 2 HC , there are constants C4 > 0 and 4 D 4 .C4 / > 0 such that for any fixed 2 .0; 4 , the following estimates hold for any t > 0. ku.t /  uN k2H 4 C ku.t /  uN k2W 3;1 C kut .t /k2H 3 C kut t .t /k2H 1 C kv.t /k2H 4 Ckv.t /k2W 3;1 C kvt .t /k2H 2 C kvt t .t /k2 C k.t /  N k2H 4 C k.t /  N k2W 3;1 Ckt .t /k2H 2 C kt t .t /k2  C4 e  t ;

(228)

2378

Z 0

Y. Qin

t

e  .ku  uN k2H 4 C ku  uN k2W 3;1 C kut k2H 4 C kut t k2H 2 C kut t t k2

]

Ckvk2H 5

N 2 5 C k.t /  N k2 4;1 C kt k2 3 Ckv.t /k2W 4;1 C kvt k2H 3 C kvt t k2H 1 C k  k H W H Ckt t k2H 1 /. /d   C4 :

(229)

Corollary 3. Under assumptions of Theorem 3, estimate (228) implies that semi4 ; that is, for any fixed 2 .0; 4  and for group S .t / is exponentially stable on HC any t > 0, k.u.t /; v.t /; .t //  .Nu; 0; N /k2H 4 D kS .t /.u0 ; v0 ; 0 /  .Nu; 0; N /k2H 4  C4 e  t : C

C

(230) Moreover, .u.t /; v.t /; .t // is the classical solution verifying that for any fixed 2 .0; 4  and for any t > 0, k.u.t /  uN ; v.t /; .t /  N /k2C 3C1=2 C 3C1=2 C 3C1=2  C4 e  t :

(231)

To show Theorem 3, it suffices to establish the estimates of .u; v;  / in H 3 and H 4 , respectively. First, estimate initial terms kvtx .x; 0/k; ktx .x; 0/k and kvt t .x; 0/k; kt t .x; 0/k by using Eqs. (13), (14), and (15) (see Lemma 27). Second, to derive the estimates on .v;  / in H 4 , first estimate terms vxt ; xt and vt t ; t t in L2 which yield estimates vxxx ; xxx and vxxxx ; xxxx in L2 , respectively, by Eqs. (14) and (15) (see Lemmas 27 and 28). For this process, it is necessary to use the delicate interpolation techniques to deal with boundary terms of vxt ; xt at endpoints x D 0; 1 and eventually to establish estimates of .v;  / in H 4 by choosing a suitable small parameter  > 0. Fourth, to get estimates of uxxx ; uxxxx in L2 , use (194) to obtain (242) and (243) and hence obtain the estimate on u in H 4 (see Lemmas 29 and 30). Fifth, to study the global attractors, it is necessary to prove the existence of the C0 semigroup S .t / on H 4 (see Lemma 31). Sixth, to show the exponential stability of solutions in H 4 , only go along the same line of the proof of global existence of solutions in H 4 just multiplying the exponential function e t with small > 0 (see Lemmas 32 33, 34). The next lemma concerns initial terms of kvtx .x; 0/k; ktx .x; 0/k; kvt t .x; 0/k; kt t .x; 0/k which are needed in obtaining estimates in H 4 . 4 and for Lemma 27. Under assumptions of Theorem 3, for any .u0 ; v0 ; 0 / 2 HC any t > 0, and  2 .0; 1/ small enough, we have

kvtx .x; 0/k C ktx .x; 0/k  C3 ;

(232)

kvt t .x; 0/k C kt t .x; 0/k C kvtxx .x; 0/k C ktxx .x; 0/k  C4 ;

(233)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

Z

2

2

kvt t .t /k C kt t .t /k2 C

Z

t

t

ktxx . /k2 d  ;

kvt tx . /k d   C4 C C4 Z

0

0 t

kt tx . /k2 d   C4  3 C C2  1

Z

0

Z

2379

(234)

t

ktxx . /k2 d  0

t

.kvt tx k2 C kvtxx k2 /. /d  :

CC1 

(235)

0

Proof. See Qin [59, 60] for details. The next lemma derives estimates in H 3 by first establishing estimates of .vtx ; tx / in L2 , then estimates of .vxxx ; xxx / in L2 . 4 Lemma 28. Under the assumptions of Theorem 3, for any .u0 ; v0 ; 0 / 2 HC , the following estimates hold for any t > 0 and for any  2 .0; 1/ small enough, Z t Z t kvtxx . /k2 d   C3  6 CC1  2 .ktxx k2 Ckvt tx k2 /. /d  ;(236) kvtx .t /k2 C 0

ktx .t /k2 C

Z

0

t

ktxx . /k2 d   C3  6 CC2  2

0

Z

t

.kvtxx k2 C kt tx k2

0

Ckxxx k2 ktx k2 /. /d  :

(237)

Proof. See Qin [59, 60] for details. The next result is concerned with estimates of .v;  / in H 4 by first deriving kvtx k; ktx k; kvt t k; kt t k, then using Eqs. (13), (14), and (15) to obtain kvxxx k; kxxx k; kvxxxx k; kxxxx k. To get the estimate of u in H 4 , two key equalities (242) and (243) are required. 4 Lemma 29. Under the assumptions of Theorem 3, for any .u0 ; v0 ; 0 / 2 HC , there holds that for any t > 0, Z t 2 2 2 2 .kvt tx k2 C kvtxx k2 kvt t .t /k C kvtx .t /k C kt t .t /k C ktx .t /k C 0 2

2

Ckt tx k C ktxx k /. /d   C4 ; (238) Z t .kuxxx k2H 1 C kuxx k2W 1;1 /. /d   C4 ; (239) kuxxx .t /k2H 1 C kuxx .t /k2W 1;1 C 0

kvxxx .t /k2H 1

C kxxx .t /k2H 1 C kxx .t /k2W 1;1 C kutxxx .t /k2 Z t Ckvtxx .t /k2 C ktxx .t /k2 C .kvt t k2 C kt t k2 C kvxx k2W 2;1 C kxx k2W 2;1 C

kvxx .t /k2W 1;1

0

Cktxx k2H 1 C kvtxx k2H 1 C ktx k2W 1;1 C kvtx k2W 1;1 C kutxxx k2H 1 /. /d   C4 ; (240)

2380

Z

Y. Qin

t

.kvxxxx k2H 1 C kxxxx k2H 1 /. /d   C4 :

0

(241)

Proof. Differentiating (192) with respect to x, and using (13), one gets 0

@  uxxx  pu uxxx D E1 .x; t / @t u

(242)

with E1 .x; t / D vtxx C Ex .x; t / C pux uxx C 0

u u xx x : u2 t

Similarly, differentiating (242) with respect to x, one finds 0

@  uxxxx  pu uxxxx D E2 .x; t / @t u

(243)

where E2 .x; t / D E1x .x; t / C pux uxxx C 0

@  uxxx ux : @t u2

For the remaining part of the proof, see Qin [59, 60] for details. The next lemma summarizes estimates of solution .u; v;  / in H 4 . 4 , one Lemma 30. Under the assumptions of Theorem 3, for any .u0 ; v0 ; 0 / 2 HC has, for any t > 0,

ku.t /  uN k2H 4 C kut .t /k2H 3 C kut t .t /k2H 1 C kv.t /k2H 4 C kvt .t /k2H 2 C kvt t .t /k2 Z t 2 2 2 N Ck.t /   kH 4 C kt .t /kH 2 C kt t .t /k C .ku  uN k2H 4 C kvk2H 5 C kvt k2H 3 0

Ckvt t k2H 1 Z

0

C k  N k2H 5 C kt k2H 3 C kt t k2H 1 /. /d   C4 ;

(244)

t

.kut k2H 4 C kut t k2H 2 C kut t t k2 /. /d   C4 :

(245)

Proof. Exploiting (47), Lemmas 27, 28, and 29, and Theorems 1 and 2, one can easily obtain (244) and (245). The proof is complete. By Lemmas 27, 28, 29, and 30, one has proved the global existence of strong 4 solutions to problem (13), (14), (15), (16), and (17) in HC with arbitrary initial 4 4 datum .u0 ; v0 ; 0 / 2 HC and the uniqueness of strong solutions in HC follows from 1 2 that of strong (generalized) solutions in HC or in HC .

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2381

To study the universal attractors in H 4 , one needs to prove the existence of a 4 . This is done in the next lemma. nonlinear C0 semigroup S .t / on HC 4 Lemma 31. The global strong solution .u.t /; v.t /; .t // in HC to problem (13), 4 (14), (15), (16), and (17) defines a nonlinear C0 semigroup S .t / on HC (also 1 and denoted by S .t / by the uniqueness of a strong (generalized) solution in HC 2 4 HC ) such that for any .u0 ; v0 ; 0 / 2 HC , we have

kS .t /.u0 ; v0 ; 0 /kH 4 D k.u.t /; v.t /; .t //kH 4  C4 ; 8t > 0;

(246)

4 S .t /.u0 ; v0 ; 0 / D .u.t /; v.t /; .t // 2 C .Œ0; C1/I HC /; 8t > 0:

(247)

C

C

Proof. See Qin [59, 60] for details. Now one begins to study the exponential stability of .v;  / in H 4 . 4 Lemma 32. Under the assumptions of Theorem 3, for any .u0 ; v0 ; 0 / 2 HC , there .1/ .1/ exists a positive constant 4 D 4 .C4 /  2 .C2 / such that for any fixed 2 .1/ .0; 4 , the following estimates hold for any t > 0 and  2 .0; 1/ small enough,

e t kvt t .t /k2 C e t kt t .t /k2 C

Z Z

t

e  kvt tx . /k2 d   C4 C C4

0 t

e  kt tx . /k2 d   C1  0

CC4 

3

C C2 

1

Z

t

e  ktxx . /k2 d  ;

(248)

0

Z

t

e  .kvtxx k2 C kvt tx k2 /. /d  0

Z

t

e  ktxx . /k2 d  :

(249)

0

Proof. The proofs of (248) and (249) are basically the same as those of (234) and (235). The difference here is to estimate (248) and (249) with a weighted exponential function e t . For the remaining part of the proof, see Qin [59, 60] for details. The next result derives the exponential stability of .v;  / in H 3 by estimating vtx ; tx in L2 . 4 Lemma 33. Under the assumptions of Theorem 3, for any .u0 ; v0 ; 0 / 2 HC , there .2/ .1/ .2/ is a positive constant 4  4 such that for any fixed 2 .0; 4 , the following estimates hold for any t > 0 and any  2 .0; 1/ small enough,

e t kvtx .t /k2 C

Z

t

e  kvtxx . /k2 d  0

 C3  6 C C2  2

Z

t

e  .ktxx k2 C kvt tx k2 /. /d  ; 0

(250)

2382

Y. Qin

t

2

Z

e ktx .t /k C

t

e  ktxx . /k2 d 

0

 C3  6 C C2  2

Z

t

e  .kvtxx k2 C kt tx k2 /. /d  ; 0

e t .kvtx .t /k2 C ktx .t /k2 / C  C3 

6

C C2 

2

Z

(251)

t

e  .kvtxx k2 C ktxx k2 /. /d  0

Z

t

e  .kt tx k2 C kvt tx k2 /. /d  :

(252)

0

Proof. See Qin [59, 60] for details. The next lemma concerns the exponential stability of .u; v;  / in H 4 by first deriving the exponential stability of .vt t ; t t / in L2 which yields the exponential stability of .vxxx ; xxx / and .vxxxx ; xxxx / in L2 by Eqs. (14) and (15), then obtaining the exponential stability of .uxxx ; uxxxx / in L2 via equalities (242) and (243). 4 Lemma 34. Under the assumptions of Theorem 3, for any .u0 ; v0 ; 0 / 2 HC , there .2/ is a positive constant 4  4 such that for any fixed 2 .0; 4 , the following estimates hold for any t > 0.

n o e t kvt t .t /k2 C kvtx .t /k2 C kt t .t /k2 C ktx .t /k2 Z t n o e  kvt tx k2 C kvtxx k2 C kt tx k2 C ktxx k2 . /d   C4 ; C

(253)

0

n o e t kuxxx .t /k2H 1 C kuxx .t /k2W 1;1 Z t n o e  kuxxx k2H 1 C kuxx k2W 1;1 . /d   C4 ; C

(254)

0

n e t kvxxx .t /k2H 1 C kvxx .t /k2W 1;1 C kxxx .t /k2H 1 C kxx .t /k2W 1;1 C kutxxx .t /k2 o Z t n 2 2 Ckvtxx .t /k C ktxx .t /k C e  kvt t k2 C kvxxxx k2H 1 C kvtxx k2H 1 0

2

Ckt t k C

kxxxx k2H 1

Cktx k2W 1;1

C ktxx k2H 1 C kvxx k2W 2;1 C kvtx k2W 1;1 C kxx k2W 2;1 o C kutxxx k2H 1 . /d   C4 : (255)

Proof. See Qin [59, 60] for details. Remark 3. The content of Section 3.3 is chosen from Sections 2.1–2.4 of Qin [60] and Qin [55–58] and Qin, Ma, and Cavalcanti [68].

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2.4

2383

Universal Attractors in H1 , H2 , and H4

This subsection is further concerned with the existence of universal (maximal) attractors for problem (13), (14), (15), (16), and (17). One may refer to Zheng and Qin [82,83], Qin [59,60], and Qin and Muñoz Rivera [69] for some mathematical difficulties in studying the dynamics of problem (13), (14), (15), (16), and (17). Note that H i .i D 1; 2; 4/ are not complete, so it is necessary to construct the complete closed subspaces of H i . To this end, let, for any given constants ıi .i D 1;    ; 7/ with ı D .ı1 ;    ; ı7 /, Z 1 n i Hıi WD .u; v;  / 2 HC W .E.u;  / C v 2 =2/dx  ı1 ; ı6 0

Z

1

.e.u;  / C v 2 =2/dx  ı7 ;

 0

Z

1

ı2 

udx  ı3 ; ı4    ı5 ; ı2 =2  u  2ı3 ; ı1 2 R; 0 < ı2 < ı3 ;

0

0 < ı4 < ı5 ; 0 < ı6 < ı7 ; 0 < ı4 < max

2Œı2 ;ı3 ;e2Œı6 ;ı7 

O e/ < ı5 .;

min

2Œı2 ;ı3 ;e2Œı6 ;ı7 

O e/; .;

o

i D 1; 2; 4 where E.u;  / DW ‰.u;  /  ‰.1; 1/  ‰u .1; 1/.u  1/  ‰ .u;  /.  1/

(256)

O e/ is the unique inverse function of the function e D e.;  / for and O D .; any fixed  2 Œı2 ; ı3 , which is a monotone increasing function in e for any fixed O e/ follows from assumption (30).  2 Œı2 ; ı3 . The unique existence of O D .; i Obviously, Hıi .i D 1; 2; 4/ is a sequence of closed subspaces of HC .i D 1; 2; 4/: It is shown later that the first three constraints are invariant, whereas the last two constraints are not invariant. These two constraints are just introduced to i overcome the difficulty that the original spaces HC are incomplete. Note also that it is very crucial to prove that the orbit starting from any bounded set of Hıi will re-enter Hıi and stay there after a finite time. Constant Cı (sometimes Cı0 ) denotes the universal constant depending only on ıi .i D 1;    ; 7/, but independent of initial data. Ciı .i D 1; 2; 4/ denotes the universal constant depending on both ıj .j D 1;    ; 7/, H i norm of initial data, min 0 .x/, and min u0 .x/. C denotes the generic absolute positive constant x2Œ0;1

x2Œ0;1

independent of ı and the initial data. We now introduce some definitions of global attractors (universal (maximal) attractors; see Ghidaglia [19], Temam [75]).

2384

Y. Qin

Definition 1. Let X be a Banach space and T .t / be a C0 semigroup on X . If there is a bounded subset B0  X such that for any bounded subset B  X , there exists some time tB D t .B/ > 0 such that for all t  tB , T .t/B  B0 , then B0 is called an absorbing set of semigroup T .t /. Definition 2. If the following conditions hold for semigroup T .t/ defined on a Banach space X , (1) A is invariant under T .t/; that is, T .t /A D A; (2) A is a compact set in X ; (3) A is an attracting set in the sense that for any bounded subset B  X , lim d i stX .T .t /B; A/ D 0;

t!C1

where d i stX .B1 ; B2 / D supx2B1 infy2B2 d .x; y/ is the Hausdorff semidistance between subsets B1 and B2 in X with d being the metric in X , then A is called a global (maximal, universal) attractor for T .t/. In practice, the above condition (2) can sometimes be weakened as “A is an asymptotic compact (smooth) set or A is a weak compact set in X (at this time, it is called a weak global attractor)” (see [60]). To establish the universal (global) attractors, it usually suffices to prove the following three points (Temam [75]): (1) semigroup T .t / W X ! X is a (nonlinear) continuous operator for any fixed t > 0; (2) there exists a bounded absorbing set B0  X ; (3) semigroup T .t / is compact in some sense (such as uniformly compact, asymptotically compact, weakly compact, etc.). Under T these S conditions, it follows that the omega limit set of an absorbing set B0 , A D ts s0 T .t/B0 , is the unique universal (maximal) attractor for T .t/. In the present case, the weak compactness from Ghidaglia’s framework [19] is used. i Theorem 4. The nonlinear C0 semigroup S .t / on HC .i D 1; 2; 4/ obtained in i Theorems 1, 2, and 3 maps each of HC .i D 1; 2; 4/ into itself. Moreover, for any ıi .i D 1;    ; 7/, it possesses a universal (maximal) attractor Ai;ı .i D 1; 2; 4/ in Hıi .

Remark 4. The set Ai D

[

Ai;ı .i D 1; 2; 4/ is a global noncompact attractor

ı1 ; ;ı7 i i in the sense that it attracts any bounded sets of HC with in the metric space HC constraints u  u;    with u;  being any given positive constants.

The existence of an absorbing ball in Hı1 is proved and it is temporarily assumed that the initial data belong to a bounded set B of Hı1 . First, one has to prove that the orbit starting from any bounded set in Hı1 will re-enter Hı1 and stay there after

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2385

a finite time which should be uniform with respect to all orbits starting from that bounded set. Lemma 35. If .u0 ; v0 ; 0 / 2 Hı1 , then the following estimates hold for any t > 0. Z

Z

1

ı2 

1

u.x; t /dx D

u0 .x/dx  ı3 ;

0

Z

0

Z

1 2

ı6  Z

1

.e.u0 ; 0 / C v02 =2/.x/dx  ı7 ; (258)

.e.u;  / C v =2/.x; t /dx D 0

0

1

.E.u;  / C v 2 =2/.x; t /dx C

Z tZ

0

0

Z

(257)

1



0

k.u;  /x2 0 vx2 C 2 u u

 dxd 

1

D 0

.E.u0 ; 0 / C v02 =2/.x/dx  ı1 :

(259)

Proof. Estimates (257) and (258) have already been obtained in (47) and (45) from the definition of Hı1 . Note that ‰.u;  / D e.u;  /   .u;  / is the Helmholtz free energy function. Recalling (256), the definition of E D E.u;  /, noting that e .u;  / D ‰ .u;  /, by (13), (14), (15), (16), (17), and (25), (26), (27), one deduces after a direct calculation that

0 vx2 k.u;  /x2 @t E.u;  / C v 2 =2 C C u u 2   .  1/k.u;  /x D . v/x C p.1; 1/vx C : u x

(260)

Integrating (260) over Qt WD .0; 1/  .0; t/ and using (14) and (15), one obtains Z

Z tZ

1 2

.E.u;  / C v =2/.x; t /dx C 0

0

Z

0

1



 0 vx2 k.u;  /x2 dxds C u u 2

1

D 0

.E.u0 ; 0 / C v02 =2/dx

which gives (259). Lemma 36. If .u0 ; v0 ; 0 / 2 Hı1 , then the following estimates hold for any t > 0, Z

1

. rC1 Cv 2 /.x; t /dxC

0

Z tZ 0

1 0 < C1ı 

Z

0

1



.1 C  q /x2 v2 C x 2 u u

.x;  /dxd   C1ı ; (261)

1

.x; t /dx  C1ı : 0



(262)

2386

Y. Qin

Proof. The proof is similar to those of (45) and (46) and the inequalities in Remark 1. The proof is complete. Lemma 37. If .u0 ; v0 ; 0 / 2 Hı1 , then the following inequalities hold,   N    ; 0< where  D

1 C1ı

min

 .x; t /;

u2Œı2 ;ı3 ;e2Œı6 ;ı7 

(263) 8.x; t / 2 Œ0; 1  Œ0; C1/;

O e/ and   D .u;

max

O e/. .u;

max

O e/: .u;

u2Œı2 ;ı3 ;e2Œı6 ;ı7 

(264)

Proof. It suffices first to show min

u2Œı2 ;ı3 ;e2Œı6 ;ı7 

O e/  N  .u;

u2Œı2 ;ı3 ;e2Œı6 ;ı7 

(265)

In fact, it follows from (257) and (258) that ı6  eN WD e.Nu; N /  ı7 ; ı2  uN  ı3 O u; eN / and (265). Thus (263) follows. In fact, if the assertion which implies that N D .N in (264) is not true, then there exists a sequence of solutions .un ; vn ; n / with the initial data .un0 ; vn0 ; n0 / 2 Hı1 converging weakly in H 1 , strongly in C Œ0; 1 to .u0 ; v0 ; 0 / 2 Hı1 such that for the corresponding solution .u; v;  / to .u0 ; v0 ; 0 /, inf  D 0. Thus there is .xn ; tn / 2 Œ0; 1  Œ0; C1/ such that as n ! C1, x2Œ0;1;t0

.xn ; tn / ! 0:

(266)

If the sequence ftn g has a subsequence, denoted also by tn , converging to C1, then by Theorem 1 and (263), one knows that as n ! C1, .xn ; tn / ! N   > 0 which contradicts (266). If the sequence ftn g is bounded, that is, there exists a constant M > 0, independent of n, such that for any n D 1;    ; 0 < tn  M . Thus there exists a point .x  ; t  / 2 Œ0; 1  Œ0; M  such that .xn ; tn / ! .x  ; t  / as n ! C1. On the other hand, by (266) and the continuity of solutions in Theorem 1, one concludes that .xn ; tn / ! .x  ; t  / D 0 as n ! C1, which contradicts (76). Thus the proof is complete. Next, it is necessary to estimate the pointwise positive lower bound and upper bound for u whose proof is similar to that of Lemma 6. Lemma 38. If .u0 ; v0 ; 0 / 2 Hı1 , then the estimate holds:

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . . 1 0 < C1ı  u.x; t /  C1ı ; 8.x; t / 2 Œ0; 1  Œ0; C1/:

2387

(267)

Proof. One may use a similar idea to that of the proof of Lemma 6. But one should note the dependence of the constants C1ı depending only on the parameters ı1 ;    ; ı7 . Here one only uses Lemmas 35, 36, and 37 and replaces constant C1 by constant C1ı in the proof of Lemma 6 to finish the proof. The proof is complete. Lemma 39. For initial data belonging to an arbitrary fixed bounded set B of Hı1 , there is a time t0 > 0 depending only on boundedness of this bounded set B such that for all t  t0 , x 2 Œ0; 1, ı4  .x; t /  ı5 ;

ı2 =2  u.x; t /  2ı3 :

(268)

Proof. Suppose that the assertion in Lemma 39 is not true. Then there is a sequence tn ! C1 such that for all x 2 Œ0; 1, sup .x; tn / > ı5 ;

(269)

where sup is taken for all initial data in a given bounded set B of Hı1 . In the same manner as for the proof of Lemma 37, there exists .u0 ; v0 ; 0 / 2 B such that for the corresponding solution .u; v;  /, .x; tn /  ı5 ; 8x 2 Œ0; 1; which, along with the definition of Hı1 , yields N  ı5 :

(270)

This contradicts (263) and the definition of Hı1 . Similarly, one can prove other parts of (268). The proof is complete. Remark 5. It follows from Lemmas 35 and 39 that for initial data belonging to a given bounded set B of Hı1 , the orbit will re-enter Hı1 and stay there after a finite time. In the sequel, the existence of an absorbing ball in Hı1 is proved. Inasmuch as it has been assumed that the initial data .u0 ; v0 ; 0 / belong to an arbitrarily bounded set B of Hı1 , there is a large positive constant B such that k.u0 ; v0 ; 0 /kH 1  B. Here constant CB;1ı denotes generic positive constants depending on B, ıi ; .i D 1;    ; 7/, and C1ı . Lemma 40. For any initial data .u0 ; v0 ; 0 / 2 Hı1 , the unique global strong solution .u.t /; v.t /; .t // to problem (13), (14), (15), (16), and (17) satisfies the estimates:

2388

Y. Qin

v2 v2 1 .juNuj2 Cjj N 2 /  E.u; v; /  CCB;1ı .juNuj2 Cjj N 2 /: CCB;1ı 2 2

(271)

Proof. By Lemmas 35, 36, 37, 38, and 39, one can use similar argumentation to the proof of Lemma 18 to show this lemma. Lemma 41. There exists a positive constant 10 D 10 .CB;1ı / > 0 such that for any fixed 2 .0; 10 , the following estimate holds, e t .kv.t /k2 C ku.t /  uN k2 C k.t /  N k2 C kux .t /k2 C kx .t /k2 / Z t C e  .kux k2 C kx k2 C kx k2 C kvx k2 /. /d   CB;1ı ; 8t >0: (272) 0

Proof. By Lemmas 35, 36, 37, 38, 39, and 40, one can use similar argumentation to the proof of Lemma 19 to show this lemma. Lemma 42. There exists a positive constant 1 D 1 .CB;1ı /  10 such that for any fixed 2 .0; 1 ; the following estimate holds, e t .kvx .t /k2 C kx .t /k2 / C

Z

t

e  .kvxx k2 C kxx k2 C kvt k2 C kt k2 /. /d  0

 CB;1ı ;

8t > 0

(273)

which, together with Lemma 41, implies that for any fixed 2 .0; 1 , k.u.t /  uN ; v.t /; .t /  N /k2H 1  CB;1ı e  t ; 8t > 0:

(274)

C

Proof. By Lemmas 35, 36, 37, 38, 39, 40, and 41, one can use similar argumentation as in the proof of Lemma 19 to show this lemma. Thus the following result on the existence of an absorbing set in Hı1 follows immediately from Lemma 42. q Lemma 43. Let R1 D R1 .ı/ D 2 ı32 C .  /2 and B1 D f.u; v;  / 2 Hı1 ; k.u; v;  /kH 1  R1 g. Then B1 is an absorbing ball in Hı1 ; that is, there C

exists some t1 D t1 .CB;1ı / D maxf 11 logŒ2.ı32 C .  /2 /=CB;1ı ; t0 g  t0 such that when t  t1 ; k.u.t /; v.t /; .t //k2H 1  R12 . C

Now it suffices to prove the existence of an absorbing set in Hı2 and temporarily assume that the initial data belong to an arbitrarily fixed bounded set B in Hı2 , that is, k.u0 ; v0 ; 0 /kH 2  B with B being a given positive constant.

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2389

The next two lemmas are concerned with the existence of an absorbing set in Hı2 . Constant CB;2ı denotes the generic constant depending only on B, ıi .i D 1; 2;    ; 7/ and C2ı . Lemma 44. There exists a positive constant 20 D 20 .CB;2ı /  1 such that for any fixed 2 .0; 20 , the following estimate holds, e t .kt .t /k2 C kvt .t /k2 C k.t /  N k2H 2 C kv.t /k2H 2 / C Ckxt k2 /. /d   CB;2ı ;

Z

t

e  .kvxt k2 0

8t > 0:

(275)

Proof. The proof is similar to that of Lemma 22. Lemma 45. There exists a positive constant 2 D 2 .CB;2ı /  20 such that for any fixed 2 .0; 2 , the following estimate holds, ku.t /  uN k2H 2  CB;2ı e  t

(276)

which, together with Lemma 44, implies that for any fixed 2 .0; 2  and for all t > 0, ku.t /k2H 2 C k.t /k2H 2 C kv.t /k2H 2  2.ı32 C .  /2 / C CB;2ı e  t :

(277)

Proof. The proof is similar to that of Lemma 37. Now define t2 D t2 .CB;2ı /  max.t1 .CB;1ı /;  21 log.2.ı32 C .  /2 /=CB;2ı /; then estimate (275) implies that for any t  t2 .CB;2ı /, ku.t /k2H 2 C k.t /k2H 2 C kv.t /k2H 2  4.ı32 C .  /2 /: q Taking R2 D 2 ı32 C .  /2 , one immediately concludes the following lemma. Lemma 46. The ball B2 D f.u; v;  / 2 Hı2 ; k.u.t /; v.t /; .t //k2H 2  R22 g is an C

absorbing ball in Hı2 , that is, when t  t2 , k.u.t /; v.t /; .t //k2H 2  R22 : C

Now it is necessary to establish the existence of an absorbing set in Hı4 and from now on, assume temporarily that the initial data belong to any given bounded set B in Hı4 , and so there exists a sufficiently large positive constant B such that k.u0 ; v0 ; 0 /kH 4  B. Here CB;4ı stands for the generic constant depending only C on B, ıi .i D 1; 2;    ; 7/ and C4ı . By Lemmas 25, 26, 35 and 39, and repeating

2390

Y. Qin

the same argument as the proof of Lemma 34, one can easily derive the following 4 . lemma that yields the existence of an absorbing set in HC Lemma 47. There exists a positive constant 4 D 4 .CB;4ı /  2 .CB;4ı / such that for any fixed 2 .0; 4 , it holds that for any t > 0; n N 2 4 C kut .t /k2 3 C kut t .t /k2 1 e t ku.t /  uN k2H 4 C kv.t /k2H 4 C k.t /  k H H H o Z t n Ckvt .t /k2H 2 C kvt t .t /k2 C kt .t /k2H 2 C kt t .t /k2 C e  ku  uN k2H 4 0

Ckvk2H 5

C k 

N k2H 5

C

kvt k2H 3

C

kvt t k2H 1

o Ckut t k2H 2 C kut t t k2 . /d   CB;4ı ;

C

kt k2H 3

C kt t k2H 1 C kut k2H 4

which implies ku.t /k2H 4 C kv.t /k2H 4 C k.t /k2H 4  2.Nu2 C N 2 / C CB;4ı e  t  R2 .ı/ C CB;4ı e  t ;

(278)

with R2 .ı/ D 2Œı32 C .  /2 . Now by taking   R2 .ı/ ; t4 D t4 .CB;4ı / D max t2 .CB;2ı /;  41 ln CB;4ı then the next lemma immediately follows from Lemma 47. o n Lemma 48. The ball BO 4 D .u; v;  / 2 Hı4 ; k.u; v;  /k2H 4  R42 is an absorbing ı

set in Hı4 , that is, when t  t4 .CB;4ı /, k.u.t /; v.t /; .t //k2H 4  R42

(279)

ı

with R42 D 2R2 .ı/: Having proved the existence of absorbing balls in Hı1 ; Hı2 and Hı4 , one can use the abstract framework established by Ghidaglia [19] to conclude the following. Lemma 49. The sets

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

!.Bi / D

\[ S .t /Bi ; i D 2; 4;

2391

(280)

s0 ts i where the closures are taken with respect to the weak topology of HC .i D 2; 4/, are included in Bi .i D 2; 4/, nonempty, and invariant by S .t /; that is,

S .t /!.Bi / D !.Bi /; i D 2; 4;

8t > 0:

(281)

Remark 6. If one takes B a bounded set in Hı2 , one can also define !.B/ by (280) and when B is nonempty, !.B/ is also included in Bi .i D 2; 4/; nonempty and invariant. Because Bi .i D 2; 4/ are absorbing balls, it is clear that !.B/ !.B4 / !.B2 /. This shows that !.Bi / .i D 2; 4/ are maximal in the sense of inclusion. Lemma 50. The sets Ai;ı D !.Bi /; i D 2; 4

(282)

Ai;ı is bounded and weakly closed in Hıi ;

(283)

satisfy

S .t /Ai;ı D Ai;ı ;

8t  0;

(284)

lim d w .S .t /B; Ai;ı / D 0:

(285)

for every bounded set B in Hıi , t!C1

Moreover, it is the maximal set in the sense of inclusion that satisfies (283), (284), and (285). Proof. The proofs of Lemmas 49 and 50 follow from the abstract framework in Ghidaglia [19], and use the facts that S .t / is continuous on Hı1 , Hı2 , and Hı4 , respectively; Hı4 is compactly embedded in Hı2 , Hı2 is compactly embedded in Hı1 , and Bi .i D 1; 2; 4/ are absorbing balls in Hıi .i D 1; 2; 4/, respectively. Following Ghidaglia [19], one also calls Ai;ı the universal attractor of S .t / in Hıi .i D 2; 4/. In order to discuss the existence of a universal attractor in Hı1 , it is necessary to prove the following lemma. Lemma 51. For every t  0, the C0 semigroup S .t / is continuous on bounded sets of Hı1 for the topology induced by the norm in L2  L2  L2 .

2392

Y. Qin

Proof. Suppose that .u0j ; v0j ; 0j / 2 Hı1 ; k.u0j ; v0j ; 0j /kH 1  R; .uj ; vj ; j / D S .t /.u0j ; v0j ; 0j / .j D 1; 2/, and .u; v;  / D .u1 ; v1 ; 1 /  .u2 ; v2 ; 2 /, respectively. Subtracting the corresponding Eqs. (13), (14), and (15) satisfied by .u1 ; v1 ; 1 / and .u2 ; v2 ; 2 /, one obtains (136), (137), (138), and (139). By Lemmas 1 and 35, 36, 37, and 38, one knows that for any t > 0 and j D 1; 2, k.uj .t /; vj .t /; j .t //k2H 1

Z

t

C 0

.kujx k2 C kvj k2H 2 C kjx k2H 1 C kjt .t /k2

Ckvjt k2 /. /d   CR;1ı ;

(286)

where CR;1ı > 0 is a generic constant depending only on R; ı, and C1ı . Multiplying (136), (137), and (138) by u; v, and  , respectively, adding them up and integrating the result over Œ0; 1, and using (34), (286), the Cauchy inequality, the embedding theorem, the mean value theorem, and inequalities k.t /k2L1  C .k.t /kkx .t /k C k.t /k2 /; kv.t /kL1  kvx .t /k; it follows that for any small  > 0,  0 vx2 2 C k.u1 ; 1 /x dx u1 0 p  .kvx .t /k2 C kx .t /k2 / C CR;1ı ./H .t /.ku.t /k2 C k e .u1 ; 1 /.t /k2

p 1 d .ku.t /k2 C kv.t /k2 C k e .u1 ; 1 /.t /k2 / C 2 dt

Z

1



Ckv.t /k2 / which, together with Lemmas 35, 36, and 38 and (66), gives p d .ku.t /k2 C kv.t /k2 C k e .u1 ; 1 /.t /k2 / C .CR;1ı /1 .kvx .t /k2 C kx .t /k2 / dt p  CR;1ı H .t /.ku.t /k2 C k e .u1 ; 1 /.t /k2 C kv.t /k2 / (287) where, by (287), H .t / D k1t .t /k2 C k2t .t /k2 C kv1xx .t /k2 C kv2xx .t /k2 C k1xx .t /k2 C k2xx .t /k2 C 1 satisfies for any t > 0, Z

t

H ./d   CR;1ı .1 C t /:

(288)

0

Therefore the assertion of this lemma follows from Gronwall’s inequality, (287), and (288) and (45). The proof is complete. Now one can again use the abstract framework in Ghidaglia [19] to obtain the following result on the existence of a universal attractor in Hı1 . Lemma 52. The set

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

A1;ı D

\[

S .t /B1

2393

(289)

s0 ts

is the (maximal) universal attractor in Hı1 where the closures are taken with respect 1 to the weak topology of HC . Remark 7. Because one has obtained three attractors A4;ı , A2;ı , and A1;ı which satisfy that A4;ı is bounded in Hı4 . Hı2 Hı1 / and A2;ı is bounded in Hı2 . Hı1 /, so A4;ı is bounded in both Hı1 and Hı2 and A2;ı is bounded in Hı1 , and by the invariance property (284), one has A4;ı A2;ı A1;ı :

(290)

On the contrary, if one knew that A1;ı is bounded in Hı2 or/and A2;ı is bounded in Hı4 , then one knows that A1;ı D A2;ı and/or A2;ı D A4;ı . The proof of Theorem 4 is finished. Remark 8. Theorems 1, 2, 3, and 4 also hold for the boundary conditions: v.0; t / D v.1; t / D 0; .0; t / D .1; t / D T0 > 0 .constant/:

(291)

Remark 9. The content of Section 3.4 is chosen from Qin [59, 60] and Qin and Muñoz Rivera [69]. Remark 10. Kawohl [36] succeeded in globally solving the system (13), (14), and (15) with the boundary conditions (16) or Q.0; t / D Q.1; t / D 0;  .0; t / D  .1; t / D 0 with r 2 Œ0; 1; q  2r C 2, whereas Jiang [30] also established the results on the asymptotic behavior of global large solutions for the boundary conditions (16) and (291) with r 2 Œ0; 1; q  r C 1. In addition, Jiang [31] established the global existence with basically the same constitutive relations as those in Kawohl [36] for the boundary conditions (16) or (291) or Q.0; t / D Q.1; t / D 0;  .0; t / D v.0; t /;  .1; t / D v.1; t / or .0; t / D .1; t / D T0 ;  .0; t / D v.0; t /;  .1; t / D v.1; t /:

3

A 1D System of Viscous Polytropic Gas

In this section, we investigate the global existence and asymptotic behavior of solutions to an initial boundary value problem and the Cauchy problem of the compressible Navier-Stokes equations that include polytropic viscous gas.

2394

3.1

Y. Qin

An Initial Boundary Value Problem

In this subsection, we discuss an initial boundary value problem of the compressible Navier-Stokes equations that include polytropic viscous gas.

3.1.1

Global Existence and Exponential Stability of Strong (Generalized) Solutions in Hi .i D 1; 2; 4/ In Section 2.3, it was discussed that problem (13), (14), (15), (16), and (17) with more general constitutive relations (19), (20), and (21), (25), (26), and (27), and the exponents q; r satisfying (28) and (29). However, assumptions (28) and (29) cannot cover the important case of q D r D 0, to which the compressible Navier-Stokes equations of the polytropic viscous gas belong; see, for example, (24). Assume that: (i) For any 0 < u < C1 and 0   < C1, p.u;  /; e.u;  /; q.u;  ; x /, and k.u;  / satisfy e.u;  / D CV  C F2 .u/; F2 .u/  0; 0 vx 0 vx  .u; v; vx / D p.u;  / C  1 .u;  / C ; u u K0 x Q.u;  ; x / D  ; u

(292) (293) (294)

where 1 .u;  /  p.u;  / D f1 .u/ C f2 .u/; K.u;  /  K0 > 0 and CV ; 0 ; K0 are positive constants. (ii) f1 .u/; f2 .u/ 2 C 2 .R/ and there exist constants c1 > 0; d1 > 0; c2  0; d2  0 such that for any u > 0, there holds that  ci  ufi .u/  di .i D 1; 2/;

f10 .u/ > 0;

f20 .u/  0

(295)

with Fi0 .u/ D fi .u/ .i D 1; 2/. (iii) For any u > 0, there are some positive constants N1 .u/; N2 .u/, and N3 .u/ such that for any u  u, F2 .u/  N1 .u/;

(296)

fi0 .u/  N2 .u/; i D 1; 2I jf1 .u/j  N3 .u/:

(297)

1 . (iv) The initial data .u0 ; v0 ; 0 / 2 HC

Here the notations in Sects. 2.1 and 2.4 are used. Theorem 5. Assume that (i)–(iv) hold. Then Theorem 1 holds for the problem (13), (14), (15), (16), and (17) with constitutive relations (292), (293), (294), (295),

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2395

R1 R1 (296), and (297) where u D 0 u0 .x/dx and N D CV1 .E0  0 F2 .u0 /dx/; R1 E0  0 .CV 0 C F2 .u0 .x// C v02 .x/=2/dx. Remark 11. The assumptions (i)–(iii) correspond to the case of q D r D 0. The assumptions (i)–(iii) include the model of ideal gas whose constitutive relations take the form of (24); that is, f1 .u/ D  Ru ; f2 .u/ D F2 .u/ D 0: Moreover, the results in Theorem 1 can cover those in Kazhikhov [37] and Kazhikhov and Shelukhin [40]. To prove Theorem 5, one can pursue the line of the proof of Theorem 1, but one still needs to establish the key estimate (298) in Lemma 53. Lemma 53. The following estimates hold for any t > 0. Z tZ 0

1



x2

Z

t

0

 .x; s/dxds  C1 ; 8t > 0; ˛2.0; 1;

(298)

k ˛ .s/  Q˛ .s/k2L1 ds  C1 ; 8t > 0; ˛2Œ0; 1/;

(299)

 1C˛

0

v2 C x˛ 

Z

t

0

2ı1 Q k.s/  .s/k L1 ds  C1

Z

t

kx .s/k2 ds  C1 ; 8t > 0;

0

ı1 2Œ1; 2; where Q˛ .t / D

R1 0

(300)

 ˛ .x; t /dx:

Proof. If ˛ D 1, then (298) is a direct result of Lemmas 2 and 3. If 0 < ˛ < 1, R 1 1 and integrating the result over Qt  then multiplying (15) by  ˛ 0  1˛ dx Œ0; 1  Œ0; t, one gets Z t Z 0

1

 1˛ dx

1 Z

0



1 0

Z tZ

 dxds

e

1

jf1 .u/vx . 1˛   1˛ /jdxds

 C1 C C1 0

˛K0 x2 0 vx2 C 1C˛ u u ˛

0

Z

t

 C1 C C1

k

1˛

 e k

1˛ 2 L1 ds

0

Z t Z  C1 C C1

1

 0

0

˛

ds

0

ds 0

!1=2

2

1

jvx jdx 0

!1=2 Z Z t

2 x dx

1=2 Z t Z

0

1

vx2 dxds ˛

1=2 :

(301)

(1) When 1=2  ˛ < 1; it follows from (301), the Young inequality, and Lemma 2 and 3 that

2396

Y. Qin

Z tZ 0

1

0

1  C1 C 2  C1 C

1 2



x2

 1C˛ Z tZ 1 0

0

Z tZ 0

0

1



v2 C x˛ 

dxds

vx2 dxds C C1 ˛

Z t Z 0

1 0

x2 dx 2

 Z

1



22˛

 dx ds

0

vx2 dxds ˛

which gives (298). (2) When when ˛ 2  1 10 < ˛ < 1=2, the induction argument is used. Assume that 1 ; ; 1 ; then .n D 2; 3;    /, (298) is valid. Now suppose that ˛ 2 Œ 2n 2n1 2nC1 2n by (301) and induction assumption,  vx2 C ˛ dxds  1C˛  0 0  Z 1  Z t Z 1 Z tZ 1 2 vx x2 1 1C 21n 2˛  C1 C dxds C C1 dx  dx ds 1 2 0 0 ˛ 0 0  1C 2n 0 Z Z 1 t 1 vx2 dxds  C1 C 2 0 0 ˛

Z tZ

1



x2

which yields (298). R1 Because 0 . ˛  Q˛ /dx D 0; for any t > 0 there is a point b.t / 2 Œ0; 1 such that  ˛ .b.t /; t / D Q˛ .t / which implies Z

t

k .s/  Q˛ .s/k2L1 ds  ˛

0

Z t ˇZ ˇ ˇ ˇ 0

x

ˇ2 ˇ ˛ Q .   /y dy ˇˇ ds ˛

b.t/

Z t Z  C1 0

0

1

 Z

x2 

dx 2˛

1

  ˛ dx ds  C1 ;

0

if 0 < ˛ < 1. Similarly, Q /kı11  C1 k.t /  .t L

Z

1

j.  Q /ı1 1 x jdx

0

Z  C1

1

2.ı1 1/

.   / 0

which leads to (300). The proof is complete.

1=2 dx

kx .t /k  C1 kx .t /k;

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2397

Now with (298), one can prove a crucial estimate (302) in Lemma 54 that has the same role as (82), but now (82) does not work for the case q D r D 0 (see also Remark 12). See Qin [54, 60] for its detailed proof. Lemma 54. There holds that for any t > 0, kux .t /k2 C

Z

t

.kux k2 C k 1=2 ux k2 /.s/ds  C1 .1 C sup k.s/kL1 /˛ ; 0st

0

8t > 0; 8 ˛ 2 .0; 1:

(302)

Remark 12. It is easy to see that ˇ D 1 if q D r D 0 in (82) in Lemma 7. So this lemma has reduced the order of  and later on it is shown that ˇ D ˛ D 1 does not work for the case discussed in this section (q D r D 0). This is why the estimate (302) must be established. Now one can pursue the same lines of the proof of Lemmas 9, 10, and 11 by using Lemmas 53 and 54 to prove the next two lemmas; see Qin [59, 60] for details. Lemma 55. There holds that for any t > 0,

2

Z

!1C˛

t 2

kvx .t /k C

2

.kvt k C kvxx k /.s/ds  C1 1 C sup k.s/kL1 8˛ 2 .0; 1;

kx .t /k2 C

;

0st

0

Z

t

(303) !2.1C˛/

kxx .s/k2 ds  C1 1 C sup k.s/kL1

;

0st

0

80 < ˛  1:

(304)

Similarly to Lemma 12, the next result follows immediately. Lemma 56. The following estimates hold for any t > 0. k.t /kL1  C1 ; kx .t /k2 C kvx .t /k2 C kux .t /k2 C

(305) Z

t

.kx k2 C kvvx k2 C kvx k2 0

Ckux k2 C kt k2 C kvt k2 C kvxx k2 C kxx k2 /.s/ds  C1 :

(306)

Proof of Theorems 5 and 6. Note that assumptions (i)–(iv) of Theorems 5 and 6 imply that e; p;  ;  with special relations (292), (293), (294), (295), (296), and (297) satisfy the assumptions of Theorems 1, 2, and 3, respectively, thus repeating

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Y. Qin

the same arguments as those in the proof of Theorems 2 and 3. The proof follows immediately.

3.1.2 Universal Attractors in Hi .i D 1; 2; 4/ This subsection is concerned with the existence of universal (maximal) attractors in i HC for problem (13), (14), (15), (16), and (17) with (292), (293), (294), and (295) and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0 (the polytropic ideal gas), which describes the motion of a one-dimensional viscous polytropic ideal gas, where u; v;  are the specific volume, velocity, and absolute temperature, respectively;  is the stress, and ; CV , and K0 are positive constants. Without loss of generality, assume that CV D R D  D K0 D 1: Let ˇi .i D 1;    ; 5/ be any given constants such that ˇ1 ˇ1 2 R; ˇ2 > 0; ˇ4  eˇ2 > ˇ3 > 0; 0 < ˇ5 < ˇ2 , and let Hˇi

n

i

Z

Z

1

WD .u; v;  / 2 H W

.log. / C log.u//dx  ˇ1 ; ˇ5  0

Z

1

ˇ3 

1

. C v 2 =2/dx  ˇ2 ;

0

o udx  ˇ4 ; ˇ5 =2    2ˇ2 ; ˇ3 =2  u  2ˇ4 ; i D 1; 2; 4;

0

with ˇ D .ˇ1 ;    ; ˇ5 /. Clearly, Hˇi is a sequence of closed subspaces of H i .i D 1; 2; 4/: Based on the results in Theorems 5 and 6, repeating the same reasoning as that in the proof of Theorem 4, one can easily prove the next theorem. Theorem 6. The nonlinear semigroup S .t / defined by the solution to problem (13), (14), (15), (16), and (17) with (292), (293), (294), (295), (296), and (297) and i f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0 maps each of HC .i D 1; 2; 4/ into itself. ˇ1 Moreover, for any ˇi .i D 1;    ; 5/ with ˇ1 < 0; ˇ2 > 0; ˇ4  eˇ2 > ˇ3 > 0; 0 < ˇ5 < ˇ2 , it possesses a maximal attractor Ai;ˇ .i D 1; 2; 4/ in Hˇi : Remark 13. The set Ai D

[

Ai;ˇ .i D 1; 2; 4/ is a global noncompact

ˇ1 ;ˇ2 ;ˇ3 ;ˇ4 ;ˇ5 i HC in the sense

i attractor in the metric space that it attracts any bounded sets of HC with constraints u  1 ;   2 with 1 ; 2 being any given positive constants.

Remark 14. The content of this subsection comes from Qin [54], Zheng and Qin [82], and Chapter 3 of Qin [60].

3.2

The Cauchy Problem

3.2.1 Global Existence in H2 .R/ This subsection concerns the regularity, continuous dependence on initial data, and large-time behavior of H i  strong (generalized) solutions .i D 1; 2; 4/ to the

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2399

Eqs. (13), (14), (15), (16), and (17) with (292), (293), (294), (295), (296), and (297) and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0 for the compressible NavierStokes equations of a one-dimensional viscous polytropic ideal gas in Lagrangian coordinates with the initial conditions: 8x 2 R:

.u.x; 0/; v.x; 0/; .x; 0// D .u0 .x/; v0 .x/; 0 .x//;

(307)

The following definitions of H i  strong solutions .i D 2; 4/ are introduced. Definition 3. For a fixed constant T > 0 and some positive constants uN and N , we call .u.t /; v.t /; .t // an H 2  strong solution to the Cauchy problem (13), (14), (15), (16), and (17), (307) with (292), (293), (294), (295), (296), and (297) and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0 if it is in the following function class of functions, u  uN ; v;   N 2 L1 .Œ0; T ; H 2 .R//;

(308)

ut 2 L1 ..0; T /; H 1 .R// \ L2 ..0; T /; H 2 .R//; 1

2

2

1

(309)

vt ; t 2 L ..0; T /; L .R// \ L ..0; T /; H .R//;

(310)

ux 2 L2 ..0; T /; H 1 .R//; vx ; x 2 L2 ..0; T /; H 2 .R//:

(311)

Furthermore, in addition to (308), (309), (310), and (311), if u  uN ; v;   N 2 L1 .Œ0; T ; H 4 .R//; 1

3

2

(312) 2

ut 2 L ..0; T /; H .R// \ L ..0; T /; H .R//;

(313)

vt ; t 2 L1 ..0; T /; H 2 .R// \ L2 ..0; T /; H 3 .R//;

(314)

1

1

2

2

ut t 2 L ..0; T /; H .R// \ L ..0; T /; H .R//;

(315)

vt t ; t t 2 L1 ..0; T /; L2 .R// \ L2 ..0; T /; H 1 .R//;

(316)

2

3

ux 2 L ..0; T /; H .R//;

(317)

vx ; x 2 L2 ..0; T /; H 4 .R//; ut t t 2 L2 ..0; T /; L2 .R//;

(318)

then .u.t /; v.t /; .t // is called an H 4  strong solution to the Cauchy problem (13), (14), and (15), (307) with (292), (293), (294), (295), (296), and (297) and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0: Kazhikhov and Shelukhin [40] proved that if for some positive constants uN ; N ; u0  uN ; v0 ; 0  N 2 H 1 .R/ and u0 .x/; 0 .x/ > 0 on R, then there exists a unique global (large) solution .u.t /; v.t /; .t // with positive u.x; t / and .x; t / to the Cauchy problem (13), (14), and (15), (307) with (292), (293), (294), (295), (296), and (297) and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0 on R  Œ0; C1/ such that for any T > 0,

2400

Y. Qin

u  uN ; v;   N 2 L1 .Œ0; T ; H 1 .R//; ut 2 L1 ..0; T /; L2 .R//;

(319)

vt ; ux ; t ; uxt ; vxx ; xx 2 L2 ..0; T /; L2 .R//:

(320)

Now .u.t /; v.t /; .t // verifying (319) and (320) is called an H 1 generalized solution to the Cauchy problem (13), (14), and (15), (307) with (292), (293), (294), and (295), and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0: Next we discuss the global existence and continuous dependence on initial data of H i .R/ .i D 1; 2; 4/ (global) solutions for large initial data and the large-time behavior of this H i .R/ .i D 2; 4/ solution for small initial data. Put k  k D k  kL2 .R/ and denote by Ci .i D 1; 2; 3; 4/ the universal constant depending only on min u0 .x/, min 0 .x/, the H i .R/ .i D 1; 2; 3; 4/ norm of .u0  uN ; v0 ; 0  N / x2R

x2R

(for some positive constants uN ; N ) and e0 or E0 ; E1 (see, e.g., Theorem 9), but independent of any length of time T > 0. Theorem 7. Assume that for some positive constants uN ; N ; u0  uN ; v0 ; 0  N 2 H 2 .R/ and u0 .x/ > 0; 0 .x/ > 0 on R and the compatibility conditions hold. Then for any but fixed constant T > 0, the Cauchy problem (13), (14), and (15), (307) with (292), (293), (294), (295), (296), and (297), and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0 admits a unique H 2  strong global solution .u.t /; v.t /; .t // on QT verifying (308), (309), (310), and (311) and the following estimates hold for any t 2 Œ0; T . 0 < C11 .T /  .x; t /  C1 .T /; 8 .x; t / 2 R  Œ0; T ; 0
0; 0j .x/ > 0 on R and the compatibility conditions .j D 1; 2/: This property implies the uniqueness of the H i  global strong (generalized) solution .i D 1; 2/: Theorem 8. Assume that for some positive constants uN ; N ; u0  uN ; v0 ; 0  N 2 H 4 .R/ and u0 .x/ > 0; 0 .x/ > 0 on R and the compatibility conditions hold. Then for any but fixed constant T > 0, the Cauchy problem (13), (14), and (15), (307) with (292), (293), (294), (295), (296), and (297), and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0 admits a unique H 4 global strong solution .u.t /; v.t /; .t // on QT satisfying (312), (313), (314), (315), (316), (317), (318) and (321), (322), and the following estimates hold for any t 2 Œ0; T . ku.t /  uN k2H 4 C ku.t /  uN k2W 3;1 C kut .t /k2H 3 C kut t .t /k2H 1 C kv.t /k2H 4 Ckv.t /k2W 3;1 C kvt .t /k2H 2 C kvt t .t /k2 C k.t /  N k2H 4 Z 0

Ck.t /  N k2W 3;1 C kt .t /k2H 2 C kt t .t /k2  C4 .T /;

(325)

t

.kux k2H 3 C kut k2H 4 C kut t k2H 2 C kut t t k2 C kux k2W 2;1 C kvx k2H 4 C kvt k2H 3 Ckvt t k2H 1 C kvx k2W 3;1 C kx k2H 4 C kt k2H 3 C kt t k2H 1 Ckx k2W 3;1 /. /d   C4 .T /:

(326)

Moreover, the H 4  global strong solution is continuously dependent on initial data in the sense of (324) with i D 4. The proofs of Theorems 7 and 8 are similar to those of Theorems 1 and 2, but the difference here is that now the constant depends on T , any given length of time. Remark 15. Recall that the H 2 -global strong solution .u.t /; v.t /; .t // obtained in Theorem 7 is not a classical one. By the embedding theorem, u0  uN ; v0 ; 0  N 2 1 C 1C 2 .R/. If one imposes on the higher regularities of v0 ; 0  N 2 C 2C .R/; 2 .0; 1/, then the global existence of classical solutions is as obtained in [37]. Remark 16. From Remark 15 it follows that the H 2 -global strong solution .u.t /; v.t /; .t // obtained in Theorem 7 can be understood as a global strong solution between the global classical solution and the H 1 generalized global solution. Theorem 9. Assume that for some positive constants uN ; N ; u0  uN ; v0 ; 0  N 2 H i .R/ .i D 2; 4/ and u0 .x/ > 0; 0 .x/ > 0 on R and the compatibility conditions hold. Define

2402

Y. Qin

e0 WD ku0  uN k2L1 C with ˛ >

1 2

Z

1 0

.1 C x 2 /˛ Œ.u0 .x/  uN /2 C v02 .x/ C .0 .x/  N /2 C v04 .x/dx;

being an arbitrary but fixed constant, and

El D k.log.0 =/; N log.v0 /; log.0 =N //kH l ; .l D 0; 1/; 0 D 1=u0 ; N D 1=Nu: Then there exists a constant 0 2 .0; 1 such that if e0  0 or E0 E1  0 , then the H i global strong solution .u.t /; v.t /; .t // .i D 2; 4/ obtained in Theorems 7 and 8 to the Cauchy problem (13), (14), and (15), (307) with (292), (293), (294), (295), (296), and (297), and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0 satisfying 0 < C11  .x; t /  C1 ;

8 .x; t / 2 R  Œ0; C1/;

(327)

0 < C11  u.x; t /  C1 ;

8 .x; t / 2 R  Œ0; C1/;

(328)

and for i D 2, besides (308), (309), (310), and (311) with T D C1, we have ku.t /  uN k2H 2 C ku.t /  uN k2W 1;1 C kut .t /k2H 1 C kv.t /k2H 2 C kv.t /k2W 1;1 Z t ˚ kux k2H 1 Ckvt .t /k2 C k.t /  N k2H 2 C k.t /  N k2W 1;1 C kt .t /k2 C 0

Ckux k2L1

C

kut k2H 2

C

kvx k2H 2

C

Ckt k2H 1 . /d   C2 ; 8t > 0;

kvx k2W 1;1

C

kvt k2H 1

C kx k2H 2 C kx k2W 1;1 (329)

and for i D 4, in addition to (327), (328), (329), and (312), (313), (314), (315), (316), (317), (318) with T D C1, we have ku.t /  uN k2H 4 C ku.t /  uN k2W 3;1 C kut .t /k2H 3 C kut t .t /k2H 1 C kv.t /k2H 4 Ckvt .t /k2H 2 C kvt t .t /k2 C kvx .t /k2W 3;1 C k.t /  N k2H 4 C k.t /  N k2W 3;1 Z 0

Ckt .t /k2H 1 C kt t .t /k2  C4 ; 8t > 0; t

˚

(330)

kux k2H 3 C kut k2H 4 C kut t k2H 2 C kut t t k2 C kux k2W 2;1 C kvx k2H 4 C kvt k2H 3

Ckvt t k2H 1 C kvx k2W 3;1 C kx k2H 4 C kt k2H 3 C kt t k2H 1 C kx k2W 3;1 . /d   C4 ; 8t > 0:

(331)

Moreover, the H i global strong (generalized) solutions .i D 1; 2; 4/ are continuously dependent on initial data in the sense that k.u1 .t /  u2 .t /; v1 .t /  v2 .t /; 1 .t /  2 .t //kH i  Ci k.u01 .t /  u02 .t /; v01 .t /  v02 .t /; 01 .t /  02 .t //kH i ; i D1; 2; 4; (332) where .uj .t /; vj .t /; j .t // .j D 1; 2/ has the same sense as in (324).

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2403

Finally, for the H 2 global strong solution .u.t /; v.t /; .t //, as t ! C1, kut .t /kH 1 C kut .t /kL1 C kvt .t /k C kt .t /k ! 0; (333) k.u.t /; v.t /; .t //  .Nu; 0; N /kW 1;1 C k.ux .t /; vx .t /; x .t //kH 1 ! 0; (334) and for the H 4 global strong solution .u.t /; v.t /; .t //, as t ! C1, k.ux .t /; vx .t /; x .t //kH 3 C kut .t /kH 3 C kut .t /kW 2;1 C kvt .t /kH 2 Ckvt .t /kW 1;1 C kt .t /kH 2 C kt .t /kW 1;1 ! 0;

(335)

kut t .t /kH 1 C kvt t .t /k C kt t .t /k C k.ux .t /; vx .t /; x .t //kW 2;1 ! 0: (336) Corollary 4. The H 4 global strong solution .u.t /; v.t /; .t // obtained in Theorem 9 is a classical one. Moreover, under the assumptions in Theorem 9, the following large-time behavior of the classical solution .u.t /; v.t /; .t // holds: as t ! C1, k.ux .t /; vx .t /; x .t //kC 2C1=2 C kut .t /kC 2C1=2 C k.vt .t /; t .t //kC 1C1=2 Ckut t .t /kC 1=2 ! 0:

(337)

Remark 17. Xin [77] proved the blow-up result of smooth solutions to the Cauchy problem of the isentropic compressible Navier-Stokes equations with compact support in 3D.

3.2.2

Large-Time Behavior of Global Strong (Generalized) Solutions in Hi .i D 1; 2; 4/ In this subsection, the proof of Theorem 9 is finished. In order to study the large-time behavior of the H i global strong solutions .i D 2; 4/, obviously all the estimates in the proofs of Theorems 7 and 8 will no longer work because those estimates depend heavily on T > 0, any given length of time. Thus one has to derive the uniform estimates in H i .R/ .i D 1; 2; 4/ in which all the constants depend only on min u0 .x/, min 0 .x/, the H i .R/ .i D 1; 2; 4/ norm of .u0  uN ; v0 ; 0  N / (and e0 x2R x2R or E0 ; E1 (see, e.g., Theorem 9)), but independent of any length of time T > 0. Now it suffices first to use some H 1 estimates given in [32, 40, 53] to establish uniform H 1 estimates in the following lemma. Lemma 57. Assume that some constants uN > 0; N > 0; u0  uN ; v0 ; 0  N 2 H 1 .R/ and u0 .x/ > 0; 0 .x/ > 0 on R, and the compatibility conditions hold. Then there exists a constant 0 2 .0; 1 such that (I) If E0 E1  0 , then, in addition to (319) and (320) with T D C1, the H 1 global generalized solution .u.t /; v.t /; .t // to the Cauchy problem (13), (14), and (15), (307) with (292), (293), (294), (295), (296), and (297) and f1 .u/ D

2404

Y. Qin

R=u; F2 .u/ D f2 .u/ D 0 and satisfies that for any .x; t / 2 R  Œ0; C1/, 0 < C11  .x; t /  C1 ;

(338)

0 < C11  u.x; t /  C1 ;

(339)

and for any t > 0, ku.t /  uN k2H 1 C kv.t /k2H 1 C k.t /  N k2H 1 Z t C .kvx k2H 1 C kx k2H 1 C kux k2 C kvt k2 C kt k2 /. /d   C1 ; (340) 0

ku.t /  uN k2L1 C kv.t /k2L1 C k.t /  N k2L1 Z t C .kut k2H 1 C kvx k2L1 C kx k2L1 /. /d   C1 ;

(341)

0

and as t ! C1, k.u.t /  uN ; v.t /; .t /  N /kL1 C k.ux .t /; vx .t /; x .t //k ! 0;

(342)

or (II) If e0  0 , then, in addition to (319) and (320) with T D C1 and (338), (339), (340), (341), and (342), the H 1  global generalized solution .u.t /; v.t /; .t // satisfies that for any .x; t / 2 R  Œ0; C1/, ju.x; t /  uN j C .t/j.x; t /  N j
0 is a positive constant independent of any length of time, (341) easily follows from (340). Case II: Recall that from Jiang [32] (see, e.g., Theorem 1.1 (ii)) there is a constant 2 2 .0; 1 such that if e0  2 , then estimates (342) and (343) and N 2C ku.t /  uN k C kv.t /k C k.t /  k 2

2

Z 0

t

.kvx k2 C kx k2 /. /d   C1 ; 8t > 0 (344)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2405

hold. Clearly, (339) is the direct result of (343). By (343), one gets that for any t  1, 0 < C11  .x; t /  C1 ;

8x 2 R:

(345)

Moreover, from the proofs in Kazhikhov and Shelukhin [40], one derives that C11 e C1 t  .x; t /  C1 e C1 t ;

8 .x; t / 2 R  Œ0; C1/:

Note that this estimate is not enough to derive (338), but combining it with (345) can yield (338). The rest of the proof is similar to the estimates in H 1 (see Lemmas 7 and 8, Qin [60], and Qin, Wu, and Liu [71]). Inasmuch as the uniform H 1 estimates in Lemma 57 have been established, it is only necessary to repeat the same arguments as in the proof of Theorem 1 to be able to reach estimates (327), (328), (329), (330), and (331) in Theorem 9. Now all constants in these estimates will no longer depend on T > 0, any length of time; that is, Ci .C1/ D Ci .i D 1; 2; 4/. In order to finish the proof of Theorem 9, it suffices to prove the results on the large-time behavior of the H i .i D 2; 4/global strong solutions in Theorem 9. The next two lemmas concern the large-time behavior of H i global strong solutions .i D 2; 4/, respectively. To prove them, it suffices to use Lemma 2.6.1 in Zheng [81] to establish some differential inequalities for vt ; t and vt t ; t t in L2 ; see Qin [60], and Qin, Wu, and Liu [71] for details. Lemma 58. Under the assumptions in Theorem 9 with i D2, if e0 0 or E0 E1  0 , then the H 2 global strong solution .u.t /; v.t /; .t // obtained in Theorem 7 to the Cauchy problem (13), (14), and (15), (307) with (292), (293), (294), (295), (296), and (297) and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0 satisfies (333) and (334). Lemma 59. Under the assumptions in Theorem 9 with i D 4, if e0  0 or E0 E1  0 , then the H 4 global strong solution .u.t /; v.t /; .t // obtained in Theorem 8 to the Cauchy problem (13), (14), and (15), (307) with (292), (293), (294), and (295), and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0 satisfies (335) and (336). The proof of Theorem 9 is now complete. Proof of Corollary 4. Applying the embedding theorem, one readily gets estimate (337) and hence completes the proof from Theorem 9. Remark 18. For the Cauchy problem (13), (14), and (15), (307) with (292), (293), (294), (295), (296), and (297), and f1 .u/ D R=u; F2 .u/ D f2 .u/ D 0, Itaya [29], Kanel [34], and Kazhikhov [37–39] obtained the global existence and large-time behavior (only for v;  ) of H 1 global generalized solutions. In this case, Okada and Kawashima [53] established the global existence and large-time behavior of the classical (or H 1 ) generalized solution with small initial data and Jiang [32] proved

2406

Y. Qin

the large-time behavior of the H 1 global generalized solution with weighted small initial data. Qin, Wu, and Liu [71] established the existence and asymptotic behavior of global H i global strong solutions .i D 2; 4/. Remark 19. Hoff and Ziane [25, 26] established the existence of compact attractors for the Navier-Stokes equations of one-dimensional isentropic compressible flow. Remark 20. For the spherically symmetric viscous polytropic ideal gas, the existence and exponential stability and global attractors in H i .i D 1; 2; 4/ of global strong (generalized) solutions have been established (see Chapter 4 in Qin [60], Qin and Song [70], and Zheng and Qin [83]). Remark 21. Duan, Fang, Huang, Liu, Qin, Vong, Xin, Yang, Yao, Zhang, and Zhu [10–15, 33, 42, 43, 62, 64, 65, 76, 78–80] proved the well-posedness of solutions to the 1D isentropic compressible Navier-Stokes equations with vacuum, densitydependent viscosity, and degenerate viscosity coefficient. See also references therein. Remark 22. Ducomet and Zlotnik [8, 9] also established the global existence and asymptotic behavior of solutions for the 1D viscous compressible heat-conducting real gas (13), (14), and (15) with nonmonotone state function (pressure p D p.u;  //. Remark 23. The content of Sect. 4.2 is adapted from Qin, Wu, and Liu [71] and Chapter 3 of Qin [60].

4

Examples of Results in Other Functional Settings

4.1

Initial Boundary Value Problems

In the previous sections, we gave a systematic overview of the theory of onedimensional compressible heat-conducting Navier-Stokes equations in an H i setting. In this section, to complete the picture, we provide some examples in different functional settings. One of the first existence and uniqueness theorems for the polytropic ideal gases (i.e., Eqs. (13), (14), (15), (16), and (17) with (292), (293), (294), (295), (296), and (297) and f1 .u/ D R=u; f2 .u/ D F2 .u/ D 0 (i.e., (24)) is that of Kazhikhov and Shelukhin [40], and Kazhikhov [37]. They established the global existence and regularity results stated in the next theorem. Theorem 10. If .u0 .x/; v0 .x/; 0 .x// 2 H 1 ./;  D .0; 1/; v0 .0/ D v0 .1/ D 0; Z 1 u0 .x/dx D 1; (346) 0

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2407



  m WD min inf 0 .x/; inf 0 .x/ > 0; M WD max sup 0 .x/; x2

x2

 sup 0 .x/ < C1;

x2

(347)

x2

with u0 .x/ D 1=0 .x/, then problem (13), (14), (15), (16), and (17) with (292), (293), (294), (295), (296) and (297) and f1 .u/ D R=u; f2 .u/ D F2 .u/ D 0 (i.e., (24)) admits a unique H 1 generalized global solution .u.x; t /; v.x; t /; .x; t // such that there exists a constant C > 0 (independent of t ) such that k.u.t /; v.t /; .t //kH 1  C; 8 t 2 Œ0; C1/;

(348)

u.x; t/ > 0; .x; t/ > 0; 8 .x; t / 2 Œ0; 1  Œ0; C1/

(349)

where .x; t / D 1=u.x; t /. Moreover, if initial data .0 ; v0 ; 0 / with 0 D 1=u0 have higher regularity: 0 .x/ 2 C 1C˛ ./; .v0 .x/; 0 .x// 2 C 2C˛ ./ and compatibility conditions v0 .x/ jxD0;1 D 00 .x/ jxD0;1 D 0; .v00 .x/=u0 .x//0  K0 .0 .x/=u0 .x//0 jxD0;1 D 0; hold, then the global generalized solution ..x; t /; v.x; t /; .x; t // is a classical one such that .x; t / 2 C 1C˛;1C˛=2 .QT /; .v.x; t /; .x; t // 2 C 2C˛;1C˛=2 .QT /; with QT D .0; 1/  .0; T /. This regularity result also holds for the boundary conditions: .vx  K0  / jxD0;1 D 0; x jxD0;1 D 0; with compatibility conditions v0 .x/  K0 0 .x/ D 0; 00 .x/ D 0 at endpoints x D 0; 1; see [37] for details. For the boundary conditions:  jxD0;1 D v jxD0;1 D 0; also see [37]. Stra˘skraba and Zlotnik [73, 74, 84–89] investigated in detail the global wellposedness of equations for the so-called viscous barotropic gases. Here we present one existence result from [87] and some results on the asymptotic behavior from [87] and [73]. Now consider the following 1D viscous compressible barotropic fluid equations in Euler coordinates, t C .u/x D 0;

(350)

.u/t C .u2 /x  .ux  P .//x D f;

(351)

2408

Y. Qin

in the domain Q D   RC WD .0; 1/  .0; C1/ subject to the boundary and initial conditions u jxD0;1 D 0;  jtD0 D 0 .x/; u jtD0 D u0 .x/ i n ;

(352)

where ; u are the density and velocity of the fluid. A pair .; u/ is called a regular generalized solution to (350), (351) and (352) if .; u/ 2 W .QT /  W22;1 .QT / such that  > 0 in QT , and (350) and (351) hold in L2 .QT /, and (352) is satisfied in the sense of C .QT /. The global existence result is due to [87], which is stated in the following theorem. Theorem 11. Assume that the pressure P is nonincreasing and satisfies the local Lipschitz condition on RC D .0; C1/ (the second condition is equivalent to the C C C relation P 0 2 L1 loc .R /). Let the function f .x; t/ be measurable on R  R , continuous with respect to x  0 for almost all t, such that sup jf .x; t/j  fN .t/ on RC :

(353)

x0

Moreover, assume that 0 2 H 1 ./; u0 2 H01 ./; fN 2 Lr .RC / f or r D 1; 2;

(354)

and there exists a constant N > 0 such that N 1  0 .x/  N on ; kDx 0 kL2 ./ C kDx u0 kL2 ./  N; kfN kLr .RC /  N f or r D 1; 2:

(355)

Then problem (350), (351) and (352) admits a regular generalized solution .; u/ 2 W .QT /  W22;1 .QT / for all T > 0 satisfying C 1 .N /  .x; t /  C .N /; i n Œ0; 1  Œ0; C1/I kkW .Q/ C kukW 2;1 .Q/  C .N / 2 (356) and C

Dx ; Dx u 2 C .R ; L2 .//

(357)

with a universal constant C .N / > 0 depending on N . The next result, also due to [87], concerns the asymptotic behavior of the regular generalized solution. Theorem 12. Under the assumptions of Theorem (11), a regular generalized solution .; u/ of problem (350), (351) and (352) has the stabilization property, as t ! C1,

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

.; t / ! s ./ i n C ./; u.; t / ! 0 i n H 1 ./

2409

(358)

and the limit function s D 1=s 2 H 1 ./ satisfies the relation P .s .x// D P .VN / RL with VN D L1 0 0 .x/dx being the mean of the function 0 D 1=0 over R1 Œ0; L; L D 0 0 .x/dx. The next result, due to [73], is a corollary of Theorem 12. Corollary 5. Assume that the external force takes the form f .x; t/ D f1 .x/ C f .x; t /

(359)

with, for all T > 0, f1 2 L1 ./; f D f1 C f2 ; f1 2 L1;1 .Q/ \ L1;2 .QT /

(360)

and f2 2 L1;2 .Q/

(361)

with QT D   .0; T /. Assume also the initial data 0 ; u0 2 H 1 ./

(362)

0 < 0  0 ; u0 jxD0;l D 0;

(363)

and

and the state function P .r/ is continuous and increasing on Œ0; C1/ such that C 0 C P .0/ D 0; P .C1/ D C1; P 0 2 L1 loc .R /; rP .r/ D O.1/ as r ! 0 ;

P .r/ D O.r 0 /

(364)

as 0 ! 0C for some 0 < 0  1. Suppose also that the viscosity coefficient  D const ant > 0:

(365)

(Clearly, these conditions (359), (360), (361), (362), (363), (364) and (365) are satisfied for the most popular state functions P .r/ D p1 r with constants p1 > 0 and > 0, and also satisfy the assumptions of Theorem 12. Note that (359), (360) and (361) clearly satisfy (353)). Moreover, if the following conditions hold for some constant N > 0,

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Y. Qin

0 < 0  N; ku0 kL2 ./  N; kf1 kL1 ./  N;

(366)

kf1 kL1;1 .Q/ C kf2 kL1;2 .Q/  N; kP .0 /kL1 ./  N;

(367)

then p k ukL2;1 .Q/ C kP ./kL1;1 .Q/ C kux kL2 .Q/  K1 .N /; p .x; t /  N WD K2 .N /; k ukL2 ./ ! 0 as t ! C1:

(368) (369)

In [4] Hoff and coauthors investigated the existence and uniqueness of discontinuous solutions to an initial boundary value problem emanating from discontinuous initial data, and the propagation of jumps in the case of initial data with jumps. We give two theorems concerning these topics. The first theorem from [4] deals with the system: ut  vx D 0;  v x vt C p.u; e/x D  ; u x     v2 vvx C ex eC C .vp.u; e//x D ; 2 t u x

(370) (371) (372)

subject to the boundary conditions v.x; t / jxD0;1 D ex .x; t / jxD0;1 D 0;

(373)

.u; v;  / jtD0 D .u0 .x/; v0 .x/; e0 .x//;

(374)

and initial conditions

where .x; t / 2 .0; 1/  Œ0; C1/; and u; v; e, and p stand for the specific volume, velocity, specific internal energy, and pressure;  and are fixed positive viscosity constants; and x is a Lagrangian coordinate so that x D const ant corresponds to a particle trajectory. Notice that (370), (371), (372), (373) and (374) is in fact system (13), (14) and (15) with  D .u; e/ D e and k.u;  / D D const ant . Theorem 13. Assume that u; e;  , and p are related by the equation of state of an ideal polytropic fluid (23) and there is a constant C0 > 0 such that 0 < C01  u0 .x/  C0 ; e0 .x/  C01 ; kv0 kL4 C ke0 kL2 C T VŒ0;1 .u0 /  C0 : (375) Then there is a global solution .u; v; e/ to problem (370), (371), (372), (373), and (374) such that .u.x; t /; v.x; t /; e.x; t // 2 C .Œ0; C1/; L2 .0; 1//;

(376)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2411

with e.; t / * e0 weakly in L2 .0; 1/ as t ! 0C . Moreover, there is a constant M D M .C0 / > 0 depending on C0 > 0, but independent of t such that M 1  u.x; t /  M; M 1  e.x; t /  M  1 .t /; 8.x; t / 2 Œ0; 1  Œ0 C 1/; T VŒ0;1 .u.; t //  M; ku.; t /  u.; t 0 /kL2 .0;1/  M jt  t 0 j1=2 ; kvx .; t /kL2 .0;1/  M  1=2 .t /; kex .; t /kL2 .0;1/  M  1 .t /; kv.; t /kL1 .0;1/  M  1=4 .t /; E.t / C F .t /  M;  E.t / D sup .s/kvx .; s/k2L2 .0;1/ C  2 .s/kex .; s/k2L2 .0;1/ 0st

C

Z t 0

kvx .; s/k2L2 .0;1/ C kex .; s/k2L2 .0;1/ C .s/kvt .; s/k2L2 .0;1/

C 2 .s/ket .; s/k2L2 .0;1/ ds;  F .t/ D sup  2 .s/kvt .; s/k2L2 .0;1/ C  3 .s/ket .; s/k2L2 .0;1/ 0st

C

Z t 0

 2 .s/kvxt .; s/k2L2 .0;1/ C  3 .s/kext .; s/k2L2 .0;1/ ds;

and .t / D min.1; t/, and T VŒ0;1 .u.; t // stands for the total variation of u. Finally, the solution .u; v; e/ tends to a constant state as t ! C1 in the sense that k.u  u1 /.; t /kL1 .0;1/ C k.v; e  e1 /.; t /kH 1 .0;1/ ! 0; where u1 D

4.2

R1 0

u0 .x/dx and e1 D

R1 0

.e0 .x/ C v02 .x/=2/dx.

The Cauchy Problems

In [21–24] Hoff and coauthors investigated the existence and uniqueness of discontinuous solutions to the Cauchy problem emanating from the discontinuous initial data, and the propagation of jumps in the case of initial data with jumps. They also observed the existence of the so-called initial layer (see estimates including function t 7! min.1; t/ in Theorem 14). The context of the second theorem corresponding to papers by Hoff [21–24] is the following. Consider the Cauchy problem (370), (371), and (372), (374) with ex replaced by x in (372) where p D p.u; e/;  D .u; e/. Assume first that p and  satisfy the conditions under which Hoff’s local existence theory [22] applies; thus there exists a set K1 of the form Œu; uN   Œe; C1/ in the positive quadrant of the .u; e/plane, and a constant C > 0 such that for any .u; e/ 2 K1 ,

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Y. Qin

C 1   e  C ; C 1  e .u; e/;

(377)

je .u; e/; u .u; e/; pe .u; e/j  C; 0 < p  C ; C   pu .u; e/  C: (378) (One can easily check that these conditions (377) and (378) hold for ideal gases (24) as well as for Van der Waals fluids.) As in [22], it is easy to find that if u0 is piecewise smooth, having isolated jumps as points y1 <    < yN , then discontinuous in u; e; p; vx and  convect along the corresponding particle paths x D yi , and satisfy the jump conditions vx i D 0; p.u; e/  u

h



x u

 D 0;

(379)

where Œw denotes a jump in a given function across x D yi , Œw.yi ; t / D w.yi C 0; t /  w.yi  0/:

(380)

In addition, the quantities v; ; vx =u  p and x =u become smoothed out, and in fact are Hölder continuous in t > 0. A linear ODE for the quantity ŒL, where L D log u follows from (379). In fact, the standard divided difference notation can be defined as ( g.z2 /g.z1 / ; z1 ¤ z2 ; z1 z2 gŒz1 ; z2  D (381) z1 D z2 ; g 0 .z1 /; for functions g.y/; and for functions f .L; e/, the partial functions f L and f e are defined by f e .L/ D f L .e/ D f .L; e/: Assume that (379) holds along the particle path x D yi : Let e˙ D e.yi ˙ 0; t /, and so on, and, abusing notation slightly, regard p and  as functions of L and e. Equations (379) and (370), (371), and (372) (with ex replaced by x in (372) where p D p.u; e/;  D .u; e/) then show h u i h v i t x D D Œp ŒLt D u u L

D pLe ;LC ŒL C pLC;LC Œe: Note that Œ D 0 implies that Œe D ˇi ŒL; L

where ˇi D Le ;LC =eC;eC , which gives

(382)

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

ŒLt D

˛i ŒL; 

2413

(383)

where L

˛i D pLe ;LC C ˇi peC;eC :

(384)

Thus it follows that 

1

Z

ŒL.yi ; t / D ŒL.yi ; 0/ exp 



t

˛i .s/ds :

(385)

0

Now (377) and (378) show that ˛i  C so that ŒL can grow at most exponentially in time. On the other hand, in order to control u and e pointwise for all time, one requires that .u; e/ eventually becomes confined to a set in which ˛i is negative, so that ŒL decays exponentially in time. By (377) and (379), the same will then be true for the jumps in e; p; vx , and x . Now introduce the following definition. Definition 4. Let K2 be a bounded set of the form Œu; uN   Œe; eN  contained in the positive quadrant of the .u; e/plane. Then p and  are said to satisfy the conditions of a near ideal gas in K2 if, in addition to conditions (377) and (378), there is a positive constant C such that, first, given any two points .u ; e /; .uC ; eC / 2 K2 , the quantity ˛i defined by (384) satisfies ˛i  1=C ; and second, there is a smooth function S W K2 ! R such that @S 1 @S p D ; D ; S 00  1=C i n K2 : @e  @u 

(386)

One can easily check that such an ideal gas (24) also satisfies the conditions of Definition 5.1 with ˇi D 0; ˛i D @p=@L D upu < 0 and S .u; e/ D const: log uuQ C R e de eQ .e/ : In this case, because  is an invertible function of e, e and  will have identical regularity properties. Assume next that we fix a constant vQ and a point .Qu; eQ / in the positive quadrant of the .u; e/plane, and let y1 <    < yN be distinguished points at which discontinuities in u0 may occur. Define C0 D k.u0 ; v0 ; e0 /k2 C ku00 k2 C

N X

!2 jŒu0 .yi /j

i D1

CT VŒ0;1 .v0 /2 C ku0 kL1 C ke0 kL1 :

(387)

Here u D u  uQ , and so on, and k  k denotes L2 .R/ norm, and k  ¬ is the R yi C1 PN the 2 2 piecewise L norm defined by k w ¬ D w2 dx with y0 D 1 and iD0 yi yN C1 D C1.

2414

Y. Qin

To describe various quantities in (370), (371), and (372) (with ex replaced by x in (372) where p D p.u; e/;  D .u; e/), and their derivatives, fix a number

2 .0; 1/ and define the functionals A; B, and F as 2 A.t/ D sup 4k.u; v; e/k2 C k ux ¬2 C 0st

Z

N X

!2 3 jŒu.yi ; s/j 5

i D1

t

.kvx .s/k2 C kx .s/k2 /ds;

C 0

  B.t / D sup .s/1=2 kvx .; s/k2 C .s/kx .; s/k2 0st

Z

t



.s/1=2C kvt .; s/k2 C k .vx =u/x .; s/ ¬2 ds

t



.s/ kvx2 .; s/k2 C kt .; s/k2 C k.x =u/x .; s/k2 ds;

C Z

0

C h

0



F .t/ D .s/3=2C kvt .; s/k2 C k .vx =u/x .; s/ ¬2

i C.s/3 kt .; s/k2 C k.x =u/x .; s/k2 Z t   C .s/3=2C kvxt .; s/k2 C .s/3 kxt .; s/k2 ds; 0

with .s/ D min.1; s/. Then the following result due to Hoff [23] can be stated as follows. Theorem 14. Let K0 ; K1 ; K2 be closed rectangles in the positive quadrant of the .u; p/plane with K2 bounded, and K0  i nt .K1 /: Assume that p and  satisfy the conditions (377) and (378) of a general gas in K1 , and the conditions of Definition 5.1 for a near-ideal gas in K2 . Fix a number 2 .0; 1/ and a constant state .Qu; v; Q Q / with .Qu; eQ / 2 i nt .K0 \ K2 /. Then there are positive constants . ; K0 ; K1 ; K2 /;  . ; K0 ; K1 /, and C . ; K1 ; K2 / such that if C0   and .u0 ; e0 / 2 K0 a.e., then problem (370), (371), and (372), (374) with ex replaced by x in (372) where p D p.u; e/;  D .u; e/ admits a global weak solution .u; v;  / such that (1)  .u; e/ 2

(2) For all t > 0,

K1 ; 0  t   ; K2 ;   t:

42 Well-Posedness and Asymptotic Behavior for Compressible Flows in One. . .

2415

A.t/ C B.t / C F .t/  C C0 : (3) The following regularity results are (some of the) consequences of (2): 1=2;1=2 =4

1=2

< u >.yi ;yi C1 /Œ0;C1/  C C0 ; i D 0;    ; N I 1=2;1=4

.t /1=4C =2 < v >RŒt;C1/ C.t /1=2C =4 kvx .; t /kL1 C.t /3=4C =2
RŒt;C1/  C C0 ; u

1=2;1=4

.t /1=2 <  >RŒt;C1/ C.t /kx .; t /kL1 C.t /3=2
 C C0 ; u RŒt;C1/

1=2;1=4

1=2

.t /1=2 < e >.yi ;yi C1 /Œt;C1/  C C0 : (4) The quantities u.; t /; e.; t /; p.; t /; vx .; t /, and x .; t / have one-sided limits at each x D yi for all t > 0 and i D 1;    ; N , and the jump conditions (379) hold pointwise. Moreover, the relations (382), (383), (384), and (385) hold at each .yi ; t / for all t > 0, so that 1=2

j Œ.u; e; p; vx ; x /.yi ; t / j C C0

exp.t =C /:

(5) The solution .u; v; e/ tends to .Qu; v; Q Q / as t ! C1 in the sense that lim k.v.; t /; .; t //kL1 .R/ D 0;

t!C1

and lim ku.; t /kL1 .I / ! 0;

t!C1

for each bounded interval I .

5

Conclusion

In this conclusion, we refer to some literature that may be useful to the reader in the study of the topics related to this chapter. For the 1D case, Antontsev, Kazhikhov, and Monakhov [2] discussed a number of important problems including the compressible viscous heat-conducting gas with some boundary conditions; Hsiao [27] investigated the quasilinear hyperbolic systems and dissipative systems, and Hsiao and Jiang [28] studied the nonlinear hyperbolic-parabolic systems including the thermoviscoelastic models; Qin and Huang [61] and Qin, Liu, and Wang [67] investigated the global well-posedness

2416

Y. Qin

of real gases with shear viscosity, pth power gases, reactive viscous radiative gases, and other fluid models including compressible radiative MHD, viscous micropolar fluid models, and non-Newtonian fluid models. For the multidimensional case, Lions [41] published two volumes of monographs that present various mathematical results on fluid mechanics models such as Navier-Stokes equations both in the incompressible case and in the compressible case; Novotný and Stra˘skraba [52] provided a comprehensive introduction to the mathematical theory of compressible flow describing both inviscid and viscous compressible flow that are governed by the Euler and Navier-Stokes equations, respectively; Feireisl [16] also presented a complete mathematical theory for the 3D full system of the Navier-Stokes equations of viscous compressible and heatconducting flows, and Feireisl and Novotný [18] discussed the singular limits of weak solutions to the system governing the flow of thermally conducting compressible viscous fluids. One can also refer to references [3, 6, 7, 20, 63]

6

Cross-References

 Concepts of Solutions in the Thermodynamics of Compressible Fluids  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Finite Time Blow-Up of Regular Solutions for Compressible Flows  Global Existence of Classical Solutions and Optimal Decay Rate for Compress-

ible Flows via the Theory of Semigroups  Symmetric Solutions to the Viscous Gas Equations  Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stabil-

ity, and Longtime Behavior  Weak Solutions for the Compressible Navier-Stokes Equations in the Intermediate

Regularity Class  Well-Posedness of the IBVPs for the 1D Viscous Gas Equations Acknowledgements This work was supported in part by the National Natural Science Foundation (NNSF) of China with contract numbers 11271066 and 11671075.

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32. S. Jiang, Large-time behavior of solutions to the equations of a viscous polytropic ideal gas. Ann. Mate. Pura Appl. CLXXV, 253–275 (1998) 33. S.Jiang, Z. Xin, P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. Methods Appl. Anal. 12, 239–252 (2005) 34. K.I. Kanel, Cauchy problem for the equations of gas dynamic viscosity. Sib. Math. J. 20, 208– 218 (1979) 35. S. Kawashima, T. Nishida, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases. J. Math. Kyoto. Univ. 21, 825–837 (1981) 36. B. Kawohl, Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas. J. Differ. Equ. 58, 76–103 (1985) 37. A.V. Kazhikhov, Sur la´solubilite´globole des problémes monodimensoinnels aux valeurs initiales-limitées pour les équations dún gaz visqueux et calorifére. C. R. Acad. Sci. Paris Ser. A 284, 317–320 (1977) 38. A.V. Kazhikhov, To a theory of boundary value problems for equations of one-dimensional nonstationary motion of viscous heat-conduction gases. Boundary value problems for hydrodynamical equations (in Russian). Inst. Hydrodyn. Sib. Branch Akad. USSR. 50, 37–62 (1981) 39. A.V. Kazhikhov, Cauchy problem for viscous gas equations. Siberian Math. J. 23(1), 44–49 (1982) 40. A.V. Kazhikhov, V.V. Shelukhin, Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41, 273–282 (1977) 41. P.L. Lions, Mathematical Topics in Fluid Dynamics. Vol. 1, Incompressible Models, Vol. 2, Compressible Models (Oxford Science Publication, Oxford, 1998) 42. T. Liu, T. Yang, Compressible flow with vacuum and physical singularity. Methods Appl. Anal. 7(3), 495–509 (2000) 43. T. Liu, X. Xin, T. Yang, Vacuum states for compressible Navier-Stokes flow. Disc. Contin. Dynm. Syst. 4A(1), 1–32 (1998) 44. A. Matsumura, T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Jpn. Acad. Ser. A 55, 337–341 (1979) 45. A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto. Univ. 20, 67–104 (1980) 46. A. Matsumura, T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Contemp. Math. 17, 109–116 (1983) 47. A. Matsumura, T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89, 445–464 (1983) 48. T. Nagasawa, On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary. J. Differ. Equ. 65, 49–67 (1986) 49. T. Nagasawa, On the asymptotic behavior of the one-dimensional motion of the polytropic ideal gas with stress-free condition. Quart. Appl. Math. 46, 665–679 (1988) 50. T. Nagasawa, On the outer pressure problem of the one-dimensional polytropic ideal gas. Jpn. J. Appl. Math. 5, 53–85 (1988) 51. T. Nagasawa, Global asymptotics of the outer pressure problem with free boundary. Jpn. J. Appl. Math. 5, 205–224 (1988) 52. A. Novotný, I. Stra˘skraba, Introduction to the Mathematical Theory of Compressible Flow (Oxford University Press, Oxford/New York, 2004) 53. M. Okada, S. Kawashima, On the equations of one-dimensional motion of compressible viscous fluids. J. Math. Kyoto Univ. 23, 55–71 (1983) 54. Y. Qin, Global existence and asymptotic behavior of solutions to a system of equations for a nonlinear one-dimensional viscous, heat-conducting real gas. Chin. Ann. Math. 20A, 343–354 (1999) (in Chinese) 55. Y. Qin, Global existence and asymptotic behavior for the solutions to nonlinear viscous, heatconductive, one-dimensional real gas. Adv. Math. Sci. Appl. 10, 119–148 (2000)

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56. Y. Qin, Global existence and asymptotic behavior for a viscous, heat-conductive, onedimensional real gas with fixed and thermally insulated endpoints. Nonlinear Anal. TMA 44, 413–441 (2001) 57. Y. Qin, Global existence and asymptotic behavior for a viscous, heat-conductive, onedimensional real gas with fixed and constant temperature boundary conditions. Adv. Differ. Equ. 7, 129–154 (2002) 58. Y. Qin, Exponential stability for a nonlinear one-dimensional heat-conductive viscous real gas. J. Math. Anal. Appl. 272, 507–535 (2002) 59. Y. Qin, Universal attractor in H 4 for the nonlinear one-dimensional compressible NavierStokes equations. J. Differ. Equ. 207, 21–72 (2004) 60. Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors. Advanced Partial Differential Equations, vol. 184 (Springer/Birkhauser, London/Basel, 2008) 61. Y. Qin, L. Huang, Global Well-Posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems (Birkhauser/Springer, Basel, 2012) 62. Y. Qin, L. Huang, S. Deng, Z. Ma, X. Su, X. Yang, Regularity of the compressible NavierStokes equations with degenerate viscosity coefficient and vacuum. Discr. Contin. Dyn. Syst.-S 2(1), 163–192 (2009) 63. Y. Qin, L. Huang, Z. Ma, Global existence and exponential stability in H 4 for the nonlinear compressible Navier-Stokes equations. Commun. Pure Appl. Anal. 8(6), 1991–2012 (2009) 64. Y. Qin, L. Huang, Z. Yao, Regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. J. Differ. Equ. 245, 3956–3973 (2008) 65. Y. Qin, L. Huang, Z. Yao, A remark on regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. J. Math. Anal. Appl. 351, 447–508 (2009) 66. Y. Qin, H. Liu, C. Song, Global attractor for a nonlinear thermoviscoelastic system in shape memory alloys. Proc. R. Soc. Edingb. 138A(5), 1103–1135 (2008) 67. Y. Qin, X. Liu, T. Wang, Global Existence and Uniqueness of Nonlinear Evolutionary Fluid Equations (Birkhauser/Springer, Basel, 2015) 68. Y. Qin, T. Ma, M.M. Cavalcanti, D. Andrade, Exponential stability in H 4 for the NavierStokes equations of viscous and heat conductive fluid. Commun. Pure Appl. Anal. 4, 635–664 (2005) 69. Y. Qin, J.E. Muñoz Rivera, Universal attractors for a nonlinear one-dimensional heatconductive viscous real gas. Proc. R. Soc. Edinb. A 132, 685–709 (2002) 70. Y. Qin, J. Song, Maximal attractors for the compressible Navier-Stokes equations of viscous and heat conductive fluid. Acta Math. Sci. 30B(1), 289–311 (2010) 71. Y. Qin, Y. Wu, F. Liu, On the Cauchy problem for one-dimensional compressible Navier-Stokes equations. Port. Math. 64(1), 87–126 (2007) 72. G.R. Sell, Global attractors for the three-dimensional Navier-Stokes equations. J. Dyn. Differ. Equ. 8, 1–33 (1996) 73. I. Stra˘skraba, A.A. Zlotnik, On a decay rate for 1D-viscous compressible barotropic fluid equations. J. Evol. Equ. 2, 69–96 (2002) 74. I. Stra˘skraba, A.A. Zlotnik, Global behavior of 1  d viscous compressible barotropic fluid with a free boundary and large data. J. Math. Fluid Mech. 5, 119–143 (2003) 75. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied mathematical sciences, vol. 68 (Springer, New York, 1988) 76. S. Vong, T. Yang, C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum (II). J. Differ. Equ. 192, 475–501 (2003) 77. Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equations with compact support. Commun. Pure Appl. Math. 51, 229–240 (1998) 78. T. Yang, Z. Yao, C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Commun. Partial Differ. Equ. 26, 965–981 (2001) 79. T. Yang, H. Zhao, A vacuum problem for the one-dimensional compressible Navier-Stokes equations with densitydependent viscosity. J. Differ. Equ. 184, 163–184 (2002) 80. T. Yang, C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Commun. Math. Phys. 230, 329–363 (2002)

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81. S. Zheng, Nonlinear Evolution Equations. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 133 (CRC Press, Boca Raton, 2004) 82. S. Zheng, Y. Qin, Maximal attractor for the system of one-dimensional polytropic viscous ideal gas. Quart. Appl. Math. 59, 579–599 (2001) 83. S. Zheng, Y. Qin, Universal attractors for the Navier-Stokes equations of compressible and heat-conductive fluid in bounded annular domains in Rn . Arch. Ration. Mech. Anal. 160, 153–179 (2001) 84. A.A. Zlotnik, On equations for one-dimensional motion of a viscous barotropic gas in the presence of a body force. Sib. Math. J. 33, 798–815 (1992) 85. A.A. Zlotnik, Uniform estimates and stabilization of solutions to equations of one-dimensional motion of a multicomponent barotropic mixture. Math. Notes 58, 885–889 (1995) 86. A.A. Zlotnik, On stabilization for equations of symmetric motion of a viscous barotropic gas with large mass force. Vestn. Moskov. Energ. Inst. 4, 57–69 (1997) 87. A.A. Zlotnik, Stabilization of solutions of a certain quasilinear system of equations with a weakly monotone nonlinearity. Differ. Equ. 35, 1423–1428 (1999) 88. A.A. Zlotnik, Uniform estimates and the stabilization of symmetric solutions to one system of quasilinear equations. Differ. Equ. 36, 701–716 (2000) 89. A.A.Zlotnik, V.V. Shelukhin, Equations of one-dimensional motion of a viscous barotropic gas in the presence of a mass force. Sib. Math. Zh. 33, 62–79 (1992)

Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

43

Alexander Zlotnik

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation and Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Well-Posedness of the IBVPs for the Viscous Barotropic Gas 1D Equations . . . . . . . . . 3.1 Weak Solutions, Their Global Existence, and Regularity . . . . . . . . . . . . . . . . . . . . 3.2 The Uniqueness and Lipschitz Continuous Dependence on Data for Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Well-Posedness of the IBVPs for the Viscous Heat-Conducting Gas 1D Equations . . . . 4.1 Additional Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Weak Solutions and Their Global Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Uniqueness and Lipschitz Continuous Dependence on Data for Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Regularity of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2422 2423 2429 2429 2451 2458 2458 2460 2481 2487 2491 2491 2491

Abstract

The inhomogeneous initial-boundary value problems (IBVPs) are posed for the Navier-Stokes systems of equations describing the viscous barotropic and heatconducting gas 1D flow in the Lagrangian mass coordinates. Weak solutions are studied without any restrictions on the magnitude of norms of data. Assumptions on the data are genuinely general, in particular, the initial data are taken from the Lebesgue spaces, the contact problems for different gases are covered, etc. Both the global in time existence of the weak solutions as well as their uniqueness

A. Zlotnik () Faculty of Economic Sciences, Department of Mathematics, National Research University Higher School of Economics, Moscow, Russia e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_33

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and Lipschitz continuous dependence on data are proved thus ensuring the wellposedness of the IBVPs. The regularity issue is studied as well.

1

Introduction

In this chapter, initial-boundary value problems (IBVPs) are considered for the Navier-Stokes systems of quasilinear equations describing the viscous barotropic gas (in Sect. 3) and heat-conducting gas (in Sect. 4) 1D in space flow in the Lagrangian mass coordinates. The most well-known results concerning the systems belong to A.V. Kazhikhov (with collaborators) who studied global strong solutions in the case of the initial and boundary data in the Sobolev spaces. For the heat-conducting case, they are well presented in [16, Chapter 2]; for the barotropic case, see also [23, 24]. The case of global in time weak solutions for discontinuous data and without imposing any restrictions on the magnitude of norms of data was treated under various assumptions on data in [6–9, 13, 19–22, 30–38], etc. The well-posedness of the IBVPs is established in the relevant classes of weak solutions by proving both global existence theorems and theorems on the uniqueness and Lipschitz continuous dependence on data (in the same classes). The genuinely general and physically important case is covered where the initial density is in L1 and strictly positive; the initial total energy is finite and, in the heat-conducting case, the initial temperature is non-strictly positive; and the initial total entropy is finite as well. In addition, some internal and up to the boundary regularity theorems and, as corollaries, the existence of strong solutions for the data in the Sobolev spaces are presented. Also several nonhomogeneous boundary conditions are posed containing the discontinuous boundary data: the velocity of pistons of bounded variation as well as the external pressures and heat fluxes in the Lebesgue spaces. All the boundary conditions are treated in the maximally unified way. The momentum and energy equations are inhomogeneous too and contain the body force and heat source from the Lebesgue spaces too; moreover, they are allowed to depend on the Eulerian coordinate as well. The general non-monotone pressure law (and the viscosity dependence) in the barotropic case and the perfect polytropic gas equations of state in the heatconducting case are treated. Moreover, in both cases, the physical constants are allowed to depend on the Lagrangian mass coordinate in L1 way thus covering the contact problems between different gases as well. The full proofs of the main results are given (for brevity, omitting proofs of some auxiliary results on weak solutions to the 1D linear parabolic IBVPs). Concerning the global existence theorems, the proofs are based on constructing suitable semidiscrete (finite difference in space) methods for our IBVPs, proving required bounds for their solutions independently of the discretization parameter h > 0 and passing to the limit as h ! 0 globally in time. Thus the proofs are selfcontained; moreover, once the approach is successful, the semidiscrete methods take

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

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the value for computations. Though often proofs of a priori bounds for semidiscrete problems are more complicated and lengthy than for the original ones, the special efforts are accomplished to make them close so that derivatives and integrals are in use (but not finite differences and sums). The presentation is mainly based on papers [8, 9] in Section 3 and [13, 36] in Section 4; for the regularity results, see also [12, 14], and, in addition, see [39].

2

Notation and Auxiliary Results

We introduce notation that will be used throughout the chapter. Let  D .0; X /, Q D QT WD   .0; T /, and Q WD f0; Xg  .0; T /. Denote Dw D @w , Dt w D @x Rt @w , and .It w/.x; t / D 0 w.x;  / d  for functions w depending on x 2  and/or @t t 2 .0; T /. The classical Lebesgue space Lq .G/ (where G is a bounded domain) together with its anisotropic version Lq;r .Q/ equipped with the norm kwkLq;r .Q/ D    kw.; t /kLq ./  r for q; r 2 Œ1; 1 is used. Set for brevity .w; v/G D L .0;T / R R 0 q wv dG for w 2 L .G/ and v 2 Lq .G/ as well as .w; v/Q D Q wv dxdt G 0

0

for w 2 Lq;r .Q/ and v 2 Lq ;r .Q/, where, for example, q10 C q1 D 1. In the proofs, the abbreviation k  kG D k  kL2 .G/ is used. For a Banach space B, let Lq .GI B/ be the space of the strongly measurable functions w: G ! B having the finite norm kwkLq .GIB/ D kkw.; t /kB kLq .G/ and C .Œ0; T I B/ be the space of continuous functions w: Œ0; T  ! B equipped with the norm kwkC .Œ0;T IB/ D kkw.; t /kB kC Œ0;T  . For a pair .w1 ; w2 / of functions in B, let k.w1 ; w2 /kB D kw1 kB C kw2 kB . The standard Sobolev spaces W 1;q .G/, with H 1 .G/ WD W 1;2 .G/, are utilized. We introduce the class (not the space) N .Q/ of functions w 2 L1 .Q/ such that w > 0, 1=w 2 L1 .Q/, and Dt w 2 L2 .Q/, and set > > > > > >w> >N .Q/ D kwkL1 .Q/ C k1=wkL1 .Q/ C kDt wkL2 .Q/ : Let Vq .Q/ and W .Q/ be the Banach spaces equipped with the norms kwkVq .Q/ D kwkLq;1 .Q/ C kDwkLq .Q/ ; kwkW .Q/ D kwkL2 .Q/ C kDt wkL2 .Q/ C kDwkL2;1 .Q/ : Recall that for all q 2 Œ1; 2, r 2 Œ4; 1 such that .2q/1 C r 1 D 1=4 one has [27] kwkLq;r .Q/  ckwkV2 .Q/ ;

(1)

kwjxD0 kL4 .0;T / C kwjxDX kL4 .0;T /  ckwkV2 .Q/

(2)

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A. Zlotnik h1;1=2i

for any w 2 V2 .Q/. We define also the Banach space V2 norm

h0;1=2i

kwkV h1;1=2i .Q/ D kwkV2 .Q/ C kwk2;2

.Q/ equipped with the

; with

2

h0;1=2i

kwk2;r

D sup  1=2 k. / wkL2;r .QT  / ; 0 1 be an arbitrarily large parameter. We impose the following conditions: ˛; ~ 2 L1 .Q/; N 1  ˛  N; N 1  ~  N; .˛v/0 2 L2 ./; v 2 V Œ0; T ; s 2 L4=3 .0; T /:

2 L2 .Q/; f 2 F m .Q/; (8) (9)

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Following [27], the function v 2 V2 .Q/ is called a weak V2 .Q/-solution (i.e., the solution from the energy class) to problem Lm provided that: (a) the integral identity Z



˛vDt ' C .~Dv 

 /D'  f ' dxdt

Q

Z D

0

Z

T

.˛v/ dx C 

0

ˇ˛DX .s˛ 'jxD˛ /ˇ˛D0 dt

(10)

holds for any ' 2 C 1 .Q/ such that 'jtDT D 0, with 'jxD0;X D 0 for m D 1 or 'jxDX D 0 for m D 2; (b) the Dirichlet boundary conditions vjxD0 D v0 for m D 1 and vjxDX D vX for m D 1; 2 are valid in L2 .0; T / in the sense of traces of v 2 V2 .Q/. Proposition 1. Let conditions (8)–(9) be valid. v 2 V2 .Q/ satisfies identity (10) if and only if there exists DIt s 2 L1;1 .Q/ and the following equalities hold: ˛v D .˛v/0 C DIt s C It f in L1;1 .Q/; .It s/jxD0 D It s0 for m D 2; 3 and .It s/jxDX D It sX for m D 3; in L1 .0; T /: The proof is rather easy; see [9, Lemma 2.1]. Proposition 2. Let conditions (8)–(9), kDt ˛kLq1 ;r1 .Q/  N , for some q1 ; r1 2 Œ1; 1, .2q1 /1 C r 1 D 1, and .˛v/0 D ˛jtD0 v 0 with v 0 2 L2 ./ be valid. Then the weak V2 .Q/-solution v to problem Lm exists, is unique, and satisfies the generalized energy bound  kvkV h1;1=2i .Q/  K.N / kv 0 kL2 ./ C k kL2 .Q/ C kf kF m .Q/ 2  Ckv kV Œ0;T  C ks kL4=3 .0;T / :

(11)

Moreover, if k. / f kF m .QT  / ! 0 as  ! 0C , then v 2 C .Œ0; T I L2 .// and its value at t D 0 satisfies vjtD0 D v 0 . Hereafter K.N / (probably with indices) are positive nondecreasing functions of N ; they can depend on X and T as well and are nondecreasing in T . Proposition 2 follows from a version [11, 12] of the well-known results for parabolic equations [27]. Its semidiscrete counterpart is proved below; see Proposition 6. The weak V1 .Q/-solution v to problem Lm is defined quite similarly to the above definition but using the spaces L1 .0; T / and V1 .Q/ in its item (b). Let us present a bound for a weak norm of v by weak norms of the data [9].

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

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Proposition 3. Let conditions (8)–(9), kDt ~kL2;4=3 .Q/  N , 2 L2;1 .Q/, f 2 L1 .Q/, .˛v/0 2 L1 ./, v 2 L4=3 .0; T /, and s 2 L1 .0; T /. If the function v is a weak V1 .Q/-solution to problem Lm , then it satisfies the bound  k˛vkL1 ..0;T /IH 1Im .// C kvkL2 .Q/  K.N / k.˛v/0 kH 1Im ./

 m m Ck ke F ~ .Q/ C kIf ke F ~ .Q/ C ı3;m khf i kL1 .0;T / C kv kL4=3 .0;T / C ks kL1 .0;T / :

Next we discuss the internal regularity and regularity up to the boundary of v. Let  be a cutoff or weight function satisfying the bounds kkC .Q/ C kDkL1 .Q/ C kDt kL1;2 .Q/ C kD.˛ 1 D/kL1;2 .Q/  N; kDt 0 kL4=3 .0;T /  N for m D 1; kDt X kL4=3 .0;T /  N for m D 1; 2; hereafter 0 D jxD0 , X D jxDX , and  0 D jtD0 , and one of the three conditions .a/ .˛ 1 D/jSm D 0 for m D 1; 2; 3I .b/ jSm D 0 for m D 1; 2; 3I .c/ jxDa D .˛ 1 D/jxDXa D 0 for m D 1; where a D 0 or a D X; where S1 D f0; Xg  .0; T /, S2 D fX g  .0; T /, and S3 D ;. Proposition 4. Let conditions (8)–(9), kDt ˛kL2 .Q/ C kDt ~kL2;1 .Q/  N , and .˛v/0 D ˛jtD0 v 0 with v 0 2 L2 ./ be valid. Let also Dt . =~/ 2 L2;1 .Q/, f 2 L2 .Q/,  0 v 0 2 H 1 ./, 0 v0 ; X vX 2 W 1;4=3 .0; T /, 0 s0 , X sX 2 V Œ0; T , and the matching conditions . 0 v 0 /.0/ D .0 v0 /.0/ for m D 1 and . 0 v 0 /.X / D .X vX /.0/ h1;1=2i for m D 1; 2 be valid. Then v 2 W .Q/, s 2 V2 .Q/, and the following bound holds:  kvkW .Q/ C kskV h1;1=2i .Q/  K.N / kD. 0 v 0 /kL2 ./ C kDt . =~/kL2;1 .Q/ 2

Ckf kL2 .Q/ C kDt .0 u0 /kL4=3 .0;T / C kDt .X uX /kL4=3 .0;T /  Ck0 s0 kV Œ0;T  C kX sX kV Œ0;T  C d 0 ; where d 0 is the sum of the norms on the right in the energy bound (11). A function v 2 W .Q/ is called the almost strong solution to problem Lm provided that there exists Ds 2 L2 .Q/, the equation Dt .˛v/ D Ds C f is valid in L2 .Q/, the initial condition vjtD0 D v 0 and the boundary conditions vjxD0 D v0 for m D 1 and vjxDX D vX for m D 1; 2 are valid in the classical sense, as well as

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the boundary conditions sjxD0 D s0 for m D 2 and sjxDX D sX for m D 2; 3 are valid in the sense of traces of functions in V2 .Q/. Proposition 5. Let conditions (8)–(9), kDt ˛kL2 .Q/ C kDt ~kL2;1 .Q/  N , and .˛v/0 D ˛jtD0 v 0 with v 0 2 H 1 ./ be valid. Let also Dt . =~/ 2 L2;1 .Q/, f 2 L2 .Q/, u 2 W 1;4=3 .0; T /, s 2 V Œ0; T , and the matching conditions v 0 .0/ D v0 .0/ for m D 1 and v 0 .X / D vX .0/ for m D 1; 2 be valid. Then the h1;1=2i weak solution v 2 V2 .Q/ to problem Lm is almost strong, has s 2 V2 .Q/, and satisfies the bound  kvkW .Q/ C kskV h1;1=2i .Q/  K.N / ku0 kH 1 ./ C k kL2 .Q/ C kDt . =~/kL2;1 .Q/ 2  Ckf kL2 .Q/ C kDt v kL4=3 .0;T / C ks kV Œ0;T  : Propositions 4–5 follow from more general results in [12]. The Gronwall-Bellman lemma (e.g., see [16]) is often used below. Lemma 1. Let a; f 2 L1 .0; T /, a  0, and y 2 L1 .0; T / satisfy the inequality 0  y.t /  C0 C It .ay C f / on .0; T /; with C0 D const  0:   Then kykL1 .0;t/  expfkakL1 .0;t/ g C0 C kf kL1 .0;t/ for any 0 < t  T . The next result [9, 36] replaces the Gronwall-Bellman lemma in some situations. Lemma 2. Let functions yi 2 Lri .0; T /, 1  i  n, satisfy the inequality X

kyi kLri .0;t/  C0 C

1in

X

kbi yi kLsi .0;t/ for all t 2 .0; T ;

1in

where C0 D const  0, kbi kL r i .0;T /  N with some 1  si < ri  1,  1 and r i D .si1  ri1 /1 , 1  i  n. Then the following bound holds: X

kyi kLri .0;T /  KT;r 1 ;:::;r n .N /C0 ;

1in

where KT;r 1 ;:::;r n .N / depends also only on either for > 1 or the function max1in !r i Œbi  for D 1. Here !r Œb./ WD sup0tT  kbkLr .t;tC / for 0 <  < T.

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3

Well-Posedness of the IBVPs for the Viscous Barotropic Gas 1D Equations

3.1

Weak Solutions, Their Global Existence, and Regularity

We consider a system of quasilinear partial differential equations describing the viscous barotropic gas 1D flow Dt D Du;

(12)

Dt u D D C gŒz;  D Œ  Du  pŒ ; D 1= ;

(13)

Dt xe D u

(14)

in the domain Q. Here the triple of the sought functions z D . ; u; xe / depends on the Lagrangian mass coordinates .x; t / 2 Q and Œ  D . .x; t /; x/, pŒ  D p. .x; t/; x/, and gŒz.x; t/ D g.z.x; t /; x; t /. We supplement the system of equations with the initial and boundary conditions . ; u; xe /jtD0 D . 0 ; u0 ; xe0 / on ;

(15)

ujxD0 D u0 .t / .or  jxD0 D p0 .t //; ujxDX D uX .t / .or  jxDX D pX .t //; (16) Rx with xe0 .x/ WD 0 0 ./ dx and 0 < t < T . We set m D 1; 2; 3 if, respectively, ujxD0;X ,  jxD0 and ujxDX , or  jxD0;X are given on Q . We define the boundary data pairs u D .u0 ; uX / and p D .p0 ; pX /. We assume that the functions u0 ; uX ; p0 , and pX equal zero if they do not appear in the mth boundary condition (16). We denote the IBVP (12), (13), (14), (15), and (16) by Pm . We recall the physical meaning of the involved quantities: , u, and xe are the specific volume, the velocity, and the Eulerian coordinate of a gas; , p, and  are the density, pressure, and stress; is the viscosity coefficient; g is the density of the body forces; u˛ and p˛ , for ˛ D 0; X, are the velocity of piston and external pressure; and X is the total mass of the gas. We define the function .; x/ D .; x/= and the primitive functions in



ƒ. ; x/ D s .; x/ d ; L. ; x/ D s 1

1

p



.; x/ d ; E. ; x/ D s Œp.; x/ d  1

on … WD RC  . Let E˙ . ; x/ D maxf˙E. ; x/; 0g. We list the main assumptions on the functions , p, and g. A1 . . ; x/ > 0 and p. ; x/  0 are measurable on …, continuous in , for almost all x 2 , and satisfy k kL1 .…a / C k1= kL1 .…a / C kpkL1 .…a /  C .a/ and kD pkL1 .…a /  C .a/ for all a > 1, where …a WD .a1 ; a/  .

2430

A. Zlotnik

A2 . ƒ. / ƒ. ; x/ƒ. / on …, with lim ƒ. / DC1 and lim ƒ. / D1.

!C1

!0C   A3 . . ; x/  c 0 EC . ; x/ C L2 . ; x/ C maxf ; 1g  on …. A4 . p. ; x/  c 0 EC . ; x/ C L2 . ; x/ C maxf ; 1g on …. A5 . E . ; x/  "L2 . ; x/ C C ."1 / on .1; 1/   for any " 2 .0; 1/. A6 . g. ; u; xe ; x; t / is measurable on RC  R  R  Q, continuous with respect to the triple . ; u; xe /, for almost all .x; t / 2 Q, and satisfies jg. ; u; xe ; x; t /j  g0 .t /juj C g1 .x; t / with kg0 kL1 .0;T / C kg1 kL2;1 .Q/  c 0 . Hereafter C ./ are some positive quantities. We introduce also two  other assumptions.  A03 . = . ; x/  c 0 EC . ; x/ C L2 . ; x/ C 1 on …. A04 . p. ; x/  c 0 Œƒ. ; x/ C 1 on .1; 1/  . Remark 1. 1. In particular, if 2 L1 .…/, ess inf… > 0 and p. ; x/ D a.x/ .x/ , with some nonnegative a;  2 L1 ./, then assumptions A2 –A5 are knowingly valid. 2. Due to assumption A1 and the Cauchy-Schwarz inequality, L2 . ; x/  ƒ. ; x/ and EC . ; x/ D 0 on .1; 1/  . Therefore assumption A4 implies A04 . 3. Assumption A1 implies that ; p; ƒ 2 L1 .I C Œa1 ; a/ for all a > 1. 4. We define the nondecreasing positive functions ƒ.1/ ./ D sup f > 0I ƒ. / .1/ ./ D inf f > 0I   ƒ. /g. Then under assumption A2 ,  g and ƒ .1/ .0 /   ƒ.1/ .1 /. inequality 0  ƒ. ; x/  1 implies that 0 < ƒ We remind that the possibility to take variable discontinuous rather than constant in x functions and p allows to cover some contact problems as well. We suppose that the initial and boundary data satisfy the following assumptions: N 1  0 ; k 0 kL1 ./ C ku0 kL2 ./  N; ku kV Œ0;T  C kp kL2 .0;T /  N; (17) N 1  k 0 kL1 ./ C It .uX  u0 / on .0; T / for m D 1:

(18)

We remind that N > 1 is the arbitrarily large parameter. A triple of functions z D . ; u; xe / 2 N .Q/  V2 .Q/  S21;1 W .Q/ is called a weak solution to problem Pm provided that: (a) equations Dt D Du and Dt xe D u are valid in L2 .Q/; (b) the integral identity Z Z T Z    ˇ˛DX uDt ' C  D'  gŒz' dxdt D u0 'jtD0 dx  p˛ 'jxD˛ ˇ˛D0 dt Q



0

(19) ˇ holds, with  D Œ  Du  pŒ , for any ' 2 C 1 .Q/ such that ' ˇtDT D 0 as well as 'jxD0;X D 0 for m D 1 or 'jxDX D 0 for m D 2; (c) the initial conditions

jtD0 D 0 , xe jtD0 D xe0 and the boundary conditions ujxD0 D u0 for m D 1; ujxDX D uX for m D 1; 2

(20)

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2431

hold in the sense of traces of functions, respectively, in C .Œ0; T I L2 .//, C .Q/, and V2 .Q/. Now we state the global with respect to time and data existence theorem for a weak solution to problem Pm . Let K.N / (possibly with indexes) be positive functions of N ; they can also depend on X , T , C , ƒ, ƒ, and c 0 (but not directly on , p, or g). The argument N is omitted in the proofs. Theorem 1. Let the functions , p, and g satisfy assumptions A1 –A6 for m D 1 or A1 –A2 , A03 –A04 , and A5 –A6 for m D 2; 3. Let the initial and boundary data satisfy (17)–(18). Then problem Pm has a global weak solution z D . ; u; xe / obeying the following bound and properties: > > > > > > > >N .Q/ C kukV h1;1=2i .Q/ C kxe kS 1;1 W .Q/  K.N /; 2

2

1

2 C .Œ0; T I L .//; u 2 C .Œ0; T I L2 .//; It  2 W .Q/:

(21)

Moreover, for any q 2 Œ1; 1, the following bound for holds:  kı kC .Œ0;T ILq .//  K.N / kı 0 kLq ./ C kı ƒkLq .IC Œa1 ;a/  Ckı pkLq .IC Œa1 ;a/ C ı ˛q ; 0 < ı < X;

(22)

where a D K0 .N /, ˛q D minf1; 1=2 C 1=qg, and ı w.x/ D w.x C ı/  w.x/ for 0 < x < X  ı or ı w.x/ D 0 otherwise. Now we turn to regularity theorems and state results on the internal regularity and regularity up to the boundary. We introduce one more assumption. A7 . kD kL1 .…a /  C .a/ for all a > 1. Let  be a cutoff or weight function satisfying the bounds and one of the three conditions (a)–(c) mentioned just before Proposition 4, in the case ˛  1. Theorem 2. Let the hypotheses of Theorem 1 and assumption A7 be valid. Let also kg0 kL1;2 .Q/ C kg1 kL2 .Q/  N in assumption A6 ,  0 u0 2 H 1 ./, 0 u0 ; X uX 2 W 1;4=3 .0; T /, 0 p0 , X pX 2 V Œ0; T , and the matching conditions . 0 u0 /.0/ D .0 u0 /.0/ for m D 1 and . 0 u0 /.X / D .X uX /.0/ for m D 1; 2 be valid. Then the h1;1=2i weak solution from Theorem 1 has the properties u 2 W .Q/ and  2 V2 .Q/ and satisfies the bound kukW .Q/ C k kV h1;1=2i .Q/  K.N /. 2

Some comments on a similar result are given in Sect. 4.4 below. A weak solution z D . ; u; xe / 2 N .Q/  W .Q/  S21;1 W .Q/ to problem Pm is called the almost strong solution provided there exists D 2 L2 .Q/, the equation Dt u D D C gŒz is valid in L2 .Q/, the initial condition ujtD0 D u0 and the boundary conditions (20) are valid in the classical sense, as well as the boundary

2432

A. Zlotnik

conditions jxD0 D p0 for m D 2 and jxDX D pX for m D 2; 3 are valid in the sense of traces of functions in V2 .Q/. In this definition, the properties of , , and p are the same as above. We emphasize that in contrast with the definition of the strong solution in [23], it is not assumed that D 2 L2;1 .Q/ and D 2 u 2 L2 .Q/ (or even that these derivatives exist). But this is no barrier to consider D 2 L2 .Q/ and thus the equation Dt u D D C gŒz in L2 .Q/. Theorem 3. Let the hypotheses of Theorem 1 and assumption A7 be valid. Let also g0 2 L2 .0; T / and g1 2 L2 .Q/ in assumption A6 , u0 2 H 1 ./, u 2 W 1;4=3 .0; T /, p 2 V Œ0; T , and the matching conditions u0 .0/ D u0 .0/ for m D 1 and u0 .X / D uX .0/ for m D 1; 2 be valid. Then the weak solution from Theorem 1 is almost strong and satisfies the bound kukW .Q/ C kkV h1;1=2i .Q/  K.N /. 2

Finally, let us cover the much simpler case of differentiable (in x) 0 , , and p. Corollary 1. 1. Let the hypotheses of Theorem 1 be valid, kD 0 kLq ./  N and kD kLq .IBa / C kDpkLq .IC Œa1 ;a/  C .a/, with Ba D L1 .a1 ; a/, for some q 2 Œ1; 2 and all a > 1. Then the bound kD kLq;1 .Q/  K.N / holds. 2. Let the hypotheses of Theorem 3 and the previous Item 1 be valid, with Ba D C Œa1 ; a. Then the bound kD 2 ukLq;2 .Q/  K.N / holds. Notice that Item 1 easily follows from bound (22) as well as Item 2 easily follows from Theorem 3 and Item 1 due to the formula Du D . C pŒ / = Œ . For q D 2, Corollary 1 ensures the existence of the standard strong solution to problem Pm .

3.1.1 Properties of Weak Solutions to the Barotropic Problems Pm Let the hypotheses of Theorem 1 be valid throughout the subsection. We establish several properties of any weak solution z D . ; u; xe / to problem Pm . Without loss of generality, one can assume that z satisfies the bounds N

1

  N ; kDt kL2 .Q/  N ; kukV2 .Q/  N ; kxe kC .Q/  N

(23)

for some N > N . For z from Theorem 1, one R can take N D K.N /. We introduce the total volume V .t/ WD  .x; t / dx D k .; t /kL1 ./ . Lemma 3. V D k 0 kL1 ./ C It .uX  u0 / on Œ0; T , for m D 1. Proof. It suffices to integrate equation (12) over Qt , t 2 Œ0; T  and take into account the initial and boundary conditions jtD0 D 0 , ujxD0 D u0 , and ujxDX D uX . The formula explains the role of condition (18) (that is a well-known fact [16]).

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2433

Lemma 4. The function u is a weak V2 .Q/-solution to the linear parabolic problem D pŒ , Lm (see (5), (6), and (7)) with the data ˛  1, ~ D Œ  D Œ  , f D gŒz, v 0 D u0 , v D u , and s D p . Moreover, u 2 C .Œ0; T I L2 .// and its value at t D 0 satisfy ujtD0 D u0 . 1

Proof. Owing to bounds (23), assumptions A1 and A6 imply that N 1  ~  N 1 with N 1 WD C .N /N , k kL1 .Q/  K1 .N / and kf kL2;1 .Q/  c 0 N . Thus the result follows from Proposition 2. Corollary 2. The property It  2 W .Q/, the equalities DIt  D ug WD u  u0  It gŒz;

(24)

.It  /jxD0 D It p0 for m D 2; 3; .It  /jxDX D It pX for m D 3; (25) and the bound k kL2 .Q/ C kIt kW .Q/  K.N / hold. Proof. Lemma 4 and Proposition 1 ensure equalities (24) and (25). Due to assumption A6 , the inequality kug kL2;1 .Q/  kukL2;1 .Q/ C ku0 k C c 0 kukL2;1 .Q/ C c 0  2N .c 0 C 1/: holds. Then equality (24) implies that It  2 W .Q/ and the following bound holds: kIt  kW .Q/ D kIt kQ C kkQ C kDIt kL2;1 .Q/    .T C 1/ N 1 kDukQ C kpŒ kQ C 2N .c 0 C 1/  K1 .N /: Lemma 5. The following formulas hold: ƒŒ  D ƒŒ 0  C It  C It pŒ ;

(26)

It  D I hmi ug C It ;m ;

(27)

It  D I ug  .I ug ; =V /   C It V 1 . ; /  V 1 .ug ; u  u;1 / for m D 1; in Q, where ;1 D h i , ;2 D p0 and ;3 D .1  u;1 WD .1  V 1 I /u0 C V 1 .I /uX .

x /p0 X

(28) 

x p X X

and

Proof. Since Du D Dt and Œ  Dt D Dt ƒŒ , the formula  D Dt ƒŒ   pŒ  holds. Applying the operator It to the last equality leads to (26). By applying the operator I hmi to (24) and using formulas (25), one derives (27) as well.

2434

A. Zlotnik

For m D 1, the solution to equation (24) having the zero mean value with the weight =V over  can be written in the form It  D .It  ; =V / C I ug  .I ug ; =V / : Then it suffices to transform the first term on the right as follows (using the easily verified equality Dt . =V / D D.u  u;1 /=V and (24)): .It  ; =V / D It .; =V / C It .It  ; Dt . =V //   D It V 1 . ; / C V 1 .It  ; D.u  u;1 //   D It V 1 . ; /  V 1 .ug ; u  u;1 / : Corollary 3. 2 C .Œ0; T I L1 .// and its value at t D 0 satisfy jtD0 D 0 . Proof. Owing to formula (26) and the property It  2 C .Q/, one obtains ƒŒ  2 C .Œ0; T I L1 .//. This property and the inequality j. / j  N 1 j. / ƒŒ j in QT  imply that 2 C .Œ0; T I L1 .//. Lemma 6. xe D xe0 C It u, Dt xe D u, Dxe D , and Dt Dxe D Du. Proof. The equalities easily follow from equations (14) and (12) and initial condition (15) for and xe . To study problem P1 , we introduce the function ƒh1i : R   ! RC such that ƒ .; x/ is the inverse function of ƒ.; x/ for almost all x 2 . Assumptions on and ƒ imply that ƒh1i exists and h1i

ı ƒh1i .0; x/ D 1; D ƒh1i .; x/ D 1 Œƒh1i .; x/

(29)

with Œƒh1i  WD .ƒh1i .; x/; x/; the measurability of ƒh1i can be derived from (29). Let B hmi Œz WD ƒŒ 0  C It pŒ  C I hmi ug . Lemma 7. For all t 2 Œ0; T  and almost all x 2 , the following equality holds: Z

  ƒh1i ˆŒz.x; x 0 ; t /; x 0 dx 0 D V .t / for m D 1;



with ˆŒz.x; x 0 ; t / WD ƒŒ .x; t /  B h1i Œz.x; t/ C B h1i Œz.x 0 ; t /. Proof. Owing to equalities (26) and (27), the function ƒŒ .x; t /  B h1i Œz.x; t/ of x. Thus ˆŒz.x; x 0 ; t / D ƒŒ .x 0 ; t / or equivalently ƒh1i is independent 0 ˆŒz.x; x ; t /; x 0 D .x 0 ; t /. Integrating over x 0 2  completes the proof.

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2435

Lemma 8. 1. Let the hypotheses of Theorem 2 be valid. Then z has the properties h1;1=2i u 2 W .Q/,  2 V2 .Q/ and satisfies the bound  kukW .Q/ C k kV h1;1=2i .Q/  K.N / kD. 0 u0 /kL2 ./ C kg0 kL1;2 .Q/ 2

Ckg1 kL2 .Q/ C kDt .0 u0 /kL4=3 .0;T / C kDt .X uX /kL4=3 .0;T /  Ck0 p0 kV Œ0;T  C kX pX kV Œ0;T  C 1 : 2. Let the hypotheses of Theorem 3 be valid. Then z is almost strong and satisfies the bound  kukW .Q/ C k kV h1;1=2i .Q/  K.N / kDu0 kL2 ./ C kg0 kL2 .0;T / C kg1 kL2 .Q/ 2  CkDt u kL4=3 .0;T / C kp kV Œ0;T  C 1 : Proof. Let us return to Lemma 4 and its proof. Now the following bounds hold: kDt ~kQ  ckD kL1 .…N / kDt kQ  K1 .N /, kf kQ  kg0 kL1;2 .Q/ N C kg1 kQ and    kDt . =~/kL2;1 .Q/  kkC .Q/ CkDt kL2;1 .Q/ k =~kL1 .Q/ CkDt . =~/kL2;1 .Q/    K2 .N / kD .p= /kL1 .…N / kDt kQ Ckp= kL1 .…N /  K3 .N /: Therefore Item 1 follows from Proposition 4. Item 2 follows from the above bounds with   1 and Proposition 5.

3.1.2 An Auxiliary Semidiscrete Parabolic Problem I We introduce additional notation. We set h D X=n for n  2 and define two grids, with nodes xi D ih, 0  i  n and xi1=2 D i  12 h, 1  i  n, respectively. We introduce the related elements 1=2 D Œ0; x1 /; i1=2 D .xi1 ; xi /; 2  i  n  1; n1=2 D .xn1 ; X ; 0 D Œ0; x1=2 /; i D .xi1=2 ; xiC1=2 /; 1  i  n  1; n D .xn1=2 ; X  and the corresponding spaces of piecewise constant functions Sh

D fwI w.x/ D wi on i ; 0  i  ng;

S1h

D fvI v.x/ D vi1=2 on i1=2 ; 1  i  ng:

Let h D .x1=2 ; xn1=2 /. We introduce the integration operators Z .Ih w/.x/ D 0

xi 1=2

w./ d  on i1=2 ; 1  i  n; for w 2 S h ;

2436

A. Zlotnik

Z .Ih v/.x/ D 0

xi

v./ d  on i ; 0  i  n; for v 2 S1h :

O 2 C ./, w O be linear on i1=2 , 1  i  n, and w.x O i / D wi , For w 2 S h , let w h 0  i  n. For v 2 S1 , let vO 2 C ./, vO be linear on i , 0  i  n, and v.x O i1=2 / D vi1=2 , 1  i  n; moreover, let vO be defined by some way at the points O / D vn1=2 . x D 0; X and, if the other way is not prescribed, v.0/ O D v1=2 and v.X The following integration by parts formula and two-sided bound hold: ˇxDX .w; D v/ O  D .wv/ O ˇxD0  .D w; O v/ for any w 2 S h ; v 2 S1h :

(30)

kwk O Lq ./  kwkLq ./  21=q kwk O Lq ./ for any w 2 S h :

(31)

O xD0 D vjxD0 and The bound is valid for v 2 S1h too provided that vj vj O xDX D vjxDX . For 2 L1 ./, let s h 2 S h (resp. s1h 2 S1h ) be equal on i , 0  i  n (resp. on i1=2 , 1  i  n), to the mean value of on this set. We notice the identities .s h ; w/ D . ; w/ for any

2 L1 ./; w 2 S h ;

(32)

.s1h ; v/ D . ; v/ for any

2 L1 ./; v 2 S1h :

(33)

The functions s h f and s1h f are defined similarly for f 2 L1 .I B/. Lemma 9. Let f 2 Lq .I B/ for some 1  q < 1 and f h D s h f or s1h f . Then [8] lim kf  f h kLq .IB/ D 0; lim sup kı f h kLq .IB/ D 0:

h!0

ı!0 h

We introduce the following auxiliary semidiscrete parabolic IBVP: Dt v D D sO C f on Q; with s WD ~D vO  vjtD0 D v 0 on ;

on Qh WD h  .0; T /;

(34) (35)

vjxD0 D v0 .t / .or sO jxD0 D s0 /; vjxDX D vX .t / .or sO jxDX D sX / on .0; T /:(36) The sought function is v 2 W 1;1 .0; T I S h /, i.e., v; Dt v 2 L1 .0; T I S h /. We assume that ~ 2 L1 .0; T I S1h /, N 1  ~  N , 2 L2 .0; T I S1h /, f 2 L1 .0; T I S h /, 0 h 1;1 v 2 S , v WD .v0 ; vX / 2 W .0; T /, and s WD .s0 ; sX / 2 L1 .0; T /. Here m D 1; 2; 3 is set like above for problem Lm ; see (5), (6), and (7). Let also the matching conditions v 0 jxD0 D v0 .0/ for m D 1 and v 0 jxDX D vX .0/ for m D 1; 2 be valid.

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2437

According to equation (34) and the boundary conditions (36), one gets sO jxD0 D sjxDx1=2  .h=2/.Dt v0  f jxD0 / for m D 1;

(37)

sO jxDX D sjxDxn1=2 C .h=2/.Dt vX  f jxDX / for m D 1; 2:

(38)

Notice that hereafter semidiscrete problems are treated closely to the original ones and deal with the space derivatives and integrals rather than finite differences and sums. The function v takes values vi .t / on i  Œ0; T , 0  i  n. With respect to them, problem (34), (35), and (36) is the Cauchy problem for the system of linear first-order ordinary differential equations (ODE). For example, for m D 1, it takes the form Dt vi D

1 viC1  vi vi  vi 1  ~iC1=2  ~i 1=2 h h h iC1=2  i1=2  C fi on Œ0; T ; h

vi .0/ D vi0 ; 1  i  n  1; with given vjiD0 D v0 and vjiDn D vX . Clearly under the above conditions, the solution v to problem (34), (35), and (36) exists and is unique. We prove the semidiscrete counterpart of bound (11). Proposition 6. The following bound holds:  kvk O V h1;1=2i .Q/  K.N / kv 0 kL2 ./ C k kL2 .Q/ C kf kL2;1 .Q/ 2  Ckv kW 1;1 .0;T / C ks kL4=3 .0;T / : h

h

Proof. We set v;1 D .1  xX /v0 C xX vX , v;2 D vX , and v;3 D 0. We rewrite equation (34) for w WD v  v;m in the form O Dt w D D sO C f 0 on Q; with s D ~D w

0

on Qh ;

(39)

where 0 D  ~D vO ;m and f 0 D f  Dt v;m . Clearly w has the zero Dirichlet boundary values wjxD0 D 0 for m D 1 and wjxDX D 0 for m D 1; 2. We multiply equation (39) by w, integrate over Qt , and then integrate by parts by formula (30) using the boundary conditions (36) (rewritten for w) and get 1 kwk2 2

C .~D w; O D w/ O Qt D 12 kwjtD0 k2 C . ˇaDX C.f 0 ; w/Q C .sa ; wj O xDa /.0;t/ ˇ : t

aD0

0

; D w/ O Qt

2438

A. Zlotnik

Consequently max

˚1 2

kwk2L2;1 .Q/ ; N 1 kD wk O Q  12 kwjtD0 k2 C k

0

kQ kD wk O Q

Ckf 0 kL2;1 .Q/ kwkL2;1 .Q/ C ks kL4=3 .0;T / kwk O L1;4 .Q/ : Using inequalities (31) and (1) leads to  kwk O V2 .Q/  K1 kwjtD0 k C k

0

 kQ C kf 0 kL2;1 .Q/ C ks kL4=3 .0;T /  K2 d; (40)

where d denotes the sum of norms on the right in the bound under proof. Next, by applying the operator . / It ,  2 .0; T /, to equation (39), one gets the equality ./ w D D. / It sO C . / It f 0 in QT  . By taking the L2 .QT  /-inner product of the equality by . / w, one finds O QT  k./ wk2QT  D ˛ C ˇ ; ˛ WD .. / It s; . / D w/ C .. / It f 0 ; . / w/QT  ;

(41)

ˇaDX with ˇ WD ../ It sa ; . / wj O xDa /.0;T  / ˇaD0 . To bound ˛ , first by the Cauchy-Schwarz and generalized Minkowski inequalities, one gets     O Q C 2. / It kf 0 k L1 .0;T  / kwkL2;1 .Q/ : j˛ j  2./ It ksk .0;T  / kD wk Then by applying the inequality k./ It 'kLr .0;T  /   11=r1 C1=r k'kLr1 .0;T / ; 1  r1  r  1;

(42)

for r D r1 D 2 and r D r1 D 1, inequality (31), and bound (40), one derives p j˛ j  2kskQ kD wk O Q C 2 2 kf 0 kL2;1 .Q/ kwk O L2;1 .Q/  K3 d 2  : To bound ˇ , we apply inequalities (42) for r D r1 D 4=3 and (2) to get jˇ j  2k./ It s kL4=3 .0;T  / kwk O L1;4 .Q/  2c0 ks kL4=3 .0;T / kwk O V2 .Q/  K4 d 2  : Now from inequality (31) and equality (41), one obtains O 2QT   k. / wk2QT   .K3 C K4 /d 2  : k. / wk Consequently kwk O V h1;1=2i .Q/  K5 d . 2

Also kvO ;m kV h1;1=2i .Q/  ckv kW 1;1 .0;T /  cd since kvO ;m kV2 .Q/  c1 kv kC Œ0;T  2 and k./ vO ;m kQT   c2 k. / It Dt v kL2 .0;T  /  c2 kDt v kL1 .0;T /  1=2

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2439

once again due to inequality (42) for r D 2 and r1 D 1. Thus the triangle inequality O V h1;1=2i .Q/ C kvO ;m kV h1;1=2i .Q/  Kd completes the proof. kvk O V h1;1=2i .Q/  kwk 2

2

2

3.1.3 A Semidiscrete Method for Solving the Barotropic Problem Pm We construct the following semidiscrete method for solving the system of equations (12), (13), and (14): Dt h D D uO h in Q;

(43)

Dt uh D D O h C g h ;  h D h Πh D uO h  p h Πh  in Q;

(44)

Dt xeh

(45)

h

D u in Q:

The sought functions are h 2 W 1;1 .0; T I S1h /, h > 0, and uh ; xeh 2 W 1;1 .0; T I S h /. Here h D s1h , p h D s1h p, h Œ h .x; t/ D h . h .x; t /; x/, p h Œ h .x; t/ D p h . h .x; t /; x/, and g h D s h .gŒQz h /, with zQ h D . h ; uO h ; xO h /. Also uO h D ubh , e

xO eh

D xbeh , and O h D bh . Further we approximate the initial and boundary conditions (15)–(16) as follows: . h ; uh ; xeh /jtD0 D . 0;h ; u0;h ; xe0;h /;

(46)

. /

. /

uh jxD0 D u0 .or O h jxD0 D p0 /; uh jxDX D uX .or O h jxDX D pX /:(47) We set 0;h D s1h 0 and xe0;h D Ih 0;h ; moreover, u0;h D s h u0 but redefine u0;h .x/ D ./ . / u0 .0/ on 0 for m D 1 and u0;h .x/ D uX .0/ on n for m D 1; 2. Also m D 1; 2; 3 like above for problem Pm , and the average '

. /

1 .t / D 

Z

tC

'.t 0 / dt 0

t

is used for ' D u˛ , where 0 <   N 2 , and the extension u˛ .t / D u˛ .T / for t > T , ˛ D 0; X is applied. We denote problem (43), (44), (45), (46), and (47) by Pmh . According to equation (44) and the boundary conditions (47), we define   . / O h jxD0 D O h jxDx1=2  .h=2/ Dt u0  g h jxD0 for m D 1;   . / O h jxDX D O h jxDxn1=2 C .h=2/ Dt uX  g h jxDX for m D 1; 2: We notice the simple formulas s1h . Œ h / D h Œ h ; s1h .pŒ h / D p h Œ h : We state a semidiscrete counterpart of Theorem 1.

(48)

2440

A. Zlotnik

Proposition 7. Let the hypotheses of Theorem 1 be valid. Then problem Pmh has a solution zh D . h ; uh ; xeh / satisfying the following bound: > > > > > > h > >N .Q/ C kOu h kV h1;1=2i .Q/ C kxO eh kS 1;1 W .Q/  K.N /: 2

2

(49)

Moreover, for any q 2 Œ1; 1, the following bound for h holds:  kı h kC .Œ0;T ILq .//  K.N / kı 0;h kLq ./ C kı ƒh kLq .IC Œa1 ;a/  (50) Ckı p h kLq .IC Œa1 ;a/ C .ı C h/˛q 1=q ı 1=q ; 0 < ı < X; where ƒh D s h ƒ and a D K0 .N /; remind that ˛q D minf1; 1=2 C 1=qg. Actually Pmh is the Cauchy problem for a system of the nonlinear ODE (after the algebraic elimination of the given uh0 for m D 1 and uhn for m D 1; 2). The existence of its solution on a sufficiently small time segment Œ0; T0   Œ0; T  follows from the classical Carathéodory theorem. The possibility to extend the solution to the whole segment Œ0; T  preserving the property h > 0 is justified by a priori bound (49). So the point in the proof of Proposition 7 is the derivation of uniform in h a priori bounds (49)–(50) (both bounds are required below for the limit transition). Let V h .t / D k h .; t /kL1 ./ be the semidiscrete total volume of gas. Lemma 10. The following relations hold:  . / . /  .2N /1  V h D k 0;h kL1 ./ C It uX  u0  .X C T /N for m D 1; (51) h 2 h V h  XN C "1 kDt LŒ h k2L2 .Qt / C "1 1 c 0 It kEC Œ  C L Œ  C 1kL1 ./

(52)

for m D 2; 3, with any "1 > 0. Proof. For m D 1, from the equation Dt h D D uO h , the initial and boundary . / conditions h jtD0 D 0;h and uh jxD˛ D u˛ , ˛ D 0; X, one gets the equality for V h in (51), cp. with Lemma 3. Since / k 0;h kL1 ./ D k 0 kL1 ./  XN; ku. ˛ kC Œ0;T   sup ju˛ .t /j; ˛ D 0; X; Œ0;T 

the right bound (51) holds. Notice also that  . / . /  V h D k 0 kL1 ./ CIt uX u0 Cr; r.t/ D

Z

 0

 ˇ˛DX .1= / u˛ .tC /u˛ .t / ˇ˛D0 d  :

Therefore the left bound (51) follows from assumptions (17)–(18) and the bound  jr.t /j  varŒ0;T  uX C varŒ0;T  u0 =2  .2N /1 .

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2441

For m D 2; 3, we transform and bound V h are as follows: q   V h .t / D k 0;h kL1 ./ C .Dt h ; 1/Qt D k 0;h kL1 ./ C Dt LΠh ; h = Πh  Qt h h  NX C "1 kDt LΠh k2L2 .Qt / C "1 1 It k = ΠkL1 ./ :

By applying assumption A03 , one derives inequality (52). The derivation of the energy bound for zh is based on the next result. ˇX Lemma 11. Let  h WD .O h uh /ˇ0 . The following energy balance equation holds:     Dt .EŒ h ; 1/ C 12 kuh k2L2 ./ C kDt LŒ h k2L2 ./ D  h C gŒQz h ; uh  : (53) Proof. We take the L2 ./-inner product of the equation Dt uh D D O h C g h by uh , apply formula (30) and property (32), and get Dt

1 2

   kuh k2 C . h ; D uO h / D  h C gŒQz h ; uh  :

Due to equation Dt h D D uO h , one can rewrite  h as  h D h Πh Dt h  p h Πh :

(54)

To complete the proof, by using the same equation, properties (48) and (33) and the definitions of L and E, one accomplishes the following transformations: . h ; D uO h / D . h Πh Dt h  p h Πh ; Dt h / D kDt LΠh k2 C Dt .EΠh ; 1/ : . /

. /

We transform  h and bound It  h . For m D 1, we set uh;1 D .1`h /u0 C`h uX . /

. /

with `h WD Ih h =V h . We notice that uh;1 jxD0 D u0 , uh;1 jxDX D uX , 0  `h  1 and . /

. /

h Dt uh;1 D .1  `h /Dt u0 C `h Dt uX C dV .uh  uh;1 /; D uO ;1 D dV h ;

(55)

 ./ . /  with dV WD uX  u0 =V h . Lemma 12. The function  h can be rewritten in the forms 8   ˆ Dt .uh ; uh;1 / .uh ; Dt uh;1 /  gŒQz h ; uh;1  CdV . h ; h / for mD1 ˆ <   h . /    h . /   h D Dt uh ; u./ z ; uX  .p0 ; Dt h / for mD2 X   u ; Dt uX   gŒQ ˆ ˆ   :  .pX  p0 /=X; uh  C.;3 ; Dt h / for mD3

(56)

2442

A. Zlotnik

with ;3 .x; t / WD .1  holds:

x /p0 .t / X



x p .t /. X X

Consequently the following bound

jIt  h j  "2 kDt LŒ h k2L2 .Qt / ˚   h 2 h CK.N / kuh kL2;1 .Qt / C "1 2 It ~m kEC Œ  C L Œ  C 1kL1 ./ C 1

(57)

for any "2 2 .0; 1/, with ~1 D 1, ~2 D p02 , and ~3 D p02 C pX2 . Proof. For m D 1, we take the L2 ./-inner product of the equation Dt uh D D O h C g h by uh;1 , apply the integration by parts formula (30), and get

b

  .Dt uh ; uh;1 / D  h   h ; D uh;1  C .gŒQz h ; uh;1 / : By using the right formula (55), one derives formula (56).  . /  For m D 2, we first notice that  h D D O h ; uX   .p0 ; D uO h / . By virtue of the equalities D O h D Dt uh  g h and D uO h D Dt h , one obtains  . /  . /  h D Dt uh ; uX   .gŒQz h ; uX /  .p0 ; Dt h / that implies formula (56). For m D 3, apply the integration by parts formula .;3 ; D uO h / D  h  .D;3 ; uO h / . Here .D;3 ; uO h / D .D;3 ; uh / since D;3 D .pX  p0 /=X is independent of x. Applying also the equality D uO h D Dt h , one derives formula (56). Next we derive bound (57). We first notice that owing to assumption A6 , the bound   kgŒQz h kL2;1 .Qt /  c 0 kOu h kL2;1 .Qt / C 1

(58)

holds, and owing to formulas (54), (48), and (32), one gets   . h ; h / D h Πh Dt h  h pΠh ; 1  :

(59)

For m D 1, from formula (56), the left formula (55), and bound (58), one obtains   jIt  h j  K1 kuh kQt C It j. h ; h / j C 1 ;

(60)

where the bound .V h /1  2N , see (51), is also applied. Now we prove the inequality   h 2 h It j. h ; h / j  "2 K11 kDt LΠh k2Qt C "1 (61) 2 K2 It kEC Π C L ΠkL1 ./ C1

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2443

p for any "2 2 .0; 1/. Due to formulas (59) and Œ h Dt h D .Dt LŒ h / h Œ h , one gets j. h ; h / j  "kDt LŒ h k2Qt C ."1 C 1/k h Œ h  C h pŒ h kL1 ./ for any " > 0: Now by using assumptions A3 –A4 and the inequality maxf h ; 1g  h C 1 and setting " D "2 K11 , one derives bound (61). Its inserting into (60) leads to (57). The case m D 2; 3 is simpler. From formulas (56) one gets     jIt h j  K3 kOu h kL2;1 .Qt / C 1 C It ~m1=2 ; jDt h j  : This bound implies bound (57); see the above derivation of bound (52). Now the energy bound for zh can be proved. Lemma 13. The following energy bound holds: kEC Œ h  C L2 Œ h kL1;1 .Q/ C kuh k2L2;1 .Q/ C kDt LŒ h k2L2 .Q/  K.N /: (62) Proof. We apply the operator It to the energy balance equation (53). Since   w2 .t /  cT w2 .0/ C kDt wk2L2 .0;t/ ; 0 < t  T; for any w 2 H 1 .0; T /; (63) by taking w D LŒ h , one gets y.t / WD kEC Œ h kL1 ./ C 12 kuh k2 C 12 cT1 kLŒ h k2 C 12 kDt LŒ h k2Qt  kE Œ h kL1 ./ C .E.Œ 0;h /; 1/ C 12 ku0;h k2 C 12 kLŒ 0;h k2 C It  h C.gŒQz h ; uh /Qt : Then owing to assumptions A1 , A6 , and (17), one obtains   y.t /  kE Œ h kL1 ./ C K1 C It  h C It g0 kuh k2 C c 0 kuh kL2;1 .Qt / : (64) According to assumption A5 , one has kE Œ h kL1 ./  "kLŒ h k2 C C ."1 /V h : Let " WD 1=.4cT /. For V h , we use bounds (51)–(52) with "1 WD 1=Œ8C ."1 /. For It  h , we apply bound (57) with "2 WD 1=8. Thus one obtains 1 1 kL2 Œ h kL1 ./ C kDt LŒ h k2Qt 4cT 4 ˚ h   h CK2 ku kL2;1 .Qt / C It .1 C ~m /kEC Œ  C L2 Œ h kL1 ./ C 1 :

kE Πh kL1 ./ C jIt  h j 

2444

A. Zlotnik

Inserting this bound into the right-hand side of bound (64), one finds y.t /  2.K1 C c 0 /kuh kL2;1 .Qt / C K2 It Œ.1 C ~m C g0 /y C 2.K1 C K2 /: Let Y .t / WD max0t y./. Since kuh kL2;1 .Qt / 

p 2Y .t/, the inequality

Y .t /  K3 It Œ.1 C ~m C g0 /Y  C K4 ; 0 < t  T; holds. Owing to the bound k1 C ~m C g0 kL1 .0;T /  K5 and the Gronwall-Bellman lemma (see Lemma 1), one derives Y .T /  K6 that completes the proof. Next we turn to derivation of the two-sided uniform bound for h . Lemma 14. The following formula and bound hold: DIt O h D uhg WD uh  u0;h  It g h ;

(65)

kDIt O h kL2;1 .Q/ D kuhg kL2;1 .Q/  K.N /:

(66)

Proof. By applying the operator It to the equation Dt uh D D O h C g h , one derives (65) since uh jtD0 D u0;h . Then due to bound (58) and the energy one (62), one gets kuhg kL2;1 .Q/  kuh kL2;1 .Q/ C ku0;h k C c 0 .kOu h kL2;1 .Q/ C 1/  K: Lemma 15. The following formulas hold: ƒh Œ h  D ƒh Œ 0;h  C It  h C It p h Œ h ;

It  h D

8   .1/ ˆ I uh C It .V h /1 . h ; h /  .V h /1 .uhg ; uh  uh;1 / ˆ ˆ < h g .2/

Ih uhg  It p0 ˆ ˆ ˆ :I .3/ uh C I  h

g

(67)

for m D 1 for m D 2 (68) for m D 3

t ;3

  .1/ .2/ .3/ with Ih w D Ih w  Ih w; h =V h  , Ih D Ih , and Ih w D Ih .w  hwi /. Proof. By rewriting equality (54) as  h D Dt ƒh Œ h   p h Œ h  and applying the operator It to it, one obtains (67). .m/ Next, we apply the operator Ih to equality (65). For m D 1, one first get .1/

It  h D .It  h ; h =V h / C Ih uhg :

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2445

We transform the term .It  h ; h =V h / using the integration by parts first in t and h next in x (using the formula Dt . h =V h / D D.Ou h  uO ;1 /=V h that follows from the equation Dt h D D uO h and the boundary conditions (47)):   .It  h ; h =V h / D It . h ; h =V h / C It It  h ; Dt . h =V h /    D It .V h /1 . h ; h /  .V h /1 .uhg ; uh  uh;1 / ; that leads to the first formula (68). For m D 2; 3, formulas (68) arise directly due to the boundary conditions (47). Notice that formula (68) for m D 2 is also valid for m D 3. Lemma 16. The following bounds hold: kIt  h kC .Œ0;T IL1 .//  K.N /; kIt O h kC .Q/  K.N /:

(69)

Proof. We first derive the left bound (69). For m D 1, from formula (68) and the relations .2N /1  V h D k h kL1 ./ (see (51)), one gets kIt  h kC .Œ0;T IL1 .//  X 1=2 kuhg kL2;1 .Q/ C NIt j. h ; uh / j C2N T kuhg kL2;1 .Q/ kuh  uh;1 kL2;1 .Q/ : Then it suffices to apply inequality (61), the energy bound (62), bound kuhg kL2;1 .Q/  K (see (66)), and the inequality ˚ . / . / kuh;1 kL2;1 .Q/  X 1=2 max ku0 kC Œ0;T  ; kuX kC Œ0;T   X 1=2 N: For m D 2; 3, from formula (68) one straightforwardly obtains kIt  h kC .Œ0;T IL1 .//  X 1=2 kuhg kL2;1 .Q/ C kp0 kL1 .0;T /  K: To derive the right bound (69) from the left one, we apply bounds (66) and kIt O h kC .Q/  kIt  h kC .Œ0;T IL1 .// C h1=2 kDIt O h kL2;1 .Q/ : Lemma 17. The uniform two-sided bound K0 .N /1  h  K0 .N / holds. Proof. We turn to formula (67). Owing to assumptions (17) and A1 , one gets K11  ƒh Œ 0;h   K1 . Using also the left bound (69) and the property p h Œ h   0, one has  K2  ƒh Œ h   It p h Œ h  C K2 :

(70)

2446

A. Zlotnik .1/

Assumption A2 leads to K2  ƒh Œ h   ƒŒ h . Thus K31 WD ƒ .K2 /  h ; see Remark 1, Item 4. Owing to this bound and assumption A1 , the inequalities 0  p h Œ h   h h;1 p Œ  C K4 hold with h;1 WD maxf h ; 1g. Now the right inequality (70) and assumption A04 imply ƒh Œ h;1   It p h Œ h;1  C K5  c 0 It ƒh Œ h;1  C K6 : The Gronwall-Bellman lemma and assumption A2 lead to ƒŒ h;1   ƒh Œ h;1   K7 and therefore h  h;1  K9 WD ƒ.1/ .K7 /. Corollary 4. K1 .N /1  h Œ h   K1 .N / and 0  p h Œ h   K2 .N /. These uniform bounds are valid due to assumption A1 . Now we complete the proof of bound (49). Lemma 18. The following bounds hold: kDt h kL2 .Q/  K.N /; kOu h kV h1;1=2i .Q/  K.N /; k h kL2 .Q/  K.N /; (71) kxO eh kS 1;1 W .Q/  K.N /:

(72)

2

Proof. uh can be considered as a solution to the semidiscrete parabolic problem like . / above Lhm with the data ~ D h Œ h , D p h Œ h , f D g h , v 0 D u0;h , v D u , and s D p . Owing to Proposition 6, Corollary 4, assumptions (17) and A6 , and bound kOu h kL2;1 .Q/  K1 , one derives kOu h kV h1;1=2i .Q/  K.N /. The relations Dt h D D uO h and k h kQ  k h Œ h kL1 .Q/ kD uO h kQ C kp h Œ h kQ imply two other bounds (71). Bound (72) follows from the bound kOu h kV2 .Q/  K.N / since Dt xO eh D uO h and h xO e D xO e0;h C It uO h according to the equation Dt xeh D uh . To prove bound (50), an auxiliary inequality is needed. Lemma 19. If w 2 S1h satisfies D wO D ‰ in h , then the following bound holds: kı wkLq ./  maxfX; 1g.ı C h/˛q 1=q ı 1=q k‰kL2 .h / for any q 2 Œ1; 1: Proof. For almost all x 2 .0; X  ı/, the following formula holds: ı w.x/ D

X 1in1

i;ı .x/‰i h;

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2447

where i;ı .x/ D 1 for x < xi < x C ı or i;ı .x/ D 0 otherwise. The functions satisfy X

i;ı .x/  ı C h; ki;ı kLp ./  ı 1=p for any p 2 Œ1; 1:

1in1

By virtue of the left inequality and the Hölder one, the bound jı w.x/js  .ı C h/s1

X

i;ı .x/j‰i js h for any s 2 Œ1; 1:

1in1

holds. This bound and the above inequality ki;ı kLp ./  ı 1=p further imply kı wkLps ./  .ı C h/11=s ı 1=.ps/ k‰kLs .h / : One completes the proof setting p WD maxf1; q2 g and s WD inequality k‰kLs .h /  maxfX; 1gk‰kL2 .h / .

q p

and using the

Lemma 20. Bound (50) holds. Proof. Owing to formula (67), one gets ƒh Œ h C ı h   ƒh Œ h  D .ı ƒh /Œ h C ı h  Cƒh Œ 0;h C ı 0;h   ƒh Œ 0;h  C .ı ƒh /Œ 0;h C ı 0;h  CIt fp h Œ h C ı h   p h Œ h g C It f.ı p h /Œ h C ı h g C ı It  h ; (73) where .ı '/Œw.x; t/ WD '.w.x; t /; x C ı/  '.w.x; t /; x/ for .x; t / 2 .0; X  h/  Œ0; T . Lemma 17 and assumption A1 imply the bounds K11 jı h j  jƒh Œ h C ı h   ƒh Œ h j  K1 jı h j; jp h Œ h C ı h   p h Œ h j  K1 jı h jI the assumption on D p is essential here. Owing to them from equality (73), one obtains  kı h kLq ./  K2 kı 0;h kLq ./ C It kı h kLq ./  Ckı ƒh kLq .IC Œa1 ;a/ C kı p h kLq .IC Œa1 ;a/ C kı It  h kLq ./ : (74) The previous lemma and bound (66) ensure the bound

2448

A. Zlotnik

kı It  h kLq;1 .Q/  maxfX; 1g.ıCh/˛q 1=q ı 1=q kuhg kL2;1 .Q/  K3 .ıCh/˛q 1=q ı 1=q : Using it in (74) and applying the Gronwall-Bellman lemma, one completes the proof.

3.1.4 Limit Transition in the Barotropic Case Owing to Lemma 9, 0;h ! 0 in Lq ./, for any 1  q < 1, thus xe0;h ! xe0 in C ./ as well as u0;h ! u0 in L2 ./ as h ! 0. First notice that from bound (50) with q D 2, it follows that lim sup kı h kL2;1 .Q/ D 0:

ı!0C

(75)

h

Here Lemma 9 is applied for f D 0 2 L2 ./ and f D ƒ; p 2 L2 .I C Œa1 ; a/. We perform the limit transition in problem Pmh as h ! 0. Set  D  .h/ ! 0 as h ! 0. The corresponding sequence zh D . h ; uh ; xeh / of solutions to problems Pmh from Proposition 7 satisfies the uniform in h bound (49) and property (75). Applying the M. Riesz criterion for compactness of sets in L2 .Q/, the weak compactness of closed balls in L2 .Q/, the –weak compactness of closed balls in L2;1 .Q/, and the compactness of the embedding S21;1 W .Q/  C .Q/, one can select a subsequence of zh (hereafter not relabeled) such that

h ! in L2 .Q/; Dt h ! Dt weakly in L2 .Q/; h

2

uO ! u in L .Q/ and weakly in L

2;1

.Q/;

D uO h ! Du weakly in L2 .Q/; xO eh

weakly in

S21;1 W

.Q/ and strongly in C .Q/:

(76) (77) (78) (79)

Once again passing to a subsequence, one can assert that h ! almost everywhere (a.e.) in Q and uO h .; t / ! u.; t / in L2 ./ for almost all t 2 .0; T /:

(80)

In addition, uh ! u in L2 .Q/ due to the inequality kOu h  uh kQ  hkD uO h kQ  Kh and the first property (77). The limit triple of functions z D . ; u; xe / belongs to N .Q/V2 .Q/S21;1 W .Q/ and satisfies bound (21). Lemma 21. h ! in C .Œ0; T I Lq .// for any q 2 Œ2; 1/. Proof. From the multiplicative inequality 1=2 1=2  kwkC .Œ0;T IL2 .//  cT kwkQ kwkQ C kDt wkQ

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2449

for w D h  and properties (76), the result follows for q D 2. By applying the inequality 2=q

12=q

kwkC .Œ0;T ILq .//  kwkC .Œ0;T IL2 .// kwkC .Œ0;T IL1 .// for any q 2 Œ2; 1; for w D h  and 2 < q < 1, and the bound k h  kC .Œ0;T IL1 .//  K1 (see Corollary 3), one completes the proof. Lemma 22. Bound (22) holds for ı . Proof. We denote by Mq and Mqh , respectively, the right-hand sides of bounds (22) and (50). For any q 2 Œ1; 1/, by virtue of Lemma 9, one has 0;h ! 0 in Lq ./, ƒh ! ƒ, and p h ! p in Lq .I C Œa1 ; a/. Therefore Mqh ! Mq as h ! 0 and the limit transition in bound (50) using Lemma 9 leads to the bound kı kC .Œ0;T ILq .//  Mq . By passing to the limit in the last bound as q ! 1, one establishes it for q D 1 as well. Let us prove that z D . ; u; xe / is a weak solution to problem Pm . Lemma 23.  h D h Œ h  h D uO h  p h Œ h  !  D Œ  Du  pŒ  weakly in L2 .Q/. Proof. Owing to the bound K01  h  K0 , the inequality k' h Œ h   'Œ h kC .Œ0;T IL2 .//  k' h  'kL2 .IC Œa1 ;a/ for a D K0 holds for ' D ; p. By virtue of Lemma 9, its right-hand side tends to 0. Consequently h Œ h  h  Œ h  h ! 0 in measure on Q and p h Œ h   pŒ h  ! 0 in L2 .Q/. Owing to assumption A1 , the above property h ! a.e., in Q and the bound K01  h  K0 imply that Œ h  h ! Œ  and pŒ h  ! pŒ  a.e. in Q as well as k Œ h  h kL1 .Q/ C kpŒ h kL1 .Q/  K1 . Thus Œ h  h ! Œ  in measure on Q and pŒ h  ! pŒ  in L2 .Q/. Applying property (78) and the Lebesgue dominated convergence theorem, one derives the result. Lemma 24. It O h ! It  in C .Q/ and It  h ! It  in C .Œ0; T I L1 .//. Proof. Let Q h D O h on Qh and Q h D  h on Q n Qh . By taking into account bound (66), one gets kIt Q h kW .Q/  ck h kQ C kDIt O h kL2;1 .Q/  K1 . Owing to this bound and inequality (3), the sequence It Q h is relatively compact in C .Q/. To complete the proof, we notice that It  h ! It  weakly in L2 .Q/ (according to Lemma 23) and the following bound holds:

2450

A. Zlotnik

kIt Q h It O h kC .Q/ CkIt Q h It  h kC .Œ0;T IL1 .//  2kDIt O h kL2;1 .Q/ h1=2  K2 h1=2 : Lemma 25. gŒQz h  ! gŒz in L2;1 .Q/. Proof. Properties (79) and (80), Lemma 21, and assumption A6 imply that gŒQz h .; t / ! gŒz.; t / in L2 ./ due to the continuity of the Nemytskii operator (e.g., see [25]). Also assumption A6 and the bound kOu h kC .Œ0;T IL2 .//  K ensure the bound kgŒQz h .; t /k  g.t / WD Kg0 .t / C kg1 .; t /k with g 2 L1 .0; T /. Application of the Lebesgue dominated convergence theorem completes the proof. Lemma 26. The function u satisfies the integral identity (19) and the boundary conditions (20). Proof. Let ' be the same as in (19) and in addition Dt D' 2 L1 .Q/. By multiplying equation (44) by ', integrating over Q and integrating by parts, one gets

 .uh ; Dt '/Q  .It O h ; Dt D'/Q D .u0;h ; 'jtD0 / ˇ˛DX C .gŒQz h ; s h '/Q  .p˛ ; 'jxD˛ /.0;T / ˇ˛D0 : We pass to the limit as h ! 0 in this identity using the properties uh ! u in L2 .Q/, It O h ! It  in C .Q/ (see (77) and Lemma 24), and Lemma 25 as well as u0;h ! u0 in L2 ./ and s h ' ! ' in L1 .Q/. Since also .It  ; Dt D'/Q D . ; D'/Q , identity (19) is derived. The above assumption Dt D' 2 L1 .Q/ is easily removed by a limit transition using the averages ' . / . Since uO h ! u in L2 .Q/ and D uO h ! Du weakly in L2 .Q/, one gets . / h uO jxD˛ ! ujxD˛ weakly in L2 .0; T / for ˛ D 0; X. But also uO h jxD˛ D u˛ ! u˛ 2 in L .0; T /, where ˛ D 0 for m D 1 and ˛ D X for m D 1; 2, so that the boundary conditions (20) hold. Lemma 27. The equations Dt D Du and Dt xe D u hold in L2 .Q/. Also

jtD0 D 0 and xe jtD0 D xe0 in the sense of traces of functions in C .Œ0; T I L2 .// and C .Q/. Proof. To get both equations, we pass in the equations Dt h D D uO h and Dt xO eh D uO h (the latter one follows from (45)) to the weak limit in L2 .Q/. To get both initial conditions, one can pass in the initial conditions h jtD0 D 0;h and xO eh jtD0 D xO e0;h to the limit, respectively, in L2 ./ and C ./ owing to the properties h ! in C .Œ0; T I L2 .//, 0;h ! 0 in L2 ./, xO eh ! xe in C .Q/, and xO e0;h ! xe0 in C ./. This completes the proof of Theorem 1.

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

3.2

2451

The Uniqueness and Lipschitz Continuous Dependence on Data for Weak Solutions

3.2.1 Statement of Results in the Barotropic Case We introduce the additional assumption on g. A8 . For all zk D . k ; uk ; xe;k / 2 Œa1 ; a  R  Œa; a, k D 1; 2, and almost all .x; t / 2 Q, the inequality    jg.z1 ; x; t /  g.z2 ; x; t /j  C .a/u 5 C b0 .t / j 1  2 j C jxe;1  xe;2 j  C.C .a/u 4 C d0 .t / ju1  u2 j for any a > 1; with u WD max fju1 j; ju2 jg; holds, with b0 D b0;a  0, d0 D d0;a  0 and kb0 kL1 .0;T / C kd0 kLr0 .0;T /  C1 .a/ for r0 D 1. Let assumption A08 arise from A8 after replacing the last inequality by   jg.z1 ; x; t /  g.z2 ; x; t /j  C .a/u 5 C b0 .t / j 1  2 j    C C .a/u 6 C b0 .t / jxe;1  xe;2 j C .C .a/u q0 C b0 .t / ju1  u2 j for any a > 1 with some q0 2 Œ1; 3, the above u, b0 and d0 but r0  2. These assumptions are versions of the local Lipschitz condition on g with respect to z (which are obviously satisfied, in particular, if g do not depend on z). The statement of the uniqueness theorem (but not its proof) is short. Theorem 4. Let the hypotheses of Theorem 1 and assumptions A7 –A8 (or A7 and A08 with q0 D 3 and r0 D 2) be valid. Then the weak solution to problem Pm from the class N .Q/  V2 .Q/  S21;1 W .Q/ (defined in Sect. 3.1) is unique. Theorem 4 is proved together with the Lipschitz continuous dependence on data of the weak solutions. To state the corresponding theorem, we define additional notation. .1/ .2/ Let us consider two problems, Pm and Pm , of the form Pm . In the problem .`/ Pm , ` D 1; 2, the functions , p, and g and the initial and boundary data 0 ; u0 ; u , .`/ and p are supplied with the superscript .`/ and written as 0;.`/ , 0;.`/ , u , etc. The .`/ .`/ .`/ .`/ .`/ weak solution to the problem Pm is denoted by z D . ; u ; xe /. We introduce the difference ˆ WD ˆ.1/  ˆ.2/ so that  D .1/  .2/ ,  0 D     .1/ .2/

0;.1/  0;.2/ , u D u u , z D z.1/ z.2/ , etc. We set g z.2/ WD g .1/ z.2/    g .2/ z.2/ . Theorem 5. Let the hypotheses of Theorem 1 and assumptions either (a) A7 –A8 or (b) A7 and A08 with some q0 2 Œ1; 3/ and r0 > 2 be valid for the data of .1/ .2/ both problems Pm and Pm . Then the following bounds for the norm of difference between their weak solutions by the sum of norms of the corresponding data

2452

A. Zlotnik

differences respectively hold: kzk.1/ WD k kC .Œ0;T IL1 .// C kukV2 .Q/ C kxe kS 1;1 W .Q/ 2

  K.N / k 0 kL1 ./ C ku0 kL2 ./ C ku kV Œ0;T  C kp kL4=3 .0;T /      C kg z.2/ kF m .Q/ C kI hmi It g z.2/ kL1 .Q/ C k. ; p/kL1 .IC Œa1 ;a/ ; (81) kzk.0/ WD k kC .Œ0;T IL2 .// C kukL2 .Q/ C kukC .Œ0;T IH 1Im .// C kxe kW .Q/   Kq0 ;r0 .N / k 0 kL2 ./ C ku0 kH 1Im ./ C ku kL4=3 .0;T / C kp kL1 .0;T /    .2/    m i kL1 .0;T / C kI hmi It g z.2/ kL2;1 .Q/ C kI g z.2/ ke F .Q/ C ı3;m khg z ~

C k. ; p/kL2 .IC Œa1 ;a/



(82)

with a D K 0 .N /, ~ D .1/ Π.1/  and . ; p/ being the pair of differences. .1/

.2/

In this theorem, the weak solutions to problems Pm and Pm are unique according to Theorem 4. The norm kzk.1/ is like in the definition of the weak solution and is bounded by the rather strong norms of data differences. The norm kzk.0/ is weaker but it is correspondingly bounded by the essentially weaker norms of data differences.

3.2.2 Proofs .1/ .2/ Let the data of problems Pm and Pm satisfy the hypotheses of Theorem 1 and .1/ and g .1/ satisfy assumptions A7 and either (a) A8 or (b) A08 for some q0 2 Œ1; 3/ and r0 > 2. .`/ For ` D 1; 2, we take some weak solutions z.`/ D . .`/ ; u.`/ ; xe / to problems .`/ Pm and assume that (without loss of generality) N

1

 .`/  N ; kDt .`/ kL2 .Q/  N ; ku.`/ kV2 .Q/ C kxe.`/ kC .Q/  N

(83)

with the parameter N > N . The main goal is to establish the bounds 0Im kzk.1/  K.N /1Im a ; kzk.0/  Kq0 ;r0 .N /a

(84)

and 0Im denote the sums of norms of the data with a D K 0 .N /, where 1Im a a differences on the right-hand sides of, respectively, bounds (81) and (82). We notice that for q0 D 3 and r0 D 2, the right bound (84) is also  valid but with C Kq0 ;r0 .N / depending also on the function !./ WD sup0tT  ku.1/ k6L6 .Q t;t C /  2 .2/ 6 ku kL6 .Q C kd0 kL2 .t;tC / . / t;t C

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2453

Below in Lemmas 28, 29, 30, 31, and 32, auxiliary bounds for the differences ƒh1i (for m D 1),  , xe , g .1/ Œz, and u are sequentially derived. Then in Lemma 33, bounds (84) are proved. Finally it is shown that bounds (84) directly imply Theorems 4–5. Note that clearly f .1/ . .1/ /  f .2/ . .2/ / D f .1/ . .1/ /  f .1/ . .2/ / C f .1/ . .2/ /  .2/ .2/ f . /; i.e., shortly, Œf ./ D f .1/ ./ C .f /. .2/ /:

(85)

Let ƒ.`/ and ƒh1i;` be the functions arising from ƒ and ƒh1i for ` D 1; 2. Lemma 28. For any a > 1 and almost all x 2 , the following bound holds: kƒh1i .; x/kC Œa;a  K.a/k .; x/kC Œa1 ;a1  with a1 D K1 .a/: 1

Proof. Owing to properties (29) for almost all x 2  and any  2 R, the formula    ƒh1i .; x/ D s  1= Œƒh1i  .0 ; x/ d 0 0

holds. By using formula (85) and assumption A7 for D .1/ , one gets the inequality jƒh1i .; x/j ˇ  ˇ ˇ ˇ ˇ ˇ  K2 .a/ ˇs ˇƒh1i .0 ; x ˇ d 0 ˇ C k .; x/kC Œa1 ;a1  with a1 D K1 .a/ 1 0

for any jj  a. Application of the Gronwall-Bellman lemma (in ) leads to the result. Lemma 29. For any q 2 Œ1; 1, the following bound holds:  lk kC .Œ0;tILq .//  K.N / kukL1 ..0;t/IW 1;qIm .// Ck 0 kLq ./ C ku0 kW 1;qIm ./ C kIt u kC Œ0;t C kIt p kC Œ0;t    Ckg .1/ ŒzkL1 .Qt / C kI hmi It g z.2/ kLq;1 .Qt / C k. ; p/kLq .IC Œa1 ;a/ (86) for t 2 .0; T , with a D K 0 .N /. Proof. From Lemmas 7 and 5, one has Z

    ƒh1i ˆŒz.x; x 0 ; t /; x 0 dx 0 D V .t / for m D 1; 

(87)

2454

A. Zlotnik

ƒŒ  D B hmi Œz C It ;m for m D 2; 3:

(88)

Remind that ˆŒz.x; x 0 ; t / WD ƒŒ .x; t /  B h1i Œz.x; t/ C B h1i Œz.x 0 ; t / and B hmi Œz WD ƒŒ 0  C It pŒ  C I hmi ug . Owing to the first bound (83) and the above assumptions on .1/ and p .1/ , one gets K3 .N /1 j j  jƒ.1/ Œ j  K3 .N /j j; jp .1/ Œ j  K3 .N /j j: (89) Application of formula (85) leads to the equalities   B hmi Œz D ƒ.1/ Œ 0  C ƒŒ 0;.2/  C It p .1/ Œ  C pŒ .2/  C I hmi ug ;   I hmi ug D I hmi u  u0  It g .1/ Œz  It gŒz.2/ : Consequently with the help of bounds (89), one obtains  kB hmi ŒzkLq ./  K4 .N / k 0 kLq ./ C k. ; p/kLq .IC ŒN 1 ;N / CIt k kLq ./ C kukW 1;qIm ./ C ku0 kW 1;qIm ./    Ckg .1/ ŒzkL1 .Qt / C kI hmi It g z.2/ kLq ./ :

(90)

For m D 1, we use formulas (85) and (29) to rewrite equality (87) in the form Z



 M .ˆ.1/ ; ˆ.2/ ; x 0 /ˆ C ƒh1i .ˆ.2/ ; x 0 / dx 0 D V;

 1 1 d ˛; M .ˆ.1/ ; ˆ.2/ ; x 0 / WD s .1/ .˛ˆ.1/ C .1  ˛/ˆ.2/ ; x 0 / 0

ˆ.`/ WD ˆŒz.`/ .x; x 0 ; t /: By setting M WD

R 

M .ˆ.1/ ; ˆ.2/ ; x 0 / dx 0 , one obtains

M ƒ.1/ Œ  D M ƒŒ .1/  C M B h1i Œz Z Z  M .ˆ.1/ ; ˆ.2/ ; x 0 /B h1i Œz.x 0 ; t / dx 0  ƒh1i .ˆ.2/ ; x 0 / dx 0 C V: (91) 



Owing to bounds (83), the assumptions N 1  0;.`/  N and ku0;.`/ k  N , and the assumptions on .`/ , p .`/ , and g .`/ , the following bounds hold: jˆ.`/ j  K5 .N /; K6 .N /1  M .ˆ.1/ ; ˆ.2/ ; x 0 /  K6 .N /; XK6 .N /1  M  XK6 .N /:

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2455

Therefore using the left bound (89) and Lemma 28, from formula (91), one gets  j j  K7 .N / k kC Œa1 ;a C k kL1 .IC Œa1 ;a/  CjB h1i Œzj C kB h1i ŒzkL1 ./ C jV j with a D K 0 .N /: Application of bound (90) and the equality V D . 0 ; 1/ C It .uX  u0 / that follows from Lemma 3 leads to the inequality   k kLq ./  K8 .N / It k kLq ./ C d hmi

(92)

for m D 1. Here d hmi denotes the sum of norms on the right-hand side of bound (86). The case m D 2; 3 is much simpler. From equality (88) and the left bound (89), one gets   j j  K9 .N / k kC ŒN 1 ;N  C jB hmi Œzj C kIt p kC Œ0;T  : This bound and bound (90) imply inequality (92) for m D 2; 3 as well. Application of the Gronwall-Bellman lemma to it completes the proof. Corollary 5. The following bounds hold on .0; T :   k kC .Œ0;tIL1 .//  K1 .N / kukL2;1 .Qt / C kg .1/ ŒzkL2;1 .Qt / C 1Im ; (93) a   k kC .Œ0;tIL2 .//  K1 .N / kukW .1;0/Im .Q / C kg .1/ ŒzkL1 .Qt / C 0Im : (94) a 2;1

t

Lemma 30. The following equalities and bounds hold: xe D xe0 C It u; Dt xe D u; Dxe D  ; Dt Dxe D Du; (95) kxe .; t /kC ./  k 0 kL1 ./ C k .; t /kL1 ./ C X 1 kukL1 .Qt / ; 0 < t  T;





kxe kW .Q/  c k kC .Œ0;T IL2 .// C kukL2 .Q/ ;   kxe kS 1;1 W .Q/  c k kC .Œ0;T IL2 .// C kukL2 .Q/ C kDukL2 .Q/ : 2

(96) (97) (98)

Proof. Lemma 6 implies equalities (95). The first of them leads to the bound kxe .; t /kL1 /  X k 0 kL1 ./ C kukL1 .Qt / By applying the inequality kwkC ./  X 1 kwkL1 ./ C kDwkL1 ./ and the formula Dxe D  , one derives bound (96). Other two bounds clearly follow from equalities (95).

2456

A. Zlotnik

Lemma 31. In the cases (a) and (b), the following bounds, respectively, hold:    (99) kg .1/ ŒzkL2;1 .Qt / K.N / k 0 kL1 ./ CIt b2 k kL1 ./ Cd2 kukL2 ./ ;    kg .1/ ŒzkL1 .Qt /  K.N / k 0 kL1 ./ C It b1 k kL2 ./ C d1 kukL2 ./ ; (100) for any t 2 .0; T , where kb2 kL1 .0;T / C kd2 kL1 .0;T /  K20 .N /; kb1 kL1 .0;T / C kd1 kLr1 .0;T /  K10 .N / (101) with r1 WD minfr0 ; 6=q0 g. Proof. Due to the bounds N one gets

1

.`/

 .`/  N and kxe kC .Q/  N , in the case (a),

  kg .1/ ŒzkL2;1 .Qt /  K1 .N /It b2 k kL1 ./ C b2 kxe kC ./ C d2 kuk with b2 D ku.1/ k5L10 ./ Cku.2/ k5L10 ./ C1Cb0 and d2 D ku.1/ k4L1 ./ Cku.2/ k4L1 ./ C 1 C d0 . Since ku.`/ kV2 .Q/  N and V2 .Q/  L1;4 .Q/ \ L10;5 .Q/ (see (1)), the left bound (101) holds. In the case (b), similarly one has   kg .1/ ŒzkL1 .Qt /  K1 .N /It b1 k k C b1 kxe kC ./ C d1 kuk q

q

with b1 D ku.1/ k6L6 ./ Cku.2/ k6L6 ./ C1Cb0 and d1 D ku.1/ kL02q0 ./ Cku.2/ kL02q0 ./ C 1 C d0 . Since also V2 .Q/  L6 .Q/, the right bound (101) holds as well. Finally application of bound (96) in both cases completes the proof. Lemma 32. The following bounds hold:   kukV2 .Qt /  K.N / kd  kL1;2 .Qt / C kg .1/ ŒzkL2;1 .Qt / C 1Im ; N  kukL1 ..0;t/IH 1Im .// C kukL2 .Qt /  K.N / kd  kL2 .Qt /  Ckg .1/ ŒzkL1 .Qt / C 0Im ; N

(102)

(103)

for any t 2 .0; T , where kd kL2 .0;T /  N C T 1=2 . Proof. Let ~ .`/ WD .`/ Œ .`/ . According to Lemma 4, u is a weak V2 .Q/-solution to problem Lm with the data ˛  1; ~ D ~ .1/ ;

D .~/Du.2/ CpŒ .2/ Cp .1/ Œ ; f D g .1/ ŒzCgŒz.2/ 

as well as v 0 D u0 , v D u , and s D  .

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2457

The first and second bounds (83) together with the assumptions on .`/ and p .`/ imply the bounds K1 .N /1  ~ .`/  K1 .N /, kDt ~ .1/ kQ  K1 .N /, and  k kLq;2 .Q/  K2 .N / k kLqQ .IC ŒN 1 ;N / kDu.2/ kQ C kpkLq .IC ŒN 1 ;N /    Cd k kLqQ ./ .0;T / ; for 1  q  2; with qQ WD 2q=.2  q/ and d WD kDu.2/ kL2 ./ C 1. Clearly kd k.0;T /  N C T 1=2 . By taking q D 2 and applying Proposition 2 and the inequality kf kF m .Qt /  ckf kL2;1 .Qt / for f D g .1/ Œz (see Sect. 2), one derives bound (102). m By taking q D 1 and applying Proposition 3 and the inequality kIf ke F .1/ .Qt /  ~

ckf kL1 .Qt / (that follows from (4)) for f D g .1/ Œz, one derives bound (103) as well.

Lemma 33. In the cases (a) and (b), respectively, the left and right bounds (84) hold. Proof. We multiply bound (93) for  by Œ2K1 .N /1 and add to bound (102) for u. By applying bound (99) for g .1/ Œz and the inequality   2    kbk kL1 ./ It bk k k2LqQ ./ It bk k kLqQ ./    K 0 .N /It bk k k2LqQ ./

(104)

for k D 2 and qQ D 1, one obtains the inequality  ˚  2 Y2 .t / WD k k2C .Œ0;tIL1 .// Ckuk2V2 .Qt /  K2 .N / It .d 2 Cb2 Cd2 /Y2 C.1Im a / : 2 with a D K3 .N /. Then Y2 .T /  K3 .N /.1Im a / according to the GronwallBellman lemma. By applying also bound (98) for xe , one derives the left bound (84). Next, we multiply bound (94) for  by Œ2K1 .N /1 and add to bound (103) for u. By applying bound (100) for g .1/ Œz, inequality (104) for k D 1 and qQ D 2, and the Hölder inequality, one obtains

Y1 .t / WD k k2C .Œ0;tIL2 .// C kuk2L1 ..0;t/IH 1Im .// C kuk2Qt   ˚  2 2  K4 .N / It .d 2 C b1 /Y1 C It .d1 kuk / C .0Im a / : The Gronwall-Bellman lemma leads to the inequality ˚ 2 2 kuk2Qt  Y1 .t /  K5 .N / It .d1 kuk / C .0Im a / :

(105)

2458

A. Zlotnik

By omitting the intermediate inequality and applying Lemma 2 for y1 D kuk (n D 1), one gets kukQ  K6;q0 ;r0 .N /0Im a . Now the right inequality (105) gives 2 / . Finally application of bound (97) completes the proof Y1 .T /  K7;q0 ;r0 .N /.0Im a of the right bound (84). Notice that in the case r1 D 2, the quantities K6;q0 ;r0 and K7;q0 ;r0 depend on d1 , more precisely, on !1 . / D sup0tT  kd1 kL2 .t;tC / . Bounds (84) are established. To prove Theorem 4 (on uniqueness), it suffices .1/ .2/ to take Pm D Pm D Pm so that z.1/ and z.2/ are simply two weak solutions to problem Pm . In this case a1Im D 0Im D 0 and the bounds lead to z.1/ D z.2/ . a By reminding Theorem 1 (on existence), this means that under the hypotheses .1/ .2/ of Theorems 5, problems Pm and Pm have unique weak solutions that satisfy bounds (83) with N D K.N /. Therefore bounds (81)–(82) immediately follow from (84) (remind that u 2 C .Œ0; T I L2 .// due to Lemma 34).

4

Well-Posedness of the IBVPs for the Viscous Heat-Conducting Gas 1D Equations

4.1

Additional Auxiliary Results

We go back to the linear parabolic IBVP Lm ; see (5), (6), and (7). In what follows the Riemann-Stieltjes integrals are marked by .S /. Proposition 8. Let the hypotheses of Proposition 2 including k. / f kF m .QT  / ! 0 as  ! 0C be valid. Then the weak solution v 2 V2 .Q/ of problem Lm satisfies the integral identity Z

 Q

 12 ˛v 2 Dt '

C

1 .Dt ˛/v 2 ' 2

Z

T

.S / 0

Z



C sD.v'/  f ' dxdt D 

ˇaDX Z ˇ .It s/jxDa d .va 'jxDa /ˇ C aD0

0

T

1 ˛j .v 0 /2 'jtD0 2 tD0

dx

ˇaDX ˇ sa vjxDa 'jxDa dt ˇ aD0

for any ' 2 C 1 .Q/, 'jtDT D 0. The result follows from [11, Proposition 2.3] (the similar argument is given also in [13, Lemma 5.4]). We assume that below in the subsection ˛; ~ 2 L1 .Q/, N 1  ˛  N , N 1  ~  N and kDt ~kL2;4=3 .Q/  N . It is essential that smoothness (or continuity) of ˛ and ~ in x is not assumed. Now we treat weaker then V2 .Q/-solutions. Proposition 9. Let kDt ˛kL2;4=.32"˛ / .Q/  N for some "˛ 2 .0; 1/. Let also D 0, f 2 L1 .Q/, v D 0, and s 2 L1 .0; T /. Then there exists a unique solution

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2459

v 2 V1 .Q/ to problem Lm . Moreover, v; ˛v 2 C .Œ0; T I L1 .// and the following bounds hold: k˛vkC .Œ0;T IL1 .//  k˛jtD0 v 0 kL1 ./ C kf kL1 .Q/ C ks kL1 .0;T / ;   kvkLq0 ;r0 .Q/ C kDvkLq1 ;r1 .Q/  K" .N / kv 0 kL1 ./ C kf kL1 .Q/ C ks kL1 .0;T / (106) for all q0 ; r0 2 Œ1; 1, q1 ; r1 2 Œ1; 2 such that .2q0 /1 Cr01 D .1C"/=2, .2q1 /1 C r11 D .1 C "/=4 and " 2 .0; "˛ . Also vjtD0 D v 0 in the sense of values of v 2 C .Œ0; T I L1 .//. In addition, It s 2 C .Œ0; T I W 1;1 .// and the following bound holds:   kIt skC .Œ0;T IW 1;1 .//  K.N / kv 0 kL1 ./ C kf kL1 .Q/ C ks kL1 .0;T / : The semidiscrete counterpart of this result is proved below; see Lemmas 38 and 43 below. Next results will be required to study the uniqueness and Lipschitz continuous dependence on data. Let problem L0m be the special case of problem Lm for 2 L6=5 .Q/, f D 0, v 0 D 0, and s D 0. In the spirit of [27], the function v 2 L2 .Q/ is called a weak L2 .Q/-solution to problem L0m if it satisfies the integral identity Z

  v ˛Dt ' C D.~D'/ dxdt D

 Q

Z D' dxdt Q

for all ' 2 H 1 .Q/ such that ~D' 2 V2 .Q/, 'jtDT D 0 as well as 'jxD0;X D 0 for m D 1, 'jxDX D 0 and .~D'/jxD0 D 0 for m D 2, or .~D'/jxD0;X D 0 for m D 3. Problem L0m has a unique solution v from L2 .Q/ and the following bound holds: kvkL2 .Q/  K.N /k kL6=5 .Q/ :

(107)

But it will be essential to have additional bounds for v in the case 2 Lp .Q/, p 2 .6=5; 2/. We state such results; for simplicity, assume that ˛ D ˛.x/. First, we present bounds v in Vp .Q/ for p > 2 and Dv in Lp .Q/ for p < 2 but, in general, only for p close to 2 in both cases. Proposition 10. 1. Let v be V1 .Q/-solution to problem L0m with 2 Lp .Q/. There exists p D 2 C ı0 with ı0 D 1=K1 .N / such that if p 2 Œ2; p, then v 2 Vp .Q/ \ C .Œ0; T I Lp .// and the following bound holds kvkVp .Q/  K.N /k kLp .Q/ :

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A. Zlotnik

2. Let v be L2 .Q/-solution to problem L0m with 2 Lp .Q/ and p D 0 p maxfp ; 6=5g. If p 2 Πp; 2, then Dv 2 L .Q/ and the following bound holds kvkL2 .Q/ C kDvkLp .Q/  K.N /k kLp .Q/ :

(108)

Note that Claim 1 is really valid without the above condition on Dt ~. Second, bound v in Lq;r .Q/ and separately in L1;r .Q/. Proposition 11. Let v be L2 .Q/-solution to problem L0m with

2 Lp .Q/.

1. Let q; r 2 Œ2; 1, .2q/1 C r 1 D .1  ı/=2, jıj < 1=2, and 0 < " < 1=2  ı. Then kvkLq;r .Q/  K1;ı;" .N /k kLp .Q/ for p D 3=.2  ı  "/: 2. Let r D 1 C 2=p00 with p0 2 .p; 2/ and 0 < "0 < 1  r 1 . Then kvkL1;r .Q/  K2;r;"0 .N /k kLp .Q/ for p 1 D .5=6/.1r 1 "0 /Cp01 .r 1 C"0 /: All Propositions 9, 10, and 11 (as well as a result much more general than bound (107)) were proved in [36, Section 2]. Here we notice only the following. The proof of bound (106) exploits L1 .Q/-bound of the well-known type [27] (see also [11]) for the solution to the problem dual to problem Lm . Claim 1 of Proposition 10 is deduced by some nonstandard duality argument [12] from the related Kruzhkov theorem [26] concerning W 2;1I p .Q/solvability of nondivergent parabolic IBVPs with the coefficients in L1 .Q/. Claim 2 follows from Claim 1 by the standard duality argument. The bounds in Proposition 11 are derived from bound (107), the energy bound kvkV2 .Q/  K.N /k kL2 .Q/ , and bound (108) by methods of the theory of real interpolation of Banach spaces [17].

4.2

Weak Solutions and Their Global Existence

We consider a system of quasilinear partial differential equations describing the viscous perfect polytropic gas 1D flow Dt D Du;

(109)

Dt u D D C gŒxe ;  D Du  p; p D k ; D 1= ;

(110)

cV Dt  D D C  Du C f Œxe ;  D  D ; Dt x e D u

(111) (112)

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2461

in the domain Q. Here the sought functions ; u; , and xe depend on the Lagrangian mass coordinates .x; t / 2 Q and F Œxe .x; t/ D F .xe .x; t /; x; t / for F D g; f . We supplement the system of equations with the initial and boundary conditions . ; u;  ; xe /jtD0 D . 0 ; u0 ;  0 ; xe0 / on ;

(113)

ujxD0 Du0 .t / .or jxD0 D  p0 .t //; ujxDX DuX .t / .or  jxDX D  pX .t //; (114) jxD0 D 0 .t /; jxDX D X .t /

(115)

Rx with xe0 .x/ WD 0 0 ./ dx, and 0 < t < T . m D 1; 2; 3 is set if, respectively, ujxD0;X ;  jxD0 and ujxD0 , or  jxD0;X are given on Q . We assume that the functions u0 ; uX ; p0 , and pX equal zero if they do not appear in the mth boundary condition (114). Denote IBVP (109), (110), (111), (112), (113), (114), and (115) by Pm . We recall the physical meaning of the involved quantities: , u; and  are the specific volume, velocity, and absolute temperature of a gas; xe is the Eulerian coordinate; and p are the density and pressure;  and  are the stress and heat flux; g and f are the densities of the body forces and heat sources; u˛ , p˛ , and ˛ , for ˛ D 0; X, are the velocity of piston, external pressure, and external heat flux; and X is the total mass of the gas. cV , , k, and  are physical coefficients independent of the solution. Now we list our conditions on the data. Let N > 1 be an arbitrarily large parameter, and u D .u0 ; uX /, p D .p0 ; pX /, and  D .0 ; X / be the boundary data pairs. C1 . The coefficients ; k; cV ;  2 L1 ./ are such that N 1   N , 0  k  N , N 1  cV  N , and N 1    N: C2 . The initial functions 0 2 L1 ./, u0 2 L2 ./, and  0 2 L1 ./ are such that 1 N  0  N ,  0 > 0, and ke 0 kL1 ./ Cks 0 kL1 ./  N , where e 0 D 12 .u0 /2 CcV  0 and s 0 D k log 0 C cV log  0 are the initial energy and entropy. C3 . The free terms g and f are Carathéodory functions on R  Q; moreover, jg.; /j  g./ and 0  f .; /  f ./ on Q for all  2 R, with kgkL2;1 .Q/ C kf kL1 .Q/  N . C4 . The boundary data u 2 V Œ0; T , p 2 L1 .0; T /, and  2 L1 .0; T / are such that p˛  0, ˛  0, for ˛ D 0; X, and ku kV Œ0;T  C kp kL1 .0;T / C k kL1 .0;T /  N . Furthermore, for all  2 .0; T / and almost all t 2 .0; T   /; the inequality holds Z p˛ .t C /  p˛ .t /  a .t /

tC

p˛ .t 0 / dt 0 ; ˛ D 0; X;

(116)

t

with some a 2 L1 .0; T /, a  0 and sup0 0 and  h > 0. We further approximate the initial and boundary conditions (113), (114), and (115) as follows: . h ; uh ;  h ; xeh /jtD0 D . 0;h ; u0;h ;  0;h ; xe0;h /; ./

. /

(146) . /

. /

uh jxD0 Du0 .or O h jxD0 D  p0 /; uh jxDX DuX .or O h jxDX D  pX /; (147)  h jxD0 D 0 ;  h jxDX D X :

(148)

Set 0;h D s1h 0 ,  0;h D s1h  0 , and xe0;h D Ih 0;h ; moreover, u0;h D s h u0 but redefine ./ . / u0;h .x/ D u0 on 0 for m D 1 and u0;h .x/ D uX .0/ on n for m D 1; 2. Here . / . / the averages in time u˛ and p˛ are used; see Sect. 3.1.3, for 0 <   N 2 , with the extensions u˛ .t / D u˛ .T / and p˛ .t / D 0 for t > T , ˛ D 0; X. We denote problem (142), (143), (144), (145), (146), (147), and (148) by Pmh . The aim is to prove a semidiscrete counterpart of Theorem 6, Claims 1 and 3. Proposition 12. 1. Let conditions C1 – C4 be valid. Then problem Pmh has a solution satisfying the uniform in h bounds > > > > > > h > >N .Q/ C kOu h kV2 .Q/ C kO h kV1 .Q/ C kxO eh kS 1;1 W .Q/  K.N /; (149) 2

Oh

h

k kLq0 ;r0 .Q/ C kD  kLq1 ;r1 .Q/  K" .N /;

(150)

k log  h kL1;1 .Q/ C kD log  h kL2 .Q/  K.N /;

(151)

h0;1=2i

kOu h k2;2

1

h0;1=2i

C k h k2;1

 K.N /

(152)

for the same .q0 ; r0 / and .q1 ; r1 / as in Theorem 6. 2. If in addition N 1   0 , then ŒK.N /1   h . Actually Pmh is the Cauchy problem for a system of nonlinear ODE (after the algebraic elimination of the given boundary data). The existence of its solution on a sufficiently small time segment Œ0; T0   Œ0; T  follows from the classical Carathéodory theorem. The possibility to extend the solution to the whole segment Œ0; T  preserving the properties h > 0 and  h > 0 is justified by a priori bound (149) and the estimate  h  ı h D expfK.N /=hg > 0; see (151). So the point in the proof of Proposition 12 is the derivation of uniform in h a priori bounds in Claims 1 and 2. We first note that condition C4 implies the inequalities . / ku./ ˛ kC Œ0;T  C kDt u˛ kL1 .0;T /  ku˛ kV Œ0;T   N; ˛ D 0; X;

(153)

kp˛./ kC Œ0;T   kp˛ kL1 .0;T /  N; Dt p˛. /  a p˛. / ; ˛ D 0; X:

(154)

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A. Zlotnik

By definition of  h , equation (142) can be rewritten in the two forms: Dt h D . h /1  h h C . h /1 k h  h ;

(155)

 h h D Dt . h h /  k h  h :

(156)

Let V h .t / D k h .; t /kL1 ./ be the semidiscrete total volume of gas. Lemma 45. The following relations hold: . /

. /

.2N /1  V h D k 0 kL1 ./ C It .uX  u0 /  .X C T /N for m D 1:

(157)

The proof repeats one for bound (51) in Lemma 10. We recall that conditions (117) and   N 2 are used namely here. h To handle the nonhomogeneous boundary data for u, we set vm WD uh  uh;m , where . /

. /

. /

uh;1 D .1  `h /u0 C `h uX with `h D Ih h =V h ; uh;2 D uX ; uh;3 D 0: . /

We remind that 0  `h  1 and the formulas (55) are valid with dV D .uX  ./ u0 /=V h . Lemma 46. The following balance equations hold: Z Dt 

1 h 2 .v / 2 m

Z dx C

R 1

h

 D uO dx D 

Dt E h D mh C where E h WD

h

h 2  2 .vm /

Z



mh

Z C 

h .gŒxO eh   Dt uh;m /vm dx; (158)

˚

h .gŒxO eh   Dt uh;m /vm C f ŒxO eh  dx C 0 C X ; (159)

 C cVh  h dx and

8 R  R  h h h h h h ˆ ˆ ˆDt  dV dx   .Dt dV / C dV k  dx < R R h m D Dt  p0. / h dx C  .Dt p0. / / h dx ˆ ˆ ˆ :D R  ./ h dx  R .D  . / / h C X 1 .p . /  p . / /uh  dx t  ;3 t ;3  X 0

with ;3 .x; t / D .1 

x /p0 .t / X



for m D 1 for m D 2 (160) for m D 3

x p .t /. X X

h Proof. We rewrite equation (143) in the form Dt vm D D O h C g h  Dt uh;m . By h taking its L2 ./-inner product by vm , one gets the balance equation (158) with

mh WD

Z 

ˇ h h ˇxDX  h D uO ;m dx C .O h vm / xD0 :

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

1h

2473

h For m RD 1, by the property vm jxD0;X D 0 and the right formula (55), the formula h h D dV   dx arises. By applying equation (156), one derives (160). For m D 2, the equalities v2h jxDX D 0 and Duh;2 D 0 hold and thus

. / ˇxDX . / 2h D .O h v2h /jxD0 D p0 uh ˇxD0 D p0

. /

. /

Z

D uO h dx D 



Z 

. /

p0 Dt h dx:

. /

For m D 3, since D;3 D X 1 .pX  p0 / is x-independent, one has  . / ˇxDX 3h D ;3 uO h ˇxD0 D Z D 



Z 



 . / . / .D;3 /Ou h  ;3 D uO h dx

 . / . / .D;3 /uh  ;3 Dt h dx:

These formulas imply (160) for m D 2; 3. By integrating equation (144) over  and using boundary conditions (148), one gets Z Z  h   D uO h C f ŒxO eh  dx C 0 C X : cVh  h dx D Dt 



Summing up this equality and (158) proves the balance equation (159) as well. Lemma 47. The energy bound kuh kL2;1 .Q/ C k h kL1;1 .Q/  K.N / holds. Proof. We apply the operator It to the balance equation (159) and get h E h .t / D E h .0/ C It mh C .gŒxO eh   Dt uh;m ; vm /Qt C .f ŒxO eh ; 1/Qt C It .0 C X / h kC .Œ0;T IL2 .//  N k 0 kL1 ./ C ku0 k2 C kuh;m jtD0 k2 C It mh C kgkL2;1 .Q/ kvm   h CIt kDt uh;m k kvm k C kf kL1 .Q/ C k kL1 .0;T / :

By using conditions C2 – C4 , one finds   h h E h .t /  It mh C N kvm kC .Œ0;T IL2 .// C It kDt uh;m k kvm k C K1 :

(161)

For m D 1, we apply relations k h h kL1 ./  N V h , k h  N 2 cVh , and Dt dV D ./ ./  Dt .uX  u0 / =V h  dV2 together with inequalities (157) and (153) and obtain



  jIt 1h j  K2 kdV kC Œ0;T  C kDt dV kL1 .0;T / C kdV kC Œ0;T  It kcVh  h kL1 ./  K3 It E h C K4 :

2474

A. Zlotnik

Then frompinequality (161), we applying also the left formula (55) and the estimate h k  2E h , one gets the inequality kvm 1=2 Y  It .bY / C K5 Ymax C K6 with Ymax WD kY kC Œ0;T 

(162)

for Y WD E h , where b D 2jdV j C K3 . For m D 2; 3, with the help of bounds (153) and (154), the inequality of the R . / same form (162) arises as well, for Y WD E h C  p0 h dx in the case m D 2 or R Y WD E h   ;3 h dx in the case m D 3, with b D a . Application of the Gronwall-Bellman lemma to inequality (162) leads to the bound Ymax  K7 and thus the energy bound too. Solving equation (155) with respect to h leads to the important formula ˚ ˚

h D 0;h C . h /1 k h It Œ. h /1  h   h with  h WD exp . h /1 It  h (163) similar to (130). Lemma 48. kDIt O h kL2;1 .Q/  K.N / and kIt  h kC .Œ0;T IL1 .//  K.N /. Proof. By applying the operator It to equation (143), one gets DIt O h D uhg WD uh  u0;h  It g h . Then by the energy bound, the following relations hold: kDIt O h kL2;1 .Q/ D kuhg kL2;1 .Q/  kuh kL2;1 .Q/ C ku0;h k C kgkL2;1 .Q/  K1 : For m D 1, we rewrite and apply formula (68): h =V h / : It  h D Ih uhg  .Ih uhg ; h =V h / C It . h ; h =V h /  It .uhg ; vm

By using formula (156), one can transform It . h ; h =V h / D . h ; h =V h /  . h ; 0;h =V h .0//   CIt dV . h ; h =V h /  It .k h  h ; 1=V h / : Since k h kL1 ./ D V h , 0  h  N , 0  k h  N and .2N /1  V h , one has p . / kIt  h kC .Œ0;T IL1 .//  XK1 C N C 2N 2 ku kL1 .0;T /   h C2N T N k h kC .Œ0;T IL1 .// C K1 kvm kC .Œ0;T IL2 .// : Now the energy bound leads to the bound kIt  h kC .Œ0;T IL1 .//  K2 . For m D 2; 3, the situation is much simpler since from DIt O h D uhg and the left . /

. /

boundary condition O h jxD0 D p0 , one immediately gets It O h D Ih uhg  It p0 . Thus p . / kIt  h kC .Œ0;T IL1 .//  X kuhg kC .Œ0;T IL2 .// C kp0 kL1 .0;T /  K3 :

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2475

Corollary 8. The following two-sided inequalities for h and bounds for V h hold: ŒK.N /1  h  K.N /.1 C It  h /;

(164)

ŒK.N /1  V h  K.N /:

(165)

Proof. Inequalities (164) follow from formula (163), Lemma 48, and the bounds N 1  0;h  N . Since k h kL1;1 .Q/  K, for m D 2; 3 integrating of (164) over  leads to bounds (165). For m D 1, see (157). Lemma 49. The following semidiscrete law of nondecreasing entropy holds: Z



Dt Z



 cVh log  h C k h log h dx

D

b



 h Q h .D O h /D.1= h / C . h .D uO h /2 C f h /= h dx



C0 = h jxD0 C X = h jxDX  0: Proof. We take the L2 ./-inner product of equation (142) and k h = h as well as of equation (144) and 1= h and integrate by parts using boundary conditions (148). Summing up the arising equalities leads to the law. Corollary 9. The following auxiliary bounds hold:

1

q  kˇ h kL1 .0;T /  K.N /; k log  h kL1;1 .Q/ C  Q h D log  h L2 .Q/  K.N /; (166)  R  h h h h O where ˇ WD   Q .D  /D.1= / dx.

b

Proof. We apply It to the law of nondecreasing entropy, use the formula log  h D j log  h j C 2.log  h /C , and omit the sign-definite terms. Also we apply the inequalities .log  h /C   h ; log h  h ;  log  0;h  s1h .log  0 /;  log 0;h  log N; where the two latter ones follow from the Jensen inequality. Then the inequalities     N 1 k log  h kL1 ./ C It ˇ h  2cVh .log  h /C C k h log h ; 1     cVh log  0;h ; 1/ C .k h log 0;h ; 1  C It . 0 C X /  2N k h kL1 ./ C N V h  .cVh log  0 ; 1 / C .k h log N; 1/ C N

2476

A. Zlotnik

hold. Applying the bounds k h kL1 ./  K1 , V h  K2 and the condition k log  0 kL1 ./  K3 leads to the left bound (166) together with k log  h kL1;1 .Q/  K4 . Usage of the inequality .D log  h /2  .D O h /D.1= h / (that follows from the number inequality .log blog a/2  .ba/.b 1 a1 )) and the Hölder inequality completes the proof.

1

b

 Lemma 50. k h kL1 ./  X 1 k h kL1 ./ C kD O h kL1 .h /  K.N / ˇ h k h kL1 ./  C1 . Proof. The left inequality is elementary. By applying the formula D O h D

b

h h .C/ D.1= h / on Qh and the bound k h kL1 ./  K, one gets ./

b

 1=2  1=2 h h kD O h kL1 .h /  k h kL1 ./   Q h ./ .C/ D.1= h / L1 .h / 1=2 1 h   ˇ h k h kL1 ./ k h kL1 ./ k h kL1 ./  2 k kL1 ./ C Kˇ h k h kL1 ./ : Now taking into account the left inequality leads to the right one as well. Now it is possible to derive the bounds in Proposition 12. Lemma 51. The following bounds for h ;  h , p h , and uh hold: ŒK.N /1  h  K.N /; kO h kV1 .Q/  K.N /; kp h kL2 .Q/  K.N /; kD uO h kL2 .Q/  K.N /: Consequently, bound (151) in Proposition 12, Claim 1, is valid. Proof. Inequalities (164) and Lemma 50 lead to the inequality     k h kL1 ./  K1 It ˇ h k h kL1 ./ C 1 : The Gronwall-Bellman lemma together with the left bound (166) gives the upper bound h  K2 . The lower bound has been proved above; see (164). Now Lemma 50 implies the bounds k h kL1;1 .Q/  ckb  h kV1 .Q/  K3 . Then since k h k2Q  k h kL1;1 .Q/ k h kL1;1 .Q/ , one also derives kp h kQ  N k h kL1 .Q/ k h kQ  K4 . Since K51  Q h , now the right bound (166) implies bound (151) as well.

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2477

Lemma 52. Bounds (149), (150), and (152) in Proposition 12, Claim 1, and k h kL2 .Q/  K.N /; k h kLq1 ;r1 .Q/  K" .N /;

(167)

hold, for .q1 ; r1 / like in Theorem 6. Moreover, Proposition 12, Claim 2, holds too. Proof. uh can be considered as the solution to the semidiscrete linear parabolic problem (34), (35), and (36) with the data ~ D h h , D ph , f D gh , 0 0;h v D u , v D u , and s D p . By using the bounds K11  ~  K1 , kp h k  K2 kg h kL2;1 .Q/  K3 , inequality (153), and the left inequality (154), due to Proposition 6, one gets the bound kOu h kV h1;1=2i .Q/  K4 . 2

Consequently the bound k h kQ  N k h kL1 .Q/ kD uO h kQ C kp h kQ  K6 holds. The formulas Dt h D D uO h ; xO eh D xO e0;h C It uO h ; Dt xO eh D uO h ; D xO eh D h ; DDt xO eh D D uO h (cp. with Lemma 6) ensure the bound   kDt h kQ C kxO eh kS 1;1 W .Q/  c kOu h kV2 .Q/ C k 0 k C k h kQ  K5 2

that completes the proof of bound (149). Next,  h is the solution to the semidiscrete linear parabolic problem (133), (134), and (135) with ˛ D cVh , ~ D h Q h , D 0, f D  h D uO h C f h , and v 0 D  0;h . Note that N 1  cVh  N; K21  ~  K2 ; k 0;h kL1 ./  N; k kL1 .0;T /  N: Moreover, k h D uO h C f h kL1 .Q/  k h kQ kD uO h kQ C kf kL1 .Q/  K3 . Therefore h0;1=2i

 K4 and thus (152) as Lemmas 43 and 44 imply the bounds (150) and k h k2;1 well. Consequently the bound k h kLq1 ;r1 .Q/  N k Q h kL1 .Q/ kD O h kLq1 ;r1 .Q/  K5;" holds. Finally,  h is also the solution to another semidiscrete linear parabolic problem (139) and (134)–(135) with f D h h .D uO h /2 C f h  0 and b D k h h D uO h . Since kbkQ  K6 , if N 1   0;h , then K71   h according to Lemma 42.

4.2.4 Limit Transition in the Heat-Conducting Case We accomplish the limit transition in problem Pmh , m D 1; 2; 3, as h ! 0 to prove Theorem 6. Set  WD .h/ ! 0. Due to Proposition 12, there exists the sequence zh D . h ; uh ;  h ; xeh / of solutions to problems Pmh satisfying the uniform in h bounds (149), (150), (151), and (152). Then there exists a subsequence (hereafter not relabeled) such that

2478

A. Zlotnik

h ! weakly in L1 .Q/; Dt h ! Dt weakly in L2 .Q/; (168) uO h ! u in L2 .Q/ and weakly in L1 .0; T I L2 .//; h

2

(169)

D uO ! Du weakly in L .Q/;

(170)

O h !  in Lq0 ;r0 .Q/; D O h ! D weakly in Lq1 ;r1 .Q/;

(171)

xO eh ! xe weakly in

S21;1 W

.Q/ and strongly in C .Q/:

(172)

for at least any .q0 ; r0 /, r0 < 1 and .q1 ; r1 / from Theorem 6. We notice that the h0;1=2i bounds kO h kV1 .Q/  K and kO h k2;1  K (remind inequality (31)) guarantee the precompactness of O h in L1 .Q/. Then bound (150) implies properties (171). Due to the bound kuh Ou h kL2 .Q/  kD uO h kL2 .Q/ h  K1 h, one can assert (passing to a subsequence) that properties (169) are valid for uh in the role of uO h . In particular, O h !  in L1;1 .Q/. Also  h !  in L1;1 .Q/ due to the bound p p k h  O h kL1;1 .Q/  kD O h kL2;1 .Q/ h  K1 h: By passing to a subsequence, one can assert that  h .; t / ! .; t / in L1 ./ for almost all t 2 .0; T /. Due to  h > 0 the property   0 holds. According to the Fatou theorem, the bounds k h .; t /kL1 ./  K2 and k log  h .; t /kL1 ./  K2 imply that k.; t /kL1 ./  K2 and k log .; t /kL1 ./  K2 for almost all t 2 .0; T /. Consequently k kL1;1 .Q/  K2 and k log  kL1;1 .Q/  K2 . The latter bound

1

implies that  > 0 a.e., in Q. Now the bound kD log  h kL2 .Q/ allows to assert that

1

D log  h ! D log  weakly in L2 .Q/ (after passing to a subsequence). Clearly the limiting vector function z WD . ; u;  ; xe / belongs to N .Q/V2 .Q/ V1 .Q/  S21;1 W .Q/ and satisfies all the bounds (121), (122), (123), and (124) from Theorem 6, Claim 1. Thus the aim is to prove in several steps that z is a weak solution to problem Pm . Due to bounds (167), one can choose a subsequence of zh with additional properties  h !  weakly in L2 .Q/;  h !  weakly in Lq1 ;r1 .Q/

(173)

for the same .q1 ; r1 / as in (171). Moreover, similarly to Lemma 24, one can assert that It O h ! It  in C .Q/; It  h ! It  in C .Œ0; T I L1 .//;

(174)

where the first bound in Lemma 48 and the left property (173) are essential. Now we turn to formula (163). By utilizing properties

0;h ! 0 ; . h /1 ! 1 ; . h /1 k h ! . /1 k in Lq ./ for any 1  q < 1;

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2479

 h !  in L1;1 .Q/ and (174), one deduces that

h ! ; h ! ; Q h ! in C .Œ0; T I Lq .// for any 1  q < 1; (175)  ˚  and the formula D 0 C 1 kIt . 1  /  holds, with  WD exp 1 It  , cp. with formula (130). One can pass in the equalities  h D h h D uO h  k h h  h and  h D h Q h D O h to the weak limit in L1 .Q/ using properties h ! , k h ! k and h !  in L4 ./ and (175) for q D 4 together with (170),  h !  in L2;1 .Q/ and D O h ! D weakly in L2;1 .Q/ (that follows from (171)). Consequently  D   Du  k ;  D    D :

(176)

Next, by (172) the properties gŒxO eh  ! gŒxe  and f ŒxO eh  ! f Œxe  a.e. in Q hold. Therefore condition C3 and the Lebesgue dominated convergence theorem lead to gŒxO eh  ! gŒxe  in L2;1 .Q/; f ŒxO eh  ! f Œxe  in L1 .Q/:

(177)

Lemma 53. The integral identity (118) and the boundary conditions (120) hold. Proof. In fact the proof repeats the argument for Lemma 26. The same properties of the sequences uh , It O h , gŒxO eh , u0;h , and s h ' are used. Lemma 54. The integral identity (132) is valid. Proof. We take 2 C 1 .Q/, jtDT D 0. Set h D s h , 0;h D s h 0 , and h h jxD˛ , ˛ D 0; X. We multiply equation (143) by uh h and equation (144) ˛ D h h by s1 , sum up the results, and integrate over Q. By integrating by parts in x, one gets   1 h 2 Dt 2 .u / ;

h

 Q

 C cVh Dt  h ; s1h

h

 Q

1/

 C  h ; D.uh

h

Q

D . h D uO h C f h ; s1h C.O h jxD˛ ; uh jxD˛

ˇ˛DX h ˇ ˛ /.0;T / ˛D0

h

1/

 C  h ; D.s1h

/Q C .g h uh ;

 C O h jxD˛ ; .s1h

h

/jxD˛



h

h

Qh

/Q C

ˇ˛DX ˇ : .0;T / ˛D0

We further transform the result integrating by parts in t, using boundary conditions (147)–(148) and identities (32)–(33) as well as the formulas

1/ D .s

D.uh

h

h 1

h

/D uO h C .s1h uh /D O

Consequently the integral identity

h

1/ D s .D O

in Q; D.s1h

h

h

h

/ in Qh :

2480

A. Zlotnik

 D

1 2

1 2

.uh /2 C cVh  h ; Dt

.u0;h /2 C cVh  0;h ; ZT

.S /

h

 Q

0;h

 

/ .It O h /jxD˛ d .u. ˛

    C  h uh ; D O h Q C  h ; D O h Qh

C .gŒxO eh uh ; ˇ˛DX h ˇ / ˛ ˇ ˛D0

h

/Q C .f ŒxO eh ; s1h

 .p˛. / ; uh jxD˛

h

/Q

ˇ˛DX h ˇ ˛ /.0;T / ˛D0

0

 C 0 ; .s1h

h

/jxD0

 .0;T /

 C X ; .s1h

is valid. The following convergence properties of h

!

; s1h

h ˛

!

˛

h

!

; Dt

in C 1 Œ0; T ; .s1h

h

h

/jxDX

 .0;T /

(178)

as h ! 0 hold:

! Dt ; D O h ! D in C .Œ0; T I L1 .//; ˇ h ˇ / xD˛ ! ˛ in C Œ0; T ; ˛ D 0; X:

h

Recall also the properties uh ! u in L2 .Q/ and – weakly in L1 .0; T I L2 .//,  h !  in L1;1 .Q/ as well as properties (173), (176), and (177). Moreover, cVh ! cV in measure on , 0  cVh  N , u0;h ! u0 in L2 ./ and  0;h !  0 in L1 ./. In addition, one has / .It O h /jxD˛ ! .It  /jxD˛ in C Œ0; T ; u. Q ˛ .t / WD u˛ .tC0/ for all t 2 Œ0; T ; ˛ .t / ! u

for ˛ D 0; X; remind also (153). Thus the following limit relations hold: ZT lim .S /

.It O

h!0

h

/ /jxD˛ d .u. ˛

0

h ˛/

ZT D .S /

.It  /jxD˛ d .Qu˛

˛/

0

ZT .It  /jxD˛ d .u˛

D .S /

˛/

0

by the first Helly theorem [29, Ch. IX, §7]. The latter equality is valid since uQ ˛ and u˛ differ on at most countable set in .0; T / and .It  /jxD˛ .0/ D 0 and ˛ .T / D 0 (according to a property of the Riemann-Stieltjes integral [29, Ch. VI, §6.4]). Passing in (178) to the limit as h ! 0 with the help of the all listed properties leads to identity (132). We remind that this is equivalent to (119) due to Lemma 37. Since clearly Lemma 27 remains valid, Theorem 6, Claim 1, is established. We remind that Claim 2 was derived from Claim 1 in Section 4.2.1. Claim 3 immediately follows from the corresponding Proposition 12, Claim 2. The proof of Theorem 6 is complete.

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

4.3

2481

The Uniqueness and Lipschitz Continuous Dependence on Data for Weak Solutions

4.3.1 Statement of Results in the Heat-Conducting Case It is well known that, for quasilinear evolution problems in continuum mechanics, the problem of uniqueness for weak solutions can be more complicated than the existence one. In a sense this is true for problem Pm under consideration. The first uniqueness theorem was presented, for not so weak solutions as above, under the additional assumptions on the initial data u0 2 L1 ./ and  0 2 L4 ./; see [6]. Only much later in [36], the uniqueness theorem was established, by another approach, under natural conditions on the data. Introduce the additional condition.   C5 . There exist (weak) derivatives D g and D f such that kD gkL1 .a;a/ L2;1 .Q/   CkD f kL1 .a;a/ L1 .Q/  C .a/ for all a > 1. Clearly this is a version of the local Lipschitz condition on g and f with respect to  (it is automatically satisfied if g and f are independent of ). The statement of the uniqueness theorem (but not its proof) is simple. Theorem 7. Let conditions C1 – C5 be valid. Then the weak solution to problem Pm from the class N .Q/  V2 .Q/  V1 .Q/  S21;1 W .Q/ (defined in Sect. 4.2) is unique. Theorem 7 is proved together with the Lipschitz continuous dependence on data (namely, the initial data, boundary data, and free terms) for the weak solutions. To state the corresponding theorem, we introduce additional notation. .1/ .2/ We consider two problems, Pm and Pm , of the form Pm . In the problem .`/ Pm , ` D 1; 2, the initial data 0 ; u0 , and  0 ; the boundary data u ; p , and  ; and the free terms g and f are supplied with the superscript .`/ and written as .`/ .`/

0;.`/ , u , g .`/ ; etc. We denote the weak solution to the problem Pm by z.`/ D .`/ . .`/ ; u.`/ ;  .`/ ; xe /. We define the difference ' D ' .1/  ' .2/ so that  0 D 0;.1/  0;.2/ , u D  .2/   .2/   .2/  .1/ .2/ D F .1/ xe  F .2/ xe for u  u ,  D .1/  .2/ , etc. Set F xe .2/ F D g; f . Below the decomposition f Œxe  D f 0 C f 00 is in use with some f 0 ; f 00 2 L1 .Q/ (in particular, with f 0 D 0 or f 00 D 0). .1/

Theorem 8. Let conditions C1 – C5 be valid for the data of both problems Pm .2/ and Pm . Then the following bound for the norm of difference between the weak solutions to the problems by the sum of norms of the corresponding data differences holds:

k kC .Œ0;T IL1 .// C kukV2 .Q/ C k kLq;r .Q/ C kxe kS 1;1 W .Q/ 2  0 0 0 1  K"1 ; " .N / k kL ./ C ku kL2 ./ C k kL1 ./ C ku kV Œ0;T 

2482

A. Zlotnik

    C kp kL4=3 .0;T / C k kL1 .0;T / C kg xe.2/ kF m .Q/ C kI hmi It g xe.2/ kL1 .Q/  C kI h3i f 0 kL3=.2C"1 "/ .Q/ C khf 0 i kL1 .0;T / C kf 00 kL1 .Q/ (179) for all q 2 Œ2; 1/, r 2 Œ2; 4/ such that .2q/1 Cr 1 D .1C"1 /=2 as well as q D 1 and r D 2=.1 C "1 / with any 0 < " < "1 < 1=2. .1/

.2/

In this theorem, the weak solutions to problems Pm and Pm are unique according to Theorem 7.

4.3.2 Proofs The derivation relies heavily on few results concerning bounds for various weak solutions to linear parabolic IBVPs with discontinuous data; see above Propositions 2, 9, 10, and 11. .`/ .`/ We take some weak solution z.`/ D . .`/ ; u.`/ ;  .`/ ; xe / to problem Pm for ` D 1; 2. It is convenient to assume (without loss of generality) that the bound > > > > > >N .Q/ C ku.`/ kV2 .Q/ C k .`/ kV1 .Q/ C kxe.`/ kC .Q/  N ; ` D 1; 2; > .`/ >

(180)

is valid for some N > N . According to bound (129) and Lemma 36, the following bounds hold: kIt  .`/ kC .Q/  ckIt  .`/ kW .Q/  K.N /;

(181)

k .`/ kLq0 ; r0 .Q/ C kD .`/ kLq1 ; r1 .Q/  K.N /;

(182)

with .q0 ; r0 / and .q1 ; r1 / as in Theorem 6. We establish a collection of auxiliary inequalities between norms of the components of z D . ; u;  ; xe /. Let below in the subsection t be any from .0; T . Lemma 55. The following inequality holds:  k kL1 .Qt /  K.N / kI hmi ukL1 .Qt / C kDukL1 .Qt / C k kL1;1 .Qt /  (183) C kbxe kL1 .Qt / C1;m CkI hmi u0 kL1 ./ CkIt p kL1 .0;T / ; where b D b.x; t/, kbkL2;1 .Q/  K 0 .N / and 1;m WD k 0 kL1 ./ C  hmi   I It g xe.2/  1 : L .Q/ Proof. By virtue of formula (130), the formula n  o ˚  D  0 C 1 kIt . /. .1/ /1   .2/ . .1/  .2/ /1   .1/

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2483

  ˚ C 0;.2/ C 1 kIt  .2/ . .2/ /1  with  .`/ WD expf 1 It  .`/ g holds. By using the bounds 0  k  N , N 1   N è K1 .N /1   .`/  K1 .N / for ` D 1; 2 (see (181)), k .2/ kL1;1 .Q/  cN , and k 0;.2/ kL1 ./  N , one gets   j j  K2 .N / j 0 j C It jj C j jmax C j j ; with j jmax .; t / WD max0t j.;  /j. Since j j  K3 .N /jIt  j, one then obtains   k kL1 .Qt /  K4 .N / k 0 kL1 ./ C kkL1;1 .Qt / C kIt  kL1 .Qt / :

(184)

Next, by virtue of formula (131), one can write It  D I hmi u  I hmi u0 C I hmi It .gŒxe / C It ;m : By using also the elementary formula   .gŒxe / D g .1/ Œxe  C g xe.2/ ;

(185)

applying the bound jI hmi It wj  kwkL1 .Qt / and condition C5 for g D g .1/ , one passes from inequality (184) to   k kL1 .Qt /  K4 .N / Y0 .t / C 1;m C kIt ;m kL1 .Qt / :

(186)

Here it is set Y0 .t / WD kI hmi ukL1 .Qt / C kkL1;1 .Qt / C kbxe kL1 .Qt / with b.x; t/ WD kD g .1/ .; x; t /kL1 .N ;N / . For m D 2; 3, applying the bound jIt ;m j  jIt p j completes the proof. For m D 1 the situation is more complicated. To bound It ;m , one writes  D ~ .1/ .Du  1 k/  ~ .1/ ~ .2/ 1 . /.Du.2/  1 k .2/ /; (187) with ~ .`/ WD .`/ , ` D 1; 2. Clearly N

2

2

 ~ .`/  N ; therefore,

kIt ;1 kL1 .Qt /  X 1 k kL1 .Qt /     K5 .N / kDukL1 .Qt / C k kL1 .Qt / C It d1 k kL1 ./ ; with d1 .t / WD kDu.2/ .; t /kL1 ./ C k .2/ .; t /kL1 ./ , and kd1 kL1 .0;T /  cN by virtue of (180). Now from (186) one obtains    k kL1 .Qt /  K6 .N / Y0 .t / C kDukL1 .Qt / C 1;m C It d1 k kL1 ./ : Applying the Gronwall-Bellman lemma leads to inequality (183) for m D 1 too.

2484

A. Zlotnik

Lemma 56. The following inequality holds:   kukV2 .Qt /  K.N / kd  kL1;2 .Qt / C kkL2 .Qt / C kbxe kL2;1 .Qt / C 2;m ; (188) where d D d .t /, kd kL2 .0;T /  K 0 .N / and 2;m WD ku0 kL2 ./ C ku kV Œ0;T  C   .2/  kp kL4=3 .0;T / C g xe F m .Q/ : Proof. According to Lemma 34 and formulas (187) and (185), the function v D u is a weak V2 .Q/-solution of the IBVP like Lm for the linear parabolic equation Dt v D D.~ .1/ Dv 

  / C g .1/ Œxe  C g xe.2/ in Q

supplemented with the initial condition vjtD0 D u0 and the boundary conditions (7) with v D .u0 ; uX / and s D .p0 ; pX /. Here s WD ~ .1/ Du 

;

  WD ~ .1/ 1 k C ~ .1/ ~ .2/ 1  /.Du.2/  1 k .2/ I

see (187). By applying Proposition 2 (see bound (11)), one gets   kukV2 .Qt /  K1 .N / 2;m C k kL2 .Qt / C kg .1/ Œxe kF m .Qt / : The proof is completed by the bounds   k kL2 .Qt /  K2 .N / k kL2 .Qt / C kd  kL1;2 .Qt / ; k g .1/ Œxe  kF m .Qt /  k g .1/ Œxe  kL2;1 .Qt /  kbxe kL2;1 .Qt / ; where d .t / WD kDu.2/ .; t /k C k .2/ .; t /k and thus kd k.0;T /  K 0 .N /. Now we combine the results of the two last lemmas. Lemma 57. The following combined inequality holds:    k kL1 .Qt / C kukV2 .Qt /  K N / k kL1;1 .Qt / C k kL2 .Qt / C 1;m C 2;m : (189) Proof. By coarsening inequality (183), one has  k kL1 .Qt /  K1 .N / kukV2 .Qt / C k kL1;1 .Qt /  Ckbxe kL2;1 .Qt / C 1;m C 2;m :

(190)

We bound the term kbxe kL2;1 .Qt / . Set b.t / WD kb.; t /kL2 ./ and note that  1=2 2 b  D kbkL2;1 .Q/  K 0 .N /. Clearly inequality (96) remains valid: .0;T /

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2485

kxe kC ./  k 0 kL1 ./ C k kL1 ./ C X 1 kukL1 .Qt / :

(191)

By applying it and the Hölder inequality, one gets     kbxe kL2;1 .Qt /  kbkL2;1 .Q/ k 0 kL1 ./ C X 1 It kukL1 ./ C It bk kL1 ./   1=2  K2 .N / k 0 kL1 ./ C kukL2 .Qt / C kb  kL1;2 .Qt / : (192) We multiply inequality (190) by .2K1 .N //1 and sum up with inequality (188). By applying inequality (192) and squaring both sides of the result, one obtains  2 Y .t / WD k kL1 .Qt / C kukV2 .Qt / 2     K3 .N / It .d2 Y / C k kL1;1 .Qt / C k kL2 .Qt / C 1;m C 2;m ; with d2 WD d 2 C b C 1 satisfying kd2 kL1 .0;T /  K4 .N /. Applying the GronwallBellmann lemma completes the proof. Next we derive a preliminary inequality to bound  . Lemma 58. The following inequality holds:  k kLq;r .Qt /  K"1 ; " .N / k kL1 .Qt / C kukV2 .Qt /  CkkL2 .Qt / C 3;"1 ;" ;

(193)

where 3;"1 ;" D k 0 kL1 ./ Ck 0 kL1 ./ Ck kL1 .0;T / CkI h3i f 0 kL3=.2C"1 "/ .Q/ Ckhf 0 i kL1 .0;T / C kf 00 kL1 .Q/ and q, r, "1 , ", f 0 , and f 00 are as in Theorem 8. Proof. Owing to Lemma 36,  is a weak V1 .Q/-solution to the linear parabolic IBVP cV Dt  D D.ˇ .1/ D  / C ˆ in Q; ˇ ˇ  ˇtD0 D  0 ; .ˇ .1/ D  /ˇxD0 D 0 ; .ˇ .1/ D 

ˇ /ˇxDX D X ;

cp. with the problem for u in the proof of Lemma 56. Here ˇ .`/ WD  .`/ ; clearly 2 2 4 N  ˇ .`/  N and kDt ˇ .1/ kQ  K1 .N / D N . Also we utilize formulas   .2/ .f Œxe / D f .1/ Œxe  C f xe (cp. with (185)), f 0 D DI h3i f 0 C hf 0 i and I h3i f 0 jxD0;X D 0 to set ˝ ˛ WD 1 ˇ .1/ ˇ .2/ . /D .2/  I h3i f 0 ; ˆ WD . Du/ C f .1/ Œxe  C f 0  C f 00 :

2486

A. Zlotnik

We apply the decomposition  D  0 C  00 , where  0 and  00 are weak solutions from V1 .Q/ of the last IBVP, respectively, with (1) ˆ D 0 and zero initial and boundary data and (2) D 0. According to Proposition 9, the solution  00 exists, is unique, and satisfies the bound   k 00 kLq;r .Qt /  K2;"1 .N / k 0 kL1 ./ C k kL1 .0;T / C kˆkL1 .Qt / : (194) We bound the last term on the right. It is easy to get the bound kˆkL1 .Qt /  k kQt kDu.2/ kQ C k .1/ kQ kDukQt   ˝ ˛ Cf .1/ Œxe L1 .Qt / C X k f 0  kL1 .0;T / C kf 00 kL1 .Q/ : 2

Since k~ .`/ kL1 .Q/  N , by formula (187), the following bound holds: k kQt  N

2

   5 kDukQt CN 2 k kQt CN k kL1 .Qt / kDu.2/ kQ CN 2 k .2/ kQ :

Next, by using condition C5 for f D f .1/ and inequality (191) for xe , one obtains   f .1/ Œxe 

 kD f .1/ kL1;1 ..N ;N /Q/ kxe kC .Qt /    C .N / k 0 kL1 ./ C X 1 kukL1 .Qt / C k kL1;1 .Qt / : L1 .Qt /

Therefore it can be concluded that  kˆkL1 .Qt /  K3 .N / k kL1 .Qt / C kukV2 .Qt / C k kQt  ˝ ˛ Ck 0 kL1 ./ C k f 0  kL1 .0;T / C kf 00 kL1 .Q/ :

(195)

Clearly  0 D    00 2 V1 .Q/ and  0 is a weak L2 .Q/-solution of the above mentioned parabolic IBVP. According to Proposition 11, Items 1 and 2, one gets k 0 kLq;r .Qt /  K4;"1 ;" .N /k kL3=.2C"1 "/ .Qt / :

(196)

In more detail, for q 2 Œ2; 1/ and r 2 Œ2; 4/, Item 1 with ı D "1 is applied. For q D 1 and r D 2=.1 C "1 /, one can apply Item 2 with p < p0 < 2, p01 D 3=2  1=.1C"1 / and "0 D ".1C"1 /=.12"1 / (then r D 1C2=p00 and p D 3=.2C"1 "/) as well as take into account that in bound (196) the general case 0 < " < "1 < 1=2 is reduced to the case of sufficiently small "1 > 0 and (for the fixed "1 ) sufficiently small " > 0 as well. 2 By using bounds kˇ .`/ kL1 .Q/  N and kD .2/ kL3=.2C"1 "/ .Q/  K5;"1 ;" .N / (see (182)), one gets

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2487

k kL3=.2C"1 "/ .Q/  K6;"1 ;" .N /k kL1 .Qt / C kI h3i f 0 kL3=.2C"1 "/ .Q/ :

(197)

Since k kLq;r .Qt /  k 00 kLq;r .Qt / C k 0 kLq;r .Qt / , inequality (193) immediately follows from bounds (194)–(195) and (196)–(197). Inequality (193) is crucial since one can take q D 1 and some r > 1 as well as q D 2 and some r > 2 there. Lemma 59. Bound (179) is valid with N replacing N . Proof. We sum up the combined inequality (189) and inequality (193) multiplied by .2K"1 ;" .N //1 to get k kL1 .Qt / C kukV2 .Qt / C kkLq;r .Qt /    K1;"1 ;" .N / k kL1;1 .Qt / C k kL2 .Qt / C 1;m C 2;m C 3;"1 ;"

(198)

We choose .q; r/ D .1; 2=.1 C "0 / and .q; r/ D .2; 4=.1 C 2"0 // with fixed "0 2 .0; 1=2/; apply Lemma 2 with yi .t / WD k.; t /kLqi ./ , q1 D 1, and q2 D 2, for n D 2; and obtain the bound kkL1;2=.1C"0 / .Q/ C k kL2;4=.1C2"0 / .Q/  K2;"1 ;" .1;m C 2;m C 3;"1 ;" /: By using it on the right in (198), one gets k kL1 .Q/ C kukV2 .Q/ C k kLq;r .Q/  K3;"1 ;" .N /.1;m C 2;m C 3;"1 ;" /: Applying in addition equation Dt  D Du and the inequality kxe kS 1;1 W .Q/  2   c k kL2;1 .Q/ C kukQ C kDukQ , see (98), completes the proof. .1/

.2/

We complete the proof of both Theorems 7 and 8. Let first problems Pm and Pm .1/ .2/ be the same, i.e., Pm D Pm D Pm . Then z.1/ and z.2/ are simply two any weak solutions to problem Pm . In this case the right-hand side of bound (179) equals 0, and thus z.1/ D z.2/ , i.e., Theorem 7 is valid. .1/ .2/ In the general case, problems Pm and Pm have unique solutions z.1/ and z.2/ according to Theorems 6 and 7. Assumption (180) is valid for N D K1 .N /, and now Theorem 8 follows from Lemma 59.

4.4

Regularity of Weak Solutions

Now we turn to regularity theorems for a weak solution to problem Pm (under the same conditions C1 on the coefficients). First, we analyze the internal regularity for u and  together with  and  (for simplicity, with respect to t only) under the same broad conditions C2 on the initial

2488

A. Zlotnik

data. Let  2 H 1 .0; T / be a cutoff function such that kDt kL2 .0;T /  N and jtD0 D 0. Theorem 9. 1. Let conditions C1 – C4 as well as the conditions kgkL2 .Q/ C k 2 f kL2 .Q/  N; kDt .u /kL4=3 .0;T / C kp kV Œ0;T  C k 2  kV Œ0;T   N be valid. Then, for the weak solution from Theorem 6, the additional regularity h1;1=2i Dt .u/, Dt . 2  / 2 L2 .Q/, and  ;  2  2 V2 .Q/ take place, and the bounds kDt .u/kL2 .Q/ C kDt . 2  /kL2 .Q/  K.N /; k kV h1;1=2i .Q/ C k 2 kV h1;1=2i .Q/  K.N / 2

2

hold. As a consequence, u;  2  2 C .Q/ and D.u/; D. 2  / 2 Lq;r .Q/ and the bounds kukC .Q/ C k 2  kC .Q/  K.N /; kD.u/kLq;r .Q/ C kD. 2  /kLq;r .Q/  K.N /K hold for all q; r 2 Œ1; 1 such that .2q/1 C r 1 D 1=4. Moreover, the equations Dt .u/ D D. / C gŒxe  C .Dt /u; cV Dt . 2  / D D. 2 / C  2  Du C  2 f Œxe  C cV .Dt . 2 // are satisfied in L2 .Q/. 2. If, in addition, the following conditions on g, f , and u are valid k 2 gkL2q2 ; 2r2 .Q/ C k 3 f kL2q3 ; 2r3 .Q/  N; kDt . 2 u /  .Dt . 2 //u kLr" .0;T /  N for some qi ; ri 2 Œ1; 1; i D 2; 3; such that .2qi /1 C ri1 D 1  " and r" D 2=.1  "/, with some " 2 .0; 1, then also  2  ;  3  2 L1 .Q/ and the bound k 2  kL1 .Q/ C k 3 kL1 .Q/  K" .N / hold. To illustrate Theorem 9, one can fix 0 < t1 < t2  T and arbitrarily small ı 2 .0; t1 /, choose .t / D 0 on Œ0; t1  ı and .t / D 1 on J D .t1 ; t2 /, and apply the theorem in the case T D t2 . Let Q0 D   J and Qı0 D   Jı with Jı D .t1  ı; t2 /. Then, under conditions C1 – C4 and g; f 2 L2 .Qı0 /, Dt u 2 L4=3 .Jı /, p ;  2 V .J ı /, the regularity properties h1;1=2i

Dt u; Dt  2 L2 .Q0 /;  ;  2 V2

0

.Q0 /; u;  2 C .Q /; u; D 2 Lq;r .Q0 /

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

2489

are valid, for .q; r/ like in Theorem 9. Moreover, equations (110)–(111) are satisfied not only in the weak sense but in L2 .Q0 / as well. If, in addition, g 2 L2q2 ; 2r2 .Qı0 /, f 2 L2q3 ; 2r3 .Qı0 /, and Dt u 2 Lr" .Jı /, for qi ; ri , and r" like in Theorem 9, then also  ;  2 L1 .Q0 /. We emphasize that the properties D D D. Du  p/ 2 L2 .Q0 / and D D D. D / 2 L2 .Q0 / are somewhat striking since in general the functions ; Du; p, and D do not have the space weak derivative in Q0 due to the lack of regularity of

0 , , k, and . Next we consider the regularity up to the moment t D 0. In this respect introduce a new definition of solution. A vector function z D . ; u;  ; xe / 2 N .Q/  H 1 .Q/  H 1 .Q/  S21;1 W .Q/ is called the almost strong solution to problem Pm , if D ; D 2 L2 .Q/,  > 0, and all the equations (109), (110), (111), and (112) are satisfied in L2 .Q/, and the initial and boundary conditions (113), (114), and (115) are satisfied in the sense of traces. In this definition, the properties of are the same as above. In contrast with the definition of the strong solution in [16], it is not assumed that D 2 L2;1 .Q/ and D 2 u; D 2  2 L2 .Q/ (or even that these derivatives exist). But this is no barrier for us to consider D ; D 2 L2 .Q/ and thus equations (110) and (111) in L2 .Q/. Theorem 10. 1. Let conditions C1 – C4 , the regularity conditions kDu0 kL2 ./ C kD 0 kL2 ./  N; kgkL2 .Q/ C kf kL2 .Q/  N; kDt u kL4=3 .0;T / C kp kV Œ0;T  C k kV Œ0;T   N and the matching conditions u0 .0C / D u0 .0/ for m D 1 and uX .0C / D u0 .X / for m D 1; 2 be valid. Then the weak solution from Theorem 6 is almost strong and satisfies the bounds kDt ukL2 .Q/ C kDt  kL2 .Q/  K.N /; kkV h1;1=2i .Q/ C kkV h1;1=2i .Q/  K.N /; 2

2

kukC .Q/ C k kC .Q/  K.N /; kDukLq;r .Q/ C kDkLq;r .Q/  K.N / for all q; r 2 Œ1; 1 such that .2q/1 C r 1 D 1=4. 2. If, in addition, the following stronger conditions on u0 ;  0 , g; f , and u are valid kDu0 kL1 ./ C kD 0 kL1 ./  N; kgkL2q2 ;2r2 .Q/ C kf kL2q3 ;2r3 .Q/  N; kDt u kLr" .0;T /  N for some qi ; ri 2 Œ1; 1, i D 2; 3; such that .2qi /1 C ri1 D 1  " and for r" D 2=.1  "/, with some " 2 .0; 1, then the following L1 .Q/-bound holds: k kL1 .Q/ C kkL1 .Q/  K.N /:

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3. Let the conditions in Items 1 and 2, the continuity properties  0 WD 0 Du0  k 0  0 2 C ./;  0 WD  0 D 0 2 C ./; p 2 C Œ0; T ; and the matching conditions  0 .0/ D p0 .0/ for m D 2; 3 and  0 .X / D pX .0/ for m D 3 as well as  0 .0/ D 0 .0/ and  0 .X / D X .0/ be valid. Then  ;  2 C .Q/. We emphasize once again that, in the above theorems, the coefficients ; k; cV ; , and initial function 0 can belong to L1 ./ and be strictly positive only. To prove Theorems 9 and 10, one can establish their counterparts for semidiscrete problem Pmh and then pass to the limit as h ! 0; details can be found in [14]. Finally, we cover the much simpler case of differentiable 0 , , k, and . Corollary 10. 1. Let the hypotheses of Theorem 6 be valid, kD 0 kLq ./  N and kD kLq ./ C kDkkLq ./  N for some q 2 Œ1; 2. Then the bound kD kLq;1 .Q/  K.N / holds. 2. Let the hypotheses of Theorem 10, Item 1, and of the previous Item 1 be valid. Then the bound kD 2 ukLq;2 .Q/  K.N / holds. If in addition kDkLq ./  N for some q 2 Œ1; 2, then also kD 2  kLq;2 .Q/  K.N /. Notice that Item 1 easily follows from Theorem 6 due to formula (130) as well as Item 2 easily follows from Theorem 10 and Item 1 due to formulas Du D . C p/= and D D =. For q D 2, Corollary 10 ensures the existence of the standard strong solution to problem Pm . Remark 2. 1. Theorem 6 was proved in [13] in the case of more general than (115) Robin boundary conditions . C 0  /jxD0 D 0 .t / and . C X  /jxDX D X .t /, with k 0 kLs .0;T / C k X kLs .0;T /  N for some s > 2 as well as 0  0 and X  0. 2. Other theorems on the Lipschitz continuous dependence on data, for the constant coefficients and not so weak solutions as in Theorem 8, can be found in [6,22,37]. 3. Well-posedness of the combustion problem for a viscous heat-conducting gas was proved in [40]. In the case of a real gas, the existence of global weak solutions (without their uniqueness) was studied in [1]. 4. A rich theory of finite-difference schemes for the IBVP (109), (110), (111), (112), (113), (114), and (115) and the more general IBVP taking into account the magnetic field was developed before constructing the above semidiscrete methods, in particular, in [2–5]; see also [15] (the case of the IBVP (12), (13), (14), (15), and (16) was covered too in other papers).

43 Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

5

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Conclusion

In this chapter, the inhomogeneous initial-boundary value problems (IBVPs) have been considered for the compressible Navier-Stokes equations for both the viscous barotropic and viscous heat-conducting gas 1D flows in the Lagrangian mass coordinates. The broad classes of discontinuous data have been treated including the initial density in L1 and strictly positive, the finite initial total energy and, in the heat-conducting case, the non-strictly positive initial temperature and the finite initial total entropy too. No any restrictions have been imposed on the magnitude of norms of data. The contact problems between different gases have been covered too. The well-posedness of the IBVPs has been proved in the suitable classes of weak solutions including theorems on the global-in-time existence and the uniqueness and Lipschitz continuous dependence on data. The internal and up to the boundary regularity results have been presented as well involving the existence of strong solutions for the data in the Sobolev spaces. This work was supported by the Russian Science Foundation, project no. 14-1100549.

6

Cross-References

 Concepts of Solutions in the Thermodynamics of Compressible Fluids  Symmetric Solutions to the Viscous Gas Equations  Weak and Strong Solutions of Equations of Compressible Magnetohydrodynam-

ics  Weak Solutions for the Compressible Navier-Stokes Equations with Density

Dependent Viscosities  Well-Posedness and Asymptotic Behavior for Compressible Flows in One Dimen-

sion

References 1. A.A. Amosov, The existence of global generalized solutions of the equations of onedimensional motion of a real viscous gas with discontinuous data. Differ. Equs. 36, 540–558 (2000) 2. A.A. Amosov, A.A. Zlotnik, A study of finite-difference method for the one-dimensional viscous heat conductive gas flow equations. Part I: a priori estimates and stability. Sov. J. Numer. Anal. Math. Model. 2, 159–178 (1987) 3. A.A. Amosov, A.A. Zlotnik, A study of finite-difference method for one-dimensional viscous heat-conducting gas flow equations. Part II: error estimates and realization. Sov. J. Numer. Anal. Math. Model. 2, 239–258 (1987)

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4. A.A. Amosov, A.A. Zlotnik, A difference scheme on a non-uniform mesh for the equations of one-dimensional magnetic gas dynamics. U.S.S.R. Comput. Math. Math. Phys. 29(2), 129–139 (1989) 5. A.A. Amosov, A.A. Zlotnik, Two-level finite-difference schemes for one-dimensional equations of magnetic gas dynamics (viscous heat-conducting case). Sov. J. Numer. Anal. Math. Model. 4, 179–197 (1989) 6. A.A. Amosov, A.A. Zlotnik, Global generalized solutions of the equations of the onedimensional motion of a viscous heat-conducting gas. Sov. Math. Dokl. 38, 1–5 (1989) 7. A.A. Amosov, A.A. Zlotnik, Solvability “in the large” of a system of equations of the onedimensional motion of an inhomogeneous viscous heat-conducting gas. Math. Notes 52, 753– 763 (1992) 8. A.A. Amosov, A.A. Zlotnik, Global solvability of a class of quasilinear systems of composite type with nonsmooth data. Differ. Equs. 30, 545–558 (1994) 9. A.A. Amosov, A.A. Zlotnik, Uniqueness and stability of generalized solutions for a class of quasilinear systems of composite type equations. Math. Notes 55, 555–567 (1994) 10. A.A. Amosov, A.A. Zlotnik, Quasi-averaged equations of the one-dimensional motion of a viscous barotropic medium with rapidly oscillating data. Comput. Math. Math. Phys. 36, 203– 220 (1996) 11. A.A. Amosov, A.A. Zlotnik, Remarks on properties of generalized solutions in V2 .Q/ to onedimensional linear parabolic problems. MPEI Bull. 3(6), 15–29 (1996) [Russian] 12. A.A. Amosov, A.A. Zlotnik, Properties of generalized solutions of one-dimensional linear parabolic problems with nonsmooth coefficients. Differ. Equs. 33, 83–96 (1997) 13. A.A. Amosov, A.A. Zlotnik, Semidiscrete method for solving equations of a one-dimensional motion of a nonhomogeneous viscous heat-conducting gas with nonsmooth data. Russian Math. (IzVUZ) 41, 3–19 (1997) 14. A.A. Amosov, A.A. Zlotnik, Semidiscrete method for solving equations of the onedimensional motion of a viscous heat-conducting gas with nonsmooth data. Regularity of solutions. Russian Math. (IzVUZ) 43, 12–25 (1999) 15. A.A. Amosov, A.A. Zlotnik, A finite-difference scheme for quasi-averaged equations of onedimensional viscous heat-conducting gas flow with nonsmooth data. Comput. Math. Math. Phys. 39, 564–583 (1999) 16. S.N. Antontsev, A.V. Kazhikhov, V.N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids (Nauka, Novosibirsk, 1983) [Russian. English translation NorthHolland, Amsterdam (1990)] 17. J. Bergh, J. Löfström, Interpolation spaces. An introduction (Springer, Berlin, 1976) 18. O.V. Besov, V.P. Il’in, S.M. Nikol’skii, Integral representations of functions and imbedding theorems, vol. 1 (V.H. Winston, Washington, DC, 1978) 19. H. Fujita-Yashima, M. Padula, A. Novotny, Équation monodimensionnelle d‘un gas visqueux et calorifére avec des conditions initiales moins restrictives. Ricerche Mat. 42, 199–248 (1993) 20. D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Am. Math. Soc. 303, 169–181 (1987) 21. D. Hoff, Global well-posedness of the Cauchy problem for nonisentropic gas dynamics with discontinuous data. J. Differ. Equs. 95, 33–74 (1992) 22. D. Hoff, Continuous dependence on initial data for discontinuous solutions of the NavierStokes equations for one-dimensional, compressible flow. SIAM J. Math. Anal. 27, 1193–1211 (1996) 23. A.V. Kazhikhov, Correctness “in the large” of the mixed boundary value problems for the model viscous gas system of equations. Dyn. Contin. (Novosib.) 21, 18–47 (1975) [Russian] 24. A.V. Kazhikhov, V.B. Nikolaev, On the correctness of boundary value problems for the equations of a viscous gas with nonmonotone state function, Numer. Meth. Mech. Contin. (Novosib.) 10, 77–84 (1979) [Russian. English translation in Am. Math. Soc. Transl. 125, 45– 50 (1985)] 25. M.A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations (Pergamon Press, Oxford, 1964)

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26. S.N. Kruzhkov, Quasilinear parabolic equations and systems with two independent variables. Trudy Semin. Petrovskogo 5, 217–272 (1979) [Russian. English translation in (1985) Topics in Modern Math. Consultant Bureau, New York] 27. O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uralt’seva, Linear and quasi-linear equations of parabolic type (American Mathematical Society, Providence, 1968) 28. O.A. Ladyzhenskaya, N.N. Uralt’seva, Linear and quasilinear elliptic equations (Academic Press, New York, 1968) 29. I.P. Natanson, Theory of functions of a real variable, 3rd edn. (Moscow, Nauka, 1974) [Russian. English translation of 1st edition Vol. I and II. F. Ungar, New York (1960–1961)] 30. D. Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible. C.R. Acad. Sci. Paris Sér. I Math. 303, 639–642 (1986) 31. D. Serre, Sur l‘équation monodimensionnelle d‘un fluide visqueux, compressible et conducteur de chaleur. C.R. Acad. Sci. Paris Sér. I Math. 303, 703–706 (1986) 32. V.V. Shelukhin, Motion with a contact discontinuity in a viscous heat conducting gas. Dyn. Contin. Medium (Novosib.) 57, 131–152 (1982) [Russian] 33. V.V. Shelukhin, Evolution of a contact discontinuity in the barotropic flow of a viscous gas J. Appl. Math. Mech. 47, 698–700 (1983) 34. V.V. Shelukhin, On the structure of generalized solutions of the one-dimensional equations of a polytropic viscous gas. J. Appl. Math. Mech. 48, 665–672 (1984) 35. A.A. Zlotnik, A.A. Amosov, Global generalized solutions of the equations of the onedimensional motion of a viscous barotropic gas. Sov. Math. Dokl. 37, 554–558 (1988) 36. A.A. Zlotnik, A.A. Amosov, On stability of generalized solutions to the equations of onedimensional motion of a viscous heat-conducting gas. Sib. Math. J. 38, 663–684 (1997) 37. A.A. Zlotnik, A.A. Amosov, Stability of generalized solutions to equations of one-dimensional motion of a viscous heat conducting gases. Math. Notes 63, 736–746 (1998) 38. A.A. Zlotnik, A.A. Amosov, Correctness of the generalized statement of initial boundary value problems for equations of a one-dimensional flow of a nonhomogeneous viscous heat conducting gas. Dokl. Math. 58, 252–256 (1998) 39. A.A. Zlotnik, A.A. Amosov, Weak solutions to viscous heat-conducting gas 1D-equations with discontinuous data: global existence, uniqueness, and regularity, in The Navier-Stokes Equations: Theory and Numerical Methods, ed. by R. Salvi. Lecture Notes in Pure and Applied Mathematics, vol. 223 (M. Dekker, New York, 2002), pp. 141–158 40. A.A. Zlotnik, S.N. Puzanov, Correctness of the problem of viscous gas burning in the case of nonsmooth data and a semidiscrete method of its solution. Math. Notes 65, 793–797 (1999)

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

44

Akitaka Matsumura

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Generalized Burgers’ Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Global Asymptotics Toward Rarefaction Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Asymptotic Stability of Viscous Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Viscous Contact Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Isentropic/Isothermal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Dissipative Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Global Asymptotics Toward Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Asymptotic Stability of Viscous Shock Wave, a Density-Dependent Viscosity Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Ideal Polytropic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Viscous Contact Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Asymptotic Stability of Viscous Contact Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2496 2499 2500 2507 2514 2514 2516 2517 2518 2522 2524 2530 2537 2538 2539 2541 2543 2546 2546 2546

Abstract

This short article focuses on several one-dimensional model systems which often appear in the field of compressible viscous fluids, and discusses their Cauchy problems with prescribed far-field states. In particular, it describes the asymptotic A. Matsumura () Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka, Japan e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_60

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behavior of the solutions in relation to the Riemann problems for the hyperbolic parts with the same far-field states. It has been expected that the solutions tend toward various asymptotic wave patterns as time goes to infinity, that is, various combinations of viscous shock, rarefaction, and contact waves. Many cases have been mathematically justified, but many others still remain open. The intent of this article is to give the reader introductory insights into how various primitive energy methods have contributed to the mathematical justifications, through some specific topics.

1

Introduction

In the mathematical study of one-dimensional motions of compressible viscous fluids, there often appears a system of nonlinear partial differential equations, known as “viscous conservation laws,” which takes the form: zt C f .z/x D .B.z/zx /x

.t > 0; x 2 R/;

(1)

where z is a vector valued conserved quantity, f .z/ represents its convective flux, and B.z/zx its diffusive flux given by a viscous effect. In what follows, the convective flux function f .z/ is simply referred to as “flux function.” Basically, the flux function f .z/ is supposed to satisfy that the inviscid system zt C f .z/x D 0 is strictly hyperbolic on a domain  in the phase space of z under consideration, and the coefficient matrix B.z/ of the viscosity term is also supposed to satisfy that the whole system does have a dissipative structure for z 2 . A general class of such dissipative systems of viscous conservation laws is well known as “KawashimaShizuta systems,” whose characterization is given by [20, 24]. The typical examples often seen in concrete problems are Burgers’ equation and its variants (generalized Burgers’ equation), an isentropic/isothermal model system for viscous gas, and an ideal polytropic model system for viscous and heat-conductive gas. One of the fundamental mathematical problems for the system (1) is the Cauchy problem with prescribed initial and far-field conditions z.0; x/ D z0 .x/

.x 2 R/;

lim z.t; x/ D z˙

x!˙1

.t  0/:

(2)

In particular, two important issues both from the mathematical and physical points of view are the existence of a solution for large time and its asymptotic behavior with respect to the time, in connection with the flux function f and the far-field states z˙ . On this subject, it has been known that even if the system (1) is fully parabolic, the asymptotic aspect of wave propagation should be closely related to that of the weak solution of the following Cauchy problem for the inviscid part (hyperbolic part) of (1):

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

(

zt C f .z/x D 0

.t > 0; x 2 R/;

z.0; x/ D zR 0 .x/

.x 2 R/;

2497

(3)

where the initial data zR 0 is defined by zR 0 .x/ D



z .x < 0/; zC .x > 0/:

The Cauchy problem (3) is the so-called “Riemann problem” and its solution the “Riemann solution” (cf. [26, 46]). Over the last half century, much research has been performed on solutions of the viscous system that corresponds to the various wave patterns of the Riemann solution to the hyperbolic system. Of particular note is the pioneering work by Il’in-Oleinik [14], where they studied the scalar case with a genuinely nonlinear flux function and showed that if the Riemann solution admits a rarefaction wave (resp. a shock wave), the global solution in time of the Cauchy problem (1), (2) tends toward the rarefaction wave (resp. a corresponding smooth traveling wave solution of (1), “viscous shock wave”). Since the main tool for the proofs was the maximum principle, there was no progress to systems until Matsumura-Nishihara [39] and Goodman [5] independently found in the mid-1980s that an L2 -energy method is applicable even to some systems for the study of asymptotic stability of viscous shock wave. Since then, many progress have been made concerning the asymptotic behavior of the solution for the viscous problem, in connection with the wave patterns of the Riemann solution. In particular, various energy methods have played essential roles to establish “a priori estimates.” This article aims to give the reader introductory insights into how various L2 energy methods have played important roles to obtain a priori estimates, through some specific topics on the generalized Burgers’ equation in Sect. 2, the isentropic/isothermal model system of viscous gas in Sect. 3, and the ideal polytropic model system of viscous and heat-conductive gas in Sect. 4. No attempt is made here to present the entire mathematical theory of viscous conservation laws. The rest of the introduction devotes some further remarks to a better understanding of the following sections. First is a heuristic argument on how the asymptotic behavior of the solution z of (1), (2) is related with the Riemann solution zR of (3). Here and in what follows, the Riemann solution of (3) will be written as zR whenever needed to distinguish it from the solution z of the viscous system. If the variables .t; x/ are rescaled to . ; / as .t; x/ D .="; ="/ and the new unknown variable zQ is defined by zQ D zQ" . ; / D z.="; ="/ for any " > 0, the Cauchy problem for z is rewritten as (

zQ C f .Qz/ D ".B.Qz/Qz /

. > 0;  2 R/;

zQ.0; / D z0 .="/

. 2 R/:

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Hence, it can be expected that zQ D zQ" ! zR ." ! C0/, which also suggests that for almost every  2 R, the asymptotic value of z.t; t / D zQ1=t .1; / as t ! 1 is given by zR .1; /. This shows an aspect of the relationship between the Riemann solution and the asymptotic behavior of the solution of (1) and (2). Next, some remarks are made on the strategies which have been employed in many works to show the existence of the global solution in time and its asymptotic behavior. For this important issue, it is hoped that the unique global existence would be generally established for any smooth and bounded initial data, and then the asymptotic behavior would be investigated in each case. However, there has been no such general theory on the global existence, except for some special cases: the scalar case (the maximum principle can be used) and the isothermal model of viscous gas (cf. [15]). Thus, the strategies employed in many practical works are as follows: • Predict the asymptotic behavior from mathematical/physical considerations or from inspiration. • Construct an approximate solution Z with the desired asymptotic behavior, as in the form Zt C f .Z/x  .B.Z/Zx /x D R

.t > 0; x 2 R/;

where R represents a residual term with suitable decay properties. • Rewrite the problem around the approximate solution Z in terms of the deviation  D zZ, and look for the global solution in time  with the asymptotic behavior  ! 0 .t ! 1/ for a suitable class of the initial data, making use of whatever specific properties of Z. If this process fortunately works, not only the global existence in time but also the asymptotic behavior can be simultaneously shown in a neighborhood of Z. The trivial example is the case where z D zC .DW zN/, that is, the constant state zN is expected as the asymptotic state. In this case, there have been many works, including the global asymptotic stability for some concrete models and the asymptotic stability for more general systems like Kawashima-Shizuta system (cf. [20, 24]), and even much deeper results on the higher level of the approximation (arguments on diffusion waves) also have been known (cf. [19, 20, 28, 45]). Since there aren’t enough pages to step further into this topic, only some remarks shall be given in the following sections. Now, in the cases z ¤ zC which are the main subjects in this article, it should be remarked firstly on how the suitable approximate solution Z is constructed, in connection with the corresponding Riemann solution. As for the Riemann problem, it is known that under a proper assumption on the flux function, there are basic three kinds of simple wave solution, “shock wave” or “rarefaction wave’’ along the genuinely nonlinear characteristic field and “contact discontinuity” along the linearly degenerate characteristic field, and then the Riemann solution generally forms a multi-wave pattern given by a various linear combination of these three simple waves (cf. [26, 46]). How the viscous effect influences these three

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

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simple waves is inferred as follows. As for the rarefaction wave, the viscous effect is expected to make the rarefaction smooth, but not to be strong enough to influence the asymptotic behavior because the rarefaction wave is spreading much faster than the diffusion process. In fact, a smoothed rarefaction wave with the same asymptotic behavior can be approximately constructed. As for the shock wave, because the discontinuity of the shock is formed by compression of waves (concentration of particle passes), the physically reasonable viscous system is expected to have a corresponding smooth traveling wave solution with a thin transition layer, which is called a “viscous shock wave.” On the other hand, for the contact discontinuity, since there are no nonlinear interactions across the discontinuity, the viscous system is expected to have an approximate solution which relaxes the discontinuity by a linear diffusion process, which is called a “viscous contact wave”. Then, the asymptotic behavior of z is predicted, and also the approximate solution Z is constructed by replacing the every shock wave by the corresponding viscous shock wave, contact discontinuity by viscous contact wave, and rarefaction wave by smoothed rarefaction wave in the wave pattern of zR . Finally, here is a word on how to show the global solution in time for the reformulated problem in terms of the deviation  D z  Z. In all the topics in this article, the global solution is constructed by combining the unique existence of the local solution in time, together with the a priori estimate, which is obtained by various energy methods according to the individual features of the problem. Some Notations. Denote by C generic positive constants unless they need to be distinguished. For function spaces, L2 D L2 .R/ and H k D H k .R/ denote the usual Lebesgue space of square integrable functions and k-th order Sobolev space on the whole space R with norms jj  jj and jj  jjk , respectively.

2

Generalized Burgers’ Equation

This section picks up some topics on the asymptotic behavior of global solutions in time of the generalized Burgers’ equation, which often appears as a basic scalar model in fluid dynamics. Through the topics, the basic strategy and techniques of various energy methods to show the global existence in time and the asymptotic behavior are presented. The Cauchy problem under consideration is described by 8 ˆ ut C f .u/x D uxx ˆ ˆ < u.0; x/ D u0 .x/ ˆ ˆ ˆ : lim u.t; x/ D u x!˙1

˙

.t > 0; x 2 R/; .x 2 R/;

(4)

.t  0/;

where u D u.t; x/ is the unknown scalar function of t  0 and x 2 R, the so-called conserved quantity, f D f .u/ is the flux function,  is the viscosity coefficient, u0 is the given initial data, and u˙ 2 R are the prescribed far-field states. The

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A. Matsumura

flux function f .u/ is assumed to be a given smooth function and  a positive constant. Also the far-field states u˙ are assumed to satisfy u < uC without loss of generality, and the initial data satisfy u0 .˙1/ D u˙ as the compatibility condition. As stated in the introduction, it has been known that the asymptotic behavior should be closely related to the corresponding Riemann solution, say uR , of the Riemann problem (

ut C f .u/x D 0 .t > 0; x 2 R/;

(5)

u.0; x/ D uR 0 .x/ .x 2 R/; where the initial data uR 0 is given by ( uR 0 .x/

D

uR 0 .x

I u ; uC / D

u .x < 0/; uC .x > 0/:

In fact, when the smooth flux function f is genuinely nonlinear on the whole R, i.e., f 00 .u/ ¤ 0 .u 2 R/, Il’in-Oleinik [14] showed the following by using the maximum principle: if f 00 .u/ > 0 .u 2 R/, that is, the Riemann solution consists of a single rarefaction wave solution, the global solution in time of the Cauchy problem (4) tends toward the rarefaction wave; if f 00 .u/ < 0 .u 2 R/, that is, the Riemann solution consists of a single shock wave solution of the so-called Lax type, the global solution of the Cauchy problem (4) tends toward the corresponding smooth traveling wave solution (“viscous shock wave”) of (4). The Sect. 2.1 picks up the case f 00 .u/ > 0 .u 2 R/ with some extra nonlinearity conditions and shows how to get the global asymptotics toward the rarefaction wave by using only an elementary energy method, without using the maximum principle. The Sect. 2.2 picks up more general case where the Riemann solution consists of a single shock wave solution of the so-called Oleinik type, including the Lax type, and shows the asymptotic stability of the corresponding viscous shock wave by introducing a technical weighted energy method. In the Sect. 2.3, a viscous contact wave in the case the flux function is linearly degenerate is introduced, and finally, further remarks are given in the Sect. 2.4.

2.1

Global Asymptotics Toward Rarefaction Wave

In this subsection, the flux function f .u/ is further assumed to satisfy for some p>1 f 00 .u/ > 0 .u 2 R/;

jf .u/j  jujp ; jf 0 .u/j  jujp1

.u ! ˙1/:

(6)

The typical example f .u/ D u2 =2, Burgers’ equation, satisfies (6) with p D 2. It is shown that the global solution in time of the viscous problem (4) tends toward the

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2501

rarefaction wave of the inviscid part by using only an elementary energy method. It is well known that under the assumptions f 00 .u/ > 0 .u 2 R/ and u < uC , the Riemann solution of (5), which is both mathematically and physically reasonable, is given by 8 .x  .u / t /; ˆ < u R r 1 u D u .x=t I u ; uC / D  .x=t/ ..u / t  x  .uC / t /; ˆ : .x  .uC / t /; uC

(7)

where .u/ D f 0 .u/ (cf. [46]). This simple wave, which consists of two constant states and one centered rarefaction wave, is called a “rarefaction wave” connecting far-field states u˙ . Then the main theorem in this subsection is as follows. Theorem 1. Assume the flux f to satisfy (6), the far-field states u < uC , and the 2 2 initial data u0  uR 0 2 L and u0;x 2 L . Then, the Cauchy problem (4) has a unique global solution in time u, satisfying (

2 u  uR 0 2 C .Π0; 1/I L /;

ux 2 C .Œ 0; 1/I L2 / \ L2loc .0; 1I H 1 /; and the asymptotic behavior ˇ ˇ sup ˇ u.t; x/  ur .x=t I u ; uC / ˇ ! 0

.t ! 1/:

x2R

In order to show Theorem 1, the several preparations concerning the rarefaction wave are needed. In particular, it is needed to construct a smooth approximate solution for the rarefaction wave (7), because the non-smoothness of ur at the edges of the centered rarefaction causes a trouble to handle the second derivative of the solution in the process of the a priori estimate. Start with the the rarefaction wave to the Riemann problem for the inviscid Burgers’ equation, with the given far-field states w˙ 2 R .w < wC /: 8  1 2 ˆ < wt C 2 w x D 0

.t > 0; x 2 R/;  w .x < 0/; R ˆ : w.0; x/ D w0 .x I w ; wC / D wC .x > 0/:

(8)

In this simplest case, the rarefaction wave is exactly given by 8 w .x  w t /; ˆ ˆ <  r w .x=t I w ; wC / D x=t .w t  x  wC t /; ˆ ˆ : wC .x  wC t /:

(9)

2502

A. Matsumura

Under the assumptions f 00 .u/ > 0 .u 2 R/ and u < uC , it is well known that the rarefaction wave solution of the Riemann problem (5) is given by   ur .x=t I u ; uC / D 1 wr .x=t I  ; C /

(10)

which is nothing but the definition (7), where ˙ D .u˙ / D f 0 .u˙ /. Here, following the arguments in [40], a smooth approximation W .t; xI w ; wC / of wr .x=t I w ; wC / is defined by the solution of the Cauchy problem (

Wt C

1 2

W2

 x

D0

.t > 0; x 2 R/;

W .0; x/ D W0 .xI w ; wC / WD

wC Cw 2

C

wC w 2

.x 2 R/:

tanh x

(11)

Then, corresponding to the formula (10), a smooth approximation U r .t; xI u ; uC / of ur .x=t I u ; uC / is defined by   U r .t; x I u ; uC / D 1 W .t; x I  ; C / :

(12)

The next lemma shows the basic properties of U r . Lemma 1. 1. U r is the smooth global solution in time of the Cauchy problem 8 r .t  0; x 2 R/; U C f .U r /x D 0 ˆ ˆ < t     r 1 C C C C 2  tanh x U .0; x/ D  2 ˆ ˆ : lim U r .t; x/ D u˙ .t  0/:

.x 2 R/;

x!˙1

2. u < U r .t; x/ < uC and Uxr .t; x/ > 0 .t  0; x 2 R/. 3. For q 2 Œ1; 1, there exists a positive constant Cq such that 1

kUxr .t /kLq  Cq .1 C t /1C q r .t /kLq kUxx

1

 Cq .1 C t /

.t  0/; .t  0/:

ˇ ˇ 4. lim sup ˇ U r .t; x/  ur .x=t/ˇ D 0: t!1 x2R

The proof of the lemma is given by elementary calculations. In fact, it is proved by the facts that the method of characteristic curve gives a representation formula of the solution W of (11) as (

W .t; x/ D W0 .x0 .t; x// D

 CC 2

x D x0 .t; x/ C W0 .x0 .t; x// t;

C

C  2

tanh.x0 .t; x//;

(13)

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2503

and the Jacobian of the variable transformation between x and x0 is given by 1 @x0 D > 0: @x 1 C W00 .x0 / t

(14)

The details are omitted (refer to [40]). Then, from Lemma 1, U r is expected to be a suitable approximate solution with the desired asymptotics for the viscous problem (4) as r Utr C f .U r /x  Uxx D R;

(15)

r where the residual term R is given by R D Uxx . Now, by setting

 D u  Ur and using (15), the problem (4) is reformulated in terms of  as (

.t > 0; x 2 R/;

t C .f .U r C /  f .U r //x   xx D R

.x 2 R/;

r

.0; x/ D 0 .x/ WD u0 .x/  U .0; x/

(16)

and the global solution in time  2 C .Π0; 1/I H 1 / is looked for. The corresponding theorem for  to Theorem 1 is as follows. Theorem 2. For any initial data 0 2 H 1 , the Cauchy problem (16) has a unique global solution in time , satisfying  2 C .Π0; 1/I H 1 /;

x 2 L2 .0; 1I H 1 /;

and the asymptotic behavior sup j .t; x/ j ! 0 .t ! 1/: x2R

The proof of Theorem 2 is given by combining the unique existence of the local solution in time, together with the a priori estimate. To state the local existence precisely, the Cauchy problem at general initial time   0 with the given initial data  2 H 1 is formulated: (

t C .f .U r C /  f .U r //x   xx D R

.t >  ; x 2 R/;

.; x/ D  .x/

.x 2 R/:

(17)

2504

A. Matsumura

Proposition 1 (local existence). For any M > 0, there exists a positive constant t0 D t0 .M / not depending on  such that if  2 H 1 and k k1  M , then the Cauchy problem (17) has a unique solution  on the time interval Œ;  C t0  satisfying 8 <  2 C .Œ;  C t0 I H 1 / \ L2 . ;  C t0 I H 2 /; :

sup k.t/k1  2M: t 2Œ; Ct0 

The proof of Proposition 1 is very standard, so omitted. Proposition 2 (a priori estimate). For any initial data 0 2 H 1 , there exists a positive constant C0 .0 / depending only on k0 k1 such that if the Cauchy problem (16) has a solution  2 C .Π0; T I H 1 / \ L2 .0; T I H 2 / for some T > 0, then it holds that k.t/k21

Z C 0

t

p .k Uxr k2 C kx k21 / d   C0 .0 /

.t 2 Π0; T /:

(18)

Once Propositions 2 is established, by combining the local existence Proposip tion 1 with M D M0 WD C0 .0 /,  D nt0 .M0 /, and  D .nt0 .M0 // .n D 0; 1; 2; : : : / together with the a priori estimate with T D .n C 1/t0 .M0 / inductively, the unique solution of (17)  2 C .Œ0; nt0 .M0 /I H 1 / for any n 2 N is easily constructed, that is, the global solution in time  2 C .Œ0; 1/I H 1 /. Then, the a priori estimate again asserts that Z

1

sup k.t/k1 < 1; t0

0

p .kx . /k21 C k Uxr ./k2 / d  < 1:

(19)

By using (19) and the equation, it is easy to see Z

1 0

ˇ d ˇ ˇ ˇ kx . /k2 ˇ d  < 1: ˇ d

(20)

Hence, it follows from (19) and (20) that kx .t /k ! 0 .t ! 1/: Due to the Sobolev’s inequality, the desired asymptotic behavior in Theorem 2 is thus obtained as sup j .t; x/ j  x2R

p 1 1 2 k .t/ k 2 k x .t / k 2 ! 0 .t ! 1/:

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2505

Now Proposition 2 is shown for p  2 for simplicity. The case p 2 .1; 2/ is similarly proved with a slight modification. By multiplying (16) by  and integrating the resultant formula with respect to x and t , it holds after integration by parts that Z tZ

1 k.t/k2 C 2

.f .U r C /  f .U r /  f 0 .U r / /Uxr dxd 

0

Z tZ

C 0

1 jx j dxd  D k0 k2 C 2 2

Z tZ 0

(21) r Uxx

dxd  :

By using the Sobolev’s and Young’s inequalities, the second term in the right hand side of (21) is estimated as follows: ˇ ˇZ tZ Z t ˇ p ˇ 1 1 r r ˇ  2 ˇ U dxd  kk 2 kx k 2 kUxx kL1 d  xx ˇ ˇ 0 0 Z Z Z t  t 4 2  jx j dxd  C C .kk2 C 1/.1 C  / 3 d  ; 2 0 0 (22) r in Lemma 1 is used. Then, plugging (22) to (21) where the decay estimate of Uxx and using the Grownwall’s inequality and the assumptions on the nonlinearity (6) give

k.t/k2 C

Z tZ 0



 .jj2 C jjp /Uxr C jx j2 dxd   C .1 C k0 k2 /:

(23)

By multiplying the equation of (16) by xx and using the Young’s inequality, the estimate of the derivative x is obtained as Z tZ

1 kx .t /k2 C 2

 jxx j2 dxd  2 0 Z tZ 1 2 r 2 .j.f .U r C /  f .U r //x j2 C jUxx j / dxd  :  k0;x k C C 2 0

(24)

The second term in the right hand side of (24) is estimated as follows: Z tZ 0

j.f .U r C /  f .U r //x j2 dxd  Z tZ

C



 jf 0 .U r C /j2 jx j2 C jf 0 .U r C /  f 0 .U r /j2 jUxr j2 dxd 

0

Z tZ C 0



 .1 C jj2.p1/ /jx j2 C .jj2 C jj2.p1/ /jUxr j2 dxd  ; (25)

2506

A. Matsumura

and Z tZ 0

r 2 jUxx j

Z

Z

t r 2 kUxx kL2

dxd   0

1

.1 C  /2 d  < 1:

d  C

(26)

0

In order to dispose the right hand side of (25) without using the maximum principle for u D U r C , multiplying (16) by jj2.p1/  yields as in (21) 1 2p

Z

j.t/j2p dx C .2p  1/

Z tZ  Z 0



.f 0 .U r C s/

0



 f 0 .U r //jsj2.p1/ ds Uxr dxd  C .2p  1/ 1 D 2p

Z j0 j

2p

Z tZ dx C 0

Z tZ

jj2.p1/ jx j2 dxd 

0

r jj2.p1/ Uxx dxd  :

(27)

As in (22), due to the Sobolev’s and Young’s inequalities, the second term in the right hand side of (27) can be estimated as follows: ˇ Z tZ ˇ Z tZ 2p1 ˇ ˇ 2.p1/ r r jj .jjp / p jUxx U dxd  j dxd  ˇ ˇ xx 0

0

Z t p 2p1 2p1 r kjjp k 2p k.jjp /x k 2p kUxx kL1 d   2 0

Z Z .2p  1/ t  jj2.p1/ jx j2 dxd  2 0  Z Z t 4p 2p 1 C jj dx .1 C / 2pC1 d  : CC

(28)

0

Hence, by plugging (28) to (27) and using the Grownwall’s inequality together with the assumption (6), it holds Z

j.t/j2p dxC

Z tZ 0

C

..jj2p C jj3p2 /Uxr C jj2.p1/ jx j2 / dxd 

Z j0 j

2p



(29)

dx C 1 :

Thus, combining all the estimates (23)(26) and (29) completes the desired a priori estimate (18).

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2.2

2507

Asymptotic Stability of Viscous Shock Wave

This subsection picks up a topic on the asymptotic stability of viscous shock wave for the Cauchy problem (4). Through this topic, another type of a priori estimate and a technical weighted energy method are presented, which are useful to treat small initial perturbations. As stated in the introduction, when the Riemann solution consists of a single shock wave ( R

s

u D u .x  sN tI u ; uC / WD

u . x  sN t /; uC . x  sN t /;

which is mathematically and physically reasonable, the equation with viscosity is expected to have a corresponding smooth traveling wave solution with the same shock speed sN and the far-field states u˙ . Here, the shock speed sN is determined by the so-called the Rankine-Hugoniot condition, and for the solution to be unique, some additional condition (e.g., the so-called “Lax’s shock (entropy) condition”, “Oleinik’s shock (entropy) condition”) depending on the nonlinearity of f is imposed. In this subsection, apart from the Riemann solution for a moment, it starts with studying the necessary and sufficient conditions on the existence of the traveling wave solution for (4). First, assume that a smooth traveling wave solution of the form u D U ./;

 D x  st

(30)

satisfies the equation and the far-field conditions in (4), where s 2 R is a propagating speed. Plugging (30) to (4) gives (

.sU C f .U /  U 0 /0 D 0 . 2 R/; U .˙1/ D u˙ ;

(31)

where 0 D d =.d /. In order for the solution of (31) to exist, it is easy to see that it necessarily holds U 0 .˙1/ D 0 and  sU C f .U /  U 0 D su˙ C f .u˙ /

. 2 R/;

(32)

which implies the “Rankine-Hugoniot condition”  s.uC  u / C .f .uC /  f .u // D 0:

(33)

Then, under the “Rankine-Hugoniot condition” (33), it follows from (32) that (

U 0 D f .U /  f .u˙ /  s.U  u˙ / DW h.U / U .˙1/ D u˙ ;

. 2 R/;

(34)

2508

A. Matsumura

which is equivalent to (31). In order for the solution of (34) to exist, it is again easy to see that when u < uC (resp. u > uC ) it necessarily holds



h.u/ D f .u/  f .u˙ /  s.u  u˙ / > 0

.u 2 .u < u < uC //

 .u 2 .uC < u < u // ; (35) which is nothing but the “Oleinik’s shock condition” (or “Oleinik’s entropy condition”). It is noted that in the case where f 00 .u/ ¤ 0 .u 2 R/, the condition (35) is equivalent to f 0 .u / > f 0 .uC / which is known as the “Lax’s shock condition” (or “Lax’s entropy condition”). Thus it should be emphasized that the RankineHugoniot and Oleinik’s shock conditions which are usually imposed for the Riemann solution of the inviscid equation to uniquely exist are naturally obtained as the necessary ones for the existence of a traveling wave solution of the viscous equation. Conversely, it can be proved that the Rankine-Hugoniot and Oleinik’s shock conditions are sufficient for the existence of a traveling wave solution of (4). In fact, under these two conditions, the existence of the solution of (34) which is unique up to shift of  can be proved by standard theory of ordinary differential equations. This traveling wave solution is called a “viscous shock wave” connecting far-field states u˙ . Now, for a fixed viscous shock wave U , the asymptotic stability is the next problem to be considered. It firstly should be noted that the usual definition of asymptotic stability needs to be modified a bit because for any ˛ 2 R, the spatially shifted function U .x  st C ˛/ also gives a viscous shock wave of (4). In fact, if the initial data is taken as u0 .x/ D U .x C ˛/, which is a perturbed one from U .x/ by U .xC˛/U .x/, then the corresponding perturbed solution u D U .xst C˛/ never tends toward the unperturbed one U .x  st /, no matter how ˛ is small. It suggests that for general small perturbations, the solution may tend toward a spatially shifted U . Here a heuristic argument is shown to see how the shift is determined. To do that, it is assumed that the perturbation u  U˛ has every nice decay property with respect to t and x, where U˛ ./ D U . C ˛/. Since U˛ is an exact solution of (4), it holds resp. h.u/ D f .u/  f .u˙ /  s.u  u˙ / < 0

.u  U˛ /t C .f .u/  f .U˛ //x  .u  U˛ /xx D 0:

(36)

Integration of (36) with respect to x gives an expectation Z

Z .u  U˛ /.t; x/ dx D

.u0 .x/  U .x C ˛// dx ! 0

.t ! 1/;

which implies Z .u0 .x/  U .x C ˛// dx D 0:

(37)

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2509

Thus, by an elementary calculation for (37), the shift ˛ is expected to be given by R ˛D

.u0 .x/  U .x// dx ; uC  u

(38)

and the solution is expected to eventually tends toward the spatially shifted U˛ . In what follows, it shall be proved true for small initial perturbations. For the proof, assume the initial data to satisfy u0  U 2 H 1 \ L1 ;

(39)

and the ˛ is given by (38) so that (37) holds. By setting Z

x

0 .x/ D

.u0  U˛ /.y/ dy;

(40)

1

further assume  0 2 L2 :

(41)

Note that (41) is equivalent to 0 2 H 2 under the condition (39). Then the main theorem in this subsection is stated as follows. Theorem 3. Suppose the Rankine-Hugoniot condition (33) and the Oleinik’s shock condition (35) with f 0 .u˙ / ¤ s, and U .x  st/ to be a viscous shock wave of (4) connecting the far-field states u˙ . Furthermore, suppose the initial data u0 satisfy (39) and (41) with the shift ˛ given by (38). Then there exists a positive constant "0 such that if k0 k2  "0 , the Cauchy problem (4) has a unique global solution in time u, satisfying u  U˛ 2 C .Œ0; 1/I H 1 / \ L2 .Œ0; 1/I H 2 /; and the asymptotic behavior sup ju.t; x/  U .x  st C ˛/j ! 0

.t ! 1/:

x2R

Before the proof of Theorem 3, a heuristic explanation is given on how to treat this problem by an energy method. For a moment, u < uC and f 00 .u/ ¤ 0 .u 2 R/ are assumed for simplicity. Then, Lax’s shock condition and (34) yields f 00 .u/ < 0

.u 2 R/;

u < U ./ < uC ; U 0 ./ > 0

. 2 R/:

(42)

Also assume that the deviation v D u  U˛ from U˛ is small and has every nice decay properties as needed. Then, it is expected that the asymptotic behavior of v is

2510

A. Matsumura

described by the linearized problem at U˛ ( vt C .f 0 .U˛ /v/x  vxx D 0 v.0; x/ D v0 .x/ WD u0 .x/  U˛ .x/

.t > 0; x 2 R/; .x 2 R/:

(43)

As for the the L2 -estimate for v, as in (21), multiplying the equation of (43) by v and integration by parts easily give Z tZ Z tZ 1 0 00 1 1 jvx j2 dxd  D kv0 k2 : kv.t /k2 C U˛ f .U˛ /jvj2 dxd  C  2 2 2 0 0 (44) Since U˛ 0 f 00 .U˛ / < 0 from (42), the second term in (44) causes a difficulty to have a dissipative property of v by standard energy estimate. This difficulty can also be explained by a fact that U˛0 is an exact solution of the linearized equation in (43), which suggests that unless an extra condition on the v0 is used, v never tends toward zero. To overcome the difficulty, recall ˛ is chosen so that the integral of v0 is zero which induces Z Z v.t; x/ dx D v0 .x/ dx D 0 .t  0/: (45) This property suggests that v has a form v D u  U˛ D x

.t  0/

(46)

for a function  which also has nice properties as needed. In fact, for the Fourier image of v, say v./, O the property (45) asserts v.0/ O D 0 which makes the O / have a sense. By plugging the representation v./ O D i.v./=.i O // D i  .i form (46) into (43) and integrating it with respect to x once, it results in ( t C f 0 .U˛ /x  xx D 0 .t > 0; x 2 R/; (47) .x 2 R/; .0; x/ D 0 .x/ where 0 is as in (40). Again, multiplying the equation of (47) by  gives Z tZ Z tZ 1 1 0 00 1 k.t/k2  U˛ f .U˛ /jj2 dxd  C  jx j2 dxd  D k0 k2 : 2 2 2 0 0 (48) Since the second term in (48) does have the right sign this time, it is expected that the L2 -energy method works well for an integrated problem (47) and even for the original nonlinear problem. This basic idea to apply the L2 -energy method for the antiderivative  of v under the integral zero condition on v was first proposed in Matsumura-Nishihara [39] and Goodman [5] independently to investigate the asymptotic stability of viscous shock waves for some systems of viscous conservation laws. This method is often referred as to “antiderivative method.”

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2511

Now, turn to the proof of Theorem 3. As suggested in the above heuristic argument, the solution of (4) is also expected to be given in the form u D U˛ C x :

(49)

Plugging (49) into (36) and integrating it with respect to x give the following Cauchy problem in terms of : (

t C .f .U˛ C x /  f .U˛ //   xx D 0

.t > 0; x 2 R/;

.0; x/ D 0 .x/

.x 2 R/:

(50)

Once the problem (50) is formulated even by heuristic argument, the small global solution in time  2 C .Œ 0; 1/I H 2 / of (50) with the asymptotic behavior kx .t /kL1 ! 0 .t ! 1/ is looked for. Then u is newly defined by (49) for the existence of the solution of (4) with the desired asymptotic behavior, and lastly the uniqueness of uU˛ in the class C .Œ0; 1/I H 1 /\L2 .Œ0; 1/I H 2 / is confirmed. Thus, the proof of Theorem 3 can be completed via the following theorem in terms of . Theorem 4. Under the same assumptions in Theorem 3, there exists a positive constant "0 such that if k0 k2  "0 , the Cauchy problem (50) has a global solution in time , satisfying  2 C 0 .Œ0; 1/I H 2 / \ L2loc .Œ0; 1/I H 3 /; and the asymptotic behavior sup jx .t; x/j ! 0

.t ! 1/:

x2R

The proof of Theorem 4 is also given by combining the unique existence of the local solution for the given initial data at general initial time   0, together with the a priori estimate, as in the Sect. 2.1. The description of the local existence theorem corresponding to Proposition 1 is omitted, because it is the same except replacing the initial data class by H 2 and the solution class by C .Œ ;  C t0 I H 2 / \ L2 . ;  C t0 I H 3 /. The a priori estimate to obtain the small global solution in time is as follows. Proposition 3 (a priori estimate). There exist positive constants ı0 and C0 depending only on the shape of the flux function f , the viscous coefficient  and the far-field states u˙ such that if the Cauchy problem (50) has a solution  2 C .Œ 0; T I H 2 / \ L2 .0; T I H 3 / for some T > 0 and sup k.t/k2  ı0 ; t2Œ0;T 

2512

A. Matsumura

then it holds that k.t/k22 C

 Z t q k jU˛0 jk2 C kx k22 d   C0 k0 k22

.t 2 Œ0; T /:

(51)

0

This type of a priori estimate to obtain small global solutions in time was first well formulated and applied to multidimensional quasilinear dissipative wave equations by Matsumura [35] and later applied to the system for multidimensional compressible and heat-conductive gas by Matsumura-Nishida p [38]. Once the a priori estimate Proposition 3 is established, choosing "0 D ı0 =.2 C0 / and combining the local existence together with the a priori estimate Proposition 3 with M D M0 WD ı0 =2 and T D nt0 .M0 / .n 2 N / step by step give the small global solution in time  2 C .Œ0; 1/I H 2 /, and the desired asymptotic behavior is shown in the same way as in the last subsection. Here the proof of Proposition 3 is given by following the arguments in [42]. Assume u < uC without loss of generality, and then assume the Oleinik’s shock condition (35). Then it follows from (34) that u < U ./ < uC ; U 0 ./ > 0

. 2 R/:

(52)

Next, rewrite the equation of (50) so that all the linearized terms at U˛ are collected on the left hand side and nonlinear term on the right as in the form t C f 0 .U˛ /x  xx D N .x /;

(53)

where the nonlinear term N is given by N .x / D .f .U˛ C x /  f .U˛ /  f 0 .U˛ /x /: In the case f satisfies the Lax’s shock condition, simple multiplication of (53) by  gives a nice estimate as in the previous discussion. However, since a more general condition, the Oleinik’s condition, is assumed here, a weighted energy method is newly proposed, where the weight function w is chosen as a positive function of the target asymptotic state U˛ itself, that is, w D w.U˛ /. This way of choosing weight function was first proposed in Matsumura-Nishihara [39] to study the asymptotic stability of the viscous shock wave for an isentropic/isothermal model of viscous gas. By multiplying the equation (53) by w and integrating with respect to x and t , it holds after integration by parts Z

Z tZ 1 wj.t/j2 dx C 2 0 Z tZ C .w0 U˛0 x 0

1 ..s  f 0 /w/0 jj2 U˛0 dxd  2 Z Z tZ 1 2 2 wN dxd : C wjx j / dxd  D wj0 j dx C 2 0 (54)

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2513

From U 0 D h.U /, f 0 .U /s D h0 .U / and taking integration by parts with respect to x again, it follows that Z Z tZ 1 1 wj.t/j2 dx  .hw/00 jj2 U˛0 dxd  2 2 0 (55) Z Z tZ Z tZ 1 2 2 wjx j dxd  D wN dxd : C wj0 j dx C 2 0 0 Now define w.u/ by 8 .u  u /.u  uC / ˆ ˆ 1=2; " > 0, and q are the constants chosen by q 0 .1Cy 2 /q dy D 1. This choice of the initial data is introduced so that it can give a better decay rate of kWxx .t /k than that in Lemma 1 and also can control the derivative Wx small by taking " small. Then, corresponding to the formula (83), a smooth approximation

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2525

Z1 D t .V1 ; U1 / (resp. Z2 D t .V2 ; U2 /) of the rarefaction wave zr1 D t .v1r ; ur1 / (resp. zr2 D t .v2r ; ur2 /) is constructed as follows: 8   1 ˆ < V1 .t; xI z ; zC / D 1 W .t; xI 1 .v /; 1 .vC // ; Z V1 .t;x/ ˆ 1 .s/ ds : U1 .t; xI z ; zC / D u  v

8 1 ˆ < V2 .t; xI z ; zC / D 2 .W .t; xI 2 .v /; 2 .vC /// ;   Z V2 .t;x/ resp. : ˆ 2 .s/ ds : U2 .t; xI z ; zC / D u 

(94)

v

Next, for zC 2 R1 R2 .z /, define the smooth approximation Z D t .V; U / of the Riemann solution zr .x=t/ in (92) by .V; U / D .V1 ; U1 /.t; xI z ; zm / C .V2 ; U2 /.t; xI zm ; zC /  .vm ; um /:

(95)

Then, it turns out that .V; U / approximately satisfies the viscous system (70) as 8 < Vt  Ux D 0;  U  (96) : Ut C p.V /x   x D R; V x where the residual term R is given by  U    x : R D p.V /  p.V1 /  p.V2 / C p.vm / x   V x Like Lemma 1, the basic properties of .V; U / can be obtained as follows, by choosing q suitably large if needed. Lemma 2. (1) Vt D Ux > 0; jVx j  C jVt j  C " .t  0; x 2 R/. (2) For p 2 Œ1; 1 and " > 0 , there exists a positive constant Cp;" such that 1

k.V; U /x .t /kLp  Cp;" .1 C t /1C p

.t  0/;

p1

k.V; U /xx .t /kLp  Cp;" .1 C t /1 2pq Z

.t  0/:

1

kR.t /k dt < 1.

(3) 0

(4) lim sup j.V; U /.t; x/  .v r ; ur /.x=t/j D 0: t!1 x2R

Now, by setting .v; u/ D .V C ; U C

/

(97)

2526

A. Matsumura

and using (96), the problem (70), (71) is rewritten as in the form 8 t  x D 0; ˆ ˆ ˆ   < Ux C x Ux C .p.V C /  p.V //    D R .t > 0; x 2 R/; t x V C V x ˆ ˆ ˆ : .; /.0; x/ D .0 ; 0 /.x/ WD .v0 .x/  V .0; x/; u0 .x/  U .0; x// .x 2 R/; (98) where note that .0 ; 0 / 2 H 1 and inf .0 C V .0; // D inf v0 > 0 under the assumptions of Theorem 5. Thus, for the proof of Theorem 5, it suffices to look for the global solution in time .; / 2 C .Œ0; 1/I H 1 / with the desired asymptotic behavior k.; /.t /kL1 ! 0 .t ! 1/. The solution is obtained by combining the unique existence of the local solution in time together with the a prior estimate in the same way as in the previous sections. To do that, the Cauchy problem (98) is reformulated as usual to one at the general initial time t D   0 with the given initial data .; /. / D . ;  / 2 H 1 satisfying inf . C V . ; // > 0. Denote the generalized problem by (98) . With keeping in mind that the system becomes singular as v D  C V ! C0, the solution space for any positive constants M; m, and interval I  R is introduced by ˚ XM;m .I / D .; / 2 C .I I H 1 / j x 2 L2 .I I L2 /; sup k.; /.t /k1  M; t2I

2 L2 .I I H 1 /;

inf .V C /.t; x/  m ; x

t2I;x2R

and in particular, for M D 1 and m D 0, ˚ X1;0 .I / D .; / 2 C .I I H 1 / j x 2 L2 .I I L2 /;

2 L2 .I I H 1 /;

inf .V C /.t; x/ > 0 : x

t2I;x2R

Proposition 4 (local existence). For any M; m > 0, there exists a positive constant t0 D t0 .M; m/ not depending on  such that if . ;  / 2 H 1 , k. ;  /k1  M and infx2R .V . ; x/ C  .x//  m, then the Cauchy problem (98) has a unique solution .; / 2 X2M;m=2 .Œ;  C t0 /. The proof of Proposition 4 is given by the standard iteration method by constructing the approximate Cauchy sequence . .n/ ; .n/ / 2 X2M;m=2 .Œ;  C t0 / .n 2 N / by 8 .n/ .n/ ˆ  D x ; ˆ ˆ t ˆ     < .n/ .n/ U  .n1/ x   V .V C D .p.V C  .n1/ /  p.V //x   V .VxC .n1/ /  R .n1/ / t x x ˆ ˆ ˆ ˆ : .n/ .n/ /./ D . ;  /; . ;

.t > /;

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2527

with . .0/ ; .0/ / D . ;  /. The details are omitted. Now, the a priori estimate to show the desired global solution in time is as follows. Proposition 5 (a priori estimate). For any initial data .0 ; 0 / 2 H 1 satisfying infx2R .V .0; x/ C 0 .x// > 0, there exists a positive constant C0 D C0 .0 ; 0 / such that if the Cauchy problem (98) has a solution .; / 2 X1;0 .Œ0; T / for some T > 0, then it holds that C01  V .t; x/ C .t; x/  C0

.t 2 Œ0; T ; x 2 R/;

Z t p k Ux k2 C kx k2 C k C

2 x k1

(99)

and k.;

/.t /k21

0



d   C0

.t 2 Π0; T /: (100)

In what follows, a sketch of the proof of Proposition 5 is given. First, multiplying the second equation in (98) by and using also the first equation give  Z tZ Z  ˇt Z tZ 1 2 ˇ Q dxd  D  R dxd  ; (101) C ˆ.v; V / dx ˇ C 0 2 0 0 where Z ˆ.v; V / D p.V /.v  V / 

v

p.s/ds;

v D V C ;

(102)

Ux x  C .p.v/  p.V /  p 0 .V //Ux : vV

(103)

V

and QD

2 x

v



it should be noted that if E.z/ denotes the physical total energy juj2 =2 C RHere v p.s/ ds of gas, then j j2 =2 C ˆ.v; V / is represented by the formula 1 2 j j C ˆ.v; V / D E.z/  E.Z/  rz E.Z/.z  Z/; 2

(104)

which may allow the form j j2 =2 C ˆ.v; V / be called a “relative total energy” to the background state Z. This kind of relative total energy is very often used for many other systems to obtain the basic energy estimate (e.g., see the Sect. 4.4). On the other hand, the term Q somehow represents the dissipation of the total energy due to the effect of viscosity and the rarefaction property Ux > 0. To see the nonnegativity of Q, set f D f .v; V / D

p.v/  p.V /  p 0 .V /.v  V / ; .v  V /2

2528

A. Matsumura

and regard Q as the quadratic form of QD

 p

x

2

p v



p .

p

x=

v/ and

p f Ux  as

p  p   p 2 Ux p p x  p p f Ux  C f Ux  : v V vf

Since the discriminant of the quadratic form is given by DD

Ux  4; V 2 vf

and 1=.V 2 vf .v; V // is uniformly bounded for v > 0, the quadratic form Q turns out to be strictly positive definite if " is suitably chosen small (use (1) of Lemma 2). Next, the right hand side of (101) is estimated as ˇZ tZ ˇ Z t Z ˇ ˇ 1 t ˇ ˇ R dxd  k kkRk d   .k k2 C 1/kRk d  : ˇ ˇ 2 0 0 0 Thus, due to Lemma 2 and the Gronwall’s inequality for (101), it holds the basic energy estimate Z k .t /k2 C ˆ.v; V / dx Z tZ  C 0

 C .0 ;

ˇ Ux x  ˇ ˇ C .p.V C /  p.V /  p 0 .V //Ux Cˇ v vV 2 x

 dxd 

0/

(105) for some positive constant C .0 ; 0 / depending only on the initial data. Note that this estimate corresponds to (74) in the Sect. 3.1. To proceed to the stage of (75), (76) and employ the Kanel’s method [18], rewrite the basic estimate (105) by setting vQ D v=V and plugging the concrete formula p.v/ D av  as Z 2 Q v/ k .t /k C V 1 ˆ. Q dx Z tZ  C 0

 C .0 ;

2 x

v

Cj

Ux x  Ux j C  .vQ   1 C .vQ  1// vV V

 dxd 

(106)

0 /;

where ( Q v/ ˆ. Q D

vQ  1  ln vQ . D 1/; 1 .vQ 1  1/ . > 1/: vQ  1 C 1

(107)

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2529

Since it holds         vt V  vVt vQ t vQ x Ux C x Ux  D  D  D  ;  V C V x vV vQ x vQ t x the second equation in (98) can be rewritten as  vQ x   vQ

 C t

 vQ x  Vx 1  vQ  C D R: V  vQ C1 V C1 vQ 

(108)

Then, multiply (108) by vQ x =vQ to have Z tZ Z   vQ x  ˇˇt  jvQ x j2   vQ x 2 dx ˇ C  dxd  0 2 vQ vQ V  vQ C2 0 Z tZ D 0

C

 Vx 1  vQ  vQ x dxd  V C1 vQ  vQ

Z tZ  j

xj

2

v

0

Ux x    dxd  C vV

(109) Z tZ R 0

vQ x dxd ; vQ

which corresponds to (76). Here the first term in the right hand side of (109) can be estimated as ˇ Z tZ ˇ ˇ

Z tZ ˇ  Vx 1  vQ  vQ x jvQ x j2 ˇ dxd  dxd  

ˇ V C1 vQ  vQ vQ C2 0 0 Z tZ Z tZ  v Q x 2 C C Ux .vQ   1 C .vQ  1// dxd  C C jVx j  1 dxd  vQ 0 0 (110)

for any > 0, where the last term in (110) is not needed for  D 1, and the estimate 



  1 holds for  > 1 from Lemma 2. Hence, plugging (110) kVx .t /kL 1 1  C .1Ct / to (109), choosing suitably small, and combining (109) and (105) with the help of the Gronwall’s inequality imply

 vQ x 2  .t / C vQ

Z tZ 0

jvQ x j2 dxd   C .0 ; vQ C2

0 /:

(111)

Now, it can be shown that (105) and (111) give the point-wise estimate of v both from below and above by using the idea of Kanel [18]. To do that, define ‰.v/ Q by Z

vQ

‰.v/ Q D 1

Q 1=2 d : ˆ. /

(112)

2530

A. Matsumura

Q (107), it is easy to see that ‰.v/ By the definition of ˆ, Q is monotonically increasing for vQ > 0 and ‰.v/ Q ! 1 .vQ ! C0/ and ‰.v/ Q ! C1 .vQ ! C1/. Then, by using (105) and (111), ‰.v/ Q is estimated as follows ˇZ ˇ j‰.v.t; Q x//j Dˇ





ˇ  ˇ ˇZ x @  vQ x ˇ ˇ ˇ Q v.t; ‰.v.t; Q y// dy ˇ D ˇ ˆ. Q y//1=2 .t; y/ dy ˇ @y v Q 1

x 1

Z

Q v/ ˆ. Q dx

1=2 

Z

ˇ vQ x ˇ2 1=2 ˇ ˇ dx  C .0 ; vQ

0 /:

(113) Thus, by the property of ‰.v/, Q (113) implies the uniform boundedness of vQ (that is, v) from both below and above as C .0 ;

0/

1

 v.t; x/ D .V C /.t; x/  C .0 ;

for some positive constant C .0 ; and (111) that 2

k .t /k C

k.t/k21

Z

t

C

0 /.

0/

(114)

By (114), it immediately follows from (105)

p .k Ux k2 C k.x ;

x /k

2

/ d   C .0 ;

0 /:

(115)

0

Finally, once the estimates (114) and (115) are established, by multiplying the second equation in (98) by  xx as in (77), it is easily shown that k

2 x .t /k C

Z

t

k

2 xx k

d   C .0 ;

0 /:

(116)

0

Thus, by the estimates (114), (115), and (116), the proof of Proposition 5 is completed.

3.5

Asymptotic Stability of Viscous Shock Wave, a Density-Dependent Viscosity Case

This subsection picks up a topic studied in Matsumura-Wang [43] where the asymptotic stability of large amplitude viscous shock wave for the isentropic model with a density-dependent viscosity is investigated and introduces another technique of weighted energy method. First, recall the isentropic model with density-dependent viscosity has the form 8 ˆ < vt  ux D 0; ux ut C p.v/x D ..v/ /x ˆ v : p.v/ D av  :

.t > 0; x 2 R/;

(117)

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2531

Here, an assumption on the dependency of viscous coefficient to the density  D 1=v should be made so that it has a physically reasonable background. According to the Chapman-Enskog expansion theory in rarefied gas dynamics (cf. [2, 22]), the viscosity coefficient is given by a function of the absolute temperature  , depending on the assumptions on molecular interaction. The typical two examples are given as follows: ( 1  D N  2 ; Hard sphere Model; 1 2 C s1 2 ; Cut  off inverse power force Model;  D N  where s . 5/ and N .> 0/ are some constants. Then, the above two models are unified as  D N  ˇ

.ˇ 

1 /: 2

(118)

On the other hand, since the model is isentropic, it holds p D R  D R

 D a v  v

.R W gas constant/;

which implies D

a . 1/ v : R

(119)

Thus, by (118) and (119), it is assumed that the viscosity coefficient .v/ has the form  D .v/ D 0 v ˛

.˛ 

1 .  1/; 0 > 0 W constants/: 2

(120)

Now, under the assumption (120), the Cauchy problem to (117) is considered with initial and far-field conditions 8 < .v; u/.0; x/ D .v0 ; u0 /.x/ .x 2 R/; (121) : lim .v; u/.t; x/ D .v˙ ; u˙ / .t  0/: x!˙1

Before the precise statement of the theorem, the argument on the existence of the viscous shock wave .V; U /.x  st / is recalled. Set  D x  st . Plugging .v; u/ D .V; U /./ into (117) gives the system of ordinary differential equations 8 0 for V taking the value between v and vC , in order for the problem (126) to have the solution, it must hold for the case s > 0 that v < vC ;

.i:e:; u > uC /:

(128)

The assumption (128) is also well known as the “entropy condition.” Conversely, if .v˙ ; u˙ / and s > 0 satisfy the Rankine-Hugoniot and entropy conditions (124), (128), it is easily shown that the solution .V; U / of (126) uniquely exists up to the shift of , satisfying V ./ > 0;

v < V ./ < vC

. 2 R/:

(129)

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2533

Now, for a fixed viscous shock wave .V; U /, the Cauchy problem (117), (121) is considered around a neighborhood of .V; U /. As for the initial data, it is assumed that .v0  V; u0  U / 2 H 1 \ L1 ; inf v0 .x/ > 0; x2R Z Z .v0  V /.x/ dx D .u0  U /.x/ dx D 0: And, by setting Z 0 .x/ D

Z

x

.v0  V /.y/ dy;

x

0 .x/ D

1

(130)

.u0  U /.y/ dy;

(131)

1

it is further assumed that .0 ;

0/

2 L2 :

(132)

Note that (132) is equivalent to .0 ; 0 / 2 H 2 under the condition (130). Then the main theorem in this subsection is stated as follows. Theorem 6. Suppose the initial data satisfy (130), (132), and ˛  12 .  1/. Then there exists a positive constant "0 such that if k.0 ; 0 /k2  "0 , the Cauchy problem (117), (121) has a unique global solution in time .v; u/, satisfying .vV; uU / 2 C .Œ0; 1/I H 1 /; vV 2 L2 .Œ0; 1/I H 1 /; uU 2 L2 .Œ0; 1/I H 2 /; and the asymptotic behavior sup j.v; u/.t; x/  .V; U /.x  st /j ! 0 .t ! 1/: x2R

For the proof of the theorem, antiderivative method is employed as in the arguments in the Sect. 2.2, that is, set v D V C x ; u D U C

x:

(133)

Plugging the relation (133) into (117) and integrating it with respect to x with the aid of the equation (122) for .V; U / give the following system in terms of .; /: 8 < t  :

t

x

D 0;

C p.V C x /  p.V / D 0

 .U C / Ux  x x  .V C x /˛C1 V ˛C1

.t > 0; x 2 R/: (134)

Then, the Cauchy problem to (134) is considered with initial data .; /.0/ D .0 ;

0/

2 H 2;

(135)

2534

A. Matsumura

where .0 ; 0 / is defined by (131), and the small global solution in time .; / is looked for. To do that, for any interval I  R, the solution space X .I / is defined by X .I / D f .; /2C .I I H 2 / j x 2L2 .I I H 1 /;

x 2L

2

.I I H 2 /; sup k.; /.t /k2 t2I

1  v g: 2 As in the previous arguments, the small global solution in X .Œ0; 1// is constructed by the combination of the local existence and the a priori estimate. Since the local solution is well understood in the previous works, only the a priori estimate is stated as follows. Proposition 6 (a priori estimate). Suppose ˛  12 .  1/, and .0 ; 0 / 2 H 2 . Then there exist positive constants ı0 and C0 such that if .; / 2 X .Œ0; T / is a solution of the Cauchy problem (134), (135) for some T > 0 and sup k.; /.t /k2  ı0 ; t2Œ0;T 

it holds that k.; /.t /k22 C

Z

t 0

.kx k21 C k

2 x k2 / d 

 C0 k.0 ;

2 0 /k2

.t 2 Œ0; T /:

Once Proposition 6 is obtained, the following global existence theorem can be shown, and it implies Theorem 6 by defining v D V C x ; u D U C x . Theorem 7. Suppose ˛  12 .  1/, and .0 ; 0 / 2 H 2 . Then there exists a positive constant "0 such that if k.0 ; 0 /k2  "0 , the Cauchy problem (134), (135) has a unique global solution in time .; / 2 X .Œ0; 1// satisfying the asymptotic behavior sup j.; /x .t; x/j ! 0 .t ! 1/: x2R

For the proof of the a priori estimate, it is assumed that .; / 2 X .Œ0; T / is a solution of the Cauchy problem (134), (135) for some T > 0 and ˛  12 . 1/, and the system (134) is rewritten so that all the linearized terms at .; / D .0; 0/ are collected on the left hand side and nonlinear terms on the right as in the form (

t  t

x

D 0;

 K.V /x 

0 V ˛C1

xx

D G;

(136)

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2535

where K.V / D p 0 .V / C .˛ C 1/

h.V / ; V

and G stands for the nonlinear terms. Here, recall that s > 0;

v < V < vC ;

Vx D

V ˛C1 h.V / > 0; 0 s

p 0 .V / D 

p.V / ; V

(137)

and note that (137) easily implies K.V /  

p.vC / : vC

Now it is ready to show the following basic energy estimate. Lemma 3. There exists a positive constant C such that it holds 2

Z

t

k.; /.t /k C

.k

2 xk

p C k Vx k2 / d 

0



 C k.0 ;

0 /k

2

Z tZ j jjGj dxd 

C



(138) .t 2 Œ0; T /:

0

Since Lemma 3 is the most essential, only the proof of Lemma 3 is given, and the estimates for the higher derivatives and the nonlinear terms are omitted. For details, refer to [43]. Proof of Lemma 3. Following the idea in [6,37], the positive weight functions 1 D 1 .V / and 2 D 2 .V / are introduced, which are properly determined later, so that Q Q / by the variable .; / is renormalized to .; Q  D 1 .V /;

D 2 .V / Q :

(139)

Plugging (139) into (136) yields 8 0; x 2 R/ x 2 t v v x where the unknown functions v > 0, u,  > 0, e > 0 and p are the specific volume, fluid velocity, internal energy, absolute temperature, and pressure, respectively, while the constants  > 0 and > 0 denote the viscosity and heat conduction

2538

A. Matsumura

Coefficients, respectively. Here the ideal and polytropic gas is studied, that is, p and e are given by the state equations pD

R ; v

eD

R   1

where  > 1 is the adiabatic exponent and R > 0 is the gas constant. The Cauchy problem to the system (150) is considered with initial and far-field conditions 8 < .v; u;  /.0; x/ D .v0 ; u0 ; 0 /.x/

.x 2 R/;

: lim .v; u;  /.t; x/ D .v˙ ; u˙ ; ˙ /

.t  0/;

x!˙1

(151)

where v˙ .> 0/; u˙ 2 R; ˙ .> 0/ are the given far-field states and .v0 ; u0 ; 0 / .˙1/ D .v˙ ; u˙ ; ˙ / is assumed as compatibility conditions. As in Sects. 2 and 3, the asymptotic behavior of global solutions in time of the Cauchy problem (150), (151) is investigated in relation to the corresponding Riemann problem for the hyperbolic part of (150) : 8 ˆ ˆ vt  ux D 0; ˆ ˆ ˆ ˆ < ut C p2x D 0; .e C u2 /t C .pu/x D 0 .t > 0; x 2 R/; ˆ  ˆ ˆ ˆ .v ; u ;  / .x < 0/; ˆ R R R ˆ : .v; u;  /.0; x/ D .v0 ; u0 ; 0 /.x/ WD .v ; u ;  / .x > 0/: C C C

(152)

Section 4 is organized as follows. After the Riemann problem (152) is briefly recalled in the Sect. 4.1, a survey of the known results on the asymptotic behaviors in time is given in the Sect. 4.2. Then, a topic on the asymptotic stability of viscous contact wave is picked up, which is the major characteristic of the full system compared with the isentropic/isothermal case. The Sect. 4.3 shows how to construct a suitable approximate solution which has the desired properties of viscous contact wavelike (61) with  D 0, and finally the Sect. 4.4 presents rough ideas how to obtain the asymptotic stability.

4.1

Riemann Problem

The system of conservation laws in (152) has three distinct real eigenvalues for positive v and  p 1 D   p=v < 0;

2 D 0;

3 D 1 > 0;

which implies the first and third characteristic fields are genuinely nonlinear and the second field is linearly degenerate. Then it is known that for any fixed z there exists a neighborhood O of z such that if zC 2 O, the Riemann solutions of (152)

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2539

consist of the various combinations of three elementary nonlinear waves, that is, either shock or rarefaction wave along each genuinely nonlinear characteristic field and contact discontinuity along the linearly degenerate one (in total, 17 cases, except the trivial case z D zC ) (cf. [46]). In what follows, the abbreviations z D .v; u;  /; zQ D .v; u; E/; z˙ D .v˙ ; u˙ ; ˙ /, etc., are used where E is the total energy E D e C juj2 =2. Also in all the cases, the strength of the Riemann solution is assumed suitably weak, that is, jzC  z j is assumed suitably small.

4.2

Historical Remarks

First, in the trivial case z D zC .DW zN/, Kazhikhov [25] first showed the existence of the unique global solution in time of (150), (151) for any initial data satisfying z0  zN 2 H 1 with inf v0 > 0; inf 0 > 0. However, the estimates of the solution are local in time, so any information of the asymptotic behavior of the solution was not obtained. Later, Jiang [16, 17] showed the uniform boundedness of the density from below and above with respect to both the space and time variables and then showed the solution tends toward some constant as time goes to infinity locally in space. Recently, Li-Liang [27] obtained a conclusive result that the temperature is also uniformly bounded from below and above, and the solution uniformly tends toward the constant state zN as time goes to infinity, by using only energy method. It would suggest possibilities of obtaining the global asymptotic stability of even nontrivial nonlinear waves. Second, in the case where the Riemann solution consists of a single rarefaction wave zri .x=t /.i D 1; 3/ corresponding to the i -characteristic field, KawashimaMatsumura-Nishihara [23] and also Liu-Xin [30] showed that in a suitably small neighborhood of the smoothed rarefaction wave, the global solution in time of (150), (151) exists and asymptotically tends toward the rarefaction wave zri .x=t/ of the hyperbolic part. For the proof, they fully made use of a relative total energy and the fact that the velocity component of the smoothed rarefaction wave is monotonically increasing as in the Sect. 3.4. The case where the Riemann solution has a multiwave pattern consisting of two rarefaction waves zr1 .x=t/ and zr3 .x=t/ can be treated similarly, and the global solution in time is proved to tend toward the linear combination zr1 .x=t/ C zr3 .x=t/  zm as in the Sect. 3.4. Here zm is the uniquely determined intermediate constant state so that zr1 .x=t/ connects z to zm and zr3 .x=t/ connects zm to zC . Third, in the case where the Riemann solution consists of a single shock wave zsi .x  si t /.i D 1; 3/ connecting z˙ with the shock speed si .s1 < 0 < s3 /, corresponding to the i -characteristic field, it is known that the system (150) has the viscous shock wave Zi .x  si t /, and it is expected that in a small neighborhood of Zi .x/ the global solution in time of (150), (151) exists and tends toward the Zi .x  si t C ˛i / with a suitable shift ˛i . Kawashima-Matsumura [21] first showed the asymptotic stability provided the integral of the initial perturbation in terms of zQ is zero, where zQ D .v; u; e C juj2 =2/. By taking into account that the velocity component of the viscous shock wave is monotone decreasing, the proof is given by

2540

A. Matsumura

the antiderivative method as in the Sect. 2.4. For more general initial perturbation whose integral in terms of zQ is not necessarily zero, Szepessy-Xin [47] replaced the viscous term by some artificial one and showed the asymptotic stability. Liu and his collaborators [29, 32, 33] and also Zumbrun and his collaborators [34, 49– 51] extended the result to the original physical system (150). They developed so much deep analysis on the linearized system at viscous shock wave (e.g., point-wise estimates by constructing the approximate Green function, or spectral analysis for the eigenvalue problem by Evans function arguments) together with energy estimates to close the a priori estimates (see their latest papers [26, 43] and references therein). However, it still seems very meaningful to give a simpler proof, in order to attack many other open problems. Fourth, in the case where the Riemann solution has a multi-wave pattern consisting of two shock waves zs1 .x s1 t / and zs3 .x s3 t /, the global solution in time of (150), (151) is expected to tend toward a linear combination of the corresponding combination of viscous shock waves Z˛1 ;˛3 D Z1 .x  s1 t C ˛1 / C Z3 .x  s3 t C ˛3 /  zm with suitable shifts ˛1 and ˛3 . Here zm is the uniquely determined intermediate constant state so that zs1 .x  s1 t / connects z to zm and zs3 .x  s3 t / connects zm to zC . Huang-Matsumura [9] showed that this asymptotic stability does hold in a small neighborhood of Z0;0 , provided the strengths of the two shock waves are small with the same order. The proof is technically given by constructing a good approximation of the linear diffusion wave around the constant state zm and combining the arguments by Liu [28] on how the shifts ˛1 ; ˛3 , and the strength of the diffusion wave are determined, together with the antiderivative method used in Kawashima-Matsumura [21]. Fifth, the case where the Riemann solution has a multi-wave pattern including both shock and rarefaction waves is entirely open as in the isentropic/isothermal case. Last, in the case where the Riemann solution consists of a single contact discontinuity corresponding to the two-characteristic field, Huang-Matsumura-Shi p [10] first introduced a corresponding viscous contact wave Z vc .x= 1 C t / which approximately satisfies the system (150), and Huang-Matsumura-Xin [11] showed that in a suitably small neighborhood of Z vc , the global solution in time of (150), (151) exists and tends toward Z vc provided the integral of the initial perturbation in terms of zQ is zero, by using the antiderivative method. Then, Huang-Xin-Yang [12] extended the result to more general initial perturbation whose integral in terms of zQ is not necessarily zero. Note that since they use the antiderivative method, their arguments can’t be applied to the case the Riemann solution has a multi-wave pattern consisting of contact discontinuity and rarefaction waves. To attack this case without using the antiderivative method, the main difficulty lies on the fact that the velocity component of the viscous contact wave is not monotone, contrary to the rarefaction wave. Huang-Li-Matsumura [8] overcame this difficulty by introducing a new weight function technique and succeeded in showing the asymptotic stability

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2541

of the multi-wave pattern consisting of contact discontinuity and rarefaction waves. Since the viscous contact wave is a main feature of the full system, compared with the isentropic/isothermal model, the following Sect. 4.3 presents how the viscous contact wave is constructed, and the Sect. 4.4 shows the essential ideas of the proof. The cases where the Riemann solution has a multi-wave pattern consisting of the contact discontinuity and shock waves are interesting open problems.

4.3

Viscous Contact Wave

This subsection presents how a viscous contact wave is constructed. It is known that the contact discontinuity of the Riemann problem (152) is given in the form z D zc .x/ D .v c ; uc ;  c /.x/ WD



.v ; u ;  / .t > 0; x < 0/; .vC ; uC ; C / .t > 0; x > 0/;

(153)

provided that u D uC .DW uN /;

p D pC .DW p/; N

(154)

where p˙ D R˙ =v˙ (cf. [46]). A corresponding viscous contact wave Z vc D .V; U; ‚/ which has the similar diffusive property as in the Sect. 2.3 is constructed as follows. Since the pressure is expected to be almost constant, set R‚ D p; N V

(155)

and ignore the second equation in (152) because Ut and .Ux =V /x are expected to have better decay estimates. Then, the first and third equations in (152) are reduced to 8 R ˆ ˆ < ‚t  Ux D 0; pN (156) ‚  jUx j2 R ˆ x ˆ N x D C ‚t C pU : :  1 V x V By ignoring the last term jUx j2 again and inserting the first equation to the second one in (156), a quasilinear diffusion equation for ‚ is obtained with far-field condition: 8   ˆ < ‚t D a ‚x .t > 0; x 2 R/; ‚ x (157) ˆ : ‚.t; ˙1/ D  .t > 0/; ˙

0. It is known that (157) has a self-similarity-type where a D p. N  1/=. R2 / >p solution ‚ D ‚./; . D x= 1 C t / which is unique among the solutions of

2542

A. Matsumura

the form ‚ D ‚./ (cf. [3, 7]). Furthermore, it is known that there exist positive N it holds that N ; C such that for ı WD jC   j  ı, constants ı;

x 2

.1 C t /j‚xx j C .1 C t /1=2 j‚x j  C ıe  1Ct

.t > 0; x 2 R/;

(158)

and

x 2

j‚  ˙ j  C ıe  1Ct

.t  0; x ? 0/:

(159)

Once ‚ is determined, .V; U / is determined by (155) and (156) as follows: V D

R ‚; pN

U D uN C

.  1/ ‚x : R ‚

(160)

Thus, a “viscous contact wave” of (150), Z vc , corresponding to the contact discontinuity zc is defined by Z vc D .V; U; ‚/, which was first introduced in [10]. Here it should be emphasized that U is not monotone with respect to x, contrary to the cases of smoothed rarefaction wave or viscous shock wave, which causes a difficulty later. Then, the viscous contact wave Z vc turns out to approximately satisfy (150) as 8 ˆ Vt  Ux D 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ U  < x D R1 ; Ut C Px   (161) x V ˆ ˆ ˆ ˆ ˆ  R  ‚ ˆ 1  U Ux  x ˆ ˆ ‚ C U 2 C .P U /x  C D R2 ; : t  1 2 V V x where P D R‚=V D p, N and the residual terms R1 and R2 satisfy the following point-wise estimates from (158) and (159)

x 2

jR1 .t; x/j; jR2 .t; x/j  C ı.1 C t /3=2 e  1Ct :

(162)

Note that due to (162), the residual terms have good estimates Z

1

.kR1 . /k C kR2 . /k/ d   C ı:

(163)

0

Finally, a remark is given on the quasilinear equation (157). If  D C .WD N /, the solution of (157) is expected to be further approximated by the self-similar solution, say ‚D , of the linear heat equation ‚t D .a=N /‚xx with the farN Then, it should be noted that the “diffusion wave” field condition ‚.˙1/ D . R D D D D D .V ; U ; ‚ / WD . pN ‚ ; uN C .1/ ‚D x ; ‚ / played an essential role in the case RN

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

2543

where the Riemann solution consists of two shock waves (refer to the last remark in the Sect. 4.2).

4.4

Asymptotic Stability of Viscous Contact Wave

This last subsection presents some essential ideas to prove the asymptotic stability of the viscous contact wave Z vc constructed in the last subsection. As in the previous sections, set .; ; / D .v  V; u  U;   ‚/;

(164)

and rewrite the Cauchy problem (150), (151) as in the form 8 t  x D 0; ˆ ˆ u ˆ ˆ Ux  x ˆ ˆ N x  D R1 ;  ˆ t C .p  p/ < v V x  x ‚x  ˆ R ˆ   C pu  pU N  D R2 C UR1 ; ˆ x x 1 t ˆ ˆ v V x ˆ ˆ : .; ; /.0/ D .0 ; 0 ; 0 / WD .v0  V ./; u0  U ./; 0  ‚.// 2 H 1 : (165) In order to obtain the small global solution in time .; ; / 2 C .Œ0; 1/I H 1 / with the desired asymptotic behavior k.; ; /.t /kL1 ! 0 .t ! 1/, it suffices to establish the following a priori estimate which is of similar type as Propositions 3 and 6. Proposition 7 (a priori estimate). For any fixed .v ; u ;  /, there exist positive constants "0 ; ı0 and C0 such that if .; ; / 2 C .Œ0; T I H 1 /; . ; / 2 L2 .0; T I H 1 /; is a solution of (165) for some T > 0, and sup k.; ; /.t /k21  "0 ;

ı D jC   j  ı0 ;

0tT

then it holds that, for t 2 Œ0; T , k.; ; /.t /k21 C

Z 0

t

.kx k2 C k.

2 x ; x /k1 / d 

 C0 .k.0 ;

2 0 ; 0 /k1

C ı 1=2 /: (166)

The most essential estimate to show Proposition 7 is the basic L2 -energy estimate as in the proofs of Propositions 4 and 5. In what follows, only some essential ideas to have the basic L2 -energy estimate are presented, and the arguments on the higher derivatives are omitted. By multiplying the first equation by P .1V =v/, the second one by , and the third one by .1  ‚= /, and then adding the resultant formulas

2544

A. Matsumura

together, it holds after integration by parts that Z

Z tZ   v    ˇˇt R  1 2 j j CR‚ˆ C ‚ˆ dx ˇ C j x j2 C jx j2 dxd  0 2 V  1 ‚ V V‚ 0 Z Z Z tZ  v t p  P Ux ˆ / C ˆ. / dxd  D G dxd  ; C V P 0 0 (167)

where ˆ.y/ D y  1  log y  0 .y > 0/, and G in the right hand of (167) represents all the residual terms which can be eventually controlled by choosing "0 and ı0 suitably small and using the estimate (162). Here it should be noted that if the entropy s is introduced by the first thermodynamical law de D  ds  p d v R (s D 1 log  C R log v C const: for the ideal polytropic gas) and the total energy R  C juj2 =2 is regarded as a function of zO D .v; u; s/, then E D e C juj2 =2 D 1 due to the relation rzO e D .p; 0;  /, the relative total energy to a background state ZO D .V; U; S / is given by O  rzO E.Z/.O O z  Z/ O E.z/ O  E.Z/ D

1 2 R j j C .  ‚/ C P .v  V /  ‚.s  S / 2  1

D

  v R 1 2 j j C R‚ˆ C ‚ˆ 2 V  1 ‚

(168)

which is nothing but the integrand in the first term of (167). It also should be noted that if the background state is a smoothed rarefaction wave, the third term in the left side of (167) has the right sign because of Ux > 0, which shows how the basic energy estimate for the rarefaction wave case is well obtained; if the background state is a viscous shock wave, the sign is opposite because of Ux < 0, which suggests the antiderivative method is suitable instead; if the background state is the viscous contact wave under consideration, because the sign of Ux changes, is needed a new technique to overcome it . Since it follows from the estimates (158), (159) for ‚ and the fact ˆ.y/  jy  1j2 .y ! 1/ that Z tZ 0

p  P jUx j ˆ / C ˆ. / dxd   C ı V P 

v

Z tZ 0

x 2

e  1C .jj2 C jj2 / dxd  .1 C  / (169)

for a suitable small "0 , it suffices to estimate the right hand side of (169). To do that, the next key lemma plays an essential role, which was first announced in [8]. Lemma 4. Suppose 2 C .Œ0; T I H 1 /, t 2 L2 .0; T I L2 /, and let a weight function ! be defined by

44 Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves

Z !D

p

x p 2.1Ct/

2545

2

e y dy

1

for a positive constant . Then, it holds that, for t 2 Œ0; T , Z

x 2 Z Z e  1C 2 1 2 2 ˇˇt 1 t ! dx ˇ C

dxd  0 2 8 0 .1 C  / Z Z Z tZ t j x j2 dxd  C ! 2

t dxd  : 

0 0

(170)

The proof is very elementally obtained by integration by parts and the Young’s inequality as follows: d  dt

Z

Z

1 2 2  ! dx D 2

Z

2

!!t dx C Z

1 D 2

! 2

t dx Z

2

! 2

t dx

!!xx dx C

D

1 2



1 4

Z Z

j!x j2 2 dx C

1

j!x j2 2 dx C



Z

Z !!x

x dx C Z

j x j2 dx C

Z

! 2

t dx

! 2

t dx; (171)

where the facts !t D

1 ! 2 xx

and 0  !  j!x j2 D

p are used. Thus, noting

x 2

e  .1Ct/ ; 2.1 C t /

and integrating (171) with respect to t complete the proof. Since there already has the nice estimate of dissipation for k. x ; x /k2 in the left hand side of (167), Lemma 4 suggests that under the estimate of dissipation for kx k2 which is restored later by the dissipative structure as explained in the Sect. 3.2, the right hand side of (169) could be controlled by taking a suitable

in Lemma 4 and choosing ı0 suitably small. In fact, this process is realized, and the following basic estimate eventually holds for suitably small "0 and ı0 (for the details, refer to [8]): k.; ; /.t /k2 C

Z

t

k.

x ; x /k

2

d

0



 C k.0 ;

2

0 ; 0 /k

Cı

1=2

Cı

1=2

Z

t 2



kx k d  : 0

(172)

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A. Matsumura

Thus, by starting with the estimate (172) and combining the estimates for the higher derivatives, the proof of Proposition 7 can be completed.

5

Conclusion

Only basic topics on the asymptotic wave patterns of the global solutions in time of the Cauchy problems have been discussed through some specific examples, in particular, in connection with the Riemann problems. Many other related interesting problems (e.g., on decay rate toward asymptotic state, higher approximation via diffusion waves, extension to initial boundary value problems, extension to other physical models, etc.) are omitted because of short pages. In particular, there is no room to discuss the initial boundary value problems on the half space, where not only the wave patterns discussed in the Cauchy problem but also their interactions with the boundary have to be taken into account (refer to an introductory survey [36]).

6

Cross-References

 Concepts of Solutions in the Thermodynamics of Compressible Fluids  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Well-Posedness and Asymptotic Behavior for Compressible Flows in One Dimen-

sion  Well-Posedness of the IBVPs for the 1D Viscous Gas Equations

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Existence of Stationary Weak Solutions for Isentropic and Isothermal Compressible Flows

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Song Jiang and Chunhui Zhou

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Isentropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Isothermal Case in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Nonuniqueness of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Regularity of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Domains with Noncompact Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2550 2554 2556 2577 2582 2586 2586 2588 2588 2590 2590 2591 2591

Abstract

The aim of this chapter is to study the existence of weak solutions to the multidimensional steady isentropic and isothermal compressible Navier-Stokes equations with large external forces. In the past decades, significant progress has been made on the existence of large weak solutions. In this chapter, a brief review of recent existence results on the existence of (renormalized) stationary weak solutions with large external forces will be presented. Different boundary value problems, such as the spatially periodic, slip and Dirichlet boundary value

S. Jiang () Institute of Applied Physics and Computational Mathematics, Beijing, China e-mail: [email protected]; [email protected] C. Zhou Department of Mathematics, Southeast University, Nanjing, China e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_63

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problems, will be investigated, and some related topics such as nonuniqueness, regularity, etc., will also be discussed. The ideas and developed techniques used in analysis will be presented and analyzed, and some open problems will be addressed.

1

Introduction

In continuum physics, material bodies are modeled as continuous media whose motion and equilibrium are governed by balance laws and constitutive relations. The mathematical description of the state of a moving fluid is effected by means of functions which give description of the fluid velocity u.t; x; y; z/ and of any two thermodynamic quantities pertaining to the fluid, for instance, the pressure p.t; x; y; z/ and the density .t; x; y; z/. All the thermodynamic quantities are determined by the values of any two of them, together with equations of state. Hence, if five quantities are given in three dimensions, for example, namely, the three components of the velocity u, the pressure p, and the density , the state of the moving fluid in R3 is completely determined. As far as the internal friction (viscosity) and thermal conduction are concerned, the complete system of equations of fluid dynamics in RN can be expressed by @t  C div.u/ D 0;

(1)

@t .u/ C div.u ˝ u/ C rp D div S C f C g;

(2)

@t E C div..E C p/u/ C div q D div.Su/:

(3)

Here  and u D .u1 ; u2 ;    ; uN / denote the density and velocity of the fluid, respectively, E WD 12 juj2 C e denotes the total energy, e is the specific internal energy, and q is the heat flux, p is the pressure, f D .f1 ; f2 ;    ; fN / and g D .g1 ; g2 ;    ; gN / denote the external volume and non-volume forces, S denotes the viscous stress tensor. In this chapter, one will consider the viscous stress tensor in the form of a Newtonian fluid: S D .ru C ruT / C divu I; where  and  are constant viscosity coefficients which have to satisfy thermodynamic constraints  > 0;  C 2=N  0; and 12 .ru C ruT / is the deformation tensor. The system (1), (2) and (3) is called the Navier-Stokes equations for compressible heat-conducting fluids in RN . According to the first law of thermodynamics, the Gibbs identity can be regarded as a system of differential equations for five quantities: the entropy S , the density , the pressure p, the temperature  , and the internal energy e. One can choose any two of these variables and obtain the other three (as a functions of selected couple) using the Gibbs equation. A general solution to these equations contains, however, an arbitrary functions. In order to fix them, the extra state equations are introduced.

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

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This chapter is mainly concerned with the physical situation where e; p satisfy the equations of state e D cV  ;

p D R .polytropic ideal gas/;

(4)

R and cV are positive constants. The heat flux obeys the Fourier’s law: q D r with  being a positive constant. The relations (4) lead to the caloric equation of state p D a ;   1;

(5)

in which the coefficient a depends on the entropy S , and the adiabatic exponent  is a constant between 1 and 5=3 for media most usually occurring. Air at moderate temperatures may be considered with  D 1:4, and the critical case  D 1 corresponds to the so-called isothermal flow. Neglecting heat exchange between different parts of the fluid (and also, of course, between the fluid and bodies adjoining it) means that the motion is adiabatic throughout the fluid. In adiabatic motion, the entropy of any particle of fluid remains constant as that particle moves about in space. If the entropy is constant throughout the volume of the fluid at some initial instant, it remains everywhere the same constant value at all times and for any subsequent motion of the fluid. Such a motion is said to be isentropic. The case of isentropic flows can be considered as a good approximation of system (1), (2) and (3) and a very interesting mathematical model. In the case of isentropic motion, the coefficient a in (5) is a constant, and thus, the pressure is a function of the density only. Then, the equations (1), (2) become an independent system, i.e., once .; u/ is known, (3) is an independent equation to determine the temperature field. Thus, one obtains the following N -dimensional compressible isentropic Navier-Stokes equations for the unknowns  and u: @t  C div.u/ D 0;

(6)

Q u C f C g; @t .u/ C div.u ˝ u/ C rp D 4u C rdiv

(7)

where Q D  C . Moreover, if the motion is assumed to be steady, i.e., the flow velocity u as well as all of the thermodynamic quantities depend only on the spatial variables but not on t, the system (6), (7) reduces to the steady compressible isentropic Navier-Stokes equations (in RN ): div.u/ D 0; div.u ˝ u/  4u  rdiv Q u C rp D f C g:

(8) (9)

This chapter is mainly devoted to the discussion of the existence of weak solutions .; u/ to boundary value problems for the system (8), (9) which is a nonlinear hyperbolic-elliptic coupled system.

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The mathematical study of the compressible Navier-Stokes equations goes back to the work by Nash [39] in the last 1960s. In [39] Nash proved the local existence of smooth solutions to the Navier-Stokes equations of compressible heat-conducting fluids. Since then, significant progress has been made on mathematical topics, such as the global existence and the time-asymptotic behavior of solutions. However, a number of important questions, for example, the existence of global solutions in the case of heat-conducting gases and the uniqueness of weak solutions, still remain open for large data. The existence theory for the steady compressible Navier-Stokes equations has been investigated by a number of researchers. The existence of strong solutions to the steady compressible Navier-Stokes equations with homogenous or inhomogenous boundary conditions has been studied in [7, 11, 22, 28–32, 40, 46, 48, 50, 51, 58, 59, 63, 64], for example; all these results, however, require certain “small” assumption in some sense, i.e., it is required that one or more than one of the data like the velocity, the density, the external forces, the Reynolds number, and the Mach number are small, or the data are a small perturbation of a constant state. For general large data, the situation becomes quite complex. A global existence result for the system (8), (9) for general large data was obtained first by Lions [36], in which he used the method of weak convergence and delicate techniques to establish the existence of global renormalized weak solutions, provided the adiabatic constant  is appropriately large, for example,   53 for N D 3. Roughly speaking, the condition on  comes from the integrability of the density  in Lp .˝/. The higher integrability of  has, the smaller  can be allowed. On the other hand, for the nonsteady system (6), (7), Feireisl Novotný and Petzeltová [13–15] delicately used the curl-div lemma to derive certain compactness and adapted Lions’ idea [36] as well as a technique from [26] to extend Lions’ existence result on the system (6), (7) to the case  > N2 , N D 2; 3. Then, adapting the approach from [15] to the case of the steady system (8), (9), Novotný and Novo [42] proved the existence of weak solutions in three dimensions provided that  > 32 and f is potential, or  > 53 for any f 2 L1 .˝/. After that, Frehse, Goj and Steinhauer, and Plotnikov and Sokolowsk independently obtained L1 -estimates of the quantity 41 P and proposed several techniques to improve estimates of the density in [16,56,57], where the authors assumed a priori L1 -boundedness of the kinetic energy 12 u2 which was not shown to hold unfortunately. Then, by combining the L1 -estimate of 41 P with boundedness of the energy and density, Bˇrezina and Novotný [5] were able to show the existence of weak solutions to the three-dimensional spatially periodic p problem (8), (9) and (13)–(14) for any  > .3 C 41/=8 when f is potential, or for any  > 1:53 when f 2 L1 .˝/, without assuming the boundedness of the kinetic energy 12 u2 in L1 .˝/. Recently, Frehse, Steinhauer, and Weigant [17–19] developed a new approach to treat the case  > 43 in three dimensions (and   1 in two dimensions) for the Dirichlet boundary value problem (as well as the slip boundary condition in two dimensions). More recently, for any  > 1; Jiang and Zhou [27] obtained the spatially periodic weak solutions to the system (8), (9) in three dimensions by establishing a new coupled estimate in the Morrey space of both kinetic energy and pressure. Then, their existence result was generalized to the

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

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case of the slip boundary condition (11) by Jesslé and Novotny [25] and to the case of the Dirichlet boundary condition (10) by Plotnikov and Weigant [55]. There are also results by Lions, Novotný, Novo, Pokorný, et al. on the existence of weak solutions in exterior domains or bounded domains with different geometry of boundary. The reader can refer to [12, 36, 40, 43–46] and the references cited therein. The main issue in this chapter is to give a review on the recent existence results of weak solutions to the system (8), (9) with large external forces in bounded domains. As the end of this section, we introduce some notations used throughout this chapter. NOTATIONS: Let ˝ be a domain in RN or a periodic cell. One denotes by k;p p L .˝/ (p  1) the Lebesgue spaces, by W k;p .˝/ or W0 .˝/ (k 2 N) the usual k Sobolev spaces, by H .˝/ (k 2 N) the Sobolev spaces W k;p .˝/ with p D 2, by p p Lloc .˝/ (abbr. Lloc ) the space of measurable locally p-integrable functions in ˝. p;˛ By L .˝/ (1  p < 1; ˛  0) the Morrey spaces with u 2 Lp .˝/ satisfying ( kukLp;˛ .˝/ ,

sup

jD.x; /j

x2˝;d >0

˛

) 1=p

Z

p

ju.y/j dy

< 1;

D.x;/

by BMO (resp. BMOloc ) the space of functions of bounded (resp. locally bounded) mean oscillation with u 2 L1 .˝/ satisfying Z Z ˇ ˇ 1 1 ˇ ˇ kukBMO.˝/ D sup udx ˇdx; ˇu  jD.x; /j D.x;/ x2˝;d >0 jD.x; /j D.x;/ where d D diam˝, D.x; / D B.x; /\˝ (see [9]). By C k .˝/ (resp. C k .˝/) the space of kth-times continuously differentiable functions in ˝ (resp. ˝). By D.˝/ the set of functions having continuous derivatives of order k for any k < 1 and with compact support in ˝, by D0 .˝/ the dual space of D.˝/. One uses H1 .˝/ to denote the Sobolev space of vector fields belonging to H 1 .˝/ with zero normal trace at the boundary. k;p The space of symmetric functions will be also used. .Wsym .˝//3 denotes the k;p 3 space of vector functions in .W .˝// which possess the symmetry (13) and (14), p while Lsym .˝/ stands for the space of functions in Lp .˝/ with symmetry (13) and (14). k  kpI˝  k  kLp .˝/ stands for the norm in the Lebesgue space Lp .˝/ and k;p k  kk;pI˝  k  kW k;p .˝/ for the norm in the Sobolev space W k;p .˝/ or W0 .˝/. BR .a/ WD fx 2 R2 W jx  aj < Rg denotes the open ball centered at a with radius R. The distance function d .x/ is given by N d .x/ D dist.x; @˝/ for x 2 ˝;

d .x/ D dist.x; @˝/ for x 2 R3 n ˝:

For c > 0, one denotes by Ac and ˝c the annuluses Ac D fx 2 R3 W dist.x; @˝/ < cg;

˝c D Ac \ ˝:

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Now, for y 2 ˝c with c sufficiently small, one denotes by .y/ its projection to @˝, namely, .y/ 2 @˝;

j .y/  yj D dist.y; @˝/:

In this chapter, if not stated explicitly, C and L will denote generic positive constants which may depend on various parameters of the problem and may take different values in different formulas.

2

Definition of Weak Solutions

This section will give the definition of weak solutions in a bounded domain ˝ 2 RN (N > 1) with different boundary conditions, i.e., the Dirichlet boundary condition uD0

on @˝;

(10)

and the slip (Navier) boundary condition u  n D 0; .S.u/  n/  n D 0

on @˝;

(11)

as well as the periodic boundary conditions. It should be pointed out that in general, there could be no solution for arbitrary f and g in the case of periodic boundary conditions, since for a (smooth) solution, which is periodic in x with period, e.g., 2 , f, and g have to satisfy the necessary condition: Z .fi C gi /dx D 0; for 1  i  N: (12) ˝

Here in the periodic case, one denotes by ˝ the periodic cell . ; /N . However, if one considers f and g with symmetry fi .x/ D  fi .Yi .x//;

and

fi .x/ D fi .Yj .x//

if i ¤ j;

i; j D 1; 2; : : : ; N; (13)

where Yi .   ; xi ;    / D .   ; xi ;    /; then u will have the same symmetry and  with the symmetry .x/ D .Yi .x//

for i D 1; 2;    ; N;

and the condition (12) is satisfied automatically. Moreover, u satisfies Z ui .x/dx D 0 ˝

for all 1  i  N:

(14)

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

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So, in this chapter, one will consider the external forces f and g that have the symmetry (13) in the periodic case. Suppose for a moment that .; u/ is a classical solution to (8), (9) with one of the boundary conditions (10), (11) or the periodic condition (13), and let b 2 C 1 .0; 1/. Multiplying the continuity equation (8) by b 0 ./, one obtains the renormalized continuity equation divŒb./u C Œb 0 ./  b./div u D 0:

(15)

To keep this equation valid even for a weak solution  2 L .˝/, u 2 H 1 .˝/, one requires that (15) is satisfied in the sense of distributions D0 .˝/ for any b 2 C 0 .Œ0; 1// \ C 1 .0; 1/ satisfying jb 0 .z/zj C jb.z/j  C for all z > 0. Similarly, taking a scalar product of the momentum equations (9) with u and integrating over ˝, one obtains, by using the continuity equation (8) and any of the boundary conditions above or the spatially periodic condition, the energy equality Z



 jruj2 C jdiv Q uj2 dx D

Z .f C g/  udx:

˝

˝

For weak solutions, due to the presence of weakly lower semicontinuity, one could only expect the energy inequality Z



 jruj2 C jdiv Q uj2 dx 

˝

Z .f C g/  udx:

(16)

˝

Finally, it should be remarked here that solutions of (8) and (9) are nonunique. Indeed, on one hand, one can always take  D 0 and solve the elliptic system for u (u  0 if g  0). On the other hand, if f is a potential force, i.e., f D r , and g D 0, then one can take u D 0 to simply solve ar D r ;

  0 in ˝;

(17)

whence, D

1   1   1 CC C a

is also a solution of (17) for any C 2 R, where ./C WD maxf; 0g. In other words, the system (8), (9) supplemented with boundary conditions (e.g., Dirichlet boundary condition uj@˝ D 0) is underdetermined. From the physical point of view, this is to expect since one has to prescribe the total mass of the gas. So one may prescribe Z .x/dx D M; ˝

0

in ˝;

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where M > 0. The case M D 0, i.e.,  D 0 is easily settled since it leads to the problem of elliptic system: 4u  rdiv Q u D g: Now, let us define a renormalized (finite energy) weak solution .; u/ to the system (8), (9) in a bounded domain ˝ with Dirichlet boundary condition (10) as follows. Definition 1. Suppose that   1 and M > 0 are given constants, one calls .; u/ a renormalized finite energy weak solution to the boundary value problem (8), (9) and (10), if Z ˝

.x/dx D M;   0 in ˝;  2 L .˝/; u 2 H01 .˝/;

and the continuity equation (8) and the momentum equations (9) hold in the sense of distributions D0 .˝/. Moreover, the energy inequality (16) is satisfied, and the renormalized continuity equation (15) holds in the sense of distributions D0 .˝/ for any b 2 C 0 .Œ0; 1// \ C 1 .0; 1/ satisfying jb 0 .z/zj C jb.z/j  C for all z > 0. For other boundary conditions, a weak solution can be defined similarly. For a renormalized bounded energy weak solution .; u/ under the periodic conditions (13) and (14), in addition to the conditions in Definition 1, (; u) should  1 satisfy  2 Lsym .˝/ and u 2 .Hsym .˝//3 , while in the case of the slip boundary condition (11), the assumption u 2 H01 .˝/ in Definition 1 is changed to u 2 H1 .˝/.

3

The Isentropic Case

This section will focus on the developments in the study of the existence of weak solutions to the steady isentropic compressible Navier-Stokes equations with  > 1 in bounded domains with general large data. The first existence result for the system (8), (9) with general large data f and g is due to the work of P.-L. Lions [36] in which Lions introduces most of the principal ideas needed for further developments in the existence theory. The crucial observation on the compactness of the effective viscous flux and the results on the weak compactness were first announced in the note [35] where the more complicated nonsteady case was investigated. His complete proof was published in [36] where the importance of the renormalized solutions to transport equations in the existence theory of compressible flows was also addressed. Under the assumption that  > 1 in two dimensions and  > 5=3 in three dimensions, Lions [36] proved the existence of weak solutions to (8), (9) and (10) for arbitrary large data f; g 2 L1 .˝/. The existence theorem established in [36] reads as follows.

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

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Theorem 1. Assume that  > 1 if N D 2,   53 if N D 3, and  > N2 if N  4 and that f; g 2 L1 .˝/, M  0. One requests in addition if N D 3,  D 53 that 2 M 3 kfkL1 is smaller than an absolute positive constant which depends only on a, , , Q and ˝. Then, there exists a weak solution .; u/ of the problem (8), (9) and (10) satisfying Z dx D M; ˝

u 2 H01 .˝/;

 2 Lp .˝/;

with p D 2 if  > 1, N D 2, or if   3, N D 3; p D NN2 .  1/ otherwise. Furthermore, if  > 1, N D 2, or if   3, N D 3, .; u/ satisfies 1 1  2 L1 loc ; div u 2 Lloc ; curl u 2 Lloc ; Du 2 BMOloc ; a  /; Dcurl u 2 Ltloc for all 1  t < 1: D.div u  C

As mentioned in the introduction, most of the physically reasonable values of the adiabatic exponent  are between 1 and 53 . However, in Lions’ proof [36], the condition  > 53 marks the limit of the applicability of DiPerna-Lions’ transport theory to the steady continuity equation where the condition  2 L2 .˝/ is needed. So it is an interesting and important problem to study the existence of weak solutions to the system (8), (9) under the condition 1    53 . According to the proofs in [13, 15], the process of the weak convergence method is still valid if the density  has higher integrability than  , i.e.,  2 LC" .˝/ for some small " > 0. In fact, the condition  2 L2 .˝/ in Lions’ proof is replaced by a weaker condition: the uniform boundedness of the quantity kTk .ı /Tk ./k0;C1 , where Tk is defined in (44). This is one of the crucial steps to show the strong convergence of the density. Adapting the approach for nonsteady flows [13,15] to the steady flow case, Novotný and Novo [42] improved Lions’ existence result. More precisely, they proved Theorem 2. Assume that ˝  R3 is a bounded domain of class C 2 , f; g 2 L1 .˝/, M > 0. Then, for any  > 32 , and curl f D 0 if   53 , there exists a renormalized bounded energy weak solution .; u/ to the problem (8), (9) and (10), such that Z dx D M; ˝

with s. / D 3.  1/ if

3 2

 2 Ls./ .˝/;

u 2 H01 .˝/;

<  < 3, s. / D 2 if   3.

Proof of Theorem 2. The proof of Theorem 2 is broken up into several steps.

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Step I. Approximate system with the artificial pressure and dissipation To construct a renormalized bounded energy weak solution to the problem (8), (9) and (10), one considers the following approximate system with parameters ˛; ı; " > 0: ˛.  h/ C div.u/   D 0; in ˝;

(18)

1 Q u C rpı D f Cg in ˝; ˛.h C /u C .div.u ˝ u/ C u  ru/4u  rdiv 2 (19) with mollified boundary conditions @ j@˝ D 0; uj@˝ D 0; @n

(20)

where pı ./ D a C ıˇ ; ˇ > maxf ; 3g; R and h is a smooth function satisfying h  0 in ˝ and ˝ h D M . One can take, for M . In the proof that follows, one needs a technical assumption that example, h D j˝j the adiabatic component in the pressure should be larger than 3. This assumption is not optimal but convenient for the proof. This is the reason to modify the pressure p./ D a by considering pı ./ D a C ıˇ with ˇ > maxf ; 3g. The quantity pı is called the artificial pressure. Since the continuity equation (8) is of degenerate hyperbolic type, it is natural C to relax its degeneracy by adding to the the left-hand side the term ˛ (˛ ! R 0 ). At the same time, one has to add to the right-hand side the term h with ˝ h D M . After these arrangements, it is natural to use elliptic regularization, by adding to the left-hand side of (8) the term "4 (" ! 0C ) to obtain the existence of solutions to the approximate system. Due to the conservation of mass, one has to @ complete this modified continuity equation with Neumann boundary condition @n D 0 on @˝. This elliptic problem can be treated by a standard method and will yield a nonnegative density, provided h  0. Then one has to modify the momentum equations in order to maintain the energy inequality. This is done by adding to its left-hand side the term ˛.h C /u and considering the convective term div.u ˝ u/ in the form 12 div.u ˝ u/ C 12 u  ru. Then, the modified system can be solved by the Leray-Schauder fixed point theorem. Then by taking the limits " ! 0 and then ˛ ! 0, one can obtain a weak solution in the framework of the weak convergence method. More precisely, in the process of taking to the limits, one will confront with the problem of the lack of compactness in the density sequence. Fortunately, this lack of compactness will be compensated by the weak compactness of the effective viscous flux and then the problem can be

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

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solved by using the transport theory of DiPerna-Lions [10] and some elements of convex analysis as well as the theory of monotone operators. The existence and uniqueness of solutions to the approximate problem (18), (19) and (20) can be established by using the following Leray-Schauder fixed point theorem, see [47, Section 1.4.11.8], for example. Theorem 3. Let X be a real Banach space and D  X a bounded open set. Let H W D  Œ0; 1 ! X be a homotopy of compact transformations, which means that H is a compact mapping for every t 2 Œ0; 1 and that it is uniformly continuous in t on any bounded set B  D. Let !  H .!; t / ¤ 0;

8 t 2 Œ0; 1; 8 ! 2 @D:

If there is an !0 2 D such that H .!0 ; 0/ D !0 , then, for any t 2 Œ0; 1, there exists an !t 2 D, satisfying H .!t ; t / D !t as well. With the help of Theorem 3, one has the following existence result and a priori estimates: Lemma 1. Suppose that all the assumptions in Theorem 2 are satisfied. Then there exists a solution ." ; u" / to the system (18), (19) and (20), such that for any 1 < p < 1, u" 2 W 2;p .˝/; " 2 W 2;p .˝/; "  0 in ˝;

Z " dx D M; ˝

and the following estimates hold. ku" k1;2  C .˝; f; g; h/.1 C k" k0;2ˇ /; "kr" k20;2

 C .˝; ı; f; g; h/.1 C k" k0;2ˇ /;

k" k0;2ˇ  C .˝; ı; f; g; h/;

(21) (22) (23)

where C is a positive constant independent of " and ˛. Proof. For t 2 Œ0;R 1, v 2 W01;1 .˝/, 2 W 1;1 .˝/ such that kvk1;1  K for some K > 0 and ˝ dx D M , one first defines Tt . / D t as the solution of the problem (

t D t .˛.  h/ C div. v//; in ˝: R @t j D 0; ˝ t dx D M: @n @˝

Using the standard theory of elliptic operators and Theorem 3, one can obtain a fixed point  D T1 ./ , S .v/ 2 W 2;p .˝/, 1 < p < 1, which is a unique solution of

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the equation (18) with u replaced by v. Similarly, one can define ut D ˘t .v/ as the solution of the Lamé system (

4ut  rdiv Q ut D tF .S .v/; v/

in ˝;

uj@˝ D 0; where 1 F .; v/ D ˛.h C /u C .div.u ˝ u/ C u  ru/ C rpı ./  f  g: 2 Applying the classical theory of elliptic systems (see [41]) and Theorem 3 again, one can obtain a fixed point u" D ˘1 .u" / 2 W 2;p .˝/; 1 < p < 1, and completes thus the proof of existence with " D S .u" /. Hence, the couple ." ; u" / solves the approximate problem (18), (19) and (20). The estimates (21), (22) can be established directly by employing the energy estimates. To show (23), one has to use the bounded linear operator   Z 1;p B W f 2 Lp .˝/j f dx D 0 7! W0 .˝/; 1 < p < 1;

(24)

˝

satisfying divB.f / D f in ˝;

B.f /j@˝ D 0:

Then, due to Bogovskij [4], see also Galdi [20] one gets kB.f /kWp1 .˝/  C kf kLp .˝/ : ˇ

Now, taking D B."  equations (19), one obtains Z ˝

"Cˇ Cı

Z

1 j˝j

R ˝

ˇ

" dx/ as a test function to test the momentum

ˇC1

˝

2ˇ2

ˇ

2ˇ1 "2ˇ C .˝; f; g; h/Œk" k0;2ˇ .1Ck" k0;2ˇ /2 Ck" k0;2ˇ .1Ck" k0;2ˇ /;

which implies (23) immediately. Finally, to obtain the non-negativity of the density, one can test the continuity equation (18) by  D ." C l/ , where " is the negative part of " satisfying " D " a.e. in f" < 0g while " D 0 a.e. in f"  0g, l > 0 and  2 .0; 1/ will be fixed later. A straightforward calculation shows that Z ˛ ˝

" ." C l/ dx 

Letting l ! 0C , one gets

 2  1=.C1/ k" C lk C1 : K k" C lkC1 C1 C lKj˝j 2"

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .



˛

2561

 2   C1 K k" kC1  0: 2"

˚  The choice  < min 1; 2"˛ gives " D 0. This completes the proof of Lemma 1. K2  Step II. Limits as " ! 0 and then ˛ ! 0. In this step, one lets the artificial dissipation terms represented by the "-quantities and the relaxation term represented by the ˛-quantities tend to zero. This is a delicate process due to the difficulty induced by the lack of compactness in the density sequence. The lack of compactness will be compensated by the weak compactness of the effective viscous flux. Then, the difficulty can be circumvented by using Lions’ approach whose main ingredients include the transport theory and the theory of convex analysis and monotone operators. First, one will take the limit " ! 0 in the approximate system (18), (19) and (20). From the estimates (21), (22) and (23), one can extract a subsequence of ." ; u" /, still denoted by ." ; u" /, such that u" * u

weakly in W 1;2 .˝/;

u" ! u

strongly in Lp .˝/;

" * 

1  p < 6;

2ˇ

weakly in L .˝/; 2ˇ

" * 

weakly in L  .˝/;

"ˇ * ˇ

weakly in L2 .˝/;

"r" ! 0

strongly in L2 .˝/;

" u" * u

weakly in Lr .˝/;

" u" ˝ u" * u ˝ u

(25)

r r C  1; 2ˇ 6 p p weakly in Lp .˝/; C  1: 2ˇ 3

Hence, letting " ! 0 in (18), R (19) and applying (25), one finds that the weak limit .; u/ of ." ; u" / satisfies ˝ dx D M and div.u/ D ˛.h  /

in D0 .˝/;

(26)

1 Q u ˛.h C /u C .div.u ˝ u/ C u  ru/ C arp  u  rdiv 2 D f C g in D0 .˝/; (27) 

ˇ

where pN is the weak limit of a" C ı" in L2 .˝/. The next goal is to show that pN D a C ıˇ :

(28)

2562

S. Jiang and C. Zhou

which is equivalent to the strong convergence of " in L1 .˝/. Now, if extending  2 L2ˇ .˝/; h 2 C 1 .˝/; u 2 H01 .˝/ to zero outside ˝, still denoted by ; h; u, one obtains by Lemma 14 in Appendix, that the continuity equation still holds in the sense of D0 .R3 /, i.e., in D0 .RN /:

div.u/ D ˛.h  /

(29)

Then Theorem 10 in Appendix implies that divŒb./u C Œb 0 ./  b./div u D ˛.h  /b 0 ./

in D0 .R3 /;

(30)

for any b 2 C 0 .Œ0; 1// \ C 1 .0; 1/ satisfying jb 0 .z/zj C jb.z/j  C for all z > 0. Next, one introduces the so-called effective viscous flux H" D a" C ı"ˇ  . C /div Q u" * H D PN  . C /div Q u; which enjoys several remarkable properties (see, e.g., [24, 36, 60]). As aforementioned, the lack of compactness in the density sequence will be compensated by the weak compactness of the effective viscous flux. More precisely, one has the following lemma. Lemma 2. For any 2 C01 .˝/, it holds that Z lim

"!0 ˝

Z .x/H" " dx D

.x/H dx:

(31)

˝

Remark 1. It was first shown by Serre [60] in one dimension. Then, Hoff [24] also studied the regularity of the effective viscous flux. The result on the effective viscous flux in the multidimensional case was first obtained by Lions [36], while in [13], Feireisl gave a simplified proof based on the Div-Curl lemma (Lemma 15 in Appendix) and the compactness lemma on commutators involving the Riesz transform (Lemma 16 in Appendix). Actually, in the proof of Lemma 2, one only applies Lemma 15 with div fn D 0 and curl gn D 0. Next, let 0 <  < 1 and " be prolonged by 0 outside ˝. Thus there is a subsequence, such that 2ˇ

" *  weakly in L  .R3 /;

 D 0 in R3 n˝:

and 2ˇ

pı ." /" * pı ./ weakly in L ˇC .R3 /;

pı ./ D 0 in R3 n˝:

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2563

Then, by virtue of the transport theory for the continuity equation (30), and the compactness of the effective viscous flux (31) as well as a delicate analysis, one can deduce that Lemma 3. For 0 <  < 1 and  ; pı ./; pı ./ defined above, one has 1

1

˛. /  Cdiv.. /  u/˛hC

 1  1 pı ./ pı ./  . /  1 .2 C /

(32)

in D0 .R3 / provided h and u are prolonged by 0 outside ˝. Remark 2. For the proof of Lemma 3, the reader can refer to [36, 47], for example. Now, subtracting the equation (29) from (32), one infers that in D0 .R3 /

˛r C div.ru/  f

(33)

where ( rD

1

. /  0;

  in ˝;

otherwise;

f D

8
0g;

which combined with (25) and Theorem 12 in Appendix implies that " converges strongly to  in LˇC .f > 0g/. Finally, on the set f D 0g, we obviously see that " converges strongly to 0 in L1 .f D 0g/. Therefore, one has proved that " converges strongly to  in L1 .˝/, and (28) follows immediately. Next, one will take the limit ˛ ! 0 in the approximate system (26), (27) with p in (27) replaced by a C ıˇ . One will again confront with the problem of the lack of compactness in the density sequence. This lack of compactness is again compensated by the weak compactness of the effective viscous flux. This is quite similar to the process in taking to the limit as " ! 0 above, and hence the details will be omitted here. Finally, one has to establish the energy inequality and estimates of  independent of the artificial pressure. This can be achieved by testing the momentum equations with the test function Z   1 D B pı  pı dx ; j˝j ˝ where B is the Bogovskij operator defined in (24),  2 .0; 1 is a proper (small) constant. Summarizing the above analyses, the main result in this step reads as follows. Lemma 4. Assume that all the assumptions in Theorem 2 are satisfied, then there exists at least one weak solution .ı ; uı / to the problem div.ı uı / D 0

in ˝;

4uı  rdiv Q uı C div.ı uı ˝ uı / C rPı .ı / D ı f C g uı D 0

in ˝;

on @˝;

(37) (38) (39)

with the following properties .1/ ı 2 L2ˇ .˝/;

Z ˝

ı dx D M; uı 2 W01;2 .˝/; ı  0 a:e: in ˝I (40)



.2/ divŒb.ı /uı  C b 0 .ı /ı  b.ı / divuı D 0

in D0 .˝/I

1

.3/ kuı k1;2 C kı k0;s./ C ı ˇ kı k0;s./ˇ  L.˝; f; g; m/; where s. / D 3.  1/ if

3 2

<  < 3, s. / D 2 if   3.

(41) (42)

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2565

Step III. Limit as ı ! 0 In this step, one will complete the proof of the existence of a renormalized weak solution to the problem (8), (9) and (10) by taking ı ! 0 in the system (37), (38) and (39). Still, one will confront with the problem of the lack of compactness in the density sequence. Since the limiting density does not belong to L2 .˝/ when  < 53 , the DiPerna-Lions transport theory could not be applied. In particular, it is not clear whether the renormalized continuity equation holds. One therefore applies Feireisl’s approach [13], where the missing condition L2 .˝/ is replaced by the uniform boundedness of the quantity kTk .ı /  Tk ./k0;C1 , to show that the renormalized continuity equation still holds . In view of the estimate (42), one can extract a subsequence of .ı ; uı /, still denoted by .ı ; uı /, such that ˇ

ıı ! 0

in D0 .˝/;

uı * u

weakly in W 1;2 .˝/;

uı ! u

strongly in Lp .˝/;

ı * 

weakly in Ls./ .˝/;



ı *   ı uı * u

weakly in L

s. / 

1  p < 6; (43)

.˝/:

r r C  1; s. / 6 p p C  1: weakly in Lp .˝/; s. / 3

weakly in Lr .˝/;

ı uı ˝ uı * u ˝ u

Consequently, by taking ı ! 0, the limit .; u/ satisfies div.u/ D 0

in D0 .˝/;

Q u D f C g div.u ˝ u/ C ar  u  rdiv divŒb.ı /uı  C Œb 0 .ı /ı  b.ı /div uı D 0

in D0 .˝/;

in D0 .˝/:

So, to show that .; u/ is a weak solution of the problem (8), (9) and (10), one has to prove  D  a:e: on ˝; which is equivalent to the strong convergence of ı in L1 .˝/. In order to show  D  , one needs that the limit .; u/ is a renormalized solution of the continuity equation (8), i.e., (15) holds. If  > 53 , then (42) implies ı 2 L2 .˝/. Thus, by applying DiPerna-Lions’ transport theory, .; u/ satisfies the renormalized continuity equation (15) in D0 .˝/.

2566

S. Jiang and C. Zhou

If 32 <   53 , one can use the following cutoff function techniques, due to Feireisl, Novotný, and Petzeltová [15], to show that .; u/ is a renormalized solution by replacing  with Tk ./ and letting then k ! 1. z Tk .z/ D kT . /; k

k D 1; 2;    ;

(44)

where T .z/ 2 C 1 .R/, concave; T .z/ D z for z  1, T .z/ D 2 for z  3. Since .ı ; uı / is a renormalized solution (recalling ı 2 L2 .˝/), one can take b.z/ D Tk .z/ in the definition of the renormalized solutions to get div .Tk .ı /uı / C ŒTk0 .ı /ı  Tk .ı /div uı D 0

in D0 .˝/:

Letting ı ! 0 in the above identity, one deduces that div .Tk ./u/ C ŒTk0 ./  Tk ./div u D 0

in D0 .˝/:

(45)

Analogously to Step II, one can use Lemmas 15 and 16 in Appendix as well as (45) to derive the following identity involving the effective viscous flux.  Π . C /divuT Q Q k ./div u k ./ D  Tk ./  . C /T

in ˝:

(46)

Then in the spirit of [26], one manages to control the oscillations of the density in the following way: lim kTk .ı /  Tk ./k0;C1  L.˝; f; g; m/;

ı!0

(47)

which means, roughly speaking, that the amplitude of possible oscillations in the density sequence is square integrable. The control of the density oscillations makes it possible to keep the renormalized continuity equation (15) valid for the limit (; u) even if the density is not known to be square integrable. Finally, introducing a family of functions ( Lk .z/ D

0zk

z log z; z log k C z

Rz

Tk .s/ k s 2 ds;

zk

2 C 1 .RC / \ C 0 Œ0; 1/

and making use of (46) and (47), one can deduce that lim kı  kL1 .˝/ D 0;

ı!0

which, by (43) and the interpolation theory, implies that ı ! 

strongly in Lp .˝/;

8 1  p < S . /:

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2567

Consequently,  D  ;

a:e:

Thus, the proof of Theorem 2 is complete.  Later, Bˇrezina and Novotný in [5] used the L1 -estimate for the inverse Laplacian of the pressure introduced in [16, 56], together with the nonlinear potential theory due to Adams, Hedberg [2], to prove the existence of weak solutions in a rectangular domain with spatially periodic conditions. Their result is stated as follows: p Theorem 4. Let f; g 2 L1 .˝/ satisfy (13),  > 13 .1 C 13/ 1:53, or let p f be potential and  > 18 .3 C 41/ 1:175. Then, there exists a renormalized bounded energy weak solution .; u/, satisfying (13) and (14), to the spatially periodic problem of the system (8), (9). Sketch of the proof. To prove Theorem 4, one works with the standard approximate system (37), (38) and (39) by introducing an artificial pressure term Pı ./ WD a C ı3 , and then tries to deduce the uniform-in-ı estimates for the solution .ı ; uı /, which will be used in passing to the limit as ı ! 0. First, testing the momentum equations (38) by .x/ D .x  y/.jx  yj/ with .t / D 1r  R1 on Œ0; r/, .t / D 1t  R1 on Œr; R/ and .t / D 1r  R1 on ŒR; 1/; one arrives at the following estimate that for any y 2 ˝, Z ˝

  pı dx  C 1 C kpı kL1 .˝/ C kı juı j2 kL1 .˝/ C kuı kH 1 .˝/ ; jx  yj

(48)

which implies, by Lemma 17 in Appendix, that kpı u2ı kL1 .˝/  C kuı k2H 1 .˝/ .1Ckpı kL1 .˝/ Ckı juı j2 kL1 .˝/ Ckuı kH 1 .˝/ /:

(49)

Then the uniform-in-ı estimates for the solution .ı ; uı / will be established by a bootstrap argument. To this end, one first tests the momentum equations (38) with uı to obtain the energy inequality Z

jruı j2 dx C Q

 ˝

Z ˝

jdiv uı j2 dx 

Z ı f  uı dx  kfkL1 .˝/ kı uı kL1 .˝/ ; ˝

(50) which implies by Korn’s inequality, Young’s inequality and Sobolev’s imbedding theorem that kuı kH 1 .˝/  C kfk1 kı k 6 : 5

(51)

2568

S. Jiang and C. Zhou

q Then for R any 2 L .˝/; 1 < q < 1, testing the momentum equations (38) with B.  ˝ dx/, where B is the Bogovskii operator defined in (24), one gets

Z

pı dx  C .kuı kH 1 .˝/ C kı juı j2 kLq .˝/ Ckı k

6

L 5 .˝/

˝

kfkL1 CkgkL1 /k kLq0 ; (52)

where q 0 is the conjugate to q. For  q > 65 , Young’s inequality together with (51) and (52) yields      b=q c=q kpı kLq  C 1 C kı juı j2 kLq .˝/  C 1 C kı juı j2 kL1 .˝/ kuı kL6 ;

(53)

3 provided q D  b; 2q D 2b C c; b C c=6  1, or equivalently q  C2 . Then, with the help of (49), (51), and (53), Hölder’s inequality and the imbedding L6 .˝/ ,! W 1;2 .˝/, one obtains

    kpı juı j2 k1  C 1 C kı juı j2 kL1C" .˝/ kı k26  C 1 C kı k 3 C" kı k26 kı k26 5

2

5

5

(54) where " 2 .0; "0 / and "0 is sufficiently small. Taking into account (53) and (54), and utilizing the interpolation inequality, one sees that    b 2b C 2q q q zCb" kpı kLq  C 1 C kı kq ; (55) C ; zD q  1 3 6 which yields q

kpı kLq  C;

(56)

p provided  q > z and thus  > 13 .1 C 13/. R If the volume force f is potential, then the term ˝ ı f  uı dx is zero by a partial integration, and one obtains, instead of (51), an a priori bound for kuı kH 1 .˝/ . 3 Similarly to the above arguments, one can obtain (56) for every 1 < q  C2 ( > 1). Summarizing all the above estimates, one concludes 1

kı kL q .˝/ Ckı juı j2 kLq .˝/ Ckuı kH 1 .˝/ Cı ˇ kı kLˇq .˝/  C .˝; M; f; g/

(57)

p uniformly with respect to ı, provided  > 13 .1 C 13/, or  > 1 and f is potential. p The condition  > 18 .3 C 41/ when f is potential in Theorem 4 comes from the limiting process of the convection term div.ı uı ˝ uı /, where the estimate

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . . 1

1

kı uı kLr .˝/  kı kL2  q kı juı j2 kL2 q .˝/  C with some r >

2569

6 5

(58)

is used (in fact, this can be relaxed to  > 1 by utilizing Lemma 6). Having had the uniform a priori estimate (57), one can employ arguments similar to those used in Theorem 2 to complete the proof of Theorem 4 by taking to the limit as ı ! 0.  Around 2010, Freshe, Steinhauer, and Weigant [19] introduced a new estimate to treat the case  > 43 in three dimensions for the Dirichlet boundary value problem. Their results rely on introducing the quantity A D kpı juı j2 kL1 .˝/ , using the potential estimates for the pressure and an ingenious bootstrap argument different from those employed in [5]. The result is stated as follows: Theorem 5. Assume that ˝  R3 is a bounded domain of class C 2 , f; g 2 L1 .˝/, and  > 43 . Then, there exists at least one weak solution .; u/ 2 Lr .˝/  W01;2 .˝/ of the Dirichlet boundary value problem (8), (9) and (10), such that Z 6 r dx D M; juj2 2 Lr .˝/; u 2 L  rC6 .˝/;   0; ˝ 3b where r D minf2; bC2 g with b 2

4 3

 ; .

Recently, Jiang and Zhou [27] improved Theorem 5 by establishing the existence of weak solutions to the spatially periodic problem (8), (9), and (13)–(14) for ˇ any  > 1 by introducing the quantity A D kPı juı j2 C ı juı j2C2ˇ kL1 .˝/ and establishing a new weighted estimate on both pressure and kinetic energy as well as utilizing a different bootstrap argument than those in [5, 19]. More precisely, they proved the following existence theorem: Theorem 6. Let f; g 2 L1 .R3 / satisfy (13). Then for  > 1, there exists a renormalized bounded energy weak solution .; u/, satisfying (13) and (14), to the spatially periodic problem of the system (8), (9). Sketch of the proof. Step I. The uniform estimates for the approximate system To prove Theorem 6, one first considers the standard approximation (37), (38) and (39) by introducing an artificial pressure term pı ./ WD a C ı3 and tries to derive the uniform-in-ı estimates for .ı ; uı / which will be used in passing to the limit as ı ! 0 to get a weak solution of the system (8), (9). One defines ˇ

A D kpı juı j2 C ı juı j2C2ˇ kL1 .˝/ ; and has

0 < ˇ < 1;

(59)

2570

S. Jiang and C. Zhou

Lemma 5. For A defined by (59), it holds that for any 1 < r < 2  1= , A C kuı kH 1 .˝/ C kpı kLr .˝/ C kı juı j2 kLr .˝/ C kı uı kLr .˝/  C; Z ı ı6 dx ! 0 as ı ! 0; ˝

where the constant C depends only on kfkL1 .˝/ , , , Q M ,  , and ˇ (but not on ı). Proof. First, one tests the momentum equations (38) with uı to obtain the basic energy inequality Z

jruı j2 dx C Q

 ˝

Z

jdiv uı j2 dx 

˝

Z ı f  uı dx  kfkL1 .˝/ kı uı kL1 .˝/ ; ˝

which implies by Hölder’s inequality and Sobolev’s imbedding theorem that  ˇ

kuı kH 1 .˝/  CA 4.ˇC 2ˇ/ :

(60)

Now, let  2 C01 .R/ be a cutoff function satisfying 0  .t /  1, jDj  2, and .t / D 1 if jt j  1; .t / D 0 if jt j  2. Let D . 1 ; 2 ; 3 / with i .x/ D

.x  x0 /i .jx  x0 j/; jx  x0 jˇ

i D 1; 2; 3:

Testing (38) with , one obtains after a straightforward calculation that Z

Z pı ı juı j2 dx C .1  ˇ/ dx ˇ ˇ B1 .x0 / jx  x0 j B1 .x0 / jx  x0 j    C 1 C kpı kL1 .˝/ C kı juı j2 kL1 .˝/ C kuı kH 1 .˝/ :

If one applies Hölder’s inequality, it is easy to see that for any 0 < ˇ < 1, Z B1 .x0 /

  pı C .ı juı j2 /ˇ dx  C 1 C kpı kL1 .˝/ C kı juı j2 kL1 .˝/ C kuı kH 1 .˝/ ; jx  x0 j (61)

where the constant C depends on kfkL1 .˝/ ; ; ; Q M;  , and ˇ, but not on ı. Then by virtue of Lemma 17 in Appendix, the inequality (61) implies that A  C kuı k2H 1 .˝/ .1 C kpı kL1 .˝/ C kı juı j2 kL1 .˝/ C kuı kH 1 .˝/ /:

(62)

Next, oneR takes the operator B defined in (24) on both sides of (9) with f D 1 s1 pıs1  j˝j ˝ pı dx to derive a higher integrability estimate of ı , where s 2

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2571

.1; ˇ C 1  ˇ= . To start with, one finds after a straightforward calculation that   kpı ksLs .˝/  C 1 C kuı ksW 1;2 .˝/ C kı juı j2 ksLs .˝/ ;

(63)

where the last term can be bounded as follows, using Hölder’s and Sobolev’s inequalities: kı juı j2 ksLs .˝/ D

Z

 sˇ

˝

ıs juı j2s dx  CA ˇC 2ˇ :

(64)

The estimate (64), together with (60) and (63), gives then   sˇ  kpı ksLs .˝/  C 1 C A ˇC 2ˇ :

(65)

Finally, recalling that (65) is true for any s 2 .1; ˇ C 1  ˇ= , one can write s D 1 C ", where " will be chosen small enough later on, and use (60), (62), (64), and (65) to infer that   A  C kuı k2H 1 .˝/ 1 C kuı kH 1 .˝/ C kı juı j2 kL1 .˝/ C kpı kL1 .˝/    ˇ  ˇ  sˇ 1  CA 2.ˇC 2ˇ/ 1 C A 4.ˇC 2ˇ/ C A .ˇC 2ˇ/  1C   3. ˇ/  C 1 C A 2.ˇC 2ˇ/ CO./ : Choosing ˇ sufficiently close to 1, one concludes that A  C , and therefore, Lemma 5 follows immediately. Step II. Vanishing artificial pressure as ı ! 0 To take to the limit as ı ! 0 for the approximate problem (37), (38) and (39) for any  > 1, one cannot directly use the arguments in Theorem 2, because one just has ı 2 Lr .˝/ with r being very close to 1 when  is close to 1, while in Theorem 2, ı uı 2 Lp .˝/ (p > 6=5) is required. To overcome this difficulty, one can use the estimates, established in Lemmas 5 and 6 below that can be shown by using Egoroff’s theorem, to get the weak compactness of the effective viscous flux. Lemma 6 ([27, Lemma 3.1]). Let 1 < p1 ; p2 ; p < 1, p  p1 , and ˝ be a bounded domain in R3 . Suppose that fn * f weakly in Lp1 .˝/, gn ! g strongly in Lp2 .˝/; and fn gn are uniformly bounded in Lp .˝/; then there is a subsequence of fn gn (still denoted by fn gn ), such that fn gn * fg Remark 3. The condition

1 p1

C

1 p2

weakly in Lp .˝/:

 1 is not needed in Lemma 6.

2572

S. Jiang and C. Zhou

The arguments in Step III of the proof of Theorem 2 are still applicable to show that the limit .; u/ of the weak solutions to the approximate system (37), (38) and (39) is indeed a weak solution of (8), (9), except that one has to verify the weak compactness of the effective viscous flux when  is small. This can be in fact achieved by utilizing the estimates established in Lemmas 5 and 6, together with the arguments used in Step III in the proof of Theorem 2.  More recently, Plotnikov and Weigant [55] were able to prove the existence of weak solutions to the Dirichlet boundary value problem for any  > 1, and their proof is based on an interesting observation that instead of the quantity A defined R in (59), one can first obtain R the bounds of two “smaller” quantities defined as A D ˝ juj2.2/ 2ˇ dx, B D ˝  ˇ dx through a bootstrap argument, which implies the boundedness of kukW 1;2 .˝/ C kpkLq .˝/ where  , q, and ˇ are defined 0 in (67) below. Then, by a uniform weighed estimate of the pressure in the Morrey space, one obtains the estimate of the kinetic energy density kjuj2 kLs .˝/ which yields the desired a priori estimates. The result in [55] reads as follows. Theorem 7. Let ˝ be a bounded domain in R3 with C 2 -boundary and let  > 1. Then for any f; g 2 L1 .˝/, there exists a renormalized bounded energy weak solution to the system (8), (9) and (10), such that kukW 1;2 .˝/ C kkLq .˝/ C kjuj2 kLs .˝/  C; 0

(66)

where the exponents q > 1 and s > 1 depend only on  , the constant C depends only on ˝; f;  ; M; , and . Q Before giving the proof sketch of Theorem 7, some notations are needed. For  > 1, denote by , ˇ, s, and q the quantities s D 1 C 2 2 ;  D

 ˇ.s  1/ 3.1  8 2 / 1 1   1 ; ˇ D ; qD1C : (67) 8 2.3  8 2 / ˇ C .1  ˇ/s

It is easy to see that 0 0, such that '.x/ equals the signed distance function in A2t . Introduce the vector field n.x/ D r'.x/; n.x/ 2 C 1 .AN 2t /; jn.x/j D 1; in AN 2t ; see [21, Chapter 14.6] on the existence of n.x/. Fix an arbitrary ˛ 2 .0; 1/ and x0 2 At , and define the vector field 

.x/ D

 '.x/  '.x0 / '.x/ C '.x0 / n.x/; C  .x; x0 /˛ C .x; x0 /˛

(82)

where ˙ .x; x0 / D j'.x/ ˙ '.x0 /j C jx  x0 j2 . Furthermore, assuming that N & 2 C 1 .˝/;

&  0 in ˝; & D 0 in ˝n˝t=2 ;

(83)

one can verify that & 2 W01;2 .˝/. Testing the momentum equations (38) with & , one infers that Z ˝

& p.x/ dx  C .1 C kukW 1;2 .˝/ C kpkL1 .˝/ C kjuj2 kL1 .˝/ /: 0 jx  x0 j˛

(84)

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2577

Next, for any nonnegative function  2 C01 .˝/ and x0 2 ˝, one may test the momentum equations (38) with in D jx  x0 j1 .x  x0 / to get Z ˝

p.x/ dx  c./.1 C kukW 1;2 .˝/ C kpkL1 .˝/ C kjuj2 kL1 .˝/ /; 0 jx  x0 j˛

(85)

which combined with (84) implies (79). Finally, the proof of Theorem 7 is completed by taking the limit ı ! 0 for the approximate problem (37), (38) and (39) and employing the same arguments as in Step II in the proof of Theorem 6.  Jesslé and Novotný [25] also considered the steady compressible Navier-Stokes system in a bounded three-dimensional domain with slip boundary conditions (11). They proved the following existence of renormalized weak solutions for any  > 1. Theorem 8. Let ˝ 2 C 2 be a bounded domain in R3 that is not axially symmetric, f; g 2 L1 .˝/ and  > 1. Then, there exists a renormalized bounded energy weak solution to the system (8), (9) with slip boundary conditions (11). Sketch of the proof. Roughly speaking, the proof of Theorem 8 is based on a bootstrap argument suggested in [27] to control the density and a potential-type estimate up to the boundary obtained through testing the momentum equations by a specially constructed test function. The main novelty in the proof lies in constructing the special test function, which is intrinsically related to the slip boundary conditions (one can also use & from Theorem 7 as a test function, where

and & are defined in (82) and (83), respectively). Then, by employing the method of weak convergence, one can prove the existence of a weak solution .; u/ to the system (8), (9), and (11).

4

The Isothermal Case in Two Dimensions

This section is mainly concerned with the existence of weak solutions to the compressible Navier-Stokes equations with adiabatic exponent  D 1, which corresponds to the critical case from the mathematical point of view and the socalled isothermal flows in physics. In this case, the system (8), (9) becomes div.u/ D 0; 4u  rdiv Q u C div.u ˝ u/ C ar D f C g:

(86) (87)

In this situation, the pressure term in the momentum equations do not cause any trouble in weak convergence, while the difficulties are induced by the convection term div.u ˝ u/. To the best of our knowledge, up to now, there are only four contributions to this problem. The first contribution is Lions’ existence result [36],

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S. Jiang and C. Zhou

in which Lions establishes the existence of a weak solution to the following twodimensional (modified) system: ˛ C div.u/ D h; ˛u  4u  rdiv Q u C div.u ˝ u/ C ar D f C g:

(88) (89)

The above system contains additional terms with a positive parameter ˛ which are motivated by the time-dependent case via discretization with respect to time. For the system (88) and (89), the energy estimate R implies the ˛-dependent boundedness of u in H 1 -norm and the kinetic energy ˝ juj2 dx as well as k ln.1 C /kL1 .˝/ , and this boundedness along with the imbedding (H01 ,! expL2 in 2D) yields a bound for div.u ˝ u/ in L1 , but not in Lp for some p > 1. This will lead to a concentration difficulty in the compactness proof of solutions to (88), (89), and Lions circumvented the difficulty by using the method of concentration compactness. The second contribution is due to Plotnikov and Sokolowski [57] in which they investigated the shape optimization problem of minimizing the drag with respect to an admissible obstacle. This problem becomes a boundary value problem with homogeneous Dirichlet boundary condition on the boundary of the flow region in case that there is no obstacle. To cope with the concentration difficulty, they introduced a special notion of generalized solutions to the Navier-Stokes equations in which the density is not anymore a L1 -function but a Borel measure in ˝. In [57] they proved the existence of such a generalized solution to the Dirichlet boundary value problem with no-slip boundary condition. The last two contributions concerning the system (86), (87) are due to Frehse, Steinhauer, and Weigant [17, 18] where they consider the existence of weak solutions to (86), (87) with slip and Dirichlet boundary conditions or spatially periodic condition in two dimensions. In these cases, the energy inequality does not imply a bound for the density and the kinetic energy, but this is settled by employing a weighted estimate of the pressure in the Morrey space and a bootstrap argument similar to the proof of Theorem 5 as well as exploiting the special structure of the system in two dimensions. This section is devoted to the existence of weak solutions to the system (86), (87) with Dirichlet boundary condition (10) [18]. It should be remarked here that the method to deal with the system (86), (87) with Dirichlet boundary condition also works for the case with periodic and slip boundary conditions. The detailed existence result reads as follows. Theorem 9 ([18]). Assume that ˝  R2 is a bounded domain of class C 2 , f; g 2 L1 .˝/. Then there exists at least one weak solution .; u/ 2 Ls1 .˝/  W01;2 .˝/ for any s1 2 Œ1; 2/ of the equations (86), (87) with boundary condition (10), such that Z .x/dx D M > 0;

  0; ˝

juj2 2 Ls2 .˝/

for all s2 2 Œ1; 2/:

(90)

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2579

To prove Theorem 9, in the same way as before, one introduces an artificial pressure term pı ./ D a C ı2 for 0 < ı  1 and considers the approximate problem: div.ı uı / D 0;

(91)

Q uı C div.ı uı ˝ uı / C rpı D ı f C g; 4uı  rdiv uı D 0

on @˝:

(92) (93)

The next aim is to derive the following uniform estimate for the above approximate problem: Proposition 1. Let .ı ; uı / be a weak (renormalized) solution of (91), (92) and (93), then kuı kW 1;2 .˝/ C kı juı j2 kLs1 .˝/ C kı kLs2 .˝/  C

for all s1 ; s2 2 Œ1; 2/;

where the constant C depends only on kfkL1 .˝/ , , , Q M , and  . The proof of Proposition 1 will be broken up into several lemmas. For the simplicity of presentation, the subscript ı will be dropped throughout the proof of Proposition 1. One begins with the following lemma. Lemma 9. Let .; u/ be a weak solution of (91), (92) and (93), then kukH 1 .˝/ C kjuj2 kL1 .˝/  C; where the constant C depends only on kfkL1 .˝/ , kgkL1 .˝/ , , , Q M and  . Proof. Testing (92) with u, one obtains the energy inequality kuk2H 1 .˝/

Z 

.f C g/  udx  C .kukL1 .˝/ C kukL1 .˝/ /; ˝

which implies kuk2H 1 .˝/  C kukL1 .˝/ : One introduces a stream function @1

D u2 ; @2

with u D .u1 ; u2 /, to have

(94)

in the following way

D u1 in ˝I

D 0 on @˝

(95)

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S. Jiang and C. Zhou

Z

juj2 dx D ˝

Z .u1 @2 Z

 u2 @1 /dx

˝

D

.@1 u2  @2 u1 /dx  2k kL2 .˝/ krukL2 .˝/ : ˝

Therefore, 1=2

1=2

1=2

1=2

kukL1 .˝/  kkL1 .˝/ kjuj2 kL1 .˝/  .2M /1=2 k kL2 .˝/ krukL2 .˝/ 1=2

1=4

 C k kL2 .˝/ kukL1 .˝/ :

(96)

On the other hand, the equations (95) imply in the two-dimensional situation that k kL2 .˝/  C kr kL1 .˝/ D C kukL1 .˝/ :

(97)

Thus, Lemma 9 easily follows from (94), (96), and (97). Lemma 10. For r 2 .1; 2/ it holds that  kpkLr .˝/  C 1 C k k



2r L 2r

.˝/

.recall p D a C ı2 /

;

where the constant C depends only on kfkL1 .˝/ , kgkL1 .˝/ , , , Q M , and ˝. Proof. Testing (92) ! D B.f / where the operator B is defined in (24) with R with 1 r1 f D p r1  j˝j dx, one finds that ˝p Z

p r dx D ˝

1 j˝j

Z

p r1 dx 

˝

Z ˝

Z

C

Z pdx C

.u  r/u  !dx ˝

Z

ru W r!dx C . C / Q ˝

Z div u div !dx 

˝

 r1  C kpkr1 Lr .˝/ kpkL1 .˝/ C kpkLr .˝/ k k

f  !dx ˝

krukL2 .˝/  C krukL2 .˝/ kr!kL2 .˝/ C M kfkL1 .˝/ k!kL1 .˝/   ˇ r1 C kpk k k  C kpkLr .˝/ C kpkr1 ; 2r r r L .˝/ L .˝/ 2r 2r

L 2r .˝/

L

where 0 < ˇ D

2r 2 2rC1 2r1

(98)

< r. Thus Lemma 10 follows from (98) immediately.

Lemma 11. For any x0 2 ˝ and ˛ 2 .0; 1/, it holds that Z ˝

p.x/  C .1 C kpkL1 .˝/ /; jx  x0 j˛

(99)

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2581

where the constant C depends only on kfkL1 .˝/ , , , Q M ,  and ˝. The inequality (99) can be derived in the same manner as to get (79) in Lemma 8, and hence, its proof will be omitted here. The following lemma is a reformulation of Lemma 5.4.1 in [37], and the reader can refer to [37] for a proof. Lemma 12. Suppose u 2 W01;2 .˝/; q 2 L1 .˝/ and q satisfies Z

jq.x/jdx  C r ˛

for all Br .x0 / and ˛ 2 .0; 1/:

˝\Br .x0 /

Then q  u 2 L1 .˝/ and Z

jq.x/  ujdx  C krukL2 .˝/ r 

for all 0 <  < ˛;

˝\Br .x0 /

where the constant C depends only on ˝; ˛, and . The following lemma is a version of Sobolev’s embedding theorem for the Morrey spaces (with n D 2 and ˛ 2 .0; 1/). The reader can refer to [1, 6] for a proof. Lemma 13. Let ˝  Rn be a bounded open set. If v 2 W01;1 .˝/ and rv satisfies Z

jrvjdx  C r ˛

for allBr .x0 / and ˛ 2 .0; n  1/;

˝\Br .x0 /

then v belongs to the Morrey space Lq;˛ .˝/ for all q < estimate Z jvjq dx  C r ˛ :

n˛ n˛1

and satisfies the

˝\Br .x0 /

In particular, v 2 Lq .˝/ and v is bounded from above in Lq .˝/ by rv in the Morrey space L1;˛ .˝/. Proof of Proposition 1. First, it can be easily seen that (99) implies Z

jp.x/jdx  CR˛ .1 C kpkL1 .˝/ / ˝\BR .x0 /

for all ˛ 2 .0; 1/, 0 < R < R0 , and x0 2 ˝. Then, one gets from Lemma 12 that

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S. Jiang and C. Zhou

Z

pjujdx  CR .1 C kpkL1 .˝/ /

(100)

˝\BR .x0 /

for all  2 .0; 1/, 0 < R < R0 and x0 2 ˝. Recalling the definition of the stream function deduces by (100) that Z

and the fact pı  aı , one

jr jdx  CR .1 C kpkL1 .˝/ / ˝\BR .x0 /

for all  2 .0; 1/, 0 < R < R0 and x0 2 ˝. Then by virtue of Lemma 13, for any r 2 .1; 2/ one can find 4.r1/ <  < 1, such that 3r2 k k

2r

L 2r

 C .1 C kpkL1 .˝/ /:

(101)

Combining (10) with (101), one concludes that k k

2r

L 2r

C kpkLr .˝/  C for all r 2 .1; 2/;

(102)

whence, kkLr .˝/ C kjuj2 kLr .˝/  C for all r 2 .1; 2/; which together with the boundedness of ru in L2 .˝/ gives Proposition 1.

(103) 

Having had the uniform estimate Proposition 1, one can employ arguments similar to those in the last section to prove Theorem 9 by passing to the limit as ı ! 0 in the approximate system (91), (92) and (93). Therefore, the proof of Theorem 9 is complete. Remark 4. It is an interesting open problem to establish the existence of weak solutions in the isothermal flow case (i.e.,  D 1) in three dimensions.

Appendix Lemma 14 (Prolongation of the continuity equation). Let N  2 and let ˝  RN be a bounded Lipschitz domain. Let  2 L2 .˝/; u 2 W01;2 .˝/, and f 2 L1 .˝/. Suppose that div.u/ D f in D0 .˝/: Then, prolonging ; u, and f by zero outside ˝ and denoting the new functions again by ; u, one has

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2583

div.u/ D f in D0 .RN /: Lemma 14 is the steady version of the same result shown for the nonsteady case which follows from the transport theory due to DiPerna-Lions [10]. Theorem 10 (Renormalized solutions of the continuity equation [10, 47]). Let ˇ 1;2 N  2, 2  ˇ < 1, and  2 Lloc .RN /,   0 a.e. in RN , u 2 Wloc .RN /, 1 N f 2 Lloc .R /. Suppose that div.u/ D f

in D0 .RN /:

(104)

Then, for any b 2 C 1 .Œ0; 1// satisfying b 0 .z/ D 0 when z tends to infinity, there holds divŒb./u C Œb 0 ./  b./div u D f b 0 ./

inD0 .R3 /;

(105)

Lemma 15 (Div-Curl lemma). Let 1 < p1 ; p2 ; q1 ; q2 < 1 (1=p1 C 1=p2 D 1=p < 1), and ˝ be a domain in R3 . Assume that fn * f weakly in Lp1 .˝/;

gn * g weakly in Lp2 .˝/;

div fn ! div f strongly in W 1;q1 .˝/;

curl gn ! curl g strongly in W 1;q2 .˝/:

Then, fn  gn * f  g weakly in Lp .˝/:

(106)

The Div-Curl lemma is a classical statement of the compensated compactness due to Murat [38] and Tartar [62]. Lemma 15 is one of its general formulation. For the proof of Lemma 15, the reader may refer to, for example, the reference [8, 47]. Lemma 16 ([15]). Assume .vn ; !n / * .v; !/ weakly in Lp .Rn /  Lq .Rn /, 1 D 1r < 1. Then, q

1 p

C

vn Rij Œ!n   !n Rij Œvn  * vRij Œ!  !Rij Œv weakly in Lr .Rn /; where Rij Œv D @xj 1 @xi v. The following theorem is a result from convex analysis, and the reader may refer to [47] for example. Theorem 11. Let 1  p < 1 and let G be a domain of RN .

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S. Jiang and C. Zhou

(i) Suppose that F W Lp ! R [ f1g is a convex lower semicontinuous functional on Lp .G/. Then F is weakly lower semicontinuous. In particular, F .u/  lim inf F .un / n!1

whenever un * u weakly in Lp .G/:

(ii) Suppose that F W Lp ! R [ f1g is a concave upper semicontinuous functional on Lp .G/. Then F is weakly upper semicontinuous. In particular, whenever un * u weakly in Lp .G/:

F .u/  lim sup F .un / n!1

(iii) Let I be an interval in R and f a convex lower semicontinuous (resp. concave upper semicontinuous) function on I . Suppose that un is a sequence of nonnegative functions from Lp .G/ with values in I , such that un * u

weaklyin Lp .G/

and f .un / * f .u/ weaklyin L1 .G/: Then, f .u/  f .u/

.resp:f .u/  f .u//

a:e: in G:

If a strictly convex function commutes with weak convergence, this weak convergence may be, in fact, strong. An overall discussion of this phenomenon can be found, e.g., in Visintin [65]. More precisely, it holds that Theorem 12. Let 1  p < 1, G be a domain in RN and I an interval in R, and f W I ! R a strictly convex function. Let un be a sequence of functions in Lp .G/ with values in I . If un * u weakly in L1 .G/ and f .un / * f .u/ weakly in L1 .G/, then un ! u

strongly in L1 .G/:

The following theorem is a result from the theory of monotone operators, and the reader may refer to [47] for a proof. Theorem 13. Let G be a domain in RN , I an interval in R, and f W I ! R a nondecreasing function. Let un be a sequence of functions in L1 .G/ with values in I such that un * u weakly in Lp .G/

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2585

f .un / * f .u/ weakly in L1 .G/: f .un /u * f .u/u weakly in L1 .G/: f .un /un * f .u/u weakly in L1 .G/: f .u/.un  u/ * 0 weakly in L1 .G/: Then, f .u/u  f .u/u

a:e: in G:

By applying Theorem 13, one arrives at the following corollary. Corollary 1. Let G be a domain in RN . Suppose that 1  s < 1 and 0 <  < 1 and that n is a nonnegative sequence in LsC .G/ satisfying n *  weakly in L ns * s weakly in L

sC 

sC 

.G/

.G/

nCs * Cs weakly in L

;

; and

sC 

.G/

;

Then, sC  s 

a:e: in G:

Lemma 17. Let ˝  R3 be a bounded domain of C 2 boundary. Let f 2 L2 .˝/ satisfy Z f  0; ˝

f .x/ dx  E jx  x0 j

for all x0 2 ˝:

(107)

Then, there is a C > 0, depending only on ˝, such that Z ˝

juj2 f dx  CEkuk2H 1 .˝/

for all u 2 H01 .˝/:

0

Proof. Let h 2 H 2 .˝/ be the unique solution of the elliptic problem: (

4h D f hD0

in ˝; on

Then, the following representation for h holds.

@˝:

(108)

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S. Jiang and C. Zhou

Z h.x/ D

G.x; y/f .x/dx; ˝

where G denotes Green’s function for the above Dirichlet boundary value problem and satisfies jG.x; y/j  C

1 jx  yj

for all x; y 2 ˝; x ¤ y:

Therefore, h 2 L1 .˝/ is bounded from above by khkL1 .˝/  CE: For u 2 H01 .˝/, setting Z

f juj2 dx;

KD

(109)

˝

one has by integrating by parts and using Cauchy-Schwarz’s inequality that Z Z K D 4hjuj2 dx D 2 .uru/  rhdx  C kukH 1 .˝/ kjujjrhjkL2 .˝/ : (110) ˝

0

˝

On the other hand, an integration by parts results in Z Z Z 2 2 juj jrhj dx D 2 h.uru/  rhdx C h.4h/juj2 dx ˝

Z

˝

˝

.jhj jrhj juj jruj C jhj j4hj juj2 /dx

C ˝

 C khkL1 .˝/ .K C kjuj jrhjkL2 .˝/ kukH 1 .˝/ /: 0

(111)

Thus, the inequalities (110) and (111) imply that K  C kuk2H 1 .˝/ khkL1 .˝/  CEkuk2H 1 .˝/ ; 0

which proves the lemma.

5

Related Problems

5.1

Unbounded Domains

(112)

0



In the case when ˝ is unbounded, the behavior of .; u/ at infinity has to be prescribed. One reasonable option to prescribe the behavior of .; u/ at infinity is to require u ! 0; .x/ ! 1 as jxj ! 1; x 2 ˝;

(113)

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2587

where 1 is a given nonnegative constant. For the definition of (renormalized) weak solutions to the system (8), (9) and (10) in an unbounded domain, one has to add the condition   1 2 Lr .˝/ for some 1  r < 1 in addition to the conditions in Definition 1. Considering unbounded domains including (i) ˝ D R3 , or (ii) ˝ c D R3 n˝ where ˝ is a bounded smooth connected domain in R3 , or (iii) ˝ D R  ! and ! is a bounded smooth connected domain in R2 , Lions [36] proved the existence of the renormalized bounded energy weak solutions to the system (8), (9) (with Dirichlet boundary condition in the last two cases) satisfying (113) and   1 2 L3 .R3 / \ L2 .R3 /, provided that f; g 2 L1 .˝/ \ L1 .˝/, 1 > 0, and  > 3. The question whether the problem (8), (9), (10) and (113) is well posed in a twodimensional exterior domain still remains open, see [12] where the two-dimensional problem is considered in the case of small data. Another difficult open problem is to show the existence of weak solutions to (8), (9) and (10) in an exterior domain ˝ c with the following conditions at infinity: lim u D u1 ¤ 0;

jxj!1

lim  D 1 > 0:

jxj!1

In general, there may not exist a nontrivial renormalized bounded energy weak solution with positive finite mass of the system (8), (9) in an exterior domain ˝ c or ˝ D RN under the condition that  and u vanish at infinity. The following example is taken from [36]. Assume that 1 D 0; g D 0; f D r ; 2 C 1 .˝/; < 0;

lim

jxj!1; x2˝

D 0;

and for any c > 0 the set fx 2 ˝; > cg is connected and unbounded. Suppose that .; u/,  ¤ 0 is a renormalized energy weak solution of (8), (9), (10) and (113) with above data. Due to the energy inequality, one has ru D 0, and from uj@˝ D 0, it follows that u D 0. Now,  satisfies the ordinary differential equation r D r

in D0 .˝/:

All solutions to this equation are  D 0;

  c D

1 i  1 h  1 . C c/ C ; c > 0: 

Since c 2 C 1 .f > cg/ and since f > cg is an unbounded domain, one has 1 h   1 i  1 D c ;  jxj!1; x2f >cg

lim

which contradicts the condition  2 Lr .˝/.

2588

5.2

S. Jiang and C. Zhou

Nonuniqueness of Weak Solutions

In general, the weak solutions constructed above are not unique. A simple example of nonuniqueness can be constructed by the following example which is taken from [47]. Let ˚ 2 W 1;1 .˝/ be a function with properties: 1. There are s; t 2 R; s < t, such that for any k 2 .s; t / the level sets fx 2 .k/ .k/ .k/ ˝I ˚.x/ > kg D O1 [ O2 , where Oi (i D 1; 2) are nonempty disjoint domains. .t/ .k / .k / 2. There holds that Oi D ; and if k1 ; k2 2 .s; t /; k1 < k2 , then Oi 2  Oi 1 , i D 1; 2. In this case, the couple ik .x/ D

1 i  1 h  1 . .x/  k/C IO.k/ ; u D 0; k 2 .s; t / i 

(114)

is a bounded energy renormalized weak solution to the problem (8), (9) and (10) R .k/ .k/ .k / with g D 0 and f D r and has the mass mi D ˝ i . Functions k ! mi i .t/ (i D 1; 2) are continuous decreasing functions on .s; t / and mi D 0: Therefore, .s/ for any m 2 .0; mc / where mc D miniD1;2 mi , there is just one ki 2 .s; t /, such .k / that mi i D m. By this procedure, for any m 2 .0; mc /, one has constructed two different bounded energy renormalized weak solutions to the problem (8), (9) and (10) with g D 0 and f D r . These solutions are .k /

. D 1 1 ; u D 0/

and

.k /

. D 2 2 ; u D 0/:

Solutions of type (114) are called rest states or equilibria

5.3

Regularity of Weak Solutions

Regardless the value of the adiabatic component   1 and whatever is the regularity of ˝; f; g, one cannot expect the density  to be continuous or in W 1;p for any p > 1 and the velocity field to be C 1 or in W 2;p for any p > 1. In particular, the density may exhibit discontinuities over two-dimensional surfaces in ˝. This fact is illustrated by the following example which is taken from [36, 47]. We take ˝ D B1 ,  D 1B1=2 , u D v if x 2 B1=2 , and u D w if x 2 B1 nB1=2 , where v 2 D.B1=2 /, div v D 0, and w solves the system 42 w D 0 in B1 nB1=2 ; w D 0 in @B1 [ @B1=2 ; x x x   rw C div Q w D in @B1 [ @B1=2 : jxj jxj jxj

(115) (116)

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

2589

The problem (115), (116) is an elliptic system of the Agmon-Douglis-Nirenberg type, and due to the general theory of elliptic equations, it admits a unique solution w 2 C 1 . B1 nB1=2 /; see [3, 41]. Thus, one has .; u/ 2 L1 .˝/  W 1;1 .˝/ but 1;1 2;1 not .; u/ 2 Wloc .˝/  Wloc .˝/. Now set f 2 D.B1 /, g 2 C 1 .B1 / satisfying ( fD

v  rv  4v  rdiv Q v in B1=2 ; 0 in B1 nB1=2 ;

and ( gD

0 in B1=2 ; 4w  rdiv Q w in B1 nB1=2 :

By construction, one can verify that .; u/ satisfies the renormalized continuity equation (15), the momentum equations (9), and the (basic) energy inequality (16). Thus, the couple .; u/ is a renormalized bounded energy weak solution to the system (8), (9) and (10). In the above construction, the lost of regularity is related to the possible presence of vacuum. In fact, one can prove that if there is no vacuum, i.e., inf ess  > 0;

(117)

and there are no boundary conditions, i.e., in the periodic case or in the case of the whole space, the density  is (Hölder-) continuous, and the velocity field is at least of class C 1 if the following assumptions are satisfied: 1;p

 2 L1 .RN /; u 2 Wunif .RN /; a 1;p 1;p div u  2 Wunif .RN /; curl u 2 Wunif .RN /; for all 1  p < 1; (118)  C Q 1;p

1;p

p

p

where Wunif .RN / D fu 2 Wloc .RN /, Ru 2 Lunif .RN /; Du 2 Lunif .RN /g and p p Lunif .RN / D fu 2 Lloc .RN /, supy2RN . B1 .y/ jujp dx/1=p < 1g. More precisely, one has the following theorem. Theorem 14 ([36]). Under the conditions (117) and (118),  and Du are bounded a and (uniformly) continuous. Furthermore, denoting  D C .inf ess / kDuk1 L1 ,  Q k D Œ (the integer part of  ) and ˛ D   k 2 Œ0; 1/, one has  2 Cbk;˛ ; k1;ˇ

 2 Cb

u 2 CbkC1;˛ if ˛ ¤ 0I ;

k;ˇ

u 2 Cb

for all ˇ 2 .0; 1/ if ˛ D 0I

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a  2 CbkC1;˛ ; curl u 2 CbkC1;˛ if ˛ ¤ 0I  C Q a k;ˇ k;ˇ div u   2 Cb ; curl u 2 Cb for all ˇ 2 .0; 1/ if ˛ D 0:  C Q

div u 

5.4

Domains with Noncompact Boundaries

In [40] Novo, Novotný, and Pokorný studied (8), (9) in the domains with conical or superconical exits and proved the existence of weak solutions, which was originally given in [42] for the domains with conical exits only. The result in [40] generalizes Lions’ related work [36] where the generalizations of exterior domains to the case of domains with several conical exits and to the case of nonzero fluxes and density drops were investigated. Also in [40], some results known for the incompressible Navier-Stokes equations [20, 23, 33, 34, 52–54, 61] are generalized to the case of the compressible isentropic Navier-Stokes equations.

6

Conclusion

The global existence of renormalized finite energy weak solutions .; u/ to the system (8), (9) with large external forces, supplemented with any of the boundary conditions (10), (11) and the periodic conditions (13)–(14), has been established for any specific heat ratio   1 in two dimensions and any  > 1 in three dimensions. Concerning the existence of weak solutions for large external forces, only the case  D 1 in dimension three is left open, which corresponds to the isothermal fluid case. It is clear to see that when  D 1, the techniques of deriving the a priori estimates described in the above sections fail, and new ideas have to be introduced to deal with this case. Another interesting open problem is whether the renormalized bounded energy weak solutions are unique when the mass M is given and the norms kfk0;p , kgk0;p are “small.” If the homogeneous Dirichlet boundary condition uj@˝ D 0 is replaced by the inhomogenous boundary conditions uj@˝ D u0 ;

where u0  n ¤ 0 on @˝

(119)

and  D 0 on the inflow portion of the boundary, i.e.,  D 0 on in D fx 2 @˝I u0  n < 0g;

(120)

the question on the existence of weak solutions still remains a difficult open problem. Some partial results about the existence of strong solutions with inhomogenous boundary conditions can be found in [28–32, 49–51, 58].

45 Existence of Stationary Weak Solutions for Isentropic and Isothermal. . .

7

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Cross-References

 Concepts of Solutions in the Thermodynamics of Compressible Fluids  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows  Symmetric Solutions to the Viscous Gas Equations  Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stabil-

ity, and Longtime Behavior  Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical

Cases

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Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

46

Piotr Bogusław Mucha, Milan Pokorný, and Ewelina Zatorska

Contents 1 2 3 4 5 6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weak and Variational Entropy Solutions: Main Results . . . . . . . . . . . . . . . . . . . . . . . . . A Priori Estimates for  > 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Approximate System Level 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Limit Passage N ! 1 Toward Approximate System Level 3 . . . . . . . . . . . . . 6.3 Limit Passage  ! 0C Toward Approximate System Level 2 . . . . . . . . . . . . . . 6.4 Limit Passage " ! 0C Toward Approximate System Level 1 . . . . . . . . . . . . . . 7 Estimates Independent of ı: Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 7.1 Estimates Based on Entropy Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Estimates of the Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Estimates Independent of ı: Navier Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 8.1 Estimates of the Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Limit Passage ı ! 0C Toward the Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Limit Passage Due to Uniform Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Effective Viscous Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Oscillation Defect Measure and Renormalized Continuity Equation . . . . . . . . . 10 Weak Solutions with Bounded Density and Internal Energy Balance . . . . . . . . . . . . . .

2596 2598 2602 2606 2608 2610 2610 2614 2615 2616 2619 2619 2620 2629 2629 2634 2635 2636 2638 2641

P.B. Mucha () Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland e-mail: [email protected] M. Pokorný Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic e-mail: [email protected] E. Zatorska Department of Mathematics, Imperial College London, London, UK Department of Mathematics, University College London, London, UK e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_64

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11 Weak Solutions in Two Space Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Steady Flow of a Compressible Radiative Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Steady Flow of Chemically Reacting Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2647 2654 2654 2656 2660 2660 2661

Abstract

The steady compressible Navier-Stokes-Fourier system is considered, with either Dirichlet or Navier boundary conditions for the velocity and the heat flux on the boundary proportional to the difference of the temperature inside and outside. In dependence on several parameters, i.e., the adiabatic constant  appearing in the pressure law p.%; #/  % C %# and the growth exponent in the heat conductivity, i.e., .#/  .1 C # m /, and without any restriction on the size of the data, the main ideas of the construction of weak and variational entropy solutions for the three-dimensional flows with temperature-dependent viscosity coefficients are explained. Further, the case when it is possible to prove existence of solutions with bounded density is reviewed. The main changes in the construction of solutions for the two-dimensional flows are mentioned, and finally, results for more complex systems are reviewed, where the steady compressible Navier-Stokes-Fourier equations play an important role.

1

Introduction

This chapter is devoted to the study of weak and variational entropy solutions to the system of partial differential equations describing the steady flow of a heatconducting compressible Newtonian fluid, i.e., we consider the steady compressible Navier-Stokes-Fourier system. The fact we want to consider solutions for arbitrary large data results in necessity of dealing with weak (or variational entropy) solutions; the strong or classical solutions are not known to exist, even for arbitrarily regular data. More information about the existence of classical solutions can be found in chapter  “Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows”. However (see [4]), strong solutions can be constructed even in some cases, when the force is not small. But a smallness condition on some of the data must be assumed. Note further that we must be more careful with the choice of correct boundary conditions (b.c.). Recall that we have to allow the energy exchange through the boundary as for thermally and mechanically insulated boundary the steady solutions may not exist. More precisely, considering the evolutionary system with mechanically and thermally insulated boundary and a time-independent external force, either the energy of the system grows to infinity or the force is potential and the velocity tends to zero, the temperature to a constant, and the density solves a certain simple first-order partial differential equation, see [11].

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2597

The steady compressible Navier-Stokes-Fourier system attracted more attention in the last few years. Even though the existence of weak solutions to the steady compressible Navier-Stokes system (i.e., the isentropic system or the system, where the exchange of the heat is negligible with respect to other processes) has been studied already in the seminal monograph [19] by the end of the last century (for more details see chapter  “Existence of Stationary Weak Solutions for Isentropic and Isothermal Compressible Flows”), the first results for the system studied here go back to the end of the first decade of this century. Indeed, P.L. Lions in his monograph considered the system of equations describing the steady flow of a heatconducting compressible fluid; however, he assumed that the density of the fluid is bounded a priori in some Lp -space for p sufficiently large. As we shall see later, to prove the bound of the density in a better space than L1 is one of the main difficulties for our system of equations. The L1 -norm, i.e., the total mass of the fluid, is a quantity which must be known, and hence, in a physically reasonable model, it is the only given bound for the density we may expect. The first existence result for such formulation appeared in 2009 in the paper [23]. The proof was based on the technique developed for the stationary Navier-Stokes equations in the papers [22] and [36], which for the Navier boundary conditions for the velocity and sufficiently large adiabatic exponent  allowed to prove existence of solutions with bounded density and “almost bounded” velocity and temperature gradients, see chapter  “Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows” for more details. Note that in this case even the internal energy balance is valid. This result was later extended in [24] to a larger interval for  and also for the homogeneous Dirichlet boundary conditions (replacing the internal energy balance by the total one); however, the value of  still remained far above the largest physically reasonable value  D 5=3, i.e., the monoatomic gas model. Note also that both abovementioned results were proved for a threedimensional domain and for viscosities which depend neither on the temperature nor on the density of the fluid. The corresponding result in the two-dimensional case can be found in [32]. Later on, in [27] and [28], the authors observed that using directly the estimates from the entropy inequality, one can obtain much better results, especially when additionally the viscosity coefficients depend on the temperature as .1 C #/. The existence of a solution for the Navier boundary conditions and any  >1 was shown in [15]. Note that the recently published paper [41] claims the existence of weak solutions for the homogeneous Dirichlet boundary conditions for the velocity under the same conditions which guarantee the existence of weak solutions in the case of the Navier boundary conditions. However, the authors of this paper are strongly convinced that the proof contains a gap in the part concerning the estimates of the density near the boundary. Analogous problems (for the Dirichlet boundary conditions) in two space dimensions were studied in [29] and [35] in the context of Orlicz spaces for the density. Finally, the situation when the viscosity behaves as .1 C # ˛ /, ˛ 2 Œ0; 1/ is the subject of the ongoing research. Some partial results in this direction can be found in [17], where only the case  > 32 has been studied.

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The chapter is organized as follows. In the next section, we introduce the model, the rheological relations, as well as the thermodynamical concept that we use. Then we introduce the notions of weak and variational entropy solutions and present the main existence results in the case of the three-dimensional domains. The next section contains the a priori estimates for our system (only for the adiabatic constant  > 32 ) to demonstrate why it is reasonable to consider two different definitions of the solutions. The following section contains all the necessary mathematical tools to deal with our problem. Next we present four approximation levels for our problem and briefly explain how to prove existence for the last one and how to pass through several levels to the first one. In the subsequent section, we show the a priori estimates independent of the first approximation parameter, and the following section contains the ideas of the limit passage to the original problem. Note that we restrict ourselves to the viscosity coefficients proportional to .1 C #/, i.e., to ˛ D 1. The next part of the chapter is devoted to the presentation of existence results for several related systems. First we discuss some ideas for proof of the existence of more regular solutions for  >3 in three space dimensions for constant viscosity coefficients and the Navier boundary conditions. Further we comment on the results in two space dimensions. Finally, we briefly mention few results for more complex system as, e.g., the steady flow with radiation or the steady flow of chemically reacting gaseous mixture. We also emphasize that results for the steady-state solutions are an important step in the analysis of time periodic solutions. Due to estimates constructed in [23] and [27], it was possible to prove existence of a weak time periodic solutions to systems (1), (2), and (3) in the paper [9]. The result has been generalized in [1]. In the whole chapter, we use standard notation for the Lebesgue space Lp .˝/ endowed with the norm k  kp;˝ and Sobolev spaces W k;p .˝/ endowed with the norm k  kk;p;˝ . If no confusion may arise, we skip the domain ˝ in the norm. The vector-valued functions will be printed in bold face, the tensor-valued functions with a special font. Moreover, we will use notation % 2 Lp .˝/, u 2 Lp .˝I R3 /, and S 2 Lp .˝I R33 /. The generic constants are denoted by C or c, and their values may change even in the same formula or in the same line. We also use P summation convention over twice repeated indices, from 1 to N , e.g., ui vi means N i D1 ui vi , where N D 2 or 3.

2

The Model

The steady flow of a compressible heat-conducting fluid in a bounded domain ˝  RN , N D 2 or 3, with sufficiently smooth boundary, can be described as follows: div .%u/ D 0;

(1)

div .%u ˝ u/  div S C rp D %f;

(2)

div .%Eu/ D %f  u  div .pu/ C div .Su/  div q:

(3)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2599

Here, %  0 is the density of the fluid, u is the velocity field, S is the viscous part of the stress tensor, p is the pressure, f is the vector of specific external forces, E is the specific total energy, and q is the heat flux. More details on systems (1), (2), and (3) can be found in chapter  “Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids”. Systems (1), (2), and (3) will be endowed with the boundary conditions on @˝: u D 0;

(4)

(i.e., the homogeneous Dirichlet boundary conditions for the velocity), or u  n D 0;

.Sn/   C u   D 0

(5)

(i.e., the Navier slip boundary conditions for the velocity, where   0 is the slip coefficient and  denotes the tangent vector to the boundary), and  q  n C L.#/.#  0 / D 0

(6)

(i.e., the Newton type boundary conditions for the temperature; here 0 > 0 is a given temperature outside ˝). We also assume that the total mass is given by Z % dx D M > 0: (7) ˝

In what follows we specify the constitutive laws for our gas. We will assume that the viscous part of the stress tensor obeys the Stokes law for Newtonian fluids, namely, h i 2 S D S.#; ru/ D .#/ ru C .ru/T  div uI C .#/div uI N

(8)

with ./, ./ continuous functions such that c1 .1 C #/˛  .#/  c2 .1 C #/˛ ;

0  .#/  c2 .1 C #/˛

(9)

with some 0  ˛  1. Moreover, the function ./ is additionally globally Lipschitz on R0C . The heat flux satisfies the Fourier law, i.e., q D .#/r#;

(10)

where ./ 2 C .Œ0; 1//;

c3 .1 C # m /  .#/  c4 .1 C # m /;

(11)

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with m>0. The coefficient L.#/ in (6) satisfies L./ 2 C .Œ0; 1//;

c5 .1 C #/l  L.#/  c6 .1 C #/l ;

l 2 R:

(12)

The specific total energy reads E.%; #; u/ D

1 2 juj C e.%; #/; 2

(13)

where e.; / is the specific internal energy. We will consider a gas law in the form p.%; #/ D .  1/%e.%; #/;

where  > 1:

(14)

This constitutive relation includes at least two physically relevant cases: if  D 5=3 it is the generic law for the monoatomic gases, while if  D 4=3 it describes the so-called relativistic gas, cf. [5]. In agreement with the second law of thermodynamics, we postulate the existence of a differentiable function s.%; #/ called the specific entropy which is (up to an additive constant) given by the Gibbs relation:  1  1 De.%; #/ C p.%; #/D D Ds.%; #/: # %

(15)

Due to (15) and (1), (2), and (3), the specific entropy obeys the entropy equation: div.%su/ C div

q #

D

S W ru q  r#  : # #2

(16)

It is easy to verify that the functions p and e are compatible with the existence of entropy if and only if they satisfy the Maxwell relation: @e.%; #/ 1 @p.%; #/  D 2 p.%; #/  # : @% % @#

(17)

Consequently, if p 2 C 1 ..0; 1/2 /, then it has necessarily the form 

p.%; #/ D #  1 P

  1

#  1 where P 2 C 1 .0; 1/.

;

(18)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2601

We will assume that P ./ 2 C 1 .Œ0; 1// \ C 2 .0; 1/; P .0/ D 0; P 0 .0/ D p0 > 0; P 0 .Z/ > 0; Z > 0; P .Z/ D p1 > 0; lim Z!1 Z  1  P .Z/  ZP 0 .Z/ 0<  c7 < 1; Z > 0:  1 Z

(19)

For more details about (18) and about physical motivation for assumptions (19), see, e.g., [10, Sections 1.4.2 and 3.2]. The consequences of these assumptions used throughout the chapter are listed below. Exactly the same results, modulo minor modifications in the proofs, can be obtained with the constitutive laws p.%; #/ D % C %#;

e.%; #/ D

1 %1 C cv #; with cv > 0;  1

(20)

whose physical relevance is discussed in [8]. We will need several elementary properties of the functions p.%; #/ and e.%; #/ and the entropy s.%; #/. They follow more or less directly from (14), (15), (16), (17), (18), and (19). We will only list them referring to [10, Section 3.2] for more details. Therein, the case  D 53 is considered; however, the computations for general  > 1 are exactly the same. We have for K a fixed constant 1

c8 %#  p.%; #/  c9 %#; %  K#  1 ; ( for  1 #  1 ; for %  K#  1 ; c10 %  p.%; #/  c11 1 for % > K#  1 : % ;

(21)

Further @p.%; #/ >0 @% p D d % C pm .%; #/;

d > 0;

in .0; 1/2 ; with

@pm .%; #/ >0 @%

in .0; 1/2 : (22)

For the specific internal energy defined by (14), it follows 9 1 p1 %1  e.%; #/  c12 .%1 C #/; > =  1 in .0; 1/2 : @e.%; #/ > 1 ; C #/ %  c13 .% @%

(23)

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Moreover, for the specific entropy s.%; #/ defined by the Gibbs law (15), we have 1  p.%; #/ 1 @p.%; #/ @e.%; #/  @s.%; #/ D  D 2 ; C 2 @% # % @% % @# 1  %  @s.%; #/ 1 @e.%; #/ 1 #  1   %  % 0 P D D P  > 0: 1 1 1 @# # @#  1 % #  1 #  1 #  1 (24) We also have for suitable choice of the additive constant in the definition of the specific entropy js.%; #/j  c14 .1 C j ln %j C j ln #j/ js.%; #/j  c15 .1 C j ln %j/ s.%; #/  c16 > 0 s.%; #/  c17 .1 C ln #/

3

in in in in

.0; 1/2 ; .0; 1/  .1; 1/; .0; 1/  .1; 1/; .0; 1/  .0; 1/:

(25)

Weak and Variational Entropy Solutions: Main Results

In this section we present definitions of weak and variational entropy solutions to our problem. They differ in the following way: for the weak solution, we require that our functions .%; u; #/ fulfill all equations of systems (1), (2), and (3) in the distributional sense, while for the variational entropy solutions, we do not require (3) to hold. Indeed, in some situations (we shall demonstrate this in the following section), we do not have sufficient regularity; hence the term %juj2 u from the total energy balance may not be integrable. One possible remedy is to consider the internal energy balance. We shall comment on this later; here let us only mention that the internal energy balance contains term like S.#; ru/ W ru which is possible to control only in L1 .˝/, and thus any limit passage in this term is difficult to perform. Therefore we shall use another possibility, namely, we replace the total energy balance by the entropy inequality. The reason why we cannot expect the entropy balance to hold is the fact that we are not able to keep equality in the limit passages in two terms, and we are obliged to use the weak lower semicontinuity therein. At the first glance, it looks like we generalized the definition of a solution too much. On the other hand, if we add to the entropy inequality the identity called the global total energy balance which is the total energy balance integrated over ˝ (here, the unpleasant term %juj2 u disappears), we end up with a system for which it is possible to show that any regular solution fulfilling three equalities (weak formulation for the continuity equation and for the balance of momentum and the global total energy balance) together with one inequality (the entropy one) is in fact a classical solution to (1), (2), and (3), i.e., the weak–strong compatibility holds. More details about the corresponding situation in the time-dependent case can be found in chapter  “Concepts of Solutions in the Thermodynamics of Compressible Fluids”.

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2603

In order to simplify the situation, we shall assume that our domain ˝ in the case of the Navier boundary conditions is not axially symmetric. It is connected with the form of Korn’s inequality valid in this case. If ˝ is axially symmetric, we have to assume that >0 in (5); the results in this situation can be found in [15]. We shall comment on them later. We have also to distinguish between the solution to the Dirichlet boundary conditions (4) and the Navier boundary conditions (5). Moreover, we mostly consider only the case N D 3. Finally, we take ˛ D 1 in (9). We have Definition 1 (weak solution for the Dirichlet b.c.). The triple .%; u; #/ is called a weak solution to systems (1), (2), (3), (4), (6), (7), (8), (9), (10), (11), (12), (13), (14), R 6 (15), (16), (17), (18), and (19), if % 2 L 5 .˝/, ˝ % dx D M , u 2 W01;2 .˝I R3 /, 6 # 2 W 1;r .˝/\L3m .˝/\LlC1 .@˝/, r>1 with %juj2 2 L 5 .˝/, %u# 2 L1 .˝I R3 /, S.#; ru/u 2 L1 .˝I R3 /, # m r# 2 L1 .˝I R3 /, and Z %u  r

dx D 0

8

2 C 1 .˝/;

(26)

˝

Z



˝

  %.u ˝ u/ W r'  p.%; #/div ' C S.#; ru/ W r' dx Z %f  ' dx 8' 2 C01 .˝I R3 /; D

(27)

˝

Z ˝

Z    %juj2 C %e.%; #/ u  r dx D %f  u C p.%; #/u  r dx 2 ˝ Z    S.#; ru/u  r C .; #/r#  r dx  R˝  @˝ L.#/.#  0 / dS 8 2 C 1 .˝/:

1



(28)

We denote Wn1;p .˝I R3 / D fu 2 W 1;p .˝I R3 /I u  n D 0 in the sense of traces.g Similarly the space Cn1 .˝I R3 / contains all differentiable functions with zero normal trace at @˝. Then we have Definition 2 (weak solution for the Navier b.c.). The triple .%; u; #/ is called a weak solution to systems (1), (2), (3), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), R 6 (15), (16), (17), (18), and (19), if % 2 L 5 .˝/, ˝ % dx D M , u 2 Wn1;2 .˝I R3 /, 6 # 2 W 1;r .˝/\L3m .˝/\LlC1 .@˝/, r>1 with %juj2 2 L 5 .˝/, %u# 2 L1 .˝I R3 /, S.#; ru/u 2 L1 .˝I R3 /, # m r# 2 L1 .˝I R3 /. Moreover, the continuity equation is satisfied in the sense as in (26), and

2604

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P.B. Mucha et al.

 ˝

  %.u ˝ u/ W r'  p.%; #/div ' C S.#; ru/ W r' dx C  Z %f  ' dx 8' 2 Cn1 .˝I R3 /; D

Z u  ' dS @˝

˝

Z

(29) Z     %juj C %e.%; #/ u  r dx D %f  u C p.%; #/u  r dx 2 ˝ ˝ Z    S.#; ru/u  r C .; #/r#  r dx  ˝ Z Z L.#/.#  0 / dS   juj2 dS 8 2 C 1 .˝/:  1

2

@˝

@˝

(30)

Definition 3 (variational entropy solution for the Dirichlet b.c.). The triple .%; u; #/ is called a variational entropy solution to systems (1), (2), (3), (4), (6), (7), (13), (14), (15), (16), (17), (18), and (19), if % 2 L .˝/, R(8), (9), (10), (11), (12), 1;2 3 1;r .˝/ \ L3m .˝/ \ LlC1 .@˝/, r>1, with ˝ % dx D M , u 2 W0 .˝I R /, # 2 W 6 2 L1 .@˝/, %u 2 L 5 .˝I R3 /, %# 2 L1 .˝/, # 1 S.#; ru/u 2 L1 .˝/, L.#/; L.#/ # 2

.#/ jr#j 2 L1 .˝/, and .#/ r# 2 L1 .˝I R3 /. Moreover, equalities (26) and (27) # #2 are satisfied in the same sense as in Definition 1, and we have the entropy inequality Z  Z jr#j2  L.#/ S.#; ru/ W ru C .#/ 2 0 dS dx C # # # ˝Z @˝ Z   r#  r  %s.%; #/u  r dx L.#/ dS C  .#/ # @˝ ˝ for all non-negative

(31)

2 C 1 .˝/, together with the global total energy balance Z

Z L.#/.#  0 / dS D @˝

%f  u dx:

(32)

˝

Similarly as above we have Definition 4 (variational entropy solution for the Navier b.c.). The triple .%; u; #/ is called a variational entropy solution to systems (1), (2), (3), (5), (6), (7), (13), (14), (15), (16), (17), (18), and (19), if % 2 L .˝/, R(8), (9), (10), (11), (12), 1;2 3 1;r .˝/ \ L3m .˝/ \ LlC1 .@˝/, r>1, with ˝ % dx D M , u 2 Wn .˝I R /, # 2 W 6 2 L1 .@˝/, %u 2 L 5 .˝I R3 /, %# 2 L1 .˝/, # 1 S.#; ru/u 2 L1 .˝/, L.#/; L.#/ # 2

2 L1 .˝/, and .#/ r# 2 L1 .˝I R3 /. Moreover, equalities (26) and (29) .#/ jr#j # #2 are satisfied in the same sense as in Definition 2, we have the entropy inequality (28) in the same sense as in Definition 3, together with the global total energy balance Z

juj2 dS C

 @˝

Z

Z L.#/.#  0 / dS D @˝

%f  u dx: ˝

(33)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2605

Remark 1. As mentioned above, any solution in the sense of Definitions 3 or 4 which is sufficiently smooth is actually a classical solution to the corresponding problem. It can be shown exactly as in the case of the evolutionary system, and we refer to [10, Chapter 2] for more details. Indeed, the same holds also for the weak solutions, i.e., for Definitions 1 and 2, where the proof is straightforward. We will also need the notion of the renormalized solution to the continuity equation Definition 5 (renormalized solution to the continuity equation). Let u 2 6

1;2 5 Wloc .R3 I R3 / and % 2 Lloc .R3 / solve

div .%u/ D 0 in D0 .R3 /: Then the pair .%; u/ is called a renormalized solution to the continuity equation, if   div .b.%/u/ C %b 0 .%/  b.%/ div u D 0 in D0 .R3 /

(34)

for all b 2 C 1 .Œ0; 1// \ W 1;1 .0; 1/ with zb 0 .z/ 2 L1 .0; 1/. The main results read Theorem 1 (Dirichlet boundary conditions; Novotný, Pokorný, 2011). Let ˝ 2 3 C 2 be a bounded domain in R3 , f 2 ˚ L1 .˝I R  /, 0  K0 >0 a.e. at @˝, 2 2 1 0 2 L .@˝/. Let  >1, m > max 3 ; 3.1/ , l D 0. Then there exists a variational entropy solution to (1), (2), (3), (4), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19) in the sense of Definition 3. Moreover, %  0, #>0 a.e. in ˝ and .%; u/ is a renormalized solution to the continuity equation. 2 In addition, if m > maxf1; 3.34/ g and  > 43 , then the solution is a weak solution in the sense of Definition 1. Theorem 2 (Navier boundary conditions; Jesslé, Novotný, Pokorný, 2014). Let 3 1 3 ˝ 2 C 2 be a bounded domain in  .˝I R /, 0  K0 >0 a.e. at @˝, 0 2 ˚ 2R , f2 2 L 1 L .@˝/. Let  >1, m > max 3 ; 3.1/ , l D 0. Then there exists a variational entropy solution to (1), (2), (3), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19) in the sense of Definition 4. Moreover, %  0, #>0 a.e. in ˝ and .%; u/ is a renormalized solution to the continuity equation. In addition, if m>1 and  > 54 , then the solution is a weak solution in the sense of Definition 2. Remark 2. The same holds for the problems (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), and (20) (i.e., with either the Dirichlet or the Navier boundary condition) with the specific entropy defined by the Gibbs relation (15).

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Remark 3. If ˝ is an axially symmetric domain and >0 in (5), then the variational entropy solutions to problems (1), (2), (3), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19) exist under the assumptions of Theorem 2. 6 However, for the existence of weak solutions, we need additionally m > 1516 for  2 . 54 ; 43 and m >

186 97

for  2 . 43 ; 53 /. More details can be found in [15].

A Priori Estimates for  >

4

3 2

In this section we present a priori estimates for our problem with both the homogeneous Dirichlet and the Navier boundary conditions for the velocity. Let us emphasize that these estimates will not be optimal in most of the cases. They just illustrate that for some values of  and m, one may get estimates which indicate that the weak solution is available; however, in some situations the only hope is the variational entropy solution. The subsequent computations also indicate how one may obtain estimates for the approximate problems. Moreover, we assume that l D 0, more precisely L D const. For simplicity we take in the case of the Navier boundary conditions  D 0 and assume that ˝ is not axially symmetric. We start with the entropy inequality (31) (note that for sufficiently smooth solutions, it can be deduced from the total energy balance; in the case of the existence proof, a certain version is available for the approximation), where we use as test function D 1. We have Z 

.#/

˝

Z Z  1 L0 jr#j2 C L dS: S.#; ru/ W ru dx C dS  #2 # @˝ # @˝ D 1 and get

Next we test the total energy balance by Z

Z

Z

L# dS D @˝

(35)

%u  f dx C ˝

L0 dS:

(36)

@˝

Using Korn’s inequality (see Lemma 1 in the next section), we have from (35) kuk21;2 C kr.# m=2 /k22 C k ln #k21;2  C;

(37)

while (35) and (36) together with the Sobolev embedding theorem yield k#k3m  C .1 C kuk6 k%k 6 kfk1 /  C .1 C k%k 6 /: 5

5

(38)

It remains to estimate the density. In order to simplify the situation as much as possible at this moment, we use the estimates based on the application of the Bogovskii operator (see Lemma 2 below). To this aim we apply as test function in (27) or (29) a solution to

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

Z 1 %˛ dx j˝j ˝ ' D 0 on @˝:

div ' D %˛ 

2607

in ˝

We have Z Z Z ˛ p.%; #/% dx D  %.u ˝ u/ W r' dx C S.%; ru/ W r' dx ˝ ˝ ˝ Z Z Z 4 X 1  %f  ' dx C p.%; #/ dx %˛ dx D Ii : j˝j ˝ ˝ ˝ i D1

(39)

Recalling that the density is bounded in L1 .˝/ (the prescribed total mass) and using Lemma 2 below, it is not difficult to check that the most restrictive terms are I1 and I2 leading to bounds (the details can be found in [27]) n 3m  2 o ˛  min 2  3;  ; 3m C 2

>

3 2 ;m> : 2 3

(40)

Hence, under assumption (40), we have kuk1;2 C kr.# m=2 /k2 C k ln #k1;2 C k#k3m C k%kC˛  C:

(41)

Therefore we see that we have all quantities in the weak formulation integrable (i.e., in particular, the density is bounded in L2C" .˝/, and the term %juj3 is integrable in L1C" .˝/) if >

5 ; 3

m  1;

(42)

while all terms in the variational entropy formulation are integrable if >

3 ; 2

m>

2 : 3

(43)

Thus, under these assumptions, we may try to construct a solution to our problems. As we shall see later (cf. [27]), the limit passage requires one more condition, namely, m>

2 ; 3.  1/

(44)

which comes into play for small  ’s. Under assumptions (43) and (44), we may prove existence of variational entropy solutions, while under assumptions (42) and (44), we could prove existence of weak solutions, see [27]. In what follows, using finer density estimates, we weaken the assumptions on  and m, i.e., we prove Theorems 1 and 2.

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Mathematical Tools

In this section we present several well-known results needed later in the proof of the existence of weak and variational entropy solutions. We first have Lemma 1 (Korn’s inequality). Let #>0 and S.#; ru/ satisfy (8) and (9) with ˛ D 1. (i) Let u 2 W01;2 .˝I R3 /. Then Z

S.#; ru/ W ru dx  C kuk21;2 ; # Z˝ S.#; ru/ W ru dx  C kuk21;2 :

(45)

˝

(ii) Let ˝ 2 C 0;1 and u 2 Wn1;2 .˝I R3 /. Then Z

Z S.#; ru/ W ru dx C juj2 dS  C kuk21;2 ; # ˝ @˝ Z Z S.#; ru/ W ru dx C juj2 dS  C kuk21;2 : ˝

(46)

@˝

If ˝ is in addition not axially symmetric, then also (45) holds. Proof. The proof of the first statement is nothing but integration by parts, see, e.g., [27]. The proof of the second statement can be found, e.g., in [14] or [15]. Further we need special solutions to the following problem: div ' D f in ˝; ' D 0 on @˝:

(47)

We have (see, e.g., [30]) R Lemma 2 (Bogovskii operator). Let f 2 Lp .˝/, 1 < p < 1, ˝ f dx D 0, ˝ 2 C 0;1 . Then there exists a solution to (47) and a constant C >0 independent of f such that k'k1;p  C kf kp :

(48)

Moreover, the solution operator B: f 7! ' is linear. Looks from new perspectives at solutions to system (47) can be found in [3] or in [31].

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2609

Next we recall several technical results needed in the part dealing with the strong convergence for the density. We denote for v a scalar function i h  i j F.v/./ ; jj2

(49)

h  i i j F.u /./ ; j jj2

(50)

.RŒv /ij D ..r ˝ r/1 /ij v D F 1 and for u a vector-valued function .RŒu /i D ..r ˝ r/1 /ij uj D F 1

with F./ the Fourier transform. We have (see [10, Theorems 10.27, 10.28, and 10.19]) Lemma 3 (Commutators I). Let Uı * U in Lp .R3 I R3 /, vı * v in Lq .R3 /, where 1 1 1 C D < 1: p q s Then vı RŒUı  RŒvı Uı * vRŒU  RŒv U in Ls .R3 I R3 /. Lemma 4 (Commutators II). Let w 2 W 1;r .R3 /, z 2 Lp .R3 I R3 /, 1 < r < 3, 1 < p < 1, 1r C p1  13 < 1s < 1. Then for all such s, we have kRŒwz  wRŒz ka;s;R3  C kwk1;r;R3 kzkp;R3 ; where a3 D 1s C 13  p1  1r . Here, kka;s;R3 denotes the norm in the Sobolev-Slobodetsky space W a;s .R3 /. Lemma 5 (Weak convergence and monotone operators). Let the couple of nondecreasing functions .P; G/ be in C .R/  C .R/. Assume that %n 2 L1 .˝/ is a sequence such that 9 P .%n / * P .%/; = in L1 .˝/: G.%n / * G.%/; ; P .%n /G.%n / * P .%/G.%/ Then P .%/ G.%/  P .%/G.%/ a.e. in ˝.

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We also have (see, e.g., [15, Lemma 2.8]) Lemma 6. Let ˝ be bounded, fn * f in L1 .˝/, gn ! g in L1 .˝/, and fn gn * h in L1 .˝/. Then h D fg. We need the following version of the Schauder fixed-point theorem (for the proof see, e.g., [7, Theorem 9.2.4]). Lemma 7. Let T W X ! X be a continuous, compact mapping and X be a Banach space. Let for any t 2 Œ0; 1 the fixed points t T u D u be bounded. Then T possesses at least one fixed point in X .

6

Approximation

6.1

Approximate System Level 4

Let us now introduce the approximating procedure. For simplicity we consider the Dirichlet boundary conditions. The proof for the Navier boundary conditions is basically the same. We also set immediately l D 0 and assume L to be constant. Recall that we have ˛ D 1 in (9). The approximation for l ¤ 0 and ˛ < 1 can be done similarly. We fix N a positive integer and ", ı and >0 (we pass subsequently N ! 1,  ! 0C , " ! 0C and finally ı ! 0C , thus the assumption " sufficiently small with respect to ı does not cause any problems) and denote by XN D span fw1 ; : : : ; wN g  W01;2 .˝I R3 / 1;2 3 with fwi g1 iD1 an orthonormal basis in W0 .˝I R /. Due to the smoothness of i 2;q ˝, we may additionally assume that w 2 W .˝I R3 / for all 1  q < 1 (we may take, e.g., the eigenfunctions of the Laplace operator with the homogeneous Dirichlet boundary conditions). Similarly, if we may proceed for the slip boundary conditions, we only replace the eigenfunction to the Laplace operator by, e.g., the eigenfunctions to the Lamé system with the Navier boundary conditions. We look for a triple .%N;;";ı ; uN;;";ı ; #N;;";ı / (denoted briefly .%; u; #/) such that % 2 W 2;q .˝/, u 2 XN , and # 2 W 2;q .˝/, 1  q < 1 arbitrary, where

Z   1 1 %.u  ru/  wi  %.u ˝ u/ W rwi C S .#; u/ W rwi dx 2 ˝ 2Z Z   ˇ 2 i p.%; #/ C ı.% C % / div w dx D %f  wi dx  ˝

˝

(51)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2611

for all i D 1; 2; : : : ; N , "%  "% C div .%u/ D "h

a.e. in ˝;

(52)

and     "C# r# C div %e.%; #/u div . .#/ C ı# B C ı# 1 / # D S .#; u/ W ru C ı# 1  p.%; #/div u C ı"jr%j2 .ˇ%ˇ2 C 2/ a.e. in ˝; (53) with ˇ  maxf8; 3 ; 3mC2 g, B  2m C 2, B  6ˇ  8, 3m2 S .#; u/ D

i  .#/ h  .#/ 2 ru C .ru/T  div uI C div uI: 1 C # 3 1 C #

M ,  ,  , and  are suitable regularizations of , In the above formulas, h D j˝j , and , respectively, that conserve (9) and (11) and that converge uniformly on compact subsets of Œ0; 1/ to , , and , respectively. We consider systems (51), (52), and (53) together with the following boundary conditions on @˝

@% D 0; (54) @n  " C # @#    C L C ı# B1 /.#  0 / C " ln # D 0;  .#/ C ı# B C ı# 1 # @n (55) 



with 0 a smooth approximation of 0 such that 0 is strictly positive at @˝. The no-slip boundary condition for the approximate velocity is included in the choice of XN . We have Proposition 1. Let ", ı, , and N be as above, ˇ  maxf8; 2 g and B  2m C 2. Let " be sufficiently small with respect to ı. Under the assumptions of Theorem 1 and the assumptions made above in this section, there exists a solution to systems (51), R (52), (53), (54), and (55) such that % 2 W 2;q .˝/ 8q < 1, %  0 in ˝, ˝ % dx D M , u 2 XN , and # 2 W 2;q .˝/ 8q < 1, #  C .N />0. The detailed proof of the proposition is in [27]. Let us only recall the main steps here. We consider a mapping T W XN  W 2;q .˝/ ! XN  W 2;q .˝/ with T .v; z/ D .u; r/;

2612

P.B. Mucha et al.

where Z

Z  1 1 %.v ˝ v/ W rwi  %.v  rv/  wi S .e ; u/ W rw dx D 2 2 ˝ ˝    z ˇ C p.%; e / C ı.% C %2 / div wi C %f  wi dx z

i

(56)

8i D 1; 2; : : : ; N ,     div . .ez / C ıezB C ıez /." C ez /rr D div %e.%; ez /v C S .ez ; v/ W rv Cıez  p.%; ez /div v C ı"jr%j2 .ˇ%ˇ2 C 2/

a.e. in ˝; (57)

with %, a unique solution to (see Lemma 8 below) "%  "% C div .%v/ D "h @% D0 at @˝; @n

in ˝; (58)

together with the boundary conditions on @˝ 

   @r C L C ıe.B1/z .ez  0n / C "r D 0:  .ez / C ıezB C ıez ." C ez / @n

(59)

Note that the fixed point of T (provided it exists) corresponds to r D ln # in (51), (52), (53), (54), and (55). We now apply Lemma 7. For fixed v 2 XN , we can find a unique solution to the approximate continuity equation (58). More precisely, we have (see, e.g., [30, Proposition 4.29]) M Lemma 8. Let ">0, h D j˝j . Let v 2 XN . Then there exists a unique solution R 2;p to (58) such that % 2 W .˝/ for all p < 1, ˝ % dx D M , and %  0 in ˝. Moreover, the mapping S W v 7! is continuous and compact from XN to W 2;p .˝/.

Both (56) and (57) with (59) are linear elliptic problems. Hence their unique solvability as well as regularity of the solution is straightforward. Using also Lemma 8, we get (see [23, Lemma 3] for a similar result) the following lemma. Lemma 9. Under the assumptions of Proposition 1, for p>3, the operator T is a continuous and compact operator from XN  W 2;p .˝/ into itself. To fulfill the assumptions of Lemma 7, we need to verify boundedness of possible fixed points to t T .u; r/ D .u; r/, t 2 Œ0; 1 . As this is the most difficult part of the proof of Proposition 1, we give more details here. The full proof can be found in [27].

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2613

Lemma 10. Let the assumptions of Proposition 1 be satisfied. Let p>3. Then there exists C >0 such that all solutions to t T .u; r/ D .u; r/

(60)

fulfill kuk2;p C krk2;p C k#k2;p  C; where # D er and C are independent of t 2 Œ0; 1 . Idea of the proof: (i) Testing (51) (in the form of (60)) by u (i.e., linear combinations of wi ) yields Z

S .#; u/ W ru dx D t

˝

Z  

  p.%; #/Cı.%ˇ C%2 / div uC%fu dx:

(61)

˝

(ii) Integrating (53) (in the form of (60)) over ˝, together with (52), (55), and (61) implies Z

Z     B1 t L C ı# .#  0 / C " ln # dS C .1  t / S .#; u/ W ru dx @˝ ˝ Z   ˇ %ˇ C 2%2 dx C"ıt ˇ  1 ˝ Z   ˇ Dt h%ˇ1 C 2"ıh% C ı# 1 dx: %f  u C "ı ˇ1 ˝ (62)

(iii) Integration over ˝ of the entropy version of the approximate energy balance (53) (i.e., (53) divided by #, again in the form of (60)), after slightly technical computations, yields Z



 " C # jr#j2 dx # #2 ˝ Z  Z     1  1 Ct S .#; u/ W ruCı# 2 dx C t L C ı# B1 0  " ln # dS ˝ # @˝ #  .#/ C ı# B C ı# 1

Z

  1 jr%j2 ˇ%ˇ2 C 2 dx  t ˝ # Z " ˇ %ˇ dx C C t ": Ct 2ˇ1 ˝

Z

C t "ı



 L C ı# B1 dS

@˝

(63)

2614

P.B. Mucha et al.

(iv) Combing identities from steps (ii)–(iii), we get Z



 " C # jr#j2 dx # #2 ˝ Z Z   1 S .#; u/ W ru C ı# 2 dx C .1  t / Ct S .#; u/ W ru dx ˝ # ˝ Z  Z  1 1 ˇ ˇ 2 % C 2% dx C t "ı jr%j2 .ˇ%ˇ2 C 2/ dx C "ıt 2 ˇ  1 # ˝ ˝ Z   ˇ ˇZ    0  ˇ ˇ B %f  u dx ˇ : C L dS  C t 1 C ˇ t L# C ı# C "j ln #j C t # @˝ ˝ (64)  .#/ C ı# B C ı# 1

(v) Estimates (61) and (64) lead to kuk1;2 C k#k3B C kr#k2 C k%kˇ  C

(65)

with C independent of t (and also of  and N ). (vi) Properties of XN and standard regularity results for elliptic equations imply kuk2;p C k%k2;p  C .N /:

(66)

(vii) Standard tools as Kirchhoff transform and regularity results for elliptic problems finally yield krk2;p C k#k2;q  C .N /

(67)

which finishes the proof of Lemma 10 as well as of Proposition 1.

Limit Passage N ! 1 Toward Approximate System Level 3

6.2

From the proof of Proposition 1 above, we deduce the following uniform estimates: kuN k1;2 C k%N kˇ C k#N k3B C k#N k1;2 C k#N2 k1 C k#N1 k1;@˝ Ck#N4 jr#N j2 k1 C k%N k2;2  C ."; ı/:

(68)

Thus, extracting suitable subsequences if necessary, we get a triple .%; u; #/ being a limit of .%Nk ; uNk ; #Nk / in spaces given by estimates (68) and solving Z   1 1 %.u  ru/  '  %.u ˝ u/ W r' C S .#; u/ W r' dx 2 ˝ 2 Z Z   p.%; #/ C ı%ˇ C ı%2 div ' dx D %f  ' dx 8' 2 W01;2 .˝I R3 /;  ˝

˝

"%  "% C div .%u/ D "h

a.e. in ˝;

(69) (70)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2615

with @% D0 @n

a.e. at @˝;

(71)

and Z  

 " C #  .#/ C ı# B C ı# 1 r#  r  %e.%; #/u  r dx # ˝ Z      dS L C ı# B1 .#  0 / C " ln # C @˝ Z  ˇ  dx; D S .#; u/ W ru C ı# 1  p.%; #/div u C "ı ˇr%j2 .ˇ%ˇ2 C 2 ˝

(72)

for all 2 C 1 .˝/. Note that in order to get (72), we need S .#N ; uN / W ruN ! S .#; u/ W ru in L1 .˝/. This fact does not follow from (68), but we may show it realizing that we can use as a test function in (69) the limit function u, together with the limit passage in (46) with wi replaced by uN . Last but not least, we can also get the entropy inequality Z 

 " C # jr#j2   # 1 S .#; u/ W ru C ı# 2 C  .#/ C ı# B C ı# 1 dx # #2  Z    " C # r#  r  %s.%; #/u  r dx  .#/ C ı# B C ı# 1  # # ˝ Z   B1 L C ı#  C .#  0 / C " ln # dS C F" ; # @˝ (73)

˝

for all

2 C 1 .˝/, non-negative, with F" ! 0 as " ! 0C .

6.3

Limit Passage  ! 0C Toward Approximate System Level 2

We use again (68) to pass to the limit in the approximate continuity equation, momentum equation, and entropy inequality (switching to subsequences if necessary) Z   1 1 %.u  ru/  '  %.u ˝ u/ W r' C S.#; u/ W r' dx 2 Z  ˝ 2    1; 6B ˇ 2 8' 2 W0 3B2 .˝I R3 /; C p.%; #/ C ı.% C % / div ' C %f  ' dx ˝

Z

Z

"

.% ˝

(74)

Z

C r%  r / dx 

%u  r ˝

dx D "h

dx ˝

8

2W

1; 65

.˝/; (75)

2616

P.B. Mucha et al.

Z 

 " C # jr#j2   # 1 S.#; u/ W ru C ı# 2 C .#/ C ı# B C ı# 1 dx # #2  ˝ Z   " C # r#  r   %s.%; #/u  r dx  .#/ C ı# B C ı# 1 # # ˝ Z   B1 L C ı# C dS C F" ; .#  0 / C " ln # # @˝ (76)

for all 2 C 1 .˝/, non-negative, with F" as above. The main difficulty in this step appears in the limit passage in the energy balance. We are not anymore able to guarantee the strong convergence of ru in L2 .˝I R33 /, and thus we are not able to recover in the limit the balance of the internal energy. However, we may consider instead of it the balance of the total energy which we get summing the approximate balance of the internal energy (72) and the approximate momentum equation (69) tested by u (i.e., the balance of the kinetic R energy). Doing so we may now pass C – as the most difficult term dx is replaced by with  ! 0 ˝ S .u ; # / W ru R S .u ; # /u  r dx – and here the information from (68) is sufficient. We get     ˝ Z   ˝

 " C #   1 r#  r dx  %juj2  %e.%; #/ u  r C .#/ C ı# B C ı# 1 2Z  Z #    L C ı# B1 .#  0 / C " ln # dS D %f  u dx C ˝ Z @˝     S.#; u/u C p.%; #/u C ı.%ˇ C %2 /u  r C ı# 1 dx C ˝ Z   1 Cı "ˇh%ˇ1 C %ˇ u  r  "ˇ%ˇ dx ˝ ˇ1 Z   dx 8 2 C 1 .˝/: Cı 2"h% C %2 u  r  2"%2 ˝

(77)

Note that due to bounds (68), the temperature is positive a.e. in ˝ and a.e. on @˝.

6.4

Limit Passage " ! 0C Toward Approximate System Level 1

From the entropy inequality (76) and the total energy balance (77), together with a version of Korn’s inequality (see Lemma 1), we can deduce the following estimates independent of ": 1

ku" k21;2 C k#" kB3B C k#" k21;2 C kr.#" 2 /k22 C k#"2 k1 C k#" kBB;@˝ C k#"1 k1;@˝  C .1 C k%" k26 /;

(78)

5

with C D C .ı/, but independent of ". The estimates above do not contain any bound on the density. To deduce it, we apply the Bogovskii-type estimates; meaning we employ as test function in (74) a vector field ˚, a solution to

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

1 j˝j ˚D0

div ˚ D %.s1/ˇ  "

Z ˝

%.s1/ˇ dx "

in ˝;

2617

(79)

on @˝

with s

sˇ

k˚k1;s1s  C k%" ksˇ ; s1

1 < s < 1;

see Lemma 2. After straightforward calculations, we get k%" k 5 ˇ  C

(80)

3

and thus, using also the approximate continuity equation with the test function %" , we obtain

D

1

ku" k1;2 C k#" k3B C k#" k1;2 C k#" 2 k1;2 C k ln #" k1;2 C k#"1 k1;@˝ p C k%" k 5 ˇ C "kr%" k2  C 3

(81)

with C independent of ". Note that we still miss an information providing the compactness of the sequence approximating the density. Passing to the limit " ! 0C , we get (switching to subsequences, if necessary) Z

30ˇ

%u  r

dx D 0

8

2 W 1; 25ˇ18 .˝/;

(82)

˝

Z 

Z    %.u˝u/ W r'CS.#; u/ W r' p.%; #/ C ı%ˇ C ı%2 div ' dx D %f' dx

˝

˝

(83)

1; 52

for all ' 2 W0 .˝I R3 /. Here and in the sequel, g.%; u; #/ denotes the weak limit of a sequence g.%" ; u" ; #" /. Further, we obtain Z  

    1  %juj2  %e.%; #/ u  r C .#/ C ı# B C ı# 1 r#  r dx 2 Z ˝ Z   %f  u dx L C ı# B1 .#  0 / dS D C Z  @˝ ˝      C  S.#; u/u C p.%; #/ C ı%ˇ C ı%2 u  r C ı# 1 dx ˝ Z   1 %ˇ C %2 u  r dx Cı 8 2 C 1 .˝/; ˇ  1 ˝ (84)

2618

P.B. Mucha et al.

and the entropy inequality Z 

 jr#j2   # 1 S.#; u/ W ru C ı# 2 C .#/ C ı# B C ı# 1 dx #2  ˝ Z    r#  r  %s.%; #/u  r .#/ C ı# B C ı# 1 dx  # ˝ Z B1 L C ı# .#  0 / dS C # @˝

(85)

for all non-negative 2 C 1 .˝/. In order to show the strong convergence of density (which is sufficient to remove the bars in (83), (84), and (85)), we combine technique introduced in [19] with some of techniques from [10, Chapter 3]. Using, roughly speaking, as test function in (74) ' WD r1 %" and in (83) ' WD r1 %, passing to the limit " ! 0C , together with several deep results from the harmonic analysis, we end up with  4  .#/ C .#/ %div u p.%; #/ C ı%ˇ C ı%2 %  3     4 .#/ C .#/ %div u D p.%; #/ C ı%ˇ C ı%2 %  3 

(86)

a.e. in ˝. This fact, together with the theory of renormalized solutions to continuity equation and standard properties of weakly convergent sequences, leads to 

   p.%; #/ C ı%ˇ C ı%2 % D p.%; #/ C ı%ˇ C ı%2 %

a.e. in ˝;

(87)

in particular %ˇC1 D %ˇ %; which implies the strong convergence of the density. The reasoning above is somewhat similar (and simpler) than that one needed for the passage ı ! 0C . The latter is described in more details below. As a conclusion, (82), (83), (84), and (85) can be rewritten as Z %u  r

dx D 0

(88)

˝ 30ˇ

for all Z 

2 W 1; 25ˇ18 .˝/, 

ˇ

2





Z

%.u˝u/ W r'CS.#; u/ W r' p.%; #/Cı% Cı% div ' dx D

˝

%f' dx ˝

(89)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2619

1; 5

for all ' 2 W0 2 .˝I R3 /, Z  

    1  %juj2  %e.%; #/ u  r C .#/ C ı# B C ı# 1 r#  r dx 2 Z ˝ Z   L C ı# B1 .#  0 / dS D %f  u dx C @˝ ˝ Z       dx C  S.#; u/u C p.%; #/ C ı%ˇ C ı%2 u  r C ı# 1 ˝ Z   1 Cı %ˇ C %2 u  r dx ˇ  1 ˝ (90)

for all

2 C 1 .˝/, and

Z 

  jr#j2  # 1 S.#; u/ W ru C ı# 2 C .#/ C ı# B C ı# 1 dx #2  ˝ Z    r# W r  %s.%; #/u  r dx .#/ C ı# B C ı# 1  # ˝ Z L C ı# B1 C .#  0 / dS # @˝

(91)

for all non-negative 2 C 1 .˝/. More details concerning all estimates and limit passages performed above are contained in [27].

7

Estimates Independent of ı: Dirichlet Boundary Conditions

We now present basic estimates independent of ı for the solutions to systems (88), (89), (90), and (91). The first part (up to few details) is the same as in the section devoted to the a priori estimates; however, the estimates of the density are different.

7.1

Estimates Based on Entropy Inequality

We first aim at showing the following estimates with constants independent of ı:

k#ı k3m

kuı k1;2  C; ˇZ ˇ  ˇ ˇ C 1Cˇ %ı uı  f dx ˇ :

(92) (93)

˝

We proceed as in the case of the formal a priori estimates. We use as test functions in the approximate entropy inequality (91) and in the total energy balance (90) 1, which leads to

2620

P.B. Mucha et al.

Z

Z    jr#ı j2 1 2 dx .#ı / C ı#ıB C ı#ı1 dx C S.# ; u / W ru C ı# ı ı ı ı #ı2 ˝ #ı Z Z   L C ı#ıB1 C L C ı#ıB1 dS; 0 dS  # ı @˝ @˝ (94)



˝

and Z



L#ı C

@˝

ı#ıB



Z

Z %ı uı  f dx C

dS D ˝



LC

@˝

ı#ıB1

Z



0 dS C ı ˝

#ı1 dx: (95)

We can get rid of the ı-dependent terms on the right-hand sides (r.h.s.) (more details are given in [27]) and hence deduce for ˇ and B sufficiently large m

kuı k1;2 C kr#ı2 k2 C kr ln #ı k2 C k#ı1 k1;@˝ B   1 2 Cı kr#ı2 k22 C kr#ı 2 k22 C k#ı kB2 3B C k#ı k1  C:

(96)

Estimate (96) with (95) leads to (92) and (93).

7.2

Estimates of the Pressure

As shown in the formal a priori estimates, the method based on the pressure estimates by means of the Bogovskii operator has a natural limitation, namely,  > 32 . To avoid this, we apply a different idea based on local pressure estimates. The method was developed in the context of the compressible Navier-Stokes system in the following three papers [2, 33], and [12]. We follow mostly the approach from the last one; some details can be found in chapter  “Existence of Stationary Weak Solutions for Isentropic and Isothermal Compressible Flows”. Note that unlike the heat-conducting case, the method gives existence of weak solutions to the compressible Navier-Stokes system only for  > 43 . This problem has been removed in [16] for the spatially periodic case and in the recent paper [34] for the Dirichlet boundary conditions, using a slightly different technique, which, however, leads in the heat-conducting case to more severe restrictions than the original method; see chapter  “Existence of Stationary Weak Solutions for Isentropic and Isothermal Compressible Flows”. We denote for b  1 Z AD %bı juı j2 dx: (97) ˝

Using Hölder’s inequality, one may easily deduce for any b  1 kuı k1;2  C;  1  k#ı k3m  C A 6b4 C 1 :

(98)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2621

We also need the following estimate based on the application of the Bogovskii operator from Lemma 2 3b , bC2

Lemma 11. We have for 1 < s  Z ˝

Z

s %ı

dx C

Z

C

˝



%ı juı j2

.s1/

%ı s

s

6m , 2C3m

m > 23 , and b  1

p.%ı ; #ı / dx Z

ˇC.s1/

dx C ı

˝

˝

%ı

 4s3  dx  C 1 C A 3b2 :

(99)

Proof. We use as test function in (89) solution to (47) with 1  j˝j

.s1/

f D%

Z

%.s1/ dx;

˝

we get Z

.s1/

˝

%ı

Z

.s1/  ˇ %ı

p.%ı ; #ı / dx C ı

%ı

˝

C %2ı / dx

Z Z Z Z 1 ı .s1/ ˇ .s1/ 2 D p.%ı ; #ı / dx %ı dx C .%ı C %ı / dx %ı dx j˝j ˝ j˝j ˝ ˝ ˝ Z Z %ı .uı ˝ uı / W r' dx C S.uı ; #ı / W r' dx  Z

˝



˝

%ı f  ' dx D I1 C I2 C I3 C I4 C I5 : ˝

The most restrictive terms are I3 leading to the first restriction on s and I4 which gives the second restriction as well as m > 23 . More details can be found in the paper [28]. We are now coming to the most important (and also the most difficult) part of the estimates. We aim at proving that for some ˛>0, Z sup x0 2˝

˝

p.%ı ; #ı / dx  C jx  x0 j˛

with C independent of ı. The goal is to get ˛ as large as possible under some conditions on m and  which still allow the values for these quantities as small as possible. Indeed, this might be sometimes contradictory. Here, the situation for the Dirichlet and the Navier boundary conditions differs; as in the latter, we have larger class of possible test functions (only the normal component vanishes on @˝).

2622

P.B. Mucha et al.

We distinguish three possible situations. In the first one, the point x0 is “far” from the boundary; in the second one x0 2 @˝ and in the last one, x0 is “close” to @˝, but does not belong to it. The first case is independent of the boundary conditions and we have Lemma 12. Let x0 2 ˝, R0 < 13 dist .x0 ; @˝/. Then Z

ˇ

p.%ı ; #ı / C ı.%ı C %2ı / dx jx  x0 j˛ BR0 .x0 /  ˇ  C 1 C kp.%ı ; #ı /k1 C kuı k1;2 .1 C k#ı k3m / C k%ı juı j2 k1 C ık%ı kˇ ; (100) provided ˛ < min

n 3m  2 2m

o ;1 :

(101)

Proof. We use as test function in (89) 'i .x/ D

.x  x0 /i 2 jx  x0 j˛

with 1 in BR0 .x0 /, R0 as above, 0 outside B2R0 .x0 /, jr j  The important observation is that

C . R0

3˛ 2 C g1 .x/; jx  x0 j˛  ı .x  x0 /i .x  x0 /j  2 ij @i 'j D C g2 .x/  ˛ jx  x0 j˛ jx  x0 j˛C2 div ' D

with g1 , g2 in L1 .˝/. Then Z

ˇ

p.%ı ; #ı / C ı.%ı C %2ı / .3  ˛/ 2 dx ˛ jx  x j 0 ˝ Z  .uı  .x  x0 //2  2 %ı juı j2  ˛%ı C dx ˛ jx  x0 j˛C2 ˝ jx  x0 j Z   .x  x0 /  r 2 ˇ p.%ı ; uı / C ı.%ı C %2ı / dx D jx  x0 j˛ ˝ Z Z  xx  x  x0 0 2 C dx C S.uı ; #ı / W r S.uı ; #ı / W r 2 dx ˛ jx  x j jx  x0 j˛ 0 ˝ ˝ Z Z x  x0 2 x  x0 %ı f  dx  %ı .uı ˝ uı / W r 2 dx: (102)  ˛ jx  x0 j jx  x0 j˛ ˝ ˝

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2623

We easily see that ˇ xx ˇ C 0 ˇ ˇ ; ˇ ˇr ˛ jx  x0 j jx  x0 j˛ hence ˇZ x  x0 2 ˇˇ ˇ S.uı ; #ı / W r dx ˇ  C .1 C k#ı k3m /kruı k2 ˇ jx  x0 j˛ ˝ provided 1 1 1 ˛ D1  > ; q 2 3m 3 for m > 23 . The second integral on the l.h.s. leading to the restriction ˛ < 3m2 2m of (102) is non-negative; it even gives a certain information about %ı juı j2 ; however, we are not able to recover it (in the case of the Dirichlet b.c.) for x0 near or on the boundary. As ˛  1, the other terms on the r.h.s. of (102) are evidently bounded. Next, we consider the situation when x0 2 @˝. In this case we may use the following test function: '.x/ D d .x/rd .x/.d .x/ C jx  x0 ja /˛

(103)

2 with a D 2˛ , x0 2 @˝, and d .x/ is a function which behaves as the distance function near the boundary and is smoothly (C 2 ) extended to the whole ˝. Then due to the properties of distance function, as ˝ 2 C 2 , the function d 2 C 2 .˝/. Moreover, in a certain neighborhood of @˝, rd .x/ D x.x/ , where .x/ 2 @˝ is d .x/ the closest point to x, cf. [42, Exercise 1.15]. Thus there exist c1 , c2 positive such that:

(i) d .x/ 2 C 2 .˝/, d .x/>0 in ˝, d .x/ D 0 at @˝ (ii) jrd .x/j  c1 >0, x 2 ˝ with dist .x; @˝/  c2 (iii) d .x/  c1 >0, x 2 ˝ with dist .x; @˝/  c2 The main properties of ' are (see [28, Lemma 3.5]) 1;q

Lemma 13. The function ', defined by (103), belongs to W0 .˝I R3 / for 1  q < 3˛ . Moreover, ˛

2624

P.B. Mucha et al.

@j 'i .x/ D

d .x/@2ij d .x/ .d .x/ C jx  x0

j a /˛

C

.1  ˛/d .x/ C jx  x0 ja @i d .x/@j d .x/ 2.d .x/ C jx  x0 ja /1C˛

C

.1  ˛/d .x/ C jx  x0 ja .@i d .x/  i .x//.@j d .x/  j .x// 2.d .x/ C jx  x0 ja /1C˛

C

˛d .x/Œ@j d .x/@i .jx  x0 ja /  @i d .x/@j .jx  x0 ja /

2.d .x/ C jx  x0 ja /1C˛



˛ 2 d 2 .x/@i .jx  x0 ja /@j .jx  x0 ja /  ; 2.d .x/ C jx  x0 ja /1C˛ .1  ˛/d .x/ C jx  x0 ja

(104)

where 1  i .x/ D ˛d .x/ .1  ˛/d .x/ C jx  x0 ja @i .jx  x0 ja /; i D 1; 2; 3. We now use ' from (103) as a test function in (89). It yields Lemma 14. Under assumptions above, we have for ˛ < R0 sufficiently small (but uniformly with respect to x0 ) Z

9m6 , 9m2

x0 2 @˝, and

ˇ

 p.%ı ; #ı / C ı.%ı C %2ı / dx  C 1 C kp.%ı ; #ı /k1 ˛ jx  x j 0 BR0 .x0 /\˝ ˇ C .1 C k#ı k3m /kuı k1;2 C k%ı juı j2 k1 C ık%ı kˇ :

(105)

Proof. We have Z ˝

Z  ˇ p.%ı ; #ı / C ı.%ı C %2ı / div ' dx C %ı .uı ˝ uı / W r' dx Z ˝ Z S.uı ; #ı / W r' dx  %ı f  ' dx: D



˝

From (104) we see that (recall that a D

(106)

˝ 2 ) 2˛

d .x/d .x/ .1  ˛/d .x/ C jx  x0 ja C jrd .x/j2 .d .x/ C jx  x0 ja /˛ 2.d .x/ C jx  x0 ja /˛C1 .1  ˛/d .x/ C jx  x0 ja C jrd .x/  .x/j2 2.d .x/ C jx  x0 ja /˛C1 C1 jrd .x/j2 ˛ 2 d 2 .x/jrjx  x0 ja j2     C2 : 2.d .x/ C jx  x0 ja /˛C1 .1  ˛/d .x/ C jx  x0 ja .d .x/ C jx  x0 ja /˛ div ' D

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2625

Thus, as d .x/ C jx  x0 ja  C jx  x0 j, d .x/ is continuously differentiable near the boundary and d .x0 / D 0, together with a>1 Z

ˇ

˝

.p.%ı ; #ı / C ı.%ı C %2ı //div ' dx  C1 Z

Z

ˇ

BR0 .x0 /\˝

p.%ı ; #ı / C ı.%ı C %2ı / dx jx  x0 j˛



 C2 ˝

 ˇ p.%ı ; #ı / C ı.%ı C %2ı / dx:

Next Z

Z

%ı juı j2 dx;

%ı .uı ˝ uı / W r' dx  C ˝

˝

as the skew symmetric part of r' has zero contribution. Note that the positive part of %ı .uı ˝ uı / W r' does not provide any useful information. The last term on the r.h.s. of (106) can be estimated by kfk1 k%ı k1 k'k1 , and ˇ ˇZ ˇ ˇ S.uı ; #ı / W r' dx ˇ  C kruı k2 .1 C k#ı k3m / ˇ ˝

provided

1 q

D

1 2



1 3m

is such that q
"; '2 .x/D ˛  ˛ ˆ jx  x0 j 2 .jx  x0 j C "/ 2  ˆ  ˆ 2 ˆ 1 1 ˆ : .x  x0 / ; jx  x0 j > "; d .x/  ": ˛  ˛ jx  x0 j 2 .jx  x0 j C d .x// 2 (107) 1;q

It is easy to verify that '2 2 W0 .˝I R3 / with the norm bounded independently of " for all 1  q < ˛3 . Moreover, due to properties mentioned above, we can verify that Z ˝

Z C K2 fxId .x/"g

Z

%ı juı j2 dx ˛ B" .x0 / jx  x0 j Z %ı .uı  rd /2 dx  K %ı juı j2 dx; 3 .d .x/ C jx  x0 ja /˛ ˝

%ı .uı ˝ uı / W r'2 dx  K1

and, Z



˝

 ˇ p.%ı ; #ı / C ı.%ı C %2ı / div '2 dx  K1 Z

 K3 ˝



ˇ

Z B" .x0 /

ˇ

p.%ı ; #ı / C ı.%ı C %2ı / dx jx  x0 j˛



p.%ı ; #ı / C ı.%ı C %2ı / dx:

Hence, taking as a test function in (89) ' D K'1 C '2 for a sufficiently large K>0, we end up with Z sup

ˇ

p.%ı ; #ı / C ı.%ı C %2ı / dx jx  x0 j˛

x0 2˝ ˝   ˇ  C 1 C kp.%ı ; #ı /k1 C kuı k1;2 .1 C k#ı k3m / C k%ı juı j2 k1 C ık%ı kˇ ; (108) . provided ˛ < 9m6 9m2

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2627

We can deduce from (108) , and ˛ > 3b2 . Then Lemma 15. Let 1  b <  , ˛ < 9m6 9m2 b Z  AD %bı juı j2 dx  C kuı k21;2 1 C kp.%ı ; #ı /k1 ˝

C kuı k1;2 .1 C k#ı k3m / C k%ı juı j2 k1

 b

:

(109) 

Proof. Take b <  and D ˛b < 3 (i.e., ˛ > 3b2 ). As %ı  Cp.%ı ; #ı /, b b we have Z  Z   b  1 b %bı %ı 1 dx D dx ˛ jx  x0 j ˝ jx  x0 j ˝ jx  x0 j  b  (110)  C 1 C kp.%ı ; #ı /k1 C kuı k1;2 .1 C k#ı k3m / C k%ı juı j2 k1 : Let h be the unique solution to h D %bı > 0 in ˝; hD0 at @˝:

(111)

Then Z h.x/ D ˝

G.x; y/%bı .y/ dy

C for all x; with G.; / the Green function to problem (111). As jG.x; y/j  jxyj 1 y 2 ˝, x ¤ y, we get from (108), (109), and (110) that h 2 L .˝/ with

Z khk1  C sup x0 2˝

Therefore AD

Z

˝

%bı .x/ dx: jx  x0 j

(112)

Z

2

h.x/juı .x/j dx D 2 ˝

ruı W .uı rh/ dx; ˝

and A  C kruı k2

Z

juı j2 jrhj2 dx

 12

:

(113)

˝

Now Z

juı j2 jrhj2 dx D 

DD ˝

Z ˝

hrh  rjuı j2 dx  1

 C khk1 .A C kruı k2 D 2 /:

Z ˝

juı j2 hh dx

2628

P.B. Mucha et al.

Thus D  C .Akhk1 C kruı k22 khk21 /: Returning to (113), Young’s and Friedrichs’ inequalities imply A  C kruı k22 khk1 which finishes the proof. Now, combining Lemmas 11 and 15 b  .s1/ 2.4s3/   1 4s3 1 1 A  C 1 C A 3b2 s C A 6b4 C A 6b4 . .s1/ C1 C .s1/ C1 / ;

i.e.,  b  1 8s7 4s3 1 A  C 1 C A 3b2 s C A 6b4 .1C .s1/ C1 / :

(114)

Therefore, assuming 4s  3 1 b < 1; s 3b  2 

1  8s  7  b 1C 1 such that Lemma 16. Let  >1, m > 23 , and m > 29 1 s %ı is bounded in L .˝/ and p.%ı ; #ı /, %ı juı j, and %ı juı j2 are bounded in Ls .˝/. Moreover, if  > 43 , and

m>1 m>

for

2 3.3  4/

we can take s > 65 . Proof. The details can be found in [28].

for

12 ; 7  4 12 i 2 ; ; 3 7

>

(116)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2629

Estimates Independent of ı: Navier Boundary Conditions

8

Note that we get in the case when ˝ is not axially symmetric estimates (92), (93), and (96) exactly as for the Dirichlet boundary conditions. Therefore, we only deal here with the estimates of the pressure, where we closely follow the papers [14] and [15].

8.1

Estimates of the Pressure

We define now for 1  a   and 0 < b < 1 BD

Z

 ˝

 %aı juj2 C %bı juj2bC2 dx;

(117)

where 1  a   and 0 < b < 1. Employing Hölder’s inequality, we easily have Lemma 17. Under the assumptions on a and b, there exists C independent of ı, such that ab

k%ı uı k1  C B 2.abCa2b/ :

(118)

1 Lemma 18. Under the assumptions on a and b and for 1 < s < 2a (if a < 2), a 0 < .s  1/ a1 < b < 1, there exists C , independent of ı, such that ab=s

k%ı juı j2 ks  C B abCa2b :

(119)

Next, using also the Bogovskii-type estimate, we get as in Lemma 11 1 a Lemma 19. Let 1  a   , 0 < b < 1, 1 < s < 2a (if a < 2), 0 < .s  1/ a1 < 6m 2 b < 1, s  3mC2 , m > 3 . Then there exists a constant C independent of ı such that

Z

 ˝

s

.s1/

% ı C %ı

ˇC.s1/ 

p.%ı ; #ı / C .%ı juı j2 /s C ı%ı

sab

dx  C .1 C B abCa2b /:

As in the previous case, we need to estimate the pressure. We proceed similarly as for the Dirichlet boundary conditions; however, since we have more freedom, i.e., we have a larger class of the test functions for the momentum equation, we can get better results here. First of all, in the case when x0 is far from the boundary, we have again Lemma 12. However, it is now possible to use the information from the second term on the l.h.s. as it is possible to recover it for x0 close and on the boundary. The

2630

P.B. Mucha et al.

difference appears near the boundary. The situation is more complex here. For the sake of simplicity, we restrict ourselves to the case of the flat boundary @˝. The general case can be treated using the standard change of variables which flattens the curved boundary. This is the reason why we require the boundary to be C 2 . More details are given in [14] or in [15]. Let us hence assume that we deal with the part of boundary of ˝ which is flat and is described by x3 D 0, i.e., z.x 0 / D 0, x 0 2 O  R2 with the normal vector n D .0; 0; 1/ and  1 D .1; 0; 0/,  2 D .0; 1; 0/ the tangent vectors. Consider first that x0 lies on the boundary of ˝, i.e., .x0 /3 D 0. Then it is possible to use as the test function in the approximate momentum equation w.x/ D v.x  x0 /; where v.x/ D

1 .x1 ; x2 ; x3 / D .x 1 / 1 C.x 2 / 2 C..0; 0; x3 z.x 0 //n/n; jxj˛

x3  0:

Note that if .x0 /3 D 0, we get precisely what we need, i.e., estimate (121) (but with supx0 2@˝ instead of supx0 2˝ /. However, if x0 is close to the boundary but not on the boundary, i.e., .x0 /3 >0, small, we loose control of some terms for 0 < x3 < .x0 /3 . In this case, as for the Dirichlet boundary conditions, we must modify the test functions; recall that we require that only the normal component (i.e., in our case the third component) of the test function vanishes on @˝. We first consider v1 .x/ 8   1 ˆ ˆ x3  .x0 /3 =2; < jx  x j˛ .x  x0 /1 ; .x  x0 /2 ; .x  x0 /3 ; 0 D   x32 1 ˆ ˆ : .x  x ; 0 < x3 < .x0 /3 =2: / ; .x  x / ; 4.x  x / 0 1 0 2 0 3 jx  x0 j˛ j.x  x0 /3 j2 Nonetheless, using v1 as test function, we would still miss control of some terms from the convective term, more precisely of those, which contain at least one velocity component u3 , however, only close to the boundary, i.e., for x3 < .x0 /3 =2. Hence we further consider 8 .0; 0; x3 / ˆ ˆ ; jx  x0 j  1=K; < .x3 C jx  x0 jj ln jx  x0 jj1 /˛ v2 .x/ D .0; 0; x3 / ˆ ˆ : ; jx  x0 j > 1=K .x3 C 1=Kj ln Kj1 /˛ for K sufficiently large (but fixed, independently of the distance of x0 from @˝). 1;q Note that both functions belong to Wn .˝I R3 /, and their norms are bounded uniformly (with respect to the distance of x0 from @˝) provided 1  q < ˛3 . Thus we finally use as the test function in the approximate momentum balance

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

' D v1 .x/ C K1 v2 .x/

2631

(120)

with K1 suitably chosen (large). Note that the choice of K and K1 is done in such a way that the unpleasant terms from both functions are controlled by those from the other one which provide us a positive information. This is possible due to the fact that the unpleasant terms from v2 are multiplied by j ln jx  x0 jj1  j ln Kj1 1. Similarly as in the case of Dirichlet boundary conditions, we therefore verify that Z

ˇ

sup ˝

x0 2˝

p.%ı ; #ı / C ı.%ı C %2ı / C .1  ˛/%ı juı j2 dx jx  x0 j˛ ˇ

 C .1 C ık%ı kˇ C kp.%ı ; #ı /k1 C .1 C k#ı k3m /kuı k1;2 C k%ı juı j2 k1 /; (121) g, m > 23 , and, moreover, the test function (120) provided 0 < ˛ < maxf1; 3m2 2m 1;p 3 belongs to W .˝I R / for 1  p < ˛3 with the norm bounded independently of the distance of x0 from @˝. Applying (121) we have to distinguish two cases. First, for m  2 we have 3m2  1; hence the only restriction on ˛ is actually ˛ < 1. In the other case, if 2m . Therefore, if m  2, we get (passing m 2 . 23 ; 2/, we have the restriction ˛ < 3m2 2m with ˛ ! 1 , by Fatou’s lemma) Z ˝

  p.%ı ; #ı / ˇ dx  C 1Cık%ı kˇ Ckp.%ı ; #ı /k1 C.1Ck#ı k3m /kuı k1;2 Ck%ı juı j2 k1 : jx  yj

Next, for 0 < b < 1  % ju j2 b %bı juı j2b 1 ı ı ;  ˛ jx  yj jx  yj jx  yj1b˛ thus Z ˝

 %bı juı j2b dx  jx  yj

Z ˝

b  %ı juı j2 dx ˛ jx  yj

Z

1

˝

jx  yj

1b˛ 1b

dx

1b

:

(122)

Hence we get (note that we may take a D  in (117), see below)  Lemma 20. Let b 2 ..s  1/ 1 ; 1/, 1 < s


3a2 , a

we have

Z

 a %aı ˇ dxC 1Cık%ı kˇ Ckp.%ı ; #ı /k1 C.1Ck#ı k3m /kuı k1;2 Ck%ı juı j2 k1  : ˝ jx  yj (125) Further, proceeding as in (122) Z

 b %bı juı j2b ˇ dxC 1Cık%ı kˇ Ckp.%ı ; #ı /k1 C.1Ck#ı k3m /kuı k1;2 Ck%ı juı j2 k1 ; ˝ jx  yj (126) 2 however, now for 1b˛ < 3, i.e., (if b > ) 1b 3 ˛>

3b  2 : b

(127)

Altogether we have  2 Lemma 21. Let b 2 ..s  1/ 1 ; 1/, 1 < s < 2 , ˛ > maxf 3a2 ; 3b2 g, m 2 a b 2 . 3 ; 2/. Then there exists C independent of ı such that

Z sup x 2˝

˝

%aı C .%ı juı j2 /b dx jx  x0 j

0 a  ˇ  C 1 C ık%ı kˇ C kp.%ı ; #ı /k1 C .1 C k#ı k3m /kuı k1;2 C k%ı juı j2 k1  b  ˇ CC 1 C ık%ı kˇ C kp.%ı ; #ı /k1 C .1 C k#ı k3m /kuı k1;2 C k%ı juı j2 k1 : (128)

We now consider the problem h D %aı C %bı juı j2b 

1 j˝j

Z ˝

@h j@˝ D 0: @n

.%aı C %bı juı j2b / dx;

(129)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2633

The unique strong solution admits the following representation: Z h.x/ D ˝

G.x; y/.%aı C %bı juı j2b / dy 

1 j˝j

Z

Z G.x; y/ dy ˝

˝

.%aı C %bı juı j2b / dxI (130)

1

since G.x; y/  C jxyj , we get due to Lemmas 20, 21 together with Lemmas 17 and 18 • m2  b=s

khk1  C .1 C B b C 2b /;

(131)

provided 10 and K./ as above. Our approximate problem reads 9 > > > > > > > 1 1 > div .K.%/%u ˝ u/ C K.%/%u  ru  div S.ru/ C rP .%; #/ D %K.%/f = 2 2 in ˝; >       Z > "C# > > div .#/ K.t/ dt # C div K.%/%u # r# C div u > > > # 0 > ; CK.%/%u  r#  #K.%/u  r% D S.ru/ W ru (162) where Z % Z Z % 1 P .%; #/ D  t K.t/ dt C # K.t/ dt D Pb .%/ C # K.t/ dt; (163) "% C div .K.%/%u/  "% D "hK.%/

0

0

0

M . and h D j˝j This system is completed by the boundary conditions on @˝

1 @# C L.#/.#  0 / C " ln # D 0; # @n k D 1; 2;  k  .S.ru/n/ C u   k D 0;

.1 C # m /." C #/ u  n D 0;

(164)

@% D 0: @n Recall that the reason for terms of the form ln # to appear in the approximate problem is that we in fact solve the approximate problem for the “entropy” s D ln # instead of the temperature itself. It provides straightaway that the temperature of the approximate problem is positive a.e. in ˝, and we keep this information throughout the limit passages. Moreover, the form of the function K./ ensures that the approximate density is bounded between 0 and k; this can be easily seen if we integrate equation (162)1 over the set, where % < 0, and over the set, where % > k. The main problem in the limit passage " ! 0C is to verify that %  k  1. This ensures that for the limit problem (which is our original problem), we have K.%/ 1. For the approximate problems, we can prove the following:

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2643

Proposition 2. Let the assumptions of Theorem 3 be satisfied. Moreover, let ">0 and k>0. Then there exists a strong solution .%; u/ to (162), (163), and (164) such that % 2 W 2;p .˝/; u 2 W 2;p .˝I R3 / and ln # 2 W 2;p .˝/ for all 1  p < 1: Moreover 0  %  k in ˝, kuk1;3m C

R ˝

% dx  M and

p "kr%k2 C kr#kr C k#k3m  C .k/;

(165)

3m g, and the r.h.s. of (165) is independent of ". where #>0, r D minf2; mC1

The proof is based on suitable linearization and application of a version of the Schauder fixed-point theorem, similarly as above for the temperature-dependent viscosities. However, the a priori estimates are obtained differently, see [23, Theorem 2] for more details. We may therefore pass with " ! 0C . Estimates (165) from Proposition 2 guarantee us existence of a subsequence " ! 0C such that u" * u in W 1;3m .˝I R3 /; %" * % in L1 .˝/; K.%" /%" * K.%/% in L1 .˝/; Z Z %"  K.t/ dt * 0

u" ! u in L1 .˝I R3 /; Pb .%" / * Pb .%/ in L1 .˝/; K.%" / * K.%/ in L1 .˝/; %

K.t/ dt

in L1 .˝/;

0

3m g; #" * # in W 1;r .˝/ with r D minf2; mC1 q #" ! # in L .˝/ for q < 3m:

(166) Passing to the limit in the weak formulation of our problem (162), we get (all equations are fulfilled in the weak sense) div .K.%/%u/ D 0; 

K.%/%u  ru  div 2D.u/ C .div u/I  Pb .%/I  #

(167) Z 

% 0



Z

 div ..#/r#/ C # div u 0

%

 K.t/ dt I D K.%/%f; (168)



K.t/ dt C div .K.%/%#u/ D 2jD.u/j2 C .div u/2

(169) together with the corresponding boundary conditions for the velocity and the temperature. Recall that (167), (168), and (169) are satisfied in the weak sense. In what follows we study the dependence of the a priori bounds on k.

2644

P.B. Mucha et al.

Lemma 24. Under the assumptions of Theorem 3 and Proposition 2, we have  3m2 m

k%" k1  k and ku" k1;3m  C .1 C k 3

/:

(170)

Proof. The estimate for the density follows directly from the properties of the function K./, and the estimate of the velocity follows from the momentum equation, using estimates of the temperature and the L1 -estimate of the density, which are independent of k. More details can be found in [23]. A crucial role in the proof of the strong convergence of the density is played by a quantity called the effective viscous flux. To define it in this context, we use the Helmholtz decomposition of the velocity u D r C rot A;

(171)

where the divergence-less part of the velocity is given as a solution to the following elliptic problem: rot rot A D rot u D ! in ˝; div rot A D 0 in ˝; rot A  n D 0 on @˝:

(172)

The potential part of the velocity is given by the solution to  D div u in ˝; @ D 0 on @˝; @n

Z  dx D 0:

(173)

˝

The classical theory for elliptic equations, see, e.g., the papers [37] and [25], gives us for 1 < q < 1 krrot Akq  C k!kq ; kr 2 rot Akq  C k!k1;q ; kr 2 kq  C kdiv ukq ; kr 3 kq  C kdiv uk1;q : We now use that the velocity satisfies the Navier boundary conditions. We have  !" D rot K.%" /%" f  K.%" /%" u"  ru"  1  1 1  "hK.%" /u" C "%" u"  rot "%" u" WD H1 C H2 in ˝; 2 2 2 !"   1 D .22  =/u"   2 on @˝; !"   2 D .21  =/u"   1 on @˝; div !" D 0 on @˝;

(174)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2645

where k are the curvatures associated with the directions  k . For the proof of relations (174)2;3 see, e.g., [21] or [26]. We may write !" D !0" C !1" C !2" ;

(175)

where !1" D H1 ; !2" D H2 in ˝; !0" D 0;   1 D .22  =/u"   2 ; !1"   1 D 0; !2"   1 D 0 on @˝; 0 !"   2 D .21  =/u"   1 ; !1"   2 D 0; !2"   2 D 0 on @˝; div !0" D 0; div !1" D 0; div !2" D 0 on @˝: (176)

!0"

Using elliptic estimates and Lemma 24, we can prove Lemma 25. For the vorticity !" written in the form (175) we have k!2" kr  C .k/"1=2

for 1  r  2; 4

2

k!0" k1;q C k!1" k1;q  C .1 C k 1C. 3  q / /

(177) for 2  q  3m:

We now introduce the effective viscous flux which is in fact the potential part of the momentum equation. Using the Helmholtz decomposition in the approximate momentum equation, we have r..2 C /" C P .%" ; #" // D rot A" C K.%" /%" f 1 1 1 K.%" /%" u"  ru"  "hK.%" /u" C "%" u"  "%" u" : 2 2 2 We define G" D .2 C /" C P .%" ; #" / D .2 C /div u" C P .%" ; #" /

(178)

and its limit version G D .2 C /div u C P .%; #/:

(179)

R R Note that we are able to control integrals ˝ G" dx D ˝ P .%" ; #" /dx and R %  R R ˝ Gdx D ˝ P .%; #/dx, where P .%; #/ D Pb .%/ C # 0 K.t/ dt . Using the results presented above, we may show (see [23] for more details)

2646

P.B. Mucha et al.

Lemma 26. We have, up to a subsequence " ! 0C , G" ! G strongly in L2 .˝/

(180)

and 2

kGk1  C ./.1 C k 1C 3 C /

for any  > 0:

(181)

The following two results form the core of the method. First, we show that %  .k  3/ a.e. in ˝, i.e., K.%" / ! 1, and next we get that %" ! % strongly in any Lq .˝/. More precisely, Lemma 27. There exists a sufficiently large number k0 >0 such that for k > k0 k3 .k  3/  kGk1  1 k

(182)

and for a subsequence " ! 0C , it holds lim jfx 2 ˝ W %" .x/ > k  3gj D 0:

(183)

"!0C

In particular it follows: K.%/% D % a.e. in ˝. Lemma 28. We have Z Z Z Z P .%; #/% dx  G% dx and P .%; #/% dx D G% dxI ˝

˝

˝

(184)

˝

consequently, P .%; #/% D P .%; #/% and up to a subsequence " ! 0C %" ! % strongly in Lq .˝/ for any q < 1:

(185)

Recall that from Lemma 27 and due to the strong convergence of the temperature, it follows P .%" ; #" / ! p.%; #/ strongly in L2 .˝/; hence (180) implies div u" ! div u

strongly in L2 .˝/:

(186)

Additionally, we have already proved that rot u" ! rot u

strongly in L2 .˝I R3 /;

(187)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2647

since we observed that the vorticity can be written as sum of two parts, one bounded in W 1;q .˝I R3 /, i.e., !0" C!1" , and the other one going strongly to zero in L2 .˝I R3 /, i.e., !2" . Whence S.ru" / W ru" ! S.u/ W ru strongly at least in L1 .˝/

(188)

which finishes the proof of Theorem 3. Remark 4. Similar results as above in the case of the two-dimensional flow can be found in the paper [32]. The existence of weak solutions with similar properties as in Theorem 3 was proved there for  >2 and m D l C 1 > 1 . 2

11

Weak Solutions in Two Space Dimensions

We consider our system of equations (1), (2), and (3) with the boundary conditions (4), (5), and (6) and the total mass (7) in a bounded domain ˝  R2 . We assume the viscous part of the stress tensor in the form (8) .N D 2/ with (9) for ˛ D 1 and the heat flux in the form (10) with (11). Moreover, we take L D const in (6). We assume the pressure law in the form (20), or for  D 1 we take p D p.%; #/ D %# C

%2 ln˛ .1 C %/ %C1

(189)

with ˛>0, the corresponding specific internal energy is e D e.%; #/ D

ln˛C1 .1 C %/ C cv #; ˛C1

cv D const > 0;

(190)

and the specific entropy is s.%; #/ D ln

# cv C s0 : %

(191)

We consider weak solutions to the problem above defined similarly as in Definition 1 with the corresponding modifications for the pressure law (189). This problem was studied in [29] for both (20) and (189), and the result for the latter was improved in [35]. More precisely, we have the following results: Theorem 4 (Novotný, Pokorný, 2011; Pokorný, 2011). Let ˝ 2 C 2 be a bounded domain in R2 , f 2 L1 .˝I R2 /, 0  K0 >0 a.e. on @˝, 0 2 L1 .@˝/, L>0. (i) Let  >1, m > 0. Then there exists a weak solution to our problem with the pressure law (20).

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(ii) Let ˛>1 and ˛  maxf1; m1 g, m>0. Then there exists a weak solution to our problem with the pressure law (189). Moreover, .%; u/, extended by zero outside of ˝, is a renormalized solution to the continuity equation. As the proof for  >1 is easy, we only refer to [29] and consider the pressure law (189). We need to work here with a class of Orlicz spaces. We therefore recall some of their properties, referring to [18] or [20] for further details. Let ˚ be the Young function. We denote by E˚ .˝/ the set of all measurable functions u such that Z ˚.ju.x/j/ dx < C1; ˝

and by L˚ .˝/ the set of all measurable functions u such that the Luxemburg norm Z  o n 1 kuk˚ D inf k > 0I ˚ ju.x/j dx  1 < C1: k ˝ We say that ˚ satisfies the 2 -condition if there exist k>0 and c  0 such that ˚.2t /  k˚.t/

8t  c:

If c D 0, we speak about the global 2 -condition. Note that we have for all u 2 E˚ .˝/ Z kuk˚ 

˚.ju.x/j/ dx C 1; ˝

while E˚ .˝/ D L˚ .˝/ only if ˚ fulfills the 2 -condition. For ˛  0 and ˇ  1, we denote by Lzˇ ln˛ .1Cz/ .˝/ the Orlicz spaces generated by ˚.z/ D zˇ ln˛ .1Cz/. In the range mentioned above, zˇ ln˛ .1Cz/ fulfills the global 2 -condition. Recall that 1=˛ the complementary function to z ln˛ .1 C z/ behaves as ez ; however, this function does not satisfy the 2 -condition. We denote by Ee.1=˛/ .˝/ and Le.1=˛/ .˝/ the corresponding sets of functions. It is well known that W 1;2 .˝/ ,! Lez2 1 .˝/, and thus kuke.2/  C .kuk1;2 C 1/:

(192)

Further, the generalized Hölder inequality yields kuvk1  kukz ln˛ .1Cz/ kvke.1=˛/

(193)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2649

as well as kuvkz ln˛ .1Cz/  C kukzp ln˛ .1Cz/ kvkzp0 ln˛ .1Cz/ ;

(194)

for any ˛ > 0 and p1 C p10 D 1, 1 < p; p 0 < 1. The definition of the Luxemburg norm immediately yields for ˇ  1, ˛  0 Z   ˇ1 kukzˇ ln˛ .1Cz/  1 C ju.x/jˇ ln˛ .1 C ju.x/j/ dx

(195)

˝

as well as for ı>0 kjujı k˚.z/ D kukı˚.zı / I hence, especially for ı>0   kjujı ke.˛/  C kukıe.ı˛/ C 1 ;

(196)

  kjujı kz ln˛ .1Cz/  C .ı/ kukızı ln˛ .1Cz/ C 1 :

(197)

and for ı  1

Finally, let us consider the Bogovskii operator, i.e., the solution operator to (47) for f with zero mean value. For Orlicz spaces such that the Young function ˚ satisfies the global 2 -condition and for certain  2 .0; 1/ the function ˚  is p quasiconvex, we get a similar result as for the R L -spaces, see [38]. Hence, especially for ˛  0 and ˇ > 1, we have (provided ˝ f dx D 0) the existence of a solution to (47) such that kjr'jkzˇ ln˛ .1Cz/  C kf kzˇ ln˛ .1Cz/ :

(198)

To construct a weak solution to our problem, we use the same approximation scheme as in the three space dimensions, i.e., we have (88), (89), (90), and (91) for any ı>0. As in three space dimensions, we get m

kuı k1;2 C kr#ı2 k2 C kr ln #ı k2 C k#ı1 k1;@˝ B   1 C ı kr#ı2 k22 C kr#ı 2 k22 C k#ı kB2 C k#ı2 k1  C; r

(199)

r < 1, arbitrary, and m 2

2 m



m 2

2 m

k#ı k1;2  C 1 C k#ı k1;@˝ C kr#ı k2 with C independent of ı.



ˇ ˇZ ˇ ˇ %ı f  uı dx ˇ C 1Cˇ 

˝

(200)

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As it is the case in three space dimensions, it is more difficult R s to prove the 1 estimates for the density. We use Lemma 2 with f D %sı  j˝j ˝ %ı dx for some 0 < s < 1 and use the corresponding ' as test function in (89). It reads Z  2Cs Z   ˇCs  %ı 1Cs ˛ %ı C %2Cs dx ln .1 C %ı / C %ı #ı dx C ı ı ˝ 1ZC %ı ˝ Z   2 %ı 1 ˇ D ln˛ .1 C %ı / C %ı #ı C ı.%ı C %2ı / dx %sı dx ˝Z %ı C 1 Z Zj˝j ˝ %ı f  ' dx C S.#ı ; uı / W r' dx  %ı .uı ˝ uı / W r' dx  ˝

˝

(201)

˝

D J 1 C J2 C J3 C J4 : The estimates of J1 and J2 are easy; hence we concentrate ourselves only on the remaining two terms. We have due to (192), (193), and (195) for ˛  0 Z

  .1 C #ı /jruı jjr'j dx  C kruı k2 kr'k 1Cs2 1 C k#ı k 2.1Cs2 / s ˝ .1s/2 Z ˇZ ˇ  %2Cs 1 ˇ ˇ s ˛ ı  C k%ı k1Cs 2 1 C ˇ %ı f  uı dx ˇ  C C ln .1 C %ı / dx 4 ˝ 1 C %ı ˝

jJ3 j  C

and the last term can be shifted to the left-hand side (l.h.s.). Note that we needed here s < 1. Finally, using (193), (194), (195), (196), (197), and (198), for ˛>1, Z

%ı juı j2 jr'j dx  C kjuı j2 ke.1/ k%ı jr'jkz ln.1Cz/

jJ4 j  ˝

   C kjuı jk2e.2/ C 1 k%ı kz1Cs ln.1Cz/ kjr'jk 1Cs  C .1 C k%ı k1Cs / z1Cs ln.1Cz/ z s ln.1Cz/ Z Z   %2Cs 1 ı C 1C %1Cs ln.1 C % / dx  C C ln˛ .1 C %ı / dx ı ı 4 ˝ 1 C %ı ˝ (hint: consider separately %ı  K and %ı  K for K sufficiently large). Thus we have shown the estimate Z ˝

%1Cs ln˛ .1 C %ı / dx  C .s/ ı

(202)

with C .s/ ! C1 for s ! 1 . Remark 5. In [29], the authors used the same test function with s D 1. This leads to an L2 -estimate of the density (and hence the limit .%; u/ is immediately a renormalized solution to the continuity equation). However, this method also requires additional restriction on ˛ and m. Note that here, we are able to get the

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2651

estimates for any m>0 and ˛>1; nevertheless, a certain restriction on ˛ in terms of m appears later, when proving the strong convergence of the density. We can now pass to the limit in the weak formulation of the approximate system (note that we still do not know whether the density converges strongly) as in three space dimensions. The main task is to get strong convergence of the density which is based on the effective viscous flux identity and validity of the renormalized continuity equation, which is connected with the boundedness of the oscillation defect measure. As the proof is similar to the three-dimensional solutions, we will only mention steps which are different here. First of all, we may get the effective viscous flux identity in the form   p.%; #/Tk .%/  .#/ C .#/ Tk .%/div u   D p.%; #/ Tk .%/  .#/ C .#/ Tk .%/div u:

(203)

Next, we introduce the oscillation defect measure defined in a more general context of the Orlicz spaces osc˚ ŒTk .%ı /  Tk .%/ D sup lim sup kTk .%ı /  Tk .%/k˚ :

(204)

k2N ı!0C

In what follows, we show that there exists  >0 such that oscz2 ln .1Cz/ ŒTk .%ı /  Tk .%/ < C1I

(205)

further we verify that this fact implies the renormalized continuity equation to be satisfied. Note that to show the latter, we cannot use the approach from the book [10] (or [27]) as there; it is required that ˚ D z2C for  >0 which we are not able to verify here. Lemma 29. Under the assumptions of Theorem 4 (particularly, for ˛>1 and ˛  1 ), we have (205). m Proof. As g.t / D

t2 ln˛ .t C 1/ t C1

(206)

is for ˛>1 convex on R0C , we get for z > y  0 Z z Z z y2 z2 g 0 .t / dt  g 0 .t  y/ dt ln˛ .1 C z/  ln˛ .1 C y/ D 1Cz 1Cy y y .z  y/2 ˛ 1 ˛ D ln .1 C z  y/  .z  y/ ln .1 C z  y/  ln˛ 2 1fzy1g : 1Czy 2 (207)

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Moreover,

lim sup ı!0C

Z h

i p.%ı ; #ı /Tk .%ı /  p.%; #/ Tk .%/ dx

˝

D lim sup

hZ

ı!0C

Z C

  .g.%ı /  g.%// Tk .%ı /  Tk .%/ dx

˝

 i %ı #ı Tk .%ı /  %#Tk .%/ dx C



˝

Z



  g.%/  g.%/ Tk .%/  Tk .%/ dx:

˝

(208) As z 7! Tk .z/ is concave and z 7! g.z/ is convex, we have (using also Lemma 5 in the second integral and the fact that g./ is increasing on R0C ) Z

jTk .%ı /  Tk .%/j ln˛ .1 C jTk .%ı /  Tk .%/j/ dx 1 C jT .% /  T .%/j k ˝ Zk ı    lim sup .g.%ı /  g.%// Tk .%ı /  Tk .%/ dx ı!0CZ ˝ i h  lim sup p.%ı ; #ı /Tk .%ı /  p.%; #/ Tk .%/ dx;

lim sup ı!0C

ı!0C

(209)

˝

and also Z

1 jTk .%ı /  Tk .%/j ln˛ .1 C jTk .%ı /  Tk .%/j/ dx .#/ C .#/ 1 C jTk .%ı /  Tk .%/j ˝ ı!0C Z i h 1 p.%ı ; #ı /Tk .%ı /  p.%; #/ Tk .%/ dx:  lim sup ˝ .#/ C .#/ ı!0C (210)

lim sup

Due to (207) and Lipschitz continuity of Tk ./ with Lipschitz constant 1, together with (195), we arrive at lim sup kTk .%ı /  Tk .%/k2z2 ln˛ .zC1/ Z ı!0C  1 C lim sup jTk .%ı /  Tk .%/j2 ln˛ .1 C jTk .%ı /  Tk .%/j/ dx C ˝ Z ı!0 h i   C 1 C lim sup p.%ı ; #ı /Tk .%ı /  p.%; #/ Tk .%/ dx : ı!0C

(211)

˝

Denote now Gkı .x; z/ D jTk .z/  Tk .%.x//j2 ln˛ .1 C jTk .z/  Tk .%.x//j/. Hence, (203) implies     Gk .; %/  C ..#/ C .#// Tk .%/div u  Tk .%/div u C 1

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2653

for all k  1. Then Z

  .1 C #/1 Gk .; %/ dx  C sup kdiv uı k2 lim sup kTk .%ı /  Tk .%/k2 C 1 ˝ ı>0 ı!0C    C lim sup kTk .%ı /  Tk .%/k2 C 1 : ı!0C

(212) On the other hand, take  >0, s>1 and compute Z Z

jTk .%ı /  Tk .%/j2 ln .1 C jTk .%ı /  Tk .%/j/ dx ˝

jTk .%ı /  Tk .%/j2 ln .1 C jTk .%ı /  Tk .%/j/.1 C #/s .1 C #/s dx

 ˝

C k1C# s ke. ms / kjTk .%ı /  Tk .%/j2 ln .1 C jTk .%ı /  Tk .%/j/.1 C #/s kz ln ms .1Cz/ Z   C 1C jTk .%ı /  Tk .%/j2 ln˛ .1 C jTk .%ı /  Tk .%/j/.1 C #/1 dx ˝

provided ˛ > m1 and  >0, s  1>0 are sufficiently small with respect to ˛  m1 . We have shown that for a certain  >0, it holds, due to (212), Z

jTk .%ı /  Tk .%/j2 ln .1 C jTk .%ı /  Tk .%/j/ dx Z  12   ; jTk .%ı /  Tk .%/j2 dx  C 1 C lim sup

lim sup ı!0C

˝

ı!0C

˝

and thus Z

jTk .%ı /  Tk .%/j2 ln .1 C jTk .%ı /  Tk .%/j/ dx  C < C1

lim sup ı!0C

(213)

˝

with C independent of k. Next we have to show that (205) is sufficient to guarantee that .%; u/ verifies the renormalized continuity equation. It holds Lemma 30. Under the assumptions of Theorem 4, the pair .%; u/ is a renormalized solution to the continuity equation. Proof. As the proof is similar (even slightly easier) to the evolutionary case, we give only the main steps here; for details see also [6, Lemma 4.5]. First we mollify the limit form of the renormalized continuity equation div .Tk .%/u/ C .Tk .%/  Tk0 .%/%/div u D 0

in D0 .R2 /

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P.B. Mucha et al.

to get   div Sm ŒTk .%/ u C Sm Œ.Tk .%/  Tk0 .%/%/div u D rm

in D0 .R2 /

(214)

with Sm Œ the standard mollifier and rm ! 0 in L2 .˝I R/ as m ! 1 for any fixed 0 k 2 N. Let b 2 C 1 .R/ satisfy  b .z/  0 for all z 2 R sufficiently large, say z  M . 0 Next multiply (214) by b Sm ŒTk .%/ and letting m ! 1, we deduce     div b.Tk .%//u C b 0 .Tk .%//Tk .%/  b.Tk .%// div u   D b 0 .Tk .%// .Tk .%/  Tk0 .%/%/div u in D0 .R2 /:

(215)

Now, exactly as in [6, Lemma 4.5], we may pass with k ! 1, employing (205) to get the renormalized form of the continuity equation for any b as above. Note that we basically need to control Tk .%ı /  Tk .%/ in a better space than just L2 .˝/; the logarithmic factor is enough. By suitable approximation we finally get (34) for any b as in Definition 5. The last step, i.e., that the validity of the renormalized continuity equation, the effective viscous flux identity, and estimates above imply the strong convergence of the density, can be shown similarly as in three space dimensions; thus, we skip it. More details can be found in [35].

12

Further Results

In the last section, we briefly mention some results, where the steady compressible Navier-Stokes equations are incorporated in some more general systems and the methods explained in the first part of this chapter are used to get existence of a solution.

12.1

Steady Flow of a Compressible Radiative Gas

The modeling of a radiative gas is a complex problem. We are not going into details of its modeling; more information can be found, e.g., in [17] and references therein. We consider the following system of equations in a bounded ˝  R3 : div .%u/ D 0;

(216)

div .%u ˝ u/  div S C rp D %f  sF ;

(217)

div .%Eu/ D %f  u  div .pu/ C div .Su/  div q  sE ; I C !  rx I D S;

(218) (219)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2655

where the last equation describes the transport of radiative intensity denoted by I . The r.h.s. S is a given function of I , !, and u, see [17] for more details. The quantity sF denotes the radiative flux and sE is the radiative energy. The viscous part of the stress tensor is taken in the form (8) with the viscosity coefficients as in (9) for 0  ˛  1. The pressure is considered in the form (14) and the heat flux fulfills (10) and (11); L is a bounded function (i.e., l D 0 in (12)). The system is completed by the homogeneous Dirichlet boundary conditions for the velocity (4) and the Newton boundary condition for the heat flux (6). We finally prescribe the total mass of the fluid (7). The main result reads as follows: Theorem 5 (Kreml, Neˇcasová, Pokorný, 2013). Let ˝ 2 C 2 be a bounded domain in R3 , f 2 L1 .˝I R3 /, 0  K0 > 0 a.e. at @˝, 0 2 L1 .@˝/, M >0. Moreover, let ˛ 2 .0; 1

r n3 1˛ 1 4.1  ˛/ .1  ˛/2 o  > max ; 1 C C C 2 6˛ 2 3˛ 9˛ 2 n .1  ˛/2 1 C ˛ .1  ˛/ ; ; ; m > max 1  ˛; 3 2  3 3.  1/2 ˛  .1  ˛/ o 1 C ˛ C .1  ˛/ 1˛ ; : 6.  1/˛  1 3.  1/

(220)

Then there exists a variational entropy solution to our system. Moreover, the pair .%; u/ is a renormalized solution to the continuity equation. If additionally n5 2 C ˛ o ; 3˛ n3 .3  1/.1  ˛/ .3  1/.1  ˛/ C 2 .1  ˛/..2  3˛/ C ˛/ o ; ; ; m > max 1; 3  5 3.  1/ ˛.6 2  9 C 5/  2 (221) then this solution is a weak solution.  > max

Remark 6. For special values of ˛, formulas (220) and (221) yield the following restrictions: For ˛ D 1, n2 o 3 2 > and m > max ; (222) 2 3 3.  1/ for the variational entropy solutions, and additionally > for the weak solutions.

5 3

and

m>1

(223)

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For ˛ D >

1 2

(physically more relevant),

7C

p 13 6

and

m > max

n1

o   ; 2 2 4  6 6  14 C 6 ;

(224)

for the variational entropy solutions, and additionally n 3  1 o  C1 ; m > max 1; 2.  1/ 6  10

(225)

for the weak solutions. The proof is similar to the case without radiation with two additional difficulties. One is connected with radiation, especially with compactness properties of the transport equation, and we are not going to comment on this issue here; the other one is connected with the fact that for ˛ < 1, we loose the nice structure of the a priori estimates coming from the entropy inequality, and the situation becomes more complex. More precisely, dropping the ı-dependent terms (they can be treated as above), the entropy inequality (94) provides us only p

3m.2p/

kuı k1;p  C k#ı k3m 2

;

(226)

6m where #ı fulfills (95) and p D 3mC1˛ (i.e., p D 2 if ˛ D 1). This complicates technically the situation; on the other hand, the values of ˛ below 1 are physically more realistic. More details can be found in [17].

12.2

Steady Flow of Chemically Reacting Mixtures

We finally review the results of the paper [13]; see also [39] for the study of the isothermal case. We consider the following system of equations in ˝  R3 : div .%u/ D 0; div .%u ˝ u/  div S C r D %f; div .%Eu/ C div .u/ C div Q  div .Su/ D %f  u; div .%Yk u/ C div Fk D mk !k ; k 2 f1; : : : ; ng:

(227)

The above system describes the flow of a chemically reacting gaseous mixture of ncomponents. It is assumed that the molar masses of the components are comparable, which is assumed, e.g., by a mixture of isomers. We denote by Yk D %k =% the mass fraction, %k is the density of the k-th constituent. More details about models of chemically reacting mixtures and corresponding analysis can be found in chapter  “Solutions for Models of Chemically Reacting Compressible Mixtures”.

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2657

The system is completed by the boundary conditions at @˝ u D 0;

(228)

Fk  n D 0;

(229)

 Q  n C L.#  0 / D 0;

(230)

the given total mass Z % dx D M > 0;

(231)

˝

and the following assumptions on the form of: • the pressure law .%; #/ D c .%/ C m .%; #/;

(232)

with m obeying the Boyle law m D

n X

%Yk # D %#

(233)

kD1

and the so-called “cold” pressure c D % ;

 > 1I

the corresponding form of the specific total energy is E.%; u; #; %1 ; : : : ; %n / D

1 2 juj C e.%; #; Y1 ; : : : ; Yn /; 2

where the internal energy takes the form e D ec .%/ C em .#; Y1 ; : : : ; Yn / with ec D

1 %1 ;  1

em D

n X kD1

Yk ek D #

n X

cvk Yk ;

kD1

where cvk is the mass constant-volume specific heat. The constant-pressure specific heat, denoted by cpk , is related (under assumption on the equality of molar masses) to cvk in the following way:

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cpk D cvk C 1;

(234)

and both cvk and cpk are assumed to be constant (but possibly different for each constituent). • the specific entropy sD

n X

Yk s k

(235)

kD1

with sk the specific entropy of the k-th constituent. The Gibbs formula has the form #Ds D De C D

X n 1  gk DYk ; %

(236)

kD1

with the Gibbs functions gk D hk  #sk ;

(237)

where hk D hk .#/, sk D sk .%; #; Yk / denote the specific enthalpy and the specific entropy of the k-th species, respectively, with the following exact forms: hk D cpk #;

sk D cvk log #  log %  log Yk :

The cold pressure and the cold energy correspond to isentropic processes; therefore using (236), one can derive an equation for the specific entropy s n

div .%su/ C div

Q X gk  Fk # #

! D ;

(238)

kD1

where  is the entropy production rate g  S.#; ru/ W ru Q  r# X k    F  r D k # #2 # n

kD1

Pn

kD1

gk ! k

#

:

(239)

• the viscous stress tensor  2 S D S.#; ru/ D .#/ ru C r T u  div uI C .#/.div u/I; 3 with .#/  .1 C #/;

0  .#/  .1 C #/

(240)

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

2659

• the heat flux QD

n X

hk Fk C q;

q D .#/r#;

(241)

kD1

where  D .#/  .1 C # m / is the thermal conductivity coefficient. • the diffusion flux Fk D Yk

n X

Dkl rYl ;

(242)

lD1

where Dkl D Dkl .#; Y1 ; : : : ; Yn /, k; l D 1; : : : ; n are the multicomponent diffusion coefficients; we consider D D D t ; N .D/ D RY; R.D/ D Y? ; D is positive semidefinite over Rn ;

(243)

where we assumed that Y D .Y1 ; : : : ; Yn /t >0, and N .D/ denotes the null-space of matrix D, R.D/ its range, U D .1; : : : ; 1/t , and U ? denotes the orthogonal complement of RU . Furthermore, we assume that the matrix D is homogeneous of a non-negative order with respect to Y1 ; : : : ; Yn and that Dij are differentiable functions of #; Y1 ; : : : ; Yn for any i; j 2 f1; : : : ; ng such that jDij .#; Y/j  C .Y/.1 C # a / for some a  0. • the species production rates !k D !k .%; #; Y1 ; : : : ; Yn / are smooth bounded functions of their variables such that !k .%; #; Y1 ; : : : ; Yn /  0 whenever Yk D 0:

(244)

Next, in accordance with the second law of thermodynamics, we assume that 

n X

gk !k  0;

(245)

kD1

where gk is specified in (237). Note that thanks to this inequality and properties of Dkl , together with (240) and (241) yield that the entropy production rate defined in (239) is non-negative.

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We consider weak only weak solutions defined in the standard way. We have the following result: Theorem 6 (Giovangigli, Pokorný, Zatorska, 2015). Let  > 53 , M >0, m > 1, . Let ˝ 2 C 2 be a bounded domain in R3 . Then there exists at least one a < 3m2 2 weak solution to our problem above. Moreover, .%; u/ is the renormalized solution to the continuity equation. The proof is based on a complicated approximation procedure, where the most difficult part is to deduce the correct form of the approximate entropy inequality and to estimate all additional terms that appear there due to approximation. The reason for the bounds  > 5=3 and m>1 is, roughly speaking, the convective term in the total energy balance. To reduce the assumptions on  and m (both using improved estimates of the pressure and consider variation entropy solutions) is the work in progress.

13

Conclusions

The known existence results for the steady compressible Navier-Stokes-Fourier equations for large data were reviewed. It is well known that strong solutions may not exist. Therefore two different notions of a solution are proposed: the weak and the variational entropy one, where the former includes the weak formulation of the total energy balance, while in the latter, the total energy balance is replaced by the weak formulation of the entropy inequality and the global total energy balance. More details in the existence proof for the three-dimensional flows were presented, subject to either the homogeneous Dirichlet or the Navier boundary conditions for the velocity. The main ideas behind the proof of existence of more regular solution in the case of the Navier boundary conditions and  >3 were explained. In this case even the internal energy balance is fulfilled. The two-dimensional flows for  almost one were also studied. Finally, few results for more complex models were presented, where the Navier-Stokes-Fourier system is combined with other equations.

14

Cross-References

 Concepts of Solutions in the Thermodynamics of Compressible Fluids  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows  Existence of Stationary Weak Solutions for Isentropic and Isothermal Compress-

ible Flows  Solutions for Models of Chemically Reacting Compressible Mixtures  Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical

Cases

46 Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows

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Acknowledgements The work of P.B. Mucha has been partly supported by Polish NCN grant No 2014/13/B/ST1/03094. The work of M. Pokorný has been partially supported by the Czech Science Foundation, grant No. 16-03230S.

References 1. Š. Axmann, M. Pokorný, Time-periodic solutions to the full Navier-Stokes-Fourier system with radiation on the boundary. J. Math. Anal. Appl. 428(1), 414–444 (2015) 2. J. Bˇrezina, A. Novotný, On weak solutions of steady Navier–Stokes equations for monoatomoc gas. Comment. Math. Univ. Carol. 49, 611–632 (2008) 3. R. Danchin, P.B. Mucha, The divergence equation in rough spaces. J. Math. Anal. Appl. 386(1), 9–31 (2012) 4. C. Dou, F. Jiang, S. Jiang, Y.-F. Yang, Existence of strong solutions to the steady NavierStokes equations for a compressible heat-conductive fluid with large forces. J. Math. Pures Appl. 103(5), 1163–1197 (2015) 5. S. Elizier, A. Ghatak, H. Hora, An Introduction to Equations of States, Theory and Applications (Cambridge University Press, Cambridge, 1996) 6. R. Erban, On the existence of solutions to the Navier–Styokes equations of a two-dimensional compressible flow. Math. Methods Appl. Sci. 26, 489–517 (2003) 7. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998) 8. E. Feireisl, Dynamics of Viscous Compressible Fluids (Oxford University Press, Oxford, 2004) 9. E. Feireisl, P.B. Mucha, A. Novotný, M. Pokorný, Time-periodic solutions to the full NavierStokes-Fourier system. Arch. Ration. Mech. Anal. 204(3), 745–786 (2012) 10. E. Feireisl, A. Novotný, A Singular Limits in Thermodynamics of Viscous Fluids. Advances in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2009) 11. E. Feireisl, D. Pražák, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics (AIMS, Springfield, 2010) 12. J. Frehse, M. Steinhauer, W. Weigant, The Dirichlet problem for steady viscous compressible flow in 3-D. J. Math. Pures Appl. 97, 85–97 (2009) 13. V. Giovangigli, M. Pokorný, E. Zatorska, On the steady flow of reactive gaseous mixture. Analysis 35(4), 319–341 (2015) 14. D. Jesslé, A. Novotný, Existence of renormalized weak solutions to the steady equations describing compressible fluids in barotropic regime. J. Math. Pures Appl. 99, 280–296 (2013) 15. D. Jesslé, A. Novotný, M. Pokorný, Steady Navier–Stokes–Fourier system with slip boundary conditions. Math. Models Methods Appl. Sci. 24, 751–781 (2013) 16. S. Jiang, C. Zhou, Existence of weak solutions to the three-dimensional steady compressible Navier-Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(4), 485–498 (2011) 17. O. Kreml, Š. Neˇcasová, M. Pokorný, On the steady equations for compressible radiative gas. Z. für angewandte Math. und Phys. 64, 539–571 (2013) 18. A. Kufner, O. John, S. Fuˇcík, Function Spaces (Academia, Praha, 1977) 19. P.L. Lions, Mathematical Topics in Fluid Dynamics, Vol.2: Compressible Models (Oxford Science Publication, Oxford, 1998) 20. L. Maligranda, Orlicz Spaces and Interpolation (Campinas SP, Brasil, 1989) 21. P.B. Mucha, On cylindrical symmetric flows through pipe-like domains. J. Differ. Eqn. 201, 304–323 (2004) 22. P.B. Mucha, M. Pokorný, On a new approach to the issue of existence and regularity for the steady compressible Navier–Stokes equations. Nonlinearity 19, 1747–1768 (2006) 23. P.B. Mucha, M. Pokorný, On the steady compressible Navier–Stokes–Fourier system. Comm. Math. Phys. 288, 349–377 (2009)

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24. P.B. Mucha, M. Pokorný, Weak solutions to equations of steady compressible heat conducting fluids. Math. Models Methods Appl. Sci. 20, 1–29 (2010) 25. P.B. Mucha, M. Pokorný, The rot-div system in exterior domains. J. Math. Fluid Mech. 16(4), 701–720 (2014) 26. P.B. Mucha, R. Rautmann, Convergence of Rothe’s scheme for the Navier-Stokes equations with slip conditions in 2D domains. ZAMM Z. Angew. Math. Mech. 86(9), 691–701 (2006) 27. A. Novotný, M. Pokorný, Steady compressible Navier–Stokes–Fourier system for monoatomic gas and its generalizations. J. Differ. Eqn. 251, 270–315 (2011) 28. A. Novotný, M. Pokorný, Weak and variational solutions to steady equations for compressible heat conducting fluids. SIAM J. Math. Anal. 43, 270–315 (2011) 29. A. Novotný, M. Pokorný, Weak solutions for steady compressible Navier-Stokes-Fourier system in two space dimensions. Appl. Math. 56, 137–160 (2011) 30. A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow (Oxford University Press, Oxford, 2004) 31. G. Panasenko, K. Pileckas, Divergence equation in thin-tube structures. Appl. Anal. 94(7), 1450–1459 (2015) 32. P. Pecharová, M. Pokorný, Steady compressible Navier-Stokes-Fourier system in two space dimensions. Comment. Math. Univ. Carolin. 51, 653–679 (2010) 33. P. I. Plotnikov, J. Sokolowski, On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier–Stokes equations. J. Math. Fluid Mech. 7, 529–573 (2005) 34. P.I. Plotnikov, W. Weigant, Steady 3D viscous compressible flows with adiabatic exponent  2 .1; 1/. J. Math. Pures Appl. 104, 58–82 (2015) 35. M. Pokorný, On the steady solutions to a model of compressible heat conducting fluid in two space dimensions. J. Part. Differ. Eqn. 24(4), 334–350 (2011) 36. M. Pokorný, P.B. Mucha, 3D steady compressible Navier–Stokes equations. Cont. Discret. Dyn. Syst. S 1, 151–163 (2008) 37. V.A. Solonnikov, Overdetermined elliptic boundary value problems. Zap. Nauch. Sem. LOMI 21, 112–158 (1971) 38. R. Vodák, The problem div v D f and singular integrals in Orlicz spaces. Acta Univ. Olomuc. Fac. Rerum Nat. Math. 41, 161–173 (2002) 39. E. Zatorska, On the steady flow of a multicomponent, compressible, chemically reacting gas. Nonlinearity 24, 3267–3278 (2011) 40. E. Zatorska, Analysis of semidiscretization of the compressible Navier-Stokes equations. J. Math. Anal. Appl. 386(2), 559–580 (2012) 41. X. Zhong, Weak solutions to the three-dimensional steady full compressible Navier-Stokes system. Nonlinear Anal. 127, 71–93 (2015) 42. W.P. Ziemer, Weakly Differentiable Functions (Springer, New York, 1989)

Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

47

Ondˇrej Kreml, Piotr Bogusław Mucha, and Milan Pokorný

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Homogeneous Boundary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Inhomogeneous Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Unbounded Two-Dimensional Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Unbounded Three-Dimensional Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter contains a survey of results in the existence theory of strong solutions to the steady compressible Navier-Stokes system. In the first part, the compressible Navier-Stokes equations are studied in bounded domains, both for homogeneous (no inflow) and inhomogeneous (inflow) boundary conditions. The solutions are constructed in Sobolev spaces. The next part contains the results for unbounded domains, especially for the exterior domains. Here, not only

O. Kreml () Institute of Mathematics, Czech Academy of Sciences, Praha, Czech Republic e-mail: [email protected] P.B. Mucha Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland e-mail: [email protected] M. Pokorný Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_65

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the question of existence and uniqueness is considered, but also the asymptotic structure near infinity is studied. Due to the different nature of the problems, the two- and three-dimensional problems are treated separately.

1

Introduction

The initial value problem for the compressible Navier-Stokes(-Fourier) system being successfully solved at the end of 1970s of the last century (see [26,27] and also chapter  “Local and Global Existence of Strong Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Maximal Regularity”), a natural question appeared, whether similar results are available also for the steady flows. In the first half of 1980s, the affirmative answers appeared first in the papers by M. Padula [46,47] or A. Valli [67,68] in the context of energy method and in the second half of 1980s in the papers by H. Beirão da Veiga [4] and M. Padula [50] in the Lp setting. Later, in 1980s and in the first half of 1990s, many further results appeared, considering different situations as bounded or unbounded domains, different methods of proof (energy method, Lp -estimates, or method of decomposition), and/or different boundary conditions. These results will be described below. Later on, after publishing the existence result of P.L. Lions, the main attention moved from the properties of strong solutions to the properties of weak solutions which can be obtained without any restriction on the size of the data. See chapters  “Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows” and  “Existence of Stationary Weak Solutions for Isentropic and Isothermal Compressible Flows” for more information about the recent results concerning the weak solutions. Still many interesting questions are left open in the field of regular solutions; among them one can find problems with nonzero inflow condition which seem to be a bridge between the weak and regular theory. These problems have also attracted more attention in the last years. This chapter is divided into two parts. In the first part, the results in bounded domains are considered, for both zero and nonzero inflow conditions. The second part is devoted to unbounded domains, where the existence and spatial asymptotics is studied separately in two and three space dimensions.

2

Bounded Domains

The issue of regular solutions to the steady flows is related to the small data case, similarly as for the evolutionary system for multidimensional models. The main difference is the following obstacle: a lack of suitable a priori estimates. In particular, there is no natural estimate for the density of the fluid. For some problems, the average of the density is even required to be fixed. The class of results is naturally divided into two parts: the first one is for the homogeneous boundary data case, where we assume no flow through the considered boundary, and the second one are inhomogeneous boundary problems admitting

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nontrivial inflow conditions. In this section, we restrict ourselves to the case of bounded domains. In the first group of problems, a constraint to determine the total density has to be chosen – even for small velocities. In the second group, we touch closer the hyperbolic–elliptic character of the system. To determine completely the required set of boundary relations, we need to prescribe the density at the part of the boundary, where inflow of the fluid appears. This assumption is natural to control the well posedness of the transport equations by prescribing the value of the solution at the part of the boundary where characteristics come into the domain. In both cases, all known techniques are based on the analysis of the stationary transport equation. The problems for homogeneous boundary conditions (i.e., no inflow) and the problems for inhomogeneous boundary conditions (i.e., nonzero inflow conditions) will be considered separately. In this section, we work in the elementary functional setting, in the Lebesgue spaces of type Lp .˝/ with their natural generalization on Sobolev spaces of the class W m;p .˝/ with the classical integer m. The corresponding norms will be denoted by k  kp and k  km;p , respectively. There is no difference in the notation of scalar function spaces and vector- and tensor-valued function spaces; however, the vector- and tensor-valued functions are printed in boldface.

2.1

Homogeneous Boundary Problems

We start with the basic problem in a bounded domain. Examine the compressible Navier-Stokes system div .%v/ D 0 in ˝; %v  rv  v  rdiv v C rp.%/ D %f in ˝

(1)

with the zero Dirichlet (no-slip) boundary conditions vD0

on @˝:

(2)

Here sought quantities are the velocity v: ˝ ! Rn , and density %: ˝ ! RC [ f0g. The viscous coefficients  and  are assumed to be constant and positive. We concentrate our attention on the constitutive equation of the form p.%/ D % with   1, but it can be replaced by any strictly monotone smooth function of the density. Looking at the formulation of system (1)–(2), we are not able to determine the density. Since we are interested in the regular solutions, the considerations are restricted to the analysis of small perturbations of the trivial flow: .%; v/ D .1; 0/:

(3)

As an easy observation, we point out that the system can be considered in an unbounded domain as well, adding some necessary conditions at the infinity; see the next section.

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We present a sketch of the proof of the basic existence result. We follow the classical idea of Hugo Beirão de Veiga [4]. The original result examined the NavierStokes-Fourier system; here we consider only a simplified version. The result is stated as the following theorem. Theorem 1 ([4]). Let n  2, p > n, ˝ be a bounded n-dimensional domain with a smooth boundary and f be a smooth vector function with small norms. Then there exists a unique solution to system (1)–(2) in the class of small perturbation of the generic flow (3) such that kr 2 v; r%kp  C kfkp

(4)

and 1 j˝j

Z % dx D 1:

(5)

˝

The considerations are split into the following parts. The first one is the a priori estimate. Here we find a great idea of application of the effective viscous flux and the stationary transport equation. The second one is the sketch of construction of solutions which provides us the existence part of the theorem. The analysis below is performed for system (1) but with the slip boundary conditions n  T.v; p/   C ˛v   D 0;

n  v D 0 at @˝

(6)

instead of the Dirichlet condition (2). It will simplify the computation, keeping the main idea of the original construction. Note that we assume the friction coefficient ˛ > 0 to be constant. Here T.v; p/ D .rv C rvT / C .div v  p/Id with Id the identity tensor. A priori estimate. The first task is to construct the a priori estimate fulfilling the following constraints: kvk2;p C k%  1k1;p  1;

(7)

with zero average of %  1, and kfkp is small enough. The basic estimate comes from the energy estimate, testing (1)2 by v we find

(8)

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

Z



jDvj2 C  .div v/2 / dx C

˝

Z

˛jvj2 d D

@˝

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Z %f  v dx:

(9)

˝

Provided (7), we find kvk1;2  C kfk2 :

(10)

Considerations for the exterior domain have to be slightly modified, as well as assumptions concerning the external force f; see the following section. Observe that information coming from (10) is not sufficient; moreover, the approach is not efficient, either. There is a need to get a better information about the density. Here we support our method by the basic quantity coming from the theory of weak solutions to the compressible Navier-Stokes system, namely, we will work with the effective viscous flux:  . C /div v C p.%/:

(11)

In order to recover this quantity, we apply the Helmholtz decomposition (see chapter  “The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties” for more details) to reduce the momentum equation only to its potential part. Let Pr : Lp .˝/ ! W 1;p .˝/ be such that Pr h D :

(12) p

A complement of Pr is the Helmholtz projector PH : Lp .˝/ ! Ldiv .˝/ on the divergence-free vector field. Then h D PH h C r

(13)

div h D ;

(14)

such that

with zero Neumann boundary conditions, then indeed div PH h D 0, and to fix this decomposition we require that n  PH h D 0 on @˝:

(15)

Note that since div PH h D 0, the trace of n  PH h is a well-defined functional from W 1=p;p .@˝/. In this part, we try to avoid introducing more complex function space setting. Thus, in order to skip problems with definitions of the space W 1=p;p .@˝/, we think here about this quantity as a distribution defined on the boundary. In other words, PH v fulfills

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rot PH v D rot v in ˝; in ˝; div PH v D 0 n  PH v D 0 at @˝:

(16)

Now due to the slip boundary conditions, one can write the system for the vorticity rot PH v rot PH v D rot .%v  rv C %f/ in ˝; on @˝; rot PH v D .2  ˛=/v  

(17)

where  is the curvature of @˝; see [30]. Based on the results for the maximal regularity for systems (16) and (17), we obtain   kr 2 PH vkp  C k%v  rvkp C k%fkp : (18) We remember the density is close to one. Hence the rotational part of the velocity is under control. The potential part of the velocity is defined by the following relation:  . C /div v C p.%/  fp.%/g D Pr Œ%v  rv  Pr ŒPH v C Pr .%f/:

(19)

Here the average of the left-hand side is zero. We denoted fp.%/g D

1 j˝j

Z p.%/ dx: ˝

Indeed we obtain the gradient of (19); however, since we control the average, we can state (19). We restate the continuity equation in the form div v C v  r% D adiv v;

(20)

where we denoted a D %  1. Adding (19) and (20) with suitable multiplications related to . C /, we obtain p.%/  fp.%/g C . C /.v  r% C adiv v/ D Pr Œ%v  rv  Pr ŒPH v C Pr .%f/: (21) Indeed it is better to consider from the very beginning the higher order of regularity of the perturbation of the density. The gradient version of (21) reads p 0 .%/ra C . C /v  rra D . C /.rv  ra C radiv v  ardiv v/   C r Pr Œ%v  rv  Pr ŒPH v C Pr Œ%f : (22) Testing the kth component of Eq. (22) by j@xk ajp2 @xk a, we find that

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

  krakp  C krvk1 krakp C kak1 kr 2 vkp C k%fkp :

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(23)

This elementary approach works due to the value of the normal component of the velocity at the boundary n  v D 0. The boundary term coming from the integration of the transport part on the left-hand side is zero; hence, we have the following identity: Z ˝

v  r@xk aj@xk ajp2 @xk a dx D 

1 p

Z ˝

div vj@xk ajp dx:

(24)

To recover the full information about the sought solution, we note that Eq. (19) gives information about the gradient of the divergence of the velocity; hence the potential part of the flow is controlled, too. More precisely, there holds . C /rdiv v D p 0 .%/ra C Œ:: :

(25)

In the brackets Œ:: , one can find better (lower-order) terms. Hence we have obtained   krdiv vkp C krakp  C krvk1 krakp C kak1 kr 2 vkp C k%fkp :

(26)

Altogether, we have constructed the following inequality on the norms of small solutions to the steady compressible Navier-Stokes system: kr 2 vkp C krakp  C .kr 2 vkp C krakp /2 C C k%fkp :

(27)

But remembering we are interested here only in perturbation of the flow (3), we conclude the a priori estimate: kvk2;p C kak1;p  C kfkp :

(28)

The Sobolev embedding theorem requires to restrict our analysis to the case p > n and sufficiently small right-hand side of (28). This choice guarantees that the space W 1;p .˝/ is an algebra; in particular the gradient of the velocity rv and the density % are point-wisely bounded. Construction of solutions. Of course, the above a priori estimate does not end our considerations. We have to find/construct solutions which fulfill estimate (28). A hyperbolic-elliptic character of the system does not allow to use a simple procedure. On the other hand, the small data case makes our considerations easier than for the case of weak solutions. Nevertheless, we have to support ourselves by the technique from the theory of weak solutions for the compressible NavierStokes system. Here we think about a need of regularization of the continuity equation [3, 5]. Given > 0, we look for a solution to the following problem:

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h in ˝; j˝j %v  rv  v  rdiv v C rp.%/ D %f in ˝;

a C div .%v/  a D

(29)

keeping in mind that we look for % D 1 C a with jaj  1:

(30)

Here h  0 is a given constant. The system is supplemented with the original boundary condition and the following extra one: @a D0 @n

on @˝:

(31)

We look for solutions to our problem as a fixed point to the following map T W X ! X in an appropriate function space X such that T .v; a/ N D .v; a/, where .v; a/ is the solution to h in ˝; j˝j 0 .1 C a/v N  rv  v  rdiv v C p 0 .1/ra D %fCr.p.1 C a/p N .1/a/ N in ˝ (32) with the boundary conditions as above. Solvability of the above problem is unquestionable. The problem is that we obtain the estimates highly depending on . Indeed, first we solve the equation N v a D

a C v  r aN C adiv

a  a D

h  v  r aN  adiv N v: j˝j

(33)

The main operator is strongly elliptic with the homogeneous Neumann boundary condition and the following constraint: Z a dx D h:

(34)

˝

In other words, the average of a is controlled by h. We can put h D 0 to keep the original requirement that the average of the density of the fluid is assumed to be one. Thus we conclude kak2;p  C . /krv; r ak N p:

(35)

Then we solve the second equation in the form  v  rdiv v D %f C r.p.1 C a/ N  p 0 .1/a/ N  .1 C a/v N  rv  p 0 .1/ra

(36)

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with homogeneous Dirichlet boundary conditions. The classical theory delivers the following estimate: N p: kr 2 v; rakp  C . /kf; rv; r ak

(37)

This way we find solutions to (32) and also show that the map T is compact. Now, we have to construct the estimate independent of . We apply the LeraySchauder fixed-point theorem; hence, we find an a priori bound on all solutions to the identity T .v; %/ D .v; %/ with  2 Œ0; 1 , whence the existence of solutions to the approximate system (29). The estimate allows to pass with to zero, too. Since this part belongs to the mathematical folklore, we present just main points of the idea of the proof. The same way as in the construction of the a priori estimate, we find that kr 2 PH vkp  C .kv  rvkp C k%fkp /

(38)

and using the properties of the Helmholtz projection, we get in the same as for (19)– (20) the following relations: p.%/  fp.%/g C . C /.v  ra  a C adiv v/ D Pr Œ%v  rv  Pr ŒPH v C Pr Œ%f : (39) The energy method yields Z kakp C

jraj2 jajp2 dx

1=p

   C kjvj2 kp C k%fkp :

(40)

˝

Here due to the presence of a, we cannot repeat directly the approach with differentiation by parts. Instead we can use localization techniques with a transformation into the half space. Then the modified problem has sufficiently good properties for the energy method to work. We are allowed then to differentiate with respect to the tangential directions, preserving the normal vector derivative to be zero at the boundary. Finally, the regularity with respect to the normal component can be identified as the Dirichlet condition obtained from the Neumann ones. This tedious technique will finally lead to the following estimate:  krakp  C kvk22;p C k%fkp :

(41)

The terms with are omitted, since their form is not too simple in the original coordinates, but of course, they stay positive. Next we recompute the estimates for the div v and we finally obtain kr 2 v; rakp  C kfkp :

(42)

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The above information gives sufficient information independent of which implies existence of solutions to the approximate system (29). Due to (42), we are allowed to pass to the limit ! 0C and we obtain the solution to the original problem (1), too. The level of regularity is so high that there is no need to analyze the behavior of nonlinear terms. The uniqueness of solutions is not immediate, too. It is required to repeat most of the steps of the construction of the a priori estimate to get the H 1 –L2 estimate on differences of the velocities and densities. The version of Theorem 1 with the slip boundary conditions is proved. Before looking at the results for nontrivial inflow conditions, let us at least briefly mention other results in this field not mentioned above. The first results concerned uniqueness, without being able to prove existence of such results; see, e.g., [48]. Existence result for Navier boundary conditions can be found in [10] and some other results for the homogeneous Dirichlet conditions in [32] or [36]. Similarly as in the unbounded domains discussed below, the decomposition method allowed to deal effectively with separate linear problems so that existence of a solution for the nonlinear problem was an easy application of the Banach fixedpoint theorem. This method is mostly connected with names A. Novotný and M. Padula. The decomposition method, with different improvements, was applied to different situations, e.g., in [44] (large potential forces with a small non-potential perturbation), [31] (corner domains), [9] (Schauder estimates and existence of a solution in Hölder continuity classes), and [16] (free boundary problems). Some improvements of the decomposition method can be found in [15] or [1]. In [2] this method was used to develop a numerical scheme. In the recent paper [6], the authors were able to prove existence of a solution for the Dirichlet boundary conditions assuming only small Mach number, the force may be large. Last but not least, let us mention the overview paper [54] and the monograph [45], where the reader can find further comments and results.

2.2

Inhomogeneous Boundary Value Problems

The Navier-Stokes system considered with inhomogeneous boundary data is a challenging problem. Even stability results are not well understood. We shall point out here series of papers of Kweon, Kellogg, and Piasecki [17, 18, 22, 24, 25, 29, 57– 59] as well as results of Plotnikov and Sokolowski [60,62] as well as the monograph [61] and the current paper [14]. The main idea is the following. We examine the solution in a vicinity of a given trivial solution of the type .v; %/ D ..1; 0; 0/; 1/:

(43)

The key obstacles are the following: – there are problems to obtain a basic estimate; boundary data do not allow to test by the solution to get a good information;

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– elliptic-hyperbolic type of the system causes some difficulties; we have to subtract the potential part of the momentum equation; here the approaches are very close as in the case of the theory of weak solution to the compressible Navier-Stokes equations; – geometry of the domain leads to singularities in the system; we meet obstacle as characteristics or stream lines are tangential to the boundary of the data. This point explains why the most of papers study polygonal domains. Let us start with the description of results from papers of Kweon et al. [17, 18, 23]. We consider div .%v ˝ v/  v C r% D 0 div .%v/ D 0 v D vb % D %i n

in ˝; in ˝; on @˝; on i n :

(44)

The subset of the boundary i n denotes the area of the inflow, i.e., i n D fx 2 @˝ W vb .x/  n E < 0g:

(45)

We concentrate on the result from [17]. The main theorem says ( denotes as above the tangent vector to @˝). Theorem 2 ([17]). Let ˝ be a smooth bounded two-dimensional domain. Moreover, we require that at boundary points such that   .0; 1/ D 0 the curvature of the boundary is strictly positive. Let vb , %i n be sufficiently smooth and kvb  .1; 0/kC 2 C k%i n  1kC 2 < c

(46)

for a small c > 0. Then there exists a regular solution to (44) such that v 2 W 2;p .˝/; % 2 W 1;p .˝/

(47)

kv  .1; 0/k2;p C k%  1k1;p  C c:

(48)

and

We require 2 < p < 3. The key element is the condition on the curvature at points where the characteristic is tangent to the boundary. Strict positivity of the curvature says that streamlines do not income into the domain too flatly. The method of the proof bases on analysis of the transport equation describing the density. At the point where the inflow is tangent to the boundary, we obtain a type of singularity and the assumption on the

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curvature allows to control this difficulty. Let us mention that the regularity of the given data is not optimal; however, for such type of problems, it is not so important; see Theorem 3. Here we present methods for a slightly different system with slip boundary conditions. The approaches are similar to ones from [17], but the slip conditions give easier way to present the steps of the proof. Here we base on results of Piasecki and coauthors [29, 59]. We examine the model case div .%v ˝ v/  v C r% D 0 div .%v/ D 0 n  T.v; p/   C ˛v   D 0; n  v D d % D %in

in K; in K; on @K; on in ;

(49)

where K D Œ0; 1 2 , and 8 < 1 C b for fx1 D 0g  Œ0; 1 ; dD 0 for Œ0; 1  f0; 1g; : 1 C b for fx1 D 1g  Œ0; 1 ;

(50)

where b is a given small smooth function. In this setting in D fx1 D 0g  Œ0; 1 ;

out D fx1 D 1g  Œ0; 1 ;

0 D Œ0; 1  f0; 1g: (51)

Theorem 3 ([29]). Let ˛j 0  0, ˛j in > 1, b and %in 1 be smooth small functions defined at proper parts of the boundary. Then there exists a regular solution to system (44) such that v 2 W 2;p .K/; % 2 W 1;p .K/

(52)

and kv  .1; 0/k2;p C k%  1k1;p  C .kbkC 2 C k%in  1kC 1 /:

(53)

We require p > 2. Note that the regularity of the data in Theorems 2 and 3 is not optimal; also Theorem 3 is a simplified version of results proved in [29]. Proof. Let us give main ideas of the proof of Theorem 3. The first point is the reformulation of the system. The needs of our technique require to consider the momentum equation with homogeneous boundary data. Note, the conditions on the friction coefficient assumed on 0 allows to consider the flow v D .1; 0/ and % D 1 as

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

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a special solution to (49) with (50). The geometry of the boundary and the structure of boundary conditions allow to put different constants on each of part of @K. We restate the system as follows @u  u C  rw D F.u; w/ @x1 @w C .u0 C u/  rw D G.u; w/ @x1 n  T.u; w/   C ˛u   D 0; n  u D 0 w D %in

in

K;

in

K;

(54)

on @K; on in ;

where u D v  u0  v and w D %  %;

(55)

and u0 a suitable extension of the boundary data of the velocity. The issue of existence is not immediate. The a priori estimate does not lead to a procedure pointing out a possible construction of solutions. The obstacle is the hyperbolic equation and the term u  rw from (54)2 . An idea to overcome this difficulty is to use a change of coordinates of Lagrangian type, in order to “remove” this term. The first point of our method is the basic information coming from the energy estimate. Testing by u the momentum system, using the continuity equation we obtain Z

jruj2 dx C K

Z

˛juj2 dS C 

@K

Z

Z  in

Z

w2 dS

out

w2in

Z dS C in

juj2 dS C C kF; Gk22 :

(56)

juj2 dS is a bad part coming from the integral

The term in

Z

Z

2

juj2 dS 

ux1 u dx D K

out

Z

juj2 dS:

(57)

in

Here we meet the condition that ˛j i n > 1; then the bad term can be put on the left-hand side of (56). Thanks to this choice, inequality (56) implies in particular kruk2  C .DATA C kF; Gk2 /;

(58)

where DATA denotes a quantity depending only on perturbation boundary data. Quantity DATA will always mean a small number. Hence the term ux1 on the lefthand side of (54)1 can be put to the right-hand side. We will be able to control it

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by some interpolation results. Thus we reduce our considerations to the following system: u C  rw D F.u; w/ @w C .u0 C u/  rw D G.u; w/ @x1 n  T.u; w/   C f u   D 0; n  u D 0 w D %in  1

in

K;

in

K;

(59)

on @K; on in :

In F and G, the information coming from the extensions of the boundary data is hidden. The second step is the change of the coordinates. The goal is to simplify the form of Eq. (59)2 . We want to construct such coordinate system that @w @w C .u0 C u/  rw D @x1 @z1

(60)

for some .z1 ; z2 /, a sought coordinate system. But here we meet the first obstacle, namely, there is a need to control the domain where we consider our system. In the general case, such transformation may change our square K into a different domain. However, to use our technique and to have a chance to obtain uniqueness, we would prefer to preserve this square K. A solution of this problem is the following: we will not change the first variable and work just on the second one by looking for the new system with the following constraint: 

1C

u10

1

Cu





u20 C u2 @ @ C 1 1 @x1 1 C u0 C u @x2



  @ D .1 C u10 C u1 / ı .z/ : @z1

(61)

Analysis of the above relation leads to the following system: dx1 D 1; dz1 u20 C u2 dx2 D dz1 1 C u10 C u1 with x1 jz1 D0 D 0, x2 jz1 D0 D z2 .

(62)

u2 Cu2

Boundary conditions keep 1Cu0 1 Cu1 to be zero on 0 . Together with the fact 0 x1 D z1 , it guarantees that the length of the streamline does not change. Hence the transformation .x1 ; x2 / $ .z1 ; z2 / does not change the domain. The square K is preserved. Introducing functions U.z/ D u.x.z// and W .z/ D w.x.z//;

(63)

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2677

we obtain the following modification of system (59): z U C  rz W D .z  x /U C .rz  rx /W C Fx .u; w/ .1 C U01 C U 1 / @W D Gx .u; w/ @z1 nz  Tz .U;  W /   C ˛U   D nz  Tz .U;  W /    nx  Tx .U;  W /   nz  U D .nz  nx /  U W D %in  1

in K; in K; on @K; on @K; on in : (64)

Note that the change of the coordinates is well defined; we will look for the solutions in such class that U 2 W 2;p .K/ with p > 2 with a small norm kUkW 2;p .K/ . It guarantees the well posedness; in addition the smallness of U and W provides possibility the system (64) can be solved by the simple Banach iteration. There is no hyperbolic transport equation; we have just simple ODE instead of it. In order to derive the estimate, let us first note that   k.z  x /U C .rz  rx /W kp  C kUk2;p kUk2;p C kW k1;p :

(65)

We have the same for the boundary term   knz  Tz .U;  W /    nx  Tx .U;  W /  k11=p;p;@K  C kUk2;p kUk2;p C kW k1;p ; (66) and k.nz  nx /  Uk21=p;p;@K  C kUk22;p :

(67)

Here we use (62) in order to obtain the normal vector nz in terms of the x coordinates. It allows to compute the change of normal vectors in terms of the norm kUk2;p . Next, we shall underline that the Laplace or even the full Lamé system is possible to be considered in the square, since we are working with slip boundary conditions. In this case the method of symmetry works. This is a fine property of these boundary relations, in comparison to the Dirichlet conditions. Summing up we obtain kUk2;p  C .kUk2;p .kUk2;p C kW k1;p / C kFkp /

(68)

and directly from (64)2 we get kW; Wz1 k1;p  C kGk1;p :

(69)

Taking (68) and (69), remembering about (58), we find the a priori bound for solutions to (64): kUk2;p C kW k1;p  DATA:

(70)

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Returning to the original system (54), we obtain kuk2;p C kwk1;p  DATA:

(71)

Since the construction of solutions to system (64) does not require any nonstandard preparations, we leave it for a kind reader. The technique of the construction of the solution is based on the Banach procedure, which implies the uniqueness for our system, too. Theorem 3 is proved.

3

Unbounded Domains

The studies of regular solutions for steady flow of a viscous compressible fluid in unbounded domains (either in the isothermal/barotropic regime or including the equation for the internal energy) were very popular in the 1990s of the last century. The first result in this direction (based on energy method, i.e., in Hilbert spaces), however, without proof, can be found in [28]. Some uniqueness results (e.g., [49]) are even older; however, the solutions with the given regularity are not constructed there. The most important results are connected with names like A. Novotný, M. Padula, K. Pileckas, and others. In what follows, the results in two and three space dimensions will be discussed separately. It may sound surprising, but the problems in two space dimensions are more difficult than in three dimensions; it is connected with the fact that a function with its gradient in L2 in three space dimensions in a certain sense decays to a constant (which may often be taken as zero), while in two space dimensions, such a function (even continuous) may still grow logarithmically. A closely related fact is the well-known Stokes paradox. For more details about these phenomena, see [12] or chapter  “Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions”. Due to this situation, the two-dimensional case will be discussed in more details, while in three space dimensions, only the most important steps will be commented. The usual idea for the “strong solution” results (see also the previous part, where the bounded domains were considered) is that the nonlinear problem is suitably linearized to get a set of “standard” systems of equations. The perturbations of the simple solutions (often zero or constants) are then sought as fixed points of some nonlinear mappings using these linear problems. Often, the Banach fixed theorem provides the solutions which leads to a natural assumptions that the perturbations must be in some sense small. As will be seen later, the most interesting part of the velocity (its potential part) solves the Oseen or Stokes problem. The properties of solutions to these problems will be fundamental for solving the nonlinear problem, especially for the choice of the function spaces (with or without weights). More details about estimates of these problems can be found in [12]; a certain generalization of the Oseen problem was considered in the paper [11]. Another

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2679

approach to the Oseen problem which is a more geometric measure way of looking at it can be found in [19]; this approach was applied to the study of the decay to the second-grade fluid in exterior three-dimensional domains in [20]. Let us now introduce the linearization. The compressible Navier-Stokes system in ˝  RN , N D 2; 3, where ˝ D RN n K with K a smooth (C 2 at least) simply connected compact subset of RN , reads 1 V  .1 C 2 /rdiv V C r .%/ D div .%V ˝ V/ C %f; div .%V/ D 0

(72)

with Vj@˝ D 0;

V ! v1 ;

%!1

as jxj ! C1:

(73)

Without the loss of generality, the velocity and the density can be written as a perturbation of the constant flow: V D v1 C v;

% D 1 C :

(74)

Furthermore, it is possible to look for the velocity v in the form of the Helmholtz decomposition (for more information about this issue, see chapter  “The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties”) v D u C r';

(75)

where div u D 0

in ˝;

unD0

on @˝:

(76)

For simplicity, it is assumed that .%/ D % (the general case can be treated similarly) and (without loss of generality) v1 D v1 e1 with v1 a given nonnegative constant. System (72) can be rewritten in the following form for the unknowns: .v;  / 1 v  .1 C 2 /rdiv v C r C v1

@v D F. ; v/; @x1

div v C div . .v C v1 // D 0

(77)

v ! 0;

(78)

with vj@˝ D v1 ; and

 !0

as jxj ! C1;

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  @. v/ F. ; v/ D div .1 C  /v ˝ v  v1 C .1 C  /f: @x1

(79)

Furthermore, system (77) with conditions (78) can be rewritten as three standard problems in ˝: the Oseen problem (or the Stokes problem if v1 = 0) for .u; ˘ / 1 u C r˘ C v1

@u D F D div F C .1 C  /f; div u D 0; @x1 u D v1 r on @˝; u ! 0; ˘ ! 0 as jxj ! C 1I (80)

the transport equation for   C .21 C 2 /div . .w C v1 // D ˘  v1  !0

@ ; @x1

(81)

as jxj ! C1I

and the Neumann problem for the Laplace equation with the unknown r', the potential part of the velocity: ' D div . .w C v1 //; @' D v1  n @n jr'j ! 0

on @˝;

as jxj ! C1:

(82)

Indeed, taking   F D F. ; uCr'/ D .1C /.uCr'/˝.uCr'/  .u C r'/T ; 0T ;

(83)

and  D  D ';

w D u C r';

(84)

the pair .v;  / = .u C r';  / is a solution to problem (77), (78), and (79) which easily generates a solution to the original problem (72)–(73). From now on, the two- and the three-dimensional cases will be treated separately.

3.1

Unbounded Two-Dimensional Domains

Due to the reasons explained above, the only well-studied problem in the twodimensional exterior domains is the case with nonzero velocity prescribed at infinity, where the leading linear problem for the velocity (more precisely, for its non-potential part) is the Oseen problem. Its solution possesses better properties (decays faster) at infinity than the solution to the Stokes problem which allows

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2681

to obtain a solution for the nonlinear problem. Indeed, there is a certain price to pay. The asymptotic structure at infinity is anisotropic (due to the anisotropy of the fundamental Oseen tensor, see below), and therefore the corresponding function spaces are more complex than in the three-dimensional situation. Unlike the three-dimensional situation, the mathematical results concerning existence and asymptotics of solutions to the compressible Navier-Stokes equations in exterior two-dimensional domains are quite rare. In fact, the authors are aware of only two results in this direction: in [13] the authors consider the existence of solutions in the Lp -setting, while in [8] also the weighted estimates, leading to the expected spatial asymptotics, are proved. The aim of this part is to present the results from the latter. The former is only slightly different, involves also the external force (in the latter, it would require a few small changes to include it), but does not study the decay of the solutions. Note that the case when ˝ D R2 was considered in [7]. The problem in the full space is slightly easier, but the problems connected with spatial decay are similar as in the exterior domain situation. As far as the authors know, there is no result (in the weighted spaces) in the case of the Navier-Stokes-Fourier system (i.e., the system with the heat equation); anyway, in the case of regular solutions (small data) in exterior domains, one may expect that the result concerning the spatial decay of the velocity and density will be similar as in the isothermal case. For the sake of simplicity, in the part devoted to the two-dimensional flows, it is assumed that f 0. Indeed, one could formulate conditions on f under which the results of this part remain valid; see also [13] for an existence result (but without the decay properties). It is possible to look for a solution to (80), (81), and (82) (with (83)–(84)) in two steps. In the first one (the linear one), functions F and w are fixed and the following two mappings are considered: M W  7! ';

and

L W  7! ':

(85)

The operator M is defined as follows. To the given , .u; ˘ / is a solution to (80) (recall f 0); then taking ˘ from (80),  is a solution to (81); and, finally, for  from (81), the function ' is a solution to (82). The aim is to show the existence of a fixed point of M , which enters into the second part. There, the operator L is defined in the following way. Taking  D ' in (80), the pair .u; ˘ / is a solution to this Oseen system. With this ˘ , the function  is a solution to (81), and finally, with this  , the function ' is a solution to (82). Again, it is aimed at showing a fixed point of the operator L. Having shown successively the existence of fixed points to M and L, the corresponding .u; ';  / solves (80), (81), and (82) for  D  D '. Then it is possible to consider the nonlinear problem. Define the operator: N W . ; ; z/ 7! . ; '; u/;

(86)

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O. Kreml et al.

where . ; '; u/ is a solution to (80), (81), and (82) with  D  D ', w D zCr, and F D F. ; zCr/ as in (83). Indeed, the fixed point of N solves (77), (78), and (79). Hence it is important to study carefully all linear problems appearing in the formulation above. However, before doing so, the function spaces needed later are introduced. Note that similar spaces are considered also in the three space dimensions case; however, due to several small differences between the two- and the three-dimensional situations, it is better to introduce them separately.

3.1.1 Function Spaces Without loss of generality, it is assumed that the compact domain K contains the unit ball, and the boundary of ˝ is contained in a ball with radius R0 =4 for some 4 < R0 < 6. Under this assumption, for an R > 0 such that @˝ is contained in BR , one denotes ˝R D ˝ \ BR .0/ and ˝ R D ˝ n BR .0/. Furthermore, as above, n denotes the outer normal vector to @˝. Standard notation for the Lebesgue space Lp .˝/ endowed with the norm k  kp;˝ and Sobolev spaces W k;p .˝/ endowed with the norm k  kk;p;˝ is used. If no confusion may arise, the domain ˝ in the norm is skipped. The vector-valued functions are printed in boldface; however, there is no difference in the notation for the spaces of scalar-, vector-, or tensor-valued functions. Due to the nature of the problem, i.e., exterior domain (possibly with decay at infinity), it is necessary to use also more complicated spaces. First, denote by b 1;p .˝/ D C 1 .˝/krkp H 0 0

(87)

the homogeneous Sobolev space; here 1  p < 1. It is well known that for 1  p 0 such that   krkq C kr 2  kk;p  c jf j1;q C kf kk;p :

(89)

Neumann problem for Laplace equation. Next, consider ˇ' D div f; @' ˇˇ D  C f  n; @n ˇ@˝ r' ! 0 as jxj ! C1

(90)

with the compatibility condition Z  dS D 0: @˝

It holds (see [65])

(91)

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Lemma 2. Let k D 0; 1; : : : , 2 < s; p < 1, 1 < q < 1, ˝ 2 C kC3 and 1 f 2 Lp .˝/ \ Ls .˝/ with div f 2 W 2;q .˝/ \ W kC1;p .˝/, f  n 2 W 3 q ;q .@˝/ \ 1 1 1 1 W kC2 p ;p .@˝/. Let  2 W 1 s ;s .@˝/ \ W 3 q ;q .@˝/ \ W kC2 p ;p .@˝/. Then there exists a solution to problem (90)–(91) p

4;q

' 2 Lsloc .˝/ \ Lloc .˝/ \ Wloc .˝/; r' 2 Ls .˝/ \ W kC2;p .˝/;

r 2 ' 2 W 2;q .˝/

(92)

satisfying the estimate kr'ks C kr'kkC2;p C kr 2 'k2;q   c kfks C kfkp C kdiv fk2;q C kdiv fkkC1;p C kk1 1 ;s;@˝ s

 C kk3 1 ;q;@˝ C kkkC2 1 ;p;@˝ C kf  nk3 1 ;q;@˝ C kf  nkkC2 1 ;p;@˝ : q

p

q

p

(93)

Moreover, any other solution in the same class differs from ' by a constant. Oseen problem. Next, consider the Oseen problem 1 u C r˘ C 

@u D f; @x1

div u D 0; uj@˝ D u ; u ! 0;

˘ !0

as jxj ! C1:

(94)

Define also  Tij .u; ˘ / D 1

@uj @ui C @xj @xi

  ˘ ıij ;

Ti .u; ˘ / D 

Z

2 X

Tij .u; ˘ /nj dS:

@˝ j D1

(95) The following results (and many more concerning the Oseen problem in exterior domains) can be found in [12]. Lemma 3. Let ˝  R2 be an exterior domain of the class C 2 , f D 0, u 2 R 1 W 2 q ;q .@˝/, 1 < q < 32 and let @˝ u  n dS D 0. Then problem (94) admits unique solution in the class 3q

3q

u 2 L 32q .˝/; @u 2 Lq .˝/; @x1

ru 2 L 3q .˝/ \ L2 .˝/;

u2 2 L

2q 2q

.˝/;

˘ 2L

2q 2q

r 2 u 2 Lq .˝/;

.˝/;

r˘ 2 Lq .˝/:

(96)

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2687

Moreover, there exists 0 > 0 and c > 0 independent of  and u such that we have for all 0 <   0 the estimate jT .u; ˘ /j  cj ln j1 ku k2 1 ;q;@˝ :

(97)

q

Lemma 4. Let k D 0; 1; : : : and ˝  R2 be an exterior domains of the class 3q  p < C1,   0 , where 0 is the same as in C maxf3;kC2g . Let 1 < q < 65 , 32q 1

1

3 q ;q 1;q k;p Lemma .@˝/ \ W kC2 p ;p .@˝/ such R 3. Let f 2 W .˝/ \ W .˝/, u 2 W that @˝ u  n dS D 0. Then there exists a unique solution to the Oseen problem (94) such that 3q

3q

p

ru 2 L 3q .˝/ \ W kC1;p .˝/; @u ; ru2 2 Lq .˝/; r 2 u 2 W 1;q .˝/; @x1 2q 2q u2 2 L 2q .˝/; ˘ 2 L 2q .˝/ \ W kC1;p .˝/; r˘ 2 W 1;q .˝/: u 2 L 32q .˝/ \ Lloc .˝/;

(98)

Moreover, the following estimates hold  @u  2      C ku2 k 2q C kru2 kq C  3 kuk 3q 2q 32q q @x 1  1 C  3 kruk 3q C kr 2 uk1;q C krukkC1;p C kr˘ k1;q C k˘ kkC1;p C k˘ k 2q 3q   2q 1   c kfkk;p C kfk1;q C j ln j1 2.1 q / ku k3 1 ;q;@˝ C ku kkC2 1 ;p;@˝ ; q p (99) and  1  2.1 q / kr 2 uk1;q C kr 2 ukk;p C kr˘ k1;q C kr˘ kk;p   1  c kfkq C 2.1 q / .kfk1;q C kfkk;p / :

(100)

Steady transport equation. The last linear problem is the transport equation in ˝ in the general form h C w  r C a D g;

(101)

with w  n D 0 on @˝ and a, g given functions and the constant h > 0. It holds (see [34] for the proof; note that in [63] it was shown, using the Calderón extension theorem, that the existence result formulated below holds true for ˝ only a Lipschitz domain) Lemma 5. (i) Let 1 < p < 1, k D 1; 2; : : : , ˝  R2 be an exterior domain of class C k and g 2 W k;p .˝/. Let w 2 C k .˝/;

w  nj@˝ D 0;

a 2 C k1 .˝/;

ra 2 W k1;p .˝/

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O. Kreml et al.

with kp > 2. Then there exists #0 > 0 such that if #0 .jrwjC k1 C jajC k1 C krakk1;p / WD 0 < h; then there exists unique solution  to (101). Moreover, it holds k kk;p  1;p

kgkk;p : h  0

(102)

1;p

If g 2 W0 .˝/ and wj@˝ D 0, then  2 W0 .˝/. (ii) Together with the assumptions from part (i), let 1 < q < 2, l D 1; 2; : : : , l;q ˝ 2 C l and g 2 Wloc .˝/ with rg 2 W l1;q .˝/. Let also r l a 2 L2 .˝/. 0 Then there exists #0 > 0 such that if 0 C #00 .jrwjC l1 C jajC l1 C kr l ak2 / D 0 C 00 < h; then the solution from part (i) also satisfies r 2 W l1;q .˝/, and kr kl1;q 

krgkl1;q : h  0  00

(103)

If a D 0, then (103) holds for any 1 < q < 1. (iii) Let 1 < t < C1, ˝  R2 exterior domain of class C 1 and g 2 Lt .˝/. Let w 2 C 1 .˝/;

w  nj@˝ D 0;

a 2 C 0 .˝/:

Then there exists #000 > 0 such that for a solution  2 Lt .˝/ of (101) we have hk kt  kgkt C #000 .krwkC 0 C kakC 0 /k kt :

(104)

3.1.3 Fundamental Solutions and Weighted Estimates While the previous section contained all the necessary material for the proof of existence of a solution in the Sobolev spaces, this part will deal with the weighted estimates. The main tool will be the weighted estimates for integral operators containing fundamental solutions to the Oseen problem and to the Laplace equation. First, recall their asymptotic behavior. Denote E the fundamental solution to the Laplace equation, i.e., E D ı0

in D0 .R2 /:

It is well known that E.x/ D

1 ln jxj; 2

(105)

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2689

i.e., jr i E.x/j 

c ; jxji

i D 1; 2; : : : :

Moreover, it is well known that r 2 E is a singular kernel of the Calderón-Zygmund type; see [66] for more information. Next, consider the fundamental solution to the Oseen system, i.e., the solution to 1 Oij  

@Oij @Pi C D ıij ı0 ; @x1 @xj 2 X @Oij j D1

@xj

D0

(106)

in the sense of distribution, i; j D 1; 2. The asymptotic structure was carefully studied in [12]; see also [63] for some complementary information. It holds O.xI / D O.xI 1/;

r ˛ O.xI / D ˛ r ˛ .xI 1/;

˛ 2 N:

The main feature of the tensor O is that its asymptotic structure is anisotropic and it contains the so-called wake region. More specifically, recalling s.x/ D jxj  x1 ; it holds for jxj > 1 jO11 .xI /j 

1 2



c

jxj 1 C s.x//

1 2

;

ˇ @O .xI / ˇ c ˇ ˇ 11  ; ˇ ˇ @x2 jxj 1 C s.x// ˇ @O .xI / ˇ c ˇ ˇ ij ; ˇ ˇ 3 1 @xk jxj 2 1 C s.x// 2 jr 2 O.xI /j 

c2 3

jOij .xI /j 

c ; jxj

.i; j / ¤ .1; 1/;

.i; j; k/ ¤ .1; 1; 2/;

 32 ;

(107)

jxj 2 1 C s.x/ and for jxj < 1 jO.xI /j  c ln jxj;

jr k O.xI /j  cjxjk ; k D 1; 2 : : : :

(108)

Recall further that Pj .x/ D

@E.x/ : @xj

(109)

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Next consider solutions to problem (90)–(91). The following integral representation formula holds true r E rE .x/ C S1I' .x/ C Rf1I' .x/; r'.x/ D V1I' 2

where r EIf V1I' .x/ D 2

rE S1I' .x/

Z

Z ˝

(110)

rx2 E.x  y/f.y/ dy;

(111)



D @˝

 r'.y/rx E.x  y/  rx2 E.x  y/'.y/  n

C rx E.x  y/f.y/  n dSy ;

(112)

Rf1I' .x/ D c  f.x/:

(113)

Note that the volume integral in (111) is taken in the sense of the principal value. Similarly (for more details concerning the precise form of the terms, see [8]) r EIr r k '.x/ D VkI' 2

k2 div f

r E .x/ C SkI' .x/ C Rr kI' k

k2 div f

.x/;

k D 2; 3;

(114)

where again the first integral is considered in the sense of principal value, the second (the surface) integral decays at least as r k E.x/, and the third term contains r k2 div f at x. Similar formulas can be obtained for the Oseen problem. Let .u; ˘ / be a solution to (94) with f D div F. Then rO O ui .x/ D V0Iu .x/ C S0Iu .x/ C RO 0Iui .x/; i i

(115)

where rO .x/ V0Iu i

O S0Iu .x/ i

Z D

2  X

Z D

2 X @Oij .x  yI / Fj k .y/ dy; @xk

(116)

˝ j;kD1

   uj .y/Tj k Oi .x  yI /; Pi .x  y/

@˝ j;kD1

   Oij .x  yI /  Oij .xI / Tj k .u.y/; ˘.y//

C uj .y/Oij .x  yI /ı1k  Oij .x  yI /Fj k .y/ nk dSy ; RO 0Iui .x/ D

2 X j D1

Oij .xI /Tj .u; ˘ /:

(117)

(118)

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2691

Similarly @ui .x/ rO rO D V1I@ .x/ C S1I@ .x/ C RrO 1I@k ui .x/ k ui k ui @xk

(119)

with analogous meaning as above. Finally, rP P ˘.x/ D V0I˘ .x/ C S0I˘ .x/ C RF 0I˘ .x/;

(120)

where rP .x/ D V0I˘

P S0I˘ .x/

Z D

Z

2 X @Pi .x  y/ Fij .y/ dy; @xj ˝ i;j D1

(121)

2  X

Pi .x  y/Fij .y/ C Tij .u; ˘ /.y/Pi .x  y/ 

@˝ i;j D1

ui .y/Pi .x  yI /ı1j C Pi .x  y/ı1i uj .y/  21

@Pi .x  y/ ui .y/ nj dSy ; @xj (122)

and RF 0I˘ .x/ D

2 X

cij Fij .x/:

(123)

i;j D1

The volume integral is considered in the sense of the principal value. Similarly as above rP rP r F r k ˘.x/ D VkI˘ .x/ C SkI˘ .x/ C RkI˘ .x/; k

k D 1; 2:

(124)

More details are again in [8]. Indeed, all formulas are in the first step deduced for smooth compactly supported right-hand sides in the equations and then, by density argument, extended for less regular functions. For the weighted estimates of the volume integral operators with CalderónZygmund-type integral kernels (considered in the principal value sense), the following can be proved: Lemma 6. Let Z.x/ D

Z

r 2 E.x  y/v.y/ dy ˝

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in the sense of the principal value. Let 1 < q < 1,  q2 < ˛ < jxj˛ v 2 Lq .˝/. Then jxj˛ Z 2 Lq .˝/ and we have

2 , q0

q0 D

kjxj˛ Zkq  ckjxj˛ vkq :

q q1

and let

(125)

The proof can be found, e.g., in [66]. Next consider the integral operator with derivatives of the fundamental Oseen tensor. In the following, two types of integral operators will be needed: .X1 v/.x/ D

Z R2

@O11 .x  yI /v.y/ dy .X2 v/.x/ @x2

R2

@Oij .x  yI /v.y/ dy; @xk

Z D

where in the latter .i; j; k/ ¤ .1; 1; 2/. Denote for a; b 2 R ba .xI / D jxja .1 C s.x//b :

(126)

The following estimates hold (see [8] or [21]) Lemma 7. Let 0 <  < 12 , 12 < ı < 1   , 0 <  13 . Then for all functions v sufficiently smooth and sufficiently fast decaying at infinity it holds 3

1

k ı2 .xI /X1 vk1;R2  c1 k ı 2 .xI /vk1;R2 ; 2

2

1 2 

k0

.xI /X1

1

 1 2

1 2 ı 2

1C 2

vk1;R2  c k0

k .xI /X2 vk1;R2 C k0

.xI /vk1;R2 ;

.xI /X2 vk1;R2 ;

1

C k02 .xI /X2 vk1;R2  c1 kı1 .xI /vk1;R2 

1 2

k0

2

.xI /X2 vk1;R2  c1 k0  1 2

1 2 ı 2

k .xI /X2 vk1;R2 C k0

.xI /vk1;R2 ;

.xI /X2 vk1;R2 3

1

C k02 .xI /X2 vk1;R2  c1 k ı 2 .xI /vk1;R2 ; 2

1 2 

k0

1 2 ı 2

1C

2

.xI /X2 vk1;R2 C k0 1 2

1C 2"

.xI /X2 vk1;R2  c1 k0 3  22

k .xI /X2 vk1;R2 C k0 .xI /X2 vk1;R2  c1 k0

.xI /vk1;R2 ;

.xI /vk1;R2 : (127)

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2693

Next consider the boundary terms. For v a sufficiently smooth function define I01 .x/ D I02 .x/ D

Z jO11 .x  yI /jjv.y/j dSy ; @˝

Z jOij .x  yI /jjv.y/j dSy ; Z

ˇ @O .x  yI / ˇ ˇ 11 ˇ ˇ ˇjv.y/j dSy ; @x 2 @˝ Z ˇ ˇ ˇ @Oij .x  yI / ˇ 2 I1 .x/ D ˇ ˇjv.y/j dSy ; @xk @˝ Z jr 2 O.x  yI /jjv.y/j dy: I2 .x/ D I11 .x/ D

.i; j / ¤ .1; 1/;

@˝

.i; j; k/ ¤ .1; 1; 2/; (128)

@˝

The following result holds true. Lemma 8. Let a 2 C .@˝/,  2 .0; 0 /, R0 2 .4; 6/ be such that @˝  B R0 .0/. 4

For any x 2 ˝ R0 it holds 1 1 ;0ˇ ; 2 2

kˇ˛ .xI /I01 k1;˝ R0  ckakC .@˝/ ;

0 0 in ˝ and g 2 W k;p .˝/ implies g 2 W k;p .˝/. Let  2 W k;p .˝/ be a solution to (101) guaranteed by Lemma 5. Then there exists #1 > 0 such that if #1 .krwkC k1 C kakC k1 C kw  r ln kC k C 0 / WD 1 < h;

2694

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then  2 W k;p .˝/ and k kk;p 

kgkk;p : h  1

(130)

(ii) Let t , ˝, a, w and g satisfy assumptions of Lemma 5 (iii). Let  be a weight such that  2 C .˝/,  > 0 in ˝ and g 2 Lt .˝/ implies g 2 Lt .˝/. Let  be a solution to (101) guaranteed by Lemma 5. Then there exists #100 > 0 such that if #100 .krwkC 0 C kakC 0 C kw  r ln kC 0 C 000 / WD 100 < h; then  2 Lt .˝/ and k kt 

kgkt : h  100

(131)

3.1.4 Existence for the Linearized Problem in Sobolev Spaces Fixing F 2 L, w 2 V and  2 ˚ (recall that f 0), one aims first at showing the existence of a fixed point for the operator M , i.e., showing the existence of a solution to (80), (81), and (82) with  D '. Note that, without loss of generality, it can be assumed that 0 < v1 < 1. The following result holds. Proposition 1. Let k D 1; 2; : : : , ˝ 2 C kC3 be an exterior domain in R2 , 1 < q < 6 , 3q  p < C1, v1 D .v1 ; 0/ and let w 2 V , F 2 L, f D 0 and  2 ˚ . 5 32q Then there exists a positive constant ˛1 < 1 such that if   2 2 3  ˛1 ; v13 kwkV;v1 C v1

(132)

then there exists exactly one solution .˘ ;  ; '; u/ of systems (80), (81), and (82) with  D '. Moreover, ˘ ;  ; div . .w C v1 // 2 G; .'; u/ 2  ;

(133)

and we have the following estimates: k'k˚  c.kFk L C v1 kk˚ C v1 /; k˘ kG;v1 C k kG;v1 C kukV;v1  c kFkL C

2.1 1 /C1 v1 q j ln v1 j1 .1

 C kk˚ / : (134)

The constant c is independent of v1 .

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2695

Proof. One applies successively the estimates for each linear problem. Lemma 4 yields for (80)   2.1 1 / kukV;v1 C k˘ kG;v1  c kFkL C v1 q j ln v1 j1 .kk˚ C v1 / :

(135)

Next, Lemma 5 applied on (81) gives k kG;v1 C kdiv . .w C v1 //kG;v1   4 2.1 q1 /C1 1 3  c kFkL C v1 j ln v1 j .1 C kk˚ / C v1 kk˚ ;

(136)

1

provided v13 kwkV;v1 C v1 is sufficiently small. Furthermore, taking the divergence in the Oseen problem (80) and then the Laplace operator in the transport equation (81), one may deduce for Œ D kkq C kkk1;p that   2 ŒC div . .w C u1 //Œ c kFkL C v1 kk˚ C v13 kwkV;v1 k kG;v1 : (137) Finally, applying Lemma 2 on the Neumann problem for the Laplace equation (82)  k'k˚  c k .w C u1 /k

3q 32q

C k .w C u1 /kp

 C kdiv . .w C u1 //kkC1;p C kdiv . .w C u1 //k2;q C v1 :

(138)

To estimate r 2 div . .w C u1 //, we may use that div . .w C u1 // is zero on @˝, and therefore it is possible to control the second derivatives by the Laplacean (see Lemma 1), for which estimate (137) is available. Hence, one finally has

 k'k˚  c

 2 2.1 1 / 2 3 kFkL C v1 q j ln v1 j1 .kk˚ C v1 / v13 kwkV;v1 C v1   4 3 kk˚ C v1 kk˚ C kFkL C v1 : C v1

(139) Therefore, the mapping M is a contraction on ˚ , provided (indeed, the contraction can be shown in this step under less restrictive assumptions)   2 2 3 < 1: c v13 kwkV;v1 C v1 Estimates (134) follow from estimates stated above. Proposition 1 is proved.

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Next, concerning a fixed point for the operator L (i.e., a fixed point of  7! ' in (80), (81), and (82) with  D '), it holds: Proposition 2. Let k D 1; 2; : : : , ˝ 2 C kC3 be an exterior domain of R2 , 1 < q < 6 , 3q  p < C1, v1 D .v1 ; 0/ and let w 2 V , F 2 L, f D 0. Then there exists 5 32q a constant ˛2 < 1 (independent of v1 ) such that if 

2

2



3 v13 kwkV;v1 C v1

 ˛2 ;

(140)

then there exists a unique solution .˘ ;  ; '; u/ to (80), (81), and (82) (with  D  D '). Moreover, ˘ ;  ; div . .w C v1 // 2 G; .'; u/ 2  ;

(141)

and we have the following estimates: k'k˚  c.kFkL C v1 /; k˘ kG;v1 C k kG;v1 C kukV;v1  c kFkL C

!  2 1 q1 C1 1 : v1 j ln v1 j

(142)

Furthermore, we also have k'k˚ 0  c.kFkL0 C v1 /; k˘ kG 0 ;v1 C kkG 0 ;v1 C kukV 0 ;v1  c kFkL0 C

!  2 1 q1 C1 1 v1 j ln v1 j ; (143)

where the spaces ˚ 0 , L0 and V 0 are the same as ˚ , L and V , respectively, with k 1 instead of k. The constants c in (142)–(143) are independent of v1 . Proof. The proof can be done similarly as the proof of Proposition 1 and it is in fact a consequence of estimates (134). The contractivity of L follows provided cv1 < 1 (see (134)1 ) which is, for ˛1 small enough, ensured by (132). Proposition 2 is proved.

3.1.5 Existence for the Linearized Problem in Weighted Spaces In what follows, the solution for the linearized problem in the weighted spaces is constructed. The main result reads.

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2697

Proposition 3. Let k D 2; 3 : : : , ˝ 2 C kC3 be an exterior domain in R2 and let in addition to the assumptions of Proposition 2 F 2 L.p/ \ L.1/ for 0 < < 2 , 0 <  < 12 , max exists ˛3  ˛2 such that if

n

3q ;2 32q

o



2 v13 kwkV;

< p < 1,

2 3

C v1

1 2

< ı < 1   . Then there

  ˛3 ;

then the solution constructed in Proposition 2 satisfies ˘ ;  ; div . .w C v1 // 2 G;

' 2 F;

u 2 U:

(144)

Moreover, the solution fulfills  7 1  6 C2.1 6 /

j ln v1 j1 ; k'kF;v1  c kFkL.p/ ;v1 C kFkL C v1

(145)

and k˘ ;  kG;v1 C kukU;v1  c kFkL.1/ ;v1 C kFkL.p/ ;v1 C

 2 1 q1 v1 kFkL

! C v1 j ln v1 j

1

; (146)

where c is independent of v1 . Proof. The aim is to show that the operator L has also a fixed point in the spaces with decay. Hence it is necessary to establish estimates in the weighted spaces such that the operator L maps closed balls in these weighted spaces into themselves. Then, due to the uniqueness of the fixed point for L established in Proposition 2, the fixed point will also belong to those weighted spaces. The estimates can be divided into two parts: in a bounded domain near the boundary @˝ and far from it, i.e., @ui near the infinity. In what follows, in the estimates of the derivatives of u, @x always k means that .i; k/ ¤ .1; 2/.

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Estimates near the boundary. Due to Proposition 1 and Sobolev embedding theorem, one easily sees that 1

1



1

k ı2 .xI v1 /u1 k1;˝R0 C k0 2 .xI v1 /u2 k1;˝R0 C k02 .xI v1 /u2 k1;˝R0 2  1  1 @u1  @ui     

 C

C 02 .xI v1 / C 02 .xI v1 /   @x2 1;˝R0 @xk 1;˝   R0 1



6  cv1 1

1C2.1 q1 /

kFkL C v1

j ln v1 j1 .1 C kk˚ / ;



1C

1

2 jxj ˘ k1;p;˝R0 C kjv1 xj 2 r 2 ˘ kp;˝R0  kv1  1 1C2.1 q1 / 6 

kFkL C v1  cv1 j ln v1 j1 .1 C kk˚ / ; 1

1



6  cv1



2 jxj1 r'k2;p;˝R0 kv1



1C2.1 q1 /

kFkL C v1

 j ln v1 j1 .1 C kk˚ / :

(147)

Weighted estimates for boundary integrals. Using estimates (129) and combining them with estimates from Proposition 1 it holds 1



1 2

O k k ı2 .xI v1 /S0Iu R C k 0 1 1;˝ 0 2

1

1

O O 2 .xI v1 /S0Iu k R C k .xI v1 /S0Iu k1;˝ R0 0 2 1;˝ 0 2 1C"



rO O 2 .xI k1;˝ R0  v1 /S1I@2 u71 k1;˝1R0 C k0 .xI v1 /S0I@k ui  C2.1 /

6 q 1  c kFkL C v1 j ln v1 j .1 C kk˚ / ;

C k02

(148)

and 1

2



1 2

k ı2 .xI v1 /RO 0Iu1 k1;˝ R0 C k0 1

C k02



1

O 2 .xI v1 /RO 0Iu2 k1;˝ R0 C k0 .xI v1 /R0Iu2 k1;˝ R0 1C"

2 .xI v1 /RrO .xI v1 /RO 1I@2 u1 k1;˝ R0 C k0 0I@k ui k1;˝ R0  cjT .u; ˘ /j: (149)

In order to estimate the right-hand side of (149), one writes u D u1 C u2 , where u1 solves the Oseen problem with zero right-hand side, and u2 solves the Oseen problem with zero boundary conditions at @˝. Then due to Lemma 3 and Sobolev embedding estimates 2.1 q1 /

jT .u; ˘ /j  cj ln v1 j1 .v1 C k'k˚ / C cv1

kFkL

which leads to the estimate 1

2



1 2

k ı2 .xI v1 /RO 0Iu1 k1;˝ R0 C k0 1

Ck02



1

O 2 .xI v1 /RO 0Iu2 k1;˝ R0 C k0 .xI v1 /R0Iu2 k1;˝ R0 1C"

2 .xI v1 /RrO .xI v1 /RO 1I@2 u1 k1;˝ R0 C k0 0I@k ui k1;˝ R0  2.1 q1 /   c v1 kFkL C j ln v1 j1 .v1 C k'k˚ / :

(150)

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2699

Similarly, for the pressure from the Oseen problem and the potential part of the velocity 1

1





P 2 2 jxj1 S0I˘ kp;˝ R0 C kv1 jxj1 RF kv1 0I˘ kp;˝ R0 1

1





rP 2 2 C kv1 jxj1 S1I˘ kp;˝ R0 C kv1 jxj1 RrF 1I˘ kp;˝ R0 1C

1C

1C

1C

rP r F C kv12 jxj 2 S2I˘ kp;˝ R0 C kv12 jxj 2 R2I˘ kp;˝ R0   7 1 6 C2.1 q /

1  c kFkL C kFkL.p/ C v1 j ln v1 j .1 C kk˚ / ; 2

(151)

and 1

1





rE 2 2 jxj1 S1I' kp;˝ R0 C kv1 jxj1 RF kv1 1I' kp;˝ R0 1



1



1



r E F 2 2 C kv1 jxj1 S2I' kp;˝ R0 C kv1 jxj1 Rdiv 2I' kp;˝ R0 2

1

1



1

r E F 2 2 2 C kv1 jxj 2  S3I' kp;˝ R0 C kv1 jxj 2  Rrdiv kp;˝ R0 3I'   7 1 6 C2.1 q /

1  c k ; div . .w C v1 //kG;v1 C kFkL C v1 j ln v1 j .1 C kk˚ / ; 3

(152) where p >

2 .

Weighted estimates of volume integrals for u. Formulas (115), (116), (117), (118), and (119) together with Lemma 7 yield 1  1 rO 1 k ı2 .xI v1 /V0Iu kı .xI v1 /F11 k1 k  cv1 1 1 2

3 3  C k ı 2 .xI v1 /F12 k1 C k0 2 .xI v1 /F22 k1 ; 2



1 rO k0 2 .xI v1 /V0Iu k 2 1 3

 1 1 kı .xI v1 /F11 k1  cv1 2

C k ı 2 .xI v1 /F12 k1 C k0

 .xI v1 /F22 k1 ;

2

 1 rO 1 k  cv1 k0 .xI v1 /V0Iu kı .xI v1 /F11 k1 2 1 1 2

3 3  C k ı 2 .xI v1 /F12 k1 C k0 2 .xI v1 /F22 k1 ; 2

1 2 

k0

1C

2

k0

1C 2

rO 1 .xI v1 /V1I@ k  cv1 k0 2 u1 1

.xI v1 /div Fk1 ;

1C 2

rO 1 .xI v1 /V1I@ k  cv1 .k0 k ui 1

.xI v1 /div Fk1 :

(153)

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Hence, combining (147), (148), (150) and (153)   2.1 1 / 1 kFkL.1/ C j ln v1 j1 .v1 C k'k˚ / : kukU;v1  c v1 q kFkL C v1 (154) Weighted estimates of volume integrals for ˘ . Lemma 6 together with (121), (122), (123), and (124) yields 2 ; 1C

1 1 2 rP 2 

2 

kv1 ; jxj1 V1I˘ kp  ckv1 jxj1 div Fkp ; p > 1C

1C

1C

1C

1C

4 rP kv12 jxj 2 V2I˘ : kp  ckv12 jxj 2 div div Fkp ; p > 3

1

1





rP 2 2 jxj1 V0I˘ kp  ckv1 jxj1 Fkp ; kv1

p>

(155)

Hence, due to (147), (151) and (155) 

7 1 6 C2.1 q /

k˘ kG;v1  c kFkL C kFkL.p/ C v1



1

j ln v1 j .1 C kk˚ / : (156)

Weighted estimatesfor the transport equation. Applying Lemma 9, it is possible  1

to show (recall that v13 kwkV;v1 C v1 is “sufficiently small”)

k kG;v1 C kdiv . .w C v1 //kG;v1



7 1 6 C2.1 q /

 c kFkL C kFkL.p/ ;v1 C v1 kkF;v1 C v1

j ln v1 j

1



(157)

:

Weighted estimates of the potential part of the velocity. Lemma 6 yields 2 ; 1C

1 1 2 2 



r E 2 2 jxj1 V2I' kp  ckv1 jxj1 div . .w C v1 /kp ; p > kv1 ; 1C

1 1 2 r2E 2 

2 

kv1 : jxj1 V3I' kp  ckv1 jxj1 rdiv . .w C v1 //kp ; p > 1C

(158) Hence, using also (147), (152) and (157) 1



1



r E 2 2 jxj1 V1I' kp  ckv1 jxj1  .w C v1 /kp ; kv1 2



p>

7 1 6 C2.1 q /

k'kF;v1  c kFkL C kFkL.p/ ;v1 C v1 kkF;v1 C v1

j ln v1 j

1

 :

(159)

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2701

Conclusion. Taking v1 sufficiently small, estimates (159) and (134) together with the linearity of the problem yield the contraction of L in F \ ˚. Hence, the fixed point  D ' exists in these spaces. Due to the uniqueness and smallness of w, we have that . ; '; u/ belongs not only to G  ˚  U but also to G  F  U (and ˘ to G \ G).

3.1.6 Existence for the Nonlinear Problem This section contains the proof of the main results of this section, namely, that the mapping N from (86) possesses a fixed point in the spaces .G  ˚  U / \ .G  F  U/ which implies that problem (77), (78), and (79) has a solution. Hence, the original problem (72)–(73) has a solution with “perturbations” decaying at infinity. More precisely. Theorem 4 ([13]). Let k D 1; 2 : : : and ˝  R2 be an exterior domain in C kC3 , 3q 1 < q < 65 , 32q  p < 1. Then there exists 0 > 0 such that if 0 < v1 < 0 ; then there exists a constant ˛4 > 0 such that in the closed ball B˛4 D f. ; '; u/ 2 G  ˚  U; k. ; '; u/kG˚ U  ˛4 g there exists just one triplet . ; '; u/ solving (77), (78), and (79) (with f 0 and v D u C r'). Moreover, it holds k. ; '; u/kG˚ U 

 1C2 1 q1 cv1 j ln v1 j1

(160)

with c independent of v1 . The proof of this theorem can be found in [13]. As the proof is just a combination of the estimates for the linear problem and similar ideas as presented below for the weighted spaces, the proof will not be presented here separately. However, a similar result, under slightly stronger assumptions, will be shown during the proof of the following theorem. The main result in the spaces with decay reads as follows. Theorem 5 ([8]). Let k D 2; 3 : : : and ˝  R2 be annexteriorodomain in C kC3 , 3q 1 < q < 65 , 0 <  < 12 , 12 < ı < 1   , 0 < < 2 , max 32q ; 2  p < 1. Then there exists 1 > 0 such that if 0 < v1 < 1 ;

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then there exist positive constants ˛4 and ˛5 such that in the closed ball S˛4 ;˛5 D f. ; '; u/ 2 .G  ˚  V / \ .G  F  U/; k. ; '; u/kG˚ V;v1  ˛4 ; k. ; '; u/kGFU;v1 g  ˛5 g there exists just one triplet . ; '; u/ solving (77), (78), and (79) (with f 0 and v D u C r'). Moreover, it holds k. ; '; u/kGFU;v1  cv1 j ln v1 j1

(161)

with c independent of v1 . Proof. Recall that the aim is to prove the existence of a fixed point for the operator N : .G  ˚  V / \ .G  F  U/ ! .G  ˚  V / \ .G  F  U/ which assigns to the triple . ; ; z/ a triple . ; '; u/, where . ; '; u/ solves (80), (81), and (82) with  D  D ', w D z C r' and F D F. ; z C r/ with F as in (83). Taking 0 < v1 < 1 it is possible to verify that kFkL 

 12 1 q1 cv1



 1 1 C v13 k kG;v1 k. ; ; z/k2G˚ V;v1 ;

 1 kFkL.1/  c .1 C v13 k kG;v1 /k.; z/k2FU;v1 C

1 k.; z/kFU;v1 k kG;v1 .v13 kzkV;v1

 C kk˚ C v1 / ;

h 1   2 2 1 1 3 kFkL.p/  c v12 v1p .1 C v13 k kG;v1 /kwk2C 0 C v1 k kG;v1 .v13 kzkV;v1  3

1

C kk˚ / C v1 2 .v13 kzkV;v1 C kk˚ /k.; z/kFU;v1 k kG;v1 1

C .1 C v13 k kG;v1 /k.; z/k2FU;v1 C v1 k.; z/kFU;v1 k kG;v1   13 1 C .v1 kzkV;v1 C kk˚ / .1 C v13 k kG;v1 /k.; z/kFU;v1 C

1 .v13 kzkV;v1

C kk˚;v1 /k kG;v1 C v1 k kG;v1

i



(162)

with c independent of v1 . Propositions 2 and 3 guarantee existence of . ; '; u/ in .G ˚ V /\.GFU/ 2

2

3 provided .v13 kwkV;v1 C v1 / is sufficiently small; the smallness is given by a number independent of v1 . Moreover, there exist positive constants a and  such

1C2.1 q1 /

that for ˛4 D av1

, ˛5 D av1 , and 0 < v1 < , the mapping N maps

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2703

S˛4 ;˛5 D f. ; '; u/ 2 .G  ˚  V /  .G  F  U/; k. ; '; u/k.G˚U /;v1  ˛4 ; k. ; '; u/k.GFU/  ˛5 g into itself. It remains to verify that the mapping is a contraction in a suitable topology. Recall that we denoted by L0 , G 0 , ˚ 0 , D 0 , and U 0 the spaces with the same structure as L, G, ˚ , D, and U , respectively, with k  1 instead of k. Let . 1 ; ' 1 ; u1 / and . 2 ; ' 2 ; u2 / be two solutions of (80), (81), and (82) corresponding to . i ;  i ; zi / and Fi D div F. i ; zi C r i /, i D 1; 2. Denote . ; '; u/ D . 1 ; ' 1 ; u1 /  . 2 ; ' 2 ; u2 / and .; ; z/ D . 1 ;  1 ; z1 /  . 2 ;  2 ; z2 / and F D F1  F2 . Furthermore, denote div F D F and wi D zi C r i , i D 1; 2, and w D w1  w2 . Then the triple . ; '; u/ solves the following system of equations: 1 u C r˘ C v1

@u D F; @x1

div u D 0; u D r' on @˝; u ! 0; ˘ ! 0 as jxj ! C1;

(163)

  @'  .21 C 2 /div  2 w ; @x1 as x ! C1; (164)

 C .21 C 2 /div ..w C v1 // D ˘  v1  !0 and

    ' D div  .w1 C v1 /  div  2 w ; @' D0 on @˝; @n as jxj ! C1: jr'j ! 0

(165)

Note that system (163), (164), and   (165) differs from (80), (81), and (82) by the presence of the term div  2 w at the right-hand side of (164) and (165). For . i ;  i ; zi / 2 S˛4 ;˛5 , one can easily show that  kFk

L0

c

1 v13 ˛4 k.; ; z/kX;v1  43 2.1 q1 /

C v1

C

12.1 q1 / v1

˛42 k.; ; z/kX;v1

 1C

1 v13 ˛4



˛4 k. ; ; z/kX;v1  1 2 C v1 ˛5 k. ; ; z/kX;v1 ; (166)

where X D f. ; '; u/I  2 G 0 ; ' 2 ˚ 0 ; u 2 U 0 g;

(167)

endowed with the standard norm. Further, it also holds 2

k2 wkD 0  cv13 ˛4 k.; ; z/kX;v1 :

(168)

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The estimates for system (163), (164), and (165) are the same as for (80), (81), and (82), except for the additional term k 2 wkD 0 at the right-hand side, cf. [8, Theorem 4.2]. Therefore, it holds 12.1 q1 /

k. ; '; u/kX;v1  cv1



 1 1 C v13 ˛4 ˛4 k. ; ; z/kX;v1 :

(169)

Thus, under the assumptions of Theorem 5, there exist a > 0 and 1 > 0 such that 1C2.1 q1 /

for ˛4 D av1

, ˛5 D av1 , and 0 < v1 < 1 , it holds 12.1 q1 / cv1

 1C

1 v13 ˛4

 ˛4 < 1

which yields that the mapping N is a contraction in X . Since N maps closed balls in S˛4 ;˛5 into itself, also the fixed point in X must belong to this closed ball. Theorem 5 is proved.

3.2

Unbounded Three-Dimensional Domains

As already mentioned before, the three-dimensional problem is somewhat easier. It also had been solved earlier than the two-dimensional case and the number of results is higher than in two space dimensions. In what follows, the proofs are much less detailed than in two space dimensions, as they are similar to the two-dimensional situation and in a certain sense also easier. Note, however, that unlike the twodimensional case, two kind of results are available: either v1 is zero or nonzero (in this case, as above, without loss of generality, it is assumed that v1 D .v1 ; 0; 0/ with v1 > 0). The text below is based on two papers, [37] and [40]. However, before going into more details, let us briefly touch other results. The first result, in the full R3 space, can be found in [51]. The existence of solutions (without decay) was proved without the use of the decomposition method, similarly as in the paper [33], where several unbounded domains were considered (full space, half space, or exterior domain). A preliminary version of the paper [37] is the proceedings paper [52]. Some improved results for the decay for nonzero velocity at infinity can be found in [38, 39]. The existence of solution in exterior domains in Hölder continuous functions was established in [43]. The flow in a pipe under different conditions was considered in papers [55, 56], and [41]. The only result concerning spatial decay of solution to the compressible Navier-Stokes-Fourier system can be found in unfortunately finally unpublished preprint [42]. Some comments and further results can be also found in the overview papers [53, 54] and last but not least in the book [45].

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2705

3.2.1 Function Spaces As in the two-dimensional case, we assume that ˝ D R3 n K, where the compact simply connected domain K contains the unit ball and the boundary of ˝ is contained in a ball with radius R0 =4 for some 4 < R0 < 6. Under this assumption, ˝R D ˝ \ BR .0/ and ˝ R D ˝ n BR .0/ for an R > 0 such that @˝  BR .0/. Furthermore, n denotes the outer normal vector to @˝. The same convention is used for the Lebesgue and Sobolev spaces. The other spaces, introduced below, are sometimes similar to those introduced in two space dimensions. However, often some minor changes occur and thus the nonstandard spaces are introduced separately. b 1;p .˝/ D C 1 .˝/jj1;p with The homogeneous Sobolev spaces are defined as H 0 0 krkk1;p k;p 1 b .˝/ D C .˝/ b 1;p .˝/ the norm jj1;p D krk0;p and similarly H \H 0 0 0 with the norm krkk1;p . jj1;p

1 b 1;p Next H 1 .˝/ D C0 .˝/

b k;p H 1 .˝/

with the norm jj1;p D krk0;p and similarly

krkk1;p

D C01 .˝/ with the norm krkk1;p . ˚

p p b 1;p Finally, H .˝/ D u 2 L  loc .˝/; ru 2 L .˝/; uj@˝ D 0 with the norm jj1;p ˚

p k p b k;p and H  .˝/ D u 2 Lloc .˝/; ru; : : :; r u 2 L .˝/; uj@˝ D 0 with the norm krkk1;p . 0 b 1;p .˝/ is denoted Using the convention 1=p C 1=p 0 D 1, the dual space to H 0 0 b 1;p .˝/ with the norm jj1;p . The dual space to W 1;p .˝/ is denoted by by H 0

0

b 1;p W 1;p .˝/ with the norm kk1;p . Finally, the dual space to H 1 .˝/ is denoted by   0 1;p b 1 .˝/ with the norm jj;p . H For purposes of this section, it is useful to introduce composite Banach spaces. In what follows,  is a parameter which takes values 0 or 1 and  D 0 if v1 D 0 and  D 1 if v1 ¤ 0. For 1 < q < 3 < p < C1, k D 1; 2; : : : and l D 2; 3; : : :, the following notation is introduced: • Wk;q;p .˝/ D W 1;q .˝/ \ W k;p .˝/ with the norm kf kWk;p;q D kf k1;q C kf kk;p I • l;q;p

D

2;q

l;p

b .˝/ \ H b .˝/ .˝/ D H  

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with the norm kf kDl;q;p D krf k1;q C krf kl1;p I • 1

n o l;q;p l;q;p D;0 .˝/ D uI u 2 D .˝/; div uj@˝ D 0

with the norm kkDl;q;p ; • 1 n o l;q;p Kvl;q;p .˝/ D uI u 2 D .˝/; uj D v @˝ 1 .v /;0 1 1 l;q;p

l;q;p

is a convex closed subset of D.v1 /;0 .˝/ and in particular it holds K0

.˝/ D

l;q;p D0;0 .˝/; kC1;q;p

• for any u 2 Kv1

.˝/

˚

k;q;p WuCv1 .˝/ D I  2 Wk;q;p .˝/; div . .u C v1 // 2 Wk;q;p .˝/ is a Banach space endowed with the norm k kWk;q;p

uCv1

D kkWk;q;p C kdiv . .u C v1 //kWk;q;p I

• lC1;q;p

.˝/ D Wk;q;p .˝/  .D1;0 QkC1;q;p 1

4q

.˝/ \ L 4q .˝//

for v1 ¤ 0

and kC1;q;p

Q0

lC1;q;p

.˝/ D Wk;q;p .˝/  D0;0

.˝/

for v1 D 0

are Banach spaces with the norm 1=4 . ; v/ ΠkC1;q;p;v1 D kkWk;q;p C kvkDkC1;q;p C v1 kvk

4q 4q

• o n l;q;p Mvl;q;p .˝/ D . ; v/I . ; v/ 2 Q .˝/; vj D v @˝ 1 / .v 1 1 l;q;p

is a closed convex subset of Q.v1 / .˝/ and in particular l;q;p

M0

l;q;p

.˝/ D Wl1;q;p .˝/  D0;0 .˝/I

I

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

• 1

kC1;q;p

S0

2707

n kC1;q;p .˝/ D . ; v/I  2 Wk;q;p .˝/; v 2 D0;0 .˝/; o jxj v 2 L1 .˝/; jxj div v 2 Lq .˝/ \ Lp .˝/

is a Banach space with the norm . ; v/kC1;q;p D . ; v/ Œ kC1;q;p;0 C kjxj vk1 C kjxj div vkq C kjxj div vkp I • 1

b 1;q .˝/ \ Lq .˝/ \ W k;p .˝/ Lk;q;p .˝/ D H

is a Banach space with the norm kf kLk;q;p D jf j1;q C kf kq C kf kk;p I • for 1  q1
0. Denote (similarly as in two space dimensions) s.x/ D jxj  x1 : and

n 0 G D I  2 W lC1;q .˝/ \ W kC1;p .˝/; jxj1C  2 W 2;r .˝/; o 0 jxj2C =2 .1 C s 1=2 .x//r 2 Lr .˝/ ; n lC2;q b 1;q b 1;p ˚ D 'I ' 2 H .˝/ \ W kC2;p .˝/; 1 .˝/ \ H 1 .˝/; r' 2 W o 0 0 jxj1C r' 2 W 2;r .˝/; jxj2C =2 .1 C s 1=2 .x//r 2 ' 2 W 1;r .˝/ ;

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n lC1;q b 1;q b 1;p U D uI u 2 H .˝/ \ W kC1;p .˝/; 1 .˝/ \ H 1 .˝/; ru 2 W jxj .1 C s.x/ /u 2 L1 .˝/; jxj3=2 .1 C s.x/3=2 /ru 2 L1 .˝/; o 0 jxj1C r 2 u 2 Lr .˝/ ; G˚U D f. ; '; u/I  2 G; ' 2 ˚; u 2 Ug are Banach spaces with norms ( > 0)   0   kkG D kklC1;q C kkkC1;p C jxj1C   2;r   0   C jxj2C =2 .1 C 3=2 s 1=2 .x//r  ; r

  0   k'k˚ D kr'klC2;q C kr'kkC2;p C jxj1C r'  2;r   0   C jxj2C =2 .1 C 3=2 s 1=2 .x//r 2 '  ; 1;r

  0   kukU D kruklC1;q C krukkC1;p C jxj1C r 2 u r     C kjxj .1 C .s.x// /uk1 C jxj3=2 .1 C .s.x//3=2 /ru

1

;

k. ; '; u/kG˚U D kkG C k'k˚ C kukU : Note that the value of  plays a role of a scaling factor in the norm. However, if k kG is finite for some 0 > 0, then it is finite for any  > 0; similarly for the spaces ˚ and U. Finally, in order to describe spaces of functions with suitable boundary conditions, the following notation is used: n G˚U D . ; '; u/I  2 G; ' 2 ˚; u 2 U; div u D 0 o @' j@˝ D v1  n; 'j@˝ D 0; .u C r'/j@˝ D v1 @n Notice that G˚U is a closed convex subset of G˚U.

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2709

3.2.2 Existence for the Linearized System in Sobolev Spaces In three space dimensions, the situation is easier and we may study, instead of two separate operators M and L as in (85), the following linearization of (77) directly: @v D F; @x1 div v C div . .w C v1 // D 0:

1 v  .1 C 2 /rdiv v C r C v1

(170)

We consider this system with the boundary conditions (78). The main existence result reads as follows. Proposition 4 (Theorem 4.1 in [37]). Let C kC3 . Let

3 2

< q < 3 < p, k D 0; 1; : : :, ˝ 2

kC2;q;p

.˝/  D.v1 /;0 .˝/; w 2 KvkC2;q;p 1

F 2 Lk;q;p .˝/:

Then there exists a positive constant ˛1 depending only on k; q; p and @˝ such that if kwkDkC2;q;p C v1  ˛1 ; then there exists a unique solution kC2;q;p

. ; v/ 2 MvkC2;q;p .˝/  Qv1 1

.˝/

of problem (170) with boundary conditions (78) satisfying the inequality . ; v/ ΠkC2;q;p;v1  C .kFkLk;q;p C v1 /: The main idea of the proof of Proposition 4 is similar as in two space dimensions; however, as mentioned above, the proof can be done more directly. Again, the velocity part v of the solution is sought in the form of its Helmholtz decomposition v D u C r';

(171)

where div u D 0, u  n D 0 on @˝ and @' D v1  n on @˝. Therefore, the potential @n part of the velocity ' solves problem (82). Next, the velocity u solves the Oseen (or Stokes for v1 D 0) problem (80) with  D ', and, finally, the density perturbation  solves (81) with  D '. Hence, one considers the linear map L W  7!  as follows: • For a given , one solves a Neumann Laplace problem: ' D div . .w C v1 //; ˇ @' ˇˇ D v1  n; r'.x/ ! 0 as jxj ! 1 @n ˇ@˝

(172) (173)

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for an unknown function '. • Next, one solves for a given F the Oseen (or Stokes) problem (80) with  D ' for unknown functions .u; ˘ /. • Finally, having ˘ , one solves the transport equation (81) with  D ' for an unknown function  . It is clear that, similarly as in two space dimensions, one needs to have proper existence and uniqueness theorems for the linear problems involved in the decomposition described above. We have Lemma 10. Let the assumptions of Proposition 4 be satisfied and let  2 kC1;q;p WwCv1 .˝/ with k D 0; 1; : : : Then there exists a unique solution ' 2 b kC3;p .˝/ of problem (172)–(173) such that b 3;q H 1 .˝/ \ H 1 j'j1;

4q 4q

C kr'k2;q C kr'kkC2;p (174) h i  C jjwCv1 ;k1;q;p C kkWkC1;q;p .kwkDkC2;q;p C v1 / C v1

with j jwCv1 ;k1;q;p D jdiv . .w C v1 //j1;q C jdiv . .w C v1 //j1;p

(175)

C k kdiv . .w C v1 //kk1;p ; where 0 D 0 and k D 1 for k  1. Proof. See Lemma 6.1 in [37]. Having obtained ' using Lemma 10, one continues with a lemma concerning the solution to the Oseen (or Stokes for v1 D 0) system. Lemma 11. Let the assumptions of Proposition 4 be satisfied and let ' be the solution of problem (172)–(173) given by Lemma 10. Then there exists a unique kC2;q;p solution u 2 D1;0 .˝/, ˘ 2 WkC1;q;p .˝/ of problem (80) with  D '. 4q

Moreover, if v1 ¤ 0, then u 2 L 4q .˝/. The solution satisfies the estimate 1=4 v1 kuk

4q 4q

h C kukDkC2;q;p C k˘ kWkC1;q;p  C kdiv FkLk;q;p C j jwCv1 ;k1;q;p i C k kWkC1;q;p .kwkDkC2;q;p C v1 / C v1 (176)

with jjwCv1 ;k1;q;p defined in Lemma 10. Proof. See Lemma 6.2 in [37].

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2711

Finally, the following Lemma holds concerning solutions to the steady transport equation. Lemma 12. Let the assumptions of Proposition 4 be satisfied, let ' be the solution of problem (172)–(173) given by Lemma 10 and let .u; ˘ / be a solution of problem (80) with  D ' given by Lemma 11. Then there exists ˛1 > 0 depending on q; p; @˝ such that if kwkDkC2;q;p C v1  ˛1 ; kC1;q;p

then there exists a unique solution  2 WwCv1 with b 1;q .˝/ \ H b 1;p .˝/ \ W k1;p .˝/  ; div . .w C v1 // 2 H of problem (81) with  D ' satisfying the estimate h k kWkC1;q;p  C kFkLk;q;p C jjwCv1 ;k1;q;p C v1 wCv1

i C k kWkC1;q;p .kwkDkC2;q;p C v1 / ;

(177)

h jjwCv1 ;k1;q;p  C kdiv FkLk;q;p C v1 .j jwCv1 ;k1;q;p C v1 /

(178) i

C .kkWkC1;q;p C kkWkC1;q;p /.kwkDkC2;q;p C v1 / with jjwCv1 ;k1;q;p defined in Lemma 10. Proof. See Lemma 6.3 in [37].

Using Lemmas 10, 11, and 12, one can prove that the chain of maps  7! ' 7! .u; p/ 7!  defines linear map L such that kC1;q;p

kC1;q;p

L W WwCv1 .˝/ ! WwCv1 .˝/ kC1;q;p

for a given w 2 Kv1

(179)

satisfying

kwkDkC2;q;p C v1  ˛ 0

.0 < ˛ 0  ˛1 /;

(180)

where ˛1 is defined in Lemma 12. It is not difficult to observe that if there exists a fixed point  of L in kC1;q;p WwCv1 .˝/, then  and the corresponding v D u C r' solve the original system (170) with the boundary conditions (78) for fixed F and w.

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To conclude the proof, one shows the contractivity of the map L:  7!  in space kC1;q;p WwCv1 .˝/. This is a consequence of estimates for solutions of linear problems involved in the decomposition, stated in Lemmas 10, 11, and 12. For more details, see Section 6 in [37].

3.2.3 Existence for the Nonlinear System: The Case v1 D 0 The main theorem of this section is as follows (see [37, Theorem 4.2]). Theorem 6 ([37]). Let 1  q1 < 32 < q2 < 3 < p, k D 0; 1; : : :, ˝ 2 C kC3 k;p and let f 2 Lq1 ;q2 .˝/. Then for any 32 < q  2 there exist positive constants 0 ; 1 depending only on k; q; p; q1 ; q2 and @˝ such that if kfkLk;p  1 ; q1 ;q2

then in the ball n o kC2;q;p D . ; v/ 2 S .˝/; . ; v/   BkC2;q;p kC2;q;p 0 0 0 there exists a unique solution . ; v/ of problem (77), (78), and (79) with v1 D 0. Moreover this solution satisfies the estimate . ; v/kC2;q;p  C kfkLk;p : q1 ;q2

Proof. The proof relies heavily on the results from the previous section, namely, on Proposition 4. The contraction principle is applied again, this time to the map N defined formally as N W . ; w/ 7! F. ; w/ 7! . ; v/;

(181)

where . ; v/ is a solution to the linear system (77)–(78) with F D F. ; w/ given by Proposition 4. Here, F. ; w/ is defined by (79). It holds: kC2;q;p

k;p

into itself for sufficiently small ˛0 and the Lq1 ;q2 -norm (i) N maps the ball B0 of f. kC2;q;p b 1;2 .˝/ (ii) N is a contraction in B0 with respect to the topology of L2 .˝/  H 0 provided q  2. To prove (i), one first needs an estimate of F. ; w/ in terms of norms of  and w. It holds h kF. ; w/kLk;q;p  C .1 C kkWkC1;q;p / kjxj fk0;q1 C kfkk;p i C .kwkDkC2;q;p C kjxj wk0;1 /2 ; (182)

47 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows

2713

see Lemma 7.1 in [37]. Using this estimate, one can apply Proposition 4 to show the existence of the unique solution to the linear system (77)–(78) with F D F. ; w/. However, one needs to prove also an estimate of jxj v in L1 .˝/. To this aim, it is possible to use the properties of the fundamental solution of the Stokes problem .S; P/. The pair .S; P/ solves (106) with  D 0 in the sense of distributions. The explicit formulas for the fundamental solution are Sij .x/ D 

xi x j 1  ıij C ; 8 jxj jxj3

Pi D

@E 1 xi D ; @xi 4 jxj3

where E is, similarly to the two-dimensional case, the fundamental solution to the Laplace equation. It is easy to see that jS.x/j  C jxj1 ;

jrS.x/j  C jxj2 ;

jP.x/j  C jxj2 ;

(183)

and that r 2 S and rP represent singular integral kernels in the sense of CalderónZygmund. Using the representation formula for v in terms of the fundamental solution .S; P/ (similarly as in two space dimensions) together with estimates (183), one obtains the estimate for jxj v in L1 .˝/. For more details, see Lemma 7.2 in [37]. It is not difficult to finish the proof (point (ii)) by verifying that the mapping N b 1;2 .˝/; see Lemma 7.4 is a contraction with respect to the topology of L2 .˝/  H 0 in [37].

3.2.4 Existence for the Nonlinear System: The Case v1 ¤ 0 In the case of the nonlinear system (77), (78), and (79) with v1 ¤ 0, there is an existence result for the strong solutions in Sobolev spaces due to [37]. The authors proved the following (see [37, Theorem 4.3]) Theorem 7 ([37]). Let p > 3, 1 < r  65 , k D 0; 1; : : :, ˝ 2 C kC3 and let f 2 3r  q  2. Then there exists a positive constant 0 < 1, Lr .˝/ \ W k;p .˝/. Let 3r which depends only on r; q; k; p and @˝ such that for any v1 with 0 < v1 < 0 there exist positive constants 0 ; 1 depending only on r; k; q; p; @˝ and v1 with the following property. If kfk0;r C kfkk;p  1 ; then in the set ˚

MvkC2;q;p .0/ D . ; v/ 2 MvkC2;q;p .˝/; . ; v/ Œ kC2;q;p;v1  0 1 ;0 1

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there exists a unique solution . ; v/ of problem (77), (78), and (79) with v1 ¤ 0. Moreover this solution satisfies the estimate . ; v/ Œ kC2;q;p;v1  C .kfk0;r C kfkk;p C v1 /: Proof. The method of the proof of Theorem 7 is similar to the one of Theorem 6. The same mapping N defined in (181) is considered, and for fixed v1 with 0 < v1 < , it can be proved that: kC2;q;p

.˝/ (with the corre(i) N is well defined in certain closed convex set Mv1 sponding norm bounded by 0 ) provided , 1 .v1 / and 0 .v1 / are sufficiently kC2;q;p .˝/ into itself. small and that N maps this ball in Mv1 kC2;q;p (ii) N is a contraction in the ball in Mv1 .˝/ with respect to the topology of  b 1;2 .˝/ \ L4 .˝/ . L2 .˝/  H 0 One starts again with the estimate of F. ; w/. It holds (184) kF. ; w/kLk;q;p  C .1 C kkWkC1;q;p /.kfk0;r C kfkk;p /  C C 1 C jv1 j1=2 .1 C k kWkC1;q;p / . ; w/ Œ kC2;q;p;v1 ; for details see Lemma 8.1 in [37]. Point (i) is then an easy consequence of this estimate, while point (ii) is similarly as in the case v1 D 0 not difficult, for the complete proof see Section 8 in [37]. Later, the same authors proved the existence result involving also the information about the asymptotic structure of the solution; see [40, Theorem 5.3]. Theorem 8 ([40]). Let l D 1; : : :; k, k D 3; : : :, 0

6 , R0 1 0

3 2

< q < 3 < p, @˝ 2 C kC3 , 3q 3q

l;q

1 2


> 0 and v1 ¤ 0. Let f 2 L .˝/\W .˝/\W .˝/ such that supp f  BR0 .0/. Then there exist positive constants ˛00 ; ˛10 dependent on l; k; p; r; @˝; R0 ;  ;  0 but independent of v1 such that if jfj1;q C kfkl;q C kfkk;p C v1 < ˛10 then in the set ˚

B˛00 D . ; '; u/ 2 G˚U ; k. ; '; u/kG˚U  ˛00 there exists a unique . ; '; u/ such that . ; v D u C r'/ solves the nonlinear system (77), (78), and (79) with v1 ¤ 0. Moreover, the following estimate holds k. ; '; u/kG˚U  C .jfj1;q C kfkl;q C kfkk;p C v1 /

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with a constant C independent of v1 . In particular, v exhibits the wake region, i.e., jxj .1 C s.x/ /v 2 L1 .˝/;

 3 3 jxj 2 1 C s.x/ 2 rv 2 L1 .˝/

and 0

jxj1C  2 L1 .˝/: Proof. The proof is based on the decomposition to auxiliary linear problems and a contraction argument, similarly in the other theorems in this chapter. The main difficulty is to obtain the asymptotic behavior of the solution. To this end, one has to work in particular with fundamental solution to the Oseen problem. The fundamental solution .O; P/, a solution to (106) in D.R3 /, has the following properties. The pressure is again given by Pj .x/ D

@E 1 xj D @xj 4 jxj3

and the fundamental Oseen tensor enjoys the following properties O.xI / D O.xI 1/; ˇ ˇ i ˇr O.xI /ˇ  C

iC1 ; .1 C jxj/1Ci=2 . C s.x//1Ci=2

jxj  1;

for i D 0; 1; 2 and some fixed positive . Near the origin the fundamental Oseen tensor behaves like ˇ ˇ i ˇr O.xI /ˇ  C jxji 1 : As in the two-dimensional case, one may write the representation formulas for the solution .u; ˘ / to the Oseen problem (80). The technical core of the proof then lies in estimates of arising singular and weakly singular integrals in anisotropic weighted spaces which are similar to the two-dimensional case studied in the previous part. All necessary details are explained thoroughly in [40].

4

Conclusions

The chapter contains an overview of results concerning the existence of strong solutions to the compressible Navier-Stokes(-Fourier) system in bounded or unbounded domain. There exist a large number of results in this direction. It is not possible to explain in details all of them; therefore, only four results are commented in details (existence of strong solutions in bounded domains in the Lp setting for

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homogeneous boundary conditions, existence of a solution in bounded domains for nonzero inflow condition, and existence as well as spatial decay estimates in twoand three-dimensional exterior domain based on the decomposition method). Other results are only briefly mentioned.

5

Cross-References

 Local and Global Existence of Strong Solutions for the Compressible

Navier-Stokes Equations Near Equilibria via the Maximal Regularity  Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions  The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties Acknowledgements The second author (PBM) has been partly supported by the National Science Centre grant 2014/14/M/ST1/00108 (Harmonia). The work of the first (OK) and the third author (MP) was partially supported by the Grant Agency of the Czech Republic, Grant No. 16-03230S.

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Low Mach Number Limits and Acoustic Waves

48

Ning Jiang and Nader Masmoudi

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Dimensionless Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Mathematical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Compressible Navier-Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Dimensionless Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Low Mach Number Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Low Mach Number Limits: Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Well-Prepared Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 General Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Long Time Existence for the Compressible System . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Convergence in Critical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Low Mach Number Limits: Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Local Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Periodic Case: Averaged System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Boundary Layers and the Damping of Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Isentropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Non-isentropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Slip Boundary Condition and Moving Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Dispersion of Acoustic Waves: The Whole Space Case . . . . . . . . . . . . . . . . . . . . . . 6 Convergence Toward the Euler System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Whole Space Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Periodic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Non-isentropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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N. Jiang School of Mathematics and Statistics, Wuhan University, Wuhan, China e-mail: [email protected] N. Masmoudi Department of Mathematics, New York University in Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates Courant Institute of Mathematical Sciences, New York University, New York, NY, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_69

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7 Anelastic Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Anelastic Approximation of the Compressible Navier-Stokes System . . . . . . . . . . 7.2 Anelastic Approximations of Euler-Type Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2760 2761 2763 2766 2767 2767

Abstract

This review is devoted to the low Mach number limits of the Navier-Stokes equations for the compressible fluids in the context of weak solutions. Acoustic waves play a crucial role in these limits. However, in the torus, whole space, and bounded domain cases, the behaviors of acoustic waves are significantly different.

1

Introduction

Any physical system can be described by a system which governs the evolution of the different physical quantities such as the density, the velocity, the temperature, etc. The unknowns usually involve several physical units such as .m; kg; s; et c:/. Introducing some length scale, time scale, velocity scale, etc., the system of equations can always be written in a dimensionless form. This dimensionless form contains some ratios between the different scales such as the Reynolds number, the Mach number, or the ratio between two length scales. Indeed, the system may have different length scales. For instance, it may have a vertical length scale and a horizontal one.

1.1

Dimensionless Parameters

Writing the system in its dimensionless form allows one to compare the relative influence of the several terms appearing in the equations. Moreover, it allows one to compare different systems. For instance two incompressible flows with the same Reynolds number have very similar properties, even if the length scales, the velocity scales, and viscosities are very different. The only important factor of comparison is the ratio Re D U0L where U is the velocity scale, L is the length scale, and 0 is the kinematic viscosity, respectively. In hydrodynamics, asymptotic problems arise when a dimensionless parameter  goes to zero in a dimensionless system of equations describing the motion of some fluid. Physically, this allows a better knowledge of the system in the limiting regime by describing (usually by a simpler system) the prevailing phenomenon when this parameter is small. Indeed, this small parameter usually describes a physical reality. For instance, a slightly compressible flow is characterized by a low Mach number, whereas a slightly viscous flow is characterized by a high Reynolds number. Notice here that the terminology slightly compressible flow or slightly viscous flow is used instead of fluid. Indeed, this is a property of the flow rather than the fluid itself.

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However, the terminology slightly compressible fluid or slightly viscous fluid is often used to mean the properties of the flow. It is noticed that if the viscosity goes to zero, then the Reynolds number goes to infinity. But this is not the only way of getting a big Reynolds number. For instance, if L or U increases, then the Reynolds number also increases and we get the same properties as when the viscosity goes to zero. This is of course very important from a physical point of view since it is much easier to change L or U in a physical experiment than to change the viscosity. This shows the importance of the dimensionless parameters. So, when the inviscid limit is mentioned, this should be understood as the limit when the Reynolds number goes to infinity. Moreover, in many cases, there may have different small parameters (it can be in presence of a slightly compressible and slightly viscous fluid at the same time). Depending on the way these small parameters go to zero, different systems can be recovered at the limit. For instance, if ; ı; ;   1, the limit system can depend on the magnitude of the ratio of =ı or =    . This again shows the importance of having dimensionless quantities which can be compared. The study of these asymptotic problems allows one to get simpler models at the limit, due to the fact that the limiting models usually have fewer variables or (and) fewer unknowns. This simplifies the numerical simulations. In fact, instead of solving the original system, only the limit system with a corrector is needed to be solved.

1.2

Mathematical Problems

Many mathematical problems are encountered in justifying the passage to the limit. These problems are mainly due to the change of the type of the equations, the presence of many spatial and temporal scales, the presence of boundary layers (usually the same boundary conditions are no longer allowed to be imposed for the initial system and the limit one), the presence of oscillations in time at high frequency, and many other effects. Usually, a singular limit means that there is a change of the type of the equations. For instance, in the inviscid limit (Reynolds number going to infinity), it goes from a parabolic equation to a hyperbolic equation. However, this terminology seems a little bit restrictive since; it can be seen from the examples that it is not usually easy to give a type to each system of equations. Indeed, a singular limit becomes more involved if there is a reduction of the number of variables or unknowns due to more restrained dynamics. Different types of questions can be asked: (1) What do the solutions of the initial system .S / converge to? Is the convergence strong or weak? (2) In the case of weak convergence, can a more detailed description of the sequences of solutions be given? Can the time oscillations for instance be described? (3) Can some properties of the limit system be used to deduce properties for the initial system when the parameters are small?

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In this review article, some of these questions will be answered by studying the low Mach number limits, more specifically, the incompressible limits of the compressible Navier-Stokes equations, and some related topics, such as anelastic approximation. In the next subsection, first the physical equations of fluid dynamics will be recalled and then several dimensionless parameters will be introduced.

1.3

The Compressible Navier-Stokes System

In this subsection, the compressible Navier-Stokes (CNS) system for a Newtonian fluid is recalled, and several dimensionless parameters used in the later sections are introduced. The CNS reads 8 @ ˆ ˆ < @t C div.u/ D 0 ;   0; @u (1) C div.u ˝ u/  div.2D.u//  r.divu/ C rp D f; @t ˆ ˆ @" : C div.u"/ C pdivu  div.kr / D 2jD.u/j2 C .divu/2 : @t

In the above system, t  0 is time variable, the divergence operator div and the gradient operator r only act in the x variable and x 2 RN ; N  2. Moreover , u, p, ", and  are, respectively, the density, the velocity, the pressure, the internal energy by unit mass, and the temperature of the fluid. Besides,  and  are the socalled Lamé viscosity coefficients and satisfy the relation   0, N C2  0. The coefficient k is the thermal conduction coefficient and satisfies k  0. In general ; , and k can depend on the thermodynamical functions and their gradients. Finally, f is the force term. For geophysical flows, we will consider a force which is the sum of the gravitational force and the Coriolis force, namely, f D g C ˝e  u where ˝ is the rotation frequency and e is the direction of rotation. It is also denoted by g D jgj. The system (1) must be closed by the thermodynamic state equations, namely, p D p.; / and " D ".; /. For an ideal gas, these functions are given by (

" D cv 

(2)

p D R

where R > 0 is the ideal gas constant, cv > 0, and cp D R C cv . The constant cv and cp are, respectively, the specific heats at constant volume and constant pressure. The adiabatic constant is defined as D cp =cv . The system formed by (1) and (2) is closed. There is an other important thermodynamical function, namely, the entropy. It is defined by the following thermodynamic relation  d D

@" d C @



@" p  2 @ 

 d:

(3)

48 Low Mach Number Limits and Acoustic Waves

For an ideal gas, (3) yields

@

@

D

cv 

and

@

@

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D  R . Hence is given by D

cv log  1 . In particular, the third equation of (1) can be replaced by an equation for the entropy, namely, @

1 2jD.u/j2 C .divu/2 C div.u / D div.kr / C : @t  

(4)

If  D  D 0 and k D 0, then (4) reduces to a transport equation and that if the entropy is constant initially, namely, D 0 , then it remains constant at later

0

times. In this case,  D e cv  1 and p D Re cv  . This yields the compressible isentropic Euler system. Another model that will be mainly concerned in this review is the isentropic compressible Navier-Stokes system (13). It corresponds to the case kD0, is constant and the variation of due to the viscous effects will be neglected. However, (13) can not be rigorously derived from (1) in any asymptotic regime.

1.4

Dimensionless Parameters

Now some different dimensionless parameters are defined as the following: t , L, U , , and P denote, respectively, the characteristic time scale, the characteristic length scale, the characteristic velocity scale, the characteristic density scale, and the characteristic pressure scale. This means that the time or length can be made dimensionless by dividing it by t or L, i.e., dimensionless time and dimensionless length can be defined by tQ D tt and xQ D Lx . All the other quantities can also be done in the same way. Moreover, the characteristic values of  and k are denoted by  and k. The Strouhal number and Reynolds number are defined by St D

L tU

(5)

Re D

LU : =

(6)

A small Strouhal number S t corresponds to the longtime behavior of a system, while a large Reynolds number Re corresponds to small viscous effects. The acoustic waves propagate at the sound speed which is given in the isentropic D Rt . Consequently the Mach number is defined as the ratio case by c 2 D @p @ between U and c, namely, Ma D

U U : Dp c Rt

(7)

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When M a < 1, the flow is subsonic and, when M a > 1, the flow is supersonic. Both the velocity and the temperature satisfy a diffusion equation with a  diffusivity given, respectively, by  and Ckv  . The ratio between these two numbers is the Prandtl number Pr D

1.5

Cv  k

D

Cp  k

:

(8)

Low Mach Number Limits

It is well-known from a fluid mechanics viewpoint that one can derive formally incompressible models such as the incompressible Navier-Stokes or Euler system from compressible models, namely, compressible Navier-Stokes system (CNS) when the Mach number goes to 0 and the density becomes constant. There are several mathematical justifications of this derivation. One can put these works in two categories depending on the type of solutions considered. Indeed, one viewpoint consists on looking at local strong solutions and trying to prove existence on some time interval independent of the Mach number and then studying the limit when the Mach number goes to zero. This was initiated by Klainerman and Majda [40] (see also Ebin [15]). The second point of view consists on retrieving the Leray global weak solutions [43, 44] of the incompressible Navier-Stokes system starting from global weak solutions of the compressible Navier-Stokes system (see [48]). There have been many works about this limit during the last 20 years and that there are many review papers about it (see for instance [3, 11, 25, 51, 54, 62]). First, we recall the general setup for such asymptotic problems. For the simplicity of the presentation and, on the other hand, for that the existence theory for the full compressible Navier-Stokes system is not complete so far, we will present for the isentropic system. The unknowns .; Q v/ are, respectively, the density and the velocity of the fluid (gas) and solve on .0; 1/  RN (

@Q C div .v/ Q @t @v Q C div .v Q @t

D 0;

Q  0 ;

Q ˝ v/  v Q  rdivv C ar Q D 0 ;

(9)

where N  2, Q > 0 , Q C Q > 0; a > 0, and > 1 are given. From a physical viewpoint, the fluid should behave (asymptotically) like an incompressible one when the density is almost constant, and the velocity is small relative to sound speed. So the isentropic compressible Navier-Stokes system (9) should be scaled at large time scale. More precisely,  and v (and thus p) are scaled in the following way

48 Low Mach Number Limits and Acoustic Waves

2727

Q D .t; x/; v D u.t; x/

(10)

and the viscosity coefficients  and are also assumed small and are scaled like Q D  ; Q D  

(11)

where  2 .0; 1/ is a “small parameter” and the normalized coefficient  ;  satisfy  !  ;  !

as  goes to 0C :

(12)

It shall always be assumed that either  > 0 and  C > 0 or  D 0. With the preceding scalings, the system (9) yields (

@ C div. u / D 0 ; @t @ u C div. u ˝ u / @t

  0;   u   rdivu C

a r 2

D0:

(13)

The following is the explanation of the heuristics which lead to incompressible models. First of all, the second equation (for the momentum  u ) indicates that  should be like N C O. 2 / where N is a constant. Of course, N  0 and it is always assumed that N > 0 (in order to avoid the trivial case N D 0). Obviously, it needs to assume this property holds initially (at t D 0), and by a simple (multiplicative) scaling, it may always be assumed without loss of generality that N D 1. Since  goes to 1, it is expected that the first equation in (13) yields at the limit: div u D 0. By writing r D r.  1/, it can be deduced from the second equation in (13) that in the case when  > 0 @u C div.u ˝ u/  u C r D 0 @t

(14)

@u C div.u ˝ u/ C r D 0 @t

(15)

or when  D 0



where is the “limit” of 1 2 . In other words, the incompressible Navier-Stokes equations (14) or the incompressible Euler equations (15) are recovered, and the hydrostatic pressure appears as the limit of the “renormalized” thermodynamical pressure 1 2 . In this review paper, the derivation of (14) (or (15)) will be justified in the context of both strong solutions and weak solutions. But we will focus more on weak solutions. In particular, it should be emphasized that for the different boundary conditions, say periodic domain, the whole space, and boundary domain, the significantly different behaviors of acoustic waves result the different types of convergence.

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2

Low Mach Number Limits: Strong Solutions

2.1

Well-Prepared Initial Data

The first mathematical justification of the incompressible limit was due to Ebin [15]. By using Lagrangian coordinates and a geometric description of the equations, he proved that “slightly compressible fluid motion can be described as a motion with a strong constraining force, while incompressible fluid flow is the analogous constrained motion.” The first justification using PDE methods was done by Klainerman and Majda [40, 41] using the theory of singular limits of symmetric hyperbolic systems. The work of Kreiss [42] about problems with different time scales should also be mentioned. His work required the control of more time derivatives at time t D 0. The compressible Euler system can be recovered from (13), by taking  D  D 0, ( @ C u  r C  divu D 0 ;   0; @t  (16)   @u C u  ru C 12 rp D 0 @t

where p and  are related by p D a where a > 0 and  1 are given constants. In this section the above system is considered in the torus or the whole space, i.e., ˝ D TN or ˝ D RN with the following initial data u .t D 0; x/ D u0 .x/;

p .t D 0; x/ D p0 .x/:

(17)

Notice that the initial data for  can be retrieved from the initial data for p . Here k:ks will denote the H s norm and s0 D ΠN2  C 1. Theorem 1. Assume the initial data (17) satisfy 1 ku0 .x/ks C kp0 .x/  pks  C0 

(18)

for some constants p > 0 and C0 and some s  s0 C 1. Then there exists an 0 and a fixed time interval Œ0; T  with T depending only upon ku0 .x/ks0 C1 C 1 kp0 .x/  pks0 C1 and a constant Cs such that for  < 0 , a classical solution  of the compressible Euler system exists on Œ0; T   ˝ and satisfies 1 @u @p sup ku ks C kp0  pks C k ks1 C k ks1  Cs :  @t @t 0tT

(19)

Moreover, if the initial data satisfy the additional condition u0 .x/ D u0 .x/ C u1 .x/;

divu0 D 0;

p0 .x/ D p C  2 p 1 .x/; 1

1

ku .x/ks C kp .x/ks  C0 ;

(20)

48 Low Mach Number Limits and Acoustic Waves

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then, on the same time interval Œ0; T , sup k 0tT

@u @p ks1 C  1 k ks1  Cs1 @t @t

(21)

and as  goes to 0, u converges weakly in L1 .Œ0; T I H s / and uniformly in Cloc .Œ0; T   ˝/ to u1 where u1 satisfies the incompressible Euler system 

@u1 @t 1

C u1  ru1 C rp 1 D 0 u .t D 0; x/ D u0 .x/; divu1 D 0:

(22)

The condition (20) means that the flow is initially almost incompressible and that the density is initially almost constant. These data are called “well-prepared” initial data. The more general condition (18) will be called general initial data or “ill-prepared” initial data. It is noticed that p0  p is still needed to be assumed of order . This is because the change of variable q D  1 .p  p/ is needed so that the system can be written in a form which is suitable for energy estimates. It will be denoted by q0 D  1 .p0  p/. Idea of the proof. The system is rewritten in terms of the new unknowns .u ; q / where q D  1 .p  p/, (

@q @t @u @t

C u  rq C  .p C q /divu D 0 ; 1 C u  ru C .pCq rq D 0 : /1=

  0;

(23)



To prove (19), it is sufficient to prove H s -estimates on some time interval Œ0; T  which is independent of . For each , it is denoted by Es .t / D

Z X j˛jDs

1 j