Fluids Under Control (Advances in Mathematical Fluid Mechanics) 303147354X, 9783031473548

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Advances in Mathematical Fluid Mechanics

Tomáš Bodnár Giovanni P. Galdi Šárka Necˇ asová  Editors

Fluids Under Control

Advances in Mathematical Fluid Mechanics Series Editor Giovanni P. Galdi, Dept. Mechanical Engineering, University of Pittsburgh, Pittsburgh, PA, USA

The Advances in Mathematical Fluid Mechanics series is a forum for the publication of high-quality, peer-reviewed research monographs and edited collections on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations and other significant viscous and inviscid fluid models. Titles in this series consider theoretical, numerical, and computational methods, as well as applications to science and engineering. Works in related areas of mathematics that have a direct bearing on fluid mechanics are also welcome. All manuscripts are peer-reviewed to meet the highest standards of scientific literature.

Tomáš Bodnár • Giovanni P. Galdi • Šárka Neˇcasová Editors

Fluids Under Control

Editors Tomáš Bodnár Department of Technical Mathematics Czech Technical University Prague, Czech Republic

Giovanni P. Galdi University of Pittsburgh Pittsburgh, PA, USA

Šárka Neˇcasová Institute of Mathematics Czech Academy of Sciences Prague, Czech Republic

ISSN 2297-0320 ISSN 2297-0339 (electronic) Advances in Mathematical Fluid Mechanics ISBN 978-3-031-47354-8 ISBN 978-3-031-47355-5 (eBook) https://doi.org/10.1007/978-3-031-47355-5 Mathematics Subject Classification: 76-xx, 76Dxx, 76D05 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The book is based on lectures of invited speakers at a summer school “Fluids under Control,” held in Prague (Czech Republic) in August 23–27, 2021, and organized by Tomáš Bodnár (Czech Technical University in Prague), Šárka Neˇcasová (Institute of Mathematics of the Czech Academy of Sciences), and Giovanni Paolo Galdi (University of Pittsburgh, USA). The summer school followed previous schools on various aspects of the mathematical fluid mechanics, held in Prague in past decade. It represented the sixth continuation of the series called the Prague-Sum. The aim of the summer school “Fluids under Control” was to offer the participants (i.e., graduate and postgraduate students, young scientists, and other interested specialists) a comprehensive series of lectures on various topics and problems related to fluids flows control and its role in mathematical analysis and numerical simulation. The summer school was organized as a multidisciplinary event and the presented lectures covered a wide range of topics, starting from mathematics up to physics and technical applications. A special attention was paid to models heading toward environmental, biomedical, and industrial applications. Prague, Czech Republic Pittsburgh, PA, USA Prague, Czech Republic

Tomáš Bodnár Giovanni P. Galdi Šárka Neˇcasová

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Contents

1

On the Stabilization Problem by Feedback Control for Some Hydrodynamic Type Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. V. Fursikov 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Setting of the Problem and the Main Idea of the Method. . . . . . . . . . . . 1.2.1 Stabilization Problem for a Simple Parabolic Equation . . . . 1.2.2 Setting of the Stabilization Problem . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Feedback Control: Previous Remarks . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 The Main Idea of Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Oseen Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Structure of Rk with k < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Holomorphic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Unique Continuation Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (k) 1.3.5 On Linear Independence of εl (x, −λj ) . . . . . . . . . . . . . . . . . . . 1.4 Stabilization of Oseen Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Setting of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Theorem on Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Result on Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Stabilization of 3D Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Invariant Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Extension Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Theorem on Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 On Non-local Stabilization of Hydrodynamic Type System. Semi-linear Parabolic Equation of Normal Type . . . . . . . . . . . 1.6.1 Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Helmholtz Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Derivation of Normal Parabolic Equations (NPEs) . . . . . . . . 1.6.4 Explicit Formula for Solution of NPE . . . . . . . . . . . . . . . . . . . . . .

1 1 6 6 9 10 11 13 13 15 16 17 18 19 19 20 26 27 27 30 33 34 35 36 36 38

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1.7

Stabilization of Solution for NPE by Starting Control . . . . . . . . . . . . . . . 1.7.1 Formulation of the Main Result on Stabilization . . . . . . . . . . . 1.7.2 Formulation of the Main Preliminary Result . . . . . . . . . . . . . . . 1.7.3 Intermediate Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.4 Proof of the Stabilization Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Burgers Equation and Corresponding Semi-linear Parabolic Equation of Normal Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Derivation of the Normal Parabolic Equation (NPE). . . . . . . 1.8.2 Explicit Formula for a Solution of NPE . . . . . . . . . . . . . . . . . . . . 1.8.3 Dynamic Structure Generated with NPE . . . . . . . . . . . . . . . . . . . 1.8.4 Formulation of the Main Result on Stabilization . . . . . . . . . . . 1.9 Differentiated Burgers Equation and Functional-Polar Coordinates (fpc) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 On Solvability of the Differential Burgers Equation for Small Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Functional-Polar Coordinates (fpc) . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.3 Contour Lines of the Functional Ψ on the Sphere Σ(1) . . . 1.10 Construction of a Stabilizing Impulsive Control for Differentiated Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Unique Continuation Properties of Static Over-determined Eigenproblems: The Ignition Key for Uniform Stabilization of Dynamic Fluids by Feedback Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Triggiani 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A First (Informal) Quantitative Description of the Strategy for Uniform Stabilization of Linear Unstable Parabolic Dynamics on a Bounded Multidimensional Domain [97] . . . . . . . . . . 2.3 Part I: Kalman Rank Conditions [48, 49] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Second-Order Elliptic Operators. Linear Independence of Interior Localized Eigenfunctions. . . . . . . . 2.3.2 Linear Independence of Boundary Traces of Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Implications of Linear Independence of Interior Localized Eigenfunctions to the Problem of Dirichlet Boundary Feedback Stabilization of Parabolic Problems. Verification of Kalman Rank Condition [48, 49] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Implications of Linear Independence of Localized Boundary Traces of Eigenfunctions to the Problem of Neumann Boundary Feedback Stabilization of Parabolic Problems. Verification of Kalman Rank Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38 38 39 41 43 46 46 48 49 50 51 51 51 54 56 59

63 63

65 68 68 71

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Relevant Unique Continuation Properties for Over-determined Oseen Eigenvalue Problems. Part IIA: Emphasis on the Localized Interior Case . . . . . . . . . . . . . . . . . . 2.4.1 Oseen Eigenproblems. Unique Continuation of the Oseen Equations from an Arbitrary Interior Subdomain. Linear Independence of Interior Localized Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Proof of Theorem 2.4.1 [102]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Implications of the UCP of Theorem 2.4.1: (i) Linear Independence of Interior Localized Eigenfunctions; (ii) Verification of the Corresponding Kalman Rank Condition . . . . . . . . . . . . . . . . . . . . 2.4.4 Feedback Stabilization of the Unstable Oseen Dynamical System by Finite-Dimensional Localized Interior Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Relevant Unique Continuation Properties for Over-determined Oseen Eigenvalue Problems. Part IIB: Emphasis on the Localized Boundary Case . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 A First “Minimally Invasive” Attempt by Use of a Tangential Boundary Control Action on an Arbitrarily Small Portion  𝚪 of the Boundary (Implementable by Jets of Air) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Second Attempt: “Minimal” Extra Condition to be Added to the Boundary Control v Acting on  𝚪 ⊂ 𝚪 .... 2.5.3 Helmholtz Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Implication of the UCP of Lemma 2.5.1 on the Validity of the Corresponding Kalman Rank Condition for Problem (2.144) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Justification of (2.149) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Part III: The Boussinesq Problem, d = 2, 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Localized Interior Feedback Control Pair {u, v} . . . . . . . . . . . 2.6.3 Main UCP Results for Both the Adjoint Eigenproblem, i.e., the Operator A∗q ; and the Original Eigenproblem, i.e., the Operator Aq . . . . . . . . . . . . . . 2.6.4 Proof of Theorem 2.6.3 via Pointwise Carleman Estimates [104] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Localized {interior = u, ¯ boundary = v} Pair of Feedback Controls, v ∈  𝚪 ⊂ 𝚪, u¯ ∈ ω, Supported by  𝚪 [69] . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3

80

81 82

97

98

107

107 109 114

114 118 122 122 124

127 129 143 150

Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 I. Lasiecka and J. T. Webster 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

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3.2

3.3

3.4

3.5 3.6

3.7

3.8

3.9

3.1.1 Motivation and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Flutter Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 More on Flutter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Mathematical Challenges in Studying Flutter . . . . . . . . . . . . . . 3.1.5 Overcoming the Challenges and Overall Strategy . . . . . . . . . . PDE Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Obtaining the PDE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Setup and Well-posedness of Weak and Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 System Under Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Energies and Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Well-posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long-Time Behavior of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Statement of the Main Result and Qualitative Demonstrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Further Discussion of Dissipation and the Effects of Rotational Inertia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline of the Stability Proofs in Theorems 3.4.1 and 3.4.2. . . . . . . . . Plate-to-Flow Mapping: Properties and Discussion . . . . . . . . . . . . . . . . . . 3.6.1 The Neumann Wave Equation on the Half-space. . . . . . . . . . . 3.6.2 Flow Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Dynamical Systems Framework for the Plate . . . . . . . . . . . . . . Discussion of Past Stabilization Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Re-statement of Main Stabilization Result . . . . . . . . . . . . . . . . . 3.7.2 Weak Stability and Smooth Data for α = 0 . . . . . . . . . . . . . . . . 3.7.3 Other Past Stability Results with α = 0 . . . . . . . . . . . . . . . . . . . . 3.7.4 General Approach to Stability in the Remainder of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . α > 0: Rotational Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Existence of Attracting Set for the Structure: Statement . . . 3.8.2 Attractor Construction: Dissipativity of Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Attractor Construction: Asymptotic Smoothness Through Quasi-stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.4 Boundedness and Finiteness of Dissipation Integral . . . . . . . 3.8.5 Plate Convergence: Weak and Strong . . . . . . . . . . . . . . . . . . . . . . . 3.8.6 Lifting from Plate to Flow Directly . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.7 Strong Convergence to Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . α = 0; Non-Rotational Stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1 Existence of Structural Attractor: Statement . . . . . . . . . . . . . . . 3.9.2 Additional Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 158 160 163 165 166 166 167 169 169 170 170 171 173 176 177 180 181 182 182 183 187 189 190 191 193 198 199 199 200 201 204 205 209 210 214 214 216

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3.9.3

Attractor Construction: Dissipativity of Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.4 Attractor Construction: Smoothness Through Compensated Compactness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.5 Exploiting Compactness of the Structural Attractor for Quasi-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.6 Proof Strategy for Stabilization to Equilibria: α = 0. . . . . . . 3.9.7 Strong Plate Convergence and Weak Convergence for the Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.8 Weak Convergence to Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.9 Improving from Weak to Strong . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.1 Dynamical Systems and Attractors- Fundamental Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.2 Microlocal Regularity of the Hyperbolic Neumann-Dirichlet Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

218 221 223 226 227 230 232 236 236 243 253

Turbulence Control: From Model-Based to Machine Learned . . . . . . . . . Nan Deng, Guy Y. Cornejo Maceda, and Bernd R. Noack 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fluidic Pinball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Proximity Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Cluster-Based Network Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Machine Learning Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259

Design Through Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Ji, M. Möller, and H. M. Verhelst 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Spline Primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Truncated Hierarchical B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Non-uniform Rational B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Multi-patch Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Creation of Analysis-Suitable Parameterizations . . . . . . . . . . . . . . . . . . . . 5.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Classification of Parameterization Methods . . . . . . . . . . . . . . . . 5.3.3 Optimization-Based Parameterization Methods . . . . . . . . . . . . 5.3.4 PDE-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Experiments and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Isogeometric Kirchhoff–Love Shell Analysis . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The Isogeometric Kirchhoff–Love Shell Element . . . . . . . . . . 5.4.2 Benchmark Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions and Outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

On the Stabilization Problem by Feedback Control for Some Hydrodynamic Type Systems A. V. Fursikov

1.1 Introduction The main hydrodynamic system that we study here is three-dimensional (3D) Navier–Stokes system. This chapter is divided into two parts: investigation of the cases of local and non-local stabilization. Local Stabilization Problem So in the first part of the chapter we consider the boundary value problem for (3D) Navier–Stokes equations: ∂t v(t, x) − Δv(t, x) + (v(t, x), ∇)v(t, x) + ∇p(t, x) = f (x), .

div v(t, x) = 0, t ∈ R+ , x ∈ Ω, v(t, x)|t=0 = v0 (x),

.

v(t, x ' )|x ' ∈∂Ω = u(t, x ' ),

(1.1) (1.2)

where .v(t, x) = (v1 (t, x), v2 (t, x), v3 (t, x)) is the velocity of the fluid flow, defined for each time moment .t > 0 and each point .x = (x1 , x2 , x3 ) belonging to a bounded domain .Ω ⊂ R3 with a smooth boundary .∂Ω, .p(t, x) is the pressure of the fluid, and .f (x) is the external force. Note that the boundary condition ' ' .u(t, x ), t ∈ R+ , x ∈ ∂Ω, being equal to fluid’s velocity on the boundary .∂Ω, is the control vector function. Recall that in (1.1) ∂t v(t, x) =

.

∂v(t, x) , ∂t

Δv(t, x) =

3  ∂ 2 v(t, x) j =1

∂xj2

,

A. V. Fursikov () Moscow State University, Moscow, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Bodnár et al. (eds.), Fluids Under Control, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-031-47355-5_1

1

2

A. V. Fursikov

.

(v(t, x), ∇)v(t, x) =

3 

vk (t, x)∂xk v(t, x),

k=1

∇p(t, x) = (∂x1 p(t, x), ∂x2 p(t, x), ∂x3 p(t, x)),

.

div v(t, x) =

3 

∂xj vj (t, x).

j =1

Note that to study (1.1) and other equations we will use functional spaces of Sobolev’s type. That is why here and almost everywhere below we understand derivatives of functions in a weak sense (see, for instance, Evance [8] ). For example, expression .div v can be defined (in a weak sense) as follows:  div v(x)ϕ(x)dx = −

.

Ω

  3 Ω j =1

vj (x)∂xj ϕ(x)dx

∀ϕ(x) ∈ C01 (Ω),

where .C01 (Ω) is the space of all continuously differentiated functions with compact support in the domain .Ω. In other words the function .div v(x) is understood (in a week sense) as the functional on the space of test functions. Rigorous mathematical notions of weak derivatives, weak solution of differential equation, defined as a certain functionals have been proposed by Sobolev in 30 years of the last century (see [32]). Suppose that .v(x) ˆ is the solution of the steady-state Navier–Stokes equations (i.e., Eqs. (1.1) with omitted term .∂t v from the first of them) and with the boundary condition .v(x ˆ ' )|x ' ∈∂Ω = u(x ˆ ' ), where .u(x ˆ ' ), x ' ∈ ∂Ω is a given vector function. The stabilization problem is formulated as follows: Given .σ > 0, find the boundary control .u(t, x ' ) in (1.1) and (1.2) such that −σ t ‖v(t, ·) − v(·)‖ ˆ , V 1 (Ω) ⩽ ce

.

as

t → ∞,

(1.3)

where .V 1 (Ω) is the space of divergence-free vector fields belonging to .(L2 (Ω))3 together with all their derivatives of the first order, and .c > 0 is a constant. A ˆ V 1 (Ω) ⩽ ε, where .ε is small stabilization problem is called local one if .‖v0 − v‖ enough and depends on .σ from (1.3). Recall that in the first part of this chapter just local stabilization problems will be considered. We require that the control .u = u(t, x ' ), t > 0, x ' ∈ ∂Ω, has to possess the following important property: u is a feedback control. This means that for each instant .tu(t, ·) is defined by fluid flow velocity vector field .v(t, ·) taken at the same instant t, and therefore the control u can react on unpredictable fluctuations of v suppressing their negative influence on fluid flow. This physical notion of feedback, and some of its mathematical formalization, has been proposed by J.C. Maxwell in the second half of nineteenth century when he studied some control problems in electrodynamics. These problems are described by ordinary differential equations. Generalizing this formalization, series of stabilization results

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

3

for equations described incompressible fluid flow were obtained: for instance, stabilization of 2D Navier–Stokes equations by distributed control supported on the whole .Ω and written in an abstract form (Barbu and Sritharan [2]) and stabilization by boundary control of 2D Euler equations for incompressible fluid flow (Coron [5, 6]). In this chapter we use a certain other formalization of feedback notion, which was proposed in [11, 12], and [13], and gave opportunity to solve the local stabilization problem by boundary feedback control in the cases of quasi-linear parabolic equation and 2D and 3D Navier–Stokes systems. The main point of this approach is to construct a special operator of extension for solenoidal vector fields from .Ω to a certain including domain G. The construction of these extension operators is based on a property of linear independence to finite systems of eigenfunctions and associated functions for adjoint steady-state Oseen system1 when these functions are regarded over arbitrary sub-domain .ω ⊂ G. This property is proved with the help of a Carleman estimate that was the main tool in our preceding investigations, the exact controllability property for Navier–Stokes equations (see [18] and references therein). The solution of the local stabilization problem for 3D Navier–Stokes system by boundary control is described below in Sects. 1.2–1.5. It is necessary to mark that up to now there exists extensive literature on the local stabilization of Navier–Stokes system in the neighborhood of a stationary point (see, for example, Barbu et al. [3, 4], Raymond [29, 30, 31], and [17, 20], as well as literature listed in these articles ). Non-local Stabilization Problem In the second part of this chapter some results connected with non-local stabilization problem are considered. Note that for some equations of fluid dynamics there are certain non-local stabilization results: for Burgers equation where the exact formula of its solution was used (see Krstic [26]) and for 2D Euler system (see Coron [5, 6]) where the construction is based on such properties of its solutions which Navier–Stokes system does not possess. It should be noted that the non-local stabilization problem for 3D Navier– Stokes system is not solved yet: we made only some steps in its solution. Indeed, stabilization problem can be solved only in the functional space where the solution of the corresponding equation is uniquely determined by its right-hand side initial and boundary conditions. For 3D Navier–Stokes equation, it can be, for instance, the space V 1,2 (R+ × Ω) = {v(t, x) ∈ L2 (R+ ; V 2 (Ω)) : ∂t v(t, x) ∈ L2 (R+ ; V 0 (Ω))}, (1.4) where .

V 0 (Ω) = {v(x) ∈ (L2 (Ω))3 : div v(x) = 0}, .

1 That

V 2 (Ω) = {v(x) ∈ (H 2 (Ω))3 : div v(x) = 0},

is, linearized Navier–Stokes system.

(1.5)

4

A. V. Fursikov

and .H 2 (Ω) is Sobolev’s space of functions belonging to .L2 (Ω) together with all their derivatives of the first and second orders. That is why the phase space of this stabilization problem connected with the space (1.4) should be the space 1 .V (Ω) defined in the next line after the formula (1.3). Just the spaces like these are used in the sections where the local stabilization problem is considered. But as well known the solvability of (non-local) boundary value problem for the 3D Navier–Stokes system is not established yet,2 and this is a very serious obstacle to solve stabilization problem. Our hope to overcome this difficulty is based on essential difference between the nature of stabilization (i.e., control) problem and the boundary value problem. Indeed, in the paper [19] based on [18] and [7] the existence of non-local solution of exact controllability problem for 3D Navier– Stokes system with distributed control supported in subdomain has been proved. Recall that the exact controllability problem for Navier–Stokes system is ill-posed one and its certain regularization is just the stabilization problem, i.e., settings of exact controllability and stabilization problems are related in some sense. We make the following simplifications in the setting of non-local stabilization problem: Assume that right-hand side of the stabilized equation equals to zero, the solution should tend to zero with growing time, and the boundary conditions of the solution are periodic on spatial variables .x = (x1 , x2 , x3 ). In other words spatial variables of the solutions belong not to domain but to three-dimensional torus .T3 = (R/2π Z)3 . We did not discuss yet what kind of control function should be used in this statement (we will do this later). It should be noted that if in the problem’s formulation written above there is no control function we get the example of the boundary value problem for 3D Navier–Stokes system whose solvability in class of smooth functions for each smooth enough initial condition was proposed to prove by Clay Institute as millennium problem. In other words simplifications in the setting of non-local stabilization problem indicated above coincide with the simplifications that have been made by Clay Institute. We made just the same simplifications because the idea to omit all unessential complications, leaving only the most difficult part of the problem, is very reasonable. To formulate one non-local result on stabilization, we recall some definitions and give certain information. (a) If the velocity vector field .v(t, x) is a solution of the Navier–Stokes system (see (1.1)), then vector field ω(t, x) = curl v(t, x) = (∂x2 v3 − ∂x3 v2 , ∂x3 v1 − ∂x1 v3 , ∂x1 v2 − ∂x2 v1 )

.

is a solution of the equations that are called the Helmholtz system (see Eq. (1.144) in Sect. 1.6.2). It was marked above that the phase space for solution .v(t, x) of the non-local stabilization problem for the Navier–Stokes system 2 This

is one of the millennium problems.

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

5

should be the space .V 1 (T3 ). Hence the formula written two lines above implies that the phase space for solution .ω(t, x) of the non-local stabilization problem for the Helmholtz system should be the space .V 0 (T3 ) defined in (1.5). Of course to use this space is much more convenient than .V 1 (T3 ) and that is why we pass from Navier–Stokes to Helmholtz system in order to study non-local stabilization problem. (b) It is easy to see (using Eq. (1.147)) that the image .B(ω(t, x)) of the nonlinear operator B, generated by nonlinear terms of 3D Helmholtz equations, is not orthogonal in .(L2 (T3 ))3 to the vector .ω(t, x), i.e., it contains a component .Ф(ω)ω(t, x), that is unidirectional to .ω(t, x) (here .Ф(ω) is a homogeneous functional of the first order). (c) To make this sentence more clear we have to do the following changes: operator .B(ω) on its component .Ф(ω)ω then obtained system of equations is called semilinear normal parabolic equations (NPEs). (d) We get from (b) and (c) that nonlinear operator .B(ω) can be written in the form B(ω) = Ф(ω)ω + Bτ (ω),

.

where

Bτ (ω) ⊥ ω

in (L2 (T3 ))3

(1.6)

(as well as nonlinear operator .(v(t, x), ∇)v(t, x) from 3D Navier–Stokes system is orthogonal to .v(t, x)). So our goal is to study the non-local stabilization problem for Navier–Stokes type system. To simplify the problem we consider the system with right-hand side .f (x) ≡ 0, and the solution of this system is stabilized to zero as .t → ∞. Moreover, we consider stabilized system with a periodic boundary conditions (on space variables), and for stabilization we use starting of impulse control. At last, we stabilize not Navier–Stokes but Helmholtz system. The reason is that stabilizing the Navier–Stokes system we are forced to use .V 1 (T3 ) as a phase space, but stabilizing Helmholtz system we can use as phase space .V 0 (T3 ) that is much more convenient by many technical reasons. By virtue of assertion (d) written above, the first term .Ф(ω)ω in decomposition (1.6) should contain the most difficulties in solution stabilization problem for Helmholtz system. Indeed, if we omit this term from nonlinear operator, the obtained system will satisfy energy estimate, and its solution will tend to zero exponentially in time without any control.3 That is why there is reason to decompose solution of stabilization problem for Helmholtz system on two parts: (1) to solve stabilization problem for semi-linear parabolic equations, defined in (c) (i.e., we omit the second term in decomposition (1.6)) and (2) to solve stabilization problem for Helmholtz system (i.e., to understand interaction of the first and second terms of decomposition (1.6) during dynamic process).

3 Because

we assume that right side .f (x) = 0.

6

A. V. Fursikov

The first result of this part of the chapter is devoted to the construction of the nonlocal stabilization theory by starting control for the semi-linear normal parabolic equations (NPEs) connected with the three-dimensional Helmholtz system. The presentation of this theory is given in Sects. 1.6 and 1.7. These results with complete proofs had been published in [15, 22]. The second step of non-local stabilization theory construction is not made yet in the case of Helmholtz system. We did it only in the case of its onedimensional analog, i.e., for differentiated Burgers equation. After the formulation of the simplified version of results from Sects. 1.6 and 1.7, the problem had been written in functional polar coordinates that made situation clearer, and some version of feedback stabilization had been proposed. See below Sects. 1.8, 1.9, and 1.10. This result with complete proofs has been published in [21].

1.2 Setting of the Problem and the Main Idea of the Method The main aim of the first part of this chapter is to study the local stabilization problem for 3D Navier–Stokes system by feedback boundary control. In the first subsection we begin to study this problem in the case of linear parabolic equation.

1.2.1 Stabilization Problem for a Simple Parabolic Equation Consider the simple parabolic equation .

∂y(t, x) ∂ 2 y(t, x) − − αy(t, x) = 0, ∂t ∂x 2

(1.7)

y(t, x)|t=0 = y0 (x),

(1.8)

.

where .α > 1, x ∈ (−π/2, π/2), t ∈ R+ , y0 (x) ∈ L2 (−π/2, π/2). We include in (1.7) the term with .α > 1 in order Eqs. (1.7) and (1.8) possess exponentially growing solutions. Really, if .y0 (x) = cos x, then .y(t, x) = cos xe(α−1)t satisfies (1.7) and (1.8). Setting of the Problem Given .σ > 0, find boundary conditions .u+ (t) and .u− (t) for (1.7) and (1.8): y(t, −π/2) = u− (t), y(t, π/2) = u+ (t),

.

(1.9)

such that the solution .y(t, x) of the problem (1.7)–(1.9) decays with the rate .σ : ‖y(t, ·)‖L2 (− π2 , π2 ) ⩽ ce−σ t ,

.

(1.10)

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

7

where the constant .c > 0 depends on .σ and .y0 (·). To solve stabilization problem (1.7)–(1.10), we reduce it to the solution of a stabilization problem by start control. For this we extend the solution .y(t, x) of (1.7)–(1.9) from rectangle .{(t, x) ∈ R+ × (−π/2 × π/2)} on cylinder .(t, x) ∈ R+ × T, where .T = R/2π Z is the circumference of length .2π . This circumference is equivalent to the interval .(−π, π ) with identified ends .−π and .π . In other words we will consider (1.7) for .(t, x) ∈ R+ × (−π, π ) with periodic boundary condition: .

∂z(t, x) ∂ 2 z(t, x) − − αz(t, x) = 0, ∂t ∂x 2

(1.11)

z(t, x + 2π ) = z(t, x),

(1.12)

z(t, x)|t=0 = z0 (x) := Ey0 (x).

(1.13)

.

.

Here .z(t, x) is the extension of .y(t, x) and .E is the following extension operator:  Ey0 (x) =

.

y0 (x), x ∈ (−π/2, π/2), u(x), x ∈ (−π, −π/2) ∪ (π/2, π ),

(1.14)

and the function .u(x) is uniquely defined by .y0 ; .u(x) is called the start control. Setting of the Stabilization Problem by Feedback Start Control Given .y0 (x) ∈ L2 (−π/2, π/2), σ > 0, find the start control .u(x) ∈ L2 ((−π, −π/2)∪(π/2, π )) such that the solution of problem (1.11)–(1.13) satisfies the estimate ‖z(t, ·)‖L2 (−π,π ) ⩽ ce−σ t ,

.

t → ∞.

(1.15)

Note that by definition a start control .u(x) is called feedback one if it is uniquely defined by .y0 . Thus the feedback property is connected with the existence and uniqueness of the operator E defined above.4 (We suppose that all functions in the problem (1.11)–(1.15) written above are real-valued.) We solve the stabilization problem (1.11)–(1.15). Let .z(t, x) be the solution of (1.11)–(1.13) where theinitial condition .z0 (x) is given. The substitution of the Fourier series .z(t, x) = zˆ (t, k)eikx into (1.11)–(1.13) implies that Fourier  πk∈Z 1 coefficients .zˆ (t, k) := 2π −π z(t, x)e−ikx dx are defined by the formula zˆ (t, k) = e−(k

.

2 −α)t

zˆ 0 (k),

(1.16)

4 This is a very previous definition of the feedback notion. This notion will be discussed below in a detailed way.

8

A. V. Fursikov

where .zˆ 0 (k) are Fourier coefficients of the initial condition .z0 (x). Multiplying both parts of Eq. (1.16) on their complex conjugate ones and summarizing them on .k ∈ Z we get using Parseval’s equality and notation.‖z‖ = ‖z‖L2 (−π,π ) . As a result we get ‖z(t, ·)‖2 = 2π



.

|ˆz(t, k)|2 = 2π

k∈Z



|ˆz0 (k)|2 e−2(k

2 −α)t

.

(1.17)

k∈Z

Relation (1.17) implies that inequality (1.15) holds if and only if zˆ 0 (k) = 0, ∀k ∈ Z : |k|
0 there exists a unique extension operator E defined in (1.14) such that for each .y0 (x) ∈ L2 (π/2, π/2) 

π

.

−π

e−ikx Ey0 (x)dx = 0 ∀k : |k|
0, and therefore restrictions z(t, x)|(t,x)∈R+ ×(−π/2,π/2) = y(t, x)

.

and

z(t, x)|(t,x)∈R+ ×{±π/2} = u± (t)

are well defined. Since .z(t, x) satisfied (1.11)–(1.15), functions .(y(t, x), u+ (t), u− (t)) satisfy (1.7) and (1.10). In the other part of this section we begin to discuss some questions connected with stabilization of 3D Navier–Stokes system.

.

1.2.2 Setting of the Stabilization Problem Let .Ω ⊂ R3 be a bounded connected domain with .C ∞ -boundary .∂Ω. We set Q = R+ × Ω,

.

Σ = R+ × ∂Ω.

(1.25)

10

A. V. Fursikov

In space-time cylinder Q we consider the following boundary value problem for the Navier–Stokes equations: ∂t v(t, x) − Δv(t, x) + (v(t, x), ∇)v(t, x) + ∇p(t, x) = f (x),

.

.

div v(t, x) :=

3  ∂vj (t, x) j =1

v(t, x)|t=0 = v0 (x),

.

∂xj

(t, x) ∈ Q, (1.26)

= 0,

(1.27)

x = (x1 , x2 , x3 ) ∈ Ω,

(1.28)

v|Σ = u,

(1.29)

.

where .v = (v1 , v2 , v3 ) and p are velocity vector field and pressure of a fluid, .f = (f1 , f2 , f3 ) is an external force, .v0 = (v0,1 , v0,2 , v0,3 ) is an initial condition, and .u = (u1 , u2 , u3 ) is a control defined on the boundary .Σ. We suppose also that a steady-state solution .(v(x), ˆ ∇ p(x)) ˆ of the Navie–Stokes system with the same right-hand side .f (x) as in (1.26) is given: .

− Δv(x) ˆ + (v(x), ˆ ∇)v(x) ˆ + p(x) ˆ = f (x),

div v(x) ˆ = 0,

v| ˆ ∂Ω = u. ˆ

.

x ∈ Ω,

(1.30) (1.31)

Let .σ > 0 be given. The problem of stabilization with the rate .σ is to look for a control .u(t, x ' ), x ' ∈ ∂Ω such that the solution .(v, p) of problem (1.26)–(1.29) with the boundary value u satisfies the inequality ‖v(t, ·) − v‖ ˆ (H 1 (Ω))3 ⩽ ce−σ t as t → ∞

.

(1.32)

with some constant c.

1.2.3 Feedback Control: Previous Remarks Our important additional requirement is that a stabilization problem should be solved with the help of feedback control. From the physical point of view the feedback notion means that the control function .u(t, ·) at the instant t should be defined with help of the state function .v(t, ·) taken just at the same time moment t. This has to give the possibility for the control to react on unpredictable fluctuations of the state function in order to suppress all undesirable effects of these fluctuations. There is a well-known mathematical formalization of this physical feedback notion Which was proposed in the control theory for ordinary differential equations: the control function .u(t) should be defined by the formula .u(t) = R(v(t)), where .v(t) is the state function and R is a continuous map.

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

11

If stabilization problem by feedback control should be solved only for initial conditions .v0 belonging to a certain neighborhood of .v, ˆ then it is called a local stabilization problem. The classical setting of stabilization problem by feedback control was successfully applied not only for controlled ordinary differential equations but also for certain controlled PDE including 2D Navier–Stokes equations with distributed control supported on the whole domain containing liquid (see [2]). Nevertheless this approach does not give the possibility to solve stabilization problem for general quasi-linear parabolic equation or for Navier–Stokes system with feedback control supported on the boundary of the domain as the setting proposed in [11, 12, 13].

1.2.4 The Main Idea of Construction Recall the construction of the boundary feedback control proposed in [11, 12, 13]. Let .ω ⊂ R3 be a bounded domain such that Ω ∩ ω = ∅,

.

Ω ∩ ω = ∂Ω.

(1.33)

We set G = Int(Ω ∪ ω)

(1.34)

.

(the notation .Int A means, as always, the interior of the set A). We suppose that .∂G is a two-dimensional surface belonging to the smoothness class .C ∞ . We extend the problem (1.26)–(1.29) from .Q = R+ × Ω to .Θ = R+ × G. For this end we forget for a while about the boundary condition in (1.29) and write this extended problem as follows: ∂t w(t, x) − Δw + (w, ∇)w + ∇q(t, x) = g(x),

.

div w(t, x) = 0,

(1.35)

w(t, x)|t=0 = w0 (x),

(1.36)

w|S = 0,

(1.37)

.

with additional condition .

where .S = R+ × ∂G. Moreover we assume that solution .(v, ˆ ∇ p) ˆ of (1.30), (1.31) is extended on G in a pair .(a(x), ∇ q(x)), ˆ .x ∈ G such that .

− Δa(x) + (a, ∇)a + ∇ q(x) ˆ = g(x), a|∂G = 0,

.

div a(x) = 0, x ∈ G,

(1.38) (1.39)

12

A. V. Fursikov

where the right side .g(x) is the same as in (1.35). (We show below how to construct such an extension.) Note that, actually, .w0 from (1.36) will be a special extension of .v0 in (1.28) from .Ω to .G : w0 = Extσ v0 . More precisely, .w0 should belong to the stable manifold .Mσ which is invariant with respect to the semigroup generated by the Navier–Stokes problem (1.35)–(1.37) and which contains solutions .w(t, ·) tending to a with the rate .σ (as in (1.32)). A more detailed definition of .Mσ will be given later. We introduce the following space of solenoidal vector fields: V k (G) = {v(x) ∈ (H k (G))3 : div v(x) = 0},

.

(1.40)

where .H k (G) is the Sobolev space of functions .f (x), x ∈ G, belonging to .L2 (Ω) together with their derivatives of order not more than k. For the definition of .H k (G) with fractional or negative k, see [28]. .(H k (G))3 is the Sobolev space of threedimensional vector fields .f (x) = (f1 (x), f2 (x), f3 (x)) with .fi (x) ∈ H k (G). For vector fields defined on G, we introduce the operator .γΩ of restriction on .Ω and the operator .γ∂Ω of restriction on .∂Ω: γΩ : V k (G) −→ V k (Ω), k ⩾ 0,

.

γ∂Ω : V k (G) −→ (H k−1/2 (𝚪))3 , k > 1/2. (1.41)

As is well known (see, for instance, [28]), operators (1.41) are bounded. Definition 1.2.1 A control .u(t, x) in stabilization problem (1.26)–(1.29) is called feedback if the solution .(v(t, x), u(t, x)) of (1.26)–(1.29) is defined by the equality: (v(t, x), u(t, x)) = (γΩ w(t, ·), γ∂Ω w(t, ·)),

.

(1.42)

where .w(t, x) satisfies to (1.35)–(1.37), and .γΩ and .γ∂Ω are operators of restriction of a function defined on G to .Ω and .∂Ω, respectively. This definition of feedback control is basic for us: we will use only it below. Let .S(t, w0 ) be the semigroup generated by the boundary value problem (1.35)– (1.37), i.e., if .w(t, ·) is a solution to (1.35)–(1.37) with an initial condition .w0 , then .S(t, w0 ) = w(t, ·). Then the map that acts initial condition .v0 from (1.28) to the solution .v(t, ·) of (1.26)–(1.28) is defined as follows:5 v(t, ·) = γΩ S(t, Extσ v0 ).

.

(1.43)

Using (1.43) we define the following extension operator from .Ω to G: E(t, v(t, ·)) = S(t, Extσ v0 ),

.

5 Note

(1.44)

that the analog of this extension operator .Extσ v0 from .Ω to G is the extension operator (1.14) from the stabilization problem described in Sect. 1.2.1.

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

13

where .v0 is the initial condition of .v(t, ·). Note that operator .E(t, ·) depends on t because it is defined on the set of functions that belong to the image of the operator .S(t, Extσ ·) and this set depends on t. Now we can define the operator R which acts as solution .v(t, ·) to the correspondent control function .u(t, ·): u(t, ·) = R(t, v(t, ·)) ≡ γ∂Ω E(t, v(t, ·)).

.

(1.45)

1.3 Oseen Equations We begin the investigation of stabilization problem from the case of linearized Navier–Stokes equations, i.e., from the Oseen equations. Note that the results of this section connected with 3D Oseen equations as well as their proofs are absolutely identic to analogous results and proofs for 2D Oseen equations. That is why we give here only their short formulation. The detailed exposition of these results can be found in [12].

1.3.1 Preliminaries Let G be domain (1.34). We consider in .R+ ×G the Oseen equation which is written as follows: ∂t w(t, x) − Δw + (a(x), ∇)w + (w, ∇)a + ∇p(t, x) = 0,

.

(1.46)

div w(t, x) = 0,

(1.47)

w(t, x)|t=0 = w0 (x).

(1.48)

.

.

Moreover, we impose on w the zero Dirichlet boundary condition w|S = 0,

.

(1.49)

where .S = R+ × ∂G. We assume that  3 a(x) ∈ V 2 (G) ∩ H01 (G) ,

.

(1.50)

where, recall, Sobolev spaces .V k (G) and .H k (G) were defined above (see (1.40) and few lines below it), and .H01 (G) = {f (x) ∈ H 1 (G) : f (x)|x∈∂G = 0}. In the case .k = 0 we define V00 (G) = {v(x) ∈ V 0 (G) : v · ν|∂Ω = 0},

.

(1.51)

14

A. V. Fursikov

where .ν(x) is the vector field of outer normals to .∂G. In [33] it is established that the restriction .v · ν|∂Ω is well defined for .v ∈ V 0 (G). Denote by π : (L2 (G))3 −→ V00 (G)

(1.52)

.

the operator of orthogonal projection. We consider the Oseen steady-state operator Av ≡ −π Δv + π [(a(x), ∇)v + (v, ∇)a] : V00 (G) −→ V00 (G),

.

(1.53)

where the vector field .a(x) satisfies (1.50). This operator is closed and it has the domain: D(A) = V 2 (G) ∩ (H01 (G))3

(1.54)

.

which is dense in .V00 (G). Assuming that spaces in (1.52) and (1.54) are complex we denote by .ρ(A) the resolvent set of operator A, i.e., the set of .λ ∈ C such that the resolvent operator R(λ, A) ≡ (λI − A)−1 : V00 (G) −→ V00 (G)

.

(1.55)

is defined and continuous. Here I is the identity operator. Denote by .Σ(A) ≡ C1 \ ρ(A) the spectrum of operator A. As is well known, Oseen operator (1.53) is sectorial, i.e., there exist .ϕ ∈ (0, π/2), .M ⩾ 1, a ∈ R such that Sa,ϕ = {λ ∈ C :

.

ϕ ⩽ | arg(λ − a)| ⩽ π,

λ /= a} ⊂ ρ(A)

(1.56)

and .‖(λI − A)−1 ‖ ⩽ M/|λ − a|, ∀λ ∈ Sa,ϕ . Besides, for .λ ∈ ρ(A), resolvent (1.55) is a compact operator, and the spectrum .Σ(A) consists of a discrete set of points. We decompose the resolvent .R(λ, −A) in a neighborhood of .−λj ∈ Σ(−A): R(λ, −A) =

∞ 

.

k=−m

(λ+λj )k Rk ,

Rk = (2π i)−1



(λ+λj )−k−1 R(λ, −A)dλ.

|λ+λj |=ε

(1.57) Note that .m < ∞. Let us consider the adjoint operator .A∗ to Oseen operator (1.53): A∗ w ≡ −π Δw − π [(a(x), ∇)w − (∇a)∗ w] : V00 (G) −→ V00 (G),

.

where

(1.58)

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

(∇a)∗ w = ((∂1 a, w), (∂2 a, w), (∂3 a, w)),

.

(∂i a, w) =

3 

15

∂i aj wj .

j =1

Evidently, .A∗ is a closed operator with domain .D(A∗ ) = V 2 (G) ∩ (H01 (G))3 . Moreover, .A∗ is sectorial with a compact resolvent and ρ(A∗ ) = ρ(A)

.

and

R(λ, A)∗ = R(λ¯ , A∗ )

∀ λ ∈ ρ(A).

(1.59)

(Here the line above means complex conjugation.) Below we always assume that the vector field a(x) from (1.50), (1.53), and (1.58) is real-valued.

.

(1.60)

That is why we have .ρ(A) = ρ(A) = ρ(A∗ ) = ρ(A∗ ).

1.3.2 Structure of Rk with k < 0 Let .−λj ∈ Σ(−A) be an eigenvalue of .−A and .e /= 0, e ∈ ker(λ0 I + A) be an eigenvector. Vector .ek is called an associated vector of order k with e if .ek satisfies (λ0 I + A)e = 0,

.

e + (λ0 I + A)e1 = 0, . . . ,

ek−1 + (λ0 I + A)ek = 0.

We say that .e, e1 , e2 , . . . form a chain of associated vectors. The maximal order m of vectors associated with e is finite, and the number .m + 1 is called the multiplicity of the eigenvector e. Definition 1.3.1 The set of eigenvectors and associated vectors (k) e(k) (−λj ), e1(k) (−λj ), . . . , em (−λj ) k

.

(k = 1, 2, . . . , N(−λj )),

(1.61)

corresponding to an eigenvalue .−λj , is called canonical system if: (i) Vectors .e(k) (−λj ), k = 1, 2, . . . , N(−λj ), form a basis in the space of eigenvectors corresponding to the eigenvalue .−λj . (ii) .e(1) (−λj ) is an eigenvector with maximal possible multiplicity. (iii) .e(k) (−λj ) is an eigenvector which cannot be expressed by a linear combination of .e(1) (−λj ), .. . . , e(k−1) (−λj ) and the multiplicity of .e(k) (−λj ) achieves a possible maximum. (iv) Vectors (1.61) with fixed k form a complete chain of associated elements. Besides canonical system (1.61) which corresponds to an eigenvalue .−λj of operator .−A, we consider a canonical system (k)

(k) ε(k) (−λj ), ε1 (−λj ), . . . , εm (−λj ) k

.

(k = 1, 2, . . . , N(−λj )),

(1.62)

16

A. V. Fursikov

which corresponds to the eigenvalue .−λj of the adjoint operator .−A∗ . The definition of canonical system (1.62) is absolutely analogous to Definition 1.3.1 of canonical system (1.61). We define the canonical system (1.62) by .E ∗ (−λj ). Theorem 1.3.1 Let .Rk be operators defined in (1.57). Then R−k x = 0,

.

∀k = 1, 2, . . . , m

if and only if ∀εl(k) (−λj ) ∈ E ∗ (−λj ).

〈x, εl(k) (−λj )〉 = 0

.

This assertion follows immediately from one result of Keldysh [24] on structure of the main part of Laurent series for .R(λ, −A). For the proof of Theorem 1.3.1, see [12].

1.3.3 Holomorphic Semigroups We regard the boundary value problem (1.46)–(1.49) for Oseen equations written in the form dw(t) + Aw(t) = 0, dt

.

w|t=0 = w0 ,

(1.63)

where A is an operator (1.53). Then for each .w0 ∈ V00 (G) the solution .w(t, ·) of (1.63) is defined by equalities .w(t, ·) = e−At w0 and e−At = (2π i)−1



(λI + A)−1 eλt dλ,

.

(1.64)

γ

where .γ is a contour belonging to .ρ(−A) such that .arg λ = ±θ for .λ ∈ γ , |λ| ⩾ N for certain .θ ∈ (π/2, π ) and for sufficiently large N. Moreover, .γ surrounds .Σ(−A) from the right. Such contour .γ exists, of course, because we can choose .γ belonging to set .−Sa,ϕ from (1.56). Let .σ > 0 satisfy Σ(−A) ∩ {λ ∈ C : Reλ = −σ } = ∅.

.

(1.65)

The case when there are certain points of .Σ(−A) placed righter than the line .{Reλ = −σ } will be interesting for us. By .γσ we denote the continuous contour that is placed in .{λ ∈ C : Reλ ⩽ −σ } and constructed from an interval of the line .{Reλ = −σ } and from two branches of contour .γ that transform to .{arg λ = θ } and .{arg λ = −θ }, θ ∈ (π/2, π ) for sufficiently large .|λ|.

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

17

By virtue of the Cauchy theorem, we reduce integration over .γ in (1.64) to integration over .γσ and integration around poles .−λj from (1.57) for .λj satisfying .Reλj < σ . After calculating the corresponding residues, we transform (1.64) to the equality:

e

.

−At

= (2π i)

−1

 (λE + A)

−1 λt

e dλ +

m(−λj )



e

−λj t

Reλj 0 satisfies (1.65). Then for each .w0 ∈ V01 (G) that satisfies 〈w0 , εl (−λ¯ j ) >= 0, (k)

.

∀l = 0, 1, . . . , mk , k = 1, 2, . . . , N(−λj ), Re(λj ) < σ

(k)

(here by definition .ε0 (−λj ) = ε(k) (−λj )), the following inequality holds: ‖eAt w0 ‖V 1 (G) ⩽ ce−σ t ‖w0 ‖V 1 (G)

.

0

0

for t ⩾ 0.

(1.67)

For the proof see [11] and [12].

1.3.4 Unique Continuation Property To solve the stabilization problem we will use a unique continuation property for the solution of adjoint Oseen equation. So we consider equality .(μ0 − A∗ )w = 0, x ∈ G, where .μ0 is an eigenvalue of the operator .A∗ defined in (1.58) and w is a corresponding eigenvector. Note that generally speaking .μ0 is a complex number and w is a complex-valued vector field. As usual, the bar over the notation of a complex number means the operation of complex conjugation. By (1.58) and by the definition of operator .π , this equality can be rewritten as follows: Δw(x) + (a(x), ∇)w(x) − (∇a(x))∗ w(x) + μ0 w + ∇ p(x) ˜ = 0,

.

divw = 0, (1.68)

where .x ∈ G, .a(x) ∈ V 2 (G) ∩ V01 (G), and .w(x) satisfies the boundary condition w|∂G = 0.

.

(1.69)

18

A. V. Fursikov

It is easy to see that if .(v(x), p(x)) ˜ ∈ V01 (G) × L2 (G) satisfies (1.68) and (1.69), then .(w(x), p(x)) ˜ ∈ (V01 (G) ∩ H 3 (G)) × H 2 (G). ˜ ∈ (V01 (G) ∩ H 3 (G)) × Theorem 1.3.3 Suppose that a solution .(w(x), p(x)) H 2 (G) of (1.68) and (1.69) satisfies the condition w(x) ≡ 0 for

.

x ∈ ω,

(1.70)

˜ ≡ const for .x ∈ G. where .ω is a subdomain of G. Then .w(x) ≡ 0 and .p(x) This result is based on one Carleman estimate. For the proof of this result and the Carleman estimate, see [13].

1.3.5 On Linear Independence of εl(k) (x, −λj ) We set some strengthening of well-known result on linear independence of eigenvectors and associated vectors for operator .A∗ which is defined in (1.58). To prove this result we use Theorem 1.3.3. Theorem 1.3.4 Consider the set Eσ∗ ≡



.

E ∗ (−λj )

Reλj 0 satisfies (1.65). Then for each .w0 ∈ V00 (G) satisfying  (w0 (x), εj (x))∂x = 0,

.

j = 1, ..., K,

G

with .εj from (1.73), inequality (1.67) is true.

1.4 Stabilization of Oseen Equations 1.4.1 Setting of the Problem As in Sect. 1.2 we suppose that .Ω ⊂ R3 is a bounded connected domain with .C ∞ boundary .∂Ω, which is decomposed on several parts: ∂Ω =

J

.

∂Ωj ,

j =1

where .∂Ωj are closed connected components of .∂Ω.

(1.74)

20

A. V. Fursikov

Let .Q = R+ × Ω and .Σ = R+ × ∂Ω. In space-time cylinder Q, we consider the Oseen equations ∂t v(t, x) − Δv + (a(x), ∇)v + (v, ∇)a + ∇p(t, x) = 0,

.

.

div v(t, x) = 0

(1.75) (1.76)

with initial and boundary conditions v(t, x)|t=0 = v0 (x),

(1.77)

v|Σ = u,

(1.78)

.

.

where .a(x) = (a1 (x), a2 (x), a3 (x)) is a solenoidal vector field (.div a = 0) and u = (u1 , u2 , u3 ) is a control. In addition to the solenoidality of initial condition .v0 (x), we assume that  . (v0 (x ' ), n(x ' ))dx ' = 0, j = 1, . . . , J. (1.79) .

∂Ωj

Similarly to Sect. 1.2 stabilization problem for Oseen equations is formulated as follows: Given .σ > 0, find a control u on .Σ such that the solution .v(t, x) of problem (1.75)–(1.78) satisfies the inequality ‖v(t, x)‖L2 (Ω) ⩽ ce−σ t ,

.

(1.80)

where .c > 0 depends on .v0 and .σ . Moreover, we require that this control u satisfies the feedback property in the meaning analogous to Definition 1.2.1: firstly we extend by a special way problem (1.75)–(1.78) to the problem (1.46)–(1.49) defined on a domain .G ⊃ Ω and solve the last problem, and after that we define the solution .(v, u) of stabilization problem (1.75)–(1.80) by the formula (1.42) The details of this definition will be given simultaneously with the construction of feedback control.

1.4.2 Theorem on Extension First of all we define the set .ω from (1.33) and (1.34) which is used to extend the domain .Ω to the set G. Note that being closed each component .∂Ωj of .∂Ω separates 3 .R on two parts: .Ωj − and .Ωj + . By definition points of .Ωj − which are close enough to .∂Ωj belong to .Ω. Taking a sufficiently small magnitude .κ > 0, we define the set .ω as follows:

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

ωj = {x ∈ Ωj + : dist(x, ∂Ωj ) < κ};

.

l

ω=

ωj .

21

(1.81)

j =1

Now we define the domain G by the formula G = Int(Ω ∪ ω).

(1.82)

.

We introduce the following spaces: V01 (G) = {u(x) = (u1 (x), u2 (x), u3 (x)) ∈ V 1 (G) : u|∂G = 0},

.

Vˆ k (G) = {u(x) = (u1 (x), u2 (x), u3 (x)) ∈ V k (G) :  . (u(x ' ), n(x ' )dx ' = 0, j = 1, . . . , J },

.

(1.83)

∂Gj

where k is a nonnegative integer, .∂Gj , j = 1, . . . , J , are all connected components of the boundary .∂G, and n is the outer normal to .∂G. Recall that operation of restriction onto the boundary for .(u(x), n(x)) is well defined for .u ∈ V k (G) with each .k ⩾ 0 (see [33]). Below we use the well-known operator .curl which is defined by the formula

curlv(x) =

.

 ∂v2 (x) ∂v3 (x) ∂v3 (x) ∂v1 (x) ∂v1 (x) ∂v2 (x) . , − , − − ∂x2 ∂x1 ∂x3 ∂x2 ∂x1 ∂x3

It is clear that for each gradient vector field .∇p(x), p ∈ H 1 (G) the inclusion .∇p(x) ∈ ker curl is true. Note that, generally speaking, for the functions space 0 0 .V (G) defined in (1.51) the inequality .V (G) ∩ ker curl /= {0} holds. Indeed, the 0 0 following orthogonal decomposition with respect to the scalar product in .L2 (G) is true (see, e.g., [9] and [33, Appendix 1,pp.458–471]): V00 (G) = W 0 (G)

.



Hc ,

(1.84)

where .Hc = V00 (G) ∩ ker curl is a finite-dimensional space of .C ∞ -vector fields isomorphic to the space of the first cohomologies of G. .Hc consists of vector fields .∇p(x), where .p(x) are multi-valued functions satisfying .Δp = 0 and 0 .(∂p/∂n)|∂G = 0; for details see [33, Appendix 1]. Now the functions space .W (G) is well defined by equality (1.84). For each integer .k ⩾ 0, we define W k (G) = W 0 (G) ∩ (H k (G))3 ,

.

(1.85)

where .(H k (G))3 is the usual Sobolev space of vector-valued functions of smoothness k.

22

A. V. Fursikov

Lemma 1.4.1 Let .k ⩾ 0. Then the operator curl : W k+1 −→ Vˆ k (G)

(1.86)

.

is an isomorphism. For the proof of this lemma, see [33, Appendix 1] and [23]. Lemma 1.4.2 The operator curl−1 : V01 (G) −→ H 2 (G)

(1.87)

.

satisfying .curl curl−1 v = v ∀v ∈ V01 (G) is well defined on .V01 (G). Proof By virtue of Lemma 1.4.1 operator (1.86) has the inverse operator .curl−1 . Restriction of this operator on .V01 (G) ⊂ Vˆ 1 (G) defines the right inverse operator (1.87). ⨆ ⨅ We formulate now the extension theorem. In the space of real-valued vector fields V01 (G), we introduce the subspace

.

 1 .Xσ (G)

= {v(x) ∈

:

V01 (G)

v(x) · εj (x) dx = 0,

j = 1, . . . , K},

(1.88)

G

where .εj (x) are functions (1.73). Let also Vˆ 1 (Ω) = {v ∈ V 1 (Ω) :



(v(x ' ), n(x ' ))dx ' = 0, j = 1, . . . , J },

.

(1.89)

∂Ωj

where .∂Ωj are closed connected components of .∂Ω, and .n(x ' ) is the vector fields of outer normals to .∂Ω. Theorem 1.4.1 There exists a linear bounded extension operator Eσ1 : Vˆ 1 (Ω) → Xσ1 (G)

.

(1.90)

(i.e., .Eσ1 (v)(x) ≡ v(x) for .x ∈ Ω). Proof Step 1. Recall firstly that there exists a linear continuous extension operator L : Vˆ 1 (Ω) → V01 (G).

.

(1.91)

Indeed let .v ∈ Vˆ 1 (Ω). Then .curl−1 v ∈ H 2 (Ω) is the vector field well defined by virtue of Lemma 1.4.2. The existence of bounded extension operator E : H 2 (Ω) −→ H 2 (G)

.

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

23

is well known (see, e.g., [28]). Set Ωε = {x ∈ G : dist(x, Ω) < ε},

.

where .dist(x, Ω) is the distance from x to .Ω. Suppose that .ε is so small that for each ¯ .0 ⩽ ψ(x) ⩽ 1, j = 1, . . . , J ωj \ Ωε /= ∅. Let .ψ(x) ∈ C ∞ (G),

.

ψ(x) =

1,

.

0,

x ∈ G ∩ Ωε/2 ,

(1.92)

x ∈ G \ Ωε .

Then we denote operator (1.91) by the formula .L = curl ◦ ψ ◦ curl−1 . ˆ = G \ Ωε/2 . We look for an extension Step 2. Introduce now an open subset .Ω operator .Eσ1 in a form Eσ1 v(x) = (Lv)(x) + w(x), ˆ

(1.93)

.

where L is an operator (1.91) and .w(x) ˆ is a vector field which satisfies wˆ ∈ V01 (G),

.

ˆ = G \ Ωε/2 . supp wˆ ⊂ Ω

(1.94)

1 v ⊂ X 1 (G), we have to assume that By virtue of (1.88) to establish inclusion .EK K



 εk (x)w(x) ˆ dx = −

.

G

εk (x)(Lv)(x) dx,

(1.95)

G

where .k = 1, . . . , K. At last, to determine .wˆ uniquely we suppose that ‖w‖ ˆ 2V 1 (G) = inf ‖w(·)‖2V 1 (G)

.

0

w∈A

0

where A = {w : w satisfies (1.94), (1.95)}.

(1.96) (Recall that .‖v‖V 1 (G) = ‖∇v‖L2 (G) .) 0 Step 3. We have to show that there exists a unique vector field .wˆ that satisfies (1.96). To do this we define the operator R by the formula ˆ → RK , R : V01 (Ω)

Rv =

.



 ε1 (x)v(x) dx, . . . ,

G

 εK (x)v(x) dx .

G

(1.97) RK .

We claim that .Im R = Indeed, if this is not true, there exists a vector .p = (p1 , . . . , pK ) /= 0 such that   K .

G j =1

pj εj (x)v(x) dx = 0

ˆ ∀v ∈ V01 (Ω).

24

A. V. Fursikov

This equality implies that K  .

pj εj (x) = ∇q(x),

ˆ x ∈ Ω.

(1.98)

j =1

ˆ Therefore, Since by virtue of (1.50) .εj (x) ∈ V 3 (G), we get that .q(x) ∈ H 4 (Ω). equality .εj |∂G = 0 implies that .∇q|∂G\𝚪0 = 0. As a result we obtain q|(∂G\𝚪0 )∩ωj = cj ,

.

∂n q|(∂G\𝚪0 )∩ωj = 0,

(1.99)

where .cj are constants and .∂n is the derivative with respect to the vector field n of outer normals to .∂G. Applying to both parts of (1.98) operator .div, we get that Δq(x) = 0,

.

ˆ x ∈ Ω.

(1.100)

By virtue of uniqueness of solution for Cauchy problem (1.100) and (1.99), .q|ωj = cj , and therefore (1.98) implies K  .

pj εj (x) = 0,

ˆ x ∈ Ω.

j =1

This equality and Lemma 1.3.1 imply .pj = 0, j = 1, . . . , K, that contradicts to the assumption .(p1 , . . . , pK ) /= 0. Since .Im R = RK , the set .A of admissible elements for problem (1.96) is not empty. By virtue of definition (1.96) .A is a closed convex subset of .V01 (G), and (1.96), actually, is the problem to determine the distance from origin to the set 1 .A in the Hilbert space .V (G). As well known, this problem has a unique solution 0 .w(x). ˆ Step 4. The existence and uniqueness of solution for problem (1.96) implies that the operator .E1 that transforms a vector field .v ∈ Vˆ 1 (Ω) to the solution .wˆ of problem (1.96), .E1 v = w, ˆ is well defined. To finish the proof of the theorem, we have to show that the operator ˆ E1 : Vˆ 1 (Ω) → V01 (Ω)

.

(1.101)

is linear and bounded. We derive optimality system for minimization problem (1.96) with the help of Lagrange principle. As one can see, for instance, in [10], the relation .Im R = RK which was proved for the operator R defined in (1.97) guarantees that we can apply to (1.96) Lagrange principle when Lagrange multiplier before minimized functional equals one. This Lagrange function has a form:

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

L(w, p1 , . . . , pK ) =

.

25

1 ‖w‖2V 1 (Ω) ˆ 2 0 

  K  + pj (εj (x), w(x)) dx + (εj (x), Lv(x)) dx . j =1

ˆ Ω

G

By Lagrange principle for solution .wˆ of (1.96), there exists a vector .p = ˆ (p1 , . . . , pK ) such that for each .h(x) ∈ V01 (Ω)

.

< L'w (w, ˆ p1 , . . . , pK ), h >=

 [(∇w, ∇h) +

K 

pj (εj , h)] dx = 0.

(1.102)

j =1

ˆ Ω

ˆ we consider the Stokes problem: In the set .Ω .

− Δw(x) + ∇p(x) = v(x),

div w(x) = 0,

ˆ x ∈ Ω;

w|∂ Ωˆ = 0.

ˆ there exists a unique solution .w ∈ V 1 (Ω) ˆ ∩ As is well known, for each .v ∈ V 0 (Ω) 0 2 ˆ V (Ω) of this problem. We denote the resolving operator of this problem as follows: −1 ˆ to G by zero .(−π Δ) v = w. We also denote the extension of .(−π Δ)−1 v from .Ω ˆ ˆ Ω

Ω

as .(−π Δ)−1 v. Evidently, .(−π Δ)−1 v ∈ V01 (G). ˆ ˆ Ω Ω means that .wˆ is the solution of the Stokes problem with right side Since (1.102) K .v = − p ε j j , we get j =1 wˆ = −(−π Δ)−1 ˆ

.

Ω

K 

(1.103)

pj εj .

j =1

Substitution of (1.103) into (1.95) yields the linear system of equations: K  .

 akj pj = bk ,

where akj =

j =1

(εk , (−π Δ)−1 ε ) dx, bk = ˆ j

 (εk , Lv) dx.

Ω

ˆ Ω

G

(1.104) Relations (1.104) and (1.103) imply that operator .E1 from (1.101) is linear. To prove the boundedness of operator (1.101), we show that the matrix .A = ‖akj ‖ is positively defined. Note that  akj =

.

εk (x) · [(−π Δ)−1 ε (x)] dx = ˆ j

 (∇wk (x)) : (∇wj (x)) dx,

Ω

ˆ Ω

ˆ Ω

(1.105)

26

A. V. Fursikov

where .wj (x) = (−π Δ)−1 ε (x) and .: is the sign of scalar product between two ˆ1 j Ω  tensors. Let .α = (α1 , · · · , αK ), f (x) = K j =1 αj wj (x). Then (Aα, α) =





K 

∇wk (x)) : ∇wj (x) dx =

αk αj

.

k,j =1

w1

Moreover, if for some .α (Aα, α) =

|∇f (x)|2 dx ⩾ 0. ω1



|∇f (x)|2 dx = 0, then .Δf (x) =

ω1

ˆ By the definition of .wj (x) we have .−Δwj (x) = div ∇f (x) = 0 for .x ∈ Ω. ˆ (to see this ˆ where .qj (x) is a harmonic function in .Ω εj (x) − ∇qj (x), x ∈ Ω, one can apply theoperator .div to both parts of the previous equality). Hence, K ˆ .0 = −Δf (x) = j −1 αj εj − ∇q(x), where .q(x) is a harmonic function in .Ω, and by the previous equality we get that .∇q|∂ Ω∩∂G = 0 (because . ε | = 0). j ∂ Ω∩∂G ˆ ˆ By virtue of uniqueness for the solution to Cauchy problem for Laplace operator, K ˆ Hence . j −1 αj εj = 0 and by Lemma 1.3.1 we get that .∇q(x) = 0 for .x ∈ Ω. .α1 = · · · = αK = 0. So, .det A /= 0. By virtue of (1.102) and (1.103) wˆ = E1 v = −(−π Δ)−1 (A−1 b, ε), ˆ

.

Ω

where .b = (b1 , . . . , bK ) (see (1.103)), .ε = (ε1 , . . . , εK ), and therefore operator (1.101) is bounded. ⨆ ⨅ Remark 1.4.1 In fact in Theorem’s 1.4.1 proof we showed that operator (1.90) can be defined by formulas (1.93) and (1.103), where .(p1 , . . . , pK ) is the solution of the system from (1.104). This definition is equivalent to the definition of operator (1.90) given in the proof of Theorem 1.4.1.

1.4.3 Result on Stabilization We prove now the main theorem of this section on stabilization of 3D Oseen equations by feedback boundary control. Theorem 1.4.2 Let domains .Ω and G satisfy (1.81), (1.82). Then for each initial value .v0 (x) ∈ Vˆ 1 (Ω) and for each .σ > 0, there exists a feedback control u defined on .Σ such that the solution .v(t, x) of (1.75)–(1.78) satisfies the inequality ‖v(t, ·)‖(H 1 (Ω))2 ⩽ ce−σ t as t → ∞.

.

(1.106)

Proof We can assume that .σ satisfies to condition (1.65), otherwise we will make it a little bit more. We act to initial condition .v0 ∈ Vˆ 1 (Ω) by the operator 1 1 1 .Eσ from (1.90), and by Theorem 1.4.1 we obtain that .w0 = Eσ v0 ∈ Xσ (G).

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

27

Since .Xσ1 (G) ⊂ V01 (G) ⊂ V00 (G), the solution .w(t, x) of problem (1.46)–(1.49) can be written in the form .w(t, ·) = e−At w0 , where A is an operator (1.53). By Theorem 1.3.2 .w(t, ·) satisfies estimate (1.67). Now we define the solution ' .(v(t, x), u(t, x )) of stabilization problem for (1.75)–(1.78) by formula (1.42). Then (1.67) implies (1.106). ⨆ ⨅

1.5 Stabilization of 3D Navier–Stokes Equations In this section we study the problem of stabilization a solution to the Navier–Stokes equations which is formulated in Sect. 1.2.2. In other words we consider the case of the nonlinear equations here. We do this stabilization with the help of control determined on the lateral surface .Σ = R+ × ∂Ω of cylinder Q, and we consider only feedback control in the meaning of Definition 1.2.1.

1.5.1 Invariant Manifolds Let .g(x) from (1.35) satisfy the condition: g(x) ∈ (L2 (G))2 .

.

(1.107)

Then, as is well known (see, for instance, [34]), Eq. (1.35) is equivalent to the following equation with respect to one unknown vector field .w(t, x): ∂t w(t, x) − π Δw + π(w, ∇)w = πg(x),

.

(1.108)

where .π is orthoprojector (1.52) on .V00 (G) (see (1.51)). We set an initial condition for this equation: w(t, x)|t=0 = w0 (x).

.

(1.109)

We look for a solution w of (1.108) (as well as the solution w of (1.35)) in the space V01,2 (ΘT ) ≡ {w(t, x) ∈ L2 (0, T ; V 2 (G) ∩ (H01 (G))3 ) : ∂t w ∈ L2 (0, T ; V00 (G)} (1.110)

.

for each .T > 0, where .ΘT = (0, T ) × G. Note that we can rewrite (1.38) in the form analogous to (1.108): .

− π Δa(x) + π(a, ∇)a = πg, a(x) ∈ V 2 (G) ∩ V01 (G).

(1.111)

28

A. V. Fursikov

It is known (see [27, 34]) that if .a(x) is a solution of (1.111) with a certain g(x) ∈ (L02 (G))2 and an initial condition .w0 (x) ∈ V01 (G), then for each .T > 0 if .‖a − w0 ‖V 1 (G) < ε with small enough .ε = ε(T ), there exists a unique solution

.

0

w(t, x) ∈ V01,2 (QT ) of problem (1.108), (1.109). We denote the solution .w(t, x) of (1.108), (1.109) taken at time moment t as .S(t, w0 )(x):

.

w(t, x) = S(t, w0 )(x).

(1.112)

.

Since embedding .V 1,2 (QT ) ⊂ C(0, T ; V01 (G)) is continuous, the family of operators .S(t, w0 ) is a continuous semigroup on the space .V01 (G) : S(t + τ, w0 ) = S(t, S(τ, w0 )). Since .a(x) is a steady-state solution of (1.108), .S(t, a) = a for each .t ⩾ 0. We can decompose semigroup .S(t, w0 ) in a neighborhood of a in the form S(t, w0 + a) = a + Lt w0 + B(t, w0 ),

(1.113)

.

where .Lt w0 = Sw' (t, a)w0 is the derivative of .S(t, w0 ) with respect to .w0 at point a, and .B(t, w0 ) is a nonlinear operator with respect to .w0 . Differentiability of .S(t, w0 ) is proved, for instance, in [1, Ch. 7. Sect. 5]. Therefore B(t, 0) = 0, Bw' (t, 0) = 0.

(1.114)

.

Moreover in [1, Ch. 7. Sect. 5] it is proved that .B ' (t, w) belongs to class .C α with 1 .α = 1/2 with respect to w. This means that for each .w0 ∈ V (G), ‖a−w0 ‖ 1 V0 (G) ⩽ 0 ε(t) ‖Bw' (t, w0 )‖C α

.

≡

‖Bw' (t, u) − Bw' (t, w0 )‖V 1 (G) 0

sup

‖u−w0 ‖V 1 (G) ⩽1, ‖u−a‖V 1 (G) ⩽ε(t) 0

‖u − w0 ‖α 1

0

< ∞,

V0 (G)

and the left side is a continuous function with respect to .w0 . We study now semigroup .Lt w0 = Sw' (t, a)w0 of linear operators. First of all note that .w(t, x) = Lt w0 is the solution of problem (1.46)–(1.49) in which the coefficient a is the solution of (1.111). Therefore Lt w0 = e−At w0 ,

.

(1.115)

where A is Oseen operator (1.53). Below we suppose that .r0 ∈ (0, 1) satisfies the property: {ζ ∈ C : |ζ | = r0 } ∩ Σ(e−At0 ) = ∅,

.

where, recall, .Σ(e−At ) is the spectrum of operator (1.115).

(1.116)

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

29

It is clear that .ζj ∈ Σ(e−At0 ) if and only if .ζj = e−λj t0 and .−λj ∈ Σ(−A). That is why condition (1.116) is equivalent to condition (1.55) where .σ = − ln r0 /t0 . Besides, if .|ζj | > r0 , then .−Reλj > −σ . The following assertion holds: Theorem 1.5.1 A family of operators .e−At : V01 (G) → V01 (G), where A is an operator (1.53), is well defined for each .t ⩾ 0. Let σ+ = {ζ1 , . . . , ζN : ζj ∈ Σ(e−At0 ), |ζj | > r0 , j = 1, . . . , N },

.

(1.117)

where .r0 ∈ (0, 1) and satisfies (1.116). Let .X+ ⊂ V01 (G) be the invariant subspace for .e−At0 corresponding to .σ+ , .Π+ : V01 (G) → X+ be the projector on .X+ (i.e., 1 1 2 .Π+ V (G) = X+ , .Π+ = Π+ ), and .X− = (I − Π+ )V (G) be complementary 0 0 + −At0 | invariant subspace. Let .Lt0 = e−At0 |X+ : X+ → X+ , .L− X− : t0 = e + + −1 X− → X− . Then operator .Lt0 has inverse operator .(Lt0 ) . For some .t0 there exist constants .rˆ , .ε+ , .ε− ∈ (0, 1) such that ‖L− t0 ‖ ⩽ rˆ (1 − ε− ),

.

−1 −1 ‖(L+ t0 ) ‖ ⩽ rˆ (1 − ε+ ).

(1.118)

The proof of this theorem is absolutely the same as in the case of space dimension two (see [12]). Generally speaking eigenvalues of operators A and .e−At are complex-valued. That is why all spaces in Theorem 1.5.1 are complex. But to apply obtained results to (nonlinear) Navier–Stokes equation, we need to have analogous results for the real spaces of the same type. Actually, for this we have to define the projector of .Π+ in real spaces. Lemma 1.5.1 Restriction of operator .Π+ on the real space .V01 (G) can be written in the form (Π+ v)(x) =

K 

.

j =1

 ej (x)

v(x)εj (x) dx,

(1.119)

G

where .{εj } is the set of functions (1.73) which are suitably renumbered and renormalized functions (1.72) and .{ej } is a set of Real and Imaginary parts of functions (1.61) analogously renumbered and renormalized. The proof of this simple lemma one can find in [11]. Lemma 1.5.2 For an arbitrary sub-domain .ω ⊂ G vector fields .{ej (x), .j = 1, . . . , K} from (1.119) restricted on .ω are linearly independent over .R. To prove this lemma we first establish analog of Theorem 4.1 in [13] for functions (1.61). After that we derive Lemma 1.5.2 from this theorem by the same way as in [13], Lemma 1.3.1 was derived from Theorem 4.1.

30

A. V. Fursikov

Using (1.119) we can easily restrict spaces .X+ and .X− as well as operators .L+ t0 1 and .L− t0 defined in formulation of Theorem 1.5.1 on the real subspaces of .V0 (G). − We denote this new real spaces and operators also by .X+ , .X− , .L+ t0 , .Lt0 . This will not lead to misunderstanding because below we do not use their complex analogs. In a neighborhood of steady-state solution a of (1.111), we establish the existence of a manifold .M− which is invariant with respect to semigroup .S(t, w) (i.e., .∀w ∈ M− ∀t > 0 S(t, w) is well defined and for each .t > 0, S(t, w) ∈ M− . This manifold can be represented as the graph: M− = {u ∈ V01 (G) : u = a + u− + g(u− ), u ∈ X− ∩ O},

.

(1.120)

where .O is a neighborhood of origin in .V01 (G), .g : X− ∩ O → X+ is an operatorfunction of class .C 3/2 , and g(0) = 0, g ' (0) = 0.

.

(1.121)

Note that condition (1.121) means that manifold (1.120) is tangent to .X− at point a. The following theorem is true. Theorem 1.5.2 Let a satisfy (1.111), .σ > 0 satisfy (1.65), and .O = Oε = {v ∈ V01 (G) : .‖v‖V 1 (G) < ε}, and .ε is sufficiently small. Then there exists a unique 0

operator-function .g : X− ∩ O → X+ of class .C 3/2 satisfying (1.121) such that the manifold .M− defined in (1.120) is invariant with respect to resolving semigroup .S(t, w0 ) of problem (1.108), (1.109). There exists a constant .c > 0 such that ‖S(t, w0 ) − a‖V 1 (G) ⩽ c‖w0 − a‖V 1 (G) e−σ t as t ⩾ 0

.

0

0

(1.122)

for each .w0 ∈ M− . This theorem follows form results of [1, Ch. 5, Sect. 2; Ch. 7,Sect. 5] and from Theorem 1.5.1 and Lemma 1.5.1.

1.5.2 Extension Operator Here we construct an extension operator for Navier–Stokes equations. This operator is nonlinear analog of the extension operator (1.90) constructed for Oseen equations. Recall that the domain .Ω and its extension G satisfy (1.81), (1.82). Besides, the space .Vˆ 1 (Ω) is defined in (1.89). Theorem 1.5.3 Suppose that .a(x) is a steady-state solution of (1.111), .v(x) ˆ = γΩ a, and .M− is the invariant manifold constructed in a neighborhood .a + O of a

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

31

in .V01 (G) in Theorem 1.5.2. Let .Bε1 = {v0 ∈ Vˆ 1 (Ω) : .‖v0 − v‖ ˆ V 1 (Ω) < ε1 }. Then for sufficiently small .ε1 there exists a continuous operator .

Extσ : vˆ + Bε1 → M− ,

(1.123)

which is an operator of extension for vector fields from .Ω to G: (Extσ v)(x) ≡ v(x),

.

x ∈ Ω.

(1.124)

Proof Let .L : Vˆ 1 (Ω) → V01 (G) be the extension operator constructed in Step 1 of Theorem’s 1.4.1 proof. Similarly to (1.93) we introduce the following operator of extension: Qv(x) = Lv(x) + w(x),

(1.125)

.

ˆ = G \ Ωε/2 which is constructed where .w(x) is a vector field concentrated in .Ω by .v(x). We describe its construction below. At last we define the desired operator .Extσ by the formula .

Extσ v = Π− Qz + g(Π− Qz) + a, with z = v − a,

(1.126)

where .Π− = I − Π+ , .Π+ is an operator (1.119) of projection on .X+ = Π+ V01 (G), 1 .X− = Π− V (G), and .g : X− → X+ is the operator constructed in Theorem 1.5.2. 0 By definition (1.120) of .M− we have .Extσ v ∈ M− . Hence we have to ensure that the equality (Extσ v)(x) ≡ v(x),

.

x∈Ω

(1.127)

is true, which shows that .Extσ is an extension operator. By (1.119) .{ej (x)} generates X+ , and therefore the map .g(u) can be written in the form

.

g(u) =

K 

.

ej gj (u).

j =1

That is why taking into account (1.119) we can rewrite (1.126) in the form

.

Extσ v = a(x) + Qz(x) −

K  j =1

 Qz(y)εj (y) dy +

ej (x) Q

K 

ej (x)gj (Π− Qz)

j =1

(1.128) (z = v − a). By virtue of Lemma 1.5.2, .{ej (x), .x ∈ Ω} are linearly independent, and therefore (1.127) and (1.128) imply

.

32

A. V. Fursikov

 Qz(x)εj (x) dx = gj (Π− Qz),

.

j = 1, . . . , K.

(1.129)

G

Similarly to (1.103) we look for the vector field .w(x) from (1.125) in the form w = −(−π Δ)−1 ˆ

.

Ω

K 

(1.130)

pj εj .

j =1

To find coefficients .(p1 , . . . , pK ) ≡ p, → we substitute (1.130) into (1.129) taking into account (1.125) and (1.119). As a result we get   z→ − Ap→ = g→ Lz − (p, → (−π Δ)−1 ε→) − (→e, z→ − Ap) → , ˆ

.

(1.131)

Ω

where L is the extension operator from (1.125), .z→ = (z1 , . . . , zK ), .A = ‖aj k ‖, and  zj =

 (Lz(x), εj (x)) dx, aj k =

.

G

Ω

G .

.

((−π Δ)−1 ε (x), εj (x)) dx, ˆ k

g→(u) = (g1 (u), . . . , gK (u)),

ε→ = (ε1 (x), . . . , εK (x)), e→ = (e1 (x), . . . , eK (x)), (→ c, e→) =

K 

cj ej .

j =1

We showed in Step 4 of Theorem’s 1.4.1 proof that matrix .A = ‖aj k ‖ is positive definite and therefore it is invertible. Applying to both parts of (1.131) the matrix .−A−1 , we get the equality p→ = Gz (p), →

(1.132)

.

where the map .Gz : RK → RK is defined by the relation Gz (p) → = A−1 z − A−1 g→(Lz − (→e, z→) + (p, → (−π Δ)−1 ε→) + (→e, Ap)). → ˆ

.

Ω

(1.133)

By virtue of Theorem 1.5.2 the map .A−1 g→ : RK → RK belongs to the class and .A−1 g→(0) = 0, .A−1 g→' (0) = 0. Therefore for sufficiently small .‖p→1 ‖RK , .‖p →2 ‖RK , .‖z‖V 1 (G) , we derive from (1.133) that 1+1/2 .C

0

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

33

‖Gz (p→1 ) − Gz (p→2 )‖ ⩽ sup ‖A−1 g→' (𝚪z − (→e, z→)+ β∈[0,1]

+ (β p→1 + (1 − β)p→2 , (−π Δ)−1 ε→) ˆ Ω

.

− (→e, A[β p→1 + (1 − β p→2 )]))‖ · ‖p→1 − p→2 ‖ ⩽ ⩽ γ (z, p→1 , p→2 )‖p→1 − p→2 ‖, where γ (z, p1 , p2 ) 1/2 V01 (G)

⩽ γ1 (‖z‖

1/2

1/2

+ ‖p1 ‖RK + ‖p2 ‖RK ),

and .γ1 > 0 is a constant. Therefore the map .Gz is a contraction one. Hence by the contraction mapping principle [25], equation (1.132) has a unique solution .p→ = (p1 , . . . , pK ) if .‖z‖V 1 (G) is sufficiently small. For these .‖z‖V 1 (G) , the operator .Extσ 0 0 defined in (1.126), (1.125), and (1.130) is the desired extension operator. ⨆ ⨅

1.5.3 Theorem on Stabilization We set Vˆ 2 (Ω) = V 2 (Ω) ∩ Vˆ 1 (Ω),

.

(1.134)

where .Vˆ 1 (Ω) is space (1.89). Proposition 1.5.1 Let .f ∈ (L2 (Ω))3 , and a pair .(v(x), ˆ ∇ p(x)) ˆ belongs to ˆ 2 (Ω) × (L2 (Ω))3 and satisfies equations (1.30) and (1.31). Then there exist an .V extension .g(x) ∈ (L2 (G))2 of .f (x) from .Ω to G and an extension .(a(x), ∇q(x)) ∈ (V 2 (G) ∩ V01 (G)) × (L2 (G))2 of .(v(x), ˆ ∇ p(x)) ˆ from .Ω to G such that the pair .(a(x), ∇q(x)) is a solution of (1.38), (1.39). The proof of this simple assertion is absolutely the same as in two-dimensional case (see Proposition 5.1 in [12]). We are now in a position to formulate the main result of this chapter connected with the local stabilization theory. Theorem 1.5.4 Let .Ω ⊂ R3 be a bounded domain with .C ∞ -boundary .∂Ω. Suppose that an extension .G ⊂ R3 of .Ω satisfies (1.81), (1.82). Let .f (x) ∈ (L2 (Ω))3 and .(∇ v(x), ˆ ∇ p(x)) ˆ ∈ Vˆ 2 (Ω) × (L2 (Ω))3 satisfy (1.30), (1.31). Then for an arbitrary ˆ 1 (Ω0 ) .σ > 0, there exists a sufficiently small .ε1 > 0 such that for each .v0 ∈ V satisfying ‖vˆ − v0 ‖V 1 (Ω) < ε1 ,

.

(1.135)

34

A. V. Fursikov

there exists a feedback boundary control .u(t, x), (t, x) ∈ Σ ≡ R+ × ∂Ω that stabilizes the Navier–Stokes boundary value problem (1.26)–(1.29) with the rate (1.32), i.e., the solution v of (1.26)–(1.29) satisfies (1.32). Proof Using Proposition 1.5.1 we extend .v(x) ˆ to .a(x) ∈ V 1 (G) and .f (x) to 2 .g(x) ∈ (L2 (G)) . As a result we get the boundary value problem (1.35)–(1.37) (with certain .w0 ) and steady-state solution .(a(x), ∇q(x)) of this problem. We can suppose that .σ > 0 satisfies (1.65): otherwise we increase .σ a little bit and get (1.65). By virtue of Theorem 1.5.2 in a neighborhood of a there exists a manifold .M− which is invariant with respect to resolving semigroup .S(t, w0 ) of problem (1.108), (1.109), and for each .w0 ∈ M− inequality (1.122) holds. Let .ε1 be so small that it satisfies condition of Theorem 1.5.3. Then we apply an extension operator .Extσ constructed in Theorem 1.5.3 to the initial condition .v0 of problem (1.26)–(1.29) and take .w0 = Ext v0 as an initial condition for problem (1.35)–(1.37) or for Eq. (1.108) (which is equivalent). Then since .w0 ∈ M− , .S(t, w0 ) ∈ M− for each .t ⩾ 0, and estimate (1.122) holds. We define the solution .(v, u) of stabilized problem (1.26)–(1.29) by formula (1.42), where .w(t, x) = S(t, w0 ) is the solution of (1.108), (1.109). Then (1.32) follows from (1.42), (1.122). ⨆ ⨅

1.6 On Non-local Stabilization of Hydrodynamic Type System. Semi-linear Parabolic Equation of Normal Type From this section the second part of the chapter begins: we pass to study non-local stabilization of some hydrodynamic type system. Recall some elements of the plan to study this domain that was discussed in introduction. Since to use .V 0 (T3 ) as phase space is more conveniently than .V 1 (T3 ) by some technical reasons, we pass from studies stabilization problem for Navier–Stokes system with phase space .V 1 (T3 ) to stabilization problem for Helmholtz system with phase space .V 0 (T3 ) (see details in this section). As has been written in the introduction in the first step of investigating the nonlocal stabilization problem, we omit some part of the nonlinear term in Helmholtz system such that obtained nonlinear term has the form .B(ω) = Ф(ω)ω where .Ф(ω) is the functional homogeneous of the first order with respect to .ω (see below Eq.(1.152)). Such system has been titled normal parabolic equation (NPE). The next two sections are devoted to normal parabolic equations. In this section we recall the basic information on parabolic equations of normal type corresponding to 3D Navier–Stokes system: their derivation, explicit formula for solutions, the theorem on the existence and uniqueness of solution for normal parabolic equations, and the structure of their dynamics. These results have been obtained in [14, 15].

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

35

1.6.1 Navier–Stokes Equations Let us consider a 3D Navier–Stokes system ∂t v(t, x) − Δv(t, x) + (v, ∇)v + ∇p(t, x) = 0, div v = 0,

.

(1.136)

with periodic boundary conditions v(t, . . . , xi , . . .) = v(t, . . . , xi + 2π, . . .), i = 1, 2, 3

.

(1.137)

and an initial condition v(t, x)|t=0 = v0 (x),

(1.138)

.

where .t ∈ R+ , .x = (x1 , x2 , x3 ) ∈ R3 , .v(t, x) = (v1 , v2 , v3 ) is the velocity vector field of fluid .∇p is the gradient of pressure, .Δ is the Laplace operator, and flow, 3 .(v, ∇)v = v j =1 j ∂xj v. Periodic boundary conditions (1.137) mean that Navier– Stokes eqautions (1.136) and initial conditions (1.138) are defined on torus .T3 = (R/2π Z)3 . For each .m ∈ Z+ = {j ∈ Z : j ≥ 0}, we define the space  V

.

m

= V (T ) = {v(x) ∈ (H (T )) : divv = 0, m

3

m

3

3

T3

v(x)dx = 0},

(1.139)

where .H m (T3 ) is the Sobolev space. It is well known that the nonlinear term .(v, ∇)v in problem (1.136)–(1.138) satisfies the relation  . (v(t, x), ∇)v(t, x) · v(t, x)dx = 0. T3

Therefore, multiplying (1.136) scalarly by v in .L2 (T3 ), integrating by parts by x, and then integrating by t, we obtain the well-known energy estimate  .

T3

|v(t, x)|2 dx + 2

 t 0

T3

 |∇x v(τ, x)|2 dxdτ ≤

T3

|v0 (x)|2 dx,

(1.140)

which allows to prove the existence of weak solution for (1.136)–(1.138). But, as is well known, the scalar multiplication of (1.136) by v in .V 1 (T3 ) does not result into an analog of estimate (1.140).

36

A. V. Fursikov

1.6.2 Helmholtz Equations Let us derive from the problem (1.136)–(1.138) for fluid velocity v the similar problem for the curl of velocity: ω(t, x) = curl v(t, x) = (∂x2 v3 − ∂x3 v2 , ∂x3 v1 − ∂x1 v3 , ∂x1 v2 − ∂x2 v1 ).

.

(1.141)

It is well known from the vector analysis that (v, ∇)v = ω × v + ∇

.

|v|2 , 2

curl(ω × v) = (v, ∇)ω − (ω, ∇)v, if div v = 0, div ω = 0,

.

(1.142) (1.143)

where .ω × v = (ω2 v3 − ω3 v1 , ω3 v1 − ω1 v3 , ω1 v2 − ω2 v1 ) is the vector product of .ω and v, and .|v|2 = v12 +v22 +v32 . Substituting (1.142) into (1.136), applying curl operator to both sides of the obtained equation, and taking into account (1.141), (1.143), and formula .curl ∇F = 0, we obtain the Helmholtz equations: ∂t ω(t, x) − Δω + (v, ∇)ω − (ω, ∇)v = 0

.

(1.144)

with initial conditions ω(t, x)|t=0 = ω0 (x) := curl v0 (x)

.

(1.145)

and periodic boundary conditions.

1.6.3 Derivation of Normal Parabolic Equations (NPEs) Using decomposition into Fourier series v(x) =



.

v(k)e ˆ

i(k,x)

, v(k) ˆ = (2π )

k∈Z3

−3

 T3

v(x)e−i(k,x) dx,

(1.146)

where .(k, x) = k1 · x1 + k2 · x2 + k3 · x3 , .k = (k1 , k2 , k3 ), and the well-known formula .curl curl v = −Δv, if .div v = 0, we see that the inverse operator to curl is well defined on space .V m and is given by the formula curl−1 ω(x) = i

.

 k × ω(k) ˆ ei(k,x) . 2 |k| 3

k∈Z

(1.147)

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

37

Therefore, operator .curl : V 1 I→ V 0 realizes isomorphism of the spaces, and thus, a sphere in .V 1 for (1.136)–(1.138) is equivalent to a sphere in .V 0 for (1.144)–(1.145). Let us denote the nonlinear term in Helmholtz system by B: B(ω) = (v, ∇)ω − (ω, ∇)v,

(1.148)

.

where v can be expressed in terms of .ω using (1.147). Multiplying (1.148) scalarly by .ω = (ω1 , ω2 , ω3 ) and integrating by parts, we get the expression  (B(ω), ω)V 0 = −

3 

.

T3 j,k=1

ωj ∂j vk ωk dx,

(1.149)

which, generally speaking, is not zero. Hence, energy estimate for solutions of 3D Helmholtz system is not fulfilled. In other words, operator B allows decomposition B(ω) = Bn (ω) + Bτ (ω),

(1.150)

.

where vector .Bn (ω) is orthogonal to sphere .Σ(‖ω‖V 0 ) = {u ∈ V 0 : ‖u‖V 0 = ‖ω‖V 0 } at the point .ω, and vector .Bτ is tangent to .Σ(‖ω‖V 0 ) at .ω. In general, both terms in (1.150) are non-zero. Since the presence of .Bn , and not .Bτ , prevents the fulfillments of the energy estimate, it is plausible that just .Bn generates the possible singularities in the solution. Therefore, it seems reasonable to omit the .Bτ term in Helmholtz system and in the first steps of our investigation to study the system (1.144) with nonlinear operator .B(ω) replaced with .Bn (ω). We will call the obtained system the system of normal parabolic equations (NPEs). Let us derive the NPE system corresponding to (1.144)–(1.145). Since summand .(v, ∇)ω in (1.148) is tangential to vector .ω, the normal part of operator B is defined by the summand .(ω, ∇)v. We shall seek for it in the form .Ф(ω)ω, where .Ф is the unknown functional, which can be found from the equation  .

T3

 Ф(ω)ω(x) · ω(x)dx =

T3

(ω(x), ∇)v(x) · ω(x)dx.

(1.151)

According to (1.151),  Ф(ω)ω =

.

T3 (ω(x), ∇) curl

−1

ω(x) · ω(x)dx/

 T3

|ω(x)|2 dx, ω /= 0, 0, ω ≡ 0, (1.152)

where .curl−1 ω(x) is defined in (1.147) . Thus, we arrive at the following system of normal parabolic equations corresponding to Helmholtz equations (1.144): ∂t ω(t, x) − Δω − Ф(ω)ω = 0,

.

div ω = 0,

(1.153)

38

A. V. Fursikov

ω(t, . . . , xi , . . .) = ω(t, . . . , xi + 2π, . . .), i = 1, 2, 3,

.

(1.154)

where .Ф is the functional defined in (1.152). Further we study problem (1.153), (1.154) with initial condition (1.145).

1.6.4 Explicit Formula for Solution of NPE In this subsection we remind the explicit formula for the NPE solution. Lemma 1.6.1 Let .S(t, x; ω0 ) be the solution of the following Stokes system with periodic boundary conditions: ∂t z − Δz = 0; .

(1.155)

z(t, . . . , xi + 2π, . . .) = z(t, x), i = 1, 2, 3; .

(1.156)

z(0, x) = ω0 ,

(1.157)

.

i.e., .S(t, x; ω0 ) = z(t, x). Then the solution of problem (1.153) with periodic boundary conditions and an initial condition (1.145) has the form ω(t, x; ω0 ) =

.

1−

t 0

S(t, x; ω0 ) Ф(S(τ, x; ω0 ))dτ

.

(1.158)

For the proof of this lemma, one can see [14] and [15]. Note that in [14] and [15] analytical and geometrical structure of sets .M− , M+ , Mg that composed the phase space .V 0 (T3 ) are studied. We do not formulate these results here because they will not be used below.6

1.7 Stabilization of Solution for NPE by Starting Control 1.7.1 Formulation of the Main Result on Stabilization We consider semi-linear parabolic equations (1.153): ∂t y(t, x) − Δy(t, x) − Ф(y)y = 0,

.

6 The

(1.159)

definition and properties of sets .M− , M+ , Mg are given below in Sect. 1.8.3 in the case of differentiated Burgers equation.

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

39

where ⎧ curl−1 y(x)·y(x)dx ⎨ T3 (y(x),∇)  , y= / 0, 2 T3 |y(x)| dx .Ф(y) = ⎩0, y ≡ 0,

(1.160)

with the periodic boundary condition y(t, . . . xi + 2π, . . .) = y(t, x), i = 1, 2, 3,

.

(1.161)

and an initial condition y(t, x)|t=0 = y0 (x) + u0 (x).

.

(1.162)

Here .y0 (x) ∈ V 0 is an arbitrary given initial datum and .u0 (x) ∈ V 0 is a control. Phase space .V 0 is defined in (1.139). We assume that .u0 (x) is supported on .[a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ] ⊂ T3 = (R/2π Z)3 : supp u0 ⊂ [a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ].

.

(1.163)

Our goal is to find for every given .y0 (x) ∈ V 0 a control .u0 ∈ V 0 satisfying (1.163) such that there exists a unique solution .y(t, x; y0 + u0 ) of (1.159)– (1.162) and this solution satisfies the estimate ‖y(t, ·; y0 + u0 )‖0 ⩽ α‖y0 + u0 ‖0 e−t

.

∀t > 0

(1.164)

with a certain constant .α > 1. The following main theorem holds: Theorem 1.7.1 Let .y0 ∈ V 0 be given. Then there exists a control .u0 ∈ V 0 satisfying (1.163) such that there exists a unique solution .y(t, x; y0 +u0 ) of (1.159)– (1.162), and this solution satisfies the bound (1.164) with a certain constant .α > 1. For the complete proof of this theorem, one can see [22]. The rest part of this section is devoted to the explanation of the main steps of this theorem proof.

1.7.2 Formulation of the Main Preliminary Result To rewrite condition (1.163) in a more convenient form, let us first perform the change of variables in (1.159)–(1.162): x˜i = xi −

.

ai + bi , i = 1, 2, 3, 2

40

A. V. Fursikov

and denote 

a2 + b2 a3 + b3 a1 + b1 , x˜2 + , x˜3 + , y(t, ˜ x) ˜ = y t, x˜1 + 2 2 2

 a2 + b2 a1 + b1 a3 + b3 , ˜0 (x) .y ˜ = y0 x˜1 + , x˜2 + , x˜3 + 2 2 2

 a3 + b3 a2 + b2 a1 + b1 . u˜0 (x) ˜ = u0 x˜1 + , x˜3 + , x˜2 + 2 2 2

(1.165)

Then substituting (1.165) into relations (1.159)–(1.162) and (1.164) and omitting the tilde sign leave these relations unchanged, while inclusion (1.163) transforms into supp u0 ⊂ Ω, Ω = [−ρ1 , ρ1 ] × [−ρ2 , ρ2 ] × [−ρ3 , ρ3 ],

.

(1.166)

bi − ai where .ρi = ∈ (0, π ), .i = 1, 2, 3. 2 Below we consider the stabilization problem (1.159)–(1.162), (1.164) with condition (1.166) instead of (1.163). We look for a starting control .u0 (x) in a form u0 (x) = u1 (x) − λu(x),

.

(1.167)

where the component .u1 (x) and the constant .λ > 0 will be defined later, and the main component .u(x) is defined as follows. For given .ρ1 , ρ2 , ρ3 ∈ (0, π ), we choose .p ∈ N such that .

π ⩽ ρi , i = 1, 2, 3, p

(1.168)

and denote by .χ πp (α) the characteristic function of interval .(− πp , πp ):

χ πp =

.

1, |α| ≤ πp , 0, πp < |α| ≤ π.

(1.169)

Then u(x) = curl curl(χ πp (x1 )χ πp (x2 )χ πp (x3 )w(px1 , px2 , px3 ), 0, 0),

.

(1.170)

where w(x1 , x2 , x3 ) =

3 

.

i,j,k=1 i βe−18t

∀t ⩾0 (1.174)

with a positive constant .β. The proof of Theorem 1.7.2 is complicated and one can see it in [22].

1.7.3 Intermediate Control To avoid certain difficulties with the proof of Theorem 1.7.1, we propose to include additional control that eliminates some Fourier coefficients in a given initial condition .y0 of our stabilization problem. We will use techniques developed in the local stabilization theory (see Sects. 1.2–1.5 above as well as [15, 17] and the references therein). Let us define the following subspaces .V+ , V− of the phase space .V 0 : V+ = {v(x) ∈ V 0 : v(x) =



.

vk ei(k,x) , vk ∈ R3 },

(1.175)

0 cλ3 e−18t − c2 (λ2 e−22t + λe−38t + e−54t ),

.

(1.195)

which completes the proof of estimate (1.188). The denominator of .Ф(S(τ, ·; z0 − λu)) can be estimated from below as follows:   2 |S(t, z0 −λu)| dx = |S(t, z0 ) − λS(t, u)|2 dx ≥ T3

.



 T3

T3

 1 2 λ |S(t, u)|2 − |S(t, z0 )|2 dx. 2

(1.196)

According to (1.186), the second term in (1.196) can be estimated from above as  .

T3



|S(t, z0 )|2 dx ≤

|ˆz0 (k)|2 · e−18t < ‖y0 ‖0 · e−18t ,

(1.197)

k∈Z3 ,|k|2 ≥18

and the first term can be estimated from below as    2 i(k,x) |S(t, u)|2 dx = | u(k)e ˆ · e−|k| t |2 dx = T3

⎛

 .

T3

⎝ T3

(2π )3



i(k,x) u(k)e ˆ ·e

k∈Z3 −|k|2 t

·

k∈Z3



k∈Z3



⎞ i(m,x) u(m)e ˆ ·e

m∈Z3 −2|k| t (u(k) ˆ · u(−k))e ˆ = (2π )3 2



−|m|2 t

⎠ dx =

(1.198)

2 −2|k| t |u(k)| ˆ e ≥ 2

k∈Z3

8π 3 (|u(1, ˆ 1, 0)|2 + |u(1, ˆ 0, 1)|2 + |u(0, ˆ 1, 1)|2 )e−4t . So, finally, from (1.196)–(1.198), we get the following estimate for the denominator:

46

A. V. Fursikov

 T3

.

|S(t, z0 − λu)|2 dx >



‖z0 ‖ ˆ 1, 0)|2 + |u(1, ≥ e−4t · 8π 3 λ2 |u(1, ˆ 0, 1)|2 + |u(0, ˆ 1, 1)|2 − |λ|2

4π 3 λ2 · e−4t (|u(1, ˆ 1, 0)|2 + |u(1, ˆ 0, 1)|2 + |u(0, ˆ 1, 1)|2 ) =: γ · λ2 · e−4t (1.199) √ 2‖y0 ‖ for .|λ| > λ0 =  . The last equality |u(1, ˆ 1, 0)|2 + |u(1, ˆ 0, 1)|2 + |u(0, ˆ 1, 1)|2 in (1.199) is the definition of .γ > 0. It follows from (1.188) and (1.199) that 



c1 · e−18τ λc0 (1 − e−14t ), dτ = 1 + −4τ 14γ γe 0 0 (1.200) which completes the proof of Theorem 1.7.1. 1−

.

t

Ф(S(τ, ·; z0 − λu))dτ > 1 + λ

t

1.8 Burgers Equation and Corresponding Semi-linear Parabolic Equation of Normal Type Below we recall the main information on normal parabolic equation (NPE) corresponding to Burgers equation: its derivation, an explicit formula for its solution, and the dynamic structure generated by NPE.

1.8.1 Derivation of the Normal Parabolic Equation (NPE) Let us consider Burgers equation ∂t v(t, x) − ∂xx v(t, x) − ∂x v 2 (t, x) = 0

(1.201)

.

with a periodic boundary condition and an initial datum v(t, x + 2π ) = v(t, x),

v(t, x)|t=0 = v0 (x).

.

(1.202)

As is known, orthogonality in .L2 (0, 2π ) of the quadratic term .∂x v 2 (t, x) from Eq. (1.201) to the function .v(t, x) implies energy inequality for a solution of Burgers equation: 

2π

.

0

v (t, x)dx + 2 2

 t 0

2π 0



2π

(∂x v(t, x)) dxdt ⩽ 2

0

v02 (x)dx.

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

47

But a similar inequality is not right if the function .v(t, x) will be changed on .∂x v(t, x). Indeed, after differentiating (1.201) with respect to x, we get the differentiated Burgers equation ∂t y(t, x) − ∂xx y(t, x) − B(v, y) = 0,

(1.203)

B(v, y) = 2y 2 + 2v∂x y.

(1.204)

.

where .y = ∂v/∂x and .

Multiplying (1.204) on y scalar in .L2 (T), where .T = R/2π Z is the circumference, and integrating by parts, we get that 

2π

.



2π

B(v, y)ydx =

0

 (2y 3 + 2vy∂x y)dx =

0

2π

y 3 dx /= 0.

(1.205)

0

Let us introduce the space important for us:  L02 (T) = {v(x) ∈ L2 (T) :

2π

.

v(x)dx = 0}

(1.206)

0

and decompose in this space the operator .B(v, vx ) on normal and tangential components: B(v, y) = Bn (v, y) + Bτ (v, y),

(1.207)

.

where the normal component is .Bn (v, y) = Ф(v, y)y, where .Ф(v, y) is a functional, and the vector .Bτ (v, y) is orthogonal to the vector y in .L02 (T): 

2π

.

Bτ (v, y)ydx = 0.

(1.208)

0

To define the functional .Ф(v, y), we substitute (1.207) into (1.205) and use (1.208). As a result we get 

2π

.

 y 3 dx =

0

2π



2π

Ф(v, y)y 2 dx = Ф(v, y)

0

y 2 dx.

0

Hence, the functional .Ф does not depend on v and depends only on y, and for .y /= 0  Ф(y) =

.

0

2π

 y 3 dx

2π

y 2 dx.

0

Evidently, for .y ≡ 0 we can take .Ф equal to zero by continuity, i.e., .Ф(0) = 0. Changing in Eq. (1.203) the term .B(v, y) on .Ф(y)y, we get

48

A. V. Fursikov

∂t y(t, x) − ∂xx y(t, x) − Ф(y)y = 0,

(1.209)

.

where Ф(y) =

.

⎧  ⎨ 2π y(x)3 dx  2π y(x)2 dx,

y /= 0,

⎩0,

y ≡ 0.

0

0

(1.210)

Equation (1.209) is called semi-linear parabolic equation of normal type or normal parabolic equation (NPE). We will consider it with a periodic boundary condition y(t, x + 2π ) = y(t, x)

(1.211)

y(t, x)|t=0 = y0 (x).

(1.212)

.

and an initial condition .

For the problem (1.209)–(1.212), theorems on the existence and uniqueness of a solution in corresponding functional spaces have been proved (see [14]), as well as for NPE generated with the 3D Helmholtz equation (see [15]). For briefness we will not formulate them here but call our attention the property of NPE, which is key to study equations of this type (in particulary existence and uniqueness of a solution immediately imply from this property).

1.8.2 Explicit Formula for a Solution of NPE The following lemma is true: Lemma 1.8.1 Let .S(t, x; y0 ) be the solution on the heat equation: ∂t S − ∂xx S = 0,

S|t=0 = y0 (x)

.

(1.213)

with a periodic boundary condition. Then the solution of the problem (1.209)– (1.212) can be written as follows: y(t, x; y0 ) =

.

1−

t 0

S(t, x; y0 ) Ф(S(τ, x; y0 ))dτ

.

(1.214)

This lemma has been proved in [14]. Its proof is reduced to the substitution of indicated formula to (1.209) and direct verification.

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

49

1.8.3 Dynamic Structure Generated with NPE Recall the basic properties of dynamic flow generated by the problem (1.209)– (1.212). More exactly we decompose the phase space .L02 (T) of this dynamical system on three parts, where behavior of dynamical flow is essentially different. Definition 1.8.1 The set .M− ⊂ L02 (T) of initial conditions .y0 such that the solution .y(t, x; y0 ) of the problem (1.209)–(1.212) exists for all times .t > 0 and satisfies inequality ‖y(t, ·; y0 )‖ ⩽ α‖y0 ‖e−t

∀t > 0

.

(1.215)

is called the stability set. Here .‖ · ‖ is the norm of the phase space .L02 (T):  ‖y0 ‖ := 2

.

‖y0 ‖2L0 (T) 2

2π

:=

|y0 (x)|2 dx,

(1.216)

0

and .α = α(y0 ) > 1 is a certain fixed number that depends on .y0 . Definition 1.8.2 The set .M+ ⊂ L02 (T) of initial conditions .y0 of the problem (1.209)–(1.212) such that corresponding solution .y(t, x; y0 ) exists only on a finite times interval .t ∈ (0, t0 ), t0 > 0, that depends on .y0 and blows up at .t = t0 7 is called the set of explosions. By virtue of explicit formula (1.214) for a solution .y(t, x; y0 ),  M+ = {y0 ∈ H 0 (T) : ∃t0 > 0

.

t0

Ф(S(τ, ·; y0 ))dτ = 1}.

(1.217)

0

The minimal value of the numbers set .{t0 } that satisfies (1.217) is called the moment of explosion. Definition 1.8.3 The set .Mg ⊂ L02 (T) of initial conditions .y0 for problem (1.209)– (1.212), such that the corresponding solution .y(t, x; y0 ) exists for all .t ∈ R+ and .‖y(t, ·; y0 )‖0 → ∞ as .t → ∞, is called the set of growth. The following statement holds: Theorem 1.8.1 The sets of stability, explosions, and growth are not empty: M− /= ∅,

.

7 That

M+ /= ∅,

is, .‖y(t, ·; y0 )‖0 → ∞ as .0 < t < t0 and .t → t0 .

Mg /= ∅.

50

A. V. Fursikov

Moreover, M− ∪ M+ ∪ Mg = L02 (T).

.

This theorem had been proved in [14, 15]. Note that in [14, 15] is studied not only analytical but also geometrical structure of the sets .M− , M+ , Mg . We do not formulate here many of these results, since they do not used below.8 Note that for instance Theorem 1.8.1 formulated above is necessary at least to justify that the theorem on stabilization of NPE by start control is reach of content.

1.8.4 Formulation of the Main Result on Stabilization Let us consider boundary value problem for NPE (1.209)–(1.212), Where the initial condition (1.212) has been changed on y(t, x)|t=0 = y0 (x) + v(x),

.

x ∈ T = R/2π Z,

(1.218)

where .y0 (x) is the given initial condition, and .v(x) is the start control. We assume that the segment .[a, b] ⊂ T is given, and by condition the support of the control .v(x) belongs to .[a, b]: supp v ⊂ [a, b].

(1.219)

.

The stabilization problem is formulated as follows: For a given .y0 (x) ∈ L02 (T), find a control .v ∈ L02 (T), satisfying (1.219), such that the solution .y(t, x; y0 + v) of the problem (1.209), (1.211), (1.218) is defined for all .t > 0 and satisfies the estimate ‖y(t, ·; y0 + v)‖ ⩽ α‖y0 + v‖e−t

.

∀t > 0

(1.220)

with a certain constant .α > 1. Note that the posed problem is full of content only for .y0 ∈ M+ or for .y0 ∈ Mg , since for .y0 ∈ M− by virtue of (1.215) for stabilization, it is enough to take the control .v ≡ 0. The following main theorem is true: Theorem 1.8.2 For any given .y0 (x) ∈ M+ ∪ Mg , there exists a control .v ∈ L02 (T), satisfying (1.219), that solves the stabilization problem.

8 The

most complete results on geometrical structure of the sets .M− , M+ , Mg had been obtained in [15] in the case NPE, generated with 3D Helmholtz system. The same results for NPE corresponding differentiated Burgers equation can be obtained absolutely similarly.

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

51

1.9 Differentiated Burgers Equation and Functional-Polar Coordinates (fpc) In this section some information on differentiated Burgers equation is gathered, which is necessary to solve the problem of its non-local stabilization.

1.9.1 On Solvability of the Differential Burgers Equation for Small Initial Conditions Let us consider the boundary value problem for differentiated Burgers equation ∂t y(t, x) − ∂xx y(t, x) − B(v, y) = 0,

.

y(t, x + 2π ) = y(t, x),

.

(1.221)

y(t, x)|t=0 = y0 (x),

(1.222)

with ∂x v(t, x) = y(t, x).

(1.223)

where B(v, y) = 2y 2 + 2v∂x y,

.

We will need later the following well-known statement: Theorem 1.9.1 There exists a sufficiently small number .r > 0 such that for .‖y0 ‖ ⩽ r there exists a unique solution .y(t, x) ∈ L2 (R+ ; L02 (T)) ∩ C ∞ (R+ × T) of the problem (1.221), (1.222). Moreover this solution satisfies the following estimate: ‖y(t, ·)‖ ⩽ c‖y0 ‖e−t

.

∀t ∈ R+ ,

(1.224)

where .c > 0 is a certain positive constant.

1.9.2 Functional-Polar Coordinates (fpc) As it has been indicated in (1.207), the bilinear operator (1.223) from Eq. (1.221) is the sum of normal and tangential operators: .B = Bn + Bτ , where the image of the normal operator .Bn (y) is collinear to y, and the image of the tangential operator .Bτ (v, y) is orthogonal to y (see (1.208). Because of this property it is convenient to study differentiated Burgers equation in so cold functional polar coordinates, which are defined as follows. Let us define unit sphere in the space .L02 (T) (introduced in (1.206)): Σ = {ϕ(x) ∈ L02 (T) : ‖ϕ‖ = 1}.

.

(1.225)

52

A. V. Fursikov

Definition 1.9.1 Function-polar coordinates (fpc) of a vector .y(x) ∈ L02 (T) are the pair .(ρ, ϕ(x)) ∈ R+ × Σ, where .ρ = ‖y‖, ϕ(x) = y(x)/ρ ∈ Σ. Since fpc will be used to write differential equations, they will be considered not in .L02 (T), but in .L02 (T) ∩ H k (T), where .H k (T) is the Sobolev space of functions square-integrable together with their derivatives up to order k. In particular, instead of the sphere defined in Eq. (1.225) the following sets will be used: Σ(k) = {ϕ(x) ∈ L02 (T) : ‖ϕ‖ = 1} ∩ H k (T).

.

(1.226)

Lemma 1.9.1 The differentiated Burgers equation (1.221), (1.222) has the following form in the functional polar coordinates: ∂t ρ(t) + ‖∂x ϕ(t, ·)‖2 ρ(t) − ρ 2 (t)Ψ(ϕ(t, ·)) = 0,

ρ|t=0 = ρ0 := ‖y0 ‖, (1.227) 2 .∂t ϕ(t, x) − ∂xx ϕ(t, x) − ‖∂x ϕ‖ ϕ − ρBτ (w, ϕ) = 0, ϕ|t=0 = ϕ0 := y0 /ρ0 , (1.228) where .w(t, x) = v(t, x)/ρ(t). .

Proof Let us rewrite (1.221) in the form Ψ(y) y − Bτ (v, y) = 0, ‖y‖2

(1.229)

Bτ (v, y) = 2y 2 + 2v∂x y − Ψ(y)/‖y‖2 .

(1.230)

∂t y(t, x) − ∂xx y(t, x) −

.

where 

2π

Ψ(y) =

.

y 3 (x)dx,

0

Taking scalar product of (1.229) and .y(t, x) in .L2 (T) and taking into account that Bτ (v, y) ⊥L2 (T) y and .‖y(t, ·)‖ = ρ(t), we will have

.

ρ(t)∂t ρ(t) + ‖∂x y(t, ·)‖2 − Ψ(y) = 0.

.

(1.231)

Using homogeneity of the function .Ψ on y of the third degree and relation .y = ρϕ, we get Eq. (1.227) after division of (1.231) on .ρ(t). Differentiating on t the equality .ϕ = y/ρ and expressing .∂t y and .∂t ρ in the right side of obtained equality from Eqs. (1.229) and (1.227), we will have ∂t ϕ =

.

y 1 (∂xx y +Ψ(y)y/‖y‖2 +Bτ (v, y))− 2 (−‖∂x ϕ‖2 ρ +ρ 2 Ψ(ϕ)). ρ ρ

(1.232)

Making in (1.232) the change .y = ρϕ, we get Eq. (1.228) after simple transformations. ⨆ ⨅

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

53

Remark 1.9.1 A remarkable property of functional polar coordinates is that normal and tangential components of quadratic operator in differentiated Burgers equation became in different equations of the system (1.227), (1.228). This property will be used later. Remark 1.9.2 NPE (1.209)–(1.212) in functional polar coordinates is written as follows: ∂t ρ(t) + ‖∂x ϕ(t, ·)‖2 ρ(t) − ρ 2 (t)Ψ(ϕ(t, ·)) = 0,

.

∂t ϕ(t, x) − ∂xx ϕ(t, x) − ‖∂x ϕ‖2 ϕ = 0,

ϕ|t=0

.

ρ|t=0 = ρ0 := ‖y0 ‖, (1.233) = ϕ0 := y0 /ρ0 . (1.234)

The proof is exactly the same as in Lemma 1.9.1. Note that an explicit formula of NPE solution (1.209)–(1.212) in fpc has the form: ρ0 ‖S(t, ·; ϕ0 )‖ , t 1 − ρ0 0 Ф(S(τ, ·; ϕ0 ))dτ

S(t, x; ϕ0 ) , ‖S(t, ·; ϕ0 )‖ (1.235) where .S(t, x) is the resolving operator of the problem (1.213). To get the proof is not difficult using the explicit formula (1.214) for NPE. ρ(t; ρ0 ϕ0 ) =

.

ϕ(t, x; ϕ0 ) =

Using the structure of the system (1.227), (1.228), indicated in Remark 1.9.1, we prove the following important assertion: Lemma 1.9.2 Let the initial data .(ρ0 , ϕ0 ) of the problem (1.227), (1.228) satisfy the conditions: ρ0 ⪢ 1,

.

Ψ(ϕ0 ) = −α1 < 0,

(1.236)

where .Ψ is the functional defined in (1.230). Suppose that .t0 > 0 is such a moment of time that ∀t ∈ [0, t0 ] Ψ(ϕ(t, ·; ϕ0 )) ⩽ −α,

.

where

0 < α < α1 .

(1.237)

Then the component .ρ(t) of the solution of problem (1.227), (1.228) satisfies the estimate ρ(t) ⩽

.

e−t 1 ρ0

+ α(1 − e−t )

∀t ∈ [0, t0 ].

(1.238)

Proof Since by the definition of the component .ϕ of solution of problem (1.227), (1.228) .‖ϕ(t, ·)‖ = 1, and ‖∂x ϕ(t, ·)‖2 =



.

k∈Z

k 2 |ϕˆk (t)|2 ⩾

 k∈Z

|ϕˆ k (t)|2 = ‖ϕ(t, ·)‖ = 1,

54

A. V. Fursikov

where .ϕˆk (t) is the k-th Fourier coefficient of the function .ϕ(t, x), and then taking into account condition (1.237), we get from (1.228) that the component .ρ(t) of solution of the problem (1.227), (1.228) satisfies to the estimate ∂t ρ(t) + ρ ⩽ −αρ 2 ,

ρ|t=0 = ρ0 .

.

⨆ ⨅

Solving this inequality, we get that .ρ(t) satisfies to the estimate (1.238).

1.9.3 Contour Lines of the Functional Ψ on the Sphere Σ(1) Since according to Lemma 1.9.2 the value of the functional .Ψ(ϕ) in (1.227) determines the behavior of the function .ρ(t), we need to study the contour lines of .Ψ on the unit sphere. To ensure the continuity of .Ψ, we use the unit sphere .Σ(1), defined by relation (1.226) (with .k = 1). For any number .α ∈ R the contour line .Ψ[α] of functional .Ψ is defined by the formula Ψ[α] = {ϕ ∈ Σ(1) : Ψ(ϕ) = α}.

.

(1.239)

Lemma 1.9.3 For each .α ∈ R the contour line .Ψ[α] is a continuous hypersurface on the sphere .Σ(1). Proof Let .ϕ ∈ Ψ[α] , and hence .ϕ ∈ Σ(1), Ψ ' (ϕ)[x] = 3ϕ 2 (x), where the “gradient” . 3ϕ 2 (x) is known to be the direction of steepest ascent of the functional .Ψ(ϕ). To assign the exact meaning to this assertion, we need to use the Riesz theorem about the representation of a linear functional and replace the functional 0 2 2 . 3ϕ (x) by the element . 3ϕ (x) − cϕ ∈ L (T1 ), realizing this functional, where 2  −1 2π 3ϕ 2 (x)dx. Set .cϕ = (2π ) 0 KerΨ ' (ϕ) = {u ∈ H 1 (T) ∩ L02 (T) :



.

T

ϕ 2 (x)u(x)dx = 0}.

(1.240)

Obviously, .KerΨ ' (ϕ) is a hypersurface in .H 1 (T) ∩ L02 (T). We denote the ball of radius .ϵ in the space .H 1 (T) by Bϵ (H 1 ) = {u(x) ∈ H 1 (T) : ‖u‖H 1 ⩽ ϵ}

.

(1.241)

and define the functional λ(·) : KerΨ ' (ϕ) ∩ Bϵ (H 1 ) → R,

.

which is determined from the relation

(1.242)

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

 .

T

(ϕ(x)+u(x)+λ(u)(3ϕ 2 (x)−cϕ ))3 dx = α, where

55

u(x) ∈ KerΨ ' (ϕ)∩Bϵ (H 1 ). (1.243)

Note that, by virtue of the inclusion .ϕ ∈ Ψ[α] , relation (1.243) is satisfied for .u(x) ≡ 0, whenever .λ(u)|u=0 = 0. Since . 3ϕ 2 (x)−cϕ , is the direction of steepest ascension of the functional .Ψ at the point .ϕ, we see that . 3ϕ 2 (x) − cϕ will be a direction of growth (but not the fastest growth) of functional .Ψ at point .ϕ + u, as well if 1 .u ∈ Bϵ (H ) and .ϵ is essentially small. This follows from the continuity of the functional .Ψ and its derivatives with respect to .ϕ. Note that directions lying in the hyperplane .KerΨ ' (ϕ) are the directions of least growth of .Ψ near .ϕ + Bϵ (H 1 ), and, therefore, .Ψ(ϕ +u) differs little from .α. Hence for each .u(x) ∈ KerΨ ' (ϕ)∩Bϵ (H 1 ) with a sufficiently small .ϵ, there exists a unique .λ(u) such that relation (1.243) holds. Let us prove the continuity of the functional .λ(u) in u. Replacing .λ(u) in the left side of (1.243) with .λ, we obtain an increasing function of .λ, because its derivative satisfies the inequality:  .

T

(ϕ + u + λ(3ϕ 2 − cϕ ))2 (3ϕ 2 − cϕ )dx ⩾ c > 0,

(1.244)

where c does not depend on .u ∈ KerΨ ' (ϕ) ∩ Bϵ (H 1 ) and .λ ∈ [minu λ(u), .maxu λ(u)]. Let us subtract relation (1.243) with .u = u2 the same relation with .u = u1 to obtain  .0 = (R(u1 ) − R(u2 ))3 dx T

 =

T

[R(u1 ) − R(u2 )][R 2 (u1 ) + R(u1 )R(u2 ) + R 2 (u2 )]dx

(1.245)

where we have used the notation R(u) = (ϕ(x) + u(x) + λ(u)(3ϕ 2 (x) − cϕ )).

.

(1.246)

Assume that .λ(u1 ) > λ(u2 ). Transforming the first factor on the right-hand side in Eq. (1.245) with the use of Eq. (1.246), we arrive at the inequality  |λ(u1 ) − λ(u2 )| .

T

(3ϕ 2 (x) − cϕ )[R 2 (u1 ) + R(u1 )R(u2 ) + R 2 (u2 )]dx ⩽

sup |u1 (x) − u2 (x)| x

Note that

 T

[R 2 (u1 ) + R(u1 )R(u2 ) + R 2 (u2 )]dx. (1.247)

56

A. V. Fursikov

R 2 (u1 ) + R(u1 )R(u2 ) + R 2 (u2 ) =

.

1 2 [R (u1 ) + (R(u1 ) + R(u2 ))2 + R 2 (u2 )]dx 2 (1.248)

with R(u1 ) + R(u2 ) = 2(ϕ +

.

u1 + u2 λ(u1 ) + λ(u2 ) + (3ϕ 2 (x) − cϕ )). 2 2

(1.249)

Substituting Eq. (1.248) into the left side of inequality (1.247), taking into account (1.246), (1.249) and inequality (1.244), we obtain the following lower bound for the integral from the left side of (1.247):  .

T

1 2

(3ϕ 2 (x) − cϕ )[R 2 (u1 ) + R(u1 )R(u2 ) + R 2 (u2 )]dx =

 T

(3ϕ 2 (x) − cϕ )[R 2 (u1 ) + (R(u1 ) + R(u2 ))2 + R 2 (u2 )]dx ⩾ 3c > 0. (1.250)

Using definition (1.246), it is possible to prove the existence of the constant .C(ϵ), independent of .u ∈ KerΨ ' (ϕ) ∩ Bϵ (H 1 ), such that  .

T

(R 2 (u1 ) + R(u1 )R(u2 ) + R 2 (u2 ))dx ⩽ C(ϵ).

(1.251)

Substituting (1.250) in the left part and (1.251) in the right part of the inequality (1.247)and using the Sobolev embedding theorem, we get that |λ(u1 ) − λ(u2 )| ⩽

.

C(ϵ) ‖u1 − u2 ‖H 1 3c

∀ u1 , u2 ∈ KerΨ ' (ϕ) ∩ Bϵ (H 1 ). (1.252)

Thus we have extended the point .ϕ ∈ Ψ[α] to the continuous hypersurface in .H 1 (T) defined by the formula .{R(u) : u ∈ KerΨ ' (ϕ)∩Bϵ (H 1 ), where .R(u) is the mapping defined in (1.247), (1.247), and (1.247). The intersection of this hypersurface with the unit sphere .Σ(1) gives the desired representation of the contour line .Ψ[α] : Ψ[α] = {R(u) : u ∈ KerΨ ' (ϕ) ∩ Bϵ (H 1 )} ∩ Σ(1).

.

(1.253) ⨆ ⨅

The proof of the lemma is complete.

1.10 Construction of a Stabilizing Impulsive Control for Differentiated Burgers Equation Consider the stabilization problem (1.221) ∂t y(t, x) − ∂xx y(t, x) − B(v, y) =

N 

.

k=0

uˆ k (x)δ(t − tk )

(1.254)

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

57

for Eq. (1.221) with the boundary and initial conditions (1.222), where .B(v, u) is the operator (1.223). It is well known that the statement of problem (1.254), (1.222) for .t ∈ [tk , tk+1 ] is equivalent to the following initial boundary value problem (periodic boundary conditions are assumed but are not explicitly indicated for brevity): ∂t y(t, x) − ∂xx y(t, x) − B(v, y) = 0,

.

y|t=tk = y(tk , x) + uk (x), t ∈ (tk , tk+1 ), (1.255)

where .y(tk , x) is the solution of problem (1.254), obtained on the previous time interval. For the zero impulse .uˆ k , we take the impulse constructed in the papers [22, 16], where the stabilization of the normal parabolic equation associated with the Burgers equation was studied. In the case of the normal parabolic equation obtained from ˆ y) = Ф(y)y + Bτ (v, y) with Eq. (1.255) by replacing the nonlinear term .A(v, .Ф(y)y, one impulse .u ˆ 0 is sufficient for stabilization; i.e., in other words, the solution .y(t, x) of Eq. (1.221) with the initial condition .y|t=0 = y0 (x) + u ˆ 0 (x) decays exponentially as t tends to infinity. For the full Eq. (1.222), the presence of the tangential operator .Bτ (v, y) completely changes the situation. Since operator .Bτ in fact acts on the sphere .Σ(1), defined in (1.226), it is useful to rewrite (1.255) in the functional polar coordinates (1.229), (1.230): ∂t ρ(t) + ‖∂x ϕ(t, ·)‖2 ρ(t) − ρ 2 (t)Ψ(ϕ(t, ·)) = 0, ρ|t=tk = ρ(tk ),

.

∂t ϕ(t, x) + ∂xx ϕ − ‖∂x ϕ‖2 ϕ − ρ(t)Bτ (w, ϕ) = 0, .

ϕ|t=tk = ϕ(tk , ·) + uk (·), t ∈ [tk , tk+1 ].

(1.256) (1.257)

Consider the problem (1.256), (1.257) with .k = 0, i.e., with .t0 = 0. Initial conditions for (1.256), (1.257) have the form ρ|t=0 = ρ0 ,

.

ϕ|t=0 = ϕ0 + u0 ,

(1.258)

Where .ρ0 ϕ0 = y0 , .y0 is the initial condition (1.222), and .ρ0 u0 = uˆ 0 , where .uˆ 0 is the impulse in Eq. (1.254). In this case .Ψ(ϕ0 + u0 ) = −α, where .α is some positive number, i.e., .ϕ0 + u0 belongs to the intersection of the contour line .Ψ[−α] with unit sphere .Σ(1) (see definition (1.226)). Let .Ψ[−α(t)] be the contour line to which the solution .ϕ(t, x) belongs at time t. Should the inequality .α(t) ≥ α be satisfied for any .t > 0, by Lemma 1.9.2, the coordinate .ρ(t) would decay as .t → ∞ with the rate indicated in the estimate (1.238) starting from .t0 = 0; i.e., no additional impulses would be required for stabilization. In reality, .α(t) may decrease with increasing t, dropping below .α. Let us choose the critical value .0 < αc < α and at the moment .t1 , when .α(t1 ) = αc , apply to .ϕ(t1 , x) the impulse .u1 (x) such that .Σ(1) ϶ ϕ(t1 , ·) + u1 ∈ Ψ[−β] with some .β > α. In fact, it is possible to construct the impulse with a weaker property, defined in the following theorem:

58

A. V. Fursikov

Theorem 1.10.1 Let .ϕ(x) ∈ Ψ[−αc ] , where .0 < αc < α, .[0, ω] ∈ (−π, π ) := T.9 There exists .n0 > 0, depending on .ϕ(x), .x ∈ [0, ω], such that for each .n ≥ n0 one can construct function .u(x) ∈ H 1 (T) supported in .[0, ω] and satisfying conditions:  .



ω

 u(x)dx = 0,

T

0

(ϕ(x) + u(x))2 dx = 1 +

1 , n

1 (ϕ(x) + u(x)) dx ≤ −(1 + )α. n T

(1.259)

3

Theorem 1.10.1 has been proved in [21]. Here we will show how it can be used for the stabilization of the solution of problem (1.254). Note that for us it suffices to ensure that the estimate .‖y(t, ·)‖ ≤ r is satisfied for some .tˆ > 0, where .r > 0 is the small number in Theorem 1.9.1, because, according to this theorem, the solution exponentially decays with increasing .t > tˆ, this time without applying any impulses. Let us switch to constructing the impulsive control in Eq. (1.254). Let .uˆ 0 be the first impulse, constructed in [14, 22], and .t1 is the time moment when the solution reaches the counter line .Ψ[−αc ] as a result of evolving after the initial impulse; let .ϕ(x) := ϕ(t1 , x), u1 (x) be the impulse constructed in Theorem 1.10.1 at .n = n0 . According to conditions (1.259), we have  .

T

(ϕ + u1 )2 dx = 1 +

1 , n0

and hence this impulse, the norm of the solution will change from   after applying 1 .ρ(t1 ) to . 1 + n0 ρ(t1 ). After the normalization of the solution, i.e., after writing it correctly in functional polar coordinates, we continue the evolution of the system. Let .t2 > t1 be the second time instant of reaching the contour line .Ψ[−αc ] . Let us use Theorem 1.10.1 to construct impulse .u2 with .n = n20 . Then the 

the second solution norm at time .t2 will be . 1 +

1 n20

ρ(t2 ). We derive an estimate for .ρ by

using a fairly coarse corollary for a version of inequality (1.238), in which the initial condition is set not at .t = 0 but at .t − τ : ρ(t) ≤

.

e−(t−τ ) 1 ρ(τ )

+ α(1 − e−(t−τ ) )

≤ ρ(τ )e−(t−τ ) .

(1.260)

As a consequence of the estimate (1.260), at the time of the first impulse we obtain

9 Relation .[−π, π ]

of length .2π .

:= T means that the segment .[−π, π ] is identified with the unit circumference

1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic. . .

59



  1 1 . 1+ ρ(t1 ) ≤ 1 + ρ0 e−t1 . n0 n0 Likewise, at time .t2 of the second impulse, we have  

1 1 ρ(t2 ) ≤ 1 + 1 + 2 ρ(t1 )e−(t2 −t1 ) ≤ n0 n0   1 1 1 + 2 + 3 ρ0 e−t2 . 1+ n0 n0 n0



.

1 1+ 2 n0



(1.261)

Furthermore, we will construct impulses so that the coefficients of .p0 in the estimates of the form Eq. (1.261) may be sums of the first several terms of a decreasing geometric progression. To this end, at the third time .t3 of reaching the contour line .Ψ[−αc ] , we must take .n = n40 , at the time .t4 we must put .n = n80 , and so on. As a result, at time .tm we obtain the estimate  .

1+

1 nk0m

 ρ(tm ) ≤

∞  1 i=0

ni0

ρ0 e−tm =

n0 ρ0 e−tm . n0 − 1

Obviously, in finitely many impulses we reach the estimate .

n0 ρ0 e−tm ≤ r, n0 − 1

which, when satisfied, will ensure by Theorem 1.9.1 that the solution of the stabilized problem exponentially decays for .t > tm without further impulses. Recall that result of stabilization the Burgers equation has been obtained by M. Krstic [26]. The purpose of Sect. 1.9 is to propose such stabilization method for the Burgers equation that can be generalized to the case of 3D Helmholtz system. Hope that such generalization will be possible to do.

References 1. A.V. Babin, M.I. Vishik, Attractors of Evolution Equations (North-Holland, Amsterdam, London, 1992) 2. V. Barbu, S.S. Sritharan, H ∞ -control theory of fluid dynamics. Proc. R. Soc. Lond. A 454, 3009–3033 (1998) 3. V. Barbu, I. Lasiecka, R. Triggiani, Abstract setting of tangential boundary stabilization strategies of Navier-Stokes equations by high-and low-gain feedback controllers. Nonlinear Analy. 64(12), 2704–2746 (2006) 4. V. Barbu, I. Lasiecka, R. Triggiani, Local exponential stabilization strategies of NavierStokes equations, d=2,3 via feedback stabilization of its linearization, International Series of Numerical Mathematics, vol. 155 (Birkhäuser, Verlag, 2007), pp. 13–46

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5. J.M. Coron, On null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domains. SIAM J. Control Optim. 37(6), 1874–1896 (1999) 6. J.M. Coron, Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136 (AMS, Providence, 2007) 7. J.M. Coron, A.V. Fursikov, Global exact controllability of the 2D Navier-Stokes equations on manifold without boundary. J. Russian Math. Phys. 4(3), 1–20 (1996) 8. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (AMS, Providence, 2002) 9. C. Foias, R. Temam, Remarques sur les équations de Navier-Stokes et les phénomènes successifs de bifurcation. Annali Scuola Norm. Sup. di Pisa Series IV V, 29–63 (1978) 10. A.V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications. Translations of Mathematical Monographs, vol. 187 (American Mathematical Society, Providence, 2000) 11. A.V. Fursikov, Stabilizability of quasi linear parabolic equation by feedback boundary control. Sbornik Mathematics 192(4), 593–639 (2001) 12. A.V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of boundary feedback control. J. Math. Fluid Mech. 3, 259–301 (2001) 13. A.V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete Cont. Dyn. Syst. 10, 289–314 (2004) 14. A.V. Fursikov, The simplest semilinear parabolic equation of normal type. Math. Control Rel. Fields 2(N2), 141–170 (2012) 15. A.V. Fursikov, On parabolic system of normal type corresponding to 3D Helmholtz system, advances in mathematical analysis of PDEs. AMS Transl. Ser. 2 232, 99–118 (2014) 16. A.V. Fursikov, Stabilization of the simplest normal parabolic equation by starting control. Commun. Pure Appl. Analy. 13(5), 1815–1854 (2014) 17. A.V. Fursikov, A.V. Gorshkov, Certain questions of feedback stabilization for Navier-Stokes equations. Evolut. Eq. Control Theory 1(1), 109–140 (2012) 18. A.V. Fursikov, O.Y. Imanuvilov, Controllability of Evolution Equations. Lecture Notes Series (Global Anan. Res. Ctr.), vol. 34, (Seoul National University, Seoul, 1996), pp. 1–163 19. A.V. Fursikov, O.Y. Immanuvilov, Exact controllability of Navier-Stokes and Boussinesq equations. Russian Math. Surv. 54(3), 565–618 (1999) 20. A.V. Fursikov, A.A. Kornev, Feedback stabilization for Navier-Stokes equations: Theory and calculations, in Mathematical Aspects of Fluid Mechanics. LMS Lecture Notes Series, vol. 402 (Cambridge University Press, Cambridge, 2012), pp. 130–172 21. A.V. Fursikov, L.S. Osipova, One method for the nonlocal stabilization of a burgers-type equation by an impulsive control. Diff. Eq. 55(5), 1–15 (2019). Pleiades Publishing, Ltd. (2019) 22. A.V. Fursikov, L.S. Shatina, Nonlocal stabilization of the normal equation connected with Helmholtz system by starting control. Discrete Cont. Dyn. Syst. 38, 1187–1242 (2018) 23. A.V. Fursikov, M.D. Gunzburger, L.S. Hou, Trace theorems for three- dimensional timedependent solenoidal vector fields and their applications. Trans. Amer. Math. Soc. 354, 1079–1116 (2002) 24. M.V. Keldysh, On completeness of eigenfunctions for certain classes of not self-adjoint linear operators. Russian Math. Surveys 26(4) , 15–41 (1971) (in Russian) 25. A.N. Kolmogorov, S.V. Fomin, Introductory Real Analysis (Dover Publications, New York, 1975) 26. M. Krstic, On global stabilization of Burgers’ equation by boundary control. Syst. Control Lett. 37, 123–141 (1999) 27. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids (Gordon and Breach, New York, 1963). (SIAM, Philadelphia, 1989) 28. J.-L. Lions, R. Magenes, Problems Aux Limites Non Homogenes et Applications, vol. 1 (Dunod, Paris, 1968) 29. J.-P. Raymond, Feedback boundary stabilization of the two-dimensional incompressible Navier-Stokes equations. SIAM J. Control Optim. 45(3), 790–828 (2006) 30. J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations. J. Math. Pures Appl. 87(6), 627–669 (2007)

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31. J.-P. Raymond, A family of stabilization problems for Oseen equations, in Control of Coupled Partial Differential Equations. International Series of Numerical Mathematics, vol. 155,, 269– 291 (2007) 32. S.L. Sobolev, Méthode nouvelle á résoudre le probléme de Cauchy pour les équations linéaries hyperboliques normales. Matematicheskii Sbornik 1(43), 39–72 (1936) 33. R. Temam, Navier-Stokes Equations-Theory and Numerical Analysis, 3rd edn. (revised) (Elsevier Science Publishers B.V., Amsterdam, 1984) 34. M.I. Vishik, A.V. Fursikov, Mathematical Problems of Statistical Hydromechanics (Kluwer Academic Publishers, Dordrecht, 1988)

Chapter 2

Unique Continuation Properties of Static Over-determined Eigenproblems: The Ignition Key for Uniform Stabilization of Dynamic Fluids by Feedback Controllers R. Triggiani

2.1 Introduction The focus of the present author’s four zoom lectures given at the Summer School “Fluid under Control” held in Prague, Czech Republic, August 23–27, 2021, was chosen to lie on recent advances in the general area of feedback stabilization of parabolic dynamical fluids, defined on a bounded 2D or (more critically) 3D domain, such as the Navier–Stokes equations or the Boussinesq system, in the vicinity of a chosen unstable equilibrium solution. “Minimally invasive” control action was the goal: thus, the feedback controls were sought to be either localized (i.e., with arbitrarily small support) interior controls, or else localized boundary-based controls; and, moreover, finite-dimensional, of “minimal” dimension. Additional insight and results followed one year later in an invited zoom talk by the present author at the Conference “Mathematical Fluid Mechanics in 2022,” held at the Institute of Mathematics, Czech Academy of Sciences, Prague, August 22–26, 2022. The subject under consideration is a long-awaited successor of the original research efforts for classical (linear and semi-linear) unstable parabolic equations defined on a bounded multidimensional domain that were carried out approximately during the period 1973–83; possibly with feedback controllers defined in terms of finite-dimensional boundary actuators and boundary controls [59–62, 97–100]. The conceptual and technical strategy that was proposed for the stabilization of these parabolic unstable dynamical equations was introduced in 1973 [97]. This, in particular, identifies the preliminary, critical obstruction that one must confront at the very first step of the overall stabilization analysis (“the ignition key” of the analysis): It consists in establishing the so-called complete stabilization property (equivalently,

R. Triggiani () University of Memphis, Memphis, TN, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Bodnár et al. (eds.), Fluids Under Control, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-031-47355-5_2

63

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the so-called pole assignment property) for the finite-dimensional linear component of the entire infinite-dimensional unstable dynamics as projected on the generalized eigenspace .Xu of the unstable eigenvalues. To describe it, let such finite-dimensional projected dynamical component be described by the pair .{Au , Bu } of finitedimensional operators, with .Au the free dynamic operator projected on .Xu and .Bu the corresponding control operator, likewise projected on .Xu . The required soughtafter goal is then to find, if possible, a (static) finite-dimensional feedback operator F , defining [control = F (state)], such that the new finite-dimensional feedback dynamics on .Xu described by the (feedback) dynamic operator .[Au + Bu F ] is (uniformly) stable on .Xu , with an arbitrarily large decay rate. A classical control theory result of the early 70s states that such desired property is possible precisely if the original dynamical pair .{Au , Bu } is controllable [110, 111] and therefore satisfies the Kalman’s algebraic rank condition. We hasten to emphasize that in the present context of unstable parabolic dynamics, particularly with localized interior or boundary feedback controllers, the above goal is not a matrix theory problem. Rather, the key tool called for to verify the required Kalman’s controllability condition on .Xu is actually a Unique Continuation Property for a suitably over-determined adjoint eigenproblem, corresponding to the unstable eigenvalues. More precisely, the technical issue amounts to ascertain that the linear independent eigenvectors corresponding to each of the unstable eigenvalues of the adjoint problem possess the following properties depending on the preassigned action of the controllers: (i) In the case of interior controls localized to an arbitrarily small interior support “.ω” that such eigenvectors that are linearly independent on all of .Ω continue to be linear independent when restricted on such “.ω” [11, 102]. (ii) In the case of boundary Neumann controls (respectively, Dirichlet boundary controls) localized to the arbitrarily small boundary support . 𝚪 of the full boundary .𝚪, that they have their Dirichlet boundary (respectively, Neumann) traces linearly independent, when restricted on . 𝚪 [100–103]. Thus verifying the required Kalman rank condition at each unstable eigenvalue of the adjoint problem consists in verifying the UCP of the corresponding overdetermined adjoint eigenproblem. Such property may be true or may be false, depending on the specific setting. When true, its validity is not a routine exercise, let alone in matrix theory. The goal of the present chapter is precisely to examine the validity of each UCP that arises in a specific stabilization problem for a fluid equation, be it the Navier–Stokes equations or else the Boussinesq system, with either localized interior controls or localized boundary-based controls. Useful general references are [75, 95, 107].

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

65

2.2 A First (Informal) Quantitative Description of the Strategy for Uniform Stabilization of Linear Unstable Parabolic Dynamics on a Bounded Multidimensional Domain [97] The projection method: Decomposition of the state space into the direct sum of the finite-dimensional unstable eigenspace and the infinite-dimensional stable subspace Boundary feedback stabilization of unstable, classical parabolic equations at first linear defined on a general, multidimensional, bounded domain was intensively investigated in the period, say 1973–1983. A general strategy was proposed in a PhD thesis, U of Minnesota, 1973 ([97] published in JMAA in 1975), and is informally described below. For sake of illustration, let the PDE problem be written in abstract form as wt = Aw + Bu ,

.

w|t=0 = w0 on the state space W .

(2.1a)

u = forcing term = open loop control

(2.1b)

where the free dynamic operator A (uncontrolled dynamics, u ≡ 0) is the generator of a s.c. analytic semigroup (parabolicity) and has compact resolvent (bounded domain). Hence, its spectrum is only point spectrum (discrete). Moreover, B is the control operator, whose technical features (“bounded” or “unbounded”) are not critical at the level of this informal presentation (Fig. 2.1). The preliminary assumption (for the stabilization problem to make sense) is that the operator A has the point spectrum as follows: Step 1. Introduce a projection operator PN :  1 .PN = − (λI − A)−1 dλ : W onto WNu . 2π i C W = WNu ⊕ WNs ; AuN = PN A = A|WNu ;

WNu ≡ PN W ;

(2.2)

WNs ≡ (I − PN )W ;

dim WNu = N . (2.3)

AsN = (I − PN )A = A|WNs .

(2.4)

Step 2. Split the linearized w-dynamic into two components: λN λ1

Fig. 2.1 The spectrum of the unstable A

λN +1 •

•

•

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Fig. 2.2 Defining the projection PN

λN +1 •

•λN

λ1 • C

‘ Fig. 2.3 Scheme of the closed-loop system

wN

u {AuN , B1 }

F

w = PN w + (I − PN )w,      

.

wN

(2.5)

ζN

where wN = finite-dimensional unstable part defined on the generalized eigenspace WNu corresponding to the unstable eigenvalues. (I − PN )w = infinite-dimensional component defined on the stable part WNs of the spectrum, on the left of λN +1 (Fig. 2.2). The finite-dimensional problem on WNu is ' wN = AuN wN + PN Bu ,

.

PN B = B1 .

(2.6)

The infinite-dimensional system on WNs is ζN' = AsN ζN + (I − PN )Bu.

.

(2.7)

Step 3. (Static State Stabilization) The preliminary obstruction: Seeking stabilization with arbitrarily fast (exponential) decay rate of the finite-dimensional unstable dynamical component wN in (2.6) by a static (finite-dimensional) state feedback operator F : Equivalence to controllability of the unstable component (2.6) (Fig. 2.3). We want to stabilize the finite-dimensional system (2.6); that is, the pair {AuN , B1 } of finite-dimensional operator AuN and B1 by a static state feedback F with an arbitrary large decay rate (surely > |ReλN +1 |): Find—if possible—a feedback finite-dimensional operator F such that with u = F wN , the corresponding closedloop system on WNu :

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . . ' wN = (AuN + B1 F )wN ,

.

wN (0) = wN,0

67

(2.8)

is exponentially stable  u    ‖wN (t)‖ = e(AN +B1 F )t wN,0  ≤ Ce−ρt ‖wN,0 ‖,

.

t ≥ 0,

(2.9)

ρ arbitrary large, at least > | ReλN +1 |. A necessary and sufficient condition for the above property (2.9) to hold true (“complete stabilization” [110, 111], “pole allocation”) has been known in finitedimensional control theory since the early 70s [20, 52]: It is the property of controllability (Kalman algebraic rank condition) of the system wN in (2.6), i.e., of the pair {AuN , B1 }. Depending on where the control function u is chosen to be located in the original PDE problem (arbitrary small set ω of the interior, arbitrary small set  𝚪 of the boundary, etc.), this condition may well be FALSE; or, if true, it may be challenging to establish. Ultimately, it depends on a suitably over-determined Unique Continuation Property of the (adjoint) eigenproblem (elliptic system) corresponding to the unstable eigenvalues of the adjoint (AuN )∗ . All this is the central topic of the present paper. Output stabilization The solution of the boundary stabilization (boundary actuation/control and boundary observation) problem of the unstable classical parabolic problem in Sect. 2.3.4 requires the following output stabilization problem. We will state in the context of the state model (2.6) augmented by an output equation .

' = AuN wN + B1 u, wN

(2.10a)

h = CwN .

(2.10b)

One seeks, if possible, to represent the control u as a feedback of the output h (not of the state wN ) by u = F h, so that the resulting system ' wN = (AuN + B1 F C)wN

(2.11)

.

is uniformly stable:  u    ‖wN (t)‖ = e(AN +B1 F C)t wN,0  ≤ Ce−ρt ‖wN,0 ‖,

.

t ≥0

(2.12)

ρ arbitrarily large. (The output stabilization problem is much more demanding and challenging than the state stabilization problem [20, 52]. It is still not fully understood [29].)

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2.3 Part I: Kalman Rank Conditions [48, 49] Before addressing the topic of UCP critical for the uniform stabilization of unstable parabolic fluid models—the case of the present paper—we find it unavoidable to present first the classical parabolic case. This will serve as an introductory avenue of both scientific and historic values toward the main core of the present article.

2.3.1 Second-Order Elliptic Operators. Linear Independence of Interior Localized Eigenfunctions Throughout this section, let .Ω be a bounded open domain in .Rn with boundary .𝚪 = ∂Ω assumed to be an (.n − 1)-dimensional manifold with .Ω locally on one side of .𝚪. Let .𝚪1 be an open, connected subset of .𝚪 of positive surface measure. Let .A(x, ∂) be a uniformly strongly elliptic operator of order two in .Ω, canonically .A = (−Δ), of the form A(x, ∂) ≡



.

aα (x)∂ α ,

x ∈ Ω;

(2.13)

|α|≤2

x = [x1 , . . . , xn ] with smooth real coefficients .aα ( · ), where the symbol .∂ denotes differentiation.

.

Remark 2.3.1 To ensure the validity of the critical unique continuation theorem (Carleman–Aronszajn–Cordes uniqueness theorem) to be invoked in the proof of Theorems 2.3.2(D) (or 2.3.3(N)) below, we may just assume that (i) the coefficients .aα are real-valued Lipschitz continuous functions for .|α| = 2, and (ii) .aα are bounded for .|α| < 2 [44, Vol. III, p. 3], [73, p. 60], [15]. To the differential expression .A(x, ∂) in (2.13), we now associate suitable boundary conditions. We distinguish two basic cases. Dirichlet B.C. case Let .AD denote the (closed) realization of .(−A(x, ∂)) (canonically, .−A = Δ) on .L2 (Ω) with Dirichlet boundary conditions: AD f = −A(x, ∂)f, D(AD ) = H 2 (Ω) ∩ H01 (Ω) = {f ∈ H 2 (Ω) : f |𝚪 = 0}; . (2.14a)

.

AD : L2 (Ω) ⊃ D(AD ) → L2 (Ω).

(2.14b)

Since .Ω is bounded in .Rn , the closed operator .AD has compact resolvent on .L2 (Ω). Let D {λD i , ϕij },

.

D D D i = 1, 2, . . . ; j = 1, . . . , 𝓁D i , AD ϕij = λi ϕij ,

(2.15)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

69

be the eigenvalues and corresponding (normalized) .L2 (Ω)-linearly independent eigenfunctions of .AD . Thus, in this section, .𝓁D i denotes the geometric multiplicity of the eigenvalue .λD . i Neumann case Let .AN be now the (closed) realization of .(−A(x, ∂)) (canonically, Δ) on .L2 (Ω), this time with Neumann boundary conditions:

.



 ∂f 2 .AN f ≡ −A(x, ∂)f, D(AN ) = f ∈ H (Ω) : = 0 ;. ∂ν 𝚪 AN : L2 (Ω) ⊃ D(AN ) → L2 (Ω).

(2.16a) (2.16b)

The closed operator .AN has likewise compact resolvent on .L2 (Ω). Let N {λN i , ϕij }, i = 1, 2, . . . ;

.

j = 1, . . . , 𝓁N i ;

N AN ϕijN = λN i ϕij ,

(2.17)

be the eigenvalues and corresponding (normalized) .L2 (Ω)-linearly independent eigenfunctions of .AN on .L2 (Ω). Thus, here, .𝓁N i denotes the geometric multiplicity . of the eigenvalue .λN i Theorem 2.3.1 ([11]) Let .{λi , ϕij } be the {eigenvalues, normalized eigenfunctions} pair either of the operator .AD in (2.14a)–(2.14b), as in (2.15) or else of the operator .AN in (2.16a)–(2.16b), as in (2.17). Let .ω be an arbitrarily small open connected smooth subset of the interior .Ω, .ω ⊂ Ω, of positive measure. Consider M distinct eigenvalues .λ1 , λ2 , . . . , λM , so that the corresponding (normalized) eigenfunctions .{ϕij , i = 1, . . . , M; j = 1, . . . , 𝓁i } are linearly independent on 2 .L (Ω), where .𝓁i = geometric multiplicity of .λi . Then: (i) For each .i, i = 1, . . . , M, the corresponding eigenfunctions {ϕij }𝓁ji=1 are linearly independent on L2 (ω).

.

(2.18)

(ii) In fact, more generally, the following eigenfunctions 𝓁i 2 {ϕij }M, i=1, j =1 are linearly independent on L (ω).

.

(2.19)

(iii) Still more generally, for .i = 1, . . . , M; j = 1, . . . , Ni , the system of generalized eigenf unctions is linearly independent on L2 (ω), (2.20) .Ni = algebraic multiplicity. .

Proof (i) Step 1. Call .−λi a fixed eigenvalue of A (either .AD or .AN ) with corresponding (true) eigenfunctions .ϕij : Aϕij = −λi ϕij , 1 ≤ j ≤ 𝓁i , .𝓁i geometric multiplicity, which are linearly independent on .L2 (Ω). To establish (2.18), we must show that, with “i” fixed:

70

R. Triggiani 𝓁i

.

αj ϕij ≡ 0 on ω =⇒ αj ≡ 0, j = 1, . . . , 𝓁i .

(2.21)

j =1

To this end, define the function .ϕ (we suppress its dependence on “i”) on .Ω by ϕ=

𝓁i

.

αj ϕij

on Ω

(2.22)

j =1

so that .ϕ ≡ 0 on .ω by (2.21). Moreover, .ϕ is an eigenfunction of A with the same eigenvalue .−λi , being a linear combination of the eigenfunctions .ϕij : Aϕij = −λi ϕij . Thus, specifically recalling (2.13), (2.14) for .AD and (2.16) for .AN , we obtain the PDE version ⎧ ⎪ ⎨A(x, in Ω ∂)ϕ = λi ϕ A(x, ∂)ϕ = λi ϕ in Ω ; . ∂ϕ ⎪ = 0 on 𝚪; ϕ ≡ 0 in ω ϕ 𝚪 = 0 on 𝚪; ϕ ≡ 0 in ω ⎩ ∂ν 𝚪

(2.23) for the operator .A = AD and .A = AN , respectively. Step 2. The crux of the matter is that each over-determined eigenproblem in (2.23) implies ϕ≡0

.

in L2 (Ω).

(2.24)

The desired uniqueness property .ϕ ≡ 0 in .L2 (Ω), implied in (2.24) by the overdetermined elliptic eigenvalue problem (2.23) even without BCs, is a corollary of well-known results (see, e.g., a detailed discussion in [73, Chapter III, Sect. 19, pp. 59–61]: Either the Aronszajn–Cordes uniqueness theorem [44, Vol. III, p. 3] or Carleman’s theorem [17], [15, p. 162; p. 262], [73, pp. 59–61]), under regularity of the coefficients as in Remark 2.3.1, states that a solution .u ∈ H 2 (Ω) of a secondorder elliptic problem that vanishes (of infinite order at some point, in particular) identically in an open interior subset .ω of .Ω: .u ≡ 0 in .ω must vanish identically in .Ω. Then, using (2.24) in (2.22) yields .αj ≡ 0, j = 0, . . . , 𝓁i as desired in (2.21), since the eigenfunctions .{ϕij } corresponding to the same eigenvalue .−λi are linearly independent in .L2 (Ω). Case (i) is proved. (ii) Consider now the case of two distinct (true) eigenvalues, say .λ1 and .λ2 of the , 𝓁2 operator A (either .AD or .AN ). We must show that the set .{ϕ1j , ϕ2k }𝓁j1=1, k=1 of all the corresponding eigenfunctions are linearly independent in .ω, that is, 𝓁1

j =1

.

=⇒

αj ϕ1j +

𝓁2

βk ϕ2k ≡ 0 in ω

k=1

αj ≡ 0, j = 1, . . . , 𝓁1 ; βk ≡ 0, k = 1, . . . , 𝓁2 ,

(2.25)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

Aϕ1j = λ1 ϕ1j ,

71

Aϕ2k = λ2 ϕ2k .

.

(2.26)

We multiply the first of these identities by .αj and sum up over .j = 1, . . . , 𝓁1 . We multiply the second of these identities by .βk and sum up over .k = 1, . . . , 𝓁2 . A

𝓁1

.

αj ϕ1j = λ1

j =1

𝓁1

αj ϕ1j ,

j =1

A

𝓁2

βk ϕ2k = λ2

k=1

𝓁2

(2.27)

βk ϕ2k .

k=1

We sum up the two expressions in (2.27) and invoke (2.25) twice: ⎡ A⎣

𝓁1

.

αj ϕ1j +

j =1

= λ1

𝓁1

𝓁2

βk ϕ2k ⎦ = A(0) = 0

k=1

αj ϕ1j + λ2

j =1

= (λ1 − λ2 )

⎤

𝓁2

βk ϕ2k = λ1

j =1

k=1 𝓁1

αj ϕ1j = 0

𝓁1

(2.28)

on ω. ⎡

αj ϕ1j + λ2 ⎣−

𝓁1

⎤ αj ϕ1j ⎦ .

(2.29)

j =1

on ω;

(2.30)

j =1

 hence, . 𝓁j1=1 αj ϕ1j ≡ 0 on .ω, since .λ1 /= λ2 . Then, by part (i) .{ϕ1j }𝓁j1=1 are linearly independent on .L2 (ω) and thus .αj ≡ 0, j = 1, . . . , 𝓁1 . Then (2.25) yields 𝓁2 . k=1 βk ϕ2k ≡ 0 in .ω, and this yields .βk ≡ 0, k = 0, . . . , 𝓁2 , since by part (i) 𝓁2 2 .{ϕ2k } k=1 are linearly independent on .L (ω). One can then proceed by inductions, following the proof above, details as in [11, p. 1460]. (iii) General case of generalized eigenfunctions. The computations for this case are more involved, and we refer to [11, p. 1460, Case B] for details. ⨆ ⨅

2.3.2 Linear Independence of Boundary Traces of Eigenfunctions In the focus is the linear  this section,   independence of the Neumann traces ∂ D D in the Dirichlet case as in (2.15), as well ϕ of the eigenfunctions . ϕ ij ∂ν ij 𝚪1   as on the linear independence of the Dirichlet traces . ϕijN 𝚪 of the eigenfunctions 1   N . ϕ ij in the Neumann case in (2.17). Here .𝚪1 is a portion of the boundary .𝚪 = ∂Ω of the domain .Ω, to be specified in the statements below. .

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Theorem 2.3.2 (D) Consider M distinct eigenvalues .λ1 , . . . , λi , . . . , λM with geometric multiplicity .𝓁D i of the operator .AD in (2.14) so that the corresponding M, 𝓁D

(normalized) eigenfunctions .{ϕijD }i=1, ij =1 are linearly independent on .L2 (Ω). Let .𝚪1 be an open connected portion of the boundary .𝚪 of positive measure, which is assumed to be, say, of class .C 2 . Then, for each .i, i = 1, . . . , M,  𝓁Di the system ∂ν ϕijD 𝚪

.

is linearly independent in L2 (𝚪1 ).

j =1

1

(2.31)

Similarly: Theorem 2.3.3 (N) Consider now M distinct eigenvalues .λ1 , . . . , λi , . . . , λM with geometric multiplicity .𝓁N i of the operator .AN in (2.16) so that the corresponding M, 𝓁N

(normalized) eigenfunctions .{ϕijN }i=1, ij =1 are linearly independently on .L2 (Ω). Let .𝚪1 be as in Theorem 2.3.2(D). Then,  𝓁N i the system ϕijN 𝚪

.

1

j =1

is linearly independent in L2 (𝚪1 ).

(2.32)

Remark 2.3.2 Theorem 2.3.3(N) admits a perfect counterpart with the Neumann boundary conditions associated to .A(x, ∂) replaced by the Robin boundary conditions. Remark 2.3.3 The proof in (ii) of Theorem 2.3.1 breaks down in the case of traces. In fact, reference [59, p. 335-336, and p. 344] shows two distinct cases: The Laplacian .Δ with Neumann B.C. defined on a 2-D sphere has the Dirichlet traces on .𝚪 that are not linearly independent. Instead, the Laplacian with Neumann B.C. defined on a 2-D rectangle, with sides that are linearly independent over the integers, has the Dirichlet traces that are linearly independent. Proof of Theorems 2.3.2(D) and 2.3.3(N) We write explicitly only the proof of Theorems 2.3.2(D). The proof of Theorems 2.3.3(N) is then the perfect counterpart, mutatis mutandis. We drop the superscript “D” for simplicity of notation. To prove Theorems 2.3.2(D), let as in [11], [12] 𝓁i

.

⎛ αj ∂ν ϕij = ∂ν ⎝

j =1

𝓁i

⎞ αj ϕij ⎠ = ∂ν ϕ ≡ 0 on 𝚪1 ,

(2.33)

j =1

i = fixed, where we have defined the function .ϕ on .Ω (we suppress its dependence on “i”) by ϕ≡

𝓁i

.

j =1

αj ϕij ∈ L2 (Ω).

(2.34)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

73

We must show that αj ≡ 0, j = 1, . . . , 𝓁i .

.

(2.35)

To this end, we note that, by (2.15) and (2.34), .ϕ is an eigenfunction of .AD with the same eigenvalue .λi : AD ϕ = λi ϕ. Thus, .ϕ satisfies homogeneous Dirichlet boundary conditions on all of .𝚪 : ϕ|𝚪 = 0, since so do the .ϕij ’s. Moreover, .∂ν ϕ|𝚪1 = 0 by (2.33). Thus, .ϕ satisfies the following over-determined elliptic eigenproblem: ⎧ ⎨ A(x, ∂)ϕ = −λi ϕ .

⎩ ϕ| ≡ 0; 𝚪

in Ω;

∂ν ϕ|𝚪1 ≡ 0.

(2.36a) (2.36b)

It is then well known that for a boundary .𝚪 of class, say, .C 2 , and coefficients as in Remark 2.3.1, as assumed, the over-determined elliptic eigenvalue problem (2.36a)– (2.36b) only has the zero solution: ϕ≡

𝓁i

.

αj ϕij ≡ 0 in L2 (Ω);

hence αj = 0, j = 1, . . . , 𝓁i .

(2.37)

j =1

The last implication on the coefficients .αj follows, since the .ϕij ’s are linearly independent on .L2 (Ω). Thus (2.35) is established. The desired uniqueness property .ϕ ≡ 0 in .L2 (Ω) implied in (2.37) by the over-determined elliptic eigenvalue problem (2.36a)–(2.36b) is a corollary of the well-known results already invoked and discussed in obtaining the uniqueness property in Step 2 of the proof of Theorem 2.3.1; again see, e.g., a detailed discussion in [73, Chapter III, Sect. 19, pp. 59–61]: Either the Aronszajn–Cordes uniqueness theorem [44, Vol. III, p. 3] or Carleman’s theorem [17], [15, p. 162; p. 262], [73, pp. 59–61], under regularity of the coefficients as in Remark 2.3.1, states that a solution .u ∈ H 2 (Ω) of a secondorder elliptic problem that vanishes (of infinite order at some point, in particular) identically in an open interior subset .ω of .Ω: .u ≡ 0 in .ω must vanish identically in .Ω. From this basic result, one then deduces the desired conclusion .ϕ ≡ 0 for problem (2.36a)–(2.36b) (with boundary .𝚪 of class, say, .C 2 ) in a classical way: By extending .ϕ by zero across .𝚪1 outside .Ω into a set .ωext , showing by integration by parts using the zero Cauchy data in (2.36b) that the extended function defined by 2 .ϕ = solution of (2.36a)–(2.36b) in .Ω, and .ϕ ≡ 0 in .ωext satisfies .ϕ ∈ H (G), where .G = Ω ∪ ωext [53, p. 75]. Then, the aforementioned Carleman’s theorem applies on G and yields .ϕ ≡ 0 in .Ω, as desired. The proof is complete. (Another approach to obtaining uniqueness of the over-determined elliptic problem with zero Cauchy data on a portion of the boundary from the uniqueness theorem for solutions vanishing of infinite order at one point is given in [83], under some smoothness of the coefficients: One flattens the boundary locally and performs an odd reflection across the boundary.) ⨆ ⨅

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Remark 2.3.4 The proof of Theorems 2.3.3(N) yields instead the following overdetermined eigenvalue problem: .

A(x, ∂)ϕ = −λi ϕ ϕ|𝚪1 ≡ 0;

in Ω;

∂ν ϕ|𝚪 ≡ 0,

(2.38a) (2.38b)

again with zero Cauchy data on the portion .𝚪1 , which again allows for an extension by zero across .𝚪1 .

2.3.3 Implications of Linear Independence of Interior Localized Eigenfunctions to the Problem of Dirichlet Boundary Feedback Stabilization of Parabolic Problems. Verification of Kalman Rank Condition [48, 49] In the present section, we point out the relevance of the results of Sect. 2.3.1 on the linear independence of interior localized eigenfunctions to a problem of boundary feedback stabilization of a classical unstable parabolic equation, with boundary controls and interior localized actuators. We keep the bounded open domain .Ω in n .R as well as the elliptic operator .A(x, ∂) of Sect. 2.3.1, Eqs. (2.13)–(2.14). The unstable closed-loop parabolic system As in [99, 100], we consider the following closed-loop parabolic problem in the unknown .y(t, x) defined over .Ω: ⎧ ∂y ⎪ (t, x)= − A(x, ∂)y(t, x) ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ y(0, x)=y0 (x) . ⎪ ⎪ ⎪ ⎪ K ⎪

⎪ ⎪ ⎪ ⎪ (y(t, · )|𝚪 , wk )L2 (ω) gk ⎩ y Σ =f (t, ·) =

in Q = (0, ∞] × Ω; (2.39a) in Ω;

(2.39b)

in Σ = (0, ∞) × 𝚪. (2.39c)

k=1

Here f is the Dirichlet boundary control written in a feedback form as to generate a closed-loop feedback control system. Here .wk and .gk are fixed vectors in 2 2 .L (ω) and .L (𝚪), respectively, where .ω ⊂ Ω is an arbitrarily small open, connected, smooth subset of the interior of positive measure. The boundary vectors K are assumed to be linearly independent. Well-posedness of the closed-loop .{gk } k=1 problem (2.39a)–(2.39c) is well known [61, 99] (even in the case where .𝚪 may have finitely many conical points [54]). We recall that the operator .AD , defined as in (2.14), generates a s.c. analytic (holomorphic) semigroup on .L2 (Ω) [28, Example p. 101].

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

75

Theorem 2.3.4 ([62, 99, 100]) The closed-loop feedback problem (2.39a)–(2.39c) is well posed in the following sense: Its closed-loop feedback solutions .y(t; y0 ) can be expressed as .y(t; y0 ) = SF (t)y0 , .y0 ∈ L2 (Ω), .t ≥ 0, where .SF (t) defines a (feedback) strongly continuous semigroup that is analytic and compact on .L2 (Ω) for .t > 0, and whose generator has compact resolvent on .L2 (Ω). Actually, such 1 feedback semigroup .SF (t) has the same properties on all spaces .H 2 −ϵ (Ω), .0 < 1 ϵ ≤ 2. Henceforth in this section, we shall omit the superscript D(=Dirichlet). We shall next proceed according to the (informal) quantitative description of Sect. 2.2, as it applies to the present case of model (2.39). The most significant version of the problem at hand is when the free open-loop system .(f ≡ 0) is unstable, in the sense that there are I unstable distinct eigenvalues .λ1 , λ2 , . . . , λI of the .L2 (Ω)-realization .AD of (2.14a)–(2.14b) associated with .(−A(x, ∂)): .

· · · Re λI +2 ≤ Re λI +1 < 0 ≤ Re λI ≤ · · · Re λ2 ≤ Re λ1 ,

(2.40)

where the eigenvalues of .AD are numbered in order of decreasing real parts. The operator .AD is the generator of a s.c., analytic semigroup on .L2 (Ω) with compact resolvent. Boundary feedback stabilization problem This may be stated as follows: Find, if possible: (i) Appropriate boundary vectors .gk ∈ L2 (𝚪), as well as their minimum number, and (ii) The weakest possible conditions on the vectors .wk ∈ L2 (ω), .k = 1, . . . , K, to guarantee that the feedback semigroup .SF (t) of Theorem 2.3.4 describing problem (2.39) satisfies ‖SF (t)y0 ‖L2 (Ω) ≤ Me−δt ‖y0 ‖L2 (Ω) ,

.

∀y0 ∈ L2 (Ω), t ≥ 0,

(2.41)

for some .M ≥ 1 and .δ > 0 (perhaps, preassigned), or more generally, ‖SF (t)y0 ‖H 2σ (Ω) ≤ Me−δt ‖y0 ‖H 2σ (Ω) ,

.

0≤σ
0, depending on .σ ).

A solution of this problem is provided by [61, 99, 100]; see also [59, 62]. The key issue is the validity of Theorem 2.3.1 that affirms the linear independence of the eigenfunctions .{ϕij∗ }𝓁ji=1 on .L2 (ω), thus allowing one to verify the Kalman rank condition. For simplicity of exposition, we shall assume the following further hypothesis, which is typically satisfied in classical parabolic equations where .AD (or .AN ) are self-adjoint, or normal operators:

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The distinct unstable eigenvalues {λi }Ii=1 are semisimple;

.

(2.43)

that is, the restriction of the free dynamics generator .AD in (2.14a)–(2.14b) is diagonalizable (or semisimple [50, p. 43]) when restricted on the finite-dimensional unstable subspace .Yu of .L2 (Ω) generated by the eigenvectors of the unstable eigenvalues .{λi }Ii=1 in (2.40). This means that for each distinct unstable eigenvalue .λi , .i = 1, . . . , I , the algebraic and geometric multiplicity .𝓁i coincide. Thus, each such eigenvalue .λi has .𝓁i , linearly independent (normalized) eigenfunctions .{ϕij }𝓁ji=1 in .L2 (Ω). Moreover, the unstable subspace .Yu is .span{ϕij , i = 1, . . . , I ; j = 1, . . . , 𝓁i }. Thus, let AD ϕij = λi ϕij ;

.

A∗D ϕij∗ = λi ϕij∗ , i = 1, . . . , I ; j = 1, . . . , 𝓁i ,

(2.44)

𝓁i = geometric and algebraic multiplicity of .λi , or .λ¯ i , be the eigenvalues/eigenvectors of .AD and of its .L2 (Ω)-adjoint .A∗D . In view of (2.43), it is well known [50, p. 51] that .{ϕij } and .{ϕij∗ } can be chosen to form bi-orthogonal sequences; that is,

.

∗ .(ϕij , ϕhk )L2 (Ω)

=

1 if i = h; j = k;

(2.45)

0 otherwise.

Thus, any vector .y ∈ Yu admits the unique expansion Yu ϶ y =

.

(y, ϕij∗ )L2 (Ω) ϕij ,

i = 1, . . . , I ;

j = 1, . . . , 𝓁i .

(2.46)

i,j

In this case, a condition imposed in [61, p. 311] on the vectors .wk to achieve uniform feedback stabilization of problem (2.39a)–(2.39c) is that the rank of the .K × 𝓁i matrix: ⎡ ⎤ ∗ ) , (w , ϕ ∗ ) , . . . (w , ϕ ∗ ) (w1 , ϕi1 ω 1 i2 ω 1 i𝓁i ω ⎢ ⎥ ⎢ ∗ ) , (w , ϕ ∗ ) , . . . (w , ϕ ∗ ) ⎥ ⎢ (w2 , ϕi1 ⎥ ω 2 ω 2 ω i𝓁i i2 ⎢ ⎥ ⎥ .Wi = ⎢ (2.47) ⎢ ⎥ .. .. .. ⎢ ⎥ . . . ⎢ ⎥ ⎣ ⎦ ∗ ) , (w , ϕ ∗ ) , . . . (w , ϕ ∗ ) (wK , ϕi1 ω K ω K ω i𝓁i i2 be full, where .K ≥ 𝓁i has been preliminarily chosen, that is, rank Wi = 𝓁i ,

.

i = 1, . . . , I.

(2.48)

The algebraic condition (2.48) can only be fulfilled if, for each i, .i = 1, . . . , I , the eigenfunctions .{ϕij∗ }𝓁ji=1 are linearly independent on .L2 (ω). Theorem 2.3.1(i) (as

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

77

applied to .A∗D ) guarantees that this is always possible, in which case the vectors .{w1 , . . . , wK } can be chosen in infinitely many ways. Its proof relies on an established UCP (2.23) .−→ (2.24). Condition (2.48) is the Kalman rank condition in the present case. Remark 2.3.5 The assumption that .AD be semisimple is not critical, in which case one has to invoke the more general Theorem 2.3.1 (iii) for .A∗D . This will be seen in the context of the linearized Navier–Stokes equation of Sect. 2.5.

2.3.4 Implications of Linear Independence of Localized Boundary Traces of Eigenfunctions to the Problem of Neumann Boundary Feedback Stabilization of Parabolic Problems. Verification of Kalman Rank Condition In the present section, we point out the relevance of the results of Sect. 2.3.2 on the linear independence of boundary traces of eigenfunctions. We keep the bounded open domain .Ω in .Rn as well as the elliptic operator .A(x, ∂) of Sect. 2.3.1, Eq. (2.13). We assume, however, dim .Ω = n ≥ 2, as we shall need a full function space .L2 (𝚪) to “counteract” the possibly large number of original unstable eigenvalues. The unstable closed-loop parabolic system As in [60], we consider the following closed-loop parabolic problem in the unknown .y(t, x) defined over .Ω: ⎧ ∂y ⎪ (t, x) = −A(x, ∂)y(t, x) in Q = (0, ∞] × Ω; ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ y(0, x) = y0 (x) in Ω; . ⎪ ⎪ ⎪ ⎪ K ⎪

⎪ ∂y ⎪ ⎪ ⎪ (y(t, · )|𝚪1 , wk )𝚪1 gk in Σ = (0, ∞) × 𝚪. = f = ⎩ ∂ν

(2.49a) (2.49b)

(2.49c)

k=1

Here, f is the Neumann boundary control written in a feedback form as to generate a closed-loop feedback control system. .wk and .gk are fixed boundary vectors in .L2 (𝚪1 ) and .L2 (𝚪), respectively; the symbol .( · , · )𝚪1 denotes the duality pairing in .L2 (𝚪1 ). The boundary vectors .{gk }K k=1 are assumed to be linearly independent. Well-posedness of the closed-loop problem (2.49a)–(2.49c) is well known (even in the case where .𝚪 may have finitely many conical points [54]). Theorem 2.3.5 ([62]) The closed-loop feedback problem (2.49a)–(2.49c) is well posed in the following sense: Its closed-loop feedback solutions .y(t; y0 ) can be expressed as .y(t; y0 ) = SF (t)y0 , .y0 ∈ L2 (Ω), .t ≥ 0, where .SF (t) defines a

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(feedback) strongly continuous semigroup that is analytic and compact on .L2 (Ω) for .t > 0, and whose generator has compact resolvent on .L2 (Ω). Remark 2.3.6 The proof of [62] actually shows that the feedback semigroup SF (t) has the same properties listed in the statement of the theorem on all spaces 3 −ϵ .H 2 (Ω), .0 < ϵ ≤ 32 . The boundary stabilization results of [59], [60], [61] are topologically consistent with the described regularity of feedback solutions. We also refer to [99] for regularity results, obtained by different techniques, that complement, but neither fully imply, nor are fully implied by, the above Theorem 2.1 [99, Remark 2.3]. Previously, stabilization problems for parabolic equations were given in [98, 99]. .

Henceforth in this section, we shall omit the superscript N (= Neumann). We shall next proceed according to the (informal) quantitative description of Sect. 2.2, as it applies to the present case. Thus, the most significant version of the problem at hand is when the free open-loop system .(gk ≡ 0) is unstable, in the sense that there are I unstable distinct eigenvalues .λ1 , λ2 , . . . , λI of the .L2 (Ω)realization .AN of (2.16a)–(2.16b) associated with .(−A(x, ∂)): .

· · · Re λI +2 ≤ Re λI +1 < 0 ≤ Re λI ≤ · · · Re λ2 ≤ Re λ1 ,

(2.50)

where the eigenvalues of .AN are numbered in order of decreasing real parts. The operator .AN is the generator of a s.c., analytic semigroup on .L2 (Ω), with compact resolvent. Boundary feedback stabilization problem This may be stated as follows: Find, if possible: (i) Appropriate boundary vectors .gk ∈ L2 (𝚪), as well as their minimum number, and (ii) The weakest possible conditions on the vectors .wk ∈ L2 (𝚪), .k = 1, . . . , K, to guarantee that the feedback semigroup .SF (t) of Theorem 2.3.1 satisfies ‖SF (t)y0 ‖L2 (Ω) ≤ Me−δt ‖y0 ‖L2 (Ω) ,

.

∀y0 ∈ L2 (Ω), t ≥ 0,

(2.51)

for some .M ≥ 1 and .δ > 0 (perhaps, preassigned), or more generally, ‖SF (t)y0 ‖H 2σ (Ω) ≤ Me−δt ‖y0 ‖H 2σ (Ω) ,

.

0≤σ
0, depending on .σ ).

A solution of this problem is provided by [4, 60]; see also [59, 61]. It requires the solution of an output stabilizability problem (definitely more demanding than the state stabilizability problem. Recall Sect. 2.2, (2.10)–(2.12)). One of the con-

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

79

ditions required—given below—represents the link with the problem of linear independence of the Dirichlet boundary traces .{ϕij∗ 𝚪 }, j = 1, . . . , 𝓁i , of the 1 eigenvectors .ϕij∗ of the adjoint operator .A∗N (linearly independent on .L2 (Ω)) corresponding to the I distinct unstable eigenvalues. Such linear independence of 𝓁i ∗ .{ϕ } , i = 1, . . . , I , characterizes the validity of the Kalman rank condition ij 𝚪1 j =1 over the unstable generalized eigenspace. Such linear independence was asserted by Theorem 2.3.3(N), see (2.32). For simplicity of exposition, we shall assume the following further hypothesis, which is typically satisfied in classical parabolic equations where .AN is a self-adjoint, or normal operators: The distinct unstable eigenvalues {λi }Ii=1 are semisimple;

.

(2.53)

that is, the restriction of the free dynamics generator .AN in (2.16a)–(2.16b) is diagonalizable (or semisimple [50, p. 43]) when restricted on the finite-dimensional unstable subspace .Yu of .L2 (Ω) generated by the eigenvectors of the unstable eigenvalues .{λi }Ii=1 in (2.17). This means that for each distinct unstable eigenvalue .λi , .i = 1, . . . , I , the algebraic and geometric multiplicity .𝓁i coincide. Thus, each such eigenvalue .λi has .𝓁i , linearly independent (normalized) eigenfunctions .{ϕij }𝓁ji=1 in .L2 (Ω). Moreover, the unstable subspace .Yu is .span{ϕij , i = 1, . . . , I ; j = 1, . . . , 𝓁i }. Thus, let A∗N ϕij∗ = λi ϕij∗ , i = 1, . . . , I ; j = 1, . . . , 𝓁i ,

AN ϕij = λi ϕij ;

.

(2.54)

𝓁i = geometric and algebraic multiplicity of .λi , or .λ¯ i , be the eigenvalues/eigenvectors of .AN and of its .L2 (Ω)-adjoint .A∗N . It is well known [50, p. 51] that .{ϕij } and .{ϕij∗ } can be chosen to form bi-orthogonal sequences; that is,

.

∗ (ϕij , ϕhk )L2 (Ω) =

1 if i = h; j = k;

.

(2.55)

0 otherwise.

Thus, any vector .y ∈ Yu admits the unique expansion Yu ϶ y =

.

(y, ϕij∗ )L2 (Ω) ϕij ,

i = 1, . . . , I ;

j = 1, . . . , 𝓁i .

(2.56)

i,j

In this case, a condition imposed in [60, p. 311] on the vectors .wk to achieve uniform feedback stabilization of problem (2.49a)–(2.49c) is that the rank of the .K × 𝓁i matrix:

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⎡

∗ | ) , (w , ϕ ∗ | ) , . . . (w , ϕ ∗ | ) (w1 , ϕi1 𝚪1 𝚪1 1 i2 𝚪1 𝚪1 1 i𝓁i 𝚪1 𝚪1

⎤

⎢ ⎥ ⎢ ⎥ ∗ | ) , (w , ϕ ∗ | ) , . . . (w , ϕ ∗ | ) ⎢ (w2 , ϕi1 ⎥ 𝚪 𝚪 2 𝚪 𝚪 2 𝚪 𝚪 1 1 1 1 1 1 i𝓁i i2 ⎢ ⎥ ⎥ .Wi = ⎢ ⎢ ⎥ .. .. .. ⎢ ⎥ . . . ⎢ ⎥ ⎣ ⎦ ∗ | ) , (w , ϕ ∗ | ) , . . . (w , ϕ ∗ | ) (wK , ϕi1 𝚪1 𝚪1 K 𝚪 𝚪 K 𝚪 𝚪 i𝓁i 1 1 i2 1 1

(2.57)

be full, where .K ≥ 𝓁i has been preliminarily chosen, that is, rank Wi = 𝓁i ,

.

i = 1, . . . , I.

(2.58)

The algebraic condition (2.58) can only be fulfilled if, for each i, .i = 1, . . . , I , the Dirichlet traces .{ϕij∗ |𝚪1 }𝓁ji=1 of the eigenfunctions of .A∗N are linearly independent. Theorem 2.3.3(N) (as applied to .A∗N ) guarantees that this is always possible, in which case the vectors .{w1 , . . . , wK } can be chosen in infinitely many ways. Condition (2.58) is a Kalman rank condition. We point out that the full uniform stabilization result as stated in [4, 60] for the feedback problem (2.49a)–(2.49c) requires additional assumptions, as to fulfill, in addition, an output stabilizability condition, as recalled at the end of Sect. 2.2. This is beyond the point in the present paper.

Part II: The Stokes/Oseen operators Part I has shown that the linear independence of interior localized eigenfunctions of second-order elliptic operators as well as the linear independence of localized boundary traces of these eigenfunctions critically hinges on unique continuation properties of corresponding over-determined (adjoint) eigenproblem (2.23) or (2.36a)–(2.36b) [or (2.38a)–(2.38b)], respectively. These results—particularly the localized interior one—have been well known for second-order elliptic operators for many year, as noted in the proof of Theorem 2.3.1 or Theorem 2.3.2(D) and Theorem 2.3.3(N). In contrast, regarding just the Stokes problem, there is a revealing, simple counterexample on the 2-D half-space case, regarding localized boundary traces, see Sect. 2.5.1 below, due to [26], 1996. We also refer to [21, 22, 38, 40, 41, 46, 78–80, 84–89, 93].

2.4 Relevant Unique Continuation Properties for Over-determined Oseen Eigenvalue Problems. Part IIA: Emphasis on the Localized Interior Case In Sect. 2.4, we assemble a comprehensive account of unique continuation problems for Oseen eigenproblems, as they pertain to the problem of controllability of finite-dimensional projected systems of the linearized Oseen dynamical PDEs that

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

81

arises in the linearization of the Navier–Stokes equations. This was explained in Sect. 2.2. The goal, as usual, consists in seeking to verify—if possible—the corresponding Kalman algebraic condition, in the study of uniform stabilization of the (linearized) Oseen dynamical PDE, with either localized interior controls or else localized boundary-based controls. A positive solution, or lack thereof, of this finitedimensional problem is a key step, or obstruction, for the analysis of the uniform stabilization of the Navier–Stokes equations. With this section, we initiate the study in the title of the present paper: analysis of the Unique Continuation Properties that are critical—the ignition key—in the analysis of the uniform stabilization of dynamical fluid equations, in the vicinity of an unstable equilibrium solution. Part I was a preparatory presentation focused on the uniform stabilization of classical parabolic equations, as they were obtained during 1973–1983.

2.4.1 Oseen Eigenproblems. Unique Continuation of the Oseen Equations from an Arbitrary Interior Subdomain. Linear Independence of Interior Localized Eigenfunctions Statement of the Problems Let .Ω be an open, bounded domain in .Rd , .d = 2, 3, the physically relevant dimensions. Let .ω be an open, connected, smooth subset of .Ω, of positive measure: .ω ⊂ Ω. We consider the following eigenvalue/vector-type problem of the Oseen operator in the unknown u (velocity field) and p (pressure): ⎧ (−ν0 Δ)u + Le (u) + ∇p = λu ⎪ ⎪ ⎪ ⎪ ⎨ . div u ≡ 0 ⎪ ⎪ ⎪ ⎪ ⎩ u≡0

in Ω;

(2.59a)

in Ω;

(2.59b)

in ω;

(2.59c)

with over-determination in (2.59c), which would be an over-determined eigenproblem if boundary conditions were also imposed on .𝚪 = ∂Ω. Here, .u = [u1 , . . . , ud ] ∈ Rd , p is a scalar; .λ ∈ C; the constant .ν0 > 0 is the viscosity coefficient; moreover, Le (u) ≡ (ye · ∇)u + (u · ∇)ye .

.

(2.60)

In (2.60), .{ye , pe } ∈ ((H 2 (Ω))d ∩ V ) × H 1 (Ω) is a steady-state (equilibrium) solution to

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⎧ −ν0 Δye + (ye · ∇)ye + ∇pe = fe ⎪ ⎪ ⎪ ⎪ ⎨ div ye ≡ 0 . ⎪ ⎪ ⎪ ⎪ ⎩ ye ≡ 0

in Ω;

(2.61a)

in Ω;

(2.61b)

on 𝚪,

(2.61c)

with .fe an external force. The steady-state (equilibrium) solution is known to exist for assigned .fe for .d = 2, 3 [19, Theorem 7.3, p. 59]. Here, as usual, [19, p. 9], [96, p. 18] V ≡ {y ∈ (H01 (Ω))d : div y ≡ 0 in Ω}

.

(2.62a)

with norm  ‖y‖V =

.

1 2 |∇y(x)|2 dΩ .

(2.62b)

Ω

We assume that the boundary .∂ω of .ω is of class .C 2 . The critical Unique Continuation result is: Theorem 2.4.1 ([102]) Let λ ∈ C. Let {p, u}, with p ∈ H 1 (Ω), u ∈ (H 2 (Ω))d , div u ≡ 0 in Ω be any solution of problem (2.59a)–(2.59c). Then: u ≡ 0 in Ω and p ≡ const in Ω,

.

(2.63)

and we can take p ≡ 0 in Ω, as p is identified only up to a constant.

2.4.2 Proof of Theorem 2.4.1 [102] Case 1 We write initially the proof for the case where .ω is at a positive distance from .∂Ω : ∂ω ∩ ∂Ω = ∅ (Figs. 2.4 and 2.5). Step 1. Since u ≡ 0 in ω, then (2.59a), (2.60) yield ∇p ≡ 0 in ω, hence p ≡ const in ω. We may, and will, take p ≡ 0 in ω, as p is only identified up to a constant. Then we have ∂u ∂p .u|∂ω ≡ 0; ≡ 0; p| ≡ 0; ≡ 0, (2.64) ∂ω ∂ν ∂ω ∂ν ∂ω ∂ where ∂ν denotes the normal derivative (ν = unit inward normal vector with respect to ω). Step 2. The cut-off function χ . Let χ be a smooth, non-negative, cut-off function defined as follows:

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

83

D Γ = ∂Ω

ω χ≡1 Ω1 ; χ ≡ 1

χ≤1

Ω∗ Ω0

χ=0

Fig. 2.4 Case 1: G = Ω1 ∪ Ω∗

χ≡1

 

Ω0

-

Ω∗

-

Ω1

-  ω - Ω 1

Ω∗

Ω

-

Ω0

-

Fig. 2.5 Case 1: the cut-off function χ

χ≡

.

⎧ ⎨1 in Ω1 ∪ ω ⎩

;

supp χ ⊂ [Ω1 ∪ ω ∪ Ω∗ ],

(2.65)

0 in Ω0

while monotonically decreasing from 1 to 0 in Ω∗ , with χ ≡ 0 also in a small layer of Ω∗ bordering Ω0 (Figs. 2.4 and 2.5). Here: (i) Ω1 is a smooth subdomain of Ω that surrounds and contains ω in its interior (Fig. 2.4). (ii) In turn, Ω∗ is a smooth subdomain of Ω that surrounds and contains Ω1 in it interior (Fig. 2.4). (iii) In turn, Ω0 is a smooth subdomain of Ω : Ω0 ≡ Ω \ {ω ∪ Ω1 ∪ Ω∗ }.

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R. Triggiani

Step 2. The (χ u)-problem. Multiply Eq. (2.59a) by χ . We then get the following equation: (−ν0 Δ)(χ u) + Le (χ u) + ∇(χp) = λ(χ u) + Fχ in Ω,

.

(2.66)

with forcing term expressed by the resulting commutators Fχ ≡ Fχ (u, p) ≡ ν0 [χ , Δ]u + [Le , χ ]u + [∇, χ ]p.

.

= first order in u; zero order in p; supp Fχ ⊂ Ω∗ ,

(2.67a) (2.67b)

since Fχ ≡ 0 for χ ≡ 1, where [ , ] denotes commutator. Since we are dealing with homogeneous symbols, say s1 and s2 , then: order of [s1 , s2 ] = (order of s1 ) + (order of s2 ) − 1. Moreover, let (Fig. 2.4) D = ∂ω ∪ {external boundary of Ω∗ } = ∂[Ω1 ∪ Ω∗ ].

.

(2.68)

Since χ ≡ 0 on Ω0 and in a small layer of Ω∗ bordering Ω0 , then (χ u) and (χp) have zero Cauchy data on the [external boundary of Ω∗ ] = [interior boundary of Ω0 ]. Moreover, since u ≡ 0 in ω and p ≡ 0 in ω by Step 1, then (χ u) and (χp) have zero Cauchy data on ∂ω. Thus: ∂(χ u) ∂(χp) .(χ u)|D ≡ 0; ≡ 0; (χp)|∂ω ≡ 0; ≡ 0, ∂ν D ∂ν ∂ω

(2.69)

where ν denotes a unit normal vector outward with respect to [Ω∗ ∪ Ω1 ] (Fig. 2.4). Step 3. A pointwise Carleman estimate. We shall invoke the following pointwise Carleman estimate for the Laplacian from [72, Corollary 4.2, Eq. (4.15), p. 73]. Theorem 2.4.2 The following pointwise estimate holds true at each point x of a bounded domain G in Rd for an H 2 -function w, where ϵ > 0 and 0 < δ0 < 1 are arbitrary  ϵ 2τ ψ(x) δ0 2ρτ − |∇w(x)|2 + [4ρk 2 τ 3 (1 − δ0 ) + O(τ 2 )]e2τ ψ(x) |w(x)|2 e 2 ! " 1 2τ ψ(x) ≤ 1+ |Δw(x)|2 + div Vw (x), x ∈ G. (2.70) e ϵ

.

Here: ψ(x) is any strictly convex function over G, with no critical points in G, to be chosen below in Step 11 when G = Ω1 ∪ Ω∗ ; ρ > 0 is a constant, defined by Hψ (x) ≥ ρI , x ∈ G, where Hψ denotes the (symmetric) Hessian matrix of ψ(x) [72, Eq. (1.1.6), p. 45]; k > 0 is a constant, defined by: inf |∇ψ(x)| = k > 0, where the inf is taken over G [72, Eq. (1.1.7), p. 45]; and τ is a free positive parameter, to be chosen sufficiently large. For what follows, it is not critical to recall what div Vw (x) is, only that, via the divergence theorem, we have

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .



85

 div Vw (x)dx =

.

G

Vw (x) · ν dσ = 0,

(2.71)

∂G

whenever the Cauchy data of w vanish on its boundary ∂G: w|∂G ≡ 0; ∇w|∂G ≡ 0. In (2.71), ν is a unit normal vector outward with respect to G. Step 4. Pointwise Carleman estimates for (χ u). Next, we apply estimate (2.70) with w = (χ u) solution of (2.66). For definiteness, we select δ0 = 12 , ϵ = 12 . We obtain # ρτ −

.

$ 1 2τ ψ(x) |∇(χ u)(x)|2 + [2ρk 2 τ 3 + O(τ 2 )]e2τ ψ(x) |(χ u)(x)|2 e 8 ≤ 3e2τ ψ(x) |Δ(χ u)(x)|2 + div V(χ u) (x),

x ∈ G.

(2.72)

Next, we integrate (2.72) over the domain G ≡ [Ω1 ∪ Ω∗ ] (Fig. 2.4), thus obtaining # .

1 ρτ − 8

$ Ω1 ∪Ω∗

e2τ ψ(x) |∇(χ u)(x)|2 dx 

+ [2ρk τ + O(τ )] 2 3

2

Ω1 ∪Ω∗

e2τ ψ(x) |(χ u)(x)|2 dx



 ≤3 Ω1

∪Ω∗

e2τ ψ(x) |Δ(χ u)(x)|2 dx + ∂[Ω1

∪Ω∗ ]

:· ν dD,  (x) V(χu) 

(2.73)

where, on the RHS of (2.73), the boundary integral over D ≡ ∂[Ω1 ∪ Ω∗ ] = the boundary of Ω1 ∪ Ω∗ , see (2.68) and Fig. 2.4, vanishes in view of (2.71) with w = (χ u) having null Cauchy data on D, by virtue of (the LHS of) (2.69). Step 5. Here, we estimate the RHS of (2.73). Returning to (2.66), we rewrite it and multiply by eτ ψ(x) across ν0 Δ(χ u) = Le (χ u) − λ(χ u) + ∇(χp) − Fχ (u, p);

.

(2.74)

ν0 eτ ψ(x) Δ(χ u) = eτ ψ(x) Le (χ u) − λeτ ψ(x) (χ u) + eτ ψ(x) ∇(χp) − eτ ψ(x) Fχ . (2.75)

.

Recalling (2.60), we obtain from (2.75): with ye ∈ (H 2 (Ω))d , as assumed below (2.60):   ν02 e2τ ψ(x) |Δ(χ u)(x)|2 ≤ Cλ,ye e2τ ψ(x) [|∇(χ u)(x)|2 + |(χ u)(x)|2

.

+ 2e2τ ψ(x) |∇(χp)(x)|2 + 4e2τ ψ(x) |Fχ (u, p)(x)|2 , x ∈ G.

(2.76)

86

R. Triggiani

Thus, integrating (2.76) over G ≡ [Ω1 ∪ Ω∗ ] as required by (2.73) yields  2 .ν0

Ω1

∪Ω∗

e2τ ψ(x) |Δ(χ u)(x)|2 dx 

≤ Cλ,ye

Ω1

∪Ω∗

e2τ ψ(x) [|∇(χ u)(x)|2 + |(χ u)(x)|2 ]dx

 +2

 Ω1 ∪Ω∗

e2τ ψ(x) |∇(χp)(x)|2 dx + 4

Ω1 ∪Ω∗

e2τ ψ(x) |Fχ (u, p)(x)|2 dx. (2.77)

We now recall from (2.67a)–(2.67b) that Fχ (u, p) is an operator that is first order in u; zero order in p; and moreover, that supp Fχ ⊂ Ω∗ . Thus, (2.77) becomes explicitly  2 .ν0

Ω1 ∪Ω∗

e2τ ψ(x) |Δ(χ u)(x)|2 dx  ≤ Cλ,ye

Ω1

∪Ω∗

e2τ ψ(x) [∇(χ u)(x)|2 + |(χ u)(x)|2 ]dx

 +2

Ω1 ∪Ω∗

e2τ ψ(x) |∇(χp)(x)|2 dx

 + cχ

Ω∗

e2τ ψ(x) [|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx,

(2.78)

which is the sought-after bound in the last term on the RHS of (2.73). In (2.78), cχ is a constant depending on χ . Step 6. (Final estimate for (χ u)-problem (2.66).) We substitute (2.78) into the RHS of inequality (2.73) and obtain

.

# $  1 ρτ − e2τ ψ(x) |∇(χ u)(x)|2 dx + 2ρk 2 τ 3 8 Ω1 ∪Ω∗  2 + O(τ ) e2τ ψ(x) |(χ u)(x)|2 dx Ω1 ∪Ω∗

1 ≤ 2 Cλ,ye ν0

 Ω1 ∪Ω∗

e2τ ψ(x) [|∇(χ u)(x)|2 + |(χ u)(x)|2 ]dx

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

+

+

cχ ν02

6 ν02

87

 Ω1

∪Ω∗

e2τ ψ(x) |∇(χp)(x)|2 dx

 Ω∗

e2τ ψ(x) [|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx,

(2.79)

writing Cλ,ye for 3Cλ,ye and cχ for 3cχ . Moving the first integral term on the RHS of inequality (2.79) to the LHS of such inequality then yields for τ sufficiently large: # .

% $ Cλ,ye 1 e2τ ψ(x) |∇(χ u)(x)|2 dx − 2 ρτ − 8 ν0 Ω1 ∪Ω∗ & + 2ρk τ + O(τ ) − 2 3

2

6 ≤ 2 ν0 +

cχ ν02

Cλ,ye ν02

' Ω1

∪Ω∗

e2τ ψ(x) |(χ u)(x)|2 dx

 Ω1 ∪Ω∗

e2τ ψ(x) |∇(χp)(x)|2 dx

 Ω∗

e2τ ψ(x) [|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx.

(2.80)

Inequality (2.80) is our final estimate for the (χ u)-problem in (2.66), (2.67), (2.69). Step 7: The (χp)-problem. We need to estimate the first integral term on the RHS of inequality (2.80). This will be accomplished in (2.89). To this end, we need to obtain preliminarily the PDE problem satisfied by (χp) on G ≡ Ω1 ∪ Ω∗ . This task will be accomplished in this step. Accordingly, we return to Eq. (2.59a), take here the operation of “div” across, use (2.59b), and obtain as usual Δp = −div Le (u)

.

in Ω,

(2.81a)

where, actually [101, Eq. (5.21)], [102, Eq. (3.24)], div Le (u) = 2{(∂x ye ·∇)u} = 2{∂x u·∇)ye } is a first-order differential operator on u, (2.81b) by (2.59b). Next, multiply (2.81a) by χ. We obtain (2.82a), (2.82c) below

.

⎧ .Δ(χp) = −div Le (χ u) + Tχ (u, p), in Ω; ⎪ ⎪ ⎪ ⎨ ∂(χp) ⎪ ⎪ ⎪ = 0, (χp)|D = 0, D = ∂[Ω1 ∪ Ω∗ ]. ⎩ ∂ν D

(2.82a) (2.82b)

88

R. Triggiani

Tχ (u, p) ≡ [Δ, χ ]p + [div Le , χ ]u .

∗

(2.82c)

= first order in p; zero order in u; supp Tχ ⊂ Ω , since again Tχ (u, p) ≡ 0 where χ ≡ 1, while the B.C. (2.82b) on the boundary D defined by (2.68) follows for 2 reasons: (i) the RHS of (2.69) on (χp) on ∂ω; (ii) χ ≡ 0 up to the external boundary of Ω∗ and a small layer of Ω∗ bordering Ω0 , = 0, on such external boundary of Ω∗ . Thus, (2.82b) is so that (χp) = 0, ∂(χp) ∂ν justified. Next, we apply the pointwise Carleman estimate (2.70) to problem (2.82a)– (2.82b), that is for w = (χp). We obtain with G = Ω1 ∪ Ω∗ :  ϵ 2τ ψ(x) δ0 2ρτ − |∇(χp)(x)|2 e 2

.

 + 4ρk 2 τ 3 (1 − δ0 ) + O(τ 2 ) e2τ ψ(x) |(χp)(x)|2 !

" 1 2τ ψ(x) ≤ 1+ |Δ(χp)(x)|2 + div V(χp) (x), e ϵ

x ∈ G.

(2.83)

Again, it is not critical to recall what div V(χp) (x) is, only the vanishing relationship (2.71) (for w = (χp)) on an appropriate bounded domain G. Indeed, we shall take again G = Ω1 ∪ Ω∗ , integrate inequality (2.83) over with G ≡ Ω1 ∪ Ω∗ (after selecting again δ0 = 12 , ϵ = 12 ), and obtain # .

1 8

ρτ −

$ Ω1 ∪Ω∗

e2τ ψ(x) |∇(χ u)(x)|2 dx 

+ [2ρk 2 τ 3 + O(τ 2 )]

Ω1 ∪Ω∗



 ≤3

Ω1 ∪Ω∗

e2τ ψ(x) |(χ u)(x)|2 dx

e

2τ ψ(x)

|Δ(χ u)(x)| dx + 2

∂[Ω1 ∪Ω∗ ]

:· ν dD,  (x) V(χu) 

(2.84)

where, on the RHS of (2.84), the boundary integral over D ≡ ∂[Ω1 ∪ Ω∗ ] = [∂ω ∪ external boundary of Ω∗ ], see (2.68) and Fig. 2.4, again vanishes in view of (2.82b) with w = (χp). Thus, the vanishing of the last integral term of (2.84) is justified. Step 7. Here we now estimate the last integral term on the RHS of (2.84). We multiply Eq. (2.82a) by eτ ψ(x) , thus obtaining eτ ψ(x) Δ(χp) = −eτ ψ(x) div Le (χ u) + eτ ψ(x) Tχ (u, p)

.

(2.85)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

89

e2τ ψ(x) |Δ(χp)(x)|2 ≤ 2e2τ ψ(x) |div Le (χ u)(x)|2 + 2e2τ ψ(x) |Tχ (u, p)(x)|2 . (2.86) Thus, integrating (2.86) over G ≡ [Ω1 ∪ Ω∗ ] yields .

 .



Ω1 ∪Ω∗

e2τ ψ(x) |Δ(χp)(x)|2 dx ≤ 2

Ω1 ∪Ω∗

e2τ ψ(x) |div Le (χ u)(x)|2 dx

 +2

Ω1 ∪Ω∗

e2τ ψ(x) |Tχ (u, p)(x)|2 dx.

(2.87)

Recall now from (2.81b) that [div Le ] is a first-order operator, and accordingly, from (2.82c), that Tχ is an operator that is first order in p, zero order in u, and that Tχ has support in Ω∗ . We then refine (2.87) as  .

 e2τ ψ(x) |Δ(χp)(x)|2 dx ≤ 2Cye

Ω1 ∪Ω∗

e2τ ψ(x) [∇(χ u)(x)|2 +|(χ u)(x)|2 ]dx

Ω1 ∪Ω∗

 + 2Cχ

Ω∗

e2τ ψ(x) [|∇p(x)|2 + |p(x)|2 + |u(x)|2 ]dx,

(2.88)

with constant Cχ depending on χ. Step 8. (Final estimate of (χp)-problem) We now substitute (2.88) into the RHS of (2.84), divide across by [ρτ − 18 ] > 0 for τ large, and obtain  .

Ω1 ∪Ω∗

e

2τ ψ(x)

2ρk 2 τ 3 + O(τ 2 )  |∇(χp)(x)| dx + ρτ − 18

6Cye ) ≤( ρτ − 18 +(



2

e2τ ψ(x) |(χp)(x)|2 dx

 Ω1

∪Ω∗

e2τ ψ(x) [|∇(χ u)(x)|2 + |(χ u)(x)|2 ]dx



6Cχ ρτ −

Ω1 ∪Ω∗

1 8

)

Ω∗

e2τ ψ(x) [|∇p(x)|2 + |p(x)|2 + |u(x)|2 ]dx.

(2.89)

Inequality (2.89) is our final estimate on the (χp)-problem (2.82). Step 9. (Combining) We return to estimate (2.80) and add to each side the term 2ρk 2 τ 3 + O(τ 2 )  . ρτ − 18 to get

 Ω1 ∪Ω∗

e2τ ψ(x) |(χp)(x)|2 dx

90

R. Triggiani

% $ Cλ,ye 1 ρτ − − 2 e2τ ψ(x) |∇(χ u)(x)|2 dx ∗ 8 ν0 Ω1 ∪Ω

# .

& + 2ρk τ + O(τ ) − 2 3

2

Cλ,ye

2ρk 2 τ 3 + O(τ 2 )  + ρτ − 18 

6 ≤ 2 ν0

e

Ω1 ∪Ω∗

2τ ψ(x)

'

ν02

Ω1 ∪Ω∗

e2τ ψ(x) |(χ u)(x)|2 dx

 Ω1 ∪Ω∗

e2τ ψ(x) |(χp)(x)|2 dx

2ρk 2 τ 3 + O(τ 2 ) |∇(χp)(x)| dx+  ρτ − 18

+



2

cχ

e2τ ψ(x) |(χp)(x)|2 dx

Ω1 ∪Ω∗



ν02

Ω∗

e2τ ψ(x) [|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx.

(2.90)

Next, we substitute inequality (2.89) for the first two integral terms on the RHS of (2.90) and obtain % $ Cλ,ye 1 ρτ − − 2 e2τ ψ(x) |∇(χ u)(x)|2 dx ∗ 8 ν0 Ω1 ∪Ω

# .



 Cλ,y + 2ρk 2 τ 3 + O(τ 2 ) − 2 e ν0 2ρk 2 τ 3 + O(τ 2 )  + ρτ − 18 * ≤

% Ω1 ∪Ω∗

e2τ ψ(x) |(χ u)(x)|2 dx

 Ω1 ∪Ω∗

e2τ ψ(x) |(χp)(x)|2 dx

+  6Cye 6 ( ) + 1 e2τ ψ(x) [|∇(χ u)(x)|2 + |(χ u)(x)|2 ]dx 1 ∗ ν02 Ω1 ∪Ω ρτ − 8

* +

+  6Cχ 6 ) e2τ ψ(x) [|∇p(x)|2 + |p(x)|2 + |u(x)|2 ]dx +1 ( ν02 ρτ − 1 Ω∗ 8

+

cχ ν02

 Ω∗

e2τ ψ(x) [|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx.

(2.91)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

91

Step 10. (Final estimate of problem (2.59a)–(2.59c)) Finally, we combine integral terms with the same integrand on the LHS of (2.91) and obtain the final sought-after estimate which we formalize as a lemma. Lemma 2.4.1 The following inequality holds true for all τ sufficiently large: ⎧ ⎨#

⎫ * + $  Cλ,ye 6Cye ⎬ 6 1 ) ρτ − − 2 − e2τ ψ(x) |∇(χ u)(x)|2 dx +1 ( . ⎩ 8 ν0 ν02 ρτ − 1 ⎭ Ω1 ∪Ω∗ 8

⎧ ⎫ * + ⎨ ⎬ C 6C 6 y λ,y e e 2 3 2 ) +1 ( e2τ ψ(x) |(χ u)(x)|2 dx + 2ρk τ +O(τ ) − 2 − ⎩ 1 ⎭ Ω1 ∪Ω∗ ν0 ν02 ρτ − 8

+ * ≤

2ρk 2 τ 3 + O(τ 2 )  ρτ − 81

 Ω1 ∪Ω∗

e2τ ψ(x) |(χp)(x)|2 dx

+  6Cχ 6 ( ) e2τ ψ(x) [|∇p(x)|2 + |p(x)|2 + u(x)|2 ]dx + 1 1 ∗ ν02 Ω ρτ − 8

+

cχ ν02

 Ω∗

e2τ ψ(x) [|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx.

(2.92)

As already noted, (2.92) is the ultimate estimate regarding the original problem (2.59a)–(2.59c). Step 11. (The choice of weight function ψ(x)) We now choose the strictly convex function ψ(x) as follows (see Figs. 2.6 and 2.7): ψ(x) ≥ 0 on Ω1 where χ ≡ 1, so that e2τ ψ(x) ≥ 1 on Ω1 ; .

(2.93)

ψ(x) ≤ 0 on Ω0 ∪ Ω∗ ; where χ < 1, so that e2τ ψ(x) ≤ 1 on Ω∗ ,

(2.94)

.

in such a way that ψ(x) has no critical point in Ω \ ω, as required: that is, the critical point(s) of ψ will fall on ω, outside the region G = Ω1 ∪ Ω∗ , where we have integrated (Fig. 2.8). Having chosen ψ(x) as in (2.93), (2.94) with no critical points in Ω \ ω—i.e., no critical points on G = Ω1 ∪ Ω∗ —we return to the basic estimate (2.92), with τ sufficiently large. On the LHS of (2.92), we retain only integration over Ω1 , where ψ ≥ 0; hence, e2τ ψ ≥ 1 and χ ≡ 1, so that (χ u) ≡ u on Ω1 . On the RHS of (2.92), we have ψ ≤ 0 on Ω∗ ; hence, e2τ ψ ≤ 1 on Ω∗ . We thus obtain from (2.92)

92

R. Triggiani

Fig. 2.6 Case 1: construction of Ω1 and Ω∗

ω Ω1 ; χ ≡ 1; ψ ≥ 0 Ω∗ : ψ ≤ 0

χ≡1

  

Ω0

-

Ω∗

-  Ω1

-

ω

-

Ω

ψ Fig. 2.7 Case 1: choice of ψ

Fig. 2.8 Case 1: from {u, p} = 0 on ω to {u, p} = 0 on Ω1

ω u≡0 p≡0 Ω1 u ≡ 0; p ≡ 0

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

93

⎧ ⎨#

⎫ * + $  6Cye ⎬ Cλ,ye 6 1 ( ) . + 1 |∇u(x)|2 dx − 2 − ρτ − ⎩ 1 ⎭ Ω1 8 ν0 ν02 ρτ − +

⎧ ⎨ ⎩

8

2ρk 2 τ 3 + O(τ 2 ) −

+

Cλ,ye ν02

* −

+

⎫ ⎬

6Cye 6 ) +1 ( ν02 ρτ − 18 ⎭

2ρk 2 τ 3 + O(τ 2 )  ρτ − 18

|u(x)|2 dx

Ω1

 |p(x)|2 dx Ω1

+  6Cχ 6 ( ) +1 [|∇p(x)|2 + |p(x)|2 + |u(x)|2 ]dx ν02 ρτ − 1 Ω∗

* ≤

8

+

cχ ν02

 Ω∗

[|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx.

(2.95)

For τ sufficiently large, inequality (2.95) is of the type ! .

τ − const −

1 τ

" |∇u(x)|2 dx Ω1

" !  1 |u(x)|2 dx + (τ 2 ) |p(x)|2 dx + τ 3 − const − τ Ω1 Ω1 c ≤ τ

 Ω∗

[|∇p(x)|2 + |p(x)|2 + |u(x)|2 ]dx 

+ const ≤

Ω∗

[|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx.

c C1 (p, u; Ω∗ ) + const C2 (p, u; Ω∗ ). τ

(2.96)

(2.97)

In going from (2.96) to (2.97), we have emphasized in the notation that we are working with a fixed solution {u, p} of problem (2.59a)–(2.59c), so that the integrals on the RHS of (2.96) are fixed numbers C1 (p, u; Ω∗ ) and C2 (p, u; Ω∗ ), depending on such fixed solution {u, p} as well as Ω∗ . Inequality (2.96)/(2.97) is more than we need. On its LHS, we may drop the ∇u-term over Ω1 ; and alternatively either keep only the u-term over Ω1 , and divide the remaining inequality across by (τ 3 −const− 1 τ ) for τ large; or else keep only the p-term over Ω1 and divide the corresponding inequality across by τ 2 . We obtain, respectively,

94

R. Triggiani

!

 |u(x)| dx ≤ 2

.

Ω1

!

 |p(x)| dx ≤ 2

Ω1

" C 1 const C1 (p, u; Ω∗ ) + 3 C2 (p, u; Ω∗ ) → 0, . τ3 τ τ

(2.98)

" C 1 const C1 (p, u; Ω∗ ) + 2 C2 (p, u; Ω∗ ) → 0 τ2 τ τ

(2.99)

as τ → +∞. We thus obtain u(x) ≡ 0 in Ω1 ;

.

p(x) ≡ 0 in Ω1 .

(2.100)

Finally, we can now push the external boundary of Ω1 as close as we please to the boundary ∂Ω of Ω, and thus we finally obtain u ≡ 0 in Ω,

.

p ≡ 0 in Ω.

(2.101)

Indeed, as u ∈ (H 2 (Ω))d , then u ∈ (C(Ω))d , d = 2, 3. Thus, if it should happen that u(x1 ) /= 0 at a point x1 ∈ Ω near ∂Ω; hence, u(x) /≡ 0 in a suitable neighborhood N of x! ; then it would suffice to take Ω1 as to intersect such N to obtain a contradiction. Theorem 2.4.1 is proved at least in the Case 1 (Fig. 2.4). Case 2. Let ω be a collar of boundary 𝚪 = ∂Ω [Fig. 2.9 (total collar of 𝚪) and Fig. 2.11 (partial collar of 𝚪)]. Then, the above proof of Case 1 can be carried out with sets Ω1 and Ω∗ indicated in Fig. 2.9, with ψ as in Fig. 2.10 or in Fig. 2.11. Theorem 2.4.1 admits a generalization to the Riemannian setting (Fig. 2.12).

ω

Ω1

χ≡1

χ≡1

Ω∗

Ω0 ; χ ≡ 0

ν

ν

D Fig. 2.9 Case 2: ω is a collar of 𝚪; G = Ω1 ∪ Ω∗ ; ∂G = D = [internal boundary of ω] ∪ [internal boundary of Ω∗ ]

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

95

χ≡1

χ≡0 

ω -

Ω1

-  Ω∗ - 

Ω0

ψ

Fig. 2.10 Case 2: the choice of ψ for Fig. 2.9

D 
 
 
 
 
 
 Ω0 
  ) χ≡0 ω

Ω∗

Ω1 ; χ ≡ 1

Fig. 2.11 Case 2: ω is a collar of a portion of the boundary

Fig. 2.12 Spectrum of the Oseen operator A

λN λ1 λN +1 •

•

•

96

R. Triggiani

Theorem 2.4.3 The same result holds true in the case Ω is an open bounded set in a complete, d-dimensional Riemannian manifold of class C 3 , with C 3 -metric g: (M, g). In particular, this holds true in case problem (2.59a)–(2.59c) is still defined in a Euclidean setting, but the differential operators are now of smooth (say C 3 )-variable d d coefficients in space. Here, (M, g) = (R , g), where g = i,j =1 gij dxi dxj , −1 where {gij (x)} = {aij (x)} is a positive symmetric matrix defined in terms of the coefficients aij (x) = aj i (x) of the second-order uniformly elliptic partial  differential operator A = di,j ∂x∂ i (aij (x) ∂x∂ j ) in place of Δ. Proof of Theorem 2.4.3 The proof in the Riemannian setting is essentially the same mutatis mutandis. The Riemannian version of the critical Theorem 2.4.3 is now available from [105, Cor. 4.2, Eq. (4.12), p. 368], [106, Cor. 4.2, Eq. (4.22), p. 345]. ⨆ ⨅ Literature A version of Theorem 2.4.1 has already appeared in the recent literature, however stated under the additional condition u|𝚪 = 0, 𝚪 = ∂Ω, which, in fact, is not needed. We believe, however, that the present proof has a number of additional advantages and is far more transparent and direct, as we shall discuss below. Theorem 2.4.1 (stated under the additional condition u|𝚪 = 0) was proved in [11, Lemma 3.7, p. 1466] under the same a priori regularity assumptions {u, p} ∈ (H 2 (Ω))d × H 1 (Ω). The proof in [11] is based on the following main steps: (1) transformation of the original problem on a bounded domain Ω ⊂ Rd , with ω ⊂ Ω, into a corresponding half-space problem, via partitions of unity; (2) further transformation of the half-space problem into a “bent” half-space problem, where the boundary is the parabola ϕ(y) = |y ' |2 − yn = 0, y = {y ' , yn }, y ' = tangential variable, yn = normal variable; (3) selection of local coordinates in the passage from a ball covering ∂ω to a corresponding rectangular domain in the half-space, in such a way that the original (Euclidean) Laplacian takes the Melrose–Sjöstrand form on the latter (that is, with no second-order mixed derivatives in y ' and yn ); (4) application of the Carleman estimate (in integral form) as in [44, Thm. 8.3.1, p. 190], on the bent half-space for compactly supported solutions of a second-order elliptic operator, with a suitable strongly pseudo-convex function obtained as a quadratic perturbation of ϕ. We much prefer the present proof—which is carried out entirely on Ω and uses the pointwise Carleman estimate of Theorem 2.4.2—over the one given in [11], as being much more transparent, direct, and self-contained. Under stronger a priori regularity assumptions for {u, p} (more regular by one Sobolev unit) and the boundaries involved, a similar unique continuation result (stated under the additional condition u|𝚪 = 0) is given also in the contemporaneous and independent paper [30, Section 4, Theorem 4.2]. [30] notes that: “[A] unique continuation property for the Stokes equations has been established in [26] with the help of Carleman estimates derived in [44]. That Stokes equation differs from the Oseen equation. That is why we give here a complete proof of the unique

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

97

continuation property of a solution [of the Oseen problem]. As in [44], [26], to do this, we use Carleman estimates but our technology differs from the techniques of [44]. . ..” We likewise note that the technology of the present paper is different from that of paper [30]: Compare, in particular, the Carleman estimate in [30, Thm. 4.1, p. 297] with our present version of Lemma 2.4.1.

2.4.3 Implications of the UCP of Theorem 2.4.1: (i) Linear Independence of Interior Localized Eigenfunctions; (ii) Verification of the Corresponding Kalman Rank Condition The following section generalizes to Oseen systems the results of Sect. 2.3 for second-order elliptic operators. The supporting pillar of the entire analysis is of course the delicate and non-trivial Theorem 2.4.1 about the UCP. Linear independence of interior localized eigenfunctions Let .{λi }∞ i=1 be the eigenvalues of the Oseen problem (2.59a)–(2.59b) (excluding (2.59c)) along with the Dirichlet boundary condition .u|𝚪 = 0, .𝚪 = ∂Ω. For any positive integer M (fixed), let .{ϕij }, .i = 1, . . . , M, .j = 1, . . . , 𝓁i be an associated set of generalized eigenfunctions corresponding to the eigenvalue .λi with geometric multiplicity .𝓁i or algebraic multiplicity .Ni . Then, as is well known, we can choose the .ϕij eigenfunctions so that: Ni 2 d the set {ϕij }M i=1, j =1 is linearly independent on (L (Ω)) .

.

(2.102)

The next result is the generalization to the Oseen system of Theorem 2.3.1 for second-order elliptic operators. Theorem 2.4.4 In the above notation, we have: (i) For each .i = 1, . . . , M, the corresponding eigenfunctions {ϕij }𝓁ji=1 are linearly independent in (L2 (ω))d .

.

(2.103)

(ii) In fact, more generally, the following eigenfunctions 𝓁i 2 d {ϕij }M, i=1, j =1 are linearly independent in (L (ω)) .

.

(2.104)

(iii) Still more generally, .

the set of all generalized eigenf unctions {ϕij } f or i = 1, . . . , M; j = 1, . . . , Ni is linearly independent in (L2 (ω))d . (2.105)

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R. Triggiani

Proof The proof that UCP Theorem 2.4.1 implies Theorem 2.4.4 is given in [11, pp. 1458–1463]. It proceeds as in the proof of Theorem 2.3.1, with the difference that problem (2.23) is replaced now by the Oseen eigenproblem, say .

−ν0 Δϕ + Le (ϕ) + ∇p = λϕ divϕ ≡ 0 in Ω; ϕ = 0 on 𝚪; 𝚪

ϕ ≡ 0 in ω.

(2.106) ⨆ ⨅

2.4.4 Feedback Stabilization of the Unstable Oseen Dynamical System by Finite-Dimensional Localized Interior Controls We shall proceed accordingly to the (informal) quantitative description of Sect. 2.2, as it applies to the model (2.107) below. Introduction: Dynamic Navier–Stokes equations under localized internal control Let, at first, .Ω be an open connected bounded domain in .Rd , d = 2, 3 with sufficiently smooth boundary .𝚪 = ∂Ω. More specific requirements will be given below. Let .ω be an arbitrarily small open smooth subset of the interior .Ω, .ω ⊂ Ω, of positive measure. Let m denote the characteristic function of .ω: .m(ω) ≡ 1, m(Ω\ω) ≡ 0. Consider the following controlled Navier–Stokes equations with no-slip Dirichlet boundary conditions, where .Q = (0, ∞) × Ω, Σ = (0, ∞) × 𝚪: yt (t, x) − νΔy(t, x) + (y · ∇)y + ∇π(t, x) = m(x)u(t, x) + f (x)

.

in Q. (2.107a)

div y = 0

in Q. (2.107b)

y=0

on Σ. (2.107c)

y(0, x) = y0 (x)

in Ω (2.107d)

Notation As already done in the literature, for the sake of simplicity, we shall adopt the same notation for function spaces of scalar functions and function spaces of vector-valued functions. Thus, for instance, for the vector-valued (d-valued) velocity field y or external force f , we shall simply write say .y, f ∈ Lq (Ω) rather than .y, f ∈ (Lq (Ω))d or .y, f ∈ Lq (Ω). This choice is unlikely to generate confusion. By way of orientation, we state at the outset two main points. For the linearized w-problem (2.115) below in the feedback form (2.131), the corresponding well-posedness and global feedback uniform stabilization result, Theorem 2.4.7,

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

99

holds in general for .1 < q < ∞. Instead, the final, main well-posedness and feedback uniform, local stabilization result, [66], for the nonlinear problem (2.111) corresponding to the original problem (2.107) will require .q > 3, see (2.129b), in the case .d = 3, hence .1 < p < 65 ; and .q > 2, in the case .d = 2, hence .1 < p < 43 . Let .u ∈ Lp (0, T ; Lq (Ω)) be the control input and .y = (y1 , . . . , yd ) be the corresponding state (velocity) of the system. Let .ν > 0 be the viscosity coefficient. The function .v(t, x) = m(x)u(t, x) can be viewed as an interior controller with support in .Qω = (0, ∞)×ω. The initial condition .y0 and the body force .f ∈ Lq (Ω) are given. The scalar function .π is the unknown pressure. The stabilization problem by open loop controls was pioneered by A. Fursikov [30–32]. Stationary Navier–Stokes equations The following result represents our basic starting point. Theorem 2.4.5 Consider the following steady-state Navier–Stokes equations in .Ω .

− νΔye + (ye .∇)ye + ∇πe = f

in Ω.

(2.108a)

div ye = 0

in Ω.

(2.108b)

ye = 0

on 𝚪.

(2.108c)

Let .1 < q < ∞. For any .f ∈ Lq (Ω), there exists a solution (not necessarily unique) 2,q (Ω) ∩ W 1,q (Ω)) × (W 1,q (Ω)/R). .(ye , πe ) ∈ (W 0 For the Hilbert case .q = 2, see [19, Thm 7.3 p 59]. For the general case .1 < q < ∞, see [6, Thm 5.iii p 58]. Remark 2.4.1 It is well known [56, 58, 96] that the stationary solution is unique when “the data is small enough, or the viscosity is large enough” [96, p 157; Chapt ‖f ‖ 2], that is, if the ratio . 2 is smaller than some constant that depends only on ν .Ω [33, p 121]. When non-uniqueness occurs, the stationary solutions depend on a finite number of parameters [33, Theorem 2.1, p 121] asymptotically in the timedependent case. Remark 2.4.2 The case where .f (x) in (2.107a) is replaced by .∇g(x) is noted in the literature as arising in certain physical situations, where f is a conservative vector field. Main goal of the present paper A first difficulty one faces in extending the local exponential stabilization result for the interior localized problem (2.107) from the Hilbert-space setting in [12, 65] to the .Lq setting is the question of the existence of a Helmholtz (Leray) projection for the domain .Ω in .Rd . More precisely: Given an open set .Ω ⊂ Rd , the Helmholtz decomposition answers the question as to whether q q .L (Ω) can be decomposed into a direct sum of the solenoidal vector space .Lσ (Ω) q and the space .G (Ω) of gradient fields. Here,

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R. Triggiani

Lqσ (Ω) = {y ∈ Cc∞ (Ω) : div y = 0 in Ω}

‖·‖q

= {g ∈ Lq (Ω) : div g = 0; g · ν = 0 on ∂Ω}, for any locally Lipschitz domain Ω ⊂ Rd , d ≥ 2 [36, p 119] 1,q

Gq (Ω) = {y ∈ Lq (Ω) : y = ∇p, p ∈ Wloc (Ω)} where 1 ≤ q < ∞.

(2.109)

Both of these are closed subspaces of .Lq . Assumption 1 Let .1 < q < ∞ and .Ω ⊂ Rn be an open set. We say that the Helmholtz decomposition for .Lq (Ω) exists whenever .Lq (Ω) can be decomposed into the direct sum (non-orthogonal) Lq (Ω) = Lqσ (Ω) ⊕ Gq (Ω).

.

(2.110)

The unique linear, bounded, and idempotent (i.e., .Pq2 = Pq ) projection operator q q q q .Pq : L (Ω) −→ Lσ (Ω) having .Lσ (Ω) as its range and .G (Ω) as its null space is called the Helmholtz projection. See also [57]. This is an important property in order to handle the incompressibility condition div y ≡ 0. For instance, if such decomposition exists, the Stokes equation (say the linear version of (2.107) with control .u ≡ 0) can be formulated as an equation in the .Lq setting. Here below we collect a subset of known results about Helmholtz decomposition. We refer to [42, Section 2.2], in particular to the comprehensive Theorem 2.2.5 in this reference, which collects domains for which the Helmholtz decomposition is known to exist. These include the following cases:

.

(i) Any open set .Ω ⊂ Rd for .q = 2, i.e., with respect to the space .L2 (Ω), in which case the decomposition is orthogonal [19, Prop. 1.9, p.8] (ii) A bounded .C 1 -domain in .Rd [25], .1 < q < ∞ [36, Theorem 1.1 p 107, Theorem 1.2 p 114] for .C 2 -boundary, [34, 35] (iii) A bounded Lipschitz domain .Ω ⊂ Rd (d = 3) and for . 32 − ϵ < q < 3 + ϵ sharp range [25] (iv) A bounded convex domain .Ω ⊂ Rd , d ≥ 2, 1 < q < ∞ [25] Henceforth in this section, we assume that the bounded domain .Ω ⊂ Rd under consideration admits a Helmholtz decomposition for the values of .q, 1 < q < ∞, here considered at first, for the linearized problem (2.115) below. The final result for the nonlinear problem (2.115) will require .q > d, see [66], in the case of interest .d = 2, 3. A counter example is in [74]. Translated nonlinear Navier–Stokes z-problem: reduction to zero equilibrium We return to Theorem 2.4.5 that provides an equilibrium pair .{ye , πe }. Then, as in [11, 12, 64], we translate by .{ye , pe } the original N-S problem (2.107). Thus we introduce new variables z = y − ye ,

.

χ = π − πe

(2.111a)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

101

and obtain the translated problem zt − νΔz + (ye · ∇)z + (z · ∇)ye + (z · ∇)z + ∇χ = mu

.

in Q. (2.111b)

div z = 0

in Q. (2.111c)

z=0

on Σ. (2.111d)

z(0, x) = y0 (x) − ye (x)

on Ω. (2.111e)

We shall accordingly study the local null feedback stabilization of the zproblem (2.111), that is, feedback stabilization in a neighborhood of the origin. As usual, we next apply the projection .Pq below (2.110) to the translated NS problem (2.111) to eliminate the pressure .χ . We thus proceed to obtain the corresponding abstract setting for the problem (2.111) as in [11] except in the .Lq setting rather than in the .L2 setting as in this reference. Note that .Pq zt = zt , since q .z ∈ Lσ (Ω) in (2.109). Abstract nonlinear translated z-model and its linearized w-model First, for .1 < q q < ∞ fixed, the Stokes operator .Aq in .Lσ (Ω) with Dirichlet boundary conditions is defined by [39, p 1404], [42, p 1] Aq z = −Pq Δz,

.

1,q

D(Aq ) = W 2,q (Ω) ∩ W0 (Ω) ∩ Lqσ (Ω).

(2.112)

The operator .Aq has a compact inverse .A−1 q on .Lσ (Ω); hence, .Aq has a compact q resolvent on .Lσ (Ω). See also [21, 22]. q

Next, we introduce the first-order operator .Ao,q , Ao,q z = Pq [(ye . ∇)z + (z . ∇)ye ],

.

1

D(Ao,q ) = D(Aq2 ) ⊂ Lqσ (Ω).

(2.113)

Thus, the Navier–Stokes translated problem (2.111), after application of the Helmholtz projector .Pq in Definition 1 and use of (2.113), can be rewritten as q the following abstract equation in .Lσ (Ω): ⎧ dz ⎪ ⎪ ⎪ + νAq z + Ao,q z + Pq [(z · ∇)z] = Pq (mu) . ⎪ ⎪ ⎪ ⎨ dt dz q − Aq z + Nq z = Pq (mu) in Lσ (Ω) or ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ q ⎩ z(x, 0) = z0 (x) = y0 (x) − ye in Lσ (Ω).

(2.114a) (2.114b)

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R. Triggiani

While, still for .1 < q < ∞, the linearized system of the translated model (2.111) is given by ⎧ ⎪ ⎪. dw + νA w + A w = P (mu) ⎪ q o,q q ⎪ ⎪ ⎪ ⎨ dt dw or − Aq w = Pq (mu) in Lqσ (Ω) ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎩ w (x) = y (x) − y in Lq (Ω). 0

0

Aq = −(νAq + Ao,q ),

e

(2.115a) (2.115b)

σ

D(Aq ) = D(Aq ) ⊂ Lqσ (Ω).

.

(2.116) q

q

Equation (2.116) is actually considered in the complexified space .Lσ (Ω) + iLσ (Ω), q which will be still denoted by .Lσ (Ω). The Oseen operator .A has compact resolvent q on .Lσ (Ω). It follows that .A has a discreet point spectrum .σ (A) = σp (A) consisting of isolated eigenvalues .{λj }∞ j =1 , which are repeated according to their (finite) geometric multiplicity .𝓁j . However, since .A generates a .C0 analytic semigroup on q ∞ lie in a triangular sector of a well-known type. .Lσ (Ω), its eigenvalues .{λj } j =1 The case of interest in stabilization occurs where .A has a finite number, say N, of eigenvalues .λ1 , λ2 , λ3 , . . . , λN on a complex half-plane .{λ ∈ C : Re λ ≥ 0} which we then order according to their real parts, so that .

. . . ≤ Re λN +1 < 0 ≤ Re λN ≤ . . . ≤ Re λ1 ,

(2.117)

each .λi , i = 1, . . . , N , being an unstable eigenvalue repeated according to its geometric multiplicity .𝓁i . Let M denote the number of distinct unstable eigenvalues .λj of .A, so that .𝓁i is equal to the dimension of the eigenspace corresponding to .λi . M

Instead, .N = Ni is the sum of the corresponding geometric multiplicity .Ni of i=1

λi , where .Ni is the dimension of the corresponding generalized eigenspace.

.

Kalman rank condition is satisfied due to the UCP of Theorem 2.4.1 For i = 1, . . . , M, we now denote by .{ϕij }𝓁ji=1 , .{ϕij∗ }𝓁ji=1 the normalized linearly

.

'

independent eigenfunctions (on .Lσ (Ω)) and .(Lqσ (Ω))' = Lqσ (Ω), respectively, 1 1 + ' = 1 corresponding to the unstable distinct eigenvalues .λ1 , . . . , λM of .Aq . q q and .λ¯ 1 , . . . , λ¯ M of .A∗q , respectively: q

1,q

Aq ϕij = λi ϕij ∈ D(Aq ) = W 2,q (Ω) ∩ W0 (Ω) ∩ Lqσ (Ω) ∈ Lq (Ω).

.

'

1,q '

A∗q ϕij∗ = λ¯ i ϕij∗ ∈ D(A∗q ) = W 2,q (Ω) ∩ W0

'

(2.118a)

'

(Ω) ∩ Lqσ (Ω) ∈ Lq (Ω). (2.118b)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

103

Finite-Dimensional Spectral Assumption (FDSA) We henceforth assume in this section that for each of the distinct eigenvalues .λ1 , . . . , λM of .A, algebraic and geometric multiplicity coincide. This assumption is only for simplicity of exposition. The Kalman rank condition—written in (2.120) below in the present case—holds in full generality. The treatment however is much more computationally extensive [67]. For each .i = 1, . . . , M, consider the corresponding eigenfunctions .{ϕij∗ }𝓁ji=1 , ¯ i of .A∗ as in (2.118b). Define .𝓁i = geometric multiplicity of the unstable eigenvalue .λ  (f, g)ω =

f g¯ dω and .K = sup{𝓁i , i = 1, . . . , M}. Then, for each distinct

.

ω

unstable eigenvalue .λ¯ i , consider the matrix .Ui ⎡

∗) ⎤ . . . (uK , ϕi1 ω ∗) ⎥ . . . (uK , ϕi2 ω⎥ ⎥ : 𝓁i × K. .. .. ⎦ . . ∗ ∗ (u1 , ϕi𝓁i )ω . . . (uK , ϕi𝓁i )ω

∗) (u1 , ϕi1 ω ⎢ (u1 , ϕ ∗ )ω i2 ⎢ .Ui = ⎢ .. ⎣ .

(2.119)

The following is the main result of the present section. Theorem 2.4.6 Assume the FDSA. It is possible to select vectors .u1 , . . . , uK ∈ q Lσ (ω), q > 1, K = sup {𝓁i : i = 1, . . . , M}, such that the matrix .Ui of size .𝓁i × K in (2.119) satisfies ∗ ) . . . (u , ϕ ∗ ) ⎤ (u1 , ϕi1 ω K i1 ω ⎢ (u1 , ϕ ∗ )ω . . . (uK , ϕ ∗ )ω ⎥ i2 i2 ⎥ ⎢ .rank[Ui ] = f ull = 𝓁i or rank ⎢ ⎥ = 𝓁i ; .. .. ⎦ ⎣ . . ∗ ∗ (u1 , ϕi𝓁i )ω . . . (uK , ϕi𝓁i )ω

⎡

𝓁i × K for each i = 1, . . . , M.

(2.120)

Proof Step 1. By selection, see (2.118) and statement preceding it, the set of vectors q' ∗ ∗ ' .ϕ , . . . , ϕ i𝓁i is linearly independent in .Lσ (Ω), and .q is the Hölder conjugate of i1 1 1 ∗ ∗ .q, q + q ' = 1, for each .i = 1, . . . , M. Next, if the set of vectors .{ϕi1 , . . . , ϕi𝓁i } q'

were linearly independent in .Lσ (ω), .i = 1, . . . , M, the desired conclusion (2.120) for the matrix .Ui to be full rank would follow for infinitely many choices of the q vectors .u1 , . . . , uK ∈ Lσ (Ω). Indeed: ' ∗ , . . . , ϕ ∗ } is linearly independent on .Lq (ω), for each .i = Claim: The set .{ϕi1 σ i𝓁i 1, . . . , M, i.e., 𝓁i

.

j =1

'

αj ϕij∗ ≡ 0 in Lqσ (ω) =⇒ αj ≡ 0, j = 1, . . . , 𝓁i .

(2.121)

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R. Triggiani

The proof will critically depend on the unique continuation result of Theorem 2.4.1 [102], see also [11, Lemma 3.7 p.1466]. To this end, define the following function q' (depending on i) in .Lσ (Ω) ∗

ϕ =

.

#

𝓁i

αj ϕij∗

$

'

∈ Lqσ (Ω),

i = 1, . . . , M

(2.122)

j =1

so that .ϕ ∗ ≡ 0 on .ω by (2.121). As each .ϕij∗ is an eigenfunction of .A∗ corresponding to the eigenvalue .λ¯ i , see (2.118b), so is the linear combination .ϕ ∗ by (2.122). Thus, ∗ ∗ .ϕ satisfies the following eigenvalue problem for the operator .A : ¯ ∗ , div ϕ ∗ = 0 in Ω; A∗ ϕ ∗ = λϕ

.

ϕ ∗ = 0 in ω, by (2.121).

(2.123)

But the linear combination .ϕ ∗ in (2.122) of the eigenfunctions .ϕij∗ ∈ D(A∗ ) satisfies itself the Dirichlet B.C. . ϕ ∗ ∂Ω = 0. Thus the explicit PDE version of problem (2.123) is ⎧ ⎪ (2.124a) −νΔϕ ∗ − (Le )∗ ϕ ∗ + ∇p ∗ = λ¯ i ϕ ∗ in Ω; ⎪ ⎪ ⎨ ∗ (2.124b) div ϕ = 0 in Ω; ⎪ ⎪ ⎪ ⎩ (2.124c) ϕ ∗ |∂Ω = 0; ϕ ∗ = 0 in ω; ϕ ∗ ∈ D(A∗ );

.

(Le )∗ ϕ ∗ = (ye .∇)ϕ ∗ + (ϕ ∗ .∇)∗ ye ,

(2.125)

d

where .(f.∇)∗ ye is a d-vector whose i th component is . (Di yej )fj [12, p 55]. j =1

Step 2. The critical point is now that the over-determined problem (2.124) implies the following unique continuation result. '

ϕ ∗ = 0 in Lqσ (Ω); or by (2.122)

∗ ∗ ∗ α1 ϕi1 + α2 ϕi2 +· · · +α𝓁i −1 ϕi𝓁 + α𝓁i ϕi𝓁i = 0. i −1 (2.126) q' ∗ ∗ Since the set .{ϕi1 , . . . , ϕi𝓁i } is linearly independent on .Lσ (Ω), then (2.126) implies .α𝓁i ≡ 0, i = 1, . . . , 𝓁i , and (2.121) is proved. The required unique continuation result is Theorem 2.4.1 [11, 102, Lemma 3.7 p.1466]. The original proof is done in the Hilbert setting, but we may invoke the same result because .ϕ ∗ has more regularity and integrability than required since .ϕ ∗ is an eigenfunction of .A∗ . Thus the claim is established. In conclusion: it is possible q to select, in infinitely many ways, interior functions .u1 , . . . , uK ∈ Lσ (Ω) such that the algebraic full-rank condition (2.120) holds true for each .i = 1, . . . , M. ⨆ ⨅ .

Local well-posedness and uniform (exponential) null stabilization of the translated linear w-problem (2.115) by means of a finite-dimensional explicit,

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

105

spectral-based feedback control localized on .ω The uniform stabilization problem is reported next for the linearized w-problem as a consequence of the verification of the Kalman rank condition in (2.120). It is obtained in the context of Besov spaces of low regularity, “close” to .L3 (Ω) for .d = 3. s on domains of class .C 1 as real interpolation of Definition of Besov spaces .Bq,p Sobolev spaces Let m be a positive integer, .m ∈ N, 0 < s < m, 1 ≤ q < ∞, 1 ≤ p ≤ ∞, and then we define [39, p 1398] s Bq,p (Ω) = (Lq (Ω), W m,q (Ω)) ms ,p .

.

(2.127a)

This definition does not depend on .m ∈ N [108, p. xx]. This clearly gives s W m,q (Ω) ⊂ Bq,p (Ω) ⊂ Lq (Ω)

.

and

‖y‖Lq (Ω) ≤ C ‖y‖Bq,p s (Ω) .

(2.127b)

We shall be particularly interested in)the following special real interpolation spaces ( of .Lq and .W 2,q . m = 2, s = 2 − p2 : / 0 2− 2 Bq,pp (Ω) = Lq (Ω), W 2,q (Ω) 1− 1 ,p .

.

(2.128)

p

Our interest in (2.128) is due to the following characterization [5, Thm 3.4], [39, p 1399]: If .Aq denotes the Stokes operator introduced in (2.112), then ( .

Lqσ (Ω), D(Aq )

) 1− p1 ,p

  2− 2 = g ∈ Bq,pp (Ω) : div g = 0, g|𝚪 = 0 if

( .

Lqσ (Ω), D(Aq )

) 1− p1 ,p

2 1 0, that is, orthogonal to .τ (ξ ) = τ (ξ + hν(η)) = τ (η). Thus, via (2.184c) and .ν(ξ ) = ν(η): # $ (ϕ(ξ + hν(η)) − ϕ(ξ )) ϕ(ξ + hν(ξ )) − ϕ(ξ ) · ν(η) ν(η) = lim . lim h↘0 h↘0 h h ! =

" ∂ϕ(ξ ) · ν(η) ν(η), ∂ν

and thus (2.185), i.e., part (i), is established. ) Part (ii) now follows by replacing .ϕ(ξ ) with . ∂ϕ(ξ ∂ν , which by part (i) satisfies the ⨅ ⨆ counterpart of assumption (2.184). ν(η)

Fig. 2.17 Normal vectors ϕ(ξ+hν(ξ)) ϕ(ξ) • ξ

•η

• ξ+hν(ξ)

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R. Triggiani

2.6 Part III: The Boussinesq Problem, d = 2, 3 2.6.1 Introduction In the present Part III, we consider the problem of uniform stabilization of the Boussinesq system, near an unstable equilibrium solution. The Boussinesq system couples the Navier–Stokes equations with a heat-conducting equation. It models heat transfer in a viscous, incompressible, heat-conducting fluid over the domain .Ω, of dimension .d = 2, 3. Uncontrolled Dynamic Boussinesq system Let .Q ≡ (0, T ) × Ω, Σ ≡ (0, T ) × ∂Ω, T > 0. The uncontrolled dynamic Boussinesq system is given by ⎧ ¯ ⎪ ⎪.y t − νΔy + (y · ∇)y − γ (θ − θ)ed + ∇π = f (x) ⎪ ⎪ ⎪ ⎪ ⎪ θt − κΔθ + y · ∇θ = g(x) ⎪ ⎨ div y = 0 ⎪ ⎪ ⎪ ⎪ ⎪ y = 0, θ = 0 ⎪ ⎪ ⎪ ⎩ y(0, x) = y 0 , θ (0, x) = θ0

in Q

(2.188a)

in Q

(2.188b)

in Q

(2.188c)

on Σ

(2.188d)

on Ω,

(2.188e)

where .y = {y1 , . . . , yd } = the fluid velocity; .θ = fluid temperature; .π = (scalar) g¯ pressure; .ν = kinematic viscosity coefficient; .κ = thermal conductivity; .γ = θ¯ where .g¯ = gravitational acceleration, .θ¯ = reference temperature; .ed = the vector .(0, . . . , 0, 1), the d-vector-valued function .f (x) = external force acting on the Navier–Stokes equation; scalar function .g(x) = heat source density acting on the heat equation. Stationary Boussinesq system: Equilibrium The starting point is the following result: Theorem 2.6.1 (Acevedo, Amarouche, Conca (2016), (2019) for .p /= 2 [1, 2]) Consider the following steady-state Boussinesq system in .Ω .

− νΔy e + (y e · ∇)y e − γ (θe − θ¯ )ed + ∇πe = f (x)

in Ω.

(2.189a)

−κΔθe + y e · ∇θe = g(x)

in Ω.

(2.189b)

div y e = 0

in Ω.

(2.189c)

y e = 0, θe = 0

on ∂Ω.

(2.189d)

Let .1 < p < ∞. For any .f , g ∈ Lp (Ω), Lp (Ω), there exists a solution (generally 1,p 1,p not unique) .(y e , θe , πe ) ∈ (W 2,p (Ω) ∩ W 0 (Ω)) × (W 2,p (Ω) ∩ W0 (Ω)) × 1,p (W (Ω)/R).

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

123

For the problem of feedback stabilization to be relevant, we assume throughout that the equilibrium solution .{ye , θe } be unstable. Instability means that: the free dynamics linearized operator .Aq of the linearized problem # $ # $ d wf wf . = Aq . wh dt wh

(2.190)

See below in (2.192), which is the generator of a s.c. analytic semigroup with compact resolvent in the appropriate functional setting—has N unstable eigenvalues .λ1 , λ2 , · · · λN on the complex half-plane .{λ ∈ C; Re λ ≥ 0}, each .λj being an unstable eigenvalue repeated according to its geometric multiplicity .𝓁j , so that .

. . . ≤ Re λN+1 < 0 ≤ Re λN ≤ . . . Re λ1 .

We also let M denote the number of distinct unstable eigenvalues .λj , with 𝓁1 + 𝓁2 + . . . 𝓁M = N.

.

Qualitative Goal As usual, we seek to introduce controls, possibly in feedback form, possibly finite-dimensional, “minimally invasive” that “stabilize” the Boussinesq system in the vicinity of an unstable equilibrium pair .{y e , θe }, in fact with a uniform exponential decay (Fig. 2.18). To this end, we pursue two strategies: (1) Localized Interior Feedback Control: A pair .{u, v} of interior localized controls, acting on the fluid and the thermal equation, respectively, with common support on an arbitrarily small subset .ω (of positive measure) of the domain .Ω. (2) Localized {interior, boundary} feedback control: A pair .{u, v} of controls, where v is a localized (scalar) boundary control acting on the thermal equation, compactly supported on an arbitrarily small portion . 𝚪 of the boundary .𝚪, and .u in a localized (vector-valued) control acting on the fluid equation, compactly supported on an arbitrarily small subset .ω (of positive measure) of .Ω, where .ω is a local collar supported by . 𝚪 . See Fig. 2.19. Fig. 2.18 Spectrum of .Aq

λN λN +1 •

•

•

λ1

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R. Triggiani

Ω ω Γ Fig. 2.19 The localized interior set .ω

2.6.2 Localized Interior Feedback Control Pair {u, v} Consider the Boussinesq controlled problem ⎧ ⎪ .y t − νΔy + (y · ∇)y − γ (θ − θ¯ )ed + ∇π = m(x)u(t, x) + f (x) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θt − κΔθ + y · ∇θ = m(x)v(t, x) + g(x) ⎪ ⎪ ⎨ div y = 0 ⎪ ⎪ ⎪ ⎪ ⎪ y = 0, θ = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y(0, x) = y 0 , θ (0, x) = θ0

in Q

(2.191a)

in Q

(2.191b)

in Q

(2.191c)

on Σ

(2.191d)

on Ω.

(2.191e)

.m = characteristic function of .ω, with interior pair .{u, v} localized on a subdomain ω: As in the case of the stabilization of classical parabolic problems and more recently of Navier–Stokes stabilization, a technical step in the strategy [97] explained in Sect. 2.2 arises at the level of examining the controllability property of the finite-dimensional projected unstable system. This, as in the case of classical parabolic and the N-S, rests on a Unique Continuation Property. There, however, is an interesting difference between the Navier–Stokes and the Boussinesq analysis of the finite-dimensional projected system. First, as already noted before in Sect. 2.2, the Kalman controllability test of any projected system is actually related to Unique Continuation Property of the adjoint system, not the original problem:

.

• In the Navier–Stokes case, what counts is the Unique Continuation Property of the adjoint Oseen eigenproblem, or the adjoint Oseen operator .A∗q ; see (2.116). However, for Navier–Stokes

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

.

the UCP of the Oseen Aq

⇐⇒

125

UCP . of the Oseen A∗q

(2.192)

• Instead, in the case of the Boussinesq system in taking the adjoint .A∗q of the # $ Aq −Cγ linearized operator .Aq = , the coupling operators .Cγ and .Cθe switch −Cθe Bq places. The adjoint .A∗q has a more favorable expression due to the vector .ed = ⎡ ⎤ 0

⎢ .. ⎥ ⎢ . ⎥ (d-coordinates). The conclusion is that the UCP for the adjoint .A∗ has less q ⎣0⎦ 1 q

conditions. Here, for .1 < q < ∞ fixed, the Stokes operator .Aq in .Lσ (Ω) with Dirichlet boundary conditions is defined by Aq z = −Pq Δz,

.

1,q

D(Aq ) = W 2,q (Ω) ∩ W 0 (Ω) ∩ Lqσ (Ω),

(2.193)

while the first-order operator .Ao,q is given by Ao,q z = Pq [(y e · ∇)z + (z · ∇)y e ],

.

1

D(Ao,q ) = D(Aq2 ) ⊂ Lqσ (Ω)

(2.194)

to describe the N-S equations. We next define the differential operator of the heat component Bq f = −κΔf + y e · ∇f,

.

1,q

D(Bq ) = W 2,q (Ω) ∩ W0 (Ω).

(2.195)

Finally, we define the coupling linear terms as bounded operators on q Lq (Ω), Lσ (Ω), respectively, .q > d:

.

[from . the NS equation] [from the heat equation]

Cγ h = −γ Pq (hed ), Cγ ∈ L(Lq (Ω), Lqσ (Ω)), . (2.196) Cθe z = z · ∇θe , Cθe ∈ L(Lqσ (Ω), Lq (Ω)).

(2.197)

The Kalman algebraic condition for the adjoint eigenproblem, or operator .A∗q For each unstable eigenvalue .λ¯ i of .A∗q , i = 1, . . . , M, we denote by .{Ф∗ij }𝓁ji=1 the normalized, linearly independent eigenfunctions of .A∗q , say, ¯i λ 𝓁i = geometric multiplicity ϕ∗i1 ,

.

ϕ∗i2 , . . .

ϕ∗i𝓁i

(2.198)

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R. Triggiani

A∗q ϕ ∗ij = λ¯ i ϕ ∗ij ∈ D(A∗q )

(2.199)

.

with ∗(1)

∗(2)

∗(d−1)

∗ ∗ ϕij = {ϕij , ψij∗ } = {ϕij , ϕij , . . . , ϕij

.

∗(d)

, ϕij , ψij∗ } , a (d + 1)-vector. (2.200)

∗ ∗ ij ϕ = { ϕij , ψij∗ } = {ϕij∗(1) , ϕij∗(2) , . . . ϕij∗(d−1) , ψij∗ } , a d-vector.

(2.201)

∗ obtained from .ϕ ∗ by omitting the d-component .ϕ ∗(d) of the vector .ϕ ∗ . ij ϕ ij ij ij

.

Next, we construct the following matrix .Ui of size .𝓁i × K, K = sup{𝓁i : i = 1, . . . , M} ⎡

∗ ) . . . (u , 4 ∗ (u1 , 4 ϕi1 ω K ϕi1 )ω

⎢ ∗) ⎢ (u1 , 4 ϕi2 ω Ui = ⎢ .. ⎢ ⎣ . ∗ ) ϕi𝓁 (u1 , 4 i ω

.

⎤

∗) ⎥ . . . (uK , 4 ϕi2 ω⎥ ⎥ : 𝓁i × K. .. ⎥ .. . ⎦ . ∗ ) . . . (uK , 4 ϕi𝓁 i ω

(2.202)

Here we have set uk = [u1k , u2k ] = [(u1k )(1) , (u1k )(2) . . . (u1k )(d−1) , u2k ]

.

4 qσ )u ⊂ 4 Lqσ (Ω) × Lq (Ω); ∈ (W N

(2.203)

4 Lqσ (Ω) ≡ the space obtained from Lqσ (Ω) after omitting the

.

d-coordinate from the vectors of 4 Lqσ (Ω);

(2.204)

4 qσ )u ≡ the space obtained from (Wqσ )u after omitting the (W N N

.

d-coordinate from the vectors of (Wqσ )uN .

(2.205)

The required Kalman algebraic controllability condition of the finitedimensional projected .wN -equation is given by ⎡

∗ ) . . . (u , 4 ∗ (u1 , 4 ϕi1 ω K ϕi1 )ω

⎤

⎢ ⎥ ∗ ) . . . (u , 4 ∗ ⎢ (u1 , 4 ϕi2 ω K ϕi2 )ω ⎥ ⎢ ⎥ = full = 𝓁i , .rank Ui = rank .. .. ⎢ ⎥ .. . ⎣ ⎦ . . ∗ ∗ ϕi𝓁i )ω . . . (uK , 4 ϕi𝓁i )ω (u1 , 4

i = 1, . . . , M,

(2.206)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . . ∗ ∗ i.e., linear independence of vectors {4 ϕi1 ,··· ,4 ϕi𝓁 } on ω. i

127

(2.207)

.

The key question is then: Is the controllability property (2.206) of the finitedimensional projected unstable system true? The answer is affirmative The crux of this chapter is that the validity of (2.206), i.e., of (2.207), is established by an appropriate Unique Continuation Result.

2.6.3 Main UCP Results for Both the Adjoint Eigenproblem, i.e., the Operator A∗q ; and the Original Eigenproblem, i.e., the Operator Aq Henceforth, for easiness of notation in the remainder of Sect. 2.6, the vector .ϕ ∈ Rd will not be boldfaced. The UCP for the adjoint operator .A∗q The required UCP for Kalman controllability purposes involves the adjoint operator .A∗q . It is given by: Theorem 2.6.2 ([104] The required UCP for problem (2.206). First version, adjoint problem) Let .ω be an arbitrary, open, connected, smooth subset of .Ω, thus of positive measure (Fig. 2.19). Let .{ϕ, h, p} ∈ (W 2,q (Ω))d × W 2,q (Ω) × W 1,q (Ω), .q > d solve the adjoint of problem ⎧ ∗ . − ν0 Δϕ + Le (ϕ) + ∇p + h∇θe = λϕ ⎪ ⎪ ⎨ −κΔh + ye · ∇h − γ ϕ · ed = λh ⎪ ⎪ ⎩ divϕ = 0 Le (ϕ) ≡ (ye · ∇)ϕ + (ϕ · ∇)ye

in Ω,

(2.208a)

in Ω,

(2.208b)

in Ω.

(2.208c)

in Ω.

(2.208d)

L∗e (ϕ) = (ye · ∇)ϕ + ∇ ⊥ ye · ϕ,

(2.208e)

.

.

along with the over-determination condition h ≡ 0,

{ϕ1 , ϕ2 , . . . , ϕd−1 } ≡ 0

.

in ω.

(2.208f)

Then, ϕ ≡ 0,

.

h ≡ 0,

p ≡ const

in Ω.

(2.209)

The proof, given in [104], is based on the following UCP of the direct eigenproblem.

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R. Triggiani

The UCP of the original operator .Aq Theorem 2.6.3 (UCP, original [104]) Let .ω be an arbitrary, open, connected, smooth subset of .Ω, thus of positive measure (Fig. 2.19). Let .{ϕ, h, p} ∈ (W 2,q (Ω))d × W 2,q (Ω) × W 1,q (Ω), .q > d solve the original eigenproblem ⎧ . − ν0 Δϕ + Le (ϕ) + ∇p − γ hed = λϕ ⎪ ⎪ ⎨ −κΔh + ye · ∇h + ϕ · ∇θe = λh ⎪ ⎪ ⎩ div ϕ = 0

in Ω,

(2.210a)

in Ω,

(2.210b)

in Ω,

(2.210c)

along with the over-determination condition ϕ ≡ 0,

.

h≡0

in ω.

(2.211)

Then, ϕ ≡ 0,

.

h ≡ 0,

p ≡ const

in Ω.

(2.212)

The technical proof is based on pointwise Carleman-type estimates for the Laplacian operator given in the next Sect. 2.6.4. Below, we deduce Theorem 2.6.2 from Theorem 2.6.3 as in [104]. Proof We recall that .ed is the d-vector .ed = {0, . . . , 0, 1}. Then .h ≡ 0 in ω, as assumed in (2.208f), implies .ϕd ≡ 0 in .ω, by (2.208b). Combining this with (2.208f), we then obtain

.

h ≡ 0 in ω,

.

ϕ ≡ 0 in ω.

(2.213)

Remark 2.6.1 We thank a referee for pointing out to us paper [77], which gives a related UCP for a linearized Boussinesq system for a 2D domain with overdetermination on a small interior set .ω that is moreover assumed to be at a finite distance from the boundary. In contrast, our result covers also the 3D domain, and the little interior set .ω may well be a boundary set (this is what we call Case 2 with pictures in Figs. 2.9, 2.10, and 2.11 as in the Oseen case). (By the way, our results are also valid in the Riemannian case, as noted in Theorem 2.4.3 in the Oseen case. See Remark 2.7.3.) Then the same proof of Theorem 2.6.2 applies and yields the conclusion (2.209). This is so since the differential term .L∗e (ϕ) in (2.208e) is first order as is .Le (ϕ) in (2.208d), while the term .γ ϕ · ed in (2.208b) is of zero order as the term .ϕ · ∇θe in (2.210b). Thus the same estimates of the proof of Theorem 2.6.2 apply. ⨆ ⨅

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

129

2.6.4 Proof of Theorem 2.6.3 via Pointwise Carleman Estimates [104] Step 0. Without loss of generality, we may normalize the constants .ν0 = κ = γ ≡ 1. Via (2.208d), we can then rewrite Eqs. (2.210a)–(2.210b) combined as in (2.214a) below, along with (2.210c) and the over-determination (2.211) ⎧ ⎡ ⎤ ⎪ 0 ⎪ ⎪ ⎪ ⎢ .. ⎥ ⎪ & ' & ' & ' & ' & ' ⎪ ⎥ ⎢ ⎪ ⎪ ⎢.⎥ ⎪ ϕ ϕ ϕ ye ∇p ⎪ ⎥ ⎢ ⎪ = λ . (−Δ) + (y · ∇) + (ϕ · ∇) + − e ⎪ ⎢0⎥ ⎪ h h h 0 θe ⎨ ⎢ ⎥ ⎣ h⎦ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ in Ω ⎪ ⎪ ⎪ ⎪ ⎩ divϕ ≡ 0 in Ω and ϕ ≡ 0, h ≡ 0 in ω.

(2.214a) (2.214b)

The above problem (2.214a)–(2.214b) is not quite the over-determined Oseen problem in Sect. 2.4, Eq. (2.59) in the variable .u = {ϕ, h}. We shall apply the Carleman estimate approach and techniques employed in Sect. 2.4.2 for the Oseen problem, with appropriate modifications. Case 1 We write initially the proof for the case where .ω is at a positive distance from .∂Ω: dist.(∂Ω, ∂ω) > 0 (Figs. 2.4, 2.5, and 2.19). Step 1. Since .u = {ϕ, h} ≡ 0 in .ω by (2.214b), then (2.214a) yields .∇p ≡ 0 in .ω; hence, .p = const in .ω. We may then take .p ≡ 0 in .ω, as p is only identified up to a constant. Then we have # $ # $ ∂u ∂ ϕ ϕ ≡ 0; ∂p ≡ 0. ≡ 0; p = .u = ≡ 0; ∂ω ∂ω h ∂ω ∂ν h ∂ω ∂ν ∂ω ∂ν ∂ω (2.215) Step 2. The cut-off function .χ . Let .χ be a smooth, non-negative, cut function defined as follows:

χ≡

⎧ .1 ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0

on ω ∪ Ω1

(2.216a) ;

on Ω0

supp χ ⊂ [Ω1 ∪ Ω∗ ∪ ω], (2.216b)

while monotonically decreasing from 1 to 0 in .Ω∗ , with .χ ≡ 0 also in a small layer within .Ω∗ bordering .Ω0 (Figs. 2.4, 2.5, and 2.6). Here: (i) .Ω1 is a smooth subdomain of .Ω surrounding .ω, and .∂ω is the interface between .ω and .Ω1 (Fig. 2.4). Thus, .∂ω = internal boundary of .Ω1 .

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(ii) In turn, .Ω∗ is a smooth subdomain of .Ω surrounding .Ω1 and the external boundary of .Ω1 is the interface between .Ω∗ and .Ω1 . Thus, [external boundary of .Ω1 ] .= [internal boundary of .Ω∗ ]. (iii) In turn, .Ω0 is a smooth subdomain of .Ω: .Ω0 ≡ Ω\{ω ∪ Ω1 ∪ Ω∗ }. Step 3. The (.χ ϕ)-problem. Multiply the .ϕ-equation in (2.214a) (i.e., (2.210a)) by χ and obtain via (2.208d)

.

(−Δ)(χ ϕ) + Le (χ ϕ) + ∇(χp) = λ(χ ϕ) + Fχ in Ω. ⎤ ⎡ 0 Fχ = Fχ (ϕ, p, h) = Fχ1,0 (ϕ, p) + ⎣ 0 ⎦ (χ h)

(2.217)

Fχ1,0 (ϕ, p) = [χ , Δ]ϕ − [χ , Le ]ϕ + [∇, χ ]p.

(2.219a)

= first order in ϕ; zero order in p.

(2.219b)

supp Fχ1,0 ⊂ Ω∗ .

(2.219c)

.

.

(2.218)

Notice that (2.219c) holds true since .χ ≡ 1 on .Ω1 , on .ω, and on a small layer within Ω∗ , so that on the union of these three sets we have that .Fχ1,0 ≡ 0. We recall that the commutator .[χ , Δ] is of order .0 + 2 − 1 = 1; the commutator .[χ , Le ] is of order .0 + 1 − 1 = 0; the commutator .[∇, χ ] is of order .1 + 0 − 1 = 0. Step 4. The .(χ h)-problem. Next, we multiply the h equation in (2.214a) = (2.210b) by .χ and obtain .

(−Δ)(χ h) + (ye · ∇)(χ h) = λ(χ h) + Gχ (h, ϕ)

.

in Ω.

(2.220)

Gχ = Gχ (h, ϕ) = G1χ (h) − (χ ϕ) · ∇θe

(2.221)

G1χ (h) = [χ , Δ]h − ye · [χ , ∇]h = first order in h.

(2.222a)

.

supp G1χ ⊂ Ω∗ .

(2.222b)

Notice that (2.222b) holds true, since as in the case for (2.219c), we have that .G1χ ≡ 0 on .ω ∪ Ω1 ∪ (a small layer within .Ω∗ ), since .χ ≡ 1 on such union. Step 5. The .(χ u)-problem, .χ u = {χ ϕ, χ h}. We combine Step 3 and Step 4 and obtain recalling (2.208d): 0 $ # $ ! # $" ! # $" #/ (χ ϕ) · ∇ ye ∇(χp) ϕ ϕ + (ye · ∇) χ + + (−Δ) χ 0 h h 0

.

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

⎤⎤ 0 ⎢ .. ⎥⎥ ⎢ . ⎥⎥ ! # $" ⎢F 1,0 (ϕ, p) + ⎢ ⎥⎥ ⎢ ⎢ χ ϕ ⎦⎥ in Ω. ⎣ 0 +⎢ =λ χ ⎢ ⎥ h (χ h) ⎥ ⎢ ⎣ ⎦ ⎡

131

⎡

(2.223)

G1χ (h) − (χ ϕ) · ∇θe Moreover, let (Fig. 2.4) D = ∂ω ∪ {external boundary of Ω∗ } = ∂[Ω1 ∪ Ω∗ ].

.

(2.224)

Since .χ ≡ 0 on .Ω0 and in a small layer of .Ω∗ bordering .Ω0 (Figs. 2.4 and 2.5), then .(χ u) = {(χ ϕ), (χ h)} and .(χp) have zero Cauchy data on the [external boundary of ∗ .Ω ] .= [interior boundary of .Ω0 ]. Moreover, since .u ≡ {ϕ, h} ≡ 0 in .ω and .p ≡ 0 in .ω by Step 1, then .(χ u) = {(χ ϕ), (χ h)} and .(χp) have zero Cauchy data on .∂ω, see (2.215). Thus, recalling D in (2.224) and .u ≡ {ϕ, h}: ∂(χ u) ≡ 0, ∂(χp) ≡ 0, .(χ u) ≡ 0, ≡ 0, (χp) (2.225) D ∂ω ∂ν D ∂ν ∂ω where .ν denotes here the unit vector outward with respect to .[Ω∗ ∪ Ω1 ], Fig. 2.4. Step 6. A pointwise Carleman estimate. We shall invoke the following pointwise Carleman estimate for the Laplacian from [72, Corollary 4.2, Eq. (4.15), p. 73], [71, Corollary 4.3, p. 254]. Theorem 2.6.4 The following pointwise estimate holds true at each point x of a bounded domain G in .Rd for an .H 2 -function w, where .ϵ > 0 and .0 < δ0 < 1 are arbitrary  ϵ 2τ ψ(x) δ0 2ρτ. − |∇w(x)|2 + [4ρk 2 τ 3 (1 − δ0 ) + O(τ 2 )]e2τ ψ(x) |w(x)|2 e 2 ! " 1 2τ ψ(x) ≤ 1+ |Δw(x)|2 + divVw (x), e ϵ

x ∈ G.

(2.226)

Here: .ψ(x) is any strictly convex function over G, with no critical points in .G, see Fig. 2.7, to be chosen below in Step 15 where .G = Ω1 ∪ Ω∗ ; .ρ > 0 is a constant, defined by .Hψ (x) ≥ ρI , .x ∈ G, where .Hψ denotes the (symmetric) Hessian matrix of .ψ(x) [72, Eq. (1.1.6), p. 45]; .k > 0 is a constant, defined by: .inf |∇ψ(x)| = k > 0, where the inf is taken over G [72, Eq. (1.1.7), p. 45]; and .τ is a free positive parameter, to be chosen sufficiently large. For what follows, it is not critical to recall what div .Vw (x) is, only that, via the divergence theorem, we have

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R. Triggiani



 divVw (x)dx =

.

G

Vw (x) · ν dσ = 0,

(2.227)

∂G

whenever the Cauchy data of w vanish on its boundary .∂G: .w|∂G ≡ 0; ∇w|∂G ≡ 0. In (2.227), .ν is a unit normal vector outward with respect to G. Step 7. Pointwise Carleman estimates for .(χ u), u = {ϕ, h}. Next, we apply estimate (2.226) with .w = (χ u) solution of problem (2.223). For definiteness, we select .δ0 = 12 , .ϵ = 12 . We obtain # $ 1 2τ ψ(x) ρτ − . e |∇(χ u)(x)|2 + [2ρk 2 τ 3 + O(τ 2 )]e2τ ψ(x) |(χ u)(x)|2 8 ≤ 3e2τ ψ(x) |Δ(χ u)(x)|2 + divV(χ u) (x),

x ∈ G.

(2.228)

Next, we integrate (2.228) over the domain .G ≡ [Ω1 ∪ Ω∗ ] (Fig. 2.4), thus obtaining #

1 ρτ − . 8

$ Ω1 ∪Ω∗

e2τ ψ(x) |∇(χ u)(x)|2 dx 

+ [2ρk τ + O(τ )] 2 3

2

Ω1 ∪Ω∗

e2τ ψ(x) |(χ u)(x)|2 dx





≤3 Ω1

∪Ω∗

e2τ ψ(x) |Δ(χ u)(x)|2 dx + ∂[Ω1

∪Ω∗ ]

:· ν dD, (2.229)  (x) V(χu) 

where, on the RHS of (2.229), the boundary integral over .D ≡ ∂[Ω1 ∪ Ω∗ ] = the boundary of .[Ω1 ∪ Ω∗ ], see (2.224) and Fig. 2.4, vanishes in view of (2.227) with .w = (χ u) having null Cauchy data on D, by virtue of (the LHS of) (2.225). Step 8. (Bound on the RHS of (2.229)) Here, we estimate the RHS of (2.229). Returning to the .(χ u)-problem (2.223), we rewrite it over .G = [Ω1 ∪ Ω∗ ] as 0 $ # ! # $" ! # $" #/ $ ∇(χp) ϕ ϕ (χ ϕ) · ∇ ye + Δ χ = (ye · ∇) χ + 0 0 h h ⎡ ⎤⎤ ⎡ 0 ⎢ .. ⎥⎥ ⎢ . ⎥⎥ ! # $" ⎢F 1,0 (ϕ, p) + ⎢ ⎢ ⎥⎥ ⎢ χ ϕ ⎣ ⎦⎥ 0 ⎢ −λ χ −⎢ ⎥ h (χ h) ⎥ ⎢ ⎦ ⎣

.

G1χ (h) − (χ ϕ) · ∇θe

(2.230)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

133

and multiply across by .eτ ψ(x) to get 0 $ ! # $" ! # $" #/ τ ψ(x) (χ ϕ) · ∇ ye ϕ ϕ e eτ ψ(x) Δ χ = (eτ ψ(x) ye · ∇) χ + 0 h h ⎡ ⎤⎤ ⎡ 0 ⎢ .. ⎥⎥ ⎢ . ⎥⎥ ⎢F 1,0 (ϕ, p) + ⎢ ! # $" $ # τ ψ(x) ⎢ ⎥⎥ ⎢ χ ∇(χp) ϕ e ⎣ ⎦⎥ . τ ψ(x) τ ψ(x) ⎢ 0 χ − λe + −e ⎢ ⎥ 0 h (χ h) ⎥ ⎢ ⎣ ⎦

.

G1χ (h) − (χ ϕ) · ∇θe (2.231) Recalling .ye ∈ (W 2,q (Ω))d , θe ∈ W 2,q (Ω) by Theorem 2.6.1, as well as the embedding .W 1,q (Ω) ͨ→ C(Ω) for .q > d, [3, p 97, for .Ω having cone property] [51, p. 79, requiring .C 1 -boundary], we have .|∇ye (x)| + |∇θe (x)| ≤ Cye ,θe , x ∈ Ω, for .q > d, as assumed. In view of this, we return to (2.231) and obtain

e

.

! # $" 2 % ! # $" 2 2 ϕ ϕ 2τ ψ(x) ∇ χ (x) + (χ ϕ)(x) Δ χ h (x) ≤ ce e h

2τ ψ(x)

⎡ ⎤ F 1,0 (ϕ, p)(x) 2 ! # $" 2 χ ϕ 2 ⎦ , (x) +e2τ ψ(x) ∇(χp)(x) +e2τ ψ(x) ⎣ + cλ e2τ ψ(x) χ h G1 (h)(x) χ

x ∈ G,

(2.232)

ce = a constant depending on .ye and .θe , .cλ = |λ|2 + 1. Thus, # $integrating (2.232) ϕ over .G ≡ [Ω1 ∪ Ω∗ ] as required by (2.229) yields with .u = h

.



 e2τ ψ(x) |Δ(χ u)(x)|2 dx ≤ Cλ,e

.

Ω1

∪Ω∗

 +

Ω1 ∪Ω∗

e2τ ψ(x) [|∇(χ u)(x)|2 +|(χ u)(x)|2 ]dx

Ω1 ∪Ω∗

e2τ ψ(x) |∇(χp)(x)|2 dx

⎡ ⎤ F 1,0 (ϕ, p)(x) 2 χ ⎦ dx, + e2τ ψ(x) ⎣ Ω1 ∪Ω∗ G1χ (h)(x) 

Cλ,e = a constant depending on .λ, ye and .θe .

.

(2.233)

134

R. Triggiani

We now recall from (2.219) and (2.222) that .Fχ1,0 (ϕ, p) is an operator that is first order in .ϕ and zero order in p, while .G1χ (h) is first order in h; and moreover, that their support is in .Ω∗ : supp Fχ1,0 ⊂ Ω∗ , supp G1χ ⊂ Ω∗ . Thus, (2.233) becomes # $ ϕ explicitly, still with .u = : h  .

 e2τ ψ(x) |Δ(χ u)(x)|2 dx ≤ Cλ,e

e2τ ψ(x) [∇(χ u)(x)|2 +|(χ u)(x)|2 ]dx

Ω1 ∪Ω∗

Ω1 ∪Ω∗

 +

Ω1 ∪Ω∗

e2τ ψ(x) |∇(χp)(x)|2 dx

 + cχ

Ω∗

e2τ ψ(x) [|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx,

(2.234)

which is the sought-after bound on the last term of the RHS of (2.229). In (2.234), cχ is a constant depending on .χ . # $ ϕ .) We substitute (2.234) Step 9. (Final estimate for .(χ u)-problem (2.223), .u = h into the RHS of inequality (2.229) and obtain

.

.

# $ 1 ρτ − e2τ ψ(x) |∇(χ u)(x)|2 dx 8 Ω1 ∪Ω∗   + 2ρk 2 τ 3 + O(τ 2 )

Ω1 ∪Ω∗

e2τ ψ(x) |(χ u)(x)|2 dx

 ≤ Cλ,e

Ω1 ∪Ω∗

e2τ ψ(x) [|∇(χ u)(x)|2 + |(χ u)(x)|2 ]dx

 + 3

Ω1 ∪Ω∗

e2τ ψ(x) |∇(χp)(x)|2 dx

 + cχ

Ω∗

e2τ ψ(x) [|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx.

(2.235)

Moving the first integral term on the RHS of inequality (2.235) to the LHS of such inequality then yields for .τ sufficiently large:

.

 # $ 1 ρτ − − Cλ,e e2τ ψ(x) |∇(χ u)(x)|2 dx ∗ 8 Ω1 ∪Ω

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

  + 2ρk 2 τ 3 + O(τ 2 ) − Cλ,e Ω1

∪Ω∗

135

e2τ ψ(x) |(χ u)(x)|2 dx

 ≤3 Ω1

∪Ω∗

e2τ ψ(x) |∇(χp)(x)|2 dx

 + cχ

Ω∗

e2τ ψ(x) [|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx.

(2.236)

Inequality (2.236) is our final estimate for the .(χ u)-problem in (2.223), (2.219), (2.222). Step 10. The .(χp)-problem. We need to estimate the first integral term on the RHS of inequality (2.236). This will be accomplished in (2.244). To this end, we need to obtain preliminarily the PDE problem satisfied by .(χp) on .G ≡ Ω1 ∪ Ω∗ . This task will be accomplished in this step. Accordingly, we return to the .ϕ—Eq. (2.214a) = (2.210a), take here the operation of “div” across, use .divϕ ≡ 0 from (2.214b) = (2.210c), and obtain, recalling .Le (ϕ) in (2.208d), ⎤ 0 ⎢ .. ⎥ ⎢ . ⎥ ⎥ ⎢ .Δp = −divLe (ϕ) + ⎢ 0 ⎥ in Ω, ⎥ ⎢ ⎣ ∂h ⎦ ∂xd ⎡

(2.237a)

where, actually [101, Eq. (5.21)], [102, Eq. (3.24)], divLe (ϕ) = 2{(∂x ye · ∇)ϕ} = 2{∂x ϕ · ∇)ye }

.

is a first-order differential operator in ϕ.

(2.237b)

The proof of (2.237b) uses .divϕ ≡ 0 and .divye ≡ 0 in .Ω from (2.214b) = (2.210c) and (2.189c). Next, multiply (2.237a) by .χ . We obtain ⎧ ⎡ ⎤ ⎪ ⎪ 0 ⎪ ⎪ .. ⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ⎪ . ⎪ ⎢ ⎥ ⎪ ⎪ ⎥ + T 0,0,1 (ϕ, h, p) in Ω; ⎪ . Δ(χp) = −divLe (χ ϕ) + ⎢ ⎪ 0 χ ⎢ ⎥ ⎨ ⎢ ⎥ ⎣ ∂ ⎦ ⎪ (χ h) ⎪ ⎪ ∂xd ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂(χp) ⎪ ⎪ ⎪ = 0, (χp)|D = 0, D = ∂[Ω1 ∪ Ω∗ ]. ⎩ ∂ν D

(2.238a)

(2.238b)

136

R. Triggiani

⎤ 0 . .. ⎥ ⎢ ⎥ ⎢ 0,0,1 ⎢ Tχ (ϕ, h, p) ≡ [Δ, χ ]p + [divLe , χ ]ϕ + ⎢ 0 $ ⎥ ⎥ = zero order in ϕ # ⎦ ⎣ . ∂ χ, h ∂xd ⎡

by (2.237b); zero order in h; first order in p; supp Tχ0,0,1 ⊂ Ω∗ , (2.238c) while the B.C.s (2.238b) on the boundary D defined by (2.224) follow for 2 reasons: (i) the RHS of (2.225) on .(χp) on .∂ω; actually, the RHS of (2.215) since .χ ≡ 1 on ∗ ∗ .ω; (ii) .χ ≡ 0 up to the external boundary of .Ω and a small layer of .Ω bordering ∂(χp) ∗ .Ω0 , so that .(χp) = 0, . ∂ν = 0, on such external boundary of .Ω . Thus, (2.238b) is justified. In (2.238b), the reason for supp .Tχ0,0,1 ⊂ Ω∗ is the same as in (2.219c) and (2.222b). Next, we apply the pointwise Carleman estimate (2.226) to problem (2.238a)– (2.238b), that is for .w = (χp). We obtain with .G = Ω1 ∪ Ω∗ :   ϵ δ0 2ρτ − e2τ ψ(x) |∇(χp)(x)|2 + 4ρk 2 τ 3 (1−δ0 )+O(τ 2 ) e2τ ψ(x) |(χp)(x)|2 2

.

" ! 1 2τ ψ(x) e |Δ(χp)(x)|2 + divV(χp) (x), ≤ 1+ ϵ

x ∈ G.

(2.239)

Again, it is not critical to recall what div .V(χp) (x) is, only the vanishing relationship (2.227) (for .w = (χp)) on an appropriate bounded domain G. Indeed, we shall take again .G = Ω1 ∪ Ω∗ , integrate inequality (2.239) over with .G ≡ Ω1 ∪ Ω∗ (after selecting again .δ0 = 21 , .ϵ = 12 ), and obtain # $ 1 . ρτ − e2τ ψ(x) |∇(χp)(x)|2 dx 8 Ω1 ∪Ω∗   + 2ρk 2 τ 3 + O(τ 2 ) e2τ ψ(x) |(χp)(x)|2 dx Ω1 ∪Ω∗

 ≤3



Ω1 ∪Ω∗

e

2τ ψ(x)

|Δ(χp)(x)| dx + 2

∂[Ω1 ∪Ω∗ ]

:· ν dD,  (x) V(χp)  

(2.240)

where, on the RHS of (2.240), the boundary integral over .D ≡ ∂[Ω1 ∪ Ω∗ ] = [∂ω ∪ external boundary of .Ω∗ ], see (2.224) and Fig. 2.4, again vanishes in view of (2.238b) for .w = (χp). Thus, the vanishing of the last integral term of (2.240) is justified. Step 11. Here we now estimate the last integral term on the RHS of (2.240).

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

137

We multiply Eq. (2.238a) by .eτ ψ(x) , thus obtaining ⎡ e

.

τ ψ(x)

Δ(χp) = −e

τ ψ(x)

divLe (χ ϕ) + e

⎤

0 .. . 0

⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ + eτ ψ(x) Tχ0,0,1 (ϕ, h, p) ⎢ ⎥ ⎣ ∂ ⎦ (χ h) ∂xd (2.241)

τ ψ(x) ⎢

 e2τ ψ(x) |Δ(χp)(x)|2 ≤ ce2τ ψ(x) |divLe (χ ϕ)(x)|2

.

2

∂ 0,0,1 2 (χ h)(x) + |Tχ (ϕ, h, p)(x)| , + ∂x

x ∈ G.

d

(2.242)

We now integrate (2.242) over .G ≡ [Ω1 ∪ Ω∗ ]. In doing so, we recall from (2.237b) that .[divLe ] is a first-order operator, and accordingly, from (2.238c), that 0,0,1 (ϕ, h, p) is an operator that is zero order in .ϕ and h; and first order in p; .Tχ and that .Tχ0,0,1 (ϕ, h, p) has support in .Ω∗ . We thus obtain from (2.242)  .

Ω1 ∪Ω∗

e2τ ψ(x) |Δ(χp)(x)|2 dx &



≤ Cy e

Ω1 ∪Ω∗

 + Cχ

Ω∗

e

2τ ψ(x)

2 ' ∂ (χ h)(x) dx |∇(χ ϕ)(x)| + |(χ ϕ)(x)| + ∂x 2

2

d

 e2τ ψ(x) |∇p(x)|2 + |p(x)|2 + |ϕ(x)|2 + |h(x)|2 dx

(2.243)

with constant .Cχ depending on .χ . Step 12. (Final estimate of the .(χp)-problem.) We now substitute (2.243) into the RHS of (2.240), divide across by .[ρτ − 18 ] > 0 for .τ large, and obtain  .

Ω1

≤(

∪Ω∗

e

2τ ψ(x)

1 8

)

 Ω1

∪Ω∗

e2τ ψ(x) |(χp)(x)|2 dx

&



Cy e ρτ −

[2ρk 2 τ 3 + O(τ 2 )]  |∇(χp)(x)| dx+ ρτ − 18 2

Ω1 ∪Ω∗

e

2τ ψ(x)

2 ' ∂ (χ h)(x) dx |∇(χ ϕ)(x)| + |(χ ϕ)(x)| + ∂x 2

2

d

138

R. Triggiani

+ (



Cχ ρτ −

1 8

)

Ω∗

 e2τ ψ(x) |∇p(x)|2 + |p(x)|2 + |ϕ(x)|2 + |h(x)|2 dx. (2.244)

Inequality (2.244) is our final estimate on the .(χp)-problem (2.238a). Step 13. (Combining the (.χ u)-estimate (2.236) with the (.χp)-estimate (2.244)) We return to estimate (2.236) and add to each side the term [2ρk 2 τ 3 + O(τ 2 )]  . ρτ − 18

 Ω1 ∪Ω∗

e2τ ψ(x) |(χp)(x)|2 dx

to get #

$

 1 e2τ ψ(x) |∇(χ u)(x)|2 dx − Cλ,e ρτ −. 8 Ω1 ∪Ω∗ 



+ 2ρk τ + O(τ ) − Cλ,e 2 3

2

Ω1 ∪Ω∗

e2τ ψ(x) |(χ u)(x)|2 dx

5

6 2ρk 2 τ 3 + O(τ 2 )  + e2τ ψ(x) |(χp)(x)|2 dx 1 ∗ Ω ∪Ω 1 ρτ − 8  ≤3

Ω1 ∪Ω∗

e2τ ψ(x) |∇(χp)(x)|2 dx

5 6 2ρk 2 τ 3 + O(τ 2 )  + e2τ ψ(x) |(χp)(x)|2 dx 1 ∗ Ω ∪Ω 1 ρτ − 8  + cχ

Ω∗

e2τ ψ(x) [|∇u(x)|2 + |u(x)|2 + |p(x)|2 ]dx.

(2.245)

Next, we substitute inequality (2.244) for the first two integral terms on the RHS of (2.245) and obtain .

$ #

 1 e2τ ψ(x) |∇(χ u)(x)|2 dx − Cλ,e ρτ − 8 Ω1 ∪Ω∗ +

  2ρk 2 τ 3 + O(τ 2 ) − Cλ,e

Ω1 ∪Ω∗

e2τ ψ(x) |(χ u)(x)|2 dx

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

139

6 2ρk 2 τ 3 + O(τ 2 )  + e2τ ψ(x) |(χp)(x)|2 dx 1 ∗ Ω ∪Ω 1 ρτ − 8 5

≤(

ρτ −

+ (

1 8

)

Ω1 ∪Ω∗



Cχ ρτ − 

+ cχ

&



C ye

Ω∗

1 8

)

Ω∗

e2τ ψ(x)

2 ' ∂ (χ h)(x) dx |∇(χ ϕ)(x)|2 + |(χ ϕ)(x)|2 + ∂x d

 e2τ ψ(x) |∇p(x)|2 + |p(x)|2 + |ϕ(x)|2 + |h(x)|2 dx

 e2τ ψ(x) |∇u(x)|2 + |u(x)|2 + |p(x)|2 dx.

Recalling that .u =

(2.246)

# $ ϕ , we rewrite (2.246) explicitly as h

$ #

  1 . e2τ ψ(x) |∇(χ ϕ)(x)|2 + |∇(χ h)(x)|2 dx − Cλ,e ρτ − 8 Ω1 ∪Ω∗ +

  2ρk 2 τ 3 + O(τ 2 ) − Cλ,e

Ω1 ∪Ω∗

 e2τ ψ(x) |(χ ϕ)(x)|2 + (χ h)(x)|2 dx

5 6 2ρk 2 τ 3 + O(τ 2 )  + e2τ ψ(x) |(χp)(x)|2 dx Ω1 ∪Ω∗ ρτ − 18 ≤(

ρτ −

+ (

1 8

)

Ω1 ∪Ω∗



Cχ ρτ −

 + cχ

Ω∗

&



C ye

1 8

)

Ω∗

e

2τ ψ(x)

2 ' ∂ (χ h)(x) dx |∇(χ ϕ)(x)| + |(χ ϕ)(x)| + ∂x 2

2

d

 e2τ ψ(x) |∇p(x)|2 + |p(x)|2 + |ϕ(x)|2 + |h(x)|2 dx

 e2τ ψ(x) |∇ϕ(x)|2 + |∇h(x)|2 + |ϕ(x)|2 + |h(x)|2 + |p(x)|2 dx. (2.247)

Step 14. (Final estimate of problem (2.214a)–(2.214b).) Finally, we combine the integral terms with the same integrand on the LHS of (2.247) and obtain the final sought-after estimate which we formalize as a lemma.

140

R. Triggiani

Lemma 2.6.1 The following inequality holds true for all .τ sufficiently large: ⎧ ⎫ $  ⎨#  Cy e ⎬ 1 ) ρτ − −Cλ,e − ( e2τ ψ(x) |∇(χ ϕ)(x)|2 + |∇(χ h)(x)|2 dx . ⎩ 8 ρτ − 1 ⎭ Ω1 ∪Ω∗ 8

 + 2ρk 2 τ 3 + O(τ 2 ) − Cλ,e Cy e

) −( ρτ − 18

 Ω1 ∪Ω∗

 e2τ ψ(x) |(χ ϕ)(x)|2 + (χ h)(x)|2 dx

5 6 2ρk 2 τ 3 + O(τ 2 )  + e2τ ψ(x) |(χp)(x)|2 dx 1 ∗ Ω ∪Ω 1 ρτ − 8 ≤(



Cχ ρτ − 

+ cχ

Ω∗

1 8

)

Ω∗

 e2τ ψ(x) |∇p(x)|2 + |p(x)|2 + |ϕ(x)|2 + |h(x)|2 dx

 e2τ ψ(x) |∇ϕ(x)|2 + |∇h(x)|2 + |ϕ(x)|2 + |h(x)|2 + |p(x)|2 dx. (2.248)

We note explicitly 2 critical features of estimate (2.248): The integral terms on its LHS are over .[Ω1 ∪ Ω∗ ], while the integral terms on its RHS are over .Ω∗ . As already noted, (2.248) is the ultimate estimate regarding the original problem (2.214a)–(2.214b). Step 15. (The choice of weight function .ψ(x).) We now choose the strictly convex function .ψ(x) as follows (Figs. 2.6 and 2.7, as well as Fig. 2.5): .

ψ(x) ≥ 0 on Ω1 where χ ≡ 1 by (2.216a), so that e2τ ψ(x) ≥ 1 on Ω1 ; .

(2.249)

ψ(x) ≤ 0 on Ω0 ∪ Ω∗ ; where χ < 1, so that e2τ ψ(x) ≤ 1 on Ω∗ ,

(2.250)

in such a way that .ψ(x) has no critical point in .Ω \ ω, as required by Theorem 2.6.4 (.ψ no critical points on .G = Ω1 ∪ Ω∗ ): that is, the critical point(s) of .ψ will fall on ∗ .ω, outside the region .G = Ω1 ∪ Ω , where we have integrated. Having chosen .ψ(x) as in (2.249), (2.250) with no critical points in .Ω \ ω—i.e., no critical points on .G = Ω1 ∪ Ω∗ —we return to the basic estimate (2.248), with .τ sufficiently large (Fig. 2.7). On the LHS of (2.248), we retain only integration over 2τ ψ ≥ 1 and .χ ≡ 1 by (2.216a), so that .(χ u) ≡ u on .Ω1 , where .ψ ≥ 0; hence, .e .Ω1 , that is, .(χ ϕ) ≡ ϕ on .Ω1 and .(χ h) ≡ h on .Ω1 . On the RHS of (2.248), we

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

141

have .ψ ≤ 0 on .Ω∗ ; hence, .e2τ ψ ≤ 1 on .Ω∗ . We thus obtain from (2.248) for .τ sufficiently large ⎫ ⎧ $ ⎬  ⎨# Cy e 1 ) ρτ − − Cλ,e − ( |∇ϕ(x)|2 + |∇h(x)|2 dx . ⎩ 1 ⎭ Ω1 8 ρτ − 8 ⎧ ⎫   ⎨ 6Cye ⎬ ) 2ρk 2 τ 3 + O(τ 2 ) − Cλ,e − ( |ϕ(x)|2 + |h(x)|2 dx + ⎩ 1 ⎭ Ω1 ρτ − 8 5 6 2ρk 2 τ 3 + O(τ 2 )  + |p(x)|2 dx 1 Ω1 ρτ − 8 ≤(



Cχ ρτ −

1 8

)

 + cχ

Ω∗

 Ω∗

 |∇p(x)|2 + |p(x)|2 + |ϕ(x)|2 + |h(x)|2 dx

|∇ϕ(x)|2 + |∇h(x)|2 + |ϕ(x)|2 + |h(x)|2 + |p(x)|2 dx. (2.251)

For .τ sufficiently large, inequality (2.251) is of the type .

≤

"  ! 1 τ − const − |∇ϕ(x)|2 + |∇h(x)|2 dx τ Ω1 "  !  1 |ϕ(x)|2 + |h(x)|2 dx + (τ 2 ) + τ 3 − const − |p(x)|2 dx τ Ω1 Ω1 c τ

 Ω∗

 |∇p(x)|2 + |p(x)|2 + |ϕ(x)|2 + |h(x)|2 dx 

+ const

Ω∗

 |∇ϕ(x)|2 + |∇h(x)|2 + |ϕ(x)|2 + |h(x)|2 + |p(x)|2 dx (2.252a)

or setting as usual .u = {ϕ, h}, we rewrite (2.252a) equivalently as .

! " 1 τ − const − |∇u(x)|2 dx τ Ω1 " !  1 |u(x)|2 dx + (τ 2 ) |p(x)|2 dx + τ 3 − const − τ Ω1 Ω1

142

R. Triggiani

≤

c τ

 Ω∗

 |∇p(x)|2 + |p(x)|2 + |u(x)|2 dx 

+ const ≤



Ω∗

|∇u(x)|2 + |u(x)|2 + |p(x)|2 dx.

c C1 (p, u; Ω∗ ) + const C2 (p, u; Ω∗ ). τ

(2.252b)

(2.252c)

In going from (2.252b) to (2.252c), we have emphasized in the notation that we are working with a fixed solution .{u, p} of problem (2.214a)–(2.214b), so that the integrals on the RHS of (2.252b) are fixed numbers .C1 (p, u; Ω∗ ) and ∗ ∗ .C2 (p, u; Ω ), depending on such fixed solution .{u, p} as well as .Ω , .u = {ϕ, h}. Inequality (2.252) is more than we need. On its LHS, we may drop the .∇u-term over .Ω1 ; and alternatively either keep only the u-term over .Ω1 , and divide the remaining inequality across by .(τ 3 − const − τ1 ) for .τ large; or else keep only the p-term over 2 .Ω1 and divide the corresponding inequality across by .τ . We obtain, respectively, 

!

" C 1 const . C1 (p, u; Ω∗ ) + 3 C2 (p, u; Ω∗ ) → 0, (2.253) τ3 τ τ

!

" C 1 const C1 (p, u; Ω∗ ) + 2 C2 (p, u; Ω∗ ) → 0 τ2 τ τ

|u(x)|2 dx ≤

.

Ω1

 |p(x)|2 dx ≤ Ω1

(2.254)

as .τ → +∞. We thus obtain u(x) ≡ {ϕ(x), h(x)} ≡ 0 in Ω1 ;

.

p(x) ≡ 0 in Ω1 ,

(2.255)

and recalling (2.214b) and Step 1 u(x) ≡ {ϕ(x), h(x)} ≡ 0,

.

p(x) ≡ 0 in ω ∪ Ω1 .

(2.256)

# $ ϕ . h Finally, we can now push the external boundary of .Ω1 as close as we please to the boundary .∂Ω of .Ω, and thus we finally obtain The implication: Step 1 .=⇒ (2.256) is illustrated by Fig. 2.8, with .u =

u(x) ≡ {ϕ(x), h(x)} ≡ 0 in Ω,

.

p(x) ≡ 0 in Ω.

(2.257)

Indeed, we have .u ≡ {ϕ, h} ∈ (W 2,q (Ω))d × (W 1,q (Ω))d and .p ∈ W 1,q (Ω). Moreover, .W 1,q (Ω) ͨ→ C(Ω) for .q > d [51, p. 78] as assumed, and more generally .W m,q (Ω) ͨ→ C k (Ω) for .qm > d, .k = m − dq [51, p. 79]. A fortiori, d .u ∈ (C(Ω)) , p ∈ C(Ω), .q > d, as assumed. Thus, if it should happen that

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

143

u(x1 ) /= 0 at a point .x1 ∈ Ω near .∂Ω, hence .u(x) /≡ 0 in a suitable neighborhood N of .x1 , then it would suffice to take .Ω1 as to intersect such N to obtain a contradiction. Theorem 2.6.3 is proved at least in the Case 1 (Figs. 2.4, 2.5, and 2.19).

.

Case 2 Let .ω be a full collar of boundary .𝚪 = ∂Ω (Figs. 2.9 and 2.10). Then, the above proof of Case 1 can be carried out with sets .Ω1 , .Ω∗ , and .Ω0 , as indicated in Fig. 2.9. Let now .ω be a partial collar of the boundary .𝚪 = ∂Ω. Then, the above proof of Case 1 can be carried out with sets .Ω1 , .Ω∗ , and .Ω0 , as indicated in Fig. 2.11. ⨆ ⨅ Remark 2.6.2 The proof in the Riemannian setting is essentially the same mutatis mutandis. The Riemannian version of the critical Theorem 2.6.4 is now available from [105, Cor. 4.2, Eq. (4.12), p. 368], [106, Cor. 4.2, Eq. (4.22), p. 345].

¯ boundary = v} Pair of Feedback 2.7 Localized {interior = u,   [69] Controls, v ∈ 𝚪 ⊂ 𝚪, u¯ ∈ ω, Supported by 𝚪 The controlled model Let .Ω in .Rd be a bounded connected region .Ω in .Rd with sufficiently smooth boundary .𝚪 = ∂Ω. More specific requirements will be given below. Let .Q ≡ (0, T ) × Ω and .Σ ≡ (0, T ) × ∂Ω where .T > 0. Further, let .ω be an arbitrary small open smooth subdomain of the region .Ω, .ω ⊂ Ω, thus of positive measure that is a local collar supported by a corresponding connected arbitrary small 𝚪 of the boundary .𝚪 = ∂Ω, Fig 2.20. portion . Let m denote the characteristic function of ω: m(ω) ≡ 1, m(Ω/ω) ≡ 0. On Ω, we consider the Boussinesq system under the action of a control pair {v, u} localized on { 𝚪 , ω}. Here v is a scalar Dirichlet boundary control for the thermal equation acting on  𝚪 , while u is a d-dimensional vector interior control acting as m(x)u(t, x) on ω: Fig. 2.20 Pair of {w,  𝚪 } for the control strategy

ω

•  Γ

•

Ω

Γ

144

⎧ ⎪ .y t − νΔy + (y · ∇)y − γ (θ − θ¯ )ed + ∇π = m(x)u(t, x) + f (x) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θt − κΔθ + y · ∇θ = g(x) ⎪ ⎪ ⎨ div y = 0 ⎪ ⎪ ⎪ ⎪ ⎪ y = 0, θ = v ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y(0, x) = y 0 , θ (0, x) = θ0

R. Triggiani

in Q

(2.258a)

in Q

(2.258b)

in Q

(2.258c)

on Σ

(2.258d)

on Ω.

(2.258e)

In the Boussinesq approximation system, y = {y1 , . . . , yd } represents the fluid velocity, θ the scalar temperature of the fluid, ν the kinematic viscosity coefficient, and κ the thermal conductivity. The scalar function π is the unknown pressure. g¯ The term ed denotes the vector (0, . . . , 0, 1). Moreover, γ = , where g¯ is the θ¯ acceleration due to gravity and θ¯ is the reference temperature. The d-vector-valued function f (x) and scalar function g(x) correspond to an external force acting on the Navier–Stokes equations and a heat source density acting on the heat equation, respectively. They are given along with the I.C.s y 0 and θ0 , which are assumed of low regularity. Note that y · ∇θ = div (θy). The Boussinesq system models heat transfer in a viscous incompressible heatconducting fluid. It consists of the Navier–Stokes equations (in the vector velocity y) [96] coupled with the convection–diffusion equation (for the scalar temperature θ ). The external body force f (x) and the heat source density g(x) may render the overall system unstable in the technical sense described below. Our starting point is the same Theorem 2.6.1 of the corresponding stationary problem. As in Sect. 2.6, we assume henceforth, for the uniform stabilization problem to be significant, that the equilibrium solution {ye , θe } considered is unstable, in the sense specified below (2.190). The goal is the same as in Sect. 2.6: To exploit the localized controls u on ω and v on  𝚪 , sought to be finite-dimensional and in feedback form, in order to stabilize the overall system near the unstable equilibrium solution {ye , θe }. Their minimal number will be equal to the maximal geometric multiplicity of the unstable eigenvalues. As an additional benefit of our investigation, the feedback fluid component of u will be of reduced dimension (d − 1) rather than d, to include necessarily the d th component of u, while the feedback heat component of v will be 1-dimensional. This is a consequence of the Unique Continuation Inverse Theory type of property expressed by Theorem 2.6.2. The Kalman algebraic condition for the adjoint eigenproblem, or the operator A∗q

As in Sect. 2.6.1, for each i = 1, . . . , M, we denote by {ϕ ij }𝓁ji=1 , {ϕ ∗ij }𝓁ji=1 the normalized, linearly independent eigenfunctions of Aq , respectively, A∗q , say, on W qσ (Ω) ≡ Lqσ (Ω) × Lq (Ω) and

.

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . . '

145

1 1 + ' = 1, q q (2.259)

'

(W qσ (Ω))∗ ≡ (Lqσ (Ω))' × (Lq (Ω))' = Lqσ (Ω) × Lq (Ω),

corresponding to the M distinct unstable eigenvalues λ1 , . . . , λM of Aq and q λ1 , . . . , λM of A∗q , respectively, on W σ (Ω): Aq ϕ ij = λi ϕ ij ∈ D(Aq )

.

1,q

1,q

= [W 2,q (Ω) ∩ W 0 (Ω) ∩ Lqσ (Ω)] × [W 2,q (Ω) ∩ W0 (Ω)]; . A∗q ϕ ∗ij

=

λ¯ i ϕ ∗ij

∈

(2.260)

D(A∗q ) 1,q '

'

'

1,q '

'

= [W 2,q (Ω) ∩ W 0 (Ω) ∩ Lqσ (Ω)] × [W 2,q (Ω) ∩ W0

(Ω)]. (2.261)

It is the adjoint problem (2.261) that is of our interest: We recall from Sect. 2.6.2 #

$ Aq −Cγ : W qσ (Ω) ≡ Lqσ (Ω) × Lq (Ω) ⊃ D(Aq ) = D(Aq ) × D(Bq ) −Cθe −Bq

Aq =

.

1,q

1,q

= (W 2,q (Ω) ∩ W 0 (Ω) ∩ Lqσ (Ω)) × (W 2,q (Ω) ∩ W0 (Ω)) −→ W qσ (Ω). (2.262) &

∗ .Aq

A∗q −Cθ∗e = −Cγ∗ Bq∗

' '

'

: W qσ (Ω) = Lqσ (Ω) × Lq (Ω) ⊃ D(A∗q ) = D(A∗q ) × D(Bq∗ ) 1,q '

'

'

1,q '

'

= (W 2,q (Ω) ∩ W 0 (Ω) ∩ Lqσ (Ω)) × (W 2,q (Ω) ∩ W0

'

(Ω)) → W qσ (Ω), (2.263)

where the adjoints Cθ∗e and Cγ∗ of the operators Cθe and Cγ in (2.196), (2.197) are '

Cθ∗e ψ ∗ = Pq ' (ψ ∗ ∇θe ) ∈ Lqσ (Ω),

.

Cγ∗ ϕ ∗ = −γ (Pq ' ϕ ∗ ) · ed , '

(

'

Cγ∗ ∈ L(Lqσ (Ω), Lq (Ω))..

)

(2.264)

A∗q = − νA∗q + A∗o,q , A∗q f = −Pq ' Δf , 1,q '

'

'

D(A∗q ) ≡ W 2,q (Ω) ∩ W 0 (Ω) ∩ Lqσ (Ω). (2.265) /

Ao.q

0∗

= A∗o,q = Pq ' (Le )∗ : W

1,q '

q'

(Ω) −→ Lσ (Ω).

(2.266)

146

R. Triggiani 1,q '

'

Bq∗ ψ = −κΔψ − y e · ∇ψ, D(Bq∗ ) = W 2,q (Ω) ∩ W0

(Ω).

(2.267)

Notice that in passing from Aq to A∗q the operators Cγ and Cθe switch places with their adjoints. This fact has a key implication on the needed Unique Continuation Property. With ϕ ∗ = [ϕ ∗ , ψ ∗ ], the explicit version of A∗q ϕ = λϕ ∗ is by (2.263) ⎧ ⎨ .A∗q ϕ ∗ − Cθ∗e ψ ∗ = λϕ ∗

(2.268a)

⎩−B ∗ ψ ∗ − C ∗ ϕ ∗ = λψ ∗ , q γ

(2.268b)

while its PDE version is, by virtue of (2.263)–(2.267), ⎧ ∗ ∗ ∗ ∗ ∗ . − νΔϕ + Le (ϕ ) + ψ ∇θe + ∇π = λϕ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −κΔψ ∗ − y e · ∇ψ ∗ − γ ϕ ∗ · ed = λψ ∗

in Ω

(2.269a)

in Ω

(2.269b)

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

div ϕ = 0

in Ω

(2.269c)

ϕ ∗ = 0, ψ ∗ = 0

on 𝚪.

(2.269d)

∗

For simplicity and space constraints, we provide here explicitly only the conceptually and computationally more amenable case, where the operator Auq,N in (2.4) is 0u / semisimple [50] on the space W qσ N of dimension N = 𝓁1 +· · ·+𝓁M . How to treat the general case is known [64–66, 68]. It is much more computationally intensive. Assumption Thus assume henceforth that 0u / Auq,N is semisimple on W qσ N :

.

(2.270)

that is, for the unstable eigenvalues λ1 , . . . , λM , geometric and algebraic multiplicity coincide. ⨆ ⨅ With reference to the direct and adjoint eigenproblem (2.260), (2.261), denote $ ϕij , .ϕ ij = ψij #

&

ϕ ∗ij

' ∗ ϕij = . ψij∗

(2.271)

Introduce the 𝓁i × K matrix Wi and the 𝓁i × K matrix Ui , i = 1, . . . , M 2 2 ⎤ 1 ⎡1 ∗| ∗| f1 , ∂ν ψi1 · · · fK , ∂ν ψi1 𝚪 𝚪  𝚪 𝚪 1 1 2 2 ⎢ f1 , ∂ν ψ ∗ |𝚪 · · · fK , ∂ν ψ ∗ |𝚪 ⎥   ⎢ i2 i2 𝚪 𝚪⎥ ⎥ : 𝓁i × K 〈 , 〉 = 〈 , 〉 q  q '  .. .. .Wi = ⎢ 𝚪 L (𝚪 ),L (𝚪 ) ⎢ ⎥ . . ⎣7 8 8 ⎦ 7 ∗ ∗ f1 , ∂ν ψi𝓁i |𝚪 · · · fK , ∂ν ψi𝓁i |𝚪   𝚪 𝚪 (2.272)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

147

2 2 ⎤ 1 ⎡1 u1 , ϕ ∗i1 ω · · · uK , ϕ ∗i1 ω 1 2 2 1 ∗ ⎥ ⎢ u1 , ϕ ∗ ⎢ i2 ω · · · uK , ϕ i2 ω ⎥ ⎥ : 𝓁i × K 〈 , 〉 ω = 〈 , 〉 q ⎢ . . .Ui = Lσ (ω) .. .. ⎥ ⎢ ⎣7 7 8 8 ⎦ u1 , ϕ ∗i𝓁i · · · uK , ϕ ∗i𝓁i ω

(2.273)

ω

⎤ W1 7 8  ⎢ W2 ⎥ ⎥   ⎢ ∗ .W =  fr , ∂ν ψij |𝚪  = ⎢ . ⎥ ; N×K,  𝚪 ⎣ .. ⎦ ⎡

⎤ U1 7 8  ⎢ U2 ⎥  ⎥  ⎢ U =  ur , ϕ ∗ij  = ⎢ . ⎥ : N×K. ω ⎣ .. ⎦ ⎡

WM

UM (2.274)

Kalman condition We then obtain the following counterpart of [68, Theorem 6.1]. / q 0u Theorem 2.7.1 Assume (2.270); that is, that Auq,N is semisimple on W σ N . The Kalman condition on the controllability of the finite-dimensional unstable projection (Sect. 2.2) holds in case rank [Wi , Ui ] = 𝓁i , i = 1, . . . , M.

(2.275)

.

Indeed, the rank condition (2.275) holds true: In fact, with reference to (2.146), (2.147d), u=

K

.

/ 0u μk (t)uk , uk ∈ W qσ N ⊂ Lqσ (Ω).

(2.276)

k=1

v=

K

νk (t)fk ∈ F ⊂ W

2− q1 ,q

( 𝚪 ),

(2.277)

k=1 1

2− ,q 𝚪 ) with it is possible to select boundary vectors f1 , . . . , fK ∈ F ⊂ W q ( q support on  𝚪 , and interior vectors u1 , . . . , uK ∈ Lσ (ω) with support on ω such that (2.275) holds true, specifically

rank [Wi , Ui ] = 2 2 1 ⎡1 ∗| f , ∂ ψ ∗ | 𝚪 · · · fK , ∂ν ψi1 𝚪  𝚪 1 1 1 ν i1∗ 𝚪 2 2 ⎢ f1 , ∂ν ψ |𝚪 · · · fK , ∂ν ψ ∗ |𝚪   ⎢ i2 i2 𝚪 𝚪 .. .. rank ⎢ ⎢ . . ⎣7 7 8 8 ∗ ∗ | f1 , ∂ν ψi𝓁i |𝚪 · · · fK , ∂ν ψi𝓁 𝚪 i  𝚪

 𝚪

1 2 2 ⎤ 1 u , ϕ∗ · · · uK , ϕ ∗i1 ω 1 1 1 ∗i1 2ω 2 u1 , ϕ i2 ω · · · uK , ϕ ∗i2 ω ⎥ ⎥ ⎥ .. .. ⎥ . . 7 7 8 8 ⎦ u1 , ϕ ∗i𝓁i · · · uK , ϕ ∗i𝓁i ω

= 𝓁i , i = 1, . . . , M.

ω

(2.278)

148

R. Triggiani

Proof This is the counterpart of the proof [67, Theorem 6.1]. As in that paper, the present proof rests on a suitable Unique Continuation Property. This is in fact Theorem 2.6.2 of Sect. 2.6, see below. In general, in seeking that the 𝓁i rows (of length 2K) in matrix (2.278) be linearly independent, we see that the full-rank statement (2.278) will hold true if and only if we can exclude that each of the two sets of vectors 9 .

∗ ∗ ∂ν ψi1 , . . . , ∂ν ψi𝓁 i

:

in Lq ( 𝚪)

9 ∗ : ϕ i1 , . . . , ϕ ∗i𝓁i in Lqσ (ω)

and

(2.279)

is linearly independent with the same linear independence relation in the two cases; that is, if and only if we can establish that we cannot have simultaneously ∗ ∂ν ψi𝓁 = i

𝓁

i −1

.

αj ∂ν ψij∗ in Lq ( 𝚪)

and

j =1

ϕ ∗i𝓁i =

𝓁

i −1

αj ϕ ∗ij in Lqσ (ω)

(2.280)

j =1

with the same constants α1 , . . . , α𝓁i −1 in both expansions [67]. Claim: Statement (2.280) is false By contradiction, suppose that both linear combinations in (2.280) hold true. Define the (d + 1)-vector ϕ ∗ = {ϕ ∗ , ψ ∗ } q (depending on i) on Lσ (Ω) × Lq (Ω) by setting & ' & ' 𝓁 −1 $ 𝓁

i −1 i

αj ϕ ∗ij ϕ ∗i𝓁i ϕ∗ ≡ .ϕ ≡ αj ϕ ∗ij − ϕ ∗i𝓁i , i = 1, . . . , M; q ≥ 2. − = ∗ ψ∗ αj ψij∗ ψi𝓁 i ∗

#

j =1

j =1

(2.281) Thus, in view of (2.280), we obtain ϕ ∗ ≡ 0 in ω,

.

∂ν ψ ∗ | 𝚪 ≡ 0.

(2.282)

Moreover, since ϕ ∗ij = {ϕ ∗ij , ψij∗ } is an eigenvector of the operator A∗q,N corresponding to the (unstable) eigenvalue λ¯ i , then so is ϕ ∗ : A∗q,N ϕ ∗ = λ¯ i ϕ ∗ . Then ϕ ∗ = {ϕ ∗ , ψ ∗ } satisfies the corresponding PDE version of the eigenvector identity as given by (2.269), ⎧ ∗ ∗ ∗ ∗ . − νΔϕ + Le (ϕ ) + ψ ∇θe + ∇π ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −κΔψ ∗ − y e · ∇ψ ∗ − γ ϕ ∗ · ed ⎪ ⎪ div ϕ ∗ ⎪ ⎪ ⎪ ⎩ ϕ ∗ |𝚪 = 0, ψ ∗ |𝚪

= λ¯ i ϕ ∗

in Ω

(2.283a)

= λ¯ i ψ ∗

in Ω

(2.283b)

=0

in Ω

(2.283c)

=0

on 𝚪

(2.283d)

and in addition, the over-determined conditions in (2.282): ϕ ∗ ≡ 0 in ω,

.

∂ν ψ ∗ | 𝚪 ≡ 0.

(2.283e)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

149

Write ϕ ∗ = {ϕ ∗(1) , ϕ ∗(2) , . . . , ϕ ∗(d) } for the d-components of the vector ϕ ∗ . Since ed = {0, . . . , 0, 1}, the property ϕ ∗d = 0 in ω, contained in (2.283e), used in (2.283b) implies:

κΔψ ∗ + y e · ∇ψ ∗ = −λ¯ i ψ ∗

.

∗ ψ ∗ | 𝚪 = 0, ∂ν ψ | 𝚪 =0

in ω

(2.284a)

on  𝚪

(2.284b)

recalling the boundary conditions for ψ ∗ in (2.283d) and (2.283e) for the ψ ∗ problem defined on ω, with  𝚪 ⊂ ∂ω, as in Fig 2.20. It is then a standard result [17], [73, Sect 19, pp 59-61], [44, Vol III, p.3], [53] that the over-determined problem (2.284) implies ψ ∗ ≡ 0 in ω.

.

(2.285)

Then (2.285) and the full strength of (2.283d) give the over-determination ϕ ∗ ≡ 0 in ω, ψ ∗ ≡ 0 in ω

.

(2.286)

for problem (2.283). By means of such over-determination, we can apply the Unique Continuation Property of Theorem 2.6.2 and conclude that ϕ ∗ = {ϕ ∗ , ψ ∗ } ≡ 0 in Ω,

.

π ≡ const. in Ω,

(2.287)

or recalling (2.281) ϕ ∗i𝓁i =

𝓁

i −1

.

αj ϕ ∗ij in W qσ (Ω) = Lqσ (Ω) × Lq (Ω);

(2.288)

j =1

i.e., the set {ϕ ∗i1 , . . . , ϕ ∗i𝓁i } is linearly dependent on W qσ (Ω). But this is false by the very selection of such eigenvectors, see above (2.259). Thus the two conditions in (2.280) cannot hold simultaneously. The claim is proved. Hence, it is possible to select, in infinitely many ways, boundary functions f1 , . . . , fK ∈ F ⊂ Lq (𝚪) for q q ≥ 2 and interior d-vectors u1 , . . . , uK in Lσ (Ω), such that the Kalman algebraic full-rank conditions (2.278) hold true. Indeed, they may be chosen independent of i. Start with (2.278) for ı¯ such that 𝓁ı¯ = K = max {𝓁i , i = 1, . . . , M} yielding f1 , . . . , fK , which are linearly independent. They also work in other i’s. ⨆ ⨅ Remark 2.7.1 The UCP in of Theorem 2.6.2 refers to problem (2.283a)–(2.283c), without the B.C. (2.283d), with the a priori over-determination ψ ∗ ≡ 0 in ω, and {ϕ ∗(1) , . . . , ϕ ∗(d−1) } ≡ 0 in ω, thus not involving the last d-component ϕ ∗(d) ≡ 0. In contrast, in the setting of proving the above Claim, we need ϕ ∗(d) ≡ 0 in ω in order to deduce ψ ∗ ≡ 0 in ω. The present proof on the validity of the rank

150

R. Triggiani

conditions (2.278), ultimately relying on the UCP of Theorem 2.6.2, requires the full strength of the fluid vectors u1 , . . . , u𝓁i , each possessing d-components. No geometrical assumptions are involved for the pair {ω,  𝚪 }, except for ω being an interior subdomain touching the boundary  𝚪 as in Fig 2.20. In [104], we report an improvement requiring only (d − 1) components from the fluid vectors u1 , . . . , u𝓁i to include necessarily the d th components. Remark 2.7.2 We have established Theorem 2.7.1 under the simplifying semisimple assumption (2.270). However, Theorem 2.7.1 holds true in full generality. The corresponding proof is lengthy and technical. It may be given by following the scheme given in [64, 65, 68] for the Navier–Stokes equations. It requires use of the controllability criterion for a finite-dimensional pair {A, B} with A in Jordan canonical form [10, 16]. Remark 2.7.3 In response to a referee’s query, we note that if v is now a Neumann ∂θ control for the thermal equation: = v on Σ in place of (2.258d), then the matrix ∂ν Wi in (2.272) will have the Dirichlet trace ψij∗ | 𝚪 instead of the normal derivative ∂ν ψij∗ | . In the classical parabolic case of Part I, Theorem 2.3.2 and Theorem 2.3.3 𝚪 describe the two cases, Dirichlet and Neumann.

References 1. P. Acevedo, C.Amrouche, C. Conca, Boussinesq system with non-homogeneous boundary conditions. Appl. Math. Lett. 53, 39–44 (2016) 2. P. Acevedo, C.Amrouche, C. Conca, Lp theory for Boussinesq system with Dirichlet boundary conditions. Appl. Anal. 98(1–2), 272–294 (2019) 3. R.A. Adams, Sobolev Spaces (Academic Press, Cambridge, MA, 1975), p. 268 4. H. Amann, Linear and Quasilinear Parabolic Problems: Abstract Linear Theory, vol. I (Birkhauser, Basel, Switzerland, 1995), p. 338 5. H. Amann, On the Strong Solvability of the Navier-Stokes Equations. J. Math. Fluid Mech. 2, 16–98 (2000) 6. C. Amrouche, M.A. Rodriguez-Bellido, Stationary Stokes, Oseen and Navier-Stokes equations with singular data. M. Á. Arch. Ration. Mech. Anal. 597–651 (2011). https://doi.org/10. 1007/s00205-010-0340-8 7. G. Avalos, R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system. J. Evol. Equs. 9(2) 341–370 (2009) 8. A.V. Balakrishnan, Applied Functional Analysis. Applications of Mathematics Series, 2nd edn. (Springer, Berlin/Heidelberg, Germany, 1981), p. 369 9. V. Barbu, Stabilization of Navier–Stokes Flows (Springer, 2011), p. 276 10. G. Basile, G. Marro, Controlled and Conditioned Invariants in Linear System Theory (Prentice Hall, Englewood Cliffs, 1992), p. 464 11. V. Barbu, R. Triggiani, Internal stabilization of Navier-Stokes equations with finitely many controllers. Indiana Univ. Math.J. 53(5), 1443–1494 (2004) 12. V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier-Stokes equations. Mem. AMS 181(852), 128 (2006) 13. V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers. Nonlinear Anal. 64, 2705–2746 (2006)

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14. V. Barbu, I. Lasiecka, R. Triggiani, Local exponential stabilization strategies of the NavierStokes equations, d = 2, 3, via feedback stabilization of its linearization, in Optimal Control of Coupled Systems of Partial Differential Equations, Oberwolfach, ISNM, vol. 155 (Birkhauser, Basel, Switzerland, 2007), pp. 13–46 15. L. Bers, F. John, M. Sclechter, Partial Differential Equations (Interscience Publishers, Geneva, 1964) 16. C.T. Chen, Linear Systems Theory and Design (Oxford University Press, Oxford, UK, 1984), p. 334 17. T. Carleman, Sur le problème d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. 26B, 1–9 (1939) 18. M.D. Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Hoboken, New Jersey, 1976), p. 503 19. P. Constantin, C. Foias, Navier-Stokes Equations (University of Chicago Press, Chicago, London, 1989) 20. E. Davison, S. Wang, On pole assignment in linear multivariable systems using output feedback. IEEE Trans. Autom. Control 20(4), 516–518 (1975) 21. P. Deuring, The Stokes resolvent in 3D domains with conical boundary points: nonregularity in Lp -spaces. Adv. Diff. Equs. 6(2), 175–228 (2001) 22. P. Deuring, W. Varnhorn, On Oseen resolvent estimates. Diff. Integral Equs. 23(11/12), 1139– 1149 (2010) 23. G. Dore, Maximal regularity in Lp spaces for an abstract Cauchy problem. Adv. Diff. Equs. 5(1–3), 293–322 (2000) 24. L. Escauriaza, G. Seregin, V. Šverák, L3,∞ -Solutions of Navier-Stokes Equations and Backward Uniqueness. (1991) Mathematical subject classification (American Mathematical Society, Providence, RI): 35K, 76D 25. E. Fabes, O. Mendez, M. Mitrea, Boundary layers of Sobolev-Besov Spaces and Poisson’s Equations for the Laplacian for the Lipschitz Domains. J. Func. Anal 159(2), 323–368 (1998) 26. C. Fabre, G. Lebeau, Prolongement unique des solutions de l’équation de Stokes. Commun. Part. Diff. Equs. 21(3–4), 573–596 (1996) 27. C. Fabre, G. Lebeau, Regularité et unicité pour le problème de Stokes. Commun. Part. Diff. Equs. 27(3–4), 437–475 (2002) 28. A. Friedman, Partial Differential Equations (reprint) (R.E. Krieger Publication, Huntington, NY, 1976) 29. M. Fu, Pole placement via static output feedback is NP-hard. IEEE Trans. Autom. Control 49(5), 855–857 (2004) 30. A. Fursikov, Real processes corresponding to the 3D Navier-Stokes system, and its feedback stabilization from the boundary, in Partial Differential Equations, American Mathematics Society Translation, Series 2, vol. 260 (AMS, Providence, RI, 2002) 31. A. Fursikov, Stabilizability of two dimensional Navier–Stokes equations with help of a boundary feedback control. J. Math. Fluid Mech. 3, 259–301 (2001) 32. A. Fursikov, Stabilization for the 3D Navier–Stokes system by feedback boundary control. DCDS 10, 289–314 (2004) 33. C. Foias, R. Temam, Determination of the solution of the Navier-Stokes Equations by a set of nodal volumes. Math. Comput, 43(167), 117–133 (1984) 34. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Nonlinear Steady Problems, vol. I (Springer, New York, 1994), p. 465 35. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Linearized Steady Problems, vol. II (Springer, New York, 1994), p. 323 36. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations (Springer, New York, 2011) 37. H. Guggenheimer, Differential Geometry (McGraw-Hill Book Company Inc., New York, 1963), p. 378 3 38. I. Gallagher, G.S. Koch, F. Planchon, A profile decomposition approach to the L∞ t (Lx ) Navier-Stokes regularity criterion. Math. Ann. 355(4), 1527–1559 (2013)

152

R. Triggiani

39. M. Geissert, K. Götze, M. Hieber, Lp -theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids. Trans. Am. Math. Soc. 365(3), 1393–1439 (2013) 40. Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr spaces. Math.Z. 178(3), 279–329 (1981) 41. Y. Giga, Domains of fractional powers of the Stokes operator in Lr spaces. Arch. Ration. Mech. Anal. 89(3), 251–265 (1985) 42. M. Hieber, J. Saal, The Stokes Equation in the Lp -setting: well posedness and regularity properties, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (Springer, Cham, 2016), pp. 1–88 43. L. Hörmander, Linear Partial Differential Operators (Springer, Berlin/Heidelberg, Germany, 1969) 44. L. Hörmander, The Analysis of Linear Partial Differential Operators III (Springer, Berlin/Heidelberg, Germany, 1985) 45. V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edn. (Springer, Berlin/Heidelberg, Germany, 2006) 46. H. Jia, V. Šverák, Minimal L3 -initial data for potential Navier-Stokes singularities. SIAM J. Math. Anal. 45(3), 1448–1459 (2013) 47. A. Jones Don, E.S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes Equations. Indiana Univ. Math. J. 42(3), 875–887 (1993). www.jstor.org/stable/24897124 48. R.E. Kalman, Contributions to the theory of optimal controls. Bol. Soc. Mat. Mexicana 5(2), 102–119 (1960) 49. R.E. Kalman, On the general theory of control systems, in Proceedings First International Conference on Automatic Control, Moscow, USSR (1960), pp. 481–492 50. T. Kato, Perturbation Theory of Linear Operators (Springer, Berlin/Heidelberg, Germany, 1966), p. 623 51. S. Kesavan, Topics in Functional Analysis and Applications (New Age International Publisher, Delhi, India, 1989), p. 267 52. M. Kimura, Pole assignment by gain output feedback. IEEE Trans. Autom. Control 20(4), 509–516 (1975) 53. V. Komornik, Exact Controllability and Stabilization—The Multiplier Method (Masson, Paris, 1994) 54. V.A. Kondratiev, Boundary problems for elliptic equations in domains with conical or angular points. Transl. Moscow Math. Soc. 16, 227–313 (1967) 55. P.C. Kunstmann, L. Weis Maximal Lp -regularity for parabolic equations, fourier multiplier theorems and H ∞ -functional calculus, in Functional Analytic Methods for Evolution Equations, Lecture Notes in Mathematics, vol. 1855 (Springer, Berlin/Heidelberg, 2004), pp. 65–311 56. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. (Gordon and Breach, New York, 1969). English translation 57. J. Leray, Sur le mouvement d’un liquide visquex emplissent l’espace. Acta Math. J. 63, 193– 248 (1934) 58. J.L. Lions, Quelques Methodes de Resolutions des Problemes aux Limites Non Lineaire (Dunod, Paris, 1969) 59. I. Lasiecka, R. Triggiani, Structural assignment of Neumann boundary feedback parabolic equations: the case of trace in the feedback loop. Annali Matem. Pura Appl. (IV) 132, 131– 175 (1982) 60. I. Lasiecka, R. Triggiani, Stabilization of Neumann boundary feedback parabolic equations: The case of trace in the feedback loop. Appl. Math. Optim. 10, 307–350 (1983). (Preliminary version in Springer Lecture Notes 54, 238–246 (1983)) 61. I. Lasiecka, R. Triggiani, Stabilization and structural assignment of Dirichlet boundary feedback parabolic equations. SIAM J. Control Optim. 21, 766–803 (1983)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

153

62. I. Lasiecka, R. Triggiani, Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations. J. Diff. Eqns. 47, 246–272 (1983) 63. I. Lasiecka, R. Triggiani, Control theory for partial differential equations: continuous and approximation theories, vol. 1, in Encyclopedia of Mathematics and Its Applications Series (Cambridge University Press, Cambridge, UK, 2000) 64. I. Lasiecka, R. Triggiani, Uniform stabilization with arbitrary decay rates of the Oseen equation by finite-dimensional tangential localized interior and boundary controls, in Semigroups of Operators -Theory and Applications (Springer, Berlin/Heidelberg, 2015), pp. 125–154 65. I. Lasiecka, R. Triggiani, Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers, in Nonlinear Analysis (Elsevier, Amsterdam, 2015), pp. 424–446 66. I. Lasiecka, B. Priyasad, R. Triggiani, Uniform stabilization of Navier-Stokes equations in critical Lq -based Sobolev and Besov spaces, by finite dimensional interior localized feedback controls. Appl. Math. Optim. 83(3), 1765–1829 (2021) 67. I. Lasiecka, B. Priyasad, R. Triggiani, Uniform stabilization of 3-d Navier-Stokes Equations in critical Besov spaces with finite dimensional, tangential-like boundary, localized feedback controllers. Arch. Ration. Mech. Anal. 241, 1575–1654 (2021) 68. I. Lasiecka, B. Priyasad, R. Triggiani, Uniform stabilization of Boussinesq systems in critical Lq -based Sobolev and Besov spaces by finite dimensional, interior, localized feedback controls. DCDS-B 25(10), 4071–4117 (2020) 69. I. Lasiecka, B. Priyasad, R. Triggiani, Finite dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory. J. Inverse Ill-Posed Prob. 30(1), 35–79 (2022). https://doi.org/10.1515/jiip-2020-0132 70. I. Lasiecka, B. Priyasad, R. Triggiani, Maximal Lp -regularity for an abstract evolution equation with applications to closed-loop boundary feedback control problems. J. Diff. Equs. 294, 60–87 (2021) 71. I. Lasiecka, R. Triggiani, Z. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: global uniqueness and observability. AMS Contemp. Math. 268, 227–326 (2000) 72. I. Lasiecka, R. Triggiani, Z. Zhang, Global uniqueness, observability and stabilization of non-conservative Schrödinger equations via pointwise Carleman estimates. Part I: H 1 (Ω)estimates. J. Inverse Ill-Posed Prob. 12(1), 1–81 (2004) 73. C. Miranda, Partial Differential Equations of Elliptic Type (Springer, Berlin/Heidelberg, Germany, 1970) 74. V. Maslenniskova, M. Bogovskii, Elliptic boundary values in unbounded domains with non compact and non smooth boundaries. Rend. Sem. Mat. Fis. Milano 56, 125–138 (1986) 75. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, 1983) 76. J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics vol. 105 (Birkhüuser Basel, Basel, Switzerland, 2016), p. 609 77. M. Ramaswamy, J.-P. Raymond, A. Roy, Boundary feedback stabilization of the Boussinesq system with mixed boundary conditions. J. Diff. Equs. 266(7), 4268–4304 (2019) 78. W. Rusin, V. Sverak, Minimal initial data for potential Navier-Stokes singularities. J. Funct. Anal. 260(3), 879-891 (2009). https://arxiv.org/abs/0911.0500 79. J.P. Raymond, Feedback boundary stabilization of the two dimensional Navier-Stokes equations. SIAM J. Control 45, 790–828 (2006) 80. J. Saal, Maximal regularity for the Stokes system on non-cylindrical space-time domains. J. Math. Soc. Japan 58(3), 617–641 (2006) 81. C. Sadosky, Interpolation of Operators and Singular Integrals (Marcel Dekker, New York City, 1979), p. 375 82. C. Schneider, Traces of Besov and Triebel-Lizorkin spaces on domains. Math. Nach. 284(5– 6), 572–586 (2011) 83. G. Schmidt, N. Weck, On the boundary behavior of solutions to parabolic equations. SIAM J. Control Optim. 16, 533–598 (1978)

154

R. Triggiani

84. J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9, 187 (1962). https://doi.org/10.1007/BF00253344 85. J. Serrin, The initial value problem for the Navier-Stokes equations, 1963 in Nonlinear Problems, Proceedings of Symposia, Madison, Wisconsin (1962), pp. 69–98. University of Wisconsin Press, Madison, Wisconsin. 35.79 86. V.A. Solonnikov, tEstimates of the solutions of a nonstationary linearized system of NavierStokes equations. A.M.S. Transl. 75, 1–116 (1968) 87. V.A. Solonnikov, Estimates for solutions of non-stationary Navier-Stokes equations. J. Sov. Math. 8, 467–529 (1977) 88. V.A. Solonnikov, On the solvability of boundary and initial-boundary value problems for the Navier-Stokes system in domains with noncompact boundaries. Pac. J. Math. 93(2), 443–458 (1981). https://projecteuclid.org/euclid.pjm/1102736272 89. V.A. Solonnikov, On Schauder estimates for the evolution generalized Stokes problem. Ann. Univ. Ferrara 53, 137–172 (1996) 90. V.A. Solonnikov, Lp -estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain. J. Math. Sci. 105(5), 2448–2484 (2001) 91. H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach (Modern Birkhauser Classics, Basel, Switzerland, 2001), p. 377 92. J. Stoker, Differential Geometry (Wiley-Interscience, Hoboken, NJ, 1969), p. 404 93. Y. Shibata, S. Shimizu, On the Lp − Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain. Adv. Stud. Pure Math. 47, 347–362 (2007) 94. J. Sokolowski, J.P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Series in Computational Mathematics, vol. 16 (Springer, Berlin, 1992), p. 250 95. A.E. Taylor, D. Lay, Introduction to Functional Analysis, 2nd edn. (Wiley Publication, 1980). ISBN-13: 978-0471846468 96. R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications, vol. 2 (North-Holland Publishing Company, Amsterdam, 1979) 97. R. Triggiani, On the stabilizability problem of Banach spaces. J. Math. Anal. Appl. 55, 303– 403 (1975) 98. R. Triggiani, On Nambu’s boundary stabilizability problem for diffusion processes. J. Diff. Eqns. 33, 189–200 (1979) 99. R. Triggiani, Well-posedness and regularity of boundary feedback parabolic systems. J. Diff. Eqns. 36, 347–362 (1980) 100. R. Triggiani, Boundary feedback stabilizability of parabolic equations. Appl. Math. Optimiz. 6, 201–220 (1980) 101. R. Triggiani, Linear independence of boundary traces of eigenfunctions of elliptic and Stokes Operators and applications, invited paper for special issue. Appl. Math. 35(4), 481–512 (2008). Institute of Mathematics, Polish Academy of Sciences 102. R. Triggiani, Unique continuation from an arbitrary interior subdomain of the variable coefficient Oseen equation. Nonlinear Anal. Theory Appl. 71(10), 4967–4976 (2009) 103. R. Triggiani, Unique continuation of boundary over-determined Stokes and Oseen problems, Spring 2008. Discrete Cont. Dyn. Syst., Series 5 2(3), 645–678 (2009) 104. R. Triggiani, X. Wan, Unique continuation properties of over-determined static Boussinesq Eigenproblems with application to uniform stabilization of dynamic Boussinesq Systems. Appl. Math. Optim. 84, 2099–2146 (2021) 105. R. Triggiani, X. Xu, Pointwise Carleman estimates, global uniqueness, observability and stabilization for non-conservative Schrödinger equations on Riemannian manifolds at the H 1 (Ω)-level (with X. Xu). AMS Contemp. Math. 426, 339–404 (2007) 106. R. Triggiani, P.F. Yao, Carleman estimates with no lower order terms for general Riemannian wave equations. Global uniqueness and observability in one shot. Appl. Math. Optim. 46(2– 3), 331–375 (2002) 107. H. Triebel, Interpolation theory, function spaces, differential operators. Bull. Amer. Math. Soc. (N.S.) 2(2), 339–345 (1980)

2 Unique Continuation Properties of Static Over-determined Eigenproblems:. . .

155

108. W. von Whal, The Equations of Navier-Stokes and Abstract Parabolic Equations (Springer Fachmedien Wiesbaden, Vieweg+Teubner, 1985) 109. L. Weis, A new approach to maximal Lp -regularity, in Evolution Equation and Application of Physical Life Science, Lecture Notes Pure and Applied Mathematics, vol. 215 (Marcel Dekker, New York, 2001), pp. 195–214 110. W. Wonham, On pole assignment in multi-input controllable linear systems. IEEE Trans. Autom. Control 12(6), 660–665 (1967) 111. J. Zabczyk, Mathematical Control Theory: An Introduction (Birkhauser, Basel, Switzerland, 1992)

Chapter 3

Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction I. Lasiecka and J. T. Webster

3.1 Introduction 3.1.1 Motivation and Applications The aim of these lectures is to present a mathematical theory for a class of control problems arising in flow-structure interactions. From the modeling point of view, these are coupled problems of two dynamics at an interface. The equations of interest are an unstable linearization of the compressible Euler equation and a nonlinear plate dynamics. The latter accounts for the effects of possibly large displacements, which are typically modeled by the scalar or vectorial von Karman equations of large deflections. The coupled model is a hybrid interaction between the flow—defined on a 3D flow domain—and the plate or shell—defined on a 2D manifold. The unperturbed flow moves in 3D with normalized velocity .U > 0 and excites the structure through the distributed pressure, acting through traces on the lower dimensional manifold. As a consequence, the structural displacements then perturb the flow. The communication (feedback) between these two dynamics lies in the heart of the mathematical problem. The described interaction is ubiquitous in nature, with a multitude of physical applications. Some central examples arise in Fluid Mechanics, Aerospace Applications, Structural Engineering, and Biological Applications.

I. Lasiecka () Department of Mathematical Sciences, University of Memphis, Memphis, TN, USA Polish Academy of Sciences, IBS, Warsaw, Poland e-mail: [email protected] J. T. Webster University of Maryland, Baltimore County, Baltimore, MD, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Bodnár et al. (eds.), Fluids Under Control, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-031-47355-5_3

157

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• Aerodynamics: Control of the flow-structure instability known as flutter; determination of the flutter speed for projectiles and vehicles in subsonic, supersonic and transonic regimes [12–15, 19, 38, 39, 46, 54, 61] • Large Space Structures: Large, thin, and highly oscillatory structures transported into and acting in space; these include: solar panels, deformable mirrors, and large antennae [16, 64–66, 108] • Medical Science: Applications such as the treatment of snoring and sleep apnea, the flow of blood through the arteries, and the treatment of vascular diseases [30–33, 40, 49–52, 108] • Engineering and Design: Prediction and suppression of the oscillation of bridges and buildings under mechanical loading, earthquakes, and in the presence of wind; post-flutter analysis for the harvesting of mechanical energy through piezoelectric energy harvesters or windmills [16, 20, 56–59, 66, 69, 87, 113, 114, 118, 121, 122, 124, 125] Due to practical importance of this class of problems, there has been voluminous relevant work in the past; this is particularly true at the levels of modeling, experiment, and numerical analysis. Controlling turbulence, oscillations, and flutter is indeed one of the most technologically advanced problems in the modern science of prediction and design. However, in order to do this, one must to understand the relevant problems at the phenomenological level, with guidance from physical and mathematical principles. The starting point will be in selecting relevant models, in particular, models which will reproduce empirically known behaviors and established experimental studies. As is often the case in applied mathematics, models may be corrupted by some “nice” simplifications which permit certain mathematical tools to be applied. This may lead to the development of new mathematical methods but does not always provide a solution to the given physical problem. One of the purposes in these lectures is to illustrate a situation when a certain phenomenology cannot be obtained for—what are seemingly—similar physical models with nice mathematical features. For this very reason one of the most important preliminary steps is a viable attempt at modeling the phenomenon in question—here, we focus on fluttering plates. The modeling may depend on a specific objective one wishes to achieve, and this will become clear from context.

3.1.2 Flutter Basics Thin structures can be excited by an adjacent or surrounding airflow [1, 21, 61, 62, 82]. As large elastic deflections bring about changes in dynamic flow pressure, the distributed body force on the structure changes, inducing an elastic response. This brings about the possibility of structural self-excitation via a dynamic feedback instability [38, 64]. In certain situations—depending on system parameters, plate boundary conditions, and flow velocity—a bifurcation can occur [83, 84] (the “onset of flutter”). Nonlinear behaviors then emerge, such as limit cycle oscillations (LCOs) or chaotic oscillations [60, 123]. By LCO we mean temporally-periodic

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

159

elastic deformations, described by closed [35], bounded trajectories in the relevant state space. LCOs are of interest for several reasons, but the goal is often prevention or suppression. In flight structures and bridges, LCOs should not be overlooked due to their potentially disastrous structural effects through fatigue failure or large amplitude response [63, 64, 104]. In others circumstances, oscillations might be fruitful and should be optimized, e.g., piezoelectric energy harvesters [71, 91]. The field of aeroelasticity is concerned with (i) producing models that describe the flutter phenomenon, (ii) gaining insight into the mechanisms of flow-structure coupling, (iii) predicting the behavior of a flow-structure system based on its configuration, and (iv) determining appropriate control mechanisms and their effect in the prevention or suppression of instability in the flow-structure system. Flutter considerations are paramount in the transition from the subsonic to the supersonic regime, with the renewed interest in supersonic flight [104, 135]. From a design point of view flutter cannot be overlooked, owing to its potentially disastrous effects on the structure due to sustained fatigue or large amplitude response. While there is an abundant engineering literature on the subject, involving predominantly experimental and computational studies, relatively little is understood about these complex and nonlinear flow-structure interactions from the point of view of mathematical analysis. The aim of this a chapter is to partially fill this gap by analyzing a prominent, yet simple, model of flutter and the ensuing nonlinear LCOs and demonstrate a rigorous condition which ensures flutter can be controlled or even eliminated. This discussion leads us to the study of a 3D compressible and barotropic gas, moving as an irrotational flow with a velocity .|U | > 01 above a nonlinear panel embedded in the flow boundary. In fact, we will utilize an inviscid potential flow approach (e.g., by linearizing Euler’s equation [89], or by beginning with the compressible Navier-Stokes equations, linearizing, and then invoking zero viscosity). We will utilize a perturbation variable .ϕ (the so-called velocity potential of the flow) about a steady state for the flow .U ≡ 〈−U, 0, 0〉. This means that the flow velocity is given by .v = U + ∇ϕ [62, 89] (see also [7, 8]). This leads to the following model, which has been prevalent in the aeroelasticity since the 1950s [21, 38, 62–64], describing the flow-plate dynamics on 3D half-space .R3+ (consider 3 .Ω ⊂ ∂R+ ): ⎧    ⎪ utt + Δ2 u + fV (u) = p0 + ∂t + U ∂x ϕ Ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨u = ∂ν u = 0 .

(∂t + U ∂x = Δϕ ⎪ ⎪ ⎪ ⎪ ∂z ϕ = (∂t + U ∂x )u ⎪ ⎪ ⎪ ⎩∂ ϕ = 0 z )2 ϕ

in Ω × (0, T ), on ∂Ω × (0, T ), in R3+ × (0, T ),

(3.1)

on Ω × (0, T ), c

on Ω × (0, T ).

simplicity we will restrict our attention to .U ⩾ 0, but there is no problem “reversing” the flow with .U < 0

1 For

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Above, the clamped von Karman plate describing transverse deflections of the plate (represented at equilibrium by the set .Ω) is coupled to the aforementioned 3D inviscid and subsonic potential flow, where .U > 0 is the unperturbed flow velocity in the x-direction, with .U < 1 indicating that the flow is subsonic. Remark 3.1.1 We note that, in general, .U > 1 is permitted above and represents a viable mathematical model for supersonic flows [45, 89]. This is not the focus here, as the supersonic panel system is not expected to have stationary end behavior. The latter is the main target in stabilization theory. [45, 63, 85]. Our main focus here is on the stability properties of a flow-structure interaction governed by (3.1) when .0 ⩽ U < 1. See [26, 36, 46, 89] for recent mathematical perspectives on this model. The dynamic pressure driving the plate, resulting from the potential flow through the aeroelastic potential, is given by  p(x, t) = p0 (x) + (∂t + U ∂x )ϕ Ω .

.

(3.2)

The specific model above is referred to a panel flutter model, in that it is used to capture the LCOs of a post-flutter panel in a potential flow of sufficient velocity [62, 63, 65, 130, 131]. The onset of this instability emanates from the linear portion of the coupled model, and the presence of the nonlinear elastic restoring forces ensures that dynamics remain globally bounded [42, 89], resulting in LCOs. The model (3.1) also captures aeroelastic buckling—a static (nonlinear) instability. In summary, the flow-panel model above can be used for all flow velocities U in describing stable and unstable dynamics.

3.1.3 More on Flutter From the modeling point of view, we will focus on distributed systems modeled by evolutionary PDEs. The presence of the flow can be represented by non-conservative PDE terms [39, 89], which will be seen to destabilize the linear portion of a plate model [83]. In the post-onset regime, linear models exhibit exponential growth, according to destabilized eigenvalues [85]. The inclusion of physical, nonlinear elastic restoring forces (such as those of von Karman) will ensure boundedness of trajectories and permit qualitative studies of the resulting plate dynamics [38, 44, 133, 134]. In short: the onset of self-excitation is a linear problem, but the study of the qualitative properties of the LCOs is requisitely nonlinear (see Fig. 3.1). The physical large deflection plate models are typically semi or quasi-linear, 4th order PDE systems, and often nonlocal in nature. Such systems are critically affected by geometry and boundary considerations. Specifically, the mixed boundary condition framework, with corners, as well as the high order free type boundary conditions, are notoriously challenging (e.g., the associated elliptic regularity). The nonlinear PDE framework will yield compact global attractors (containing the LCOs), which

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction 106 U = 2Ucrit

U = 1.01Ucrit

100 U = Ucrit

E(t)

Fig. 3.1 Energy plots [85] for linear (T) and nonlinear (B) damped plate models with flow U . Below bifurcation, .Ucrit , energies decay; above, the system is unstable. When nonlinear forces are active (B), energies plateau and LCO results

161

U =0

10−6

U = 0.99Ucrit

U = 0.5Ucrit

10−12

0

2

4

6

8

10 t

12

14

16

18

20

18

20

106 U = 2Ucrit U = 1.01Ucrit

100 E(t)

U = Ucrit U =0

10−6

U = 0.99Ucrit U = 0.5Ucrit

10−12

0

2

4

6

8

10 t

12

14

16

can be analytically and numerically studied through the lens of modern dynamical systems. Of course, there will be several critical choices concerning how the plateflow feedback is captured, i.e., the representation of surrounding or adjacent flows. This presentation centers on LCOs and nonlinearly elastic behaviors, rather than the immensely challenging, fully nonlinear, fluid-structure systems based on the Navier-Stokes equations. Even when utilizing simplified flow systems, which neglect viscosity and/or vorticity, we are able to capture a broad range of elastic behaviors. For several key research inquiries described above, potential flow theory is physically justified. We also emphasize that our focus is on self-destabilization in contrast to traditional resonance—a periodic response to a periodic forcing. With that said, there are many studies of elastic responses to periodic forcing, and the topic of nonlinear resonance is well-studied, e.g., [23]. We do note that elastic LCOs give rise to periodic behaviors in the flow; such periodicity may bring about secondary resonant effects. Let us consider a simplified evolutionary beam system (omitting initial and boundary conditions) in the transverse displacement .w(x, t) defined for .x ∈ [0, L]. The parameter .D > 0 measures structural stiffness, while .k > 0 captures some frictional or aerodynamic damping in the elastic system. The nonlinearity .f (w) is of Woinowsky-type [17, 70, 87], namely semilinear, nonlocal, and cubic, and the flag .δ = 0 or 1 permits turning the nonlinearity on and off. Finally, the parameter .U ∈ R captures the strength of an over-body potential flow in the .−e1 direction. wtt + D∂x4 w + kwt + δf (w) = U wx ,

.

  f (w) ≡ − ||wx ||2L2 (0,L) wxx .

(3.3)

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This system has been studied often—in mathematics and engineering literature— for some time, as well as higher dimensional analogues [18, 86]; our reference [85] provides a broad overview of the theory, and in-depth computational studies in several configurations (from which the discussion below is drawn). The above system admits an energy equality; let .Ebeam (t) capture the nonlinear elastic energy for (3.3),

t

Ebeam (t) + k

.

0

||wt ||2L2 (0,L) dt

= Ebeam (0) + U

t 0

L

[wx ][wt ]dξ dτ .

(3.4)

0

With both damping and non-dissipative flow “pollution,” we can concisely address the nature of flutter: there exists a .Ucrit corresponding to the onset of instability. In the modal approach for linear stability, one assumes simple harmonic motion of the flow-perturbed solution: w(x, t) =



.

ωt qn (t)sn (x) ≈ e−i

n



αn sn (x),

n

where .sn (x) are the unperturbed eigenfunctions (modes) associated with the operator .D∂x4 (with chosen boundary conditions). This reduces the onset problem to a (linear) eigenvalue computation in . ω: instability occurs when .I m(ω) ˜ > 0. The method works well [85, 87, 130] for predicting onset. When .U < Ucrit , we expect the dynamics to converge exponentially to equilibria (.δ = 0 or 1), owing to the presence of damping .k > 0. For .U > Ucrit , linear dynamics (.δ = 0) exhibit exponential growth (see Fig. (3.2)) and nonlinear dynamics (.δ = 1) exhibit postflutter behavior characterized by a low mode LCO (e.g., harmonic motion of a superposition of the first two modes). Note that the energy .Eplate (t) in Fig. (3.2) remains bounded for all supercritical (unstable) velocities .U > Ucrit when .δ = 1, demonstrating the stability induced by nonlinear restoring forces. Since each trajectory remains bounded in time, we can ask after the geometry of the LCO. Figure 3.2 shows a plot of the midpoint displacement (.x = L/2) for a beam as it leaves the transient regime and enters the LCO [86]. Having established a Hadamard well-posed PDE system giving a dynamical system .(St , Y ) in both subsonic and supersonic cases [43, 45, 89, 133], we may consider long-time behavior of PDE solutions to (3.1). We note that, in the subsonic Fig. 3.2 Beam entering LCO (midpoint displacements) u(𝓁/2, t)

0.1

0

−0.1 0

0.1

0.2

0.3

0.4

0.5 t

0.6

0.7

0.8

0.9

1

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case owing to the (Lyapunov) stability of the semigroup observed on Y , we can consider .t → ∞ for the full flow-plate system. For supersonic flows, since the semigroup does not have a viable global-in-time bound, we must consider reduced methods. Thus, in the supersonic scenario, we will consider only reduced plate dynamics and discuss the behavior as .t → ∞. Our efforts are directed toward studies of these (and related) dynamical systems; in this context, ultimate dissipativity, asymptotic compactness, and global attractors (and their properties) are of primary interest—these objects being associated with the structural dynamics, and not the full flow-plate dynamics. We also note that, since the entire system cannot be ultimately dissipative without a major intervention on .R3+ , the initial pursuit of a theory of attractors for the structure only is reasonable. We complete this subsection by informally outlining how to “embed” a fluttering plate dynamics within the context of a dissipative dynamical system, using a rigorous reduction result (Theorem 3.6.4 below). The first step is to show that under specified conditions, we have a dynamical system .(St , Y ) composed of the entire structure and flow coupled system. Our ultimate goal is to show that such dynamical system can be stabilized (in the strong sense) to an equilibrium. However, as per the discussion in the previous paragraph, to achieve this it will be necessary to obtain “good” uniform convergence/attractiveness of the structural dynamics while under the influence of the flow. So our main results will actually focus on a reduced plate dynamical system. By invoking Theorem 3.6.4 below, we will obtain a plate equation with finite memory term, encapsulating the flow response over a characteristic time, in the form of a temporal integral which functions as distributed plate forcing. This memory term is at critical regularity for finite energy plate dynamics. The reduction provides a closed plate system, valid for all .U /= 1, with the property that solutions to the full flow-plate dynamics (3.6) have their plate dynamics satisfying (3.37) (in the strong, mild, or weak sense, respectively). The tradeoff, however, is that by invoking the reduction, we destroy the gradient structure of the full flow-plate dynamics of .(St , Y ), which of course is the backbone of strong stability results. However, in this way, we may operate on the reduced system, using established techniques for nonlinear plates—including the recent quasi-stability theory [29, 42]. Our final results will provide a compact global attractor, both smooth and finite dimensional, for the plate dynamics. Subsequently, we may use this information to say something further about the long-time behavior of the full flow-plate system (3.6), under some additional assumptions. This latter part turns out quite challenging and necessarily exploits several technical areas of PDE analysis and dynamical systems.

3.1.4 Mathematical Challenges in Studying Flutter Flow-structure models have attracted considerable attention in the past mathematical literature, see, e.g., [13, 15, 24–26, 36, 42, 43, 62, 133] and the references therein. However, as mentioned above, the literature on this topic is predominantly

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engineering-focused; see, for instance, [12, 21, 62, 64], and also the survey [101] and the literature cited there. Many mathematical studies have been based on linear, two dimensional plate models with specific geometries, where the primary goal was to determine the flutter point (i.e., the flow speed at which flutter occurs) [12, 21, 64, 101]. See also [13, 15, 120] for the recent studies of linear models with a one dimensional flag-type structure (beams). This line of work has focused primarily on spectral properties of the system, with particular emphasis on identifying aeroelastic eigenmodes corresponding to the associated Possio integral equation (addressed classically by [129]). We emphasize that these investigations have been linear, as their primary goal is to predict the flutter phenomenon and isolate aeroelastic modes. Given the difficulty of modeling coupled PDEs at an interface [93, 94], theoretical results have been sparse. Additionally, post-flutter LCOs and stabilization are inherently nonlinear; although the flutter point (the flow velocity for which the transition to periodic or chaotic behavior occurs) can be ascertained within the realm of linear theory, predicting the magnitude of the instability requires a nonlinear model of the structure (and potentially for the flow as well) [62, 63, 67, 68]. The results presented herein demonstrate that flutter models can be studied from infinite dimensional analysis point of view, and moreover that meaningful statements can be made about the physical mechanisms in flow-structure interactions strictly from the PDE model.The fundamental challenges associated with long-time behavior of flow-plate (flutter) dynamics are: • A lack of “good” topological properties exhibited by the global energy functional for full flow-plate dynamics (as well as for reduced plate dynamics). This relates to the mismatch of regularity between the two types of dynamics— flow and the plate—which are coupled in a hybrid way. • A lack of “full” dissipation in the energy relation for flow-plate dynamics, as the flow has no inherent dissipation. However, there is a “hidden” dissipation that needs to be uncovered through the reduction result and flow-plate coupling conditions (see Theorem 3.6.4 below). • The regularity of downwash (coupling) term does not reconstruct finite energy flow solutions, if one considers the composite dynamics in a componentwise sense. • The structure has no active dissipation, unless the reduction result is invoked, destroying the gradient structure of the full flow-plate dynamics. • The presence of the flow, driving the structure, contributes forcing acting at or above the plate’s finite energy level. This relates to the model containing boundary trace terms of .L2 (Ω) functions, which will ultimately be defined by flow-plate solutions but, of course, are not generally defined a priori. This chapter will rigorously address the fully interactive dynamics between a nonlinear plate and a surrounding potential flow [21, 62]. As we have seen, the description of physical phenomena such as flutter and divergence translate into mathematical questions related to existence of nonlinear semigroups (representing a given dynamical system on a particular energy space), asymptotic stability of

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trajectories, and convergence to equilibria or to compact attracting sets. Interestingly enough, different model configurations lead to an array of diverse mathematical issues that involve not only classical PDEs, but subtle questions in non-smooth elliptic theory, harmonic analysis, and singular operator theory. For more details concerning the mathematical theory developed for other flutter models discussed above, see [38, 39, 89]. Our goal is to understand the interaction between the flow and the structure. Of particular mathematical interest are questions such as: • Does a flow have any stabilizing effect on the structure? • How one can control and stabilize the structure (in the strong topology of the energy space) to equilibria? Bringing the dynamics to an equilibrium corresponds to establishing strong stabilization for the relevant dynamics. While typically the strong stabilization property is easier to prove than an attraction in a uniform topology, this is not the case for the flutter model under the considerations. Persistence of structural oscillations contributing .ut within the coupled dynamics prevents us from concluding strong convergence to a stationary set. Weak convergence, instead, follows from the gradient property of the full flow-plate system. In order to resolve the disparity, one needs to address first uniform attraction of the structural dynamics. Thus, strong stability of the coupled system will require uniform stability of one component of that system. The latter is connected with compactness properties—which are absent in an explicit form in the flow dynamics.

3.1.5 Overcoming the Challenges and Overall Strategy To overcome these challenges, as discussed above, we shall reduce the flow dynamics to a delay potential driving the structural dynamics. This is accomplished using Huygen’s principle and the Kirchhoff solution to the wave equation in the context of the half-space .R3+ , contributing a delay representation of the flow acting upon structure after some sufficient waiting time. We will construct a dynamical system for plate dynamics by accounting for the delay representation of plate dynamics. We will have to show both asymptotic smoothness and ultimate dissipativity of this dynamical system; the latter will be accomplished using an appropriately constructed Lyapunov functional on a delayed state space. The former will be accomplished in two different ways, depending on the value of a regularization parameter .α ⩾ 0 (the plate’s rotational inertia parameter), with the plate equation given by:    (1 − αΔ)utt + (k − k1 αΔ)ut + Δ2 u + fV (u) = p0 + ∂t + U ∂x ϕ Ω .

.

(3.5)

We will cover both cases .α > 0 and .α = 0 in these notes, with .α > 0 representing a regularized model and .α = 0 representing the physical model, as presented in the

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engineering literature. We will be able to clearly and directly demonstrate several features of the dynamics in the .α > 0 case, which will provide context for the more difficult, physical case of .α = 0. In the case .α = 0, a compensated compactness property of the delay potential will be demonstrated, made apparent by a formal integration by parts in the nondissipative energy relation. In the final step for the .α = 0 case, we achieve the quasi-stability estimate on the resulting global attractor with a new technique exploiting only the compactness of the attractor, noting that without a gradient structure we cannot use the powerful technique of backward smallness of velocities on the attractor [41, 90]. Lastly, we return to the full flow-plate dynamics (with .U < 1) and transfer stability properties of the plate to the flow—this will require considering dissipation acting upon the structure. From a philosophical standpoint, the discussion above reflects the limitations of flow-plate modeling. For the full supersonic flow-plate interaction, we can say nothing of long-time stability, owing to the energy relation (3.14). In the case of subsonic flows, the boundedness of the semigroup .(St , Y ) does allow a discussion as .t → ∞, but there are no inherent damping mechanisms for the plate or the flow in the coupled dynamics. However, if we consider the reduced dynamics, focusing only on the structural component, we then can address all .U /= 1 at once, and we see (in this delay representation) that the flow by itself does contribute weak damping. As discussed, this comes at the cost of destroying the gradient structure of the problem, and we must address the associated non-dissipative and non-compact terms. In our analysis of this reduced system, we will obtain an attractor for the structure only (with no imposed damping in the model when .α = 0), while our results on strong stability of the full flow-plate model will require us to restrict our attention to .U < 1 and impose structural damping. In some sense, there is a correspondence between our results, and what is expected physically. For a panel (as considered here), it is well established that flutter does not occur subsonically [38, 39, 63, 64]. On the other hand, flutter is expected supersonically. Thus, it is reasonable that stabilization to equilibria results (as we present later) are only possible for .U < 1—our result below critically depends on the subsonic nature of the flow.

3.2 PDE Modeling 3.2.1 Setup Here we shall outline some principles which lead to the model under consideration—mostly to provide some orientation. • We consider a thin, flexible plate defined on a domain .Ω ⊂⊂ R2 , moving with a normalized velocity U . (In general, one considers the quantity .M = U local speed of sound , known as the Mach number, but here we simply take .U = 1 to be the normalized speed of sound.)

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• .Ω is a smooth domain embedded in .∂R3+ . • The unperturbed flow is given by .U = 〈−U, 0, 0〉 which a constant field in .R3+ in the negative x-direction. z y U

Ω x As described above .ϕ(·; t) : R3+ → R represents the perturbation velocity potential at an instance of time, .u(·, t) : Ω → R represents the vertical displacement of the plate, which is permitted to be large.

3.2.2 Obtaining the PDE Model We now provide an overview for obtaining the flutter model of interest. We note that there are alternative approaches to obtaining this model, see, e.g., [7, 39, 64, 89]. Below, we begin with Euler’s equation for the flow, given in variables for density .ρ, pressure p, and velocity .v defined on some domain .D with .Ω ⊂ ∂D and outward unit normal .n. • .ρt + div(ρv) = 0inD ⊆ R3 • .ρ[vt + (v · ∇)v] + ∇p = 0 inD • .utt − αΔutt + k(1 − αΔ)ut + Δ2 u + fV (u) = p|Ω in Ω. Above, the quantity .α ≥ 0 represents possible rotational inertia in the filaments of the plate, and the parameter .k ⩾ 0 is a damping parameter; when .α = 0, .k > 0 represents frictional damping, and when .α > 0, the damping is of square-root type [87]. The nonlinearity is the scalar von Karman nonlinearity [42, 132], represented by .fV (u), which will be described in more detail below. We now linearize the inviscid flow around unstable profile .U = 〈−U, 0, 0〉 and assume that the flow is irrotational, and thus .v = U + ∇ϕ for a potential function .ϕ, linearized-barotropic so .p = γρ, which leads to the following hyperbolic-like system: • (1) .[∂t + U ∂x ]ρ + div(v) = 0, inD ∂v • (2) .[∂t + U ∂x ](vt + U ∂x + ∇p) = 0inD

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Then: (2) → (3) ≡ (∂t + U ∂x )[ϕt + U ϕx + γρ] = 0inD

.

(1) + (3) = (∂t + U ∂x )[ϕt + U ϕx ] − γ Δϕ = 0inD We then define the aeroelastic potential .ψ. ψ ≡ (∂t + U ∂x )ϕ, v = U + ∇ϕ

.

Then, the flow-structure interaction can be written in the variables .ϕ, ψ, and u (normalizing non-critical constants): • .ψ ≡ (∂t + U ∂x )ϕ • .(∂t + U ∂x )ψ − Δϕ = 0 • .p = p0 (x) + ψ|Ω We now note two central boundary conditions possible for the flow-plate coupling on .Ω and off .Ω, with .n as the unit outward normal to .D: (∂t + U ∂x )u on Ω ∂ϕ • Flow-tangency (Neumann-type) Conditions: . ∂n = 0 offΩ; ∂ϕ = (∂t + U ∂x )u on Ω • Kutta-Joukowski Conditions [95]: . ∂n ψ =0 off Ω. For the structure, we have many boundary conditions possible along .𝚪 = ∂Ω: • Clamped: .u = ∇u · ν = 0, on𝚪 • Free: .Dν Δu + (1 − μ)Dτ Dν Dτ u = Δu + (1 − μ)[Dτ2 − div(ν)Dν ]u = 0on 𝚪 • Hinged: .u = Δu + kDτ2 u = 0, on𝚪, where above .ν and .τ are the unit normal and tangential, associated with .𝚪 = ∂Ω; μ ∈ (0, 1) is the Poisson modulus of the plate, and k is the curvature. In typical applications, as described above, the following combinations of boundary conditions [80] are often considered:

.

Bridge [3, 5, 20, 22, 75–77]: Flow–Neumann + Structure (rectangular)—Hinged, Free, Hinged, Free Flag [4, 56–58, 85, 87, 88, 115]: Flow–Neumann + Structure (rectangular)— Clamped, Free, Free, Free Panel [38, 39, 42, 43, 61–64, 67, 85, 86, 98, 104, 133]: Flow–Neumann + Structure—Clamped Wing [12–16, 55, 74, 118, 119]: Flow—Kutta-Joukowsky + Structure—Hinged/ Clamped, Free Remark 3.2.1 Experiments (for instance at NASA, UCLA’s AFOSR Lab, and Duke) demonstrate the flow exhibiting a stabilizing effect on the structure in the absence of any imposed damping mechanism.

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In these notes, we focus on the Neumann-tangency boundary conditions imposed for the flow and clamped boundary conditions imposed on the plate. This leads to the so-called panel model. Nonlinear effects can be represented by various nonlinear structural models (Berger, von Karman, full von Karman), but we shall focus on the scalar von Karman model here. Thus, nonlocal effects will be represented by Airy’s stress function [42, 53]. We provide the details of the nonlinear model .fV below. • .fV (u) = −[u, v(u) + F0 ]. • .[g, h] ≡ gxx hyy + gyy hxx − 2gxy hxy is the von Karman bracket. • .v(u) is the Airy Stress function, i.e., .

Δ2 v = −[u, u] inΩ v=

∂v ∂ν

= 0on𝚪.

• .F0 is the static in-plane forcing term for the plate, taken in .H 4 (Ω), which can contribute to buckling or even chaos. In this case, .fV (u) is nonlocal, and of cubic-type on .H 2 (Ω). It models large deflections of thin plates and shells. This was the nonlinearity in the original flowplate investigations [21, 62] and is the principal nonlinearity in our analysis. Above, parameters such as mass, density, thickness, and stiffness have been scaled out. The remaining parameters are those centrally relevant to this mathematical analysis: .U, α, ⩾ 0. Here, .α ≥ 0 corresponds to the accommodation of rotational inertia of plate filaments [92], as described above. The function .p0 (x) corresponds to a stationary pressure on the top surface of the plate.

3.3 Functional Setup and Well-posedness of Weak and Strong Solutions 3.3.1 Notation and Conventions In this paper we utilize the standard notation and conventions for .Lp (O) spaces and Sobolev spaces of order .s ∈ R, .H s (O) where .O is some domain. The space s ∞ s .H (Ω) denotes the completion of the test functions .C (Ω) in the .H (Ω) norm with 0 0 −s dual .H (Ω). For our norm notation, we will denote .|| · ||H s (O ) = || · ||s , where the spatial domain will be clear from context; we will identify .|| · ||L2 (O ) = || · ||, omitting .s = 0. Inner products on .R3+ will be denoted by .(·, ·) := (·, ·)L2 (R3 ) and on +

∂R3+ we utilize the notation .〈·, ·〉 := (·, ·)L2 (Ω) . The trace operator on .H 1 (O) spaces will be denoted by .γ [·] with range in .H 1/2 (∂O). We denote an open ball of radius R in a Banach space X by .BR (X). Throughout most of these notes we consider .U ∈ [0, 1), in particular, for our main stabilization to equilibria result. .

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3.3.2 System Under Consideration We will consider the system: ⎧ 2 ⎪ ⎪ tt + k(1 − αΔ)ut + Δ u + fV (u) = ⎪(1 − αΔ)u ⎪    ⎪ ⎪ ⎪p0 + ∂t + U ∂x ϕ Ω ⎪ ⎪ ⎪ ⎪ ⎪ u(x, y; 0) = u0 (x, y); ut (x, y; 0) = u1 (x, y) ⎪ ⎪ ⎪ ⎨u = ∂ u = 0

.

ν

⎪ (∂t + U ∂x )2 ϕ = Δϕ ⎪ ⎪ ⎪ ⎪ ⎪ϕ(x, y, z; 0) = ϕ0 (x, y, z); ϕt (x, y, z; 0) = ϕ1 (x, y, z) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂z ϕ = (∂t + U ∂x )u ⎪ ⎪ ⎪ ⎩∂ ϕ = 0 z

in Ω × (0, T ), in Ω on ∂Ω × (0, T ), in R3+ × (0, T ), in R3+ on Ω × (0, T ), c

on Ω × (0, T ). (3.6)

We will distinguish between the two cases, .α = 0 (non-rotational plate) and α > 0 (rotational plate), as well as between the undamped .k = 0 and damped .k > 0 cases. Note: the strength of damping is scaled to the presence of .α > 0, as will be discussed in more detail below.

.

3.3.3 Energies and Spaces The energetic constraints for solutions manifest themselves through natural topological requirements, namely, for .L2α (Ω) given by || · ||2L2 (Ω) := α||∇ · ||2L2 (Ω) + || · ||2L2 (Ω) ,

.

α

finite energy solutions should have the properties: u ∈ C(0, T ; H02 (Ω)) ∩ C 1 (0, T ; L2α (Ω)); .

(3.7)

ϕ ∈ C(0, T ; W1 (R3+ )) ∩ C 1 (0, T ; L2 (R3+ )),

where .W1 (R3+ ) denotes the homogeneous Sobolev space of order 1. Here,

 W1 (R3+ ) = ϕ ∈ L2loc (R3+ ) : ∇ϕ ∈ L2 (R3+ ) ,

.

which is to say the space topologized by the gradient norm .||∇ϕ||L2 (R3 ) without L2 (R3+ ) norm control.

.

+

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We can consider a base space corresponding to the energy norm:     Y = Yf l × Ypl ≡ W1 (R3+ ) × L2 (R3+ ) × H02 (Ω) × L2α (Ω) .

.

(3.8)

We will also consider a stronger space for well-posedness on finite time intervals: Ys ≡ H 1 (R3+ ) × L2 (R3+ ) × H02 (Ω) × L2α (Ω).

.

(3.9)

The above norm will be relevant for stability considerations as well. The energies corresponding to finite energy solutions of (3.6), and the above space Y , are given below. Epl =

 1 1 ||ut ||2L2 (Ω) + ||Δu||2 + ||Δv(u)||2 − 〈F0 , [u, u]〉 + 〈p0 , u〉, . α 2 2 (3.10)

Ef l =

 1 ||ϕt ||2 + ||∇ϕ||2 − U 2 ||∂x ϕ||2 , 2

.

.

Eint = 2U 〈γ [ϕ], ∂x1 u〉, .

(3.11) (3.12)

E =Epl + Ef l + Eint .

(3.13)

It will be shown that all subsonic weak solutions to the flow-plate system, with k = 0, satisfy the so-called energy balance which is

.

E(t) = E(s), t ≥ s.

.

(3.14)

On the other hand, it can be easily noticed that the energy functions do not always code topological properties of the solution. This is the case in a subsonic regime .U > 1 where .Ef l can be even negative.

3.3.4 Solutions First, let us consider the case .0 ⩽ U < 1. The pair .(ϕ, u) as in (3.7) is said to be a strong solution to (3.6) on .[0, T ] if:   • .(ϕt , ut ) ∈ L1 a, b; H 1 (R3+ ) × H 2 (Ω) ∩ L2α (Ω) for any .(a, b) ⊂ [0, T ].   • .(ϕtt , utt ) ∈ L1 a, b; L2 (R3+ ) × H01 (Ω) for any .(a, b) ⊂ [0, T ]. • .ϕ(t) ∈ H 2 (R3+ ) and .Δ2 u(t) + k(1 − Δ)ut (t) ∈ [L2α (Ω)]' a.e. .t ∈ [0, T ]. • The equation .(1 − αΔ)utt + Δ2 u + k(1 − αΔ)ut + fV (u) = p0 + rΩ γ [ϕt + U ϕx ] holds in .[L2α (Ω)]' a.e. .t > 0. • The equation .(∂t + U ∂x )2 ϕ = Δϕ holds a.e. .t > 0 and a.e. .x ∈ R3+ .

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• The boundary conditions in (3.6) hold a.e. .t ∈ [0, T ] and a.e. .x ∈ 𝚪, .x ∈ R2 , respectively. • The initial conditions are satisfied point-wisely; that is .ϕ(0) = ϕ0 , ϕt (0) = ϕ1 , u(0) = u0 , ut (0) = u1 . Strong solutions are point-wise. The pair .(ϕ, u) is said to be a generalized solution to problem (3.6) on the interval .[0, T ] if there exists a sequence of strong solutions .(ϕ n (t); un (t)) with some initial data .(ϕ0n , ϕ1n ; un0 ; un1 ) such that .(ϕ n , un ) converge to .(ϕ, u) in the sense of .C([0, T ]; Ys ) as .n → ∞. Such solutions correspond to semigroup solutions for initial data in Y , rather than the domain of the generator [42, 109]. Lastly, the pair .(u, ϕ), with

    u ∈ WT ≡ u ∈ L∞ 0, T ; H02 (Ω) , ∂t u(x, t) ∈ L∞ 0, T ; L2α (Ω)

.

    ϕ ∈ VT ≡ ϕ ∈ L∞ 0, T ; H 1 (R3+ ) , ∂t ϕ(x, t) ∈ L∞ 0, T ; L2 (R3+ ) ,

.

is said to be a weak solution to (3.6) on .[0, T ] if • .u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and .ϕ(x, 0) = ϕ0 (x), ϕt (x, 0) = ϕ1 (x) •

T

.

   〈(1 − αΔ)∂t u(t), ∂t w(t) − k〈(1 − αΔ))∂t u(t), w(t)

0

    − Δu(t), Δw(t) − fV (u(t)) − p0 , w(t)   − rΩ γ [ϕ(t)], ∂t w(t) + U ∂x1 w(t) dt   = u1 − rΩ γ [ϕ0 ], w(0) L2 (Ω) for all test functions .w ∈ WT with .w(T ) = 0. • .

0

T



   (∂t + U ∂x1 )ϕ(t), (∂t + U ∂x1 )ψ(t) − ∇ϕ(t), ∇ψ(t)

    + (∂t + U ∂x1 )u(t), rΩ γ [ψ(t)] dt = ϕ1 + U ∂x1 ϕ0 , ψ(0)

for all test functions .ψ ∈ VT such that .ψ(T ) = 0. It is clear that strong solutions are generalized, and we assert that generalized solutions are in fact weak. The central point here concerns the von Karman nonlinearity—see the discussion of abstract second order equations in [42].

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3.3.5 Well-posedness Results Well-posedness of Weak and Strong Solutions Theorem 3.3.1 (Weak Solutions) Let U ⩾ 0, U /= 1. For finite energy initial data in the space (ϕ0 , ϕ1 , u0 , u1 ) ∈ Ys ≡ H 1 (R3+ ) × L2 (R3+ ) × H02 (Ω) × L2α (Ω)

.

and for all T > 0 there exists unique finite energy, global-in-time solution (ϕ, ϕt , u, ut ) ∈ C([0, T ]; Ys ). When U < 1 solutions are bounded in time t > 0 in the norm of Y , as described above. The proofs utilize semigroup methods but vary depending on whether U > 1, and whether α > 0. See [43, 133] for α = 0 and U < 1, and [45, 89] for α = 0 and U > 1. For the case when α > 0, see the recent [10] as well as older literature, nicely summarized in [42]. Reformulating the above result in terms of dynamical systems, one has the following: dynamical system for the flow-structure interaction with clamped boundary data Corollary 3.3.2 (Dynamical System) For all U /= 1, U ⩾ 0, the system described by (u, ϕ) above generates a (nonlinear) strongly continuous semigroup (St , Y ) which is bounded on Y by Meωt , with ω = 0 when U < 1. Let us now make more precise the notion of strong solutions (as defined generally, above), with α = 0 and distinguishing between permitting all 0 ⩽ U /= 1. We note the well-posedness results below can be improved when α > 0, but we forgo that discussion here—see [25, 26]. We introduce the space:  Y1 ≡ (ϕ, ϕ1 , u, u1 ) ∈ Ys : ϕ1 ∈ H 1 (R3+ ), u1 ∈ H02 (Ω), Δϕ−U 2 ϕxx ∈ L2 (R3+ ),

.

ϕz ∈ H 1 (R2 ), Δ2 u − U γ [ϕx ] ∈ L2 (Ω)



.

Theorem 3.3.3 (Strong Solutions) Let α = 0. Let 0 ⩽ U /= 1. If the initial data (ϕ0 , ϕ1 , u0 , u1 ) ∈ Ys and satisfy the natural compatibility conditions at the boundary ∂R3+ , then the corresponding weak solution (as described above) and becomes strong, with (ϕ, ϕt , u, ut ) ∈ C([0, ∞); Y1 ). Now suppose 0 ⩽ U < 1; if the initial data (ϕ0 , ϕ1 , u0 , u1 ) ∈ Ys and satisfy the natural compatibility conditions then one, in fact, obtains (ϕ, ϕt , u, ut ) ∈ C([0, ∞); Y2 ), where Y2 ≡ H 2 (R3+ ) × H 1 (R3+ ) × (H 4 (Ω) ∩ H02 (Ω)) × H02 (Ω).

.

In addition, solutions are bounded t ∈ (0, ∞) in the norm of Y .

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Remark 3.3.1 We remark briefly about regularity loss associated with the transition from sub- to supersonic flows. • Note that in the supersonic case U > 1 there is a loss of differentiability for strong solutions, as the space Y1 ⊇ Y2 . In particular, u ∈ H 7/2 (Ω) ∩ H02 (Ω) and ϕ ∈ H 3/2 (R3+ ), thus there is a loss of 1/2 derivative [45]. • In the case when 0 < U < 1, the proof of the theorem given in [133] employs a renormalization of the norm of Ys and a construction of a suitable inner product in order to achieve dissipative system. • In the case when rotational inertia are included, with α > 0 in plate model, the proof of well-posedness is much simpler due to the increased regularity of the structural velocity: ut ∈ H01 (Ω) which appears in the Neumann condition ∂z ϕ Ω = ut + U ux ∈ H01 (Ω). This, in turn, implies that aerodynamic pressure ψ|Ω is well defined as a boundary trace restriction to Ω. This is not the case in the non-rotational case—α = 0.

Discussion of Well-posedness Proofs We shall list several points when determining the well-posedness of the system in (3.6). As stated before, the main challenge is the case .α = 0. When .α > 0 wellposedness is achieved by a straightforward application of a semigroup argument— see [10], and the earlier [24–26] for a Galerkin approach. The simplification is due to the regularizing effect of rotational inertia which produces a “subcritical” nature of the nonlinearity .fV and viable coupling traces on .Ω. This is no longer the case when .α = 0. Below we shall list the main points to carry out the proof of the most general case .0 ⩽ U /= 1 with .α = 0—the details are found in [45]. • Change of State Variable: Consider .ϕt I→ ψ = U ϕx + ϕt ; set up the system in the variables .(ϕ, ψ; u, ut ) ∈ Ys • The lack of ellipticity of the spatial flow operator, .Δ − U 2 ∂x2 , when .U > 1. • Maximal Dissipativity for a dissipative generator .A0 , where  the full flow-plate  dynamics are encoded in .A = A0 + P , with .P u = 0, 0, γ ∗ [U ux ], 0 an unbounded perturbation on the state space. • One must also address the unclosable perturbation P via Ball’s Theorem [17] and a hidden regularity result for these hyperbolic dynamics. • Hidden regularity: – Microlocal (hidden) trace regularity [106, 107, 112], which does not follow from the standard trace theory and given finite energy interior regularity – Sharp regularity of the trace .ψ|Ω = [ϕt + U ϕx ]|Ω ∈ L2 (H −1/2 (Ω)) • Compensated regularity—reconstruction in the limit—to prove uniqueness. • Exploiting nonlinear terms and global a priori bounds on solutions. – The so-called sharp regularity of the Airy stress function [42, 43, 73] is critical, as expressed through the estimates below:

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

||v(u)||W 2,∞ (Ω) ≤ C||u||22.Ω.   [v(u), u] ≤ C||u||3 . 2,Ω   [v(u), u] − [v(w), w] ≤ C(R)||u − w||2,Ω .

175

(3.15) (3.16) (3.17)

∀u, w ∈ BR (H 2 (Ω)). – Showing that super-nonlinearity provides a lower bound for the energy functional .E. This prevents the physical (possibly non-positive) energy to go to .−∞. The latter follows from critical property of the von Karman nonlinearity, in the context of long-time behavior, is its “good” potential energy (namely, control of low frequencies) [42], which we state below as a proposition. To conclude this section, we state—formally—some of the central facts about the von Karman nonlinearity, bracket, and Airy stress function which yield the estimates above in (3.15)–(3.17). These will also be utilized throughout below, in particular with respect to the construction of an absorbing ball and in compensated compactness for the .α = 0 case. Theorem 3.3.4 Assume that .ui ∈ C(s, t; H 2 (Ω)) ∩ C 1 (s, t; L2 (Ω)), then we have that .

− 〈f (u1 ) − f (u2 ), zt 〉 =

1 1 d Q(z) + P (z), 2 4 dt

where Q(z) = 〈v(u1 ) + v(u2 ), [z, z]〉 − ||Δv(u1 + u2 , z)||2

.

and P (z) = −〈u1t , [u1 , v(z)]〉−〈u2t , [u2 , v(z)]〉−〈u1t +u2t , [z, v(u1 +u2 , z)]〉.

.

(3.18)

Moreover,  t   1  t    1 2 2 P (z)dτ  . 〈f (u (τ )) − f (u (τ )), zt (τ )〉dτ  ⩽ C(R) sup ||z||2−η +  2 s s τ ∈[s,t] (3.19) for some .0 < η < 1/2, provided .ui (τ ) ∈ BR (H02 (Ω)) for all .τ ∈ [s, t]. The above bounds rely on the decomposition f (u1 ) − f (u2 ) = [z, v(u) + F0 ] + [u2 , v(u1 ) − v(u2 )]

.

and on the so-called sharp regularity of Airy’s stress function .v(u) [42, pp.44–45]:

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Remark 3.3.2 Note that the operator Q enjoys subcritical regularity. This fact will be critical for the development. Lemma 3.3.5 Airy stress function regularity. For all .u1 , u2 ∈ H 2 (Ω) ||Δ−2 [u1 , u2 ]||W 2,∞ ≤ C||u1 ||2,Ω ||u2 ||2,Ω .

.

Proposition 3.3.6 For any .η, ϵ > 0 there exists .Mϵ,η such that   ‖u‖22−η,Ω ≤ ϵ ‖Δu‖2 + ||Δv(u)||2 + Mη,ϵ , ∀ u ∈ H 2 (Ω) ∩ H01 (Ω).

.

(3.20)

The Lemma (3.20) is obtained through a compactness uniqueness argument that exploits superlinearity of .fV and a maximum principle associated with the MongeAmpere equations [42]. Remark 3.3.3 The inequality in Proposition 3.3.6 is valid for plates with clamped or hinged boundary conditions. However, free plate boundary conditions are not covered by the standard argument. The regularity reported in (3.17) is referred to as “sharp.” Indeed, the standard regularity of Airy’s stress function [2, 3] does not allow one to conclude the .W 2,∞ -regularity for finite energy plate solutions.

3.4 Long-Time Behavior of Weak Solutions In order to understand the evolution for large times it is essential to determine the precise invariants of the dynamics. We refer here to the so-called energies as defined above in (3.10). We recall that in the case of .0 ⩽ U < 1 with .k = 0 (no damping) we have the energy balance .E(t) = E(s). Roughly speaking, the main results pertaining to long-time behavior are the following: • Global attraction of the plate dynamics to a “structural set” when .U /= 1; in the case when .α = 0, no additional damping is required. • Strong stability to equilibria of the full flow-plate interaction when .U < 1 and .k > 0 (this includes both cases, .α > 0 and .α = 0). The first result is valid for both subsonic and supersonic frequencies. The second result refers to the full interaction, in the subsonic regime and under feedback control. It states that a well calibrated feedback control is capable of eliminating flutter—which is the main strong stabilization result. Note that there is a major distinction in the strength of damping, depending on whether or not rotational inertia is present. This will be discussed in more detail below.

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

177

3.4.1 Statement of the Main Result and Qualitative Demonstrations In this section we focus on the most challenging case .α = 0, for which the main result is completely new [11]. We note that the analogous result for .α > 0 was known earlier, with a proof sketch given in [42] and fleshed out in [10]. Further discussion of past stabilization results will be given below. The first result focuses on the plate dynamics alone, making reference to the reduction result which will be given in detail below. Theorem 3.4.1 (Structural Attracting Set) Suppose .α = 0 and .k ⩾ 0 in (3.6). Let .0 ⩽ U /= 1, .F0 ∈ H 4 (Ω) and .p0 ∈ L2 (Ω). Then there exists a compact set 2 2 .U ⊂ H (Ω) × L (Ω) of finite fractal dimension (in that topology) such that 0 .

lim

inf

t→∞ (v0 ,v1 )∈U

  ||u(t) − v0 ||22 + ||ut (t) − v1 ||2 = 0

for any weak solution .(u, ut ; ϕ, ϕt ) to (3.6) with initial data (u0 , u1 ; ϕ0 , ϕ1 ) ∈ H02 (Ω) × L2 (Ω) × H 1 (R3 ) × L2 (R3 )

.

which are localized in .R3+ (i.e., .ϕ0 (x) = ϕ1 (x)  = 0 for .|x| > Rfor some .R > 0). We have the additional regularity .U ⊂ H 4 (Ω) ∩ H02 (Ω) × H 2 (Ω) and the attracting set is a bounded set in this higher topology. The second main result refers to strong stability of the overall interaction in the subsonic case, .U < 1. In order to formulate it, we introduce the set of stationary solutions. Let .N denote the set of (weak) stationary solutions to (3.6). We will assume .N is finite. (This fact is generically true—see [42, 89]; this assumption can likely be eliminated using Lojasiewicz theory—see [38] and references therein). For this result, we must consider the plate structural equation with some feedback control, with intensity measured by a damping constant .k > 0. We consider (3.6) with .α = 0 and .k > 0, resulting in the plate equation: utt + kut + Δ2 u + fV (u) = p0 (x) + (∂t + U ∂x )ϕ|Ω .

.

(3.21)

Theorem 3.4.2 (Stability for .U < 1 and .α = 0 with .k > 0) Let .U < 1 with .α = 0 and .k > 0 in (3.6). Then any weak solution .(ϕ(t), ϕt (t); u(t), ut (t)) ∈ Ys with compactly supported flow initial data (as in the previous theorem) stabilizes to the stationary set .N . Assuming .N is finite, we have: There exist .(u0 , u1 , Ф0 , Ф1 ) ∈ N such that for any .R > 0. .

.

lim ||u(t) − u0 ||22,Ω + ||ut (t) − u1 ||20,Ω = 0

t→∞

lim ||∇[ϕ(t) − Ф0 ]||20,B(R) + ||ϕt (t) − Ф1 ||20,B(R) = 0,

t→∞

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where .B(R) denotes a ball of radius R in .L2 (R3+ ) here. As a consequence we assert that dynamic flow-plate instability, flutter, can be eliminated by applying a light damping to the structure only, in the case of subsonic flows. Nonlinear effects as well as the subsonic nature of the flow are here critical. We now illustrate the stability results with a few simulations and figures (most appearing in [86] originally). Below, take .E(t) (linear) and .E(t) (nonlinear) to represent a generic plate/beam energy, and .u(x, t) to represent a generic plate displacement. We denote by .Ucrit the velocity at which flutter occurs. Beyond .Ucrit , linear dynamics grow exponentially according to a destabilized eigenvalue for the flutter system; nonlinear dynamics, on the other hand, are seen to enter an LCO— characterized by bounded energies. Lyapunov stability with a damping: Linear (L) and nonlinear (R) energies at various multiples of .Ucrit 109

ˆcrit U = 2U

ˆcrit U = 2U

104

105

ˆcrit U =U

E(t)

E(t)

ˆcrit U = 1.01U

ˆcrit U = 0.99U

101

ˆcrit U = 1.01U

102 100

ˆcrit U = 0.5U

ˆcrit U =U ˆcrit U = 0.99U

U =0

ˆcrit U = 0.5U

U =0

10−3

10−2 0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

5

Convergence to a given LCO: Effect of damping coefficient on convergence to the “stable” LCO k=1 k=2 k=6

2

u(𝓁/2, t)

1 0 −1 −2 0

1

2

3

4

5 t

6

7

8

9

10

Convergence to an equilibrium: Effect of damping coefficient on convergence

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction 200

k=1 k=2 k = 10

150 E(t)

179

100 50 0

0

2

4

6

t

8

10

12

14

A non-trivial steady state: The stationary state for a convergence to equilibrium

u(x, t)

1

0.5

0 0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

Two different stationary states: Convergence to particular equilibria affected by damping coefficient 4

u(𝓁/2, t)

2 k=1 k=2

0 −2 −4 0

1

2

3

4

5 t

6

7

8

9

10

Convergence to non-trivial LCO: Transient period transitioning to LCO; onset of flutter

u(𝓁/2, t)

0.1

0

−0.1 0

0.1

0.2

0.3

0.4

0.5 t

0.6

0.7

0.8

0.9

1

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3.4.2 Further Discussion of Dissipation and the Effects of Rotational Inertia The main results stated above in Theorems 3.4.1 and 3.4.2 have been shown in the case of “rotational inertia” with .α > 0 present in the plate model. See [10] for a recent treatment, as well as [42] (and many references therein). In what follows we shall provide complete proofs for both .α = 0 and .α > 0. In this latter case, it is mathematically clear that strong damping must be added to the plate—as above— with .k, α > 0 so the plate dynamics take the form  (1 − αΔ)utt + k(1 − αΔ)ut + Δ2 u + fV (u) = p0 + ψ Ω .

.

In this case, the damped plate alone produces an exponentially stable semigroup, which is strong enough to yield the main results above without “harnessing” any dissipation from the flow. (This fact is also reflected by the energy balance, which in t

the case of .α > 0, exhibits a dissipation integral of the form .αk 0

||∇ut ||2Ω ds.)

Looking ahead, the “reduction” result which has been described above is given in Theorem 3.6.4. It shows that, upon rewriting the flow-plate dynamics as a closed (though non-dissipative) plate dynamics with memory, we inherit some dissipation from the flow itself via the Neumann-tangency conditions. It is, then, an interesting question to ask: To what extent is this flow-inherited dissipation viable for stabilizing the dynamics? Indeed, experimental studies have indicated that the flow has some “stabilizing effect” on the plate, even in the case there is no dissipation explicitly accounted for in the structural dynamics. On the other hand, the flow has the possibility of destabilizing the structure, through the flutter phenomenon. The battle between these two effects is at the heart of flow-structure (in)stability. Clearly, the challenge is to show that, in the total absence of structural dissipation, (.α = 0) the effect of aerodynamic pressure, .ψ = ϕt + U ϕx , has a hidden dissipation effect. To see mathematically, one can write the structural dynamics using Theorem 3.6.4 to obtain (for .α, k ⩾ 0): (1 − αΔ)utt + k(1 − αΔ)ut + Δ2 u + fV (u) = p0 − ut + U ux + q(ut ),

.

(3.22)

where .q(ut ) denotes memory effects over some time .τ ∈ (t − t ∗ , t). However, this form suggests that the term .ut contributed by the flow does exert a stabilizing effect on the structure, but at the topological level of .L2 (Ω). While the presence of the term .ut provides some hope, this hope is challenged by the non-dissipative “pollution” coming in the form of .U ux on the RHS, as well as the delay term .q(ut ), which we will see is at the critical topological level .H 2 (Ω) for the plate. One should mention, then, that in the case .α > 0 the flow stabilizing contribution .ut is not sufficiently strong to lead to stable dynamics. This can be seen from a simple Fourier’s modes decomposition of the plate dynamics. Hence, by accounting

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

181

for rotational inertia in the model (a mathematically regularizing effect), we lose the stabilizing effect of the flow through the reduction result. This is why, when .α > 0, strong damping of the form .−kαΔut must be imposed on the structure in order to have any sort of stability results. Fortunately, the physical model as presented in engineering literature [21, 62] focuses on thin plates in the various flutter applications and thus chooses .α = 0. We summarize by saying that the game to play, here, is precisely to understand and balance the two opposite effects described above: damping and non-dissipative flow pollution. In fact, the final—recent [11]—resolution of the problem requires the introduction of special microlocal spaces, denoted by .Xθs , which are able to resolve some of the purely PDE questions in the area of hyperbolic boundary traces [126]. This is at the heart of the central technical results we present and prove below.

3.5 Outline of the Stability Proofs in Theorems 3.4.1 and 3.4.2 The remaining part of these lectures is devoted to establishing the validity of Theorem 3.4.1 and Theorem 3.4.2. For pedagogical reasons we provide the details for both rotational .α > 0 case and irrotational .α = 0 case. The former being more straightforward provides context for the latter, which is much more involved and challenging. • We will begin with an account of several preliminary results. A critical ingredient in handling the dynamics at the interface is so-called plate-to-flow map described in Sect. 3.6, which describes topological properties of the flow when excited by the displacements of the structure. This aspect is of independent interest, as it relates to the regularity of the hyperbolic Neumann map and associated Neumann-to-Dirichlet mapping. It is known that such mappings are of compromised regularity in dimensions higher than one. This is to say that .H s data on the boundary does not propagate .H s solution traces, as it is the case when considering a purely Dirichlet hyperbolic problem. In the case of Neumann boundary conditions, .L2 (Σ) data on the boundary does not lead to finite energy solutions; there is a loss of at least .1/3 derivative, or .1/2 derivative in a tangential direction. This is the content of Theorem 3.6.1. Since the coupling on interface is the main actor in the stability problem at hand, a good understanding of boundary regularity is of major importance to this work. • The next step is to produce representation formulas for the structure, as driven by the flow. These formulas—valid after some time (Huygen’s Principle), Theorem 3.6.4—reveal a “hidden” stabilizing effect of the flow. However, the stabilizing effect is associated with a “destabilizing” action coming from a (non-

182

•

• • •

•

•

I. Lasiecka and J. T. Webster

conservative full material derivative .[∂t + U ∂x ] and delayed terms), due to the aerodynamic pressure. Thus the centerpiece of the game here is to “play-out” both effects. Based on the representation formula and regularity of the hyperbolic Neumann map, it will become clear that the rotational .α > 0 and non-rotational .α = 0 problems are bound to produce different quantitative results. In the case .α > 0 there is an additional smoothness of solutions which allows more standard and direct methods to obtain. Section 3.7 provides a discussion of the main results and comparisons with previous work for .α ≥ 0. Section 3.8 provides the details of the analysis in the rotational case .α > 0. The case .α = 0 is the game changer and this will be presented in Sect. 3.9. First of all, this case does not necessitate the introduction of any strong damping. Thus long-time behavior can be first considered without any added damping—a fact of essential value in physical applications. However, from the mathematical standpoint, the analysis of the rotational model is inadequate for non-rotational model and new mathematical technologies must to enter. This includes a recent microlocal methods applied to the delicate interface coupling which will become the main protagonist for the problem. It shall turn out that the physical, nonrotational model reveals very special cancelations which are essential to carrying the analysis. In addition, the search for compactness to “go from weak-to-strong” characterizes the latter stages of the work where the critical embeddings are not compact. Methods of compensated compactness must be employed throughout the analysis. The appendix provides: (a) general results of relevance to these lectures from the area of infinite dimensional dynamical systems; (b) microlocal estimates critical in the proof of strong stability, in non-rotational case.

3.6 Plate-to-Flow Mapping: Properties and Discussion 3.6.1 The Neumann Wave Equation on the Half-space In studying flow-structure interactions, a critical role is played by the interface between the dynamics of the 3D wave and the 2D plate. This interaction occurs via the boundary (Neumann wave boundary conditions) and the boundary traces of the aeroelastic potential acting as a forcing on the plate. A natural approach to the problem is to consider two decoupled dynamics and resolve the interaction via some sort of fixed point. However, there are several things to notice.

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183

• The boundary data perturbing the flow involve finite energy solutions of the plate. These are generally .L2 (Ω) elements—for instance, .ut when .α = 0; when rotational inertia are included .ut ∈ H 1 . • As described above, .L2 (R3+ ) on the boundary in Neumann problem do not produce finite energy solutions. • Hence, a decoupling approach is bound to fail, unless .α > 0. • There is another predicament, even if .α > 0. The aeroelastic potential on the boundary is a trace of .L2 (R3+ ) function—not necessarily a priori defined. Here, however, for .α > 0, the dynamics of the plate can be forced by .H −1 (Ω) forcing. So, the decoupling strategy has a chance to succeed in this case. In fact, this was shown in [10] and will be reported below. • However, the non-rotational case was clearly open. With respect to the well-posedness, one must search (as usual) for some cancelation of “bad” (ill-defined) terms [81]. And this indeed happens, leading to a relatively painless (semigroup-based) proof of subsonic well-posedness [133]. The supersonic case requires additional elements of distributional trace regularity theory [45]. But when it comes to long-time behavior, particularly strong stability, one needs to discover certain properties of the flow as perturbed by the structure. The key to the strategy is first to look at the structure forced by the flow. Here we have at our disposal Huygen’s principle and semi-explicit formulas for the structural displacements after some time (owing to the flow being defined on the half-space 3 .R+ ). This strategy leads to characterization of long-time behavior of structural displacements, which in fact works for all .α ⩾ 0. However, transferring stability from the boundary—2D plate stability to 3D flow—is quite challenging. Decoupling is the natural approach and does indeed work well when .α > 0. But for .α = 0 one is stacked with same loss of derivatives discussed above, in an attempt to close the argument. And this is where the specific structure of the boundary interaction plays a critical role. It allows us to use microlocal spaces [11, 126], which enable more precise measurements of the (plate) boundary inputs entering into the Neumann hyperbolic map. In the section following, we discuss the properties of the classic hyperbolic Neumann map.

3.6.2 Flow Formulae The energy balance for the entire system already displays that any stability result must depend on a “hidden dissipation” which is not displayed through energetic considerations. Thus, a nicely wrapped “black box” must be unwrapped in order to account for some “anomalous” dissipation. The first step toward this goal is to study the effect of the downwash on the flow, i.e., the response of dynamic Neumann map to the excitations caused by tangential forcing at the boundary. This leads to an analysis of the flow map lifting the boundary data via Neumann solver.

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To this end, let us consider the “decoupled” Neumann flow problem: ⎧ 2 ⎪ ⎪  U ∂x ) ϕ = Δϕ ⎨(∂t +  ∂z ϕ  = h(x, y, t) . z=0 ⎪ ⎪ ⎩ ϕ(t0 ) = ϕ0 ; ϕt (t0 ) = ϕ1

in R3+ in R2

(3.23)

for which we have [25, 42, 106]: 1 3 2 3 Theorem .U ∈ R, .U /= ±1; take .(ϕ0 , ϕ1 ) ∈ H (R ) × L (R ). If  3.6.1 Assume  1/2 2 .h ∈ C [t0 , ∞); H (R ) then (3.23) is well-posed (in the weak sense) with

    ϕ ∈ C [t0 , ∞); H 1 (R3+ ) , ϕt ∈ C [t0 , ∞); L2 (R3+ ) .

.

Remark 3.6.1 Finite energy .H 1 (Ω) × L2 (Ω) solutions would be obtained if .h ∈ H 1/3 ((0, T ) × R2 ) [126]. Now, let us denote .ϕ ∗ by the solution to (3.23) with .h ≡ 0, and .ϕ ∗∗ as the solution to (3.23) with .ϕ0 = ϕ1 ≡ 0. We may look at .ϕ ∗ and .ϕ ∗∗ separately via linearity of the flow equation. With .(ϕ0 , ϕ1 ) ∈ H 1 (R3+ ) × L2 (R3+ ) one obtains [45, 106]: ∗ ∗ 1 3 2 3 .(ϕ (t), ϕt (t)) ∈ H (R+ ) × L (R+ ). Thus, by the established well-posedness of the full flow-plate system for .ϕ = ϕ ∗ + ϕ ∗∗ , we also have that .(ϕ ∗∗ (t), ϕt∗∗ (t)) ∈ H 1 (R3+ ) × L2 (R3+ ). In this section, we will look at .ϕ ∗ and .ϕ ∗∗ separately and establish results regarding each of them. These results will be used to obtain useful estimates in the next section. For the analysis of .ϕ ∗ we use the tools developed in [24, 25], namely, the Kirchhoff type representation for the solution .ϕ ∗ (x, t) in .R3+ (see, e.g., [42, Theorem 6.6.12]). We conclude that if the initial data .ϕ0 and .ϕ1 are localized in the ball .Kρ ≡ R3+ ∩ Bρ (R3 ), then by finite dependence on the domain of the signal in 3-D (Huygen’s principle), one obtains for any .ρ˜ that .ϕ ∗ (x, t) ≡ 0 for all .x ∈ Kρ˜ and .t ⩾ tρ˜ . Thus .ϕ ∗ tends to zero in the sense of the local flow energy, i.e., ‖∇ϕ ∗ (t)‖2L2 (K ) + ‖ϕt∗ (t)‖2L2 (K

.

ρ˜

ρ˜ )

→ 0, t → ∞,

(3.24)

for all fixed .ρ˜ > 0. Also, in this case,  .

 ∂t + U ∂x1 γ [ϕ ∗ ] ≡ 0, x ∈ Ω, t ⩾ tρ˜ .

(3.25)

Now, let .x = (x, y, z) = (x1 , x2 , x3 ) (as dictated by convenience). Then, for .ϕ ∗∗ as above, we have the following result [42]:

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

185

Theorem 3.6.2 let h(x, y, t) = [ut (x, y, t) + U ux (x, y, t)]ext ,

.

there exists a time .t ∗ (Ω, U ) such that, for all .t > t ∗ , we have the following representation for the weak solution: ϕ ∗∗ (x, t) = −

.

χ (t − x3 ) 2π

t x3

2π 0

(u†t (x, t, s, θ )+U u†x1 (x, t, s, θ ))dθ ds,

(3.26)

where .χ (s) is the Heaviside function, and the definition of .u† is given below in (3.30). The time .t ∗ is given by: t ∗ = inf{t : x(U, θ, s) ∈ / Ωfor all(x1 , x2 ) ∈ Ω, θ ∈ [0, 2π ], ands > t},

.

(3.27)

with x(U, θ, s) = (x1 − (U + sin θ )s, x2 − s cos θ ) ⊂ R2 .

.

(3.28)

We may use the elementary formulae d † h (x, t, s, θ ) − U ∂x1 h† (x, t, s, θ ) ds s − [Mθ h† ](x, t, s, θ ), 2 2 s − x3

∂t h† (x, t, s, θ ) = − .

(3.29)

where .Mθ = sin θ ∂x1 + cos θ ∂x2 and     2 2 2 2 x1 − U s + s − x3 sin θ, x2 − s − x3 cos θ, t − s , .u (x, t, s, θ ) = [u]ext †

(3.30) in computing partials of .ϕ ∗∗ . We recall that .Ω is a compact subset of .R2 , and thus .[u]ext vanishes for spatial arguments .(x1 , x2 ) ∈ / Ω. We also note that, owing to the definition of the escape time .t ∗ as above, and the definition of .u† , the integral in (3.26) can be truncated (or in subsequent partial derivative calculations) at .t = t ∗ , since .u† (the integrand) will vanish for .s > t ∗ .

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For .ϕt∗∗ , observing several cancelations due to (3.29) we have: 1 = 2π

ϕt∗∗ (x, t)

 0



.

2π

t∗

+

dθ u†t (x, t, t ∗ , θ ) −

ds 

x3

s s 2 − x32





2π

dθ u†t (x, t, x3 , θ )

0

 dθ [Mθ u†t ](x, t, s, θ ) .

2π

0

(3.31)

Similarly, the spatial partials for .i = 1, 2 are: ϕx∗∗i (x, t) =

.

1 = 2π



t∗

x3



1 2π 2π



t∗

x3



2π

0

[∂t + U ∂x1 ]u†xi (x, t, s, θ )dθ ds

U ∂x1 u†xi (x, t, s, θ )dθ ds

0

1 + 2π



t∗

x3



2π

0

(3.32)

∂t u†xi (x, t, s, θ )dθ ds.

Differentiation in .x3 is direct, and so with cancelation, it yields: ∂x3 ϕ ∗∗ (x, t) = (∂t + U ∂x1 )u(x1 − U x3 , x2 , t − x3 ) (3.33) 2π t∗   x3 1  + dθ (∂t + U ∂x1 )[Mθ u]† (x, t, s, θ ). 2π x3 s 2 − x32 0

.

From these direct calculations, bounds on solutions can be obtained directly [110, Lemma 8] using interpolation: Lemma 3.6.3 For (3.23), taken with .h(x, t) = (ut + U ux )ext , we have ‖∇ϕ ∗∗ (t)‖2η,Kρ + ‖ϕt∗∗ (t)‖2η,Kρ  ⩽ C(ρ) ‖u(·)‖2 s+η ∗

.

H

2+η

(t−t ,t;H0

(Ω))

+ ‖ut (·)‖2

 1+η

H s+η (t−t ∗ ,t;H0

(Ω))

(3.34) for .s, η ⩾ 0, .0 < s + η < 1/2 and .t > t ∗ (U, Ω). From here, for smooth solutions, we can explicitly solve for the needed Dirichlet trace of the material derivative appearing on the RHS of (3.6) in the plate equation in terms of the Neumann data .h = [ut + U ux1 ]. Considering the term rΩ γ

.



     ∂t + U ∂x1 ϕ = rΩ γ ∂t + U ∂x1 ϕ ∗∗

for .t > tρ by (3.25), where again .ρ corresponds to the .supp(ϕ0 ), supp(ϕ1 ) ⊂ Kρ . Using the above expressions for .∂t ϕ ∗∗ (3.31) and .∂x1 ϕ ∗∗ (3.32), we obtain [owning to Hughens ] the expression in terms of the plate displacement only.

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

    rΩ ∂t + U ∂x1 γ ϕ ∗∗ = −(∂t + U ∂x1 )u − q(ut ),

.

187

(3.35)

for .t ≥ max{t ∗ , tρ } (.t ∗ as defined above in (3.27)) with q(ut ) =

.

1 2π





t∗

2π

ds 0

0

  dθ Mθ2 [u]ext (x(U, θ, s)) ,

(3.36)

and .x(U, θ, s) as in (3.28).   The notation above for .ut indicates the entire set . u(t + s) : s ∈ (−t ∗ , 0) , where .t ∗ is the fixed delay time given in (3.27) depending only on .Ω and U ; this notation is used in considerations with dynamical systems with delay/memory [28, 34, 42].

3.6.3 Dynamical Systems Framework for the Plate Huygen’s principle, along with the point-wise formulate of the previous section— including the calculation of the “Neumann-to-material-derivative trace” in (3.36)— allows us to obtain the theorem below by waiting a time .t # = max{t ∗ , tρ }. Theorem 3.6.4 Let .α, k ⩾ 0, and .(u0 , u1 ; ϕ0 , ϕ1 )T ∈ H02 (Ω)×L2α (Ω)×H 1 (R3+ )× L2 (R3+ ). Assume that there exists an .ρ such that .ϕ0 (x) = ϕ1 (x) = 0 for outside .Kρ . Then there exists a time .t # (ρ, U, Ω) > 0 such that for all .t > t # the plate solution .u(t) to (3.6) satisfies the following equation (in a weak sense): Mα utt + Δ2 u + kMα ut + fV (u) = p0 − (∂t + U ∂x1 )u − q(ut )

.

(3.37)

with q(ut ) =

.

1 2π





t∗

ds 0

0

2π

dθ [Mθ2 uext ](x1 − (U + sin θ )s, x2 − s cos θ, t − s),

with .Mθ and .t ∗ as in the previous section.

(3.38)

Proof Here we note that with .(ϕ0 , ϕ1 ) ∈ H 1 (R3+ ) × L2 (R3+ ) one obtains [45, 106] 3 3 ∗ ∗ 1 .(ϕ (t), ϕt (t)) ∈ H (R+ ) × L2 (R+ ). Thus, by the well-posedness theorems ∗ ∗∗ presented above, since .ϕ − ϕ = ϕ we also have that (ϕ ∗∗ (t), ϕt∗∗ (t)) ∈ H 1 (R3+ ) × L2 (R3+ ).

.

We can hence write           rΩ ∂t + U ∂x1 γ [ϕ] = rΩ ∂t + U ∂x1 γ ϕ ∗ + rΩ ∂t + U ∂x1 γ ϕ ∗∗ . (3.39)

.

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Applying Eqs. (3.25) and (3.35) to Eq. (3.39), we get   rΩ ∂t + U ∂x1 γ [ϕ] = −(∂t + U ∂x1 )u − q u (t), x ∈ Ω, t ⩾ max{tρ , t ∗ }. (3.40) Finally, applying Eqs. (3.40) to (3.6), we get the desired result ⨆ ⨅ .

We then have the following direct estimates on the delay potential .q(ut ) [24, 25, 42, 44]: Proposition 3.6.5 Let .q(ut ) be given by (3.38). Then ||q(ut )||2−1 ⩽ ct ∗

.



t t−t ∗

||u(τ )||21 dτ

(3.41)

for any .u ∈ L2 (t − t ∗ , t; H01 (Ω)). If .u ∈ L2loc (−t ∗ , +∞; (H 2 ∩ H01 )(Ω)) we also have t t t t 2 ∗ .||q(u )|| ⩽ ct ||u(τ )||22 dτ, ||q(uτ )||2 dτ ⩽ c[t ∗ ]2 ||u(τ )||22 dτ t−t ∗

0

−t ∗

(3.42)

for all .t ⩾ 0. With these estimates, the system given in (3.37)–(3.38) is independently wellposed as a plate equation with memory [42, 44], specifically, for initial data u(0) = u0 ∈ H02 (Ω), ut (0) = u1 ∈ L2α (Ω)u|t∈(−t ∗ ,0) = η ∈ L2 (−t ∗ , 0; H02 (Ω)).

.

Remark 3.6.2 Note that the estimates for delay term q given in Proposition (3.6.5) reveal compact behavior on the state space corresponding to .α > 0. This is because the RHS of the plate equation is permitted to reside in .H −1 (Ω). However, this will not the case when .α = 0. In the latter case, the regularity of delay terms is at the critical level. As is well-known, this particular feature provides for major challenges at the level of asymptotic compactness. In application, we will consider an initial datum .y0 ∈ Y corresponding to the full flow-plate dynamics .St (y0 ) in (3.6). We wait a sufficiently long time .t # (ρ, U, Ω) and employ the reduction result Theorem 3.6.4, and we may consider the “initial time” (.t = t0 > t # ) for the delay dynamics. At such a time, the data which is fed into (3.37) is .x0 = (u(t0 ), ut (t0 ), ut0 ), where this data is determined by the full dynamics of (3.6) on .(t0 − t ∗ , t0 ). Thus, given a trajectory .St (y0 ) = y(t) = (ϕ(t), ϕt (t); u(t), ut (t))T ∈ Y , we may analyze the corresponding delay evolution   2 2 2 −t ∗ , 0; H 2 (Ω) , with given data .x ∈ H. .(Tt , H), with .H ≡ H (Ω) × L (Ω) × L 0 0 0   We then have that .Tt (x0 ) = u(t), ut (t); ut with .x0 = (u0 , u1 , η). The natural norm is taken to be

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

||(u, v; η)||2H ≡ ||Δu||2 + ||v||2 +

.

0

−t ∗

189

||Δη(t + s)||2 ds.

Using standard multiplier methods, along with the a priori boundedness in Lemma 3.8.7 and sharp regularity of the Airy function (3.17), we obtain via Grönwall’s inequality the Lipschitz estimate below. Lemma 3.6.6 Suppose .ui (t) for .i = 1, 2 are solutions to (3.37) with different initial data and .z = u1 − u2 . Additionally, assume that ||uit (t)||2L2 (Ω) + ||Δui (t)||2 ⩽ R 2 , i = 1, 2

.

α

(3.43)

for some .R > 0 and all .t ∈ [0, T ]. Then there exists .C > 0 and .aR ≡ aR (t ∗ ) > 0 such that

2 .||zt (t)|| 2 + ||Δz(t)||2 ⩽CeaR t ||Δ(u10 − u20 )||2 + ||u11 − u21 ||21 (3.44) L (Ω) α

+

0

−t ∗

||η1 (τ ) − η2 (τ )||22 dτ



for all .t ∈ [0, T ]. A critical role in the estimate above is played by locally Lipschitz nature of the von Karman nonlinearity on the state space. While this property is obtained in a standard fashion when .α > 0, where .H −1 (Ω) regularity suffices, it is far from obvious when .α = 0. It is here when “sharp” regularity of Airy enters the picture-recall (3.17), (3.15).

3.7 Discussion of Past Stabilization Results Various special cases of our main stabilization results have been dealt with in the past, in order to circumvent intrinsic lack of compactness along with compromised dissipation. We shall briefly recall these results which are of independent interest and provide contrast and context for the results presented below. Before we go into the details let us discuss the previous mathematical work on this and closely related models. Early engineering references address panel flutter (in the comparable formulation to (3.6)) as motivated by the paneling and external layers on aircraft and projectiles [62, 63]. We make special note of the work of Bolotin [21], whose early work has the mathematical formulation of the flow-plate system here, as well as good mathematical insight into a variety of qualitative features of the dynamics. Later, the work of Chueshov et al. began to address flow-plate models in a modern PDE and dynamical systems sense [24–26]—indeed, Chueshov should be given credit as a driving force for the analysis of this and other

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models of mathematical aeroelasticity. More broadly, we mention other seminal works in mathematical aeroelasticity [12, 83, 84], as well as the surveys [38, 39] and the book chapter [89] which provide an overview of mathematical aeroelasticity, including some modeling discussions for configurations other than that of a panel. Specific to the models described in this treatment, we point to early work on the delay dynamical system as it appears here can be found in [27, 28, 34, 37, 47]. Later, the works [24–26] consider the system presented here as (3.6). Well-posedness is addressed through Galerkin constructions with good microlocal estimates [106] applied by decoupling the flow-plate system. Later, stabilizationtype results appeared for the flow-plate system when beneficial thermal effects are accounted for in the plate [110, 111]. In the monograph [42], many results appeared for attractors for the plate system, though not explicitly using the more recently developed quasi-stability theory. A proof of convergence to equilibrium for the model discussed here was outlined in both [36, 42], without details. In the case without rotational inertia—namely .α = 0—well-posedness was obtained for the first time in [133] using a semigroup approach [109, 116], and later again with boundary dissipation in [96]. Stabilization to equilibria was considered in the .α = 0 case (with only weak damping) in the sequence [97] and [98]; these results are much more complicated, the nonlinearity is not principally that of von Karman, and the results are in some sense partial accounting for a simplified Berger’s type of modeling. Von Karman nonlinear effects—fully non-local—present a major challenge to be contended with later in Sect. 1.9.

3.7.1 Re-statement of Main Stabilization Result To focus the attention we recall the formulation of the main stability result present here. We provide some more technical discussion in relation to what was already established in the literature. The main past references are [10, 42, 97, 98, 110, 111]. Theorem 3.7.1 Let .0 ⩽ U < 1, .α ≥ 0, and .k > 0. Assume .p0 ∈ L2 (Ω) and .F0 ∈ H 4 (Ω). Then any solution .(ϕ(t), ϕt (t); u(t), ut (t)) to (3.6) with Data .(ϕ0 , ϕ1 ; u0 , u1 ) ∈ Y that are spatially localized in the flow component has that .

lim

inf

t→∞ (u, ˆ ϕ)∈ ˆ N

 ˆ 2W1 (Kρ ) + ‖ϕt (t)‖2L2 (K ) = 0 ‖u(t) − u‖ ˆ 2H 2 (Ω) + ‖ut (t)‖2L2 (Ω) + ‖ϕ(t) − ϕ‖ 0

α

ρ

for any .ρ > 0, where .Kρ ≡ {x ∈ R3+ : |x| ⩽ ρ} and .N is the stationary set discussed before.

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

191

3.7.2 Weak Stability and Smooth Data for α = 0 As the first step of the long-time behavior analysis, one may consider “weak” stabilization, where the convergence to equilibria set is in a weak topology of the state space. The gradient structure of the full flow-structure interaction will not available for the reduced plate model (with delay), but we will be able to show the existence of compact attracting set for the plate. The latter allows us to establish stabilization to a stationary set for the sufficiently regular initial data regular. This, in turn, translates consistency and stability into weak convergence of trajectories for the data in the phase space. The corresponding result is provided below in Theorem 3.7.4. For finite energy solutions to the flow-plate system, a uniform-in-time bound on plate solutions in a higher topology yields the desired convergence to equilibria result in the topology .Yρ for any .ρ > 0. This result is independent of the particular structure of the plate dynamics—only global-in-time bounds in higher norms of the plate solution .(u, ut ) are needed; indeed, Theorem 3.7.2 follows from the structure of the (linear) flow-plate coupling via the Neumann flow condition, written in terms of .[ut + U ux ]ext , and appearing in Theorem 3.6.2. Theorem 3.7.2 Suppose .0 ⩽ U < 1 and .p0 ∈ L2 (Ω), and take any .k > 0. Let .(u, ϕ) be a weak solution to (3.6) with finite energy (flow-localized) initial data in ∗ .y0 ∈ Yρ0 . If there is a time .T so that .

sup

t∈[T ∗ ,∞)

 ||u(t)||24 + ||Δut (t)||20 + ||utt (t)||20 ⩽ C1 ,

(3.45)

then for any sequence of times .tn → +∞, there is a subsequence of times .tnk and a point . y = ( u, 0;  ϕ , 0) with .( u,  ϕ ) ∈ N so that .

lim dYρ (Stnk (y0 ),  y) = 0

k→∞

for any .ρ > 0. This implies the strong stability (in the topology of .Yρ ) to the equilibria set. We present this theorem as an independent result, in the spirit of what is used in [110, 111]. However, a word of caution: the assumed bound (3.45) will be valid only for “certain” solutions to the plate problem, and certainly not for every weak solution. This is in striking contrast with [110, 111] where the smoothing properties of thermoelasticity provide the additional boundedness of all plate solutions in higher topologies. Here, we can obtain the global-in-time bounds in higher topologies (as in (3.45) below) by considering smooth initial data for the flow-plate system and propagating this regularity. In order to utilize the key technical supporting result in Theorem 3.7.2, we require a propagation of regularity result for the dynamics. Specifically, we must show that for “smooth” initial data, we have “smooth” time dynamics; this is

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Theorem 3.7.3. Such a result was given and proved in [99, (2015)]. To show infinitetime propagation of the plate dynamics, we rely on the full flow-plate dynamics to achieve propagation of the initial regularity on any interval .[0, T ∗ ]. Once this is achieved, we may work on the reduced delay plate (after sufficient time has passed) and utilize sharp bounds to obtain regularity of plate trajectories on the infinite horizon (propagation on .[T ∗ , ∞)). This part of the argument depends on (i) uniform exponential decay for the nonlinear plate equation with large static and viscous damping [92], and (ii) specific properties of nonlinearity (in the von Karman case one applies the sharp regularity of Airy stress function as in (3.17)—see also [42, p. 44]). Theorem 3.7.3 Consider the dynamics .(St , Y ) corresponding to (3.6). Consider initial data .ym ∈ Y2 such that .y m ∈ BR (Ys ) and take .k > 0. Then we have that for m m the trajectory .St (y m ) = (um (t), um t (t); ϕ (t), ϕt (t))   (um (·), ϕ m (·)) ∈ C 1 0, T ; H02 (Ω) × H 1 (R3+ ) ,

.

for any T , along with the bound

sup

.

t∈[0,T )

 2 m 2 ‖ + ‖u ‖ ‖Δum t Ω tt Ω ≤ Cm,T < ∞.

(3.46)

Additionally, if we assume the flow initial data are localized and consider the delayed plate trajectory (via the reduction result Theorem 3.6.4), we will have: .

sup t∈[0,∞)

 2 m 2 ‖Δum t ‖Ω + ‖utt ‖Ω ⩽ C1 < ∞.

(3.47)

Then, by the boundedness in time of each of the terms in the (3.37), we have sup ‖Δ2 um (t)‖0 ⩽ C2 ,

.

t∈[0,∞)

where we critically used the previous bound in (3.47). In particular, this implies that, taking into account the clamped boundary conditions: .

sup ‖um (t)‖4,Ω ⩽ C3 .

(3.48)

t∈[0,∞)

Each of the terms .Ci above depends on the intrinsic parameters in the problem (including the respective loading). Putting Theorems 3.7.3 and 3.7.2 together, we obtain the main result. Theorem 3.7.4 Consider .ym ∈ Y2 ∩BR (Y ) ⊂ (H 4 ∩H02 )(Ω)×H02 (Ω)×H 2 (R3+ )× H 1 (R3+ ) ⊂ Y . Suppose that the initial flow data are supported on a ball of radius .ρ0 (as in Theorem 3.6.4). Suppose .k > 0. Then for any sequence of times .tn → ∞

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

193

there is a subsequence of times .tnk identified by .tk and a point .yˆ = (u, ˆ 0; ϕ, ˆ 0) ∈ Y, with .(u, ˆ ϕ) ˆ ∈ N , so that ‖Stk (y m ) − y‖ ˆ Yρ → 0, tk → ∞.

.

This implies the primary result: strong convergence of .(u(t), ϕ(t)) to the equilibria set .N for initial data .y0 ∈ Y2 ∩ Yρ0 .

3.7.3 Other Past Stability Results with α = 0 Improving convergence from “weak-to-strong” convergence is the central challenge. It is here where the lack of compactness of the flow-plate resolvent comes to play and becomes a main obstruction in proof approaches. To set the stage, in this section we shall provide several results where this challenge has been dealt with through regularizations or model modifications. These include:(i) additional assumptions on the size of the damping, (ii) a simpler nonlinear term (Berger’s nonlinearity), or (iii) the inclusion of thermal effects.

Large Static and Viscous Damping It is very natural to ask, after obtaining the desired strong convergence to equilibria result for smooth data, if one can extend this proof approach to finite energy data; this would be to have Theorem 3.7.1 with data (u0 , u1 ; ϕ0 , ϕ1 ) ∈ Ys

.

(of course, still spatially localized flow data component). In particular, since the result we seek is one of strong stability, a typical approach involves approximating the finite energy data .y0 ∈ Ys with smooth data .y0m ∈ Y2 and invoking Hadamard continuity of the semigroup .(St , Ys ) to push the convergence from the approximating sequence onto the data .y0 . Unfortunately, the semigroup .(St , Ys ) corresponding to solutions to (3.6) is not known to be Hadamard continuous on the infinite time horizon for .α = 0. In addition, even with large damping .k >> 0, this property is not clear—this has to do with the complex structure of the global attractor for the non-gradient reduced plate dynamics, as well as the non-uniqueness of stationary solutions to (3.6).

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However, if one considers large static and viscous damping of the form:  utt + Δ2 u + K[u + ut ] − [u, v(u) + F0 ] = p0 (x) + [ϕt + U ϕx ]Ω ,

(3.49)

.

for K sufficiently large, we can obtain the Hadamard continuity property (via the variation of parameters formula) by forcing the linear semigroup to exhibit exponential decay. This is a main strategy [117] for extending the results for smooth data in [99]. We now state the associated results: Lemma 3.7.5 Let .0 ⩽ U < 1 and .α ⩾ 0, and assume .p0 ∈ L2 (Ω). Assume .y0 = (u0 , u1 ; ϕ0 , ϕ1 ) ∈ Ys . Consider the dynamics generated by generalized solutions to (3.6) (taken with plate component (3.49)), denoted .St (y0 ). Assuming that the damping parameter is sufficiently large .K ≥ Kc > 0, the semigroup .St (·) is uniform-in-time Hadamard continuous, i.e., for any sequence .y0m → y0 in Y and any .ϵ > 0 there is an M so that for .m > M .

sup ‖St (y0m ) − St (y0 )‖Yρ < ϵ. t>0

From this, we conclude one of the main results in [99]: Theorem 3.7.6 Take the same assumptions as in Lemma 3.7.5. Then there is a minimal damping coefficient .Kc so that for .K > Kc > 0 any generalized solution .(u(t), ϕ(t)) to the system with finite energy, localized (in space) initial flow data has the property that .

lim

inf

t→∞ (u, ˆ ϕ)∈ ˆ N

‖u(t) − u‖ ˆ 2H 2 (Ω) + ‖ut (t)‖2L2 (Ω) + ‖ϕ(t) − ϕ‖ ˆ 2H 1 (K ) + ‖ϕt (t)‖2L2 (K ρ



ρ)

= 0, for anyρ > 0. Remark 3.7.1 The minimal damping coefficient .Kc depends on the invariant set for the plate dynamics, which itself depends on the loading .p0 and .F0 , as well as the domain .Ω and the constants .U, ρ0 , but is independent on the particular initial data of the system. It is also worth noting that, in some sense, increasing the static and viscous damping coefficients simultaneously (to exponentially stabilize the linear dynamics) runs the risk of destroying the structure of equilibria set. Indeed, when K is large, the equilibria set may collapse into a single point.

The Berger Nonlinearity In all of the discussions above, and below, there are other pertinent physical nonlinearities. The principal one among them is the Berger nonlinearity, as referenced earlier and described in detail below. The engineering literature (see the overview in

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

195

[79]) analyzes the validity of Berger’s plate approximation (originally appearing in [18]) and concludes that it is accurate in the case of clamped and hinged boundary conditions. The approximation (at least in 2D) is based on the (ad hoc) assumption that the second strain invariant is negligible. Mathematically, the Berger dynamics are addressed in [28, 41, 48] (in the context of aeroelasticity). Much of the abstract work in [42] (focused on the scalar von Karman nonlinearity) also applies to the Berger dynamics. The main results appearing in [43–46, 133] for .f = fV apply to the Berger nonlinearity, as well as other “physical” nonlinearities satisfying a specific set of bounds (which are referred to as nonlinearities of physical type)—see [6, 45]. Let us recall the Berger nonlinearity: fB (u) = [b − ||∇u||2 ]Δu.

.

(3.50)

The parameter .b ∈ R is a physical parameter [79] that corresponds to in-plane compression or stretching.2 The plate energy is defined as above [42, 92], with .Π(u) the potential of the nonlinear and non-conservative forces, given in this case by Π(u) = ΠB (u) =

.

1 b ||∇u||4Ω − ||∇u||2Ω − 〈p0 , u〉Ω . 4 2

(3.51)

Owing to the fact that .fB is a simplification of .fV , it is often easier to work with. Additionally, for all of the main results presented thus far, results that hold for .fV also hold with .fB replacing it (mutatis mutandis). However, whenever higher topologies are invoked (and the PDE differentiated), the structure of .fB is more amenable to certain analytic techniques that do not obtain for .fV . (This is detailed below.) In particular, when considering strong stabilization to equilibria of the full flow-plate dynamics with .U < 1, a major difference is noted. This is the focus of [97], which considers finite energy initial data and large viscous damping .k >> 0 when .α = 0, and is able to obtain the stabilization to equilibria result (as in Theorem 3.7.1). We again emphasize that this proof critically depends on the structure of .fB , and the applied techniques do not apply for .f = fV . Decomposing the Berger Dynamics When considering long-time behavior of Berger plate models [86, 87, 97], we often critically make use of a decomposition technique, whereby the nonlinear dynamics are broken up into a smooth component and an exponentially decaying component. This is now outlined, and we point out what fails for von Karman dynamics. The idea is based on one in [72]. Write the solution .u = z + w where z and w correspond to the system:

often choose the more mathematically interesting non-dissipative case, with .b ⩾ 0—this is the case that can give rise to static buckling as well as chaos, see [60].

2 We

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⎧ 2 ⎪ ⎪ ⎨ztt + kzt + βz + Δ z + F(z, u) = H (z) inΩ . F(z, u) = [κ − ||∇u||2 ]Δz ⎪ ⎪ ⎩z = ∂ z = 0 on𝚪, z(t0 ) = u(t0 ), zt (t0 ) = ut (t0 ). ν ⎧ 2 ⎪ ⎪ ⎨wtt + kwt + Δ w = p0 + H (w) + F(w, u) + βz inΩ . F(w, u) = [κ − ||∇u||2 ]Δw ⎪ ⎪ ⎩w = ∂ w = 0 on𝚪, w(t ) = 0, w (t ) = 0. ν

0

t

(3.52)

(3.53)

0

Remark 3.7.2 Since we are taking null initial data for the w portion of the decomposition, we note that what is driving the dynamics here is the presence of z (via .u = z + w) in the term .F(w, u). Given u (coming from a well-posed problem), the .z/w system is well-posed with 1 .z + w = u. Let .Eβ (z(t)) = [||Δz||2 + ||zt ||2 + β||z||2 ]. For the decomposed plate 2 dynamics, with some mild assumptions on the structure of the map H [97] we can obtain: Lemma 3.7.7 There exists .ke (R) > 0 and .βe (R) > 0 such that for .k > ke and β > βe the quantity .Eβ (z(t)) decays exponentially to zero.

.

Lemma 3.7.8 There exists a .kQ such that for any .k > kQ all .β > 0 the evolution  .(w, wt ) on .Ypl  corresponding to (3.53) has that .(w, wt ) ∈ C [t0 , ∞); H 4 (Ω) × H 2 (Ω) . In the above, R denotes dependence on the size of some invariant set from which the data are taken. Lemma 3.7.7 is proved with a Lyapunov approach. To address the w dynamics, recall .F(w, u) = [κ − ||∇u||2 ]Δw. Consider the time-differentiated w dynamics; let .w  = wt : ⎧ d ⎪ {F(w, u)} + βzt inΩ tt + k wt + Δ2 w  = H (wt ) − ⎪ ⎨w dt . w  = ∂ν w  = 0 on𝚪 ⎪ ⎪ ⎩ w (t0 ) = 0, w t (t0 ) = 0.

(3.54)

Noting that .||zt || decays exponentially and is thus bounded, we will obtain higher w||2 + || wt || (after which, elliptic regularity can be norm, global-in-time bound of .|| applied). We invoke the exponential decay of the linear, damped (static and viscous) plate equation in .w : w tt + k wt + K w  + Δ2 w  =H (wt ) −

.

d {F(w, u)} + βzt + K w . dt

(3.55)

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

197

 (t) = ( Let .W w (t), w t (t)). For .t0 sufficiently large we utilize the variation of parameters formula:      .W (t)

Ypl

     ⩽ C  W (t0 )

Ypl

+

t

 e−ω(k,K)(t−s) ||H (wt )||0

(3.56)

t0

     d   w||0 + β||zt ||0 ds. +  F(w, u) + K|| dt 0

We note that .ω → ∞ as .min[k, K] → ∞. We also have:  d         F(w, u) =(∇u, ∇ut )Δw + ||∇u||2 Δwt  0 0 dt

(3.57)

.

w|| . ⩽ ||(Δu, ut )Δw|| + ||∇u||2 ||Δ Remark 3.7.3 This is precisely the step where an argument, such as the one here, will fail for .f (u) = fV (u). The exact structure of the nonlocality in the Berger nonlinearity .fB is what permits the above step and resulting bound. Thus .

    d    F(w, u) ⩽ C(R) 1 + ||Δ w|| .   dt 0

   (t), w t (t)   w

.

Ypl

⩽C||( w(t0 ), w t (t0 ))||Ypl .   + C(R, U ) K + sup ||( w, w t )||Ypl s∈[t0 ,t]

   t (t0 )  ⩽ C  w (t0 ), w

(3.58)

t

e−ω(k,K)(t−s) ds

t0

Ypl

(3.59)

 1 − e−ω(k,K)(t−t0 )  + C(R, U ) K + sup ||( w, w t )||Ypl . ω(k, K) s∈[t0 ,t] The global-in-time bound follows by taking supremums for .t ∈ [t0 , ∞) on both sides and up-scaling .min[k, K] (and thus scaling .ω(k, K)).

Rotational Inertia and Thermal Effects—Velocity Smoothing We now mention two other closely related flow-plate (cf. (3.6)) scenarios which have been intensely studied in the literature to date: (i) the addition of thermoelastic dynamics to the plate dynamics [94] and (ii) the inclusion of rotational inertia and strong mechanical damping (.α > 0).

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The treatments in [110, 111] consider the plate (.α ⩾ 0) with the addition of a heat equation in the plate dynamics, and associated thermoelastic coupling. In this case no damping is necessary, as the analytic smoothing and stability properties of thermoelasticity provide ultimate compactness of the finite energy plate dynamics and, furthermore, provide a stabilizing effect to the flow dynamics. As for (ii) mentioned above, results on ultimate compactness of plate dynamics, as well as convergence (for subsonic flows) of full flow-plate trajectories to points of equilibria, were announced in [36], with a proof appearing in [42]. We now address the difficulties involved in showing a stabilization to equilibria result without assuming either (i) .α > 0 and strong damping of the form .−k2 Δut (as in [36, 42]) or (ii) exploiting parabolic effects in a thermoelastic plate [110, 111]. In both cases, the key task is to first show compact end behavior for the plate dynamics. This requires the use of the reduction of the full flow-plate dynamics to a delayed plate equation (Theorem 3.6.4 above), at which point one may work on this delayed system. In both case (i) and (ii) the ultimate character of the nonlinear component of the model is compact—owing to the fact that parabolic smoothing and rotational inertia both provide .∇ut ∈ L2 (Ω). The results in [44] were the first to show that dissipation could be harvested from the flow in the principal case .α = 0 (via the reduction result) in order to show ultimate compactness of the plate dynamics without imposed mechanical damping nor thermoelastic effects. The major contribution of [97, 99] is the ability to circumvent the seeming lack of natural compactness in the dynamics (particularly in the plate velocity .ut ). Specifically, the methods utilized in showing the (analogous) stabilization to equilibria result in [42, 110, 111] (and references therein) each critically use that .ut → 0 in 1 .H (Ω). This measure of compactness for the plate component is translated (albeit in different ways) to the flow component of the model (via the flow equations in Theorem 3.6.2). We note that in [111] the key to the stabilization result lies in the estimate Lemma 3.6.3 for flow trajectories wherein the flow is bounded in higher norms by the plate trajectory, also in higher norms.

3.7.4 General Approach to Stability in the Remainder of This Chapter We return to the main task of proving strong stability in general case of the model introduced in these lectures. As it has been mentioned several times already, the rotational case, .α > 0, is much simpler. In order to describe the main philosophy behind the proof of the new result for .α = 0, and, also to emphasize the differences between these two cases, we shall run first the full analysis for the rotational case .α > 0 in Sect. 1.8 and then follow it with that of .α = 0 in Sect. 1.9. This will allow see precisely what is missing from the .α = 0 case which justifies the need for a fresh and novel approach.

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

199

Concisely summarizing, we will: • Establish the attracting set for the reduced plate model. This is much more challenging for .α = 0, where there is neither structural dissipation nor compactness are apparent. • Stabilize plate through dissipation integral. • Stabilize flow through Neumann lift (harder for .α = 0 as it requires anisotropic microlocal spaces and explicit microlocal estimates. • Recover weak stationary solutions as limit points along trajectories, using compactness of the plate attractor. • Prove strong convergence to those points (thus to equilibria), which is very challenging for .α = 0 due to the explicit loss of compactness.

3.8 α > 0: Rotational Stabilization 3.8.1 Existence of Attracting Set for the Structure: Statement The main result in this section is that the reduced plate dynamical system .(Tt , H) has a compact global attractor, which has additional nice regularity properties— see the Appendix for a discussion on global attractors and notions from infinite dimensional dynamical systems. By projecting on the non-delay components of the dynamics, we will obtain an attracting set for the plate dynamics as they appear in the full-flow structure interaction. Theorem 3.8.1 (Smooth, Finite Dimensional Global Attractor) Let .α, k > 0, U /= 1, .p0 ∈ L2 (Ω), and .F0 ∈ H 4 (Ω) in (3.6). Also assume the flow data .ϕ0 , ϕ1 ∈ Yρ are localized (with supports in .Kρ , as in Theorem 3.6.4). Then the corresponding delay system .(Tt , H) has a compact global attractor .A of finite fractal dimension in .H. Moreover, .A has additional regularity: any full trajectory t 3 .y(t) = (u(t), ut (t), u ) ⊂ A, .t ⩾ 0, has the property that .u ∈ Cr (R; H (Ω) ∩ 2 2 H0 (Ω)), .ut ∈ Cr (R; H0 (Ω)), and .utt ∈ Cr (R; H) 1(Ω)). .

This can be rephrased for the full system .(St , Y ) by projecting on the first two components of .H: Corollary 3.8.2 With the same hypotheses as Theorem 3.8.1, there exists a compact set .U ⊂ H02 (Ω)×H01 (Ω) of finite fractal dimension such that for any weak solution .(u, ut ; ϕ, ϕt ) to (3.6) has .

  lim dYpl (u(t), ut (t)), U = lim

t→∞

inf

t→∞ (w0 ,w1 )∈U

  ||u(t)−w0 ||22 +||ut (t)−w1 ||21 = 0.

  We also have the additional regularity .U ⊂ H 3 (Ω) ∩ H02 (Ω) × H02 (Ω).

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The proof of Theorem (3.8.1) proceeds in two steps: the construction of an absorbing ball by Lyapunov methods, followed by the attainment of the so-called quasi-stability property on the absorbing ball.

3.8.2 Attractor Construction: Dissipativity of Dynamical System For the non-conservative plate dynamics given by (3.37), we explicitly construct the absorbing ball via a Lyapunov approach. Remark 3.8.1 Recall that the full flow-structure interaction is a gradient system, so that an absorbing ball can be constructed through the gradient structure. But such construct cannot be pushed any further due to the lack of dissipation and compactness in the flow. This is why we consider the reduced system first. Recalling the definition of .Epl from (3.10) and defining the quantities Π∗ (u) =

.

 1 1 1 ||Δu||2 + ||Δv(u)||2 , E∗ (u, ut ) = ||ut ||2L2 (Ω) + Π∗ (u), α 2 2 2

we consider the Lyapunov-type function    V Tt (x) ≡ Epl (u(t), ut (t)) + ν 〈Mα1/2 ut , Mα1/2 u〉 t∗ t  + k〈Mα1/2 u, Mα1/2 u〉 + μ Π∗ (u(τ ))dτ ds,

.

0

(3.60)

t−s

where .Tt (x) ≡ x(t) = (u(t), ut (t), ut ) for .t ⩾ 0,3 and .μ, ν are some small, positive numbers to be specified below. Using the elementary inequality

t∗



t

.

0

t−s

Π∗ (u(τ ))dτ ds ⩽ t

∗



t

t−t ∗

Π∗ (u(τ ))dτ,

  we establish the topological equivalence between .V Tt (x) and .E∗ , which is given by the following lemma. Lemma 3.8.3 With .(Tt , H) giving the delay dynamical system defined in Sect. 3.6.3 and V defined as in (3.60), we have that there exists .ν0 > 0 such that for all .0 < ν ⩽ ν0 there are .c0 (ν0 ), c1 , c(ν0 ), C > 0

3 Without

loss of generality, take .t0 = 0

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

c0 E∗ − c ⩽ V (Tt (x)) ⩽ c1 E∗ + μCt ∗



.

0 −t ∗

Π∗ (u(t + τ ))dτ + c.

201

(3.61)

d V (Tt (x)), coupled with the estimates on the dt nonlinear potential energy Lemma 3.20 and the estimate on .q(ut ) at the .L2 level in Lemma 3.6.5, produces, for .0 < ν < min {ν0 , 1}, and for .μ sufficiently small, the following lemma:

A careful but direct calculation of .

Lemma 3.8.4 For all .k > 0, there exist .μ, ν > 0 sufficiently small, and c(μ, ν, t ∗ , k), C(μ, ν, p0 , F0 ) > 0 such that

.

.

d V (Tt (x)) ⩽C − c E∗ (u, ut ) + dt



0 −t ∗

 Π∗ (u(t + τ ))dτ .

(3.62)

From this lemma and the upper bound in (3.61), we have a .δ(k, μ, ν) > 0 and a C(μ, ν):

.

.

d V (Tt (x)) + δV (Tt (x)) ⩽ C, t > 0. dt

(3.63)

The estimate above in (3.63) implies (via an integrating factor) that V (Tt (x)) ⩽ V (x)e−δt +

.

C (1 − e−δt ). δ

(3.64)

Hence, the set   C .B ≡ x ∈ H : V (x) ⩽ 1 + δ is a bounded forward invariant absorbing set for .(Tt , H). This, along with (3.61), gives that .(Tt , H) is ultimately dissipative in the sense of dynamical systems [9, 128].

3.8.3 Attractor Construction: Asymptotic Smoothness Through Quasi-stability For the discussion of quasi-stability, we begin with the standard observability and energy inequalities which follow from energy methods developed for the wave equation. The details presented for .α = 0 in [44] are unaltered here. Let us utilize the notation that .Ez (t) = ||zt (t)||2L2 (Ω) + ||Δz(t)||2 . We state the α following estimates without proof.

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Lemma 3.8.5 (Preliminary Estimates) Let ui ∈ C(0, T ; H02 (Ω)) ∩ C 1 (0, T ; L2α (Ω)) ∩ L2 (−t ∗ , T ; H02 (Ω))

.

solve (3.37) with appropriate initial conditions on .[0, T ] for .i = 1, 2, .T ⩾ 2t ∗ . Additionally, assume .(ui (t), uit (t)) ∈ BR (Ypl ) for all .t ∈ [0, T ]. Then the following estimates on z holds for some .δ ∈ (0, 2]: Ez (t) +

 Ez dτ ⩽a0 Ez (s) +

t

.

s

− a1



s

||z(τ )||22−δ dτ

+ C(T , R, δ) sup ||z||22−δ

〈f (u1 ) − f (u2 ), zt 〉dτ.

(3.65)

s−t ∗

t

τ ∈[s,t]

s

T Ez (T ) + . 2



T T −t ∗

 Ez (τ )dτ ⩽a2 Ez (0) + 

0 −t ∗

 ||z(τ )||22 dτ

+ C(T , R, δ) sup ||z||22−δ − a3

T

0

− a4

τ ∈[0,T ]



T

〈f (u1 ) − f (u2 ), zt 〉dτ ds

(3.66)

s T

〈f (u1 ) − f (u2 ), zt 〉dτ,

0

with the positive .ai independent of T and R. We note the elementary bound [applicable when .α > 0 only due to the structure of Ez (t)].

.

|〈f (u1 ) − f (u2 ), zt 〉| ⩽ Cϵ ||f (u1 ) − f (u2 )||2−δ + ϵ||zt ||2δ .

.

Remark 3.8.2 Note that in this step we additional regularity of the velocity displacements .zt is used. This can be done owning to .α > 0 here. In such case the estimate for .||f (u1 ) − f (u2 )||−δ follows from standard Airy’s estimates [42, 100]. This is not the case when .α = 0, (3.17). Utilizing the above bound directly, and invoking the locally Lipschitz nature of the von Karman nonlinearity, we see (rescaling constants) that

t

.

s

  〈f (u1 ) − f (u2 ), zt 〉dτ ⩽ ϵ

s

t

||zt ||21 + C(ϵ, R, δ, |t − s|) sup ||z(τ )||22−δ . [s,t]

(3.67)

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

203

This yields the estimate from which the quasi-stability property of .(Tt , H) can be deduced. Lemma 3.8.6 Suppose .z = u1 − u2 as before, with .y i (t) = (ui (t), ut (t)i , ut,i ) and .y i (t) ∈ B (i.e., the trajectories lie in the absorbing ball) for all .t ⩾ 0. Also, let .δ > 0 and .Ez (t) be defined as above. Then there exists a time T such that the following estimate holds: Ez (T ) +

T

T −t ∗

.

 ||z(τ )||22 dτ ≤ β Ez (0) +



0 −t ∗

||z(τ )||22 dτ ) + C(R, T , t ∗ , δ)

sup ||z(τ )||22−δ

τ ∈[0,T ]

(3.68) with .β < 1. Proof of Lemma 3.8.6 Applying (3.67) to (3.65) and (3.66) with .s = 0 and .t = T , we obtain

T

Ez (T ) +

.





Ez dτ ⩽ a0 Ez (0) +

0



sup ||z||22−η + ϵa1

τ ∈[s,t]

T

0

0 −t ∗

 ||z(τ )||22−η dτ

+ C(T , R, η, ϵ)

‖zt (τ )‖21 dτ.

(3.69)

and T Ez (T ) + . 2



T T −t ∗





Ez (τ )dτ ⩽ a2 Ez (0) +





0 −t ∗

||z(τ )||22 dτ



+ C(T , R, η, ϵ) sup ||z||22−η + CT ϵ τ ∈[0,T ]

0

T

(3.70)

‖zt (τ )‖21 dτ

for .T ≥ max{t ∗ , 1}. After adding (3.69) and (3.70) and invoking the Sobolev embeddings, we can drop suitable terms to obtain .

   T T cT T Ez (T ) + αcp − ϵ (a1 + CT ) ‖zt (τ )‖21 dτ + ‖z(τ )‖22 dτ ∗ 2 2 0 T −t (3.71)   0 ≤ A Ez (0) + ||z(τ )||22−δ dτ + C(T , R, δ, ϵ) sup ||z||22−δ , −t ∗

τ ∈[0,T ]

where .0 < c < 1 and .cp is a Poincare constant. Scaling .ϵ small enough, and T large enough, we obtain after simplifying

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Ez (T ) +

.

T T −t ∗

 ||z(τ )||22 dτ ≤ β Ez (0) +

0 −t ∗

 ||z(τ )||22 dτ + C∗ sup ||z(τ )||22−δ τ ∈[0,T ]

(3.72) with .β < 1 and .C∗ = C(R, T , t ∗ , δ).

⨆ ⨅

We note that the necessary Lipschitz estimate in (3.148) holds by Lemma 3.6.6. Via the semigroup property for the evolution .(Tt , H) (iterating on intervals of size T ), we obtain from (3.72) in a standard way the quasi-stability estimate in (3.154). Hence the dynamical system .(Tt , H) is quasi-stable on the absorbing ball .B in the sense of Theorem 3.10.9. This immediately implies the existence of a compact global attractor for this dynamical system.

Attractor and its Properties We conclude this section by pointing out the existence and discussing the regularity of the global attractor for the decoupled plate system. We first observe that Theorem 3.10.9 applies to the dynamical system .(Tt , H). Hence, we have existence of the global attractor .A ⊂⊂ H for the system .(Tt , H) and the estimate: .

 sup ‖ut (t)‖2H 2 (Ω) + ‖utt (t)‖2H 1 (Ω) ⩽ C. t∈R

0

(3.73)

0

Consequently, .ut (t) ∈ H02 (Ω) and .utt (t) ∈ H01 (Ω) for each .t ∈ R. Applying elliptic regularity to the equation Δ2 u = −Mα utt + kMα ut + [u, v(u) + F0 ] + p0 − ut − U ux − q(ut ) ∈ H −1 (Ω)

.

point-wisely in time, we obtain that .u(t) ∈ H 3 (Ω)∩H02 (Ω). Therefore, .A possesses better regularity than .H and we obtain the statement of Theorem 3.8.1.

3.8.4 Boundedness and Finiteness of Dissipation Integral Synthesizing our results up to this point, we have: Lemma 3.8.7 Any weak (and hence generalized or strong) solution to (3.6) will satisfy the bound .

   sup ‖ut ‖2L2 (Ω) + ‖Δu‖2Ω + ‖ϕt ‖2R3 + ‖∇ϕ‖2R3 ≤ C ‖y0 ‖Y < +∞. t⩾0

α

+

+

Thus, solutions are (Lyapunov) stable in time in the norm Y .

(3.74)

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

205

An immediate corollary from the energy identity, then, and the above boundedness is the finiteness of the dissipation integral, which is used critically [.k > 0] below. Corollary 3.8.8 Any solution (3.6) satisfying the energy identity with .k > 0 has the property .

0

∞

‖ut (t)‖2L2 (Ω) dt ≤ K(||y0 ||Y ) < ∞. α

Remark 3.8.3 We note that global-in-time boundedness of solutions cannot be obtained without accounting for nonlinear effects. Also, we note that any generalized solution has the properties: .y(t) ∈ C([0, T ]; Ys ) and .y(t) ∈ C([0, ∞); Y ). This is to say, on infinite time intervals we lose control of the quantity .||ϕ(t)||0 .

3.8.5 Plate Convergence: Weak and Strong Proposition 3.8.9 Let u be a generalized solution to (3.6). Then ut (t) ⇀ 0 in H01 (Ω) (i.e., weakly in H01 (Ω)) as t → ∞. Proof of Proposition 3.8.9 We will show that Mα ut ⇀∗ 0 in H −1 (Ω) and rely on the continuity of Mα : H01 (Ω) → H −1 (Ω). Multiplying the plate equation in (3.6) by a test function w ∈ C0∞ (Ω), we obtain 〈Mα utt , w〉 = − 〈Δu, Δw〉 + 〈p0 , w〉 − k〈Mα ut , w〉 + 〈[u(t), v + F0 ], w〉 (3.75)   + 〈rΩ γ (ϕt + U ϕx ) , w〉.

.

The first and second terms on the RHS of the equality are uniformly bounded in time by Lemma 3.8.7. For the third term, we have .

|k〈Mα ut , w〉| ≤ k(||ut || · ||w||) + kα(||∇ut || · ||∇w||).

(3.76)

For the fourth term, Theorem 1.4.3 and Corollary 1.4.5 from [42] provide us with the following estimates for fv : |〈[u, v + F0 ], w〉| ≤ ‖[u, v]‖‖w‖ + ‖[u, F0 ]‖−1 ‖w‖1 .

≤ C1 ‖u‖32 ‖w‖ + C2 ‖u‖2 ‖F0 ‖2 ‖w‖1 .

To estimate the fifth term above in (3.75), recall that   rΩ ∂t + U ∂x1 γ [ϕ(t)] = −(∂t + U ∂x1 )u(t) − q(ut ),

.

(3.77)

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for t ≥ t ∗ , where t ∗ and q(ut ) are defined as in Theorem 3.6.4. Also, by (3.42), .

       rΩ γ (ϕt (t) + U ϕx (t)) , w  ≤ ‖rΩ γ (ϕt (t) + U ϕx (t)) ‖ · ‖w‖.   ≤ ‖ut (t)‖ + U ‖ux (t)‖ + ‖q(ut )‖ ‖w‖ t   ≤ C ‖ut (t)‖L2α + U ‖u(t)‖2 + ct ∗ ||u(τ )||2 dτ.

(3.78) (3.79)

t−t ∗

The right hand sides of (3.76), (3.77), and (3.78) are then uniformly bounded in time by Lemma 3.8.7. Hence, by (3.75), |(Mα utt , w)| = |∂t (Mα ut , w)| is uniformly bounded in time for each w ∈ C0∞ (Ω). Next we see that ∞ ∞ ∞ 2 2 2       〈Mα ut , w〉 dτ ≤ 〈ut , w〉 dτ + α 〈∇ut , ∇w〉 dτ 0

0

.



≤ C(||w||1 ) 0

0

∞

||ut ||21,Ω

(3.80)

0. By interpolation, for δ > 0, we obtain ||u(tn ) − u|| ¯ 2−δ ≤ ||u(tn ) − u|| ¯ 1−δ ||u(tn ) − u|| ¯ δ1 . 2

.

(3.84)

Since ‖u(tn ) − u‖ ¯ 2 is bounded, from Eqs. 3.83 and 3.84, we have .

max ||u(tn + τ ) − u|| ¯ 2−δ,Ω → 0asn → ∞.

τ ∈[−a,a]

(3.85)

Using a simple contradiction argument, we can push the Sobolev index to 2. Proposition 3.8.12 Let u be a generalized solution to (3.6) and let tn → ∞. Then .

max ||u(tn + τ ) − u|| ¯ 2 → 0asn → ∞.

τ ∈[−a,a]

Proof of Proposition 3.8.12 For any fixed a, consider a subsequence  .

max ||u(tnk + τ )||2

∞

τ ∈[−a,a]

. k=1

Since t → I ‖u(t)‖2 is continuous and [−a, a] is compact, .

max ||u(tnk + τ )||2 = ||u(tnk + τk )||2

τ ∈[−a,a]

for some τk ∈ [−a, a]. From Corollary (3.8.2), we know that the sequence ˜ ∈ H02 (Ω), as {u tnk + τk }∞ k=1 has a convergent subsequence u(tnkl + τkl ) → u well as in any lower Sobolev space; then     ¯ 2−δ − ‖u¯ − u‖ ˜ 2−δ  ≤ ‖u(tnk + τk ) − u‖ ˜ 2−δ → 0 ‖u(tnk + τk ) − u‖

.

(3.86)

From (3.85) we know that ‖u(tnk + τk ) − u‖ ¯ 2−δ ≤ max ||u(tn + τ ) − u|| ¯ 2−δ → 0.

.

τ ∈[−a,a]

(3.87)

Substituting (3.87) in (3.86), we ˜ 2−δ = 0 and we identify the limits.  see ‖u¯ − u‖ Hence, any subsequence of max ||u(tn + τ ) − u|| ¯ 2 has a further convergent τ ∈[−a,a]

subsequence that converges to 0, yielding the result.

⨆ ⨅

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

209

The above result is not explicitly used below but is of independent interest and has not appeared prevalently in previous works, though was mentioned in [10, 11].

3.8.6 Lifting from Plate to Flow Directly t ∗ 2π 1 Let .ϕ(x, ¯ t) = ds dθ U ∂x1 u¯ † (x, t, s, θ ). 2π x3 0 For .t > tρ , .ϕ = ϕ ∗∗ , thus we can replace .ϕ with .ϕ ∗∗ in (3.26); we then have the following estimates: dx|ϕ(x, tn ) − ϕ(x, ¯ tn )|

.

Kρ

  dx 



1 = 2π

Kρ



t∗

2π

ds 0

x3

  dθ (∂t + U ∂x1 )(u (x, tn , s, θ ) − u¯ (x, tn , s, θ )). †

†

(3.88) ≤t

∗



 max ‖ut (τ )‖1 + U

τ >t−t ∗

max

τ ∈[−t ∗ ,t ∗ ]

‖u(tn + τ ) − u‖ ¯ 1

(3.89)

dx Kρ

For .j = 1, 2: .

Kρ

dx|∂xj ϕ(x, tn ) − ∂xj ϕ(x, ¯ tn )| 1 = 2π

Kρ

  dx 



t∗

2π

ds x3

0

 †   dθ ∂xj (∂t + U ∂x1 )(u − u) ¯ (x, tn , s, θ ). (3.90)

≤t

∗



 max ‖ut (τ )‖1 + U

τ >t−t ∗

max

τ ∈[−t ∗ ,t ∗ ]

‖u(tn + τ ) − u‖ ¯ 2

dx

(3.91)

Kρ

and .

Kρ

dx|∂x3 ϕ(x, tn ) − ∂x3 ϕ(x, ¯ tn )|. ≤

dx|(u − u)(x ¯ 1 − U x3 , x2 , tn − x3 )| + Kρ

(3.92) 1 . 2π

(3.93)

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I. Lasiecka and J. T. Webster

Kρ

  dx 

t∗

x3



x3



s 2 − x32

2π 0

  dθ [Mθ ((∂t + U ∂x1 )(u − u)) ¯ ](x, tn , s, θ ) †

 ≤ ‖u − u‖ ¯ 1 + max ∗ ‖ut (τ )‖1 + U τ >t−t

 max ‖u(tn + τ ) − u‖ ¯ 2 ∗ ∗

τ ∈[−t ,t ]

dx. Kρ

(3.94) Also, from Eq. (3.31), we obtain   ‖ϕt (x, t)‖Kρ ≤ max ∗ ‖ut (τ )‖1 2 + t ∗ .

(3.95)

.

τ >t−t

By Proposition 3.8.11 and 3.8.12, all terms on the right hand side of estimates (3.88)–(3.95) approach zero. Hence, we obtain the convergence ‖ϕ(tn ) − ϕ(t ¯ n )‖1,Kρ + ‖ϕt (tn )‖Kρ → 0, n → ∞.

.

Remark 3.8.4 We note that this is in this moment where the presence of rotational forces is so critical in this argument. Having at our disposal only .ut ∈ L2 (Ω) prevents from using the formulas above which require the estimate of first derivative of .ut .

3.8.7 Strong Convergence to Equilibria In this step, we show that .(u; ¯ ϕ), ¯ as constructed in the previous Steps, is a weak solution to the stationary version of (3.6). We multiply the plate equation in (3.6) by smooth function .w ∈ C0∞ (Ω) in .L2 (Ω), integrate from .tn to .tn + a for some .a > 0, and integrate by parts to obtain tn +a  〈ut , w〉 + tn

.

+ kα

tn +a

tn

tn +a

tn +a  〈Δu, Δw〉 + 〈α∇ut , ∇w〉 +k

〈∇ut , ∇w〉dt −

tn

−

tn +a

tn

tn

tn +a

tn

  〈rΩ γ (ϕt + U ϕx1 ) , w〉dt = 0

Each term may be estimated:

〈ut (t), w〉dt

tn

〈[u, v + F0 ], w〉 +

tn +a

tn +a

tn

〈p0 , w〉dt

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

  

tn +a

tn

  〈Δu − Δu, ¯ Δw〉dt  ≤ a‖Δw‖ max ‖u(tn + τ ) − u(τ ¯ )‖2 τ ∈[0,a]



    〈ut (t), w〉dt ≤ a‖w‖ max ‖ut (tn + τ )‖1

tn +a

k

τ ∈[0,a]

tn

 t +a   〈ut , w〉 n  ≤ 2‖w‖ max ‖ut (τ )‖1  τ >t−t ∗ tn    t +a   〈α∇ut , ∇w〉 n  ≤ 2α‖w‖ max ‖ut (τ )‖1   ∗

.



211

τ >t−t

tn

tn +a

|〈∇ut , ∇w〉| dt ≤ akα‖∇w‖

kα

(3.96)

tn

max

τ ∈[tn ,tn +a]

‖ut (τ )‖1 .

We substitute .x3 = 0 in (3.32) and (3.31) to obtain the point-wise expression for 〈rΩ γ 〈ϕt + U ϕx1 〉 , w〉, which is then used to obtain the following estimate:

.

.

tn +a

tn



      〈rΩ γ 〈ϕt + U ϕx 〉 , w〉dt − U 〈γ ∂x ϕ¯ , w〉 ≤ 1 1

t∗ + 2 + Ut∗



(3.97) max ∗ ‖ut (τ )‖1 + U 2 t ∗

τ >t−t

max∗ ‖u(tn + τ ) − u‖ ¯ 2.

τ ∈[−t ,a]

As we have noted, .fV (u) = [u, v(u) + F0 ] is locally Lipschitz (3.17) for each F0 ∈ H 4 (Ω), yielding

.

.

|〈[u, v(u) + F0 ] − [u, ¯ v(u) ¯ + F0 ], w〉| ≤ ‖〈[u, v(u) + F0 ] − [u, ¯ v(u) ¯ + F0 ]‖ · ‖w‖   ≤ C ‖u‖, ‖w‖2 , ‖F0 ‖ ‖u − u‖ ¯ 2 (3.98)

Each term on the right hand side of (3.96)–(3.98) goes to zero as .n → ∞ by Propositions 3.8.11 and 3.8.12. Hence, by density, we obtain the following relation for all .w ∈ H02 (Ω):   〈Δu, ¯ Δw〉 − 〈[u, ¯ ∂x1 w = 〈p0 , w〉. ¯ v(u) ¯ + F0 ] , w〉 + U γ [ϕ],

.

(3.99)

Similarly, multiplying the fluid part of Eq. 3.6 with .ψ ∈ C0∞ (R3+ ) and integrating from .tn to .tn + a, we get

tn +a

(ϕtt , ψ)dt +

tn



.

+

tn +a tn

tn +a

tn

U 2 (∂x21 ϕ, ψ)dt

(2U ∂x1 ϕt , ψ)dt =



(3.100)

tn +a

(Δϕ, ψ)dt. tn

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This implies tn +a  (ϕt , ψ) − tn



.

tn +a

+

tn +a tn

U 2 (∂x1 ϕ, ∂x1 ψ)dt − 2U

(∇ϕ, ∇ψ)dt +

tn



tn +a tn

tn +a

tn

(ϕt , ∂x1 ψ)dt (3.101)

〈(∂t + U ∂x1 )u, γ [ψ]〉 = 0.

We now estimate each term. Recall that .Kρ ⊂⊂ R3+ contain the support of .ψ. Then:

tn +a

U2

tn

    ¯ ∂x1 ψ)dt ≤ aU 2 ‖∂x1 ψ‖ max ‖ϕ(tn + τ ) − ϕ‖ ¯ 1,Kρ (∂x1 ϕ − ∂x1 ϕ,

2U

τ ∈[0,a]

tn

.



τ ∈[0,a]

 tn +a    (ϕt , ∂x1 ψ)dt ≤ 2aU ‖∂x1 ψ‖ max ‖ϕt (tn + τ )‖1,Kρ

 tn +a    ¯ ∇ψ)dt ≤ a‖∇ψ‖ max ‖ϕ(tn + τ ) − ϕ‖ ¯ 1,Kρ (∇ϕ − ∇ ϕ, τ ∈[0,a]

tn tn +a

tn

   ((∂t + U ∂x )u − U ∂x u, 1 1 ¯ γ [ψ]) ≤ a max ‖ut (τ )‖1 τ >tn

+ a max ‖u(tn + τ ) − u‖ ¯ 2. τ ∈[0,a]

(3.102) Applying (3.88)–(3.95) to (3.102), and again straightforwardly invoking Propositions 3.8.11 and 3.8.12, we see that each term on the right hand side of (3.102) approaches zero. Whence we obtain (∇ ϕ, ¯ ∇ψ) − U 2 (∂x1 ϕ, ¯ ∂x1 ψ) + U 〈∂x1 u, ¯ γ [ψ]〉 = 0

.

(3.103)

for any .ψ ∈ H 1 (R3+ ). Thus .(u; ¯ ϕ) ¯ satisfies (3.99) and (3.103) and is hence a weak solution. We therefore have shown that any sequence .(u(tn ); ϕt (tn )) with .tn → ∞ contains a subsequence which converges to some stationary solution. We conclude by improving this convergence to the set .N . Proposition 3.8.13 For .(u, ϕ) a generalized solution to (3.6) where .ϕ0 , ϕ1 have localized support in .Kρ , we have: lim

inf

t→∞ {u; ¯ ϕ}∈ ¯ N

 2 2 2 2 ||u(t) − u(t)|| ¯ = 0, + ||u (t)|| + ||ϕ(t) − ϕ(t)|| ¯ + ||ϕ (t)|| t t 1 2 2 1 H (Kρ˜ ) L (Kρ˜ ) (3.104) for any .ρ˜ > 0. .

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

213

Proof of Proposition 3.8.13 Assume the statement is not true. Then there is a sequence .tn → ∞ and some .ϵ > 0 so that for all n sufficiently large inf

{u; ¯ ϕ}∈ ¯ N

.

 ||u(tn ) − u(t ¯ n )||22,Ω +||ut (tn )||21,Ω +||ϕ(tn )− ϕ(t ¯ n )||2H 1 (K ) +||ϕt (tn )||2L2 (K ) > ϵ. ρ

ρ

But for any such sequence .{tn }, we have shown that there exists a subsequence .{tnk } such that lim

inf

k→∞ {u; ¯ ϕ}∈ ¯ N

.

 ||u(tnk )− u(t ¯ nk )||22,Ω +||ut (tnk )||21,Ω +||ϕ(tnk )− ϕ(t ¯ nk )||2H 1 (K ) +||ϕt (tnk )||2L2 (K ) , ρ

.

ρ

→ 0,

which is a contradiction. Hence lim

inf

t→∞ {u; ¯ ϕ}∈ ¯ N

.

 2 2 2 2 ||u(t) − u(t)|| ¯ = 0. ¯ + ||ϕ (t)|| t 1 2 2 + ||ut (t)||1,Ω + ||ϕ(t) − ϕ(t)|| H (K ) L (K ) ρ

ρ

⨆ ⨅ With the above claim, we conclude the proof of Theorem 3.8.2. In the case that N is isolated (e.g., finite), (3.104) collapses to explicit convergence to a particular equilibrium.

.

Remark 3.8.5 Note that the strategy of proving strong stability of the flowstructure interaction to an equilibria set requires a uniform convergence of structural solutions to an appropriate attracting set for the plate only. This may sound surprising within the context of long-time behavior and stability theories. Why should one need to prove a much stronger uniform convergence result for the plate in order to conclude only strong stability for the overall interaction? The reason is that in order to upgrade weak convergence to strong convergence, without any compactness inherited from the full model, one must harvest the “strong topology” from somewhere else and this turns out to be the attractor for the delayed plate equation. To our knowledge, this is the first time where this type of strategy was used within the context of strong stability.

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3.9 α = 0; Non-Rotational Stabilization Note that in this case the flow produces damping at the natural level for the nonrotational plate. Thus, both properties, the dissipation and compactness, can initially be “recovered” from the interaction itself. This will produce the “stabilization” to the attracting set. Following this, by imposing some weak damping, we will be able to show the main stabilization result for the full flow-plate interaction, the latter being required to see some dissipative effects in the full flow-plate presentation (rather than just in the reduced dynamics). As in the rotational case, the proof of stability result follows from rewriting the dynamical system .(St , Y ) generated by (3.6) as a delayed dynamical system corresponding to weak solutions to (3.37)—we denote this again as .(Tt , H) with 2 2 2 2 ∗ .H = H (Ω) × L (Ω) × L (−t , 0; H (Ω)) in this case. The latter is again possible 0 0 for sufficiently large times by Theorem 3.6.4. For the .α = 0 case, we need not impose structural damping to obtain the attractor. Indeed, we “harvest” the damping from the flow. The discovered dissipation, however, enters the system polluted by destabilizing effects-including some of critical Sobolev exponent with respect to the state space. The central point here is the tradeoff between the flow-harvested dissipation which is however polluted by non-conservative destabilizers. We will show (through steps detailed below) that these flow effects ultimately produce a compact global attractor .A ⊂ H of finite dimension and additional regularity; we then take the needed attracting set, .U , to be the projection of .A on .H02 (Ω) × L2 (Ω), which yields Theorem 3.8.1. Thus we must now discuss the compact global attractor for the dynamical system .(Tt , H).

3.9.1 Existence of Structural Attractor: Statement The main result in this section is that the plate dynamical system .(Tt , H) has a compact global attractor which has additional nice properties. A counterpart of Theorem 3.8.1 is the following: Theorem 3.9.1 (Smooth, Finite Dimensional Global Attractor) Let .α = 0 and k ≥ 0, .U /= 1, .p0 ∈ L2 (Ω), and .F0 ∈ H 4 (Ω) in (3.6). Also assume the flow data .ϕ0 , ϕ1 ∈ Y are localized (with supports in .Kρ , as in Theorem 3.6.4). Then the corresponding delay system .(Tt , H) has a compact global attractor .A of finite fractal dimension in .H. Moreover, .A has additional regularity: any full trajectory t 4 .y(t) = (u(t), ut (t), u ) ⊂ A, .t ⩾ 0, has the property that .u ∈ Cr (R; H (Ω) ∩ 2 2 H0 (Ω)), .ut ∈ Cr (R; H0 (Ω)), and .utt ∈ Cr (R; L2 (Ω)). .

This can be rephrased for the full system .(St , Y ) by projecting on the first two components of .H: Corollary 3.9.2 With the same hypotheses as Theorem 3.8.1, there exists a compact set .U ⊂ H02 (Ω) × L2 (Ω) of finite fractal dimension such that for any weak solution

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

215

(u, ut ; ϕ, ϕt ) to (3.6) has

.

.

  lim dYpl (u(t), ut (t)), U = lim

t→∞

inf

t→∞ (w0 ,w1 )∈U

  ||u(t)−w0 ||22 +||ut (t)−w1 ||2 = 0.

  We also have the additional regularity .U ⊂ H 4 (Ω) ∩ H02 (Ω) × H02 (Ω). The proof of Theorem (3.9.1) proceeds in several steps: the construction of an absorbing ball by Lyapunov methods—a necessary step as the system is not conservative (or gradient)—followed by the attainment of the so-called quasistability property on the attractor itself. Step 1. Prove existence of compact global attractor for the delayed plate .(Tt , H). The difficulty lies in the fact that the system is not gradient, and thus asymptotic smoothness alone is not sufficient to generate the attractor result (Theorem 3.10.1). Moreover the delay term appears to be at the energy level—it is not explicitly compact. However, careful examination of term q (as discussed below) reveals the “hidden” compactness. This discovery, along with sharp regularity of Airy stress function (3.17) and suitable algebraic decomposition of the bracket (keeping the nonlinearity at the critical level), allows us to make use of the compensated compactness criterion Theorem 3.10.3 to obtain asymptotic smoothness on every forward invariant set. On the other hand, Lyapunov based arguments allow us to show directly that the dynamical system is ultimately dissipative. From which we deduce the compact global attractor .A ⊆ H, via Theorem 3.10.2 based on compensated compactness criterion. The latter is needed due to the criticality of the nonlinear term—see Theorem 3.9.7 Step 2. We then obtain the quasi-stability estimate (3.148) on the attractor A. Note that this is in contrast to the .α > 0 case, where the quasi-stability estimate was obtained on the entire absorbing ball, which is not possible here; indeed, we must use directly the compactness of the attractor. Direct estimates in this case are difficult due to critical nonlinearity and the lack of structure for the attractor, not being characterized by the unstable manifold of the equilibria, as in our case we lack this characterization of the attractor (Theorem 3.10.1) and we need to do further work. The winning idea is to use the already established compactness of the attractor, and a finite .ϵnet, covered by smoother elements—see [78]. This allows us to generate string of estimates that propagate smoothness from the finite net. See subsection 3.9.5. Step 3. From the quasi-stability estimate, we invoke the powerful tools from the theory of quasi-stable dynamical systems [29, 42], yielding finite dimensionality and smoothness of the global attractor. In this case, it suffices to apply the abstract result to the delayed framework.

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3.9.2 Additional Preliminaries The following propositions concern the properties of (3.38) in the von Karman plate model (3.37) and will be used in the following section. Note that these properties (unlocking compensated compactness from the delay term in the reduced model) are absolutely essential when .α = 0. These are refinements and stronger versions of estimates presented previously, which were obtained more directly when .α > 0. Proposition 3.9.3 Let .q(ut ) be given by (3.38). Then ||q(ut )||2−1 ⩽ ct ∗

.



t t−t ∗

||u(τ )||21 dτ

(3.105)

for any .u ∈ L2 (t − t ∗ , t; H01 (Ω)). If .u ∈ L2loc ([−t ∗ , +∞[; H 2 ∩ H01 )(Ω)) we also have t t 2 ∗ .||q(u )|| ⩽ ct ||u(τ )||22 dτ, ∀t ⩾ 0, (3.106) t−t ∗

and .

t

||q(uτ )||2 dτ ⩽ c[t ∗ ]2

0



t −t ∗

||u(τ )||22 dτ, ∀t ⩾ 0.

(3.107)

  C −t ∗ , +∞; (H 2 ∩ H01 )(Ω) , we have that .q(ut )

Moreover, if .u ∈ C 1 (R+ ; H −1 (Ω)),



‖∂t [q(ut )]‖−1 ⩽ C ||u(t)||1 + ||u(t − t ∗ )||1 +

.

0 −t ∗

∈

 ||u(t + τ )||2 dτ , ∀t ⩾ 0. (3.108)

For the proof details, see [44]. We provide a brief discussion now. First note the following formula for the time derivative of the delay potential, t .q(u ), appearing above: ∂t [q ](t) =

.

u

0

2π



  1 [M 2 u]ext x(U, θ, 0), t dθ 2π θ

(3.109)

  1 [Mθ2 u]ext x(U, θ, t ∗ ), t − t ∗ dθ 2π 0 ∗   t 2π   1 (U + sin θ ) [Mθ2 ux ]ext x(U, θ, s), t − s dθ ds + 2π 0 0 ∗  t  2π   1 + (cos θ ) [Mθ2 uy ]ext x(U, θ, s), t − s dθ ds . 2π 0 0 −

2π

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

217

The estimate (3.108) is the “hidden compactness” property of the delay potential. Now, for any .ψ ∈ H01 (Ω) we recall that .x(U, θ, s) = (x −(U +sin θ )s, y −s cos θ ); extend the integration over .Ω to all of .R2 and change spatial variables, yielding:  

   u  .〈qt (t), ψ〉 ⩽ C    +

2π R2

0

 t ∗  +   +

0

0

t∗ 0

2π



0

2π

R2

0

  [Mθ2 u](τ )ψdxdθ 

  [Mθ2 u](t − t ∗ )ψ(x(U, θ, −t ∗ ))dxdθ  R2

  (U + sin θ )[Mθ2 u]x (x, τ − s)ψ(x(U, θ, −s))dxdθ ds 

2π

R2

  cos θ [Mθ2 u]y (x, τ − s)ψ(x(U, θ, −s))dxdθ ds  .

Commuting the first order operator .Mθ and integrating by parts, we obtain u .|〈qt (t), ψ〉|

∗

⩽ C ||u(t)||1 + ||u(t − t )||1 +



0 −t ∗

 ||u(t + τ )||2 dτ ||ψ||1 .

This implies the conclusion in (3.108). Remark 3.9.1 A priori, when .ut is in .H01 (Ω), it is clear from (3.105) that .

T



T

〈q(u ), ut (τ )〉dτ ⩽ ϵ τ

0

0

||ut (τ )||21 + C(ϵ, T )

sup

τ ∈[−t ∗ ,T ]

||u(τ )||21 .

(3.110)

This is not apparent when .ut ∈ L2 (Ω), as .||q(ut )||20 has no such a priori bound from above, as in (3.105). From standard hyperbolic multipliers employed in the memory framework [44], we have the baseline observability inequality for the difference of two nonlinear trajectories .z = u1 − u2 : Lemma 3.9.4 (Observability) Let ui ∈ C(0, T ; H02 (Ω)) ∩ C 1 (0, T ; L2 (Ω)) ∩ L2 (−t ∗ , T ; H02 (Ω))

.

solve the plate equation with clamped boundary conditions and appropriate initial conditions on .[0, T ] for .i = 1, 2, .T ⩾ 2t ∗ . Additionally, assume .ui (t) ∈ BR (H 2 (Ω)) for all .t ∈ [0, T ]. Then the following estimates

218

I. Lasiecka and J. T. Webster

.

T Ez (T ) + 2



T T −t ∗

  Ez (τ )dτ ⩽a0 Ez (0) +



0

−t ∗

||z(τ )||22 dτ

(3.111)

+ C(T , R) sup ||z||22−η∗ τ ∈[0,T ]





T

− a1

T

ds

0 T

− a2

〈f (u1 ) − f (u2 ), zt 〉dτ

s

〈f (u1 ) − f (u2 ), zt 〉dτ

0

hold with .ai independent of T and R.

3.9.3 Attractor Construction: Dissipativity of Dynamical System Proof Our first task, in order to make use of Theorem 3.10.2, is to show (ultimate) dissipativity of the dynamical system (Tt , H)—namely, that there exists a bounded, forward invariant, absorbing set in the topology of H. To show this, similar to the consideration in [42, Theorem 9.3.4, p.480], let  1 1 ||ut ||20,Ω + ||Δu||20,Ω + ||Δv(u)||20,Ω − 〈F0 , [u, u]〉Ω 2 2   1 1 and Π∗ = ||Δu||2 + ||Δv(u)||2 and consider the Lyapunov type function 2 2 for (3.37) with α = 0: Epl =

.

  k+1 V (Tt y) ≡Epl (u(t), ut (t)) − 〈q(ut , t), u(t)〉 + ν 〈ut , u〉 + ||u||2 2 (3.112) t∗ t   t Π∗ (u(s))ds + ds Π∗ (u(τ ))dτ , +μ

.

t−t ∗

0

t−s

where Tt y ≡ y(t) = (u(t), ut (t), ut ) for t ⩾ 0 and μ, ν are some positive numbers to be specified below. Note: the structure of this Lyapunov function is different, and more involved, than that of the case of α > 0; this is necessary owing to the lack of compactness described above. In view of the results for the von Karman plate in [42, Section 4.1.1], and the inequality

t∗



t

.

0

t−s

Π∗ (u(τ ))dτ ds ⩽ t

∗



t

t−t ∗

Π∗ (u(τ ))dτ,

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

219

we have that c0 [||ut ||2 + Π∗ (u)] − c ⩽ V (Tt y) ⩽ c1 [||ut ||2 + Π∗ (u)] + μCt ∗ 0 . Π∗ (u(t + τ ))dτ + c

(3.113)

−t ∗

for ν > 0 small enough. Here c0 , c1 , c, C > 0 are constant. The terms c0 and c may depend on ν but do not depend on the damping parameter k. To obtain the above bound, we make direct use of our assumption on the L2 bound on the term 〈q, u〉. Additionally, we critically invoke the control of the lower frequencies given in Lemma 3.20; we will often use it to give 〈[u, F0 ] + p0 , u〉 + ||u||2 ⩽ ϵΠ∗ (u) + Cϵ,F0 ,p0 .

.

We now compute .

d V (Tt y) = − (k + 1)‖ut ‖2 + 〈p0 − U ux , ut 〉 − 〈qt , u〉 dt

(3.114)

+ ν〈utt + kut , u〉 + ν||ut ||2 + μΠ∗ (u(t)) − μΠ(u(t − t ∗ )) 0 ∗ Π∗ (u(t − τ ))dτ. + μt Π∗ (u(t)) − μ −t ∗

Invoke the PDE utt + (k + 1)ut = −Δ2 u + p0 + q(ut , t) + [u, v(u) + F0 ] − U ux

.

and simplify, yielding: .

  d V (Tt y) = − k + 1 − ν ||ut ||2 − ν||Δu||2 − ν||Δv(u)||2 dt + ν〈[u, F0 ], u〉 + ν〈q, u〉 + 〈p0 − U ux , ut 〉 + ν〈p0 − U ux , u〉 − 〈qt , u〉 + μ(t ∗ + 1)Π∗ (u(t)) − μΠ∗ (u(t − t ∗ )) 0 Π∗ (u(t + τ ))dτ. −μ −t ∗

Estimating, using Young’s inequality and Lemma 3.20 and the assumption that 0 < ν < min{1, k}, we obtain for all ϵ > 0

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I. Lasiecka and J. T. Webster

.

 μ(1 + t ∗ )  d V (Tt y) ⩽ (−k − 1 + ν)||ut ||2 + − ν ||Δu||2 dt 2 0  μ(1 + t ∗ )  Π∗ (u(t + τ ))dτ + − ν ||Δv(u)||2 − μΠ(u(t − t ∗ )) − μ 4 −t ∗   + ϵ||ut ||2 + ϵ ||Δu||2 + ||Δv(u)||2 + Cϵ,δ,p0 ,F0 + ϵ||q||2 + |〈qt , u〉|.

In the above, ϵ does not depend on μ or ν. Now, ||q(u , t)|| ⩽ C

.

t

2

t

t−t ∗

||Δu(τ )||2 dτ,

and with ψ = u: 

∗



|〈qt (u , t), u(t)〉| ⩽ ϵ ||Δu(t)|| + ||Δu(t − t )|| +

.

2

t

.

0

2

−t ∗

||Δu(t + τ )||2 dτ

+ Cϵ,t ∗ ||u(t)||2

we have for 0 < ν < min {k, 1}, and for μ and ϵ sufficiently small, the following lemma: Lemma 3.9.5 For any k > 0 there exist μ, ν > 0 and c(μ, ν, t ∗ , k) > 0 and C(μ, ν, p0 , F0 ) > 0 such that .

 d V (Tt y) ⩽ −c ||ut ||2 + ||Δu||2 + ||Δv(u)||2 dt .

∗

+ Π∗ (u(t − t )) +



0

−t ∗

(3.115)

 Π∗ (u(t + τ ))dτ + C.

From this lemma and the upper bound in (3.113), we have for some δ > 0 (again, depending on μ and ν) and a C (independent of k): .

d V (Tt y) + δV (Tt y) ⩽ C, t > 0. dt

The estimate (3.116) implies (by Gronwall’s inequality) that V (Tt y) ⩽ V (y)e−δt +

.

and hence, the set

C (1 − e−δt ), δ

(3.116)

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

221

  C .B ≡ y ∈ H : V (y) ⩽ 1 + δ is a bounded forward invariant absorbing set. Giving that (Tt , H) is ultimately dissipative. Remark 3.9.2 If the damping coefficient k is increased, the size of the absorbing set B does not increase—it is set by the value k. However, it is clear that this will decrease the time of absorption for B. ⨆ ⨅

3.9.4 Attractor Construction: Smoothness Through Compensated Compactness To show asymptotic smoothness of .(Tt , H), we will make use of an abstract Theorem 3.10.3. We see that we need to produce an estimate which bounds trajectories in .H, i.e., .||(u(t), ut (t), ut )||2H (taking the metric d to be .|| · ||H ). Such an estimate will be produced via our energy estimate in Lemma 3.9.4 and a von Karman bracket decomposition. This yields: Lemma 3.9.6 Suppose .z = u1 − u2 be the difference of two trajectories of the delayed plate (3.37) with .α = 0, with .y i (t) = (ui (t), ut (t)i , ut,i ) and .y i (t) ∈ 1 BR (H) for all .t ⩾ 0. Also, let .η > 0 and .Ez (t) = [||zt ||2 + ||Δz||2 ]. Then for 2 every .0 < ϵ < 1 there exists .T = Tϵ (R) such that the following estimate holds: Ez (T ) +

.

T T −t ∗

||z(τ )||22 dτ ⩽ ϵ + Ψϵ,T ,R (y 1 , y 2 ),

where Ψϵ,T ,R (y 1 , y 2 ) ≡C(R, T ) sup ||z(τ )||22−η + a1

.

  

0

τ ∈[0,T ]

T

  〈f (u1 (τ )) − f (u2 (τ )), zt (τ )〉dτ 

  + a2 

0

T



T s

  〈f (u1 (τ )) − f (u2 (τ )), zt (τ )〉dτ ds .

Proof It follows from (3.111) by dividing by T and taking T large enough.

⨆ ⨅

In Lemma 3.9.6 above, we have obtained the necessary estimate for asymptotic smoothness; it now suffices to show that .Ψ, as defined above, has the compensated compactness condition described in Theorem 3.10.3. Before proceeding, let us

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introduce some notation which will be used throughout the remainder of this section and in the following section. We will write l.o.t. = sup ||z(τ )||22−η , F(z) = f (u1 ) − f (u2 ).

.

τ ∈[0,T ]

(3.117)

Theorem 3.9.7 The dynamical system .(Tt , H) generated by weak solutions to the delayed plate (3.37) with .α = 0 and .k ⩾ 0 is asymptotically smooth. Proof In line with the discussion above, we aim to make use of Theorem 3.10.3. To do so, it suffices to show the compensated compactness condition for .Ψϵ,T ,R which we now write as .Ψ, with .ϵ, T , and R fixed along with the other constants given by the equation. Let B be a bounded, positively invariant set in .H, and let .{yn } ⊂ B ⊂ BR (H). We would like to show that .

lim inf lim inf Ψ(yn , ym ) = 0. m

n

More specifically, for any initial data .U01 = (u10 , u11 , η1 ), U02 = (u20 , u21 , η2 ) ∈ B (where .ηi belongs to .L2 (−t ∗ , 0; H02 (Ω))) we define   2 1 2  .Ψ (U0 , U0 ) = 

T 0

    〈F(z)(τ ), zt (τ )〉dτ  + 

T 0

s

T

  〈F(z(τ )), zt (τ )〉dτ ds , (3.118)

where the function .z = u1 − u2 has initial data .U01 − U02 . The key to compensated compactness is the following representation for the bracket [42, pp. 598–599] (and described above in Theorem 3.3.4): 〈F(z(τ )), zt (τ )〉 =

.

 1 d  − ||Δv(u1 )||2 − ||Δv(u2 )||2 + 2〈[z, z], F0 〉 4 dτ − 〈[v(u2 ), u2 ], u1t 〉 − 〈[v(u1 ), u1 ], u2t 〉.

Integrating the above expression in time and evaluating on the difference of two solutions .zn,m = w n − wm with initial data .W0n − W0m , where .w i ⇀ w, yields: .

lim lim

n→∞ m→∞ s

=

T

〈F(zn,m )(τ ), ztn,m (τ )〉dτ

(3.119)

 1 ||Δv(w(s))||2 − ||Δv(w(T ))||2 2 T   〈[v(w n ), w n ], wtm 〉 + 〈[v(w m ), w m ], wtn 〉 , − lim lim n→∞ m→∞ s

where we have used (i) the weak convergence in .H 2 (Ω) of .zn,m to 0, and (ii) compactness of .Δv(·) from .H 2 (Ω) → L2 (Ω) as quoted above. The iterated limit in (3.119) is handled via iterated weak convergence, as follows:

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

.

n→∞ m→∞ s

.



T

lim lim T

=2

223

 〈[v(w n ), w n ], wtm 〉 + 〈[v(w m ), w m ], wtn 〉

〈[v(w), w], wt 〉 =

s

1 1 ||Δv(w)(s)||2 − ||Δv(w)(T )||2 . 2 2

This yields the desired conclusion that .

lim lim

n→∞ m→∞ s

T

〈F(zn,m (τ )), ztn,m (τ )〉dτ = 0.

 2 is handled similarly. Since the term .l.o.t. above is The second integral term in .Ψ compact (below energy level) via the Sobolev embedding, we obtain  (yn , ym ) = 0. lim inflim infΨ

.

m→∞ n→∞

This concludes the proof of the asymptotic smoothness.

⨆ ⨅

Having shown the asymptotic smoothness property, we can now conclude by Theorem 3.10.2 that there exists a compact global attractor .A ⊂ H for the dynamical system .(Tt , H).

3.9.5 Exploiting Compactness of the Structural Attractor for Quasi-stability Quasi-stability and the Dimension and Smoothness of the Attractor In this section we refine our methods in the asymptotic smoothness calculation and work on trajectories from the attractor, whose existence has been established in the previous sections. Lemma 3.9.8 Suppose z = u1 − u2 as before, with y i (t) = (ui (t), ut (t)i , ut,i ) and y i (t) ∈ A for all t ⩾ 0. Also, let η > 0 and Ez (t) be defined as above. Then there exists a time T such that the following estimate holds: Ez (T ) +

.

+ with β < 1.

0 −t ∗

T

T −t ∗

 ||z(τ )||22 dτ ≤ β Ez (0)

||z(τ )||22 dτ ) + C(A, T , k, t ∗ ) sup ||z(τ )||22−η τ ∈[0,T ]

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Proof Analyzing (3.111), we may also write 



T Ez (T ) +

.

T

T −t ∗

  + C · T sup  s∈[0,T ]

s

 Ez (τ )dτ ⩽ c Ez (0) + T



0 −t ∗

||z(τ )||22 dτ



(3.120)

  〈F(z), zt 〉dτ  + C(R, T ) sup ||z||22−η , τ ∈[s,t]

where F(z) is given in (3.117). We note that c does not depend on T ⩾ min{1, 2t ∗ }. In order to prove the quasi-stability estimate (as in (3.148)), we have to handle the non-compact term 〈F(z), zt 〉. Remark 3.9.3 When the dynamical system is gradient, this term is typically estimated by backward smoothness/smallness of trajectories on the attractor. As our dynamical system is non-gradient, we must utilize a different tool. We recall that from the decomposition of the von Karman bracket, we have: if ui ∈ C(s, t; H 2 (Ω)) ∩ C 1 (s, t; L2 (Ω)) with ui (τ ) ∈ BR (L2 (Ω)) for τ ∈ [s, t], then  t   C  t    2 P (z(τ ))dτ  . 〈F(z), zt (τ )〉dτ  ⩽ C(R) sup ||z||2−η +  2 s s τ ∈[s,t]

(3.121)

for some 0 < η < 1/2. Here P (z) is given by as in Theorem 3.3.4. Let γu1 = {(u1 (t), u1t (t), [u1 ]t ) : t ∈ R} and γu2 = {(u2 (t), u2t (t), [u2 ]t ) : t ∈ R} be trajectories from the attractor A. It is clear that for the pair u1 (t) and u2 (t) satisfy the hypotheses of the estimate in (3.121) for every interval [s, t]. Our main goal is to handle the second term on the right hand side of (3.121) which is of critical regularity. To accomplish this we shall use the already established compactness of the attractor in the state space H = H02 (Ω) × L2 (Ω) × L2 (−t ∗ , 0; H02 (Ω)). Since for every τ ∈ R, the element uit (τ ) belongs to a compact set in L2 (Ω), by density of H02 (Ω) in L2 (Ω) we can assume, without a loss of generality, that for every ϵ > 0 there exists a finite set {ϕj } ⊂ H02 (Ω), j = 1, 2, . . . , n(ϵ), such that for all τ ∈ R we can find indices j1 (τ ) and j2 (τ ) so that ||u1t (τ ) − ϕj1 (τ ) || + ||u2t (τ ) − ϕj2 (τ ) || ≤ ϵ for allτ ∈ R.

.

Let P (z) be given by Theorem 3.3.4 with the pair u1 (t) and u2 (t) and "  # "  #  Pj1 ,j2 (z) ≡ − ϕj1 , u1 , v (z) − ϕj2 , u2 , v (z, z) − ϕj1   # +ϕj2 , z, v u1 + u2 , z ,

.

where z(t) = u1 (t) − u2 (t). It can be easily shown that for all j1 , j2 ≤ n(ϵ) ||P (z(τ )) − Pj1 (τ ),j2 (τ ) (z(τ ))|| ≤ ϵC(A)||z(τ )||22

.

uniformly in τ ∈ R.

(3.122)

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

225

Starting with the estimate (1.4.17) page 41 [42], ||[u, w]||−2 ≤ C||u||2−β ||w||1+β , ∀β ∈ [0, 1)

.

and exploiting elliptic regularity one obtains ||[u, v(z, w)]||−2 ≤ C||u||2−β ||[z, w]||−2 ≤ C||u||2−β ||z||2−β1 ||w||1+β1 , (3.123) where above inequality holds for any β, β1 ∈ [0, 1) Recalling the additional smoothness of ϕj ∈ H02 (Ω), along with the estimate in (3.123) applied with β = β1 = η, and accounting the structure of the Pj terms, one obtains: .

  ||Pj1 ,j2 (z)|| ≤ C(A) ||ϕj1 ||2 + ||ϕj2 ||2 ||z(τ )||22−η

.

for some 0 < η < 1. So we have .

sup ||Pj1 ,j2 (z)|| ≤ C(ϵ)||z(τ )||22−η for some 0 < η < 1,

(3.124)

j1 ,j2

where C(ϵ) → ∞ when ϵ → 0. Taking into account (3.122) and (3.124) in (3.121) we obtain t  t    2 . 〈F(z), zt 〉 ≤ C(ϵ, T , A) sup ||z(τ )||2−η + ϵ ||z(τ )||22 dτ (3.125) τ ∈[s,t]

s

s

for all s ∈ R with η > 0 and t > s. Considering (3.125) and taking T sufficiently large, we have from (3.120) that: Ez (T ) +

.

+

T T −t ∗ 0 −t ∗

 ||z(τ )||22 dτ ≤ β Ez (0)

||z(τ )||22 dτ ) + C(A, T , k, t ∗ ) sup ||z(τ )||22−η τ ∈[0,T ]

⨆ ⨅

with β < 1.

We are now in a position to obtain the quasi-stability estimate on the attractor A. By standard argument (see [42]) we finally conclude that for y(t) = (z(t), zt (t), zt ) ||y(t)||2H ⩽ C(σ, A)||y(0)||2H e−σ t + C sup ||z(τ )||22−η .

.

τ ∈[0,t]

Hence, on the strength of Theorem 3.10.5 applied to (Tt , H) we conclude that A has a finite fractal dimension in H, and that ||utt (t)||2 + ||ut (t)||22 ⩽ C for allt ∈ R.

.

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Since ut ∈ H 2 (Ω) ⊂ C(Ω), elliptic regularity theory for Δ2 u = −utt − (k + 1)ut − f (u) − U ux + q(ut )

.

with the clamped boundary conditions recovers .

||u(t)||24 ≤ C for allt ∈ R.

Thus, we can conclude additional regularity of the trajectories from the attractor A ⊂ H stated in Theorem 3.9.1. The final result, as stated in Theorem 3.4.1, follows by taking the projection of the attractor A ⊆ H onto its first two components, thus producing the attracting set U ⊆ Ypl .

3.9.6 Proof Strategy for Stabilization to Equilibria: α = 0 The main goal is to prove the results stated in Theorem 3.4.1 for the case .α = 0. The following steps will be taken: • Boundedness of trajectories and finiteness of dissipation integral are same as before in the case of .α > 0, except that the topology of the velocity is lowered to 2 2 .ut ∈ L (0, ∞; L (Ω)). • Treating the boundary data .γ1 [ϕ] = [ut + U ux ]ext globally in .L2 (R2 ) presents an insurmountable problem, as described above. This crude treatment neglects the microlocal structure of the data in an analysis of the N-to-D map. This is main purpose of Sect. 3.9.9, on which the entire recent breakthrough is based. Indeed, the loss of a fraction of a derivative for the wave occurs in the so-called characteristic sector.5 However, in the characteristic sector, the plate has the property that .ut ∼ ux by the comparability of time and space derivatives therein. Thus, since .u ∈ H 2 (Ω) globally, in the characteristic sector, we have 1 .ut ∈ H (Ω) (of course, microlocally). Thus: the critical loss of regularity in the hyperbolic Neumann mapping is (more than) compensated for by the gain in regularity associated with the microlocalization of the boundary data. We express this through a new microlocal estimate in Sect. 3.9.9 for the Neumann map, in the context of microlocally anisotropic spaces [126]. • In all, we are able to push the strong convergence of the plate to a stationary point to a strong convergence for the flow, improving from weak convergence of the flow (known) to strong convergence of the flow (to date, open). After identifying said limit point as a stationary solution, one concludes strong convergence of the flow-plate system to equilibria—hence the so-called elimination of flutter. This is

5 Where

the time and space dual variables are comparable, see the Appendix 3.10.2

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

227

done in Sect. 3.9.9 and the Appendix, constituting the main key to unlocking the difficulty in the problem. • To establish the “weak to strong” result, it is critical, that we be able to pass from .L2 based information (Sect. 3.9.9) to point-wise information in time. This requires some additional work, since we are working with several different types of temporal regularity coming from the various techniques utilized. To do this, we use interpolation. We have uniform-in-time control of the plate on intervals of finite size (Theorem 3.9.9, item 4); this critically exploits the existence and compactness of an attractor for the plate dynamics [44]. • Finally, to close out the argument, we prove a “converging together” lemma on time intervals of uniform finite size (Sect. 3.9.9). We subtract off the known stationary state (from the established weak convergence) on a subsequence for the linear flow dynamics, invoke our obtained microlocal estimate in anisotropic spaces, and then obtain convergence to zero on translated time intervals. This allows us to relate point-wise flow information using the energy identity (in a decoupled sense), as it is linear. Arbitrariness of the time interval size allows us to show that flow is converging to a constant on these intervals, and the weak-tostrong improvement is finally deduced. Remark 3.9.4 It is important to note that, if the plate were replaced by membrane (so the system would constitute wave-wave coupling) no microlocal compensation (gain of derivatives in the characteristic sector) in the 3D wave would present itself from the 2D wave in this characteristic sector. This would render the main stabilization result here impossible.

3.9.7 Strong Plate Convergence and Weak Convergence for the Flow Recall the finiteness of the dissipation of the dissipation integral as presented in Corollary 3.8.8: From the energy identity, then, and the above boundedness below of the energies: Any solution (3.6) satisfying the energy identity with .k > 0 has the property .

0

∞

‖ut (t)‖2L2 (Ω) dt ≤ K(||y0 ||Y ) < ∞.

We begin by noting that the existence of the compact attracting set .U for the plate dynamics in Theorem 3.4.1 we infer that for any initial data .y0 = (ϕ0 , ϕ1 ; u0 , u1 ) ∈ Yρ0 and any sequence of times .tn → ∞ there exists a subsequence of times .tnk → ∞ and a point .(u, ˆ w) ˆ ∈ Ypl = H02 (Ω) × L2 (Ω) such that .(u(tnk ), ut (tnk )) → (u, ˆ w) ˆ strongly in .Ypl . Indeed, by the definition of convergence in the Hausdorff distance of .(u(t), ut (t)) → U , and the compactness of the .U , we generate a

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convergent subsequence on .U . By uniqueness of limits, we obtain aforesaid point and subsequence. Additionally, by the global-in-time bound given in Lemma 3.8.7, we know that the set .{St (y0 )} is bounded in Y , and hence for any sequence .tn → ∞ there exists a ˆ u, ˆ u, subsequence .tnk , and a point .(ϕ, ˆ ψ; ˆ w) ˆ ∈ Y such that .Stnk (y0 ) ⇀ (ϕ, ˆ ψ; ˆ w) ˆ in .Yρ for any .ρ (recalling that .Yρ is the subspace of Y where all functions are localized to the ball .Kρ ). Utilizing these results in conjunction, for any sequence of times .tn → ∞ there is a subsequence such that both of the above convergences hold simultaneously. We collect the above convergences and add two additional points which will be central to the arguments that follow. Theorem 3.9.9 For any initial data .y0 = (ϕ0 , ϕ1 ; u0 , u1 ) ∈ Yρ0 and any sequence ˆ u, of times .tn → ∞ there is .tnk and a point .yˆ = (ϕ, ˆ ψ; ˆ w) ˆ so that for a generalized solution .(ϕ, ϕt ; u, ut ) we have: 1. 2. 3. 4.

Stnk (y0 ) ⇀ yˆ in .Yρ for any .ρ > 0. ‖u(tnk ) − u‖ ˆ 2,Ω → 0. .‖ut (t)‖0 → 0, t → ∞, and hence .w ˆ = 0. Along the sequence of times .tnk → ∞ with corresponding limit point .u, ˆ we have . .

.

sup ‖u(tnk + τ ) − u‖ ˆ 2−δ,Ω → 0for any fixedc > 0, δ ∈ (0, 2).

τ ∈[−c,c]

(3.126)

Proof of Theorem 3.9.9 Points (1.) and (2.) follow as described above, through compactness of the attracting set .U ⊂⊂ Ypl and bounded dynamics system dynamics (Theorem 3.4.1 and Lemma 3.8.7, respectively). To establish point (3.), we will prove that, for a given generalized solution, ∞ ' .ut (t) → 0 in .D (Ω), which is to say that for any .η ∈ C (Ω) we have 0 .

lim 〈ut (t), η〉L2 (Ω) = 0.

t→∞

Given this fact, and the strong subsequential limits given from the attractor, we will boost the pointwise-in-time convergence in .D' (Ω) to that of .L2 (Ω) and identify all subsequential limits with zero. To show .ut (t) → 0 in .D' (Ω), we will operate on the quantities .〈ut (t), η〉Ω and its time derivative for a given .η ∈ C0∞ (Ω). We first consider a strong solution, and show that the quantity .∂t 〈ut (t), η〉Ω is uniformly bounded for .t ∈ [0, ∞). We note, from the plate equation itself in (3.6), that  |〈utt , η〉Ω | =  − k〈ut , η〉Ω − 〈Δu, Δη〉Ω + 〈fV (u), η〉Ω    +〈p0 , η〉Ω +〈γ0 [ϕt + U ϕx ], η〉Ω  ⩽ ‖η‖2,Ω Epl (t) + C +|〈γ0 [ϕt + U ϕx ], η〉Ω |,

.

where all quantities on the RHS are well-defined as .L2 (Ω) inner products for strong solutions. We can invoke our reduction result, since we have assumed .|U | < 1

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

229

and that there exists a .ρ0 such that for .|x| > ρ0 , .ϕ0 (x) = ϕ1 (x) = 0. Utilizing Theorem 3.6.4, we note that for sufficiently large times .t > t # (ρ0 , Ω, U ) we may utilize the formula for the trace of the flow, namely: γ0 [ϕt + U ϕx ] = −(ut + U ux ) − q(ut ), x ∈ Ω.

.

It is then follows that for sufficiently large times .t > t # : |〈γ0 [ϕt +U ϕx ], η〉Ω | ⩽ ‖η‖0,Ω · sup

.

τ ∈[t # ,∞)

   ‖Δu‖2 + ‖ut ‖2 ⩽ C ||(u, ut )||Ypl ‖η‖,

where in the final step we have used the boundedness in Theorem 3.8.7. For smooth solutions, we thus obtain a uniform-in-time bound (on the time derivative) of the form   d     . , η〉 〈u Ω  ⩽ C ||(u, ut )||Ypl ||η||2,Ω .  dt t By density (and the definition of a generalized solution), we can extend this inequality to a generalized solution. Hence the quantity .∂t 〈ut (t), η〉Ω is bounded uniformly in t for any generalized solution satisfying the hypotheses at hand. Now, from the finiteness of the dissipation integral, Corollary 3.8.8 and the above, we conclude

∞

.

|〈ut (τ ), η〉|2 dτ < ∞.

0

!t We apply the Barbalat Lemma to the function . |〈ut , η〉Ω |2 dτ (as in [10, 110]) and conclude that .

lim k〈ut , η〉Ω = 0

t→+∞

(3.127)

for any .η ∈ C0∞ (Ω). This distributional convergence .ut to 0 in (3.127) and the strong convergence on subsequences .ut (tn ) → wˆ (from the existence of the compact attracting set) imply that the strong limit .wˆ = 0. Since every sequence of times .tn has a convergent subsequence such that .||ut (tnk )||0 → 0 as .k → ∞, we infer that . lim ||ut (t)||20 = 0, t→∞ as desired for Point (3.) in Theorem 3.9.9. Now, as described above, from the existence of the attracting set for the plate component we conclude strong convergence on a subsequence of ‖u(tnk ) − u‖ ˆ 2,Ω → 0

.

when .k → ∞. As for the lower order term, the following bound is clear

(3.128)

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I. Lasiecka and J. T. Webster

‖u(tn + ϵ) − u(tn )‖0,Ω ⩽

tn +ϵ

.

‖ut (τ )‖0,Ω dτ ⩽ ϵ ·

tn

sup

τ ∈[tn ,tn +ϵ]

‖ut (τ )‖0,Ω . (3.129)

Thus .

sup ‖u(tn + τ ) − u‖ ˆ 0,Ω ⩽ sup ‖u(tn + τ ) − u(tn )‖0,Ω + ‖u(tn ) − u‖ ˆ 0,Ω

τ ∈[−c,c]

τ ∈[−c,c]

(3.130) → 0for any fixedc > 0 along the subsequence .tnk by (3.128) and (3.129) above. More is true: by the interpolation inequality δ/2

1−δ/2

‖u(t)‖2−δ,Ω ⩽ ‖u(t)‖0,Ω ‖u(t)‖2,Ω ,

.

  and the boundedness of . ‖u(t)‖2,Ω : t ⩾ 0 we see that .

sup ‖u(tn + τ ) − u‖ ˆ 2−δ,Ω → 0for any fixedc > 0, δ ∈ (0, 2).

τ ∈[−c,c]

(3.131)

As seen in the previous section, the convergence .

sup ‖u(tn + τ ) − u‖ ˆ 2−δ,Ω → 0for any fixedc > 0, δ ∈ (0, 2],

τ ∈[−c,c]

i.e., for .δ = 0 by a contradiction argument.

(3.132) ⨆ ⨅

3.9.8 Weak Convergence to Equilibria Now, to further characterize the flow limit point, we return to the formula in Theorem 3.6.2. The key result is the identification of the limit as a stationary point. ˆ ∈ Yf l,ρ in Theorem 3.9.9 above has Lemma 3.9.10 The weak limit point .(ϕ, ˆ ψ) 2 ˆ .ψ = 0 in the sense .L (Kρ ) for any .ρ > 0. Proof The flow solution coming from the initial data, denoted .(ϕ ∗ (t), ϕt∗ (t)), tends to zero in the local flow energy sense, owing to Huygen’s principle. Thus from Point (1.) of Theorem 3.9.9 we have that .ϕt∗∗ (tnk ) ⇀ ψˆ (since .ϕ = ϕ ∗ + ϕ ∗∗ ). To identify ˆ with zero, we differentiate the flow formula (3.26) to obtain (3.31) (holding in .ψ distribution). For a fixed .ρ > 0, we multiply the function .ϕt∗∗ (x, t) by a smooth function .ζ ∈ C0∞ (Kρ ) and integrate by parts in space. This results in the bound |(ϕt∗∗ , ζ )L2 (Kρ ) | ⩽ C(ρ) sup ‖ut (t − τ )‖0,Ω ‖ζ ‖1,Kρ .

.

τ ∈[0,t ∗ ]

(3.133)

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

231

From this point, we utilize the previously established fact that .ut (t) → 0 in L2 (Ω) (Theorem 3.9.9), and hence .(ϕt∗∗ (t), ζ )L2 (Kρ ) → 0, so .ϕt∗∗ (t) → 0 in ' 2 .D (Kρ ). This gives that .ϕt (tnk ) ⇀ 0 in .L (Kρ ), and we identify the limit point  = 0 in .L2 (Kρ ). .ψ ⨆ ⨅ .

We must now show that the obtained weak limit .Stn (y0 ) ⇀ (ϕ, ˆ 0; u, ˆ 0) in .Yρ provides a weak solution to the stationary problem. The proof proceeds as in [10], but we sketch it for completeness. Lemma 3.9.11 The pair .(ϕ, ˆ u) ˆ in Theorem 3.9.9 is a weak solution to the stationary version of (3.6) with .α = 0. Proof The proof proceeds similarly as in the .α > 0 case. Consider a strong solution to (3.6). We begin by multiplying the system (3.6) by .η ∈ C0∞ (Ω) and .μ ∈ C0∞ (R3+ ) and integrate over the respective domains. This yields .

〈utt , η〉 + 〈Δu, Δη〉 + 〈ut , η〉 + 〈f (u), η〉 = 〈p0 , η〉 − 〈γ0 [ϕt + U ϕx ], η〉 (ϕtt , μ)+U (ϕtx , μ)+U (ϕxt , μ)+U 2 (ϕx , μx )=− (∇ϕ, ∇μ)+〈ut + U ux , γ0 [μ]〉. (3.134)

Now, we consider the above relations evaluated on some points .tn (identified as a subsequence for which the various convergences hold). We then integrate in the time variable from .tn to .tn + c. After temporal integration, the resulting identity can be justified on generalized solutions by the approximation definition of thereof. We then pass to limit in n, using Theorem 3.9.9. Limit passage on linear terms is clear, owing to the main convergence properties for the plate component in the Theorem 3.9.9. The local Lipschitz property of .fV allows us to pass with the limit on the nonlinear term (this is by now standard, [42]). We then arrive at the following relations: .

〈Δu, ˆ Δw〉 + 〈fV (u), w〉 = 〈p0 , w〉 + U 〈γ0 [ϕ], ˆ wx 〉 (∇ ϕ, ˆ ∇μ) − U 2 (ϕˆx , μx ) = U 〈uˆ x , γ0 [μ]〉.

(3.135)

This implies that our limit point .(u, ˆ 0; ϕ, ˆ 0) for .(ϕ(tn ), ϕt (tn ); u(tn ), ut (tn )) is a weak stationary solution to the flow-plate system (3.6). We have thus shown: any trajectory contains a sequence of times .tn → ∞ such that the restricted trajectory converges weakly to a stationary solution. ⨆ ⨅ Remark 3.9.5 Knowing that the system is gradient would allow to reach this conclusion in a more straightforward way. However, this property is known only for the unperturbed problem .U = 0. Summarizing Up to This Point Theorem 3.9.9 guarantees that for any .y0 ∈ Yρ0 and for any sequence .tn → ∞ and any .ρ > 0, there is weak convergence in .Yρ along

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ˆ u, a subsequence .tnk of the trajectory .Stnk (y0 ) to some point .yˆ = (ϕ, ˆ ψ; ˆ w) ˆ ∈ Yρ . Moreover, point (3.) of Theorem 3.9.9 guarantees that .wˆ = 0. We then note that Lemma 3.9.10 holds, and hence .ψˆ = 0. With our weak limit point satisfying the weak form of the stationary problem in Lemma 3.9.11, we conclude that we have obtained weak convergence to a weak solution of the stationary flow-plate problem. We therefore have shown that any sequence .(u(tn ); ϕt (tn )) with .tn → ∞ contains a subsequence which converges to a stationary solution. A simple contradiction argument (see, e.g., [10, 42, 110]) gives that for any .ρ0 , ρ > 0 and any .y0 ∈ Yρ0 we have .St (y0 ) ⇀ N in the sense of .Yρ . What remains to show to obtain the main result is the improvement of this convergence from weak to strong.

3.9.9 Improving from Weak to Strong This is the final portion of the proof of the main result, Theorem 3.7.1. With the technical preliminaries established, and the improved microlocal mapping properties of the Neumann-Dirichlet mapping (Sects. 3.9.9 and 3.10.2), we now boost weak convergence in Lemma 3.9.11 to a strong convergence. We outline the strategy here: • We consider a given sequence of times .tn → ∞, for which we have established a subsequential weak limit .(ϕ, ˆ 0; u, ˆ 0) that is a weak solution to the stationary problem. We consider the wave component of the PDE system on its own, where we take the difference variables .ψ = ϕ − ϕˆ and .v = u − u. ˆ Since the wave equation itself is linear, we may apply our microlocal regularity results for the resulting .(ψ, v) system. This point is the major technical departure from the strategy used in the rotational case .α > 0. The goal is to show that .(ψ(tn ), v(tn )) → (0, 0) strongly in .Yρ (perhaps on a further subsequence). • We consider the Neumann mapping bound on time intervals of the form .[−c + tn , tn + c], where .c > 0 is arbitrary; we then apply the microlocal result. Since these “sliding” intervals are all of the same length, we can invoke the semigroup property for tight control of constants that depend on T , which is to say, only the length of this interval matters. • We will then obtain that .ψ goes to zero (via  convergence of the plate dynamics) in the sense of .L2 [−c + tn , tn + c]; Yρ . We must convert this information to point-wise convergence along the sequence .tnk for the flow. We finally show a “converging together” lemma that gives .E(tn ), .E(tn + c) converge to the same value with c arbitrary; this allows us to deduce the required point-wise information.

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

233

Convergence Through the Microlocal Regularity Let .{tnk } be as in Lemma 3.9.11, with .(u, ϕ) satisfying (3.6) and .(u, ˆ ϕ) ˆ the associated solution the stationary problem to which it convergence along a given subsequence of times .tnk . By linearity of the flow equation in (3.23), the variable .ψ = ϕ − ϕ ˆ satisfies the same flow equation with the boundary data given by h = [vt + U vx ]ext = [ut + U (u − u) ˆ x ]ext .

.

Our ultimate goal is to show for any .ρ > 0 ||ϕ(tnk ) − ϕ|| ˆ 2W1 (Kρ ) + ||ϕt (tnk )||2L2 (K

.

→ 0.

ρ)

(3.136)

We arrive at this through several steps, as outlined above. We define, for a fixed .a > 0, time translates of the solutions along .tn . Consider the sequences .{fn } and .{gn } as: fk (t) =

.

vext (tnk + t), t ∈ [−a, a] 0,

otherwise

gk (t) =

[vt ]ext (tnk + t), t ∈ [−a, a] 0,

otherwise

and sequences .{pn } and .{qn } as pk (t) =

.

ψ(tnk + t), t ∈ [−a, a] 0,

otherwise

qk (t) =

ψt (tnk + t), t ∈ [−a, a] 0,

otherwise.

Now, we wait a sufficient time so as to consider the flow equation with zero initial flow data on .Kρ (Huygen’s Principle). Applying the result on the Neumann lift (with zero flow data) in Theorem 3.10.15, for any fixed .a > 0, we obtain: ||ψt ||2L2 (t

.

2 nk −a,tnk +a;L (Kρ ))

+ ||∇ψ||2L2 (t

=||qk ||2L2 (R

2 nk −a,tnk +a;L (Kρ ))

2 + ;L (Kρ )

+ ||∇pk ||2L2 (R

≤CT ||gk + U (fk )x ||2 1/2

X−1/2

. 2 + ;L (Kρ ))

(3.138)

.

 ≤ CT ||gk ||2 1/2 + U ‖(fk )x ||2 1/2 X−1/2

(3.137)

X−1/2

,

(3.139)

where above .T = 2a (independent of k). We now look at the terms on the RHS. For simplicity, we use the notation .χ (k) := χ[tnk −a,tnk +a] for the characteristic function on the set .[tnk − a, tnk + a]. Using the 1/2

definition of the space .X−1/2 (see the Appendix 3.10.2) and Parseval’s equality, we obtain

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||gk ||2 1/2 = X−1/2

+



R2

R2

dμ dμ

+ ≤2

R2

R2





+ .

dσ |gk (μ, σ )|2

√ |μ| √ 0 and .x ∈ H implies that x is a stationary point of .(St , H ). If the dynamical system has a strict Lyapunov function defined on the entire phase space, then we say that .(St , H ) is gradient. We can address attractors for gradient systems and characterize the attracting set. The following result follows from Theorem 2.28 and Corollary 2.29 in [41]. Theorem 3.10.1 Suppose that .(St , H ) is a gradient, asymptotically smooth dynamical system. Suppose its Lyapunov function .Ф(x) is bounded from above on any bounded subset of .H and the set .ФR ≡ {x ∈ H : Ф(x) ≤ R} is bounded for every R. If the set of stationary points for .(St , H ) is bounded, then .(St , H ) possesses a compact global attractor .A which coincides with the unstable manifold, i.e., A = M u (N ) ≡ {x ∈ H : ∃ U (t) ∈ H, ∀t ∈ R such that U (0) = x

.

and .

lim dH (U (t)|N ) = 0}.

t→−∞

The following if and only if characterization of global attractors is well-known [9, 42] Theorem 3.10.2 Let .(St , H ) be an ultimately dissipative dynamical system in a complete metric space A. Then .(St , H ) possesses a compact global attractor .A if and only if .(St , H ) is asymptotically smooth. For non-gradient systems, the above theorem is often the mechanism employed to obtain the existence of a compact global attractor. If one can show that an ultimately dissipative dynamical system .(St , H ) is also asymptotically smooth, one obtains the existence of a compact global attractor. In many cases, showing asymptotic smoothness can be done conveniently using the following criterion due to [90] and presented in a streamlined way in [29, 42]. An asymptotically smooth dynamical system for which there is a Lyapunov function .Ф(x) that is bounded from above on any bounded set can be thought of as one which possesses local attractors. To see this stated precisely see [41] page 33. Such a result provides an existence of local attractors, i.e., and attractor for

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any bounded set of initial data. However, these sets need not be uniformly bounded with respect to the size of the set of initial data. The latter can be guaranteed by the existence of a uniform absorbing set. However, establishing this existence of an absorbing set may be technically demanding. In some instances, there is a way of circumventing this difficulty which takes advantage of the “good” structure of a Lyapunov function. Theorem 3.10.3 Let .(St , H ) be a dynamical system, H a Banach space with norm || · ||. Assume that for any bounded positively invariant set .B ⊂ H and for all .ϵ > 0 there exists a .T ≡ Tϵ,B such that

.

||ST y1 − ST y2 ||H ⩽ ϵ + Ψϵ,B,T (y1 , y2 ), yi ∈ B

.

with .Ψ a functional defined on .B × B depending on .ϵ, T , and B such that .

lim inf lim inf Ψϵ,T ,B (xm , xn ) = 0 m

n

(3.147)

for every sequence .{xn } ⊂ B. Then .(St , H ) is an asymptotically smooth dynamical system. Remark 3.10.1 The above compactness criterion given in Theorem 3.10.3 is a more general version than typically available in the theory of dynamical systems. Note that the functional .ψ need not be explicitly compact. The .lim inf condition is satisfied when .ψ is compact, however, there are many situations with non-compact perturbations that do fi the above requirement. Typical examples involve critical nonlinearities in dynamical systems where other structural (Hamiltonian) properties compensate for this lack of compactness. For this reason we shall refer to the condition (3.147) as a compensated compactness criterion. A generalized fractal exponential attractor for the dynamics .(St , H ) is a forward invariant, compact set .Aexp ⊂ H in the phase space, with finite fractal dimension (possibly in a weaker topology), that attracts bounded sets with uniform exponential rate. When we refer to .Aexp as a fractal exponential attractor, we are simply indicating that .Aexp ⊂ H has fractal dimension in H , rather than in some weaker space. Remark 3.10.2 We include the word “generalized” to indicate that the finite fractal dimensionality could be shown in a weaker topology than that of the state space. See [29, 42] for extensive discussions. Here we define quasi-stability as our primary tool in the long-time behavior analysis. A quasi-stable dynamical system is one where the difference of two trajectories can be decomposed into a uniformly stable part and a compact part, with controlled scaling of the powers. The theory of quasi-stable dynamical systems has been developed rather thoroughly in recent years [29, 42]. This includes more general definitions of quasi-stable dynamical systems [29] than what we present

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

239

below. For ease of exposition and application in our analysis we focus on this more narrow definition. Informally, we note that: • Obtaining the quasi-stability estimate on the global attractor A implies additional smoothness and finite dimensionality A. This follows from the so-called squeezing property and one of Ladyzhenskaya’s theorems (see [42, Theorems 7.3.2 and 7.3.3]). • Obtaining the quasi-stability estimate on an absorbing ball implies the existence of an exponentially attracting set; uniform in time Hölder continuity (in some topology) yields finite dimensionality of this exponentially attracting set (in said topology). We emphasize the power of the quasi-stability inequality: essentially, one estimate—an improvement of a typical observability estimate—yield many desirable properties, depending upon which set the inequality is valid upon; one inequality does it all. We now proceed with a formal discussion of quasi-stability. Condition 1 Consider second order (in time) dynamics yielding the dynamical system .(St , H ), where .H = X×Z with .X, Z Banach, and X compactly embeds into Z. Further, suppose .y = (x, z) ∈ H with .St y = (x(t), xt (t)) where the function 1 .x ∈ C(R+ , X) ∩ C (R+ , Z). With Condition 1 we restrict our attention to second order, hyperbolic-like evolutions. Assumption 3 With Condition 1 in force, suppose that the dynamics .(St , H ) admit the following estimate for .y1 , y2 ∈ B ⊂ H : ||St y1 −St y2 ||2H ⩽ e−γ t ||y1 −y2 ||2H +Cq sup ||x1 −x2 ||2Z∗ , for some γ , Cq > 0,

.

τ ∈[0,t]

(3.148) where .Z ⊆ Z∗ ⊂ X and the last embedding is compact. Then we say that .(St , H ) is quasi-stable on B. Remark 3.10.3 As mentioned above, the definition of quasi-stability in the key references [29, 42] is much more general; specifically, the estimate in (3.148) can be replaced with: ||St y1 − St y2 ||2H ⩽ b(t)||y1 − y2 ||2H + c(t) sup [μH (St y1 − St y2 )]2 ,

.

τ ∈[0,t]

(3.149)

where: (i) .b(·) and .c(·) are nonnegative scalar functions on .R+ such that .c(t) is locally bounded on .[0, ∞) and .b ∈ L1 (R+ ) and . lim b(t) = 0; (ii) .μH is a compact t→∞ seminorm on H . In fact, this definition is recent [29] and is more general than that in [42], accommodating a broader class of nonlinear dynamical systems arising the in the long-time analysis of plate models.

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Remark 3.10.4 (User friendly version of quasi-stability estimate) Let .z(t) ≡ St y1 − St y2 and assume that the following balance of energy holds: 2 .||z(T )||H

T

+ 0

D(t)dt = ||z(0)||2H + CT [l.o.t.(z)],

(3.150)

where .D(t) ≥ 0 denotes a term controlled by the (boundary or interior) damping inherent in the problem; the notation .l.o.t (z) indicates terms which are measured in a spatial norm below the energy level. Then, a “user friendly” form of the quasistability inequality is the following improved observability inequality z:

T

.

0

||z(t)||2H ≤ C||z(0)||2H + CT

T

D(t) + CT [l.o.t (z)],

(3.151)

0

with a quadratic structure for the lower order terms and with constant C in (3.151) independent of T . Inequality (3.151) needs only to hold on a finite time interval .(0, T ). This type of inequality is generally obtained from a multiplier method. Clearly (3.150) and (3.151) taken together imply ||z(T )||2H ≤ CT

.

T

D(t) + CT [l.o.t.(z)].

0

The above inequality has a clear interpretation as an “inverse” relation: the dynamical state .z(T ) is controlled entirely from the damping (in the finite energy sense), modulo lower order terms. We now run through a handful of consequences of the type of quasi-stability described by Definition 3 above for dynamical systems .(St , H ) satisfying Condition 1. [42, Proposition 7.9.4] Theorem 3.10.4 (Asymptotic smoothness) If a dynamical system .(St , H ) satisfying Condition 1 is quasi-stable on every bounded, forward invariant set .B ⊂ H , then .(St , H ) is asymptotically smooth. Thus, if in addition, .(St , H ) is ultimately dissipative, then there exists a compact global attractor .A ⊂ H . The theorems in [42, Theorem 7.9.6 and 7.9.8] provide the following result concerning improved properties of the attractor A if the quasi-stability estimate can be shown on the attractor A. Theorem 3.10.5 (Dimensionality and smoothness) If a dynamical system .(St , H ) satisfying Condition 1 possesses a compact global attractor .A ⊂ H , and is quasistable on A, then .A has finite fractal dimension in H , i.e., .dimH f A < +∞. Moreover, any full trajectory .{(x(t), xt (t)) : t ∈ R} ⊂ A has the property that xt ∈ L∞ (R; X) ∩ C(R; Z); xtt ∈ L∞ (R; Z),

.

with bound

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

241

||xt (t)||2X + ||xtt (t)||2Z ⩽ C,

.

where the constant C above depends on the “compactness constant” .Cq in (3.148). The following theorem relates generalized fractal exponential attractors to the quasi-stability estimate [42, p. 388, Theorem 7.9.9]. Theorem 3.10.6 Let Condition 1 be in force. Assume that the dynamical system generated by solutions .(St , H ) is ultimately dissipative and quasi-stable on a  ⊃ H so that bounded absorbing set .B. We also assume there exists a space .H  for every .y ∈ B; this is to say there exists .t I→ St y is Hölder continuous in .H .0 < α ⩽ 1 and .CB,T >0 so that ||St y − Ss y||H ⩽ CB,T |t − s|α , t, s ∈ [0, T ], y ∈ B.

.

(3.152)

Then the dynamical system .(St , H ) possesses a generalized fractal exponential , i.e., .dimHAexp < +∞. attractor .Aexp whose dimension is finite in the space .H f Remark 3.10.5 Remark 7.9.10 [42, pg. 389] discusses the need for the Hölder continuity assumption above. It is presently an open question as to how “necessary” this condition is for general hyperbolic systems possessing global compact attractors. Remark 3.10.6 In addition, owing to the abstract construction of the set .Aexp ⊂ X, boundedness of .Aexp in any higher topology is not addressed by Theorem 3.10.6. In fact, this is a recognized open problem: additional smoothness of exponential attractors [105]. The proofs of Theorems 3.10.5 and 3.10.6 can be found in [29, 42] and rely fundamentally on the technique of “short” trajectories or “l” trajectories (see, e.g., [102, 103]). The above two theorems appeal to the quasi-stability property of the dynamics on the global attractor A or the absorbing set B. If one can construct a compact set K which is itself exponentially attracting, then having the quasi-stability estimate on K (along with the transitivity of exponential attraction shown below) yields a stronger result. The result below is given as [29, Theorem 3.4.8, p. 133] and proved there. Theorem 3.10.7 ([29]) Let .(St , H ) be a dynamical system, where H is a separable Banach space. Assume: 1. There exists a positively invariant compact set .F ⊂ H and positive constants C and .γ such that .

sup {dH (St x, F ) : x ∈ D} ⩽ Ce−γ (t−tD ) , t

for every bounded set .D ⊂ H and for .t ⩾ tD . 2. There exists a neighborhood .O of F and numbers .Δ1 and .α1 such that

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||St x1 − St x2 || ⩽ Δ1 eα1 t ||x1 − x2 ||.

.

3. The system .(St , H ) is quasi-stable on F for .t ⩾ t∗ for some .t∗ > 0. 4. The mapping .t I→ St x is uniformly Hölder continuous on F ; that is there exists constants .CF (T ) > 0 and .η ∈ (0, 1] such that ||St1 x − St2 x|| ⩽ CF (T )|t1 − t2 |η , ti ∈ [0, T ], x ∈ F.

.

Then there exists a fractal exponential attractor .Aexp ⊂ H for .(St , H ) whose fractal dimension is finite in H . In the previous theorem, and in some of the notes below, we will implicitly appeal to the transitivity of exponential attraction (via Theorem 3.10.7); we show it here for the sake of exposition. Loosely, in a fixed topology, the property of a set uniformly exponentially attracting bounded sets in this topology is transitive. We now state this formally—see [72] for discussion and applications to the nonlinear wave equation. Theorem 3.10.8 (Theorem 5.1, [72]) Let .(M, d) be a metric space with .dM denoting the Hausdorff semi-distance. Let a semigroup .St act on .M such that d(St m1 , St m2 ) ⩽ CeKt d(m1 , m2 ), mi ∈ M.

.

Assume further that there are sets .M1 , M2 , M3 ⊂ M such that dM (St M1 , M2 ) ⩽ C1 e−α1 t , dM (St M2 , M3 ) ⩽ C2 e−α2 t .

.

Then for .C ' = CC1 + C2 and .α ' =

α1 α2 we have K + α1 + α2 '

dM (St M1 , M3 ) ⩽ C ' e−α t .

.

Tools from Quasi-stability Theory We now proceed by discussing the specific tool we use in the construction of the attractor: quasi-stability [29, 42]. Condition 2 Consider second order (in time) dynamics .(St , H ) where .H = X × Z with .X, Z Hilbert, and X compactly embedded into Z. Further, suppose .y = (x, z) ∈ H with .St y = (x(t), xt (t)) where the function .x ∈ C(R+ , X)∩C 1 (R+ , Z). Condition 2 restricts our attention to second order, hyperbolic-like evolutions. Condition 3 Suppose the evolution operator .St : H → H is locally Lipschitz, with Lipschitz constant .a(t) ∈ L∞ loc ([0, ∞)):

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

‖St y1 − St y2 ‖2H ⩽ a(t)‖y1 − y2 ‖2H .

.

243

(3.153)

Assumption 4 With Conditions 2 and 3 in force, suppose that the dynamics .(St , H ) admit the following estimate for .y1 , y2 ∈ B ⊂ H : ‖St y1 − St y2 ‖2H ⩽ e−γ t ‖y1 − y2 ‖2H + Cq sup ‖x1 − x2 ‖2Z∗ , for some γ , Cq > 0,

.

τ ∈[0,t]

(3.154) where .Z ⊆ Z∗ ⊂ X, and the last embedding is compact. Then we say that .(St , H ) is quasi-stable on B. We now run through a handful of consequences of the type of quasi-stability described by Definition 4 above for dynamical systems .(St , H ) satisfying Condition 2 [42, Proposition 7.9.4]. Theorem 3.10.9 If a dynamical system .(St , H ) satisfying Conditions 2 and 3 is quasi-stable on an absorbing ball .B ⊂ H , then there exists a compact global attractor .A ⊂⊂ H .

 2 2 ⩽ C, . sup ‖xt (t)‖X + ‖xtt (t)‖Z t∈R

where the constant C above depends on the “compactness constant” .Cq in (3.154). Elliptic regularity can then be applied to the equation itself generating the dynamics (St , H ) to recover regularity for .x(t) in a norm higher than that of the state space X.

.

3.10.2 Microlocal Regularity of the Hyperbolic Neumann-Dirichlet Map The fundamental supporting result here—what finally allows us to overcome the issues encountered in previous stability analyses [36, 97, 98]—is a sharp microlocal estimate for the Neumann lift. We note, for the first time, that the critical loss of Sobolev regularity in the so-called characteristic sector for the wave equation is compensated for by the plate dynamics. In this characteristic sector (defined below) we have .ut ∼ ux ∈ H 1 (Ω). (Note: there is no loss of regularity for the Neumann wave equation in the hyperbolic and elliptic sectors.) We will work with microlocally anisotropic spaces, as utilized heavily in the work of Tataru for trace estimates for the wave equation. From [126, 127], we introduce the anisotropic space .Xθs . These spaces are defined via a tangential differential operator .R(x, D) defined on the space-time boundary .Σ which is associated with the interior wave operator .P (x, D). For notation, let us use .ξ = 〈ξ1 , · · · , ξn−1 , ξn 〉 and .ξ ' = 〈ξ1 , · · · , ξn−1 〉. Then, let .r(x, ξ ) be the principal part

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of R. For .Σ a non-characteristic surface, we can choose local coordinates so that Σ = {ξn = 0}. We can then write the principal symbol for .P (x, D) of the form ' ' ' 2 .p(x, ξ ) = −μn + r(x, ξ ) where .r(x, ξ ) = p(x, ξ , 0). .

Assumption 5 The space of distributions on .Σ denoted by .Xθs , .s ∈ R, θ ∈ [−1, 1] is defined as a complex interpolation .Xθs ≡ [X0s , X1s ]θ when .θ ∈ [0, 1] and s s s s s s .[X , X 0 −1 ]−θ when .θ ∈ [−1, 0]. We take .X0 ≡ H (Σ), .X1 ≡ {u ∈ H (Σ) : s ≡ H s (Σ) + R(x, D)H s+1 (Σ). R(x, D)u ∈ H s−1 (Σ)}, and .X−1 Remark 3.10.7 From these definitions, and the fact that R is a second order s consist of distributions which differential operator, one sees that elements in .X−1 are .H s−1 (Σ) outside the characteristic region of R, denoted .char(R), and .H s (Σ) near .char(R). Roughly, the notation .u ∈ Xθs means that u belongs to .H s (Σ) “near” .char(R), whereas .u ∈ H s+θ (Σ) away from .char(R). In our case this will concide with .H s regularity in (3.165) in .(c) (where time and space dual variables are comparable), and .H s+θ (Σ) regularity (.θ < 0) in the other two sectors .(e) and .(h) (also defined in (3.165)). The microlocal spaces defined above provide more accurate description of traces on the boundary of the wave maps than the classical s .H (Σ) spaces and have been heavily used in the past [127]. Let us introduce the dual variables .σ ∈ R, μ ∈ R2 so that by Fourier-Laplace transformation t →τ = β + iσ, 〈x1 , x2 〉 → iμ = i(μ1 , μ2 ),

.

(3.155)

with .β fixed and sufficiently large. In what follows we shall use .r(ξ ' ) = s ≡ |μ|2 +τ 2 which is a principal symbol of space-time tangential second order operator. Thus .char(R) ⊆ {c|μ| ≤ |σ | ≤ C|μ|} for some positive constants .c, C. We 1/2 will specifically utilize the space .X−1/2 which consists of the functions .h(x, y; t) with .H 1/2 (Σ) microlocal regularity near .char(R) and .L2 (Σ) regularity globally. In particular, e−ξ t h ∈ L2 (R2 × R+ ).

.

ˆ μ) satisfies The Fourier-Laplace transform (defined explicitly below) .h(τ,   ˆ μ) ∈ L2 R+ ; H 1/2 (R2 ) m0 (σ, μ)h(τ,

.

$0 (t, x1 , x2 ; t) for .M0 (x1 , x2 ; t) a zeroth order tangential .ΨDO with .m0 (σ, μ) = M    2 on (the boundary) .R , with the property that .supp m0 (σ, μ) ⊂ MC,c ≡ c|μ| ≤  |σ | ≤ C|μ| for some constants .c, C > 0. In addition, we assume that .m ≡ 1 in .M 3C+c 4c+C . 4 , 4 With that notation in place, we look at the following Neumann wave equation:

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

⎧ ⎪ ⎪ ⎨ηtt = Δη . η(0) = η0 ; ηt (0) = η1 ⎪ ⎪ ⎩∂ η = h∗ (x, t)

245

in R3+ × (0, T ), in R3+

(3.156)

on{z = 0} × (0, T ).

z

Our goal in this section is to determine the interior and the trace regularities of ηwhen the flow data .h∗ has microlocally anisotropic regularity. In particular, for 1 .ut as the plate component of a solution to (3.188), we will have .ut ∈ X −1 . However, this provides an overcompensation for the Neumann loss (as discussed above); thus 1/2 we only need consider Neumann data of the form .h∗ ∈ X−1/2 . We will show the following theorem: .

Theorem 3.10.10 Any weak solution to (3.156) satisfies the following a priori bound for .0 ⩽ t ⩽ T : ||γ0 [ηt ]||2L2 (0,T ;L2 (R2 )) +||γ0 [η]||2L2 (0,T ;H 1 (R2 )) +||ηt ||2L2 (0,T ;L2 (R3 ))

.

+

+||η||2L2 (0,T ;H 1 (R3 )) +

  ⩽ CT ||η(0)||2H 1 (R3 ) + ||ηt (0)||2 + ||h∗ ||2 1/2 . X−1/2

+

(3.157)

Proof of Theorem 3.10.10 To prove (3.157), we critically exploit the half-space structure of the problem in avoiding commutators and variable coefficients. We apply superposition with respect to initial and boundary data. Step 1: Consider .h∗ = 0 and let .η be the corresponding response to the initial conditions. Applying [106, Theorem 3], we obtain ||η(t)||2H 1 (R3 ) + ||ηt (t)||2 + ||γ0 [ηt ]||2L2 (0,T ;H −1/2 (R2 )) + ||γ0 [η]||2L2 (0,T ;H 1/2 (R2 )) +   ≤ CT ||η(0)||2H 1 (R3 ) + ||ηt (0)||2 . (3.158)

.

+

Step 2: Consider zero initial data and arbitrary .h∗ ∈ X−1/2 , and again let .η be the associated solution. Since .ϕ0 , ϕ1 = 0 we then take the Fourier-Laplace transform with .ξ > 0 fixed: 1/2

t →τ = ξ + iσ, (x, y) → iμ = i(μ1 , μ2 ),

.

denoting by . η = η(z, μ, σ ) the Fourier-Laplace transform of .η in x, y and t, i.e., 1 . η(z, μ, τ ) = (2π )2



R2

dxdy 0

+∞

dte−τ t · e−i(xμ1 +yμ2 ) · η(x; t).

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This yields the equation  ηzz = (|μ|2 + ξ 2 − σ 2 + 2iξ σ ) η,

.

ηz (z = 0) = h∗ (μ, τ ). with transformed boundary condition, . Solving the ODE in z and choosing the decaying solution .z → +∞, we have √ 1  η(z, μ, τ ) = − √ h∗ (μ, τ )exp(−z s), s

.

with   s ≡ |μ|2 + τ 2 = |μ|2 − σ 2 + ξ 2 + 2iξ σ ;

.

(3.159)

% √ and . s is chosen such that .Re τ 2 + |μ|2 > 0 for .Reτ = ξ > 0. On the boundary .z = 0 we have 1  η(z = 0, μ, τ ) = √ h∗ (μ, τ ). s

.

(3.160)

Taking the time derivative amounts to premultiplying by .τ = ξ + iσ , and with a slight abuse of notation, we have τ  ηt (z = 0, μ, τ ) = √ h∗ (μ, τ ), τ = ξ + iσ. s

.

(3.161)

Denoting the multiplier .

ξ + iσ ≡ m(ξ, σ, μ), √ s

(3.162)

we can infer the trace regularity of .η from .m(ξ, σ, μ):  ξ + iσ  %ξ 2 + σ 2   .|m(ξ, σ, μ)| =  √ . = (|s|2 )1/4 s % σ2 + ξ2 = 1/4 . (|μ|2 − σ 2 )2 + ξ 4 + 2ξ 2 σ 2 + 2|μ|2 ξ 2

(3.163) (3.164)

WLOG, we take each of the arguments .ξ , .σ and .|μ| to be positive and consider the partition of the first quadrant of the .(|μ|, σ )-plane into the following sectors:

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

⎧ √ ⎪ σ/ 2, ⎪ ⎨(h) |μ| ⩽ √ . (e) |μ| ⩾ 2σ, ⎪ ⎪ ⎩(c) σ/√2 < |μ| < √2σ

247

(hyperbolic) (3.165)

(elliptic) (characteristic).

Trace Regularity Our goal in this section is to prove the trace component of (3.157). Lemma 3.10.11 Let s be as in (3.159). Then, in sectors .(e) and .(h) we have the estimate 1/2  % |μ|2 1 |s| ≥ |ξ | 1 + 2 2 ξ

(3.166)

.

and in sector .(c), we have the estimate 1/4  % |μ|2 . . |s| ≥ |ξ | 1 + ξ2

(3.167)

Proof of Lemma 3.10.11 Also, from (3.163), we may write % .

1/4  |s| = (|μ|2 − σ 2 )2 + ξ 4 + 2ξ 2 σ 2 + 2|μ|2 ξ 2 .

Hence, in sectors .(e) and .(h), we obtain 1/4  % |s| ≥ (|μ|2 − σ 2 )2 + ξ 4 + 2|μ|2 ξ 2 ≥

.



|μ|4 + ξ 4 + 2|μ|2 ξ 2 4 1/2  2 |μ| + ξ2 ≥ 2  1/2 |μ|2 1 . ≥ |ξ | 1 + 2 2 ξ

1/4

(3.168) In sector .(c), we have %

.

1/4  1/4  ≥ |ξ |1/2 ξ 2 + 2σ 2 + 2|μ|2 |s| ≥ ξ 4 + 2ξ 2 σ 2 + 2|μ|2 ξ 2  1/4 ≥ |ξ |1/2 ξ 2 + 3|μ|2 2 1/4

 |μ| ≥ |ξ | 1 + 2 ξ

.

(3.169)

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⨆ ⨅

This proves (3.166).

Lemma 3.10.12 Let .m(ξ, σ, μ) be defined as in (3.162). Then, in sectors .(e) and (h)(as in (3.165)), we have the estimate

.

|m(ξ, σ, μ)| ⩽ 2, for ξ /= 0,

(3.170)

.

and in sector .(c)  |μ|2 |m(ξ, σ, μ)| ⩽ 2 1 + 2 ξ

.

1/4

for ξ /= 0.

(3.171)

Proof of Lemma 3.10.12 We consider the partition of the first quadrant of the (|μ|, σ )-plane as in (3.165). In cases .(e) and .(h) above, we can write

.

%

.

σ2 + ξ2 |m(ξ, σ, μ)| ⩽  1/4  |μ|2 − σ 2 2 + ξ 4 + 2ξ 2 σ 2 % σ2 + ξ2 ⩽ 1/4 ⩽ 2. 4 4 2 2 σ /4 + ξ + 2ξ σ

In case .(c) we have % σ2 + ξ2 1 ⩽  1/4 1/2 |ξ | ξ 2 + 2σ 2 + 2|μ|2

% σ2 + ξ2

|m(ξ, σ, μ)| ⩽  1/4 ξ 4 + 2ξ 2 σ 2 + 2|μ|2 ξ 2 %  1/4 σ2 + ξ2 1  2 1  1 2 1/4 ⩽ 1/2  ⩽ 1/2 2|μ|2 + ξ 2 . 1/4 ⩽ 1/2 σ + ξ |ξ | |ξ | |ξ | ξ 2 + 3σ 2

.

⨆ ⨅

This implies the estimate in (3.171). We now complete the proof of the trace component in the following lemma.

Lemma 3.10.13 Let .η satisfy (3.156) with zero initial data and .h∗ ∈ X−1/2 . Then, we have the following estimate 1/2

  ||γ0 [ηt ]||2L2 (0,T ;L2 (R2 )) + ||γ0 [η]||2L2 (0,T ;H 1 (R2 )) ≤ CT ||h∗ ||2 1/2 .

.

X−1/2

(3.172)

Proof of Lemma 3.10.13 We bound each term on the LHS of (3.172) separately. By the inverse Fourier-Laplace transform we have that e

.

−ξ t

1 ηt (x, y, z = 0, t) = 2π



R2

dμ

∞

−∞

2 dσ eiσ t ·ei(xμ1 +yμ ) ·m(ξ, σ, μ)h∗ (μ, ξ +iσ )

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

249

and e

.

−ξ t



1 η(x, y, z = 0, t) = 2π



R2

dμ

∞

1 2 dσ eiσ t · ei(xμ1 +yμ ) · √ h∗ (μ, ξ + iσ ). s −∞

Thus by the Parseval equality we obtain

1 2π

n(ξ ; ηt ) ≡

.

=

+∞ 0



dμ

R2

‖e−ξ t ηt (z = 0, t)‖2L2 (R2 ) dt. ∞

−∞

dσ |m(ξ, σ, μ)|2 |h∗ (μ, ξ + iσ )|2

(3.173) (3.174)

and p(ξ ; η) ≡

.



1 2π

=

+∞

0

R2



dμ

‖e−ξ t η(z = 0, t)‖2H 1 (R2 ) dt. ∞

−∞

dσ

(1 + |μ|2 ) ∗ |h (μ, ξ + iσ )|2 . |s|

(3.175) (3.176)

Proving (3.172) is equivalent to showing the RHS in (3.173) and (3.175) are bounded. We first look at (3.173). From Lemma 3.10.12, in sector .(c), we have |m(ξ, σ, μ)|2 ≤ 4(1 + ξ −2 |μ|2 )1/2 ≤ 4(1 + ξ −2 )1/2 (1 + |μ|2 )1/2 .

.

(3.177)

Using (3.177) and (3.170) and splitting the RHS of (3.173) based on sectors .(e), .(h) and .(c), we obtain

R2

dμ

∞ −∞

dσ |m(ξ, σ, μ)|2 |h∗ (μ, ξ + iσ )|2

=8

R2

dμ

|μ| 0≤σ ≤ √ 2

.

+ 8(1 + ξ −2 )1/2 +8

R2

dμ

dσ 2 |h∗ (μ, ξ + iσ )|2

R2

dμ

dσ √ σ ≥ 2|μ|

√ |μ| √ 0. Thus we obtained the desired result for the perturbed wave equation. Theorem 3.10.15 Any weak solution to (3.23) with zero initial data satisfies the following a priori bound: ||γ0 [ϕt ]||2L2 (0,T ;L2 (R2 )) + ||γ0 [ϕ]||2L2 (0,T ;H 1 (R2 )) + ||ϕt ||2L2 (0,T ;L2 (R3 )

.

+

+ ||ϕ||2L2 (0,T ;H 1 (R3 )) +

≤

CT ||h||2 1/2 . X−1/2

(3.190)

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

253

This theorem will be applied to the flow-plate solution taking .h = [ut + U ux ]ext and waiting a sufficiently long time (with respect to the support of the initial data characterized by .ρ > 0). Remark 3.10.8 We note that, from the point of view of stability (long-time behavior), the above bound may not be directly useful, owing to the constant .CT appearing on the RHS. We will only need to apply this estimate on time-translated intervals of a uniform size, i.e., .[tn − a, tn + a] for some fixed .a > 0. This is a critical point in the analysis here. Acknowledgments The first author acknowledges partial support from NSF-DMS 2205508. The second author acknowledges partial support from NSF-DMS 1907620. Finally, the authors are most pleased to acknowledge that this work is greatly influenced by a long-standing past collaboration with the late Igor Chueshov. This work is dedicated to his memory.

References 1. T. Aittokallio, M. Gyllenberg, O. Polo, A model of a snorer’s upper airway. Math. Biosci. 170(1), 79–90 (2001) 2. S. Antman, Nonlinear Problems of Elasticity, 2nd edn. (Springer, New York, NY, 2005) 3. O.H. Ammann, T. von Kármán, G.B. Woodruff, The Failure of the Tacoma Narrows Bridge: A Report to the Honorable John M. Carmody, (Administrator, Federal Works Agency, 1941) 4. M. Argentina, L. Mahadevan, Fluid-flow-induced flutter of a flag. Proc. Nat. Acad. Sci. USA 102, 1829–1834 (2005) 5. G. Arioli, F. Gazzola, Torsional instability in suspension bridges: the Tacoma Narrows Bridge case. Commun. Nonlinear Sci. Numer. Simul. 42, 342–357 (2017) 6. G. Avalos, P.G. Geredeli, J.T. Webster, Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary. Comm. Pure Appl. Anal. 15(6), 2301–2328 (2016) 7. G. Avalos, P.G. Geredeli, J.T. Webster, Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discr. Cont. Dyn. Sys. B 23(3), 1267–1295 (2018) 8. G. Avalos, P.G. Geredeli, J.T. Webster, A linearized viscous, compressible flow-plate interaction with non-dissipative coupling. J. Math. Anal. Appl. 477(1), 334–356 (2019) 9. A. Babin, M. Vishik, Attractors of Evolution Equations (North-Holland, Amsterdam, 1992) 10. A. Balakrishna, J.T. Webster, Large deflections of a structurally damped panel in a subsonic flow. Nonlinear Dynam. 103(4), 3165–3186 (2021) 11. A. Balakrishna, I. Lasiecka, J.T. Webster, Elastic stabilization of an intrinsically unstable hyperbolic flow-structure interaction on R3+ . Math. Models Methods Appl. Sci. 33(03), 505– 545 (2022) 12. A.V. Balakrishnan, Aeroelasticity—Continuum Theory (Springer, New York, Heidelberg, Dordrecht, London, 2012) 13. A.V. Balakrishnan, Nonlinear aeroelasticity, continuum theory, flutter/divergence speed, plate wing model, free and moving boundaries, in Pure Applied Mathematics, vol. 252. Lecture Notes (Chapman & Hall, FL, 2007), pp. 223–244 14. A.V. Balakrishnan, Toward a mathematical theory of aeroelasticity, in International Federation for Information Processing System Model Optimization, IFIP, vol. 166 (Kluwer Academic Publishers, Boston, MA, 2005), pp. 1–24

254

I. Lasiecka and J. T. Webster

15. A.V. Balakrishnan, M.A. Shubov, Asymptotic behaviour of the aeroelastic modes for an aircraft wing model in a subsonic air flow. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 460(2044), 1057–1091 (2004). The Royal Society 16. A.V. Balakrishnan, A.M. Tuffaha, Aeroelastic flutter in axial flow–The continuum theory. AIP Conf. Proceed. 1493(1), 58–66 (2012) 17. J.M. Ball, Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10, 31–52 (2004) 18. H.M. Berger, A new approach to the analysis of large deflections of plates. J. Appl. Mech. 22, 465–472 (1955) 19. E. Berchio, A. Ferrero, F. Gazzola. Structural instability of nonlinear plates modelling suspension bridges: mathematical answers to some long-standing questions. Nonlinear Anal.: Real World Appl. 28, 91–125 (2016) 20. R. Bisplinghoff, H. Ashley, Principles of Aeroelasticity (Dover, New York, 1975) 21. V.V. Bolotin, Nonconservative Problems of Elastic Stability (Pergamon Press, Oxford, 1963) 22. D. Bonheure, F. Gazzola, I. Lasiecka, J. Webster, Long-time dynamics of a hinged-free plate driven by a nonconservative force. Ann. l’Institut Henri Poincare (C) Analyse Non Lineaire 39(2), 457 (2022) 23. D. Bonheure, F. Gazzola, E. Moreira dos Santos, Periodic solutions and torsional instability in a nonlinear nonlocal plate equation. SIAM J. Math. Anal. 51(4), 3052–3091 (2019) 24. A. Boutet de Monvel, I. Chueshov, The problem on interaction of von Karman plate with subsonic flow of gas. Math. Methods Appl. Sci. 22(10), 801–810 (1999) 25. L. Boutet de Monvel, I. Chueshov, Oscillation of von Karman’s plate in a potential flow of gas. Izv. Vuz. Mat.+ 63, 219–244 (1999) 26. L. Boutet de Monvel, I. Chueshov, Non-linear oscillations of a plate in a flow of gas. Comptes rendus de l’Académie des Sci. Série 1, Mathématique 322(10), 1001–1006 (1996) 27. L. Boutet de Monvel, I.D. Chueshov, A.V. Rezounenko, Long—time behaviour of strong solutions of retarded nonlinear PDEs. Commun. Partial Diff. Equs. 22(9–10), 1453–1474 (1997) 28. L. Boutet de Monvel, I.D. Chueshov, A.V. Rezounenko, Long-time behaviour of strong solutions of retarded nonlinear P.D.E.s. Commun. Part. Diff. Equ. 22, 1453–1474 (1998) 29. I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems (Springer, New York, 2015) 30. I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid. Nonlinear Anal.: Theory, Methods Appl. 95, 650–665 (2014) 31. I. Chueshov, Interaction of an elastic plate with a linearized inviscid incompressible fluid. Comm. Pure Appl. Anal. 13(5), 1759–1778 (2014) 32. I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate. Math. Meth. Appl. Sci. 34, 1801–1812 (2011) 33. I. Chueshov, Introduction to the theory of infinite dimensional dissipative systems, in Acta (Kharkov, 1999), in Russian; English translation: Acta (Kharkov, Ukraine, 2002) 34. I.D. Chueshov, On a certain system of equations with delay, occurring in aeroelasticity. J. Soviet Math. 58, 385–390 (1992) 35. I. Chueshov, Remark on an elastic plate interacting with a gas in a semi-infinite tube: periodic solutions. Evol. Equs. Control Theory 5(4), 561–566 (2016) 36. I. Chueshov, Dynamics of von Karman plate in a potential flow of gas: rigorous results and unsolved problems, in Proceedings of the 16th IMACS World Congress, Lausanne, Switzerland (2000), pp. 1–6 37. I.D. Chueshov, On a certain system of equations with delay, occurring in aeroelasticity. J. Sov. Math. 58(4), 385–390 (1992) 38. I. Chueshov, E.H. Dowell, I. Lasiecka, J.T. Webster, Nonlinear elastic plate in a flow of gas: recent results and conjectures. Appl. Math. Optim. 73(3), 475–500 (2016) 39. I. Chueshov, E.H. Dowell, I. Lasiecka, J.T. Webster, Mathematical aeroelasticity: a survey. Math. Eng. Sci. Aerospace (MESA) 7(1), 1–26 (2016) 40. I. Chueshov, T. Fastovska, On interaction of circular cylindrical shells with a Poiseuille type flow. Evol. Equns. Contr. Theo. 5(4), 605–629 (2016)

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

255

41. I. Chueshov, I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, vol. 195(912) (Memoirs of the American Mathematical Society, Providence, RI, 2008) 42. I. Chueshov, I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics (Springer, New York, NY, 2010) 43. I. Chueshov, I. Lasiecka, Generation of a semigroup and hidden regularity in nonlinear subsonic flow-structure interactions with absorbing boundary conditions. J. Abstr. Differ. Equ. Appl. 3, 1–27 (2012) 44. I. Chueshov, I. Lasiecka, J.T. Webster, Attractors for delayed, nonrotational von Karman plates with applications to flow-structure interactions without any damping. Commun. Partial Diff. Equs. 39(11), 1965–1997 (2014) 45. I. Chueshov, I. Lasiecka, J.T. Webster, Evolution semigroups in supersonic flow-plate interactions. J. Diff. Equs. 254(4), 1741–1773 (2013) 46. I. Chueshov, I. Lasiecka, J.T. Webster, Flow-plate interactions: well-posedness and longtime behavior. Discrete Contin. Dyn. Syst. Ser. S, Special Volume: New Developments in Mathematical Theory of Fluid Mechanics 7(5), 925–965 (2014) 47. I. Chueshov, A. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations. Comptes rendus de l’Académie des Sci. Série 1, Mathématique, 321(5), 607–612 (1995) 48. I.D. Chueshov, A.V. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations. C. R. Acad. Sci. I-Math. 321, 607–612 (1995) 49. I. Chueshov, I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations. J. Diff. Equ. 254, 1833–1862 (2013) 50. I. Chueshov, I. Ryzhkova, A global attractor for a fluid-plate interaction model. Commun. Pur. Appl. Anal. 12, 1635–1656 (2013) 51. I. Chueshov, I. Ryzhkova, On interaction of an elastic wall with a Poiseuille type flow. Ukr. Math. J. 65(1), 158–177 (2012) 52. I. Chueshov, I. Ryzhkova, Well-posedness and long time behavior for a class of fluidplate interaction models, in IFIP Advances in Information and Communication Technology, vol. 391, ed. by D. Hömberg, F. Tröltzsch (25th IFIP TC7 Conference, Berlin, Sept 2011) (Springer, Berlin, 2013), pp. 328–337 53. P. Ciarlet, P. Rabier, Les Equations de Von Karman, vol. 826 (Springer, New York, NY, 2006) 54. J. Cole, L. Cook. Transonic Aerodynamics (North Holland, Netherlands, 1986) 55. D.G. Crighton, The Kutta condition in unsteady flow. Ann. Rev. Fluid Mech. 17, 411–445 (1985) 56. M. Deliyianni, V. Gudibanda, J. Howell, J.T. Webster, Large deflections of inextensible cantilevers: modeling, theory, and simulation. Math. Model. Nat. Phenom. 15, 44 (2020) 57. M. Deliyianni, J.T. Webster, Theory of solutions for an inextensible cantilever. Appl. Math. Optim. 84(2), 1345–1399 (2021) 58. M. Deliyianni, K. McHugh, J.T. Webster, E. Dowell, Dynamic equations of motion for inextensible beams and plates. Arch. Appl. Mech. 92(6), 1929–1952 (2022) 59. O. Doaré, S. Michelin, Piezoelectric coupling in energy-harvesting fluttering flexible plates: linear stability analysis and conversion efficiency. J. Fluids Struct. 27(8), 1357–1375 (2011) 60. E.H. Dowell, Flutter of a buckled plate as an example of chaotic motion of a deterministic autonomous system. J. Sound Vibr. 85(3), 333–344 (1982) 61. E. Dowell, Aeroelasticity of Plates and Shells (Nordhoff, Leyden, 1975) 62. E.H. Dowell, Nonlinear oscillations of a fluttering plate, I and II AIAA J. 4, 1267–1275 (1966); 5, 1857–1862 (1967) 63. E.H. Dowell, Panel flutter-A review of the aeroelastic stability of plates and shells. AIAA J. 8(3), 385–399 (1970) 64. E.H. Dowell, A Modern Course in Aeroelasticity (Kluwer Academic Publishers, Springer Nature, Switzerland, 2004) 65. E. Dowell, Some recent advances in nonlinear aeroelasticity: fluid-structure interaction in the 21st century, in 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 18th AIAA/ASME/AHS Adaptive Structures Conference 12th, (2010), p. 3137

256

I. Lasiecka and J. T. Webster

66. E. Dowell, K. McHugh (2016) Equations of motion for an inextensible beam undergoing large deflections. J. Appl. Mech. 83(5), 051007 (2016) 67. E.H. Dowell, O. Bendiksen, Panel Flutter, Encyclopedia of Aerospace Engineering (Wiley, 2010) 68. E. Dowell, O. Bendiksen, J. Edwards, T. Strganac, Transonic Nonlinear Aeroelasticity. In Encyclopedia of Aerospace Engineering (eds R. Blockley and W. Shyy) (2010). https://doi. org/10.1002/9780470686652.eae151 69. J.A. Dunnmon, S.C. Stanton, B.P. Mann, E.H. Dowell, Power extraction from aeroelastic limit cycle oscillations. J. Fluids Struct. 27(8), 1182–1198 (2011) 70. A. Eden, A.J. Milani, Exponential attractors for extensible beam equations. Nonlinearity 6(3), 457 (1990) 71. E. Erturk, D. Inman, Piezoelectric Energy Harvesting (John Wiley and Sons, UK, 2011) 72. P. Fabrie, C. Galusinski, A. Miranville, S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation. Discr. Contin. Dyn. Sys. 10(1/2), 211–238 (2004) 73. A. Favini, M. Horn, I. Lasiecka, D. Tataru, Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation. Diff. Integral Equs. 9(2), 267–294 (1996) 74. W. Frederiks, H.C.J. Hilbering, J.A. Sparenberg, On the Kutta condition for the flow along a semi-infinite elastic plate. J. Engin. Math. 20, 27–50 (1986) 75. A. Ferrero, F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges. Discrete Contin. Dyn. Syst. 35, 5879–5908 (2015) 76. A. Ferrero, F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges. Discrete Contin. Dyn. Syst.-A 35(12), 5879–5908 (2015) 77. F. Gazzola, Mathematical Models for Suspension Bridges, MS&A, vol. 15 (Springer International Publishing Switzerland, 2015) 78. P.G. Geredeli, I Lasiecka, J.T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer. J. Diff. Equs. 254(3), 1193–1229 (2013) 79. P.G. Geredeli, J.T. Webster, Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping. Nonlinear Anal. B. 31, 227–256 (2016) 80. S.C. Gibbs, I. Want, E. Dowell, Theory and experiment for flutter of a rectangular plate with a fixed leading edge in three-dimensional axial air flow. J. Fluids Struct. 34, 68–83 (2012) 81. P.J. Graber, Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions. Nonlinear Anal.: Theory, Methods Appl. 74(10), 3137– 3148 (2011) 82. S.M Han, H. Benaroya, T. Wei, Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vibr. 225(5), 935–988 (1999) 83. P.J. Holmes, Bifurcations to divergence and flutter in flow-induced oscillations: a finite dimensional analysis. J. Sound Vibr. 53(4), 471–503 (1977) 84. P. Holmes, J. Marsden, Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis. Automatica 14(4), 367–384 (1978) 85. J. Howell, K. Huneycutt, J.T. Webster, S. Wilder, A thorough look at the (in) stability of piston-theoretic beams. Math. Eng. 1(3), 614–647 (2019) 86. J.S. Howell, I. Lasiecka, J.T. Webster, Quasi-stability and exponential attractors for a nongradient system—applications to piston-theoretic plates with internal damping. Evol. Equs. Control Theory 5(4), 567 (2016) 87. J.S. Howell, D. Toundykov, J.T. Webster, A cantilevered extensible beam in axial flow: semigroup well-posedness and postflutter regimes. SIAM J. Math. Anal. 50(2), 2048–2085 (2018) 88. L. Huang, Flutter of cantilevered plates in axial flow. J. Fluids Struct. 9, 127–147 (1995) 89. B. Kaltenbacher, I. Kukavica, I. Lasiecka, R. Triggiani, A. Tuffaha, J.T. Webster, Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions (Birkhäuser, Switzerland, 2018)

3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

257

90. A.K. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation. J. Math. Analys. Appl. 318(1), 92–101 (2006) 91. H.S. Kim, J.H. Kim, J. Kim, A review of piezoelectric energy harvesting based on vibration. Int. J. Precis. Eng. Manuf. 12(6), 1129–1141 (2011) 92. J.E. Lagnese, Boundary Stabilization of Thin Plates. (Society for Industrial and Applied Mathematics, SIAM, Philadelphia, 1989) 93. I. Lasiecka, Mathematical Control Theory of Coupled PDE’s. CMBS-NSF Lecture Notes (SIAM, 2002) 94. I. Lasiecka, R. Triggiani, Control theory for partial differential equations: Continuous and approximation theories, Abstract parabolic systems. vol. 1 (Cambridge University Press, United Kingdom, 2000) 95. I. Lasiecka, J.T. Webster, Kutta-Joukowski flow conditions in flow-plate interactions: subsonic case. Nonlinear Anal.: B, Real World Appl. 17, 171–191 (2014) 96. I. Lasiecka, J.T. Webster, Generation of bounded semigroups in nonlinear subsonic flowstructure interactions with boundary dissipation. Math. Methods Appl. Sci. 36, 1995–2010 (2013) 97. I. Lasiecka, J.T. Webster, Eliminating flutter for clamped von Karman plates immersed in subsonic flows. Commun. Pure Appl. Anal. 13(5), 1935–1969 (2014) 98. I. Lasiecka, J.T. Webster, Feedback stabilization of a fluttering panel in an inviscid subsonic potential flow. SIAM J. Math. Anal. 48(3), 1848–1891 (2016) 99. I. Lasiecka, J.T. Webster, Eliminating flutter for clamped von Karman plates immersed in subsonic flows. Comm. Pure Appl. Anal. 13(5), 1935–1969 (2014) 100. J.L. Lions, Quelques méthodes de résolution des problemes aux limites non linéaires (Dunod, Paris, 1969) 101. E. Livne, Future of airplane aeroelasticity. J. Aircraft 40, 1066–1092 (2003) 102. J. Malek, J. Necas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids. JDE 127(2), 498–518 (1996) 103. J. Malek, D. Prazak, Large time behavior via the method of l-trajectories. JDE 181(2), 243– 279 (2002) 104. C. Mei, K. Abdel-Motagaly, R. Chen, Review of nonlinear panel flutter at supersonic and hypersonic speeds. Appl. Mech. Rev. 52, 321–332 (1999). NASA Conference Publication 105. A. Miranville, S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations, vol. 4 (Elsevier, Switzerland, 2008) 106. S. Miyatake, Mixed problem for hyperbolic equation of second order. J. Math. Kyoto Univ. 13(3), 435–487 (1973) 107. S. Miyatake, Neumann operator for wave equation in a half space and microlocal orders of singularities along the boundary. Seminaire Equations aux derivees partielles (Polytechnique) (1993). pp.1–6 108. M.P. Paidoussis, Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 1 (Academic Press, San Diego, 1998) 109. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 (Springer Science & Business Media, New York, NY, 2012) 110. I. Ryzhkova, Stabilization of a von Karman plate in the presence of thermal effects in a subsonic potential flow of gas. J. Math. Anal. Appl. 294, 462–481 (2004) 111. I. Ryzhkova, Dynamics of a thermoelastic von Karman plate in a subsonic gas flow. Zeitschrift Ang. Math. Phys. 58, 246–261 (2007) 112. R. Sakamoto, Mixed problems for hyperbolic equations I Energy inequalities. J. Math. Kyoto Univ. 10(2), 349–373 (1970) 113. R.H. Scanlan, The action of flexible bridges under wind, I: flutter theory, II: buffeting theory. J. Sound Vibr. 60, 187–199 & 201–211 (1978) 114. R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivatives. J. Eng. Mech. (ASCE) 97, 1717–1737 (1971)

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115. C. Semler, G.X. Li, M.P. Païdoussis, The non-linear equations of motion of pipes conveying fluid. J. Sound Vibr. 169, 577–599 (1994) 116. R.E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, vol. 49 (AMS, Philadelphia, USA, 1997) 117. M. Slemrod, Weak asymptotic decay via a “relaxed invariance principle” for a wave equation with nonlinear, nonmonotone damping. Proc. Royal Soc., Edinburgh Sect. A. 113, 87–97 (1989) 118. S.C. Stanton, A. Erturk, B.P. Mann, E.H. Dowell, D.J. Inman, Nonlinear nonconservative behavior and modeling of piezoelectric energy harvesters including proof mass effects. J. Intell. Mater. Sys. Struct. 23(2), 183–199 (2012) 119. M. Shubov, Riesz basis property of mode shapes for aircraft wing model (subsonic case). Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 462, 607–646 (2006) 120. M. Shubov, Solvability of reduced Possio integral equation in theoretical aeroelasticity. Adv. Diff. Equs. 15, 801–828 (2010) 121. M.A. Shubov, V.I. Shubov, Asymptotic and spectral analysis and control problems for mathematical model of piezoelectric energy harvester. Math. Eng. Sci. Aero. (MESA) 7(2), 249 (2016) 122. M.A. Shubov, Asymptotic representation for the eigenvalues of a non-selfadjoint operator governing the dynamics of an energy harvesting model. Appl. Math. Optim. 73(3), 545–569 (2016) 123. D.M. Tang, H. Yamamoto, E.H. Dowell, Flutter and limit cycle oscillations of two dimensional panels in three-dimensional axial flow. J. Fluids Struct. 17, 225–242 (2003) 124. D. Tang, M. Zhao, E.H. Dowell, Inextensible beam and plate theory: computational analysis and comparison with experiment. J. Appl. Mech. 81(6), 061009 (2014) 125. D. Tang, S.C. Gibbs, E.H. Dowell, Nonlinear aeroelastic analysis with inextensible plate theory including correlation with experiment. AIAA J. 53, 1299–1308 (2015) 126. D. Tataru, On the regularity of boundary traces for the wave equation. Ann. Scuola Normale. Sup. di Pisa. 26, 185–206 (1998) 127. D. Tataru, The Xθs spaces and unique continuation for solutions to the semilinear wave equation. Commun. Partial Diff. Equs. 21, 841–887 ((1996)) 128. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics (Springer, New York, NY, 1988) 129. F.G. Tricomi, Integral Equations (Interscience Publishers Inc., New York, 1957) 130. V.V. Vedeneev, Panel flutter at low supersonic speeds. J. Fluids Struct. 29, 79–96 (2012) 131. V. Vedeneev, Effect of damping on flutter of simply supported and clamped panels at low supersonic speeds. J. Fluids Struct. 40, 366–372 (2013) 132. T. von Kármán, Festigkeitsprobleme im maschinenbau. Encyclopedia der Mathematischen Wissenschaften, Leipzig vol. IV/4(C) (1910), pp. 348–352 133. J.T. Webster, Weak and strong solutions of a nonlinear subsonic flow—structure interaction: semigroup approach. Nonlinear Anal.: Theory Methods Appl. 74(10), 3123–3136 (2011) 134. J.T. Webster, Attractors and determining functionals for a flutter model: finite dimensionality out of thin air. Pure Appl. Func. Anal. 5(1), 85–119 (2019) 135. J.R. Wilson, A new boom in supersonics. AIAA J. Aerospace Am. 49(2), 30–38 (2011)

Chapter 4

Turbulence Control: From Model-Based to Machine Learned Nan Deng, Guy Y. Cornejo Maceda, and Bernd R. Noack

4.1 Introduction Flow analysis, modeling, and control are at the heart of engineering applications: aeronautics, airborne and ground transport, wind power generation, and industrial processes, to cite a few examples [6]. Fluid flows are characterized by high dimensionality, nonlinearity, multiscale, and time delays, which pose challenges to existing theories and methods for analysis, modeling, and control. Methods of big data, machine learning, and artificial intelligence are revolutionizing these fields and are going toward full automation. In this chapter, we present AI-powered automated methods to overcome the complexity of fluid flows and solve analysis, modeling, and control tasks. To tackle the challenge of high dimensionality in data analysis, we employ proximity maps based on multidimensional scaling (MDS). Proximity maps enable feature extraction for high-dimensional data and reconstruction of the dynamics in an automated way [26, 33, 11]. Modeling of complex dynamics is enabled with clustering and statistical methods of network science [3]. In particular, representative states and dynamics are extracted and reconstructed with only one parameter to tune in the Clusterbased Network Models [25, 37, 20, 32, CNM]. For control, the mapping between the outputs and inputs is learned automatically, thanks to Machine Learning Control [18, 10, MLC]. MLC has been successfully employed in dozens of experiments, often outperforming open-loop controllers. Our approach is exemplified on a lowcost, easy-to-execute benchmark configuration, the fluidic pinball, which consists of a cluster of three cylinders in an incoming flow. This simple configuration encapsulates key characteristics of real flows, such as successive bifurcations and

N. Deng · G. Y. Cornejo Maceda · B. R. Noack () HIT Shenzhen, Shenzhen, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Bodnár et al. (eds.), Fluids Under Control, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-031-47355-5_4

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Fig. 4.1 AI-powered automated methods for analysis, modeling, and control. Three data-driven methods are introduced in the chapter, including proximity maps based on multidimensional scaling, cluster-based network modeling, and machine learning control. The details are in the following sections

frequency crosstalk. In addition, the independent rotation of the three cylinders makes the fluidic pinball a simple benchmark system to explore multiple-input multiple-output control (Fig. 4.1). This chapter is organized into five sections. Section 4.2 overviews the fluidic pinball richness, including the flow configuration, description of the steady solutions, transient and post-transient dynamics, route to chaos, and an open-loop control parametric study. In Sect. 4.3, we describe the proximity map methodology for automated feature extraction for high-dimensional data. Cluster-based network modeling is detailed in Sect. 4.4 and exemplified for the automated modeling of the unforced fluidic pinball at Reynolds numbers 90 and 160. Section 4.5 describes machine learning control for automated learning of feedback control laws demonstrated on the drag reduction problem of the fluidic pinball. Finally, Sect. 4.6 summarizes the presented machine learning tools for automated analysis, modeling, and control.

4.2 Fluidic Pinball We propose the fluidic pinball as the benchmark configuration used to exemplify all the methods for automated analysis, modeling, and control. We provide here an overview of the pinball configuration, including the numerical methods to simulate

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Fig. 4.2 Configuration of the fluidic pinball, the simulation domain .[−20, 80] × [−30, 30] and evaluation domain .[−4, 20] × [−4, 4] in this work. An example vorticity field is visualized in color with .[−1.5, 1.5]

the steady and unsteady flow, the underlying instabilities, a parametric study, and automatically extracted flow features. I. Flow Configuration The pinball configuration considers a two-dimensional incompressible viscous flow with incoming velocity .U∞ past three rotatable circular cylinders. The axes of the three cylinders are parallel and located at the vertices of an equilateral triangle pointing downstream. The diameter of the cylinder is D, and the distance between the two axes is .3D/2. The Reynolds number is defined as .Re = U∞ D/ν, where .ν is the kinematic viscosity. The cylinders can rotate independently of others, so the pinball system contains four control parameters: Reynolds number and three rotation speeds. Therefore, the fluidic pinball is a multiple-input multiple-output (MIMO) flow control system, allowing to test different control designs [24, 9, 33] for wake flows. As shown in Fig. 4.2, a Cartesian coordinate system .x = (x, y) is used to describe the flow, whose c-axis is parallel to the streamwise direction and y-axis is in the spanwise direction. The origin is located at the midpoint of the two rearward cylinders. Therefore, the cylinder centers are located at .(xi , yi ): x1 = −3/2 cos(30◦ ) y1 = 0, . x2 = 0 y2 = −3/4, y3 = 3/4, x3 = 0 with .i = 1, 2, 3 for the front, bottom, and top cylinders, respectively. The velocity and pressure field are denoted respectively by .u = (u, v) and p, where u and v

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are the streamwise and spanwise components of the velocity. The forcing is exerted by rotating the three cylinders with independent circumferential velocities with an actuation command .b = [b1 , b2 , b3 ]⏉ . A positive (negative) value of the actuation command corresponds to the counter-clockwise (clockwise) rotation of the cylinders along their axis. II. Direct Numerical Simulation The flow dynamics is governed by the non-dimensionalized incompressible Navier– Stokes equations, which read 1 Δu − ∇p, . Re ∇ · u = 0.

∂t u + ∇ · u ⊗ u =

.

(4.1a) (4.1b)

All the variables have been non-dimensionalized according to the cylinder diameter D, the incoming velocity .U∞ , the convective time scale .D/U∞ , and the pressure 2 .ρU∞ , where .ρ is the density. The flow field is calculated by two-dimensional direct numerical simulations (DNS). The solution is obtained iteratively using the Newton–Raphson type approach with a prescribed residual tolerance. This work considers a larger computational domain as compared to [13, 39], which is a rectangle bounded by .[−20, 80] × [−30, 30], to simulate an unbounded flow under the free-stream condition as far as possible. An evaluation domain .[−4, 20] × [−4, 4] is used to determine and visualize the dominant dynamics of the wake. The computational domain, as shown in Fig. 4.3, is discretized on an unstructured grid with .15 258 second-order Taylor-Hood finite elements and .30 826 vertices by Finite Element Method. The time integration is based on implicit third-order time integration.

Fig. 4.3 Grid of the fluidic pinball

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The far-field boundary conditions .U∞ = ex are exerted on the inflow, upper and lower boundaries, while the outflow boundary is assumed to be stress-free, and transparent for the outgoing fluid structures. The flow is controlled by the rotating cylinders. A non-slip condition is adopted on the surface of the cylinders, and their circumferential velocity is set according to the actuation command .b = [b1 , b2 , b3 ]⏉ . We note that the blockage effect in the fluidic pinball changes the critical Reynolds numbers for the secondary and subsequent successive bifurcations [15]. Enlarging the size of the computational domain reduces the blockage effect but largely increases the computational cost to simulate similar flow dynamics. More effort can be invested to simulate an unbounded flow through ingenious design of boundary locations and boundary conditions, but this is not the focus of this work. The authors suggest choosing an optimized grid trade-off between computation cost and accuracy. A smaller grid size will be more appropriate in case the focus is on dynamics at low Reynolds numbers [13] or the number of test cases is large [9]. The initial condition to start the simulation can be defined with a snapshot containing full information of the flow state .(u, p), otherwise, a uniform crossflow .u = U∞ will be applied. The output files of unsteady DNS are snapshots equidistantly sampled at .t m = mΔt with .m = 1, . . . , M, recording the velocity flow field .um (x) = u(x, t m ) and the pressure field .pm (x) on all the grid points. In this work, the default time interval is .Δt = 0.1. III. Flow Features In this subsection, we give an overview of the flow features by flow field visualization and force measurements. We first define the configuration symmetry and the force measurements before the description of flow features. Symmetry is an important feature of the flow field, strictly required with the following properties on the velocity field .u = (u, v) and the pressure field p: u(x, −y) = u(x, y),

.

v(x, −y) = −v(x, y),

p(x, −y) = p(x, y).

(4.2)

In this case, the spanwise vorticity .ω = ∂x v − ∂y u satisfies .ω(x, −y) = −ω(x, y). The steady solution at a low Reynolds number .Re ≤ 80 is symmetric, see Fig. 4.4 for example. In summary, for a symmetric field, the u and p are even, and v and .ω are odd in y. A reflection operator .R can be defined by .R(u, v, p, ω)(x, y) ≡ (u, −v, p, −ω)(x, −y). Similarly, we define reflectional antisymmetry as .

u(x, −y) = −u(x, y), v(x, −y) = v(x, y), p(x, −y) = −p(x, y), ω(x, −y) = ω(x, y),

(4.3)

satisfying the operator .−R. For the unsteady flow, the symmetry can be also satisfied in the periodic flow, where the flow and its half-period delay satisfy the reflectional symmetry .Ru(t) = u(t + T /2), so-called spatio-temporal symmetry in [4]. In addition, symmetry and antisymmetry are usually discussed in the modal analysis, such as the spatial modes from proper orthogonal decomposition (POD).

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u− s

u+ s

us

10 20 50 80 100 120 140 160 Fig. 4.4 Steady solutions at different values of the Reynolds numbers, including the symmetric + steady solution .us and the asymmetric steady solutions .u− s and .us . Vorticity fields are plotted in color with the range .[−1.5, 1.5]

A useful indicator to show the flow dynamics is the instantaneous lift .CL and drag .CD coefficients on the cylinders, which can be measured from the velocity and pressure fields. On the boundary .𝚪 of the cylinders, .n is the unit normal pointing outward the surface element dS. For the considered two-dimensional pinball system, the .α-component .Fαν of the viscous force .F ν is derived as 



Fαν = F ν · eα = 2ν

Sα,β nβ dS,

.

𝚪

(4.4)

β=x,y,z

with .eα is the unit vector in .α-direction, .nα is the .α-component  of .n, the kinematic  viscosity .ν = 1/Re, and the strain rate tensor .Sα,β = ∂α uβ + ∂β uα /2 with .α, β = x, y. The pressure force is derived as  Fαp = F p · eα = −

nα p dS.

.

𝚪

(4.5)

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The resulting drag and lift force is defined as the projection on .ex and .ey of the pressure and viscous forces exerted on the body p

FD = Fx + Fxν ,

.

p

FL = Fy + Fyν .

(4.6)

Then, the drag and lift force coefficients after nondimensionalization are: CD = 2FD ,

.

CL = 2FL .

(4.7)

Steady Solutions The steady solution is computed by solving the following steady Navier–Stokes equations, 0 = −∇ · us ⊗ us + νΔus − ∇ps , .

(4.8a)

0 = ∇ · us .

(4.8b)

.

The boundary conditions are the same as the unsteady Navier–Stokes equations (4.1). The Newton–Raphson iteration method is used to solve the steady Navier– Stokes equations at the defined Reynolds number. The steady solution is stable only when the Reynolds number is less than all the critical Reynolds numbers of the underlying instabilities. Otherwise, the steady solution is unstable. The unstable steady solutions can be numerically computed but are rarely or only transiently observed in experiments, due to the unavoidable perturbations. The vorticity field of the steady solutions at different Reynolds numbers are shown in Fig. 4.4. At a low Reynolds number .Re ≤ 80, as shown in Fig. 4.4, only one symmetric steady solution .us can be obtained for each Reynolds number, satisfying the reflectional symmetry of the configuration with respect to the x-axis. A basebleeding jet appears between the back two cylinders, and the length of the jet increases at a higher Reynolds number. Multiple steady solutions can be found at .Re > 80, where a symmetry breaking of the symmetric steady solution .us occurs on the symmetric base-bleeding jet and leads to three steady solutions. The jet can − deflect to the top or bottom, noted as .u+ s and .us . By drawing the drag and lift coefficients of all the existing steady solutions, we find the bifurcating point is at .Re = 80, as shown in Fig. 4.5. This critical value can also be determined by linear stability analysis of the symmetric steady solution. Unsteady Flow In Fig. 4.6, all the simulations start in the vicinity of the symmetric steady solution at the corresponding Reynolds number. At .Re = 60, the infinitesimal perturbations destabilize the flow from the unstable steady solution. A Bénard-von Kármán vortex street can be found downstream of the cylinders. This periodic wake flow has spatiotemporal symmetry. At .Re = 90, the wake flow preserves the spatio-temporal symmetry first. However, this state is still unstable, and the base-bleeding jet will

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Fig. 4.5 Drag (left) and lift coefficient (right) of the steady solutions as a function of Reynolds number. The displayed data corresponds to the symmetric steady solution .us (black curve) and the + asymmetric steady solutions .u− s (blue curve), and .us (red curve)

Re

u(t1 = 0)

u(t2 = 700)

u(t3 = 1500)

60

90

120

160 Fig. 4.6 Unsteady flow states visualized with vorticity fields, resulting from DNS starting with the corresponding symmetric steady solution at different values of the Reynolds numbers. Three snapshots were selected for the initial condition, the transient flow, and the post-transient flow. Vorticity fields are plotted in color with the range .[−1.5, 1.5]

eventually deflect to the top. At .Re = 120, the symmetry breaking of the jet and the vortex shedding occur almost simultaneously. But the jet will oscillate around the upwards deflected position at a lower frequency, which modulates the vortex shedding and presents quasi-periodic dynamics. At .Re = 160, the transient dynamics are similar to the case at .Re = 120, while more interaction can be observed between the vortex shedding and the jet in the post-transient dynamics. The jet oscillates at the centering place and is highly coupled with the vortex shedding dynamics. The wake flow presents chaotic characteristics.

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Fig. 4.7 Time evolution of the drag (left) and lift (right) coefficients, starting from the symmetric steady solution .us (black curve) and the asymmetric steady solutions (red curve of .u− s , and blue curve of .u+ s ) at different Reynolds numbers, .Re = 60, 90, 120, 160 from top to bottom. The black dashed curve is the mirror-conjugated trajectory of the black curve, obtained from applying the reflection operator to the flow field

Transient and Post-transient Dynamics The transient and post-transient dynamics contain a destabilizing stage in the vicinity of the unstable steady solution (fixed point), a transient stage from the unstable solution to the destination attractor, and a post-transient regime evolving on the attractor (limit cycle, torus, or chaotic). In Fig. 4.7, we illustrate the transient and post-transient dynamics at different Reynolds numbers with the time evolution of the drag .CD and lift .CL coefficients. For all the Reynolds numbers, the DNS starts from all the possible unstable steady solutions and calculates the unsteady flow for 1500 time units. The unsteady flow will stay in the vicinity of the unstable steady solution for an extremely long time. In this case, the cumulative numerical error works as an unbiased initial small perturbation. A white-noise velocity perturbation with small kinetic energy can also be applied to start the simulation to speed up the destabilization from the steady solutions.

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At .Re = 60, the drag coefficient .CD increases and reaches a plateau quickly over the time range .t ∈ [500, 800]. The oscillation amplitude of the drag is very small and hardly seen compared to the significantly increased mean drag force. The lift coefficient .CL starts to oscillate around zero, and the amplitude has the same tendency as the drag. The asymptotic regime is a periodic state, where the frequency of .CL has the same value as the vortex shedding frequency .fVS = 0.080, and the frequency of .CD is twice the shedding frequency. At .Re = 90, we run three simulations initialized with the three steady solutions. To illustrate all possible transitions from the three unstable steady solutions to the attractors, we apply the reflection operator R to the unsteady flow starting with the symmetric steady solution. So we can get two pairs of mirror-conjugated trajectories, one pair starting from the symmetric steady solution and the other from the two asymmetric steady solutions. As shown in Fig. 4.7, the initial condition decides the transient dynamics. The symmetry of the state can be observed from the lift force. The flow starting with the asymmetric steady solution will go to the corresponding periodic state with the same deflection. However, starting with the symmetric steady solution, the flow will reach an unstable symmetric oscillating state with frequency .fVS = 0.080, then destabilize again to one of two asymmetric oscillating states with frequency .fVS = 0.088. The symmetric breaking is hardly observed in the initial stage of the transient dynamics. In summary, there exist four transients between the three unstable steady solutions, one unstable periodic state, and two stable periodic states. At .Re = 120, similar to .Re = 90, four transients can be obtained from DNS initialized with the three steady solutions. However, symmetry breaking occurs during the oscillation establishment, making the unstable symmetric state not observable. For the asymptotic regime, a secondary frequency of the jet .fJet = 0.010 appears and modulates the vortex shedding .fVS = 0.098. The quasi-periodic oscillations can be observed both in the drag and lift forces. At .Re = 160, the four transients start to oscillate with limited amplitude near the steady solutions but quickly enter a chaotic regime with zero-mean value. This indicates that symmetry-breaking instability still exists for the steady solution, yet only a symmetric chaotic attractor can exist in the asymptotic state. The four transients can be plotted in a three-dimensional forces-embedding state space, using the drag .CD (t), lift .CL (t) and time-delayed lift .CL (t − τ ) force coefficients, with .τ = 2. As shown in Fig. 4.8, we can illustrate the trajectories from the steady solutions to the asymptotic regimes. In the state space, the steady solutions are represented by fixed points, and the asymptotic regimes are characterized by limit cycle, torus, or chaotic cloud for different Reynolds numbers. The oscillating dynamics of the Bénard-von Kármán instability is well represented by the force dynamics of lift and its time delay, like in Fig. 4.8a at .Re = 60. A helix spirals out from a fixed point to a limit cycle on the top. The unsteady vortex shedding in the asymptotic regime contributes to a .21.5% increase of drag force compared to the steady flow. Therefore, controlling the airflow by stabilizing the wake will help reduce drag.

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Fig. 4.8 Scenarios of transient and post-transient dynamics, based on the drag .CD (t), lift and time-delayed lift .CL (t − τ ) coefficients. The simulations start near the unstable symmetric/asymmetric steady solutions at (a) .Re = 60, (b) 90, (c) 120, and (d) 160: the black curve is associated with the symmetric steady solution .us (.×), the black dashed curve with its mirror-conjugated trajectory, the red curve with the asymmetric steady solution .u− s (red filled circle), and the blue curve with the asymmetric steady solution .u+ s (blue filled square)

.CL (t)

At a higher Reynolds number, the symmetry breaking is observed by the multiple fixed points, as well for limit cycles and tori. The mirror-conjugated trajectories represent the vortex shedding with upward and downward-deflected jets, respectively. At .Re = 90, three limit cycles indicate the periodic vortex shedding with three different jet states. The symmetric cycle is unstable and will finally converge to one of two stable asymmetric cycles. At .Re = 120, two tori are found for the quasi-periodic vortex shedding with upward and downward-deflected jets. The secondary frequency comes from the small oscillation of the jet around the deflected place. At .Re = 160, the mirror-conjugated trajectories can only exist in the initial transient regime and then quickly enter the same chaotic cloud. The center

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of the chaotic attractor has a vanishing lift force, indicating that the jet is deflected randomly with respect to the x-axis. IV. Route to Chaos This section briefly introduces the instabilities and bifurcations [8] that occur in this configuration with increasing Reynolds numbers. Although we can numerically observe unsteady dynamics destabilizing from a fixed point, an analytical determination of the local stability of the fixed point can reveal the destabilizing mechanism. Linear Stability Analysis of the Base Flow Linear stability analysis is applied to determine the underlying instabilities of the base flow (the equilibrium) by assuming linear dynamics with infinitesimal perturbations in the vicinity of the base flow. The base flow can be steady or periodic solutions by solving the steady/unsteady Navier–Stokes equations. In the case of periodic base flow, Floquet theory is applied to consider the evolution of perturbations in one period of the base flow. We describe the linear stability analysis of the symmetric steady solution. For the steady state .q s (x) = (us (x), ps (x)), an infinitesimal perturbation .q ' = (u' , p' ) is considered: q(x, t) = q s (x) + q ' (x, t).

.

(4.9)

Substituting Eq. (4.9) into the unsteady Navier–Stokes equations (4.1) and ignoring the second-order small term .u' ⊗ u' , the dynamics of the perturbation are governed by the linearized Navier–Stokes equations:   ∂t u' = −∇ · u' ⊗ us + us ⊗ u' + νΔu' − ∇p' , .

.

'

0 = ∇ ·u.

(4.10a) (4.10b)

The boundary conditions for the fluctuation are now homogenized as the original conditions are absorbed in the steady solution. We define the linearized Navier–Stokes operator around the steady solution .us as ' .Lus and the perturbation state vector .q , the linearized system can be re-written as: ∂t q ' = L u s q ' .

.

(4.11)

Eigenvalue decomposition can be applied to Eq. (4.11) to solve .(σ + iω) qˆ = (σ +iω)t , with the eigenˆ Lus qˆ with the perturbation in the form .q ' (x, t) = q(x)e modes .qˆ and their corresponding values .σ + iω. The real part of the eigenvalues .σ is the growth rate of eigenmodes, indicating the stability of the base flow. The base flow is stable only when all the .σ are negative, otherwise, the base flow is unstable. Generally, a stable steady flow tends to become unstable with an increasing Reynolds number. Positive eigenvalues can be found by the linear stability analysis of this steady solution when the Reynolds number is larger than a critical value. The critical Reynolds number is related to a bifurcation that changes the stability of the

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Fig. 4.9 Eigenspectrum and unstable eigenmodes from the linear stability analysis of the symmetric steady solutions at (a) .Re = 20 and (b) .Re = 81. Only the real part of the complex eigenmodes is shown with the vorticity field. The displayed interior box is the evaluation domain. Negative and positive values of vorticity are colored from blue to red

steady solution. The imaginary part .ω determines the frequency of the instability at the beginning, which is zero for a pitchfork bifurcation and non-zero for a Hopf bifurcation [47]. In this work, we consider an order-nine Krylov subspace and apply the subspace iteration method [50] to obtain the leading eigenvalues of .Lus . The iterations will stop when the residual of the eigenvalue problem is less than .10−5 or the maximum number of iterations (100 by default) is reached [36]. Instabilities and Bifurcations We show the instabilities detected from the linear stability analysis of the steady solutions at Reynolds numbers slightly above the bifurcation threshold: at .Re = 20 for the primary supercritical Hopf bifurcation and .Re = 81 for the secondary supercritical pitchfork bifurcation in Fig. 4.9. A pair of complex-conjugated eigenvalues can be found with positive real parts at .Re = 20 in Fig. 4.9a. The pair of eigenvalues crosses the vertical axis of .σ = 0 at the Reynolds number changing from 18 to 19, indicating a supercritical Hopf bifurcation at the critical Reynolds number .Re1 = 18, associated with the Bénard-von Kármán instability of the vortex shedding. In this case, the three cylinders are taken as a single obstacle, forming the classic von Kármán vortex street in the wake. The corresponding eigenmodes are visualized with the real part of the vorticity field, presenting as a row of regularly

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Re1

10 18

Re2

60

Re3

120

90 80

Re4

117

132

160 Re

Fig. 4.10 Route to chaos of the fluidic pinball. Each flow regime is shown with the post-transient vorticity field at the Reynolds numbers marked in red. The critical values of the supercritical Hopf bifurcation .Re1 = 18, the supercritical pitchfork bifurcation .Re2 = 80, and the Neimark-Säcker bifurcation .Re3 = 117 before the system enters into chaos at .Re4 = 132 are marked in black on the Re-axis

arranged vortices with alternating positive and negative values. The conjugated modes are antisymmetric and have a phase shift associated with the Bénard-von Kármán instability of the vortex shedding. A real eigenvalue can be found with a positive real part at the Reynolds number changing from 80 to 81, corresponding to a supercritical pitchfork bifurcation at the critical Reynolds number .Re2 = 80. Three unstable eigenmodes can be found at .Re = 81 in Fig. 4.9b. Mode A and B are associated with the primary Hopf bifurcation but with a larger growth rate, indicating that the vortex shedding modes are more unstable than the case at a lower Reynolds number. It can be seen that the spatial distribution of vorticity is away from the cylinder. The width of the vortex street decreases along the flow direction, and the distance between the vortex centers of the upper/lower parts and the center part becomes larger. Mode C explains the changes to the unstable vortex shedding modes. The spatial mode is also reflectionantisymmetric but concentrates in the near wake behind the rear two cylinders, which is associated with the symmetry-breaking instability of the base-bleeding jet. The resulting symmetry breaking does not separate the bluff body wake but changes the wake direction downstream. Therefore, the appearance of Mode C changes the spatial distribution of vortex shedding modes. And the coupling relations and the interaction between these two instabilities lead to variant wake dynamics. See [13, 14] for more details on the stability analysis results, e.g., the Floquet stability analysis of periodic solutions. Route to Chaos According to the detected bifurcations and stable unsteady dynamics in the asymptotic regime at different Reynolds numbers, we can summarize the overall route to chaos of the fluidic pinball in Fig. 4.10. At a very low Reynolds number, the flow is stable and satisfies the reflectional symmetry. All the perturbations will vanish and converge to the corresponding steady solution. A primary supercritical Hopf bifurcation occurs at .Re1 ≈ 18. The steady solution becomes unstable beyond .Re1 , and a periodic vortex shedding street appears in the wake. The unsteady flow satisfies the spatio-temporal symmetry. A secondary supercritical pitchfork

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bifurcation occurs at .Re2 ≈ 80. Three steady solutions exist by solving the steady Navier–Stokes equations, as well as three periodic vortex shedding with different jet states. The flow undergoes a Neimark-Säcker bifurcation at .Re3 ≈ 117, as the asymptotic regime shows quasi-periodic dynamics. The jet will oscillate at the deflected place with a newly emerged low frequency, which modulates the wake downstream. The flow enters the chaotic regime beyond .Re4 ≈ 132. The jet starts to switch randomly between two deflected positions, leading to random vortex pairing in the wake. V. Parametric Study This section highlights the richness of the fluidic pinball dynamics by exploring the actuation space. Steady control laws are sampled in the actuation space described by a new set of parameters (.p1 , .p2 , .p3 ) introduced by Qixin “Kiki” Lin in her master thesis [34] and referred in the following as the Kiki parameters: p1 = b1 —Stagnation point parameter, b2 − b3 —Boat-tailing parameter, p = . 2 2 b1 + b2 + b3 —Magnus parameter. p3 = 3

(4.12)

The Kiki parameters are related to wake actuation mechanisms and allow a direct interpretation of the results. Thus, .p1 directly controls the stagnation point, .p2 measures the difference in rotation speed between the top and bottom cylinder. The fluidic pinball reproduces a boat-tailing control when .p2 is positive and a basebleeding control when .p2 is negative. The flow deflection of the flow is measured by the Magnus parameter .p3 , and the wake is oriented upwards if .p3 is positive and downwards if .p3 is negative. For this parametric study, steady control laws are sampled in the Kiki parameter space, .p1 from 0 to 2 in steps of .0.5, .p2 from .−2 to 4 in steps of .0.5 and .p3 from .−2 to 2 in steps of .0.5. To avoid numerical instabilities, the control is limited to .b1 ∈ [0; 2], .b2 ∈ [−4; 4] and .b3 ∈ [−4; 4]. All in all, 384 steady control laws are simulated. The symmetry following the center line allows us to mirror our samples in .b1 ∈ [−2; 0] with the mapping .[b1 , b2 , b3 ] I→ [−b1 , −b3 , −b2 ]; note that in this case, the sign of the lift needs to be inverted. Each control law has been evaluated over 400 time units starting from the unsteady post-transient flow at .t = 1500. Lift and drag coefficients are computed over the last 100 time units. Figure 4.11 shows the distribution of control laws in the Kiki parameter space, visualized by their flow regime, lift and drag coefficients, respectively. The steady, periodic, and chaotic regimes are marked with black, blue, and red dots in Fig. 4.11a. According to the spatial distribution of three regimes, a boundary between steady and periodic dynamics and a boundary between periodic and chaotic dynamics are embedded in the Kiki parameter space. The frontier between steady and periodic regimes is rather well-defined, while the frontier between chaotic and periodic is not regular and is expected to be fractal. The control

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Fig. 4.11 Distribution of the fluidic pinball dynamics and aerodynamic forces in the Kiki parameter space (.p1 , .p2 , .p3 ) for the 384 control cases. (a) Steady, periodic, and chaotic flows are represented by black, blue, and red points, respectively. The blue and red surfaces are the frontier between steady-and-periodic and periodic-and-chaotic dynamics, respectively. (b) Isosurfaces of lift coefficient. (c) Isosurfaces of drag coefficient

laws leading to steady state distributes on the side of .p2 > 0. For .p2 < 0, the flow is mostly periodic but tends to be chaotic as .p2 decreases. When the boat-tailing parameter .p2 increases, a stronger stabilizing effect can be found, progressively changing the flow from chaotic to steady regimes. A larger Magnus parameter .p3 in absolute value can also stabilize the flow, while the stabilizing effect of the stagnation point parameter is not obvious. Expectedly, Fig. 4.11b shows the lift coefficient is strongly anti-correlated to the Magnus parameter .p3 . When .p3 > 0, the flow is pushed upwards, resulting in a negative lift, and when .p3 < 0, the flow is pushed downwards, resulting in a positive lift. The stagnation point parameter .p1 also contributes to the overall lift as it decreases when .p1 increases and increases when .p1 decreases. The drag

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isosurfaces in Fig. 4.11c show that the drag coefficient is strongly anti-correlated to the boat-tailing parameter .p2 . For .p ≈ −2, two pockets of high drag include both periodic and chaotic regimes. These results indicate that drag reduction is mainly achieved with boat-tailing, which is consistent with the results of literature, e.g., [9, 33].

4.3 Proximity Map In a proximity map, the similarity between individuals is determined by their pairwise distances. By multidimensional scaling (MDS), the individuals in a highdimensional space are mapped into a low-dimensional space while preserving the pairwise distances as much as possible. Hence, the MDS is a very useful nonlinear dimensionality reduction for data visualization. I. Multidimensional Scaling To measure the proximity of two state vectors .a i and .a j , .i, j = 1, . . . , M, a common choice is the Euclidean distance, which is defined as an .L2 norm by   di,j = a i − a j  ,

.

then a proximity matrix can be .D constructed as ⎡

d1,1 ⎢ .. ⎢ . ⎢ .D = ⎢ di,1 ⎢ ⎢ . ⎣ .. dM,1

. . . d1,j . .. . .. . . . di,j . .. . .. . . . dM,j

⎤ . . . d1,M . ⎥ .. . .. ⎥ ⎥ . . . di,M ⎥ ⎥. .. ⎥ .. . . ⎦ . . . dM,M

For the snapshots of the velocity field collected from an experimental test or a numerical simulation, the distance between snapshots .ui and .uj is defined as di,j := ||ui − uj ||Ω ,

.

where .||u||2Ω =

(4.13)

dx u(x) · u(x) is the inner product in the observation domain .Ω.

Ω

Classical MDS uses eigen decomposition of a matrix .B = − 12 CD(2) C, which 2 ] and the centering is constructed with the squared proximity matrix .D(2) = [di,j matrix .C = IM − (1/M)JM , where M is the total number of snapshots, .IM is an .M × M identity matrix, and .JM is an .M × M matrix of all ones. The N largest eigenvalues .λ1 ≥ λ2 , . . . , ≥ λN and the corresponding eigenvectors .E = [e 1 , e 2 , . . . , e N ] of .B are preserved. Then, the coordinates .𝚪 of M snapshots can be derived from .𝚪 = EΛ1/2 , where .Λ = diag{λ1 , λ2 , . . . , λN } is a diagonal

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matrix with the N eigenvalues. As a result, an N-dimensional space spanned by the coordinate basis of the orthogonal eigenvectors is obtained. The coordinates of data satisfy .𝚪 ⏉ 𝚪 = Λ, which indicates an optimal Euclidean approximation in N dimensions of the original proximity matrix. The eigenvalue is the sum of squares of the data component in the direction of the corresponding eigenvector. In other words, the eigenvalues and eigenvectors are ordered by their size of data variance. In the process of dimensionality reduction, MDS maximally preserves the Euclidean distance relationship between snapshots, which helps to reveal the potential data structure. The pairwise distances .D of M snapshots will be represented in an N-dimensional feature space .γ = [γ1 , γ2 , . . . , γN ], where .N 0 ∀ξ ∈ supp bi;𝚵 := [ξi , ξi+d+1 ),

(5.4)

= 0 otherwise.

Looking at this property from a different perspective, in every knot span .[ξi , ξi+1 ) d d at most .d + 1 degree d basis functions are non-zero, namely, .bi−d;𝚵 , .bi−d+1;𝚵 , .. . . , d .b . The set of basis functions moreover satisfies the partition of unity property i;𝚵 n  .

d bi;𝚵 (ξ ) ≡ 1,

∀ξ ∈ [ξd+1 , . . . ξn ) .

(5.5)

i=1

It is moreover easy to show that the r-th derivative of a degree d B-spline with respect to the independent variable .ξ is given by (cf. the proof of Lemma 3.20 in [60])  r d .D b i;𝚵 (ξ )

=d

d−1 D r−1 bi;𝚵 (ξ )

ξi+d − ξi

−

d−1 (ξ ) D r−1 bi+1;𝚵

ξi+d+1 − ξi+1

 ,

(5.6)

which can be worked out recursively to arrive at an explicit expression. However, computing derivatives in that way is tedious and not to be recommended. In the following section we will discuss a matrix representation of B-splines that simplifies the evaluation of function values and derivatives significantly. Last but not least, B-spline basis functions have maximal continuity .C d−1 provided that no two knots in the knot vector are the same. If the i-th knot .ξi is

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Fig. 5.4 Quadratic B-spline basis functions (.n = 7, .d = 2) and their first derivatives defined over the open knot vector .𝚵 = {0, 0, 0, 1, 2, 3, 3, 4, 4, 4}; the fifth basis function .b52 exhibits a local .C 0 continuity around the repeated knot .ξ6 = ξ7 = 3, which causes its derivative to jump at that point. (a) Univariate B-spline basis functions. (b) Derivatives of univariate B-spline basis functions

repeated .0 < mi ⩽ d + 1 times, then the continuity reduces to .C d−mi in the vicinity of that point. In practice, knot vectors are often chosen to be open, that is, the first and last knots are repeated .d + 1 times, which reduces the continuity locally to .C −1 and causes the respective B-spline basis function to attain the function value one and all other basis functions to vanish at the two end points. Figure 5.4 depicts the .n = 7 degree .d = 2 B-spline basis functions (top) and their first derivatives (bottom) generated from the open knot vector .𝚵 = {0, 0, 0, 1, 2, 3, 3, 4, 4, 4} with a twice repeated knot .ξ6 = ξ7 = 3, which leads to a local .C 0 continuity of the fifth basis function .b52 around that point while elsewhere 2 2 1 ∗ .b and all other basis functions are .C . Consequently, .Db exhibits a jump at .ξ = 3 5 5 0 and is .C continuous elsewhere as are all other basis functions. Let us define the spline space .Sd𝚵 as the space of all linear combinations of the B-spline basis functions as defined in (5.2)–(5.3), i.e.,  d d Sd𝚵 = span b1;𝚵 , . . . , bn;𝚵 .

(5.7)

.

 =

n 

d ci bi;𝚵

: ci ∈ R, for 1 ⩽ i ⩽ n .

(5.8)

i=1

 d (ξ ) of this space is termed a spline function or just An element .f (ξ ) = ni=1 ci bi;𝚵 n a spline and the set .(ci )i=1 denotes the B-spline coefficients of f .

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To improve readability, we drop subscript .𝚵 whenever there is no ambiguity about the knot vector used. Moreover, we introduce the vectors

bd = b1d b2d · · · bnd

.

and



c = c1 c2 · · · cn

(5.9)

so that the sum in (5.8) can be written as the dot product .bd · c. Last but not least, let us define the selection operator “.[·; ·]” that selects a sub-vector, e.g., bd[j −d;j ] · c[j −d;j ] =

j 

.

ci bid .

(5.10)

i=j −d

Among all possible points where a spline function .f (ξ ) can be evaluated, the Greville abscissae play a special role. They are defined as the knot average [15] ξ¯i =

.

ξi+1 + · · · + ξi+d . d

(5.11)

In general, .ξ¯i lies near the parameter value which corresponds to a maximum of the B-spline basis function .bid [66]. In combination with the B-spline coefficients ¯i , ci ) form the so-called control polygon of the spline function f .c, the pairs .(ξ (cf. Theorem 2.8 in [60]). This is illustrated in Fig. 5.5 for the same

knot vector as before and the vector of B-spline coefficients .c = 0 2 1 1 3 1 2 . The repetition of knots causes the respective basis functions to attain the value one at the associated point and all other basis functions to vanish causing the spline function to become interpolatory, i.e., .f (ξ ∗ = 3) = c5 = 3 for the example above.

Fig. 5.5 Quadratic spline function and its control polygon constructed from .𝚵 {0, 0, 0, 1, 2, 3, 3, 4, 4, 4} with B-spline coefficients .c = [0 2 1 1 3 1 2]

=

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A Matrix Representation of B-Splines Before we proceed to multi-variate B-splines, we would like to review the construction procedure (5.2)–(5.3) and discuss an alternative representation of B-splines that is hardly discussed in the literature but has several advantages from both the theoretical and the implementation point of view. The following description is based on chapters 2 and 3 from the lecture notes by Lyche and Mørken [60]. We encourage the reader to study these excellent notes for a detailed derivation of the matrix representation and for an in-depth discussion of splines and B-splines in general. Exploiting the fact that at most .d + 1 degree d B-spline basis functions are nonzero per knot span, an element .f (ξ ) of .Sd can be written equivalently as f (ξ ) = bd[j −d;j ] (ξ ) · c[j −d;j ] ,

.

∀ξ ∈ [ξj , ξj +1 ).

(5.12)

In Theorem 2.14 of [60], Lyche and Mørken provide an alternative representation of B-spline basis functions in terms of a chain of matrix products, namely bd[j −d;j ] (ξ ) = Rd1 (ξ )Rd2 (ξ ) · · · Rdd (ξ ),

(5.13)

.

where for each positive integer .k ⩽ d the .k × (k + 1)-dimensional B-spline matrix Rdk is given by

.

⎡

ξj +1 −ξ ξ −ξj +1−k 0 ⎢ ξj +1 −ξj +1−k ξj +1 −ξj +1−k ⎢ ξ −ξj +2−k ξj +2 −ξ 0 ⎢ ξj +2 −ξj +2−k ξj +2 −ξj +2−k ⎢

Rdk (ξ ) = ⎢ ⎢ ⎢ ⎣

.

.. .

.. .

..

0

0

···

.

⎤

···

0

···

0

..

.. .

.

ξ −ξj ξj +k −ξ ξj +k −ξj ξj +k −ξj

⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(5.14)

From the plentiful examples given in chapter 2 of the lecture notes [60], we would like to replicate Examples 2.12 and 2.13 to illustrate the above construction procedure. Let us consider linear B-splines (.d = 1) and let .ξ ∈ [ξj , ξj +1 ). It follows from the local support property that the only two basis functions that are non-zero in that knot span are .bj1−1 and .bj1 . Their restriction to the interval can be given as .

   ξ +1 −ξ bj1−1 bj1 = ξjj+1 −ξj

ξ −ξj ξj +1 −ξj

 .

(5.15)

Likewise, for quadratic B-splines (.d = 2) the row vector of B-splines that are nonzero in .[ξj , ξj +1 ) can be written as the product of two matrices, namely .

   ξ +1 −ξ bj2−2 bj2−1 bj2 = ξjj+1 −ξj

ξ −ξj ξj +1 −ξj



·.

(5.16)

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⎡

ξj +1 −ξ ξ −ξj −1 0 ξj +1 −ξj −1 ξj +1 −ξj −1 ⎣ ξj +2 −ξ ξ −ξj 0 ξj +2 −ξj ξj +2 −ξj

⎤ ⎦.

(5.17)

The matrix form moreover leads to “explicit” expressions for computing (higher order) derivatives of B-splines. Let us reconsider the general case (5.13). The vector of all r-th derivatives of degree d B-spline basis functions that are non-zero in .[ξj , ξj +1 ) can be computed from (see Theorem 3.15 in [60]) D r bd[j −d;j ] (ξ ) =

.

d! Rd (ξ ) · · · Rdd−r (ξ )DRdd−r+1 · · · DRdd , (d − r)! 1

(5.18)

where the (first ordinary) derivative of the k-th B-spline matrix is given by ⎡

−1 1 0 ξ −ξ ξ −ξ ⎢ j +1 j +1−k j +1 −1j +1−k 1 ⎢ 0 ξj +2 −ξj +2−k ξj +2 −ξj +2−k ⎢

DRdk = ⎢ ⎢ ⎢ ⎣

.

.. .

.. .

..

0

0

···

.

···

0

···

0

..

.. .

.

−1 1 ξj +k −ξj ξj +k −ξj

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(5.19)

As remarked by Lyche and Mørken in [60], the derivative operator D can be applied to any of the d matrices .Rdk as long as we differentiate r of them. From a computational point of view, it makes sense to choose the trailing r matrices as this reduces the number of arithmetic operations to be performed. Recall that the k-th B-spline matrix has dimension .k × (k + 1) and features 2k nonzero entries. Each entry of .Rdk involves two subtractions and one division, whereas this reduces to one subtraction and one division for .DRdk . Accumulated over all d matrices, this yields an overall count of arithmetic operations for assembling the non-zero matrix entries of .3d 2 − 2dr + 3d + r 2 − r when the derivatives are applied to the trailing r matrices and .3d 2 + 3d − r 2 − r when the leading r matrices are differentiated. A similar cost analysis can be performed for the order in which the chain of matrix–matrix multiplications is evaluated. Assuming that the matrices are stored in dense format, the product between an .n × o and an .o × m matrix gives rise to .nm(2o − 1) arithmetic operations (o multiplications and .o − 1 additions per entry of the resulting .n × m matrix). A quick calculation shows that for .d ⩾ 2 a left-to-right evaluation requires .(4d 3 + 9d 2 − d − 12)/6 arithmetic operations, whereas a right-to-left evaluation requires .(4d 4 + d 3 − 4d 2 − 1)/6 operations. We therefore recommend to evaluate the matrix chains in (5.13) and (5.18) from left to right starting at .Rd1 and applying the derivatives (if any) to the trailing matrices. Putting it all together, a handy approach to evaluate an element f of the spline space .Sd (or its r-th derivative .D r f ) in a given point .ξ ∗ is as follows: Find the knot span .[ξj , ξj +1 ) such that .ξj ⩽ ξ ∗ < ξj +1 and compute

5 Design Through Analysis

f (ξ ∗ ) =

.

313

bd[j −d;j ] (ξ ∗ ) · c[j −d;j ]

or.

D r f (ξ ∗ ) = D r bd[j −d;j ] (ξ ∗ ) · c[j −d;j ]

(5.20) (5.21)

using the matrix representations (5.13) and (5.18), respectively.

Efficient Evaluation of B-Splines For the reader who is interested in implementing the above in their own code, we would like to remark that (5.20) and (5.21) is particularly suited for programming languages like Python or Matlab and linear algebra libraries that support tensor operations. In that case, .b’s and .c’s can be generalized to tensors, whereby the third dimension represents the different .ξ ∗ values and respective sub-vectors of coefficients. Then the spline or its derivative can be evaluated in all given points simultaneously by simple tensor contraction along the first two dimensions. Instead of assembling the matrices (5.17) and (5.19) (or their generalization to tensors) and performing the multiplications explicitly, which might be time consuming and require a lot of computer memory especially for many evaluation points at a time, we present a modified version of Algorithm 2.22 from chapter 2 of [60] in Algorithm 2. The main difference is the automated handling of repeated knots (i.e., the smart circumvention of the “.0/0 := 0” check) and the uniform treatment of functions and their derivatives, respectively. Algorithm 2 B-spline evaluation Require: Find positive integer j ⩽ n + d + 1 such that ξ ∗ ∈ [ξj , ξj +1 ) 1: b = 1 2: for k = 1, . . . , d − r do 3: t1 = ξi−k+1 . . . ξi

4: t21 = ξi+1 . . . ξi+k − t1 5: mask = (t21 < tol) ⊳ < element-wise comparison ⊳ ÷ element-wise division 6: w = (ξ ∗ − t1 − mask) ÷ (t21 − mask)

7: b = (1 − w) ⊙ b 0 + 0 w ⊙ b ⊳ ⊙ element-wise multiplication 8: end for 9: for k = d − r + 1, . . . , d do 10: t1 = ξi−k+1 . . . ξi

11: t21 = ξi+1 . . . ξi+k − t1 12: mask = (t21 < tol) 13: w = (1 − mask) ÷

(t 21 − mask) 14: b = −w ⊙ b 0 + 0 w ⊙ b 15: end for

After the execution of the algorithm, vector .b contains the values of the r-th derivatives of the .d + 1 degree d B-splines that are non-zero at the point .ξ ∗ . Like with (5.20) and (5.21), Algorithm 2 can be generalized to handle multiple points .ξ ∗

d=2

d=4

d=5

1e0 1e1 1e2 1e3 1 2.5ee44 5e4 1 2.5ee55 5e5 1e6

d=1

1e0 1e1 1e2 1e3 1e4 2.5e4 5e4 1 2.5ee55 5e5 1e6

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105 104 103 102 101

AMD EPYC 7402 (24 cores)

1e0 1e1 1e2 1e3 1 2.5ee44 5e4 1 2.5ee55 5e5 1e6

1e0 1e1 1e2 1e3 1 2.5ee44 5e4 1e5 2.5e5 5e5 1e6

100 10−1

1e0 1e1 1e2 1e3 1 2.5ee44 5e4 1 2.5ee55 5e5 1e6

Wallclock time in ns/entry

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Fujitsu A64FX (48 cores)

Tesla V100S PCIe 32GB

d=2

d=4

d=5

1e0 1e1 1e2 1e3 1 2.5ee44 5e4 1 2.5ee55 5e5 1e6

d=1

1e0 1e1 1e2 1e3 1 2.5ee44 5e4 1 2.5ee55 5e5 1e6

106

d=3

105 104 103 102 101

AMD EPYC 7402 (24 cores)

1e0 1e1 1e2 1e3 1 2.5ee44 5e4 1 2.5ee55 5e5 1e6

1e0 1e1 1e2 1e3 1 2.5ee44 5e4 1e5 2.5e5 5e5 1e6

100 10−1

1e0 1e1 1e2 1e3 1 2.5ee44 5e4 1 2.5ee55 5e5 1e6

Wallclock time in ns/entry

(a)

Fujitsu A64FX (48 cores)

Tesla V100S PCIe 32GB

(b)

Fig. 5.6 Computational performance of vectorized B-spline evaluation as given in Algorithm 2 implemented in LibTorch 2.0 and executed on different CPU and GPU architectures: (a) univariate B-splines and (b) bi-variate B-splines

simultaneously by extending all vector quantities to matrices and adjusting lines 7 and 14 to not only append/prepend a scalar “0” to the vector but an entire column of zeros to the matrix. We have implemented this approach in an in-house research code based on the C.++ API of LibTorch 2.0 and measured its computational performance for different processors (CPUs) and graphics cards (GPUs) for problem sizes between a single and a million evaluation points. A summary of the results is presented in Fig. 5.6 for B-splines of degree 1 to 5 both for the univariate case (described here) and for the case of bi-variate functions (to be addressed below). By exploiting vectorization and parallelization, the compute time for Bspline evaluation can be reduced by multiple orders of magnitude, especially when using GPUs. It is easy to see that all vectors in lines 3–6 and 10–14 of Algorithm 2 are of length k and that the .b’s in lines 7 and 14 are after the appending/prepending with zeros of length .k +1. Let us neglect this detail and estimate the cost of the loop body by 10k arithmetic operations (counting the element-wise comparison as arithmetic operation). Then the overall cost of Algorithm 2 is .5(d 2 + d) operations which is significantly cheaper than assembling the matrices .Rdk and performing the matrix– matrix multiplications in (5.13) and (5.18), respectively. Application of the vector of evaluated B-spline basis functions and their derivatives to the coefficient vector .c in (5.20) and (5.21), respectively, gives rise to the costs of a standard inner product, namely, d multiplications and .d − 1 additions. Even though one might be tempted to evaluate (5.20) and (5.21) from right to left by

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applying the trailing .Rdd matrix to the coefficient vector to obtain a smaller column vector and repeat the process until .Rd1 is reached, we advise against this approach, since in many applications the B-spline basis functions can be pre-evaluated, e.g., in the Greville abscissae (5.11), and reused throughout the simulation, whereas the control points may change, e.g., in an analysis and optimization process.

Knot Insertion The reader who is familiar with the finite element method might wonder if it is possible to “refine” the B-spline function space defined in (5.8). As .Sd𝚵 is solely determined by the degree d and the knot vector .𝚵 = (ξi )n+d+1 , any “finer” knot i=1 d (with .m > n) will generate a richer function space as .S . vector .Π = (πj )m+d+1 Π j =1 Let us consider the special case that .𝚵 ⊆ Π , that is, all knots from the original sequence are contained in the “refined” one. Moreover, let both knot vectors have common knots at the two ends, and let no knot occur with multiplicity higher than d .d +1 in which case we call the knot vectors .d +1-regular. Then a spline f in .S with 𝚵 B-spline coefficients .c = (ci )ni=1 can be represented exactly (!) in .SdΠ by calculating the B-Spline coefficients .d = (dj )m j =1 relative to the knot vector .Π by the second Oslo algorithm (see Algorithm 4.11 from chapter 4 in [60]): Algorithm 3 Knot insertion (Oslo algorithm 2; cf. [60]) 1: for j = 1, . . . , m do 2: Find positive integer i such that ξi ⩽ πj < ξi+1 3: Compute the B-spline coefficient bj as  bj =

ci

if d = 0,

Rd1 (πj +1 ) · · · Rdd (πj +d ) · c[i−d;i]

if d > 0.

4: end for

The chain of matrix–matrix products can again be computed efficiently by resorting to Algorithm 2. This approach of refining the spline space .Sd𝚵 through augmenting its underlying knot vector .𝚵 is termed knot insertion and common practice. In particular the fact that spline functions f in .Sd𝚵 have an equivalent and easy-to-compute representation in .SdΠ (with .𝚵 ⊆ Π ) makes it an amenable tool for design-through-analysis workflows as it allows to first generate a spline space for representing the geometry model with sufficient level of detail and then refine this space for the analysis while preserving the capability to represent the original geometry model exactly within the refined spline space. Next to the Oslo algorithm, there exist alternative approaches like blossoming to determine the B-spline coefficients after the insertion of knots into .𝚵 as discussed for instance in Section 4.4 of [60]. However, blossoming is mathematically more

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advanced and requires a different implementation, whereas Algorithm 3 reuses the concept of B-spline matrices (5.17). It thereby suggests itself for programming languages like Python and Matlab and linear algebra libraries that support tensor operations since, like before, all .b coefficients can be computed simultaneously.

Multi-variate B-Splines A canonical approach to generalize B-splines into multiple parametric dimensions   is by taking the tensor product of univariate ones. Let .𝚵 = 𝚵 , . . . , 𝚵 1 p denote  .d = d1 , . . . , dp the individual degrees for the set of p independent knot vectors,   each direction, .n = n1 , . . . , np the number of univariate B-spline basis functions per direction, and .p ∈ N⩾1 the number of parametric dimensions. The degree .d tensor-product B-spline basis functions are then defined as d Bi;𝚵 (ξ ) =

p 

.

bidkk;𝚵k (ξk ),

(5.22)

k=1

  where  .i = i1 , .. . , ip is a multi-index from the admissible range .1 ⩽ i ⩽ n and .ξ = ξ1 , . . . , ξp denotes the multi-component independent variable. In a practical implementation, the multi-index .i is typically “flattened” to a global index i that varies between 1 and .n = n1 n2 · · · np by imposing an ordering on the different dimensions (e.g., “smaller dimensions run faster”) and computing i := (i3 − 1)n1 n2 + (i2 − 1)n1 + i1 .

(5.23)

.

Figure 5.7 illustrates the degree .2×3 B-splines basis functions defined over the knot vectors .𝚵1 = {0, 0, 0, 1, 2, 3, 3, 4, 4, 4} and .𝚵2 = {0, 0, 0, 0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4} in .ξ1 - and .ξ2 -direction, respectively. The interpolation property carries over to those basis functions for which both univariate components evaluate to one. With definition (5.22) at hand, the corresponding spline space .Sd𝚵 is defined analogously to its univariate counterparts (5.8) as follows:  d d Sd𝚵 = span B1;𝚵 , . . . , Bn;𝚵 .

(5.24)

.

 =

n 

d ci Bi;𝚵

: ci ∈ R, for 1 ⩽ i ⩽ n .

(5.25)

i=1

We will drop subscript .𝚵 whenever there is no ambiguity about the knot vector. For the efficient evaluation of multi-variate B-spline basis functions (or their derivatives), Algorithm 2 can be applied to each univariate component independently from which the final function value can be computed as follows:

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Fig. 5.7 Tensor-product degree .2 × 3 B-spline basis functions defined over the two knot vectors = {0, 0, 0, 1, 2, 3, 3, 4, 4, 4} and .𝚵2 = {0, 0, 0, 0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4} in .ξ1 - and .ξ2 direction, respectively

.𝚵1

  1 2 f (ξ ∗ ) = bd1,[j (ξ ∗ ) ⊗ bd2,[j (ξ ∗ ) · c[j−d;j] . 1 −d1 ;j1 ] 1 2 −d2 ;j2 ] 2

.

(5.26)

Here, .[j − d; j] stands for the multi-dimensional generalization of the selection operator “.[·; ·],” that is, the operator that extracts a sub-matrix from the matrix of B-spline coefficients in the bi-variate case and a sub-tensor in the tri-variate case. In a practical implementation, the values of .c[j−d;j] are typically not stored at contiguous memory positions unless the selection operator makes a deep copy. Often, the data is contiguous in the leading dimension, say, .ξ1 with offsets of .n1 along the second direction, offsets of .n1 n2 along the third direction, etc. We therefore suggest to use Algorithm 993 [20] for the efficient computation of matrix– matrix products with matrices composed of Kronecker products. Let us drop suband superscripts for the moment. Then the tri-variate counterpart of (5.26) reads .

(b1 ⊗ b2 ⊗ b3 ) · c = (I ⊗ I ⊗ b3 ) · (I ⊗ b2 ⊗ I) · (b1 ⊗ I ⊗ I) · c.

(5.27)

This expression can be evaluated in three steps over the number of univariate directions as given in Algorithm 4. The reshape operation in line 2 assumes that matrices are stored in column-major format, that is, after the transposition in line 3 .c is a matrix with .nd rows. After the successful execution of the algorithm, .c contains the scalar value of f or its derivative in the point .ξ ∗ .

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Algorithm 4 Efficient evaluation of tensor-product B-splines (Algorithm 993 [20])

Require: Univariate B-spline basis functions b1 · · · bp evaluated in ξ ∗ and selection of B-spline coefficients c both restricted to [j − d; j] 1: for k = 1, . . . , p do 2: c = reshape(c, [·], nk ) 3: c = bk · c⏉ 4: end for

The arithmetic costs of this algorithm for the bi-variate case are .(2n1 + 1)n2 − 1 arithmetic operations and .((2n1 + 1)n2 ) + 1)n3 − 1 for the tri-variate case, respectively, assuming that reshaping and transposition are not performed explicitly but via clever memory accessing. As before, Algorithm 4 can be easily extended to a tensorized version that can evaluate f in multiple points simultaneously.

Geometry Modeling with B-Splines In much the same way as we defined the spline function spaces (5.8) and (5.25), one can define higher dimensional spaces, i.e., d,s .S 𝚵

 n 

=

d ci Bi;𝚵

: ci ∈ R , for 1 ⩽ i ⩽ n , s

(5.28)

i=1

with the main difference being that the B-spline coefficients are vector-valued data. Then, an element .f ∈ Sd,s 𝚵 realizes a mapping from the parameter space

.

ˆp = Ω

p 

ξk,dk +1 , ξk,nk ⊂ Rp ,

(5.29)

k=1

a p-dimensional hypercube, to the s-dimensional geometric or physical space .Ωs ⊂ ˆ 1 → Ωs , .1 ⩽ s ⩽ 3, defines a spline curve, .f : Ω ˆ2 → Rs . Canonically, .f : Ω s 3 3 ˆ Ω , .2 ⩽ s ⩽ 3, a spline surface, and .f : Ω → Ω a spline volume. Higher order extensions for, e.g., space-time formulations or, in general, the parametric representation of s-dimensional data are also straightforward. In what follows, we will drop the superscripts p and s if the dimensions are clear from the context.

5.2.2 Truncated Hierarchical B-Splines Since B-spline bases are typically constructed as a tensor product between the bases in each direction, tensor-product refinement of one element implies the refinement of multiple elements in one direction, see Fig. 5.9a. This implies a

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quasi-local approach for adaptive refinement, which improves the speed of multiscale simulation problems only partially. In order to provide local refinement of the spline basis, several spline basis constructions have been proposed in the literature such as hierarchical B-splines (HB-splines) [25], truncated hierarchical B-splines (THB-splines) [29, 30], T-splines [72] and their variations [49, 51, 50, 43, 52, 87], polynomial splines over hierarchical T-meshes (PHT-splines) [16], locally refined (LR) splines [17], and splines over unstructured meshes (U-splines) [78], the latter not to be confused with the unstructured splines to be presented in Sect. 5.2.4. Here, we review the (T)HB splines, and we refer the reader to the reference works for the other spline constructions. The construction of the truncated (hierarchical) B-spline basis (.H) .T is defined recursively as given in [29]: 1. Initialize .T 0 = H0 = {ϕ ∈ B 0 : supp ϕ = / ∅}, with the superscript denoting level 0, .B 0 a tensor B-spline basis on level 0, and .ϕ a basis function with non-empty support. 𝓁+1 𝓁+1 2. Recursively define .T 𝓁+1 = TA𝓁+1 ∪ TB𝓁+1 or .H𝓁+1 = HA ∪ HB for .𝓁 = 𝓁+1 0, ..., N − 2 with N the maximum level. The truncated basis .TA is defined as TA𝓁+1 = {trunc𝓁+1 τ : τ ∈ T 𝓁 ∧ supp τ /⊆ Ω𝓁+1 }

.

𝓁+1 and the hierarchical basis .HA 𝓁+1 HA = {ϕ ∈ H𝓁 : supp ϕ /⊆ Ω𝓁+1 }.

.

𝓁+1 Furthermore, the basis .TB𝓁+1 = HB is given by 𝓁+1 HB = {ϕ ∈ B 𝓁+1 : supp ϕ ⊆ Ω𝓁+1 },

.

with .Ω𝓁+1 ⊆ Ω𝓁 nested domains, .B 𝓁 the B-spline basis on level .𝓁, and .trunc𝓁 τ the truncation of .τ with respect to .B 𝓁+1 and .Ω𝓁+1 . 3. Then the final THB-spline basis is defined as .T = T N −1 and the final HB-spline basis is defined as .H = HN −1 . Figure 5.8 illustrates the principle of local refinement with B-splines using (truncated) hierarchical B-splines (HB- and THB-splines, respectively). In the top row of this figure, an initial uniform degree 2 B-spline basis with uniform knot vector .𝚵 = {0, 1/8, 2/8, ..., 7/8, 1} is presented. In the bottom row, a uniform refinement and (T)HB refinements of the indicated functions are presented. The potential of refinement splines compared to knot insertions for local is illustrated in Fig. 5.9. When a knot insertion is performed in a tensor B-spline basis to refine a marked element, the refinement automatically introduces refinement of other elements in the knot line (see Fig. 5.9a). For hierarchical splines, the basis functions are inserted only locally, resulting in the addition of degrees of freedom only in the marked element, see Fig. 5.9b.

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B-spline basis

HB-spline basis

THB-spline basis

βi

1

Q

0

βi

1

Q

0 0

0.5 ξ

1

0

0.5 ξ

1

0

0.5 ξ

1

Fig. 5.8 Principles of refinement for different spline bases. The top plots represent the basis on level 0, optionally with refined basis functions given in blue color. The bottom plots illustrate the refined bases: uniform refinement (left), hence level 1; HB-refinement (middle); THB-refinement (right) with truncated basis functions in red color. The line .Q represents the elements of the basis. The unrefined unique knot vector in all cases is .𝚵 = {0, 1/8, 2/8, . . . , 7/8, 1} and the degree of the basis is 2. All bases are generated with the open-source IGA library G+Smo [42]

(a) Refinement of a tensor-product B-spline basis. To refine the element, the knot 0.5625 is inserted in both knot vectors.

(b) Refinement of a (T)HB-spline basis. To refine the element, the basis functions with support on this element from the finer level (below) are inserted in the original basis (above) following the procedure from Fig. 5.8.

Fig. 5.9 Refinement of a two-dimensional tensor B-spline basis (a) and a (T)HB-spline basis (b) for a marked element with corners .(0.5, 0.5) and .(0.625, 0.625). The original B-spline basis has degree 2 and unique knot vector .𝚵 = {0, 1/8, 2/8, . . . , 7/8, 1} in both directions

5.2.3 Non-uniform Rational B-Splines While B-spline basis functions offer great flexibility to model geometric freeform shapes like curves, surfaces, and volumes, they fall short in accurately representing

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3

ω2

cωi

ω=1

ci

1

2

Origin 1

0 y

-1

-2

-2

-1

0

x

1

2

Fig. 5.10 Illustration of a univariate NURBS curve

conic curve surfaces such as ellipses and hyperbolas, which are widely encountered in industrial applications. To overcome this limitation, a natural progression is to generalize B-splines to Non-Uniform Rational B-Splines (NURBSs), which have become the industry standard in modern CAD systems. Essentially, NURBSs extend the concept of B-spline basis functions by incorporating rational functions, achieved through the introduction of weights .(ωi )ni=1 as an additional mechanism to modulate the shape of the geometric object. In essence, each control point .ci ∈ Rs is assigned a weight factor .ωi ∈ R+ with which we define the projected control point as follows:  cωi =

ωi ci



.

ωi

∈ Rs+1 .

A NURBS geometry in .Rs space is obtained by projecting a B-spline geometry in s+1 onto the hyperplane of .ω = 1 through a central projection transformation, as .R illustrated in Fig. 5.10. Notably, this transformation enables the accurate representation of conic curves using piecewise quadratic polynomial curves, making quadratic NURBS curves an appropriate choice for precise representation. In practice, NURBS curves (and other NURBS-based geometric objects) are not constructed via central projection transformation but by replacing the univariate Bspline basis functions (5.3) by their rational counterparts d Ni;𝚵,ω (ξ ) =

.

d (ξ ) ωi Ni;𝚵,ω n  d ωi Ni;𝚵,ω (ξ )

i=1

and defining the corresponding function space as follows:

(5.30)

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 d d Sd,s = span N , . . . , N 1;𝚵,ω n;𝚵,ω . 𝚵,ω

(5.31)

.

 =

n 

d ci Ni;𝚵,ω

: ci ∈ R , for 1 ⩽ i ⩽ n . s

(5.32)

i=1

The generalization to multi-variate NURBS spaces follows the same tensor-product construction principle as was adopted for B-splines with a single weight per basis function. Let us conclude this section by remarking that NURBSs, although being the predominant industry standard, are often not necessary in practical design-through-analysis applications and that other features like local refinability of (T)HB-splines outweigh the shortcoming of B-spline basis functions to not being able to represent conic curves and their higher dimensional extensions exactly. As before, we will drop sub- and superscripts whenever their information can be deduced from the context. Moreover, we will adopt .Ni as generic notation for B-spline and NURBS basis functions alike unless stated otherwise.

5.2.4 Multi-patch Splines Many geometries of practical interest like the one depicted in Fig. 5.11 cannot be ˆ → Ω unless represented by a single-patch B-spline or NURBS mapping .f : Ω trimming is used excessively. It is therefore common practice to combine multiple such mappings to a multi-patch spline, i.e., ˆ → Ω𝓁 , f𝓁 : Ω

.

Ω=



¯ 𝓁. Ω

(5.33)

𝓁

For the reader who is familiar with finite elements, each patch .Ω𝓁 can be considered as a kind of macro element with .f𝓁 being the push-forward operator from the ˆ into the physical space. Likewise, if the mapping .f𝓁 is bijective, reference element .Ω −1 ˆ exists and is termed the pull-back operator. its inverse .f𝓁 : Ω𝓁 → Ω While most conformal finite element formulations do not go beyond .C 0 continuity over element interfaces (i.e., continuity of the values of the basis functions but not of their derivatives), it would be beneficial to preserve at least part of the .C d−1 continuity of higher order B-splines when coupling multiple patches to a multi-patch object. To achieve this, so-called unstructured splines, i.e., splines with higher order smoothness over patch interfaces, can be constructed. In case of one-dimensional bases, the concept of patch smoothing is trivial, but illustrative for higher dimensions. The concept of interface smoothing is illustrated in Fig. 5.12 and can be interpreted as a construction where basis functions .ϕ ∈ d,1 Sd,1 𝚵 of a spline space .S𝚵 with interface smoothness 1 are represented by a linear

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Fig. 5.11 Illustration of a planar multi-patch geometry

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ψi

1

0 Aij ψj

1

0

ϕi

1

0 0

0.5

1 ξ

1.5

2

Fig. 5.12 The concept of interface smoothing between two bases of degree 2 with unique knot vector .𝚵 = {0, 1/8, 2/8, . . . , 7/8, 1} (left) and .𝚵 = {1, 9/8, 10/8, . . . , 15/8, 2} (right). The top figure provides the two bases with the dotted basis functions .ψi , the functions that have non-zero derivatives and values on the interface, and the dashed basis functions represent the functions that have zero values but non-zero derivatives on the boundary. The middle row presents scaled basis functions .Aij ψj . Here, all functions are scaled by a factor of 1, except for the dotted functions, which are scaled by a factor of .1/2. The bottom row presents the basis .ϕi = Aij ψj where the sum is evaluated over the repeated index j . The blue and red functions are constructed by taking the sum of the dashed and dotted functions in their support on the one side (resulting in the dash-dotted line) and taking the dotted line on the other side

combination of functions .ψ ∈ Sd,0 from a space .Sd,0 . For example, basis function .ϕi is represented by all basis functions .ψj weighted with coefficient .Aij : ϕi = Aij ψj ,

.

(5.34)

By applying this procedure to all basis functions .ϕi , we arrive at the relation ϕ = Aψ

.

(5.35)

with A denoting the transformation matrix. In higher dimensions, interface smoothing as illustrated in Fig. 5.12 can be performed to construct interface basis functions. However, the increased parametric dimension (see Fig. 5.13) introduces vertices where the smoothing of basis functions is non-trivial. Spline constructions that provide smoothing mappings like the matrix A are referred to as unstructured splines, providing bases with higher

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Fig. 5.13 Multi-patch decomposition of a simple domain into six patches. All the vertices on the boundary are regular since there are 1 or 2 patches joining. The vertices in the interior are irregular since their valence (i.e., the number of patches joining in the vertex) is not equal to 4. These vertices are also referred to as extraordinary vertices

analysis-suitable

IGA

parameterization

B-Rep

V-Rep

Field

structural design optimization

Fig. 5.14 Design-analysis-optimization pipeline

smoothness than .C 0 over patch interfaces and vertices. Examples of unstructured spline constructions include the D-Patch [70, 80], the Almost-.C 1 construction [77], the Approximate .C 1 basis [89, 90], the Analysis-Suitable .G1 construction [12, 22], polar spline constructions [79], and constructions based on subdivision surfaces [54, 5].

5.3 Creation of Analysis-Suitable Parameterizations In modern CAD systems, as shown in Fig. 5.14, Boundary Representations (B-Rep) are commonly used to represent CAD models. B-Rep describes the boundaries of an object through vertices, edges, and faces, organizing these elements using topological relationships. This representation provides high flexibility and efficiency in geometric modeling and design. However, before conducting simulation-based analysis, it is essential to construct a Volumetric Representation (V-Rep) for the interior of the CAD model. This process is typically known as domain parameterization or simply parameterization. Once an analysis-suitable parameterization is obtained,

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unknown inner control points ci

x(ξ) =

 i∈II



ciNi (ξ) + 

unknown



 j∈IB



cj Nj (ξ) . 



known

known boundary control points cj

Fig. 5.15 Illustration of the problem statement for the creation of analysis-suitable parameterizations

simulation-based analysis can be performed on the volumetric representation model. For visual reference, please see Fig. 5.14.

5.3.1 Problem Statement Unless noted otherwise, we will restrict ourselves to the single-patch case in what follows. For the sake of notation convenience, let .II and .IB be the index sets for the unknown inner control points and the known boundary control points, respectively. Then, the parameterization .x can be represented as follows: x(ξ ) =



.

i∈II



ci Ni (ξ ) + 

unknown





cj Nj (ξ ),

j ∈IB





(5.36)



known

where .ci , .i ∈ II are unknown inner control points, .cj , .j ∈ IB are the given boundary control points, .Ni (ξ ) and .Nj (ξ ) are the corresponding NURBS basis ˆ As depicted in Fig. 5.15, the known boundary control points functions, and .ξ ∈ Ω. .cj are represented by the blue points. Conversely, the unknown inner control points .ci , are indicated by the red points. It should be noted that the black lines that connect the control points indicate the so-called control net, whereas the actual parameterized domain .Ω is the gray quarter annulus underneath. In fact, the quality of the parameterization significantly impacts the accuracy and efficiency of subsequent analysis tasks [11, 93, 67]. First and foremost, a highquality analysis-suitable parameterization should be a bijection. Additionally, it should exhibit good orthogonality of “grid lines” and uniformity of “cell sizes”

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or, in other words, minimal distortion in terms of angles and area/volume. Strictly speaking, the technical terms “grid lines” and “cells” are anachronisms from classical grid-based methods and should be replaced by “parametric curves with all but one parameter kept fixed” and “images of the push-forward operator for ˆ respectively. At the same time, use of the tensor-product of knot spans in .Ω,” this terminology might help the reader familiar with grid-based methods like finite elements to interpret grids as the lowest order .C 0 parameterizations, where by virtue of the interpolation property the control net and the domain .Ω fall together. ˆ → Ω, the In light of the above requirements on the mapping .x : Ω parameterization problem can be formulated as follows: Given the set of boundary control points .cj , the objective is to construct the unknown inner control points .ci in such a way that the resulting parameterization .x guarantees bijectivity and minimizes angle, as well as area/volume distortion.

5.3.2 Classification of Parameterization Methods For an analysis-suitable parameterization, the bijective property plays a crucial role. In this section, we categorize the existing methods based on the approaches they employ to handle the bijectivity constraints. Algebraic Parameterization Methods These methods rely on algebraic principles and involve, if at all, the solution of a linear system of equations, which makes them computationally efficient. One notable example is the Discrete Coons method [24], which belongs to a special type of Transfinite Interpolation (TFI) methods. This explicit parameterization method does not require the solution of a linear system of equations and is generally considered highly efficient. Xu et al. extended this method to three-dimensional NURBS volumetric parameterization [93]. Several linear parameterization methods that solve a linear system, such as the spring model method and the mean value coordinates method, were discussed in [31]. Since these algebraic methods do not specifically consider the bijectivity constraint, they often yield self-intersecting parameterizations when dealing with complex domains. In these scenarios, algebraic methods are often used to compute an initial guess as a starting point for a more advanced parameterization approach. This two-step procedure, of course, requires that the downstream method is capable of turning a non-bijective parameterization into a bijective one.

Nonlinear Constrained Optimization Methods These methods inherently treat the bijection constraint as constraint terms and employ energy functions that characterize the orthogonality and uniformity of the

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parameterization as the objective function. An analysis-suitable parameterization is then generated by solving the constrained optimization problem. Xu et al. [91] utilized the observation that the Jacobian determinant of the parameterization can be expressed using higher order NURBS functions. They employed the nonlinear coefficients of the Jacobian determinant as inequality constraints to ensure bijectivity in the parameterization results. They subsequently extended this method to volumetric parameterizations [93]. Wang et al. [85] proposed an accelerated constrained optimization framework by utilizing constraint aggregation, divide and conquer, and hierarchical optimization strategies. Xu et al. [94] introduced a computational reuse method for computation domains with consistent topological structures. Ugalde et al. [1] presented a series of sufficient and necessary conditions for achieving injectivity in quadratic B-spline parameterizations. Despite the effective performance of these parameterization methods based on nonlinear constrained optimization in small-scale examples, the number of constraints significantly increases as the problem size grows, particularly in the context of volumetric parameterizations. To mitigate the computational burden, Pan et al. [65] introduced a constraint addition strategy that gradually incorporates collocation points in a coarse-to-fine resolution manner. Nevertheless, this method still incurs a substantial computational cost, necessitating the utilization of commercial optimization solvers.

Nonlinear Unconstrained Optimization Methods In the aforementioned parameterization methods based on nonlinear constrained optimization, the significant quantity of nonlinear constraints represents a challenging problem to solve. Consequently, in recent years, unconstrained optimizationbased parameterization methods have gained popularity. Xu et al. [92] introduced a parameterization method that involves minimizing the variational harmonic mapping. Nguyen and Jüttler [61] initially computed a series of harmonic mappings from the computational domain to the parameter domain, subsequently employing spline approximation for the inverse mapping. Falini et al. [21] extended this method to planar THB-spline parameterization by first calculating the harmonic mapping from the computational domain to the parameter domain through the boundary element method. They then utilized spline least squares fitting for the inverse mapping. Nian et al. [62] presented a planar parameterization method that relies on Teichmüller mapping. Pan et al. [64] introduced a low-rank parameterization method that utilizes quasi-conformal mapping and employs an alternating direction multiplier method to minimize the objective functional. They subsequently extended this method to include volumetric parameterization [63]. The Radó–Kneser–Choquet theorem guarantees the injectivity of the solution to the Winslow functional, which resulted in its widespread usage in traditional mesh generation fields, where it is known as the Most Isometric ParameterizationS (MIPS) energy in computer graphics [37]. In the field of computer graphics, a three-

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dimensional version of this energy is also recognized [26]. These energy functionals frequently necessitate an initial parameterization that possesses bijectivity. To tackle this issue, researchers have proposed various foldover elimination methods. Su et al. [76] projected the Jacobian matrix onto a space with bounded K-distortion, while Liu et al. [53] further enhanced this method by incorporating simultaneous optimization of boundary correspondence. Zheng et al. [96] recently employed this idea in THB-spline volumetric parameterization and introduced an efficient method for volumetric parameterization. Another approach involves the usage of penalty functions and Jacobian regularization techniques, which find their roots in the literature on grid distortion problems [27, 28]. Wang and Ma [84] implemented this idea in planar parameterization problems, successfully circumventing the need for extra foldover elimination steps.

Nonlinear Partial Differential Equation (PDE)-Based Methods These methods either approximate the stationary points of the Dirichlet energy while satisfying known boundary conditions or solve the corresponding Euler–Lagrange equations. Martin et al. [55] utilized discrete volumetric harmonic mappings to fit trivariate B-spline volumes. Shamanskiy et al. [75] developed analysis-oriented parameterizations through the solution of nonlinear elasticity equations using neoHookean hyperelastic material laws. Ali and Ma [2] utilized an isogeometric approach with equigeometric points to solve PDEs with boundary vector constraints in planar parameterization. Hinz et al. [35, 36, 34] proposed a series of parameterization construction methods based on nonlinear PDEs by discretizing the Laplace equation, drawing upon the principles of Elliptic Grid Generation (EGG). These elliptic parameterization methods based on EGG demonstrate favorable convergence properties and excel in slender domains with extreme aspect ratios.

5.3.3 Optimization-Based Parameterization Methods This section introduces optimization-based parameterization methods. As mentioned in Sect. 5.3.2, constrained optimization methods are computationally inefficient due to the presence of numerous nonlinear constraints. Accordingly, we will present two unconstrained optimization methods: the barrier function-based method and the penalty function-based method. These methods are derived from the authors’ two published papers: [40] and [41], respectively. When considering energy functions that characterize angle distortion and area/volume distortion, the Jacobian matrix .J plays a crucial role. The following fundamental quantities are frequently involved in this context.

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s

1 1

s

1

s

Fig. 5.16 The geometric interpretation of the singular values of the Jacobian matrix

Fig. 5.17 The geometric interpretation of the Jacobian determinant

• The Jacobian matrix of the parameterization .x:  For the 2D case:

.

For the 3D case:

J =

x1,ξ1 x1,ξ2

;. x2,ξ1 x2,ξ2 ⎤ ⎡ x1,ξ1 x1,ξ2 x1,ξ3 ⎥ ⎢ ⎥ J =⎢ ⎣x2,ξ1 x2,ξ2 x2,ξ3 ⎦ . x3,ξ1 x3,ξ2 x3,ξ3

(5.37a)

(5.37b)

First, it is crucial to consider the singular value .σi (.i = 1, 2, . . . , s, .1 ≤ s ≤ 3) of the Jacobian matrix .J . Figure 5.16 illustrates how these singular values reflect the variations in the lengths of the principal axes when locally mapping the unit sphere to an ellipsoid. Ideally, we aim for uniform singular values across the Jacobian matrix to ensure minimal angle distortion. Second, another significant quantity is the Jacobian determinant .|J | of the parameterization .x. The Jacobian determinant at a specific point provides the optimal linear approximation of the distorted parallelogram in the vicinity of that point. As depicted in Fig. 5.17, the Jacobian determinant represents the ratio

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between the area of the approximating parallelogram and that of the original square. • The metric tensor .G = J ⏉ J :  x ,ξ1 · x ,ξ1 x ,ξ1 · x ,ξ2 .For the 2D case: G= (5.38a) ;. x ,ξ2 · x ,ξ1 x ,ξ2 · x ,ξ2 ⎤ ⎡ x ,ξ1 · x ,ξ1 x ,ξ1 · x ,ξ2 x ,ξ1 · x ,ξ3 ⎥ ⎢ ⎥ For the 3D case: G=⎢ (5.38b) ⎣x ,ξ2 · x ,ξ1 x ,ξ2 · x ,ξ2 x ,ξ2 · x ,ξ3 ⎦ . x ,ξ3 · x ,ξ1 x ,ξ3 · x ,ξ2 x ,ξ3 · x ,ξ3 In particular, a parameterization .x is locally conformal at a point .ξ ∗ if and only if the metric tensor .G satisfies .G = n(ξ ∗ ) · Ip×p , where .Ip×p represents the identity matrix and .n(ξ ∗ ) > 0. In this case, the Jacobian matrix .J represents a combination of proportional scaling and rotation in each direction, leading to an orthogonal isoparametric structure for the parameterization .x. It is worth noting that under this condition, all singular values of the Jacobian matrix .J are equal. Based on the aforementioned fundamental quantities, the pointwise Most Isometric ParameterizationS (MIPS) energy [37, 26] can be employed to quantify angle distortion in the vicinity of a single point: angle .Ep

 σ1 σ2+ =

σ1 σ2

1 8

σ2 σ1 ,

+

σ2 σ1



σ2 σ3

+

σ3 σ2



σ1 σ3

+

σ3 σ1

2D case,



, 3D case,

(5.39)

where .σi represent the singular values of .J . The minimum value of .E angle occurs when .σ1 = σ2 = · · · = σp , ensuring minimal angle distortion. Furthermore, we utilize the following pointwise uniformity energy function unif. to assess the distortion in area/volume .Ep Epunif. =

.

vol(Ω) |J | + , vol(Ω) |J |

(5.40)

where .vol(Ω) denotes the area/volume of the computational domain .Ω. Recalling the angle distortion energy (5.39), in the 2D case, we have angle

Ep

.

=

σ 2 + σ22 σ1 trace(G) σ2 = 1 = + . |J | σ1 σ1 σ2 σ2

(5.41)

This energy is also known as Winslow’s functional, which may be more familiar to the mesh generation community. It is solely determined by the Jacobian determinant and the metric tensor .G of the parameterization .x, making it an intrinsic ˆ = [0, 1]2 is convex, the geometric quantity. Given that the parameter domain .Ω

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Radó–Kneser–Choquet theorem [69, 47] states that the unique minimum value of Winslow’s functional establishes a differential homeomorphism between the interior of the parameter domain and the interior of the computational domain. Moreover, as the Jacobian determinant approaches zero, which is in the denominator, the energy angle .Ep that measures angle distortion tends to infinity. This property effectively prevents self-intersections from occurring. Although there is no strict mathematical theory supporting the 3D case, a similar property holds, that is, angle Ep .

"! "! " σ2 σ1 σ1 σ2 σ3 σ3 + + + σ2 σ1 σ3 σ2 σ3 σ1      σ12 + σ22 + σ32 σ22 σ32 + σ12 σ32 + σ12 σ22 1 = −1 . 8 |J |2

1 = 8

!

(5.42)

It can be observed that the Jacobian determinant appears in the denominator, which has the capability to prevent self-intersections. Basically, this property appears to provide insight for constructing an analysissuitable parameterization by minimizing the following energy function: # E =

.

angle

ˆ Ω

λangle Ep

ˆ + λunif. Epunif. dΩ,

(5.43)

where .λangle and .λunif. are trade-off parameters used to balance the angle and area/volume distortion. However, the situation is far more intricate than it initially appears. The presence of the Jacobian determinant in the denominator creates a barrier that can potentially prevent self-intersections. Conversely, starting from an infeasible initial parameterization, which is frequently encountered in complex domain parameterization problems, it introduces an arbitrary level of complexity. In essence, a bijective parameterization must be established prior to the minimization of the energy function (5.43). In the subsequent two subsections, we will investigate approaches to tackle this challenge.

Barrier Function-Based Method In this section, we will introduce a three-step strategy known as the barrier function-based method, designed to generate high-quality parameterizations. We aim to overcome the challenges associated with numerous nonlinear constraints and achieve superior outcomes. The fundamental workflow of the barrier function-based method is illustrated in Fig. 5.18, providing a visual representation of the step-bystep process involved.

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Foldovers elimination

Quality improvement

Almost foldover-free

Initialization

Final result

Fig. 5.18 The workflow of the barrier function-based method

Initialization A common approach to solving nonlinear optimization problems is through iterative methods. Therefore, a reasonable initial guess is crucial to improve the convergence speed of subsequent solutions. As discussed in Sect. 5.3.2, algebraic parameterization methods such as the discrete Coons method [24], the spring model method [31], and the smoothness energy method [65] are commonly employed to generate an initial guess. In what follows, we adopt the smoothness energy method. Specifically, the unknown inner control points .ci , i ∈ II are obtained by solving the following quadratic programming problem: # ˆ ‖Δx‖2 d Ω. . arg min (5.44) ci , i∈II

ˆ Ω

It can be obtained by solving a sparse and symmetric linear system of equations. The preconditioned conjugate gradient method or the GMRES method with incomplete Cholesky decomposition is typically used for the solution. The initial parameterization constructed by this method is shown in the left part of Fig. 5.18. Note that this method does not guarantee a self-intersection-free parameterization, as it becomes evident from the presence of many self-intersections on the back of the duck. Foldover Elimination Generally, for complex computational domains, the initial parameterization constructed using algebraic parameterization methods does not guarantee bijectivity. Therefore, in our method, the second step is to eliminate foldovers. To ensure a bijective parameterization, it is necessary for the Jacobian determinant to be greater than zero throughout the entire computational domain. To this end, we solve the following unconstrained optimization problem: # fold ˆ . arg min E = max {0, δ − |J |} dΩ, (5.45) ci , i∈II

ˆ Ω

where .δ is a user-specified parameter value.

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The objective function in problem (5.45) clearly attains a minimum value of zero. During practical computations, the objective function is commonly evaluated using Gaussian numerical integration. However, in this scenario, a zero value of the objective function only indicates that the integrand evaluates to zero at the Gaussian integration points, which does not guarantee a bijective parameterization. Therefore, we refer to parameterizations that satisfy the condition of the Jacobian determinant being greater than zero at all Gaussian integration points as almost foldoverfree parameterizations. As depicted in the central portion of Fig. 5.18, solving the optimization problem (5.45) leads to a substantial reduction in self-intersections. Solving the problem (5.45) plays a crucial role in achieving success and improving the quality of the subsequent parameterization. However, it is important to note that addressing this problem alone is insufficient to completely eliminate selfintersections for the majority of complex computational domains. To this end, we further improve the parameterization quality in the next step. In problem (5.45), the choice of parameter .δ greatly influences the success of the problem-solving process. On the one hand, a larger value of .δ is desired as it leads to a higher-quality parameterization, which in turn enhances the convergence efficiency in improving the subsequent parameterization. On the other hand, setting .δ too large may result in the failure to solve the problem. To enhance the robustness of our method in this chapter, we employ an adaptive solving strategy that begins with a larger initial value of .δ and gradually decreases it. For a general parameterization .x, the determinant of the Jacobian at a specific parameter value represents the ratio of the area in the computational domain to that in the parameter domain near that point. Since we assume the parameter domain to be the unit square .[0, 1]2 , the Jacobian determinant should be equal to the area of the computational domain. To achieve this, we initially set .δ to 5% of the computational domain’s area. If the problem cannot be solved with this value of .δ, we progressively decrease it using a decay factor .decay_f actor. The specific steps for solving the problem are outlined in Algorithm 5.

Algorithm 5 Foldover elimination Require: x: Planar NURBS parameterization; Require: decay_f actor: Decay factor for parameter δ; Require: area(Ω): Area of the computational domain; Require: max_iter: Maximum number of iterations. Ensure: x: Parameterization after foldover elimination. 1: for k = 0, 1, . . . , max_iter do 2: Calculate δ = decay_f actor k ∗ 0.05 ∗ area(Ω); 3: Solve the unconstrained optimization problem arg min E fold ; ci , i∈II

Update the control points ci , i ∈ II ; if E fold < 100 ∗ MACHINE_PRECISION then return Parameterization x after foldover elimination; 6: end if 7: end for return “Maximum number of iterations reached!”;

4: 5:

⊳ (5.45)

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Parameterization Quality Improvement Considering that the Jacobian determinant is present in the energy function (5.43), it acts as a barrier to prevent self-intersections in the resulting parameterization. However, as the objective function exhibits discontinuous variation, the Jacobian determinant value can cross zero abruptly. In such cases, the objective function lacks a minimum value, and employing the original energy function (5.43) would lead to failure in solving. Therefore, we introduce the following modifications to the original energy function:  E =

.

E,

c

if min |J | > 0,

+∞, otherwise.

(5.46)

In other words, when the minimum value of the Jacobian determinant at the Gaussian integration points is negative, the original energy function .E is modified to become infinite. This modification serves as a penalty for invalid parameterizations. The modified energy function, denoted as .E c , acts as a barrier that distinguishes between bijective and non-bijective parameterizations. Hence, we refer to the approach presented in this subsection as the barrier function method. Modern nonlinear optimization solvers commonly employ line search techniques, such as the Armijo–Goldstein criterion, the Wolfe–Powell criterion, strong Wolfe criterion, and others, to ensure a significant decrease in the objective function value. By incorporating these techniques, the modified energy function .E c effectively prevents the occurrence of self-intersections, ensuring a smooth and valid parameterization. The resulting parameterization, depicted on the right side of Fig. 5.18, demonstrates the effectiveness of the approach.

Penalty Function-Based Method The barrier function-based method presented in the previous section requires an initial parameterization that already exhibits bijectivity. However, obtaining such a parameterization efficiently can be challenging and may necessitate additional foldover elimination steps. It is crucial to highlight that these foldover elimination steps offer limited enhancements to the parameterization quality and may lead to redundant computations (see the middle of Fig. 5.18). To tackle this issue, we present a penalty function-based approach for parameterization. This method is straightforward to implement and effectively eliminates the need for extra foldover elimination steps. The fundamental workflow of this penalty function-based method is illustrated in Fig. 5.19. To the best of our knowledge, this concept was initially introduced by Garanzha to address mesh untangling problems [27, 28]. Notably, Wang and Ma recently applied this idea to planar parameterization problems [84]. The basic idea is quite simple. In order to accommodate invalid initial parameterizations and unfold folds during the optimization process, we introduce a novel

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Penalty objective function Untangling & Minimizing distortion

Optimized parameterization

Initialization

B-Rep

Fig. 5.19 The workflow of the penalty function-based method 18

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0

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(b)

(a)

Fig. 5.20 Penalty function and Jacobian regularization techniques. (a) Jacobian determinant and different penalty functions. (b) Reciprocal square of Jacobian determinant and penalty functions

penalty function. The proposed penalty function, represented as .χ (|J | , ε, β), is defined as follows:  ε · eβ(|J |−ε) if |J | ≤ ε, .χ (|J | , ε, β) = (5.47) |J | if |J | > ε, where .ε is a small positive number, and .β is a penalty factor used to control the slope of the penalty function. As shown in Fig. 5.20a, if .|J | < ε, then .χ (|J | , ε, β) is equal to a small positive number. On the other hand, if .|J | ≥ ε, it is exactly equal to the Jacobian determinant .|J |. Therefore, intuitively, . χ 2 (|J1|,ε,β) imposes a significant penalty for negative Jacobian determinants and has a smaller value to accept positive Jacobian determinants, as shown in Fig. 5.20b. Remark 5.3.1 For mesh untangling problems, Garanzha [27] proposed a penalty function .χorg : χorg (|J | , ε) =

.

|J | +

$

ε2 + |J |2 , 2

(5.48)

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where .ε is a positive number. However, as shown in Fig. 5.20a, even when .|J | > ε, the penalty function .χorg is not strictly equal to the Jacobian determinant .|J |. This introduces additional errors, which may not be desirable in practical applications. Based on the idea of Jacobian regularization, we replace the Jacobian determinant in the denominator of the pointwise angle distortion function (5.42) with the penalty function (5.47), and we obtain angle,c . Ep

! " % %2 1 2 % −1 % ‖J ‖F %J % − 1 . = F 8

(5.49)

Similarly, we also modify the uniformity energy function (5.40) as follows: Epunif.,c =

.

χ (|J | , ε, β) vol(Ω) + . χ (|J | , ε, β) vol(Ω)

(5.50)

Eventually, the corrected weighted objective functional is expressed as follows: #   mips,c ˆ λangle Ep .E = + λunif. Epunif.,c dΩ. c

(5.51)

ˆ Ω

The resulting volumetric parameterization is depicted in Fig. 5.19 (right).

5.3.4 PDE-Based Methods The theory of harmonic mapping has garnered considerable attention in the realm of planar parameterization owing to its exceptional mathematical properties and robust theoretical underpinnings. In actuality, the optimization-based approaches elucidated in the preceding sections fundamentally strive to approximate the inverse mapping of harmonic mapping in finite-dimensional spline spaces. Figure 5.21 illustrates a cross-section of a twin-screw compressor, presenting notable geometric challenges, particularly with extreme aspect ratios between the rotor clearances. While numerous parameterization techniques perform admirably on established benchmark geometries, they often demonstrate subpar performance in practical applications. Figure 5.21a showcases the parameterization results achieved using the penalty function method outlined in Sect. 5.3.3 [41]. For such challenging geometric shapes, the precise selection of pertinent parameters becomes crucial. Moreover, during our research, we discovered an unreasonable influence of the parameter domain size on the parameterization results. In contrast, the PDE-based Elliptic Grid Generation (EGG) method [35] yields a satisfactory parameterization outcome, as depicted in Fig. 5.21b.

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(a)

(b)

Fig. 5.21 Parameterizations generated by the penalty function-based method [41] and the Elliptic Grid Generation (EGG) method [34]. (a) Penalty function-based method. (b) Elliptic Grid Generation (EGG) method

The fundamental concept behind the Elliptic Grid Generation (EGG) method is to ˆ to the computational compute a harmonic mapping .x from the parametric domain .Ω domain .Ω by solving the following set of Laplace equations:  .

Δξ(x, y) = 0 Δη(x, y) = 0

ˆ s.t. x −1 |∂Ω = ∂ Ω.

(5.52)

The problem (5.52) belongs to a specific class of Dirichlet problems. The existence of a solution is ensured if the boundary .∂Ω satisfies the .C 1,α Hölder continuity condition for some .α ∈ (0, 1), and the uniqueness of the solution is guaranteed by the maximum principle. Given the assumption that the parametric ˆ is convex, typically represented as a unit square, the unique solution .x −1 domain .Ω

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establishes a one-to-one correspondence, ensuring its non-vanishing Jacobian .J , ˆ and the computational between the interior regions of the parametric domain .Ω domain .Ω [18]. Discretization in Sobolev Space H 2 In the context of generating parameterizations for IGA, the primary focus lies on ˆ to the computational domain .Ω, the mapping .x from the parametric domain .Ω which represents the inverse of the harmonic mapping .x −1 . Consequently, the set of Laplace equations (5.52) is transformed into an equivalent problem by Xu et al. [92]. The resulting problem is a nonlinear vector-valued second-order PDE:  .

& =0 Lx & =0 Ly

s.t. x|∂ Ωˆ = ∂Ω,

(5.53)

L , g11 + g22

(5.54)

where &= L

.

with the differential operator ∂2 ∂2 ∂2 + g − 2g , 12 11 ∂ξ ∂η ∂ξ 2 ∂η2

L = g22

.

(5.55)

and .gij = x ,ξi · x ,ξj denotes the entries of the metric tensor .G in (5.38). The scaled operator (5.54) was introduced by Hinz et al. [36] to enhance convergence. This operator provides a more consistent convergence criterion for geometries with varying length scales and demonstrates improved convergence properties in numerical experiments. Let us denote by .S the spline space spanned by NURBS basis functions. Let ˆ .S0 = {Ni ∈ S : Ni | ˆ = 0} be the collection of .Ni ∈ S that vanish on .∂ Ω. ∂Ω Following the IGA setting, we have the following variational counterpart of (5.53):  ∀Ni ∈ S0 :

.

F x = 0, F y = 0,

s.t. x|∂ Ωˆ = ∂Ω,

(5.56)

where # Fx =

.

ˆ Ω

# F = y

ˆ Ω

& dΩ, ˆ . N Lx

(5.57)

& dΩ, ˆ N Ly

(5.58)

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Fig. 5.22 Duck example: The left displays the parameterization resulting from the .H 2 discretization (5.56), where some non-uniform elements can be observed inside the red circle. On the right side, the result achieved through the discretization (5.59) in the .H 1 space is displayed. The color encodes the scaled Jacobian, with white representing optimal orthogonality

and .N denotes the column collection of the NURBS basis functions .Ni ∈ S0 . Then the unknown inner control points can be determined by solving the aforementioned nonlinear system, with the known boundary control points acting as Dirichlet boundary conditions. The parameterization obtained from solving the nonlinear system (5.56) is presented on the left side of Fig. 5.22. It is evident that non-uniform elements appear near the head of the duck, highlighted by the red circle. As pointed out in [35], this issue can be partially alleviated by refining the current geometry to achieve a more accurate approximation of the harmonic mapping. However, such refinement operations introduce unnecessary control points and increase the complexity of CAD geometries, potentially leading to challenges in subsequent analyses and downstream processes. This phenomenon, widely observed in EGG [92], is an inherent characteristic. In the following section, our objective is to improve the quality of the parameterization while maintaining the same number of control points. To tackle this problem, we introduce a scale factor and propose a novel discretization for (5.52) in the Sobolev space .H 1 instead of .H 2 . Discretization in Sobolev Space H 1 In this section, to further enhance the quality of the parameterization, we introduce the following discretization in the Sobolev space .H 1 :  ∀Ni ∈ S0 :

.

F xH 1 = 0, y

F H 1 = 0,

s.t. x|∂ Ωˆ = ∂Ω,

(5.59)

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where # F xH 1 = #

.

y FH1

ˆ Ω

=

ˆ Ω

ˆ ∇x N · ∇x ξ dΩ, (5.60) ˆ ∇x N · ∇x η dΩ,

and .N denotes the column collection of the NURBS basis functions .Ni ∈ S0 . Upon solving (5.59), the resulting parameterization is depicted on the right side of Fig. 5.22. It is evident that the parameterization quality has been substantially enhanced, as observed from the improved orthogonality and uniformity, and the absence of non-uniform elements. Importantly, the cardinality of control points remains unchanged.

5.3.5 Experiments and Comparisons In this section, we embark on a comprehensive examination of the aforementioned parameterization techniques. Our main objective is to evaluate their effectiveness and applicability by utilizing the comprehensive test dataset [53] that comprises 977 planar models. Figure 5.23 show some representative examples of this dataset. By subjecting these techniques to rigorous scrutiny across a diverse range of models, we

Fig. 5.23 Planar parameterization results gallery

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aim to gain a deeper understanding of their performance and identify any potential limitations.

Quality Metrics for Parameterizations In this section, we adopt the following quality metrics to measure the quality of parameterizations: • Scaled Jacobian: .

|J |s =

|J | p '

.

(5.61)

x ,ξi

i=1

The scaled Jacobian characterizes the orthogonality of the parameterization and falls within the range of .[−1, 1]. A parameterization .x is considered bijective ˆ If the scaled Jacobian .|J |s takes on negative only when .|J |s > 0 for all .ξ ∈ Ω. values, it indicates that the parameterization is not bijective. A value close to .1.0 for the scaled Jacobian .|J |s across the entire parameter domain suggests a high degree of orthogonality. • Uniformity metric: ! unif. =

.

"2 |J | −1 , area(Ω)

(5.62)

where .area(Ω) denotes the area (or volume) of the computational domain .Ω. Ideally, the Jacobian determinant .|J | for the parameterization .x should equal the ratio of the computational domain area (or volume) to the parameter domain area (or volume). Thus, the optimal value of the uniformity index .unif. is .0.0. In our experiments, we evaluate both quality metrics using a dense sampling of 1001 × 1001 points, including the boundaries. We omit the maximum values of scaled Jacobian and the minimum values of uniformity metric in our statistics since they are attainable in most examples.

.

Effectiveness and Quality Assessment Figure 5.24 illustrates the worst-case quality metrics for the resulting parameterizations, specifically .min (|J |s ) and .max (unif.). It is worth noting that a negative value of .min (|J |s ) indicates the presence of self-intersections in the resulting parameterization. From the figure, it is evident that the optimizationbased parameterization methods exhibit greater robustness in this planar dataset compared to the PDE-based methods. Table 5.1 presents the success rates of various parameterization approaches.

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10

8

6

4

2

0 -1

-0.5

0

0.5

1

Fig. 5.24 Planar dataset: .min (|J |s ) vs. .max (unif.) Table 5.1 Success rates of parameterization approaches

Method

Success rate

Barrier function-based method

.

961 977

≃ 98.36%

Barrier function-based method

.

956 977

≃ 97.85%

PDE-H2 discretization

.

608 977

≃ 62.23%

PDE-H1 discretization

.

721 977

≃ 73.80%

Figure 5.25 presents the mean values of the scaled Jacobian .|J |s and the uniformity metric. It is evident that the two optimization methods, namely the barrier function-based method and the penalty function-based method, exhibit convergence to similar results across most models. This outcome is expected, as the primary difference between these two methods lies in their approach to handling bijectivity constraints. In comparison to the PDE-.H2 discretization method, the .H1 discretization method demonstrates improved uniformity.

Computational Time Figure 5.26 illustrates the performance of different parameterization approaches in terms of computational time, with the PDE-.H2 discretization exhibiting the best performance. In general, the PDE-based parameterization methods demonstrate faster computation times compared to the optimization-based methods.

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10

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Fig. 5.25 Planar dataset: .mean (|J |s ) vs. .mean (unif.) 10 8 6 4 2 0

0

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Fig. 5.26 Planar dataset: computational time

Volumetric Parameterizations The aforementioned parameterization approaches can be seamlessly applied to 3D volumetric parameterization problems. Figure 5.27 showcases volumetric parameterization results obtained using the penalty function-based method. It is noteworthy that the minimum value of the scaled Jacobian indicates the resulting parameterizations are bijective.

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Fig. 5.27 Volumetric parameterization results gallery

Fig. 5.28 Multi-patch volumetric parameterization

Extension to Multi-patch Parameterizations Although our previous discussion primarily focused on the single-patch scenario, it is important to emphasize that our parameterization techniques can be seamlessly extended to handle multi-patch domain parameterizations. This versatility arises when the topological layout of the quadrilateral or hexahedral elements is determined. By intelligently integrating multiple patches, our techniques enable the effective representation of complex domains. Figure 5.28 serves as a visual demonstration, showcasing a volumetric parameterization that leverages a multipatch configuration.

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Fig. 5.29 Multi-patch THB-spline parameterization

Extension to THB-Spline Parameterizations As illustrated in Fig. 5.29, it is important to highlight that our parameterization techniques exhibit impeccable compatibility with THB-spline parameterizations. This noteworthy characteristic indicates that our methods are adept at effectively managing the intricate complexities that arise in THB-spline-based parameterization. Consequently, our parameterization techniques offer a versatile and robust solution that can be applied across a broad spectrum of applications.

5.4 Isogeometric Kirchhoff–Love Shell Analysis In this section, an example of isogeometric analysis for thin shell mechanics is provided. The aim of the section is to show how the isogeometric Kirchhoff–Love shell equations can be derived from geometric and mechanics principles, employing geometric and solution representations using splines and how they can be used in the analysis. The derivation of the shell element is provided in Sect. 5.4.1. Thereafter, three benchmark studies are presented in Sect. 5.4.2.

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5.4.1 The Isogeometric Kirchhoff–Love Shell Element The isogeometric Kirchhoff–Love shell was first presented by Kiendl in [45] and has been widely used within the isogeometric analysis community. The derivation of the Kirchhoff–Love shell model in this section follows the PhD Thesis of Kiendl [44]. For basic principles of continuum mechanics, the reader is referred to [6] or other continuum mechanics textbooks. The section concludes with a benchmark problem using snapping of a hyperelastic shell, based on [82].

Geometry The Kirchhoff–Love shell theory describes the deformation of surfaces. Hence, let S(ξ1 , ξ2 ) : R2 → R3 be a surface. For this surface, the covariant basis vector .a α is defined by taking the derivatives of the surface with respect to the parametric coordinate .ξα , i.e.,

.

aα =

.

∂S , ∂ξα

α = 1, 2.

(5.63)

Using the covariant basis, the covariant metric tensor or first fundamental form is defined by aαβ = a α · a β .

.

(5.64)

Using the first fundamental form of the surface, the contravariant metric tensor is defined using the inverse of .[aαβ ], as .a αβ = [aαβ ]−1 . Furthermore, the contravariant basis .a α is defined by a α = a αβ a β .

.

(5.65)

Using the covariant basis vectors from (5.63), the surface unit normal vector is defined by aˆ 3 =

.

a1 × a2 . |a 1 × a 2 |

(5.66)

In addition to the surface gradients, the curvature of the surface is a quantity of interest, typically related to bending. In the present derivation of the Kirchhoff– Love shell theory, the curvature is included via the second fundamental form, as bαβ = aˆ 3 · a α,β = −aˆ 3,β · a α .

.

(5.67)

Here, .a α,β denotes the second derivative or Hessian of the surface and .aˆ 3,α denotes the derivative of the unit normal vector with respect to the parameter .ξα . Via

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β

Weingarten’s formula [88] it holds that .aˆ 3,α = −bα a β with .bα = a αγ bγβ . Since the second fundamental form .bαβ depends on the surface Hessian .a α,β , second derivatives of the surface description .S(ξ1 , ξ2 ) are required. Assuming the Kirchhoff hypothesis [19], i.e., no shear of the shell cross-section, orthogonality of orthogonal vectors after deformation, and no thickness change, the Kirchhoff–Love shell formulation assumes that any point in the shell can be described by its position on the surface .S(ξ1 , ξ2 ) and its position along the surface normal .a 3 as x(ξ1 , ξ2 , ξ3 ) = S(ξ1 , ξ2 ) + ξ3 aˆ 3 .

.

(5.68)

The derivatives of the coordinate system .x with respect to the parametric coordinates ξi (.i = 1, 2, 3) provide the full basis of the coordinate system used for the Kirchhoff–Love shell element. The covariant basis of .x is given by

.

gα =

∂x = a α + ξ3 a 3,α , ∂ξα

g3 =

∂x = aˆ 3 . ∂ξα

.

(5.69)

Following from the covariant basis, the first fundamental form .gij = g i · g j is defined using the first and second fundamental forms as gαβ = (a α + ξ3 a 3,α ) · (a β + ξ3 a 3,β ), = aαβ − 2ξ3 bαβ + ξ32 a α · a β , .

(5.70)

g33 = 1, gi3 = g3i = 0. The last term, quadratic in .ξ3 , can be neglected for thin or moderately thick shells [10]. The contravariant metric tensor .g ij and the contravariant basis .g i are derived like for the surface .S. Using the shell coordinate system (5.68) and the covariant basis (5.70), the kinematic relation for the Kirchhof–Love shell can be derived.

Kinematic Relation The kinematic relation relates shell displacements to strains. Let ˚ .x (ξ1 , ξ2 , ξ3 ) denote the undeformed configuration of the shell and let .x(ξ1 , ξ2 , ξ3 ) denote the deformed configuration of the shell. Then, the deformation .u(ξ1 , ξ2 , ξ3 ) of a material point is defined as

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u(ξ1 , ξ2 , ξ3 ) = x(ξ1 , ξ2 , ξ3 ) − ˚ x (ξ1 , ξ2 , ξ3 ).

.

(5.71)

Additionally, the deformation gradient .F is a tensor that maps between the g i and the deformed basis .g i , meaning that an infinitesimal line undeformed basis .˚ element .d˚ x in the undeformed configuration is defined as .dx = F · d˚ x in the undeformed configuration [6]. Accordingly, the deformation gradient .F is defined as F = gi ⊗ ˚ gi .

.

(5.72)

g i onto .g i via .g i = Fg˚i [6]. Using the Indeed, the deformation gradient maps .˚ g i ⊗˚ g j relates the deformation gradient, the Green–Lagrange strain tensor .E = Eij ˚ nonlinear relation between deformations and strains E=

.

 1 1 ⏉ F F − I = (C − I), 2 2

(5.73)

where .C is the deformation tensor. Using the definition of the deformation gradient gi ⊗ ˚ and the fact that the identity tensor .I is equal to the metric tensor .Gij on .˚ gj yields Eij =

.

 1 gij − Gij . 2

(5.74)

Using the definition of the metric tensor from (5.70), the coefficients of the strain tensor can be expressed in terms of the surface metric and the curvature:  1 aαβ − 2ξ3 bαβ − ˚ aαβ + 2ξ3˚ bαβ x] 2    1 bαβ − bαβ = εαβ + καβ . = aαβ − ˚ aαβ + ξ3 ˚ 2

Eαβ = .

(5.75)

The shear strains .Ei3 and .E3i and the normal strain .E33 vanish because of the orthogonality and unity of the basis vector .g 3 in deformed and undeformed configurations. This indeed shows that the shell formulation following from the assumed coordinate system in (5.68) yields a formulation free of cross-sectional shear and thickness change. Hence, the shell can be represented by its mid-surface only and the strain tensor is represented with respect to the first two components of g α ⊗˚ the basis, i.e., .E = Eαβ ˚ g β . The coefficients .εij and .καβ relate to the membrane α g ⊗˚ gα ⊗ ˚ strain tensor .ε = εαβ ˚ g β and the bending strain tensor .κ = καβ ˚ gβ .

Constitutive Relation In general continuum mechanics, the second Piola–Kirchhoff stress tensor .S = gi ⊗ ˚ S ij ˚ g j is energetically conjugate to the Green–Langrange strain tensor .E =

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Eij ˚ g j [6]. For a 3D continuum, the coefficients of the second Piola–Kirchhoff gi ⊗ ˚ stress tensor can be defined using a strain energy density function .Ψ : S ij = 2

.

∂Ψ . ∂Cij

(5.76)

In addition, the material tensor or elasticity tensor .C = C ij kl ˚ g j ⊗˚ g k ⊗˚ g l is a g i ⊗˚ fourth-order tensor that relates the total differentials of the second Piola–Kirchhoff stress .S and the Green–Lagrange strain .E. Its coefficients are defined by C ij kl =

.

∂Sij ∂ 2Ψ =4 , kl ∂E ∂Cij ∂Ckl

(5.77)

such that the coefficients of the total differential of the second Piola–Kirchhoff stress tensor, .dS ij , relate to the total differential of the Green–Lagrange strain tensor, .dE ij , via .

dS ij = C ij kl dE ij .

(5.78)

For linear elastic materials, stress and strain are linearly dependent such that .C has constant coefficients according to (5.77). Therefore, the following identity is valid for linear materials: S ij = C ij kl Ekl .

.

(5.79)

Furthermore, assuming small strains, through thickness deformation is neglected and .C33 = g33 = 1, which allows to use 2D constitutive models. However, when strains are large, for example, in hyperelastic material models, the plane stress assumption that .S 33 = 0 is typically violated [46], hence .C33 /= 1. To use the inplane components of the stress tensor, .S αβ in the Kirchhoff–Love shell model, static condensation of the material tensor .C needs to be performed to satisfy the plane stress condition. The formulations for the hyperelastic stress and material tensors for Kirchhoff–Love shells are provided in [46] and an extension for stretch-based material models was provided by [82].

Variational Formulation The variational formulation for the Kirchhoff–Love shell is derived based on the Principle of Virtual Work. According to this principle, the total energy in the system, represented by .W (u) = W int (u) − W ext (u), is minimized for the deformation .u if and only if its variation .δW (u, v) with respect to .u is equal to zero: δW (u, v) = δW int (u, v) − δW ext (u, v).

.

(5.80)

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Here, .v denotes the virtual displacements. Since .δW (u, v) can be nonlinear, the displacement .u can be found using the Newton–Raphson method by solving 2 δv W + δvw W Δu = 0,

(5.81)

.

2 W = δ 2 W (u, v, w) is the second variation of the where .δv W = δW (u, v) and .δvw energy in the system using virtual displacements .v and .w and .Δu is the incremental update of the displacements. The derivation of the external virtual work is rather straightforward. Assuming that the body force vector .f and the boundary force vector .g are independent of the deformation field .u, the first variation of the external work .W ext simply yields

# δv W ext =

# #

#

.

Ω✶

f · v dΩ✶ +

∂Ω✶

g · v d𝚪 =

f · v dΩ dξ3 τ

Ω

τ

∂Ω

# # +

g · vn d𝚪 dξ3 ,

(5.82)

where .Ω✶ = τ × Ω with .τ the thickness domain .τ = [−t/2, t/2] of the shell and .Ω the surface domain. For the internal virtual work, the first variation with respect to the displacements .u is given by # δv W

.

int

=

# #

#

S : δv E dΩ =

S : δv E dΩ =

✶

Ω✶

τ

Ω

N : δv ε + M : δv κ dΩ . Ω

(5.83)

Here, the definition of the strain tensors .ε and .κ from (5.75) is used and the membrane force tensor .N and the bending moment tensor .M are defined as moments of the stress tensor through thickness: # N=

S dξ3 , τ

(5.84)

#

.

M=

ξ3 S dξ3 . τ

The second variation of the energy of the system required for solving the nonlinear system of equations using the Newton–Raphson iterations (see (5.81)) solely depends on the second variation of the internal energy, assuming deformationindependent body forces. Taking the variation of the internal energy with respect to .u, the second variation becomes # 2 2 2 .δvw W (u) = δw N : δv ε + N : δvw ε + δw M : δv κ + M : δvw κ dΩ . (5.85) Ω

The variations of .N and .M can be obtained using the total differential of .S and .δS (see (5.78)). First, since

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δE = δ(ε + ξ3 κ) = δε + δ(ξ3 κ) = δε + κδξ3 + ξ3 δκ

.

(5.86)

and using the total differential of the strain, .δE, and integrating .δS through the thickness, the total differentials of .N and .M are obtained: # # δN = δS dξ3 = (C : δε + ξ3 C : δκ + δξ3 C : κ) dξ3 , τ

τ

ξ3 δS dξ3 =

δM =

(5.87)

#

#

.

τ

(ξ3 C : δε + ξ3 C : δκ + δξ3 C : κ) dξ3 . τ

From the first and second variations of the internal energy, respectively (5.83) and (5.85), it can be seen that the first and second variations of the membrane strain and bending strain tensors.

Discretization The principle of virtual work derived in (5.80) is valid for any variation of the unknown displacement field .u(ξ1 , ξ2 , ξ3 ). In order to discretize the principle of virtual work, it is assumed that the undeformed and deformed configurations ˚ .x and .x, respectively, are represented by a finite sum of basis functions .ϕk (ξ1 , ξ2 ) weighted h h by coefficients ˚ .x k and .x , i.e., k ˚ x h (ξ1 , ξ2 ) =



ϕk (ξ1 , ξ2 )˚ x hk ,

k .

x h (ξ1 , ξ2 ) =



(5.88) ϕk (ξ1 , ξ2 )x hk .

k

Here, the superscript h indicates discrete approximations of ˚ .x or .x and the index k indicates the k-th component of this representation. Since the displacement field .u is defined as the difference between ˚ .x and .x, it can similarly be expressed as a discrete field .uh and the variations in the principle of virtual work are represented by virtual displacements .uhk . As a consequence, all variations in the virtual work equation are represented by derivatives with respect to components of the virtual nodal displacements .uhk . In the following, all quantities are referred to in the discrete setting, and hence the superscript h is omitted. In the sequel, r denotes the global index of the degree of freedom .ur representing a component of one of the nodal displacement vectors. For the sake of brevity, the shorthand notation .(·),r = ∂(·) ∂ur is used to represent derivatives with respect to .ur . Using (5.71), the variation of the deformed configuration is x ,r =

.

   ˚ x k,r + uk,r = ϕk uk,r , k

k

(5.89)

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where the last equality follows from the fact that the undeformed configuration is trivially independent of the deformation field .u. Similarly, the derivatives of the covariant basis vectors .a α of the discrete deformed configuration .x h , see (5.63), are ! a α,r =

.

∂x k ∂ξα

" =

 ∂ϕk

,r

∂ξα

k

uk,r .

(5.90)

As a consequence, the variation of the surface metric tensor of the deformed configuration, .aαβ (see (5.64)), becomes   aαβ,r = a α · a β r = a α,r · a β + a α · a β,r .

.

(5.91)

Since the undeformed configuration is invariant to the deformation field .u, the first variation of the membrane strain tensor .ε from (5.75) becomes 1 aαβ,r . 2

εαβ,r =

.

(5.92)

Similarly, the second variations of the deformed configuration, the deformed surface metric tensor, and the membrane strain can be derived. Starting with the first variation of the deformed configuration from (5.89), the second variation becomes  .x ,rs = ϕk uk,rs = 0. (5.93) k

The second variation of .uk is zero since the components of these nodal weights are linear in .ur . Similarly, .a α,rs = 0. As a consequence, the second variation of the surface metric tensor in the deformed configuration, .aαβ , becomes aαβ,rs = a α,rs · a β + a α,r · a β,s + a α,s · a β,s + a α · a β,rs , .

.

= a α,r · a β,s + a α,s · a β,s .

(5.94) (5.95)

Again, since the undeformed configuration is invariant to the deformation field .u, the second variation of the membrane strain tensor becomes 1 aαβ,rs . 2

εαβ,rs =

.

(5.96)

To derive the variations of the curvature tensor, the variations of the second fundamental form .bαβ are needed, hence requiring variations of .a α,β and .aˆ 3 , see (5.67). Firstly, the variation of .a α,β with respect to .ur is ! a (α,β),r =

.

∂ 2x ∂ξα ∂ξβ

" = ,r

 ∂ 2 ϕk uk,r . ∂ξα ∂ξβ k

(5.97)

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Furthermore, using .a 3 = a 1 × a 2 , the variation of the unit normal vector .aˆ 3 is ! aˆ 3,r =

.

a3 |a 3 |

" = ,r

|a 3 |a 3,r − a 3 (|a 3 |),r . |a 3 |2

(5.98)

Here, the variation of the non-unit normal vector .a 3 is obtained by a 3,r = a 1,r × a 2 + a 1 × a 2,r ,

.

and since .|a 3 | =

(5.99)

√ a 3 · a 3 , the variation of the normalization .|a 3 | is (|a 3 |),r =

.

a 3 · a 3,r , |a 3 |

(5.100)

such that the variation of the unit surface normal vector of the undeformed configuration, .aˆ 3 , can be obtained. Together with the variation of the surface Hessian, .a (α,β),r from (5.97), the variation of the second fundamental form becomes bαβ,r = aˆ 3,r · a α,β + aˆ 3 · a (α,β),r .

.

(5.101)

From the definition of the bending strain tensor .κ in (5.75) and the fact that the undeformed configuration is invariant to the deformation field .u, the coefficients of the first variation of the bending strain tensor become καβ,r = −bαβ,r .

.

(5.102)

To obtain the second variation of the bending strain tensor .κ, the second variations of .a α,β and .aˆ3 need to be obtained in order to compute the second variation of .bαβ . Firstly, from (5.97), it follows that .a (α,β),rs = 0 since the second variation of .uk is zero. Secondly, for the second variation of the unit normal vector .aˆ 3 , the second variation of the non-unit normal vector .a 3 and its length .|a 3 | are needed. The second variation of the non-unit normal vector follows from the first variation in (5.98) and from .a (α,β),rs : a 3,rs = a 1,rs × a 2 + a 1,r × a 2,s + a 1,s × a 2,r + a 1 × a 2,rs .

.

= a 1,r × a 2,s + a 1,s × a 2,r .

(5.103) (5.104)

Furthermore, the second variation of .|a 3 | is (|a 3 |),rs

.

  |a 3 |(a 3 · a 3,r ),s − a 3 · a 3,r (|a 3 |),s = . |a 3 |2    a 3 · a 3,r a 3 · a 3,s a 3,s · a 3,r + a 3 · a 3,rs = . − |a 3 | |a 3 |3

(5.105) (5.106)

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Using the second variations of the non-unit normal .a 3 (see (5.103)) and its length |a 3 | (see (5.105)), the second variation of the unit normal vector .aˆ 3 can be derived

.

! aˆ 3,rs =

=

.

|a 3 |a 3,r − a 3 (|a 3 |),r |a 3 |2

" ,s

  |a 3 |a 3,r − a 3 (|a 3 |),r ,s |a 3 |2

  |a 3 |a 3,r − a 3 (|a 3 |),r 2|a 3 |(|a 3 |),s − |a 3 |4

a 3,s (|a 3 |),r a 3 (|a 3 |),rs a 3 (|a 3 |),r (|a 3 |),s (|a 3 |),s a 3,r a 3,rs − − − +2 . 2 2 2 |a 3 | |a 3 | |a 3 | |a 3 | |a 3 |3 (5.107) Additionally, taking the variation of .bαβ,r and using the first and second variations of .a α,β and .aˆ 3 , the second variation of the second fundamental form .bαβ can be obtained =

bαβ,rs = aˆ 3,rs · a α,β + aˆ 3,r · a (α,β),s + aˆ 3,s · a (α,β),r + aˆ 3 · a (α,β),rs , .

= aˆ 3,rs · a α,β + aˆ 3,r · a (α,β),s + aˆ 3,s · a (α,β),r .

(5.108)

From (5.108) and (5.75), it directly follows that the coefficients of the second variation of the bending strain tensor are καβ,rs = −bαβ,rs .

.

(5.109)

Besides the first and second variations of the membrane strain tensor .ε and the bending strain tensor .κ, the first variations of the membrane force tensor .N and the bending moment tensor .M also need to be obtained. Using the total differentials .dN and .dM (see (5.84)), the coefficients of the first variations of .N and .M with respect to .ur are " " !# !# αβ αβγ δ αβγ δ N,r = C dξ3 εγ δ,r + ξ3 C dξ3 κγ δ,r , τ

M,rαβ =

τ

"

!#

.

!#

ξ3 C αβγ δ dξ3 εγ δ,r + τ

τ

" 2 αβγ δ ξ3 C dξ3 κγ δ,r .

(5.110)

Note that the last term of (5.86) drops out because the variation of .ξ3 with respect to ur is zero. Using the variations with respect to the nodal displacement components .ur , the first and second variations of the energy equation in the shell following from the virtual work statement in (5.80) can be defined for each component .ur . Firstly, the first variation of the energy statement provides the components of the residual vector .R as .

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#

#

Rr (u) =

N(u) : ε ,r (u) + M(u) : κ ,r (u) dΩ −

.

Ω

# f · u,r dΩ −

Ω

g · u,r d𝚪 . ∂Ω

(5.111)

Secondly, the second variation of the energy statement from (5.81) provides the Jacobian matrix for the Newton–Raphson iterations, also known as the (tangential) stiffness matrix K, with coefficients # Krs =

N,s (u) : ε ,r (u) + N(u) : ε ,rs (u) + M,s (u) : κ ,r (u) + M(u) : κ ,rs (u) dΩ .

.

Ω

(5.112)

In case of zero displacements, i.e., .u = 0, the deformation gradient .F is an identity map and the deformation tensor .C is the identity tensor. Therefore, the stress tensor becomes the null tensor, .S = 0, making the tensors .N and .M vanish as well. In this case, the first integral of the residual vector .R is zero and the second and fourth terms drop out of the stiffness matrix K. This gives the external force vector .P and the linear stiffness matrix .K L , with coefficients #

#

Pr = −Rr (0) =

f · u,r dΩ + Ω

g · u,r d𝚪 , ∂Ω

#

.

L Krs

= Krs (0) =

(5.113)

N,s (0) : ε ,r (0) + M,s (0) : κ ,r (0) dΩ . Ω

Up to this point, all quantities have been defined to be used in the variational formulation, except for the basis functions .ϕk to define the undeformed and deformed configurations of the shell surface as well as the displacement field: ˚ .x , .x, and .u, respectively (see (5.88)). Since the Hessian of the metric tensor .aα,β is used in the definition of the second fundamental form and its variations, see (5.67), (5.101), and (5.108), the basis functions .ϕk need to be differentiable up to the second derivative. Due to the higher order continuity that can be achieved using splines, they provide a suitable basis for the Kirchhoff–Love shell. In the paradigm of using the same splines for the representation of the geometry .S(ξ1 , ξ2 ) as well as for the discrete solution of the displacement field .uh (ξ1 , ξ2 , ξ3 ), this choice of the basis introduces the isogeometric Kirchhoff–Love shell.

5.4.2 Benchmark Problems In this section, we present some example problems using the isogeometric Kirchhoff–Love shell. In the first example, adopted from [82], the collapse of a truncated cone is simulated. This example uses a hyperelastic Ogden model such that the stress and material tensors are defined by (5.76)–(5.77). In the second example, adopted from [83], a mesh adaptivity example using THB-splines is

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presented, using a basis as explained in Sect. 5.2.2. In the last example, a postbuckling problem on a multi-patch geometry is solved. The example is adopted from [23] and modelled using the Analysis-Suitable .G1 construction, which is a surface unstructured spline construction mentioned in Sect. 5.2.4.

Nonlinear Hyperelastic Shell Analysis As a first example we model the collapse of a truncated cone (or frustrum). The geometry of the cone is given in Fig. 5.30 and is represented by NURBS from Sect. 5.2.3. The original benchmark problem is adopted from [7] and the presented results are published in [82]. The problem parameters from Fig. 5.30 are as follows. The shell has a height .H = 1 [m], top radius .r = 1 [m], bottom radius .R = 2 [m], and thickness .t = 0.1 [m]. Only a quarter of the shell is modelled since the original reference [6] uses axisymmetric elements. The shell is represented by quadratic NURBS with 32 elements over the height and one over the circumference. On the boundary .𝚪1 the displacements are fixed in the x- and y-direction and free in z. Also, a uniform load p is applied on .𝚪1 , providing a uniform displacement .Δ. On .𝚪2 the displacements are fixed but rotations are free. On .𝚪3 and .𝚪4 , symmetry conditions are applied, restricting in-plane deformations normal to the boundaries and restricting rotations on the boundary by applying clamped boundary conditions as described in [45]. The corresponding material model is of the Ogden type and has the following parameters: μ1 = 6.300 [N/m2 ],

α1 = 1.3,

μ2 = 0.012 [N/m2 ],

α2 = 5.0,

μ3 = −0.100 [N/m2 ],

α3 = −2.0,

.

implying that .μ = 4.225 [N/m2 ]. For more information about the stretch-based Ogden material model inside the isogeometric Kirchhoff–Love shell, the reader is referred to [82]. The load applied on the top boundary is either applied using displacement control (DC), by incrementally increasing the displacement of the boundary, or by arc-length control by employing Crisfield’s spherical arc-length method [13] with extensions for resolving complex roots [48, 97]. If this method does not converge to an equilibrium point, the step size is bisected until a converged step is found. After this step, the step size is reset to its original value [81]. In Fig. 5.31, the results for the collapsing conical shell are presented and compared to the reference results from [6]. The displacement-controlled (DC) result

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Fig. 5.30 Geometry of the collapsing conical shell with 32 quadratic elements over the height

0.4 ALM OG

Distributed load p [N/m]

E

0.2

F

DC OG OG, Ref. [5]

H B

0 I D G

−0.2

−0.4 −0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Distributed load p [N/m]

Displacement Δ [m] ·10−2 4 J

2 0 −2 C

−4 0

0.2

0.4

0.6

0.8 1 1.2 Displacement Δ [m]

1.4

A

1.6

1.8

K

2

Fig. 5.31 Load-displacement diagram of the collapsing conical shell. The lines represent solutions obtained using the Arc-Length Method (ALM) and the markers represent solutions obtained by Displacement Control (DC) together with the reference solution by Ba¸sar and Itskov [7]. The capital letters in the diagram represent points for which the deformed geometry is plotted in Fig. 5.32 and the arrows describe the direction of the solution path

shows good agreement with the reference results, with differences that can be explained by shear locking as present in the reference results. When performing arc-length stepping on the problem, it can be seen that at the moment of snapping,

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A

B

C

D

E

F

G

H

I

J

K

Fig. 5.32 Deformed geometries corresponding to the solution path presented in Fig. 5.31 u=0

L

P

u=0

W

z y

symm sy m m

x u=0

u=0

E

=

1.0

ν

=

0.3

P

=

4 · 10

L

=

1

W

=

1

t

=

10

[MPa] [-] −7

[N] [mm] [mm]

−3

[mm]

Fig. 5.33 Geometry and parameters for a square thin plate subject to a point load P in the middle. The plate is fully constrained in every corner. Because the problem is symmetric, only a quarter of the domain is modelled. Hence, symmetry conditions are applied. On the x-aligned symmetry ∂ux z axis, this implies that .uy = ∂u ∂y = ∂y = 0, and on the y-aligned symmetry axis this implies that .ux

=

∂uz ∂x

=

∂uy ∂x

=0

around .Δ ∼ 1.9, a complex snapping phenomenon arises, coinciding with the moment of snapping in the DC path. Observing the deformations in Fig. 5.32, it can be seen that the collapsing mechanism consists of the formation of multiple waves in radial direction that invert after the loop with the highest force amplitude.

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Fig. 5.34 Deformed surface from the benchmark presented in Fig. 5.33. The result is the last solution from the adaptive meshing routine with deformation norm goal functional of which the results are presented in Fig. 5.35

Nonlinear Adaptive Shell Analysis As a next example, we consider adaptive refinement of isogeometric meshes. This example is adopted from [83]. The adaptivity iterations are performed using THBsplines and the marking of the element is driven by the Dual-Weighted Residual (DWR) method [4, 9] see [83] for the complete procedure. We consider a square membrane subject to a point load in the middle and with corners fixed in all directions, see Fig. 5.33. As a material model, a linear SaintVenant–Kirchhoff constitutive law with Young’s modulus .E = 1.0 [MP a] and a Poisson ratio .ν = 0.3 is used. The membrane is considered very thin compared to its in-plane dimensions, i.e., .t = 10−3 [mm], .L × W = 1× 1[mm] (.L/t = 1000). In the middle of the membrane, .P = 4·10−7 [N] is applied. As depicted in Fig. 5.33, the simulations are performed on a quarter of the domain, using symmetry conditions. As a goal functional for the DWR method, a displacement norm is used # L(u) =

‖u‖ dΩ .

.

(5.114)

Ω

The maximum number of levels for the THB refinement is 11, meaning that the finest level has .211 × 211 = 2048 × 2048 elements. The result of the deformed membrane for the last step of the adaptivity simulation is given in Fig. 5.34. In Fig. 5.35, the estimated error via the DWR method is given for the uniformly refined mesh as well as for the adaptively refined mesh. In addition, Fig. 5.36 provides the element errors on the meshes corresponding to the points with the marked border in Fig. 5.35, together with contour lines for the displaced geometry. From the results, it can be observed that the adaptively refined mesh is in general more efficient per degree of freedom than the uniformly refined mesh, due to localization of the error in the bottom-right corner. However, it can also be seen that the adaptive mesh is not monotonously decreasing, probably due to the nonlinearity of the problem where asymmetries in the mesh can result in large errors. From the bottom plot in Fig. 5.35, it can also be observed that the nonmonotone decrease of the error is related to the percentage of the elements in the mesh that is eligible for refinement (i.e., that did not reach the maximum level yet).

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ΔL

10−4 10−5 10−6

Ref. % of e

10

Adaptive Uniform

−7

10−1 10−4 10−7

101

102

103 #DoFs

104

105

Fig. 5.35 Estimated error convergence (top) and the percentage of the total element error e that is available for refinement (bottom) against the number of degrees of freedom (DoFs) for adaptively and uniformly refined meshes with respect to the goal displacement-based goal functionals. The markers labeled with a black border are the markers for which the mesh is plotted in Fig. 5.36

Fig. 5.36 Normalized element error values .e(k /ΔL for uniformly (top) and adaptively (bottom) refined meshes using goal function .L(u) = Ω ‖u‖ dΩ. The meshing steps increase from left to right. The contour lines represent the displacement of the membrane, with intervals of .0.1 [mm]. The bottom-right corner of the pictures indicates the fixed corner and the top-left corner is the corner where the load is applied. (a) Uniform refinement. (b) Adaptive refinement

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When this percentage is small, refinement is performed using elements that have an insignificant contribution to the total error.

Nonlinear Multi-patch Shell Analysis As a last example, a post-buckling problem on a multi-patch geometry is solved. This example is adopted from the paper [23] where the isogeometric Kirchhoff– Love shell is used on a multi-patch domain using the Analysis-Suitable .G1 basis from [22]. The geometry is given in Fig. 5.37 and is fixed on the left side and loaded with an in-plane load of .λP and an out-of-plane load .λPs using .P /Ps = 100 on the bottomright side. The dimensions of the geometry are .L = 255 [mm], .W = 30 [mm], .Lh = 55 [mm], and .Wh = 10 [mm], and the thickness is .t = 0.6 [mm]. Furthermore, the material parameters for a linear Saint-Venant–Kirchhoff material are .E = 71240 [N/mm2 ] and .ν = 0.3. Figure 5.38 shows the deformation of the L-shaped domain. The load-displacement curves are given in Fig. 5.39 for a degree .p = 4 basis with maximum regularity .r = p − 2. The example has been computed by using the penalty method [33] with the penalty parameter .α = 103 . The results show that the Analysis-Suitable .G1 basis provides a good alternative to penalty methods for multi-patch analysis of this post-buckling simulation. Fig. 5.37 Geometry of the L-shaped domain with length .L = 255 [mm] and width .W = 30 [mm] and with rectangular holes of size .Lh = 55 [mm] and .Wh = 10 [mm]. The geometry consists of 25 bilinear patches

L W Lh

Wh

λP

λPs

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Fig. 5.38 Deformed geometry of the L-shaped domain with 25 patches corresponding to Fig. 5.37 on the last point of the load-displacement curve for the Analysis-Suitable .G1 method for .p = 4 in Fig. 5.39. The color scale represents the out-of-plane displacement

p=4 Pen. α = 103

λP

1

0.5

0 0

10 20 30 40 50 Out-of-plane displacement at end point [mm]

60

Fig. 5.39 Displacements at the point where the load is applied in Fig. 5.37. All results are plotted with the out-of-plane displacement component on the horizontal axis and the load .λP on the vertical axis. The penalty parameter used for the penalty method is .α = 103

5.5 Conclusions and Outlook Design through analysis—to the authors’ best knowledge first mentioned in the literature in the 1970s in a report from the General Motors Research Laboratories has gained new impetus since the advent of Isogeometric Analysis in the early 2000s. It is obviously more than the bundling of CAD and CAE tools under a common user interface that hides (but does not cure) the tedious and errorprone back-and-forth conversion between genuinely incompatible representations of geometry and analysis models. As we tried to show in this chapter, breaking with the traditional triad of pre-processing, analysis, and post-processing and placing the entire workflow on common mathematical grounds—splines (cf. Sect. 5.2)—brings plentiful advantages. One of the most beneficial ones is the higher continuity that not only brings higher accuracy per degree of freedom but also simplifies the solution of higher order differential equations without auxiliary variables.

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At the same time, the many advantages of IGA do not come for free: preserving higher continuity in the multi-patch case requires special techniques—unstructured spline constructions. Moreover, the task of creating analysis-suitable parameterizations from CAD models can be as complicated as generating admissible computational grids. In Sect. 5.3 we have presented several computational approaches to construct high-quality analysis-suitable parameterizations from boundary descriptions. For the reader who prefers working with classical computational grids, e.g., in the context of finite elements and finite volumes, we would like to remark that it is still possible to utilize the presented approaches to construct a bijective mapping ˆ → Ω and generate both structured and unstructured meshes by “pushing .x : Ω ˆ to the physical space .Ω [59]. forward” a grid created in the parameter domain .Ω That way, the task of (i) ensuring the non-folding of elements and (ii) optimizing the shape and the size of elements gets decoupled as the former property is ensured by the bijectiveness of the push-forward operator. Section 5.4 finally gave a brief insight into the analysis capabilities of IGA at the hand of Kirchhoff–Love shell models. Let us conclude this chapter with a word of caution that is likewise meant as motivation to continue research in IGA and design through analysis. We believe that all numerical methods have their strengths and weaknesses and none of them is superior to others at large. In our opinion, breaking with the traditional triad of pre-processing, analysis, and post-processing is what makes IGA stand out at this moment and, possibly, a precursor for other numerical approaches that will enable even more sophisticated design-throughanalysis workflows in the future. Acknowledgments The first author would like to thank Prof. Dr. Chun-Gang Zhu from Dalian University of Technology for his valuable contribution to domain parameterization in this chapter, as well as Dr. Kewang Chen and Prof. Dr. Cornelis Vuik from Delft University of Technology for their contributions to the development of the .H 1 PDE-based methods. All three authors would like to express their gratitude to Dr. Angelos Mantzaflaris (Inria Sophia Antipolis Méditerranée), Dr. Andrea Farahat (Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria), Dr. Mario Kapl (ADMiRE Research Center, Carinthia University of Applied Sciences, Villach, Austria), Prof. Dr. Josef Kiendl (Institute of Engineering Mechanics Structural Analysis, Universität der Bundeswehr München, Munich, Germany), and Dr. Henk den Besten (Department of Maritime and Transport Technology, Delft University of Technology, The Netherlands) for fruitful discussions and collaboration on Kirchhoff–Love shell theory, multi-patch coupling and joint code development in the open-source IGA library G+Smo (https://github.com/gismo/gismo) [42].

References 1. I.A. Abelló Ugalde, V. Hernández Mederos, P. Barrera Sánchez, G. González Flores, Injectivity of B-spline biquadratic maps. Comput. Methods Appl. Mech. Eng. 341, 586–608 (2018) 2. Z. Ali, W. Ma, Isogeometric collocation method with intuitive derivative constraints for PDEbased analysis-suitable parameterizations. Comput. Aided Geom. Des. 87, 101994 (2021) 3. J.A. Augustitus, M.M. Kamal, L.J. Howell, Design through analysis of an experimental automobile structure. SAE Trans. 86, 2186–2198 (1977)

5 Design Through Analysis

365

4. W. Bangerth, R. Rannacher, Adaptive Finite Element Methods for Differential Equations, 1st edn. (Birkhäuser Basel, Basel, 2003). ISBN 978-3-7643-7009-1 5. P.J. Barendrecht, Isogeometric Analysis for Subdivision Surfaces (Eindhoven University of Technology, Eindhoven, 2013) 6. Y. Ba¸sar, R. Grytz, Incompressibility at large strains and finite-element implementation. Acta Mech. 168(1), 75–101 (2004) 7. Y. Ba¸sar, M. Itskov, Finite element formulation of the Ogden material model with application to rubber-like shells. Int. J. Numer. Methods Eng. 42(7), 1279–1305 (1998) 8. Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R. Hughes, S. Lipton, M.A. Scott, T.W. Sederberg, Isogeometric analysis using T-splines. Comput. Methods Appl. Mech. Eng. 199(5), 229–263 (2010). Computational Geometry and Analysis 9. R. Becker, R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10:1–102 (2001) 10. M. Bischoff, K.-U. Bletzinger, W.A. Wall, E. Ramm, Models and finite elements for thinwalled structures, in Encyclopedia of Computational Mechanics, chapter 3 (John Wiley & Sons, Ltd, Hoboken, 2004) 11. E. Cohen, T. Martin, R.M. Kirby, T. Lyche, R.F. Riesenfeld, Analysis-aware modeling: understanding quality considerations in modeling for isogeometric analysis. Comput. Methods Appl. Mech. Eng. 199(5–8), 334–356 (2010) 12. A. Collin, G. Sangalli, T. Takacs, Analysis-suitable G1 multi-patch parametrizations for C1 isogeometric spaces. Comput. Aided Geom. Des. 47, 93–113 (2016) 13. M.A. Crisfield, An arc-length method including line searches and accelerations. Int. J. Numer. Methods Eng. 19(9), 1269–1289 (1983) 14. C. de Boor, Package for calculating with B-splines. SIAM J. Numer. Anal. 14(3), 441–472 (32 pages) (1977). Published By: Society for Industrial and Applied Mathematics 15. C. De Boor, A Practical Guide to Splines. Applied Mathematical Sciences, 1 edn. (Springer, New York, 1978) 16. J. Deng, F. Chen, X. Li, C. Hu, W. Tong, Z. Yang, Y. Feng, Polynomial splines over hierarchical T-meshes. Graph. Models 70(4), 76–86 (2008) 17. T. Dokken, T. Lyche, K.F. Pettersen, Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Des. 30(3), 331–356 (2013) 18. P. Duren, W. Hengartner, Harmonic mappings of multiply connected domains. Pac. J. Math. 180(2), 201–220 (1997) 19. A. Edward, H. Love, G.H. Darwin, XVI. The small free vibrations and deformation of a thin elastic shell. Philos. Trans. R. Soc. London (A) 179, 491–546 (1997) 20. P.L. Fackler, Algorithm 993: efficient computation with Kronecker products. ACM Trans. Math. Softw. 45(2), 1–9 (2019) 21. A. Falini, J. Špeh, B. Jüttler, Planar domain parameterization with THB-splines. Comput. Aided Geom. Des. 35, 95–108 (2015) 22. A. Farahat, B. Jüttler, M. Kapl, T. Takacs, Isogeometric analysis with C1-smooth functions over multi-patch surfaces. Comput. Methods Appl. Mech. Eng. 403, 115706 (2023) 23. A. Farahat, H.M. Verhelst, J. Kiendl, M. Kapl, Isogeometric analysis for multi-patch structured Kirchhoff–Love shells. Comput. Methods Appl. Mech. Eng. 411, 116060 (2023) 24. G. Farin, D. Hansford, Discrete Coons patches. Comput. Aided Geom. Des. 16(7), 691–700 (1999) 25. D.R. Forsey, R.H. Bartels, Hierarchical B-spline refinement. ACM SIGGRAPH Comput. Graph. 22(4), 205–212 (1988) 26. X.-M. Fu, Y. Liu, B.-N. Guo, Computing locally injective mappings by advanced MIPS. ACM Trans. Graph. 34(4), 1–12 (2015) 27. V.A. Garanzha, I.E. Kaporin, Regularization of the barrier variational method. Comput. Math. Math. Phys. 39(9), 1426–1440 (1999) 28. V. Garanzha, I. Kaporin, L. Kudryavtseva, F. Protais, N. Ray, D. Sokolov, Foldover-free maps in 50 lines of code. ACM Trans. Graph. 40(4), 1–16 (2021)

366

Y. Ji et al.

29. C. Giannelli, B. Jüttler, H. Speleers, THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29(7), 485–498 (2012) 30. C. Giannelli, B. Jüttler, S.K. Kleiss, A. Mantzaflaris, B. Simeon, J. Špeh, THB-splines: an effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis. Comput. Methods Appl. Mech. Eng. 299, 337–365 (2016) 31. J. Gravesen, A. Evgrafov, D.-M. Nguyen, P. Nørtoft, Planar parametrization in isogeometric analysis, in Mathematical Methods for Curves and Surfaces: 8th International Conference, MMCS 2012, Oslo, June 28–July 3, 2012, Revised Selected Papers 8 (Springer, Berlin, 2014), pp. 189–212 32. Ch. Heinrich, B. Simeon, St. Boschert, A finite volume method on NURBS geometries and its application in isogeometric fluid–structure interaction. Math. Comput. Simul. 82(9), 1645– 1666 (2012) 33. A.J. Herrema, E.L. Johnson, D. Proserpio, M.C.H. Wu, J. Kiendl, M.-C. Hsu, Penalty coupling of non-matching isogeometric Kirchhoff–Love shell patches with application to composite wind turbine blades. Comput. Methods Appl. Mech. Eng. 346, 810–840 (2019) 34. J.P. Hinz, PDE-Based Parameterization Techniques for Isogeometric Analysis Applications. PhD thesis, Delft University of Technology (2020) 35. J. Hinz, M. Möller, C. Vuik, Elliptic grid generation techniques in the framework of isogeometric analysis applications. Comput. Aided Geom. Des. 65, 48–75 (2018) 36. J. Hinz, M. Möller, C. Vuik, Spline-based parameterization techniques for twin-screw machine geometries, in IOP Conference Series: Materials Science and Engineering, vol. 425 (IOP Publishing, Bristol, 2018), p. 012030 37. K. Hormann, G. Greiner, MIPS: an efficient global parametrization method. Technical report, Erlangen-Nürnberg University (Germany) Computer Graphics Group (2000) 38. T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39), 4135–4195 (2005) 39. K.C. Hui, Y.-B. Wu, Feature-based decomposition of trimmed surface. Comput. Aided Des. 37(8), 859–867 (2005). CAD ’04 Special Issue: Modelling and Geometry Representations for CAD 40. Y. Ji, Y.-Y. Yu, M.-Y. Wang, C.-G. Zhu, Constructing high-quality planar NURBS parameterization for isogeometric analysis by adjustment control points and weights. J. Comput. Appl. Math. 396, 113615 (2021) 41. Y. Ji, M.-Y. Wang, M.-D. Pan, Y. Zhang, C.-G. Zhu, Penalty function-based volumetric parameterization method for isogeometric analysis. Comput. Aided Geom. Des. 94, 102081 (2022) 42. B. Jüttler, U. Langer, A. Mantzaflaris, S.E. Moore, W. Zulehner, Geometry + simulation modules: implementing isogeometric analysis. PAMM 14(1), 961–962 (2014) 43. H. Kang, F. Chen, J. Deng, Modified T-splines. Comput. Aided Geom. Des. 30(9), 827–843 (2013) 44. J. Kiendl, Isogeometric Analysis and Shape Optimal Design of Shell Structures. PhD thesis, Technische Universität München (2011) 45. J. Kiendl, K.-U. Bletzinger, J. Linhard, R. Wüchner, Isogeometric shell analysis with Kirchhoff–Love elements. Comput. Methods Appl. Mech. Eng. 198(49–52), 3902–3914 (2009) 46. J. Kiendl, M.-C. Hsu, M.C.H. Wu, A. Reali, Isogeometric Kirchhoff–Love shell formulations for general hyperelastic materials. Comput. Methods Appl. Mech. Eng. 291, 280–303 (2015) 47. H. Kneser, Lösung der Aufgabe 41. Jber. Deutsch. Math.-Verein. 35, 123–124 (1926) 48. W.F. Lam, C.T. Morley, Arc-length method for passing limit points in structural calculation. J. Struct. Eng. 118(1), 169–185 (1992) 49. X. Li, M.A. Scott, Analysis-suitable T-splines: characterization, refineability, and approximation. Math. Models Methods Appl. Sci. 24(06), 1141–1164 (2014) 50. X. Li, T.W. Sederberg, S-splines: a simple surface solution for IGA and CAD. Comput. Methods Appl. Mech. Eng. 350, 664–678 (2019)

5 Design Through Analysis

367

51. X. Li, J. Zhang, AS++ T-splines: linear independence and approximation. Comput. Methods Appl. Mech. Eng. 333, 462–474 (2018) 52. L. Liu, Y.J. Zhang, X. Wei, Weighted T-splines with application in reparameterizing trimmed NURBS surfaces. Comput. Methods Appl. Mech. Eng. 295, 108–126 (2015) 53. H. Liu, Y. Yang, Y. Liu, X.-M. Fu, Simultaneous interior and boundary optimization of volumetric domain parameterizations for IGA. Comput. Aided Geom. Des. 79, 101853 (2020) 54. M. Marsala, A. Mantzaflaris, B. Mourrain, G1 – Smooth biquintic approximation of CatmullClark subdivision surfaces. Comput. Aided Geom. Des. 99, 102158 (2022) 55. T. Martin, E. Cohen, R.M. Kirby, Volumetric parameterization and trivariate B-spline fitting using harmonic functions. Comput. Aided Geom. Des. 26(6), 648–664 (2009) 56. B. Marussig, J. Zechner, G. Beer, T.-P. Fries, Stable isogeometric analysis of trimmed geometries. Comput. Methods Appl. Mech. Eng. 316, 497–521 (2017). Special Issue on Isogeometric Analysis: Progress and Challenges 57. X. Meng, G. Hu, A NURBS-enhanced finite volume solver for steady Euler equations. J. Comput. Phys. 359, 77–92 (2018) 58. X. Meng, Y. Gu, G. Hu, A fourth-order unstructured NURBS-enhanced finite volume WENO scheme for steady Euler equations in curved geometries. Commun. Appl. Math. Comput. 5(1), 315–342 (2021) 59. M. Möller, J. Hinz, Isogeometric analysis framework for the numerical simulation of rotary screw machines. I. general concept and early applications. IOP Conf. Ser. Mat. Sci. Eng. 425(1), 012032 (2018) 60. T. Lyche, K. Mørken, Spline Methods. Lecture notes from the Department of Mathematics, University of Oslo (2018). https://www.uio.no/studier/emner/matnat/math/MAT4170/v18/ pensumliste/splinebook-2018.pdf 61. T. Nguyen, B. Jüttler, Parameterization of contractible domains using sequences of harmonic maps, in International Conference on Curves and Surfaces (Springer, Berlin, 2010), pp. 501– 514 62. X. Nian, F.-L. Chen, Planar domain parameterization for isogeometric analysis based on Teichmüller mapping. Comput. Methods Appl. Mech. Eng. 311, 41–55 (2016) 63. M.-D. Pan, F.-L. Chen, Low-rank parameterization of volumetric domains for isogeometric analysis. Comput. Aided Des. 114, 82–90 (2019) 64. M.-D. Pan, F.-L. Chen, W.-H. Tong, Low-rank parameterization of planar domains for isogeometric analysis. Comput. Aided Geom. Des. 63, 1–16 (2018) 65. M.-D. Pan, F.-L. Chen, W.-H. Tong, Volumetric spline parameterization for isogeometric analysis. Comput. Methods Appl. Mech. Eng. 359, 112769 (2020) 66. L. Piegl, W. Tiller, The NURBS Book (Springer, Berlin, 1995) 67. E. Pilgerstorfer, B. Jüttler, Bounding the influence of domain parameterization and knot spacing on numerical stability in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 268, 589–613 (2014) 68. C.G. Provatidis, Precursors of Isogeometric Analysis (Springer International Publishing, Berlin, 2019) 69. T. Rado, Aufgabe 41. Jber. Deutsch. Math.-Verein. 35, 49 (1926) 70. U. Reif, A refineable space of smooth spline surfaces of arbitrary topological genus. J. Approximation Theory 90(2), 174–199 (1997) 71. I.J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions. Q. Appl. Math. 4, 45–99 and 112–141 (1946) 72. T.W. Sederberg, J. Zheng, A. Bakenov, A. Nasri, T-splines and T-NURCCs. ACM Trans. Graph. 22(3), 477–484 (2003) 73. R. Sevilla, S. Fernández-Méndez, A. Huerta, NURBS-enhanced finite element method (NEFEM). Int. J. Numer. Methods Eng. 76(1), 56–83 (2008) 74. R. Sevilla, S. Fernández-Méndez, A. Huerta, NURBS-enhanced finite element method (NEFEM). Arch. Comput. Methods Eng. 18(4), 441–484 (2011) 75. A. Shamanskiy, M.H. Gfrerer, J. Hinz, B. Simeon, Isogeometric parametrization inspired by large elastic deformation. Comput. Methods Appl. Mech. Eng. 363, 112920 (2020)

368

Y. Ji et al.

76. J.-P. Su, X.-M. Fu, L.-G. Liu, Practical foldover-free volumetric mapping construction, in Computer Graphics Forum, vol. 38 (Wiley Online Library, Hoboken, 2019), pp. 287–297 77. T. Takacs, D. Toshniwal, Almost-C1 splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems. Comput. Methods Appl. Mech. Eng. 403, 115640 (2023) 78. D.C. Thomas, L. Engvall, S.K. Schmidt, K. Tew, M.A. Scott, U-splines: splines over unstructured meshes. Comput. Methods Appl. Mech. Eng. 401, 115515 (2022) 79. D. Toshniwal, H. Speleers, R.R. Hiemstra, T.J.R. Hughes, Multi-degree smooth polar splines: a framework for geometric modeling and isogeometric analysis. Comput. Methods Appl. Mech. Eng. 316, 1005–1061 (2017) 80. D. Toshniwal, H. Speleers, T.J.R. Hughes, Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: geometric design and isogeometric analysis considerations. Comput. Methods Appl. Mech. Eng. 327, 411–458 (2017) 81. H.M. Verhelst, M. Möller, J.H. Den Besten, F.J. Vermolen, M.L. Kaminski, Equilibrium path analysis including bifurcations with an arc-length method avoiding a priori perturbations, in Proceedings of ENUMATH2019 Conference (2020) 82. H.M. Verhelst, M. Möller, J.H. Den Besten, A. Mantzaflaris, M.L. Kaminski, Stretch-based hyperelastic material formulations for Isogeometric Kirchhoff–Love Shells with application to wrinkling. Comput. Aided Des. 139, 103075 (2021) 83. H.M. Verhelst, A. Mantzaflaris, M. Möller, J.H. Den Besten, Goal-adaptive meshing of isogeometric Kirchhoff-Love shells. arXiv:2307.08356 (2023) 84. X. Wang, W. Ma, Smooth analysis-suitable parameterization based on a weighted and modified Liao functional. Comput. Aided Des. 140, 103079 (2021) 85. X. Wang, X. Qian, An optimization approach for constructing trivariate B-spline solids. Comput. Aided Des. 46, 179–191 (2014) 86. B. Wassermann, S. Kollmannsberger, S. Yin, L. Kudela, E. Rank, Integrating CAD and numerical analysis: ‘dirty geometry’ handling using the finite cell method. Comput. Methods Appl. Mech. Eng. 351, 808–835 (2019) 87. X. Wei, Y. Zhang, L. Liu, T.J.R. Hughes, Truncated T-splines: fundamentals and methods. Comput. Methods Appl. Mech. Eng. 316, 349–372 (2017) 88. J. Weingarten, Über eine Klasse auf einander abwickelbarer Flächen. J. Reinen Angew. Math. 1861(59), 382–393 (1861) 89. P. Weinmüller, T. Takacs, Construction of approximate C1 bases for isogeometric analysis on two-patch domains. Comput. Methods Appl. Mech. Eng. 385, 114017 (2021) 90. P. Weinmüller, T. Takacs, An approximate C1 multi-patch space for isogeometric analysis with a comparison to Nitsche’s method. Comput. Methods Appl. Mech. Eng. 401(Part B), 115592 (2022). ISSN 0045–7825. https://doi.org/10.1016/j.cma.2022.115592 91. G. Xu, B. Mourrain, R. Duvigneau, A. Galligo, Parameterization of computational domain in isogeometric analysis: methods and comparison. Comput. Methods Appl. Mech. Eng. 200(23– 24), 2021–2031 (2011) 92. G. Xu, B. Mourrain, R. Duvigneau, A. Galligo, Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method. J. Comput. Phys. 252, 275–289 (2013) 93. G. Xu, B. Mourrain, R. Duvigneau, A. Galligo, Optimal analysis-aware parameterization of computational domain in 3D isogeometric analysis. Comput. Aided Des. 45(4), 812–821 (2013) 94. G. Xu, T.-H. Kwok, C.C.L. Wang, Isogeometric computation reuse method for complex objects with topology-consistent volumetric parameterization. Comput. Aided Des. 91, 1–13 (2017) 95. L. Zhang, A. Gerstenberger, X. Wang, W.K. Liu, Immersed finite element method. Comput. Methods Appl. Mech. Eng. 193(21), 2051–2067 (2004). Flow Simulation and Modeling 96. Y. Zheng, F.-L. Chen, Volumetric parameterization with truncated hierarchical B-splines for isogeometric analysis. Comput. Methods Appl. Mech. Eng. 401, 115662 (2022) 97. Z. Zhou, D.W. Murray, An incremental solution technique for unstable equilibrium paths of shell structures. Comput. Struct. 55(5), 749–759 (1995)