Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents [1 ed.] 3031296699, 9783031296697

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Lecture Notes in Mathematics  2329

Alex Kaltenbach

Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents

Lecture Notes in Mathematics Volume 2329

Editors-in-Chief Jean-Michel Morel, Ecole Normale Supérieure Paris-Saclay, Paris, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Mark Policott, Mathematics Institute, University of Warwick, Coventry, UK Sylvia Serfaty, NYU Courant, New York, NY, USA László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig, Germany Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

This series reports on new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes: 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research. Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is sometimes more important than its form, which may be preliminary or tentative. Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews, and zbMATH.

Alex Kaltenbach

Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents

Alex Kaltenbach Department of Applied Mathematics University of Freiburg Freiburg, Germany

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-031-29669-7 ISBN 978-3-031-29670-3 (eBook) https://doi.org/10.1007/978-3-031-29670-3 Mathematics Subject Classification: 46E35, 76A05, 35K55, 47H05 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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 Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

In writing this book, I had two different target readerships in mind: On the one hand, this book is intended to be accessible to researchers or graduate students interested in studying the analysis of unsteady problems in variable exponent spaces deploying parabolic pseudo-monotonicity techniques. On the other hand, this book is also intended to serve as a useful reference for mathematicians already active in the area. For both audiences, a coherent treatment of the material is provided. The book includes a concise introduction of variable Lebesgue and Sobolev spaces as well as of Bochner–Lebesgue and Bochner–Sobolev spaces, which provides all definitions and results necessary to follow this book. However, for a more in-depth introduction into the general theory of variable Lebesgue and Sobolev spaces, I recommend the textbooks [42, 49] and the contribution [103], which also served as main references of this book. References providing a comprehensive introduction into the general theory of Bochner–Lebesgue and Bochner–Sobolev spaces are the textbooks [116, 149, 155, 167, 168]. The latter also provide an introduction into the general theory of pseudo-monotone operators. Apart from this shorter introductory chapter, all definitions and findings are built up gradually in the course of the book, making the content accessible to researchers and graduate students less familiar with these new concepts. At this point, I would like to mention that the present book is based on my doctoral thesis (University of Freiburg, 2021). More precisely, up to minor corrections of misprints and small changes for better understandably of the text, the content of this book is the same as in the thesis [94]. Writing such a book would not have been possible without various sources of support: First and foremost, I would like to express my graduate to my advisor Prof. Dr. Micheal R˚užiˇcka for his guidance, his support, for creating the perfect working environment, and especially for showing me the beauty of both non-linear functional analysis and fluid mechanics. I am also deeply indebted to my friends and colleagues Luca Courte, Pablo Alexei Gazca-Orozco, Mirjam Hoferichter, Julius Jeßberger, Susanne Knies, Martin

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Kˇrepela, Jakob Rotter, Nico Weber, and Marius Zeinhofer for the nice atmosphere, help, and fun we had during our time together in Freiburg. I also want to express my deep gratitude to the great love of my life, Larissa, for always backing me up. Above all, I want to express my deep graduate to my parents, Rita and Berhard Kaltenbach, without whose endless love, support, and steady encouragement I would never have got this far, and this book presumably would never have come about. Therefore, I want to dedicate this work entirely – in deep gratitude – to my parents Rita and Berhard Kaltenbach. Freiburg, Germany January 30, 2023

Alex Kaltenbach

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fluid Mechanics and Pseudo-Monotone Operator Theory: A Never-Ending Love Affair or Merely an On-Off Relationship?. . . . 1.2 Electro-Rheological Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Aims and Outline of This Manuscript. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Main Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Theory of Pseudo-Monotone Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Variable Exponent Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Classical Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Variable Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Duality in Variable Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Variable Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 The Hardy–Littlewood Maximal Operator and log-Hölder Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Mollification in Lp(·) (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Banach-Valued Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Banach-Valued Classical Function Spaces . . . . . . . . . . . . . . . . . . . . 2.3.2 Bochner–Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Bochner–Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Advanced Theory of Pseudo-Monotone Operators for Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 7 10 10 13 15 15 16 16 17 20 21 23 25 26 26 27 31 34

Part I Main Part 3

Variable Bochner–Lebesgue Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q,p q,p 3.1 The Spaces X∇ (QT ) and Xε (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q,p 3.2 Duality in Xε (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q,p 3.3 Embedding Theorems for Xε (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q,p 3.4 Smoothing in Xε (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 45 52 57 69 vii

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3.5 3.6 3.7 3.8 3.9 4

5

˚ q,p (QT ) and X ˚ q,p (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 The Spaces X ∇ ε ˚ q,p (QT )∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Generalized Time Derivative in X ε q,p Formula of Integration-by-Parts for Wε (QT ) . . . . . . . . . . . . . . . . . . . . . . 90 Abstract Existence Result for Lipschitz Domains . . . . . . . . . . . . . . . . . . . . . 98 Application to Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Solenoidal Variable Bochner–Lebesgue Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . q,p q,p 4.1 The Spaces V∇ (QT ) and Vε (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q,p 4.2 Duality in Vε (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q,p 4.3 Smoothing in Vε (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Failure of Smoothing via Bogovski˘ı Correction. . . . . . . . . . . . . . . 4.3.2 Smoothing via Transversal Expansion of Lipschitz Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˚q,p (QT ) and V ˚εq,p (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Spaces V ∇ ˚εq,p (QT )∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Generalized Time Derivative in V q,p 4.6 Formula of Integration-by-Parts for Wε,σ (QT ) . . . . . . . . . . . . . . . . . . . . . .

113 114 117 125 125

Existence Theory for Lipschitz Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Bochner Pseudo-Monotonicity, Bochner Condition (M) and Bochner Coercivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Hirano–Landes Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 C 0 -Bochner Pseudo-Monotonicity, C 0 -Bochner Condition (M) and C 0 -Bochner Coercivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Abstract Existence Theorem for Lipschitz Domains and p− ≥ 2 . . . . 5.5 Application to Model Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Unsteady p(·, ·)-Stokes Equations in a Lipschitz Domain with p− ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Unsteady p(·, ·)-Navier–Stokes Equations in a Lipschitz Domain with p− ≥ 3d+2 d+2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177

128 153 157 160

178 189 204 211 217 218 219

Part II Extensions 6

Pressure Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Pressure Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Application to Model Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Applicability of Parabolic L∞ - and Lipschitz Truncation . . . . . . . . . . . . 6.3.1 Parabolic L∞ - and Lipschitz Truncation . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Parabolic Solenoidal Lipschitz Truncation . . . . . . . . . . . . . . . . . . . .

225 225 237 240 240 241

7

Existence Theory for Irregular Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Bochner–Sobolev Condition (M). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 L1 -Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Finite Radon Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Minty–Trick Like Argument for L1 -Monotone Operators. . . .

245 246 253 255 257

Contents

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7.3 Anisotropic Variable Exponent Bochner–Lebesgue Spaces . . . . . . . . . . q,p,s 7.3.1 The Space Xε,div (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q,p,s 7.3.2 Duality in Xε,div (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q,p,s 7.3.3 Smoothing in Xε,div (QT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q,p,s 7.3.4 Generalized Time Derivative in Xε,div (QT )∗ and q,p,s Formula of Integration-by-Parts for Wε,div (QT ) . . . . . . . . . . . . 7.4 First Parabolic Compensated Compactness Principle . . . . . . . . . . . . . . . . . 7.5 Abstract Existence Result for Irregular Domains and p− ≥ 2. . . . . . . . 7.6 Application to Model Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Unsteady p(·, ·)-Stokes Equations in an Irregular Domain with p − ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Unsteady p(·, ·)-Navier–Stokes Equations in an Irregular Domain with p− > 3d+2 d+2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

260 261 262 263

Existence Theory for .p− < 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Second Parabolic Compensated Compactness Principle . . . . . . . . . . . . . . 8.2 Unsteady p(·, ·)-Navier–Stokes Equations in an Irregular 3d ................................................ Domain with p − > d+2 8.3 Unsteady p(·, ·)-Stokes Equations in an Irregular Domain 2d .......................................................... with p− > d+2

297 301

329

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Point-Wise Poincaré Inequality for the Symmetric Gradient . . . . . . . . . 9.2 Generalized Lebesgue Differentiation Theorem . . . . . . . . . . . . . . . . . . . . . . . 9.3 Portemanteau Theorem for Weak-* Convergence . . . . . . . . . . . . . . . . . . . . .

331 331 340 344

268 274 289 293 293 295

323

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Notation-Related Comments

The necessary notation to understand this work will be gradually built up. Apart from that, the following notation is employed:  Basic notation: By .N, we denote the natural numbers. By .R, .R≥0 , .R>0 , .R≤0 , and .R0 cofinal, if for each .r ∈ R>0 , there exists a .λ ∈  such that .λ ≥ r. . Constants: In calculations occurring constants are predominantly positive and denoted by .c, ck , c(k), . . . , where the subscript .· k or the call .·(k) are intended to indicate a dependence of the constant c on the quantity k. Moreover, the actual value represented by a constant may change from line to line, while the notation of the constant remains unchanged. n . Topological notation: For a set .S ⊆ R , .n ∈ N, we denote by .int(S) the interior, by .S the closure, by .∂S the (topological) boundary, and by S  = Rn \ S the complement of S. We denote by .Bεn (x0 ) := {x ∈ Rn | |x − x0 | < ε}, .n ∈ N, the n-dimensional Euclidean ball with radius .ε > 0 and center .x0 ∈ Rn and by n−1 := .S {x ∈ Rn | |x| = 1} the n-dimensional Euclidean unit sphere. The sum of sets .S1 , S2 ⊆ Rn , .n ∈ N, is defined by .S1 + S2 := {s1 + s2 | .s1 ∈ S1 , s2 ∈ S2 }. In particular, for a set .S ⊆ Rn , .n ∈ N, and .ε > 0, it holds .S + Bεn (0) = {x ∈ Rn | dist(x, S) < ε}, where .dist(x, S) := infs∈S |x − s|, that is an approximation of S from outside. On the other hand, the set .Sε := {s ∈ S | dist(s, ∂S) > ε}, where n .ε > 0, defines an approximation of S from inside. A set .S1 ⊆ R , .n ∈ N, is n compactly contained in another set .S2 ⊆ R , written .S1 ⊂⊂ S2 , if .S1 ⊆ S2 and if .S1 is compact. Furthermore, the convex hull of two sets .S1 , S2 ⊆ Rn , .n ∈ N, is defined by .conv{S1 , S2 } := {λs1 + (1 − λ)s2 | s1 ∈ S1 , s2 ∈ S2 , λ ∈ [0, 1]}. . General notation from functional analysis: For a Banach space X with the norm . · X , we denote by .X∗ its continuous dual space equipped with the dual norm . x ∗ X∗ := sup x X ≤1 x ∗ , x X , .x ∗ ∈ X∗ , where .·, · X denotes the duality .

xi

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Notation-Related Comments

pairing. All occurring Banach spaces are assumed to be real. For Banach spaces X, Y , we denote the Cartesian product by .X × Y , which forms a Banach space, if equipped with the norm . (x, y) X×Y := x X + y Y for .(x, y) ∈ X × Y . Further, we denote for Banach spaces .X, Y by .D(A) the domain of definition of an operator .A : D(A) ⊆ X → Y , by .R(A) := {Ax | x ∈ D(A)} its range, and by .G(A) := {(x, Ax) ∈ X × Y | x ∈ D(A)} its graph. For a linear, bounded operator .A : X → Y , the adjoint operator .A∗ : Y ∗ → X∗ , which is again linear and bounded, is defined via the relation .A∗ y ∗ , x X := y ∗ , Ax Y for all .x ∈ X and .y ∗ ∈ Y ∗ . We call a linear and bounded operator .A : X → Y an embedding, if it is additionally injective. For Banach spaces .X, Y such that .X ⊆ Y , the identity mapping .idX : X → Y is defined by .idX (x) := x in Y for all .x ∈ X. If .idX : X → Y is an embedding, then we often write .X → Y . If .idX : X → Y is a compact embedding, i.e., an embedding that maps bounded sets in X into relatively compact sets in Y , then we write .X →→ Y . The arrows .→, ., and ∗ . denote strong, weak, and weak-* convergence in a Banach space X and its dual space .X∗ , respectively. . Vectors and vector-valued functions: Vectors and vector-valued functions will be indicated by bold lower-case letters, i.e., by .a ∈ Rn , .n ∈ N, and n m .f : ⊆ R → R , .m ∈ N, respectively. An exception constitute points .x ∈ ⊆ Rn , .n ∈ N, if . is the domain of definition of a function. We employ for two n

vectors .a := (a1 , . . . , an ) , b := (b 1n, . . . , bn ) ∈ R , .n ∈ N, the standard Euclidean scalar product .a · b := a b , and the corresponding induced i=1 i i √ Euclidean norm .|a| := a · a. Further, for two vectors .a := (a1 , . . . , an ) ∈ Rm ,

∈ Rn , .n ∈ N, the outer product is defined by .m ∈ N, and .b := (b1 , . . . , bn ) m×n . For .n, i ∈ N, .i ≤ n, we denote by .a ⊗ b := (ai bj )i=1,...,m;j =1,...,n ∈ R

∈ Rn , where .δ := 1 if .i = j and .δ := 0 if .i = j , .ei := (δi1 , . . . , δin ) ij ij the i-th unit vector. . Tensors and tensor-valued functions: Tensors and tensor-valued functions will be indicated by bold capital letters, i.e., by .A ∈ Rm×n , .m, n ∈ N, and .F : ⊆ Rl → Rm×n , .l ∈ N. For a tensor .A := (Aij )i=1,...,m;j =1,...,n ∈ Rm×n , .m, n ∈ N, we denote by .A := (Aj i )i=1,...,n;j =1,...,m ∈ Rn×m its transpose tensor. For a tensor-valued function .F : ⊆ Rl → Rm×n , .l, m, n ∈ N, the transpose function l n×m is defined point-wise. In the particular case .m = n ∈ N,

.F : ⊆ R → R n×n the we employ for two tensors .A := (Aij )i,j =1,...,n , B := (Bij )i,j =1,...,n ∈ R Frobenius scalar product .A : B := tr(A B) = tr(B A) = ni=1 nj =1 Aij Bij ,  n×n , where .tr(C) := ni=1 Cii is the trace of a tensor .C := (Cij )i,j =1,...,n ∈ R √ and the induced Frobenius norm .|A| := A : A. Moreover, we denote by n×n n×n | A = A} the space of symmetric tensors. We denote .Msym := {A ∈ R for a tensor .A ∈ Rn×n , .n ∈ N, by .Asym := [A]sym := 12 (A + A ) ∈ Mn×n sym its symmetric part and by .det(A) ∈ R its determinant. A tensor .A ∈ Rn×n , .n ∈ N, is called invertible, if .det(A) = 0. In this case, there exists a tensor −1 ∈ Rn×n , called the inverse of .A, such that .A−1 A = AA−1 = I in .Rn×n , .A n where .In := (δij )i,j =1,...,n ∈ Mn×n sym denotes the identity matrix. .

Notation-Related Comments

xiii

 Spatial differential operators: In the notation, we do not differentiate between classical, weak, or distributional differential operators. For a scalar function n → R, .n ∈ N, we denote by .∂ f := ∂ f : ⊆ Rn → R, .f : ⊆ R i xi .i ∈ {1, . . . , n}, the i-th partial derivative. Partial derivatives of vector-valued and tensor-valued functions are defined component-wise. For a vector-valued function .f := (f1 , . . . , fm ) : ⊆ Rn → Rm , .m, n ∈ N, we call .Df := (∂j fi )i=1,...,m;j =1,...,n : ⊆ Rn → Rm×n its Jacobian and .∇f := Df = (∂i fj )i=1,...,n;j =1,...,m : ⊆ Rn → Rn×m its gradient. In the particular case 1 n n×n

.m = n ∈ N, we denote by .ε(f) := 2 (∇f + ∇f ) : ⊆ R → Msym its n symmetric gradient and by .div(f) := tr(∇f) = tr(Df) = tr(ε(f)) : ⊆ R → R its divergence. For a tensor-valued function .F : ⊆ Rn → Rn×n , .n ∈ N, we define the row-wise divergence .div(F) : ⊆ Rn → Rn for every .i = 1, . . . , n by .div(F)i := div(Fi ) : ⊆ Rn → R, where .Fi = (Fi1 , . . . , Fin ) : ⊆ Rn → Rn is the i-th row of .F. This notion of a divergence must not be confused with the column-wise divergence .∇ · F : ⊆ Rn → Rn×n , .n ∈ N. Nevertheless, whenever we consider symmetric tensor-valued mappings, i.e., .F : ⊆ Rn → Mn×n sym , then these operators agree, i.e., we have that .∇ · F = div(F) in . . . Time-dependent functions: Time-dependent functions are indicated by bold italic letters, i.e., .x : I → X, where X is an arbitrary set. Furthermore, if a function is defined on a time-space cylinder .QT := I × , where .I ⊆ R denotes an interval and . ⊆ Rn , .n ∈ N, an arbitrary set, i.e., .x : QT → Rm , .m ∈ N, then we frequently identify this function via the identification .x(t) := x(t, ·) (a.e.) in . for .t ∈ I , as a function defined on the time interval I , which takes its value in appropriate, potentially on time depending, function spaces .X(t), .t ∈ I , defined on . , i.e., we write .x(t) ∈ X(t) for (almost) every .t ∈ I . . Solenoidal vector fields: Throughout the thesis, we predominantly follow the convention of denoting solenoidal vector fields by the letters .u, v, w, and similarly time-dependent solenoidal vector fields by .u, v, w, while we employ the notation .x, y, z and .x, y, z, respectively, to indicate that the vector fields under consideration are not necessarily solenoidal. n . Miscellaneous: For a given set .S ⊆ R , .n ∈ N, the characteristic function n .χS : R → {0, 1} is defined by .χS (x) := 1 if .x ∈ S and .χS (x) := 0 if x ∈ S  . For a scalar function .f : S ⊆ Rn → R, .n ∈ N, we define its support by .supp(f ) .:= {x ∈ S | f (x) = 0}. The support of a vector-valued := (f1 , . . . , fm ) : S ⊆ Rn → Rm , .m, n ∈ N, is defined by function .f  m .supp(f) := function .F : ⊆ i=1 supp(fi ), and the support of a tensor-valued ffl l m×n R →R , .l, m, n ∈ N, is defined accordingly. By . S · · · dx, we denote the mean value integral over a (Lebesgue) measurable set .S ⊆ Rn , .n ∈ N. .

Chapter 1

Introduction

Physical models involving variable growth conditions are of interest in many engineering applications, including, e.g., smart materials like electro-rheological fluids, cf. [142, 147], micro-polar electro-rheological fluids, cf. [57, 65], synovial fluids, cf. [88, 107], and thermo-rheological fluids, cf. [8, 172], or image processing applications, cf. [1, 21, 38, 115]. Most of these models have a related structure, consisting of the generalized Navier–Stokes equations coupled to partial differential equations that describe additional physical quantities, e.g., an electric field, a temperature field, or a concentration. Here, the coupling, inter alia, takes place through a power-law index p which depends on the physical quantities described by the additional equations and is, therefore, time- and space-dependent. These models are also similar from a purely mathematical point of view: if the additional coupled equations and the described physical quantities are understood, then the remaining mathematical challenges in all models are the same: all models treat the generalized Navier–Stokes equations with a time- and space-dependent power-law index p, the so-called unsteady .p(t, x)-Navier–Stokes equations, which will be introduced in the following section (cf. Sect. 1.1). For this reason, understanding the .p(t, x)-Navier– Stokes equations is the first step that must be taken in order to understand all of the models mentioned above. The method of choice of this book to approach the .p(t, x)-Navier–Stokes equations is the theory of pseudo-monotone operators. Over the last decades the theory of pseudo-monotone operators proved itself as a reliable tool in the verification of the weak solvability of the unsteady pNavier–Stokes equations, i.e., if the power-law index p is constant rather than timeand space-dependent. On the contrary, the challenges arising in the analysis of the unsteady p-Navier–Stokes equations numerous times gave rise for new developments in theory of pseudo-monotone operators as presented in the subsequent section (cf. Sect. 1.1).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaltenbach, Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents, Lecture Notes in Mathematics 2329, https://doi.org/10.1007/978-3-031-29670-3_1

1

2

1 Introduction

Given this, the first part of this introductory chapter is intended to give a brief historical overview of the interplay between fluid mechanics and the theory of pseudo-monotone operators until the unsteady .p(t, x)-Navier–Stokes equations, where this interplay seemingly came to an end. As a real world application, this manuscript points out the model describing the unsteady motion of an incompressible electro-rheological fluid, which is briefly introduced and explained in the second part of this introductory chapter (cf. Sect. 1.2). However, it should be emphasized that intended scope of application of the methods developed in this book is not strictly limited to this particular model, but should be taken as a blue print for the analysis of related unsteady problems like the models mentioned above. In the third and last part of this introductory chapter (cf. Sect. 1.3), the content and methods used in this book are outlined.

1.1 Fluid Mechanics and Pseudo-Monotone Operator Theory: A Never-Ending Love Affair or Merely an On-Off Relationship? It was G.J. Minty who in 1960, i.e., more than 60 years ago, introduced the general concept of monotonicity for mappings defined on Banach spaces with values in their continuous dual spaces. While G.J. Minty originally initiated the study of the concept of monotonicity to aid in the abstract study of electrical networks [124] and mainly provided the first systematic results for monotone operators between Hilbert spaces and their continuous dual spaces (cf. [125, 126]), it was F.E. Browder who extended Minty’s approach to general reflexive Banach spaces (cf. [33, 36]) for the treatment of fixed point problems, for the calculus of variations, and –most notably– for non-linear elliptic and parabolic partial differential equations. This progress was accompanied by further seminal contributions by M.I. Vishik [162], J. Leray and J.–L. Lions [114], H. Brézis [26, 27, 29, 32], P. Hartmann and G. Stampacchia [83], R.I. Kaˇcurovski˘ı [97–99], M.M. Va˘ınberg [160], as well as G.J. Minty himself [127– 129], to name just a few among them. Since there exist several non-linear partial differential equations involving nonmonotone operators to which both Minty’s and Browder’s abstract theory is not applicable, H. Brézis introduced in [28] a large class of operators, which he classified as the class of pseudo-monotone operators, e.g., operators that can be decomposed into monotone and compact parts. Since its conception in 1968, the notion of pseudo-monotonicity has occurred prominently in the seminal contributions of, e.g., F.E. Browder [34–36], H. Brézis [30], R. Landes [109] and V. Mustonen [110, 111], E. Zeidler [168], and N. Kenmochi [100, 101]. The heart of the theory of pseudo-monotone operators –although nowadays there are stronger and more general results available, such as, e.g., Proposition 3.27– forms the main theorem on pseudo-monotone operators, tracing back to H. Brézis (cf. [28, Cor. 14]).

1.1 Fluid Mechanics and Pseudo-Monotone Operator Theory: A Never-. . .

3

Theorem 1.1 (Main Theorem on Pseudo-Monotone Operators) Let X be a reflexive Banach space and .A : X → X∗ bounded,1 coercive, i.e., .

Ax, xX = ∞, xX →∞ xX lim

and pseudo-monotone, i.e., from xn  x

.

in X

(n → ∞) ,

lim sup Axn , xn − xX ≤ 0 , n→∞

for every .y ∈ X, it follows that Ax, x − yX ≤ lim inf Axn , xn − yX .

.

n→∞

Then, .A : X → X∗ is surjective. Since its introduction in 1968, the theory of pseudo-monotone operators has been in a romantic long-term relationship with fluid mechanics, in the sense that several problems arising in fluid mechanics seem to exist solely to be solved by methods from the theory of pseudo-monotone operators, although admittedly the inverse is usually true. In truth, the theory of pseudo-monotone operators is subject to an ongoing development in order to cope with the diversity of problems arising in fluid mechanics. One particular application of the main theorem on pseudo-monotone operators, which was not obtainable by the pure theory of monotone operators, is the weak solvability of the steady p-Navier–Stokes equations, that describe the steady motion of an incompressible non-Newtonian fluid of constant density, which we, for simplicity, set equal to 1, in a bounded domain . ⊆ Rd , .d ≥ 2. More precisely, the steady p-Navier–Stokes equations search for a velocity vector field .u :  → Rd and a scalar kinematic pressure .π :  → R that solve the system −div(S(·, ε(u))) + div(u ⊗ u) + ∇π = f − div(F) .

div(u) = 0 u=0

in  , in  ,

(1.1)

on ∂ ,

where .f :  → Rd is a given vector field and .F :  → Md×d sym is a given symmetric tensor field, jointly describing external body forces on the fluid, and 1 d×d  .ε(u) := (∇u + ∇u ) :  → Msym is the symmetric gradient. The tensor-valued 2 d×d d×d mapping .S :  × Msym → Msym , commonly referred to as the extra stress tensor, 1 All essential notions concerning the theory of pseudo-monotone operators are briefly summarized

in Sect. 2.1.

4

1 Introduction

is supposed to have a .(p, δ)-structure (cf. (S.1)–(S.4)), where .p ∈ (1, ∞) is the constant so-called power-law index and .δ ≥ 0. To give some simple examples, d×d leaving aside the x-dependency of .S, typical realizations of .S : Md×d sym → Msym are, d×d e.g., for every .A ∈ Msym given by .

S(A) := (δ + |A|)p−2 A

or

S(A) := (δ + |A|2 )

p−2 2

A.

(1.2)

The weak solvability of the steady-state p-Navier–Stokes equations as an application of the main theorem of pseudo-monotone operators for power-law indexes 3d ∞ .p > d+2 can be found, e.g., in [116] or [104, 106]. By deploying the .L - and Lipschitz truncation techniques (cf. [46, 68, 70, 145]), it was even possible to 2d establish the weak solvability of the steady p-Navier–Stokes equations for .p > d+2 . Note that the steady p-Navier–Stokes equations merely represent a simplification of the unsteady p-Navier–Stokes equations. The latter describe the unsteady motion of an incompressible non-Newtonian fluid of constant density, which we, again, for simplicity, set equal to 1, in a bounded time-space cylinder .QT := I × , where d . ⊆ R , .d ≥ 2, is a bounded domain and .I := (0, T ), .T < ∞, a finite time interval. More precisely, the unsteady p-Navier–Stokes equations search for a velocity vector field .u : QT → Rd and a scalar kinematic pressure .π : QT → R that solve the system ∂t u − div(S(·, ·, ε(u))) + div(u ⊗ u) + ∇π = f − div(F ) div(u) = 0 .

u=0 u(0) = u0

in QT , in QT , on I × ∂ ,

(1.3)

in  ,

where the time-dependent vector field .f : QT → Rd and symmetric tensor field d×d .F : QT → Msym again describe external body forces and the extra stress tensor d×d d×d .S : QT ×Msym → Msym is supposed to have a .(p, δ)-structure (cf. (S.1)–(S.4)). Considering the system (1.3) as a prototypical example, numerous contributions, see, e.g., [16, 28, 30, 75, 92, 95, 96, 116, 117, 139, 143, 149, 154, 155, 168], to name just a few, have addressed the question of the extent to which Brézis’ result, i.e., Theorem 1.1, is transferable to the framework of abstract non-linear evolution equations. The common tenor of all these contributions, which was already present in Brézis’ pioneering work, was that the original notion of pseudo-monotonicity as in Theorem 1.1, according to H. Brézis, becomes obsolete in the context of nonlinear evolution equations and, therefore, has to evolve to survive in this field as well. Consequently, it appeared that the theory of pseudo-monotone operators and fluid mechanics entered a state of crisis when reaching the field of non-linear evolution equations. The final salvation came by the connection of pseudo-monotonicity with a generalized notion of a time derivative. This led to a generalized notion of pseudomonotonicity and to the following unsteady generalization of Theorem 1.1, tracing back to J.–L. Lions (cf. [116, p. 319]) and H. Brézis (cf. [30, Thm. 26]).

1.1 Fluid Mechanics and Pseudo-Monotone Operator Theory: A Never-. . .

5

Theorem 1.2 (Unsteady Main Theorem on Pseudo-Monotone Operators) Let .(X, Y, j ) be an evolution triple,2 .I := (0, T ), .T < ∞, .p ∈ (1, ∞), and 1,p,p de .A : We (I, X, X∗ ) ⊆ Lp (I, X) → (Lp (I, X))∗ coercive, . dt -pseudo-monotone, i.e., from xn  x

.

1,p,p

in We

(I, X, X∗ )

(n → ∞) ,

lim supAx n , x n − xLp (I,X) ≤ 0 , n→∞

for every .y ∈ Lp (I, X), it follows that Ax, x − yLp (I,X) ≤ lim inf Ax n , x n − yLp (I,X) ,

.

n→∞

and assume there exists a bounded mapping .ψ : R≥0 × R≥0 → R≥0 and a constant 1,p,p .θ ∈ [0, 1) such that, for every .x ∈ We (I, X, X∗ ), it holds d x   e  Ax(Lp (I,X))∗ ≤ ψ(xLp (I,X) , x c (0)Y ) + θ  .  dt (Lp (I,X))∗

.

1,p,p

Then, for every .x ∗ ∈ (Lp (I, X))∗ and .x0 ∈ Y , there exists .x ∈ We such that de x + Ax = x ∗ . dt x c (0) = x0

(I, X, X∗ )

in (Lp (I, X))∗ , in Y .

In fact, it has been shown, e.g., in [116, pp. 325–327] or [30, Appl. 2.], that the weak solvability of the system (1.3) for power-law indexes .p ≥ 3d+2 d+2 is a straightforward application of Theorem 1.2. Using parabolic variants of the .L∞ and Lipschitz truncation techniques (cf. [48, 164]), it was even possible to establish 2d the weak solvability of the unsteady p-Navier–Stokes equations for .p > d+2 . As a result, the theory of pseudo-monotone operators and fluid mechanics lived in harmony again. Then, however, it was discovered in laboratory experiments that in a number of non-Newtonian fluid flow problems, the power-law index p is not a fixed constant but is, rather, variable-dependent, i.e., more realistic realizations of the extra stress tensor in (1.3) than (1.2) might, e.g., be given by S(t, x, A) := (δ + |A|)p(t,x)−2 A or S(t, x, A) := (δ + |A|2 )

.

2 All

p(t,x)−2 2

notions appearing in this theorem are thoroughly introduced in Sect. 2.3.4.

A,

(1.4)

6

1 Introduction

for .(t, x) ∈ QT and .A ∈ Md×d sym , where .δ ≥ 0 and .p : QT → (1, +∞) denotes an at least measurable power-law index satisfying p− := ess inf(t,x) ∈QT p(t, x) > 1 and .p+ := ess sup(t,x) ∈QT p(t, x) < ∞. The system (1.3) with an extra stress d×d tensor .S : QT × Md×d sym → Msym of the form (1.4) is called the unsteady .p(·, ·)Navier–Stokes equations. The unsteady .p(·, ·)-Navier–Stokes equations describe the unsteady motion of an incompressible electro-rheological fluid with constant density. For more information about the physical background of these fluids, please refer to the following section. From a mathematical point of view, the first crucial question that arises when approaching the unsteady .p(·, ·)-Navier–Stokes equations is the choice of an appropriate mathematical framework. In the steady case, i.e., for the system (1.1) containing an extra stress tensor of the form (1.4) but without a time-dependency, commonly referred to as the steady .p(·)-Navier–Stokes equations, this question has already been answered and a well-developed theory is available (cf. [46, 147, 148]). Here, it turned out that the appropriate mathematical framework is not given by classical Lebesgue and Sobolev spaces, but rather variable Lebesgue and variable Sobolev spaces, and the weak solvability was proved 3d for sufficiently smooth .p :  → (1, +∞) with p− := ess infx∈ p(x) > d+2 by 2d − using pseudo-monotone operator theory in [147] and for .p > d+2 by resorting to a variable exponent generalization of the Lipschitz truncation technique in [46, 89]. For the unsteady .p(·, ·)-Navier–Stokes equations, the theory is not developed to that extent. Formal standard estimation shows that a sufficiently smooth solution d .u : QT → R of (1.3) with (1.4), where we for the moment assume that such a solution exists, has to satisfy the energy estimate  .

 |u(t, x)| dx +

ess sup t∈I

|ε(u)(t, x)|p(t,x) dtdx < ∞ .

2



(1.5)

QT

This energy estimate, more precisely, the second summand in (1.5), precludes the usage of classical Bochner–Lebesgue spaces for the treatment of the unsteady .p(·, ·)-Navier–Stokes equations and thus, in turn, the application of the advanced theory of pseudo-monotone operators for non-linear evolution equations, such as, e.g., Theorem 1.2.3 First results addressing the existence of weak solutions of the unsteady .p(·, ·)-Navier–Stokes equations trace back to M. R˚užiˇcka [146, 147], 3×3 assuming that .S : QT × M3×3 sym → Msym assumes a form similar to (1.4) with a 3+p− variable exponent .p ∈ C 1 (QT ), which satisfies .2 ≤ p− ≤ p+ < min 83 , 2(3−p −) , that . ⊆ R3 has a .C 3,1 -boundary, and that the data .u0 , .f and .F are sufficiently smooth, by employing the so-called A-approximation. It was even possible to establish the weak solvability in a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2, for an 3d at least measurable exponent .p : QT → (1, +∞) with . d+2 < p− ≤ p+ < p− d+2 d in [140, 173], using hydro-mechanical parabolic compensated compactness principles. Nevertheless, until now, it has not been possible to attack the unsteady

3A

brief overview of this theory and also of more recent methods can be found in Sect. 2.3.4.

1.2 Electro-Rheological Fluids

7

p(·, ·)-Navier–Stokes equations with the aid of the theory of pseudo-monotone operators and, thus, to get rid of the artificial relation .p+ < p− d+2 d in [140]. This is mainly due to the lack of a suitable mathematical environment allowing the theory of pseudo-monotone operators to further develop. Thus, it may seem that the long-lasting liaison between the theory of pseudo-monotone operators and fluid mechanics is coming to a sad end at the unsteady .p(·, ·)-Navier–Stokes equations. Against this serious background, this work sees its duty in providing relational counseling between these two theories. From a mathematician’s point of view, this means that we need to establish the appropriate mathematical environment that allows the theory of pseudo-monotone operators to open up for further development.

.

1.2 Electro-Rheological Fluids Most commonly, electro-rheological fluids are composed of electrically polarizable particles and a non-polarizable, isolating carrier fluid. When such fluids are exposed to an electro-magnetic field, then the polarized particles in the carrier fluid arrange themselves along the field lines of the electro-magnetic field, i.e., they form chains, see, e.g., Fig. 1.1, and the viscosity of the fluid undergoes a significant change. This behavior is sometimes referred to as the “electro-rheological effect”. Possible examples for electrically polarizable particles include metal oxides, polyurethanes, silica hybrids, and polymers containing metallic ions. These particles are typically about 1–100 µm in size and make up about 30–50% of the total weight of the electro-rheological fluid. Examples for the non-polarizable, isolating carrier

Electro-magnetic field: Off

Electro-magnetic field: On

− −

−

−

− −

+ + + + ++ − − − + − − + − − − − + + + + ++

Electrode

Electrically polarizable particle

Isolating fluid

Fig. 1.1 Diagram of the “electro-rheological effect” under an applied electro-magnetic field

8

1 Introduction Gas tank

Diaphragm

Voltage source

Electrodes

Chasing

Electro-magnetic field in duct (cf. Fig. 1)

F

V

Piston

Electorheological fluid flow

Fig. 1.2 Diagram of an electro-rheological shock absorber, according to [135]

fluid include silicone oil, paraffin, or ether, a halogenated or aromatic hydrocarbon. Here, in the case of an electro-magnetic field of approximately 1–10 .kV/mm, it is possible that the viscosity of modern electro-rheological fluids changes by a factor of about 1000 compared to the case without an applied electro-magnetic field. Such a dramatic change of the viscosity typically occurs on short time scales of about 1–5 .ms. These two observations describe two characteristics of the behavior of electro-rheological fluids: These fluids not only change their characteristic behavior extremely rapidly, but also over a relatively large range of scales. The first observations of electro-rheological fluids were reported already in 1949 by the engineer W. Winslow [163]. He analyzed the suspension of corn starch in oil. Inasmuch as early realizations of electro-rheological fluids had a highly abrasive nature, called for enormous voltage requirements for a visible change in their material properties, and also evinced a high degree of instability of the suspension, some time elapsed before electro-rheological fluids found prolific applications. Nowadays, there exist electro-rheological fluids having both the quality and the potential for an implementation in various areas. One possible field of application is, e.g., shock absorbers for cars (cf. Fig. 1.2). Conventional shock absorbers are subject to a tenfold higher response time than modern shock absorbers that deploy electro-rheological fluids. Further possible applications, e.g., include clutches, force-feedback devices, haptic displays, earthquake-proof constructions, artificial muscles and joints, inkjet printers, as well as rehabilitation and fitness equipment. There exist several possibilities for modeling the physics of electro-rheological fluids. In this work, the model originally proposed by K.R. Rajagopal and M. R˚užiˇcka in [142], and further developed by M. R˚užiˇcka in [147], is studied. This model is derived from the general balance laws of mass, linear momentum, angular momentum, energy, the second law of thermodynamics in the form of the Clausius– Duhem inequality, and Maxwell’s equations in their Minkowskian formulation. Apart from that, the interaction between the electro-magnetic field and the fluid is described on the basis of the “dipole current-loop” model, according to R.A. Grot [79] and Y.H. Pao [138]. The full model for an incompressible electro-rheological fluid in a bounded domain . ⊆ R3 over a finite time interval .I := (0, T ), .T < ∞, searches for an electro-magnetic field .E : QT → R3 , a polarization vector field

1.2 Electro-Rheological Fluids

9

P : QT → R3 , a velocity vector field .u : QT → R3 , and a scalar kinematic pressure .π : QT → R that solve the system .

div(E + P ) = 0

in QT ,

curl(E) = 0

in QT ,

E · n = E0 · n .

ρ∂t u − div(S(·, ·, ε(u))) + ρdiv(u ⊗ u) + ∇π = ρf + [∇E]P div(u) = 0 u=0 u(0) = u0

on I × ∂ , in QT , in QT , on I × ∂ , in  , (1.6)

where .ρ > 0 is the density and .f : QT → R3 a vector field describing external 3×3 body forces. In the system (1.6), the extra stress tensor .S : QT × M3×3 sym → Msym is supposed to assume the form   p(t,x)−1 2 S(t, x, A) = α1 1 + |A|2 − 1 E(t, x) ⊗ E(t, x)

.

   p(t,x)−2 2 + α2 + α3 |E(t, x)|2 1 + |A|2 A p(t,x)−2

  sym 2 AE(t, x) ⊗ E(t, x) + α4 1 + |A|2 for (t, x) ∈ QT , A ∈ M3×3 sym , where .αi > 0, .i = 1, . . . , 4, denote material constants and the variable exponent p : QT → (1, +∞) is essentially governed by the strength of the electric field, i.e., by the scalar function .|E|2 : QT → R, in the sense that there exists a function  ∈ Q , it holds .p0 : R≥0 → R≥0 such that for every .(t, x) T .

  1 < p− ≤ p(t, x) = p0 |E(t, x)|2 ≤ p + < ∞ .

.

(1.7)

To simplify the presentation, we assume for the entire thesis that .ρ = 1. Moreover, similar to [147], we impose the linear dependence .P = χ E E, where .χ E denotes the dielectric susceptibility. Then, the first two equations in (1.6) decouple from the remaining equations and form the quasi-static Maxwell’s equations, which have been widely studied in the literature and are also well-understood. For details, we refer to the overview article by A. Milani and R. Picard [123] or G. Schwarz [152]. With this in mind, because the quasi-static Maxwell’s equations, i.e., .(1.6)1,2 , are uniquely solvable,4 we can view the electro-magnetic field .E : QT → R3 and 4 For

the uniqueness, an additional condition must be imposed (cf. [147, Proposition 3.35]).

10

1 Introduction

the polarization vector field .P : QT → R3 as given quantities. This observation simplifies the analysis of (1.6), in the sense that now one solely has to establish the existence of a velocity vector field .u : QT → R3 and of a scalar kinematic pressure .π : QT → R that solve the slightly reduced system .(1.6)2,–,6 . In other words, the weak solvability of (1.6) reduces to the weak solvability of the unsteady .p(·, ·)Navier–Stokes equations.

1.3 Aims and Outline of This Manuscript This book can essentially be divided into two main parts. The first part, or main part, is primarily devoted to creating the appropriate mathematical framework that allows us to extend the classical theory of pseudo-monotone operators to develop an abstract existence theory for a class of initial-boundary value problems involving variable exponent non-linearities, which has the unsteady .p(·, ·)-Navier–Stokes equations as a guiding example and for which an application of the classical theory of pseudo-monotone operators in the framework of Bochner– Lebesgue spaces is precluded. The ultimate aim of this first part consists in proving an abstract existence result that applies on bounded Lipschitz domains and for so-called .log–Hölder continuous exponents p satisfying .p− ≥ 2, and that directly implies the weak solvability of the unsteady .p(·, ·)-Navier–Stokes equations in a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2, for a .log-Hölder continuous exponent p with .p− ≥ 3d+2 d+2 . The second part comprises extensions of the results developed in the first part, such as the extension of the abstract existence theory to irregular, i.e., non-Lipschitz, domains or .log-Hölder continuous exponents p satisfying .p− < 2, with the aim of proving the weak solvability of the unsteady .p(·, ·)-Navier–Stokes equations in an arbitrary bounded domain . ⊆ Rd , .d ≥ 2, for a .log-Hölder continuous exponent p 3d with .p− > d+2 .

1.3.1 Main Part Having recalled in Chap. 2 some fundamental definitions and facts concerning variable Lebesgue spaces and variable Sobolev spaces as well as Bochner–Lebesgue spaces and Bochner–Sobolev spaces, we develop our abstract existence theory, which has the unsteady .p(·, ·)-Navier–Stokes equations as a guiding example, in the following stages: To begin with, for understanding the influence of the symmetric gradient .ε(·) on the system (1.3), neglecting the incompressibility constraint .(1.3)3 , we will first study in Chap. 3, for a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2, a finite time

1.3 Aims and Outline of This Manuscript

11

interval .I := (0, T ), .T < ∞, and a time-space cylinder .QT := I × , as a model problem, the initial-boundary value problem ∂t x − div(S(·, ·, ε(x))) + b(·, ·, x) = f − div(F ) .

x=0 x(0) = x0

in QT , on I × ∂ ,

(1.8)

in  .

Here, .f : QT → Rd d is a given vector field, .F : QT → Md×d sym is a given symmetric tensor field, and .x0 :  → Rd an initial condition. The tensor-valued mapping d×d d×d .S : QT × Msym → Msym is supposed to have a .(p, δ)-structure (cf. (S.1)–(S.4)). The vector-valued mapping .b : QT × Rd → Rd shall satisfy r-growth conditions (cf. (B.1)–(B.3)). Starting from the energy estimate (1.5), which formally is also valid for a sufficiently smooth solution of (1.8), we will first address the elementary question of which function spaces are actually qualified for the treatment of (1.8) employing advanced pseudo-monotonicity methods. For this purpose, we will first introduce for bounded measurable variable exponents .q, p : QT → [1, +∞), two potential energy spaces, both called variable Bochner–Lebesgue space, namely q,p the space .X∇ (QT ) (cf. Definition 3.2), incorporating the full gradient .∇, which is, in fact, not new and is already well understood (cf. [6, 9–11, 51]), and the q,p space .Xε (QT ), incorporating only the symmetric gradient .ε(·), which is new and uncharted. Immediately after, we will establish the invalidity of a Korn type inequalq,p q,p ity, which implies, in particular, that the spaces .X∇ (QT ) and .Xε (QT ) do not coincide in general, regardless of the regularity of the variable exponents .q, p. This, q,p q,p in turn, rules out the space .X∇ (QT ) and, thus, suggests the space .Xε (QT ) for a usage in an existence proof for the system (1.8). Against this backdrop, the main task q,p of this chapter is to examine the space .Xε (QT ) for its specific properties, such as completeness, separability, reflexivity, and a characterization of its dual space, all with respect to an appropriate norm. Furthermore, we will prove fundamental q,p embedding and compactness results for the space .Xε (QT ), which will serve as a substitute for a Poincaré type inequality, whose invalidity we will establish, and which will turn out to be indispensable for an existence proof for the unsteady .p(·, ·)-Navier–Stokes equations, later in Chap. 5. Like in the study of abstract nonlinear evolution equations in the context of classical Bochner–Lebesgue spaces and Bochner–Sobolev spaces, the validity of a formula of integration-by-parts in time with respect to an appropriate notion of a time derivative usually opens the door to an existence analysis based on advanced pseudo-monotonicity methods. Since the proofs of such formulas of integration by parts traditionally rely on smoothing arguments, an integral task of Chap. 3 consists in constructing a suitable q,p smoothing method for .Xε (QT ), which, in particular, is intended to ascertain the density of smooth vector fields that are compactly supported at least with respect to the space variable, i.e., of .C ∞ (I , C0∞ ()d ) (cf. Proposition 3.16), in q,p .Xε (QT ). For this purpose, we will resort to the so-called space truncation smoothing method, originally developed by L. Diening, P. Nägele, and M. R˚užiˇcka

12

1 Introduction

in [51] for proving the density of .C ∞ (I , C0∞ ()d ) in .X∇ (QT ), provided .q, p are .log-Hölder continuous, written .q, p ∈ Plog (QT ), i.e., satisfy the rather weak continuity assumption q,p

ω(R) ≤

.

C − log(R)

for all R ∈ (0, 1) ,

where .ω : (0, 1) → R is the modulus of continuity of .q, p, respectively. For the proper functioning of the space truncation smoothing method for the space q,p .Xε (QT ), a point-wise Poincaré inequality involving only the symmetric gradient near .∂ is indispensable, which is proved in the appendix. Then, aided by the space truncation smoothing method, we will prove the validity of a formula of integration-by-parts with respect to an appropriate notion of a time derivative q,p living in .Xε (QT )∗ . This formula of integration-by-parts combined with a version of the main theorem on pseudo-monotone perturbations of maximal monotone mappings, due to H. Brézis [30], will then lead to an abstract existence result, which immediately implies the weak solvability of the system (1.8). Having clarified the influence of the symmetric gradient on unsteady problems of the kind (1.8), we eventually let the incompressibility constraint .(1.3)2 enter our investigations. More specifically, Chap. 4 is devoted to developing the appropriate mathematical framework for the treatment of the unsteady .p(·, ·)-Navier–Stokes equations. In this context, we will, at the same time, examine the so-called unsteady .p(·, ·)-Stokes equations, a simplification of the unsteady .p(·, ·)-Navier–Stokes equations in which the convective term, i.e., the vector field .div(u ⊗ u) : QT → Rd , is dropped. More precisely, the unsteady .p(·, ·)-Stokes equations in a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2, over a finite time interval .I := (0, T ), .T < ∞, search for a velocity vector field .u : QT → Rd and a scalar kinematic pressure .π : QT → R that solve the system ∂t u − div(S(·, ·, ε(u))) + ∇π = f − div(F ) div(u) = 0 .

u=0 u(0) = u0

in QT , in QT , on I × ∂ ,

(1.9)

in  .

The absence of the convective term in the system (1.9) slightly simplifies the mathematical analysis compared to the system (1.3). Nevertheless, the simultaneous treatment of both the peculiarities of the symmetric gradient in the framework q,p of .Xε (QT ) and the incompressibility constraint .(1.9)2 , jointly encoded in the q,p space .Vε (QT ), gives rise to a number of obstacles. First difficulties arise when q,p we try to characterize the dual space of .Vε (QT ). Here, it will turn out that an analogue of de Rham’s lemma in the framework of variable Bochner–Lebesgue spaces cannot be valid, which is why a thorough characterization of the dual space q,p q,p .Vε (QT )∗ , such as, e.g., for .Xε (QT )∗ , is not available. Further difficulties

1.3 Aims and Outline of This Manuscript

13

arise when we try to modify the space truncation smoothing operator from Chap. 3 q,p using a Bogovski˘ı correction to obtain a smoothing method for .Vε (QT ) since, as will become apparent, the Bogovski˘ı operator is not continuously extendable to q,p an operator in the context of .Xε (QT ). We circumvent this issue by resorting to another smoothing method, the so-called smoothing method via expansion of bounded Lipschitz domains along transversal vector fields in combination with a contravariant Piola transform. This smoothing method enables us to ascertain the density of smooth functions that are both compactly supported and solenoidal with ∞ ()) (cf. (4.22)), in .V q,p (Q ), respect to the space variable, i.e., of .C ∞ (I , C0,σ T ε provided the variable exponents satisfy .q, p ∈ Plog (QT ), i.e., are .log-Hölder continuous, and .q ≥ p in .QT . In analogy to Chap. 3, we are thus able to prove the validity of a formula of integration-by-parts with respect to an appropriate q,p notion of a hydro-mechanical time derivative that lives in .Vε (QT )∗ rather than q,p ∗ .Xε (QT ) , provided that the variable exponents satisfy .q, p ∈ Plog (QT ), .p− ≥ 2 and .q ≥ p in .QT . Having established the appropriate mathematical framework in Chap. 4, Chap. 5 is devoted to extending the theory of pseudo-monotone operators to a class of initial-boundary value problems governed by the symmetric gradient and with an incompressibility constraint, such as (1.3) or (1.9). As the formula of integration by parts from Chap. 4 is valid only under the condition .q ≥ p in .QT and the extra q,p q,p stress tensor .S induces an operator that maps .Vε (QT ) into .Vε (QT )∗ but is not coercive if .q ≥ p in .QT , the existence theory of Chap. 3 does not apply in q,p the context of .Vε (QT ). In [92, 95], a similar problem has already been faced in the framework of classical Bochner–Lebesgue spaces and Bochner–Sobolev spaces and has been overcome by the introduction of the notions (.C 0 -)Bochner pseudo-monotonicity, (.C 0 -)Bochner condition (M), and (.C 0 -)Bochner coercivity in combination with the so-called Hirano–Landes approach, giving general and easily verifiable sufficient conditions for these notions. We will generalize the concepts q,p of [92, 95] to the space .Vε (QT ). This enables us to prove an abstract existence q,p result in the framework of .Vε (QT ) assuming that . ⊆ Rd , .d ≥ 2, is a bounded Lipschitz domain and .q, p ∈ Plog (QT ) with .p− ≥ 2 and .q ≥ p in .QT . This abstract existence result will then, in turn, directly imply the weak solvability of the unsteady .p(·, ·)-Stokes equations for .p ∈ Plog (QT ) with .p− ≥ 2 and of the unsteady .p(·, ·)-Navier–Stokes equations for .p ∈ Plog (QT ) with .p− ≥ 3d+2 d+2 , each in a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2.

1.3.2 Extensions Since we have merely established the existence of the velocity vector field u : QT → Rd for both the unsteady .p(·, ·)-Stokes equations and the unsteady .p(·, ·)-Navier–Stokes equations in Chap. 5, Chap. 6 eventually addresses the reconstruction of the scalar kinematic pressure .π : QT → R for the systems (1.3) and (1.9) in appropriate function spaces. In doing so, it will transpire that a pressure

.

14

1 Introduction

reconstruction in full regularity, i.e., with variable exponent integrability, is not possible, which can mainly be traced back to the invalidity of de Rham’s lemma q,p in the context of .Xε (QT ), established in Chap. 4. Apart from that, in this connection, we also intend to figure out whether the well-known parabolic .L∞ and (solenoidal) Lipschitz truncation techniques, which are closely related to the q,p issue of a pressure reconstruction, admit extensions to the framework of .Xε (QT ). Chapter 7 is devoted to the extension of the existence results of Chap. 5 to irregular, i.e., non-Lipschitz, domains. Since in irregular domains the formula of integration-by-parts from Chap. 4, which guaranteed the applicability of the notions of .(C 0 -)Bochner pseudo-monotonicity and .(C 0 -)Bochner condition (M) in Chap. 5, is no longer at disposal, we will introduce the notion of Bochner–Sobolev condition (M), which enables us to prove an abstract existence result that also applies on irregular domains without using the formula of integration-by-parts from Chap. 4. The main task of Chap. 7 consists in the characterization of operators which meet this condition. In doing so, we will notice at a very early stage that the convective term induces an operator that satisfies the Bochner–Sobolev condition (M). To verify that the extra stress tensor .S induces an operator that satisfies the Bochner–Sobolev condition (M), a whole machinery must be set in motion. The idea is to apply a Minty–Trick like argument that merges the theory of monotone operators with the theory of finite Radon measures. For the applicability of this Minty–Trick like argument, a parabolic compensated compactness principle, or briefly PCCP, forms an integral part. In doing so, we will follow the ideas of V.V. Zhikov [173] and S.E. Pastukhova [140, 175]. To be more precise, we will improve their results in the sense that we will overcome the artificial relation .p+ < p− d+2 d for the variable exponent p. This will be achieved on the basis of a formula of integration-by-parts for socalled anisotropic variable Bochner–Lebesgue spaces, i.e., hybrid spaces between q,p q,p .Xε (QT ) and .Vε (QT ) that take additional integrability of the divergence into account. Last but not least, because presumably of utmost importance for real-world applications, we will turn our attention in Chap. 8 towards the question of the extension of the results of Chap. 7, in particular, in view of the weak solvability of both the unsteady .p(·, ·)-Stokes equations and the unsteady .p(·, ·)-Navier–Stokes equations, to the delicate case of .p− < 2. For these purposes, we fall back on Pastukhova’s hydro-mechanical parabolic compensated principle (cf. [140]), or briefly hydromechanical PCCP, which originates from an approach of V.V. Zhikov (cf. [173]) for 3d solving the .p(·, ·)-Navier–Stokes equations with .p− > d+2 that dispenses with any ∞ truncation arguments used, e.g., in the .L - and Lipschitz truncation techniques. While Zhikov’s approach applies exclusively to time-independent variable exponents, Pastukhova’s hydro-mechanical PCCP also covers time-dependent variable exponents but has yet to impose the restrictive relation .p+ < p− d+2 d . By combining Pastukhova’s hydro-mechanical PCCP and the methods we developed in Chap. 7, i.e., the first PCCP, we will prove a second PCCP that allows us to prove the weak solvability of the unsteady .p(·, ·)-Stokes equations for .p ∈ Plog (QT ) with − > 2d and of the unsteady .p(·, ·)-Navier–Stokes equations for .p ∈ Plog (Q ) .p T d+2 3d , each in a bounded domain . ⊆ Rd , .d ≥ 2. with .p− > d+2

Chapter 2

Preliminaries

In this chapter, we present preliminary materials that is needed in the course of this book. We will start with a brief overview of the standard notions from the classical theory of pseudo-monotone operators. Immediately afterwards, we will introduce various function spaces, including classical function spaces, variable Lebesgue spaces and variable Sobolev spaces, and describe their basic properties. In the end, we will turn towards Banach-valued function spaces, including (weakly) continuous functions, Bochner–Lebesgue spaces and Bochner–Sobolev spaces.

2.1 Theory of Pseudo-Monotone Operators For a Banach space X equipped with norm . · X , we denote by .X∗ its continuous dual space, which is equipped with the dual norm .x ∗ X∗ := supxX ≤1 x ∗ , xX for .x ∗ ∈ X∗ , where .·, ·X : X∗ × X → R, defined by .x ∗ , xX := x ∗ (x) for all ∗ ∗ .x ∈ X and .x ∈ X, denotes the duality pairing. Moreover, we denote for Banach spaces .X, Y , by .D(A) the domain of definition of .A : D(A) ⊆ X → Y , by .R(A) := {Ax ∈ Y | x ∈ D(A)} its range, and by .G(A) := {(x, Ax) ∈ X × Y | x ∈ D(A)} its graph. A sequence .(xn )n∈N in a Banach space X is called (strongly) convergent to .x ∈ X, written .xn → x in X .(n → ∞), if .limn→∞ xn − xX = 0 and weakly convergent, written .xn  x in X .(n → ∞), if .limn→∞ x ∗ , xn − xX = 0 for every ∗ ∗ .x ∈ X . All occurring Banach spaces are assumed to be real. Throughout this book, we will make frequent use of the following notions from pseudo-monotone operator theory. For a more detailed presentation of this theory, we refer to [28, 143, 149, 155, 168].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaltenbach, Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents, Lecture Notes in Mathematics 2329, https://doi.org/10.1007/978-3-031-29670-3_2

15

16

2 Preliminaries

Definition 2.1 Let X and Y be Banach spaces. An operator .A : D(A) ⊆ X → Y is said to (i) be monotone, if .Y = X∗ and .Ax − Ay, x − yX ≥ 0 for all .x, y ∈ D(A). (ii) be maximal monotone, if .Y = X∗ , .A : D(A) ⊆ X → X∗ is monotone and for ∗ ∈ X × X ∗ from .x ∗ − Ay, x − y ≥ 0 for all .y ∈ D(A), it follows .(x, x ) X that .x ∈ D(A) and .Ax = x ∗ in .X∗ . (iii) be pseudo-monotone, if .Y = X∗ , .D(A) = X and for a sequence .(xn )n∈N ⊆ X from .xn  x in X .(n → ∞) and .lim supn→∞ Axn , xn −xX ≤ 0, it follows for every .y ∈ X that .Ax, x − yX ≤ lim infn→∞ Axn , xn − yX . (iv) satisfy the condition (M), if .Y = X∗ , .D(A) = X and for a sequence ∗ ∗ .(xn )n∈N ⊆ X from .xn  x in X .(n → ∞), .Axn  x in .X .(n → ∞) ∗ ∗ and .lim supn→∞ Axn , xn X ≤ x , xX , it follows that .Ax = x in .X∗ . (v) be weakly continuous, if .D(A) = X and for a sequence .(xn )n∈N ⊆ X from .xn  x in X .(n → ∞), it follows that .Axn  Ax in Y .(n → ∞). (vi) be demi-continuous, if .D(A) = X and for a sequence .(xn )n∈N ⊆ X from .xn → x in X .(n → ∞), it follows that .Axn  Ax in Y .(n → ∞). (vii) be strongly continuous, if .D(A) = X and for a sequence .(xn )n∈N ⊆ X from .xn  x in X .(n → ∞), it follows that .Axn → Ax in Y .(n → ∞). (viii) be bounded, if for every set .M ⊆ D(A) bounded in X, the image .A(M) is bounded in Y . X (ix) be coercive, if .Y =X∗ , .D(A) is unbounded and .limxX →∞ Ax,x xX = ∞. x∈D(A)

2.2 Variable Exponent Spaces This section summarizes all the essential information concerning variable Lebesgue spaces and variable Sobolev spaces that will find use in the hereinafter discussion. These spaces have already been extensively studied by Hudzik [90], Musielak [131], Kováˇcik and Rákosník [103], and R˚užiˇcka [147], to name just a few. A thorough examination of these spaces, including all results of this section, can be found, e.g., in [42, 49, 103].

2.2.1 Classical Function Spaces Before we immediately address variable Lebesgue and variable Sobolev spaces, we first briefly review the most important classical function spaces. Here, if .G ⊆ Rn , .n ∈ N, is open and .f : G → R is k-times continuously differentiable, we denote |α| by .D α f := ∂ α1 x∂1 ···∂fαn xn : G → R, as it is customary, the partial derivative of f with respect to the multi-index .α ∈ Nn with .|α| ≤ k.

2.2 Variable Exponent Spaces

17

Definition 2.2 (Classical Function Spaces) Let .G ⊆ Rn , .n ∈ N, be open and .k ∈ N. Then, we define the spaces   C 0 (G) := f : G → R | f is continuous in G ,   C k (G) := f ∈ C 0 (G) | f is k-times continuously differentiable in G ,   C ∞ (G) := f ∈ C 0 (G) | f ∈ C k (G) for all k ∈ N ,   C 0 (G) := f ∈ C 0 (G) | f is continuously extendable to G ,   C k (G) := f ∈ C k (G) | D α f ∈ C 0 (G) for all α ∈ Nn with |α| ≤ k ,   C ∞ (G) := f ∈ C 0 (G) | f ∈ C k (G) for all k ∈ N ∪ {0} . .

2.2.2 Variable Lebesgue Spaces In what follows, for .n ∈ N, we always denote by .dx, .dy, or .dη the n-dimensional Lebesgue measure. For a (Lebesgue-)measurable set .G ⊆ Rn , .n ∈ N, we  denote by the quantity .|G| := G 1 dx ∈ R≥0 the n-dimensional Lebesgue measure and by .L0 (G) the vector space of scalar (Lebesgue-)measurable functions on G. Two functions are identical, if they agree almost everywhere in G. In particular, throughout the entire book, we will predominantly suppress the prefix Lebesgue- in this context. Definition 2.3 Let .G ⊆ Rn , .n ∈ N, be measurable and .p ∈ L0 (G) such that .p ≥ 1 almost everywhere in G. Then, p will be called a variable exponent and we define .P(G) to be the set of variable exponents. Furthermore, we define the limit exponents p− := ess infx∈G p(x) and p+ := ess supx∈G p(x). Then, by ∞ + < ∞}, we denote the set of bounded variable .P (G) := {p ∈ P(G) | p exponents. In this book, we will consider only bounded variable exponents .p ∈ P∞ (G) since they allow for a simple definition of modulars and variable Lebesgue spaces. Definition 2.4 Let .G ⊆ Rn , .n ∈ N, be measurable and .p ∈ P∞ (G). Then, we denote by .ρp(·) : L0 (G) → [0, +∞], for every .f ∈ L0 (G) defined by  ρp(·) (f ) := ρp(·),G (f ) :=

|f (x)|p(x) dx ,

.

G

the modular with respect to p. Moreover, the variable Lebesgue space is defined by   Lp(·) (G) := f ∈ L0 (G) | ρp(·) (f ) < ∞ .

.

18

2 Preliminaries

Remark 2.1 Note that for .p ∈ P∞ (G), the modular .ρp(·) : L0 (G) → [0, +∞] is well-defined since for every .f ∈ L0 (G), it holds .(x → |f (x)|p(x) ) ∈ L0 (G) (cf. [49, Lemma 2.3.10]). Next, we introduce the so-called Luxemburg norm on .Lp(·) (G). Definition 2.5 Let .G ⊆ Rn , .n ∈ N, be measurable and .p ∈ P∞ (G). Then, we denote by . · Lp(·) (G) : Lp(·) (G) → R≥0 , for every .f ∈ Lp(·) (G) defined by   f Lp(·) (G) := inf λ > 0 | ρp(·) (λ−1 f ) ≤ 1 ,

.

the Luxemburg norm with respect to p. For a constant exponent, the connection between the modular and its norm is unambiguous. For a variable exponent, the situation is not so simple. In this case, we only have the following relations between a modular and its Luxemburg norm. Lemma 2.1 (Norm-Modular Unit Ball Property) Let .G ⊆ Rn , .n ∈ N, be measurable and .p ∈ P∞ (G). Then, for every .f ∈ Lp(·) (G), the following statements apply: (i) .f Lp(·) (G) ≤ 1 if and only if .ρp(·) (f ) ≤ 1. (ii) If .f Lp(·) (G) ≤ 1, then .ρp(·) (f ) ≤ f Lp(·) (G) . If .f Lp(·) (G) > 1, then .f Lp(·) (G) ≤ ρp(·) (f ). p−

p+

(iii) .f Lp(·) (G) − 1 ≤ ρp(·) (f ) ≤ f Lp(·) (G) + 1. Proof See [49, Lemma 3.2.4. & Lemma 3.2.5.].

 

Even though the Luxemburg norm does not possess all convenient properties of the conventional .Lp (G)-norm for a constant exponent .p ∈ (1, ∞), it enables us to turn variable Lebesgue spaces into Banach spaces with all desirable properties and results: Proposition 2.1 Let .G ⊆ Rn , .n ∈ N, be measurable and .p ∈ P∞ (G). Then, p(·) (G) forms a vector space and equipped with the Luxemburg norm . ·  .L Lp(·) (G) a separable Banach space. Proof See [49, Theorem 3.2.7. & Lemma 3.4.4.].

 

Proposition 2.2 (Fischer–Riesz) Let .G ⊆ Rn , .n ∈ N, be measurable and .p ∈ P∞ (G). Then, for a sequence .(fn )n∈N ⊆ Lp(·) (G) and a function .f ∈ Lp(·) (G) from .fn → f in .Lp(·) (G) .(n → ∞), it follows the existence of a cofinal subset 1 . ⊆ N such that .fn → f .(  n → ∞) almost everywhere in G. Proof See [42, Proposition 2.67.].

 

set . ⊆ N is called cofinal, if for every .n ∈ N, there exists a .λ ∈  such that .λ ≥ n (cf. Notation comments).

1A

2.2 Variable Exponent Spaces

19

Proposition 2.3 (Convergence in Norm .⇔ Convergence in Modular) Let .G ⊆ Rn , .n ∈ N, be measurable and .p ∈ P∞ (G). Then, for a sequence .(fn )n∈N ⊆ Lp(·) (G) and a function .f ∈ Lp(·) (G), it holds .fn − f Lp(·) (G) → 0 .(n → ∞) if and only if .ρp(·) (fn − f ) → 0 .(n → ∞).  

Proof See [42, Theorem 2.58.].

Proposition 2.4 (Lebesgue’s Dominated Convergence Theorem) Let .G ⊆ Rn , ∞ p(·) (G) .n ∈ N, be measurable and .p ∈ P (G). Then, for a sequence .(fn )n∈N ⊆ L 0 and a function .f ∈ L (G) from .fn → f .(n → ∞) almost everywhere in G, p(·) (G), it follows that .supn∈N |fn | ≤ |g| almost everywhere in G, where .g ∈ L p(·) .fn → f in .L (G) .(n → ∞).  

Proof See [49, Lemma 3.2.8. (c)].

Proposition 2.5 (Vitali’s Convergence Theorem) Let .G ⊆ Rn , .n ∈ N, be bounded and measurable, .p ∈ P∞ (G), and .(fn )n∈N ⊆ Lp(·) (G) and .f ∈ Lp(·) (G) with .fn → f .(n → ∞) almost everywhere in G. Then, it holds .fn → f in .Lp(·) (G) p(·) (G)-uniformly integrable, i.e., for every .(n → ∞) if and only if .(fn )n∈N is .L .ε > 0, there exists a .δ > 0 such that for every measurable set .K ⊆ G from .|K| < δ,  it follows that .supn∈N K |fn (x)|p(x) dx < ε.  

Proof See [81, Theorem 3.8].

Remark 2.2 All above and still following definitions and results admit congruent extensions to the spaces  Lp(·) (G)n := f = (f1 , . . . , fn ) : G → Rn | fi ∈ Lp(·) (G),  i = 1, . . . , n ,  Lp(·) (G)n×n := F = (Fij )i,j =1,...,n : G → Rn×n | Fij ∈ Lp(·) (G),  i, j = 1, . . . , n , .

n ∈ N, each equipped with

.

fLp(·) (G)n :=

n 

.

fi Lp(·) (G)

i=1

if f ∈ Lp(·) (G)n and

.

FLp(·)(G)n×n :=

n 

.

Fij Lp(·) (G)

i,j =1

if F ∈ Lp(·) (G)n×n , respectively. In addition, we define the space

.

  n×n p(·) Lp(·) (G, Mn×n (G)n×n , sym ) := F : G → Msym | F ∈ L

.

which is a closed subspace of .Lp(·) (G)n×n .

20

2 Preliminaries

2.2.3 Duality in Variable Lebesgue Spaces This subsection clarifies that the description of duality for a classical Lebesgue space  p Lp (G), for a constant exponent .p ∈ (1, ∞), through .Lp (G), where .p := p−1 denotes the dual exponent of p, to a large extent is transferable to variable Lebesgue spaces, up to small bearable losses. The first step is to introduce a variable exponent version of the dual exponent.

.

Definition 2.6 Let .G ⊆ Rn , .n ∈ N, be measurable and .p ∈ P∞ (G) with .p− > 1. p(x) We define the dual variable exponent .p ∈ P∞ (G) of p by .p (x) := p(x)−1 for almost every .x ∈ G. As in the classical Lebesgue space theory, the dual exponent plays a central role  in Hölder’s inequality, which is a cornerstone in the identification of .Lp (G) with p ∗ .L (G) for constant .p ∈ (1, ∞). Fortunately, we can likewise hope for such an inequality for variable Lebesgue spaces. For this, however, we have to pay with a constant that is greater than 1, but yet independent of .p ∈ P∞ (G). Proposition 2.6 (Hölder’s Inequality) Let .G ⊆ Rn , .n ∈ N, be measurable and ∞ − > 1. Then, for every .g ∈ Lp (·) (G) and .f ∈ Lp(·) (G), it holds .p ∈ P (G) with .p gf L1 (G) ≤ 2gLp (·) (G) f Lp(·) (G) .

.

 

Proof See [49, Lemma 3.2.20.]. Rn ,

Corollary 2.1 Let .G ⊆ .n ∈ N, be measurable with .|G| < ∞ and let .q, p ∈ P∞ (G) with .q ≤ p almost everywhere in G. Then, every .f ∈ Lp(·) (G) also satisfies q(·) (G) with .f ∈ L f Lq(·) (G) ≤ 2(1 + |G|)f Lp(·) (G) .

.

i.e., it holds the embedding .Lp(·) (G) → Lq(·) (G).  

Proof See [49, Corollary 3.3.4.].

Hölder’s inequality (cf. Proposition 2.6) yields the well-definedness of the   product .(·, ·)Lp(·) (G) : Lp (·) (G) × Lp(·) (G) → R, for every .g ∈ Lp (·) (G) and p(·) (G) defined by .f ∈ L  (g, f )Lp(·) (G) :=

g(x)f (x) dx ,

.

(2.1)

G

from which we plan to make frequent use of. The products .(·, ·)Lp(·) (G)n :   Lp (·) (G)n ×Lp(·) (G)n → R and .(·, ·)Lp(·) (G)n×n : Lp (·) (G)n×n ×Lp(·) (G)n×n → R are defined accordingly. At the same time, Hölder’s inequality guarantees the well-definedness of the so-called Riesz representation operator, which eventually  enables us to identify .Lp (·) (G) with .Lp(·) (G)∗ for .p ∈ P∞ (G) with .p− > 1.

2.2 Variable Exponent Spaces

21

Proposition 2.7 (Characterization of .Lp(·) (G)∗ ) Let .G ⊆ Rn , .n ∈ N, be measurable and let .p ∈ P∞ (G) with .p− > 1. Then, the Riesz representation operator p (·) (G) → Lp(·) (G)∗ , which for every .g ∈ Lp (·) (G) and .f ∈ Lp(·) (G) is .J : L defined by J g, f Lp(·) (G) := (g, f )Lp(·) (G) ,

.

1  2 gLp (·) (G) p(·) addition, .L (G) is

is an isomorphism satisfying every .g ∈ L

p (·)

(G). In

.

≤ J gLp(·) (G)∗ ≤ 2gLp (·) (G) for reflexive.  

Proof See [49, Theorem 3.4.6. & Theorem 3.2.13].

Proposition 2.8 (.ε-Young Inequality) Let .G ⊆ Rn , .n ∈ N, be measurable and ∞ − > 1. Then, the .ε-Young inequality, i.e., .p ∈ P (G) with .p −

− 

εp p(x) ε−(p ) p (x) a + b .ab ≤ p− (p+ ) 

for a.e. .x ∈ G, all .a, b ≥ 0 and .ε ∈ (0, 1], for every .g ∈ Lp (·) (G), .f ∈ Lp(·) (G) and .ε ∈ (0, 1], implies that −

(g, f )Lp(·) (G) ≤

.

− 

εp ε−(p ) ρp(·) (f ) + ρp (·) (g) . − p (p+ )



2.2.4 Variable Sobolev Spaces Let .G ⊆ Rn , .n ∈ N, be an open set and let .p ∈ P∞ (G). Owing to the inclusion .Lp(·) (G) ⊆ L1loc (G),2 every scalar function .f ∈ Lp(·) (G), as well as every vector-valued function .f ∈ Lp(·) (G)n , possesses derivatives at least in the sense of distributions, i.e., in .D (G), the topological dual space of the locally convex Hausdorff vector space (cf. [75, Kap. II, 4. Distr.])   C0∞ (G) := f ∈ C ∞ (G) | supp(f ) ⊂⊂ G .

.

(2.2)

More precisely, the distributional gradient .∇f ∈ D (G)n of a scalar function 1 (G) for every .x ∈ C ∞ (G)n is defined by .f ∈ L loc 0  ∇f, x

.

2 .L1 (G) loc

C0∞ (G)n

:= −

f (x)div(x)(x) dx . G

denotes the space of locally Lebesgue-integrable functions.

(2.3)

22

2 Preliminaries

 n×n The distributional symmetric gradient .ε(f) ∈ D (G, Mn×n sym ), where .D (G, Msym ) denotes the topological dual space of the vector space .C0∞ (G, Mn×n sym ) := ∞ n×n n×n .{X : G → Msym | X ∈ C (G) }, which is a closed subspace of the locally con0 vex Hausdorff vector space .C0∞ (G)n×n , of a vector-valued function .f ∈ L1loc (G)n for every .X ∈ C0∞ (G, Mn×n sym ) is defined by

 ε(f), XC ∞ (G,Mn×n ) := −

f(x) · div(X)(x) dx .

.

sym

0

(2.4)

G

Using these notions, we next introduce two specialized variants of variable Sobolev spaces, where one variant already occurred in a similar form in a contribution of A. Huber (cf. [89, p. 344]). Definition 2.7 Let .G ⊆ Rn , .n ∈ N, be open and .q, p ∈ P∞ (G). Then, we define the variable Sobolev spaces   (G) := f ∈ Lq(·) (G) | ∇f ∈ Lp(·) (G)n ,   q(·),p(·) (G) := f ∈ Lq(·) (G)n | ε(f) ∈ Lp(·) (G, Mn×n Xε sym ) , q(·),p(·)

X∇

.

where .∇f and .ε(f) have to be understood in the distributional senses of (2.3) and (2.4), respectively Proposition 2.9 Let .G ⊆ Rn , .n ∈ N, be open and .q, p ∈ P∞ (G). Then, the norms  · Xq(·),p(·) (G) :=  · Lq(·) (G) + ∇ · Lp(·) (G)n ,

.

∇

 · Xq(·),p(·) (G) :=  · Lq(·) (G)n + ε(·)Lp(·) (G)n×n , ε

q(·),p(·)

turn .X∇

q(·),p(·)

(G) and .Xε

(G), respectively, into Banach spaces. q(·),p(·)

Proof We solely give a proof for .Xε (G) because all the arguments transfer q(·),p(·) q(·),p(·) (G). So, let .(fk )k∈N ⊆ Xε (G) be a Cauchy sequence, i.e., both to .X∇ q(·) (G)n and .(ε(f )) p(·) (G, Mn×n ) are Cauchy sequences. .(fk )k∈N ⊆ L k k∈N ⊆ L sym Thus, since .Lq(·) (G)n and .Lp(·) (G, Mn×n ) are Banach spaces (cf. Proposition 2.1 sym q(·) n and Remark 2.2), we obtain functions .f ∈ L (G) and .E ∈ Lp(·) (G, Mn×n sym ) such that .fk → f in .Lq(·) (G)n .(k → ∞) and .ε(fk ) → E in .Lp(·) (G, Mn×n ) . (k → ∞). sym Consequently, owing to (2.4), for every .X ∈ C0∞ (G, Mn×n ), it holds sym .

− (div(X), f)Lq(·) (G)n = lim −(div(X), fk )Lq(·) (G)n k→∞

= lim (X, ε(fk ))Lp(·) (G)n×n = (X, E)Lp(·) (G)n×n , k→∞

q(·),p(·)

i.e., .f ∈ Xε .(k → ∞).

q(·),p(·)

(G) with .ε(f) = E in .Lp(·) (G, Mn×n sym ) and .fk → f in .Xε

(G)  

2.2 Variable Exponent Spaces

23

Definition 2.8 Let .G ⊆ Rn , .n ∈ N, be open and .q, p ∈ P∞ (G). Then, we introduce the spaces ˚q(·),p(·) (G) := C ∞ (G)·X∇ X ∇ 0

q(·),p(·)

(G)

.

˚εq(·),p(·) (G) := C ∞ (G) X 0

,

q(·),p(·) (G) n ·Xε

.

Remark 2.3 In the case .q, p ∈ P∞ (G) with .q = p almost everywhere in G, q(·),p(·) q(·),p(·) (G) and .Xε (G) coincide with the standard notion of the spaces .X∇ variable Sobolev spaces, such as, e.g., in [42, 49], and we, therefore, define in p(·),p(·) 1,p(·) ˚p(·),p(·) (G), to reduce (G) and .W0 (G) := X this case .W 1,p(·) (G) := X∇ ∇ the risk of confusion. Furthermore, guided by the notation, e.g., in [24, 147], we define for .q, p ∈ P∞ (G) with .q = p almost everywhere in .G, the spaces 1,p(·) (G) := X p(·),p(·) (G) and .E 1,p(·) (G) := X ˚εp(·),p(·) (G). .E ε 0

2.2.5 The Hardy–Littlewood Maximal Operator and log-Hölder Continuity This section introduces the Hardy–Littlewood maximal operator, one of the most important tools in harmonic analysis. In this connection, we will also examine the notion of .log-Hölder continuity. Definition 2.9 Let .n ∈ N. For .f ∈ L1loc (Rn ), we define for almost every .x ∈ Rn 1 .Mn (f )(x) := sup |f (y)| dy := sup n n r>0 Br (x) r>0 |Br (x)|

 Brn (x)

|f (y)| dy .

Then, the function .Mn (f ) : Rn → [0, +∞] is called the Hardy–Littlewood maximal function of f . The resulting mapping .Mn : L1loc (Rn ) → L0 (Rn ) is called the Hardy– Littlewood maximal operator. Remark 2.4 For .n ∈ N, we define the operator .Mn : L1loc (Rn )n → L0 (Rn ) for every 1 (Rn )n by .M (f)(x) := M (|f|)(x) for almost every .x ∈ Rn . The operator .f ∈ L n n loc 1 (Rn )n×n → L0 (Rn ), .n ∈ N, is defined accordingly. .Mn : L loc Next, we will introduce a special modulus of continuity for variable exponents, which is, e.g., sufficient for the boundedness of the Hardy–Littlewood maximal operator .Mn from .Lp(·) (Rn ) into .Lp(·) (Rn ), and also for a coincidence of the spaces ˚q(·),p(·) (G)n and .X ˚εq(·),p(·) (G). .X ∇

24

2 Preliminaries

Definition 2.10 (.log-Hölder Continuity) Let .G ⊆ Rn , .n ∈ N, be open. We say that .p ∈ P∞ (G) is locally .log-Hölder continuous in G, if there exists a constant .c1 > 0 such that |p(x) − p(y)| ≤

.

c1 log(e + 1/|x − y|)

for all x, y ∈ G .

We say that .p ∈ P∞ (G) satisfies the .log-Hölder decay condition in G, if there exist constants .p∞ ∈ R and .c2 > 0 such that |p(x) − p∞ | ≤

.

c2 log(e + |x|)

for all x ∈ G .

We say that .p ∈ P∞ (G) is globally .log-Hölder continuous in G, if it is locally .log-Hölder continuous in G and satisfies .log-Hölder decay condition in G. The constants .c1 and .c2 are called the local .log-Hölder constant and the .log-Hölder decay constant, respectively. Then, the maximum .clog (p) := max{c1 , c2 } is just called the .log-Hölder constant of p. Furthermore, we define the set   Plog (G) := p ∈ P∞ (G) 

.

1 p

 is globally log -Hölder continuous .

 

Remark 2.5 For every .p ∈ P∞ (G), since . s → 1s : [p− , p+ ] → p1+ , p1− is a biLipschitz mapping, there holds .p ∈ Plog (G) if and only if p is globally .log-Hölder continuous in G. One useful property of globally .log-Hölder continuous exponents is their ability to admit extensions to the whole space .Rn , .n ∈ N, having similar characteristics, see, e.g., [49, Prop. 4.1.7.]. Proposition 2.10 Let .G  Rn , .n ∈ N, be open. If .p ∈ Plog (G), then it admits an extension .p ∈ Plog (Rn ), i.e., .p = p in G, satisfying .clog (p) = clog (p), .p− = p− and .p + = p+ . A similar extension result applies to continuous exponents on closed sets, see, e.g., [59, Prop. 3.1]. Proposition 2.11 Let .G  Rn , .n ∈ N, be closed. If .p ∈ P∞ (G) ∩ C 0 (G), then it admits an extension .p ∈ P∞ (Rn ) ∩ C 0 (Rn ), i.e., .p = p in G, satisfying .p− = p− and .p + = p+ . The next proposition is a keystone of this book and was first proved in [43, Theorem 2.10]. Proposition 2.12 Let .n ∈ N and .p ∈ Plog (Rn ) with .p− > 1. Then, there exists a constant .c > 0, depending only on .n ∈ N and .clog (p) > 0, such that for every p(·) (Rn ), it holds .f ∈ L Mn (f )Lp(·) (Rn ) ≤ cf Lp(·) (Rn ) .

.

2.2 Variable Exponent Spaces

25

The following proposition states that under the assumption of .log-Hölder continuity of the exponent, i.e., when .p ∈ Plog (G) with .p− > 1, the spaces ˚εq(·),p(·) (G) and .X ˚q(·),p(·) (G)n coincide. .X ∇

Proposition 2.13 (Korn’s Inequality) Let .G ⊆ Rn , .n ≥ 2, be a bounded domain, ∞ log .q ∈ P (G) and .p ∈ P (G) with .p− > 1. Then, there exists a constant .c > 0, ˚εq(·),p(·) (G), it depending only on .n ∈ N and .clog (p) > 0, such that for every .f ∈ X q(·),p(·) ˚ holds .f ∈ X (G)n with ∇ ∇fLp(·) (G)n×n ≤ cε(f)Lp(·) (G)n×n ,

.

˚εq(·),p(·) (G) with norm equivalence. ˚q(·),p(·) (G)n = X i.e., it holds .X ∇ Proof See [49, Theorem 14.3.21.].

 

2.2.6 Mollification in Lp(·) (Rn ) In this section, we want to examine mollification in .Lp(·) (Rn ), .n ∈ N, a standard tool to deduce the density of smooth functions. In [49, p. 93], it has been shown that the convolution operator .∗ := ((f, g) → f ∗ g) : Lp(·) (Rn ) × L1 (Rn ) .→ Lp(·) (Rn ) is not continuous in the case of a non-constant variable exponent .p ∈ P∞ (Rn ), which seemed to be disastrous in view of a possible generalization of the smoothing method via convolution, which itself undoubtedly is fundamentally based on the continuity of the convolution operator. Nevertheless, it turned out that the situation is not that serious if we restrict our attention to convolution with so-called bellshaped functions. Definition 2.11 A function .ψ : Rn → R≥0 , .n ∈ N, is called bell-shaped, if it is radially decreasing and radially symmetric, i.e., there exists a non-increasing function .η : [0, ∞) → [0, ∞] such that .ψ(x) = η(|x|) for every .x ∈ Rn . Then, we call a bell-shaped function .ψ : Rn → R≥0 a standard mollifier, if, in addition, 1 n .ψ ∈ L (R ). Furthermore, we define the set of standard mollifiers   SM(Rn ) := ψ ∈ L1 (Rn , R≥0 ) | ψ is bell-shaped .

.

Proposition 2.14 Let .p ∈ Plog (Rn ), .n ∈ N, with .p− > 1 and .ψ ∈ SM(Rn ). Then, 1 x n 1 n .(ψε )ε>0 ⊆ L (R ), defined by .ψε (x) := n ψ( ) for all .ε > 0 and .x ∈ R , satisfies ε ε n n p(·) .(ψε )ε>0 ⊆ SM(R ). Moreover, for every .f ∈ L (R ), it holds: (i) .supε>0 ψε ∗ f Lp(·) (Rn ) ≤ Kf Lp(·) (Rn ) , where .K > 0 depends only on n .ψ 1 L (R ) and .clog (p). (ii) .supε>0 |ψε ∗ f | ≤ 2ψL1 (Rn ) Mn (f ) almost everywhere in .Rn . (iii) If, in addition, .ψL1 (Rn ) = 1, then there holds .ψε ∗ f → f .(ε → 0) almost everywhere in .Rn and .ψε ∗ f → f in .Lp(·) (Rn ) .(ε → 0). Proof See [49, Lemma 4.6.3. & Theorem 4.6.4.].

 

26

2 Preliminaries

Remark 2.6 (Special Standard Mollifier) Throughout the entire book, for .n ∈ N, we will always denote by .ωn ∈ C0∞ (Rn ) .∩SM(Rn ) the special standard mollifier defined by ω (x) :=

.

n

 −1 cωn exp 1−|x| if x ∈ B1n (0) 2 0

if x ∈ B1n (0)

,

where .cωn > 0 is a constant that is supposed to guarantee that .ωn L1 (Rn ) = 1. Moreover, we define the family of scaled standard mollifiers .(ωεn )ε>0 ⊆ C0∞ (Rn )∩ 1 n x n n n .SM(R ) for every .ε > 0 and .x ∈ R by .ωε (x) := n ω ( ). ε ε

2.3 Banach-Valued Function Spaces Throughout this section, unless otherwise specified, let X be a Banach space and I ⊆ R an interval.

.

2.3.1 Banach-Valued Classical Function Spaces This section recalls definitions and basic facts about Banach-valued (weakly) continuous functions, which generalize the concept of continuity of the space .C 0 (I ) to Banach-valued functions .x : I → X, i.e., to functions taking their values in a Banach space X. The primary source of this section is [75]. Definition 2.12 We denote by   C 0 (I, X) := x : I → X | x is continuous from I into X ,

.

the space of X-valued in I continuous functions. Definition 2.13 A function .x : I → X is called differentiable in a point .t ∈ I , if there exists an element .x ∈ X such that



x(t + h) − x(t)

− x . lim

= 0.

h→0 h X t+h∈I

In this case, we define .∂t x(t) := x. A function .x : I → X is called differentiable, if it is differentiable at every point .t ∈ I . In this case, we denote by the mapping .∂t x : I → X the time derivative of .x : I → X. For .k ∈ N ∪ {∞}, a function .x : I → X is called k-times differentiable, if .∂ti x : I → X is differentiable for all i−1 i .i = 0, . . . , k − 1, where .∂t x := ∂t [∂t x] for .i = 1, . . . , k − 1 and .∂t0 x := x.

2.3 Banach-Valued Function Spaces

27

For .k ∈ N ∪ {∞}, by   C k (I, X) : = x ∈ C 0 (I, X) | ∃∂ti x ∈ C 0 (I, X) for all i = 1, . . . , k ,

.

we denote the space of X-valued in I k-times continuously differentiable functions. Proposition 2.15 (Completeness of .C k (I , X)) Let .I ⊆ R be a bounded interval k , X) forms a Banach space, if equipped with the norm and .k ∈ N ∪ {0}.

kThen, .C (I

i .x k i=0 maxs∈I ∂t x(s) X . C (I ,X) :=  

Proof See [75, Kap. IV, Satz 1.1]. Weak convergence

in .C 0 (I , X)

can be characterized as follows.

Proposition 2.16 Let .I ⊆ R be a bounded interval. For a sequence .(x n )n∈N ⊆ C 0 (I , X) and a function .x ∈ C 0 (I , X), it holds .x n  x in .C 0 (I , X) .(n → ∞) if and only if .x n (t)  x(t) in X .(n → ∞) for all .t ∈ I . Proof See [20, Theorem 4.3].

 

In this work, we will often encounter functions that satisfy the following weak continuity concept. Definition 2.14 We denote by   Cω0 (I, X) := x : I → X | x is weakly continuous from I into X ,

.

the space of X-valued in I weakly continuous functions. For the space .Cω0 (I, X), the following result regarding point-wise weak convergence in X applies. Proposition 2.17 Let X, Y be Banach spaces, where X is reflexive, and .j : X → Y an embedding, i.e., linear, injective and continuous. Furthermore, let .(x n )n∈N ⊆ Cω0 (I, X) be a sequence and let .x ∈ Cω0 (I, X) be a function such that for every .t ∈ I , it holds .supn∈N x n (t)X < ∞ and .j (x n (t))  j (x(t)) in Y .(n → ∞). Then, it holds .x n (t)  x(t) in X .(n → ∞) for all .t ∈ I . Proof We apply [95, Proposition 2.9 (iv)] for each .t ∈ I .

 

2.3.2 Bochner–Lebesgue Spaces This section recalls definitions and basic facts about the classical theory of Bochner– Lebesgue spaces, which extends the concept of classical Lebesgue spaces .Lp (I ), where .p ∈ [1, ∞], to Banach-valued functions .x : I → X. In what follows, we will always denote by .dt, .ds, or .dτ the one-dimensional Lebesgue measure. The primary sources of this section are [54, 75], and [149].

28

2 Preliminaries

Definition 2.15 (Bochner Measurability) We denote by S(I, X) :=

n 

.

s i χE i

   n ∈ N, si ∈ X, Ei ⊆ I (Lebesgue) measurable 

i=1



with |Ei | < ∞ and Ei ∩ Ej = ∅ for i = j , the vector space of X-valued simple functions in I . Then, a function .x : I → X is called Bochner-measurable, if there exists a sequence .(s n )n∈N ⊆ S(I, X) such that .s n (t) → x(t) in X .(n → ∞) for almost every .t ∈ I . The Bochner integral of a  simple function .s = ni=1 si χEi ∈ S(I, X), where .n ∈ N, .si ∈ X and .Ei ⊆ I are measurable with .|Ei | < ∞ and .Ei ∩ Ej = ∅ for .i = j , is defined by  s(t) dt :=

.

I

n 

si |Ei |

in X .

i=1

Lemma 2.2 If .x : I → X is Bochner-measurable, then .x(·)X : I → R is Lebesgue-measurable.  

Proof See [149, Kap. II, Lem. 1.7].

Lemma 2.2 allows for a general definition of the Bochner integral (cf. [149, Kap. II, Def. 1.3]). Definition 2.16 (Bochner Integrability) A function .x : I → X is called Bochnerintegrable, if there exists a sequence .(s n )n∈  N ⊆ S(I, X) such that .s n (t) → x(t) in X .(n → ∞) for almost every .t ∈ I 3 and . I x(t) − s n (t)X dt → 0 .(n → ∞). In this case, the Bochner integral is defined as the limit 

 x(t) dt := lim

.

I

n→∞ I

s n (t) dt

in X .

Definition 2.17 (Bochner–Lebesgue Space) For .p ∈ [1, ∞], we define   Lp (I, X) := x : I → X | x is Bochner measurable, x(·)X ∈ Lp (I ) ,

.

and denote by .∼X the equivalence relation with respect to equality in X almost everywhere in I . Then, the Bochner–Lebesgue space is defined as the quotient space p Lp (I, X) := L (I,X)/∼X .

3 i.e., .x

: I → X is Bochner-measurable

2.3 Banach-Valued Function Spaces

29

Remark 2.7 Elements of Bochner–Lebesgue spaces are equivalence classes [x]X := [x]∼X . Nevertheless, with the common abuse of notation, we will omit the brackets .[·]X throughout this book and call .x ∈ Lp (I, X) a function, even though it actually represents an equivalence class.

.

Proposition 2.18 For .p ∈ [1, ∞], the space .Lp (I, X) forms a Banach space, if equipped with the norm

xLp (I,X) := x(·)X Lp (I )

.

for .x ∈ Lp (I, X). If .p ∈ [1, ∞), then the space of simple functions .S(I, X) lies densely in .Lp (I, X). If .p ∈ [1, ∞) and X is separable, then .Lp (I, X) is separable. Proof See [75, Kap. IV, Satz 1.11, Lemma 1.3 & Satz 1.12] and [54, Corollaire   1.3.2]. Proposition 2.19 Let .X, Y be Banach spaces, where X is reflexive, .j : X → Y an embedding and .I ⊆ R a bounded interval. Then, each function .x ∈ L∞ (I, X) for which the function .j x ∈ L∞ (I, Y ), defined by .(j x)(t) := j (x(t)) in Y for almost every .t ∈ I , has a representation .j ω x ∈ Cω0 (I , Y ), has a representation 0 .x ω ∈ Cω (I , X).  

Proof See [158, Lem. 1.4, Ch. III, §1].

We have the following characterization of duality for Bochner–Lebesgue spaces. Proposition 2.20 Let X be reflexive and .p ∈ [1, ∞). Then, the Riesz representation   operator .JX : Lp (I, X∗ ) → (Lp (I, X))∗ , which for every .x ∗ ∈ Lp (I, X∗ ) and p .x ∈ L (I, X) is defined by JX x ∗ , xLp (I,X) :=



x ∗ (t), x(t)X dt ,

.

I

is an isometric isomorphism. In addition, if .p ∈ (1, ∞), then .Lp (I, X) is reflexive.  

Proof See [52, IV. 1. Theorem 1, Corollary 2].

Proposition 2.20 in the case .p = 1, enables us to introduce a notion of weak-* convergence for the space .L∞ (I, X). For details, please refer to [93, Lemma 3.15]. Definition 2.18 (Weak-* Convergence in .L∞ (I, X)) Let X be reflexive. A sequence .(x n )n∈N ⊆ L∞ (I, X) is said to converge weakly-* to .x ∈ L∞ (I, X), if for every .x ∗ ∈ L1 (I, X∗ ), it holds 

∗



x ∗ (t), x(t)X dt

x (t), x n (t)X dt →

.

I

(n → ∞) .

(2.5)

I ∗

With a slight abuse of notation, we always write .x n  x in .L∞ (I, X) .(n → ∞) instead of (2.5).

30

2 Preliminaries

Corollary 2.2 (Weak-* Compactness in .L∞ (I, X)) Let X be separable and reflexive. Furthermore, let .(x n )n∈N ⊆ L∞ (I, X) be bounded. Then, there exists a subsequence .(x n )n∈ ⊆ L∞ (I, X), with cofinal . ⊆ N, and a function .x ∈ ∗ L∞ (I, X) such that .x n  x in .L∞ (I, X) .(  n → ∞). Proposition 2.21 (Linear-Induced Operators) Let X and Y be Banach spaces, p ∈ [1, ∞], and let .A : X → Y be linear and bounded. Then, the linear-induced operator .A : Lp (I, X) → Lp (I, Y ), which for every .x ∈ Lp (I, X) is defined by .(Ax)(t) := A(x(t)) in Y for almost every .t ∈ I , is well-defined, linear and bounded. Furthermore, the following statements apply:   (i) . I (Ax)(t) dt = A I x(t) dt in Y for every .x ∈ L1 (I, X). (ii) If .A : X → Y is an embedding, then .A : Lp (I, X) → Lp (I, Y ) is an embedding. (iii) If .A : X → Y is an isomorphism, then .A : Lp (I, X) → Lp (I, Y ) is an isomorphism. (iv) The operator .A : Cω0 (I, X) → Cω0 (I, Y ), defined by .(Ax)(t) := A(x(t)) in Y for every .t ∈ I and .x ∈ Cω0 (I, X), is well-defined and linear. In addition, if .A : X → Y is injective, or bijective, then .A : Cω0 (I, X) → Cω0 (I, Y ) is injective, or bijective, respectively, as well. .

Proof Concerning a proof of the well-definedness, the linearity, and the boundedness, including (i), we refer to [54, Proposition 1.2.2]. The verification of (ii)–(iv) is elementary and, thus, omitted.   Smoothing via convolution with standard mollifiers can be extended to Bochner– Lebesgue spaces. Proposition 2.22 (Smoothing in Bochner–Lebesgue Spaces) Let .I := (0, T ), T < ∞, .p ∈ [1, ∞], and denote by .I : Lp (I, X) → Lp (R, X) the zero extension operator outside I , which is for every .x ∈ Lp (I, X) defined by .(I x)(t) := x(t) in X for almost every .t ∈ I and .(I x)(t) := 0 in X for almost every .t ∈ I  . For every p .x ∈ L (I, X), .h > 0, and .t ∈ R, we define the smoothing operator .

(SIh x)(t) := (ωh1 ∗ I x)(t)  := ωh1 (t − s)(I x)(s) ds R

.

in X ,

where .(ωh1 )h>0 ⊆ C0∞ (R) are the scaled standard mollifiers from Remark 2.6 in the case .d = 1.

2.3 Banach-Valued Function Spaces

31

Then, for every .x ∈ Lp (I, X), it holds: (i) .(SIh x)h>0 ⊆ C ∞ (R, X) with .supp(SIh x) ⊆ [−h, T + h]4 for every .h > 0. (ii) .suph>0 SIh xLp (I,X) ≤ xLp (I,X) . (iii) If .p < ∞, then .SIh x → x in .Lp (I, X) .(h → 0). Proof For .p ∈ [1, ∞), we refer to [54, Prop. 1.7.1 & Thm. 1.7.1]. For .p = ∞, (ii)   is obvious.

2.3.3 Bochner–Sobolev Spaces In this section, we will recall and discuss properties of the well-known concept of the Banach-valued distributional time derivative. It should be emphasized that this book is confined to so-called regular Banach-valued distributional time derivatives, i.e., Banach-valued distributional time derivatives that can be represented by Bochner integrable functions, see, e.g., [23, Sec. 5.2], since this notion is perfectly adequate for the scope of this book. For this reason, we refrain from a thorough reexamination of the general concept of Banach-valued distributions and refer for this, e.g., to [75] or [54]. Definition 2.19 Let X and Y be Banach spaces, .j : X → Y an embedding, i.e., linear, injective and continuous, and .p, q ∈ [1, ∞]. Then, a function .x ∈ Lp (I, X) has a Banach-valued distributional time derivative with respect to j in .Lq (I, Y ), if there exists a function .y ∈ Lq (I, Y ) such that for every .ϕ ∈ C0∞ (I ), it holds     .j − x(s)∂t ϕ(s) ds = y(s)ϕ(s) ds I

in Y .

Since the function .y ∈ Lq (I, Y ) is uniquely determined by (2.6),5 . defined. By 1,p,q

Wj

.

(2.6)

I dj x dt

:= y is well-

   d x  j ∈ Lq (I, Y ) , (I, X, Y ) := x ∈ Lp (I, X)  ∃ dt

we denote the Bochner–Sobolev space with respect to j . In the case .Y = X and didX dX .j = idX , we employ the abbreviations . dt := dt and 1,p,q

W 1,p,q (I, X) := WidX (I, X, X).

.

a Banach-valued function .x : I → X, where .I ⊆ R is an interval, we define .supp(x) := {t ∈ I | x(t) = 0 in X}. 5 This is an immediate consequence of the fundamental theorem for Bochner–Lebesgue functions (cf. [54, Lem. 2.1.2]). 4 For

32

2 Preliminaries

Proposition 2.23 Let X and Y be Banach spaces, .j : X → Y an embedding, and 1,p,q p, q ∈ [1, ∞]. Then, the space .Wj (I, X, Y ) forms a Banach space, if equipped with the norm

dj ·

:=  · Lp (I,X) + . ·  1,p,q . Wj (I,X,Y ) dt Lq (I,Y )

.

1,p,q

If .p, q ∈ (1, ∞) and .X, Y are reflexive, then .Wj

(I, X, Y ) is reflexive.

Proof An adaptation of the proofs of [54, Théorème 2.1.1 & Proposition 2.1.4].   The classical fundamental theorem of calculus admits an extension to Bochner– Sobolev spaces. Proposition 2.24 Let .I := (0, T ), .T < ∞, and .p, q ∈ [1, ∞]. Then, it holds: (i) First fundamental theorem of calculus: Each function .x ∈ W 1,p,q (I, X) (defined almost everywhere) possesses a unique representation .x c ∈ C 0 (I , X), which satisfies for every .t  , t ∈ I with .t  ≤ t 

x c (t) = x c (t ) +

.



t

t

dX x (s) ds dt

in X.

(2.7)

The resulting mapping .(·)c : W 1,p,q (I, X) → C 0 (I , X) is an embedding. In consequence, it holds .W 1,p,q (I, X) = W 1,∞,q (I, X) with norm equivalence. We, therefore, introduce the notation .W 1,q (I, X) := W 1,∞,q (I, X). (ii) Second fundamental theorem of calculus: The Volterra operator .X : Lq (I, X) → W 1,q (I, X), for every .x ∈ Lq (I, X) defined by 

t

(X x)(t) :=

x(s) ds

.

in X

for all t ∈ I ,

0

is a continuous right inverse of . ddtX : W 1,q (I, X) → Lq (I, X). Proof Concerning a proof of point (i), we refer to [54, Lemma 2.2.1]. A proof of point (ii), except for the continuity of .X , can be found in [75, Kap. IV, Lemma 1.8]. The verification of the claimed continuity of .X is an elementary calculation and, thus, omitted.   The following proposition is an essential component for extending the smoothing operator from Proposition 2.22 to a smoothing operator for Bochner–Sobolev spaces.

2.3 Banach-Valued Function Spaces

33

Proposition 2.25 (Extension in Time via Reflection) Let X and Y be Banach spaces, .j : X → Y an embedding, .I := (0, T ), .T < ∞, and .p, q ∈ [1, ∞]. For every function .x ∈ Lp (I, X), we define the in time extension via reflection by

 .ET x(t) :=

⎫ if t ∈ (−T , 0] ⎪ ⎪ ⎬

⎧ ⎪ ⎪x(−t) ⎨

if t ∈ I

x(t) ⎪ ⎪ ⎩x(2T − t)

⎪ ⎪ if t ∈ [T , 2T ) ⎭

in X

for every .t ∈ 3I := (−T , 2T ). Then, it holds: p p (i)  .ET : L (I, X) → L (3I, X) is well-defined, linear and Lipschitz continuous with constant 3. 1,p,q 1,p,q (ii)  .ET : W (I, X, Y ) → Wj (3I, X, Y ) is well-defined, linear and Lipsj chitz continuous with constant 3. More precisely, if .x ∈ Wj1,p,q (I, X, Y ), then

.

  dj  ET x − dj x = ET dt dt

in Lq (3I, Y ) ,

−

q q q where  .ET : L (I, Y ) → L (3I, Y ) is for every .y ∈ L (I, Y ) defined by − − .( ET y)(t) := ( ET y)(t) in Y if .t ∈ I and .( ET y)(t) := −( ET y)(t) in Y if .t ∈ 3I \ I .

 

Proof See [54, Proposition 2.3.1].

The combination of Propositions 2.25 and 2.22 results in the following smoothing method for Bochner–Sobolev spaces. Proposition 2.26 (Smoothing in Bochner–Sobolev Spaces) Let X and Y be Banach spaces, .j : X → Y an embedding, .I := (0, T ), .T < ∞, and .q, p ∈ [1, ∞). h x := For .x ∈ Wj1,p,q (I, X, Y ) and .h > 0, we define the smoothing operator .S I h 1,p,q ∞ S3I ( (I, X, Y ), it holds: ET x)|I ∈ C (I , X). Then, for every .x ∈ Wj  h − dj x q h h x)h>0 ⊆ W 1,p,q (I, X, Y ) with . dj S (i) .(S I j dt I x = S3I ET dt |I in .L (I, Y ) for every .h > 0. h x 1,p,q (ii) .suph>0 S ≤ 3xW 1,p,q (I,X,Y ) . I W (I,X,Y ) j

j

h x → x in .W 1,p,q (I, X, Y ) .(h → 0). (iii) .S I j Proof See [54, Théorème 2.3.1].

 

34

2 Preliminaries

2.3.4 Advanced Theory of Pseudo-Monotone Operators for Evolution Equations Let X be a reflexive Banach space, Y a Hilbert space with inner product .(·, ·)Y , and j : X → Y an embedding such that .R(j ) is dense in Y . Then, the triple .(X, Y, j ) is said to be an evolution triple. Denote by .RY : Y → Y ∗ the Riesz isomorphism with respect to .(·, ·)Y . Since j is a dense embedding, the adjoint operator .j ∗ : Y ∗ → X∗ and, therefore, also ∗ ∗ .e := j RY j : X → X are embeddings as well. We call e the canonical embedding of .(X, Y, j ). Note that .ex, yX = (j x, jy)Y for all .x, y ∈ X. .

Proposition 2.27 (Formula of Integration-by-Parts) Let .(X, Y, j ) be an evolution triple, .I := (0, T ), .T < ∞, and .p ∈ (1, ∞). Then, it holds: 1,p,p

(i) Given .x ∈ We (I, X, X∗ ), the function .j x ∈ Lp (I, Y ), defined by .(j x)(t) := j (x(t)) in Y for almost every .t ∈ I , has a unique representation 1,p,p 0 (I, X, X∗ ) → C 0 (I , Y ) .j c x ∈ C (I , Y ) and the resulting mapping .j c : We is an embedding. 1,p,p (ii) For every .x, y ∈ We (I, X, X∗ ) and .t, t  ∈ I with .t  ≤ t, it holds  t .

t

de x (s), y(s) dt

 X

s=t

ds = ((j c x)(s), (j c y)(s))Y s=t  −

 t t

de y (s), x(s) dt

 ds . X

Proof See [155, Chapter III.1, Proposition 1.2].

 

For an evolution triple .(X, Y, j ), .I := (0, T ), .T < ∞, and .p ∈ (1, ∞), we denote by .Lp (I, X) ∩j c C 0 (I , Y ), the subspace of .Lp (I, X) consisting of all functions .x ∈ Lp (I, X) for which the function .j x ∈ Lp (I, Y ), defined by .(j x)(t) := j (x(t)) in Y for almost every .t ∈ I , possesses a continuous rep1,p,p (I, X, X∗ ) ⊆ Lp (I, X) ∩j c C 0 (I , Y ) resentation .j c x ∈ C 0 (I , Y ). It holds .We (cf. Proposition 2.27 (i)). Furthermore, we endow .Lp (I, X) ∩j c C 0 (I , Y ) with the canonical norm  · Lp (I,X)∩j

.

cC

0 (I ,Y )

:=  · Lp (I,X) + j c (·)C 0 (I ,Y ) ,

which turns .Lp (I, X) ∩j c C 0 (I , Y ) into a Banach space (cf. [92, Remark 2.5]). For a sequence .(x n )n∈N ⊆ Lp (I, X) ∩j c C 0 (I , Y ) and a function .x ∈ Lp (I, X) ∩j c C 0 (I , Y ), it holds .x n  x in .Lp (I, X) ∩j c C 0 (I , Y ) .(n → ∞) if and only if p 0 .(x n )n∈N ⊆ L (I, X) ∩j C (I , Y ) is bounded and .(j c x n )(t)  (j c x)(t) in Y c .(n → ∞) for all .t ∈ I (cf. [92, Proposition 2.6]).

2.3 Banach-Valued Function Spaces

35

In the context of evolutionary problems, the following generalizations of the standard notions from the classical theory of pseudo-monotone operators (cf. Definition 2.1) have proved fruitful. Definition 2.20 Let .(X, Y, j ) be an evolution triple, .I := (0, T ), .T < ∞, and p ∈ (1, ∞). Then, an operator .A : Lp (I, X) ∩j c C 0 (I , Y ) → (Lp (I, X))∗ is said to

.

(i) be .C 0 -Bochner pseudo-monotone, if for .(x n )n∈N ⊆ Lp (I, X) ∩j c C 0 (I , Y ) from xn  x

.

in Lp (I, X) ∩j c C 0 (I , Y )

(n → ∞) , .

(2.8)

lim sup Ax n , x n − xLp (I,X) ≤ 0 ,

(2.9)

n→∞

for every .y ∈ Lp (I, X), it follows that Ax, x − yLp (I,X) ≤ lim inf Ax n , x n − yLp (I,X) .

.

n→∞

(ii) satisfy the .C 0 -Bochner condition (M), if for .(x n )n∈N ⊆ Lp (I, X) ∩j c C 0 (I , Y ) from (2.8), Ax n  x ∗

in (Lp (I, X))∗

(n → ∞) , .

(2.10)

lim sup Ax n , x n Lp (I,X) ≤ x , xLp (I,X) ,

(2.11)

.

∗

n→∞

it follows that Ax = x ∗ in (Lp (I, X))∗ .

.

(iii) .C 0 -Bochner coercive with respect to .x ∗ ∈ (Lp (I, X))∗ and .x0 ∈ Y , if there exists a constant .M := M(x ∗ , x0 ) > 0 such that for every .x ∈ Lp (I, X) ∩j c C 0 (I , Y ) from .

1 1 (j c x)(t)2Y + Ax − x ∗ , xχ[0,t] Lp (I,X) ≤ x0 2Y 2 2

for all t ∈ I ,

it follows that xLp (I,X)∩j

.

cC

0 (I ,Y )

≤ M.

On the basis of the concepts from Definition 2.20, the following abstract existence result applies.

36

2 Preliminaries

Theorem 2.1 Let .(X, Y, j ) be an evolution triple, .I := (0, T ), .T < ∞, and p ∈ (1, ∞). Moreover, let .A : Lp (I, X) ∩j c C 0 (I , Y ) → (Lp (I, X))∗ be bounded, 0 ∗ p ∗ .C -Bochner coercive with respect to .x ∈ (L (I, X)) and .x0 ∈ Y , and satisfying 1,p,p the .C 0 -Bochner condition (M). Then, there exists .x ∈ We (I, X, X∗ ) such that

.

 JX .

de x dt



+ Ax = x ∗

(j c x)(0) = x0

in (Lp (I, X))∗ , in Y ,



where .JX : Lp (I, X∗ ) → (Lp (I, X))∗ is the isomorphism from Proposition 2.20. Furthermore, the initial condition has to be understood in the sense of the unique continuous representation .j c x ∈ C 0 (I , Y ) (cf. Proposition 2.27 (i)).  

Proof See [92, Theorem 6.1].

In this book, we will rarely have access to the formula of integration-by-parts in Proposition 2.27. Then, the following generalization of Proposition 2.27 will often serve as a substitute. Proposition 2.28 (Non-Symmetric Formula of Integration-By-Parts) Let (X+ , Y, j+ ), .(X− , Y, j− ) be evolution triples such that .X+ → X− and − ≤ p + < ∞, .R : Y → Y ∗ the Riesz .j− |X+ = j+ , .I := (0, T ), .T < ∞, .1 < p Y ∗ ∗ ∗ and6 isomorphism, .e+ := j− RY j+ : X+ → X− , .e− := j+∗ RY j− : X− → X+ .

1,p+ ,(p− )

∗ (I, X+ , X− ),

1,p− ,(p+ )

∗ (I, X− , X+ ).

W+ (I ) := We+

.

W− (I ) := We− Then, it holds:

−

(i) Every .x ∈ W− (I ) such that .j − x ∈ Lp (I, Y ), defined by .(j − x)(t) := j− (x(t)) in Y for almost every .t ∈ I , satisfies .j − x ∈ L∞ (I, Y ), has a 0 representation .j − ω x ∈ Cω (I , Y ). Moreover, for every .y ∈ W+ (I ), the function + + p (I, Y ), defined by .(j + y)(t) := j (y(t)) in Y for almost every .j y ∈ L + + 0 .t ∈ I , has a representation .j c y ∈ C (I , Y ).

as .j+ : X+ → Y and .j− : X− → Y are dense embeddings, their adjoints operators ∗ and .j ∗ : Y ∗ → X ∗ and, thus, .e : X ∗ ∗ Y ∗ → X+ + + → X− and .e− : X− → X+ , are − − embeddings. This ensures the well-definedness of .W+ (I ) and .W− (I ) (cf. Definition 2.19).

6 Inasmuch ∗

.j+ :

2.3 Banach-Valued Function Spaces

37

(ii) For every .x ∈ W− (I ) with .j − x ∈ L∞ (I, Y ), .y ∈ W+ (I ), and .t, t  ∈ I with  .t ≤ t, it holds  t .

t

de− x (s), y(s) dt

 X+

s=t + ds = ((j − ω x)(s), (j c y)(s))Y s=t  −

 t t

de+ y (s), x(s) dt

 ds . X−

(iii) For every .x ∈ W− (I ) and .y ∈ W+ (I ) with .supp(y) ⊆ I , it holds   .

I

de− x (s), y(s) dt

 

 ds = − X+

I

de+ y (s), x(s) dt

 ds . X−

Proof ad (i) Let .x ∈ W− (I ) with .j − x ∈ L∞ (I, Y ). Then, we have that 1,p− ,(p+ ) (I, X ∗ ), where .(e x)(t) := e (x(t)) in .X ∗ for almost every .e − x ∈ W − − + + .t ∈ I , as well. Therefore, Proposition 2.24 (i) yields a continuous representation 0 ∗ 1,p− ,(p+ ) (I, X ∗ ). Thus, since .j ∗ R : Y → .(e − x)c ∈ C (I , X+ ) of .e − x ∈ W + + Y ∗ is an embedding, since .j : X → Y is a dense embedding and .R : Y → Y ∗ X+ + + Y an isomorphism, Proposition 2.19 yields a unique weakly continuous representation − − − − 0 ∞ .j ω x ∈ Cω (I , Y ) of .j x ∈ L (I, Y ), i.e., it holds .(j ω x)(t) = (j x)(t) in Y for almost every .t ∈ I . 1,p− ,(p− ) ∗ ), (I, X− , X− Next, let .y ∈ W+ (I ). Then, we have that .y ∈ We d

y

− 

e+ ∗ , with . de y = ∗ ), since where .e := j−∗ RY j− : X− → X− in .L(p ) (I, X− dt dt + − p p + − .L (I, X+ ) → L (I, X− ) due to .p ≥ p and .X+ → X− . In consequence, 0 Proposition 2.27 (i) yields a unique continuous representation .j + c y ∈ C (I , Y ) of + + p (I, Y ), i.e., it holds .(j + y)(t) = (j + y)(t) in Y for almost every .t ∈ I . .j y ∈ L c ad (ii) Let .x ∈ W− (I ) with .j − x ∈ L∞ (I, Y ) and .y ∈ W+ (I ). 1,p− ,(p− ) ∗ ), where once more (I, X− , X− Then, we have again that .y ∈ We ∗ ∗ .e := j− RY j− : X− → X− . In addition, by virtue of Proposition 2.26, we choose h x)h>0 ⊆ C ∞ (I , X− ) of .x ∈ W− (I ). Therefore, the approximation .(x h )h>0 := (S I by resorting to Propositions 2.26 (iii) and 2.22 (ii), we obtain

xh → x

(h → 0) ,.

(2.12)

sup j x h L∞ (I,Y ) ≤ 3j xL∞ (I,Y ) .

(2.13)

.

−

h>0

in W− (I ) −

38

2 Preliminaries

∗ ) .(h → 0), Due to Proposition 2.24 (i), (2.12) implies .e− x h → (e− x)c in .C 0 (I , X+ ∗ ) are defined by .(e x )(t) := e (x (t)) in .X ∗ where .(e− x h )h>0 ⊆ C ∞ (I , X+ − h − h + ∗ ) denotes the unique continuous for all .t ∈ I and .h > 0, and .(e− x)c ∈ C 0 (I , X+ − +  ∗ ) (cf. (i)). In particular, we have that representation of .e− x ∈ W 1,p ,(p ) (I, X+ ∗ in X+

(e− x h )(t) → (e− x)c (t)

.

for all t ∈ I

(h → 0) .

(2.14)

− ∗ is an embedding and .sup Since .j+∗ RY : Y → X+ h>0 j x h L∞ (I,Y ) < ∞ (cf. (2.13)), by applying Proposition 2.17 to (2.14), we find that

(j − x h )(t)  (j − ω x)(t)

in Y

.

for all t ∈ I

(h → 0) . −

(2.15)

− 

∗ ) with On the other hand, we have that .(x h )h>0 ⊆ We1,p ,(p ) (I, X− , X− − ) de x h (p ∗ . (I, X− ) for every .h > 0. Therefore, since also h ) in .L dt = e(∂t x 1,p− ,(p− ) (I, X , X ∗ ), Proposition 2.27 (ii) is applicable. It yields for .y ∈ We − − every .t, t  ∈ I with .t  ≤ t and .h > 0 that

 t t .

 s=t

de x h (s), y(s) ds = ((j − x h )(s), (j + c y)(s))Y s=t  dt X−   t de y ds . − (s), x h (s) dt t X−

(2.16)

Exploiting that, due to .e = j−∗ RY j− , .e− = j+∗ RY j− and .j− |X+ = j+ , it holds for almost every .t ∈ I  .

de x h (t), y(t) dt

 X−

= e(∂t x h (t)), y(t))X− = (j− (∂t x h (t)), j− (y(t)))Y = (j− (∂t x h (t)), j+ (y(t)))Y = e− (∂t x h (t)), y(t)X+   de− x h (t), y(t) = , dt X+

2.3 Banach-Valued Function Spaces

39 d

y

also taking into account that . ddte y = edt+ in .L(p with .t  ≤ t and .h > 0 can be rewritten as  t t

de− x h (s), y(s) dt



− )

∗ ), (2.16) for every .t, t  ∈ I (I, X−

s=t

ds = ((j − x h )(s), (j + c y)(s))Y s=t 

X+

.

−

 t t

de+ y (s), x h (s) dt



(2.17) ds. X−

Therefore, by passing for .h → 0 in (2.17), using (2.12) and (2.15) in doing so, we conclude (ii). ad (iii) For .x ∈ W− (I ), .y ∈ W+ (I ) with .supp(y) ⊆ I , and .(x h )h>0 ⊆ C ∞ (I , X− ) as in (ii), it holds, thanks to (ii) or (2.17), for every .h > 0 that   .

I

de− x h (s), y(s) dt

 

 ds = − X+

I

de+ y (s), x h (s) dt

 ds .

(2.18)

X−

On the other hand, based on Proposition 2.26 (iii), .(x h )h>0 ⊆ C ∞ (I , X− ) again satisfies (2.12). Therefore, by passing for .h → 0 in (2.18), using (2.12) in doing so, we conclude (iii).  

Part I

Main Part

Chapter 3

Variable Bochner–Lebesgue Spaces

To begin with, we consider, as a model problem, the following initial-boundary value problem: Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain, I := (0, T ), T < ∞, a finite time interval, QT := I ×  a time-space cylinder, and T := I × ∂. We are searching for a vector-valued function x : QT → Rd that solves the system ∂t x − div(S(·, ·, ε(x))) + b(·, ·, x) = f − div(F ) .

x=0 x(0) = x0

in QT , on T ,

(3.1)

in  .

Here, f : QT → Rd denotes a given vector field, F : QT → Md×d sym a given symmetric tensor field, x0 :  → Rd an initial condition, and ε(x) := 12 (∇x + ∇x ) denotes the symmetric gradient. Furthermore, the mapping S : QT × Md×d sym → d×d Msym is supposed to possess a (p(·, ·), δ)-structure, i.e., for a bounded exponent p : QT → [1, +∞) and some δ ≥ 0, the following properties are satisfied: d×d 1 (S.1) S : QT × Md×d sym → Msym is a Carathéodory mapping. (S.2) |S(t, x, A)| ≤ α(δ + |A|)p(t,x)−2 |A| + β(t, x) for every A ∈ Md×d sym and a.e.  

(·,·)  p 2 (QT , R≥0 ) . (t, x) ∈ QT α ≥ 1, β ∈ L

1 S(·, ·, A) :

d×d d×d d×d QT → Md×d sym is measurable for every A ∈ Msym and S(t, x, ·) : Msym → Msym is  continuous for almost every (t, x) ∈ QT . 2 For a measurable set G ⊆ Rn , n ∈ N, and p ∈ P∞ (G), let Lp(·) (G, R ) := {f ∈ Lp(·) (G) | ≥0 f ≥ 0 a.e. in G}.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaltenbach, Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents, Lecture Notes in Mathematics 2329, https://doi.org/10.1007/978-3-031-29670-3_3

43

44

3 Variable Bochner–Lebesgue Spaces

(S.3) S(t, x, A) : A ≥ c0 (δ + |A|)p(t,x)−2 |A|2 − c1 (t, x) for every A ∈ Md×d sym and    1 a.e. (t, x) ∈ QT c0 > 0, c1 ∈ L (QT , R≥0 ) .     (S.4) S(t, x, A) − S(t, x, B) : A − B ≥ 0 for every A, B ∈ Md×d sym and a.e.  (t, x) ∈ QT . The mapping b : QT × Rd → Rd is supposed to have the following properties: (B.1) b : QT × Rd → Rd is a Carathéodory mapping. (B.2) |b(t, x, a)| ≤ γ (1 + |a|)r + η(t, x) for every a ∈ Rd and a.e. (t, x) ∈ QT   −

γ > 0, η ∈ L(p ) (QT , R≥0 ), r < (p− )• /(p− ) .3  (B.3) b(t, x, a) · a ≥ −c2 (t, x) for every a ∈ Rd and a.e. (t, x) ∈ QT c2 ∈  L1 (QT , R≥0 ) . A system like (3.1) has, e.g., applications in non-linear elastic mechanics [171] or in the field of image processing [1, 38, 80, 82, 115], and can also be seen as a simplification of the unsteady p(·, ·)-Stokes and p(·, ·)-Navier–Stokes equations, as it does not contain an incompressibility constraint, but yet the non-linear elliptic operator S is governed only by the symmetric gradient. Similar equations also appear in Zhikov’s model for the thermistor problem in [174], or in the investigation of variational integrals with non-standard growth by Acerbi and Mingione [2–4] or by Marcellini [121]. Great progress was also made by Diening and R˚užiˇcka in [44], at least in the steady counterpart of (3.1), as they proved the validity of Korn’s inequality for variable Sobolev spaces. As usual, non-negligible difficulties in the treatment of unsteady problems such as (3.1) lie in the choice of appropriate function spaces in which we expect our solutions to exist in. A conventional procedure for determining which function spaces are actually the correct ones consists in assuming the existence of an already sufficiently smooth solution x : QT → Rd and then to formally test the problem with its expected solution, i.e., we form the inner product between the equation (3.1)1 and x, integrate over  × [0, t) for some arbitrary time slice t ∈ I , and use the (p(·, ·), δ)-structure of S, i.e., the coercivity condition (S.3), the condition (B.3) and ∂t 12 |x|2 = ∂t x · x in QT , to obtain a constant M := M(x0 , f , F , S, b) > 0, which does not depend on the solution x, such that4 

 .

|x(t, x)|2 dx +

ess sup t∈I



|ε(x)(t, x)|p(t,x) dtdx ≤ M .

(3.2)

QT

So, what are the correct function spaces for the treatment of (3.1)? In the case of a constant exponent p(·, ·) ≡ p ∈ (1, ∞), the situation is quite clear. Here, we have ∞ 2 d that ε(x) ∈ Lp (QT , Md×d sym ) and x ∈ L (I, L () ), which by recourse to the isometrically isomorphic identification, see, e.g., [149], Lp (QT ) ∼ = Lp (I, Lp ()) ,

.

  for every s ∈ (1, ∞). (·)• ∈ W 1,∞ (1, ∞) is defined by s• := min s + 2, s d+2 d 4 Here, dtdx := dt ⊗ dx denotes the (d + 1)-dimensional Lebesgue measure. 3 Here,

(3.3)

q,p

q,p

3.1 The Spaces X∇ (QT ) and Xε

45

(QT )

which is based on Fubini’s theorem, in conjunction with both Korn’s and Poincaré’s inequality for the constant exponent p ∈ (1, ∞), using the zero Dirichlet bound1,p ary condition (3.1)2 , suggests that x ∈ Lp (I, W0 ()d ) ∩ L∞ (I, L2 ()d ). These observations open the door to the framework of both Bochner–Lebesgue spaces and Bochner–Sobolev spaces. Unfortunately, in the case of a non-constant exponent p ∈ Plog (QT ), an identification of the form of (3.3) is not available anymore in general because of the definition of the Luxemburg norm on variable Lebesgue spaces (cf. Definition 2.5). Nevertheless, (3.2) still suggests that x ∈ L∞ (I, L2 ()d ), i.e., a classical Bochner– Lebesgue space structure, on the one hand, and that ε(x) ∈ Lp(·,·) (QT , Md×d sym ), i.e., a kind of variable exponent Sobolev space structure on the time-space cylinder QT involving only spatial weak derivatives, on the other hand. Therefore, the heuristic approach that motivates the introduction of the correct mathematical framework is to merge the theories of both variable Sobolev spaces (cf. Sect. 2.2.4) and Bochner– Lebesgue spaces (cf. Sects. 2.3.2–2.3.4), i.e., the appropriate function spaces for the treatment of (3.1) are hybrid spaces, so-called variable Bochner–Lebesgue spaces. In this way, we get a foot in the door of both the theory Bochner–Lebesgue spaces and variable Lebesgue and Sobolev spaces, which will often lead to synergies, as can be observed in this book. Variable Bochner–Lebesgue spaces are not new and have for the first time been proposed by Antontsev and Shmarev in [9, 10], and have frequently been used for the analysis of unsteady problems in a variable exponent setting ever since, cf. [6, 11, 51, 53, 60–64, 80, 102, 132, 141, 153, 169, 170]. The peculiarity of the problem (3.1), in comparison to these contributions, is that the main part, i.e., the mapping S, is governed only by the symmetric gradient ε(·) rather than by the full gradient ∇. In steady problems, in virtue of the validity of Korn’s inequality, this does not much affect the analysis. Nevertheless, we will find that in the unsteady level, a Korn type inequality cannot be valid, which has a far-reaching impact on the analytical treatment of the problem (3.1). So far for now, let us stop with this heuristic description and finally give a detailed introduction of variable Bochner–Lebesgue spaces.

q,p

q,p

3.1 The Spaces X∇ (QT ) and Xε

(QT )

Motivated through the previous discussion, we now introduce variable Bochner– Lebesgue spaces. Recall that the basic idea is to construct hybrid spaces that incorporate concepts from the theory of Bochner–Lebesgue spaces and the theory of variable Sobolev spaces. In order to incorporate the structure of a Bochner– Lebesgue space, we follow the common approach of reinterpreting a function x : QT → Rd , i.e., defined on a time-space cylinder, via x(t) := x(t, ·)

.

in 

for a.e. t ∈ I ,

(3.4)

46

3 Variable Bochner–Lebesgue Spaces

as a function defined on the time interval I that takes its value in appropriate, potentially on time depending, function spaces X(t), t ∈ I , defined on , i.e., that x(t) ∈ X(t) for almost every t ∈ I . We will frequently use the identification (3.4) for functions defined on the cylinder QT . Therefore, our first task is to introduce these time-dependent function spaces. Throughout the entire section, if nothing else is stated, let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and q, p ∈ P∞ (QT ). Definition 3.1 (Time Slice Spaces) We define for almost every t ∈ I , the time slice spaces ˚q(t,·),p(t,·) ()d , ˚q,p (t) := X X ∇ ∇

.

˚εq,p (t) := X ˚εq(t,·),p(t,·) () . X

Furthermore, we define the limiting time slice spaces .

+ ,p +

q,p ˚+ ˚q := X X ∇

()d ,

− ,p −

q,p ˚− ˚q := X X ∇

()d .

Remark 3.1 (i) For every q, p ∈ P∞ (QT ), we have that q(t, ·), p(t, ·) ∈ P∞ () for almost every t ∈ I . In fact, let p ∈ P∞ (QT ), i.e., p ≤ M in K  for M > 0 and measurable K ⊆ QT with |K| = 0. Then, [58, §V.1, Kor.1.6] yields |Kt | = 0 for almost every t ∈ I , where Kt := {x ∈  | (t, x) ∈ K}. (ii) In the case of time-independent exponents q, p ∈ P∞ (), we define .

˚q,p := X ˚q(·),p(·) ()d , X ∇ ∇

˚εq,p := X ˚εq(·),p(·) () . X

(iii) Recall that if p ∈ Plog (QT ) with p− > 1, by virtue of Korn’s inequality (cf. ˚q,p (t) and X ˚εq,p (t) coincide for every t ∈ I , Proposition 2.13), the spaces X ∇ with a possibly on t ∈ I depending norm equivalence. Thus, for p ∈ Plog (QT ) with p− > 1, we define for every t ∈ I .

˚εq,p (t) = X ˚q,p (t) . ˚q,p (t) := X X ∇

For time-independent exponents q ∈ P∞ () and p ∈ Plog () with p− > 1, we define .

˚q,p . ˚q,p := X ˚εq,p = X X ∇ q,p

q,p

˚+ → X ˚ (t) → (iv) For almost every t ∈ I , there hold the dense embeddings X ∇ q,p q,p ˚ ˚ Xε (t) → X− . ˚q,p (t), t ∈ I , and X ˚εq,p (t), t ∈ I , we next By means of the time slice spaces X ∇ introduce two types of variable Bochner–Lebesgue spaces.

q,p

q,p

3.1 The Spaces X∇ (QT ) and Xε

47

(QT )

Definition 3.2 We define the variable Bochner–Lebesgue spaces  q,p X∇ (QT ) := x ∈ Lq(·,·) (QT )d | ∇x ∈ Lp(·,·) (QT )d×d ,  ˚q,p (t) for a.e. t ∈ I , x(t) ∈ X ∇  q,p Xε (QT ) := x ∈ Lq(·,·) (QT )d | ε(x) ∈ Lp(·,·) (QT , Md×d sym ),  q,p ˚ε (t) for a.e. t ∈ I . x(t) ∈ X

.

Furthermore, we define the limiting Bochner–Lebesgue spaces q,p

X+ (QT ) := Lmax{q

.

+ ,p + }

q,p

q,p

X− (QT ) := Lmin{q

˚+ ) , (I, X

− ,p − }

q,p

˚− ) . (I, X

˚q,p (t), t ∈ I , and X ˚εq,p (t), t ∈ I , Remark 3.2 Note that the time slice spaces X ∇ primarily serve to encode the zero Dirichlet boundary condition (3.1)2 in the q,p q,p definition of X∇ (QT ) and Xε (QT ), but they also implicitly reflect the Bochner– q,p q,p Lebesgue character of X∇ (QT ) and Xε (QT ), in the sense of (3.4), which in many situations affords us access to the theory of Bochner–Lebesgue spaces. q,p

q,p

Before we equip X∇ (QT ) and Xε (QT ) with appropriate norms, we will address the question of whether there exists a wide range of variable exponents for q,p q,p which the spaces X∇ (QT ) and Xε (QT ) coincide, i.e., whether a distinction q,p q,p of X∇ (QT ) and Xε (QT ) becomes unnecessary, if, e.g., the variable exponents are smooth and/or not depending on time. Recall that for a log-Hölder continuous ˚q(·),p(·) ()d and X ˚εq(·),p(·) (), exponent p ∈ Plog () with p− > 1, the spaces X q,p q,p ∇ i.e., the steady counterparts of X∇ (QT ) and Xε (QT ), respectively, by virtue of Korn’s inequality (cf. Proposition 2.13), coincide. Apart from that, in the case of a constant exponent p ∈ (1, ∞), Korn’s inequality implies the norm equivalence q,p q,p ∇ · Lp (QT )d×d ∼ ε(·)Lp (QT )d×d on Xε (QT ), and the spaces X∇ (QT ) and q,p Xε (QT ) coincide again. For these reasons, one may wonder whether this also holds true in the case of a non-constant exponent p ∈ Plog (QT ) with p− > 1. Regrettably, even if the variable exponent is smooth and does not depend on time, the answer will be negative. This can be traced back to the following nerve-wracking phenomenon that occurs in variable Bochner–Lebesgue spaces and leads to the invalidity of a variety of inequalities. Proposition 3.1 (Wet Blanket5 ) Let  ⊆ Rd , d ≥ 2, be an arbitrary bounded domain and let k, l, m, n ∈ N. Furthermore, let F ⊆ C0∞ ()m×n × C0∞ ()k×l 6 be a family of function couples containing a couple (G0 , F0 ) ∈ F such that F0 = 0 and int(supp(G0 )) \ supp(F0 ) = ∅. Then, for every 1 < α < β < ∞, there

5 The

author chose this name to express his deep dislike towards Proposition 3.1 because it only proves invalidities. 6 For n ∈ N, we identify C ∞ ()n×1 ∼ C ∞ ()1×n ∼ C ∞ ()n in this proposition. = =

48

3 Variable Bochner–Lebesgue Spaces

exists a smooth exponent p ∈ C ∞ (Rd ), with p|QT ∈ Plog (QT ),7 p− = α and p+ = β, such that for every ϕ ∈ Lα (I ) \ Lβ (I ), it holds ϕF0 ∈ Lp(·) (QT )k×l and / Lp(·) (QT )m×n . In addition, we have that ϕG0 ∈ .

sup

ϕG0 Lp(·) (QT )m×n

ϕ∈C0∞ (I )

ϕF0 Lp(·) (QT )k×l

= ∞,

i.e., there is no constant c > 0 such that ϕG0 Lp(·) (QT )m×n ≤ cϕF0 Lp(·) (QT )k×l for every ϕ ∈ C0∞ (I ). Proof First, we define 1 := int(supp(F0 )) = ∅, ε1 := 1 + Bεd (0) and ε2 :=  \ ε1 for ε > 0. Owing to 1 ⊂⊂ , we readily observe for ε > 0 sufficiently small that ε1 ⊂⊂  and, thus, ε2 = ∅. In addition, on the basis of int(supp(G0 )) \ supp(F0 ) = ∅, we obtain for a possibly smaller ε > 0 that ε2 ∩ int(supp(G0 )) = ∅. Let us fix such an ε > 0 and define the variable exponent    d   d + β 1 − ωε/2 ∈ C ∞ (Rd ) , p := α ωε/2 ∗ χε/2 ∗ χε/2 1 1

(3.5)

.

where (ωεd )ε>0 ⊆ C0∞ (Rd ) denotes the family of scaled standard mollifiers from Remark 2.6. Then, it holds p(x) = α if x ∈ 1 , p(x) = β if x ∈ Rd \ ε1 and α ≤ p(x) ≤ β if x ∈ ε1 \ 1 . In particular, it holds p|QT ∈ Plog (QT ) and for every ϕ ∈ C0∞ (I ) and λ > 0 that ρp(·) (λ−1 ϕF0 ) = λ−1 ϕF0 αLα (I ×

.

1)

k×l

= λ−α ϕαLα (I ) F0 αLα (

1)

k×l

,

i.e., ϕF0 Lp(·) (QT )k×l = ϕLα (I ) F0 Lα (1 )k×l according to the definition of the Luxemburg norm (cf. Definition 2.5). On the other hand, it holds for every ϕ ∈ C0∞ (I ) and λ > 0 that ρp(·) (λ−1 ϕG0 ) ≥ λ−1 ϕG0 Lβ (I ×ε )m×n = λ−β ϕLβ (I ) G0 Lβ (ε )m×n , β

.

β

2

β

2

i.e., ϕG0 Lp(·) (QT )m×n ≥ ϕLβ (I ) G0 Lβ (ε )m×n . Next, take ϕ ∈ Lα (I ) \ Lβ (I ) 2 and a sequence (ϕj )j ∈N ⊆ C0∞ (I ) such that ϕj → ϕ in Lα (I ) (j → ∞). Then, ϕF0 ∈ Lp(·) (QT )k×l and ϕG0 ∈ / Lp(·) (QT )m×n . Furthermore, we have that ϕj Lβ (I ) → ∞ (j → ∞) and, therefore, that .

ϕj G0 Lp(·) (QT )m×n ϕj F0 Lp(·) (QT )k×l

≥

ϕj Lβ (I ) G0 Lβ (ε )m×n 2

ϕj Lα (I ) F0 Lα (1 )k×l

→ ∞ (j → ∞) ,

whereby we have made use of G0 Lβ (ε )m×n > 0, since, by construction, the 2   intersection ε2 ∩ int(supp(G0 )) = ∅ is open. 7 Here,

we extend p ∈ C ∞ (Rd ) constantly in time, i.e., p(t, x) := p(x) for all (t, x) ∈ QT .

q,p

q,p

3.1 The Spaces X∇ (QT ) and Xε

49

(QT )

To get a more in-depth insight into the underlying idea of Proposition 3.1, consider next a general operator T : C0∞ ()k×l → C0∞ ()m×n , k, l, m, n ∈ N, that does not preserve compact supports, in the sense that there exists F0 ∈ C0∞ ()k×l such that F0 = 0 and int(supp(T (F0 ))) \ supp(F0 ) = ∅. Then, Proposition 3.1 applied to the family FT := G(T ) := {(T (F), F) | F ∈ C0∞ ()k×l } yields a smooth, time-independent variable exponent p ∈ C ∞ (Rd ) that does not admit a constant c > 0 such that for every φ ∈ C0∞ (QT )k×l , it holds T (φ)Lp(·) (QT )m×n ≤ cφLp(·) (QT )k×l .

.

Note that the formulation of Proposition 3.1 with respect to a general family F ⊆ C0∞ ()k×l × C0∞ ()m×n allows us to establish the invalidity of inequalities that cannot be properly described by an operator T : C0∞ ()k×l → C0∞ ()m×n , such as, e.g., Korn’s inequality. q,p

Remark 3.3 (Invalidity of Korn’s Inequality on Xε (QT )) Let  ⊆ Rd , d ≥ 2, be an arbitrary bounded domain, I := (0, T ), T < ∞, and QT := I × . Moreover, let   F Korn := (∇x, ε(x)) | x ∈ C0∞ ()d ⊆ C0∞ ()d×d × C0∞ ()d×d .

.

Let η ∈ C0∞ () with η = 1 in G, where G ⊂⊂  is a domain, and A ∈ Rd×d \ {0} skew-symmetric.8 If we set x(x) := η(x)Ax for every x ∈ , then x ∈ C0∞ ()d with ∇x = A and ε(x) = 0 in G. In other words, (∇x, ε(x)) ∈ F Korn satisfies both ε(x) = 0 and int(supp(∇x)) \ supp(ε(x)) = ∅. Consequently, according to Proposition 3.1, there exists a smooth, time-independent exponent p ∈ C ∞ (Rd ) with p− > 1 that does not admit a constant c > 0 such that for every φ ∈ C0∞ (QT )d , it holds ∇φLp(·) (QT )d×d ≤ cε(φ)Lp(·) (QT )d×d .

.

−

p−,p

+

Apart from that, for every ϕ ∈ Lp (I ) \ Lp (I ), it holds ϕx ∈ Xε (QT ) \ − p−,p X∇ (QT ), since ϕx ∈ Lp (QT )d with ε(ϕx) = ϕε(x) ∈ Lp(·) (QT , Md×d sym ), but p−,p

p−,p

∇(ϕx) = ϕ∇x ∈ / Lp(·) (QT )d×d , i.e., we have that Xε (QT ) = X∇ (QT ). 2 (0), G := B 2 (0), x ∈ The situation is illustrated in Fig. 3.1 for  := B2.5  0.6 ∞ 2 C0 () , for every x ∈  defined by x(x) := η(x)Ax, where A := 10 −1 ∈ R2×2 , 0 η := χB 2 (0) ∗ ωε2 ∈ C0∞ () for ε := 0.4. 1

8A

tensor A ∈ Rn×n , n ∈ N, is called skew-symmetric, if A = −A in Rn×n .

50

3 Variable Bochner–Lebesgue Spaces

Fig. 3.1 Plots of |∇x| ∈ C0∞ () (blue/left), |ε(x)| ∈ C0∞ () (red/middle) and p| ∈ C ∞ () (green/right), constructed according to Proposition 3.1 for d = 2, α = 1.1 and β = 2 12

ϕ

10

12

8

8

6

6

4

4

2

2

0 −1.5

(ϕj )j=1,...,5

10

0 −1.0

−0.5

0.0

0.5

1.0

1.5 −1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

Fig. 3.2 Plots of ϕ ∈ L1.1 (I ) \ L2 (I ) (red/left) and (ϕj )j =1,...,5 ⊆ C0∞ (I ) (blue/right)

In Fig. 3.2, one can see for I := (−1.5, 1.5), the function ϕ ∈ L1.1 (I ) \ L2 (I ),   1 for every t ∈ I defined by ϕ(t) := χ(−1,1) (t) |t|− 2 − 1 , and its approximations (ϕj )j ∈N := (ϕ ∗ ω21−j )j ∈N ⊆ C0∞ (I ). According to Proposition 3.1, the sequence (φ j )j ∈N ⊆ C0∞ (QT )2 , for every (t, x) ∈ QT and j ∈ N defined by φ j (t, x) := ϕj (t)x(x), satisfies ∇φ j Lp(·) (QT )2×2 /ε(φ j )Lp(·) (QT )2×2 → ∞ (j → ∞). In consequence, even if p ∈ P∞ () ∩ C ∞ (), i.e., if the variable exponent is smooth and does not depend on time, it is mandatory to distinguish between q,p q,p the spaces X∇ (QT ) and Xε (QT ). In particular, we always have to endow the q,p q,p spaces X∇ (QT ) and Xε (QT ) with different norms to guarantee at least the well-definedness of these norms. q,p

q,p

Proposition 3.2 The spaces X∇ (QT ) and Xε (QT ) form Banach spaces, if equipped with the norms  · X q,p (QT ) :=  · Lq(·,·) (QT )d + ∇ · Lp(·,·) (QT )d×d ,

.

∇

 · Xεq,p (QT ) :=  · Lq(·,·) (QT )d + ε(·)Lp(·,·) (QT )d×d , respectively. In addition, we have that q,p

q,p

q,p

q,p

X+ (QT ) → X∇ (QT ) → Xε (QT ) → X− (QT ) .

.

q,p

q,p

3.1 The Spaces X∇ (QT ) and Xε

51

(QT ) q,p

Proof We only give a proof for Xε (QT ) because all arguments are transferable q,p to X∇ (QT ). The norm properties follow from the definition. It remains to prove the comq,p pleteness. To this end, let (x n )n∈N ⊆ Xε (QT ) be a Cauchy sequence, i.e., q(·,·) d (QT ) and (ε(x n ))n∈N ⊆ Lp(·,·) (QT , Md×d also (x n )n∈N ⊆ L sym ) are Cauchy d×d q(·,·) d p(·,·) sequences. Since L (QT ) and L (QT , Msym ) are Banach spaces (cf. Proposition 2.1 and Remark 2.2), there, thus, exist functions x ∈ Lq(·,·) (QT )d and E ∈ Lp(·,·) (QT , Md×d sym ) such that xn → x .

ε(x n ) → E

in Lq(·,·) (QT )d

(n → ∞) ,

in Lp(·,·) (QT , Md×d sym )

(n → ∞) .

(3.6)

Because norm convergence implies modular convergence (cf. Proposition 2.3), also using Fubini’s theorem, we infer from (3.6) that  ρq(t,·) (x n (t) − x(t)) + ρp(t,·) (ε(x n )(t) − E(t)) dt

.

I

= ρq(·,·) (x n − x)

(3.7)

+ ρp(·,·) (ε(x n ) − E) → 0 (n → ∞) . Fischer–Riesz’s theorem in L1 (I ) applied to (3.7) yields a subsequence (x n )n∈ ⊆ q,p Xε (QT ), with a cofinal subset  ⊆ N, such that for almost every t ∈ I ρq(t,·) (x n (t) − x(t)) + ρp(t,·) (ε(x n )(t) − E(t)) → 0

.

(  n → ∞) .

(3.8)

Since conversely modular convergence implies norm convergence, (3.8) gives for almost every t ∈ I x n (t) → x(t) .

ε(x n )(t) → E(t)

in Lq(t,·) ()d

(  n → ∞) ,

in Lp(t,·) (, Md×d sym )

(  n → ∞) ,

˚εq,p (t) is for almost every t ∈ I a Cauchy sequence. Since i.e., (x n (t))n∈ ⊆ X ˚εq,p (t) is a Banach space (cf. Definition 2.8), for almost every t ∈ I , the space X ˚εq,p (t) with ε(x)(t) = E(t) in we conclude for almost every t ∈ I that x(t) ∈ X q,p ˚ Lp(t,·) (, Md×d sym ) and x n (t) → x(t) in Xε (t) (  n → ∞). In other words, we q,p q,p have that x ∈ Xε (QT ) and x n → x in Xε (QT ) (n → ∞). The stated embedding chain is a straightforward application of Corollary 2.1 and Korn’s inequality (cf. Proposition 2.13), respectively.   q,p

Henceforth, in this book, predominantly the space Xε (QT ) will be considered since this space is of greater interest with regard to the model problem (3.1). We emphasize, nevertheless, that all following definitions and results admit congruent q,p adaptations to the space X∇ (QT ).

52

3 Variable Bochner–Lebesgue Spaces q,p

Corollary 3.1 (Fischer–Riesz for Time Slices of Xε (QT )) Let (x n )n∈N ⊆ q,p q,p Xε (QT ) be such that x n → x in Xε (QT ) (n → ∞). Then, there exists q,p a subsequence (x n )n∈ ⊆ Xε (QT ), with a cofinal subset  ⊆ N, such that ˚εq,p (t) (  n → ∞) for almost every t ∈ I . x n (t) → x(t) in X q,p

The next proposition gives us first structural properties of Xε (QT ). q,p

Proposition 3.3 The mapping ε: Xε defined by ε x := (x, ε(x))

.

(QT )→Lq(·,·) (QT )d ×Lp(·,·) (QT , Md×d sym ),

in Lq(·,·) (QT )d × Lp(·,·) (QT , Md×d sym )

q,p

for every x ∈ Xε (QT ), is an isometric isomorphism into its range R(ε ). In q,p q,p particular, Xε (QT ) is separable. In addition, if q − , p− > 1, then Xε (QT ) is reflexive. q,p

Proof Since ε : Xε (QT ) → Lq(·,·) (QT )d × Lp(·,·) (QT , Md×d sym ) is an isometry q,p and Xε (QT ) is a Banach space (cf. Proposition 3.2), R(ε ) is a closed subspace of Lq(·,·) (QT )d × Lp(·,·) (QT , Md×d sym ), i.e., R(ε ) is a Banach space and, therefore, q,p ε : Xε (QT ) → R(ε ) an isometric isomorphism. Because Lq(·,·) (QT )d and Lp(·,·) (QT , Md×d sym ) are separable (cf. Proposition 2.1 and Remark 2.2), the product Lq(·,·) (QT )d × Lp(·,·) (QT , Md×d sym ) is separable as well. Then, the range R(ε ) inherits the separability of Lq(·,·) (QT )d × Lp(·,·) (QT , Md×d sym ) and, in virtue of q,p q,p the isomorphism ε : Xε (QT ) → R(ε ), likewise Xε (QT ). If, in addition, q − , p− > 1, then the spaces Lq(·,·) (QT )d and Lp(·,·) (QT , Md×d sym ) and, therefore, d×d q(·,·) d p(·,·) also the product L (QT ) ×L (QT , Msym ), are reflexive (cf. Proposition 2.7 and Remark 2.2). Consequently, the range R(ε ) is reflexive and, in virtue of the q,p q,p isomorphism ε : Xε (QT ) → R(ε ), likewise Xε (QT ).  

q,p

3.2 Duality in Xε

(QT )

This section addresses a thorough description of the space Xε (QT )∗ , the dual q,p space of Xε (QT ). The following results are similar to those in [5, 23, 76] concerning the dual spaces of the standard Sobolev spaces W01,r (), r ∈ (1, ∞), which q,p supports the interpretation of Xε (QT ) as variable Sobolev space defined on the time-space cylinder QT involving only spatial weak derivatives. We emphasize that q,p a comprehensive description of X∇ (QT )∗ is already given in [11, 51, 132] and q,p we, therefore, refrain from a further examination of X∇ (QT )∗ and restrict our q,p ∗ attention to Xε (QT ) , which is anyway the space of greater interest, looking back to the model problem (3.1). q,p

q,p

3.2 Duality in Xε

53

(QT )

For the entire section, let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and q, p ∈ P∞ (QT ) with q − , p− > 1.



Proposition 3.4 The operator Jε : Lq (·,·) (QT )d ×Lp (·,·) (QT , Md×d → sym ) q,p Xε (QT )∗ , defined by Jε (f , F ), xXεq,p (QT ) := (f , x)Lq(·,·) (QT )d + (F , ε(x))Lp(·,·) (QT )d×d

.





q,p

for every f ∈ Lq (·,·) (QT )d , F ∈ Lp (·,·) (QT , Md×d sym ) and x ∈ Xε (QT ), is well-defined, linear and Lipschitz continuous with constant 2. In addition, for every



q,p x ∗ ∈ Xε (QT )∗ , there exist f ∈ Lq (·,·) (QT )d and F ∈ Lp (·,·) (QT , Md×d sym ) such q,p ∗ ∗ that x = Jε (f , F ) in Xε (QT ) and .

1 ∗ x Xεq,p (QT )∗ ≤ f Lq (·,·) (QT )d + F Lp (·,·) (QT )d×d 2 ≤ 2x ∗ Xεq,p (QT )∗ .

(3.9)



∗ Remark 3.4 The operator Jε : Lq (·,·) (QT )d × Lp (·,·) (QT , Md×d sym ) → Xε (QT ) ∗ is closely related to the adjoint operator ∗ε : (Lq(·,·) (QT )d × Lp(·,·) (QT , Md×d sym ))



q,p → Xε (QT )∗ . In fact, consider the mapping J q,p : Lq (·,·) (QT )d ×Lp (·,·) (QT , q(·,·) (Q )d × Lp(·,·) (Q , Md×d ))∗ , defined by Md×d T T sym ) → (L sym

J q,p (f , F ), (g, G) Lq(·,·) (Q

.

T)

q,p

d×d Msym )

d ×Lp(·,·) (Q , T

:= (f , g)Lq(·,·) (QT )d + (F , G)Lp(·,·) (QT )d×d



q(·,·) (Q )d and for every f ∈ Lq (·,·) (QT )d , F ∈ Lp (·,·) (QT , Md×d T sym ), g ∈ L d×d p(·,·) (QT , Msym ), which is, based on Proposition 2.6, well-defined, linear and G∈L Lipschitz continuous with constant 2. Furthermore, with the aid of Proposition 2.7



it is readily observed that for every f ∈ Lq (·,·) (QT )d and F ∈ Lp (·,·) (QT , Md×d sym ), it holds

1 (f , F ) Lq (·,·) (QT )d ×Lp (·,·) (QT )d×d 2 ≤ J q,p (f , F )(Lq(·,·) (Q

.

T)

d×d ∗ Msym ))

d ×Lp(·,·) (Q , T

(3.10)

≤ 2(f , F ) Lq (·,·) (QT )d ×Lp (·,·) (QT )d×d ,



q(·,·) (Q )d × Lp(·,·) i.e., J q,p : Lq (·,·) (QT )d × Lp (·,·) (QT , Md×d T sym ) → (L d×d ∗ (QT , Msym )) is an isomorphism. Moreover, we have that Jε = ∗ε ◦ J q,p , i.e., Jε and ∗ε agree up to the isomorphism J q,p .

54

3 Variable Bochner–Lebesgue Spaces

∗ Remark 3.5 Denote by div : Lp (·,·) (QT , Md×d sym ) → Xε (QT ) the distributional

(·,·) q,p d×d p (QT , Msym ) and x ∈ Xε (QT ) defined divergence, which is for every F ∈ L by q,p

div(F ), xXεq,p (QT ) := −(F , ε(x))Lp(·,·) (QT )d×d .

.

Then, Proposition 3.4 also proves that Xε (QT )∗ is isomorphic to the sum of



the Banach spaces Lq (·,·) (QT )d and R(div) = div(Lp (·,·) (QT , Md×d sym )). Here, we

endow Lq (·,·) (QT )d + R(div) with the canonical norm on sums of Banach spaces, see, e.g., [75, Kap. I, Bem. 5.13], where R(div) forms a Banach spaces, if equipped with the norm q,p

x ∗ R(div) :=

.



min

F ∈Lp (·,·) (QT ,Msym ) q,p ∗ x =div(F ) in Xε (QT )∗ d×d

F Lp (·,·) (QT )d×d .

Proof of Proposition 3.4 The well-definedness and Lipschitz continuity with con

q,p ∗ stant 2 of the mapping Jε : Lq (·,·) (QT )d × Lp (·,·) (QT , Md×d sym ) → Xε (QT ) follow from Hölder’s inequality for variable Lebesgue spaces (cf. Proposition 2.6). The linearity is evident. So, let us prove the surjectivity including (3.9). Since q,p

ε : Xε

.

(QT ) → R(ε ) ⊆ Lq(·,·) (QT )d × Lp(·,·) (QT , Md×d sym )

is an isometric isomorphism, the corresponding inverse q(·,·) −1 (QT )d × Lp(·,·) (QT , Md×d ε : R(ε ) ⊆ L sym ) → Xε

q,p

.

(QT )

is an isometric isomorphism as well. We fix an arbitrary x ∗ ∈ Xε (QT )∗ . Then, ∗ we have that x ∗ ◦ −1 ε ∈ (R(ε )) and the Hahn–Banach theorem provides a ∗ q(·,·) d p(·,·) ∗ functional x ∈ (L (QT ) × L (QT , Md×d sym )) such that q,p

(idR(ε ) )∗ x ∗ = x ∗ ◦ −1 ε

.

in (R(ε ))∗ ,

where (idR(ε ) )∗ denotes the adjoint operator of the identity mapping idR(ε ) : R(ε ) → Lq(·,·) (QT )d × Lp(·,·) (QT , Md×d sym ), and x ∗ (Lq(·,·) (Q

.

T)

∗ −1 d×d ∗ = x ◦ ε (R(ε ))∗ . Msym ))

d ×Lp(·,·) (Q , T





(3.11)

In addition, Remark 3.4 provides f ∈ Lq (·,·) (QT )d and F ∈ Lp (·,·) (QT , Md×d sym ) such that J q,p (f , F ) = x ∗

.

∗ in (Lq(·,·) (QT )d × Lp(·,·) (QT , Md×d sym )) .

q,p

3.2 Duality in Xε

55

(QT ) q,p

As a consequence, for every x ∈ Xε

(QT ), it holds

x ∗ , xXεq,p (QT ) = x ∗ ◦ −1 ε , ε xR(ε )

.

= (idR(ε ) )∗ x ∗ , ε xR(ε ) = (idR(ε ) )∗ J q,p (f , F ), ε xR(ε ) = J q,p (f , F ), ε xLq(·,·) (Q

T)

d×d Msym )

d ×Lp(·,·) (Q , T

= (f , x)Lq(·,·) (QT )d + (F , ε(x))Lp(·,·) (QT )d×d = Jε (f , F ), xXεq,p (QT ) ,



i.e., x ∗ = Jε (f , F ) in Xε (QT )∗ . Therefore, Jε : Lq (·,·) (QT )d × Lp (·,·) (QT , q,p q,p −1 ∗ Md×d sym ) → Xε (QT ) is surjective. By exploiting that ε : R(ε ) → Xε (QT ) is an isometric isomorphism, we readily deduce the norm equality q,p

∗ x ∗ ◦ −1 ε (R(ε ))∗ = x Xεq,p (QT )∗ .

(3.12)

.

Eventually, we conclude from (3.11), (3.12) and (3.10) to the norm equivalence (3.9).   q,p

˚ε In the same manner, we can characterize the dual space of X

˚εq(·),p(·) (). := X

Corollary 3.2 Let  ⊆ Rd , d ≥ 2, be a bounded domain and q, p ∈ P∞ () . ˚εq,p → Lq(·) ()d × Lp(·) (, Md×d ), for every x ∈ X ˚εq,p Then, the mapping ε : X sym defined by ε x := (x, ε(x))

.

in Lq(·) ()d × Lp(·) (, Md×d sym ) ,

˚εq,p is is an isometric isomorphism into its range R(ε ). In particular, X q,p ˚ε is reflexive and the operator separable. Furthermore, if q − , p− > 1, then X



˚εq,p )∗ , for every f ∈ Lq (·) ()d , F ∈ ) → ( X Jε : Lq (·) ()d × Lp (·) (, Md×d sym

˚εq,p defined by ) and x ∈ X Lp (·) (, Md×d sym Jε (f, F), xX ˚εq,p := (f, x)Lq(·) ()d + (F, ε(x))Lp(·) ()d×d ,

.

is well-defined, linear and Lipschitz continuous with constant 2. In addition, for ˚εq,p )∗ , there exist f ∈ Lq (·) ()d and F ∈ Lp (·) (, Md×d ) such that every x∗ ∈ (X sym ˚εq,p )∗ and x∗ = Jε (f, F) in (X .

1 ∗ ∗ x (X ˚εq,p )∗ ≤ fLq (·) ()d + FLp (·) ()d×d ≤ 2x (X ˚εq,p )∗ . 2

56

3 Variable Bochner–Lebesgue Spaces

Proposition 3.4 and Corollary 3.2 enable us to give a well-posed definition of a q,p time evaluation for functionals x ∗ ∈ Xε (QT )∗ . Remark 3.6 (Time Slices of Functionals x ∗ ∈ Xε

q,p

(QT )∗ )





(i) As, by Fubini’s theorem, for f ∈ Lq (·,·) (QT )d and F ∈ Lp (·,·) (QT , Md×d sym ),

(t,·)

(t,·) d×d q d p () and F (t) ∈ L (, Msym ) for almost it holds f (t) ∈ L ˚εq,p (t)∗ for every t ∈ I , Corollary 3.2 yields Jε (f (t), F (t)) ∈ X almost every t ∈ I . Apart from that, Fubini’s theorem implies in conjunction with Hölder’s inequality (cf. Proposition 2.6) for every x ∈ q,p ˚εq,p (t) for almost every t ∈ I , that the function Xε (QT ), i.e., x(t) ∈ X ˚εq,p (t) ) = (t → (f (t), x(t))Lq(t,·) ()d + (t → Jε (f (t), F (t)), x(t) X (F (t), ε(x)(t))Lp(t,·) ()d×d ) : I → R belongs to L1 (I ). More precisely, we have that

.

Jε (f , F ), xXεq,p (QT )  = (f (t), x(t))Lq(t,·) ()d + (F (t), ε(x)(t))Lp(t,·) ()d×d dt 

I

= I

(3.13)

Jε (f (t), F (t)), x(t)X ˚εq,p (t) dt .

(ii) At first glance, it is not clear whether x ∗ ∈ Xε (QT )∗ has time slices, in ˚εq,p (t)∗ for almost every t ∈ I . But since, due to the sense that x ∗ (t) ∈ X



Proposition 3.4, there exist f ∈ Lq (·,·) (QT )d and F ∈ Lp (·,·) (QT , Md×d sym ) q,p ∗ ∗ such that x = Jε (f , F ) in Xε (QT ) , we are apt to define q,p

x ∗ (t) := Jε (f (t), F (t))

.

˚εq,p (t)∗ in X

for a.e. t ∈ I .

(3.14)

For this purpose, we first have to clarify whether (3.14) is well-defined, i.e.,



independent of the choice of f ∈ Lq (·,·) (QT )d and F ∈ Lp (·,·) (QT , Md×d sym ).

So, let us consider another representation g ∈ Lq (·,·) (QT )d and G ∈

q,p ∗ ∗ Lp (·,·) (QT , Md×d sym ) such that x = Jε (g, G) in Xε (QT ) . Then, by testing q,p q,p ˚+ and ϕ ∈ C0∞ (I ) are chosen with x := xϕ ∈ Xε (QT ), where x ∈ X arbitrarily, also making use of the relation (3.13), we find that 

 .

I

Jε (f (t), F (t)), xX ˚εq,p (t) ϕ(t) dt =

I

Jε (g(t), G(t)), xX ˚εq,p (t) ϕ(t) dt ,

˚εq,p (t) = Jε (g(t), G(t)), x X ˚εq,p (t) for almost every i.e., Jε (f (t), F (t)), x X q,p ˚εq,p (t), we conclude ˚+ t ∈ I . Since X lies for almost every t ∈ I densely in X q,p ˚ε (t)∗ for almost every t ∈ I . that Jε (f (t), F (t)) = Jε (g(t), G(t)) in X Consequently, the time slices (3.14) are well-defined.

q,p

3.3 Embedding Theorems for Xε

57

(QT )

q,p

3.3 Embedding Theorems for Xε

(QT )

This section is concerned with fundamental embedding theorems for variable Bochner–Lebesgue spaces, which will be indispensable for proving the existence of weak solutions of both the unsteady p(·, ·)-Stokes equations and unsteady p(·, ·)Navier–Stokes equations, later in Chap. 5. To get an appreciation of where the difficulties of the proof of such embedding theorems are hidden, let us –as a starter– revisit Poincaré’s inequality in the context of variable exponent spaces. Proposition 3.5 (Poincaré’s Inequality) Let  ⊆ Rd , d ≥ 2, be a bounded domain and let p ∈ P∞ () ∩ C 0 (). Then, there exists a constant c > 0 such 1,p(·) that for every x ∈ W0 ()d , it holds xLp(·) ()d ≤ c∇xLp(·) ()d×d .

.

Proof See [103, Theorem 3.10 (vi)] or [49, Theorem 8.2.18.].

 

Proposition 3.5 can be interpreted as a minimal embedding result since it also ˚1,p(·) ()d likewise satisfies x ∈ W 1,p(·) ()d , states that every function x ∈ X ∇ 0 i.e., it additionally holds x ∈ Lp(·) ()d . The latter straightforwardly follows from ˚1,p(·) ()d . In the case of a constant Proposition 3.5 and the density of C0∞ ()d in X ∇ ˚1,p ()d ) with exponent p ∈ (1, ∞), Proposition 3.5 also yields for x ∈ L1 (I, X ∇ 1,p ∇x ∈ Lp (QT )d×d that x ∈ Lp (QT )d , i.e., x ∈ Lp (I, W0 ()d ). Moreover, Proposition 3.5 guarantees that the gradient norm ∇·Lp(·) ()d×d , and if, in addition, p ∈ Plog () with p− > 1, by virtue of Korn’s inequality (cf. Proposition 2.13), also 1,p(·) ()d . Our purpose is to gain ε(·)Lp(·) ()d×d , defines an equivalent norm on W0 an analogue of Proposition 3.5 for variable Bochner–Lebesgue spaces. Of particular importance, in this context, for p ∈ P∞ (QT ), is the space p−,p

Xε

.

(3.15)

(QT ) .

If, in addition, p− > 1, then, using Poincaré’s and Korn’s inequality with respect to the constant exponent p− ∈ (1, ∞) and also Corollary 2.1, we arrive for every p−,p x ∈ Xε (QT ) at the inequality xLp− (QT )d ≤ cp− ε(x)Lp− (QT )d×d .

≤ cp− 2(1 + |QT |)ε(x)Lp(·,·) (QT )d×d , p−,p

(3.16)

i.e., ε(·)Lp(·,·) (QT )d×d is an equivalent norm on Xε (QT ). We emphasize that inequality (3.16) already resembles an analogue of Proposition 3.5 for variable Bochner–Lebesgue spaces, although it is much weaker inasmuch as we actually desire an inequality of type (3.16) in which the constant exponent p− ∈

58

3 Variable Bochner–Lebesgue Spaces

is replaced by the variable exponent p ∈ Plog (QT ), i.e., the embedding (1, ∞) p−,p Xε (QT ) → Lp(·,·) (QT )d . Regrettably, we cannot hope for such an inequality. p−,p

Remark 3.7 (Invalidity of Poincaré’s Inequality on Xε (QT )) Let  ⊆ Rd , d ≥ 2, be an arbitrary bounded domain, I := (0, T ), T < ∞, and QT := I × . Moreover, let   F Poin := (x, ∇x) | x ∈ C0∞ ()d ⊆ C0∞ ()d × C0∞ ()d×d .

.

Let x ∈ C0∞ ()d with x = a in G, where G ⊂⊂  is a domain and a ∈ Rd \ {0}. Then, it holds (x, ∇x) ∈ F Poin with ∇x = 0 and int(supp(x)) \ supp(∇x) = ∅. In consequence, according to Proposition 3.1, there exists an exponent p ∈ C ∞ (Rd ) with p− > 1 that does not admit a constant c > 0 such that for every φ ∈ C0∞ (QT )d , it holds φLp(·) (QT )d ≤ c∇φLp(·) (QT )d×d .

.

−

+

p−,p

In addition, for every ϕ ∈ Lp (I )\Lp (I ), it holds ϕx ∈ X∇ (QT )\Lp(·) (QT )d , − / Lp(·) (QT )d , because−ϕx ∈ Lp (QT )d and ϕ∇x ∈ Lp(·) (Q−T )d×d , but ϕx ∈ p ,p p ,p p(·) d p(·) i.e.,−X∇ (QT ) ⊆−L (QT ) , and thus also Xε (QT ) ⊆ L (QT )d , owing to p ,p p ,p X∇ (QT )  Xε (QT ) (cf. Remark 3.3). 2 (0), G := B 2 (0), x := aη ∈ The situation is illustrated in Fig. 3.3 for  := B2.5 0.6 ∞ 2 2  C0 () , where a := e1 = (1, 0) ∈ R and η := χB 2 (0) ∗ ωε2 ∈ C0∞ () for 1 ε := 0.4. Fortunately, in unsteady problems, even with a variable exponent structure, the appearance of a time derivative traditionally provides some additional regularity in time, see, e.g., the estimate (3.2). In fact, one usually can expect a solution to lie additionally in the space Y 0 (QT ) := C 0 (I , Y ) ,

.

or at least

Y ∞ (QT ) := L∞ (I, Y ) ,

(3.17)

Fig. 3.3 Plots of |x| = η ∈ C0∞ () (blue/left), |∇x| = |∇η| ∈ C0∞ () (red/middle) and p| ∈ C ∞ () (green/right), constructed according to Proposition 3.1 for d = 2, α = 1.1 and β = 2

q,p

3.3 Embedding Theorems for Xε

59

(QT )

where Y := L2 ()d . Taking into account this additional regularity beforehand, we will succeed in proving an embedding result that provides the desired generalization of Proposition 3.5 to the framework of variable Bochner–Lebesgue spaces. An essential tool for the incorporation of the Y ∞ (QT )-information is the following Gagliardo–Nirenberg interpolation inequality. Lemma 3.1 (Gagliardo–Nirenberg Interpolation Inequality) Let  ⊆ Rd , d ≥ 2, be a bounded C 1 -domain and s, r ∈ [1, ∞). Then, for every x ∈ W 1,r ()d ∩ Ls ()d , it holds x ∈ Lq ()d with xLq ()d ≤ cxθW 1,r ()d x1−θ , Ls ()d

.

(3.18)

where c = c(q, r, s, ) > 0, provided that q1 = θ ( 1r − d1 ) + (1 − θ ) 1s for some θ ∈ [0, 1], unless r = d, in which (3.18) only holds for θ ∈ [0, 1). Proof For the case r = d, we can refer to [71, Theorem 10.1]. For the case r = d, we can refer to [74, 130, 136] or [137], where (3.18) is proved with an additional xLs ()d -term on the right-hand side. However, for r = d, we can then use xLs ()d = xθLs ()d x1−θ ≤ cxθW 1,r ()d x1−θ .   Ls ()d Ls ()d Since the Gagliardo–Nirenberg interpolation inequality only takes into account the full gradient, but we only have control over the symmetric part of the gradient, we need to switch locally from the full gradient to the symmetric gradient via Korn’s second inequality. Lemma 3.2 (Korn’s Second Inequality) Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain and p ∈ (1, ∞). Then, there exists a constant c = c(p, ) > 0 such that for every x ∈ E 1,p () (cf. Remark 2.3), it holds x ∈ W 1,p ()d with xW 1,p ()d ≤ cxE 1,p () .

.

Proof See [73, Lem. 3], [119, Thm. 1.10, (1.17)], [47, Thm. 5.17], or [49, Thm. 14.3.23.].   Aided by Lemmas 3.1 and 3.2, we can next prove the main result of this section. p−,p

Proposition 3.6 (Poincaré’s Inequality for Xε (QT ) ∩ Y ∞ (QT )) Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p ∈ Plog (QT−) with p− > 1. Then, there exists a constant c > 0 such that for every p ,p x ∈ Xε (QT ) ∩ Y ∞ (QT ), it holds x ∈ Lp(·,·) (QT )d with   xLp(·,·) (QT )d ≤ c ε(x)Lp(·,·) (QT )d×d + xY ∞ (QT ) ,

.

p−,p

i.e., it holds the embedding Xε

(3.19)

(QT ) ∩ Y ∞ (QT ) → Lp(·,·) (QT )d .

To be more precise, Proposition 3.6 follows from the following uniform pointwise inequality.

60

3 Variable Bochner–Lebesgue Spaces

Lemma 3.3 Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p∈C 0 (QT ) with p− > 1. Then, exist constants c := c(p, , I ), γ := γ (p, , I ) > 0 (both not depending on t ∈ I ) such that for every t ∈ I and 1,p(t,·) x ∈ E0 () ∩ Y , it holds  γ ρp(t,·) (x) ≤ c 1 + ρp(t,·) (ε(x)) + xY .

(3.20)

.

Proof We divide the proof into two main steps. Step 1: Let p ∈ C 0 (Rd+1 ) be an extension of p ∈ C 0 (QT ) with p− ≤ p ≤ p+ in Rd+1 (cf. Proposition 2.11). We construct a finite open covering of QT by cylinders (Qij )i,j =1,...,m , m ∈ N, where Qij = Ii × Bj for open intervals Ii , i = 1, . . . , m, and open balls Bj , j = 1, . . . , m, such that the local exponents − + := sup(s,y) ∈Qij p(s, y) and pij := inf(s,y) ∈Qij p(s, y) satisfy for every pij i, j = 1, . . . , m 2

+ − . 1+ pij < pij d

(3.21)

.

Since p ∈ C 0 (Rd+1 ), i.e., p : QT + B11 (0) × B1d (0) → (1, ∞) is uniformly continuous, we can find ρ ∈ (0, 1) such that for every point z := (t, x) ∈ QT , the local exponents pz+ := sup(s,y) ∈Qρz p(s, y) and pz− := inf(s,y) ∈Qρz p(s, y), ρ where Qz := Bρ1 (t) × Bρd (x), satisfy pz+ < pz− (1 + d2 ). On the other hand, since I and  are compact, there exist finitely many ti ∈ I , i = 1, . . . , m, m ∈ N, and xj ∈ , j = 1, . . . , m, such that the intervals Ii := Bρ1 (ti ), i = 1, . . . , m, form an open covering of I and the balls Bj = Bρd (xj ), j = 1, . . . , m, form an open covering of . Putting everything together, the cylinders Qij := Ii × Bj , i, j = 1, . . . , m, form an open covering of QT , such that the local exponents + − := sup(s,y) ∈Qij p(s, y) and pij := inf(s,y) ∈Qij p(s, y) satisfy (3.21). pij Step 2: We fix an arbitrary time slice t ∈ I . Then, there exists an i = 1, . . . , m 1,p(t,·) () ∩ Y be arbitrary. First, we extend x by zero with t ∈ Ii . Let x ∈ E0 outside of , i.e., we define x := x in  and x := 0 in  . Then, it holds x ∈ 1,p(t,·) d E0 (R ) ∩ L2 (Rd )d . Let us fix an arbitrary ball Bj from step 1 for some + ≤ 2. j = 1, . . . , m. There are two possibilities. First, we consider the case pij p(t,x) 2 2 ≤ (1 + a) ≤ 2 + 2a for all a ≥ 0 and x ∈ Bj Then, making use of both a and xL2 (Bj )d ≤ xY , we observe that ρp(t,·) (xχBj ) ≤ 2|Bj ∩ | + 2x2L2 (B

j)

.

≤ 2|| + 2x2Y .

d

(3.22)

q,p

3.3 Embedding Theorems for Xε

61

(QT )

+ − + d Next, we assume that pij > 2. Then, exploiting that pij > pij d+2 > − d−pij 1 (cf. (3.21)), i.e., dp− < 2 , we deduce that

2d d+2

ij

1 2

0 < θij := 1 2 .

(3.21)


0 (not depending on t ∈ I ) such that yields a constant cij = cij (pij −

x ∈ W 1,pij (Bj )d with xW 1,pij− (Bj )d ≤ cij xE 1,pij− (Bj )d .

(3.24)

.

By the Gagliardo–Nirenberg interpolation inequality (cf. Lemma 3.1), since θij ∈ (0, 1) satisfies .

 1 1 1 1 + (1 − θij ) , = θ − ij + − d 2 pij pij

− + − + we obtain constants cij = cij (pij , pij , ), θij = θij (pij , pij , ) > 0 (both not + pij d depending on t ∈ I ) such that x ∈ L (Bj ) with + pij θij

ρp+ (xχBj ) ≤ cij x

.

ij

1,p− W ij (Bj )d

p+ (1−θij )

xLij2 (B

j)

d

.

(3.25)

Inserting (3.24) in (3.25), exploiting again that xL2 (Bj )d ≤ xY in doing so, we observe that  p+ θ p+ (1−θij ) ρp. + (xχBj ) ≤ cij xLpij− (Bj )d + ε(x)Lpij− (Bj )d×d ij ij xY ij . ij

(3.26)

+ − Thanks to pij θij < pij (cf. (3.23)), we can apply the ε-Young inequality (cf. − + Proposition 2.8) with respect to ρij := pij (pij θij )−1 > 1 with the constant

cij (ε) := (ρij ε)1−ρij (ρij )−1 > 0 for all ε ∈ (0, ρij−1 ) in (3.26). In this way, using

62

3 Variable Bochner–Lebesgue Spaces −

+

−

−

−

+

+

(a + b)pij ≤ 2p (a pij + bpij ) and a pij ≤ 2p (1 + a pij ) for all a, b ≥ 0, we find that p−  p+ (1−θij )ρij

ρp+ (xχBj ) ≤ cij ε xLpij− (Bj )d + ε(x)Lpij− (Bj )d×d ij + cij (ε)xY ij

.

ij

 p+ (1−θij )ρij

+ ≤ cij ε2p ρp− (xχBj ) + ρp− (ε(x)χBj ) + cij (ε)xY ij ij ij   2p+ |Bj ∩ | + ρp+ (xχBj ) + ρp− (ε(x)χBj ) ≤ cij ε2 ij

+ pij (1−θij )ρij

+ cij (ε)xY

ij

(3.27)

.

We set c0 := max i,j =1,...,m cij and c0 (ε) := maxi,j =1,...,m cij (ε). Then, if we + + 2−2p −1 choose ε := c0 and absorb cij ε22p ρpij+ (xχBj ) ≤ 12 ρpij+ (xχBj ) in the lefthand side in (3.27), we further infer from (3.27) p+ (1−θij )ρij

ρp+ (xχBj ) ≤ || + ρp− (ε(x)χBj ) + 2c0 (ε)xY ij

.

ij

ij

+

.

(3.28)



+ If we set γ := maxi,j =1,...,m pij (1 − θij )ρij and exploit α pij (1−θij )ρij ≤ 2γ (1 + α γ ) +

for all α ≥ 0, ρp(t,·) (xχBj ∩ ) = ρp(t,·) (xχBj ) ≤ 2p (|| + ρpij+ (xχBj )), + ρpij− (ε(x)χBj ) ≤ 2p (||+ρp(t,·) (ε(x)χBj )) and ρp(t,·) (ε(x)χBj ) ≤ ρp(t,·) (ε(x)) in (3.28), we arrive at  + ρp(t,·) (xχBj ∩ ) ≤ 2p || + ρp+ (xχBj )

.

ij

p+

≤2

 p+ (1−θij )ρij  2|| + ρp− (ε(x)χBj ) + 2c0 (ε)xY ij ij

  + ≤ 2 2|| + 2p || + ρp(t,·) (ε(x))  γ  + c0 (ε)2γ +1 1 + xY . p+

(3.29)

If we sum up the inequalities (3.22) and (3.29) with respect to j = 1, . . . , m, observing that all the constants are independent of t ∈ I , we conclude for every t ∈ I the desired inequality (3.20).   p−,p

Proof of Proposition 3.6 Let x ∈ Xε (QT ) ∩ Y ∞ (QT ) be arbitrary, i.e., it holds − ˚εp ,p(t,·) ()∩Y for almost every t ∈ I . Because p ∈ Plog (QT ) with p− > 1, x(t) ∈ X − ˚p ,p(t,·) () ∩ Y for Korn’s inequality (cf. Proposition 2.13) guarantees that x(t) ∈ X ∇ almost every t ∈ I . Eventually, Poincaré’s inequality (cf. Proposition 3.5) implies

q,p

3.3 Embedding Theorems for Xε

63

(QT )

1,p(t,·)

further that x(t) ∈ W0 ()d ∩ Y for almost every t ∈ I . Therefore, Lemma 3.3 provides constants c, γ > 0 (not depending on t ∈ I ) such that for almost every t ∈ I , it holds x(t) ∈ Lp(t,·) ()d with  γ ρp(t,·) (x(t)) ≤ c 1 + ρp(t,·) (ε(x)(t)) + x(t)Y .

.

(3.30)

By the Fubini–Tonelli theorem (cf. [58, §2., Satz 2.1, (a)]),9 we have that the function (t → ρp(t,·) (x(t))) : I → R≥0 is Lebesgue-measurable. Moreover, due to (3.30), it has an L1 (I )-integrable majorant. As a consequence, we have that (t → ρp(t,·) (x(t))) ∈ L1 (I ) with  

 |x(t, x)|p(t,x) dx dt =

.

I



ρp(t,·) (x(t)) dt I

  γ ≤ c T + ρp(·,·) (ε(x)) + T xY ∞ (QT ) .

(3.31)

The Fubini–Tonelli theorem (cf. [58, §2., Satz 2.1, (c)]) applied to (3.31) then gives |x|p(·,·) ∈ L1 (QT ), i.e., x ∈ Lp(·,·) (QT )d with ρp(·,·) (x) ≤ c T + ρp(·,·) (ε(x)) +  γ p−,p T xY ∞ (QT ) . In other words, it holds Xε (QT ) ∩ Y ∞ (QT ) → Lp(·,·) (QT )d . In particular, there exists a constant c > 0 such that the inequality (3.19) holds, for which we also make use of the inequality (3.16).   Proposition 3.6 can equally be interpreted as a variable exponent variant of the well-known parabolic interpolation inequality, which likewise exploits an additional Y ∞ (QT )-information to improve parabolic embedding results using interpolation arguments. Definition 3.3 For d ∈ N and p ∈ [1, ∞), the parabolic interpolation exponent is defined by p∗ := p

.

d +2 ∈ (1, ∞) . d

(3.32)

Proposition 3.7 (Parabolic Interpolation Inequality) Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p ∈ (1, ∞) a constant. 1,p Then, there exists a constant c > 0 such that for every x ∈ Lp (I, W0 ()d ) ∩ L∞ (I, L2 ()d ), it holds x ∈ Lp∗ (QT )d with p

xL∗p∗ (Q

.

T)

2p/d

p

d

≤ cx

1,p

Lp (I,W0 ()d )

xL∞ (I,L2 ()d ) ,

(3.33)

i.e., Lp (I, W0 ()d ) ∩ L∞ (I, L2 ()d ) → Lp∗ (QT )d . In addition, if ∂ ∈ C 1 , 1,p then the statement also applies with W0 ()d replaced by W 1,p ()d . 1,p

9 Here,

we rely crucially on that ((t, x) → |x(t, x)|p(t,x) ) ∈ L0 (QT ) (cf. Remark 2.1).

64

3 Variable Bochner–Lebesgue Spaces

Proposition 3.7 is an immediate consequence of the following lemma. Lemma 3.4 Let  ⊆ Rd , d ≥ 2, be a bounded domain and p ∈ (1, ∞). Then, there 1,p exists a constant c = c(d, p, ) > 0 such that for every x ∈ W0 ()d ∩ L2 ()d , it holds x ∈ Lp∗ ()d with p

2p/d

p

xL∗p∗ ()d ≤ cx

.

1,p

W0 ()d

xL2 ()d ,

(3.34)

1,p

If ∂ ∈ C 1 , then the statement also applies with W0 ()d replaced by W 1,p ()d . Proof The Gagliardo–Nirenberg interpolation inequality for  = Rd (cf. [71, 74, Thm. 9.3] or [136]) states for r, s ∈ [1, ∞) that for every x ∈ W01,r (Rd )d ∩Ls (Rd )d , it holds x ∈ Lq (Rd )d with xLq (Rd )d ≤ c∇xθ r

.

L (R )d×d d

x1−θ , d Ls (R )d

(3.35)

where c = c(q, r, s, d) > 0, provided that q1 = θ ( 1r − d1 ) + (1 − θ ) 1s for some θ ∈ [0, 1], unless r = d, in which (3.35) only holds for θ ∈ [0, 1). If we extend 1,p 1,p x ∈ W0 ()d ∩ L2 ()d by zero outside of  to x ∈ W0 (Rd )d ∩ L2 (Rd )d , i.e.,  x := x in  and x := 0 in  , and employ (3.35) for q = p∗ , r = p, s = 2, and d θ = d+2 , then we find that p∗

2p/d

p

x

.

Lp∗ (R )d d

≤ c∇x

Lp (R )d×d d

x

L2 (R )d d

,

(3.36)

1,p

and x ∈ W0 ()d ∩ L2 ()d consequently satisfies (3.34). If ∂ ∈ C 1 , then we d can apply Lemma 3.1 with q = p∗ , r = p, s = 2, and θ = d+2 .   Proof We argue as in the proof of Proposition 3.6, but now using Lemma 3.4.

 

Needless to say, we should not be satisfied with Proposition 3.6 because it provides no higher integrability than the given variable exponent p ∈ Plog (QT ), whereas Proposition 3.7 improves the integrability at least up to the constant interpolation exponent p∗ = d+2 d p > p. Therefore, our next objective is to prove an analogue of Proposition 3.7 involving variable exponents. For this purpose, we introduce a variable exponent variant of the parabolic interpolation exponent. Definition 3.4 (Variable Parabolic Interpolation Exponent) Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p ∈ P∞ (QT ). Moreover, let (·)∗ : [1, ∞) → (1, ∞) be defined via (3.32). Then, we denote by p∗ := (·)∗ ◦ p ∈ P∞ (QT )

.

the variable parabolic interpolation exponent.

q,p

3.3 Embedding Theorems for Xε

65

(QT ) p−,p

Proposition 3.8 (Interpolation Inequality for Xε (QT ) ∩ Y ∞ (QT )) Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p ∈ Plog (QT ) with p− > 1. Then, for every ε ∈ (0, (p−−)∗ − 1], there exists a p ,p constant cε = c(ε, p, , I ) > 0 such that for every x ∈ Xε (QT ) ∩ Y ∞ (QT ), it p (·,·)−ε d (QT ) with holds x ∈ L ∗   xLp∗ (·,·)−ε (QT )d ≤ cε ε(x)Lp(·,·) (QT )d×d + xY ∞ (QT ) ,

.

p−,p

i.e., for every ε ∈ (0, (p− )∗ −1], it holds the embedding Xε Lp∗ (·,·)−ε (QT )d .

(3.37)

(QT ) ∩ Y ∞ (QT ) →

By analogy with Proposition 3.6, Proposition 3.8 follows from a uniform pointwise inequality. Lemma 3.5 Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p ∈ C 0 (QT ) with p− > 1. Then, for every ε ∈ (0, (p− )∗ − 1], there exist constants cε = c(ε, p, , I ), γε = γ (ε, p, , I ) > 0 (not depending 1,p(t,·) on t ∈ I ) such that for every t ∈ I and x ∈ E0 () ∩ Y , it holds x ∈ p (t,·)−ε d L ∗ () with  γ  γ  ρp∗ (t,·)−ε (x) ≤ cε 1 + ρp(t,·) (ε(x)) + xYε 1 + xYε .

.

(3.38)

Proof We fix an arbitrary ε ∈ (0, (p− )∗ − 1] and divide the proof into two steps: Step 1: Let p ∈ C 0 (Rd+1 ) be an extension of p ∈ C 0 (QT ) with p− ≤ p ≤ p+ in Rd+1 (cf. Proposition 2.11). Since both p : QT + B11 (0) × B1d (0) → (1, ∞) and (·)∗ : [1, ∞) → (1, ∞) are uniformly continuous, the same argumentation as in step 1 of the proof of Lemma 3.3 shows that there exist ρ ∈ (0, 1) as well as finitely many ti ∈ I , i = 1, . . . , m, m ∈ N, and xj ∈ , j = 1, . . . , m, such that the intervals Ii := Bρ1 (ti ), i = 1, . . . , m, form an open covering of I , the balls Bj = Bρd (xj ), j = 1, . . . , m, form an open covering of , and the local − + := sup(s,y) ∈Qij p(s, y) and pij := inf(s,y) ∈Qij p(s, y), where exponents pij + − Qij := Ii × Bj , satisfy (pij )∗ − ε ≤ (pij )∗ for all i, j = 1, . . . , m. Step 2: We fix an arbitrary time slice t ∈ I . Then, there exists an index i = 1, . . . , m with t ∈ Ii . Let x ∈ E01,p(t,·) () ∩ Y be arbitrary. Then, its − zero extension x ∈ E01,p(t,·) (Rd ) ∩ L2 (Rd )d satisfies x ∈ E 1,pij (Bj ) ∩ L2 (Bj )d for all j = 1, . . . , m. Let us fix an arbitrary ball Bj for j = 1, . . . , m. − − − Lemma 3.2, i.e., E 1,pij (Bj ) = W 1,pij (Bj )d , Lemma 3.4 with W01,pij (Bj )d − + replaced by W 1,pij (Bj )d , Lemma 3.3, ρpij− (xχBj ) ≤ 2p (|| + ρp(t,·) (x)) and + ε = c (ε, p − , B ), ρpij− (ε(x)χBj ) ≤ 2p (|| + ρp(t,·) (ε(x))), yield constants cij ij j ij

66

3 Variable Bochner–Lebesgue Spaces − γijε = γij (ε, pij , Bj ), γ > 0 (not depending on t ∈ I ) such that

  γε ε ρp− (xχBj ) + ρp− (∇xχBj ) xLij2 (B ρ(p− )∗ (xχBj ) ≤ cij ij

ij

j)

ij

  γε ε ρp− (xχBj ) + ρp− (ε(x)χBj ) xLij2 (B ≤ cij ij

.

≤

ε cij

d

j)

ij

d

  γε 1 + ρp(t,·) (x) + ρp(t,·) (ε(x)) xYij

(3.39)

 γε γ ε 1 + ρp(t,·) (ε(x)) + xY xYij . ≤ cij ε > 0. We set γε := maxi,j =1,...,m γijε + γ > 0 and cε := maxi,j =1,...,m 4γε cij +

Then, also making use of ρp∗ (t,·)−ε (xχBj ) ≤ 2(p )∗ (|| + ρ(pij− )∗ (xχBj )), since + − p∗ (t, x) − ε ≤ (pij )∗ − ε ≤ (pij )∗ for all x ∈ Bj (cf. step 1), and that ε γ γ γ γ ε ε max{a , a ij } ≤ 2 (1 + a ) for all a ≥ 0, we observe that

.

ρp∗ (t,·)−ε (xχBj ∩ ) = ρp∗ (t,·)−ε (xχBj )  +  γ  γ  ≤ 2(p )∗ || + cε 1 + ρp(t,·) (ε(x)) + xYε 1 + xYε .

(3.40)

Eventually, by summing up (3.40) with respect to j = 1, . . . , m, observing that all the constants are independent of t ∈ I , we conclude for every t ∈ I the desired inequality (3.38).   Proof of Proposition 3.8 We argue as in the proof of Proposition 3.6, but now with   the aid of Lemma 3.5. Next, let us prove an unsteady analogue of the well-known Sobolev embedding theorem –or Rellich’s compactness theorem– for variable Sobolev spaces, which states the following. Proposition 3.9 Let  ⊆ Rd , d ≥ 2, be a bounded domain and p ∈ C 0 () with dp 1 < p− ≤ p+ < d. Moreover, denote by p∗ := d−p ∈ C 0 () the Sobolev conjugate  

exponent. Then, for every ε ∈ 0, d , it holds the compact embedding 1,p(·)

W0

.

()d → → Lp

∗ (·)−ε

()d .

In addition, if p ∈ Plog () with 1 < p− ≤ p+ < d, then 1,p(·)

W0

.

()d → Lp

∗ (·)

()d .

Proof See [103, Theorem 3.9] and [49, Theorem 8.3.1.].

 

q,p

3.3 Embedding Theorems for Xε

67

(QT )

Motivated by a comparison of Propositions 3.8 and 3.9, we next prove a variable Bochner–Lebesgue version of Proposition 3.9. This compactness result will be based on the following –surprisingly, not particularly well-known– compactness principle due to Landes and Mustonen [108, 112]. Proposition 3.10 (Landes’ and Mustonen’s Compactness Principle) Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, and p ∈ (1, ∞) a constant. 1,p Then, for a bounded sequence (x n )n∈N ⊆ Lp (I, W0 ()d )∩L∞ (I, L1 ()d ) from 1,p

xn  x .

x n (t)  x(t)

in Lp (I, W0 ()d )

(n → ∞) ,

in L1 ()d

(n → ∞)

for a.e. t ∈ I ,

(3.41)

it follows that xn → x

.

in Lp (I, Lp ()d )

(n → ∞) . 1,p

If ∂ ∈ C 1 , then the statement also applies with W0 ()d replaced by W 1,p ()d . Proof See [108, 112, Proposition 1] or [133, Theorem 4.1].

 

By combining the parabolic interpolation inequality (cf. Proposition 3.7), Vitali’s convergence theorem (cf. Proposition 2.5) and Landes’ and Mustonen’s compactness principle, we conclude the following interpolated parabolic compactness principle. Corollary 3.3 Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p ∈ (1, ∞) a constant. Then, for a bounded sequence 1,p (x n )n∈N ⊆ Lp (I, W0 ()d ) ∩ L∞ (I, L2 ()d ) from (3.41), it follows that x n → x in Ls (QT )d (n → ∞) for all s ∈ [1, p∗ ). If ∂ ∈ C 1 , then the statement also 1,p applies with W0 ()d replaced by W 1,p ()d . Proposition 3.11 (Compactness Principle for Xε (QT ) ∩ Y ∞ (QT )) Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p,p p ∈ Plog (QT ) with p− > 1. Furthermore, let (x n )n∈N ⊆ Xε (QT ) ∩ Y ∞ (QT ) be a sequence such that p,p

xn  x

.

∗

xn  x x n (t)  x(t)

in Xε (QT )

p,p

(n → ∞) ,

in Y ∞ (QT )

(n → ∞) ,

in Y

(n → ∞)

for a.e. t ∈ I .

Then, for every ε ∈ (0, (p− )∗ − 1], it holds xn → x

.

in Lmax{2,p∗ (·,·)}−ε (QT )d

(n → ∞) .

68

3 Variable Bochner–Lebesgue Spaces −

1,p−

Proof Due to Xε (QT ) → Lp (I, W0 ()d ), Y ∞ (QT ) → L∞ (I, L1 ()d ) and Y → L1 ()d , Corollary 3.3 implies that x n → x in Rd (  n → ∞) almost everywhere in QT for a cofinal subset  ⊆ N. Owing to Proposition 3.8, the sequence (x n )n∈ ⊆ Lmax{2,p∗ (·,·)}−ε (QT )d is Lmax{2,p∗ (·,·)}−ε (QT )-uniformly integrable for every ε ∈ (0, (p− )∗ − 1].10 Thus, making use of Vitali’s convergence theorem for variable Lebesgue spaces (cf. Proposition 2.5) and the standard convergence principle (cf. [166, Prop. 10.13 (1)]), we conclude the assertion.   p,p

An attentive reader will now wonder whether Proposition 3.6, Propositions 3.8 and 3.11, similar to Poincaré’s inequality (cf. Proposition 3.5) and Rellich’s compactness theorem (cf. Proposition 3.9), admit congruent generalizations to the case of a variable exponent p ∈ C 0 (QT ) with p− > 1, i.e., which is merely uniformly continuous. Looking back to Lemmas 3.3 and 3.5, which already hold for p ∈ C 0 (QT ) with p− > 1, we have high hopes of deriving embedding theorems of this type. However, since for p ∈ C 0 (QT ) with p− > 1, we cannot rely on Korn’s inequality (cf. Proposition 2.13), we also cannot expect a function p−,p 1,p(t,·) x ∈ Xε (QT ) ∩ Y ∞ (QT ) to satisfy x(t) ∈ E0 () ∩ Y for almost every t ∈ I , which was a crucial step in the proof of Proposition 3.6. The good news is that Proposition 3.6, Propositions 3.8 and 3.11, nonetheless, admit generalizations 1,p(t,·) () ∩ Y , for variable exponents p ∈ C 0 (QT ) with p− > 1, if we replace E0 − 2,p(t,·) p ,p 2,p ˚ε t ∈ I , by X (), t ∈ I , respectively, as well as Xε (QT ) by Xε (QT ). Proposition 3.12 (Interpolation Inequality for Xε (QT ) ∩ Y ∞ (QT )) Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p ∈ C 0 (QT ) with p− > 1. Then, for every ε ∈ (0, (p− )∗ − 1], there exists a 2,p constant cε = c(ε, p, , I ) > 0 such that for every x ∈ Xε (QT ) ∩ Y ∞ (QT ), p (·,·)−ε d it holds x ∈ L ∗ (QT ) with (3.37), i.e., for every ε ∈ (0, (p− )∗ − 1], it holds 2,p the embedding Xε (QT ) ∩ Y ∞ (QT ) → Lp∗ (·,·)−ε (QT )d . 2,p

Proposition 3.12 is an immediate consequence of the following uniform pointwise inequality. Lemma 3.6 Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p ∈ C 0 (QT ) with p− > 1. Then, for every ε ∈ (0, (p− )∗ − 1], there exist constants cε = c(ε, p, , I ), γε = γ (ε, p, , I ) > 0 (both not ˚ε2,p(t,·) (), it holds depending on t ∈ I ) such that for every t ∈ I and x ∈ X p (t,·)−ε d ∗ () with x∈L  γ  γ  ρp∗ (t,·)−ε (x) ≤ cε 1 + ρp(t,·) (ε(x)) + xYε 1 + xYε .

.

(3.42)

we use that for measurable G ⊆ Rn , n ∈ N, with |G| < ∞ and p ∈ P∞ (G) with p − > 1, if (fn )n∈N ⊆ Lp(·) (G) is bounded, then (fn )n∈N ⊆ Lp(·)−δ (G) is Lp(·)−δ (G)-uniformly integrable for every δ ∈ (0, p − − 1].

10 Here,

q,p

3.4 Smoothing in Xε

69

(QT )

Proof It is sufficient to consider the case ε ∈ (0, (p− )∗ − 1) because the case + ε = (p− )∗ − 1 then follows from ρp∗ (t,·)−ε (x) ≤ 2(p )∗ (|| + ρp∗ (t,·)−δ (x)) for all x ∈ Lp∗ (t,·)−δ ()d and δ ∈ (0, (p− )∗ − 1). So, let ε ∈ (0, (p− )∗ − 1) ˚ε2,p(t,·) () (cf. and let t ∈ I be arbitrary. Due to the density of C0∞ ()d in X Definition 2.8), there exists a sequence (xn )n∈N ⊆ C0∞ ()d such that xn → x ˚ε2,p(t,·) () (n → ∞). Moreover, according to Lemma 3.5, there exist constants in X cε = c(ε, p, , I ), γε = γ (ε, p, , I ) > 0 (both not depending on t ∈ I ) such that for every n ∈ N, it holds  γ  γ  ρp∗ (t,·)−ε (xn ) ≤ cε 1 + ρp(t,·) (ε(xn )) + xn Yε 1 + xn Yε .

.

(3.43)

Thus, (xn )n∈N ⊆ C0∞ ()d is bounded in Lp∗ (t,·)−ε ()d . Owing to the reflexivity of Lp∗ (t,·)−ε ()d (cf. Proposition 2.7), where we use that p∗ − ε ≥ (p− )∗ − ε > 1 for ε ∈ (0, (p− )∗ − 1), there exists a cofinal subset  ⊆ N as well as a function x˜ ∈ Lp∗ (t,·)−ε ()d such that xn  x˜ in Lp∗ (t,·)−ε ()d (  n → ∞). Therefore, on the basis of the uniqueness of weak limits, we observe that x = x˜ ∈ Lp∗ (t,·)−ε ()d . Eventually, using ρp∗ (t,·)−ε (x) ≤ lim infn→∞ ρp∗ (t,·)−ε (xn ) (cf. [49, Thm. 3.2.9.]),   taking the limit inferior with respect to n → ∞ in (3.43) proves (3.42). Proposition 3.13 (Compactness Principle for Xε (QT ) ∩ Y ∞ (QT )) Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and 2,p p ∈ C 0 (QT ) with p− > 1. Furthermore, let (x n )n∈N ⊆ Xε (QT ) ∩ Y ∞ (QT ) be a sequence such that 2,p

xn  x

.

∗

xn  x x n (t)  x(t)

in Xε (QT )

2,p

(n → ∞) ,

in Y ∞ (QT )

(n → ∞) ,

in Y

(n → ∞)

for a.e. t ∈ I .

Then, for every ε ∈ (0, (p− )∗ − 1], it holds xn → x

.

in Lmax{2,p∗ (·,·)}−ε (QT )d

(n → ∞) .

Proof We follow the proof of Proposition 3.11, but employ Proposition 3.12.

q,p

3.4 Smoothing in Xε

 

(QT )

This section is devoted to the construction of an appropriate smoothing operator q,p for the space Xε (QT ), which is indispensable for the proof of a formula of integration-by-parts, such as, e.g., Proposition 2.27, but now in the framework of variable Bochner–Lebesgue spaces. Recall that the basic procedure of smoothing in Bochner–Lebesgue spaces consists in mollification with respect to the temporal variable, i.e., for x ∈ Lp (I, X), where X is a Banach space and

70

3 Variable Bochner–Lebesgue Spaces

p ∈ [1, ∞), we consider the family (SIh x)h>0 = (I x ∗ ωh1 )h>0 ⊆ C ∞ (R, X) (cf. Proposition 2.22). In particular, Proposition 2.22 guarantees that SIh x → x in Lp (I, X) (h → 0). However, if one tries to adjust this method to the framework of variable Bochner–Lebesgue spaces, one quickly becomes disillusioned. Indeed, this smoothing method is fundamentally based on the structure of Bochner–Lebesgue spaces, in which time and space variables are perfectly decoupled, and fails in q,p q,p the context of Xε (QT ). To be more precise, if x ∈ Xε (QT ), then it holds q,p q,p ˚− x ∈ X− (QT ) (cf. Proposition 3.2), which gives (SIh x)h>0 ⊆ C ∞ (R, X ) and q,p SIh x → x in X− (QT ) (h → 0), but we cannot ascertain that (SIh x)h>0 ⊆ q,p Xε (QT ). Even additionally mollifying with respect to the space variable, i.e., d (ωh ∗ (ηh SIh x))h>0 ⊆ C ∞ (Rd+1 )d , where (ηh )h>0 ⊆ C0∞ () is a suitable q,p family of cut-off functions that guarantees that (ωhd ∗ (ηh SIh x))h>0 ⊆ Xε (QT ), brings no salvation because we still cannot determine that ωhd ∗ (ηh SIh x) → x in q,p Xε (QT ) (h → 0). The major mistake in this situation is to appeal to the quasi q,p Bochner–Lebesgue space character of Xε (QT ) (cf. Remark 3.2), while almost all integrability comes from its interpretation as a kind of a variable Sobolev space on the time-space cylinder QT involving only spatial weak derivatives. For this reason, the correct method for smoothing variable Bochner–Lebesgue functions is motivated by the smoothing of variable Sobolev functions, i.e., instead of smoothing with respect to time and space variables separately, we smooth with respect to the entire time-space variable. We emphasize that the presented smoothing method is not new and is based on ideas from [51, 132]. However, they solely demonstrated that their smoothing q,p method applies to X∇ (QT ). An essential component of their argumentation is the following point-wise Poincaré inequality. Theorem 3.1 Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain and r(x) := dist(x, ∂) for x ∈ . Then, there exists a constant c0 = c0 () > 0 such that for every x ∈ W01,1 ()d , it holds  |x(x)| ≤ c0

.

d B2r(x) (x)∩

|∇x(y)| dy |x − y|d−1

for a.e. x ∈  .  

Proof See [51, Proposition 4.5]. q,p We extend their method to the larger space Xε (QT ).11

In doing so, we have to guarantee that an analogue of Theorem 3.1 involving only the symmetric part of a gradient holds. This will be done in the following theorem, which provides a pointwise Poincaré inequality with respect to the symmetric gradient near the boundary of a bounded Lipschitz domain, whose proof, due to its length, is outsourced to the appendix (cf. Theorem 9.1) and is based on techniques presented in [37, 40].

q,p

q,p

that, according to Remark 3.3, the space X∇ (QT ) is a proper subspace of Xε q,p q,p i.e., in general, we merely have X∇ (QT )  Xε (QT ).

11 Recall

(QT ),

q,p

3.4 Smoothing in Xε

71

(QT )

Theorem 3.2 Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain and r(x) := dist(x, ∂) for x ∈ . Then, there exist constants c0 = c0 (), h0 = h0 () > 0 such that for every x ∈ E01,1 (), it holds  |x(x)| ≤ c0

.

d B2r(x) (x)∩

|ε(x)(y)| dy |x − y|d−1

for a.e. x ∈  with r(x) ≤ h0 .

In what follows, let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain, I := (0, T ), T < ∞, QT := I × , and q, p ∈ Plog (QT ) with q − , p− > 1. Furthermore, let c0 , h0 > 0 be the constants provided by Theorem 3.2, d ) ⊆ C ∞ (), i.e., for every h > 0, h1 := h40 , (ηh )h>0 := (χ5h/2 ∗ ωh/2 0 there holds supp(ηh ) ⊆ 2h , 0 ≤ ηh ≤ 1 in , ηh = 1 in 3h and c ∇ηh L∞ (Rd )d ≤ hη , where h := {x ∈  | dist(x, ∂) > h} and cη > 0 is a constant not depending h > 0. Henceforth, we will always denote by ω := ωd+1 ∈ C0∞ (Rd+1 ) ∩ SM(Rd+1 ) the standard mollifier from Remark 2.6. Then, we define the family (ωh )h>0 ⊆ C0∞ (Rd+1 ) ∩ SM(Rd+1 ) for every (t, x) ∈ Rd+1 1 and h > 0 by ωh (t, x) := hd+1 ω( ht , xh ). Throughout the entire section, c > 0 always denotes a constant that depends only on the dimension d ∈ N, the constants cη , c0 , h0 > 0, or the mollifier ω ∈ C0∞ (Rd+1 ). Eventually, we will make frequently use of the zero extension operator QT : L1 (QT )d → L1 (Rd+1 )d , defined by QT x := x in QT and QT x := 0 in QT for every x ∈ L1 (QT )d . It is not difficult q,p q, ˜ p˜

T := I˜ × 

is an extended to see that QT : Xε (QT ) → Xε (Q T ), where Q

⊇ , and cylinder, with a bounded interval I˜ ⊇ I and a bounded domain 

T ) are extended exponents, with q| q, ˜ p˜ ∈ Plog (Q ˜ QT = q, p| ˜ QT = p, q − ≤ q˜ ≤ q +

T , is well-defined, linear and Lipschitz continuous with and p− ≤ p˜ ≤ p+ in Q q, ˜ p˜ ∗ q,p ∗ constant 1. Consequently, its adjoint operator ∗QT : Xε (Q T ) → Xε (QT ) equally is well-defined, linear and Lipschitz continuous with constant 1. Proposition 3.14 (Smoothing via Truncation in Space) For x ∈ Lq(·,·) (QT )d and h > 0, we define the space truncation smoothing operator by RhQT x := ωh ∗ (ηh QT x) ∈ C ∞ (Rd+1 )d .

.

Then, for every x ∈ Lq(·,·) (QT )d , it holds: (i) (RhQT x)h>0 ⊆ C0∞ (Rd+1 )d with supp(RhQT x) ⊆ [−h, T + h] × h for every h > 0. (ii) suph>0 |RhQT x| ≤ 2Md+1 (QT x) almost everywhere in Rd+1 .

72

3 Variable Bochner–Lebesgue Spaces

(iii) For an extension q ∈ Plog (Rd+1 )12 of q ∈ Plog (QT ), with q − ≤ q ≤ q + in Rd+1 , there exists a constant cq > 0 (depending on q ∈ Plog (Rd+1 )) such that .

sup RhQT xLq(·,·) (Rd+1 )d ≤ cq xLq(·,·) (QT )d .

h>0

(iv) RhQT x → x in Lq(·,·) (QT )d (h → 0). Remark 3.8 The smoothing operator in Proposition 3.14 admits congruent extensions to not relabeled smoothing operators for scalar functions, i.e., to RhQT : Lq(·,·) (QT ) → Lq(·,·) (Rd+1 ), and for tensor-valued functions, i.e., to RhQT : Lq(·,·) (QT )d×d → Lq(·,·) (Rd+1 )d×d, satisfying (i)–(iv) from Proposition 3.14. Proof ad (i). Due to QT x ∈ L1 (Rd+1 )d , we have that (RhQT x)h>0 ⊆ C ∞ (Rd+1 )d by the standard theory of mollification, see, e.g., [5, 31, 67]. In particular, it holds for every h > 013 supp(RhQT x) ⊆ supp(ηh QT x) + Bhd+1 (0)

.

⊆ (I × 2h ) + Bhd+1 (0) ⊆ [−h, T + h] × h , i.e., (RhQT x)h>0 ⊆ C0∞ (Rd+1 )d . ad (ii). Using Proposition 2.14 (ii), Remark 2.6, and suph>0 ηh L∞ (Rd ) ≤ 1, we observe that .

sup |RhQT x| ≤ sup 2Md+1 (ηh QT x)

h>0

h>0

≤ 2Md+1 (QT x)

a.e. in Rd+1 .

ad (iii). Let q ∈ Plog (Rd+1 ) denote an extension of q ∈ Plog (QT ) with q − ≤ q ≤ q + in Rd+1 (cf. Proposition 2.10). In particular, we have that QT x ∈ Lq(·,·) (QT )d and q − ≥ q − > 1. Hence, Proposition 2.12 provides a constant cq > 0 (depending on q ∈ Plog (Rd+1 )) such that Md+1 (QT x)Lq(·,·) (Rd+1 ) ≤ cq QT xLq(·,·) (Rd+1 )d = cq xLq(·,·) (QT )d . Using (ii), we conclude (iii).

12 The

existence of such an extension, thanks to Proposition 2.10, is always guaranteed. we exploit the inclusion supp(f ∗ g) ⊆ supp(f ) + supp(g) for functions f ∈ L1 (Rn ), n ∈ N, and g ∈ Lp (Rn ), p ∈ [1, ∞], (cf. [31, Prop. 4.18]) and the identity A + B = A + B for bounded sets A, B ⊆ Rn , n ∈ N.

13 Here,

q,p

3.4 Smoothing in Xε

73

(QT )

ad (iv). By applying Proposition 2.14 (i) & (iii) and Proposition 2.4, we obtain RhQT x − xLq(·,·) (QT )d ≤ ωh ∗ [ηh QT x − QT x]Lq(·,·) (Rd+1 )d

.

+ ωh ∗ (QT x) − QT xLq(·,·) (Rd+1 )d ≤ K(1 − ηh )QT xLq(·,·) (Rd+1 )d + ωh ∗ (QT x) − QT xLq(·,·) (Rd+1 )d → 0 (h → 0) .

 

We need another smoothing operator. Since for q ∈ Plog (QT ) with q − > 1 also q ∈ Plog (QT ) with (q )− > 1, Proposition 3.14 (iii) shows for every



h > 0 that RhQT : Lq (·,·) (QT )d → Lq (·,·) (QT )d is likewise well-defined, linear and continuous. As a consequence, for every h > 0, there exists the corresponding quasi adjoint operator (RhQT ) : Lq(·,·) (QT )d → Lq(·,·) (QT )d 14 (with respect to

(·, ·)Lq (·,·) (QT )d ), which for every x ∈ Lq(·,·) (QT )d and y ∈ Lq (·,·) (QT )d is defined by  h   (RQT ) x, y Lq (·,·) (Q

.

T)

d

  := x, RhQT y Lq (·,·) (Q

T)

d

.

(3.44)

Remark 3.9 The quasi adjoint operator (RhQT ) is related to the adjoint operator



(RhQT )∗ of RhQT : Lq (·,·) (QT )d → Lq (·,·) (QT )d via (RhQT ) = J −1 ◦(RhQT )∗ ◦J ,

where we again denote by J : Lq(·,·) (QT )d → (Lq (·,·) (QT )d )∗ the isomorphism defined in Proposition 2.7. This can be seen by simple calculations. Therefore, the quasi adjoint operator (RhQT ) : Lq(·,·) (QT )d → Lq(·,·) (QT )d is well-defined, linear and bounded. The following proposition shows that (RhQT ) has similar properties to RhQT .

Proposition 3.15 For every x ∈ Lq(·,·) (QT )d , it holds: (i) (RhQT ) x = (ωh ∗ (QT x))ηh almost everywhere in QT for every h > 0. In particular, setting (RhQT ) x := (ωh ∗ (QT x))ηh almost everywhere in QT for every h > 0, we have that ((RhQT ) x)h>0 ⊆ C0∞ (Rd+1 )d with supp((RhQT ) x) ⊆ [−h, T + h] × 2h for every h > 0. (ii) suph>0 |(RhQT ) x| ≤ 2Md+1 (QT x) almost everywhere in Rd+1 .

14 The

of

superscript  is intended to indicate that (RhQT ) is not the actual adjoint operator (RhQT )∗ .

RhQT

74

3 Variable Bochner–Lebesgue Spaces

(iii) For an extension q ∈ Plog (Rd+1 ) of q ∈ Plog (QT ), with q − ≤ q ≤ q + in Rd+1 , there exists a constant cq > 0 (depending on q ∈ Plog (Rd+1 )) such that .

sup (RhQT ) xLq(·,·) (Rd+1 )d ≤ cq xLq(·,·) (QT )d .

h>0

(iv) (RhQT ) x → x in Lq(·,·) (QT )d (h → 0). Remark 3.10 The smoothing operator (RhQT )∗ defined in (3.44) admits congruent extensions to not relabeled smoothing operators for scalar functions, i.e., to (RhQT ) : Lq(·,·) (QT ) → Lq(·,·) (Rd+1 ), and for tensor-valued functions, i.e., to (RhQT ) : Lq(·,·) (QT )d×d → Lq(·,·) (Rd+1 )d×d , satisfying (i)–(iv) from Proposition 3.15.

Proof ad (i). For every y ∈ Lq (·,·) (QT )d , by a simple change of variables, we obtain  h   (RQT ) x, y Lq (·,·) (Q

.

T)

d

  = x, ωh ∗ (ηh QT y) Lq (·,·) (Q )d T   = (ωh ∗ (QT x))ηh , y Lq (·,·) (Q )d , T

i.e., it holds (RhQT ) x = (ωh ∗ (QT x))ηh almost everywhere in QT for every h > 0. Therefore, if we define (RhQT ) x := (ωh ∗ (QT x))ηh almost everywhere in QT for every h > 0, then ((RhQT ) x)h>0 ⊆ C0∞ (Rd+1 )d with supp((RhQT ) x) ⊆ [−h, T + h] × 2h for every h > 0. ad (ii). We resort to Proposition 2.14 (ii) and exploit that suph>0 ηh L∞ (Rd ) ≤ 1. ad (iii). Follows as in Proposition 3.14 (iii) from (ii) using Proposition 2.12. ad (iv). By applying Proposition 2.14 (i) & (iii) and Proposition 2.4, we obtain (RhQT ) x − xLq(·,·) (QT )d ≤ ηh [ωh ∗ (QT x) − x]Lq(·,·) (QT )d

.

+ (1 − ηh )xLq(·,·) (QT )d ≤ ωh ∗ (QT x) − QT xLq(·,·) (Rd+1 )d + (1 − ηh )QT xLq(·,·) (Rd+1 )d →0

(h → 0) .

 

Next, let us examine RhQT for its smoothing properties with respect to the space Here, Theorem 3.2 will be the crucial component.

q,p Xε (QT ).

q,p

3.4 Smoothing in Xε

75

(QT ) q,p

Proposition 3.16 (Smoothing in Xε holds:

q,p

(QT )) For every x ∈ Xε

(QT ), it

(i) There exists a constant c > 0 (not depending on q, p ∈ Plog (QT )) such that .

sup |ε(RhQT x)| ≤ c Md+1 (QT ε(x))

a.e. in Rd+1 .

h∈(0,h1 )

(ii) For an extension p ∈ Plog (Rd+1 ) of p ∈ Plog (QT ), with p− ≤ p ≤ p+ in Rd+1 , there exists a constant cp > 0 (depending on p ∈ Plog (Rd+1 ) and c > 0) such that .

sup ε(RhQT x)Lp(·,·) (Rd+1 )d×d ≤ cp ε(x)Lp(·,·) (QT )d×d .

h∈(0,h1 )

In particular, suph∈(0,h1 ) RhQT xXεq,p (QT ) ≤ (cq + cp )xXεq,p (QT ) , where cq > 0 is from Proposition 3.14 (iii). q,p (iii) RhQT x → x in Xε (QT ) (h → 0), i.e., C ∞ (I , C0∞ ()d ) lies densely in q,p Xε (QT ), where   C ∞ (I , C0∞ ()d ) := φ ∈ C ∞ (QT )d | φ(t) ∈ C0∞ ()d for every t ∈ I .

.

Proof ad (i). We first calculate the symmetric gradient of RhQT x ∈ C0∞ (Rd+1 )d . In doing so, using differentiation under the integral sign and the product rule ε(ηx) = 1,1 d 1 d ηε(x) + [x ⊗ ∇η]sym in L1 (Rd , Md×d sym ) for every x ∈ E (R ) and η ∈ C (R ), for every h > 0, we observe that ε(RhQT x) = RhQT (ε(x)) .

+ ωh ∗ [(QT x) ⊗ ∇ηh ]sym

in Rd+1 .

(3.45)

Proposition 3.14 (ii) and Remark 3.8 provide that .

sup |RhQT (ε(x))| ≤ 2Md+1 (QT ε(x))

a.e. in Rd+1 .

(3.46)

h>0

For the remaining term, note that ωh ∗ [(QT x) ⊗ ∇ηh ]sym = 0 in R × 4h because ∇ηh = 0 in 3h . On the other hand, by applying Theorem 3.2, which is allowed  ˚εq,p (t) ⊆ E 1,1 () for almost every t ∈ I , ωh  ∞ d+1 ≤ cd+1 and since x(t) ∈ X 0 h L (R ) ∇ηh L∞ (Rd )d ≤

c h ,

we deduce for every t ∈ R, x ∈ 4h and h ∈ (0, h1 ), i.e.,

76

3 Variable Bochner–Lebesgue Spaces

r(x) = dist(x, ∂) ≤ 4h ≤ 4h1 = h0 , that |ωh ∗ [(QT x) ⊗ ∇ηh ]sym (t, x)|  ≤ ωh (t − s, x − y)|∇ηh (s, y)||(QT x)(s, y)| dsdy

.

Bhd+1 (t,x)

≤

c hd+2

≤

c hd+2

≤

c hd+1





(3.47)



|(QT ε(x))(s, z)| dz dy ds |y − z|d−1     1 |(QT ε(x))(s, z)| dy dzds d (x) d (z) |y − z|d−1 Bh1 (t) B11h B12h  |(QT ε(x))(s, z)| dsdz Bh1 (t)

Bhd (x)

d B2r(y) (y)

d+1 B11h (t,x)

≤ c Md+1 (QT ε(x))(t, x) , d d (x) and B d (x) ⊆ B d (z) for every where we exploited that B2r(y) (y) ⊆ B11h h 12h d d d  z ∈ B2r(y) (y), y ∈ Bh (x) and x ∈ 4h since for y˜ ∈ B2r(y) (y), it holds |y˜ − x| ≤ |x − y| + |y − y| ˜ < h + 2r(y) ≤ 3h + 2r(x) ≤ 11h, where we used that r(·) = dist(·, ∂) is Lipschitz continuous with constant 1, and for z˜ ∈ Bhd (x), it holds  |˜z − z| ≤ |˜z − x| + |x − z| < h + |x − y| + |y − z| < 2h + 2r(y) ≤ 12h, as well as B d (z) |y − z|1−d dy = 12hℋd−1 (Sd−1 ).15 Eventually, by combining (3.45)– 12h (3.47), we conclude (i). ad (ii). Follows as in the proof of Proposition 3.14 (iii) from (i) by means of Proposition 2.12. ad (iii). Proposition 3.14 (iv) and Remark 3.8 provide that

RhQT x → x .

RhQT (ε(x)) → ε(x)

in Lq(·,·) (QT )d

(h → 0) ,

in Lp(·,·) (QT , Md×d sym )

(h → 0) .

(3.48)

Because of ωh ∗ [(QT x) ⊗ ∇ηh ]sym = 0 in R × 4h for every h > 0 and (3.47), we find that ωh . ∗ [(QT x) ⊗ ∇ηh ]sym → 0

in Md×d sym

a.e. in QT (h → 0) ,

sup |ωh ∗ [(QT x) ⊗ ∇ηh ]sym | ≤ c Md+1 (QT ε(x)) a.e. in QT .

h∈(0,h1 )

15 Here,

ℋd−1 denotes the (d − 1)-dimensional Hausdorff measure..

(3.49)

q,p

3.4 Smoothing in Xε

77

(QT )

Since Md+1 (QT ε(x)) ∈ Lp(·,·) (Rd+1 ), due to Proposition 2.12, as |QT ε(x)| ∈ Lp(·,·) (Rd+1 ) and p− ≥ p− > 1, we conclude, with the help of Proposition 2.4, from (3.49) to ωh ∗ [(QT x) ⊗ ∇ηh ]sym → 0

.

in Lp(·,·) (QT , Md×d sym )

(h → 0) .

(3.50)

All things considered, (3.45), (3.48) and (3.50) prove that RhQT x → x

.

q,p

in Xε

(QT )

(h → 0) .

 

The same method as in the proofs of Propositions 3.14 and 3.16 can be used to ˚εq(·),p(·) () = X ˚q(·),p(·) (), provided ˚q,p = X construct a smoothing operator for X ∇ that  ⊆ Rd , d ≥ 2, is a bounded Lipschitz domain and that q, p ∈ Plog () with q − , p− > 1. Corollary 3.4 Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain, q, p ∈ Plog () ˚q,p = X ˚εq(·),p(·) () = X ˚q(·),p(·) (). For x ∈ X ˚q,p and with q − , p− > 1, and X ∇ h > 0, we define the smoothing operator Rh x := ωhd ∗ (ηh ℰ x) ∈ C ∞ (Rd )d ,

.

where (ωhd )h>0 ⊆ C0∞ (Rd ) are the scaled standard mollifiers from Remark 2.6 and ℰ x ∈ L1 (Rd )d is defined by ℰ x := x in  and ℰ x := 0 in  . Then, for every ˚q,p , it holds: x∈X (i) (Rh x)h>0 ⊆ C0∞ ()d with supp(Rh x) ⊆ h for every h > 0. (ii) There exists a constant c > 0 (not depending on q, p ∈ Plog ()) such that sup |Rh x| ≤ 2Md (ℰ x)

a.e. in Rd ,

sup |ε(Rh x)| ≤ c Md (ℰ ε(x))

a.e. in Rd .

.

h>0

h∈(0,h1 )

(iii) For extensions q, p ∈ Plog (Rd ) of q, p ∈ Plog (), resp., with q − ≤ q ≤ q + and p− ≤ p ≤ p+ in Rd , there exist constants cq , cp > 0 (depending on q, p ∈ Plog (Rd ) and c > 0, resp.) such that .

sup Rh xLq(·) (Rd )d ≤ cq xLq(·) ()d ,

h>0

sup ε(Rh x)Lp(·) (Rd )d×d ≤ cp ε(x)Lp(·) ()d×d ,

h∈(0,h1 )

In particular, suph∈(0,h1 ) Rh xX ˚q,p ≤ (cq + cp )xX ˚q,p . ˚q,p (h → 0). (iv) Rh x → x in X

78

3 Variable Bochner–Lebesgue Spaces

The following proposition demonstrates that the quasi adjoint operator (RhQT ) q,p is an equally attractive smoothing operator for the space Xε (QT ) as RhQT . q,p

Proposition 3.17 For every x ∈ Xε

(QT ), it holds:

(i) There exists a constant c > 0 (not depending on q, p ∈ Plog (QT )) such that .

sup |ε((RhQT ) x)| ≤ c Md+1 (QT ε(x))

a.e. in Rd+1 .

h∈(0,h1 )

(ii) For an extension p ∈ Plog (Rd+1 ) of p ∈ Plog (QT ), with p− ≤ p ≤ p+ in Rd+1 , there exists a constant cp > 0 (depending on p ∈ Plog (Rd+1 ) and c > 0) such that .

sup ε((RhQT ) x)Lp(·,·) (Rd+1 )d×d ≤ cp ε(x)Lp(·,·) (QT )d×d .

h∈(0,h1 )

In particular, suph∈(0,h1 ) (RhQT ) xXεq,p (QT ) ≤ (cq +cp )xXεq,p (QT ) , where cq > 0 is from Proposition 3.15 (iii). q,p (iii) (RhQT ) x → x in Xε (QT ) (h → 0). Proof ad (i). First, we calculate the symmetric gradient of (RhQT ) x ∈ C0∞ (Rd+1 )d . Again, using the product rule ε(ηx) = ηε(x) + [x ⊗ ∇η]sym in L1 (Rd , Md×d sym ) for every x ∈ E 1,1 (Rd ) and η ∈ C 1 (Rd ) and Proposition 3.15 (i), for every h > 0, we observe that ε((RhQT ) x) = (RhQT ) (ε(x)) .

+ [(ωh ∗ (QT x)) ⊗ ∇ηh ]sym

in Rd+1 .

(3.51)

Proposition 3.15 (ii) and Remark 3.10 provide that .

sup |(RhQT ) (ε(x))| ≤ 2Md+1 (QT ε(x))

a.e. in Rd+1 .

(3.52)

h>0

For the remaining term, we note that [(ωh ∗ (QT x)) ⊗ ∇ηh ]sym = 0 in R × 3h for every h > 0. On the other hand, similar to in (3.47), i.e., using Theorem 3.2,

q,p

3.4 Smoothing in Xε

79

(QT )

 ωh L∞ (Rd+1 ) ≤ hcd+1 and ∇ηh L∞ (Rd )d ≤ ch , we infer for every t ∈ R, x ∈ 3h

and h ∈ (0, h1 ), i.e., r(x) = dist(x, ∂) ≤ 3h ≤ 3h1 = 34 h1 < h0 , that |[(ωh ∗ (QT x)) ⊗ ∇ηh ]sym (t, x)|  ωh (t − s, x − y)|(QT x)(s, y)| dsdy ≤ |∇ηh (x)| .

c ≤ d+2 h

Bhd+1 (t,x)







d Bh1 (t) Bhd (x) B2r(y) (y)

|(QT ε(x))(s, z)| dz dy ds |y − z|d−1

(3.53)

≤ c Md+1 (QT ε(x))(t, x) . Eventually, if we combine (3.51)–(3.53), then we conclude (i). ad (ii). Follows as in the proof Proposition 3.14 (iii) from (i) by means of Proposition 2.12. ad (iii). Proposition 3.15 (iv) and Remark 3.10 provide that (RhQT ) x → x .

(RhQT ) (ε(x)) → ε(x)

in Lq(·,·) (QT )d

(h → 0) ,

in Lp(·,·) (QT , Md×d sym )

(h → 0) .

(3.54)

Because of [(ωh ∗ (QT x)) ⊗ ∇ηh ]sym = 0 in R × 3h for every h > 0 and (3.53), we further conclude, using Proposition 2.4, that [(ωh ∗ (QT x)) ⊗ ∇ηh ]sym → 0

.

in Lp(·,·) (QT , Md×d sym )

(h → 0) .

(3.55)

All things considered, (3.51), (3.54) and (3.55) prove that (RhQT ) x → x in q,p Xε (QT ) (h → 0).   After a modification of RhQT , we additionally obtain the density of C0∞ (QT )d in q,p Xε (QT ).   Proposition 3.18 Let (ϕh )h∈I◦ ⊆ C0∞ (I ), where I◦ := 0, T4 , be a family of cutoff functions such that for every h ∈ I◦ , it holds supp(ϕh ) ⊆ I2h ,16 where Ih := q,p (h, T − h), 0 ≤ ϕh ≤ 1 in I and ϕh → 1 (h → 0) a.e. in I . For x ∈ Xε (QT ) and h ∈ I◦ , we define the smoothing operator ˚h x := Rh (ϕh x) ∈ C ∞ (Rd+1 )d . R 0 QT QT

.

16 The

assumption h ∈ I◦ guarantees that I2h = ∅.

80

3 Variable Bochner–Lebesgue Spaces q,p

Then, for every x ∈ Xε

(QT ), it holds:

˚h x)h∈I ⊆ C ∞ (QT )d with supp(R ˚h x) ⊆ Ih × h for every h ∈ I◦ . (i) (R ◦ QT QT 0 (ii) There exists a constant c > 0 (not depending on q, p ∈ Plog (QT )) such that ˚h x| ≤ 2Md+1 (Q x) sup |R T QT

.

a.e. in QT ,

h∈I◦

sup h∈(0,h1 )∩I◦

˚h x)| ≤ c Md+1 (Q ε(x)) |ε(R T QT

a.e. in QT .

(iii) For extensions q, p ∈ Plog (Rd+1 ) of q, p ∈ Plog (QT ), resp., with q − ≤ q ≤ q + and p− ≤ p ≤ p+ in Rd+1 , there exist constants cq , cp > 0 (depending on q, p ∈ Plog (Rd+1 ) and c > 0, resp.) such that .

˚h x sup R d+1 d ≤ cq xLq(·,·) (Q )d , QT T ) Lq(·,·) (R

h∈I◦

sup h∈(0,h1 )∩I◦

˚h x) ε(R d+1 d×d ≤ cp ε(x)Lp(·,·) (Q )d×d . QT T Lp(·,·) (R )

˚h x q,p q,p In particular, suph∈(0,h1 )∩I◦ R Xε (QT ) ≤ (cq + cp )xXε (QT ) . QT q,p h ∞ d ˚ (iv) RQT x → x in Xε (QT ) (h → 0), i.e., C0 (QT ) lies densely in q,p Xε (QT ). ˚h x)h>0 ⊆ C ∞ (Rd+1 )d . To Proof ad (i). Proposition 3.14 (i) guarantees that (R QT 0 h ∞ ˚ be more precise, we have that (RQT x)h∈I◦ ⊆ C0 (QT )d . Indeed, for every h ∈ I◦ , it holds ˚h x) ⊆ supp(ϕh ηh (Q x)) + B d+1 (0) supp(R T QT h

.

⊆ (I2h × 2h ) + Bhd+1 (0) ⊆ Ih × h ⊆ QT . ad (ii).& (iii). Follow from Proposition 3.14 (ii) & (iii), Proposition 3.16 (i) & (ii), and that suph∈I◦ ϕh L∞ (R) ≤ 1. ad (iv). By applying Proposition 3.16 (ii) & (iii) and Proposition 2.4, we obtain ˚h x − x q,p R QT Xε (QT )

.

≤ RhQT [(1 − ϕh )x]Xεq,p (QT ) + RhQT x − xXεq,p (QT ) ≤ (cq + cp )(1 − ϕh )xXεq,p (QT ) + RhQT x − xXεq,p (QT ) → 0 (h → 0) .

 

q,p

3.4 Smoothing in Xε

81

(QT )

Last but not least, let us extend the smoothing operator RhQT , using Proposiq,p tion 3.4, to a smoothing operator for functionals from the dual space Xε (QT )∗ . Proposition 3.19 (Smoothing in Xε (QT )∗ ) For every h > 0, we define q,p q,p RhQT : Xε (QT )∗ → Xε (QT )∗ by q,p

   RhQT x ∗ , x X q,p (Q ) := x ∗ , (RhQT ) x X q,p (Q ) T T ε ε

 .

for every x ∗ ∈ Xε it holds:

q,p

(QT )∗ and x ∈ Xε

q,p

(3.56)

(QT ). Then, for every x ∗ ∈ Xε

q,p

(QT )∗ ,

17 (i) (RhQT x ∗ )h>0 ⊆ C∗∞ (QT ) := Jε (C ∞ (I ,C0∞ ()d )×C ∞ (I ,C0∞ (,Md×d sym ))). h ∗ ∗ (ii) suph∈(0,h1 ) RQT x Xεq,p (QT )∗ ≤ (cq + cp )x Xεq,p (QT )∗ , where cq , cp > 0 are from Propositions 3.15 (iii) and 3.17 (ii), respectively. q,p (iii) RhQT x ∗ → x ∗ in Xε (QT )∗ (h → 0), i.e., C∗∞ (QT ) lies densely in q,p Xε (QT )∗ .

Remark 3.11 The operator defined in (3.56) is the operator ((RhQT ) )∗ : Xε (QT )∗ q,p → Xε (QT )∗ , i.e., the adjoint operator of the quasi adoint operator q,p q,p h  (RQT ) : Xε (QT ) → Xε (QT ) (cf. Proposition 3.17 (iii)). Nevertheless, we employ for this operator the same notation as for the operator in Proposition 3.14, i.e., we define RhQT := ((RhQT ) )∗ . This is motivated by the q,p q,p fact that RhQT : Xε (QT )∗ → Xε (QT )∗ can be seen as an extension of q,p







RhQT : Lq (·,·) (QT )d → Lq (·,·) (QT )d , in the sense that for every x ∈ Lq (·,·) (QT )d ,   q,p it holds RhQT Jε (x, 0) = Jε (RhQT x, 0) in Xε (QT )∗ . More precisely, for every

q,p

x ∈ Lq (·,·) (QT )d and y ∈ Xε

(QT ), we have that

 .

   RhQT Jε (x, 0) , y X q,p (Q ε

T)

  = Jε (x, 0), (RhQT ) y X q,p (Q ) T ε   h  = x, (RQT ) y Lq(·,·) (Q )d T  h  = RQT x, y Lq(·,·) (Q )d T   = Jε (RhQT x, 0), y X q,p (Q ) . ε

T





Proof ad (i). Proposition 3.4 provides f ∈ Lq (·,·) (QT )d and F ∈ Lp (·,·) q,p ∗ ∗ (QT , Md×d sym ) such that x = Jε (f , F ) in Xε (QT ) . Therefore, aided by (3.51),

17 C ∞ (I , C ∞ (, Md×d )) sym 0

:= { ∈ C ∞ (QT )d×d | (t) ∈ C0∞ (, Md×d sym ) for every t ∈ I }.

82

3 Variable Bochner–Lebesgue Spaces

we infer for every φ ∈ C0∞ (QT )d and h > 0 that  .

   RhQT x ∗ , φ X q,p (Q ) = Jε (f , F ), (RhQT ) φ X q,p (Q ) T T ε ε   h h = Jε (RQT f , RQT F ), φ X q,p (Q ) T ε   sym + F , [∇ηh ⊗ (ωh ∗ (QT φ))] Lp(·,·) (Q

T)

d×d

,

i.e., for every h > 0, it holds the identity RhQT x ∗ = Jε (RhQT f .

+ ωh ∗ (QT (F ∇ηh )), RhQT F )

q,p

in Xε

(QT )∗ ,

(3.57)

owing to the density of C0∞ (QT )d in Xε (QT ) (cf. Proposition 3.18 (iv)) and, hence, apparently (i). q,p ad (ii). Using Proposition 3.17 (ii), we obtain for every x ∗ ∈ Xε (QT )∗ q,p

RhQT x ∗ Xεq,p (QT )∗ =

.

≤



sup q,p x∈Xε (QT ) xX q,p (Q ) ≤1 ε T

sup q,p x∈Xε (QT ) xX q,p (Q ) ≤1 ε T

 x ∗ , (RhQT ) x X q,p (Q ) T ε

x ∗ Xεq,p (QT )∗ (RhQT ) xXεq,p (QT )

≤ (cq + cp )x ∗ Xεq,p (QT )∗ .



ad (iii). Proposition 3.4 provides f ∈ Lq (·,·) (QT )d and F ∈ Lp (·,·) (QT , Md×d sym ) q,p such that x ∗ = Jε (f , F ) in Xε (QT )∗ . Using Proposition 3.14 (iv),



Remark 3.8, and the continuity of Jε : Lq (·,·) (QT )d × Lp (·,·) (QT , Md×d sym ) q,p ∗ → Xε (QT ) (cf. Proposition 3.4), we find that Jε (RhQT f , RhQT F ) → x ∗

.

q,p

in Xε

(QT )∗

(h → 0) .

(3.58)

Due to suph∈(0,h1 ) |[∇ηh ⊗ (ωh ∗ (QT x))]sym | ≤ c Md+1 (QT ε(x)) a.e. in Rd+1 q,p for x ∈ Xε (QT ) (cf. (3.53)), Proposition 2.12 yields a constant cp > 0 such that q,p for every x ∈ Xε (QT ), it holds sup [∇ηh ⊗ (ωh ∗ (QT x))]sym Lp(·,·) (QT )d×d

h∈(0,h1 ) .

≤ cp ε(x)Lp(·,·) (QT )d×d ≤ cp xXεq,p (QT ) .

q,p

q,p

˚ (QT ) and X ˚ (QT ) 3.5 The Spaces X ∇ ε

83

Therefore, because of [∇ηh ⊗(ωh ∗(QT x))]sym = 0 in I ×3h for every h > 0 and q,p x ∈ Xε (QT ), in particular, making use of Proposition 2.6 and Proposition 2.4, we conclude that Jε (ωh ∗ (QT (F ∇ηh )), 0)Xεq,p (QT )∗   F χI ×3h , [∇ηh ⊗ (ωh ∗ (QT x))]sym Lp(·,·) (Q = sup

.

T)

q,p

x∈Xε (QT ) xX q,p (Q ) ≤1 ε

d×d

T

≤ 2cp F χI ×3h Lp (·,·) (QT )d×d → 0 (h → 0) .

(3.59)  

Eventually, the combination of (3.57), (3.58) and (3.59) proves (iii).

˚ q,p (QT ) and X ˚ q,p (QT ) 3.5 The Spaces X ∇ ε For the introduction of a well-posed notion of a generalized time derivative for the treatment of unsteady problems in variable exponent spaces, which is, in particular, aligned with the notion of the distributional time derivative living in D (QT )d , i.e., the topological dual space of the locally convex Hausdorff vector space q,p C0∞ (QT )d (cf. [75, Kap. II, 4.]), the density of C0∞ (QT )d in Xε (QT ) becomes indispensable. Unfortunately, we have only been able of proving the density of q,p C0∞ (QT )d in Xε (QT ) by requiring Lipschitz regularity of our spatial domain d  ⊆ R , d ≥ 2, and that q, p ∈ Plog (QT ) with q − , p− > 1 (cf. Proposition 3.18 ˚ q,p (QT ) (iv)). It is precisely for this reason that we next introduce the spaces X ∇ q,p q,p q,p ˚ (QT ), subspaces of X (QT ) and Xε (QT ), respectively, that trivially and X ε ∇ possess this density property. For the entire section, let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and q, p ∈ P∞ (QT ) with q − , p− > 1. Definition 3.5 We define the closure variable Bochner–Lebesgue spaces ·X q,p ∞ ∇ . ˚∇ (QT ) := C (QT )d X 0

q,p

˚ q,p (QT ) := C ∞ (QT ) X ε 0

(QT )

· q,p d Xε (QT ) q,p

, . q,p

˚ (QT ) = X Remark 3.12 We have no information on whether X ∇ ∇ (QT ) or ˚ q,p (QT ) = Xεq,p (QT ) in general. If  ⊆ Rd , d ≥ 2, is a bounded Lipschitz X ε

84

3 Variable Bochner–Lebesgue Spaces

domain and q, p ∈ Plog (QT ) with q − , p− > 1, then, by virtue of Proposition 3.18 ˚ q,p (QT ) and X q,p (QT ), as well as (iv) and [51, Theorem 4.7],18 then the spaces X ∇ ∇ q,p q,p ˚ (QT ) and Xε (QT ), respectively, coincide, and we will, hence, always write X ε q,p ˚ q,p (QT ), or Xεq,p (QT ) instead of X ˚ q,p (QT ), respectively, X∇ (QT ) instead of X ∇ ε in this situation, to indicate that these conditions are satisfied. Proposition 3.20 The following statements apply: ˚ q,p (QT ) → Lq(·,·) (QT )d × (i) The mapping ε,◦ := ε ◦ id X˚εq,p (QT ) : X ε Lp(·,·) (QT , Md×d 3.3, is an isometric sym ), where ε denotes from Proposition ˚ q,p (QT ) is separable. In isomorphism into its range R(ε,◦ ). In particular, X ε ˚ q,p (QT ) is reflexive. addition, if q − , p− > 1, then X ε

(ii) If q − , p− > 1, then the mapping Jε,◦ := (id X˚εq,p (QT ) )∗ ◦Jε : Lq (·,·) (QT )d ×

∗ ˚ q,p ˚ q,p (QT ) )∗ denotes the adjoint Lp (·,·) (QT , Md×d sym ) → X ε (QT ) , where (id X ε q,p q,p ˚ (QT ) → Xε (QT ) and Jε the mapping is operator of id X˚εq,p (QT ) : X ε from Proposition 3.4, is linear and Lipschitz continuous with constant 2. In ˚ q,p (QT )∗ , there exist f ∈ Lq (·,·) (QT )d and F ∈ addition, for every x ∗ ∈ X ε

∗ ∗ ˚ q,p Lp (·,·) (QT , Md×d sym ) such that x = Jε,◦ (f , F ) in X ε (QT ) and 1 ∗ x X˚ q,p (Q )∗ ≤ f Lq (·,·) (QT )d + F Lp (·,·) (QT )d×d T ε 2 . ≤ 2x ∗ X˚ q,p (Q ε

T

(3.60)

. )∗

˚ q,p (QT ) is a closed subspace Proof ad (i). Follows from Proposition 3.3 because X ε q,p of Xε (QT ). ad (ii). Well-definedness, linearity and Lipschitz continuity with constant 2



q,p ∗ transfer directly from Jε : Lq (·,·) (QT )d ×Lp (·,·) (QT , Md×d sym ) → Xε (QT ) (cf. Proposition 3.4). So, let us prove the surjectivity including (3.60). Let ˚ q,p (QT )∗ be arbitrary. The Hahn–Banach theorem yields a functional x∗ ∈ X ε q,p ˚ q,p (QT )∗ and x ∗ ∈ Xε (QT )∗ such that (id X˚εq,p (QT ) )∗ x ∗ = x ∗ in X ε

x ∗ Xεq,p (QT )∗ = x ∗ X˚ q,p (QT )∗ . Proposition 3.4 provides f ∈ Lq (·,·) (QT )d

ε

∗ ∗ and and F ∈ Lp (·,·) (QT , Md×d sym ) such that x = Jε (f , F ) in Xε (QT ) 1 ∗ ∗ q,p



2x Xεq,p (QT )∗ . Putting 2 x Xε (QT )∗ ≤ f Lq (·,·) (QT )d + F Lp (·,·) (QT )d×d ≤q,p ˚ (QT )∗ and (3.60).   everything together, we have that x ∗ = Jε,◦ (f , F ) in X ε q,p

In favor of readability, we set for the entirety of this book ε := ε,◦ and Jε := Jε,◦ . ˚ q,p (QT )∗ ) In virtue of ProposiRemark 3.13 (Time Slices of Functionals x ∗ ∈ X ε tion 3.20 (ii) and Corollary 3.2, Remark 3.6 extends to the potentially larger dual ˚ q,p (QT )∗ . space X ε = 2 in QT is considered, but the procedure equally applies to arbitrary q ∈ Plog (QT ) with q − > 1.

18 More precisely, in [51, Theorem 4.7], only the case q

q,p

˚ (QT )∗ 3.6 Generalized Time Derivative in X ε

85

˚ q,p (QT )∗ 3.6 Generalized Time Derivative in X ε Since the introduction of variable Bochner–Lebesgue spaces was driven by the inability of classical Bochner–Lebesgue spaces to properly describe unsteady problems with variable exponent non-linearities, such as, e.g., the model problem (3.1), the notion of Banach-valued distributional time derivative living in Bochner– Lebesgue spaces (cf. Definition 2.19) is likewise not qualified for a thorough ˚ q,p (QT ), however, enables description of problems of this nature. The space X ε to introduce an appropriate notion of a generalized time derivative for variable exponent Bochner–Lebesgue spaces, even on irregular domains, which is supposed ˚ q,p (QT )∗ and which is aligned with the notion of the to be a functional in X ε distributional time derivative living in D (QT )d . For the entire section, let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), ˚ q,p → Y := L2 ()d , T < ∞, QT := I × , and q, p ∈ P∞ (QT ), such that X − 2d e.g., if p− ≥ d+2 or simply q − ≥ 2. q,p

Definition 3.6 A function x ∈ Xε (QT ) possesses a generalized time derivative ˚ q,p (QT )∗ such that for every ˚ q,p (QT )∗ , if there exists a functional x ∗ ∈ X in X ε ε φ ∈ C0∞ (QT )d , it holds  .

− I

In this case, we define

dx dt

(x(t), ∂t φ(t))Y dt = x ∗ , φX˚ q,p (Q ) . ε

T

(3.61)

q,p

˚ (QT )∗ . := x ∗ in X ε

Lemma 3.7 The generalized time derivative in the sense of Definition 3.6 is unique. q,p ˚ q,p (QT )∗ such that for Proof Suppose for x ∈ Xε (QT ), there exist x ∗1 , x ∗2 ∈ X ε q,p ∗ i = 1, 2, it holds (3.61). Then, we have that x 1 , φX˚ε (QT ) = x ∗2 , φX˚εq,p (QT ) for every φ ∈ C0∞ (QT )d . Because, by Definition 3.5, the space C0∞ (QT )d is dense in ˚ q,p (QT ), we conclude that x ∗ = x ∗ in X ˚ q,p (QT )∗ .   X ε ε 1 2 d from Definition 3.6 is related to Remark 3.14 The generalized time derivative dt the distributional time derivative ∂t , i.e., for x ∈ L1loc (QT )d , the distribution ∂t x ∈  D (QT )d , defined by ∂t x, φC0∞ (QT )d := − QT x(t, x) · ∂t φ(t, x) dtdx for every φ ∈ C0∞ (QT )d , in the following sense: q,p ∗ ˚ q,p If x ∈ Xε (QT ) possesses a generalized time derivative dx dt ∈ X ε (QT ) ,

(·,·)

(·,·) d×d q d p then Proposition 3.20 (ii) provides f ∈L (QT ) and F ∈ L (QT , Msym ) q,p dx ∗ ˚ such that dt = Jε (f , F ) in X ε (QT ) . Therefore, it can readily be read from (3.61) that ∂t x = f − div(F ) in D (QT )d .19

the common abuse of notation, the functional f − div(F ) ∈ D (QT )d is defined by f − div(F ), φC0∞ (QT )d := (f , φ)Lq(·,·) (QT )d + (F , ε(φ))Lp(·,·) (QT )d×d for every φ ∈ C0∞ (QT )d . 19 With

86

3 Variable Bochner–Lebesgue Spaces

Next, we introduce a variable exponent generalization of a classical Bochner– Sobolev space. Definition 3.7 We define the variable Bochner–Sobolev space q,p .Wε (QT )

   dx q,p q,p  ∗ ˚ ˚ := x ∈ X ε (QT )  ∃ ∈ X ε (QT ) . dt q,p

Proposition 3.21 The space Wε (QT ) forms a Banach space, if equipped with the norm   d ·  q,p q,p . . ·  ˚ (QT ) +  Wε (QT ) :=  · X  dt  ˚ q,p ε X ε (QT )∗ In addition, if q − , p− > 1, then Wε (QT ) is separable and reflexive. q,p

Proof q,p

1. Completeness: Let (x n )n∈N ⊆ Wε (QT ) be a Cauchy sequence, i.e., also ∗ ˚ q,p (QT ) and ( dx n ) ˚ q,p the sequences (x n )n∈N ⊆ X ε dt n∈N ⊆ X ε (QT ) are Cauchy q,p q,p ∗ ˚ (QT )∗ such ˚ (QT ) and x ∈ X sequences. Then, there exist x ∈ X ε ε q,p ˚ (QT ) (n → ∞) and dx n → x ∗ in X ˚ q,p (QT )∗ that x n → x in X ε ε dt (n → ∞), from which we infer for every φ ∈ C0∞ (QT )d , also making use ˚ q,p (QT ) → X q,p (QT ) → Lmin{q − ,p− } (I, Y ), based on Proposition 3.2 of X − ε ˚q,p → Y , that and X − n→∞

∗

x , φX˚ q,p (Q

.

ε

T)



←

  dx n , φ q,p = − (x n (t), ∂t φ(t))Y dt dt ˚ (QT ) I X ε  n→∞ → − (x(t), ∂t φ(t))Y dt , I q,p

∗ ∗ ˚ i.e., x ∈ Wε (QT ) with dx dt = x in X ε (QT ) and x n → x in Wε (QT ) (n → ∞). q,p 2. Reflexivity and separability: If q − , p− > 1, then the space Wε (QT ) inherits q,p q,p ∗ ˚ (QT ) (cf. Proposition 3.20 ˚ (QT ) × X the reflexivity and separability of X ε ε (i)), in virtue of the isometric isomorphism q,p

q,p

ε : W ε

.

q,p

q,p

q,p

˚ (QT ) × X ˚ (QT )∗ , (QT ) → R(ε ) ⊆ X ε ε q,p

which for every x ∈ Wε ˚ q,p (QT )∗ . X ε

 ˚ q,p (QT ) is defined by ε x := (x, dx dt ) in X ε (QT ) ×  

q,p

˚ (QT )∗ 3.6 Generalized Time Derivative in X ε

87

q,p q,p q,p : X q,p : X ˚+ ˚− ˚+ ˚− Since, by assumption, the mappings id X → Y and id X →Y are embeddings, which are inevitably also dense, their corresponding adjoint operq,p ∗ q,p ∗ q,p )∗ : Y ∗ → (X q,p )∗ : Y ∗ → (X ˚+ ˚− ˚+ ˚− ators (id X ) and (id X ) are embeddings as well. Consequently, also the mappings

.

q,p q,p ∗ q,p )∗ R id ˚q,p : X ˚− ˚+ ˚+ e− := (id X → (X ) , Y X−

(3.62)

q,p q,p ∗ q,p )∗ R id ˚q,p : X ˚+ ˚− ˚− → (X ) , e+ := (id X Y X+

where RY : Y → Y ∗ denotes the Riesz isomorphism with respect to Y , are embeddings. This ensures the well-posedness of the following limiting Bochner– Sobolev spaces (cf. Definition 2.19). Definition 3.8 We define the limiting Bochner–Sobolev spaces q,p

1,max{q + ,p+ },max{(q − ) ,(p− ) }

q,p

1,min{q − ,p− },min{(q + ) ,(p+ ) }

W+ (QT ) := We+

.

W− (QT ) := We−

q,p

˚+ , (X ˚− )∗ ) , (I, X q,p

q,p

q,p ˚q,p ∗ ˚− (I, X , (X+ ) ) . q,p

We emphasize that the spaces W+ (QT ) and W− (QT ) employ a different q,p concept of a time derivative than Wε (QT ), namely the Banach-valued disd e+ d e− tributional time derivatives dt and dt (cf. Definition 2.19) living in classical Bochner–Lebesgue spaces. This gives us immediate access to the following nonq,p q,p symmetric formula of integration-by-parts for W+ (QT ) and W− (QT ). Proposition 3.22 (Non-Symmetric Formula of Integration-By-Parts) The following statements apply: (i) Each x ∈ W− (QT ) ∩ Y ∞ (QT ) possesses a weakly continuous repq,p resentation x ω ∈ Cω0 (I , Y ). Moreover, each y ∈ W+ (QT ) possesses a 0 continuous representation y c ∈ Y (QT ) (cf. (3.17)) and the resulting mapping q,p (·)c : W+ (QT ) → Y 0 (QT ) is an embedding. q,p q,p (ii) For every x ∈ W− (QT ) ∩ Y ∞ (QT ), y ∈ W+ (QT ), and t , t ∈ I with

t ≤ t, it holds q,p

 t .

t

 s=t  de− x (s), y(s) ds = (x ω (s), y c (s))Y s=t

q,p dt ˚+ X   t de+ y (s), x(s) − ds . q,p dt ˚− t

X

88

3 Variable Bochner–Lebesgue Spaces q,p

q,p

(iii) For every x ∈ W− (QT ) and y ∈ W+ (QT ) with supp(y) ⊆ I × , it holds       de+ y de− x (s), y(s) (s), x(s) . ds = − ds . q,p q,p dt dt ˚+ ˚− I I X X q,p q,p ˚+ ˚− Proof Follows from Proposition 2.28 for X+ := X , X− := X , Y := L2 ()d , + + + − q,p , j q,p , p := max{q , p }, and p := min{q − , p− }. ˚+ ˚− j+ := id X   − := id X

The non-symmetric formula of integration-by-parts enables us to examine the q,p q,p exact relation between the spaces Wε (QT ) and W− (QT ). More precisely, q,p q,p Proposition 3.23 below states that Wε (QT ) ⊆ W− (QT ). Apart from that, q,p q,p by comparing the norms of Wε (QT ) and W− (QT ), it is readily seen that even q,p q,p Wε (QT ) → W− (QT ). Similar arguments as in the proof of Proposition 3.23 q,p q,p q,p also show that W+ (QT ) → Wε (QT ). In this way, the spaces W+ (QT ) and q,p W− (QT ) afford in many situations access to standard methods from the classical theory of Bochner–Sobolev spaces, such as, e.g., to the method of in time extension q,p via reflection (cf. Proposition 3.24), even though Wε (QT ) actually does not fit into this framework. q,p

Proposition 3.23 (Alternative Characterization of Wε (QT )) For a function ˚ q,p (QT ) and a functional x ∗ ∈ X ˚ q,p (QT )∗ , the following statements are x ∈X ε ε equivalent: q,p ∗ ∗ ˚ q,p (i) x ∈ Wε (QT ) with dx dt = x in X ε (QT ) . q,p ∞ ˚+ and ϕ ∈ C (I ), it holds (ii) For every y ∈ X 0

 .



−

(x(s), y)Y ∂t ϕ(s) ds = I

q,p

i.e., x ∈ W− (QT ) with

I d e− x dt

x ∗ (s), yX ˚εq,p (s) ϕ(s) ds ,

  ∗ ∗ in Lmin{(q + ) ,(p+ ) } q,p = J˚−1 q,p (idX+ (QT ) ) x X+

q,p ∗ ˚+ (I, (X ) ), where we denote by the mapping (idX+q,p (QT ) )∗ the adjoint operq,p ˚ q,p (QT ) and J˚q,p : Lmin{(q + ) ,(p+ ) } ator of idX+q,p (QT ) : X+ (QT ) → X ε X+ q,p ∗ q,p ˚+ (I, (X ) ) → X+ (QT )∗ is the isomorphism from Proposition 2.20.

Proof ˚ q,p (QT ) → X q,p (QT ) (cf. Proposition 3.2), it holds x ∈ (i) ⇒ (ii). Due to X − ε q,p X− (QT ). We take y ∈ C0∞ ()d and ϕ ∈ C0∞ (I ). Then, φ := yϕ ∈ C0∞ (QT )d . Thus, using Proposition 2.21 (i), that e− (x(t)), yX ˚q,p = (x(t), y)Y for almost +

q,p

˚ (QT )∗ 3.6 Generalized Time Derivative in X ε

89

every t ∈ I , ∂t φ = y∂t ϕ in QT , Proposition 2.20, and Definition 3.6, we find that     e− − x(s)∂t ϕ(s) ds , y  =− 

.

q,p

˚+ X

I

I

=−

e− (x(s)), yX ˚q,p ∂t ϕ(s) ds +

(x(s), ∂t φ(s))Y ds I

= x ∗ , φXεq,p (QT )    −1  ∗ ∗ = J˚q,p (idX q,p (QT ) ) x (s)ϕ(s) ds, y I

+

X+

q,p

.

˚+ X

q,p ˚+ Since, by definition, C0∞ ()d lies densely in X (cf. Definition 2.8), we infer for every ϕ ∈ C0∞ (I )

     ∗ ∗ ˚q,p ∗ e− − x(s)∂t ϕ(s) ds = J˚−1 q,p (idX q,p (QT ) ) x (s)ϕ(s) ds in (X+ ) , + X+

.

I

I

  d x q,p ∗ ∗ in Lmin{(q + ) ,(p+ ) } q,p i.e., x ∈ W− (QT ) with edt− = J˚−1 q,p (idX+ (QT ) ) x X+ q,p ∗ ˚+ (I, (X ) ), according to Definition 2.19. (ii) ⇒ (i) Let φ ∈ C0∞ (QT )d . Making use of both Proposition 3.22 (iii) and −

−

d φ q,p q,p ∗ ˚− ) ), that φ ∈ W+ (QT ) with edt+ = e+ (∂t φ) in Lmax{(q ) ,(p ) } (I, (X q,p = (x(t), ∂t φ(t))Y for almost every t ∈ I , and that at e+ (∂t φ(t)), x(t)X ˚− least supp(φ) ⊆ I × , we observe that   .− (x(s), ∂t φ(s))Y ds = − e+ (∂t φ(s)), x(s)X ˚q,p ds I

I

 



−

de+ φ (s), x(s) ds q,p dt ˚− I X    de− x (s), φ(s) ds = q,p dt ˚+ I X =−

= x ∗ , φX˚ q,p (Q ) , ε

q,p

i.e., x ∈ Wε

(QT ) with

dx dt

T

˚ q,p (QT )∗ , according to Definition 3.6. = x ∗ in X ε  

Apart from that, Proposition 3.22 (i) allows the introduction of a generalized notion of evolution equation (cf. Sect. 2.3.4) in the framework of variable Bochner– Lebesgue spaces and variable Bochner–Sobolev spaces.

90

3 Variable Bochner–Lebesgue Spaces

Definition 3.9 (Generalized Evolution Equation) Let x0 ∈ Y be an initial value, ˚ q,p (QT )∗ a right-hand side, and A : Wεq,p (QT )∩Y ∞ (QT ) → X ˚ q,p (QT )∗ x∗ ∈ X ε ε an operator. The initial value problem .

dx + Ax = x ∗ dt x ω (0) = x0

˚ q,p (QT )∗ , in X ε in Y ,

is called a generalized evolution equation. Here, the initial condition has to be understood in the sense of the weakly continuous representation x ω ∈ Cω0 (I , Y ) (cf. Proposition 3.22 (i)).

q,p

3.7 Formula of Integration-by-Parts for Wε

(QT )

This section is concerned with the proof of a formula of integration-by-parts q,p for Wε (QT ), which is a cornerstone in the existence analysis of generalized evolution equations (cf. Definition 3.9) based on advanced pseudo-monotonicity methods. For this purpose, we follow the traditional procedure of first establishing q,p the density of smooth functions, i.e., of C ∞ (I , C0∞ ()d ), in the space Wε (QT ). To be more concrete, we will modify the smoothing operator from Proposition 3.14– in analogy to Proposition 2.26– by means of the method of extension in time via reflection (cf. Proposition 2.25). It is noteworthy that, even though the construction q,p of a smoothing operator for Xε (QT ) was mainly driven by its interpretation as a kind of a variable Sobolev space on the time-space cylinder QT involving only spatial weak derivatives, it can be modified by falling back on methods from the classical theory of Bochner–Sobolev spaces to obtain a smoothing operator for q,p Wε (QT ). This circumstance, in turn, reflects the Bochner–Lebesgue character of q,p Xε (QT ) (cf. Remark 3.2), and in doing so, at the same time, the unique interplay of steady and unsteady approaches in the analysis of unsteady problems in variable exponent spaces. To start with, based on Proposition 3.23, we generalize the method q,p of extension in time via reflection to the framework of Wε (QT ). For the entire section, let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain, I := (0, T ), T < ∞, QT := I × , and q, p ∈ Plog (QT ) with q − , p− > 1,20 such q,p ˚− that X

→ Y .

20 Precisely

(iv)).

q,p

the assumptions that guarantee that Xε

q,p

˚ε (QT ) (cf. Proposition 3.18 (QT ) = X

q,p

3.7 Formula of Integration-by-Parts for Wε

91

(QT ) q,p

Proposition 3.24 (Extension in Time via Reflection) For every x ∈ Xε (QT ), we define the in time extension via reflection for every t ∈ 3I := (−T , 2T ) by ⎫ ⎧ ⎪ ⎪ x(−t) if t ∈ , 0] (−T ⎪ ⎪ ⎬ ⎨ ˚qT ,pT (t) ,  in X .(ET x)(t) := (3.63) x(t) if t ∈ I ⎪ ⎪ ⎪ ⎪ ⎩x(2T − t) if t ∈ [T , 2T ) ⎭ where qT :=  ET q ∈ Plog (Q3T ), pT :=  ET p ∈ Plog (Q3T ), and Q3T := 3I × . Then, it holds: q,p q ,p ET : Xε (QT ) → Xε T T (Q3T ) is well-defined, linear and Lipschitz (i)  continuous with constant 3. q,p q ,p ET : Wε (QT ) → Wε T T (Q3T ) is well-defined, linear and Lipschitz (ii)  q,p continuous with constant 12. More precisely, if x ∈ Wε (QT ) with

q,p dx ∗ for f ∈ Lq (·,·) (Q )d and F ∈ Lp (·,·) T dt = Jε (f , F ) in Xε (QT ) q ,p T T  (QT , Md×d (Q3T ) with sym ), then ET x ∈ Wε

.

d ET x − − = Jε ( ET f ,  ET F ) dt

q ,pT

in Xε T

(Q3T )∗ ,

(3.64)

− − where  ET f ∈ LqT (·,·) (Q3T )d is defined by ( ET f )(t) := ( ET f )(t) in − q (t,·) d q (t,·) d T T   () if t ∈ I and (ET f )(t) := −(ET f )(t) in L () if t ∈ 3I \ I , L

− ) analogously. and  ET F ∈ LpT (·,·) (Q3T , Md×d sym

Proof ad (i). Let x ∈ Xε (QT ) be arbitrary. Then, it holds  ET x ∈ LqT (·,·) (Q3T )d , d×d p (·,·) ˚qT ,pT (t) for almost ET ε(x) ∈ L T (Q3T , Msym ), and ( ET x)(t) ∈ X ε( ET x) =  q ,p every t ∈ 3I , i.e., we have that  ET x ∈ Xε T T (Q3T ) (cf. Definition 3.2) with  ET xX qT ,pT (Q3T ) ≤ 3xXεq,p (QT ) . q,p

ε





ad (ii). Let f ∈ Lq (·,·) (QT )d and F ∈ Lp (·,·) (QT , Md×d sym ) be such that q,p dx ∗ . Then, Proposition 3.23 yields that x ∈ W q,p (Q ) = J (f , F ) in X (Q ) ε T T ε − dt   d x q,p ∗ ∗ J (f , F ) in Lmin{(q + ) ,(p+ ) } (I, (X q,p ˚ with edt− = J˚−1 (id ) ) ). Hence, q,p X+ (QT ) ε + X+

q ,p using Proposition 2.25, we find that  ET x ∈ W−T T (Q3T ) with

.

⎫  de− ET x − de− x ⎪ ⎪ = ET ⎬ dt dt  ⎪ ⎪ ∗ q ,p ⎭ − − = J −1 qT ,pT (idX+ T T (Q3T ) ) Jε (ET f , ET F ) X+

in L

min{(qT+ ) ,(pT+ ) }

q ,pT ∗

˚+T (3I, (X

) ).

(3.65)

92

3 Variable Bochner–Lebesgue Spaces

q ,p Using again Proposition 3.23, we conclude from (3.65) to  ET x ∈ Wε T T (Q3T )

(·,·)

(·,·) qT ,pT q p d×d d with (3.64). Using that Jε : L T (Q3T ) × L T (Q3T, Msym )→Xε (Q3T )∗ is Lipschitz continuous with constant 2 (cf. Proposition 3.4), we deduce from (3.64) that     d   − −  ET x  ≤ 2  ET f  qT (·,·) +  ET F  pT (·,·) d d×d  dt  qT ,pT L (Q ) L (Q ) 3T 3T . (3.66) Xε (Q3T )∗   = 6 f Lq (·,·) (QT )d + F Lp (·,·) (QT )d×d .

In addition, Proposition 3.4 yields the existence of functions f 0 ∈ Lq (·,·) (QT )d and

q,p dx ∗ F 0 ∈ Lp (·,·) (QT , Md×d sym ) satisfying both dt = Jε (f 0 , F 0 ) in Xε (QT ) and    1  dx  ≤ f 0 Lq (·,·) (QT )d + F 0 Lp (·,·) (QT )d×d  2 dt Xεq,p (QT )∗ .    dx   ≤ 2 .  dt  q,p Xε (QT )∗

(3.67)



Thus, if we insert f 0 ∈ Lq (·,·) (QT )d and F 0 ∈ Lp (·,·) (QT , Md×d sym ) into (3.66), exploiting (3.67) in doing so, then we arrive at     d    ET x  ≤ 6 f 0 Lq (·,·) (QT )d + F 0 Lp (·,·) (QT )d×d  dt  qT ,pT Xε (Q3T )∗ . (3.68)    dx   . ≤ 12  dt  q,p Xε (QT )∗ Combining (3.68) and (i), we conclude that  ET : Wε Lipschitz continuous with constant 12.

q,p

q ,pT

(QT ) → Wε T

(Q3T ) is  

In analogy to Proposition 2.26, the combination of the smoothing operator from q,p Propositions 3.14 and 3.24 yields a smoothing operator for the space Wε (QT ). q,p

Proposition 3.25 For x ∈ Wε tor

(QT ) and h > 0, we define the smoothing opera-

∞ d ∞ h x := Rh ( R QT Q3T ET x)|QT ∈ C (I , C0 () ) .

.

q,p

Then, for every x ∈ Wε (i)

h x)h>0 (R QT

⊆

.

(QT ), q,p Wε (QT )

(3.69)

it holds: with

   dET x d h RQT x = ∗QT RhQ3T dt dt

in Xε (QT )∗ q,p

  q ,p q,p for every h ∈ I∧ := 0, T2 , where ∗QT : Xε T T (Q3T )∗ → Xε (QT )∗ q,p q ,p denotes the adjoint operator of QT : Xε (QT ) → Xε T T (Q3T ).

q,p

3.7 Formula of Integration-by-Parts for Wε

93

(QT )

(ii) It holds h x → x R QT

.

q,p

in Wε

(QT )

(h → 0) ,

i.e., C ∞ (I , C0∞ ()d ) lies densely in Wε (QT ). (iii) For extensions q T , p T ∈ Plog (Rd+1 ) of qT , pT ∈ Plog (Q3T ), with q − ≤ q T ≤ q + and p− ≤ pT ≤ p+ in Rd+1 , there exists a constant cq T ,pT > 0 (depending on q T , p T ∈ Plog (Rd+1 )) such that q,p

sup

.

h∈(0,h1 )∩I∧

 h  R  x  q,p QT W (Q ε

T)

≤ cq T ,pT xWεq,p (QT ) .

h x)h>0 ⊆ C ∞ (I , C ∞ ()d ), because Proof ad (i). First, we note that (R QT 0 Proposition 3.14 (i) states that (RhQ3T ( ET x))h>0 ⊆ C0∞ (Rd+1 )d with supp(RhQ3T ( ET x)) ⊆ [−T − h, 2T + h] × h for all h > 0. In particular, h x)h>0 ⊆ Wεq,p (QT ) with resorting to this regularity, it is readily seen that (R QT .

d h h x, 0) R x = Jε (∂t R QT dt QT

q,p

in Xε

(QT )∗

(3.70)

for every h > 0. Using (3.70) together with the formula of integration-by-parts in time for smooth vector fields, we observe for every φ ∈ C0∞ (QT )d and h > 0 that  .

   d h h x, φ q(·,·) RQT x, φ = ∂t R (3.71) QT L (QT )d q,p dt Xε (QT ) h x)Lq(·,·) (Q )d = −(∂t φ, R QT T   h  = − QT ∂t φ, RQ3T (ET x) LqT (·,·) (Q )d 3T  h   = − (RQ3T ) (QT ∂t φ),  ET x LqT (·,·) (Q )d 3T   h   = − ∂t [(RQ3T ) (QT φ)], ET x LqT (·,·) (Q )d 3T    h  ( ET x)(t), ∂t [(RQ3T ) (QT φ)](t) Y dt . =− 3I

Since ((RhQ3T ) (QT φ))h∈I∧ ⊆ C0∞ (Q3T )d, as ((RhQ3T ) (QT φ))h∈I∧ ⊆ C0∞ (Rd+1 )d with supp((RhQ3T ) (QT φ)) ⊆ (supp(φ) + Bhd+1 (0)) ∩ (R × 2h )

.

⊆ [−h, T + h] × 2h ⊆ Q3T

94

3 Variable Bochner–Lebesgue Spaces

for all h ∈ I∧ , due to Proposition 3.15 (i),21 we further deduce from (3.71) by q ,p means of Proposition 3.24 (ii), i.e.,  ET x ∈ Wε T T (Q3T ), Proposition 3.19, and ∞ d Definition 3.6 that for every φ ∈ C0 (QT ) and h ∈ I∧  .

    d h dET x h  R x, φ , (RQ3T ) (QT φ) = q ,p q,p dt QT dt Xε (QT ) Xε T T (Q3T )      dET x ,φ . = ∗QT RhQ3T q,p dt Xε (QT )

(3.72)

Due to the density of C0∞ (QT )d in Xε (QT ) (cf. Proposition 3.18 (iv)), (3.72) then extends to (i). q ,p ad (ii). Proposition 3.16 (iii) yields RhQ3T ( ET x) →  ET x in Xε T T (Q3T ) (h → 0), from which we, in turn, conclude that q,p

h x → x R QT

.

q,p

in Xε (QT )

(h → 0) .

(3.73)

Using (i), Proposition 3.19 (iii), and the continuity of ∗QT : Xε T q,p Xε (QT )∗ , we get

q ,pT

.

dx d ET x d h RQT x → ∗QT = dt dt dt

in Xε (QT )∗ q,p

(QT )∗ →

(h → 0) ,

(3.74)

 q,p ∗ where the identity ∗QT dEdtT x = dx dt in Xε (QT ) follows easily if we test with ∞ d arbitrary φ ∈ C0 (QT ) and then refer to Definition 3.6. Eventually, (3.73) together with (3.74) just corresponds to (ii). ad (iii). Resorting to Propositions 3.14 (iii) and 3.16 (ii), for every h ∈ (0, h1 ), we find that h  h x q,p q ,p R QT Xε (QT ) ≤ RQ3T (ET x)X T T (Q3T ) ε

.

(3.75)

≤ cq T ,pT  ET xX qT ,pT (Q3T ) . ε

Using (i), Proposition 3.19 (ii), and that ∗QT : Xε T T (Q3T )∗ → Xε (QT )∗ is Lipschitz continuous with constant 1, for every h ∈ (0, h1 ) ∩ I∧ , we observe that       d h   ∗ dET x    R  Rh  x = QT Q3T  dt QT  q,p  dt Xεq,p (QT )∗ Xε (QT )∗      h dET x    ≤ RQ3T . (3.76) dt XεqT ,pT (Q3T )∗    d ET x    . ≤ cq T ,pT  dt XεqT ,pT (Q3T )∗ q ,p

⊆ (supp(f ) + supp(g))∩supp(h) for f ∈ L1 (Rn ), q, p ∈ [1, ∞], (cf. [31, Prop. 4.18]) and that supp(ηh ) ⊆

21 Here, we exploit the inclusion supp((f ∗g)h)

n ∈ N, g ∈ and h ∈ 2h for every h > 0 (cf. Sect. 3.4). Lp (Rn ),

Lq (Rn ),

q,p

q,p

3.7 Formula of Integration-by-Parts for Wε

95

(QT )

q,p Combining (3.75), (3.76) and the Lipschitz continuity of  ET : Wε (QT ) qT ,pT → Wε (Q3T ) (cf. Proposition 3.24 (ii)), we conclude that (iii) applies.  

Proposition 3.25 is now the key element in the proof of the desired formula of q,p integration-by-parts for the space Wε (QT ). q,p

Proposition 3.26 (Formula of Integration-By-Parts for Wε (QT )) The following statements apply: q,p

(i) Each function x ∈ Wε (QT ) possesses a unique continuous representation q,p x c ∈ Y 0 (QT ) and the resulting mapping (·)c : Wε (QT ) → Y 0 (QT ) is an embedding. q,p (ii) For every x, y ∈ Wε (QT ) and t, t ∈ I with t ≤ t, it holds    t s=t  dx (s), y(s) ds = (x c (s), y c (s))Y s=t

dt ˚q,p (s) t

X . (3.77)   t dy (s), x(s) − ds . dt ˚q,p (s) t

X q,p

Proof ad (i). Let x ∈ Wε (QT ) be arbitrary. We choose the family of smooth h x)h∈(0,h1 )∩I∧ ⊆ C ∞ (I , C ∞ ()d ) (cf. approximations (x h )h∈(0,h1 )∩I∧ := (R QT 0 Proposition 3.25), i.e., in particular, it holds q,p

xh → x

in Wε

.

(QT )

(h → 0) .

(3.78)

Then, resorting to the formula of integration-by-parts in time for smooth vector fields, for every t, t ∈ I and h , h ∈ (0, h1 ) ∩ I∧ , we find that x h (t) − x h (t)2Y = x h (t ) − x h (t )2Y  t    ∂t x h (s, y) − ∂t x h (s, y) +2

.

t

(3.79)



 · (x h (s, y) − x h (s, y)) dy ds = x h (t ) − x h (t )2Y   t dx h dx h

(s) − (s), x h (s) − x h (s) +2 ds dt ˚q,p (s) t dt X ≤ x h (t ) − x h (t )2Y    dx h dx h   − x h − x h Xεq,p (QT ) , + 2  dt dt Xεq,p (QT )∗ where we used √ in the second equality (3.70), Proposition 3.4, Corollary 3.2, and Remark 3.6. As a 2 + b2 ≤ a + b for every a, b ≥ 0, we conclude from (3.79) for

96

3 Variable Bochner–Lebesgue Spaces

every t, t ∈ I and h , h ∈ (0, h1 ) ∩ I∧ x h (t) − x h (t)Y ≤ x h (t ) − x h (t )Y (3.80) 1  1 √  dx h 2 dx h  2  . + 2 x h − x h X q,p  dt − dt  q,p ε (QT ) ∗ Xε (QT )

.

If we integrate (3.80) with respect to t ∈ I and then divide by T > 0, we further obtain for every t ∈ I and h , h ∈ (0, h1 ) ∩ I∧ that x h (t) − x h (t)Y ≤ T −1 x h − x h L1 (I,Y )

(3.81)

.

1  1 √  dx h 2 dx h  2  

 − . x − x + 2 q,p h h Xε (QT ) dt dt Xεq,p (QT )∗ q,p

q,p

As t ∈ I was arbitrary in (3.81) and Xε (QT ) → X− (QT ) → L1 (I, Y ), based q,p ˚− on X

→ Y and Proposition 3.2, also making use of Hölder’s inequality for Bochner–Lebesgue functions and Corollary 2.1, for every h , h ∈ (0, h1 ) ∩ I∧ , we observe that 1

x h − x h Y 0 (QT ) ≤ T − 2 2(1 + |QT |)x h − x h Xεq,p (QT )

.

(3.82)

1  1 √  dx h 2 dx h  2   − . + 2 x h − x h X q,p  ε (QT ) dt dt Xεq,p (QT )∗ As a consequence, the sequence (x h )h∈(0,h1 )∩I∧ ⊆ Y 0 (QT ) is a Cauchy sequence, i.e., owing to the completeness of Y 0 (QT ) (cf. Proposition 2.15), there exists a function x c ∈ Y 0 (QT ) such that xh → xc

.

in Y 0 (QT )

(h → 0) .

(3.83)

Since Y 0 (QT ) → L1 (I, Y ), (3.83) also gives us that x h → x c in L1 (I, Y ) q,p (h → 0). As simultaneously x h → x in Xε (QT ) → L1 (I, Y ) (h → 0), we q,p obtain x c = x almost everywhere in QT . Therefore, each x ∈ Wε (QT ) 0 possesses a unique continuous representation x c ∈ Y (QT ), i.e., the mapping q,p (·)c : Wε (QT ) → Y 0 (QT ) is well-defined. If we next repeat the steps (3.79)– (3.82) and substitute x h − x h by x h in doing so, then we furthermore conclude for every h ∈ (0, h1 ) ∩ I∧ that 1

x h Y 0 (QT ) ≤ T − 2 2(1 + |QT |)x h Xεq,p (QT ) .

+

1  1 √  dx h  2 2  . x h X 2 q,p  dt  q,p ε (QT ) ∗ Xε (QT )

(3.84)

q,p

3.7 Formula of Integration-by-Parts for Wε

97

(QT )

Thus, by passing for h → 0 in (3.84), taking into account (3.78) and (3.83) in doing so, we find that 1

x c Y 0 (QT ) ≤ T − 2 2(1 + |QT |)xXεq,p (QT )  1 √  dx  2  + 2  dt 

.

(3.85)

1

q,p

Xε

(QT )∗

2 , xX q,p (Q ) T

ε

q,p

i.e., the resulting mapping (·)c : Wε (QT ) → Y 0 (QT ) is an embedding. q,p ad (ii). Let x, y ∈ Wε (QT ) be arbitrary. We choose families of smooth h x)h∈(0,h1 )∩I∧ ⊆ C ∞ (I , C ∞ ()d ) and approximations (x h )h∈(0,h1 )∩I∧ := (R QT 0 h y)h∈(0,h1 )∩I∧ ⊆ C ∞ (I , C ∞ ()d ) (cf. Proposition 3.25), (y h )h∈(0,h1 )∩I∧ := (R QT 0 i.e., in particular, it holds xh → x .

yh → y

q,p

in Wε in

(QT )

(h → 0) ,

q,p Wε (QT )

(h → 0) . q,p

With recourse to (i), i.e., to the continuity of (·)c : Wε apart from that, obtain

.

(3.86)

(QT ) → Y 0 (QT ), we,

x h = (x h )c → x c

in Y 0 (QT )

(h → 0) ,

y h = (y h )c → y c

in Y 0 (QT )

(h → 0) .

(3.87)

Using the formula of integration-by-parts for smooth vector fields, we obtain for every h ∈ (0, h1 ) ∩ I∧ .

 s=t (x h (s), y h (s))Y s=t

  t  = ∂t x h (s, y) · y h (s, y) + ∂t y h (s, y) · x h (s, y) dy ds t

=

 t t

(3.88)



   t dy h dx h (s), y h (s) (s), x h (s) ds + ds , dt dt ˚q,p (s) ˚q,p (s) t

X X

where we used for the second equality (3.70), Proposition 3.4, Corollary 3.2, and Remark 3.6. Eventually, by passing for h → 0 in (3.88), using (3.86) and (3.87) in doing so, we conclude (3.77).   Apart from that, through the combination of Propositions 3.26 and 3.11, we obtain the following parabolic compactness result for variable Bochner–Sobolev spaces, which also can be considered as a variable exponent generalization of the well-known Aubin–Lions lemma (cf. [12, 116]), although it is fundamentally based on Landes’ and Mustonen’s compactness principle (cf. Proposition 3.10).

98

3 Variable Bochner–Lebesgue Spaces

Corollary 3.5 For every ε ∈ (0, (p− )∗ − 1], it holds q,p

Wε (QT ) → → Lmax{2,p∗ (·,·)}−ε (QT )d .

.

q,p

q,p

Proof Let (x n )n∈N ⊆ Wε (QT ) be a sequence such that x n  x in Wε (QT ) (n → ∞). Then, Proposition 3.26 (i) yields that (x n )c  x c in Y 0 (QT ) (n → ∞), which due to the embedding Y 0 (QT ) → Y ∞ (QT ) and the characterization of ∗ weak convergence in Y 0 (QT ) (cf. Proposition 2.16), in turn, implies that x n  x in Y ∞ (QT ) and (x n )c (t)  x c (t) in Y (n → ∞) for every t ∈ I , i.e., also x n (t)  x(t) in Y (n → ∞) for almost every t ∈ I . Therefore, Proposition 3.11 proves   x n → x in Lmax{2,p∗ (·,·)}−ε (QT )d (n → ∞) for every ε ∈ (0, (p− )∗ − 1].

3.8 Abstract Existence Result for Lipschitz Domains This section addresses the extension of the unsteady main theorem on pseudomonotone operators, i.e., Theorem 1.2 from the introduction, which yields the weak solvability of conventional evolution equations, to an abstract existence result for generalized evolution equations (cf. Definition 3.9). Here, we will follow the usual approach of reinterpreting the time derivative, i.e., in our setting the generalized time d derivative dt (cf. Definition 3.6), as a maximal monotone mapping on the space of variable Bochner–Sobolev functions having zero initial trace and applying an adjusted version of the main theorem on pseudo-monotone perturbations of maximal monotone mappings. More precisely, we intend to make use of the following abstract existence result. Proposition 3.27 Let X and Y be reflexive Banach spaces, let L : D(L) ⊆ X → X∗ linear and densely defined, and let γ0 : D(L) → Y linear such that the following properties are satisfied: (i) R(γ0 ) is dense in Y and G((L, γ0 ) ) = {(x, Lx, γ0 (x)) | x ∈ D(L)} is closed in X × X∗ × Y . (ii) There exists a constant c0 > 0 such that Lx, xX + c0 γ0 (x)2Y ≥ 0 for every x ∈ D(L). (iii) L : N (γ0 ) ⊆ X → X∗ , where N(γ0 ) := {x ∈ D(L) | γ0 (x) = 0}, is maximal monotone. Furthermore, let A : D(L) ⊆ X → X∗ be coercive and pseudo-monotone with respect to L, i.e., for a sequence (xn )n∈N ⊆ D(L) and an element x ∈ D(L) from xn  x .

in X

(n → ∞) ,

sup Lxn X∗ < ∞ ,

n∈N

lim sup Axn , xn − xX ≤ 0 , n→∞

(3.89)

3.8 Abstract Existence Result for Lipschitz Domains

99

for every y ∈ X, it follows that Ax, x − yX ≤ lim inf Axn , xn − yX ,

.

n→∞

and assume that there exist a bounded function ψ : R≥0 × R≥0 → R≥0 and a constant θ ∈ [0, 1) such that for every x ∈ D(L), it holds AxX∗ ≤ ψ(xX , γ0 (x)Y ) + θ LxX∗ .

(3.90)

.

Then, R((L + A, γ0 ) ) = X∗ × Y , i.e., for arbitrary x ∗ ∈ X∗ and y0 ∈ Y , there exists x ∈ D(L) such that Lx + Ax = x ∗ in X∗ and γ0 (x) = y0 in Y .  

Proof See [30, Corollary 22].

To the best of the author’s knowledge, this approach finds its origin in [116], at least for the homogeneous case, i.e., when Y = {0}, was extended by Brézis in [30] to the non-homogeneous case, i.e., when Y = {0}, and was first applied in the framework of variable Bochner–Lebesgue spaces in the rather simplified case of monotone operators by Alkhutov and Zhikov in [6]. As already revealed, the basic idea now consists in the reinterpretation of the generalized time derivative q,p d ∗ ˚ q,p dt : Wε (QT ) → X ε (QT ) as the mapping L. In doing so, the initial q,p trace operator γ 0 : Wε (QT ) → Y ,22 defined by γ 0 (x) := x c (0) in Y for all q,p d x ∈ Wε (QT ), takes the role of γ0 . Before checking whether dt and γ 0 fall within the framework of Proposition 3.27, we first introduce a special notion of d pseudo-monotonicity that is directly coupled to the generalized time derivative dt and that is equivalent to the notion of pseudo-monotonicity in the sense of (3.89). d Definition 3.10 ( dt -Pseudo-Monotonicity) Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I ×, and q, p ∈ P∞ (QT ) with q − , p− > 1, q,p q,p ˚ q,p(QT ) → X ˚ q,p(QT )∗ ˚− such that X

→ Y . Then, an operator A : Wε (QT ) ⊆ X ε ε q,p d is said to be dt -pseudo-monotone, if for (x n )n∈N ⊆ Wε (QT ) from q,p

in Wε

xn  x

.

(n → ∞) , .

(QT )

lim sup Ax n , x n − xX˚ q,p (Q ε

n→∞

T)

(3.91)

≤ 0,

(3.92)

˚ q,p (QT ), it follows that for every y ∈ X ε Ax, x − yX˚ q,p (Q

.

ε

T)

≤ lim inf Ax n , x n − yX˚ q,p (Q ) . n→∞

ε

T

d d -Pseudo-Monotonicity ⇔ Pseudo-Monotonicity with Respect to dt ) Remark 3.15 ( dt q,p q,p ∗ ˚ (QT ) that is pseudoApparently, each operator A : Wε (QT ) → X ε d d monotone with respect to dt in terms of (3.89) is also dt -pseudo-monotone. q,p On the other hand, if (x n )n∈N ⊆ Wε (QT ) is a sequence satisfying 22 Note that γ

0:

q,p

Wε

(QT ) → Y is well-defined only under the assumptions of Proposition 3.26.

100

3 Variable Bochner–Lebesgue Spaces

dx n ˚ q,p (QT ) (n → ∞) and sup x n  x in X ˚ q,p (QT )∗ < ∞, ε n∈N  dt X ε q,p then the reflexivity of Wε (QT ) (cf. Proposition 3.21) yields a subsequence q,p q,p (x n )n∈ ⊆ Wε (QT ), with  ⊆ N, and an element x˜ ∈ Wε (QT ) such q,p q,p ˚ q,p (QT ), that x n  x˜ in Wε (QT ) (  n → ∞). Due to Wε (QT ) → X ε q,p q,p we thus find that x = x˜ ∈ Wε (QT ) and x n  x in Wε (QT ) (  n → ∞). Because this argumentation remains valid for each subsequence q,p q,p of (x n )n∈N ⊆ Wε (QT ), x ∈ Wε (QT ) is the only weak accumulation q,p point of (x n )n∈N ⊆ Wε (QT ). Therefore, the standard convergence q,p principle (cf. [166, Proposition 10.13 (4)]) yields x n  x in Wε (QT ) q,p q,p d ˚ (QT )∗ (n → ∞). In other words, each dt -pseudo-monotone A : Wε (QT ) → X ε d is also pseudo-monotone with respect to dt . d Remark 3.16 ( dt -Pseudo-Monotonicity ⇒ Pseudo-Monotonicity) Apparently, each ˚ q,p (QT )∗ is also d -pseudo˚ q,p (QT ) → X pseudo-monotone operator A : X ε ε dt monotone. Nevertheless, the converse is not true in general. To see this, it suffices to consider the case of a bounded Lipschitz domain  ⊆ Rd , d ≥ 2, and q = p = 2 in QT . Further, let I := (0, 2π ), s > 0 so large ˚2,2 , and A : X 2,2 (QT ) → X 2,2 (QT )∗ that ρ2 (ε(x)) < sρ2 (x) for some x ∈ X ε ε 2,2 defined by Ax := Jε (−sx, ε(x)) in Xε (QT )∗ for every x ∈ Xε2,2 (QT ). Then, A : Xε2,2 (QT ) → Xε2,2 (QT )∗ is not pseudo-monotone. In fact, (x n )n∈N ⊆ ˚2,2 ), defined by x n (t) := x sin(nt) in X ˚2,2 for all t ∈ I and C ∞ (I , X ∗ ∞ 2,2 23 ˚ ) (n → ∞), n ∈ N, satisfies x n  0 in L (I, X i.e., also x n  0 in  2 Xε2,2 (QT ) (n → ∞), and owing to limn→∞ I sin(nt)  dt = π , also lim supn→∞ Ax n , x n − 0Xε2,2 (QT ) = π ρ2 (ε(x)) − sρ2 (x) < 0, but likewise   lim infn→∞ Ax n , x n − yXε2,2 (QT ) = π ρ2 (ε(x)) − sρ2 (x) < 0 = A0, 0 − yXε2,2 (QT ) for every y ∈ Xε2,2 (QT ). On the other hand, A : Xε2,2 (QT ) → d -pseudo-monotone. Indeed, if (x n )n∈N ⊆ Wε2,2 (QT ) is a sequence Xε2,2 (QT )∗ is dt satisfying

(n → ∞) , .

(3.93)

lim sup Ax n , x n − xXε2,2 (QT ) ≤ 0 ,

(3.94)

xn  x

.

in Wε2,2 (QT )

n→∞

then we infer from (3.93), by resorting to Corollary 3.5, that x n → x in L2 (QT )d (n → ∞). Using this and Ax n  Ax in Xε2,2 (QT )∗ (n → ∞), which follows from (3.93) and the weak continuity of A : Xε2,2 (QT ) → Xε2,2 (QT )∗ , in (3.94), we further obtain lim supn→∞ x n 2X 2,2 (Q ) ≤ x2X 2,2 (Q ) , i.e., in total, we have that ε

T

ε

T

x n → x in Xε2,2 (QT ) (n → ∞). Therefore, it follows for every y ∈ Xε2,2 (QT ) that Ax, x − yXε2,2 (QT ) ≤ lim infn→∞ Ax n , x n − yXε2,2 (QT ) . 23 (f ) ∞ n n∈N ⊆ C (I ), I ∗ ∞ fn  0 in L (I ) (n →

:= (0, 2π ), defined by fn (t) := sin(nt) for every t ∈ I , n ∈ N, satisfies ∞).

3.8 Abstract Existence Result for Lipschitz Domains

101

Proposition 3.28 Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, q,p ˚− QT := I × , and q, p ∈ P∞ (QT ) with q − , p− > 1, such that X

→ Y . q,p q,p d ∗ ˚ If the operators A, B : Wε (QT ) → X ε (QT ) are dt -pseudo-monotone, then q,p ˚ q,p (QT )∗ is d -pseudo-monotone. the sum A + B : Wε (QT ) → X ε dt q,p

Proof Let (x n )n∈N ⊆ Wε q,p x ∈ Wε (QT ) and .

(QT ) be a sequence satisfying (3.91) with respect to

lim sup (A + B)x n , x n − xX˚ q,p (Q ε

n→∞

T)

≤ 0.

Set an := Ax n , x n − xX˚ q,p (Q ) and bn := Bx n , x n − xX˚ q,p (Q ) for every n ∈ T T ε ε N. Then, lim supn→∞ an ≤ 0 and lim supn→∞ bn ≤ 0. In fact, suppose the contrary, e.g., that lim supn→∞ an =: a > 0. Then, there exists a subsequence with ank → a (k → ∞), and thus, lim supk→∞ bnk ≤ lim supk→∞ (ank + bnk ) − limk→∞ ank d -pseudo-monotonicity ≤ −a < 0, i.e., a contradiction, because then, in turn, the dt q,p q,p ∗ ˚ of B : Wε (QT ) → X ε (QT ) implies that 0 ≤ lim infk→∞ bnk < 0. Thus, we d -pseudo-monotonicity get lim supn→∞ an ≤ 0 and lim supn→∞ bn ≤ 0, and the dt q,p q,p q,p ∗ ˚ ˚ of A, B : Wε (QT ) → X ε (QT ) yields for every y ∈ X ε (QT ) Ax, x − yX˚ q,p (Q

≤ lim inf Ax n , x n − yX˚ q,p (Q ) ,

Bx, x − yX˚ q,p (Q

≤ lim inf Bx n , x n − yX˚ q,p (Q ) .

.

ε

ε

T) T)

n→∞ n→∞

Summing these two inequalities, for every y (A + B)x, x − yX˚ q,p (Q

.

ε

T)

ε

ε

˚ q,p (QT ), ∈X ε

T

T

we conclude that

≤ lim inf (A + B)x n , x n − yX˚ ,p (Q ) . n→∞

ε

T

 

The following two propositions demonstrate that, under the assumptions of d Proposition 3.26, the generalized time derivative dt and initial trace operator γ 0 perfectly fall within the framework of Proposition 3.27. Proposition 3.29 Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain, I := (0, T ), q,p ˚−

→ Y . Then, the T < ∞, q, p ∈ Plog (QT ) with q − , p− > 1, such that X q,p initial trace operator γ 0 : Wε (QT ) → Y , defined by γ 0 (x) := x c (0) in Y for q,p every x ∈ Wε (QT ), is well-defined, linear and continuous. Furthermore, the following statements apply: (i) R(γ 0 ) is dense in Y .  q,p q,p d  , γ 0 ) ) = (x, dx (ii) G(( dt dt , γ 0 (x)) | x ∈ Wε (QT ) is closed in Xε (QT )× q,p ∗ Xdxε (Q  T ) × Y. 1 q,p (iii) dt , x X q,p (Q ) + 2 γ 0 (x)2Y ≥ 0 for every x ∈ Wε (QT ). ε

T

Proof Well-definedness, linearity, and continuity are direct consequences of Proposition 3.26 (i) together with properties of strong convergence in Y 0 (QT ). So, let us verify the remaining assertions (i)–(iii):

102

3 Variable Bochner–Lebesgue Spaces q,p

q,p

˚+ , the function x ∈ Wε (QT ), defined by x(t) := x ad (i). For arbitrary x ∈ X q,p q,p ˚ ˚+ in X+ for every t ∈ I , satisfies γ 0 (x) = x c (0) = x in Y . Hence, X ⊆ R(γ 0 ), q,p ˚+ is dense in Y . i.e., R(γ 0 ) is dense in Y , because X d   n ad (ii). Let ((x n , dx dt , γ 0 (x n )) )n∈N ⊆ G(( dt , γ 0 ) ) be a sequence such that

  dx n , γ 0 (x n ) . xn, → (x, x ∗ , x0 ) dt q,p

in Xε

q,p

(QT )Xε

(QT )∗ × Y

(n → ∞) .

(3.95)

∗  ∗ n  Apparently, (3.95) implies that (x n , dx dt ) → (x, x ) in Xε (QT )×Xε (QT ) d (n → ∞), which due to the closedness of G( dt ) (cf. Proposition 3.21) and q,p q,p q,p d ∗ ∗ D( dt ) = Wε (QT ) yields that x ∈ Wε (QT ) with dx dt = x in Xε (QT ) q,p q,p and x n → x in Wε (QT ) (n → ∞). Thus, since γ 0 : Wε (QT ) → Y is continuous, we conclude that γ 0 (x n ) → γ 0 (x) in Y (n → ∞), i.e., γ 0 (x) = x0 in Y , since at the same time γ 0 (x n ) → x0 in Y (n → ∞). Altogether, d d    (x, x ∗ , x0 ) = (x, dx dt , γ 0 (x)) ∈ G(( dt , γ 0 ) ), i.e., G(( dt , γ 0 ) ) is closed in q,p q,p ∗ Xε (QT ) × Xε (QT ) × Y .   ad (iii). A straightforward application of (3.77). q,p

q,p

Proposition 3.30 Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain, I := (0, T ), q,p ˚− T < ∞, q, p ∈ Plog (QT ) with q − , p− > 1, such that X

→ Y . Then, q,p q,p q,p d the generalized time derivative dt : Wε (QT ) ⊆ Xε (QT ) → Xε (QT )∗  q,p is linear, densely  defined and its restriction to N (γ 0 ) := x ∈ Wε (QT ) | x c (0) = 0 in Y is maximal monotone. Proof q,p

q,p

d 1. Linearity: Follows from the definition of dt : Wε (QT ) ⊆ Xε (QT ) → q,p ∗ Xε (QT ) (cf. Definition 3.6). q,p d ) 2. Density of domain of definition: Since C0∞ (QT )d ⊆ Wε (QT ) = D( dt q,p q,p d is dense in Xε (QT ) (cf. Proposition 3.18 (iv)), dt : Wε (QT ) ⊆ q,p q,p Xε (QT ) → Xε (QT )∗ is densely defined. 3. Maximal monotonicity: Making use of (3.77), for every x ∈ N(γ 0 ), we observe that   dx 1 1 ,x . = x c (T )2Y − x c (0)2Y q,p dt 2 2 Xε (QT )

=

1 x c (T )2Y ≥ 0 , 2

3.8 Abstract Existence Result for Lipschitz Domains

103

d i.e., dt : N(γ 0 ) ⊆ Xε (QT ) → Xε (QT )∗ is positive semi-definite and, therefore, monotone, as it is likewise linear. For the maximal monotonicity, we q,p q,p assume that (x, x ∗ ) ∈ Xε (QT ) × Xε (QT )∗ satisfies for every y ∈ N(γ 0 )   dy ∗ . x − ≥ 0. (3.96) ,x − y q,p dt Xε (QT ) q,p

q,p

q,p ˚+ If we choose y = xϕ ∈ N(γ 0 ) for arbitrary x ∈ X and ϕ ∈ C0∞ (I ) in (3.96), also observing that   dy 1 ,y = x2Y (ϕ(T ) − ϕ(0)) = 0 , . q,p dt 2 Xε (QT )

due to (3.77), we find that 

∗

x , x − y

.

q,p Xε (QT )

 dy ≥ . ,x q,p dt Xε (QT )

(3.97)

q,p q,p ˚+ ˚+ Choosing next x = ±τ y ∈ X for arbitrary y ∈ X and τ > 0 in (3.97), also q,p dy ∗ exploiting that dt = Jε (x∂t ϕ, 0) in Xε (QT ) and Remark 3.6, dividing by τ q,p ˚+ and passing for τ → ∞ in (3.97), provides for every y ∈ X and ϕ ∈ C0∞ (I ) that   .− (x(s), y)Y ∂t ϕ(s) ds = x ∗ (s), yX (3.98) ˚q,p (s) ϕ(s) ds . I

I

q,p

Therefore, Proposition 3.23 applied to (3.98) yields x ∈ Wε (QT ) with dx dt = q,p x ∗ in Xε (QT )∗ . However, we are still not aware of whether x ∈ N (γ 0 ), i.e., whether we, apart from that, have that γ 0 (x) = x c (0) = 0 in Y . To this end, we q,p ˚+ consider a sequence (xnT )n∈N ⊆ X that satisfies xnT → x c (T )

.

in Y

(n → ∞) ,

(3.99)

and define the sequence (y n )n∈N := (xnT ϕ)n∈N ⊆ N (γ 0 ) for ϕ ∈ C ∞ (I ) q,p ∗ ∗ with ϕ(0) = 0 and ϕ(T ) = 1. Then, using (3.77) and dx dt = x in Xε (QT ) in (3.96), for every n ∈ N, we observe that   dy dx − n , x − yn 0≤ q,p dt dt Xε (QT ) .

1 1 x c (T ) − (y n )c (T )2Y − x c (0) − (y n )c (0)2Y 2 2 1 1 = x c (T ) − xnT 2Y − x c (0)2Y . 2 2

=

(3.100)

By passing for n → ∞ in (3.100), using (3.99) in doing so, we conclude that 0 ≤ − 12 x c (0)2Y , i.e., we have that x c (0) = 0 in Y and, thus, x ∈ N (γ 0 ). In other q,p q,p d : N (γ 0 ) ⊆ Xε (QT ) → Xε (QT )∗ is maximal monotone.   words, dt

104

3 Variable Bochner–Lebesgue Spaces

Having transferred the abstract framework of Proposition 3.27 to generalized evolution equations, we eventually arrive at the following abstract existence result. Theorem 3.3 Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain, I := (0, T ), q,p ˚− T < ∞, q, p ∈ Plog (QT ) with q − , p− > 1, such that X

→ Y , and let q,p q,p q,p d ∗ A : Wε (QT ) ⊆ Xε (QT ) → Xε (QT ) be coercive, dt -pseudo-monotone and assume there exist a bounded function ψ : R≥0 × R≥0 → R≥0 and a constant q,p θ ∈ [0, 1) such that for every x ∈ Wε (QT ), it holds AxXεq,p (QT )∗

.

   dx   . ≤ ψ(xXεq,p (QT ) , x c (0)Y ) + θ   dt  q,p Xε (QT )∗

Then, for arbitrary x0 ∈ Y and x ∗ ∈ Xε such that

q,p

.

(3.101)

(QT )∗ , there exists x ∈ Wε

q,p

dx + Ax = x ∗ dt

in Xε

x c (0) = x0

in Y .

q,p

(QT )

(QT )∗ ,

Here, the initial condition has to be understood in the sense of the unique continuous representation x c ∈ Y 0 (QT ) (cf. Proposition 3.26 (i)). Proof With recourse to Remark 3.15, Propositions 3.29 and 3.30, we can directly   refer to Proposition 3.27.

3.9 Application to Model Problem Recall that we first considered for a bounded Lipschitz domain  ⊆ Rd , d ≥ 2, a finite time interval I := (0, T ), T < ∞, a time-space cylinder QT := I × , and T := I × ∂, as a model problem, the initial-boundary value problem ∂t x − div(S(·, ·, ε(x))) + b(·, ·, x) = f − div(F ) .

x=0 x(0) = x0

in QT , on T ,

(3.102)

in  .

Here, f : QT → Rd is a vector field, F : QT → Md×d sym a symmetric tensor field, and d x0 :  → R an initial condition. Moreover, the tensor-valued mapping S : QT × d×d Md×d sym → Msym is supposed to have a (p(·, ·), δ)-structure, i.e., there exist p ∈ log P (QT ) with p− > 1 and δ ≥ 0 such that (S.1)–(S.4) are satisfied, and the vectorvalued mapping b : QT × Rd → Rd is supposed to satisfy (B.1)–(B.3).

3.9 Application to Model Problem

105

We demonstrate that the mappings that occur in the model problem (3.102), i.e., S and b, induce operators that perfectly fall within the scope of the preceding section. Let us start with the monotone part, i.e., the tensor-valued mapping d×d S : QT × Md×d sym → Msym . For later applications, we will consider the more delicate situation of a possibly irregular, i.e., non-Lipschitz, domain. Proposition 3.31 Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p ∈ Plog (QT ) with p− > 1. Furthermore, let S : QT × Md×d sym → d×d Msym be a mapping satisfying (S.1)–(S.4) with respect to p. Then, the operator p−,p

˚ S: X ε

p−,p

˚ (QT ) → X ε

p−,p

˚ (QT )∗ , for every x ∈ X ε

Sx := Jε (0, S(·, ·, ε(x)))

.

(QT ) defined by

p−,p

˚ in X ε

(QT )∗ ,

is well-defined, bounded, continuous, monotone, coercive, pseudo-monotone, and satisfies the boundedness condition (3.101). Proof −

˚ p ,p (QT ) be arbitrary. Since S : QT × Md×d → 1. Well-definedness: Let x ∈ X sym ε is a Carathéodory mapping (cf. ( S.1)), we can easily see that the function Md×d sym ((t, x) → S(t, x, ε(x)(t, x))) : QT → Md×d sym is Lebesgue-measurable. Therefore, we can examine it for integrability. In doing so, making use of (S.2) and repeatedly of the estimate (a + b)s ≤ max{1, 2s−1 }(a s + bs ) for every a, b ≥ 0 and s > 0, we observe that for almost every (t, x) ∈ QT

|S(t, x, ε(x)(t, x))|p (t,x)  p (t,x) ≤ α(δ + |ε(x)(t, x)|)p(t,x)−1 + β(t, x) 





. ≤ 2p (t,x) α p (t,x) (δ + |ε(x)(t, x)|)p(t,x) + β(t, x)p (t,x)    −  −

≤ 2(p ) α (p ) 2p(t,x) δ p(t,x) + |ε(x)(t, x)|p(t,x) + β(t, x)p (t,x)   −  −

+

≤ 2(p ) α (p ) 2p δ p(t,x) + |ε(x)(t, x)|p(t,x) + β(t, x)p (t,x) ,

(3.103)

−

˚ p ,p (QT )∗ i.e., S(·, ·, ε(x)) ∈ Lp (·,·) (QT , Md×d that Sx ∈ X sym− ). We conclude ε − ˚ p ,p (QT ) → X ˚ p ,p (QT )∗ is well-defined. (cf. Proposition 3.20 (ii)), i.e., S : X ε ε 2. Boundedness: Because yX˚ p−,p (Q ) ≤ 1 implies ρp(·,·) (ε(y)) ≤ 2 for every T ε − ˚ p ,p (QT ) by the norm-modular unit ball property (cf. Lemma 2.1 (i)), y ∈X

ε

106

3 Variable Bochner–Lebesgue Spaces

also applying the ε-Young inequality− (cf. Proposition 2.8) for ε = 1, we conclude ˚ p ,p (QT ), it holds from (3.103) that for every x ∈ X ε Sx ˚ p−,p

.

Xε

≤

sup

p−,p

˚ y∈X (QT ) ε − y X ˚ p ,p (Q ) ≤1 ε

(p− )

≤2

(3.104)

(QT )∗

Sx, y ˚ p−,p Xε

(QT )

≤ ρp (·,·) (S(·, ·, ε(x))) + ρp(·,·) (ε(y))

T

   (p− ) p+  2 ρp(·,·) (δ) + ρp(·,·) (ε(x)) + ρp (·,·) (β) + 2 , α

˚p i.e., S : X ε

−,p

˚p (QT ) → X ε

−,p

(QT )∗ is bounded.

p+

3. Boundedness condition (3.101): Since ρp(·,·) (ε(x)) ≤ ε(x)Lp(·,·) (Q )d×d + 1 − T ˚ p ,p (QT ) (cf. Lemma 2.1 (iii)), (3.104) implies the boundedness for every x ∈ X ε condition (3.101). − ˚ p ,p (QT ) be a sequence such that x n → x in 4. Continuity: Let (x n )n∈N ⊆ X ε − ˚ p ,p (QT ) (n → ∞), i.e., in particular, it holds X ε ε(x n ) → ε(x)

.

in Lp(·,·) (QT , Md×d sym )

(n → ∞) .

(3.105)

Fischer–Riesz’s theorem in Lp(·,·) (QT , Md×d sym ) −(cf. Proposition 2.2) applied ˚ p ,p (QT ), with a cofinal subset to (3.105), yields a subsequence (x n )n∈ ⊆ X ε  ⊆ N, such that ε(x n ) → ε(x)

.

(  n → ∞)

Then, since S : QT × Md×d → sym (cf. (S.1)), (3.106) further leads to .

S(·, ·, ε(x n )) → S(·, ·, ε(x))

a.e. in QT .

(3.106)

Md×d sym is a Carathéodory mapping

(  n → ∞)

a.e. in QT .

(3.107)

On the other hand, due to (3.103), for any measurable set K ⊆ QT and n ∈ , we have that  (p− ) p+  α 2 ρp(·,·) (δχK )   + ρp(·,·) (ε(x n )χK ) + ρp (·,·) (βχK ) ,

ρp (·,·) (S(·, ·, ε(x n ))χK ) ≤ 2(p

.

− )





p (·,·) (Q )-uniformly intei.e., (S(·, ·, ε(x n )))n∈ ⊆ Lp (·,·) (QT , Md×d T sym ) is L grable, based on (3.105). As a result, also using (3.107), Vitali’s convergence theorem (cf. Proposition 2.5) then implies that

S(·, ·, ε(x n )) → S(·, ·, ε(x))

.



in Lp (·,·) (QT , Md×d sym )

(  n → ∞) . (3.108)

3.9 Application to Model Problem

107

Using the standard convergence principle (cf. [75, Proposition 10.13 (1)]), we deduce that (3.108) holds even if  = −N. Hence, since the mapping −

˚ p ,p (QT )∗ is continuous (cf. Jε : L(p ) (QT )d × Lp (·,·) (QT , Md×d sym ) → X ε − ˚εp ,p (QT )∗ (n → ∞), Proposition −3.20 (ii)), we conclude that Sx n → Sx in X − ˚ p ,p (QT ) → X ˚ p ,p (QT )∗ is continuous. i.e., S : X ε ε 5. Monotonicity: Results directly from the condition (S.4). − ˚ p ,p (QT ) and almost 6. Coercivity: Based on (S.3), we obtain for every x ∈ X ε every (t, x) ∈ QT S(t, x, ε(x)(t, x)) : ε(x)(t, x) ≥ c0 (δ + |ε(x)(t, x)|)p(t,x)−2 |ε(x)(t, x)|2

.

− c1 (t, x) .

(3.109)

Next, using that (δ + a)p(t,x)−2 a 2 ≥ 12 a p(t,x) − δ p(t,x)24 for all a ≥ 0 and − ˚ p ,p (QT ) and almost every (t, x) ∈ QT in (3.109), we deduce for every x ∈ X ε (t, x) ∈ QT that S(t, x, ε(x)(t, x)) : ε(x)(t, x) ≥

.

c0 |ε(x)(t, x)|p(t,x) 2 − c0 δ p(t,x) − c1 (t, x) .

(3.110)

Hence, we finally conclude from (3.110) and Lemma 2.1 (iii) that for every − ˚ p ,p (QT ), it holds x ∈X ε Sx, xX˚ p−,p (Q ε

.

T)

c0 ρp(·,·) (ε(x)) − c0 ρp(·,·) (δ) − c1 L1 (QT ) 2  c0  p− ε(x)Lp(·,·) (Q )d×d − 1 ≥ T 2

≥

(3.111)

− c0 ρp(·,·) (δ) − c1 L1 (QT ) , −

−

˚ p ,p (QT ) → X ˚ p ,p (QT )∗ is coercive since due to the estii.e., S : X ε ε mate (3.16), it holds the norm equivalence ε(·)Lp(·,·) (QT )d×d ∼  · X˚ p−,p (Q ) T ε − ˚εp ,p (QT ). on X − − ˚ p ,p (QT ) → X ˚ p ,p (QT )∗ is monotone 6. Pseudo-monotonicity: Because S : X ε ε and continuous, it is likewise pseudo-monotone (cf. [149, Kap. III, Lem. 2.6 (i)]).  

we used for a, δ ≥ 0 that if p ≥ 2, then it holds (δ + a)p−2 a 2 ≥ a p , and if 1 < p < 2,  then (δ + a)p−2 a 2 ≥ (δ + a)p−2 12 (δ + a)2 − δ 2 ≥ 12 (δ + a)p − δ p ≥ 12 a p − δ p , due to (δ + a)2 ≤ 2(δ 2 + a 2 ) and (δ + a)p−2 ≤ δ p−2 .

24 Here,

108

3 Variable Bochner–Lebesgue Spaces

Remark 3.17 (i) Proposition 3.31 remains true if only p ∈ P∞ (QT ) with p− > 1, where d×d S : QT × Md×d sym → Msym is a mapping that satisfies (S.1)–(S.4) with respect to p and δ ≥ 0. 2d (ii) If we, in addition, assume that p− ≥ d+2 , i.e., precisely the condition d that guarantees the well-definedness of the notion of dt -pseudo-monotonicity − p ,p ˚ (cf. Definition 3.10), since then X−

→ Y due to the Sobolev embedding theorem (cf. Proposition 3.9), then Proposition 3.31 and −Remark 3.16 also yield − d ˚ p ,p (QT ) → X ˚ p ,p (QT )∗ . the dt -pseudo-monotonicity of S : X ε ε Next, let us turn towards the non-monotone part of lower order, i.e., the vectorvalued mapping b : QT × Rd → Rd . Here, we restrict our attention again to Lipschitz domains to have access to the formula of integration-by-parts for q,p Wε (QT ) (cf. Proposition 3.26 (ii)). Proposition 3.32 Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain, I := (0, T ), 2d T < ∞, QT := I × , and p ∈ Plog (QT ) with p− ≥ d+2 . Furthermore, d d let b : QT ×− R → R be a mapping satisfying (B.1)–(B.3). Then, the operp ,p p−,p p−,p ator− B : Xε (QT ) ∩ Y ∞ (QT ) ⊆ Xε (QT ) → Xε (QT )∗ , for every x ∈ p ,p Xε (QT ) ∩ Y ∞ (QT ) defined by Bx := Jε (b(·, ·, x), 0)

.

is well-defined, (3.101).

d dt -pseudo-monotone

p−,p

in Xε

(QT )∗ ,

and satisfies the boundedness condition

Proof p−,p

1. Well-definedness: Let x ∈ Xε (QT ) ∩ Y ∞ (QT ) be arbitrary. Because b : QT × Rd → Rd is a Carathéodory mapping (cf. (B.1)), the function ((t, x) → b(t, x, x(t, x))) : QT → Rd is Lebesgue- measurable. In addition, using (B.2) and (a + b)r ≤ 2r (a r + br ) for all a, b ≥ 0, we observe that b(·, ·, x)L(p− ) (QT )d ≤ γ (1 + |x|)r L(p− ) (QT ) + ηL(p− ) (QT )

.

= γ 1 + |x|r r(p− )

 ≤ γ 2r |QT | p−,p

L

1 (p− )

(QT )

(3.112)

+ ηL(p− ) (QT )

+ xr r(p− )

L

 (QT )d

+ ηL(p− ) (QT ) .

−

Due to Xε (QT ) ∩ Y ∞ (QT ) → Lr(p ) (QT )d (cf. Proposition 3.7), from (3.112) that b(·, ·, x) ∈ since r(p− ) < (p− )• ≤ (p− )∗ ,25 we conclude −

p−,p ∗ L(p ) (QT )d and, therefore, that Bx ∈ X (Q T ) (cf. Proposition 3.4), i.e., ε p−,p p−,p ∞ the operator B : Xε (QT ) ∩ Y (QT ) → Xε (QT )∗ is well-defined. 25 Recall

that (·)• ∈ W 1,∞ (1, ∞) is defined by s• := min{s + 2, s∗ } for every s ∈ (1, ∞).

3.9 Application to Model Problem

2.

109

p−,p d Let (x n )n∈N ⊆ Wε (QT ) be a sequence such that dt -pseudo-monotonicity: p−,p x n  x in Wε (QT ) (n → ∞) and lim supn→∞ Bx n , x n − xXεp−,p (QT ) ≤ 0. In consequence, by resorting to Corollary 3.5, we get for all ε ∈ (0, (p− )∗ − 1]

xn → x

.

(n → ∞) ,

in Lp∗ (·,·)−ε (QT )d −

(3.113)

−

i.e., (x n )n∈N ⊆ Lr(p ) (QT )d is Lr(p ) (QT )-uniformly integrable, since r(p− ) < (p− )∗ . In particular, (3.113) gives us, by virtue of Proposition 2.2 and (B.1), that b(·, ·, x n ) → b(·, ·, x) in Rd (  n → ∞) almost everywhere in QT for a cofinal subset  ⊆ N. Proceeding similar as for the estimate (3.112), for any measurable set K ⊆ QT and n ∈ N, we observe, that 1  b(·, ·, x n )L(p− ) (K)d ≤ γ 2r |K| (p− ) + x n r r(p− )

.

L

−

 (K)d

+ ηL(p− ) (K) ,

−

i.e., (b(·, ·, x n ))n∈N ⊆ L(p ) (QT )d is L(p ) (QT )-uniformly integrable. Hence, Vitali’s convergence theorem (cf. Proposition 2.5) and the standard convergence principle (cf. [75, Prop. 10.13 (1)]) yield that b(·, ·, x n ) → b(·, ·, x)

in L(p

.

− )

(n → ∞) .

(QT )d

−

(3.114) p−,p



By the continuity of Jε : L(p ) (QT )d × Lp (·,·) (QT , Md×d ) → Xε (QT )∗ sym p−,p (cf. Proposition 3.4), (3.114) implies Bx n → Bx in Xε (QT )∗ (n → ∞), from which we conclude that Bx, x − yXεp−,p (QT ) ≤ lim inf Bx n , x n − yXεp−,p (QT )

.

n→∞

p−,p

for every y ∈ Xε (QT ). 3. Boundedness condition (3.101): Appealing to Proposition 3.7, there exist constants c = c(p− , d, ) > 0 and θ = θ (p− , d, ) ∈ (0, 2] such that, also applying Korn’s inequality (cf. Proposition 2.13) for the constant exponent p−,p p− ∈ (1, ∞), for every x ∈ Wε (QT ), it holds26 (p− )• − L(p )• (QT )d

x

p− xθY ∞ (QT ) − Lp (QT )d×d

≤ cp− ε(x)

.

≤c

p−

26 For

x∈

p − > d, we use the estimate x

1,p− W0 ()d

∩ Y.

p−

(3.115)

xX p−,p (Q ) xY ∞ (QT ) . ε

(p− )• − L(p )• ()d

θ

T

p−

≤ xL∞ ()d x2Y ≤ x

p− x2Y 1,p − W0 ()d

for all

110

3 Variable Bochner–Lebesgue Spaces

θr . Inserting (3.115) in (3.112), as r(p− ) < (p− )• , and using (p− )• −

p−,p that Jε : L(p ) (QT )d ×Lp (·,·) (QT , Md×d (QT )∗ is Lipschitz continsym ) → Xε p−,p uous with constant 2 (cf. Proposition 3.4), we obtain for every x ∈ Wε (QT )

We define σ :=

.Bx

p−,p Xε (QT )∗

p− r ! 1 (p− ) ≤ γ 2r+1 |QT | (p− ) + cp− x p−•,p

Xε

(QT )

xσY ∞ (Q ) T

+ 2ηL(p− ) (Q ) . T

"

(3.116)

√ σ   1 Using (3.85) in (3.116) and setting cσ := max T − 2 2(1 + |QT |), 2 , − p ,p we observe for every x ∈ Wε (QT ), also using that (a + b)σ ≤ 2σ (a σ + bσ ) for all a, b ≥ 0, that p− r −

(p )• xX xσY ∞ (QT ) p−,p (Q )

(3.117)

.

ε

T

 σ  σ  dx  2 2  −,p cσ 2 xX p−,p (Q ) +  x ≤ x p  dt  p−,p Xε (QT ) T ε Xε (QT )∗  σ   p− r  dx  2 σ − )• +σ   (p − ≤ cσ 2 (1 + xXεp ,p (QT ) ) 1+  − dt Xεp ,p (QT )∗     p− r  dx  +σ σ0

 + εcσ 2σ  , ≤ cσ 2σ (1 + cσ0 (ε))(1 + xXεp−,p (QT ) ) (p− )•  dt  p−,p Xε (QT )∗ p− r (p− )• p−,p Xε (QT )



σ

σ

where we have applied in the last inequality the ε-Young inequality with respect

to the exponent σ0 := σ2 , with the constant cσ0 (ε) := (σ0 ε)1−σ0 (σ0 )−1 for all ε ∈ (0, σ0−1 ), which is allowed as σ0 > 1, because σ < 2 due to (p−r ) < (p1− ) < • 1 and θ ≤ 2. Eventually, inserting (3.117) in (3.116), we conclude for sufficiently small ε > 0 the boundedness condition (3.101).   Propositions 3.31 and 3.32 allow us to conclude that (3.102) is weakly solvable. Theorem 3.4 Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain, I := (0, T ), 2d T < ∞, QT := I × , and p ∈ Plog (QT ) with p− ≥ d+2 . Furthermore, let S : d×d d×d QT × Msym → Msym be a mapping satisfying (S.1)–(S.4) with respect to p and let f ∈ b : QT × Rd → Rd be a mapping satisfying (B.1)–(B.3). Then, for arbitrary −

p−,p L(p ) (QT )d , F ∈ Lp (·,·) (QT , Md×d (QT ) sym ) and x0 ∈ Y , there exists x ∈ Wε with a representation x c ∈ Y 0 (QT ) such that .

dx + Sx + Bx = Jε (f , F ) dt x c (0) = x0

p−,p

in Xε in Y ,

(QT )∗ ,

3.9 Application to Model Problem

111

i.e., for every φ ∈ C ∞ (QT ) with supp(φ) ⊆ [0, T ) × , it holds  .

−

x(t, x) · ∂t φ(t, x) dtdx QT



+

b(t, x, x(t, x)) · φ(t, x) + S(t, x, ε(x)(t, x)) : ε(φ)(t, x) dtdx QT



= (x0 , φ(0))Y +

f (t, x) · φ(t, x) + F (t, x) : ε(φ)(t, x) dtdx . QT

Proof Appealing to − Proposition 3.31, Remark− 3.17 (ii), Propositions 3.32, p ,p p ,p d -pseudo-monotone and 3.28, S + B : Xε (QT ) ∩ Y ∞ (QT ) → Xε (QT )∗ is dt p−,p d and satisfies (3.101) with respect to dt . In addition, since S : Xε (QT ) → − − − p ,p p ,p p ,p ) → Xε (QT )∗ satisfies Xε (QT )∗ is coercive and B : Xε (QT ) ∩ Y ∞ (Q −,pT p Bx, xXεp−,p (QT ) ≥ −c2 L1 (QT ) for every x ∈ Xε (QT )∩Y ∞ (QT ) (cf. (B.3)), p−,p p−,p p−,p S + B : Xε (QT ) ∩ Y ∞ (QT ) ⊆ Xε (QT ) → Xε (QT )∗ is coercive. Then, the assertion is a direct application of Theorem 3.3.  

Chapter 4

Solenoidal Variable Bochner–Lebesgue Spaces

The aim of this chapter is to adapt the results obtained in Chap. 3 to solenoidal variable Bochner–Lebesgue spaces, i.e., to a variant of variable Bochner–Lebesgue spaces in which an additional incompressibility constraint is incorporated into the time slice spaces. In this context, the well-known unsteady .p(·, ·)-Stokes equations and the unsteady .p(·, ·)-Navier–Stokes equations will serve as guiding examples. Recall that the unsteady .p(·, ·)-Stokes equations in a time-space cylinder .QT := I ×, where . ⊆ Rd , .d ≥ 2, is a bounded domain and .I := (0, T ), .T < ∞, a finite time interval, with a homogeneous Dirichlet boundary condition on .T := I × ∂, search for a velocity vector field .u : QT → Rd and a scalar kinematic pressure .π : QT → R that solve the system ∂t u − div(S(·, ·, ε(u))) + ∇π = f − div(F ) .

in QT ,

div(u) = 0

in QT ,

u=0

on T ,

u(0) = u0

(4.1)

in  .

Here, .f : QT → Rd is a given vector field and .F : QT → Md×d sym a given symmetric tensor field, jointly describing external body forces, and .u0 :  → Rd is the d×d initial velocity vector field. The tensor-valued mapping .S : QT × Md×d sym → Msym , associated with the extra stress tensor, is supposed to possess a .(p(·, ·), δ)-structure, i.e., to satisfy (S.1)–(S.4) with respect to .p ∈ Plog (QT ) with .p− > 1 and .δ ≥ 0. Even though the system (4.1) is physically meaningful only to a limited extent, it contains the typical incompressibility constraint .div(u) = 0 in .QT (cf. .(4.1)2 ), which entails several obstacles in view of a potential adaptation of the concepts presented in Chap. 3 to solenoidal variable Bochner–Lebesgue spaces. In addition, the unsteady .p(·, ·)-Stokes equations can be regarded as a simplification of the unsteady .p(·, ·)-Navier–Stokes equations. The latter describe the unsteady motion © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaltenbach, Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents, Lecture Notes in Mathematics 2329, https://doi.org/10.1007/978-3-031-29670-3_4

113

114

4 Solenoidal Variable Bochner–Lebesgue Spaces

of an incompressible electro-rheological fluid of constant density in a bounded domain . ⊆ Rd , .d ≥ 2. To be more precise, the unsteady .p(·, ·)-Navier–Stokes equations equally search for a velocity vector field .u : QT → Rd and a scalar kinematic pressure .π : QT → R that solve the system ∂t u − div(S(·, ·, ε(u))) + div(u ⊗ u) + ∇π = f − div(F ) .

in QT ,

div(u) = 0

in QT ,

u=0

on T ,

u(0) = u0

(4.2)

in  .

The main difference between the systems (4.1) and (4.2) represents the term div(u ⊗ u) : QT → Rd , which is commonly referred to as the convective term. For a proof of the existence of weak solutions of both the unsteady .p(·, ·)Stokes equations and the unsteady .p(·, ·)-Navier–Stokes equations by resorting to advanced pseudo-monotonicity methods, such as, e.g., those methods we have already developed in Chap. 3, we follow the conventional approach of incorporating the incompressibility constraint into the energy spaces. This leads, on the one hand, to a disappearance of the pressure term, which has to be reconstructed separately (cf. Chap. 6), and, on the other hand, to the so-called solenoidal variable q,p Bochner–Lebesgue space .Vε (QT ), which will be introduced in the following section. Furthermore, this approach will also give rise to a large number of not be underestimated difficulties. For instance, it will turn out that the prior procedure q,p of Chap. 3, i.e., of treating .Xε (QT ) as a kind of a variable Sobolev space with q,p respect to the spatial variable, e.g., in the description of duality in .Xε (QT ) (cf. q,p Sect. 3.2) or in the construction of an appropriate smoothing operator for .Xε (QT ) q,p (cf. Sect. 3.4), works only in parts for .Vε (QT ). As a consequence, we are forced to modify the procedure of Chap. 3 at various places.

.

q,p

q,p

4.1 The Spaces V∇ (QT ) and Vε

(QT )

Throughout the entire section, unless otherwise specified, let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, .QT := I × , and .q, p ∈ P∞ (QT ). Moreover, for exponents .q, ˜ p˜ ∈ P∞ () and   ∞ V := C0,σ () := u ∈ C0∞ ()d | div(u) = 0 in  ,

.

we introduce the solenoidal variable Sobolev spaces q(·), ˜ p(·) ˜

· X ˜ p(·) ˜ ˚q(·), ∇ () := V V ∇

.

()d

,

q(·), ˜ p(·) ˜

· Xε ˜ p(·) ˜ ˚εq(·), V () := V

()

.

q,p

q,p

4.1 The Spaces V∇ (QT ) and Vε

115

(QT )

As already announced, the idea of solenoidal variable Bochner–Lebesgue spaces consists in incorporating the incompressibility constraint .(4.1)2 into the time slice spaces. With this in mind, we make the following definitions. Definition 4.1 We define for almost every .t ∈ I , the solenoidal time slice spaces .

˚q,p (t) := V ˚q(t,·),p(t,·) () , V ∇ ∇

˚εq,p (t) := V ˚εq(t,·),p(t,·) () . V

Furthermore, we define the limiting solenoidal time slice spaces .

˚q ˚+q,p := V V ∇

+ ,p +

() ,

˚q ˚−q,p := V V ∇

− ,p −

() .

Remark 4.1 (i) In the case of time-independent exponents .q, p ∈ P∞ (), we define .

˚q(·),p(·) () , ˚q,p := V V ∇ ∇

˚εq(·),p(·) () . ˚εq,p := V V

˚q,p (t) and .X ˚εq,p (t) coincide (ii) If .p ∈ Plog (QT ) with .p− > 1, then the spaces .X ∇ for every .t ∈ I (cf. Remark 3.1 (iii)), with a possibly on .t ∈ I depending norm ˚q,p (t) and .V ˚εq,p (t) coincide for every .t ∈ I . equivalence. Consequently, also .V ∇ Therefore, for .p ∈ Plog (QT ) with .p− > 1, we define .

˚q,p (t) := V ˚εq,p (t) = V ˚q,p (t) . V ∇

for every .t ∈ I . For time-independent .q ∈ P∞ () and .p ∈ Plog () with − .p > 1, we also define .

˚εq,p = V ˚q,p := V ˚q,p . V ∇ q,p

˚+ (iii) For almost every .t ∈ I , there hold the dense embeddings .V ˚εq,p (t)→ V ˚−q,p . .→ V

q,p

˚ (t) → V ∇

˚q,p (t), .t ∈ I , and .V ˚εq,p (t), .t ∈ I , By means of the time slice spaces .V ∇ we next introduce solenoidal variable Bochner–Lebesgue spaces, by analogy with Definition 3.2. Definition 4.2 We define the solenoidal variable Bochner–Lebesgue spaces  q,p V∇ (QT ) := u ∈ Lq(·,·) (QT )d | ∇u ∈ Lp(·,·) (QT )d×d ,  ˚q,p (t) for a.e. t ∈ I , u(t) ∈ V ∇  q,p Vε (QT ) := u ∈ Lq(·,·) (QT )d | ε(u) ∈ Lp(·,·) (QT , Md×d sym ),  ˚εq,p (t) for a.e. t ∈ I . u(t) ∈ V

.

Furthermore, we define the limiting solenoidal Bochner–Lebesgue spaces q,p

V+ (QT ) := Lmax{q

.

+ ,p + }

q,p

˚+ ) , (I, V

q,p

V− (QT ) := Lmin{q

− ,p − }

q,p

˚− ) . (I, V

116

4 Solenoidal Variable Bochner–Lebesgue Spaces

˚q,p (t), .t ∈ I , and .V ˚εq,p (t), .t ∈ I , now do not Remark 4.2 The time slice spaces .V ∇ only serve to encode the zero Dirichlet boundary condition .(4.1)3 in the definitions q,p q,p of .V∇ (QT ) and .Vε (QT ) (cf. Remark 3.2) but also the incompressibility constraint .(4.1)2 of the systems (4.1) and (4.2), which once more reflects the Bochner–Lebesgue character of these spaces. q,p

q,p

Proposition 4.1 The spaces .V∇ (QT ) and .Vε equipped with the norms

(QT ) form Banach spaces, if

· V q,p (QT ) := · Lq(·,·) (QT )d + ∇ · Lp(·,·) (QT )d×d ,

.

∇

· Vεq,p (QT ) := · Lq(·,·) (QT )d + ε(·) Lp(·,·) (QT )d×d , respectively. In addition, we have that q,p

q,p

q,p

V+ (QT ) → V∇ (QT ) → Vε

.

q,p

(QT )→ V− (QT ).

q,p

Proof We only give a proof for .Vε (QT ) because all arguments are transferable q,p to .V∇ (QT ). q,p Since we are already aware of that .Xε (QT ) forms a Banach spaces equipped q,p with the same norm (cf. Proposition 3.2), it is sufficient to establish that .Vε (QT ) q,p q,p is norm-closed in .Xε (QT ). So, let .(un )n∈N ⊆ Vε (QT ) be a sequence such q,p that .un → u in .Xε (QT ) .(n → ∞). According to Fischer–Riesz’s theorem for q,p time slices of .Xε (QT ) (cf. Corollary 3.1), this immediately yields a subsequence q,p .(un )n∈ ⊆ Vε (QT ), with a cofinal subset . ⊆ N, such that for almost every .t ∈ I un (t) → u(t)

.

q,p

˚ε (t) ( n → ∞) . in X

(4.3)

˚εq,p (t) is a norm-closed subspace of .X ˚εq,p (t) and .(un (t)) ˚q,p Since .V n∈N ⊆ Vε (t) for q,p ˚ε (t) and .un (t) → u(t) in almost every .t ∈ I , we infer from (4.3) that .u(t) ∈ V ˚εq,p (t) .(n → ∞) for almost every .t ∈ I , i.e., in total, we have that .u ∈ Vεq,p (QT ) .V q,p and .un → u in .Vε (QT ) .(n → ∞).  q,p

Corollary 4.1 (Fischer–Riesz for Time Slices of .Vε (QT )) Let .(un )n∈N ⊆ q,p q,p Vε (QT ) be such that .un → u in .Vε (QT ) .(n → ∞). Then, there exists q,p a subsequence .(un )n∈ ⊆ Vε (QT ), with a cofinal subset . ⊆ N, such that ˚εq,p (t) .( n → ∞) for almost every .t ∈ I . .un (t) → u(t) in V q,p

Proposition 4.2 The mapping .σε := ε ◦idVεq,p (QT ) : Vε (QT ) → Lq(·,·) (QT)d × p(·,·) (Q , Md×d ), where the mapping . is from Proposition 3.3, is an isometric .L T ε sym q,p isomorphism into its range .R(σε ). In particular, .Vε (QT ) is separable. In q,p addition, if .q − , p− > 1, then .Vε (QT ) is reflexive. Proof Follows along the lines of the proof of Proposition 3.3.



q,p

4.2 Duality in Vε

117

(QT )

q,p

4.2 Duality in Vε

(QT )

This section addresses a thorough description of the space .Vε (QT )∗ , the dual q,p space of .Vε (QT ). We emphasize that all following results admit congruent q,p q,p adaptations to .V∇ (QT ), and we restrict our attention to .Vε (QT ) solely for ease of presentation and since this space is anyway of greater interest with regard to the unsteady .p(·, ·)-Stokes and the unsteady .p(·, ·)-Navier–Stokes equations, i.e., the systems (4.1) and (4.2). Looking back to Chap. 3, Sect. 3.2, which was strongly inspired by the description of duality in classical Sobolev spaces, it might be of advantage to first consider the steady case in order to understand the peculiarities of q,p a characterization of .Vε (QT )∗ . For this purpose, let .q, p ∈ P∞ () with .q − , p− > 1. A well-known result from functional analysis1 yields that the mapping q,p

∗ ˚ ˚εq,p )∗ , ε : (Xε ) (V ˚εq,p )◦ → (V q,p

(4.4)

.

˚ε for every .x∗ + (V

q,p ◦ )

˚ε )∗ /(V ˚ε ∈ (X

q,p ◦ )

q,p

˚ε ε (x∗ + (V

defined by

q,p ◦

.

) ) := (idV˚εq,p )∗ x∗

q,p ∗

˚ε in (V

) ,

(4.5)

˚εq,p → X ˚εq,p , is an where .(idV˚εq,p )∗ denotes the adjoint operator of .idV˚εq,p : V isometric isomorphism. The identification (4.4) can be interpreted as follows: ˚εq,p )∗ can be realized by the restriction of a functional Any functional .u∗ ∈ (V q,p q,p ∗ ˚ε )∗ to .V ˚ε , and such a functional is unique except for perturbations by .x ∈ (X ˚εq,p )◦ . Let us follow this approach to functionals coming from the annihilator .(V q,p conclude a first characterization of .Vε (QT )∗ . Proposition 4.3 (First Characterization of .Vε (QT )∗ ) Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, ∞), .T < ∞, .QT := I × , and .q, p ∈ P∞ (QT ) with − − > 1. Then, the mapping .q , p q,p

∗ q,p  ε : Xε (QT ) V q,p (Q )◦ → Vε (QT )∗ , T ε q,p

.

a Banach space X and a closed subspace .M ⊆ X, the quotient space .X/M equipped with the quotient norm . x + M X/M := infm∈M x − m X forms a Banach space (cf. [144, Theorem 1.41 (d)]). As a consequence, since the annihilator .M ◦ := {x ∗ ∈ X∗ | x ∗ , mX = 0 for all m ∈ M} is a closed subspace of .X∗ , the quotient space .X∗ /M ◦ equipped with the quotient norm . x ∗ + M ◦ X∗ /M ◦ := infm◦ ∈M ◦ x ∗ − m◦ X∗ forms a Banach space as well. In particular, the mapping ∗ ◦ ∗ ∗ ◦ ∗ ∗ ∗ ∗ ◦ ∗ ◦ . : X /M → M , defined by .(x + M ) := (idM ) x in .M for every .x + M ∈ X /M , where .(idM )∗ denotes the adjoint operator of .idM : M → X, forms an isometric isomorphism (cf. [144, Theorem 4.9 (i)]). 1 For

118

4 Solenoidal Variable Bochner–Lebesgue Spaces

for every .x ∗ + Vε

q,p

(QT )◦ ∈ Xε (QT )∗ /Vε q,p

 ε (x ∗ + Vε

q,p

.

q,p

(QT )◦ defined by

(QT )◦ ) := (idVεq,p (QT ) )∗ x ∗

q,p

in Vε

(QT )∗ ,

where .(idVεq,p (QT ) )∗ denotes the adjoint operator of .idVεq,p (QT ) : Vε q,p Xε (QT ), forms an isometric isomorphism.

q,p

Proof A straightforward application of [144, Theorem 4.9 (i)].

(QT ) → 

Regrettably, this first characterization is rather unsatisfactory, in the sense that q,p unlike .Xε (QT )∗ , for which we have already provided a thorough description (cf. q,p Proposition 3.4), the annihilator .Vε (QT )◦ is still uncharted. The next question should therefore be whether there also exists a characterization of the annihilator q,p .Vε (QT )◦ . Again, we should first consider the steady case. Here, the well-known de Rham lemma, which requires the following space, partly provides a remedy. Definition 4.3 Let . ⊆ Rd , .d ∈ N, be a bounded domain and .p ∈ P∞ (). We introduce the space   p(·) L0 () := π ∈ Lp(·) () | (1, π )Lp(·) () = 0 .

.

Lemma 4.1 (De Rham) Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, log ˚p,p )◦ , there is a unique function .p ∈ P () with .p− > 1. Then, for every .x∗ ∈ (V  p (·) ˚p,p )∗ , where .∇ : Lp (·) () → (X ˚p,p )∗ .π ∈ L () such that .x∗ = ∇π in .(X 0 ˚p,p := −(π, div(x))Lp(·) () denotes the distributional gradient, defined by .∇π, xX  ˚p,p . Moreover, there exists a constant .c > 0 such for all .π ∈ Lp (·) () and .x ∈ X that π Lp (·) () ≤ c x∗ (X ˚p,p )∗ .

.

p (·)

In other words, .∇ : L0

˚p,p )◦2 is an isomorphism. () → (V

˚p,p )◦ , there Proof In [72, Lemma 4.8], it has been proved that for every .x∗ ∈ (V ffl  (·) p ∗ p,p ∗ ˚ exists a .π˜ ∈ L () such that .∇ π˜ = x in .(X )  . Then, .π := π˜ −  π˜ (x) dx ∈ p (·) ˚p,p )∗ , i.e., .∇ : Lp (·) () → (V ˚p,p )◦ is surjective. L0 () satisfies .∇π = x∗ in .(X 0 The claimed injectivity, continuity and continuity of the inverse follow by the negative norm theorem (cf. [49, Theorem 14.3.8., (14.3.19)]).  With the aid of de Rham’s lemma and (4.4), we obtain at least for the special case of a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2, and exponents .q, p ∈ Plog () with .q = p in . and .p− > 1, the isometrically isomorphic identification ε ˚p,p ∗ (X )    . ˚p,p )∗ ∼ (V p (·) = ∇ L0 ()

.

(4.6)

p(·) 2 We always endow .Lp(·) () with the norm . · Lp(·) () , which turns .L0 () into a Banach space. 0

q,p

4.2 Duality in Vε

119

(QT )

˚p,p )∗ possesses an The identity (4.6) expresses that every functional .u∗ ∈ (V ∗ p,p ∗ ˚ extension .x ∈ (X ) , which is unique up to perturbations by gradient vector  ˚p,p )◦ of functions .π ∈ Lp (·) (). As we have already elaborated fields .∇π ∈ (V 0 ˚p,p )∗ (cf. Corollary 3.2), it can be a thorough description of the dual space .(X deduced from the identity (4.6) that    ε p (·) ()d + div Lp (·) (, Md×d ) L . ˚p,p )∗ ∼ sym (V   p (·) = ∇ L0 ()

.

(4.7)

Looking back to the previous steps, we should attempt to prove an analogue of de Rham’s lemma for variable Bochner–Lebesgue spaces. In doing so, the space q,p ˚p,p and .Vεq,p (QT ) the role of .V ˚p,p . .Xε (QT ) is supposed to take the role of .X p (·) Nevertheless, an appropriate generalization of the space .L0 () to the framework q,p q,p of .Xε (QT ) and .Vε (QT ) is still missing. q,p In the same manner as for .Xε (QT ) (cf. Definition 3.2 and Remark 3.2) and q,p .Vε (QT ) (cf. Definition 4.2 and Remark 4.2), we, therefore, make the following definition. Definition 4.4 Let . ⊆ Rd , .d ∈ N, be a bounded domain, .I := (0, T ), .T < ∞, ∞ .QT := I × , and .p ∈ P (QT ). We introduce the space    p(·,·) p(t,·) L0, (QT ) := π ∈ Lp(·,·) (QT )  π (t) ∈ L0 () for a.e. t ∈ I .

.

Proposition 4.4 Let . ⊆ Rd , .d ∈ N, be a bounded domain, .I := (0, T ), .T < ∞, p(·,·) ∞ .QT := I × , and .p ∈ P (QT ). Then, . · Lp(·,·) (Q ) turns .L 0, (QT ) into a T Banach space. p(·,·)

Proof It is sufficient to establish that .L0, (QT ) is norm-closed in .Lp(·,·) (QT ) (cf. p(·,·) Proposition 2.1). So, let .(π n )n∈N ⊆ L0, (QT ) be a sequence such that .π n → π in .Lp(·,·) (QT ) .(n → ∞). Because norm convergence implies modular convergence (cf. Proposition 2.3), also using Fubini’s theorem, we observe that ρp(t,·) (π n (t) − π (t)) dt = ρp(·,·) (π n − π ) → 0 (n → ∞) .

.

(4.8)

I

Therefore, Fischer–Riesz’s theorem in .L1 (I ) applied to (4.8) yields a subsequence .(π n )n∈ , with cofinal . ⊆ N, such that .ρp(t,·) (π n (t) − π (t)) → 0 .( n → ∞) for almost every .t ∈ I . Thus, Proposition 2.3 implies conversely .π n (t) → π(t) p(t,·) in .Lp(t,·) () .( n → ∞) for almost every .t ∈ I . Since .L0 (), .t ∈ I , are a norm-closed subspaces of .Lp(t,·) (), .t ∈ I , respectively, we conclude that p(t,·) p(·,·) .π (t) ∈ L () for almost every .t ∈ I , i.e., .π ∈ L0, (QT ) and .π n → π in 0 p(·,·)

L0, (QT ) .(n → ∞).

.



120

4 Solenoidal Variable Bochner–Lebesgue Spaces

It is still pending to introduce the distributional gradient as a functional in q,p Xε (QT )∗ .

.

Proposition 4.5 Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, ∞), .T < ∞, ∞ − − > 1. For .π ∈ Lp (·,·) (Q ), we .QT := I × , and .q, p ∈ P (QT ) with .q , p T q,p define the distributional gradient .∇π ∈ Xε (QT )∗ by ∇π, xXεq,p (QT ) := −(π , div(x))Lp(·,·) (QT )

.

q,p

for every .x ∈ Xε (QT ). Then, it holds: 

(i) .∇ : Lp (·,·) (QT ) → Vε (QT )◦ ⊆ Xε (QT )∗ is well-defined, linear and bounded. p (·,·) q,p (ii) .∇ : L0, (QT ) → Vε (QT )◦ is injective. q,p

q,p



Proof ad (i). Let .π ∈ Lp (·,·) (QT ) be arbitrary. By applying Hölder’s inequality for q,p variable Lebesgue spaces (cf. Proposition 2.6), for every .x ∈ Xε (QT ), we find that |∇π, xXεq,p (QT ) | ≤ 2 π Lp (·,·) (QT ) div(x) Lp(·,·) (QT )

.

1

≤ 2d 2 π Lp (·,·) (QT ) ε(x) Lp(·,·) (QT )d×d , 1

where we used that .|tr(A)| ≤ d 2 |Asym | for all .A ∈ Rd×d . In other words, p (·,·) (Q ) → X q,p (Q )∗ is well-defined, linear and bounded. Moreover, .∇ : L T T ε q,p .R(∇) ⊆ Vε (QT )◦ follows right from the definition. p (·,·) q,p ad (ii). Let .π ∈ L0, (QT ) be such that .∇π = 0 in .Xε (QT )∗ , i.e., for every q,p .x ∈ Xε (QT ), it holds (π , div(x))Lp(·,·) (QT ) = 0 .

.

(4.9)

If we choose .x = xϕ ∈ Xε (QT ) in (4.9), where both .x ∈ C0∞ ()d and ∞ .ϕ ∈ C (I ) are arbitrary, then we infer by the fundamental theorem that for 0 almost every .t ∈ I and all .x ∈ C0∞ ()d it holds .(π(t), div(x))Lp(t,·) () = 0. Thus, resorting to [5, Chapter 3, Corollary 3.32], we deduce for almost every .t ∈ I that there exists a constant .c(t) ∈ R such that .π (t) = c(t) almost everywhere in .. Since .(1, π (t))Lp (t,·) () = 0 for almost every .t ∈ I , we conclude that .c(t) = 0 for almost every .t ∈ I and, therefore, .π = 0 almost everywhere in .QT , p (·,·) q,p  i.e, .∇ : L0, (QT ) → Vε (QT )◦ is injective. q,p

Since for constant exponents .q = p ∈ (1, ∞), the usual Bochner–Lebesgue space structure is available, in which isomorphisms between steady spaces inherit into isomorphisms between the corresponding Bochner–Lebesgue spaces (cf. p ˚p,p )◦ Proposition 2.21 (iii)), we infer from the isomorphism .∇ : L0 () → (V

q,p

4.2 Duality in Vε

121

(QT )

(cf. Lemma 4.1), where we assume that .∂ ∈ C 0,1 , that the distributional gradient interpreted as an operator between Bochner–Lebesgue spaces, i.e., the operator 

p



˚p,p )◦ ) , ∇ : Lp (I, L0 ()) → Lp (I, (V

.

(4.10)



 p ˚p,p )◦ for almost for every .π ∈ Lp (I, L0 ()) defined by .(∇π )(t) := ∇(π (t)) in .(V every .t ∈ I , is an isomorphism as well. Consequently, the first impression might lead one to expect that at least for .q = p ∈ Plog (QT ) with .p− > 1 and .∂ ∈ C 0,1 , the distributional gradient in Proposition 4.5, i.e.,

p (·,·)

∇ : L0, (QT ) → Vε (QT )◦ ,

.

p,p

(4.11)

is likewise an isomorphism, i.e., owing to the continuity of the inverse, there exists a p (·,·) constant .c > 0 such that for every .π ∈ L0, (QT ), the following weak Poincaré– Wirtinger type inequality applies π Lp (·,·) (QT ) ≤ c ∇π Xεp,p (QT )∗ .

(4.12)

.

Unfortunately, (4.12) cannot be valid, even if .p ∈ C ∞ (Rd ) with .p− > 1. This can be seen from the following two appended remarks. First, we will establish that a Poincaré–Wirtinger type inequality in a classical sense, i.e., with not only distributional but integrable gradient, cannot apply. p(·,·)

Remark 4.3 (Invalidity of Classical Poincaré–Wirtinger Inequality on .L0, (QT )) Let . ⊆ Rd , .d ≥ 2, be an arbitrary bounded domain, .I := (0, T ), .T < ∞, and .QT := I × . Moreover, let    ∞ ∞ .FP–W := (π, ∇π ) | π ∈ C0,0 () ⊆ C0,0 () × C0∞ ()d . Let .η ∈ C0∞ () with .η = 1 in G, where .G ⊂⊂ , and .ξ ∈ C0∞ ( \ G) with



∞ ∞ 1 . \G ξ(x) dx = 1. If .π := η − ξ  η(x) dx, then .π ∈ C0,0 () := C0 () ∩ L0 () with .π = 1 in G. In other words, we have that .(π, ∇π ) ∈ FP–W with .∇π = 0 and .int(supp(π )) \ supp(∇π ) = ∅. In consequence, according to Proposition 3.1, there exists an exponent .p ∈ C ∞ (Rd ) with .p− > 1 that does not admit a constant ∞ 3 ∞ .c > 0 such that for every .π ∈ C (I , C 0,0 ()), it holds π Lp(·) (QT ) ≤ c ∇π Lp(·) (QT )d .

.

(4.13)

2 (2.5e ), B 2 (2.5e )}, .G := The situation is illustrated in Fig. 4.1 for . := conv{B2.5 1 1 2.5 2 ∞ 2 (· − 2.5e ) ∈ C ∞ ( \ G) 2 B0.6 (2.5e1 ), .η := χB 2 (2.5e1 ) ∗ ωε ∈ C0 () and .ξ := ω4ε 1 0 1 for .ε := 0.4.

3 .C ∞ (I , C ∞ ()) 0,0

∞ () for every t ∈ I }. := {π ∈ C ∞ (QT ) | π(t) ∈ C0,0

122

4 Solenoidal Variable Bochner–Lebesgue Spaces

∞ () (blue/left) and .|∇π | ∈ L∞ () (red/middle) and .p| ∈ C ∞ () Fig. 4.1 Plots of .π ∈ C0,0  (green/right), constructed according to Proposition 3.1 for .d = 2, .α = 1.1 and .β = 2

p(·,·)

Remark 4.4 (Invalidity of Weak Poincaré–Wirtinger Inequality on .L0, (QT )) Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, .QT := I × , and .q, p ∈ Plog (QT ) with .q − , p− > 1 and .q ≥ p in .QT .4 Suppose there exists a p (·,·) constant .c > 0 such that for every .π ∈ L0, (QT ), the following weak Poincaré– Wirtinger type inequality holds π Lp (·,·) (QT ) ≤ c ∇π Xεq,p (QT )∗ .

. 

(4.14)



Since for .π ∈ Lp (·,·) (QT ) with .∇π ∈ Lp (·,·) (QT )d , we have, by Proposition 2.6 and Corollary 2.1, that ∇π Xεq,p (QT )∗ =

.

sup q,p

x∈Xε (QT ) x X q,p (Q ) ≤1 ε

=

T

sup q,p

x∈Xε (QT ) x X q,p (Q ) ≤1 ε

−(π , div(x))Lp(·,·) (QT )

(∇π , x)Lq(·,·) (QT )d

T

≤ 2 ∇π Lq  (·,·) (QT )d ≤ 4(1 + |QT |) ∇π Lp (·,·) (QT )d ,

assumption .q, p ∈ Plog (QT ) with .p − > 1 and .q ≥ p in .QT is in accordance with the assumptions of the remaining density results (cf. Proposition 4.14) and, therefore, the variable exponents of interest in this book.

4 The

q,p

4.2 Duality in Vε

123

(QT ) p (·,·)



we deduce from (4.14) that for every .π ∈ L0, (QT ) with .∇π ∈ Lp (·,·) (QT )d , it holds π Lp (·,·) (QT ) ≤ 4c(1 + |QT |) ∇π Lp (·,·) (QT )d ,

.

i.e., the classical Poincare–Wirtinger inequality (4.13), which, due to Remark 4.3, cannot be valid in general. All things considered, for .q, p ∈ Plog (QT ) with .p− > 1 and .q ≥ p in .QT , we can neither say that p (·,·)

∇ : L0, (QT ) → Vε (QT )◦

.

q,p

(4.15)

is an isomorphism, nor that it is surjective. Note that because of the open mapping theorem, a surjectivity of (4.15) would, in virtue of its injectivity and continuity, lead to an isomorphism. Eventually, we can summarize that the characterization of dualq,p ity in .Vε (QT ) works only in parts as in the steady case, in the sense that we cannot q,p give a comprehensive description of the corresponding annihilator .Vε (QT )◦ . Nevertheless, inasmuch as there is already a quite convenient characterization of q,p .Xε (QT )∗ given by Proposition 3.4, we can fall back on this characterization of q,p q,p .Xε (QT )∗ to deduce a second, more useful characterization of .Vε (QT )∗ . Proposition 4.6 (Second Characterization of .Vε (QT )∗ ) Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, ∞), .T < ∞, .QT := I × , and .q, p ∈ P∞ (QT ) with .q − , p− > 1. Then, the operator q,p





q,p ∗ Jεσ := (idVεq,p (QT ) )∗ ◦ Jε : Lq (·,·) (QT )d × Lp (·,·) (QT , Md×d sym ) → Vε (QT ) ,

.

where .(idVεq,p (QT ) )∗ denotes the adjoint operator of .idVεq,p (QT ) : Vε (QT ) → q,p Xε (QT ) and .Jε from Proposition 3.4, is linear and Lipschitz continuous with  q,p constant 2. In addition, for every .u∗ ∈ Vε (QT )∗ , there exist .f ∈ Lq (·,·) (QT )d  (·,·) q,p d×d p ∗ σ ∗ and .F ∈ L (QT , Msym ) such that .u = Jε (f , F ) in .Vε (QT ) and q,p

1 ∗ u Vεq,p (QT )∗ ≤ f Lq  (·,·) (QT )d + F Lp (·,·) (QT )d×d . 2 ≤ 2 u∗ Vεq,p (QT )∗ .

(4.16)

Remark 4.5 At first glance, looking back to Proposition 3.4, it may seem that q,p q,p Proposition 4.6 states that the dual spaces .Vε (QT )∗ and .Xε (QT )∗ coinq,p cide, which of course cannot apply because .Vε (QT ) is a proper subspace of q,p .Xε (QT ). In fact, Proposition 4.6 only states, similar to Proposition 4.3, that any q,p functional in .u∗ ∈ Vε (QT )∗ can be realized by the restriction of a functional q,p q,p ∗ .x ∈ Xε (QT )∗ to .Vε (QT ).

124

4 Solenoidal Variable Bochner–Lebesgue Spaces

Proof Well-definedness, linearity and Lipschitz continuity with constant 2 transfer   q,p ∗ directly from .Jε : Lq (·,·) (QT )d × Lp (·,·) (QT , Md×d sym ) → Xε (QT ) (cf. Propoq,p ∗ sition 3.4). So, let us prove the surjectivity including (4.16). Let .u ∈ Vε (QT )∗ . q,p ∗ ∗ The Hahn–Banach theorem yields a functional .x ∈ Xε (QT ) such that q,p ∗ ∗ ∗ .(idVεq,p (QT ) ) x = u in .Vε (QT )∗ and . x ∗ Xεq,p (QT )∗ = u∗ Vεq,p (QT )∗ . Fur



thermore, Proposition 3.4 provides .f ∈ Lq (·,·) (QT )d and .F ∈ Lp (·,·) (QT , Md×d sym ) q,p such that .x ∗ = Jε (f , F ) in .Xε (QT )∗ and . 12 x ∗ Xεq,p (QT )∗ ≤ f Lq  (·,·) (QT )d + F Lp (·,·) (QT )d×d ≤ 2 x ∗ Xεq,p (QT )∗ . Altogether, we have that .u∗ = Jεσ (f , F ) in q,p .Vε (QT )∗ and (4.16).  q,p

˚ε In the same manner, we can characterize the dual space of .V

q(·),p(·)

˚ε =V

().

Corollary 4.2 Let . ⊆ Rd , .d ≥ 2, be a bounded domain and .q, p ∈ P∞ ()  with .q − , p− > 1. Then, the operator .Jσε := (idV˚εq,p )∗ ◦ Jε : Lq (·) ()d ×  ˚q,p ∗ ˚εq,p )∗ denotes the adjoint operator of Lp (·) (, Md×d sym ) → (Vε ) , where .(idV ˚εq,p → X ˚εq,p , is linear and Lipschitz continuous with constant 2. In addi˚εq,p : V .idV ∗ ˚εq,p )∗ , there exist .f ∈ Lq  (·) ()d and .F ∈ Lp (·) (, Md×d ) tion, for every .u ∈ (V sym ˚εq,p )∗ and such that .u∗ = Jσε (f, F) in .(V .

1 ∗ u (V˚εq,p )∗ ≤ f Lq  (·) ()d + F Lp (·) ()d×d ≤ 2 u∗ (V˚εq,p )∗ . 2

Remark 4.6 (Time Slices of Functionals .u∗ ∈ Vε with .q − , p− > 1.

q,p

(QT )∗ ) Let .q, p ∈ P∞ (QT ) 

(i) The same arguments as in Remark 3.6 show for .f ∈ Lq (·,·) (QT )d and .F ∈  σ ˚q,p ∗ Lp (·,·) (QT , Md×d sym ) that .Jε (f (t), F (t)) ∈ Vε (t) for almost every .t ∈ I q,p (cf. Corollary 4.2). Moreover, for every .u ∈ Vε (QT ), the function .(t → σ q,p Jε (f (t), F (t)), u(t)V˚ε (t) ) : I → R belongs to .L1 (I ) and Jεσ (f , F ), uVεq,p (QT ) =

.

I

Jσε (f (t), F (t)), u(t)V˚εq,p (t) dt .

(4.17)

(ii) Again we intend to define time slices of functionals .u∗ ∈ Vε (QT )∗ . Based   on Proposition 4.6, there exist .f ∈ Lq (·,·) (QT )d and .F ∈ Lp (·,·) (QT , Md×d sym ) q,p ∗ σ ∗ such that .u = Jε (f , F ) in .Vε (QT ) . Therefore, we define q,p

u∗ (t) := Jσε (f (t), F (t))

.

˚εq,p (t)∗ in V

for a.e. t ∈ I .

(4.18)

The same arguments as in Remark 3.6 clarify that (4.18) is well-defined, i.e.,   independent of the choice of .f ∈ Lq (·,·) (QT )d and .F ∈ Lp (·,·) (QT , Md×d sym ) q,p with .u∗ = Jεσ (f , F ) in .Vε (QT )∗ .

q,p

4.3 Smoothing in Vε

125

(QT )

q,p

4.3 Smoothing in Vε

(QT )

This section is entirely devoted to the construction of a trace- and divergenceq,p q,p preserving smoothing operator for .Vε (QT ) and .V∇ (QT ), which is indispensable on the way of proving a formula of integration-by-parts, such as Proposiq,p q,p tion 3.26, but now in the context of .Vε (QT ) and .V∇ (QT ) and, therefore, for the existence of weak solutions of both the unsteady .p(·, ·)-Stokes equations and the unsteady .p(·, ·)-Navier–Stokes equations, i.e., (4.1) and (4.2), by resorting to advanced pseudo-monotonicity methods, similar to those in Chap. 3. We follow the approach of adapting smoothing methods for variable Sobolev spaces to the framework of variable Bochner–Lebesgue spaces, which has already proven to be quite fruitful in Chap. 3, Sect. 3.4. In this context, we will study two common approaches, both motivated by smoothing methods for solenoidal variable Sobolev spaces. The first approach represents a failing method, intended merely as a cautionary example to illustrate the difficulties in the construction of a trace- and divergence-preserving smoothing method for variable Bochner–Lebesgue spaces. Since the following subsection will lead to hardly anything, an eager reader is permitted to skip this subsection. Nonetheless, it might be recommendable to face the failure of a normally reliable tool in the context of variable Bochner–Lebesgue spaces.

4.3.1 Failure of Smoothing via Bogovski˘ı Correction A commonly used procedure for smoothing solenoidal variable Sobolev functions consists in modifying a standard smoothing operator for variable Sobolev spaces by means of a Bogovski˘ı correction, see, e.g., [76, 132, 133]. To be more precise, ˚q,p , where we assume that . ⊆ Rd , .d ≥ 2, is a bounded Lipschitz if .u ∈ V domain and that .q, p ∈ Plog () with .q − , p− > 1, then we choose a sequence hn ∞ ∞ d d .(xn )n∈N ⊆ C () , e.g., we can take .(R u)n∈N ⊆ C () from Corollary 3.4  0 0 for a suitable null sequence .(hn )n∈N ⊆ (0, ∞), such that xn → u

.

˚q,p in X

(n → ∞) .

(4.19)

Since we usually cannot expect, in advance, that .(xn )n∈N ⊆ V, e.g., in most cases we merely have .div(Rhn u) = ωhdn ∗ (∇ηhn · ℰ u) = 0 for every .n ∈ N, we artificially recover the incompressibility constraint by subtracting the so-called Bogovski˘ı correction, i.e., for every .n ∈ N, we introduce the function un := xn − B (div(xn )) ∈ V ,

.

(4.20)

∞ () → C ∞ ()d is the Bogovski˘ı operator, provided by the where .B : C0,0 0 following proposition.

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4 Solenoidal Variable Bochner–Lebesgue Spaces

Proposition 4.7 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain and log .p ∈ P () with .p− > 1. Then, there exists a linear and continuous operator p(·) ˚p,p , called Bogovski˘ı operator, satisfying .div(B f ) = f in .B : L 0 () → X p(·) p(·) ∞ ∞ d .L 0 () for every .f ∈ L0 () and .B (C0,0 ()) ⊆ C0 () . 

Proof See [89, Theorem 2] or [49, Theorem 14.3.15.]. We intend to take advantage of the continuity of precisely, we need the continuity of .E := B ◦ guaranteed by the following proposition.

p(·) .B : L 0 () ˚q,p → div : X

˚p,p . More →X q,p ˚ X , which is

Proposition 4.8 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain and log ˚q,p → X ˚q,p .q, p ∈ P () with .q − , p− > 1. Then, the operator .E := B ◦div : X is well-defined, linear, continuous, i.e., . E x Lq(·) ()d ≤ cq x Lq(·) ()d and ˚q,p , and .div(E x) = div(x) in . ∇E x Lp(·) ()d×d ≤ cp div(x) Lp(·) () for all .x ∈ X p(·) ˚q,p . In addition, .E (C ∞ ()d ) ⊆ C ∞ ()d . .L () for all .x ∈ X 0

0

0



Proof See [89, Theorem 4].

˚q,p → X ˚q,p , we infer from (4.19) that In virtue of the continuity of .E : X q,p q,p ˚ .un → u in .V .(n → ∞). Next, let us attempt the same for .u ∈ Vε (QT ). So, ∞ d we choose .(x n )n∈N ⊆ C0 (QT ) such that xn → u

.

q,p

in Xε (QT ) (n → ∞) ,

(4.21)

∞ d ˚hn u) e.g., .(R n∈N ⊆ C0 (QT ) from Proposition 3.18 for a null sequence QT .(hn )n∈N ⊆ (0, ∞). For sake of simplicity, let us initially assume that the exponents log .q, p ∈ P (QT ) are constants, i.e., we have that .q(·, ·) ≡ q, p(·, ·) ≡ p ∈ (1, ∞). Then, both linear-induced operators p

p,p

B : Lp (I, L0 ()) → Xε

.

(QT ),

˚p,p (B f )(t) := B (f (t)) in X E

:

q,p Xε (QT )

→

for a.e. t ∈ I ,

q,p Xε (QT ),

˚q,p (E x)(t) := E (x(t)) in X

for a.e. t ∈ I ,

p ˚p,p are well-defined and inherit the continuity of the operators .B : L0 () → X ˚q,p → X ˚q,p (cf. Proposition 4.7, Proposition 4.8 and Proposition 2.21 and .E : X (ii)), respectively. In addition, .E (C ∞ (I , C0∞ ()d )) ⊆ C ∞ (I , C0∞ ()d ). Thus, by analogy with (4.20), if we define for every .n ∈ N

un := x n − E x n ∈ C ∞ (I , V)   := φ ∈ C ∞ (I , C0∞ ()d ) | φ(t) ∈ V for every t ∈ I ,

.

(4.22)

q,p

4.3 Smoothing in Vε

127

(QT ) q,p

then we infer from (4.21) that .un → u in .Vε (QT ) .(n → ∞). So, what are the difficulties when we consider non-constant exponents .q, p ∈ Plog (QT ) with − − > 1? Because at least .u ∈ V q,p (Q ), the sequence .(u ) ∞ .q , p T n n∈N ⊆ C (I , V), − q,p defined as in (4.22), satisfies .un → u in .V− (QT ) .(n → ∞). In order to obtain q,p the stronger convergence .un → u in .Vε (QT ) .(n → ∞), however, the continuity of ∞ B : C ∞ (I , C0,0 ()) ⊆ L0, (QT ) → C ∞ (I , C0∞ ()d ) ⊆ Xε p(·,·)

.

p,p

(QT ) ,

(4.23)

E : C ∞ (I , C0∞ ()d ) ⊆ Xε (QT ) → C ∞ (I , C0∞ ()d ) ⊆ Xε (QT ) , q,p

q,p

is still missing. Regrettably, these operators are not necessarily continuously extendable. Remark 4.7 (Discontinuity of the Bogovski˘ı Operator .B and .E ) Let . := −1 (−3, 3)d , .d ≥ 2. Then, we consider .ω0 ∈ C0∞ (), defined by .ω0 (x) := exp 1−|x| 2

if .x ∈ B1d (0) and .ω0 (x) := 0 if .x ∈ B1d (0) , .ω1 := ω0 (· + 65 e1 ) ∈ C0∞ (), .ω2 := ω ∈ C0∞ (), i.e., we have that ω0 (· − 65 e1 ) ∈ C0∞ (), and .ω := ω1 −1 L1 () 1

∞ ∞ d .  ω(y) dy = 1. Then, the Bogovski˘ı operator .B : C0,0 () → C0 () , for every ∞ .f ∈ C 0,0 () and .x, y ∈  can be defined by an explicit representation formula due to M. Bogovski˘ı (cf. [89, Sec. 3]) (B f )(x) :=

f (y)N(x, y) dy,

.



 N(x, y) :=

(x − y)

∞ 1

ω(y + r(x − y))r d−1 dr

if x = y if x = y

0

.

  ∞ () ⊆ C ∞ ()d × C ∞ (). Next, let us consider .FBog := (B f, f ) | f ∈ C0,0 0 0,0 ∞ () with .[int(supp(B f )) ∪ There exists .f ∈C0,0 . int(supp .(ε(B f )))]\supp(f ) = ∅  ∞ (), which satisfies and .f = 0. In fact, consider the function .f := ω1 − ω2 ∈ C0,0 d .f = 0 in .B 0.2 (0) and, if one introduces the notation .N1 := N · e1 , in addition, that (B f )1 (0) = −

ω2 (y)N1 (0, y) dy > 0 ,

.

(4.24)



where we have used that .supp(ω1 ) ∩ supp(N1 (0, ·)) = ∅ and .int(supp(ω2 ))∩ int(supp(N1 (0, ·))) = ∅ (cf. Fig. 4.2). Hence, (4.24) tells us that .int(supp(B f )) \supp(f ) = ∅. In the same manner, by inspecting the origin, one also finds that .int(supp(ε(B f )))\supp(f ) = ∅. More precisely, for .d = 2, the author determined numerically by employing Scipy’s (cf. [161]) nquad quadrature method that .ε(B f )11 (0.19, 0) ≈ −0.161 with an absolute error of about .2.29e−10, i.e., d .ε(B f ) ≡ 0 in .B 0.2 (0).

128

4 Solenoidal Variable Bochner–Lebesgue Spaces

Fig. 4.2 Plots of .ω1 ∈ C0∞ () (blue/left), .ω2 ∈ C0∞ () (red/middle) and .N1 (0, ·) ∈ C ∞ (Rd \{0}) (green/right) for .d = 2

As a consequence, according to Proposition 3.1, there exists a .p ∈ C ∞ (Rd ) with > 1 that admits no constant .c > 0 such that . B (φ) Xεp,p (QT ) ≤ c φ Lp(·) (QT ) ∞ ()), i.e., .(4.23) not necessarily admits a continuous for every .φ ∈ C ∞ (I , C0,0 1 p(·,·) extension to all of .L0, (QT ). Similarly, we predict the existence of .x ∈ C0∞ ()d with .[int(supp(E x)) ∪ int(supp(ε(E x)))] \ [supp(x) ∩ supp(ε(x))] = ∅, implyq,p ∞ d ∞ ing the non-extendability of (QT ) → .E : C (I , C () ) ⊆ Xε 0 q,p ∞ d ∞ C (I , C0 () ) ⊆ Xε (QT ).

− .p

Putting everything together, even if .(un )n∈N ⊆ C ∞ (I , V), we cannot ensure that q,p .un → u in Vε (QT ) .(n → ∞).

4.3.2 Smoothing via Transversal Expansion of Lipschitz Domains As we have already noted in the previous subsection, the usual approach of constructing a smoothing method for solenoidal variable Sobolev functions by modifying a standard smoothing operator by means of a Bogovski˘ı correction does q,p not extend to the framework of .Vε (QT ). As a result, we have to construct a smoothing method without the usage of a Bogovski˘ı correction. This is remedied by the smoothing method via transversal expansion of bounded Lipschitz domains (cf. [13, 66, 151]). The idea of smoothing via transversal expansion of bounded Lipschitz domains can be traced back to the well-known smoothing method via expansion of domains that are star-shaped with respect to some ball, see, e.g., [22, Thm. 3.1] or [122, Appx.]. For this reason, to give some more in-depth insights into this smoothing method’s underlying idea, we will first present this method in the highly simplified situation of a domain that is star-shaped with respect to some ball. Remark 4.8 (Mollification via Expansion of Star-Shaped Domains) Let . ⊆ Rd , d .d ≥ 2, be a bounded domain that is star-shaped with respect to a ball .B ⊆ R . d Without loss of generality, we may assume that .B = Bρ (0) for some .ρ > 0. We

q,p

4.3 Smoothing in Vε

129

(QT )

define for .h ∈ (0, 1), the expansion mapping .ν h : Rd → Rd by ν h (x) := x + hx

in Rd

.

for all x ∈ Rd .

˚q,p ,5 where .q, p ∈ Plog () with .q − , p− > 1, and its zero extension For .u ∈ V 1 d d d ∞ d d .ℰ u ∈ L (R ) , we set .uε,h := (ωε ∗ ℰ u) ◦ ν h ∈ C (R ) for every .ε > 0 ∞ d d and .h ∈ (0, 1), where .(ωε )ε>0 ⊆ C0 (R ) denotes the family of scaled standard mollifiers from Remark 2.6. Apparently, we have that .div(uε,h ) = 0 in .Rd for every .ε > 0 and .h ∈ (0, 1). On the other hand, the star-shape of . further provides a constant .ζ > 0 (cf. [22, Lem. 2.1]) such that for any .h ∈ (0, 1) h :=

.

1  + Bζdh (0) ⊂⊂  . 1+h

Therefore, since .supp(ωεd ∗ ℰ u) ⊆  + Bεd (0), we infer that .supp(uε,h ) ⊆ (1 + h)−1 ( + Bεd (0)) ⊆ h ⊆  for all .ε ∈ (0, ζ h] and .h ∈ (0, 1). Altogether, if we introduce .Sh u := uζ h,h for every .h ∈ (0, 1), we have that .(Sh u)h∈(0,1) ⊆ V. By a change of variables, Proposition 2.14 (iii), Remark 2.6, and Lebesgue’s theorem on dominated convergence (cf. Proposition 2.4), we find that −

Sh u − u Lq − ()d ≤ (1 + h)

.

d q−

ωζdh ∗ ℰ u − ℰ u Lq − (Rd )d

+ (ℰ u) ◦ ν h − ℰ u Lq − (Rd )d → 0

(n → ∞) .

In particular, .Sh u → u .(h → 0) almost everywhere in . for at least a subsequence. On the other hand, using that . ωhd L∞ (Rd ) ≤ hcdd and .Bζdh (ν h (x)) ⊆ Bζd0 h (x) for all .x ∈  and .h ∈ (0, 1) (cf. Proposition 4.10, (4.29), below), where .ζ0 := ζ + idRd C 0 ()d > 0, we observe for almost every .x ∈  that h .|(S u)(x)| ≤ |ωζdh (y − ν h (x))(ℰ u)(y)| dy Bζdh (ν h (x))

≤

cd ζ0d ζd

Bζd h (x)

|(ℰ u)(y)| dy ≤

cd ζ0d Md (ℰ u)(x) . ζd

0

Therefore, since Proposition 2.12 implies that .Md (ℰ u) ∈ Lq(·) (Rd ), where .q ∈ Plog (Rd ) is an extension of .q ∈ Plog () with .q − ≤ q ≤ q + (cf. Proposition 2.10), Proposition 2.4 yields .Sh u → u in .Lq(·) ()d .(h → 0).6 Due to .ε(Sh u) = (1 + h)Sh (ε(u)) in . for every .h ∈ (0, 1), we conclude analogously .ε(Sh u) → ε(u) in p(·) (, Md×d ) .(h → 0), i.e., in total, .Sh u → u in .V ˚q,p .(h → 0). .L sym  5 Here, we exploit that bounded domains that are star-shaped with respect to a ball are also ˚q,p := V ˚εq,p := V ˚q,p (cf. Remark 4.1 Lipschitz domains (cf. [86]) and, thus, by definition, .V ∇ log − − (ii)), since also .q, p ∈ P () with .q , p > 1. 6 This convergence applies first to a subsequence. However, the convergence principle (cf. [166, Prop. 10.13 (1)]) then provides the convergence of the entire sequence.

130

4 Solenoidal Variable Bochner–Lebesgue Spaces

Fig. 4.3 Sketch of the transversal expansion of a bounded Lipschitz domain . in two dimensions

Needless to say, we should not be satisfied with bounded domains that are starshaped with respect to some ball. Inasmuch as such domains also represent Lipschitz domains (cf. [86, Prop. 4.22]), the next question should therefore be whether it is possible to extend this expansion method in a similar way to Lipschitz domains. Fortunately, this question can be answered in the affirmative. Proposition 4.9 (Transversal Expansion of Bounded Lipschitz Domains) Let  ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain and .B ⊆ Rd a ball with . ⊂⊂ B. We define .O := B \ . Then, there exists a vector field .k ∈ C ∞ (Rd )d , whose negative restriction to .∂O is globally transversal for .O, i.e., there exists a constant .γ > 0 such that .−n (x) · k(x) ≥ γ and .|k(x)| = 1 for almost every .x ∈ ∂O ⊇ ∂, O where .nO : ∂O → Rd denotes the normal vector field of .O pointing outward, i.e., it holds .−nO = n on .∂, where .n : ∂ → Rd denotes the normal vector field of . pointing outward. We define for .h ∈ [0, 1], the transversal expansion mapping ∞ d d .ν h ∈ C (R ) by (Fig. 4.3) .

ν h (x) := x + hk(x)

.

in Rd

for all .x ∈ Rd . Moreover, we define .κ := Lip(k| ) > 0. Then, these mappings have the following properties: (i) There exists a constant .ζ > 0 (not depending on .h ∈ [0, 1]) such that for every ζh .h ∈ (0, 1], .h := 1+κh and .h := {x ∈  | dist(x, ∂) > h }, it holds d d d .ν h (O) + B 3ζ h (0) ⊆ O, .ν h (O) + B2ζ h (0) ⊆ O and .ν h ( \ h ) + Bζ h (0) ⊆ O. (ii) There exists a constant .h2 ∈ (0, κ1 ) such that for every .h ∈ [0, h2 ), the restricted transversal expansion mapping .ν h | :  → ν h () defines a bi-Lipschitz .C ∞ -diffeomorphism satisfying .Lip(ν h| ) ≤ 1 + hκ,7 −1 −1 and . 1 ≤ det(Dν ) ≤ 2 in .. .Lip(ν h h |ν h () ) ≤ (1 − hκ) 2

a set .G ⊆ Rn , .n ∈ N, and a Lipschitz continuous function .f : G → Rm , .m ∈ N, the Lipschitz constant is defined by .Lip(f) := inf{λ ≥ 0 | |f(x) − f(y)| ≤ λ|x − y| for all x, y ∈ G}.

7 For

q,p

4.3 Smoothing in Vε

(QT )

131

(iii) For every .i ∈ N ∪ {0}, there exists a constant .κi > 0 (not depending on .h ∈ [0, h2 )) such that for every .h ∈ [0, h2 ) and the Jacobian .Jh := Dν h ∈ C ∞ (Rd )d×d , it holds Di (Jh − Id ) L∞ ()d i+2 + Di (J−1 h − Id ) L∞ ()d i+2

.

+ Di (det(Jh ) − 1) L∞ ()d i ≤ κi h . Proof ad (i). We refer to [66, (2.4), Lemma 2.2 (iii) & Lemma 4.1 (cf. proof)]. See also, e.g., [86]. 1 ad (ii). For every .h ∈ 0, 2κ , we have that .Lip([ν h − idRd ]| )Lip(idRd | ) = Lip(hk| ) = hκ ≤ 12 . Hence, the Lipschitz inverse function theorem (cf. [39, 1 p. 137]) implies for every .h ∈ 0, 2κ that .ν h = idRd + hk :  → ν h () is bi-Lipschitz with the desired Lipschitz constants. Since .det : Rd×d → R is continuous, the set .det−1 (( 12 , 2)) is open in .Rd×d and .Id ∈ det−1 (( 12 , 2)). Thus, there exists some .ε > 0 such that .Bεd×d (Id ) ⊆ det−1 (( 12 , 2)). Apparently, it holds  ε . Dν h − Id L∞ ()d×d < ε for every .h ∈ 0, 2κ , since . Dk L∞ ()d×d ≤ κ, i.e., 1 ε −1 1 .Dν h (x) ∈ det (( 2 , 2)) for every .x ∈ . Therefore, if we define .h2 := min 2κ , 2κ , then .ν h :  → ν h () is for every .h ∈ (0, h2 ) bijective and satisfies 1 . ≤ det(Dν h ) ≤ 2 in .. Eventually, by using the classical inverse function theorem, 2 we conclude for every .h ∈ (0, h2 ) that .ν h :  → ν h () is a .C ∞ -diffeomorphism. ad (iii). Apparently, we have that . Di (Jh − Id ) L∞ ()d i+2 = Di+1 k L∞ ()d i+2 h for all .h ∈ [0, h2 ) and .i ∈ N ∪ {0}. Exploiting this, the second summand in (iii) can be estimated by using induction, that . J−1 h − Id L∞ ()d×d .≤ −1 −1 Jh (Id − Jh ) L∞ ()d×d ≤ (1 − h2 κ) Dk L∞ ()d×d h for all .h ∈ [0, h2 ), since −1 −1 −1 for all .h ∈ [0, h ) (cf. (ii)), and . J 2 h L∞ ()d×d ≤ Lip(ν h |ν h () ) ≤ (1 − h2 κ) −1 −1 −1 the identity .∂j Jh = −Jh ∂j Jh Jh in . for all .h ∈ [0, h2 ) and .j = 1, . . . , d. Similarly, we can estimate the third summand in (iii) by using induction, the Taylor expansion of .det ∈ C ∞ (Rd×d ), and the identity .∂j [det(Jh )] = det(Jh )tr(J−1 h ∂j [Jh ]) in . for all .h ∈ [0, h2 ) and .j = 1, . . . , d.  We are still not aware of whether this method admits an extension to the framework of variable Bochner–Lebesgue spaces. With this in mind, in the next step, we construct a smoothing operator for variable Bochner–Lebesgue spaces in the same procedure as in Remark 4.8. In fact, if we proceed exactly as in Remark 4.8 and merely use a transversal expansion mapping provided by Proposition 4.9 instead, then we obtain a smoothing operator that does not preserve the incompressibility constraint. This will be accomplished later through the modification of the following smoothing operator using a contravariant Piola transform. To simplify the presentation, we will first examine this not divergence-preserving smoothing operator for its specific properties.

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4 Solenoidal Variable Bochner–Lebesgue Spaces

Proposition 4.10 (Smoothing via Transversal Expansion in Space) Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, let .(ν h )h∈[0,1] ⊆ C ∞ (Rd )d be the transversal expansion mappings provided by Proposition 4.9, .I := (0, T ), .T < ∞, log .QT := I × , and .q ∈ P (QT ) with .q − > 1. For .x ∈ Lq(·,·) (QT )d , .h ∈ (0, h2 )  and .(t, x) ∈ QR := R × , we define the transversal expansion smoothing operator by h (SQ x)(t, x) := (ωζ h ∗ QT x)(t, ν h (x)) T ω(s, y)(QT x)(t + ζ hs, ν h (x) + ζ hy) dsdy =

.

= =

B1d+1 (0)

Bζd+1 h (t,ν h (x))

Bζd+1 h (t,x)

ωζ h (s − t, y − ν h (x))(QT x)(s, y) dsdy

ωζ h (s − t, y − ν h (x))(QT x)(s, y) dsdy ,

0

where .ζ0 := ζ + k C 0 ()d > 0. Moreover, .ω := ωd+1 ∈ C0∞ (Rd+1 ) and .(ωh )h>0 ⊆ C0∞ (Rd+1 ) are the standard mollifiers defined as above Proposition 3.14. Then, for every .x ∈ Lq(·,·) (QT )d , it holds: h x) h ∞ d (i) .(SQ h∈(0,h2 ) ⊆ C0 (QR ) with .supp(SQT x) ⊆ [−ζ h, T + ζ h] × h for T every .h ∈ (0, h2 ). (ii) There exists a constant .c > 0 (not depending on .q ∈ Plog (QT ) (cf. Remark 4.9)) such that h sup |SQ x| ≤ c Md+1 (QT x) T

.

h∈(0,h2 )

a.e. in QR .

(iii) For an extension .q ∈ Plog (Rd+1 ) of .q ∈ Plog (QT ), with .q − ≤ q ≤ q + in d+1 , there exists a constant .c > 0 (depending on .q ∈ Plog (Rd+1 ) and .R q .c > 0) such that .

h sup SQ x Lq(·,·) (Q )d ≤ cq x Lq(·,·) (QT )d . T R h∈(0,h2 )

h x → x in .Lq(·,·) (Q )d .(h → 0). (iv) .SQ T T 



h ) : Lq (·,·) (Q )d → Lq (·,·) (Q )d , .h ∈ (v) Denote by .(SQ (0, h2 ), the quasi T T T h q(·,·) d q(·,·) d adjoint operators of .SQT : L (QT ) .→ L (QT ) , in the sense of 

Remark 3.9, which are for every .x ∈ Lq(·,·) (QT )d , .y ∈ Lq (·,·) (QT )d and .h ∈ (0, h2 ) defined via the relation  h   (SQT ) y, x Lq(·,·) (Q

.

d T)

  h := y, SQ x Lq(·,·) (Q T

T)

d

.

q,p

4.3 Smoothing in Vε

133

(QT ) 

Then, for every .y ∈ Lq (·,·) (QT )d , .h ∈ (0, h2 ) and almost every .(t, x) ∈ QT , it holds

.

h  ((SQ ) y)(t, x) T =

Bζd+1 h (t,x)∩(I ×ν h ())

=

Bζd+1 h (t,x)

ωζ h (t −s, x −y)

y(s, ν −1 h (y))

det(Jh (ν −1 h (y)))

dsdy

(4.25)

ωζ h (t − s, x − ν h (y))(QT y)(s, y) dsdy .

0



h ) y) ∞ d q (·,·) (Q )d . In particular, .((SQ T h∈(0,h2 ) ⊆ C (QT ) for every .y ∈ L T

Remark 4.9 In Proposition 4.10 and in the rest of this book, the constant c > 0 depends only on constants related to Lipschitz properties of ., such as .ζ, ζ0 , κ, k L∞ ()d , ∇k L∞ ()d×d , .κi , .i ∈ N, and so forth. In addition, it equally depends on the quantities . ω L∞ (Rd+1 ) and . ∇ω L∞ (Rd+1 )d . .

Remark 4.10 Proposition 4.10 admits congruent extensions to not relabeled h : Lq(·,·) (Q ) → Lq(·,·) (Q ), smoothing operators for scalar functions, i.e., .SQ T R T h q(·,·) (QT )d×d → Lq(·,·) (QR )d×d , and tensor-valued functions, i.e., .SQT : L satisfying (i)–(v) from Proposition 4.10. h x) ∞ d Proof of Proposition 4.10 ad (i). .(SQ h∈(0,h2 ) ⊆ C (QR ) follows using the T d chain rule as .(ν h )h∈[0,1] ⊆C ∞ (R )d (cf. Proposition 4.9) and .(ωζ h ∗ (QT x))h>0 ⊆ C ∞ (Rd+1 )d . Moreover, for every .h ∈ (0, h2 ) and .(t, x) ∈ R × ( \ h ), Proposition 4.9 (i) implies that

Bζdh (ν h (x)) ⊆ ν h ( \ h ) + Bζdh (0) ⊆ O ⊆  ,

.

(4.26)

  ∈ i.e., .Bζd+1 h (t, ν h (x)) ⊆ R ×  . On the other hand, for every .(t, x)

d  (−ζ h, T + ζ h) × , it holds .Bζ1h (t) ⊆ I  , i.e., .Bζd+1 h (t, ν h (x)) ⊆ I × R . Altogether, it holds for every .h ∈ (0, h2 ) and .(t, x) ∈ QR \ ((−ζ h, T + ζ h) × h ) = (R × ( \ h )) ∪ ((−ζ h, T + ζ h) × )8 that

        d Bζd+1 = I ×  = QT . h (t, ν h (x)) ⊆ R ×  ∪ I × R

.

(4.27)

Due to .supp(QT x) ⊆ QT , we conclude from (4.27) for every .(t, x) ∈ QR \ h x)(t, x) = 0 in .Rd . In other ((−ζ h, T + ζ h) × h ) and .h ∈ (0, h2 ) that .(SQ T h words, for every .h ∈ (0, h2 ), it holds .supp(SQT x) ⊆ [−ζ h, T + ζ h] × h ⊆ QR , i.e., (i).

we use the algebraic identity .X × Y \ (A × B) = ((X \ A) × Y ) ∪ (X × (Y \ B)) for sets ⊆ X and .B ⊆ Y .

8 Here, .A

134

4 Solenoidal Variable Bochner–Lebesgue Spaces

 ad (ii). Using . ωh L∞ (Rd+1 ) ≤ hcd+1 , we obtain for almost every .(t, x) ∈ QR and all .h ∈ (0, h2 ) h |(SQT x)(t, x)| ≤ ωζ h (s − t, y − ν h (x))|(QT x)(s, y)| dsdy

Bζd+1 h (t,ν h (x))

.

c ≤ (ζ h)d+1



(4.28)

Bζd+1 h (t,ν h (x))

|(QT x)(s, y)| dsdy .

Since for every .(t, x) ∈ QR and .h ∈ (0, h2 ), for every .(s, y) ∈ Bζd+1 h (t, ν h (x)), it holds |(s, y) − (t, x) | = |(s, y) − (t, ν h (x)) + (0, hk(x)) |

.

≤ |(s, y) −(t, ν h (x)) |+h|k(x)|

(4.29)

< (ζ + k C 0 ()d )h = ζ0 h , d+1 i.e., .Bζd+1 h (t, ν h (x)) ⊆ Bζ0 h (t, x), we further infer from (4.28) for almost every  ∈ Q that .(t, x) R .

h sup |(SQ x)(t, x)| ≤ T

h∈(0,h2 )

c ζ0d+1 d+1 h∈(0,h2 ) ζ sup

Bζd+1 h (t,x)

|(QT x)(s, y)| dsdy

0

≤ c Md+1 (QT x)(t, x) . ad (iii). By the boundedness of .Md+1 : Lq(·,·) (Rd+1 )→ Lq(·,·) (Rd+1 ) (cf. Proposition 2.12), since .q − ≥ q − > 1, and .QT x ∈ Lq(·,·) (Rd+1 ), we conclude from (ii) that .

h sup SQ x Lq(·,·) (Q )d ≤ c Md+1 (QT x) Lq(·,·) (Rd+1 ) T R ≤ cq QT x Lq(·,·) (Rd+1 )d

h∈(0,h2 )

= cq x Lq(·,·) (QT )d . ad (vi). First, assume that .x ∈ C0∞ (Rd+1 )d . Then, for every .(t, x) ∈ QR and .h ∈ (0, h2 ), it holds h |(SQT x)(t, x)−x(t, x)|. ≤ ω(s, y)|x(t +ζ hs, ν h (x)+ζ hy)−x(t, x)| dsdy B1d+1 (0)

≤

B1d+1 (0)

ω(s, y)Lip(x)|(ζ hs, hk(x) + ζ hy) | dsdy

≤ Lip(x)(ζ + k C 0 ()d )h = Lip(x)ζ0 h .

(4.30)

q,p

4.3 Smoothing in Vε

135

(QT )

On the basis the density of .C0∞ (Rd+1 )d in .Lq(·,·) (Rd+1 )d (cf. [49, Thm. 3.4.12.]), where once more .q ∈ Plog (Rd+1 ) is an extension of .q ∈ Plog (QT ), with .q − ≤ q ≤ q + in .Rd+1 , there exists a sequence .(x n )n∈N ⊆ C0∞ (Rd+1 )d such that .x n → QT x in .Lq(·,·) (Rd+1 )d .(n → ∞), i.e., for each .ε > 0, there exists an integer .n0 (ε) ∈ N such that . x n − QT x Lq(·,·) (Rd+1 )d < ε for all .n ∈ N with .n ≥ n0 (ε). Therefore, also making use of (iii) and (4.30), for every .h ∈ (0, h2 ) and .ε > 0, we observe that h h SQ x − x Lq(·,·) (QT )d ≤ SQ (x − x n0 (ε) ) Lq(·,·) (Q )d T T R

.

h + SQ x − x n0 (ε) Lq(·,·) (QT )d T n0 (ε)

+ x n0 (ε) − QT x Lq(·,·) (Rd+1 )d

(4.31)

≤ (cq + 1) x n0 (ε) − QT x Lq(·,·) (Rd+1 )d + 1 Lq(·,·) (QT ) Lip(x n0 (ε) )ζ0 h < (cq + 1)ε + 1 Lq(·,·) (QT ) Lip(x n0 (ε) )ζ0 h . h x − x Consequently, we have that .lim suph→0 SQ Lq(·,·) (QT )d ≤ (cq + 1)ε for T every .ε > 0, i.e., (iv). ad (v). By applying Fubini’s theorem, we observe for arbitrary .x ∈ Lq(·,·) (QT )d , q  (·,·) (Q )d and .h ∈ (0, h ) that .y ∈ L T 2 .

 h     h (SQT ) y, x Lq(·,·) (Q )d = y, SQ x Lq(·,·) (Q )d (4.32) T T T = y(t, x) · ω(s, y)(QT x)(t +ζ hs, ν h (x)+ζ hy) dsdy dtdx =

B1d+1 (0)

QT



B1d+1 (0)

ω(s, y) QT

y(t, x) · (QT x)(t +ζ hs, ν h (x)+ζ hy) dtdx dsdy .

For each .z ∈ Rd+1 and .h ∈ (0, h2 ), consider the mapping . hz : QT → hz .(QT ), defined by hz (t, x) := (t, ν h (x)) + ζ hz

.

in Rd+1

for all (t, x) ∈ QT .

Proposition 4.9 (ii) ensures for fixed .z ∈ Rd+1 and .h ∈ (0, h2 ) that the mapping h h d+1 , 1 . z : QT → z (QT ) is a .C -diffeomorphism. Its Jacobian for every .z ∈ R  .h ∈ (0, h2 ) and .(t, x) ∈ QT is given by  h .(D z )(t, x)

=

1 0 0 Jh (x)

 in R(d+1)×(d+1) ,

136

4 Solenoidal Variable Bochner–Lebesgue Spaces

i.e., we have that .det(D hz ) = det(Jh ) in .QT for every .h ∈ (0, h2 ) and .z ∈ Rd+1 . Therefore, if we use for almost every .z = (s, y) ∈ B1d+1 (0) and every .h ∈ (0, h2 ) the transformation formula with respect to . hz : QT → hz (QT ) in the inner integral in the last line in (4.33), then the latter for almost every .z = (s, y) ∈ B1d+1 (0) and every .h ∈ (0, h2 ) can be rewritten as .

QT

y(t, x) · (QT x)(t + ζ hs, ν h (x) + ζ hy) dtdx = =

QT

y(t, x) · (QT x)( hz (t, x))

QT

det((D hz )(t, x)) det((D hz )(t, x))

dtdx

(y ◦ ( hz )−1 )(τ, η)

dτ dη (det(D hz ) ◦ ( hz )−1 )(τ, η)    y ◦ ( hz )−1 (τ, η) dτ dη , x(τ, η) ·  h (QT ) z det(D hz ) ◦ ( hz )−1

hz (QT )

=

(QT x)(τ, η) ·

(4.33)

where we denote for every .z ∈ B1d+1 (0) by . h (QT ) : L1 ( hz (QT )) → L1 (Rd+1 ) z the zero extension operator outside . hz (QT ). We insert (4.33) into (4.33) and apply again Fubini’s theorem to arrive for every .h ∈ (0, h2 ) at9  h   (SQT ) y, x Lq(·,·) (Q )d T = ω(s, y)

(4.34)

.

B1d+1 (0)

 ·  h





x(τ, η) ·



 ×  h

(s,y) (QT )





=

x(τ, η) · QT

  ×  h (QT ) 0

9 Note

that



.

ω(s, y) y ◦ ( h(s,y) )−1

det(D h(s,y) ) ◦ ( h(s,y) )−1

Bζd+1 h (τ,η)

 (τ, η) dsdy dτ dη

ωζ h (τ −s, η−y)

y ◦ ( h0 )−1

 (s, y) dsdy dτ dη ,

det(D h0 ) ◦ ( h0 )−1

   ω(s, y)x(τ, η) ·  h (s, y) , (τ, η) →

L1 (B1d+1 (0) × QT ).

(τ, η) dτ dη dsdy

det(D h(s,y) ) ◦ ( h(s,y) )−1

B1d+1 (0)

QT



y ◦ ( h(s,y) )−1

(s,y) (QT )

=

x(τ, η)

QT



y◦( h(s,y) )−1 h h −1 (s,y) (QT ) det(D (s,y) )◦( (s,y) )



 (τ, η) ∈

q,p

4.3 Smoothing in Vε

137

(QT )

where we used for the last line the transformation formula with respect to the .C 1 diffeomorphism

h(τ,η) := ((s, y) → ( h0 ◦ ( h(s,y) )−1 )(τ, η))

.

= ((s, y) → (τ, η) − ζ h(s, y) ) : B1d+1 (0) → Bζd+1 h (τ, η) for almost every .(τ, η) ∈ QT and .h ∈ (0, h2 ), and that .det(D h(s,y) ) = det(Jh ) = det(D h0 ) in .QT for every .(s, y) ∈ B1d+1 (0) and .h ∈ (0, h2 ). Since .x ∈ Lq(·,·) (QT )d was chosen arbitrarily in (4.33), we conclude from (4.34) for almost every .(t, x) ∈ QT and every .h ∈ (0, h2 ) that h ) y)(t, x) = ((SQ T

h Bζd+1 h (t,x)∩ 0 (QT )

ωζ h (t − s, x − y)

.

y(( h0 )−1 (s, y)) ((D h0 )(( h0 )−1 (s, y))) dsdy , × det

(4.35)

 i.e., the first representation formula in .(4.25) because .( h0 )−1 (s, y) = (s, ν −1 h (y)) h h d+1  in .R for every .(s, y) ∈ 0 (QT ) = I × ν h () and .det(D 0 ) = det(Jh ) in .QT for every .h ∈ (0, h2 ). If we apply the transformation formula with respect to the .C 1 -diffeomorphism .( h0 )−1 : h0 (QT ) → QT in (4.35), also exploiting that h −1 ) = det((D h )−1 ) ◦ ( h )−1 = det(D h )−1 ◦ ( h )−1 in . h (Q ) for .det(D( ) T 0 0 0 0 0 0 every .h ∈ (0, h2 ), then we observe for almost every .(t, x) ∈ QT and every .h ∈ (0, h2 ) that

.

h  ((SQ ) y)(t, x) T =

h ( h0 )−1 (Bζd+1 h (t,x)∩ 0 (QT ))

ωζ h (t − s, x − ν h (y))y(s, y) dsdy .

(4.36)

h h −1 Also, for every .(t, x) ∈ QT , .(s, y) ∈ ( h0 )−1 (Bζd+1 h (t, x) ∩ 0 (QT )) = ( 0 )

d+1 h  (Bζd+1 h (t, x)) ∩ QT , i.e., we have that . 0 (s, y) ∈ Bζ h (t, x) and .(s, y) ∈ QT , and .h ∈ (0, h2 ), we find that

|(t, x) − (s, y) | = |(t, x) − h0 (s, y) + (0, hk(y)) |

.

< (ζ + k C 0 ()d )h = ζ0 h , d+1 d+1 h h −1 i.e., .( h0 )−1 (Bζd+1 h (t, x) ∩ 0 (QT )) = ( 0 ) (Bζ h (t, x)) ∩ QT ⊆ Bζ0 h (t, x) for every .(t, x) ∈ QT and .h ∈ (0, h2 ). Thus, since, on the other hand, for every

138

4 Solenoidal Variable Bochner–Lebesgue Spaces

(t, x) ∈ QT and .h ∈ (0, h2 ), we have that

.

supp(ωζ h (t − ·, x − ν h (·))QT y) ⊆ ( h0 )−1 (Bζd+1 h (t, x)) ∩ QT ,

.

(4.37)

we conclude from (4.36) to the second representation formula in (4.25). Eventually, the claimed regularity follows from (4.25) by invoking standard properties of mollification.  h ) of .S h Obviously, the quasi adjoint operator .(SQ QT does not possess a T handy representation formula such as, e.g., the quasi adjoint operator .(RhQT ) (cf. Proposition 3.15 (i)) of .RhQT (cf. Proposition 3.14). This entails a number of obstacles which considerably complicate the hereinafter analysis. The major h ) in comparison to .(Rh ) , which will cause the greatest effort, weakness of .(SQ QT T is its inability to preserve spatial compact supports, i.e., to form a self-mapping on ∞ ∞ d .C (I , C () ). 0 h ) According to Proposition 4.10 (i), it holds Remark 4.11 (Major Weakness of .SQ T h (C ∞ (I , C ∞ ()d )) .⊆ C ∞ (I , C ∞ ()d ) for all .h ∈ the inclusion .SQ (0, h2 ). We 0 0 T h ∞ ∞ ∞ d ∞ d have also observed that .RQT (C (I , C0 () )) ⊆ C (I , C0 () ) holds for all .h ∈ (0, h1 ) (cf. Proposition 3.14 (i)). A major strength of the smoothing operator .RhQT is that its quasi adjoint operator .(RhQT ) likewise satisfies .(RhQT ) (C ∞ (I , C0∞ ()d )) ⊆ C ∞ (I , C0∞ ()d ) for every .h ∈ (0, h1 ) (cf. Proposition 3.15 (i)), which can be read directly from the representation formula

((RhQT ) y)(t, x) = ηh (x)

.

Bhd+1 (t,x)

ωh (t − s, x − y)(QT y)(s, y) dsdy ,

for every .y ∈ C ∞ (I , C0∞ ()d ), .(t, x) ∈ QT and .h > 0, where .(ηh )h>0 ⊆ C0∞ () denotes a suitable family of cut-off functions (cf. Proposition 3.14, above) satisfying .supp(ηh ) ⊆ 2h for every .h > 0. h ) (C ∞ (I , C ∞ ()d )) ⊆ Unfortunately, we cannot hope for the inclusion .(SQ 0 T C ∞ (I , C0∞ ()d ) for every .h ∈ (0, h2 ). For instance, let . := B1d (0), .d ≥ 2, .I := (0, T ), .T < ∞, and .QT := I × . Then, we have already observed in Remark 4.8 that a possible choice of a transversal vector field is .k := idRd ∈ C ∞ (Rd )d . Thus, we obtain the expansion mappings .ν h := (1 + h)idRd ∈ C ∞ (Rd )d , .h ∈ [0, 1], and h  q  (·,·) (Q )d → Lq  (·,·) (Q )d , according to Proposition 4.10 (v), .(4.25) , .(S T T 1 QT ) : L 

for every .h ∈ [0, 1], .y ∈ Lq (·,·) (QT )d and almost every .(t, x) ∈ QT can be represented by h  ((SQ ) y)(t, x) = T

.

1 (1 + h)d

Bhd+1 (t,x)

 ωh (t − s, x − y)(QT y) s,

y  dsdy . 1+h

q,p

4.3 Smoothing in Vε

139

(QT )

Using this representation, it is easy to see that for every .h > 0, there exists h ) y ∈ ∞ ∞ d y h ∈ C ∞ (I , C0∞ ()d ) such that .(SQ h / C (I , C0 () ). In fact, if, e.g., T ∞ d d ∞ .y h := φ h e1 ∈ C (I , C () ), where .φ h (t, x) := (1 + h) ω(t, (1 + h)x) for 0 ∞ d+1 d+1  d+1 every .(t, x) ∈ R and .ω := ω ∈ C0 (R ) denotes the standard mollifier h ) y = (ω ∗ ω)e ∈ C ∞ (Rd+1 )d and, hence, defined in Remark 2.6, then .(SQ h 1 h 0 T

.

h ) y ) = B d+1 (0), i.e., .(S h ) y ∈ ∞ ∞ d supp((SQ h / C (I , C0 () ). h QT 1+h T

.

h for its approximation properties with respect to the space Next, we examine .SQ T q,p .Xε (QT ).

Proposition 4.11 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), T < ∞, .QT := I × , and .q, p ∈ Plog (QT ) with .p− > 1 and .q ≥ p in .QT . Then, q,p for every .x ∈ Xε (QT ), it holds:

.

(i) There exist error functions .(Eh (x))h∈(0,h2 ) ⊆ C0∞ (QR , Md×d sym ) such that for every .h ∈ (0, h2 ), it holds h h ε(SQ x) = SQ (ε(x)) + Eh (x) T T

in QR .

.

(ii) There exists a constant .c > 0 (not depending on .q, p ∈ Plog (QT ) (cf. Remark 4.9)) such that .

sup |Eh (x)| ≤ c Md+1 (QT x)

a.e. in QR ,

h∈(0,h2 )

h x)| ≤ c M suph∈(0,h2 ) |ε(SQ  d+1 (QT x) T .

+Md+1 (QT ε(x))

(4.38)

(4.39) a.e. in QR .

(iii) For an extension .p ∈ Plog (Rd+1 ) of .p ∈ Plog (QT ), with .p− ≤ p ≤ p+ in .Rd+1 , there exists a constant .cp > 0 (depending on .p ∈ Plog (Rd+1 ) and .c > 0) such that .

sup Eh (x) Lp(·,·) (Q )d×d ≤ cp x Lp(·,·) (QT )d , R

(4.40)

h∈(0,h2 )

h x) suph∈(0,h2 ) ε(SQ Lp(·,·) (Q )d×d ≤ cp x Lp(·,·) (QT )d T R .

+ ε(x) Lp(·,·) (QT )d×d .

(4.41)

h x q,p In particular, .suph∈(0,h2 ) SQ Xε (QT ) ≤ max{cq , 2(1 + |QT |)cp } T . x q,p , where .cq > 0 is from Proposition 4.10 (iii). Xε (QT ) q,p

h x → x in .X (iv) .SQ ε (QT ) .(h → 0). T

140

4 Solenoidal Variable Bochner–Lebesgue Spaces

h x ∈ C ∞ (Q )d Proof ad (i). We need to calculate the symmetric gradient of .SQ R 0 T for .h ∈ (0, h2 ). In doing so, using differentiation under the integral sign, applying the chain rule for the symmetric gradient, i.e., .ε(x ◦ ) = 12 (∇ (∇x ◦ ) + (Dx ◦ )D )10 in .L1 ()d×d for .x ∈ W 1,1 ( ())d and a .C 1 -diffeomorphism d . :  → () ⊆ R , also using that .ε(QT x) = QT ε(x) almost everywhere in d+1 .R , for every .(t, x) ∈ QR and .h ∈ (0, h2 ), we find that h ε(SQ x)(t, x) = T

.

1 2

B1d+1 (0)

ω(s, y)Jh (x)

(4.42)

× (QT ∇x)(t + ζ hs, ν h (x) + ζ hy) dsdy 1 + ω(s, y) 2 B1d+1 (0) × (QT Dx)(t + ζ hs, ν h (x) + ζ hy)Jh (x) dsdy h (ε(x)))(t, x) = (SQ T 1  ω(s, y) + (Jh (x) − Id ) 2 B1d+1 (0)

× (QT ∇x)(t + ζ hs, ν h (x) + ζ hy) dsdy 1 + ω(s, y) 2 B1d+1 (0) × (QT Dx)(t + ζ hs, ν h (x) + ζ hy)dsdy(Jh (x) − Id ) h (ε(x)))(t, x) + Eh (x)(t, x) . =: (SQ T

Then, from the representation formula .(4.42)4,5 it can easily be read that (Eh (x))h∈(0,h2 ) ⊆ C0∞ (QR , Md×d sym ). ad (ii). Integration-by-parts in space in .(4.42)4,5 gives for every .(t, x) ∈ QR and .h ∈ (0, h2 ) that11 1  .Eh (x)(t, x) = ω(s, y) (Jh (x) − Id ) 2 B1d+1 (0)   1 × ∇y (QT x)(t + ζ hs, ν h (x) + ζ hy) dsdy ζh

.

= (∂j ui )i,j =1,...,n : G ⊆ Rn → Rn×n , .n ∈ N, denotes the Jacobian and .∇u = (∂i uj )i,j =1,...,n : G ⊆ Rn → Rn×n the gradient of a vector-valued function .u = (u1 , . . . , un ) : G ⊆ Rn → Rn (cf. Notation-related comments). 11 Here, .∇ and .D denote the gradient and the Jacobian, respectively, with respect to the spatial y y variable y. 10 .Du

q,p

4.3 Smoothing in Vε

1 2

+

141

(QT )

B1d+1 (0)

ω(s, y)



 1 (QT x)(t + ζ hs, ν h (x) + ζ hy) dsdy (Jh (x) − Id ) ×Dy ζh 1 =− ∇ω(s, y) (4.43) (Jh (x) − Id ) 2ζ h B1d+1 (0) ⊗ (QT x)(t + ζ hs, ν h (x) + ζ hy) dsdy 1 − (QT x)(t + ζ hs, ν h (x) + ζ hy) 2ζ h B1d+1 (0) ⊗ ∇ω(s, y) dsdy (Jh (x) − Id ) 1 =− (Jh (x) − Id ) d+1 ζ h Bζ h (t,ν h (x)) × (∇ω)ζ h (s − t, y − ν h (x)) × ⊗(QT x)(s, y)

sym

dsdy .

 Using (4.43), . (∇ω)h L∞ (Rd+1 )d ≤ hcd+1 , . J h − Id L∞ ()d×d ≤ κ0 h (cf. Proposi-

d+1  ∈ Q tion 4.9 (iii)) and .Bζd+1 R and h (t, ν h (x)) ⊆ Bζ0 h (t, x) for every .(t, x)  .h ∈ (0, h2 ) (cf. (4.29)), we then conclude for almost every .(t, x) ∈ QR and every .h ∈ (0, h2 ) that

|Eh (x)(t, x)| ≤

.

J h − Id L∞ ()d×d ζh

×

Bζd+1 h (t,ν h (x))

κ0 h c ≤ ζ h (ζ h)d+1 ≤

κ0 c ζ0d+1 ζ d+2

|(∇ω)ζ h (s − t, y − ν h (x))||(QT x)(s, y)| dsdy



Bζd+1 h (t,ν h (x))

Bζd+1 h (t,x)

|(QT x)(s, y)| dsdy

|(QT x)(s, y)| dsdy

0

≤ c Md+1 (QT x)(t, x), i.e., (4.38). In addition, from (i), (4.38), Proposition 4.10 (ii), and Remark 4.10, it follows (4.39). ad (iii). As in the proof of Proposition 4.10 (iii), i.e., making use of (4.38) and Proposition 2.12, we conclude (4.40). Then, (4.41) follows from (i), (4.40), Proposition 4.10 (iii), and Remark 4.10. ad (iv). By resorting to Proposition 4.10 (iv), we immediately observe that h SQ x→x T

.

in Lq(·,·) (QT )d

(h → 0) .

(4.44)

142

4 Solenoidal Variable Bochner–Lebesgue Spaces

h x) → ε(x) in .Lp(·,·) (Q , Md×d ) .(h → 0). To It remains to establish that .ε(SQ T sym T h x ∈ C ∞ (Q )d for .h ∈ this end, we first calculate the full gradient of .SQ (0, h2 ). R 0 T In doing so, for every .h ∈ (0, h2 ), we obtain h h ∇(SQ x) = J h SQT (∇x) T

.

in QR .

(4.45)

Next, using (4.45), Proposition 4.9 (iii), Proposition 4.10 (iii) and (iv), and Remark 4.10, we find that h h ε(SQ x) − ε(x) Lp− (QT )d×d ≤ ∇(SQ x) − ∇x Lp− (QT )d×d T T

.

h ≤ J h −Id L∞ ()d×d SQT (∇x) Lp− (QT )d×d h + SQ (∇x)−∇x Lp− (QT )d×d T

(4.46)

h h ≤ κ0 h SQ (∇x) Lp− (QT )d×d + SQ (∇x) − ∇x Lp− (QT )d×d T T

→ 0 (h → 0) . As a result of (4.46), Proposition 2.2 provides a cofinal subset . ⊆ R>0 such that h ε(SQ x) → ε(x) T

.

a.e. in QT

(h−1 ∈ ; h → 0) .

(4.47)

Based on (4.39) and .Md+1 (QT ε(x)), Md+1 (QT x) ∈ Lp(·,·) (Rd+1 ) (cf. Proposih x)) p(·,·) (Q , Md×d ) tion 2.12), as .q ≥ p in .QT and .p− > 1, .(ε(SQ h∈(0,h2 ) ⊆ L T sym T has an .Lp(·,·) (QT )-integrable majorant. Thus, using (4.47), we conclude, with the aid of Lebesgue’s theorem on dominated convergence (cf. Proposition 2.4), that h p(·,·) ε(S . (QT , Md×d sym ) QT x) → ε(x) in L

(h−1 ∈ ; h → 0) .

(4.48)

The standard convergence principle (cf. [166, Prop. 10.13 (1)]) guarantees that (4.48) even holds if . = R>0 . Eventually, (4.44) and (4.48) with . = R>0 prove q,p h .S  QT x → x in Xε (QT ) .(h → 0). We emphasize that the requirement that the variable exponents .q, p ∈ Plog (QT ) satisfy .q ≥ p in .QT is solely imposed in Proposition 4.11 to be capable q,p of controlling the symmetric gradient. In fact, for the space .X∇ (QT ), this assumption can be omitted and the smoothing operator from Proposition 4.10 provides better results, even better results than the smoothing operator from Chap. 3, Sect. 3.4, in the sense that we gain convergence h x) − S h (∇x) ∈ C ∞ (Q )d×d , rates for commutativity errors .Eh (x) = ∇(SQ R QT 0 T p(·,·) (Q )d×d , which is not clear –or even possible– for .h ∈ (0, h2 ), in .L R h x) − Rh (∇x) ∈ C ∞ (Rd+1 )d×d , .h ∈ .ω h ∗ (∇ηh ⊗ x) = ∇(R (0, h1 ). QT QT 0

q,p

4.3 Smoothing in Vε

143

(QT )

Proposition 4.12 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .q − , p− > 1. Then, for every q,p .x ∈ X ∇ (QT ), it holds: (i) There exist error functions .(Eh (x))h∈(0,h2 ) ⊆ C0∞ (QR )d×d such that for every .h ∈ (0, h2 ), it holds h h ∇(SQ x) = SQ (∇x) + Eh (x) T T

in QR .

.

(ii) There exists a constant .c > 0 (not depending on .q, p ∈ Plog (QT ) (cf. Remark 4.9)) such that for every .h ∈ (0, h2 ), it holds |Eh (x)| ≤ c hMd+1 (QT ∇x)

.

a.e. in QR .

(iii) For an extension .p ∈ Plog (Rd+1 ) of .p ∈ Plog (QT ), with .p− ≤ p ≤ p+ in .Rd+1 , there exists a constant .cp > 0 (depending on .p ∈ Plog (Rd+1 ) and .c > 0) such that for every .h ∈ (0, h2 ), it holds Eh (x) Lp(·,·) (Q )d×d ≤ cp h ∇x Lp(·,·) (QT )d×d . R

.

Proof ad (i). In (4.45), we have established that for every .h ∈ (0, h2 ), it holds h h h ∇(SQ x) = SQ (∇x) + (J h − Id )SQT (∇x) T T

.

h =: SQ (∇x) + Eh (x) T

in QR .

Proposition 4.9, Proposition 4.10 (i) and Remark 4.10 ensure that .(Eh (x))h∈(0,h2 ) ⊆ C0∞ (QR )d×d . ad (ii). Taking advantage of . J h − Id L∞ ()d×d ≤ κ0 h (cf. Proposition 4.9 (iii)), Proposition 4.10 (ii), and Remark 4.10, we observe for almost every .(t, x) ∈ QR and every .h ∈ (0, h2 ) that h |Eh (x)(t, x)|. ≤ J h − Id L∞ ()d×d |SQT (∇x)(t, x)|

≤ c hMd+1 (QT ∇x)(t, x) . ad (iii). Follows as in the proof of Proposition 4.10 (iii) from (ii) by means of  Proposition 2.12.

144

4 Solenoidal Variable Bochner–Lebesgue Spaces

In the same manner as in Chap. 3, Sect. 3.4, we next modify the smoothing h operator .SQ to obtain a smoothing operator that again yields the density of T q,p ∞ d .C (QT ) in .Xε (QT ). 0 Proposition 4.13 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .p− > 1 and .q ≥ p in ∞ .QT . Furthermore, let .(ϕh )h∈I◦ ⊆ C (I ) be the family of cut-off functions from 0q,p Proposition 3.18. For every .x ∈ Xε (QT ) and .h ∈ (0, h2 ) ∩ I◦ζ , where .I◦ζ :=  T 0, 4ζ , we define the smoothing operator ˚h x := S h (ϕζ h x) ∈ C ∞ (Q )d . S 0 QT QT R

.

q,p

Then, for every .x ∈ Xε (QT ), it holds: ˚h x)h∈(0,h )∩I ζ ⊆ C ∞ (QT )d with .supp(S ˚h x) ⊆ Iζ h ×  12 for every (i) .(S h 2 QT QT 0 ◦ ζ .h ∈ (0, h2 ) ∩ I◦ . (ii) There exists a constant .c > 0 (not depending on .q, p ∈ Plog (QT ) (cf. Remark 4.9)) such that sup

.

h∈(0,h2 )∩I◦ζ

.

˚h x| |S QT

≤ c Md+1 (QT x)

a.e. in QT ,

(4.49)

+Md+1 (QT ε(x)) a.e. in QT .

(4.50)

˚h x)| ≤ c Md+1 (Q x) suph∈(0,h2 )∩I◦ζ |ε(S T QT

(iii) For extensions .q, p ∈ Plog (Rd+1 ) of .q, p ∈ Plog (QT ), resp., with .q − ≤ q ≤ q + and .p− ≤ p ≤ p+ in .Rd+1 , there exist constants .cq , cp > 0 (depending on .q, p ∈ Plog (Rd+1 ) and .c > 0, resp.) such that ˚h x q(·,·) S (Q )d ≤ cq x Lq(·,·) (QT )d , . L QT R ˚h x) p(·,·) sup ε(S (Q )d×d ≤ cp x Lp(·,·) (QT )d L QT R h∈(0,h2 )∩I◦ζ .

sup

(4.51)

h∈(0,h2 )∩I◦ζ

(4.52)

+ ε(x) Lp(·,·) (QT )d×d .

˚h x → x in Xεq,p (QT ) .(h → 0). (iv) .S QT ˚h x)h∈(0,h )∩I ζ Proof ad (i). Using Proposition 4.10 (i), we observe that .(S 2 QT ◦

 ⊆ C0∞ (QR )d . In addition, recalling (4.26), it holds .Bζd+1 h (t, ν h (x)) ⊆ R ×  for

.

12 Again, .I h

:= (h, T − h) for every .h ∈ (0,

T 2

) (cf. Proposition 3.18).

q,p

4.3 Smoothing in Vε

145

(QT )

every .(t, x) ∈ R × ( \ h ) and .h ∈ (0, h2 ) ∩ I◦ζ . On the other hand, it holds for  13 i.e., .B d+1 (t, ν (x)) ⊆ every .t ∈ Iζh and .h ∈ (0, h2 ) ∩ I◦ζ that .Bζ1h (t) ⊆ I2ζ h ζh h,   d  I2ζ h × R for every .(t, x) ∈ Iζ h × . Putting it all together, for every  ∈ Q \ (I ×  ) = (R × ( \  )) ∪ (I  × ) and .h ∈ (0, h ) ∩ I ζ , .(t, x) h h 2 ζh ◦ R ζh we have that         d = I2ζ h ×  . Bζd+1 h (t, ν h (x)) ⊆ R ×  ∪ I2ζ h × R

(4.53)

.

Because of .supp(QT xϕζ h ) ⊆ I2ζ h × , we conclude from (4.53) for every  ∈ Q \ (I ×  ) and .h ∈ (0, h ) ∩ I ζ that .(S ˚h x)(t, x) = 0 in .Rd . In .(t, x) ζh h 2 ◦ R QT ζ ˚h x) ⊆ Iζ h ×  ⊆ QT , other words, for every .h ∈ (0, h2 ) ∩ I , it holds .supp(S ◦

QT

h

˚h x)h∈(0,h )∩I ζ ⊆ C ∞ (QT )d . i.e., .(S 2 QT 0 ◦ ad (ii) and (iii). We fall back on Proposition 4.10 (ii) and (iii) as well as Proposition 4.11 (ii) and (iii), and exploit that .suph∈I◦ ϕh L∞ (R) ≤ 1. ad (iv). By applying Proposition 4.11 (iii) and (iv) and Proposition 2.4, we immediately obtain h h ˚h x − x q,p q,p q,p S QT Xε (QT ) ≤ SQT [ϕζ h x − x] Xε (QT ) + SQT x − x Xε (QT )

.

≤ max{cq , 2(1 + |QT |)cp } (1 − ϕζ h )x Xεq,p (QT ) h + SQ x − x Xεq,p (QT ) → 0 T

(h → 0) .



Remark 4.12 We emphasize that Proposition 4.13 does not represent a gain in q,p knowledge because we already determined the density of .C0∞ (QT )d in .Xε (QT ) without the assumption that .q, p ∈ Plog (QT ) are subject to the restrictive relation .q ≥ p in .QT (cf. Proposition 3.18). Nonetheless, we will later fall back on ∞ (Q )d lies densely in .V q,p (Q ), where Proposition 4.13 when we prove that .C0,div T T ε   ∞ C0,div (Q)d := φ ∈ C0∞ (Q)d | div(φ) = 0 in Q

.

(4.54)

for a domain .Q ⊆ Rd+1 , for which we will have to impose the restrictive relation .q ≥ p in .QT . h from Proposition 4.10 to obtain Next, let us modify the smoothing operator .SQ T q,p a smoothing operator for the space .Vε (QT ). This will be obtained by means of a contravariant Piola transform, whose fundamental properties are briefly summarized in the following lemma.

13 The

assumption .h ∈ I◦ζ guarantees that .I2ζ h = ∅.

146

4 Solenoidal Variable Bochner–Lebesgue Spaces

Lemma 4.2 (Contravariant Piola Transform) Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain and let .(ν h )h∈[0,1] ⊆ C ∞ (Rd )d be the transversal expansion mappings provided by Proposition 4.9. For every .h ∈ (0, h2 ), we define the tensorvalued function14  ∞ d×d Ph := det(Jh )J−1 . h = cof(Jh ) ∈ C ()

.

 1  3κ1 , κ0 (2 + d 2 ) > 015 and for every .h ∈ (0, h2 ), Then, for .cP := max 1−h2 2 κ , 1−h 2κ the following applies: (i) .div(P h ) = 0 in .. (ii) . Ph L∞ ()d×d ≤ cP . (iii) . ∇Ph L∞ ()d 3 + Ph − Id L∞ ()d×d ≤ cP h. Proof ad (i). Results from .div(P h ) = div(cof(Jh )) = div(cof(Dν h )) in . for every ∞ d .h ∈ (0, h2 ), the crucial fact that .div(cof(Dx)) = 0 in . for every .x ∈ C () d ∞ d (cf. [149, Kap. I, Lem. 2.9]), and the smoothness of .(ν h )h∈(0,h2 ) ⊆ C (R ) (cf. Proposition 4.9). ad (ii). We combine −1 −1 J−1 h L∞ ()d×d = D(ν h ) L∞ (ν h ())d×d ≤ Lip(ν h |ν h () ) ≤

.

1 1 − h2 κ

and det(Jh ) L∞ () ≤ 2

.

for every .h ∈ (0, h2 ) (cf. Proposition 4.9 (ii)). ad (iii). By applying the product rule and Proposition 4.9 (ii) and (iii), for every .h ∈ (0, h2 ), we obtain ∇Ph L∞ ()d 3 ≤ ∇ det(Jh ) L∞ ()d J−1 h L∞ ()d×d

.

+ det(Jh ) L∞ () ∇(J−1 h ) L∞ ()d 3 ≤

κ1 h + 2κ1 h 1 − h2 κ

≤

3κ1 h ≤ cP h . 1 − h2 κ

∈ Rn×n , .n ∈ N, denotes the cofactor matrix of a matrix .A ∈ Rn×n , i.e., := (−1)i+j det(Aij ) for .i, j = 1, . . . , n, where .Aij ∈ R(n−1)×(n−1) results from n×n , if we omit the i-th. row and j -th. column. In particular, for every .A ∈ Rn×n with .A ∈ R − . .det(A) = 0, it holds the relation .cof(A) = det(A)A 15 Note that .h ∈ (0, 1 ) (cf. Proposition 4.9 (ii)) and, therefore, .1 − h κ > 0. 2 2 k 14 Here, .cof(A)

.(cof(A))ij

q,p

4.3 Smoothing in Vε

147

(QT )

Using again Proposition 4.9 (ii) & (iii), for every .h ∈ (0, h2 ), we observe that Ph − Id L∞ ()d×d ≤ det(Jh ) L∞ () J−1 h − Id L∞ ()d×d

.

+ det(Jh ) − 1 L∞ () |Id | 1

≤ 2κ0 h + κ0 d 2 h ≤ cP h . Putting it all together, we conclude that the assertion (iii) holds.



Proposition 4.14 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .p− > 1 and .q ≥ p in q,p .QT . For every .u ∈ Vε (QT ) and .h ∈ (0, h2 ), we define the divergence-preserving transversal expansion smoothing operator by h h SQ u := Ph SQ u ∈ C0∞ (QR )d . T ,σ T

.

q,p

Then, for every .u ∈ Vε

(QT ), it holds:

h ∞ (Q )d with .supp(S h (i) .(SQ u)h∈(0,h2 ) ⊆ C0,div R QT ,σ u) ⊆ [−ζ h, T + ζ h]×h T ,σ for every .h ∈ (0, h2 ). (ii) There exists a constant .c > 0 (not depending on .q, p ∈ Plog (QT ) (cf. Remark 4.9)) such that .

h sup |SQ u| ≤ c Md+1 (QT u) T ,σ

a.e. in QR ,

h∈(0,h2 )

.

h suph∈(0,h2 ) |ε(SQ u)| ≤ c Md+1 (QT u) T ,σ +Md+1 (QT ε(u))

(4.55)

(4.56) a.e. in QR .

(iii) For extensions .q, p ∈ Plog (Rd+1 ) of .q, p ∈ Plog (QT ), resp., with .q − ≤ q ≤ q + and .p− ≤ p ≤ p+ in .Rd+1 , there exist constants .cq , cp > 0 (depending on .q, p ∈ Plog (Rd+1 ) and .c > 0, resp.) such that .

h sup SQ u Lq(·,·) (Q )d ≤ cq u Lq(·,·) (QT )d , T ,σ R

(4.57)

h∈(0,h2 )

h suph∈(0,h2 ) ε(SQ u) Lp(·,·) (Q )d×d ≤ cp u Lp(·,·) (QT )d T ,σ R .

+ ε(u) Lp(·,·) (QT )d×d . In particular, . u q,p Vε (QT ) .

(4.58)

h suph∈(0,h2 ) SQ u Vεq,p (QT ) ≤ max{cq , 2(1 + |QT |)cp } T ,σ

.

q,p

h (iv) .SQ u → u in Vε T ,σ q,p in .Vε (QT ).

(QT ) .(h → 0), i.e., .C ∞ (I , V) (cf. (4.22)) lies densely

148

4 Solenoidal Variable Bochner–Lebesgue Spaces

h u) ∞ d Proof ad (i). Proposition 4.10 (i) implies that .(SQ h∈(0,h2 ) ⊆ C0 (QR ) T h u) ⊆ with .supp(SQ [−ζ h, T + ζ h] × h for all .h ∈ (0, h2 ). As a consequence, T h by Leibniz’s product rule, we have that .(SQ u)h∈(0,h2 ) ⊆ C0∞ (QR )d with T ,σ h .supp(S QT ,σ u) ⊆ [−ζ h, T + ζ h] × h for all .h ∈ (0, h2 ). More precisely, using the product rule .div(Ax) = div(A ) · x + A : ∇x in .C 0 () for every .A ∈ C 1 ()d×d h u, in particular, making use of (4.45) and .x ∈ C 1 ()d , for .A = Ph and .x = SQ T and Lemma 4.2 (i), we find that h h h div(SQ u) = div(P h ) · SQT u + Ph : ∇(SQT u) T ,σ .

h = 0 + tr(P h ∇(SQT u))

(4.59)

 h = det(Jh )tr(J− h Jh SQT (∇u)) h (div(u)) = 0 = det(Jh )SQ T

in QR ,

h ∞ (Q )d . for every .h ∈ (0, h2 ), i.e., we have that .(SQ u)h∈(0,h2 ) ⊆ C0,div R T ,σ ad (ii). By applying Proposition 4.10 (ii) and Lemma 4.2 (ii), we get (4.55). For h ∞ (Q )d for (4.56), we need to calculate the symmetric gradient of .SQ u ∈ C0,div R T ,σ any .h ∈ (0, h2 ). In doing so, using the product rule .∇(Au) = (∇A)x + (∇x)A 3 in .C 0 ()d×d for every .A ∈ C 1 ()d×d and .x ∈ C 1 ()d , where .∇A ∈ C 1 ()d is defined by .(∇A)x := ((∂1 A)x, . . . , (∂d A)x) ∈ C 0 ()d×d for every .x ∈ Rd , for h .A = Ph and .x = S QT ,σ u, for every .h ∈ (0, h2 ), we observe that



h h u sym + ε(S h u) ε(SQ u) = (∇Ph )SQ QT T ,σ T .

sym h  + ∇(SQT u)(Ph − Id ) in QR .

(4.60)

Using Proposition 4.10 (ii), Proposition 4.11 (ii) and Lemma 4.2 (iii), we obtain a constant .c > 0 (not depending on .q, p ∈ Plog (QT ) (cf. Remark 4.9)) such that for every .h ∈ (0, h2 ), it holds ⎫ h h ⎪ |[(∇Ph )SQ u]sym | + |ε(SQ u)| ⎪ T T ⎪ ⎬ h h in QR . . (4.61) ≤ cP h|SQT u| + |ε(SQT u)| ⎪

⎪ ⎪ ≤ c Md+1 (QT u) + Md+1 (QT ε(u)) ⎭ It is still pending to estimate the third term in (4.60). In the same way as in the derivation of (4.43), for almost every .(t, x) ∈ QR and every .h ∈ (0, h2 ), we calculate that h ∇(SQ u)(t, x) T . Jh (x) (∇ω)ζ h (s −t, y −ν h (x)) ⊗ (QT u)(s, y) dsdy . =− ζ h Bζd+1 h (t,ν h (x))

(4.62)

q,p

4.3 Smoothing in Vε

149

(QT )

Since . J h L∞ ()d×d = Jh L∞ ()d×d = Lip(ν h ) ≤ 1 + hκ (cf. Proposition 4.9 d+1  , .Bζd+1 (iii)), . (∇ω)h L∞ (Rd+1 )d ≤ hcd+1 h (t, ν h (x)) ⊆ Bζ0 h (t, x) (cf. (4.29)), and P h − Id L∞ ()d×d ≤ cP h (cf. Lemma 4.2 (iii)), we infer from (4.62) for almost every .(t, x) ∈ QT and every .h ∈ (0, h2 ) that

.

h |∇(SQ u)(t, x)(Ph (x) − Id )| T

.

≤ cP h

(1 + h2 κ) c ζ0d+1 ζh ζ d+1

Bζd+1 h (t,x)

|(QT u)(s, y)| dsdy

(4.63)

0

≤ cP

(1 + h2 κ)c ζ0d+1 Md+1 (QT u)(t, x) . ζ d+2

By combining (4.60), (4.61) and (4.63), we then conclude (4.56). To sum up, we have verified (ii). ad (iii). Follows as in the proof of Proposition 4.10 (iii) from (ii) by means of Proposition 2.12. ad (iv). Falling back on Lemma 4.2 (iii) and Proposition 4.10 (iv), we obtain h h u SQ u − u Lq(·,·) (QT )d ≤ Ph − Id L∞ ()d×d SQ Lq(·,·) (QT )d T ,σ T h u − u + SQ Lq(·,·) (QT )d T .

h u ≤ cP h SQ Lq(·,·) (QT )d T

(4.64)

h u − u + SQ Lq(·,·) (QT )d T

→0

(h → 0) .

With the use of Lemma 4.2 (iii), Proposition 4.10 (iv), Proposition 4.11 (iv), q − ,p− Proposition 4.12,− and (4.60), because likewise .u ∈ Vε (Q T ), owing to Corolq ,p− q − ,p− lary 2.1, and .Vε (QT ) = V∇ (QT ), based on Korn’s inequality (cf. Proposition 2.13) for the exponent .p− ∈ (1, ∞), we conclude that h h ε(SQ u) − ε(u) Lp− (QT )d×d ≤ ∇Ph L∞ ()d 3 SQ u Lp− (QT )d T ,σ T

.

h + ε(SQ u) − ε(u) Lp− (QT )d×d T h + P h − Id L∞ ()d×d ∇(SQT u) Lp− (QT )d×d h ≤ cP h SQ u Lp− (QT )d T h + ε(SQ u) − ε(u) Lp− (QT )d×d T h + cP h ∇(SQ u) Lp− (QT )d×d T

→ 0 (h → 0) .

(4.65)

150

4 Solenoidal Variable Bochner–Lebesgue Spaces

h By combining (4.65) and Proposition 2.2, we obtain .ε(SQ u) → ε(u) in .Md×d sym T ,σ −1 ∈ ; h → 0) almost everywhere in .Q for a cofinal subset . ⊆ R . .(h T >0 Therefore, by virtue of the estimate (4.56), Lebesgue’s theorem on dominated convergence (cf. Proposition 2.4) gives us h ε(SQ u) → ε(u) T ,σ

.

in Lp(·,·) (QT , Md×d sym )

(h−1 ∈ ; h → 0) .

(4.66)

The standard convergence principle (cf. [166, Prop. 10.13 (1)]) guarantees that (4.66) even holds if . = R>0 . Eventually, (4.64) and (4.66) with . = R>0 prove q,p h .S  QT ,σ u → u in .Vε (QT ) .(h → 0). The same methods as in the proof of Proposition 4.14 can be used in order to con˚εq(·),p(·) () = V ˚q(·),p(·) (), pro˚q,p = V struct a smoothing operator for the space .V ∇ vided that . ⊆ Rd , .d ≥ 2, is a bounded Lipschitz domain and that .q, p ∈ Plog () ˚q,p → Lp(·) (), with .q − , p− > 1. In particular, note that, owing the embedding .V P −K q,p ˚ (cf. Proposition 3.5 i.e., . u Lp(·) () ≤ cp ε(u) Lp(·) ()d×d for every .u ∈ V and Proposition 2.13), the assumption .q ≥ p in . can be omitted in this case. Corollary 4.3 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, let ∞ d d be the transversal expansion mappings provided by .(ν h )h∈[0,1] ⊆ C (R ) ˚q,p and Proposition 4.9, and .q, p ∈ Plog () with .q − , p− > 1. For every .u ∈ V .h ∈ (0, h2 ), we define the smoothing operator  d  h ∞ d d .S,σ u := Ph (ωζ h ∗ ℰ u) ◦ ν h ∈ C (R ) , where .(ωhd )h>0 ⊆ C0∞ (Rd ) are the scaled standard mollifiers from Remark 2.6 and 1 d d  .ℰ u ∈ L (R ) is defined by .ℰ u := u in . and .ℰ u := 0 in . . Then, for every q,p ˚ , it holds: .u ∈ V (i) .(Sh,σ u)h∈(0,h2 ) ⊆ V with .supp(Sh,σ u) ⊆ h for every .h ∈ (0, h2 ). (ii) There exists a constant .c > 0 (not depending on .q, p ∈ Plog () (cf. Remark 4.9)) such that .

sup |Sh,σ u| ≤ c Md (ℰ u)

a.e. in  , .

(4.67)

a.e. in  .

(4.68)

h∈(0,h2 )



sup |ε(Sh,σ u)| ≤ c Md (ℰ u) + Md (ℰ ε(u))

h∈(0,h2 )

(iii) For extensions .q, p ∈ Plog (Rd ) of .q, p ∈ Plog (), resp., with .q − ≤ q ≤ q + and .p− ≤ p ≤ p+ in .Rd , there exist constants .cq , cp > 0 (depending on log d .q, p ∈ P (R ) and .c > 0, resp.) such that .

sup Sh,σ u Lq(·) ()d ≤ cq u Lq(·) ()d ,

h∈(0,h2 )



sup ε(Sh,σ u) Lp(·) ()d×d ≤ cp u Lp(·) ()d + ε(u) Lp(·) ()d×d .

h∈(0,h2 )

˚q,p .(h → 0). (iv) .Sh,σ u → u in V

q,p

4.3 Smoothing in Vε

151

(QT )

Note that, on the basis of Proposition 4.14 (iii), for an in time extended cylinder T := I˜ × , where .I˜ ⊇ I is a bounded interval, and for in time extended Q T ), satisfying .q − ≤ q˜ ≤ q + and .p− ≤ p˜ ≤ p+ in .Q T , exponents .q, ˜ p˜ ∈ Plog (Q q,p q, ˜ p˜  h the smoothing operators .SQT ,σ : Vε (QT ) → Vε (QT ), .h ∈ (0, h2 ), remain well-defined, linear and bounded. Hence, their corresponding adjoint operators q, ˜ p˜  ∗ q,p h ∗ ∗ .(S QT ,σ ) : Vε (QT ) → Vε (QT ) , .h ∈ (0, h2 ), defined by .

 .

 h (SQ )∗ u˜ ∗ , u V q,p (Q T ,σ

T)

ε

q, ˜ p˜

  h := u˜ ∗ , SQ u V q,˜ p˜ (Q  T ,σ

T)

ε

T )∗ , .u ∈ Vε (QT ) and .h ∈ (0, h2 ), are also well-defined, for every .u˜ ∗ ∈ Vε (Q linear and bounded. The following proposition demonstrates that these adjoint operators inherit certain approximative properties from the smoothing operators q,p q, ˜ p˜  h .S QT ,σ : Vε (QT ) → Vε (QT ), .h ∈ (0, h2 ). q,p

Proposition 4.15 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .p− > 1 and .q ≥ p in .QT . Then, q, ˜ p ˜ T )∗ , it holds: for every .u˜ ∗ ∈ Vε (Q q, ˜ p˜

T )∗ → Vε (i) If .∗QT : Vε (Q

(QT )∗ denotes the adjoint operator q,p q, ˜ p˜  Vε (QT ) → Vε (Q T ), for every .u ∈ q,p

extension operator .QT : T \ QT , then defined by .QT u := u in .QT and .QT u := 0 in .Q h (SQ )∗ u˜ ∗  ∗QT u˜ ∗ T ,σ

.

q,p

in Vε

(QT )∗

of the zero q,p

Vε

(QT )

(h → 0) .

T ), resp., with ˜ p˜ ∈ Plog (Q (ii) For extensions .q, p ∈ Plog (Rd+1 ) of .q, d+1 + − + − .q ≤ q ≤ q and .p ≤ p ≤ p in .R , there exists a constant .cq,p > 0 (depending on .q, p ∈ Plog (Rd+1 )) such that .

h sup (SQ )∗ u˜ ∗ Vεq,p (QT )∗ ≤ cq,p u˜ ∗ V q,˜ p˜ (Q  T ,σ ε

h∈(0,h2 )

T)

∗

.

q,p

Proof ad (i). Let .u ∈ Vε (QT ) be arbitrary. Then, if we use Proposition 4.14 h (iv) together with Proposition 2.2 and .supp(SQ u) ⊆ [−ζ h, T + ζ h] × h for T ,σ every .h ∈ (0, h2 ) (cf. Prop. 4.14 (i)), then we deduce the existence of a cofinal subset . ⊆ R>0 such that (h−1 ∈ ; h → 0)

T , a.e. in Q

h u) → QT ε(u) (h−1 ∈ ; h → 0) ε(SQ T ,σ

T . a.e. in Q

h SQ u → QT u T ,σ .

(4.69)

152

4 Solenoidal Variable Bochner–Lebesgue Spaces

Then, Proposition 4.14 (ii) and (4.69) in combination with Proposition 2.4, let us deduce that q, ˜ p˜

T ) in Vε (Q

h SQ u → QT u T ,σ

.

(h−1 ∈ ; h → 0) .

(4.70)

The standard convergence principle (cf. [166, Prop. 10.13 (1)]) guarantees, in addition, that (4.70) even holds if . = R>0 . Eventually, we conclude from (4.70) with . = R>0 to  .

 h (SQ )∗ u˜ ∗ , u V q,p (Q T ,σ ε

T)

  h = u˜ ∗ , SQ u V q,˜ p˜ (Q T ) T ,σ ε  ∗  → u˜ , QT u V q,˜ p˜ (Q T ) ε  ∗ ∗  = QT u˜ , u V q,p (Q ) (h → 0) , ε

T

h i.e., .(SQ )∗ u˜ ∗  ∗QT u˜ ∗ in .Vε (QT )∗ .(h → 0), because .u ∈ Vε (QT ) was T ,σ chosen arbitrarily. ad (ii). We argue as in the proof of Proposition 3.19 (ii), but now using Proposition 4.14 (iii).  q,p

q,p

∞ (Q )d in .V Eventually, we prove the density of .C0,div T ε (QT ). This will be ˚h from Propoaccomplished by a simple modification of the smoothing operator .S QT sition 4.13 by a contravariant Piola transform, or equivalently by a modification of h .S QT ,σ analogous to Proposition 4.13. q,p

Proposition 4.16 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .p− > 1 and .q ≥ p in ∞ .QT . Furthermore, let .(ϕh )h∈I◦ ⊆ C (I ) be the family of cut-off functions from 0q,p Proposition 3.18. For every .u ∈ Vε (QT ) and .h ∈ (0, h2 ) ∩ I◦ζ , where .I◦ζ := T ), we define the smoothing operator (0, 4ζ ˚h u := Ph S ˚h u = S h (ϕζ h u) ∈ C ∞ (Q )d . S 0,div QT ,σ QT QT ,σ R

.

q,p

Then, for every .u ∈ Vε

(QT ), it holds:

˚h u)h∈(0,h )∩I ζ ⊆ C ∞ (QT )d with .supp(S ˚h u) ⊆ Iζ h × for every (i) .(S h 2 QT ,σ QT ,σ 0,div ◦ ζ .h ∈ (0, h2 ) ∩ I◦ . (ii) There exists a constant .c > 0 (not depending on .q, p ∈ Plog (QT ) (cf. Remark 4.9)) such that .

sup h∈(0,h2 )∩I◦ζ

sup h∈(0,h2 )∩I◦ζ

˚h u| ≤ c Md+1 (Q u) |S T QT ,σ

a.e. in QT ,



˚h u)| ≤ c Md+1 (Q u) + Md+1 (Q ε(u)) a.e. in QT . |ε(S T T QT ,σ

q,p

q,p

˚ (QT ) and V ˚ε (QT ) 4.4 The Spaces V ∇

153

(iii) For extensions .q, p ∈ Plog (Rd+1 ) of .q, p ∈ Plog (QT ), resp., with − ≤ q ≤ q + and .p − ≤ p ≤ p + in .Rd+1 , there exist constants .c , c > 0 .q q p (depending on .q, p ∈ Plog (Rd+1 ) and .c > 0, resp.) such that ˚h u q(·,·) S (Q )d ≤ cq u Lq(·,·) (QT )d , L QT ,σ R

˚h u) p(·,·) ε(S sup (Q )d×d ≤ cp u Lp(·,·) (QT )d + ε(u) Lp(·,·) (QT )d×d . L QT ,σ R ζ h∈(0,h2 )∩I◦ sup

.

h∈(0,h2 )∩I◦ζ

˚h u → u in Vεq,p (QT ) .(h → 0), i.e., .C ∞ (QT )d lies densely in (iv) .S QT ,σ 0,div q,p .Vε (QT ). Proof ad (i)–(iii). Direct consequences of a combination of Proposition 4.13 and Proposition 4.14. ad (iv). We proceed as in Proposition 4.13 (iv) and infer, using Proposition 4.14 (iii) and (iv), that ˚h u − u q,p q,p S QT ,σ Vε (QT ) ≤ max{cq , 2(1 + |QT |)cp } (1 − ϕζ h )u Vε (QT )

.

h + SQ u − u Vεq,p (QT ) T ,σ

→ 0 (h → 0) .



˚q,p (QT ) and V ˚εq,p (QT ) 4.4 The Spaces V ∇ ∞ (Q )d Similar to Chap. 3, Sect. 3.4, we are not able to guarantee the density of .C0,div T q,p log in .Vε (QT ) in non-Lipschitz domains, or even if .q, p ∈ P (QT ) with .p− > 1 and .q ≥ p in .QT is not satisfied. However, the well-posed introduction of a generalized notion of a time derivative for the treatment of unsteady problems in variable exponent spaces with an incompressibility constraint strongly depends ∞ (Q )d in .V q,p (Q ). For this reason, similar to Chap. 3, on the density of .C0,div T T ε q,p Sect. 3.5, we next introduce a smaller space than .Vε (QT ) that trivially possesses this density property. For the entire section, let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), ∞ .T < ∞, .QT := I × , and .q, p ∈ P (QT ).

Definition 4.5 We define the solenoidal closure variable Bochner–Lebesgue spaces ˚q,p (QT ) := C ∞ (QT )d · V∇q,p (QT ) , V ∇ 0,div

.

˚εq,p (QT ) := C ∞ (QT )d · Vεq,p (QT ) . V 0,div

154

4 Solenoidal Variable Bochner–Lebesgue Spaces

Remark 4.13 In the case of a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2, q,p and .q, p ∈ Plog (QT ) with .p− > 1 and .q ≥ p in .QT , the spaces .Vε (QT ) q,p q,p q,p 16 ˚ε (QT ), as well as .V (QT ) and .V ˚ (QT ), respectively, coincide and .V ∇ ∇ (cf. Proposition 4.16 (iv)) and we will, therefore, always write in this situation q,p ˚εq,p (QT ), or .V q,p (QT ) instead of .V ˚q,p (QT ), respectively. .Vε (QT ) instead of .V ∇ ∇ q,p ˚εq,p (QT ) or Nevertheless, we have no information on whether .Vε (QT ) = V q,p q,p ˚ .V ∇ (QT ) = V∇ (QT ) in general. Looking back to Chap. 3, Sect. 3.4, we conjecture, though, that the assumption .q ≥ p in .QT is solely a by-product of the method of proof and could be omitted by more advanced approaches. A possible starting point could be a modification of the smoothing method via transversal expansion in space, i.e., of Proposition 4.11, by means of Theorem 3.2 in order to eliminate the necessity of controlling h p(·,·) (Q , Md×d ) in addition to .ε(u) ∈ Lp(·,·) (Q , Md×d ) .(ε(S T T sym sym QT ,σ u))h∈(0,h2 ) in .L q(·,·) d also by .u ∈ L (QT ) , as we did similarly in Proposition 3.16. However, this may necessitate a cut-off procedure with carefully to be chosen parameters, which lies far beyond the scope of this book. Proposition 4.17 The following statements apply: ˚εq,p (QT ) → Lq(·,·) (QT )d × ˚εq,p (QT ) : V (i) The mapping .σε,◦ := σε ◦ idV σ Lp(·,·) (QT , Md×d sym ), where .ε denotes from Proposition 4.2, is an isometric ˚εq,p (QT ) is separable. In isomorphism into its range .R(σε,◦ ). In particular, .V ˚εq,p (QT ) is reflexive. addition, if .q − , p− > 1, then .V σ := (id ˚q,p ∗ σ q  (·,·) (Q )d × (ii) If .q − , p− > 1, then the mapping .Jε,◦ Vε (QT ) ) ◦ Jε : L T  ∗ denotes the adjoint ˚εq,p (QT )∗ , where .(idV q,p ˚ ) → V ) Lp (·,·) (QT , Md×d sym ε (QT ) ˚εq,p (QT ) → Vεq,p (QT ) and .J σ is from ˚εq,p (QT ) : V operator of .idV ε Proposition 4.6, is linear and Lipschitz continuous with constant 2. In  ˚εq,p (QT )∗ , there exist .f ∈ Lq (·,·) (QT )d and addition, for every .u∗ ∈ V  (·,·) d×d p σ (f , F ) in .V ˚εq,p (QT )∗ and .F ∈ L (QT , Msym ) such that .u∗ = Jε,◦ .

1 ∗ ∗ u V ˚εq,p (QT )∗ ≤ f Lq  (·,·) (QT )d + F Lp (·,·) (QT )d×d ≤ 2 u V ˚εq,p (QT )∗ . 2

Proof Follows along the lines of the proof of Proposition 3.20 up to obvious adjustments.  In favor of readability, we set for the entirety of this book .σε := σε,◦ and σ . := Jε,◦

σ .Jε

q,p

q,p

˚ (QT ) coincide under the above condifact, we have not yet proved that .V∇ (QT ) and .V ∇ tions. But this follows from an argumentation analogous to Proposition 4.14 and Proposition 4.16.

16 In

q,p

q,p

˚ (QT ) and V ˚ε (QT ) 4.4 The Spaces V ∇

155

˚εq,p (QT )∗ ) In virtue of ProposiRemark 4.14 (Time Slices of Functionals .u∗ ∈ V tion 4.17 (ii) and Corollary 4.2, Remark 4.6 extends to the potentially larger dual ˚εq,p (QT )∗ . space .V The following two lemmas are particularly important for Chap. 7 and Chap. 8, in which we will develop a general existence theory that will also be applicable to the case of irregular domains. It should also be noted that congruent analogues q,p q,p of the following lemmas hold for the dual spaces .Vε (QT )∗ , .Xε (QT )∗ , and q,p ∗ ˚ .Xε (QT ) . The proofs all follow exactly the same argumentation. ˚εq,p (QT )∗ be such that .u∗  u∗ in Lemma 4.3 Let .q − , p− > 1 and .(u∗n )n∈N ⊆ V n q  (·,·) (Q )d and ˚εq,p (QT )∗ .(n → ∞). Then, there exist sequences .(f n ) .V ⊆ L T n∈N p (·,·) (Q , Md×d ) for every .n ∈ N satisfying .u∗ = J σ (f , F ) in .(F n )n∈N ⊆ L T n n sym n ε ˚εq,p (QT )∗ and .V 1 ∗ un V ˚εq,p (QT )∗ ≤ f n Lq  (·,·) (QT )d + F n Lp (·,·) (QT )d×d . 2 ≤

2 u∗n V ˚εq,p (QT )∗



(4.71)

,



∗ σ f ∈ Lq (·,·) (QT )d and .F ∈ Lp (·,·) (QT , Md×d sym ) satisfying .u = Jε (f , F ) in q,p ∗ ˚ε (QT ) , and a cofinal subset . ⊆ N such that .V .

fn  f

.

Fn  F



( n → ∞) ,

in Lq (·,·) (QT )d in L

p (·,·)

(QT , Md×d sym )

( n → ∞) . 

Proof Proposition 4.17 (ii) yields sequences .(f n )n∈N ⊆ Lq (·,·) (QT )d and p (·,·) (Q , Md×d ) satisfying .u∗ = J σ (f , F ) in .V ˚εq,p (QT )∗ and .(F n )n∈N ⊆ L T n n sym n ε ∗ ˚εq,p (QT )∗ we (4.71) for every .n ∈ N. From the boundedness of .(un )n∈N ⊆ V  infer by means of (4.71) the boundedness of both .(f n )n∈N ⊆ Lq (·,·) (QT )d and  p (·,·) (Q , Md×d ). As a consequence, exploiting the reflexivity of .(F n )n∈N ⊆ L T sym   q (·,·) d (QT ) and .Lp (·,·) (QT , Md×d both .L sym ) (cf. Proposition 2.7 and Remark 2.2),  there exist a cofinal subset . ⊆ N as well as functions .f ∈ Lq (·,·) (QT )d and  p (·,·) (Q , Md×d ) such that .F ∈ L T sym . 

fn  f

in Lq (·,·) (QT )d

Fn  F

in Lp (·,·) (QT , Md×d sym )



( n → ∞) , ( n → ∞) .

(4.72)

156

4 Solenoidal Variable Bochner–Lebesgue Spaces

  ∗ ˚q,p Since .Jεσ : Lq (·,·) (QT )d × Lp (·,·) (QT , Md×d sym ) → Vε (QT ) is weakly continuous (cf. Proposition 4.17 (ii)), we conclude from (4.72) that

Jεσ (f n , F n )  Jεσ (f , F )

.

˚ε (QT )∗ in V q,p

( n → ∞) ,

˚εq,p (QT )∗ , since .u∗  u∗ in .V ˚εq,p (QT )∗ .(n → ∞). i.e., .u∗ = Jεσ (f , F ) in .V n A similar



˚εq,p (QT )∗ . result applies to strong convergence in .V ˚εq,p (QT )∗ be such Let .q − , p− > 1 and .(u∗n )n∈N ⊆ V

Lemma 4.4 that .u∗n → u∗ in q,p q  (·,·) (Q )d and ˚ε (QT )∗ .(n → ∞). Then, there exist sequences .(f n ) .V T n∈N ⊆ L  q,p p (·,·) (Q , Md×d ) satisfying .u∗ = J σ (f , F ) in .V ∗ ˚ .(F n )n∈N ⊆ L (Q T n T ) for ε n sym n ε   every .n ∈ N, as well as functions .f ∈ Lq (·,·) (QT )d and .F ∈ Lp (·,·) (QT , Md×d sym ) ˚εq,p (QT )∗ such that satisfying .u∗ = Jεσ (f , F ) in .V fn → f .

Fn → F



(n → ∞) ,

in Lq (·,·) (QT )d in L

p (·,·)

(QT , Md×d sym )

(4.73)

(n → ∞) . 

Proof Proposition 4.17 (ii) provides .(g n )n∈N ⊆ Lq (·,·) (QT )d and .(Gn )n∈N ⊆  ∗ ∗ σ Lp (·,·) (QT , Md×d sym ) for every .n ∈ N satisfying .un − u = Jε (g n , Gn ) in ˚εq,p (QT )∗ and .V .

1 ∗ u − u∗ V ˚εq,p (QT )∗ ≤ g n Lq  (·,·) (QT )d + Gn Lp (·,·) (QT )d×d 2 n ≤ 2 u∗n − u∗ V ˚q,p (Q ε

i.e., in particular, it holds gn → 0 .

Gn → 0



in Lq (·,·) (QT )d 

in Lp (·,·) (QT , Md×d sym )

T)

∗

,

(n → ∞) , (n → ∞) . 

Apart from that, Proposition 4.17 (ii) provides functions .f ∈ Lq (·,·) (QT )d  ∗ σ ∗ ˚q,p and .F ∈ Lp (·,·) (QT , Md×d sym ) such that .u = Jε (f , F ) in .Vε (QT ) . Thus, if  one introduces the sequences .(f n )n∈N := (g n + f )n∈N ⊆ Lq (·,·) (QT )d and p (·,·) (Q , Md×d ), then one has both that .u∗ = .(F n )n∈N := (Gn + F )n∈N ⊆ L T sym n ˚εq,p (QT )∗ for every .n ∈ N and (4.73). Jεσ (f n , F n ) in .V 

˚ε (QT )∗ 4.5 Generalized Time Derivative in V q,p

157

˚εq,p (QT )∗ 4.5 Generalized Time Derivative in V In this section, we introduce an appropriate notion of a generalized time derivative for solenoidal variable Bochner–Lebesgue spaces. We will proceed in analogy to Chap. 3, Sect. 3.6, but with a significant difference. Now the time derivative ˚εq,p (QT )∗ . Logically, the resulting notion of a is supposed to be functional in .V time derivative will not be consistent with the standard notion of the distributional time derivative living in .D (QT )d , i.e., the topological dual space of the locally convex Hausdorff vector space .C0∞ (QT )d , but rather with a notion of a hydromechanical distributional time derivative living in .Ddiv (QT )d , i.e., the topological ∞ (Q )d (cf. (4.54)), which is a closed subspace of .C ∞ (Q )d . dual space of .C0,div T T 0 This circumstance entails several obstacles compared to the generalized time derivative we have already encountered in Chap. 3, Sect. 3.6. For the entire section, let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), ∞ ˚−q,p → H , where .T < ∞, .QT := I × , and .q, p ∈ P (QT ), such that .V H := L20,σ () := V

.

e.g., if .p− ≥

2d d+2

· Y

 · 2 d  = u ∈ C0∞ ()d | div(u) = 0 in  L () ,

(4.74)

or simply .q − ≥ 2. q,p

Definition 4.6 A function .u ∈ Vε (QT ) possesses a generalized time derivative ˚εq,p (QT )∗ such that for every ˚εq,p (QT )∗ , if there exists a functional .u∗ ∈ V in .V ∞ d .φ ∈ C 0,div (QT ) , it holds .

− I

(u(t), ∂t φ(t))H dt = u∗ , φV ˚q,p (Q ) . ε

T

(4.75)

σ u := ∗ ˚εq,p (QT )∗ . In this case, we define . ddt u in .V

Lemma 4.5 The generalized time derivative in the sense of Definition 4.6 is unique. Proof Follows analogously to the proof of Lemma 3.7, but now by using the density ∞ (Q )d in .V ˚εq,p (QT ) (cf. Definition 4.5). of .C0,div  T Remark 4.15 The generalized time derivative . ddtσ from Definition 4.6 is related to the hydro-mechanical distributional time derivative .∂tσ , i.e., for .u ∈ L1loc (QT )d ,

∞ (Q )d := .− the functional .∂tσ u ∈ Ddiv (QT )d , defined by .∂tσ u, φC0,div T QT u(t, x)· ∞ d .∂t φ(t, x) dtdx for every .φ ∈ C 0,div (QT ) , in the following sense: q,p σu ˚εq,p (QT )∗ , then ∈ V If .u ∈ Vε (QT ) has a generalized time derivative . ddt   Proposition 4.17 (ii) yields .f ∈ Lq (·,·) (QT )d and .F ∈ Lp (·,·) (QT , Md×d sym ) such q,p dσ u σ ∗ ˚ that . dt = Jε (f , F ) in .Vε (QT ) . Therefore, it can readily be read from (4.75) that .∂tσ u = f − div(F ) in .Ddiv (QT )d .17 ∞ (Q )d := (f , φ)Lq(·,·) (Q )d − div(F ) ∈ Ddiv (QT )d is defined by .f − div(F ), φC0,div T T ∞ (Q )d . +(F , ε(φ))Lp(·,·) (QT )d×d for every .φ ∈ C0,div T

17 Here, .f

158

4 Solenoidal Variable Bochner–Lebesgue Spaces

Definition 4.7 We define the solenoidal variable Bochner–Sobolev space    q,p ˚εq,p (QT )∗ . ˚εq,p (QT )  ∃ dσ u ∈ V Wε,σ (QT ) := u ∈ V dt

.

q,p

Proposition 4.18 The space .Wε,σ (QT ) forms a Banach space, if equipped with the norm    dσ ·    q,p := q,p . . · + · ˚ε (QT ) Wε,σ (QT )  dt  ˚q,p V Vε (QT )∗ q,p ˚εq,p (QT ) × V ˚εq,p (QT )∗ , for every In particular, the mapping .σε : Wε,σ (QT ) → V q,p dσ u  σ ∗ ˚q,p ˚q,p .u ∈ Wε,σ (QT ) defined by .ε u := (u, dt ) in .Vε (QT ) × Vε (QT ) , is an q,p σ − − isometric isomorphism into .R(ε ). In addition, if .q , p > 1, then .Wε,σ (QT ) is separable and reflexive.

Proof Follows along the lines of the proof of Proposition 3.21 up to obvious adjustments.  ˚+q,p → H and .idV˚q,p : V ˚−q,p → H Since, by assumption, the mappings .idV˚+q,p : V − are embeddings, which are inevitably also dense, their corresponding adjoint oper˚+q,p )∗ and .(idV˚q,p )∗ : H ∗ → (V ˚−q,p )∗ are embeddings as ators .(idV˚+q,p )∗ : H ∗ → (V − well. Consequently, also the mappings

.

σ ˚−q,p → (V ˚+q,p )∗ , := (idV˚+q,p )∗ RH idV˚−q,p : V e− σ ˚+q,p → (V ˚−q,p )∗ , := (idV˚−q,p )∗ RH idV˚+q,p : V e+

(4.76)

where .RH : H → H ∗ denotes the Riesz isomorphism with respect to H , are embeddings. This ensures the well-posedness of the following limiting Bochner– Sobolev spaces (cf. Definition 2.19). Definition 4.8 We define the limiting solenoidal Bochner–Sobolev spaces q,p

1,max{q + ,p+ },max{(q − ) ,(p− ) }

Wσ,+ (QT ) := Weσ

.

+

q,p

1,min{q − ,p− },min{(q + ) ,(p+ ) }

Wσ,− (QT ) := Weσ

−

˚+q,p , (V ˚−q,p )∗ ) , (I, V

˚−q,p , (V ˚+q,p )∗ ) . (I, V

˚ε (QT )∗ 4.5 Generalized Time Derivative in V q,p

159 q,p

q,p

Similar to Chap. 3, Definition 3.8, the spaces .Wσ,+ (QT ) and .Wσ,− (QT ) q,p employ a different concept of a time derivative than .Wε,σ (QT ), namely the σ σ d e+ d e− Banach-valued distributional time derivatives . dt and . dt (cf. Definition 2.19) living in Bochner–Lebesgue spaces. This gives us again access to the following q,p q,p non-symmetric formula of integration-by-parts for .W+ (QT ) and .W− (QT ). ∞ Furthermore, since for this formula solenoidal variants of .Y (QT ) and .Y 0 (QT ) are still missing, we introduce the spaces H∞ (QT ) := L∞ (I, H ) .

H0 (QT ) := C 0 (I , H ) ,

.

(4.77)

Proposition 4.19 (Non-symmetric Formula of Integration-by-Parts) The following statements apply: (i) Each .u ∈ Wσ,− (QT ) ∩ H∞ (QT ) possesses a weakly continuous representaq,p tion .uω ∈ Cω0 (I , H ). Moreover, each .v ∈ Wσ,+ (QT ) possesses a continuous q,p 0 representation .v c ∈ H (QT ) and the resulting mapping .(·)c : Wσ,+ (QT ) → H0 (QT ) is an embedding. q,p q,p (ii) For every .u ∈ Wσ,− (QT ) ∩ H∞ (QT ), .v ∈ Wσ,+ (QT ), and .t  , t ∈ I with  .t ≤ t, it holds q,p

t .

t

de−σ u dt

 (s), v(s)

q,p

˚+ V

ds = [(uω (s), v c (s))H ]s=t s=t  −

q,p

t t

de+σ v dt

 (s), u(s)

q,p

ds .

˚− V

q,p

(iii) For every .u ∈ Wσ,− (QT ) and .v ∈ Wσ,+ (QT ) with .supp(v) ⊆ I × , it holds    σ  σ de− u de+ v (s), v(s) (s), u(s) ds = − ds . . dt dt ˚+q,p ˚−q,p I I V V ˚+q,p , .X− := V ˚−q,p , .Y := H = L2 , Proof Follows from Proposition 2.28 for .X+ := V 0,σ σ , .e σ + := max{q + , p + }, and ˚+q,p , .j− := idV ˚−q,p , .e+ := e+ .()j+ := idV − := e− , .p − := min{q − , p − }. .p 

160

4 Solenoidal Variable Bochner–Lebesgue Spaces q,p

Proposition 4.20 (Alternative Characterization of .Wε,σ (QT )) For a function ˚εq,p (QT ) and a functional .u∗ ∈ V ˚εq,p (QT )∗ , the following statements are .u ∈ V equivalent: q,p σu ˚εq,p (QT )∗ . (i) .u ∈ Wε,σ (QT ) with . ddt = u∗ in .V ˚+q,p and .ϕ ∈ C ∞ (I ), it holds (ii) For every .v ∈ V 0

.

−

(u(s), v)H ∂t ϕ(s) ds =

I

I

u∗ (s), vV˚εq,p (s) ϕ(s) ds ,



∗ ∗ in .Lmin{(q + ) ,(p+ ) } q,p = J˚−1 q,p (idV+ (QT ) ) u V+ ˚+q,p )∗ ), where we denote by the mapping .(idV q,p (Q ) )∗ the adjoint oper(I, (V T + q,p ˚εq,p (QT ) and .J˚q,p : .Lmin{(q + ) ,(p+ ) } ator of .idV+q,p (QT ) : V+ (QT ) → V V+ ˚+q,p )∗ ) → V+q,p (QT )∗ is the isomorphism from Proposition 2.20. .(I, (V q,p

i.e., .u ∈ Wσ,− (QT ) with .

σu d e− dt

Proof Follows analogously to the proof of Proposition 3.23, but now by using Proposition 4.19.  Proposition 4.19 (i) allows us to define a solenoidal analogue of a generalized evolution equation. Definition 4.9 (Solenoidal Generalized Evolution Equation) Let .u0 ∈ H q,p ˚εq,p (QT )∗ a right-hand side, and .A : Wε,σ be an initial value, .u∗ ∈ V (QT ) ∩ q,p ∞ ∗ ˚ H (QT ) → Vε (QT ) an operator. Then, the initial value problem dσ u + Au = u∗ . dt uω (0) = u0

˚εq,p (QT )∗ , in V

(4.78)

in H ,

is called a solenoidalgeneralizedevolutionequation.Here, the initial condition has to be understood in the sense of the weakly continuous representation .uω ∈ Cω0 (I , H ) (cf. Proposition 4.19 (i)).

q,p

4.6 Formula of Integration-by-Parts for Wε,σ (QT ) This section is concerned with the proof of a formula of integration-by-parts for q,p Wε,σ (QT ). The basic procedure is the same as in Chap. 3, Sect. 3.7. However, due h ) to preserve spatial compact supports (cf. Remark 4.11), to the inability of .(SQ T in contrast to .(RhQT ) (cf. Proposition 3.15 (i)), some additional work needs to be done. First of all, we verify that the method of in time extension via reflection extends to the framework of solenoidal variable Bochner–Lebesgue spaces, i.e. to ˚εq,p (QT )∗ . the framework of the generalized time derivative . ddtσ living in .V

.

q,p

4.6 Formula of Integration-by-Parts for Wε,σ (QT )

161

Proposition 4.21 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .p− > 1 and .q ≥ p in q,p q ,p 18 .QT . Moreover, we denote again by .ET : Xε (QT ) → Xε T T (Q3T ) the in time extension via reflection operator from Proposition 3.24, where .qT := ET q, pT := ET p ∈ Plog (Q3T ), .3I := (−T , 2T ), and .Q3T := 3I × . Then, it holds: q,p

q ,p

(i) .ET : Vε (QT ) → Vε T T (Q3T ) is well-defined, linear and Lipschitz continuous with constant 3. q,p q ,p (ii) .ET : Wε,σ (QT ) → Wε,σT T (Q3T ) is well-defined, linear and Lipschitz q,p σu = continuous with constant 12. More precisely, if .u ∈ Wε,σ (QT ) with . ddt  (·,·)  (·,·) q,p σ ∗ q d p Jε (f , F ) in .Vε (QT ) for .f ∈ L (QT ) and .F ∈ L (QT , Md×d sym ), qT ,pT then .ET u ∈ Wε,σ (Q3T ) with .

dσ ET u − − = Jεσ (ET f , ET F ) dt

−

q ,pT

in Vε T

−



(Q3T )∗ ,

(4.79)



where .ET f ∈ LqT (·,·) (QT )d and .ET F ∈ LpT (·,·) (QT , Md×d sym ) are defined as in Proposition 3.24. q,p

Proof ad (i). Let .u ∈ Xε (QT ) be arbitrary. Then, using Proposition 3.24, we find q ,p that .ET u ∈ Xε T T (Q3T ) with . ET u X qT ,pT (Q3T ) ≤ 3 u Xεq,p (QT ) . On the other ε ˚qT ,pT (t) hand, it is also not difficult to see that we even have that .(ET u)(t) ∈ V qT ,pT for almost every .t ∈ 3I . Therefore, we conclude that .ET u ∈ Vε (Q3T ) (cf. Definition 4.2) with . ET u V qT ,pT (Q3T ) ≤ 3 u Vεq,p (QT ) , i.e., (i). ε

d

q,p

σ

u

ad (ii). By Proposition 4.20, we have that .u ∈ Wσ,− (QT ) with . edt− =

+  +  ∗ σ q,p ˚+q,p )∗ ). Hence, using J˚−1 in .Lmin{(q ) ,(p ) } (I, (V q,p (idV+ (QT ) ) Jε (f , F ) V+

q ,pT

T Proposition 2.25, we infer that .ET u ∈ Wσ,−

de−σ ET u .

dt

=

− ET



de−σ u

(Q3T ) with ⎫ ⎪ ⎪ ⎬



dt

⎪ ⎪ − ∗ σ − q ,p ⎭ = J −1 qT ,pT (idV+T T (Q3T ) ) Jε (ET f , ET F ) V+

in L

18 Precisely

(iv)).

min{(qT+ ) ,(pT+ ) }

q ,pT ∗

(3I, (V+T

(4.80)

) ).

q,p

the assumptions that guarantee that .Vε

q,p

˚ε (QT ) (cf. Proposition 4.16 (QT ) = V

162

4 Solenoidal Variable Bochner–Lebesgue Spaces q ,p

Using Proposition 4.20, we derive from (4.80) that .ET u ∈ Wε,σT T (Q3T ) with (4.79). Eventually, by analogy with Proposition 3.24, by exploiting the Lipschitz   qT ,pT (Q3T )∗ (cf. continuity of .Jεσ : LqT (·,·) (Q3T )d × LpT (·,·) (Q3T , Md×d sym ) → Vε Proposition 4.6), we conclude from (4.79) the Lipschitz continuity of .ET : q,p q ,p  Wε,σ (QT ) → Wε,σT T (Q3T ) with constant 12, i.e., (ii). Next, aided by Proposition 4.21, we can prove the following density result for q,p the space .Wε,σ (QT ). Proposition 4.22 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , .q, p ∈ P (QT ), with .p− ≥ 2 and .q ≥ p in .QT , and  T q,p .I∧ := 0, 2ζ . For every .u ∈ Wε,σ (QT ) and .h ∈ (0, h2 ), we define the smoothing operator h h SQ u := SQ (ET u)|QT ∈ C ∞ (I , V) . T ,σ 3T ,σ

.

q,p

Then, for every .u ∈ Wε,σ (QT ), it holds: q,p

h (i) .(SQ u)h∈(0,h2 ) ⊆ Wε,σ (QT ) with a family error functionals T ,σ q,p .(Eh (u))h∈(0,h2 )∩I∧ ⊆ Vε (QT )∗ such that for every .h ∈ (0, h2 ) ∩ I∧ , it holds

.

  dσ h h ∗ dσ ET u SQT ,σ u = (SQ + Eh (u) ) T ,σ dt dt

in Vε (QT )∗ , q,p

h where .(SQ )∗ : Vε T T (Q3T )∗ → Vε (QT )∗ denotes the adjoint operator T ,σ q,p q ,p h of the smoothing operator .SQ : Vε (QT ) → Vε T T (Q3T ) in the sense of T ,σ Proposition 4.15. (ii) For extensions .q T , p T ∈ Plog (Rd+1 ) of .qT , pT ∈ Plog (Q3T ), with − ≤ q ≤ q + and .p − ≤ p ≤ p + in .Rd+1 , there exists a constant .q T T log d+1 .cq ,p > 0 (depending on .q T , p T ∈ P (R )) such that T T q ,p

.

sup

q,p

 h S

h∈(0,h2 )∩I∧

sup h∈(0,h2 )∩I∧

 

q,p ≤ cq T ,pT u Wε,σ (QT ) ,

q,p QT ,σ u Wε,σ (QT )

Eh (u) Vεq,p (QT )∗ ≤ cq T ,pT u Vεq,p (QT ) .

(iii) It holds

.

q,p

(QT )

(h → 0) ,

q,p

(QT )∗

(h → 0) ,

q,p

(QT )∗

(h → 0) .

h SQ u→u T ,σ

in Vε

dσ u dσ h S u dt QT ,σ dt

in Vε

Eh (u)  0

in Vε

q,p

4.6 Formula of Integration-by-Parts for Wε,σ (QT )

163

h Proof ad (i). To begin with, we note that .(SQ u)h∈(0,h2 ) ⊆ C ∞ (I , V), since T ,σ h ∞ (Q )d . Using this Proposition 4.14 (i) yields that .(SQ (ET u))h∈(0,h2 ) ⊆ C0,div R 3T ,σ q,p h regularity, it is readily seen that .(SQ u) ⊆W (Q ) with h∈(0,h T ) ε,σ 2 ,σ T

.

dσ h h S u = Jεσ (∂t SQ u, 0) T ,σ dt QT ,σ

q,p

in Vε

(QT )∗

(4.81)

for every .h ∈ (0, h2 ). Next, making use of (4.81) and the formula of integration-by∞ (Q )d and .h ∈ (0, h ), parts in time for smooth vector fields, for every .φ ∈ C0,div T 2 we observe that     dσ h h S u, φ = ∂t SQ u, φ Lq(·,·) (Q )d T ,σ T q,p dt QT ,σ Vε (QT )   h = ∂t SQ u, φ L2 (QT )d T ,σ  h  = − SQT ,σ u, ∂t φ L2 (Q )d T   h . (4.82) = − Ph SQ3T (ET u), QT ∂t φ L2 (Q )d 3T  h  = − SQ (ET u), P h QT ∂t φ L2 (Q3T )d 3T   h = − ET u, (SQ ) (P h QT ∂t φ) L2 (Q3T )d 3T   h = − ET u, (SQ ) (QT ∂t φ) L2 (Q )d , 3T ,σ 3T

h ) : L2 (Q )d → L2 (Q )d denotes the quasi adjoint of .S h where .(SQ 3T 3T Q3T (cf. 3T h  : L2 (Q )d → L2 (Q )d is defined by Proposition 4.10 (v)) and .(SQ ) 3T 3T 3T ,σ h  v := (S h ) (P v) ∈ C ∞ (Q )d for all .v ∈ L2 (Q )d and .h ∈ (0, h ). .(S ) 3T T 2 Q3T ,σ Q3T h ∞ (Q )d , .(t, x) ∈ Q In particular, for every .φ ∈ C0,div and . h ∈ h we have (0, ), T 3T 2 that  h  (SQ3T ,σ ) (QT ∂t φ) (t, x) . (4.83) ωζ h (t − s, x − ν h (y))(QT [P = h ∂t φ])(s, y) dsdy . Bζd+1 h (t,x) 0

Using differentiation under the integral sign, where we, in particular, exploit that d+1   ∈ .supp(ω ζ h (t − ·, x − ν h (·))QT [P φ]) ⊆ B ζ0 h (t, x) (cf. (4.37)) for all .(t, x) h Q3T and .h ∈ (0, h2 ) to realize that we can always get rid of the .t ∈ 3I dependency in the integral domain in (4.83) by replacing .Bζd+1 (t, x) with .Rd+1 , and subse0h ∞ (Q )d , quently using integration-by-parts in time in (4.83), for every .φ ∈ C0,div T

164

4 Solenoidal Variable Bochner–Lebesgue Spaces

(t, x) ∈ Q3T and .h ∈ (0, h2 ), we find that

.

h ∂t (SQ ) (QT φ) (t, x) (4.84) 3T ,σ

d = ωζ h (t − s, x − ν h (y)) (QT [P h φ])(s, y) dsdy Bζd+1 (t,x) dt h 0

d =− ωζ h (t − s, x − ν h (y)) (QT [P h φ])(s, y) dsdy d+1 Bζ h (t,x) ds 0 ωζ h (t − s, x − ν h (y))(QT [P = h ∂s φ])(s, y) dsdy

.

Bζd+1 h (t,x) 0

h = ((SQ ) (QT ∂t φ))(t, x) . 3T ,σ

If one exploits that .(ωh )h>0 ⊆ C0∞ (Rd+1 ) ∩ SM(Rd+1 ) are radially symmetric and, hence, .(∂t ω)h (−s, y) = −(∂t ω)h (s, y) and .(∂t ω)h (s, −y) = (∂t ω)h (s, y) for any  ∈ Rd+1 and .h > 0, we deduce from .(4.84) for every .φ ∈ C ∞ (Q )d , .(s, y) T 1 0,div  ∈Q .(t, x) 3T and .h ∈ (0, h2 ) that h

∂t (SQ ) (QT φ) (t, x) (4.85) 3T ,σ 1 = (∂t ω)ζ h (t − s, x − ν h (y))(QT [P h φ])(s, y) dsdy ζ h Bζd+1 (t,x) 0h 1 =− (∂t ω)ζ h (s − t, ν h (y) − x)(QT [P h φ])(s, y) dsdy . ζ h Bζd+1 h (t,x)

.

0

∞ (Q )d and On the other hand, by inserting (4.84) in (4.82), for every .φ ∈ C0,div T .h ∈ (0, h2 ), we arrive at

 .

 dσ h u, φ S q,p dt QT ,σ Vε (QT )  h

 (ET u)(t), ∂t (SQ =− ) (QT φ) (t) Y dt . 3T ,σ

(4.86)

3I

h ) (QT φ))h∈(0,h2 ) ⊆ C ∞ (Q3T )d . Regrettably, Apparently, it holds .((SQ 3T ,σ h ∞ (Q )d (cf. we cannot guarantee that .((SQ ) (QT φ))h∈(0,h2 ) ⊆ C0,div 3T 3T ,σ h ∞ Remark 4.11). However, we can guarantee that .(SQT ,σ φ)h∈(0,h2 ) ⊆ C0,div (QR )d h with .supp(SQ φ) ⊆ [−ζ h, T + ζ h] × h ⊆ Q3T for all .h ∈ I∧ (cf. T ,σ h ∞ (Q )d . Proposition 4.14 (i)), i.e., we have that .(SQ φ)h∈(0,h2 )∩I∧ ⊆ C0,div 3T T ,σ

q,p

4.6 Formula of Integration-by-Parts for Wε,σ (QT )

165

∞ (Q )d , .(t, x) ∈ Q and .h ∈ (0, h ), by More specifically, for every .φ ∈ C0,div T 2 R definition, it holds h (SQ φ)(t, x) T ,σ . = Ph (x)

Bζd+1 h (t,x)

(4.87)

ωζ h (s − t, y − ν h (x))(QT φ)(s, y) dsdy .

0

h Therefore, we are apt to replace .(SQ ) (QT φ) for every .h ∈ (0, h2 ) ∩ I∧ 3T ,σ h with .SQ φ in (4.86). For this, however, one has to pay with the family of error T ,σ ∞ (Q )d and functionals .(Eh (u))h∈(0,h2 )∩I∧ ⊆Ddiv (QT )d , which for every .φ ∈ C0,div T .h ∈ (0, h2 ) ∩ I∧ is defined by

Eh (u), φC ∞

.

d 0,div (QT )

 h

 h := ET u, ∂t SQ φ − (SQ ) φ L2 (Q T ,σ 3T ,σ

3T )

d

.

(4.88)

To control these errors, it is necessary to have suitable formulas for the derivatives h ∞ (Q )d , e.g., similar to (4.85). By proceeding as (∂t [SQ φ])h∈(0,h2 )∩I∧ ⊆ C0,div R T ,σ ∞ (Q )d , .(t, x) ∈ Q and for .(4.84)1 , we deduce from (4.87) for every .φ ∈ C0,div T R

.

h ∈ (0, h2 ), using that .supp(ωζ h (· − t, · − ν h (x))QT φ) ⊆ Bζd+1 h (t, ν h (x)) ⊆

.

Bζd+1 (t, x) (cf. (4.29)), that 0h h ∂t [SQ φ](t, x) T ,σ

(4.89)

.

=−

1 Ph (x) ζh

Bζd+1 h (t,x)

(∂t ω)ζ h (s − t, y − ν h (x))(QT φ)(s, y) dsdy .

0



h ∞ (Q )d in (4.86) for .φ ∈ C ∞ (Q )d and We add and subtract .∂t SQ φ ∈ C0,div 3T T 0,div T ,σ qT ,pT .h ∈ (0, h2 ) ∩ I∧ , and exploit that .ET u ∈ Wε,σ (Q3T ) (cf. Proposition 4.21 (ii)), ∞ (Q )d and .h ∈ (0, h ) ∩ I that to obtain for every .φ ∈ C0,div T 2 ∧ 

.

  h

 dσ h (ET u)(t), ∂t SQ u, φ =− φ (t) H dt S T ,σ q,p dt QT ,σ 3I Vε (QT ) + Eh (u), φC ∞ (QT )d 0,div   dσ ET u h = , SQT ,σ φ q ,p dt Vε T T (Q3T ) + Eh (u), φC ∞ (QT )d 0,div     h ∗ dσ ET u = (SQ ) ,φ T ,σ q,p dt Vε (QT ) + Eh (u), φC ∞

0,div (QT )

d

,

(4.90)

166

4 Solenoidal Variable Bochner–Lebesgue Spaces

h where .(SQ )∗ : Vε T T (Q3T )∗ → Vε (QT )∗ denotes the adjoint operator from T ,σ ∞ (Q )d Proposition 4.15. Therefore, (i) follows by virtue of the density of .C0,div T q,p in .Vε (QT ) (cf. Proposition 4.16 (iv)), if we can ascertain that the functionals  (Q )d admit unique continuous extensions to func.(Eh (u))h∈(0,h2 )∩I∧ ⊆ D T div q,p tionals .(Eh (u))h∈(0,h2 )∩I∧ ⊆ Vε (QT )∗ . To this end, we need to control the h h  errors .(∂t [SQT ,σ φ − (SQ3T ,σ ) φ])h∈(0,h2 )∩I∧ , e.g., in .L2 (Q3T )d . Resorting to the ∞ (Q )d , representation formulas (4.85) and (4.89), these errors for every .φ ∈ C0,div T  .(t, x) ∈ Q3T and .h ∈ (0, h2 ) ∩ I∧ can be rewritten as q ,p

q,p

h h ∂t [SQ φ − (SQ ) φ](t, x) T ,σ 3T ,σ 1 = − Ph (x) (∂t ω)ζ h (s − t, y − ν h (x))(QT φ)(s, y) dsdy ζh Bζd+1 (t,x) 0h 1 + (∂t ω)ζ h (s − t, ν h (y) − x)(QT [P h φ])(s, y) dsdy ζ h Bζd+1 (t,x) 0h 1 [Id − Ph (x)] = (∂t ω)ζ h (s − t, y − ν h (x))(QT φ)(s, y) dsdy ζh Bζd+1 (t,x) h 0 1 + (∂t ω)ζ h (s − t, ν h (y) − x)(QT [(P h − Id )φ])(s, y) dsdy ζ h Bζd+1 (t,x) 0h 1 +

h (s, t, y, x)(QT φ)(s, y) dsdy ζ h Bζd+1 h (t,x)

.

0

=:

3 !

Ihi (φ)(t, x) ,

(4.91)

i=1

where the family .( h )h∈(0,h2 )∩I∧ ⊆ C ∞ (R2d+2 ) for every .(s, t, y, x) ∈ R2d+2 and .h ∈ (0, h2 ) ∩ I∧ is defined by

h (s, t, y, x) := (∂t ω)ζ h (s − t, ν h (y) − x) − (∂t ω)ζ h (s − t, y − ν h (x)) .

.

Our next objective is to show that for almost every .(t, x) ∈ QR , it holds 3 !

sup .

h∈(0,h2 )∩I∧ i=1



|Ihi (φ)(t, x)|

(4.92)

≤ c Md+1 (QT φ)(t, x) + Md+1 (QT ∇φ)(t, x) , where .c > 0 is a constant that depends neither on .h ∈ (0, h2 ) ∩ I∧ , nor on the q,p variable exponents .q, p ∈ Plog (QT ), nor on the function .u ∈ Wε,σ (QT ), but on

q,p

4.6 Formula of Integration-by-Parts for Wε,σ (QT )

167

the Lipschitz characteristics of . and the fixed standard mollifier .ω ∈ C0∞ (Rd+1 ) ∩ SM(Rd+1 ) in the sense of Remark 4.9.  and . Id − Ph L∞ ()d×d ≤ cP h ad .Ih1 (φ). If we use . (∂t ω)h L∞ (Rd+1 ) ≤ hcd+1 ∞ (Q )d (cf. Lemma 4.2 (iii)), then for almost every .(t, x) ∈ QR , every .φ ∈ C0,div T and .h ∈ (0, h2 ) ∩ I∧ , we find that

|Ih1 (φ)(t, x)| ≤

1 Id − Ph L∞ ()d×d (∂t ω)ζ h L∞ (Rd+1 ) ζh × |(QT φ)(s, y)| dsdy Bζd+1 h (t,x)

(4.93)

0

.

c ζ0d+1 1 ≤ cP h d+1 ζh ζ

Bζd+1 h (t,x)

|(QT φ)(s, y)| dsdy

0

≤ c Md+1 (QT φ)(t, x) . ad .Ih2 (φ). Analogously, but now with recourse to . P h − Id L∞ ()d×d ≤ cP h (cf. Lemma 4.2 (iii)), we further observe for almost every .(t, x) ∈ QR , every ∞ d .φ ∈ C ∈ (0, h2 ) ∩ I∧ that 0,div (QT ) and .h . |Ih2 (φ)(t, x)| ≤ c Md+1 (QT φ)(t, x) .

(4.94)

ad .Ih3 (φ). For every .x ∈ Rd and .h ∈ (0, h2 ) ∩ I∧ , we introduce the mapping h d d  ∈ [0, 1] × Rd is defined by .bx : [0, 1] × R → R , which for every .(λ, y) .

bhx (λ, y) := λ(ν h (y) − x) + (1 − λ)(y − ν h (x)) = ν λh (y) − ν (1−λ)h (x).

Using the mappings .bhx : [0, 1] × Rd → Rd , .x ∈ Rd , .h ∈ (0, h2 ) ∩ I∧ , and the Newton–Leibniz formula, the family .( h )h∈(0,h2 )∩I∧ ⊆ C ∞ (R2d+2 ) for every  ∈ Rd+1 , .(s, y) ∈ Q and .h ∈ (0, h ) ∩ I can be represented as19 .(t, x) 2 ∧ R

d (∂t ω)ζ h (s − t, bhx (λ, y)) dλ (4.95) 0 dλ 1   1 = (∇∂t ω)ζ h (s −t, bhx (λ, y)) · ν h (y)−x −(y −ν h (x)) dλ ζh 0 1

= Jλh (y)− ∇y (∂t ω)ζ h (s − t, bhx (λ, y)) · h(k(x) + k(y)) dλ 1

h (s, t, y, x) =

.

0



1

=h 0

19 Here, .∇

y



∇y (∂t ω)ζ h (s −t, bhx (λ, y)) · Jλh (y)−1 (k(x)+k(y)) dλ ,

denotes the gradient with respect to the spatial variable y.

168

4 Solenoidal Variable Bochner–Lebesgue Spaces



where we made use of that .∇y (∂t ω)ζ h (s − t, bhx (λ, y)) =

1 ζ h ∇y

bhx (λ, y)

(∇∂t ω)ζ h (s − t, bhx (λ, y)) together with .∇y bhx (λ, y) = (∇ν λh )(y) = Jλh (y) for every .(t, x) ∈ Rd+1 , .(s, y) ∈ QR , .λ ∈ [0, 1] and .h ∈ (0, h2 ). In particular, note that, owing to Proposition 4.9 (ii), for every .y ∈ , .λ ∈ (0, 1] and .h ∈ (0, h2 ), the −1 1 inverse .Jλh (y)−1 ∈ Rd×d exists with . J−1 λh L∞ ()d×d ≤ Lip(ν λh |ν λh () ) ≤ 1−h2 κ for all .h ∈ (0, h2 ). Consequently, if we define .(gh )h∈(0,h2 )∩I∧ ⊆ C ∞ ( × )d for every .x, y ∈  and .h ∈ (0, h2 ) ∩ I∧ by .gh (x, y) := Jh (y)−1 (k(x) + k(y)), we find, using (4.95), for every .(t, x) ∈ QR and .h ∈ (0, h2 ) ∩ I∧ that Ih3 (φ)(t, x) =

.

=

1 h ζh

1 ζh



1 



Bζd+1 h (t,x)∩QT

(4.96)



∇y (∂t ω)ζ h (s − t, bhx (λ, y))

Bζd+1 h (t,x)∩QT

0

h (s, t, y, x)φ(s, y) dsdy

0

0

  · gλh (x, y) φ(s, y) dsdy dλ 1 =− ζ 1 − ζ

1

0

0



Bζd+1 h (t,x)∩QT

1 

(∂t ω)ζ h (s − t,

bhx (λ, y))Dφ(s, y)gλh (x, y) dsdy

dλ

0

Bζd+1 h (t,x)∩QT 0

(∂t ω)ζ h (s − t, bhx (λ, y))φ(s, y)



× div(gλh )(x, y) dsdy dλ . The integration-by-parts in (4.96) produces no boundary integrals because for every .(t, x) ∈ QR and .λ ∈ [0, 1], it holds .(∂t ω)ζ h (s − t, bhx (λ, y)) = 0 if and only if .(s, ν λh (y)) ∈ Bζd+1 h (t, ν (1−λ)h (x)), and for every .x, y ∈  and d+1  ∈ B d+1 (t, ν  .λ ∈ [0, 1], .(s, ν λh (y)) (1−λ)h (x)) implies .(s, y) ∈ Bζ0 h (t, x). Thus, ζh c using (4.96), . (∂t ω)h L∞ (Rd+1 ) ≤ hd+1 , . gλh L∞ (×)d ≤ 1−h2 2 κ k L∞ ()d as . J−1 λh L∞ ()d×d ≤

1 1−h2 κ ∇k L∞ ()d×d Jλh (y)−1 : ∇k(y) 1 d 2 DJ−1 λh L∞ ()d 3 ≤

1

and . div(gλh ) L∞ (×) .≤ d 2 κ1 h2 2 k L∞ ()d +

due to .div(gλh )(x, y) = div(J− λh )(y) · k(x) + k(y) + 1 1−h2 κ ,

for every .x, y ∈  as well as . div(J− λh ) L∞ ()d 1 2

≤

1 2

d κ1 λh ≤ d κ1 h2 (cf. Proposition 4.9 (iii)), we further

∞ (Q )d and .h ∈ (0, h ) ∩ I arrive for almost every .(t, x) ∈ QR , every .φ ∈ C0,div T 2 ∧ at

|Ih3 (φ)(t, x)| ≤ c .

Bζd+1 h (t,x) 0

|(QT φ)(s, y)| + |(QT ∇φ)(s, y)| dsdy



≤ c Md+1 (QT φ)(t, x) + Md+1 (QT ∇φ)(t, x) .

(4.97)

q,p

4.6 Formula of Integration-by-Parts for Wε,σ (QT )

169

Putting everything together, we deduce, by taking into account (4.93), (4.94) and ∞ (Q )d and (4.97) in (4.91), that for almost every .(t, x) ∈ Q3T , every .φ ∈ C0,div T .h ∈ (0, h2 ) ∩ I∧ , it holds (4.92), i.e.,  h  ∂t S

QT ,σ φ

.



h − (SQ ) φ (t, x) 3T ,σ



≤ c Md+1 (QT φ)(t, x) + Md+1 (QT ∇φ)(t, x) .

(4.98)

Hence, using Proposition 2.12, Korn’s inequality with respect to the exponent 2, and that .p− ≥ 2, i.e., Corollary 2.1, we conclude from (4.98) in (4.88) for every ∞ d .φ ∈ C 0,div (QT ) and .h ∈ (0, h2 ) ∩ I∧ that |Eh (u), φC ∞

d| 0,div (QT )

.

 h

 h ≤ ET u L2 (Q3T )d ∂t SQ φ − (SQ ) φ L2 (Q )d T ,σ 3T ,σ 3T ≤ c u L2 (QT )d Md+1 (QT φ) L2 (Rd+1 )

+ Md+1 (QT ∇φ) L2 (Rd+1 ) ≤ c u L2 (QT )d QT φ L2 (Rd+1 )d (4.99)

+ QT ∇φ L2 (Rd+1 )d×d

= c u L2 (QT )d φ L2 (QT )d + ∇φ L2 (QT )d×d ≤ 4c (1 + |QT |)2 u Vεq,p (QT ) φ Vεq,p (QT ) .

∞ (Q )d in .V Hence, based on the density of .C0,div T ε (QT ) (cf. Proposition 4.16 (iv)), we obtain unique, and thus not relabeled, continuous extensions of q,p  (Q )d to functionals .(E (u)) ∗ .(Eh (u))h∈(0,h2 )∩I∧ ⊆ D 3T h h∈(0,h2 )∩I∧ ⊆ Vε (QT ) div that satisfy q,p

.

sup h∈(0,h2 )∩I∧

Eh (u) Vεq,p (QT )∗ ≤ 4c (1 + |QT |)2 u Vεq,p (QT ) .

(4.100)

∞ (Q )d in .V With the renewed exploitation of the density of .C0,div T ε (QT ) (cf. Proposition 4.16 (iv)), (4.90) then extends for every .h ∈ (0, h2 ) ∩ I∧ to q,p

  dσ h h ∗ dσ ET u S + Eh (u) u = (SQT ,σ ) . dt QT ,σ dt

q,p

in Vε

(QT )∗ .

(4.101)

ad (ii). The second inequality in (ii) is (4.100). Resorting to Proposition 4.14 (iii), we find that .

sup h∈(0,h2 )∩I∧

 h S

 

QT ,σ u Vεq,p (QT )

≤

sup h∈(0,h2 )∩I∧

 

 h S

Q3T ,σ (ET u) VεqT ,pT (Q3T )

≤ cq T ,pT ET u V qT ,pT (Q3T ) . ε

(4.102)

170

4 Solenoidal Variable Bochner–Lebesgue Spaces

On the other hand, making use of Proposition 4.15 (ii), we observe that sup h∈(0,h2 )∩I∧

     h ∗ dσ ET u  (S   QT ,σ ) dt

   dσ ET u   ≤ cq T ,pT   dt 

.

q ,pT

Vε T

q,p

Vε

(Q3T )∗

(QT )∗

(4.103)

.

q,p

q ,p

By combining (4.100)–(4.103), using that .ET : Wε,σ (QT ) → Wε,σT T (Q3T ) is Lipschitz continuous with constant 12 (cf. Proposition 4.21), we conclude that  h S

sup .

h∈(0,h2 )∩I∧

 

q,p QT ,σ u Wε,σ (QT )

(4.104)

  q,p ≤ 12cq T ,pT + 4c (1 + |QT |)2 u Wε,σ (QT ) .

q ,pT

h ad (iii). First, Proposition 4.14 (iv) yields .SQ u → ET u in .Vε T T ,σ .(h → 0), from which we, in turn, conclude that q,p

h SQ u→u T ,σ

in Vε

.

q,p

Using both (ii) and the reflexivity of .Vε that the family  σ h  ε (SQT ,σ u) h∈(0,h

=

h SQ u, T ,σ

dσ h S u dt QT ,σ

  h∈(0,h2 )∩I∧

  dσ ⊆G dt

.

(4.105)

(QT ) (cf. Proposition 4.2), we also detect

 2 )∩I∧

(h → 0) .

(QT )

(Q3T )

(4.106)

is (sequentially) weakly compact in .Vε (QT ) × Vε (QT )∗ . In addition, the famq,p h ily .(σε (SQ u))h∈(0,h2 )∩I∧ has a single weak accumulation point in .Vε (QT ) T ,σ q,p

q,p

σu  ) ∈ Vε (QT ) × Vε ×Vε (QT )∗ , that is .(u, ddt assume for an arbitrary cofinal subset . ⊆ R>0 that

q,p

q,p

h ˜ u˜ ∗ ) σε (SQ u)  (u, T ,σ

.

q,p

in Vε

q,p

q,p

(QT ) × Vε

(QT )∗ . To see this, we

(QT )∗

(h−1 ∈ ; h → 0) . (4.107)

Owing to the weak closedness of .G( ddtσ ) (cf. Proposition 4.18), (4.107) provides q,p q,p ˜ σu that .u˜ ∈ Wε (QT ) with . ddt = u˜ ∗ in .Vε (QT )∗ . On the other hand, (4.107) q,p h also implies that .SQ u  u˜ in .Vε (QT ) .(h → 0, h−1 ∈ ), which due T ,σ

σu  ˜ i.e., .(u, ddt ) ∈ Vε (QT ) × Vε (QT )∗ is to (4.105) proves that .u = u, h u))h∈(0,h2 )∩I∧ in the only weak accumulation point of the family .(σε (SQ T ,σ

q,p

q,p

q,p

4.6 Formula of Integration-by-Parts for Wε,σ (QT )

171

Vε (QT ) × Vε (QT )∗ . Therefore, the standard convergence principle [166, Prop. 10.13 (4)] yields that . = R>0 in (4.107). In other words, we have that

.

q,p

q,p

.

dσ h dσ u SQT ,σ u  dt dt

q,p

in Vε

(QT )∗

(h → 0) .

(4.108)

On the other hand, it follows from Proposition 4.15 (i) that h (SQ )∗ T ,σ

.



 dσ ET u dσ u dσ ET u  ∗QT = dt dt dt

q,p

in Vε

(QT )∗

(h → 0) , (4.109)

ET u = dσ u in .V q,p (Q )∗ follows easily if we test with where the identity .∗QT dσ dt T ε dt ∞ (Q )d and then refer to Definition 4.6. Finally, (4.108) and (4.109) in .φ ∈ C T 0,div q,p (4.101) prove .Eh (u)  0 in .Vε (QT )∗ .(h → 0).  Remark 4.16 (Lack of Strong Approximability of . ddtσ and the Assumption .p− ≥ 2) q,p q,p Recall that for .x ∈ Wε (QT ), we proved .RhQT x → x in .Wε (QT ) d .(h → 0), provided that . ⊆ R , .d ≥ 2, is a bounded Lipschitz domain and that q,p log .q, p ∈ P (QT ) with .q − , p− > 1, such that .X− → Y (cf. Proposition 3.25). In contrast to that, in Proposition 4.22, we have merely accomplished to q,p q,p h u → u in .Vε (QT ) .(h → 0) and show for .u ∈ Wε,σ (QT ) that .SQ T ,σ in .Vε (QT )∗ .(h → 0), provided that we have, in addition, with .p− ≥ 2 and .q ≥ p in .QT . In Remark 4.13, we have already mentioned that the assumption .q ≥ p in .QT could be overcome by more advanced approaches. Nonetheless, the lack of strong approximability of dσ u − ≥ 2 for .p ∈ Plog (Q ) can mainly be . T dt and the imposed lower bound .p h ) to preserve traced back to the inability of the quasi adjoint operator .(SQ 3T ,σ spatial compact supports (cf. Remark 4.11), which forced us to incorporate h ∞ (Q )d in (4.90) and, therefore, in φ)h∈(0,h2 )∩I∧ ⊆ C0,div the family .(SQ T ,σ

  h3T h turn, to deal with the error family . ∂t SQ φ − (SQ ) φ h∈(0,h )∩I in T ,σ 3T ,σ ∧ 2 the space .L2 (Q3T )d . In consequence, if one accomplishes to demonstrate that h h  2 d .∂t [S QT ,σ φ − (SQ3T ,σ ) φ] → 0 in .L (Q3T ) .(h → 0), or at least that .Eh (u) → 0 q,p q,p in .V (Q )∗ .(h → 0), and .(S h )∗ [ dσ ET u ] → dσ u in .V (Q )∗ .(h → 0), dσ u dσ h dt SQT ,σ u  dt log .q, p ∈ P (QT )

.

ε

T

h gets .SQ u T ,σ

q,p

QT ,σ dt q,p in .Wε,σ (QT ) .(h

dt

ε

T

then one →u → 0). Or even better, if one manages q,p to construct a smoothing operator for the space .Vε (QT ) with similar properties but having a quasi adjoint operator that preserves both compact supports and the solenoidality, then one is certainly in the position to prove an analogue of Proposition 4.22 without imposing the lower bound .p− ≥ 2 for .p ∈ Plog (QT ). After all this, aided by Proposition 4.22, we find ourselves in the position to prove q,p a formula of integration-by-parts for .Wε,σ (QT ). Apparently, the method of proof, similar to Proposition 3.26, will be based on a smoothing argument. Nevertheless, we have to proceed a little more cautiously because we have merely proved weak q,p approximability with respect to . ddtσ in .Vε (QT )∗ .

172

4 Solenoidal Variable Bochner–Lebesgue Spaces q,p

Proposition 4.23 (Formula of Integration-by-Parts for .Wε,σ (QT )) Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), .T < ∞, .QT := I × , and log .q, p ∈ P (QT ) with .p− ≥ 2 and .q ≥ p in .QT . Then, the following statements apply: q,p

(i) Each function .u ∈ Wε,σ (QT ) possesses a unique continuous representation 0 .uc ∈ H (QT ). q,p (ii) For every .u, v ∈ Wε,σ (QT ) and .t, t  ∈ I with .t  ≤ t, it holds t .

t

 dσ u (s), v(s) ds dt ˚q,p (s) V

= [(uc (s), v c (s))H ]s=t s=t  −

t t

 dσ v (s), u(s) ds . dt ˚q,p (s) V

q,p

(iii) The from (i) resulting mapping .(·)c : Wε,σ (QT ) → H0 (QT ) is a strongq,p weak-embedding, i.e., for a sequence .(un )n∈N ⊆ Wε,σ (QT ) from

.

q,p

(QT )

(n → ∞) ,

q,p

(QT )∗

(n → ∞) ,

un → u

in Vε

dσ un dσ u  dt dt

in Vε

(4.110)

it follows that .(un )c → uc in .H0 (QT ) .(n → ∞). In particular, q,p 0 .(·)c : Wε,σ (QT ) → H (QT ) is an embedding. q,p

Proof ad (i). Let .u ∈ Wε,σ (QT ) be arbitrary. Then, we choose smooth approxih mations .(uh )h∈(0,h2 )∩I∧ :=(SQ u)h∈(0,h2 )∩I∧ ⊆ C ∞ (I , V) from Proposition 4.22, T ,σ i.e., in particular, it holds q,p

(QT )

(h → 0) , .

(4.111)

q,p

(QT )∗

(h → 0) .

(4.112)

uh → u

in Vε

dσ u dσ uh  dt dt

in Vε

.

  If we introduce .c(u) := suph∈(0,h2 )∩I∧  dσdtuh V q,p (Q )∗ < ∞ (cf. (4.112)), and T ε apply the formula of integration-by-parts in time for smooth vector fields, then for

q,p

4.6 Formula of Integration-by-Parts for Wε,σ (QT )

173

every .h , h ∈ (0, h2 ) ∩ I∧ and .t, t  ∈ I , we find that uh (t) − uh (t) 2H = uh (t  ) − uh (t  ) 2H t    ∂t uh (s, y) − ∂t uh (s, y) +2

.

t

(4.113)



 · (uh (s, y) − uh (s, y)) dy ds = uh (t  ) − uh (t  ) 2H  t dσ uh dσ uh (s) − (s), uh (s) − uh (s) ds +2 dt dt ˚q,p (s) t V ≤ uh (t  ) − uh (t  ) 2H    dσ uh dσ uh   − uh − uh Vεq,p (QT ) + 2  dt dt Vεq,p (QT )∗ ≤ uh (t  ) − uh (t  ) 2H + 4c(u) uh − uh Vεq,p (QT ) , where we used in √ the second equality (4.81), Proposition 4.6, Corollary 4.2, and Remark 4.6. As . a 2 + b2 ≤ a + b for every .a, b ≥ 0, we infer from (4.113) for every .t, t  ∈ I and .h , h ∈ (0, h2 ) ∩ I∧ uh (t) − uh (t) H ≤ uh (t  ) − uh (t  ) H .

1

1

2 + 2c(u) 2 uh − uh V . q,p (Q )

(4.114)

T

ε

We integrate (4.114) with respect to .t  ∈ I and divide by .T > 0 to arrive for every  .t ∈ I and .h , h ∈ (0, h2 ) ∩ I∧ at the estimate uh (t) − uh (t) H ≤ T −1 uh − uh L1 (I,H ) .

1

1

2 + 2c(u) 2 uh − uh V . q,p (Q )

(4.115)

T

ε

As .t ∈ I was arbitrary in (4.115) and .p− ≥ 2, we infer by Corollary 2.1 for every  .h , h ∈ (0, h2 ) ∩ I∧ 1

uh − uh H0 (QT ) ≤ 2T − 2 (1 + |QT |) uh − uh Vεq,p (QT ) .

1 2

1 2

+ 2c(u) uh − u V q,p (Q ) . h

ε

T

(4.116)

174

4 Solenoidal Variable Bochner–Lebesgue Spaces

In consequence, using (4.111), we conclude from (4.116) that .(uh )h∈(0,h2 )∩I∧ is a Cauchy sequence in .H0 (QT ). Hence, owing to Proposition 2.15, there exists a function .uc ∈ H0 (QT ) such that .uh → uc in .H0 (QT ) → L2 (QT )d .(h → 0). q,p Since also .uh → u in .Vε (QT ) → L2 (QT )d .(h → 0), where we have made − use of .p ≥ 2 and Corollary 2.1, we conclude .uc = u almost everywhere in q,p .QT . Therefore, each .u ∈ Wε,σ (QT ) possesses a unique continuous representation 0 0 .uc ∈ H (QT ) and there holds .uh → uc in .H (QT ) .(h → 0), i.e., (i). q,p ad (ii). Let .u, v ∈ Wε,σ (QT ) be arbitrary. Then, we choose smooth approximah tions .(uh )h∈(0,h2 )∩I∧ := (SQ u)h∈(0,h2 )∩I∧ ⊆ C ∞ (I , V) and .(v h )h∈(0,h2 )∩I∧ := T ,σ h (SQT ,σ v)h∈(0,h2 )∩I∧ ⊆ C ∞ (I , V) from Proposition 4.22, i.e., in particular, it holds .

q,p

(QT )

(h → 0) ,

q,p

(QT )∗

(h → 0) ,

q,p

(QT )

(h → 0) ,

q,p

(QT )∗

(h → 0) .

uh → u

in Vε

dσ uh dσ u  dt dt

in Vε

vh → v

in Vε

dσ v h dσ v  dt dt

in Vε

(4.117)

With the same procedure as in the proof of (i), i.e., repeating (4.113)–(4.116), we find that .

u h → uc

in H0 (QT )

(h → 0) ,

vh → vc

in H (QT )

(h → 0) .

0

(4.118)

Apart from that, on the basis of the formula of integration-by-parts in time for smooth vector fields, it holds for every .t  , t ∈ I and .h ∈ (0, h2 ) ∩ I∧ .

[(uh (s), v h (s))H ]s=t s=t   t  = ∂t uh (s, y) · v h (s, y) + ∂t v h (s, y) · uh (s, y) dy ds t

=

t t

(4.119)



  t dσ uh dσ v h (s), v h (s) (s), uh (s) ds + ds , dt dt ˚q,p (s) ˚q,p (s) t V V

where we used for the second equality (4.81), Proposition 4.6, Corollary 4.2, and Remark 4.6. Therefore, by passing for .h → 0 in (4.119), having regard to both (4.117) and (4.118) in doing so, we conclude the desired formula of integration-byparts, i.e., (ii).

q,p

4.6 Formula of Integration-by-Parts for Wε,σ (QT )

175

q,p

ad (iii). Due to the linearity of .(·)c : Wε,σ (QT ) → H0 (QT ), it is sufficient q,p to consider a sequence .(un )n∈N ⊆ Wε,σ (QT ) satisfying (4.110) with respect to q,p  .u = 0 in .Wε,σ (QT ). Therefore, using (ii), we find for every .t, t ∈ I and .n ∈ N that .

(un )c (t) 2H

= (un )c (t



) 2H

+2

t t

 dσ un ds . (s), un (s) dt ˚q,p (s) V

(4.120)

  Following the proof of (i), introducing .c(u) := supn∈N  dσdtun V q,p (Q )∗ < ∞, we T ε infer from (4.120) that 1

(un )c H0 (QT ) ≤ 2T − 2 (1 + |QT |) un Vεq,p (QT )

.

1

1

2 + 2c(u) 2 un V q,p (Q ε

T)

→0

(n → ∞) ,

where we have made use of (4.110), i.e., it holds .(un )c → 0 in .H0 (QT ) .(n → ∞).  Similarly to Chap. 3, Sect. 3.8, we can deduce from the validity of the formula q,p of integration-by-parts for .Wε,σ (QT ) (cf. Proposition 4.23), by using the theory of maximal monotone operators –more precisely Proposition 3.27– a first abstract existence result for solenoidal generalized evolution equations (cf. Definition 4.9). Proposition 4.24 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .p− ≥ 2 and .q ≥ p in .QT . Then, q,p σ the initial trace operator .γ 0 : Wε,σ (QT ) → H , defined by .γ σ0 (u) := uc (0) in H q,p for every .u ∈ Wε,σ (QT ), is well-defined, linear and continuous. Furthermore, the following statements apply: (i) .R(γ σ0 ) is dense in H .   q,p q,p σu (ii) .G(( ddtσ , γ σ0 ) ) = (u, ddt , γ σ0 (u)) | u ∈ Wε,σ (QT ) is closed in .Vε (QT ) q,p ∗ × Vε (QT ) × H .  dσ u  q,p (iii) . dt , u V q,p (Q ) + 12 γ σ0 (u) 2H ≥ 0 for every .u ∈ Wε,σ (QT ). ε

T

Proof Follows analogously to the proof of Proposition 3.29, but now by using  Proposition 4.23. Proposition 4.25 (Maximal Monotonicity of . ddtσ ) Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), .T < ∞, .QT := I × , and .q, p ∈ Plog (QT ) with .p− ≥ 2 and .q ≥ p in .QT . Then, the generalized time derivative q,p q,p q,p dσ ∗ . dt : Wε,σ (QT ) ⊆ Vε (QT ) → Vε (QT ) is linear, densely defined and its q,p restriction to .N(γ σ0 ) := {u ∈ Wε,σ (QT ) | uc (0) = 0 in H } is maximal monotone. Proof Follows analogously to the proof of Proposition 3.30, but now by using Proposition 4.23. 

176

4 Solenoidal Variable Bochner–Lebesgue Spaces

Definition 4.10 (. ddtσ -Pseudo-Monotonicity) Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, .QT := I × , and .q, p ∈ P∞ (QT ) with .q − , p− > 1, q,p ˚εq,p (QT ) → V ˚εq,p (QT )∗ is ˚−q,p → H . An operator .A : Wε,σ such that .V (QT ) ⊆ V q,p said to be . ddtσ -pseudo-monotone, if for a sequence .(un )n∈N ⊆ Wε,σ (QT ) from q,p

in Wε,σ (QT )

un  u

.

(n → ∞) , .

lim sup Aun , un − uV ˚q,p (Q ε

n→∞

T)

(4.121)

≤ 0,

(4.122)

˚εq,p (QT ), it follows that for every .v ∈ V Au, u − vV ˚q,p (Q

.

ε

T)

≤ lim inf Aun , un − vV ˚q,p (Q ) . n→∞

ε

T

Theorem 4.1 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .p− ≥ 2 and .q ≥ p in .QT . Furq,p q,p q,p thermore, let the operator .A : Wε,σ (QT ) ⊆ Vε (QT ) → Vε (QT )∗ be coerdσ cive, . dt -pseudo-monotone and assume there exist a bounded function .ψ : R≥0 × q,p R≥0 → R≥0 and a constant .θ ∈ [0, 1) such that for every .u ∈ Wε,σ (QT ), it holds Au Vεq,p (QT )∗

.

   dσ u    . ≤ ψ( u Vεq,p (QT ) , uc (0) H ) + θ  dt Vεq,p (QT )∗

Then, for arbitrary .u0 ∈ H and .u∗ ∈ Vε such that

q,p

.

(4.123)

(QT )∗ , there exists .u ∈ Wε,σ (QT ) q,p

dσ u + Au = u∗ dt

in Vε

uc (0) = u0

in H .

q,p

(QT )∗ ,

Here, the initial condition has to be understood in the sense of the unique continuous representation .uc ∈ H0 (QT ) (cf. Proposition 4.23 (i)). Proof Follows from Proposition 3.27 in conjunction with Proposition 4.24 and Proposition 4.25, because if we argue as in Remark 3.15, it is readily seen that dσ . dt -pseudo-monotonicity (cf. Def. 4.10) is equivalent to pseudo-monotonicity with respect to . ddtσ in the sense of (3.89). 

Chapter 5

Existence Theory for Lipschitz Domains

This chapter is concerned with the development of an abstract existence theory for unsteady problems in variable exponent spaces on the basis of advanced pseudomonotonicity methods, which precisely corresponds to the title of this book. In this connection, we will formulate appropriate notions of continuity, growth, and coercivity that will allow us to automatize the process of proof. This chapter’s core objective, which can also be considered as the main result of this book, represents an abstract existence result on which all following existence results of this book will be built. In doing so, the model problems (4.1) and (4.2), i.e., the unsteady .p(·, ·)Stokes equations and the unsteady .p(·, ·)-Navier–Stokes equations, respectively, and all operators occurring therein, will serve as prototypical examples. This means that we will apply all the developed abstract theory of this chapter –which certainly has a wider scope of application– only to these two problems. Note also that we restrict our attention to the existence theory of weak solutions. An extensive study of the existence of strong solutions for (4.1) and (4.2) can be found, e.g., in [43, 45, 147]. Surprisingly, the existence theory concerning weak solutions for (4.1) and (4.2) is not developed to this extent. A brief review of historical milestones in the weak solvability of the unsteady .p(·, ·)-Navier–Stokes equations is deferred to the beginning of Chap. 8. We have already elaborated an abstract existence result in the previous chapter by Theorem 4.1. We emphasize, though, that Theorem 4.1 possesses an extremely limited scope of application, which can primarily be traced back to the constraints on the variable exponents, i.e., we had to require that .q, p ∈ Plog (QT ) are subject to − ≥ 2 and .q ≥ p in .Q to have access to the formula of integration-by-parts for .p T q,p .Wε,σ (QT ) (cf. Proposition 4.23). Remember that the formula of integration-byq,p parts for .Wε (QT ) (cf. Proposition 3.26) forgoes the requirements .p− ≥ 2 and − ≥ 2d or .q − ≥ 2. .q ≥ p in .QT , but gets along with .p d+2 To demonstrate this restrictive character, let us consider by analogy with Chap. 3, d×d Sect. 3.9, for a mapping .S : QT × Md×d sym → Msym satisfying the conditions (S.1)– © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaltenbach, Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents, Lecture Notes in Mathematics 2329, https://doi.org/10.1007/978-3-031-29670-3_5

177

178

5 Existence Theory for Lipschitz Domains

2d , the operator (S.4) with respect to an exponent .p ∈ Plog (QT ) with .p− ≥ d+2 p,p p,p p,p ∗ ˚ε (QT ) → V ˚ε (QT ) , for every .u ∈ V ˚ε (QT ) defined by .S : V

Su := Jεσ (0, S(·, ·, ε(u)))

.

˚ε (QT )∗ . in V p,p

(5.1)

Then, along the lines of the proof of Proposition 3.31, we can observe that ˚εp,p (QT ) → V ˚εp,p (QT )∗ is well-defined, bounded, continuous, monotone, S: V dσ coercive, and . dt -pseudo-monotone. Nevertheless, in order to find oneself in the position to apply Theorem 4.1, all these properties need to be satisfied with respect ˚εq,p (QT ), where at least .q ∈ Plog (QT ) with .q ≥ p in to a smaller energy space .V ˚εq,p (QT ) → V ˚εq,p (QT )∗ .QT . Apparently, the restricted, not relabeled operator .S : V dσ remains bounded, continuous, monotone, and . dt -pseudo-monotone. Unfortunately, ˚εq,p (QT ) → V ˚εq,p (QT )∗ loses its coercivity property. This issue indicates an .S : V imbalance between the demanded continuity and growth conditions in Theorem 4.1. More precisely, while the required . ddtσ -pseudo-monotonicity is quite general, coercivity is a restrictive assumption, which –as in this case– is often not fulfilled. In [92, 95], the same problem has already been considered in the framework of classical Bochner–Lebesgue spaces and Bochner–Sobolev spaces. The joint approach of these contributions consists in the introduction of alternative notions of pseudo-monotonicity and coercivity, which –in contrast to . ddtσ -pseudo-monotonicity and coercivity– both incorporate information from the time derivative, and are thus more balanced. The underlying idea is to weaken the notion of pseudo-monotonicity to a bearable level in order to make accessible a coercivity condition that takes into account the additional information from the time derivative. This approach led to the abstract notions of Bochner pseudo-monotonicity, Bochner condition (M), and Bochner coercivity. We will follow the joint approach of [92, 95]. In that sense, the first section of this chapter addresses the extension of Bochner pseudo-monotonicity, Bochner condition (M), and Bochner coercivity to the framework of solenoidal variable Bochner–Lebesgue spaces. .

5.1 Bochner Pseudo-Monotonicity, Bochner Condition (M) and Bochner Coercivity In this section, we generalize the notions of Bochner pseudo-monotonicity, Bochner condition (M), and Bochner coercivity, which have been introduced in [92, 95] in ˚εq,p (QT ). the setting of classical Bochner–Lebesgue spaces, to the framework of .V We emphasize that all following definitions and results admit congruent adaptations ˚q,p (QT ), .X ˚εq,p (QT ), and .X ˚q,p (QT ), and we restrict our attention to to the spaces .V ∇ ∇ q,p ˚ε (QT ) solely for ease of presentation. Moreover, throughout the entire chapter, .V unless otherwise specified, let . ⊆ Rd , .d ≥ 2, is a bounded domain, .I := (0, T ), ∞ − − > 1. .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .q , p

5.1 Bochner Pseudo-Monotonicity, Bochner Condition (M) and Bochner. . .

179

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is said to Definition 5.1 An operator .A : V ˚εq,p (QT )∩H∞ (QT ) from (i) be Bochner strongly continuous, if for .(un )n∈N ⊆ V q,p

un  u

.

∗

un  u un (t)  u(t)

˚ε (QT ) (n → ∞) , . in V

(5.2)

in H∞ (QT )

(n → ∞) , .

(5.3)

in H

(n → ∞)

for a.e. t ∈ I ,

(5.4)

it follows that ˚ε (QT )∗ in V q,p

Aun → Au

.

(n → ∞) .

˚εq,p (QT ) ∩ H∞ (QT ) from (ii) be Bochner pseudo-monotone, if for .(un )n∈N ⊆ V (5.2)–(5.4) and .

lim sup Aun , un − uV ˚q,p (Q ε

n→∞

T)

≤ 0,

(5.5)

˚εq,p (QT ), it follows that for every .v ∈ V

Au, u − vV ˚q,p (Q

.

T)

ε

≤ lim inf Aun , un − vV ˚q,p (Q ) . n→∞

ε

T

˚εq,p (QT ) ∩ H∞ (QT ) (iii) satisfy the Bochner condition (M), if for .(un )n∈N ⊆ V from (5.2)–(5.4) and Aun  u∗

.

˚ε (QT )∗ in V q,p

lim sup Aun , un V ˚q,p (Q ε

n→∞

T)

(n → ∞) , .

(5.6)

≤ u∗ , uV ˚q,p (Q ) ,

(5.7)

ε

T

it follows that ˚ε (QT )∗ . Au = u∗ in V

.

q,p

We will find in the proof of Theorem 5.1 that (5.2)–(5.7) are natural properties of ˚εq,p (QT )∩ .H∞ (QT ) coming from a suitable Galerkin a sequence .(un )n∈N ⊆ V approximation of (4.78), if .A satisfies appropriate additional assumptions. In fact, (5.2) usually is an immediate consequence of the respective notion of coercivity of .A, (5.3) stems from the generalized time derivative . ddtσ , while (5.4) and (5.5) follow directly from the Galerkin approximation. The following proposition illustrates the relations between these new notions, also in direct comparison with the standard notions from the classical theory of pseudo-monotone operators (cf. Definition 2.1).

180

5 Existence Theory for Lipschitz Domains

Proposition 5.1 The following statements apply: ˚εq,p (QT ) → V ˚εq,p (QT )∗ is pseudo-monotone, or satisfies the con(i) If .A : V dition (M), then it is Bochner pseudo-monotone, or satisfies the Bochner condition (M), respectively. ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is Bochner strongly continuous, (ii) If .A : V then it is Bochner pseudo-monotone. ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is Bochner pseudo-monotone, (iii) If .A : V then it satisfies the Bochner condition (M). ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ are Bochner pseudo(iv) If .A, B : V q,p ˚ε (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ monotone, then the sum .A + B : V is Bochner pseudo-monotone. Proof ad (i). Results by reference to the respective definitions (cf. Definition 2.1 and Definition 5.1). ˚εq,p (QT ) ∩ H∞ (QT ) be a sequence satisfying (5.2)– ad (ii). Let .(un )n∈N ⊆ V ˚εq,p (QT )∩H∞ (QT ). Inasmuch as .A : V ˚εq,p (QT )∩ (5.5) with respect to some .u ∈ V q,p ∞ ∗ ˚ε (QT ) is Bochner strongly continuous, we infer that .Aun → H (QT ) → V ˚εq,p (QT )∗ .(n → ∞), which implies that Au in .V

Au, u − vV ˚q,p (Q

.

T)

ε

= lim inf Aun , un − vV ˚q,p (Q n→∞

T)

ε

˚εq,p (QT ). for every .v ∈ V ˚εq,p (QT ) ∩ H∞ (QT ) be a sequence satisfying (5.2)– ad (iii). Let .(un )n∈N ⊆ V ˚εq,p (QT ) ∩ H∞ (QT ) (5.4), (5.6) and (5.7) with respect to some .u ∈ V q,p ∗ ∗ ˚ and .u ∈ Vε (QT ) . Apparently, (5.2), (5.6) and (5.7) imply (5.5). Thus, ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ the Bochner pseudo-monotonicity of .A : V q,p ˚ yields for every .v ∈ Vε (QT ) that

Au, u − vV ˚q,p (Q

.

ε

T)

≤ lim inf Aun , un − vV ˚q,p (Q n→∞

T)

ε

≤ lim sup Aun , un − uV ˚q,p (Q

T)

ε

n→∞

+ lim Aun , u − vV ˚q,p (Q n→∞

ε

T)

∗

≤ u , u − vV ˚q,p (Q ) , ε

T

˚εq,p (QT )∗ . i.e., .Au = u∗ in .V ad (iv). A straightforward adaptation of the proof of Proposition 3.28.



5.1 Bochner Pseudo-Monotonicity, Bochner Condition (M) and Bochner. . .

181

Using Proposition 5.1 (i), we can derive the first example of a Bochner pseudomonotone operator, which occurs in both the unsteady .p(·, ·)-Stokes equations, i.e., (4.1), and the unsteady .p(·, ·)-Navier–Stokes equations, i.e., (4.2), namely the unsteady extra stress tensor. ˚εp Proposition 5.2 The unsteady extra stress tensor .S : V defined by (5.1), is Bochner pseudo-monotone.

−,p

p−,p

˚ε (QT ) → V

(QT )∗ ,

Proof By proceeding as in the proof of−Proposition 3.31,− it is readily seen that ˚εp ,p (QT ) → V ˚εp ,p (QT )∗ is monotone, the unsteady extra stress tensor .S : V continuous and, therefore, pseudo-monotone. Consequently, the assertion follows immediately by applying Proposition 5.1 (i). 

Proposition 5.2 helps us to see that the implication in Proposition 5.1 (ii) is strict. Remark 5.1 (Bochner Pseudo-Monotonicity .⇒ Bochner Strong Continuity) For d×d a constant exponent .p ∈ (1, ∞), the mapping .S : Md×d sym → Msym , defined by d×d p−2 .S(A) := |A| A for all .A ∈ Msym , fulfills the conditions (S.1)–(S.4). Therefore, p,p ˚ ˚εp,p (QT )∗ , defined by .Su := J σ (0, S(ε(u))) in .V ˚εp,p (QT )∗ .S : Vε (QT ) → V ε p,p ˚ε (QT ), is Bochner pseudo-monotone (cf. Proposition 5.2). for every .u ∈ V ˚p,p be a However, it is not Bochner strongly continuous. In fact, let .(un )n∈N ⊆ V ˚p,p .(n → ∞), but .un → 0 in .V ˚p,p .(n → ∞). Then, sequence such that .un  0 in .V p,p ∞ ˚ ˚p,p the sequence .(un )n∈N ⊆ Vε (QT ) ∩ H (QT ), defined by .un (t) := un in .V for every .t ∈ I and .n ∈ N, satisfies (5.2)–(5.4). Now suppose that .Sun → S0 = 0 ˚εp,p (QT )∗ .(n → ∞). Then, we observe, also using Poincaré’s inequality for the in .V constant exponent .p ∈ (1, ∞) (cf. Proposition 3.5 or (3.16)), that .

p

lim sup un  ˚p,p

Vε (QT )

n→∞

≤ lim sup cp Sun , un V ˚p,p (Q ε

n→∞

T)

= 0,

˚p,p .(n → ∞). Thus, .Sun → S0 i.e., a contradiction, as then .un → 0 in .V p,p ∗ ˚εp,p (QT ) → V ˚εp,p (QT )∗ is not ˚ in .Vε (QT ) .(n → ∞). In other words, .S : V Bochner strongly continuous. The unsteady .p(·, ·)-Navier–Stokes equations, i.e., (4.2), apart from that, contain an example of a non-monotone but Bochner pseudo-monotone operator, namely the unsteady convective term. − Proposition 5.3 For .d ∈ N, we define .pC := 3d+2 d+2 .−If .p ≥ pC , then the unsteady −,p p p ,p ˚ε (QT ) ∩ H∞ (QT ) → V ˚ε (QT )∗ , defined by convective term .C : V

Cu := Jεσ (0, −u ⊗ u)

.

p−,p

˚εp in V

−,p

(QT )∗

˚ε (QT ) ∩ H∞ (QT ), is well-defined, bounded and Bochner for every .u ∈ V − − ˚εp ,p (QT ) ∩ H∞ (QT ) → V ˚εp ,p (QT )∗ pseudo-monotone. If .p− > pC , then .C : V is even Bochner strongly continuous.

182

5 Existence Theory for Lipschitz Domains

Proof 1. Well-definedness −and Boundedness: It suffices to show that the mapping ˚εp ,p (QT ) ∩ H∞ (QT ) → Lp (·,·) (QT , Md×d ) is well-defined .(u → u ⊗ u) : V sym

−

−   ˚p ,p (QT )∗ is and bounded since .Jεσ : L(p ) (QT )d × Lp (·,·) (QT , Md×d sym ) → Vε −,p p ˚ε (QT ) ∩ .H∞ (QT ), bounded (cf. Proposition 4.17). Clearly, for every .u ∈ V  the function .((t, x) → u(t, x) ⊗ u(t, x)) : .QT → Md×d sym is Lebesguemeasurable. Therefore, we can examine it for integrability. In doing so, making use of Corollary 2.1, the parabolic interpolation inequality (cf. Proposition 3.7) for the exponent .pC , and that .2pC = (pC )∗ , we observe that

u ⊗ uLp (·,·) (QT )d×d ≤ 2(1 + |QT |)u2 2p L

C (QT )d

≤ 2(1 + |QT |)cpC u2˚pC ,pC

.

Vε

≤ 8(1 + |QT |)3 cpC u2˚p−,p Vε

p−,p

(5.8)

(QT )∩H∞ (QT )

(QT )∩H∞ (QT )

,



˚ε (QT ) ∩ H∞ (QT ) → Lp (·,·) (QT , Md×d ) is welli.e., .(u → u ⊗ u) : V sym defined and bounded. − ˚εp ,p (QT ) ∩ H∞ (QT ) be a 2. Bochner pseudo-monotonicity: Let .(un )n∈N ⊆ V sequence satisfying (5.2)–(5.5). Using Corollary 3.3, for every .s ∈ [1, (p− )∗ /2), we obtain un ⊗ un → u ⊗ u

.

in Ls (QT , Md×d sym )

(n → ∞) .

(5.9)



Since, due to (5.8), the sequence .(un ⊗un )n∈N ⊆ Lp (·,·) (QT , Md×d sym ) is bounded, we infer from (5.9) that un ⊗ un  u ⊗ u

.



in Lp (·,·) (QT , Md×d sym ) (n → ∞) ,

(5.10)

−

˚εp ,p (QT )∗ .(n → ∞), due to the weak continuity of i.e., .Cun  Cu in .V − −  σ (p ) (Q )d × Lp (·,·) (Q , Md×d ) → V ˚εp ,p (QT )∗ (cf. Proposition 4.17). .Jε : L T T sym Thus, using . Cu, uV ˚εp−,p (QT ) = Cun , un V ˚εp−,p (QT ) = 0 for all .n ∈ N, we conclude from (5.10) that .

lim Cun , un − vV ˚p−,p (Q

n→∞

−

T)

ε

−

= Cu, u − vV ˚p−,p (Q ε

T) −

˚εp ,p (QT ), i.e., .C : V ˚εp ,p (QT ) ∩ H∞ (QT ) → V ˚εp ,p (QT )∗ is for every .v ∈ V Bochner pseudo-monotone. − ˚εp ,p (QT ) ∩ H∞ (QT ) be a 4. Bochner strong continuity: Let .(un )n∈N ⊆ V sequence satisfying (5.2)–(5.4). Then, making use of Corollary 3.3, Corollary 2.1

5.1 Bochner Pseudo-Monotonicity, Bochner Condition (M) and Bochner. . .

183

and that .(p− )∗ > 2(p− ) ≥ 2p holds almost everywhere in .QT for .p− > pC , we find that un ⊗ un → u ⊗ u

.



in Lp (·,·) (QT , Md×d sym ) (n → ∞) .

(5.11)

−   ∗ ˚p,p Eventually, since of .Jεσ : L(p ) (QT )d × .Lp (·,·) (QT , Md×d sym ) → Vε (QT ) is continuous (cf. Proposition 4.17), we conclude from (5.11) that .Cun → Cu in ˚εp,p (QT )∗ .(n → ∞). .V 

The unsteady convective term also represents the first example of a Bochner pseudo-monotone operator that is not pseudo-monotone, which gives us further assurance that the notion of Bochner pseudo-monotonicity is a proper generalization of the notion of pseudo-monotonicity. Remark 5.2 (Bochner Pseudo-Monotonicity ⇒ . Pseudo-Monotonicity) Let .I−:=(0,2π ) − ˚εp ,p (QT ) → V ˚εp ,p (QT )∗ and .p− > 3. Then, the unsteady convective term .C : V is well-defined, bounded, but neither strongly continuous, nor pseudo-monotone. − ˚εp ,p (QT ), In fact, let .u, v ∈ V with .(u ⊗ u, ε(v))L2 ()d×d < 0 and .(un )n∈N ⊆ V := u sin(nt) for every .t ∈ I and .n ∈ N. Then, we have that defined by .un (t) − ˚εp ,p (QT ) .(n → ∞)1 and .lim sup .un  0 in .V ˚p−,p (QT ) ≤ 0, n→∞ Cun , un − 0V p−,p ε ˚ − since for every .n ∈ N, . Cun , un V (QT ), defined ˚p ,p (Q ) = 0. But for .v ∈ Vε ε

T

by .v(t) := v for every .t ∈ I , it holds

.

lim inf Cun , un −vV ˚p−,p (Q )= π(u ⊗ u, ε(v))L2 ()d×d < 0 = C0, 0−vV ˚p−,p (Q ) , n→∞

ε

−

˚εp ,p (QT ) → V ˚εp i.e., .C : V strongly continuous.

T

−,p

ε

T

(QT )∗ is not pseudo-monotone and, therefore, not

The following remark, in addition, shows that Bochner pseudo-monotonicity is a stronger property than the Bochner condition (M), whereby the battle lines between all notions introduced in Definition 5.1 should have been drawn. Remark 5.3 (Bochner Condition (M) .⇒ Bochner Pseudo-Monotonicity) Let ˚2,2 , if equipped with the scalar I := (0, T ), .T < ∞, and .p := 2. Since .V product .(·, ·)V˚2,2 := (ε(·), ε(·))L2 ()d×d , forms a separable Hilbert space, the ˚2,2 . Then, Gram–Schmidt process provides an orthonormal basis .(un )n∈N ⊆ V 2,2 2,2 ∗ σ ˚ ˚ ˚2,2 (QT ) .A : Vε (QT ) → Vε (QT ) , defined by .Au := Jε (0, −ε(u)) in .V ε 2,2 ˚ (QT ), satisfies the Bochner condition (M), since it is weakly for all .u ∈ V ε continuous, but it is not Bochner pseudo-monotone. Indeed, the sequence ∞ ˚2,2 ), defined by .un (t) := un in .V ˚2,2 for all .t ∈ I and .n ∈ N, .(un )n∈N ⊆ C (I , V .

1 .(f ) n n∈N

⊆ C ∞ (I ), .I := (0, 2π ), defined by .fn (t) := sin(nt) for every .t ∈ I and .n ∈ N, satisfies ∗

 0 in .L2 (I ) .(n → ∞), i.e., .fn  0 in .L∞ (I ) .(n → ∞), owing to .supn∈N fn L∞ (I ) ≤ 1, as well as .fn 2L2 (I ) → π .(n → ∞), but not .fn (t) → 0 .(n → ∞) for almost every .t ∈ I .

.fn

184

5 Existence Theory for Lipschitz Domains

˚2,2 ) .(n → ∞)2 and .lim supn→∞ Aun , un − 0 ˚2,2 satisfies .un  0 in .C 0 (I , V Vε (QT )= −T < 0, but .lim infn→∞ Aun , un − vV ˚ε2,2 (QT ) = −T < 0 = A0, 0−vV ˚ε2,2 (QT ) 2,2 ˚ for every .v ∈ V (QT ). ε

Next, we introduce a relaxed notion of coercivity, which –in contrast to the classical notion of coercivity (cf. Definition 2.1 (ix))– additionally takes into account information provided by the generalized time derivative . ddtσ . ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is said Definition 5.2 An operator .A : V to be ˚εq,p (QT )∗ and .u0 ∈ H , if there exists (i) Bochner coercive with respect to .u∗ ∈ V ∗ ˚εq,p (QT )∩H∞ (QT ) a constant .M := M(u , u0 ) > 0 such that for every .u ∈ V from 1 1 2 ∗ 2 u(t) . ˚εq,p (QT ) ≤ u0 H H + Au − u , uχ[0,t] V 2 2

for a.e. t ∈ I ,

(5.12)

it follows that uV ˚q,p (Q

.

ε

T )∩H

∞ (Q ) T

≤ M.

(ii) Bochner coercive, if it is Bochner coercive with respect to every .u∗ ∈ ˚εq,p (QT )∗ and .u0 ∈ H . V Note that Bochner coercivity, similar to semi-coercivity (cf. [143, 149]) in conjunction with Grönwall’s inequality, takes into account the information of both the operator and the generalized time derivative . ddtσ . In fact, Bochner coercivity is a more general property. Bochner coercivity is phrased in the spirit of a local q,p ˚εq,p (QT ) → V ˚εq,p (QT )∗ . To be coercivity3 condition of . ddtσ + A : Wε,σ (QT ) ⊆ V more precise, suppose that the assumptions of Proposition 4.23 are satisfied and that ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is Bochner coercive with the operator .A : V q,p q,p ∗ ∗ ˚ respect to .u ∈ Vε (QT ) and .u0 ∈ H . Then, for .u ∈ Wε,σ (QT ) from uc (0)H ≤ u0 H ,

.

˚2,2 .(n → ∞) and the characterization of weak convergence in we combine .un  0 in .V ˚2,2 ) (cf. Proposition 2.16). the space .C 0 (I , V 3 An operator .A : D(A) ⊆ X → X ∗ is said to be locally coercive (cf. [168, §32.4.]) with respect to .x ∗ ∈ X∗ , if .D(A) is unbounded and there exists a constant .M > 0 such that for every .x ∈ X from . Ax, xX ≤ x ∗ , xX , it follows that .xX ≤ M, i.e., all elements whose images with respect to .A : D(A) ⊆ X → X∗ do not grow beyond the data .x ∗ ∈ X∗ in this weak sense, are contained in a fixed ball in X.

2 Here,

5.1 Bochner Pseudo-Monotonicity, Bochner Condition (M) and Bochner. . . σu i.e., . ddt , uV ˚q,p (Q

T)

ε

 .

185

≥ − 12 u0 2H , and

 dσ u + Au, uχ[0,t] ≤ u∗ , uχ[0,t] V ˚εq,p (QT ) dt ˚εq,p (QT ) V

for a.e. t ∈ I ,

(5.13)

it follows that ˚εq,p (QT )∩H∞ (QT ) ≤ M , uV

.

because (5.13) is just (5.12) (cf. Proposition 4.23 (ii)). In other words, if the image q,p of .u ∈ Wε,σ (QT ) with respect to . ddtσ and .A is bounded by the data .u0 , .u∗ in ˚εq,p (QT ) ∩ H∞ (QT ). We this weak sense, then .u is contained in a fixed ball in .V ˚εq,p (QT ) ∩ H∞ (QT ) chose (5.12) instead of (5.13) in Definition 5.2 because .u ∈ V is not admissible in (5.13). Note also that there is a relation between Bochner coercivity and the original ˚εq,p (QT ) ∩ H∞ (QT ) ⊆ notion of coercivity (cf. Definition 2.1 (ix)). In fact, if .A : V q,p q,p ∗ ˚ ˚ Vε (QT ) → Vε (QT ) is in a weak sense bounded from below by a bounded ˚εq,p (QT ) ∩ H∞ (QT ) ⊆ V ˚εq,p (QT ) → R, then Bochner coercivity mapping . : V extends the classical concept of coercivity. ˚εq,p (QT ) ∩ H∞ (QT ) ⊆ V ˚εq,p (QT ) → V ˚εq,p (QT )∗ Proposition 5.4 Let .A : V ˚εq,p (QT ) ∩ be coercive and assume that there exists a bounded mapping . : V q,p q,p ∞ ˚ε (QT ) → R such that for every .u ∈ V ˚ε (QT ) ∩ H∞ (QT ), H (QT ) ⊆ V it holds

Au, uχ[0,t] V ˚q,p (Q

.

ε

T)

≥ (u)

for all t ∈ I .

(5.14)

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is Bochner coercive. Then, .A : V ˚εq,p (QT )∩H∞ (QT ) → V ˚εq,p (QT )∗ is Bochner Proof It suffices to show that .A : V q,p ∗ ˚ε (QT ) and .u0 ∈ H . For .u∗ ∈ V ˚εq,p (QT )∗ \ {0}, coercive with respect to .0 ∈ V q,p ∗ ∞ ˚ε (QT ) ∩ H (QT ) → V ˚εq,p (QT )∗ . we consider the shifted operator .A − u : V q,p q,p ∞ ∗ ˚ε (QT ) ∩ H (QT ) → V ˚ε (QT ) is Bochner coercive with To show that .A : V ˚εq,p (QT ) ∩ H∞ (QT ) satisfies respect to .0 and .u0 , we assume that .u ∈ V .

1 1 2 u(t)2H + Au, uχ[0,t] V ˚εq,p (QT ) ≤ u0 H 2 2

for a.e. t ∈ I .

(5.15)

˚εq,p (QT ) ∩ H∞ (QT ) ⊆ V ˚εq,p (QT ) → V ˚εq,p (QT )∗ is coercive, there Since .A : V exists a constant .R > 0 such that . Av, vV ˚εq,p (QT ) ≥ vV ˚εq,p (QT ) for all .v ∈ q,p ∞ ˚ ≥ R. We set .M0 := max{R, 1 u0 2 } and Vε (QT ) ∩ H (QT ) with .v ˚q,p Vε (QT )

2

H

186

5 Existence Theory for Lipschitz Domains

assume that .uV ˚q,p (Q ε

T)

> M0 ≥ R. Thus, using the coercivity and (5.15), we find

1 2 that .M0 < uV ˚εq,p (QT ) ≤ Au, uV ˚εq,p (QT ) ≤ 2 u0 H ≤ M0 , i.e., a contradiction, ˚q,p and .uV ˚εq,p (QT ) ≤ M0 has to be valid. As a consequence, since . : Vε (QT ) ∩ q,p ∞ ˚ε (QT ) → R is bounded, we obtain a constant .λ > 0 such that H (QT ) ⊆ V ˚εq,p (QT ) ∩ H∞ (QT ) with .v ˚q,p .|(v)| ≤ λ for all .v ∈ V ≤ M0 . When the

latter is used in conjunction with .uV ˚q,p (Q ε

T)

ε

T )∩H

u2H∞ (QT ) ≤ u0 2H + 2λ, i.e., .uV ˚q,p (Q

.

Vε (QT )

≤ M0 and (5.14) in (5.15), we infer 1

∞ (Q ) T

≤ M0 + (u0 2H + 2λ) 2 .



Proposition 5.4 enables us to identify first examples of Bochner coercive operators. Proposition 5.5 Let .p ∈ P∞ (QT ) with .p− > 1. Then, it holds: −

−

˚εp ,p (QT ) → V ˚εp ,p (QT )∗ (cf. Proposition 5.2) is Bochner coercive. (i) .S : V − − − ˚εp ,p (QT ) ∩ H∞ (QT ) → V ˚εp ,p (QT )∗ (ii) If .p ≥ pC , then the sum .S + C : V (cf. Proposition 5.3) is Bochner coercive. −

˚εp ,p (QT ) → Proof ad (i). Follows from Proposition 5.4 because .S : V −,p p ∗ ˚ε (QT ) is obviously coercive (cf. Proposition 3.31 and Remark 3.17 (i)) V − ˚εp ,p (QT ) and .t ∈ I that . Su, uχ[0,t]  ˚p−,p and satisfies for every .u ∈ V Vε (QT ) ≥ (u) := −c0 ρp(·,·) (δ) − c1 L1 (QT ) (cf. (3.111)), i.e., the condition (5.14). ad (ii). Follows from Proposition 5.4 as . Cu, uχ[0,t] V ˚εp−,p (QT ) = 0 for all − p−,p ∞ ˚ ˚εp ,p (QT ) ∩ .u ∈ Vε (QT ) ∩ −H (QT ) and .t −∈ I and, thus, the sum .S + C : V ˚εp ,p (QT ) → V ˚εp ,p (QT )∗ is coercive and satisfies the condiH∞ (QT ) ⊆ V tion (5.14) with . ≡ −c0 ρp(·,·) (δ) − c1 L1 (QT ) . 

Remark 5.4 (Synergy Between Bochner Coercivity and Poincaré’s Inequality for Variable Bochner–Lebesgue Spaces) Bochner coercivity unfolds its true strength in combination with Poincaré’s inequality for variable Bochner–Lebesgue spaces (cf. log Proposition 3.6). To see this, we consider for a variable exponent .p ∈ P (QT ) with p−,p p−,p − ∞ ∗ ˚ ˚ .p > 1, an operator .A : Vε (Q ) ∩ H (Q ) → V (Q ) that is Bochner T T T ε − ˚εp ,p (QT )∗ and .u0 ∈ H , i.e., there exists a constant coercive with respect to .u∗ ∈ V − ∗ ˚εp ,p (QT ) ∩ H∞ (QT ) from .M := M(u , u0 ) > 0 such that for .u ∈ V .

1 1 2 u(t)2H + Au − u∗ , uχ[0,t] V ˚εp−,p (QT ) ≤ u0 H 2 2

for a.e. t ∈ I ,

it follows that uV ˚p−,p (Q

. −

ε

−

T )∩H

∞ (Q ) T

≤M,

(5.16) −

˚εp ,p (QT ) → V ˚εp ,p (QT )∗ or .S + C : V ˚εp ,p (QT ) ∩ H∞ (QT ) → e.g.,− .S : V p ,p ∗ − ˚ Vε (QT ) , if, in addition, .p ≥ pC (cf. Proposition 5.5).

5.1 Bochner Pseudo-Monotonicity, Bochner Condition (M) and Bochner. . .

187

Certainly, the estimate (5.16) is too weak for our methods, in the sense that we ˚εq,p (QT ) ∩ H∞ (QT )-norm, where at least must guarantee such an estimate in the .V log .q, p ∈ P (QT ) with .q ≥ p in .QT , to have access to the formula of integrationq,p by-parts for .Wε,σ (QT ) (cf. Proposition 4.23), leaving aside the assumption − .p ≥ 2. Fortunately, Poincaré’s inequality for variable Bochner–Lebesgue spaces (cf. Proposition 3.6), or more precisely Proposition 3.8, allow us to deduce from (5.16) for arbitrary .ε ∈ (0, (p− )∗−− 1] the existence of a constant .M  > 0 ˚εp ,p (QT ) ∩ H∞ (QT )) such that (depending on .ε > 0, but not on .u ∈ V uLp∗ (·,·)−ε (QT )d ≤ M  .

.

log As a consequence,  2 −  for .q := max{2, p∗ − ε} ∈ P (QT ), i.e., .q ≥ p in .QT provided that .ε ∈ 0, d p , using that

uLq(·,·) (QT )d ≤ uLp∗ (·,·)−ε (QT )d + uL2 (QT )d ,

.

we can, nonetheless, guarantee that uV ˚q,p (Q

.

ε

T )∩H

1

∞ (Q ) T

≤ M  + (1 + T 2 )M .

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ ˚εq,p (QT ) )∗ A : V Thus, the restriction .(idV q,p ˚ε (QT )∗ and ˚εq,p(QT ) )∗ u∗ ∈ V is at least Bochner coercive with respect to .(idV ∞ .u0 ∈ H , i.e., we can still retrieve some regularity by incorporating the .H (QT )dσ information, which usually comes from the generalized time derivative . dt , whereas classical coercivity ignores additional information from . ddtσ . This is the major benefit of Bochner coercivity in comparison with the classical notion of coercivity, and also reflects once again the special interplay of the theories of variable Sobolev spaces and Bochner–Lebesgue spaces in variable Bochner–Lebesgue spaces. Recall that the vector-valued mapping .b : QT × Rd → Rd satisfying (B.1)–(B.3) from Chap. 3 possesses growth conditions solely with respect to the constant limit exponents. This can primarily be traced back to the circumstance that we had to work with the classical concept of coercivity. Bochner coercivity enables us to deal with non-monotone variable exponent non-linearities, such as, e.g., the following.

188

5 Existence Theory for Lipschitz Domains

Proposition 5.6 Let .p, q := max{2, p∗ − ε}, r := max{2, p∗ } − ε ∈ C 0 (QT )4 with − > 1 and .ε ∈ (0, (p − ) − 1]. Furthermore, let .d : Q × Rd → Rd be a mapping .p ∗ T with the following properties: (D.1) .d : QT × Rd → Rd is a Carathéodory mapping. d  (D.2) .|d(t,x, a)| ≤ γ (1 + |a|)r(t,x)−1 + η(t,  x) for every .a ∈ R and a.e. .(t, x) ∈  (·,·) r QT . γ ≥ 1, η ∈ L (QT , R≥0 ) . 2 + c (t, x) for every .a ∈ Rd and a.e. .(t, x) ∈ Q (D.3) .d(t, x, a) · a ≥ −c |a| 2 3 T   1 . c2 ≥ 0, c3 ∈ L (QT ) . ˚εq,p (QT ) → V ˚εq,p (QT )∗ , defined by Then, the operator .D : V Du := Jεσ (d(·, ·, u), 0)

.

˚εq,p (QT )∗ , in V

˚εq,p (QT ), is well-defined, bounded and Bochner strongly continufor every .u ∈ V ous. Proof 1. Well-definedness and Boundedness: Due to Proposition 4.17 (ii), it suffices ˚εq,p (QT ) → Lq  (·,·) (QT )d is well-defined and to show that .(u → d(·, ·, u)) : .V q,p ˚ε (QT ) be arbitrary. Using (D.1), we deduce the Lebesgue bounded. Let .u ∈ V measurability of .((t, x) → d(t, x, u(t, x))) : QT → Rd . Then, by the aid of r  (t,x) ≤ 2(r − ) (a r  (t,x) + br  (t,x) ) and .(a + b)r(t,x) ≤ 2r + (a r(t,x) + br(t,x) ) .(a + b) for all .a, b ≥ 0 and .(t, x) ∈ QT , also making use of (D.2), we obtain  (r − )  γ ρr(·,·) (1 + |u|) + ρr  (·,·) (η)   −  −  + ≤ 2(r ) γ (r ) 2r |QT | + ρr(·,·) (u) + ρr  (·,·) (η) ,

ρr  (·,·) (d(·, ·, u)) ≤ 2(r .

− )

(5.17)

˚εq,p (QT ) → Lq  (·,·) (QT )d is well-defined and bounded, i.e., .(u → d(·, ·, u)) : V −  since, apart from (5.17), .ρq  (·,·) (d(·, ·, u)) ≤ 2(r ) (|QT | + ρr  (·,·) (d(·, ·, u))) and q + (|Q | + ρ   .ρr(·,·) (u) ≤ 2 T q(·,·) (u)) due to .q ≥ r in .QT , i.e., .r ≥ q in .QT . q,p ∞ ˚ 2. Bochner strong continuity: Let .(un )n∈N ⊆ Vε (QT ) ∩ H (QT ) be a sequence satisfying (5.2)–(5.4). Then, applying Proposition 3.13, we obtain un → u

.

4 If .p −

≥ 2 and .ε ∈ (0, d2 p − ], or .p − > .QT and, thus, .q = r in .QT .

in Lr(·,·) (QT )d

2d d+2

(n → ∞) .

(5.18)

and .ε ∈ (0, (p − )∗ − 2], then we have .p∗ − ε ≥ 2 in

5.2 The Hirano–Landes Approach

189

By proceeding as for the estimate (5.17), we also deduce for any measurable set K ⊆ QT and .n ∈ N

.

ρr  (·,·) (d(·, ·, un )χK ) ≤ 2(r

.

− )

 (r − ) r +    γ 2 |K| + ρr(·,·) (un χK ) + ρr  (·,·) (ηχK ) , 



i.e., .(d(·, ·, un ))n∈N ⊆ Lr (·,·) (QT )d is .Lr (·,·) (QT )-uniformly integrable (cf. (5.18)). Furthermore, (5.18) and (D.1) provide a cofinal subset . ⊆ N such that .d(·, ·, un ) → d(·, ·, u) in .Rd .(  n → ∞) almost everywhere in .QT . Thus, Vitali’s theorem (cf. Proposition 2.5) yields .d(·, ·, un ) → d(·, ·, u)  in .Lr (·,·) (QT )d .(  n → ∞). Finally, by exploiting the continuity of .Jεσ   (cf. Proposition 4.17 (ii)), the embedding .Lr (·,·) (QT )d → Lq (·,·) (QT )d (cf. Corollary 2.1) and the standard convergence principle (cf. [166, Prop. 10.13 ˚εq,p (QT )∗ .(n → ∞). (1)]), we conclude that .Dun → Du in .V 

5.2 The Hirano–Landes Approach This section is concerned with the so-called Hirano–Landes approach, which –in the broadest sense– can be attributed to contributions of R. Landes [108] and N. Hirano [84, 85], see also [15, 16, 19, 92, 139, 143] for more recent developments, ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is induced and states that if an operator .A : V q,p q,p ∗ ˚ ˚ by a family of operators .A(t) : Vε (t) → Vε (t) , .t ∈ I , i.e., for every .u ∈ ˚εq,p (QT ) ∩ H∞ (QT ) and .v ∈ V ˚εq,p (QT ), it holds V  := . Au, v ˚q,p

A(t)(u(t)), v(t)V˚εq,p (t) dt , (5.19) V (Q ) ε

T

I

˚εq,p (t)∗ for almost every .t ∈ I , due or equivalently .(Au)(t) := A(t)(u(t)) in .V to Remark 4.14, then the question of whether .A is Bochner pseudo-monotone, satisfies the Bochner condition (M), or is Bochner coercive, can be traced back to ˚εq,p (t) → V ˚εq,p (t)∗ , .t ∈ I , the properties of the inducing family of operators .A(t) : V in each single time slice .t ∈ I , i.e., to those properties that are usually already wellknown from the corresponding steady problem. Here, we will establish general and easily verifiable sufficient conditions on such families of operators that guarantee that the corresponding induced operators comply with these new concepts. This approach is consistent with the driving concept of this chapter to automatize the process of proof. In fact, if the corresponding steady problem is already well studied, then the Hirano–Landes approach represents an attractive alternative in determining whether an operator complies with the notions established in Sect. 5.1, rather than checking this directly, as we already did previously in Sect. 5.1.

190

5 Existence Theory for Lipschitz Domains q,p

˚ε (t) → Let us first examine which assumptions on a family of operators .A(t) :V .t ∈ I , are sufficient to ensure the well-definedness, boundedness, and ˚εq,p (QT )∩H∞ (QT ) → V ˚εq,p (QT )∗ , demi-continuity of the induced operator .A : V defined by (5.19). ˚εq,p (t)∗ , V

˚εq,p (t) → V ˚εq,p (t)∗ , .t ∈ I , be a family of operators Proposition 5.7 Let .A(t) : V with the following properties: (C.1) (C.2) (C.3)

˚εq,p (t) → V ˚εq,p (t)∗ is demi-continuous for almost every .t ∈ I . A(t) :V   ˚q,p . t → A(t)u, v ˚q,p Vε (t) : I → R is measurable for every .u, v ∈ V+ . There exist a function .α ∈ L1 (I, R≥0 ) and a non-decreasing function ˚εq,p (t), .ℬ : R≥0 → R≥0 such that for almost every .t ∈ I and every .u, v ∈ V it holds .

 A(t)u, v˚q,p ≤ ℬ(uH ) α(t)ρq(t,·) (u) + ρp(t,·) (ε(u)) Vε (t)  + ρq(t,·) (v) + ρp(t,·) (ε(v)) .

.

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ , defined Then, the induced operator .A : V by (5.19), is well-defined, bounded and demi-continuous. Proof ˚εq,p (QT ) ∩ H∞ (QT ) and .u2 ∈ V ˚εq,p (QT ). Since 1. Well-definedness: Let .u1 ∈ V q,p q,p ˚+ ) is dense in .V+ (QT ) (cf. Proposition 2.18) and .C ∞ (QT )d ⊆ .S(I, V 0,div q,p ˚εq,p (QT ) (cf. Definition 4.5), .S(I, V ˚+q,p ) is dense V+ (QT ) is dense in .V ˚εq,p (QT ). Thus, there exist sequences of simple functions .(s m ) in .V n n∈N ⊆

knm m q,p m ˚+ ), .m = 1, 2, i.e., .s (t) = m S(I, V n i=1 sn,i χEn,i (t) for a.e. .t ∈ I and q,p m m m ˚ .m = 1, 2, where .s n,i ∈ V+ , .kn ∈ N and .En,i ⊆ I are measurable knm m m ∩ E m = ∅ for .i = j , such that .s m → u with . i=1 En,i = I and .En,i m n n,j q,p ˚ in .Vε (QT ) .(n → ∞) for .m = 1, 2. Therefore, Corollary 4.1 provides subsequences .(s m n )n∈ , .m = 1, 2, with a cofinal subset . ⊆ N, such ˚q,p that for .m = 1, 2, it holds .s m n (t) → um (t) in .Vε (t) .(  n → ∞) for almost every .t ∈ I . Based on (C.1), we infer from the latter that 1 2 ˚εq,p (t) → A(t)(u1 (t)), u2 (t)V ˚εq,p (t) .(  n → ∞) for . A(t)(s n (t)), s n (t)V   1 almost every .t ∈ I . Thus, since . t → A(t)(s n (t)), s 2n (t)V˚εq,p (t) : I → R, .n ∈ N, are measurable because, for every .n ∈ N, we have the representation 1

1 2 . A(t)(s n (t)), s n (t) ˚q,p Vε (t)

=

2

kn kn i=1 j =1

A(t)s1n,i , s2n,j V˚εq,p (t) χE 1

for a.e. t ∈ I ,

2 n,i ∩En,j

(t)

5.2 The Hirano–Landes Approach

191

  and . t → A(t)s1n,i , s2n,j V˚εq,p (t) : I → R, .i = 1, . . . , kn1 , .j = 1, . . . , kn2 , .n ∈ N, are measurable (cf. (C.2)), we conclude to the measurability of   ˚εq,p (t) : I → R. Hence, we can next examine the . t → A(t)(u1 (t)), u2 (t)V   function . t → A(t)(u1 (t)), u2 (t)V˚εq,p (t) : I → R for integrability. In doing so, taking into account (C.3), we observe that  .

I

A(t)(u1 (t)), u2 (t)˚q,p dt Vε (t)   ≤ ℬ(u1 H∞ (QT ) ) ρq(·,·) (u1 ) + ρp(·,·) (ε(u1 )) (5.20)   + ℬ(u1 H∞ (QT ) ) αL1 (I ) + ρq(·,·) (u2 ) + ρp(·,·) (ε(u2 )) ,

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is well-defined. i.e., .A : V 2. Boundedness: Since .vV ˚εq,p (QT ) ≤ 1 implies .ρq(·,·) (v) + ρp(·,·) (ε(v)) ≤ 2 for q,p ˚ all .v ∈ Vε (QT ) by the norm-modular unit ball property (cf. Lemma 2.1 (i)), ˚εq,p (QT ) ∩ H∞ (QT ), it holds we deduce from (C.3) that for every .u ∈ V AuV ˚q,p (Q

.

ε

T)

∗

=

sup q,p

˚ε (QT ) v∈V vV ˚q,p (Q ) ≤1 ε

Au, vV ˚q,p (Q ε

T)

T

  ≤ ℬ(uH∞ (QT ) ) αL1 (I ) + ρq(·,·) (u) + ρp(·,·) (ε(u)) + 2 , ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is bounded. i.e., .A : V ˚εq,p (QT )∩H∞ (QT ) be a sequence such that 3. Demi-continuity: Let .(un )n∈N ⊆ V ˚εq,p (QT ) ∩ H∞ (QT ) .(n → ∞). Then, by applying Corollary 4.1, .un → u in .V we obtain a subsequence .(un )n∈ , with a cofinal subset . ⊆ N, such that ˚εq,p (t) .(  n → ∞) for almost every .t ∈ I . In con.un (t) → u(t) in .V sequence, (C.1) implies . A(t)(un (t)), v(t)V˚εq,p (t) → A(t)(u(t)), v(t)V˚εq,p (t) ˚εq,p (QT ). Apart from .(  n → ∞) for almost every .t ∈ I and every .v ∈ V   that, using (C.3), one readily sees that . A(t)(un (t)), v(t)V˚εq,p (t) n∈N ⊆ L1 (I ) is .L1 (I )-uniformly integrable. Thus, using Vitali’s convergence theorem (cf. ˚εq,p (QT ) that Proposition 2.5), we establish for every .v ∈ V 

Aun − Au, vV ˚q,p (Q

.

ε

T)

= I

A(t)(un (t)) − A(t)(u(t)), v(t)V˚εq,p (t) dt → 0

(  n → ∞) , ˚εq,p (QT )∗ .(  n → ∞). As this argumentation i.e., .Aun  Au in .V ˚εq,p (QT ) ∩ H∞ (QT ), remains valid for each subsequence of .(un )n∈N ⊆ V q,p ˚ε (QT )∗ is a weak accumulation point of each subsequence of .Au ∈ V ˚εq,p (QT )∗ . Consequently, the standard convergence principle (cf. .(Aun )n∈N ⊆ V ˚εq,p (QT )∗ .(n → ∞). 

[166, Prop. 10.13 (4)]) ensures that .Aun  Au in V

192

5 Existence Theory for Lipschitz Domains

The next proposition clarifies which assumptions on a family of operators ˚εq,p (t) → V ˚εq,p (t)∗ , .t ∈ I , are sufficient in order to guarantee the Bochner A(t) : V pseudo-monotonicity and Bochner coercivity of the induced operator.

.

Proposition 5.8 Let .p ∈ C 0 (QT ) with .p− > 1 and .q ∈ P∞ (QT ) with − .2 ≤ q ≤ max{2, p∗ − ε} almost everywhere in .QT for some .ε ∈ (0, (p )∗ − 1]. q,p q,p ∗ ˚ε (t) → V ˚ε (t) , .t ∈ I , be a family of operators satisfying Moreover, let .A(t) : V (C.1)–(C.3) and additionally satisfying the following properties: ˚εq,p (t) → V ˚εq,p (t)∗ is pseudo-monotone for almost every .t ∈ I . (C.4) .A(t) : V (C.5) There exist a constant .c0 > 0 and functions .c1 , c2 ∈ L1 (I, R≥0 ) such that for ˚εq,p (t), it holds almost every .t ∈ I and every .u ∈ V

A(t)u, uV˚εq,p (t) ≥ c0 ρp(t,·) (ε(u)) − c1 (t)u2H − c2 (t) .

.

(C.6) There exist a function .α ∈ L1 (I, R≥0 ), a non-decreasing function .ℬ : R≥0 → R≥0 , and a function .c : (0, ε˜ 0 ) → R≥0 , where .ε˜ 0 > 0, such that for almost ˚εq,p (t), it holds every .t ∈ I , every .ε˜ ∈ (0, ε˜ 0 ), and .u, v ∈ V  A(t)u, v˚q,p ≤ ℬ(uH + vH ) α(t) + ε˜ ρp(t,·) (ε(u)) Vε (t)  + c(˜ε )ρp(t,·) (ε(v)) .

.

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ , defined Then, the induced operator .A : V by (5.19), satisfies: ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is Bochner pseudo-monotone. (i) .A : V q,p ˚ε (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is Bochner coercive with respect (ii) .A : V ˚εq,p (QT )∗ , where .f ∈ Lmin{2,(p− ) } (QT )d to every .u0 ∈ H and .Jεσ (f , F ) ∈ V  and .F ∈ Lp (·,·) (QT , Md×d sym ). The core of the Hirano–Landes approach, which is also an elementary component of the proof of Proposition 5.8 (i), forms the following abstract lemma. In fact, it is formulated more abstractly than we actually need. The reason for this is that the author predicts that the following lemma and an adapted version of Proposition 5.8 may also have an impact in FSI (Fluid Structure Interaction) problems in this general form. First investigations in this direction are already in progress. Lemma 5.1 (Hirano–Landes) Let .(X(t), Y (t)), .t ∈ I , be a family of compatible couples,5 .(x n (t))n∈N ⊆ X(t) ∩ Y (t), .t ∈ I , families of sequences, .x(t) ∈ X(t) ∩ Y (t), .t ∈ I , a family of elements, .A(t) : X(t) ∩ Y (t) → (X(t) ∩ Y (t))∗ , .t ∈ I ,

5 Two Banach spaces X, Y form a compatible couple, if there is a Hausdorff vector space Z such that .X, Y → Z. Then, the intersection .X ∩ Y in Z is well-defined and forms a Banach space, if equipped with the canonical norm . · X∩Y :=  · X +  · Y (cf. [18, Theorem 1.3, p. 97]).

5.2 The Hirano–Landes Approach

193

a family of operators, and .μ ∈ L1 (I ). Moreover, we assume that for almost every .t ∈ I , the following time slice conditions are satisfied: (i) .x n (t)  x(t) in Y (t) .(n → ∞). (ii) .A(t) : X(t)∩Y (t) → (X(t)∩Y (t))∗ is pseudo-monotone and .X(t) is reflexive. (iii) . A(t)(x n (t)), x n (t) − x(t)X(t)∩Y (t) ≥ 𝒦(t)(x n (t)) − μ(t) for every .n ∈ N, where the mapping .𝒦(t) : X(t) → R≥0 is weakly coercive.6 Suppose further that for .(hn (t))n∈N := ( A(t)(x n (t)), x n (t) − x(t)X(t)∩Y (t) )n∈N , t ∈ I , the following integrability conditions are satisfied:

(iv) .(hn )n∈N ⊆ L1 (I ) with .lim supn→∞ I hn (s) ds ≤ 0.

.

Then, there exists a cofinal subset . ⊆ N (not depending on .t ∈ I ) such that for almost every .t ∈ I , it holds

A(t)(x(t)), x(t) − yX(t)∩Y (t) ≤ lim inf A(t)(x n (t)), x n (t) − yX(t)∩Y (t)

.

n→∞

for every .y ∈ X(t) ∩ Y (t). Proof We define .S := {t ∈ I | (i) − (iii) and |μ(t)| < ∞ holds for t}. Apparently, I \ S is a null set. Our first aim is to establish for every .t ∈ S that

.

.

lim inf hn (t) ≥ 0 . n→∞

(∗)t

For this purpose, we fix an arbitrary .t ∈ S and introduce the discrete set t := {n ∈ N | hn (t) < 0}. We assume without loss of generality that .t is not finite. Otherwise, .(∗)t would already be valid for this specific .t ∈ S and nothing would be left to do. But if .t is not finite, then we have that

.

.

lim sup hn (t) ≤ 0 .

t n→∞

(5.21)

Using the definition of .t in (iii), we note that .supn∈t 𝒦(t)(x n (t))≤|μ(t)| 0 for every .ε˜ ∈ ν > 1, ν ε ) := (ν ε˜ )  0, ν −1 , to the first summand in the last integral on the right-hand side in (5.31) and with respect to the exponent .p ∈ C 0 (QT )10 , with the constant .cp (˜ε) := −  (p− ε˜ )1−(p ) ((p+ ) )−1 > 0 for every .ε˜ ∈ (0, (p− )−1 ), to the second summand, for every .t ∈ I and .ε˜ ∈ (0, ε˜ 0 ), where .ε˜ 0 := min{ν −1 , (p− )−1 }, we observe that

.

 t 0

|f (s, y)||u(s, y)| + |F (s, y)||ε(u)(s, y)| dy ds 

≤ cν (˜ε )ρν  (f ) + cp (˜ε )ρp (·,·) (F )  t + ε˜ ρν (u(s)) + ρp(s,·) (ε(u)(s)) ds .

.

(5.32)

0

In (5.32) using .ρν (u(t)) ≤ u(t)2H + ρp− (u(t)) for almost every .t ∈ I and, in + addition, of .ρp− (u(t)) ≤ cp− ρp− (ε(u)(t)) ≤ cp− 2p (|| + ρp(t,·) (ε(u)(t))) for almost every .t ∈ I , based on Poincaré’s and Korn’s inequality for .p− ∈ (1, ∞), we infer for every .t ∈ I and .ε˜ ∈ (0, ε˜ 0 ) that  t 0

.

|f (s, y)||u(s, y)| + |F (s, y)||ε(u)(s, y)| dy ds



≤ cν (˜ε )ρν  (f ) + cp (˜ε )ρp (·,·) (F )  t + + ε˜ cp− 2p |QT | + ε˜ u(s)2H

(5.33)

0

 + + 1 + cp− 2p ρp(s,·) (ε(u)(s)) ds . If we set .M0 := 12 u0 2H + c2 L1 (I ) + cν (˜ε )ρν  (f ) + cp (˜ε)ρp (·,·) (F ) + + ε˜ cp− 2p |QT | > 0, .y := 12 u(·)2H ∈ L∞ (I ), .a := 2˜ε + 2|c1 (·)| ∈ L1 (I )  ε˜ 0  c0 ∈ (0, ε˜ 0 ), then we conclude from (5.33) as well as .ε˜ := min + , 2(1+cp− 2p ) 2 in (5.31) for almost every .t ∈ I to the estimate c0 .y(t) + 2

 0

t



t

ρp(s,·) (ε(u)(s)) ds ≤ M0 +

a(s)y(s) ds .

(5.34)

0

apply Proposition 2.8 with respect to the constant .ν > 1 for every .ε = (˜εν)1/ν ∈ (0, 1), where .ε˜ ∈ (0, ν −1 ). − 10 We apply Proposition 2.8 with respect to .p ∈ C 0 (Q ) for every .ε = (˜ ε p − )1/p ∈ (0, 1), where T − −1 .ε ˜ ∈ (0, (p ) ). 9 We

198

5 Existence Theory for Lipschitz Domains

If we next apply Grönwall’s inequality (cf. [23, Lemma II.4.10]) to (5.34), then we observe that u2H∞ (QT ) = 2yL∞ (I ) ≤ 2M0 exp(aL1 (I ) ) =: M1 .

.

(5.35)

Looking back to (5.34) in the special case .t = T , (5.35) then, leads to ρp(·,·) (ε(u)) ≤

.

  M1 2 M0 + aL1 (I ) =: M2 . c0 2

(5.36)

In virtue of the parabolic interpolation inequality for .Xε (QT ) ∩ Y ∞ (QT ) (cf. Proposition 3.12), also exploiting (5.35), (5.36), .2≤q≤ max{2, p∗ −ε} a.e. in .QT , Corollary 2.1, Lemma 2.1 (iii), and .uLq(·,·) (QT )d ≤uLp∗ (·,·)−ε (QT )d + uL2 (QT )d , there exists a constant .cε > 0 such that 2,p

  uLq(·,·) (QT )d ≤ cε ε(u)Lp(·,·) (QT )d×d + uH∞ (QT )  . 1 1 ≤ cε (M2 + 1) p− + M12 =: M3 .

(5.37)

Altogether, by combining (5.35)–(5.37), using again Lemma 2.1 (iii) in doing so, we conclude that uV ˚q,p (Q

.

ε

∞ T )∩H (QT )

1

1

≤ M3 + (M2 + 1) p− + M12 =: M ,

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is Bochner i.e., .A : V ˚εq,p (QT )∗ , where .f ∈ every .u0 ∈ H and .Jεσ (f , F ) ∈ V  (·,·) d×d p .F ∈ L (QT , Msym ).

coercive with respect to −  Lmin{2,(p ) } (QT )d and 

Apparently, Proposition 5.8 applies only to the case .2 ≤ q ≤ max{2, p∗ − ε} a.e. in .QT and does not cover arbitrary combinations of variable exponents ∞ − − > 1. This can be traced back to the required .q, p ∈ P (QT ) with .q , p growth conditions (C.5) and (C.6). If we impose stronger growth and coercivity conditions, then we obtain the following analogue of Proposition 5.8 treating arbitrary combinations of variable exponents .q, p ∈ P∞ (QT ) with .q − , p− > 1. Proposition 5.9 Let .q, p ∈ P∞ (QT ) with .q − , p− > 1. Moreover, let ˚εq,p (t) → V ˚εq,p (t)∗ , .t ∈ I , be a family of operators fulfilling (C.1)–(C.4) .A(t) : V and additionally satisfying the following properties:

5.2 The Hirano–Landes Approach

199

(C.5*) There exist a constant .c0 > 0 and functions .c1 , c2 ∈ L1 (I, R≥0 ) such that ˚εq,p (t), it holds for almost every .t ∈ I and every .u ∈ V  

A(t)u, uV˚εq,p (t) ≥ c0 ρq(t,·) (u) + ρp(t,·) (ε(u)) − c1 (t)u2H − c2 (t) .

.

(C.6*) There exist a function .α ∈ L1 (I, R≥0 ), a non-decreasing function .ℬ : R≥0 → R≥0 , and a function .c : (0, ε ˜ 0 ) → R≥0 , where .ε˜ 0 > 0, such ˚εq,p (t), it holds that for almost every .t ∈ I , every .ε˜ ∈ (0, ε˜ 0 ), and .u, v ∈ V    A(t)u, v˚q,p ≤ ℬ(uH + vH ) α(t)+ ε˜ ρq(t,·) (u)+ρp(t,·) (ε(u)) Vε (t)   + ℬ(uH + vH )c(˜ε ) ρq(t,·) (v) + ρp(t,·) (ε(v)) .

.

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ , defined Then, the induced operator .A : V by (5.19), is Bochner pseudo-monotone and Bochner coercive. Proof We proceed as in the proof of Proposition 5.8 except for obvious adjustments in which we again exploit Lemma 5.1 but now have to apply (C.5*) and (C.6*) instead of (C.5) and (C.6). In particular, in this case, we have to exploit that Lemma 5.1 also applies for compatible couples because we cannot rely on the ˚εq,p (t) → H for almost every .t ∈ I if only .q − , p− > 1. embedding .V 

Next, we provide a couple of important –especially in view of the model problems (4.1) and (4.2)– examples of operator families satisfying the assumptions of Proposition 5.7 and Proposition 5.8. Proposition 5.10 Let .p ∈ C 0 (QT ) with .p− > 1 and .q ∈ P∞ (QT ) with − .2 ≤ q ≤ max{2, p∗ − ε} a.e. in .QT for some .ε ∈ (0, (p )∗ − 1]. Furthermore, d×d be a mapping satisfying (S.1)–(S.4) with respect to p. let .S : QT × Md×d → M sym sym ˚εq,p (t) → V ˚εq,p (t)∗ , .t ∈ I , defined by Then, the family of operators .S(t) : V

S(t)u, vV˚εq,p (t) := Jσε (0, S(t, ·, ε(u))), vV˚εq,p (t)

.

:= (S(t, ·, ε(u)), ε(v))Lp(t,·) ()d×d , ˚εq,p (t) satisfies (C.1)–(C.6). In for almost every .t ∈ I and every .u, v ∈ V ˚εq,p (QT ) → V ˚εq,p (QT )∗ particular, the restricted, unsteady extra stress tensor .S : V from Proposition 5.2 is bounded, Bochner pseudo-monotone and Bochner ˚εq,p (QT )∗ , where coercive with respect to every .u0 ∈ H and .Jεσ (f , F ) ∈ V −   min{2,(p ) } (Q )d and .F ∈ Lp (·,·) (Q , Md×d ). .f ∈ L T T sym Proof ad (C.1) and (C.4). Well-definedness, continuity, monotonicity and, thus, ˚εq,p (t) .→ V ˚εq,p (t)∗ , .t ∈ I , follow by the same pseudo-monotonicity, of .S(t) : V arguments as in the proof of Proposition 3.31, but now we have to work in . for fixed but arbitrary time slices .t ∈ I rather than in all of .QT .

200

5 Existence Theory for Lipschitz Domains

ad (C.2). The inequality (3.103) in Proposition 3.31 in conjunction with Hölder’s inequality for variable Lebesgue spaces (cf. Proposition 2.6) ensures q,p that .((t, x) → S(t, x, ε(u)(x)) : ε(v)(x)) ∈ L1 (QT ) for all .u, v ∈ V+ .  q,p Thus, Fubini’s theorem yields  that . t → S(t)u, vV˚ε (t) = t → (S(t, ·, ε(u)), ε(v))Lp(t,·) ()d×d : I → R is Lebesgue measurable. ad (C.5). By combining (S.3) and .(δ + a)p(t,x)−2 a 2 ≥ 12 a p(t,x) − δ p(t,x) for all .a ≥ 0 and .(t, x) ∈ QT , we notice for almost every .t ∈ I and every .u ∈ ˚εq,p (t) that V

S(t)u, uV˚εq,p (t) ≥

.

c0 ρp(t,·) (ε(u)) − c0 ρp(t,·) (δ) − c1 (t, ·)L1 () . 2

ad (C.6) and (C.3). By analogy with (3.103) in Proposition 3.31, i.e., using (S.2) and repeatedly the estimate .(a + b)s ≤ 2s (a s + bs ) for all .a, b ≥ 0 and  ∈ Q and every .u ∈ V ˚εq,p (t) that .s > 0, we deduce for almost every .(t, x) T 

|S(t, x, ε(u)(x))|p (t,x) .   −  −  +  ≤ 2(p ) α (p ) 2p δ p(t,x) + |ε(u)(x)|p(t,x) + β(t, x)p (t,x) .

(5.38)

Applying the .ε-Young inequality (cf. Proposition 2.8) with respect to the + exponent .p ∈ C 0 (QT ),11 with the constant .cp (˜ε ) := ((p+ ) ε˜ )1−p (p− )−1 > 0 for every .ε˜ ∈ (0, ε˜ 0 ), where .ε˜ 0 := ((p+ ) )−1 , we observe for almost every  ∈ Q , every .ε˜ ∈ (0, ε˜ ), and .u, v ∈ V ˚εq,p (t) that .(t, x) T 0 (S(t, ·, ε(u)), ε(v))

Lp(t,·) ()d×d

.

≤ ε˜ ρp (t,·) (S(·, ·, ε(u)))

(5.39)

+ cp (˜ε)ρp(t,·) (ε(v)) .

Eventually, if we integrate (5.38) for almost every .t ∈ I with respect to .x ∈ , and use the resulting estimate in (5.39), we further find for almost every .t ∈ I , ˚εq,p (t) that every .ε˜ ∈ (0, ε˜ 0 ), and .u, v ∈ V    S(t)u, v˚q,p ≤ ε˜ 2(p− ) α (p− ) 2p+ ρp(t,·) (δ) + ρp(t,·) (ε(u)) Vε (t)  + ρp (t,·) (β(t, ·)) + cp (˜ε)ρp(t,·) (ε(v)) ,

.



i.e., condition (C.6), and, at the same time, also condition (C.3).

apply Proposition 2.8 with respect to .p  ∈ C 0 (QT ) for all .ε = (˜ε (p + ) )1/(p where .ε˜ ∈ (0, ((p + ) )−1 ).

11 We

+ )

∈ (0, 1),

5.2 The Hirano–Landes Approach

201

Proposition 5.11 Let .p ∈ P∞ (QT ) be such that .p− ≥ pC = ˚εp,p (t) → V ˚εp,p (t)∗ , .t ∈ I , defined by family of operators .C : V

3d+2 d+2 .

Then, the

Cu, vV˚εp,p (t) := Jσε (0, −u ⊗ u), vV˚εp,p (t) := −(u ⊗ u, ε(v))Lp(t,·) ()d×d ,

.

˚εp,p (t), satisfies (C.1)–(C.4) and (C.6). for almost every .t ∈ I and every .u, v ∈ V Proof ad (C.1), (C.2), and (C.4). We fix an arbitrary time slice .t ∈ I ˚εp,p (t) be arbitrary. The mapping such that .p(t, ·) ∈ P∞ ().12 Let .u ∈ V d×d .(x → u(x) ⊗ u(x)) :  → Msym is Lebesgue-measurable. By applying Corollary 2.1 and the Sobolev embedding theorem (cf. Proposition 3.9), we find that u ⊗ uLp (t,·) ()d×d ≤ 2(1 + ||)u2 2(p− )

()d

L

≤ 2(1 + ||)cp− u2V˚p,p

.

−

(5.40)

≤ 8(1 + ||)3 cp− u2V˚p,p (t) , ε

where we used that the Sobolev conjugate exponent is subject to .(p− )∗ > 2(p− )  3d . Therefore, we have that .u ⊗ u ∈ Lp (t,·) (, Md×d because .p− > d+2 sym ) and p,p p,p σ ∗ ˚ ˚ ˚εp,p (t)∗ .Cu = Jε (0, −u ⊗ u) ∈ Vε (t) (cf. Corollary 4.2), i.e., .C : Vε (t) → V ˚εp,p (t) → V ˚εp,p (t)∗ is strongly is well-defined and bounded. Apart from that, .C : V p,p ˚ ˚εp,p (t) .(n → continuous. In fact, let .(un )n∈N ⊆ Vε (t) be such that .un  u in .V ∞). Then, employing Rellich’s compactness theorem (cf. Proposition 3.9) and − ∗ −   2p (t,·) ()d .(n → .(p ) > 2(p ) ≥ 2p (t, ·) in ., we observe that .un → u in .L  ∞), i.e., .un ⊗ un → u ⊗ u in .Lp (t,·) (, Md×d sym ) .(n → ∞). Due to the continuity of  (t,·)  (t,·) σ d×d p d p ˚εp,p (t)∗ (cf. Corollary 4.2), we obtain .Jε : L () × L (, Msym ) → V ˚εp,p (t)∗ .(n → ∞), i.e., .C : V ˚εp,p (t) → V ˚εp,p (t)∗ is strongly .Cun → Cu in .V 13 ˚εp,p (t) → continuous and, thus, pseudo-monotone. All things considered, .C : V p,p ∗ ˚ε (t) , .t ∈ I , satisfies (C.1), (C.4), and trivially also (C.2). V ad (C6) and (C.3). Based on Lemma 3.4 and Korn’s inequality for .pC (cf. Proposition 2.13), there exist constants .c, γ > 0 (both not depending on .t ∈ I ) 1,p such that for every .u ∈ W0 C ()d ∩ Y , it holds γ

ρ(pC )∗ (u) ≤ cρpC (ε(u))uY .

.

(5.41)

to Remark 3.1, for every .p ∈ P∞ (QT ), .p(t, ·) ∈ P∞ () applies for almost every ∈ I. 13 Appealing to [149, Kap. 3, Lem. 2.6 (ii)], strong continuity implies pseudo-monotonicity. 12 According

.t

202

5 Existence Theory for Lipschitz Domains

Due to .(pC )∗ = 2(pC ) , applying the .ε-Young inequality (cf. Proposition 2.8) with respect to .pC > 1, with the constant .cpC (˜ε) := ((pC ) ε˜ )1−pC pC−1 for every + .ε ˜ ∈ (0, ε˜ 0 ), where .ε˜ 0 := pC−1 , using that .ρpC (ε(u)) ≤ 2p (|| + ρp(t,·) (ε(u))) ˚εp,p (t), since, by assumption, .p− ≥ pC , for almost every .t ∈ I and all .u ∈ V ˚εp,p (t) and (5.41), we deduce for almost every .t ∈ I , every .ε˜ ∈ (0, ε˜ 0 ), and .u ∈ V that Cu, v˚p,p ≤ ε˜ ρ2(p ) (u) + cp (˜ε )ρp (ε(v)) C C C Vε (t)

.

γ

≤ ε˜ cρpC (ε(u))uH + cpC (˜ε )ρpC (ε(v))  + γ ≤ ε˜ c2p || + ρp(t,·) (ε(u)) uH  + + cpC (˜ε )2p || + ρp(t,·) (ε(v)) , i.e., condition (C.6), and, at the same time, also condition (C.3).



− .p

≥ pC . Furthermore, let .S : QT × Proposition 5.12 Let .p ∈ T ) with d×d be a mapping satisfying (S.1)–(S.4) with respect to p. Then, Md×d → M sym sym ˚εp,p (t) → V ˚εp,p (t)∗ , .t ∈ I , satisfies (C.1)–(C.6). In particular, the sum .S(t) + C : V ˚εp,p (QT ) ∩ H∞ (QT ) → V ˚εp,p (QT )∗ is bounded, the induced operator .S + C : V Bochner pseudo-monotone and Bochner coercive with respect to every .u0 ∈ H and σ ˚εp,p (QT )∗ , where .f ∈ L(p− ) (QT )d and .F ∈ Lp (·,·) (QT , Md×d ). .Jε (f , F ) ∈ V sym C 0 (Q

˚εp,p (t)→V ˚εp,p (t)∗ , Proof Due to Proposition 5.10 and Proposition 5.11, .S(t), C : V p,p p,p ˚ ˚ .t ∈ I , satisfy (C.1)–(C.4) and (C.6). Hence, also .S(t) + C : Vε (t) → Vε (t)∗ , p,p p,p ˚ ˚ .t ∈ I , satisfies (C.1)–(C.4) and (C.6). Eventually, since .S(t) : Vε (t) → Vε (t)∗ , p,p p,p ˚ ˚ε (t) = 0 for every .t ∈ I and .u ∈ Vε (t), the .t ∈ I , satisfies (C.5) and . Cu, uV ˚εp,p (t) → V ˚εp,p (t)∗ , .t ∈ I , likewise satisfies (C.5). Thus, the rest sum .S(t) + C : V of the assertion is simply an application of Propositions 5.7 and 5.8. 

Proposition 5.13 Let .p, q := max{2, p∗ − ε}, r := max{2, p∗ } − ε ∈ C 0 (QT ) with .p− > 1 and .ε ∈ (0, (p− )∗ − 1]. Furthermore, let .d : QT × Rd → Rd be a mapping satisfying (D.1)–(D.3) with respect to r. Then, the family of operators ˚εq,p (t) → V ˚εq,p (t)∗ , .t ∈ I , defined by .D(t) : V

D(t)u, vV˚εq,p (t) := Jσε (d(t, ·, u), 0), vV˚εq,p (t) := (d(t, ·, u), v)Lq(t,·) ()d ,

.

q,p

˚ε for almost every .t ∈ I and every .u, v ∈ V

(t), satisfies (C.1)–(C.4) and (C.6).

Proof ad (C.1) and (C.4). Follow along the lines of the proof of Proposition 5.6, where we now have to work in . for fixed but arbitrary time slices .t ∈ I rather than in all of .QT . In doing so, we have to replace all parabolic embedding and compactness results by their steady counterparts, i.e., we replace Proposition 3.12 ˚εq,p (t) →→ Lr(t,·) ()d for and Proposition 3.13 by the compact embedding .V

5.2 The Hirano–Landes Approach

203

every .t ∈ I , which follows easily from Proposition 3.9 in conjunction with Proposition 2.5.14 ad (C.3) and (C.6). By resorting to the .ε-Young inequality (cf. Proposition 2.8) with respect to the exponent .r  ∈ C 0 (QT ), with the constant + .cr (˜ ε) := ((r + ) ε˜ )1−r (r − )−1 > 0 for every .ε˜ ∈ (0, ε˜ 0 ), where .ε˜ 0 := ((r + ) )−1 , also employing Lemma 3.6 and condition (D.2), i.e., proceeding as for (5.17), we ˚εq,p (t) that obtain for almost every .t ∈ I , every .ε˜ ∈ (0, ε˜ 0 ), and .u, v ∈ V D(t)u, v˚q,p ≤ ε˜ ρr(t,·) (d(t, ·, u)) + cr (˜ε)ρr(t,·) (v) Vε (t)   −  −  + ≤ ε˜ 2(r ) γ (r ) 2r || + ρr(t,·) (u) + ρr  (t,·) (η(t, ·))

.

+ cr (˜ε)ρr(t,·) (v)  −  −  + γ  ≤ ε˜ 2(r ) γ (r ) 2r || + cε 1 + ρp(t,·) (ε(u)) + uHε   γ  × 1 + uHε + ρr  (t,·) (η(t, ·))  γ  γ  + cr (˜ε)cε 1 + ρp(t,·) (ε(v)) + vHε 1 + vHε , i.e., condition (C.3), which follows from the second inequality since .ρr(t,·) (u) ≤ + ˚εq,p (t), and condition (C.6), which 2q (|| + ρq(t,·) (u)) for every .t ∈ I and .u ∈ V is precisely the last inequality. 

.

Proposition 5.14 Let .p, q := max{2, p∗ − ε}, r := max{2, p∗ } − ε ∈ C 0 (QT ) with − > 1 and .ε ∈ (0, (p − ) − 1]. Furthermore, let .S : Q × Md×d → Md×d be a .p ∗ T sym sym mapping satisfying (S.1)–(S.4) with respect to p and .d : QT × Rd → Rd a mapping ˚εq,p (t) → satisfying (D.1)–(D.3) with respect to r. Then, the sum .S(t) + D(t) : V q,p ˚ε (t)∗ , .t ∈ I , satisfies (C.1)–(C.6). In particular, the induced operator .S + V ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is bounded, Bochner pseudo-monotone D: V ˚εq,p (QT )∗ , and Bochner coercive with respect to every .u0 ∈ H and .Jεσ (f , F ) ∈ V −   where .f ∈ Lmin{2,(p ) } (QT )d and .F ∈ Lp (·,·) (QT , Md×d sym ). ˚εq,p (t)→V ˚q,p (t)∗ , Proof Due to Proposition 5.10 and Proposition 5.13, .S(t),D(t) :V ε q,p ˚ε (t)→V ˚εq,p (t)∗ , .t∈I , satisfy (C.1)–(C.4) and (C.6). Thus, .S(t)+D(t) : V ˚εq,p (t) → V ˚εq,p (t)∗ , .t ∈ I , .t ∈ I , satisfies (C.1)–(C.4) and (C.6). Since .S(t) : V q,p q,p ˚ε (t) → V ˚ε (t)∗ , .t ∈ I , owing to (D.3), satisfies satisfies (C.5) and .D(t) : V 2 q,p ˚ε (t) ≥ −c2 uH − c3 (t, ·)L1 () for almost every .t ∈ I and every . D(t)u, uV ˚εq,p (t), the sum .S(t) + D(t) : V ˚εq,p (t) → V ˚εq,p (t)∗ , .t ∈ I , likewise satisfies .u ∈ V (C.5). 

particular, one exploits for this embedding that .r(t, ·) = max{2, p ∗ (t, ·)} − ε in . for every ∈ I.

14 In .t

204

5 Existence Theory for Lipschitz Domains

5.3 C 0 -Bochner Pseudo-Monotonicity, C 0 -Bochner Condition (M) and C 0 -Bochner Coercivity We emphasize at this point that the notion of Bochner pseudo-monotonicity is not ˚εq,p (QT )∩H∞ (QT ). This defined with respect to the weak sequential topology of .V is mainly attributable to the fact that (5.3) together with (5.4) is strictly weaker than weak convergence in .H∞ (QT ), i.e., for a sequence .(un )n∈N ⊆ H∞ (QT ) from ∗

un  u

.

un (t)  u(t)

in H∞ (QT )

(n → ∞) ,

in H

(n → ∞)

for a.e. t ∈ I ,

it does not follow that .un  u in .H∞ (QT ) .(n → ∞), as can be observed in the following remark. Remark 5.5 Let .I := (1, 1). Due to [159, Thm. 2.9], there exists an additive measure .ω ∈ L∞ (I )∗ with .ω((− 2l1 , 0) ∪ (0, 2l1 )) = 1 for all .l ∈ N. Then, let ∞ .(fn )n>2 ⊆ L (I ) for every .n > 2 be defined by ⎧ ⎪ ⎪ ⎨0 .fn (t) := 1 ⎪ ⎪ ⎩2 − n|t|

if t = 0 or |t| ≥ if 0 < |t| ≤ if

1 n

< |t|

1 n < n2

2 n

.

It is easy to see that both .supn>2 fn L∞ (I ) ≤ 1 and .fn (t) → 0 for every .t ∈ I ∗ (n → ∞), which of course implies that .fn  0 in .L∞ (I ) .(n → ∞). Apart from that, [159, Thm. 2.8] guarantees that . ω, fn L∞ (I ) = 1 for every .n > 2. Next, fix .h ∈ H and .u ∈ V such that .(h, u)H = 1. Since for every .u ∈ H∞ (QT ), we have that .(h, u(·))H ∈ L∞ (I ) with .(h, u(·))H L∞ (I ) ≤ hH uH∞ (QT ) , the functional .h∗ ∈ H∞ (QT )∗ , for every .u ∈ H∞ (QT ) defined by

.

h∗ , uH∞ (QT ) := ω, (h, u(·))H L∞ (I ) ,

.

is well-defined, linear and continuous. We consider the sequence .(un )n>2 := (ufn (·))n>2 ⊆ H∞ (QT ). Then, we have that .supn>2 un H∞ (QT ) ≤ uH and

.

∗

un (t) → 0 in H .(n → ∞) for every .t ∈ I , i.e., it holds .un  0 in .H∞ (QT ) ∗ .(n → ∞), but . h , un H∞ (Q ) = ω, fn L∞ (I ) = 1 for every .n > 2, which lets us T preclude that .un  0 in .H∞ (QT ) .(n → ∞). .

In view of Remark 5.5, one might be reluctant to require weak convergence in H∞ (QT ) in Definition 5.1 instead of (5.3) and (5.4). However, we will see later in Theorem 5.1 that a sequence of Galerkin approximations of (4.78) satisfies (5.4) not only for almost every but for all .t ∈ I , which is equivalent to weak convergence in 0 .H (QT ) (cf. Proposition 2.16). This suggests generalizations of Bochner pseudomonotonicity and Bochner condition (M) that respect the weak sequential topology

.

5.3 C 0 -Bochner Pseudo-Monotonicity, C 0 -Bochner Condition (M) and C 0 -. . .

205

in .H0 (QT ). Having this in mind, we will first give an appropriate definition of ˚εq,p (QT ) (consisting of equivalence classes) and an intersection space between .V 0 .H (QT ) (consisting of functions). Throughout the entire section, if nothing else is stated, let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, ∞ − − > 1. By analogy with Sect. 2.3.4, .QT := I × , and .q, p ∈ P (QT ) with .q , p we define the intersection space ˚εq,p (QT ) ∩c H0 (QT ) V

(5.42)

.

˚εq,p (QT ) that possess a continuous to be the space of all functions .u ∈ V ˚εq,p (QT ) ∩c H0 (QT ) with representation .uc ∈ H0 (QT ). Moreover, we endow .V the canonical norm  · V ˚q,p (Q

.

T )∩c H

ε

0 (Q ) T

:=  · V ˚q,p (Q ε

T)

+ (·)c H0 (QT ) ,

˚εq,p (QT ) ∩c H0 (QT ) into a Banach space. It is straightforward to which turns .V verify with the aid of the characterization of weak convergence in intersection spaces and .H0 (QT ) (cf. Proposition 2.16) that for a sequence .(un )n∈N ⊆ ˚εq,p (QT )∩c H0 (QT ), it is valid .un  u ˚εq,p (QT )∩c H0 (QT ) and a function .u ∈ V V q,p 0 ˚εq,p (QT )∩c H0 (QT ) ˚ in .Vε (QT )∩c H (QT ) .(n → ∞) if and only if .(un )n∈N ⊆ V is bounded and satisfies .(un )c (t)  uc (t) in H .(n → ∞) for every .t ∈ I , see, e.g., [92, Prop. 2.6] for a proof within the scope of classical Bochner–Lebesgue spaces. ˚εq,p (QT ) ∩c H0 (QT ). Next, we generalize Definition 5.1 to the framework of .V ˚εq,p (QT ) ∩c H0 (QT ) → V ˚εq,p (QT )∗ is said to Definition 5.3 An operator .A : V q,p

˚ε (QT )∩c.H0 (QT ) (i) be .C 0 -Bochner strongly continuous, if for .(un )n∈N ⊆ V from q,p

˚ε (QT ) ∩c H0 (QT ) (n → ∞) , in V

un  u

.

(5.43)

it follows that Aun → Au

.

˚ε (QT )∗ in V q,p

(n → ∞) .

˚εq,p (QT ) ∩c H0 (QT ) (ii) be .C 0 -Bochner pseudo-monotone, if for .(un )n∈N ⊆ V from (5.43) and .

lim sup Aun , un − uV ˚q,p (Q n→∞

ε

T)

≤ 0,

(5.44)

206

5 Existence Theory for Lipschitz Domains

˚εq,p (QT ), it follows that for every .v ∈ V

Au, u − vV ˚q,p (Q

.

T)

ε

≤ lim inf Aun , un − vV ˚q,p (Q ) . n→∞

ε

T

˚εq,p (QT )∩c H0 (QT ) (iii) satisfy the .C 0 -Bochner condition (M), if for .(un )n∈N ⊆ V from (5.43) and ˚εq,p (QT )∗ (n → ∞) , . Aun  u∗ in V

.

lim sup Aun , un V ˚q,p (Q ε

n→∞

T)

∗

≤ u , uV ˚q,p (Q ) , ε

T

(5.45) (5.46)

it follows that ˚εq,p (QT )∗ . Au = u∗ in V

.

Remark 5.6 Clearly, Bochner strong continuity implies .C 0 -Bochner strong continuity, Bochner pseudo-monotonicity implies .C 0 -Bochner pseudo-monotonicity, and the Bochner condition (M) implies the .C 0 -Bochner condition (M). Note that the inverse is not valid in general. In fact, there exist .C 0 -Bochner strongly continuous operators not satisfying the Bochner condition (M). In order to see this, it suffices to treat the case of a constant exponent .p ∈ (1, ∞), a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2, and .I := (−1, 1). Moreover, let .h∗ ∈ H∞ (QT )∗ be the functional constructed in Remark 5.5, and .u∗ ∈ p,p p,p Vε (QT )∗ and .u ∈ Vε (QT ) such that . u∗ , uVεp,p (QT ) = 1. Then, the operator p,p p,p p,p .A : Vε (QT ) ∩ H∞ (QT ) → Vε (QT )∗ , for every .v ∈ Vε (QT ) ∩ H∞ (QT ) defined by Av := h∗ , vH∞ (QT ) u∗

.

in Vε (QT )∗ , p,p

is well-defined and bounded, but does not satisfy the Bochner condition (M) and, hence, is neither Bochner pseudo-monotone, nor Bochner strongly continuous. ˚p,p ) ⊆ Vεp,p (QT ) ∩ H∞ (QT ) be the sequence In fact, let .(un )n>2 ⊆ L∞ (I, V constructed in Remark 5.5. Then, it is easy to observe that .supn>2 un L∞ (I,V˚p,p ) ≤

∗ ˚p,p ) ˚p,p .(n → ∞) for all .t ∈ I , .un  uV˚p,p , .un (t) → 0 in .V 0 in .L∞ (I, V p,p .(n → ∞), and .un → 0 in .Vε (QT ) .(n → ∞). Apart from that, it holds p,p ∗ ∗ . h , un H∞ (Q ) = 1, i.e., .Aun = u in .Vε (QT )∗ , for every .n > 2, from which T we, in turn, deduce that .

lim sup Aun , un Vεp,p (QT ) = lim u∗ , un Vεp,p (QT ) = 0 = u∗ , 0Vεp,p (QT ) . n→∞

n→∞

5.3 C 0 -Bochner Pseudo-Monotonicity, C 0 -Bochner Condition (M) and C 0 -. . .

207

Overall, .(un )n>2 ⊆ Vε (QT ) ∩ H∞ (QT ) satisfies (5.2)–(5.7), but yet .A0 = 0 = p,p p,p p,p u∗ in .Vε (QT )∗ , i.e., the operator .A : Vε (QT ) ∩ H∞ (QT ) → Vε (QT )∗ cannot satisfy the Bochner condition (M). p,p Nevertheless, if .(un )n∈N ⊆ Vε (QT ) ∩c H0 (QT ) is a sequence ∗ satisfying (5.43), then, since .h0 := (idH0 (QT ) )∗ h∗ ∈ H0 (QT )∗ , where .(idH0 (QT ) )∗ denotes the adjoint operator of the identity mapping .idH0 (QT ): H0 (QT )→H∞ (QT ), we have that p,p

Aun = h∗0 , un H0 (QT ) u∗ → h∗0 , uH0 (QT ) u∗

.

in Vε (QT )∗ p,p

= Au

(n → ∞) .

In other words, .A : Vε (QT ) ∩c H0 (QT ) → Vε (QT )∗ is .C 0 -Bochner strongly continuous. p,p

p,p

In the same way as in Definition 5.3, we next introduce a generalization of ˚εq,p (QT ) ∩c H0 (QT ). Bochner coercivity, which is adjusted to the space .V ˚εq,p (QT ) ∩c H0 (QT ) → V ˚εq,p (QT )∗ is said to Definition 5.4 An operator .A : V be ˚εq,p (QT )∗ and .u0 ∈ H , if there (i) .C 0 -Bochner coercive with respect to .u∗ ∈ V ∗ ˚εq,p (QT ) ∩c exists a constant .M := M(u , u0 ) > 0 such that for every .u ∈ V 0 H (QT ) from .

1 1 2 uc (t)2H + Au − u∗ , uχ[0,t] V ˚εq,p (QT ) ≤ u0 H 2 2

for all t ∈ I ,

it follows that uV ˚q,p (Q

.

ε

T )∩c H

0 (Q ) T

≤ M.

(ii) .C 0 -Bochner coercive, if it is Bochner coercive with respect to every .u∗ ∈ ˚εq,p (QT )∗ and .u0 ∈ H . V We have seen in Remark 5.4, by using Poincaré’s inequality for variable Bochner–Lebesgue spaces (cf. Proposition 3.6), that the restriction of a ˚εp,p (QT ) ∩ H∞ (QT ) → V ˚εp,p (QT )∗ to Bochner coercive operator .A : V q,p log ˚ .Vε (QT ), where .q := max{2, p∗ − ε} ∈ P (QT ) for .ε ∈ (0, (p− )∗ − 1], i.e., ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ , where .(idV q,p ˚ε (QT ) )∗ A : V ˚εq,p (QT ) )∗ .(idV q,p p,p ˚ ˚ q,p ˚ε (QT ) : Vε (QT ) → Vε (QT ), is Bochner denotes the adjoint operator of .idV ˚εq,p (QT )∗ , where ˚εq,p (QT ) )∗ u∗ ∈ V coercive with respect to every .u0 ∈ H and .(idV p,p ∗ ∗ ˚ ˚εq,p (QT )∩ q,p ˚ε (QT ) )∗ A : V .u ∈ Vε (QT ) . The next proposition shows that .(idV q,p 0 ∗ 0 ˚ .H (QT ) → Vε (QT ) is even .C -Bochner coercive with respect to all ˚εq,p (QT )∗ and sufficiently small –with respect to appropriate norms– .u∗ ∈ V

208

5 Existence Theory for Lipschitz Domains

˚εq,p (t) → u0 ∈ H , when induced by a time-dependent family of operators .A(t) : V q,p ∗ ˚ε (t) , .t ∈ I , satisfying suitable assumptions, e.g., the conditions (C.1)–(C.4) V and a modified variant of condition (C.5). This allows us to deal with right-hand sides with low integrability, provided that their global energy is small enough. A crucial component is the following generalized version of Grönwall’s inequality.

.

Lemma 5.2 (Generalized Grönwall Inequality) Let .I := (0, T ), .T < ∞. Suppose further that for non-negative functions .y 1 ∈ C 0 (I ), .y 2 , y 3 ∈ L1 (I ), ∞ .a ∈ L (I ), and constants .A,B ≥ 0, .αi > 0, .i = 1, . . . , m, .m ∈ N, there hold for every .t ∈ I the estimates  y 1 (t) +

t

.



0 t 0

 y 2 (s) ds ≤ A + y 3 (s) ds ≤ B

t 0

 m i=1

 a(s)y 1 (s) ds +

0

 max y 1 (s)

s∈[0,t]

t

t

αi 0

y 3 (s) ds , y 1 (s) + y 2 (s) ds ,

and m .

(8AE)αi ≤

i=1

1 , 8B(1 + T )E

where .E := exp(aL1 (I ) ). Then, is valid y 1 C 0 (I ) + y 2 L1 (I ) ≤ 8AE .

.

Proof A straightforward generalization of the proof in [14, Proposition 6.2, p. 158], where the case .m = 1 is shown. 

Proposition 5.15 Let .p, q := max{2, p − ε} ∈ C 0 (QT ) with .p− > 1 and some  − ˚εq,p (t) → V ˚εq,p (t)∗ , .t ∈ I , be a .ε ∈ 0, (p )∗ − 1 . Furthermore, let .A(t) : V family of operators satisfying (C.1)–(C.4) and (C.5) with .c2 = 0 in .L1 (I ). Then,  there exists a constant .R > 0 such that for all functions .f ∈ Lq (·,·) (QT )d ,  p (·,·) (Q , Md×d ) and .u ∈ H that are subject to .F ∈ L T 0 sym f Lq  (·,·) (QT )d + F Lp (·,·) (QT )d×d + u0 H ≤ R ,

.

˚εq,p (QT ) ∩c H0 (QT ) → V ˚εq,p (QT )∗ , defined by (5.19), the induced operator .A : V 0 σ ˚εq,p (QT )∗ and .u0 ∈ H . is .C -Bochner coercive with respect to .Jε (f , F ) ∈ V

5.3 C 0 -Bochner Pseudo-Monotonicity, C 0 -Bochner Condition (M) and C 0 -. . . 

209



Proof To begin with, let .f ∈ Lq (·,·) (QT )d , .F ∈ Lp (·,·) (QT , Md×d sym ) and .u0 ∈ H q,p ˚ be arbitrary. Then, we assume that an arbitrary function .u ∈ Vε (QT ) ∩c H0 (QT ) satisfies for every .t ∈ I 1 uc (t)2H + 2



t 0

.

A(s)(u(s)) − Jσε (f (s), F (s)), u(s)V˚εq,p (s) ds (5.47)

1 ≤ u0 2H . 2 Then, by taking into account (C.5), with .c2 = 0 in .L1 (I ), in (5.47), we further obtain for every .t ∈ I .

1 uc (t)2H + c0 2

 

t 0 t

+ 0

+

ρp(s,·) (ε(u)(s)) ds ≤ |c1 (s)|u(s)2H ds

 t 0

1 u0 2H 2 (5.48)

|f (s, y)||u(s, y)| + |F (s, y)||ε(u)(s, y)| dy ds . 

Applying the .ε-Young inequality (cf. Proposition 2.8) to the first summand in the last integral in (5.48) with respect to the exponent .q ∈ C 0 (QT ), with the −  constant .cq (δ) := (q − δ)1−(q ) ((q + ) )−1 for all .δ ∈ (0, (q − )−1 ), and also to the second summand with respect to the exponent .p ∈ C 0 (QT ), with the constant − 1−(p− ) ((p + ) )−1 for all .δ ∈ (0, (p − )−1 ), also using Lemma 3.6, .cp (δ) := (p δ) ˚εq,p (QT )) such that we obtain constants .cε , γε > 0 (independent of .t ∈ I or .u ∈ V for every .δ ∈ (0, δ0 ), where .δ0 := min{(q − )−1 , (p− )−1 }, and .t ∈ I , it holds  t

|f (s, y)||u(s, y)| + |F (s, y)||ε(u)(s, y)| dy ds

.

0





≤ cq (δ)ρq  (·,·) (f ) + cp (δ)ρp (·,·) (F ) + δ

t

(5.49)

ρq(s,·) (u(s)) + ρp(s,·) (ε(u(s))) ds

0

≤ cq (δ)ρq  (·,·) (f ) + cp (δ)ρp (·,·) (F )  t  2+γ  2+γ  +δ cε 1 + ρp(s,·) (ε(u(s))) + u(s)H ε 1 + u(s)H ε 0

+ ρp(s,·) (ε(u(s))) ds ≤ cq (δ)ρ

q  (·,·)

 + δcε 0



(f ) + cp (δ)ρ

p (·,·)

(F ) + cε δT + δ

t

(1 + cε )ρp(s,·) (ε(u(s))) ds

0 t



γ

2+γε

2u(s)Hε + u(s)H

2+2γε 

+ u(s)H

  × ρp(s,·) (ε(u(s))) + u(s)2H ds .

210

5 Existence Theory for Lipschitz Domains

We set .Aδ (f , F , u0 ) := 12 u0 2H + cq (δ)ρq  (·,·) (f ) + cp (δ)ρp (·,·) (F ) + cε δT , 1 2+2γε c δ , .y := 1 u (·)2 ∈ .a := 2|c1 (·)| ∈ L (I ), .E := exp(aL1 (I ) ), .B := 2 ε 0 c 1 H 2   α3 α1 α2 c0 1 0 := ρ (ε(u)(t)) ∈ L (I ), and . y + y + y .C (I ), .y 2 := t → (y 3 1 1 1 )[y 1 + 2 p(t,·) γε 2 1 α1 + α2 . Then, if we c0 y 2 ] ∈ L (I ), where .α1 := 2 , .α2 := 1 + α1 , .α3 :=  c0 } and .t ∈ I , we observe insert (5.49) in (5.48), for every .δ ∈ 0, min{δ0 , 2(1+c ε) that  t y 1 (t) + y 2 (s) ds ≤ Aδ (f , F , u0 ) 

.

0

t

+ 0

 a(s)y 1 (s) ds + B

0

(5.50)

t

y 3 (s) ds .

Apparently, .Aδ (f , F , u0 ) → 0 as .u0 H , .f Lq  (·,·) (QT )d , .F Lp (·,·) (QT )d×d , −2 and .ρ  −2 .δ → 0, provided that .ρq  (·,·) (f ) ≤ cq (δ) p (·,·) (F ) ≤ cp (δ) . There (·,·) q (QT )d , .F ∈ fore, there exists a radius .R > 0 such that for every .f ∈ L  (·,·) d×d p (QT , Msym ) and .u0 ∈ H from L f Lq  (·,·) (QT )d + F Lp (·,·) (QT )d×d + u0 H ≤ R ,

.

(5.51)

it follows that 3 .

i=1

(8Aδ (f , F , u0 )E)αi ≤

1 . 8B(1 + T )E

The generalized version of Grönwall’s inequality (cf. Lemma 5.2) applied to (5.50) implies y 1 C 0 (I ) + y 2 L1 (I ) ≤ 8Aδ (f , F , u0 )E ,

.

i.e., .

1 c0 uc 2H0 (Q ) + ρp(·,·) (ε(u)) ≤ 8Aδ (f , F , u0 )E =: M0 , T 2 2

˚εq,p (QT )∩c from which we deduce, following the proof of Proposition 5.8, that .A : V q,p 0 ˚ε (QT )∗ is .C 0 -Bochner coercive with respect to every .J σ (f , F ) ∈ .H (QT ) → V ε q,p ∗ ˚ 

Vε (QT ) and .u0 ∈ H satisfying (5.51).

5.4 Abstract Existence Theorem for Lipschitz Domains and p − ≥ 2

211

5.4 Abstract Existence Theorem for Lipschitz Domains and p − ≥ 2 At long last, let us next turn towards a somewhat abstract, but also very useful, existence result for solenoidal generalized evolution equations (cf. Definition 4.9). The following theorem is the main result of this book, on which all subsequent existence results in the context of solenoidal variable Bochner–Lebesgue spaces will be built. It is itself fundamentally based on Theorem 2.1, which originates from the author’s master thesis (cf. [92, 93, 95]). Theorem 5.1 (Main Theorem) Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), .T < ∞, .QT := I × , and .q, p ∈ Plog (QT ) with .p− ≥ 2 q,p q,p and .q ≥ p in .QT . Furthermore, let .A : Vε (QT ) ∩c H0 (QT ) → Vε (QT )∗ be q,p 0 ∗ ∗ bounded, .C -Bochner coercive with respect to .u ∈ Vε (QT ) and .u0 ∈ H , and q,p satisfying the .C 0 -Bochner condition (M). Then, there exists .u ∈ Wε,σ (QT ) with a 0 representation .uc ∈ H (QT ) such that .

dσ u + Au = u∗ dt uc (0) = u0

in Vε (QT )∗ , q,p

in H .

Proof 0. Reduction of assumptions: It suffices to treat the case .u∗ = 0 in .Vε (QT )∗ . q,p Otherwise, we can consider the shifted operator .A − u∗ : Vε (QT ) ∩c q,p H0 (QT ) → Vε (QT )∗ , which is again bounded, .C 0 -Bochner coercive with q,p respect to .0 ∈ Vε (QT )∗ and .u0 ∈ H , and satisfies the .C 0 -Bochner condition (M). 1. Galerkin approximation: On the basis of the separability of the space ˚q,p that lies densely in .V ˚+q,p , there exists a sequence .(vi ) ˚+q,p . .V i∈N ⊆ V+ ˚+q,p in H and the Gram–Schmidt process, we Owing to the density of .V can further assume that the sequence .(vi )i∈N is dense and orthonormal in H . We define .Vn := span{v1 , . . . ,vn } if equipped with . · V˚+q,p and .Hn := span{v1 , . . . ,vn } if equipped with .(·,·)H . Then, the triple .(Vn , Hn , idVn ) forms an evolution triple (cf. Beginning of Sect. 2.3.4) with canonical embedding ∗ ∗ ∗ .en := (idVn ) Rn idVn : Vn → Vn , where .Rn : Hn → Hn denotes the Riesz ∗ isomorphism of .Hn and .(idVn ) the adjoint operator of .idVn : Vn → Hn . Moreover, we introduce q,p

Vn := Lmax{q

.

+ ,p + }

(I, Vn ) ,

1,max{q + ,p+ },min{(q + ) ,(p+ ) }

Wn := Wen

Hn := C 0 (I , Hn ) .

(I, Vn , Vn∗ ) ,

212

5 Existence Theory for Lipschitz Domains q,p

Then, it holds .Vn → Vε (QT ) and Proposition 2.27 provides an embedding .(·)c : Wn → Hn , which guarantees for every .u ∈ Wn the existence of a unique continuous representation .uc ∈ Hn , and a formula of integration-byparts for .Wn . We are searching for functions .un ∈ Wn , .n ∈ N, that solve the system of evolution equations15 .

 JVn

den un dt

 + A n un = 0

in Vn∗ ,

(un )c (0) =

in Hn .

un0

(5.52)

where for every .n ∈ N16 An := (idVn )∗ A : Vn ∩c Hn → Vn∗ ,

.

un0 :=

n

(u0 , vi )H vi ∈ Hn .

i=1

2. Existence of Galerkin solutions: We intend to apply Theorem 2.1. Thus, we have to check for every .n ∈ N, whether .An : Vn ∩c Hn → Vn∗ is bounded, n 0 ∗ .C -Bochner coercive with respect to .0 ∈ Vn and .u ∈ Hn , and satisfies the 0 0 .C -Bochner condition (M) in the sense of Definition 2.20. In fact, because q,p q,p .A : Vε (QT ) ∩c H0 (QT ) → Vε (QT )∗ is bounded, .C 0 -Bochner coercive q,p with respect to .0 ∈ Vε (QT )∗ and .u0 ∈ H in the sense of Definition 5.4, satisfies the .C 0 -Bochner condition (M) in the sense of Definition 5.3, and n .supn∈N u H ≤ u0 H , one readily sees that the same also holds in the 0 sense of Definition 2.20 for its restrictions .An : Vn ∩c Hn → Vn∗ , .n ∈ N. Consequently, Theorem 2.1 yields the existence of .un ∈ Wn , .n ∈ N, that solve (5.52). Furthermore, if we test .(5.52)1 for every .n ∈ N with .un χ[0,t] ∈ Vn , for arbitrary .t ∈ (0, T ], apply the formula of integration-by-parts for .Wn (cf. Proposition 2.27 (ii)), and exploit .(5.52)2 for every .n ∈ N as well as n .supn∈N u H ≤ u0 H , then we obtain for every .t ∈ I and .n ∈ N 0 .

1 1 (un )c (t)2H + Aun , un χ[0,t] Vεq,p (QT ) ≤ u0 2H . 2 2

+ 

+ 

(5.53)

: Lmin{(q ) ,(p ) } (I, Vn∗ ) → Vn∗ for every .n ∈ N denotes the isomorphism defined in Proposition 2.20. 16 Here, .V ∩ H for every .n ∈ N denotes the subspace of .V consisting of all functions .u ∈ n c n n Vn that possess a representation .uc ∈ Hn (cf. Sect. 2.3.4), and .(idVn )∗ the adjoint operator of q,p .idVn : Vn → Vε (QT ). 15 Here, .J Vn

5.4 Abstract Existence Theorem for Lipschitz Domains and p − ≥ 2

213 q,p

Combining (5.53) and the .C 0 -Bochner coercivity of .A : Vε (QT ) ∩c q,p q,p H0 (QT ) → Vε (QT )∗ with respect to .0 ∈ Vε (QT )∗ and .u0 ∈ H , as well as its boundedness, we establish the existence of a constant .M > 0, which does not depend on .n ∈ N, such that for every .n ∈ N un Vεq,p (QT )∩c H0 (QT ) + Aun Vεq,p (QT )∗ ≤ M .

.

(5.54)

3. Passage to the limit: 3.1. Convergence of Galerkin solutions: From the a priori estimate (5.54), q,p q,p the reflexivity of both .Vε (QT ) and .Vε (QT )∗ (cf. Proposition 4.2), and Corollary 2.2, we obtain a not relabeled subsequence .(un )n∈N ⊆ q,p q,p Vε (QT ) ∩ H∞ (QT ) as well as elements .u ∈ Vε (QT ) ∩ H∞ (QT ) q,p and .ξ ∈ Vε (QT )∗ such that un  u .

∗

un  u

in Vε (QT )

q,p

(n → ∞) ,

in H∞ (QT )

(n → ∞) ,

in Vε (QT )∗

(n → ∞) .

q,p

Aun  ξ

(5.55)

3.2. Regularity and initial condition: Let .v ∈ Vk , where .k ∈ N is arbitrary, and .ϕ ∈ C 1 (I ) with .ϕ(T ) = 0. Testing .(5.52)1 for every .n ∈ N, where .n ≥ k, with .vϕ ∈ Vk ⊆ Vn , a subsequent application of the formula of integration-by-parts for .Wn (cf. Proposition 2.27 (ii)), and exploiting .(5.52)2 for every .n ∈ N, yields for every .n ∈ N with .n ≥ k that 

Au. n , vϕVεq,p (QT ) =

I

(un (s), v)H ∂t ϕ(s) ds + (un0 , v)H ϕ(0) .

(5.56)

Passing for .n → ∞ in (5.56), using (5.55) and .un0 → u0 in H .(n → ∞) in doing so, we find that 

ξ , vϕVεq,p (QT ) =

(u(s), v)H ∂t ϕ(s) ds + (u0 , v)H ϕ(0)

.

(5.57)

I

 case .ϕ ∈ for every .v ∈ k∈N Vk and .ϕ ∈ C 1 (I ) with .ϕ(T ) = 0. In the  C0∞ (I ) in (5.57), Proposition 4.20 proves, due to the density of . k∈N Vk q,p ˚+q,p , that .u ∈ Wε,σ in .V (QT ) with a representation .uc ∈ H0 (QT ) (cf. q,p Proposition 4.23 (i)), i.e., that .u ∈ Vε (QT ) ∩c H0 (QT ) (cf. (5.42)), and .

dσ u = −ξ dt

in Vε (QT )∗ . q,p

(5.58)

214

5 Existence Theory for Lipschitz Domains

Therefore, we are able to apply the formula of integration-by-parts for q,p Wε,σ (QT ) (cf. Proposition 4.23) in (5.57) for .ϕ ∈ C 1 (I ) with .ϕ(T ) = 0 and .ϕ(0) = 1, from which, in particular, using (5.58), it follows that

.

(uc (0) − u0 , v)H = 0

.

(5.59)

  ˚+q,p and .V ˚+q,p is for every .v ∈ k∈N Vk . Because . k∈N Vk is dense in .V dense in H , we then conclude from (5.59) to the desired initial condition, i.e., uc (0) = u0

.

in H .

(5.60)

q,p

3.3. Weak convergence in .Vε (QT ) ∩c H0 (QT ): Next, we will prove that .(un )c (t)  uc (t) in H .(n → ∞) for every .t ∈ I . To do this, let us fix an arbitrary time slice .t ∈ (0, T ]. From the a priori estimate .supn∈N (un )c (t)H ≤ M (cf. (5.54)), we obtain the existence of a subsequence .((un )c (t))n∈t ⊆ H , with a cofinal subset .t ⊆ N, which initially depends on this fixed t, and an element .ut ∈ H such that (un )c (t)  ut

(t  n → ∞) .

in H

.

(5.61)

For .v ∈ Vk , where .k ∈ t is arbitrary, and .ϕ ∈ C 1 (I ) with .ϕ(0) = 0 and .ϕ(t) = 1, we test .(5.52)1 for every .n ∈ t , where .n ≥ k, with .vϕχ[0,t] ∈ Vk ⊆ Vn and use the formula of integration-by-parts for .Wn (cf. Proposition 2.27 (ii)) to arrive for every .n ∈ t with .n ≥ k at  .

Aun , vϕχ[0,t] Vεq,p (QT ) =

t

(un (s), v)H ∂t ϕ(s) ds 0

(5.62)

− ((un )c (t), v)H . Passing for  .t  n → ∞ in (5.62), using (5.55) and (5.61), we obtain for any .v ∈ k∈t Vk that 

ξ ,. vϕχ[0,t] Vεq,p (QT ) =

t 0

(u(s), v)H ∂t ϕ(s) ds − (ut , v)H .

(5.63)

Moreover, the combination of (5.63) and (5.58), and the formula of q,p integration-by-parts for .Wε,σ (QT ) (cf. Proposition 4.23 (ii)) shows that (uc (t) − ut , v)H = 0

.

(5.64)

 for every t Vk . Thanks to .Vk ⊆Vk+1 for every .k ∈ N, we have  .v ∈ k∈ that . k∈t Vk = k∈N Vk . Therefore, . k∈t Vk lies densely in H and

5.4 Abstract Existence Theorem for Lipschitz Domains and p − ≥ 2

215

(5.64) proves that .uc (t) = ut in H , i.e., owing to (5.61), we have that (un )c (t)  uc (t)

in H

.

(t  n → ∞) .

(5.65)

Inasmuch as this argumentation remains valid for each subsequence of .((un )c (t))n∈N ⊆ H , .uc (t) ∈ H is a weak accumulation point of each subsequence of .((un )c (t))n∈N ⊆ H . Thus, from the standard convergence principle (cf. [166, Prop. 10.13 (4)]), it follows that (5.65) holds even if .t = N, i.e., we have that (un )c (t)  uc (t)

in H

.

(n → ∞)

for all t ∈ I .

(5.66)

Therefore, combining .(5.55)1,2 and (5.66), we conclude, arguing analogously to [92, Prop. 2.6], that un  u

.

q,p

in Vε (QT ) ∩c H0 (QT ) (n → ∞) .

(5.67)

3.4. Identification of .Au and .ξ : According to (5.53) in the case .t = T , also using (5.60), for every .n ∈ N, it holds 1 1

Aun , un Vεq,p (QT ) ≤ − (un )c (T )2H + uc (0)2H . 2 2

(5.68)

.

The limit superior with respect to .n → ∞ on both sides in (5.68), (5.66) in the case .t = T , the weak lower semi-continuity of the . · H -norm, the q,p formula of integration-by-parts for .Wε,σ (QT ) (cf. Proposition 4.23 (ii)), and (5.58) then yield that 1 1 lim sup Aun , un Vεq,p (QT ) ≤ − uc (T )2H + uc (0)2H 2 2 n→∞   dσ u . =− ,u q,p dt Vε (QT )

(5.69)

= ξ , uVεq,p (QT ) . q,p

Eventually, by virtue of the .C 0 -Bochner condition (M) of .A : Vε (QT )∩c q,p H0 (QT ) → Vε (QT )∗ , we conclude from .(5.55)3 , (5.67) and (5.69) that Au = ξ

.

in Vε (QT )∗ q,p

holds and, looking back to (5.58) and (5.60), in doing so, to the assertion of this theorem. 

216

5 Existence Theory for Lipschitz Domains

By falling back on Propositions 5.7, 5.8, 5.9, and Remark 5.6, resp., we infer the following less abstract and thus potentially more applicable variants of Theorem 5.1. Corollary 5.1 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .p− ≥ 2 and .p ≤ q ≤ p∗ − ε in  2 − ˚q,p (t) → V ˚q,p (t)∗ , .t ∈ I , be a family .QT for .ε ∈ 0, p . Moreover, let .A(t) : V d q,p of operators satisfying (C.1)–(C.6). Then, for .u∗ := Jεσ (f , F ) ∈ Vε (QT )∗ , −   where .f ∈ L(p ) (QT )d and .F ∈ Lp (·,·) (QT , Md×d sym ), and .u0 ∈ H , there exists q,p 0 .u ∈ Wε,σ (QT ) with a representation .uc ∈ H (QT ) such that .uc (0) = u0 in H q,p and for every .v ∈ Vε (QT ), it holds   .

I

  dσ u (t), v(t) dt + A(t)(u(t)), v(t)V˚q,p (t) dt dt ˚q,p (t) I V  = u∗ (t), v(t)V˚q,p (t) dt . I

  Remark 5.7 The assumption .ε ∈ 0, d2 p− in Corollary 5.1 guarantees that log .p∗ − ε ≥ p in .QT , and thereby that there can exist an exponent .q ∈ P (QT )  2 − satisfying .p ≤ q ≤ p∗ − ε in .QT at all. Indeed, for .ε ∈ 0, d p , we have that d+2 2 − .p∗ − ε = p d − ε ≥ p + d p − ε ≥ p in .QT . Corollary 5.2 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .q, p ∈ P (QT ) with .p− ≥ 2 and .q ≥ p in .QT . q,p q,p ∗ ˚ (t) → V ˚ (t) , .t ∈ I , be a family of operators satisfying Moreover, let .A(t) : V q,p (C.1)–(C.4), (C.5*) and (C.6*). Then, for .u∗ ∈ Vε (QT )∗ and .u0 ∈ H , there exists q,p 0 .u ∈ Wε,σ (QT ) with a representation .uc ∈ H (QT ) such that .uc (0) = u0 in H q,p and for every .v ∈ Vε (QT ), it holds   .

I

  dσ u (t), v(t) dt + A(t)(u(t)), v(t)V˚q,p (t) dt dt ˚q,p (t) I V  = u∗ (t), v(t)V˚q,p (t) dt . I

Remark 5.8 (Teething Troubles of Theorem 5.1) One obvious teething trouble of Theorem 5.1, which was also already present in the formula of integration-by-parts q,p for .Wε,σ (QT ) (cf. Proposition 4.23), is the requirement that .q, p ∈ Plog (QT ) are subject to the restrictive relation .q ≥ p in .QT . This requirement is not directly related to the method of proof of Proposition 4.23, but nevertheless needed for the functioning of the smoothing method of transversal expansion in space for q,p .Vε (QT ). However, as we have already pointed out in Remark 5.4, thanks to the synergies between the notion of (.C 0 -)Bochner coercivity and Poincaré’s inequality for variable Bochner–Lebesgue spaces (cf. Proposition 3.6), this requirement is not entailing any further difficulties.

5.5 Application to Model Problems

217

The second eye-catching teething trouble of Theorem 5.1 is the demanded lower bound .p− ≥ 2 for the variable exponent .p ∈ Plog (QT ). In Remark 4.16, we already extensively explained that this demanded lower bound is immediately related to the q,p selected smoothing method for .Wε,σ (QT ) (cf. Proposition 4.22). Admittedly, in view of an application of Theorem 5.1 to the unsteady .p(·, ·)-Navier–Stokes equations, where we first have to assume that .p− ≥ pC anyway (cf. Proposition 5.3), the lower bound .p− ≥ 2 does not appear to be very restrictive. Notwithstanding, with regard to the unsteady .p(·, ·)-Stokes equations, the admissible lower bound − > 2d , or even .p − > 1, undoubtedly lies within the realms of possibility .p d+2 because the majority of results of this chapter apply for .p ∈ Plog (QT ) with − > 1. Therefore, if one manages to get rid of the assumption .p − ≥ 2 in .p Proposition 4.23, then Theorem 5.1, Corollary 5.1 and Corollary 5.2 are immediately free from this restriction. Unfortunately, we are incapable of doing this by means of Proposition 4.22 (cf. Remark 4.16). As a consequence, we will initially only prove the existence of weak solutions of the unsteady .p(·, ·)-Stokes equations with the lower bound .p− ≥ 2 and of the unsteady .p(·, ·)-Navier–Stokes equations with the lower bound .p− ≥ pC . These results will later be generalized to nonLipschitz domains in Chap. 7 with the aid of parabolic compensated compactness methods. Immediately afterwards, in Chap. 8, we modify these methods and prove the existence of weak solutions of the unsteady .p(·, ·)-Stokes equations with the 2d lower bound .p− > d+2 and of the unsteady .p(·, ·)-Navier–Stokes equations with 3d − the lower bound .p > d+2 .

5.5 Application to Model Problems In this section, we establish the existence of weak solutions of the unsteady p(·, ·)-Stokes equations with the lower bound .p− ≥ 2 and of the unsteady .p(·, ·)Navier–Stokes equations with the lower bound .p− ≥ pC = 3d+2 d+2 , each in a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2. Note that these results concerning the existence of weak solutions for the unsteady .p(·, ·)-Stokes equations and for the unsteady .p(·, ·)-Navier–Stokes equations are the first ones that do not impose any artificial upper bound on the exponent .p ∈ Plog (QT ) that depends in any sense on the lower bound .p− , although the weak solvability of the unsteady .p(·, ·)-Navier– Stokes equations was recently claimed in [102]. However, in [102], a formula of integration-by-parts was used decisively, see [102, (4.47) ff.], giving no further details, such as giving a proof for this formula of integration-by-parts. Apart from that, the verification of the a priori estimate [102, (4.22)] is only possible if a Korn type inequality applies, which is precluded due to Remark 3.3, why the author still views the unsteady .p(·, ·)-Navier–Stokes equations as hitherto not completely solved. Thoroughly proved results e.g., the contributions [140, 147, 173], typically impose the relation .p+ < (p− )• on the exponent p. This requirement is primarily attributable to the method of proof, which in [140, 173] is based on parabolic compensated compactness principles, see Chaps. 7 and 8.

.

218

5 Existence Theory for Lipschitz Domains

Arranged in ascending order of physical significance, we start with an unsteady p(·, ·)–Stokes–like system with an optional, non-monotone, lower-order variable exponent non-linearity.

.

5.5.1 Unsteady p(·, ·)-Stokes Equations in a Lipschitz Domain with p − ≥ 2 Theorem 5.2 Let  ⊆ Rd , d ≥ 2, be a bounded Lipschitz domain, I := (0, T ), T 0 such that for almost every  ∈ Q and every .A, B ∈ Md×d , it holds18 .(t, x) T sym   S(t, x, A) − S(t, x, B) : (A − B) ≥ μ0 |A − B|2 .

.

(5.72)

Then, the solution existing according to Theorem 5.3 is unique. Proof We proceed analogously to [119, Thm. 4.29, p. 254], where the  assertion is proved for the simplified case of a constant exponent .p ∈ d+2 , ∞ . In addition, 2 we restrict our attention to the case .d ≥ 3 and .p− < d because the other cases can p,p be treated similarly. Let .u1 , u2 ∈ Wε,σ (QT ) be two solutions with representations

18 The

property (5.72) is sometimes referred to as strong monotonicity property.

5.5 Application to Model Problems

221

(ui )c ∈ H0 (QT ), .i = 1, 2, respectively, of

.

.

dσ ui + Sui + Cui = Jεσ (f , F ) dt (ui )c (0) = u0

in Vε (QT )∗ , p,p

in H ,

(5.73)

i = 1, 2 .

p,p

We introduce .δu := u1 −u2 ∈ Wε,σ (QT ). Thus, if we test the difference of .(5.73)1 p,p for .i = 1, 2 with .δuχ[0,t] ∈ Vε (QT ) for arbitrary .t ∈ I , apply the formula p,p of integration-by-parts for .Wε,σ (QT ) (cf. Proposition 4.23 (ii)), use .(5.73)2 for .i = 1, 2, and exploit (5.72), then for every .t ∈ I , we obtain 1 δuc (t)2H + μ0 . 2



t 0

ε(δu)(s)2L2 ()d×d ds

(5.74)

≤ − Cu1 − Cu2 , δuχ[0,t] Vεp,p (QT ) . Integration-by-parts (cf. [119, (4.31)]), also using that .div(δu) = 0 in .Lp(·,·) (QT ), yields for any .t ∈ I

.

− Cu1 − Cu2 , δuχ[0,t] Vεp,p (QT ) = Cδu, u1 χ[0,t] Vεp,p (QT )  t ≤ δu(s)2 2(p− ) d ε(u1 )(s)Lp− ()d×d ds . L

0

(5.75)

()

Next, by means of interpolation with respect to . 2(p1− ) = ∗

1−θ 2∗

+ θ2 , where .2∗ =

2d d−2

and .θ = 1 − 2pd− , the embedding .W01,2 ()d → L2 ()d and Korn’s inequality, we find for almost every .t ∈ I that δu(t)2 2(p− ) L

d −

()d

2−

≤ δu(t)Lp 2∗ ()d δu(t)H

.

d p− L2 ()d×d

≤ cd ε(δu)(t)

d p−

2−

δu(t)H

d p−

(5.76) .

If we insert (5.76) in (5.75) and then apply the .ε- Young inequality (cf. Proposi− − tion 2.8) with respect to the exponent .σ := 2p2p− −d , i.e., .σ  = 2pd , with the constant cσ (ε) :=

.

(σ  ε)1−σ σ

for every .ε ∈ (0, σ1 ), then we arrive for every .t ∈ I at 

−. Cu1 −Cu2 , δuχ[0,t] 

p,p Vε (QT )

t

≤ cd 0

cσ (ε)δuc (s)2H ε(u1 )(s)σ p− L



+ cd 0

t

εε(δu)(s)2L2 ()d×d ds .

()d×d

ds

(5.77)

222

5 Existence Theory for Lipschitz Domains

Next, if we choose .ε := that for every .t ∈ I

μ0 2cd

> 0, we observe, taking into account (5.77) in (5.74), 

δuc (t)2H + μ0

t

.

0



t

≤ 2cd cσ (ε) 0

ε(δu)(s)2L2 ()d×d ds

δuc (s)2H ε(u1 )(s)σ p− L

()d×d

ds .

−

Eventually, owing to .ε(u1 ) ∈ Lσ (I, Lp ()d×d ), since we have .p− ≥ σ for − ≥ d+2 , we conclude, by applying Grönwall’s inequality (cf. [23, Lem. II.4.10]), .p 2 that .δuc (t) = 0 in H for every .t ∈ I , i.e., .(u1 )c = (u2 )c in .H0 (QT ) and, by the p,p p,p injectivity of .(·)c : Wε,σ (QT ) → H0 (QT ), eventually .u1 = u2 in .Wε,σ (QT ). 

Remark 5.10 In the case .δ = 0, .c1 ≡ 0 in (S.3) and .c3 ≡ 0 in (D.3), then  Theorem 5.2, in addition, applies f ∈ Lq (·,·) (QT )d , where  2for− functionsp.(·,·) log (QT , Md×d .q := p∗ − ε ∈ P (QT ) for .ε ∈ 0, d p , .F ∈ L sym ) and .u0 ∈ H ,   provided that .f Lq (·,·) (QT )d + F Lp (·,·) (QT )d×d + u0 H is sufficiently small. In fact, Theorem 5.1 also applies for .C 0 -Bochner coercive operators and, by resorting q,p q,p to Proposition 5.15, we observe that .S + D : Vε (QT )∩H∞ (QT ) → Vε (QT )∗  (·,·) 0 q d is .C -Bochner coercive respect to  all .f ∈ L (QT ) , where   log .q := p∗ − ε ∈ P (QT ) for .ε ∈ 0, d2 p− , .F ∈ Lp (·,·) (QT , Md×d sym ) and .u0 ∈ H , for which .f Lq  (·,·) (QT )d + F Lp (·,·) (QT )d×d + u0 H does not assume a too large value. Furthermore, if .δ = 0 and .c1 ≡ 0 in (S.3), then exactly the same also applies to Theorem 5.3.

Part II

Extensions

Chapter 6

Pressure Reconstruction

Until this point, we have only proved the existence of the velocity for the unsteady p(·, ·)-Stokes and the unsteady .p(·, ·)-Navier–Stokes equations, i.e., the model problems (4.1) and (4.2), respectively, which, beyond that, has hitherto solely a derivative in the hydro-mechanical sense of Definition 4.6 (cf. Remark 4.15). This chapter is concerned with the reconstruction of the scalar kinematice pressure in an appropriate function space, which will show that the velocity solves the unsteady .p(·, ·)-Stokes and the unsteady .p(·, ·)-Navier–Stokes equations, respectively, at least in the sense of distributions, i.e., in .D (QT )d . Apart from that, in this connection, we intend to figure out whether the .L∞ - and Lipschitz truncation techniques, which are closely related to the issue of a pressure reconstruction, admit generalizations to the framework of variable Bochner–Lebesgue spaces. .

6.1 Pressure Reconstruction The reconstruction of the pressure, both in steady and unsteady problems, traditionally is based on de Rham’s lemma (cf. Lemma 4.1), which we have already encountered in Chap. 4, Sect. 4.2. More precisely, one commonly exploits for a constant exponent .p ∈ (1, ∞) and a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2, that the isomorphism properties of the steady distributional gradient p ˚p,p )◦ , by virtue of Proposition 2.21 (iv), inherit into the bijectivity .∇ : L () → (V 0 of the mapping p

˚p,p )◦ ) , ∇ : Cω0 (I , L0 ()) → Cω0 (I , (V

.

(6.1)



p ˚p,p )◦ which is for every .π ∈ Cω0 (I , L0 ()) defined by .(∇π)(t) := ∇(π (t)) in .(V for every .t ∈ I . Recall that we have already observed that de Rham’s lemma does

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaltenbach, Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents, Lecture Notes in Mathematics 2329, https://doi.org/10.1007/978-3-031-29670-3_6

225

226

6 Pressure Reconstruction

not admit an analogue in the framework of variable Bochner–Lebesgue spaces (cf. Remark 4.4) but surely does in the framework of variable Lebesgue spaces and variable Sobolev spaces (cf. Lemma 4.1). Consequently, since in (6.1) time and space variables are perfectly decoupled, de Rham’s lemma also guarantees the bijectivity of the mapping (6.1) if we replace the constant .p ∈ (1, ∞) with a timeindependent variable exponent .p ∈ Plog () with .p− > 1. Lemma 6.1 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, and let .p ∈ P () with .p− > 1. Then, the unsteady distributional gradient p (·)

∇ : Cω0 (I , L0

.

p (·)

for every .π ∈ Cω0 (I , L0 .t ∈ I , is bijective.

˚p,p )◦ ) , ()) → Cω0 (I , (V

˚p,p )◦ for every ()) defined by .(∇π)(t) := ∇(π(t)) in .(V

Proof Direct consequence of the isomorphism properties of the steady distribup (·) ˚p,p )◦ (cf. Lemma 4.1) using Proposition 2.21 (iv). tional gradient .∇ : L0 () → (V 

In the case of a time-independent variable exponent .p ∈ Plog () with .p− > 1, we are also able to reinterpret the distributional divergence, which we have already q,p discussed in Chap. 3, Sect. 3.2, instead of as a mapping with values in .Xε (QT )∗ , like we did it first in Chap. 3, Remark 3.5, now as mapping with values in a Bochner– Lebesgue space. Lemma 6.2 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .p ∈ P () with .p− > 1. Then, the unsteady distributional divergence 

div : Lp (·) (QT )d×d → L(p

.

+ )

˚p,p )∗ ) , (I, (X

 ˚p,p )∗ for for every .H ∈ Lp (·) (QT )d×d defined by .(div(H ))(t) := div(H (t)) in .(X almost every .t ∈ I , is well-defined, linear and bounded. 

+ 



Proof We make use of the embedding .Lp (·) (QT )d×d → L(p ) (I, Lp (·) ()d×d )1  ˚p,p )∗ , defined by and the boundedness of the operator .div : Lp (·) ()d×d → (X  (·) p d×d ˚p,p , . div(F), x ˚p,p := −(F, ∇x)Lp(·) ()d×d for every .F ∈ L () and .x ∈ X X in conjunction with Proposition 2.21. 

Aided by Lemmas 6.1 and 6.2, we can derive a parabolic pressure reconstruction result for time-independent .log-Hölder continuous variable exponents and bounded Lipschitz domains.

(p+ )



we employ the inequality .F p (·) d×d ≤ 1 + ρp (·) (F) for every .F ∈ Lp (·) ()d×d (cf. L () Lemma 2.1 (iii)) and Pettis’ theorem in combination with Hölder’s inequality (cf. Proposition 2.6) and Fubini’s theorem.

1 Here,

6.1 Pressure Reconstruction

227

Proposition 6.1 (Pressure Reconstruction, .C 0,1 -Case) Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), .T < ∞, .QT := I × , and .q, p ∈ Plog () with .q − , p− > 1. Furthermore, let .u ∈ Cω0 (I , Y ) be such that .div(u(t)) = 0 in .(W01,2 ())∗ , i.e., .(u(t), ∇f )Y = 0 for any .f ∈ W01,2 (), for all .t ∈ I , .h ∈   ∞ (Q )d , it holds Lq (·) (QT )d and .H ∈ Lp (·) (QT )d×d such that for every .φ ∈ C0,div T  − .

u(t, x) · ∂t φ(t, x) dtdx QT



(6.2)

=

h(t, x) · φ(t, x) + H (t, x) : ∇φ(t, x) dtdx . QT 

Then, there exists .π ∈ Cω0 (I , L0s (·) ()), where .s ∈ Plog (), such that for every .φ ∈ C0∞ (QT )d , it holds 

:= max



 qd 2d d+2 , d+q , p



−

u(t, x) · ∂t φ(t, x) dtdx. = QT

h(t, x) · φ(t, x) + H (t, x) : ∇φ(t, x) dtdx QT



+

π (t, x)∂t div(φ)(t, x) dtdx .

(6.3)

QT

Proof Since .s + ≥ d is not precluded, we set .r := max + < d, and .r erσ := (idV˚r,r )∗ J : L(r ˚r,r )∗ J : L(r er := (idX .

∗ 

∗ ) (·)

σ := (idV˚s,s )∗ erσ : L(r er,s

˚s,s )∗ er : L(r er,s := (idX

∗ ) (·)



qd  2d log d+2 , d+q ∈ P (),

˚r,r )∗ , ()d → (V

˚r,r )∗ , ()d → (X

∗ ) (·)

∗ ) (·)

i.e.,

˚s,s )∗ ()d → (V

(6.4)

˚s,s )∗ , ()d → (X

∗

. where .J : L(r ) (·) ()d → (Lr (·) ()d )∗ denotes the vector-valued version of the ˚r,r )∗ denote Riesz isomorphism defined in Proposition 2.7, .(idV˚r,r )∗ and .(idX ∗ (·) r,r r d r,r ˚ →L ˚ → Lr ∗ (·) ()d , ˚r,r : X the adjoint operators of .idV˚r,r : V () and .idX respectively, which are dense embeddings due to Proposition 3.9, and .(idV˚s,s )∗ and ˚s,s →V ˚r,r and .idX ˚s,s →V ˚r,r , ˚s,s )∗ denote the adjoint operators of .idV ˚s,s : V ˚s,s : V .(idX respectively. Since the operators (6.4) are embeddings, their corresponding linearinduced operators eσr : Lρ (I, L(r er : Lρ (I, L(r .

eσr,s : Lρ (I, L(r er,s : Lρ (I, L(r

∗ ) (·)

˚r,r )∗ ) , ()d ) → Lρ (I, (V

∗ ) (·)

˚r,r )∗ ) , ()d ) → Lρ (I, (X

∗ ) (·)

˚s,s )∗ ) , ()d ) → Lρ (I, (V

∗ ) (·)

˚s,s )∗ ) , ()d ) → Lρ (I, (X

(6.5)

228

6 Pressure Reconstruction

where .ρ ∈ [1, ∞], are embeddings as well (cf. Proposition 2.21 (ii)). Then, 2d using Definition 2.19, in particular, exploiting that .(r ∗ ) ≤ 2 since .r ≥ d+2 , qd σ ∞ s,s ∗ ∗   ˚ i.e., .er,s u ∈ L (I, (V ) ) is well-defined, that .(r ) ≤ q since .r ≥ d+q , +  ˚s,s )∗ ) is well-defined, and that .s  ≤ p since .s ≥ p, i.e., i.e., .eσr,s h ∈ L(q ) (I, (V +  (s ) (I, (X ˚s,s )∗ ) is well-defined (cf. Lemma 6.2), it is not difficult to .div(H ) ∈ L ˚s,s )∗ ), i.e., .u ∈ L∞ (I, Y ) and see that (6.2) is equivalent to .u ∈ We1,∞,1 (I, Y, (V σ r,s σ u ∈ W 1,1 (I, (V ˚s,s )∗ ), with .e r,s

.

σ u der,s

=

dt

d(V˚s,s )∗ eσr,s u = eσr,s h − (idV˚s,s )∗ div(H ) dt

˚s,s )∗ ) . in L1 (I, (V

(6.6)

The fundamental theorem for Bochner–Sobolev functions (cf. Proposition 2.24 (i)), ˚s,s )∗ ) of provides a unique continuous representation .(eσr,s u)c ∈ C 0 (I ,(V σ 1,1 (I, (V ˚s,s )∗ ). Furthermore, Proposition 2.24 (ii) guarantees the well.e r,s u ∈ W definedness and continuity of the Volterra operator .X : Lρ (I, X) → W 1,ρ (I, X), where .ρ ∈ [1, ∞] and X is a Banach space, which is for every .x ∈ Lρ (I, X) t defined by .(X x)(t) := 0 x(s) ds in X for every .t ∈ I . If we apply .(V˚s,s )∗ to (6.6) and resort to the identity (2.7) in doing so, then we arrive at (eσr,s u)c − (eσr,s u)c (0) .   = (V˚s,s )∗ eσr,s h − (idV˚s,s )∗ div(H )

˚s,s )∗ ) . in C 0 (I , (V

(6.7)

˚s,s )∗ ) (cf. Proposition 2.21 (iv)). Due to .u ∈ Cω0 (I , Y ), it holds .er,s u ∈ Cω0 (I , (X ∗ ˚s,s )∗ ), we ˚s,s )∗ on .L1 (I, (X Therefore, also using .(V˚s,s )∗ (idV˚s,s ) = (idV˚s,s )∗ (X read from the identity (6.7) that   0 ˚s,s )◦ ) . ˚s,s )∗ er,s h − div(H ) ∈ Cω er,s u − (er,s u)(0) − (X (I , (V

.

(6.8)



s (·) ˚s,s )◦ ) Due to (6.8) and based on the bijectivity of .∇ : Cω0 (I , L0 ())→ Cω0 (I , (V (cf. Lemma 6.1), we are now in the position to introduce the function

   ˚s,s )∗ er,s h − div(H ) π := ∇ −1 er,s u − (er,s u)(0) − (X .

s  (·)

∈ Cω0 (I , L0

(6.9)

()) .

By construction, the function defined in (6.9) satisfies the identity

.

er,s u − (er,s u)(0) − ∇π   ˚s,s )∗ er,s h − div(H ) =  (X

˚s,s )∗ ) . in W 1,1 (I, (X

(6.10)

Eventually, testing (6.10) with .∂t φ ∈ C0∞ (QT )d , for arbitrary .φ ∈ C0∞ (QT )d , and a subsequent integration-by-parts in time on the right-hand side, as well as using the definitions of both .∇ and .div, one concludes the weak formulation (6.3). 

6.1 Pressure Reconstruction

229

Reviewing the proof of Proposition 6.1, it should appear that we did not ˚s,s )∗ exploit all regularity available. In fact, we only used .er,s u − (er,s u)(0) + (X 0 s,s ◦ ˚ ) ), although we actually even have that .u − u(0) ∈ .[e r,s h − div(H )] ∈ Cω (I , (V ˚s,s )∗ [er,s h − Cω0 (I , Y ), i.e., higher spatial regularity, on the one hand, and .(X 1,1 s,s ∗ ˚ div(H )] ∈ W (I, (X ) ), i.e., higher temporal regularity, on the other hand. To get access to this additional regularity, it is necessary to have an extension of the inverse 

˚p,p )◦ ⊆ (X ˚p,p )∗ → Lp (·) () ∇ −1 : (V 0

.

˚p,p )∗ that in addition takes into account higher regularity, to the whole dual space .(X  p (·) i.e., that maps .R(ep ) (cf. .(6.4)1 ), e.g., into .W 1,p (·) () ∩ L0 (). Here, the following variable exponent regularity result for the steady Stokes equations, which is due to L. Diening, D. Lengeler and M. R˚užiˇcka [50], see also [49, Section 14.2], provides a remedy. Proposition 6.2 Let . ⊆ Rd , .d ≥ 2, be a bounded .C 1,1 -domain and let .p ∈ Plog () with .p− > 1. Then, the following statements apply: 

˚p,p )∗ , there exist unique .u ∈ V ˚p ,p and .π ∈ Lp (·) () (i) For every .x∗ ∈ (X 0 ˚p,p )∗ . In addition, there exists a constant such that .− u + ∇π = x∗ in .(X .c = c(p, ) > 0 such that uV˚p ,p + π Lp (·) () ≤ cx∗ (X ˚p,p )∗ ,

.

(6.11)

p (·)

i.e., if we define .𝒫x∗ := π in .L0 (), then the resulting mapping  ˚p,p )∗ → Lp (·) () is well-defined, linear, bounded, and satisfies .𝒫: (X 0 ∗ ∗ ˚p,p )◦ for every .x∗ ∈ (V ˚p,p )◦ . .∇𝒫x = x in .(V  (·) p d ˚p ,p ∩ W 2,p (·) ()d 2 ()  , there exist unique .u ∈ V (ii) For every .f ∈ L  (·)  p (·) 1,p and .π ∈ W () ∩ L0 () such that .− u + ∇π = f in .Lp (·) ()d . In addition, there exists a constant .c = c(p, ) > 0 such that uW 2,p (·) ()d + π W 1,p (·) () ≤ cfLp (·) ()d .

.





(6.12)

p (·)

In other words, .𝒫◦ep : : Lp (·) ()d → W 1,p (·) ()∩L0 () is well-defined  ˚p,p )∗ is defined as in the proof of and bounded, where .ep : Lp (·) ()d → (X Proposition 6.1, i.e., as in .(6.4)1 .

2 Here, .W 2,p(·) () := {f ∈ W 1,p(·) () | ∇ 2 f := (∂ ∂ f ) p(·) ()d×d } is equipped i j i,j =1,...,d ∈ L with the canonical norm . · W 2,p(·) () :=  · W 1,p(·) () + ∇ 2 · Lp(·) ()d×d (cf. [49, Definition 8.1.2. & Remark 8.1.5.]).

230

6 Pressure Reconstruction

Proof See [50, Theorem 20 & Theorem 21]. To be more precise, .∇𝒫x∗ = x∗ ˚p,p )◦ for every .x∗ ∈ (V ˚p,p )◦ follows from the fact that for .x∗ ∈ (V ˚p,p )◦ , in .(V the unique solution of the steady Stokes equations is given by the unique velocity  ˚p ,p and a unique pressure .π ∈ Lp (·) (). In fact, if .x∗ ∈ (V ˚p,p )◦ , .u = 0 ∈ V 0 p (·) ∗ ˚p,p )∗ . then Lemma 4.1 provides a unique .π˜ ∈ L0 () with .x = ∇ π˜ in .(X ˚p,p )∗ , it follows that .− u + ∇(π − π˜ ) = 0 Then, from .− u + ∇π = x∗ in .(X p,p ∗ ˚ in .(X ) and, hence, on the basis of the uniqueness, that .u = 0 and ˚p,p )◦ . .π = π, ˜ i.e., .∇𝒫x∗ = ∇π = ∇ π˜ = x∗ in .(V 



˚p,p )∗ → Lp (·) () from PropoIn virtue of Proposition 2.21, the operator .𝒫: (X 0 sition 6.2 induces for all .ρ ∈ [1, ∞] an unsteady operator p (·)

˚p,p )∗ ) → Lρ (I, L  : Lρ (I, (X 0

.

()) ,

p (·)

defined by .(x ∗ )(t) := 𝒫 (x ∗ (t)) in .L0 () for almost every .t ∈ I and  ˚p,p )∗ ), inheriting all properties of .𝒫: (X ˚p,p )∗ → Lp (·) (), every .x ∗ ∈ Lρ (I, (X 0 collected in the following corollary. Corollary 6.1 Let . ⊆ Rd , .d ≥ 2, be a bounded .C 1,1 -domain, .p ∈ Plog () with − > 1, and .ρ ∈ [1, ∞]. Then, the following statements apply: .p p (·)

˚p,p )∗ ) → Lρ (I, L (i) . : Lρ (I, (X 0 satisfies ∇x ∗ = x ∗

.

˚p,p )◦ ) in Lρ (I, (V 

()) is well-defined, linear, bounded, and ˚p,p )◦ ) . for all x ∗ ∈ Lρ (I, (V

(6.13)

p (·)



(ii) . ◦ ep : Lρ (I, Lp (·) ()d ) → Lρ (I, W 1,p (·) () ∩ L0 ()) is well-defined, linear and bounded.  ˚p,p )∗ ) → C 0 (I , Lp (·) ()) is well-defined, linear, and satisfies (iii) . : Cω0 (I , (X ω 0 ∇x ∗ = x ∗

.

˚p,p )◦ ) in Cω0 (I , (V 

˚p,p )◦ ) . for all x ∗ ∈ Cω0 (I , (V 

p (·)

(iv) . ◦ ep : Cω0 (I , Lp (·) ()d ) → Cω0 (I , W 1,p (·) () ∩ L0 and linear.

(6.14)

()) is well-defined

 ˚p,p )∗ ) and .ep : C 0 (I , Lp (·) ()d ) → Here, .ep : Lρ (I, Lp (·) ()d ) → Lρ (I, (X ω ˚p,p )∗ ) denote the through .(6.4)1 , in the sense of Proposition 2.21, linearCω0 (I , (X induced operators (cf. .(6.5)1 ).

Proof The assertions (i) and (ii) follow from Proposition 6.2 (i) and (ii), respectively, using Proposition 2.21. The assertions (iii) and (iv) likewise follow from Proposition 6.2 (i) and (ii), respectively, but now by referring to Proposition 2.21 (iv). 

6.1 Pressure Reconstruction

231

By means of Proposition 6.2 and Corollary 6.1, we can now improve Proposition 6.1 in the case of an additional .C 1,1 -regularity of the bounded domain . ⊆ Rd , .d ≥ 2. Proposition 6.3 (Pressure Reconstruction, .C 1,1 -case) Let . ⊆ Rd , .d ≥ 2, be a bounded .C 1,1 -domain, .I := (0, T ), .T < ∞, .QT := I × , and .q, p ∈ Plog () with .q − , p− > 1. Furthermore, let .u ∈ Cω0 (I , Y ) with .div(u(t)) = 0 in .(W01,2 ())∗   for every .t ∈ I , .h ∈ Lq (·) (QT )d and .H ∈ Lp (·) (QT )d×d such that for every .φ ∈ ∞ (Q )d , it holds C0,div T  −

u(t, x) · ∂t φ(t, x) dtdx QT



.

(6.15) h(t, x) · φ(t, x) + H (t, x) : ∇φ(t, x) dtdx .

= QT

+ 



q  (·)

+ 

p (·)

Then, there exist .π 1 ∈ L(q ) (I, W 1,q (·) () ∩ L0 ()), .π 2 ∈ L(p ) (I, L0 ()) and .π h ∈ Cω0 (I , W 1,2 () .∩L20 ()) such that for every .φ ∈ C0∞ (QT )d , it holds  .

−

[u(t, x) + ∇π h (t, x)] · ∂t φ(t, x) dtdx QT

 [h(t, x) + ∇π 1 (t, x)] · φ(t, x) dtdx

=

(6.16)

QT



+

[H (t, x) + π 2 (t, x)Id ] : ∇φ(t, x) dtdx . QT

In addition, it holds . (π h (t)) = 0 in .(W01,2 ())∗ , i.e., .(∇(π h (t)), ∇f )Y = 0 for every .f ∈ W01,2 (), for every .t ∈ I , .π h (0) = 0 in .W 1,2 () ∩ L20 (), and there hold the estimates π 1 L(q + ) (I,W 1,q  (·) ()) ≤ cq hLq  (·) (QT )d , .

π 2 L(p+ ) (I,Lp (·) ()) ≤ cp H Lp (·) (QT )d×d , π h (t)W 1,2 () ≤ ch u(t) − u(0)Y

(6.17) for all t ∈ I ,

for constants .cq = c(d, , q), cp = c(d, , p), ch = c(d, ) > 0. Proof We proceed as in [48, Theorem 2.2], in which the assertion is proved for constant exponents. For alternative proving strategies, we also recommend the contributions [164, Thm. 2.6] and [165].

232

6 Pressure Reconstruction 

+ 



To begin with, based on the embeddings .Lq (·) (QT ) → L(q ) (I, Lq (·) ()) and +   .L (QT ) → L(p ) (I, Lp (·) ()), we make the following constructions p (·)

π h := ( ◦ e2 )(u(0) − u) ∈ Cω0 (I , W 1,2 () ∩ L20 ()) , .

π 1 := ( ◦ eq )(h) ∈ L(q

+ )

π 2 := ( ◦ div)(H ) ∈ L(p

q  (·)



(I, W 1,q (·) () ∩ L0

+ )

p (·)

(I, L0

(6.18)

()) ,

()) ,

where we used Corollary 6.1 for the extraction of the available regularity. Apparently, it holds .π h (0) = 𝒫(e2 (u(0) − u(0))) = 𝒫 0 = 0 in .W 1,2 () ∩ L20 () and . (π h (t)) = 0 in .(W01,2 ())∗ for every .t ∈ I since Proposition 6.2 (ii) ˚2,2 ∩ W 2,2 ()d with yields for every time slice .t ∈ I a unique function .ut ∈ V .− ut + ∇(π h (t)) = u(0) − u(t) in Y , from which one can easily conclude that 1,2 ∗ . (π h (t)) = div(∇(π h (t))) = (div(ut )) + div(u(0) − u(t)) = 0 in .(W 0 ()) . Proposition 6.2 (ii) gives us a constant .ch > 0 such that for every .t ∈ I π h (t)W 1,2 () = (𝒫◦ e2 )(u(0) − u(t))W 1,2 () ≤ ch u(t) − u(0)Y ,

.

i.e., the estimate .(6.17)3 . In addition, Corollary 6.1 (ii) provides a constant .cq > 0 such that π 1 L(q + ) (I,W 1,q  (·) ()) = ( ◦ eq )(h)L(q + ) (I,W 1,q  (·) ())

.

≤ cq hL(q + ) (I,Lq  (·) ()d ) ≤ cq hLq  (·) (QT )d , 

i.e., the estimate .(6.17)1 , where we made use of the embedding .Lq (·) (QT )d → +   L(q ) (I, Lq (·) ()d ), which again comes from Lemma 2.1 (iii). Similarly, one also deduces the estimate .(6.17)2 by resorting to Corollary 6.1 (i) and Lemma 6.2. Thus, it remains to establish (6.16). To do this, let anew .s, r ∈ Plog () be defined as in Proposition 6.1. We observe as in the proof of Proposition 6.1 that (6.15) is ˚s,s )∗ ) with equivalent to .u ∈ We1,∞,1 (I, Y, (V σ r,s

  0 ˚s,s )◦ ) . ˚s,s )∗ er,s h − div(H ) ∈ Cω er,s u − (er,s u)(0) − (X (I , (V

.

(6.19)

6.1 Pressure Reconstruction

233

Using the identity (6.14) from Corollary 6.1 (iii), we deduce from (6.19) that   ˚s,s )∗ er,s h − div(H ) er,s u − (er,s u)(0) − (X  = ∇ er,s u − (er,s u)(0)   ˚s,s )∗ er,s h − div(H ) −  (X  = ∇ ( ◦ e2 )(u − u(0))

 − Ls0 (·) () [( ◦ eq )h − ( ◦ div)(H )]   = −∇ π h + Ls0 (·) () [π 1 − π 2 ]   ˚s,s )∗ er,s ∇π 1 − div(π 2 Id ) = −er,s ∇π h + (X

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

˚s,s )◦ ) . in Cω0 (I , (V

where .( ◦ er,s )(u − u(0)) = ( ◦ e2 )(u − u(0)) and .( ◦ er,s )h = ( ◦ eq )h easily follow from the uniqueness in Proposition 6.2 (i) and we used .∇ = er,s ∇ on 1 1,(r ∗ ) (·) ()d ). Hence, it follows that .L (I, W er,s [u + ∇π h ] = (er,s u)(0)   ˚s,s )∗ ) . ˚s,s )∗ er,s (h + ∇π 1 ) − div(H + π 2 Id ) in C 0 (I , (X +  (X

.

Eventually, testing the latter with .∂t φ ∈ C0∞ (QT )d , for arbitrary .φ ∈ C0∞ (QT )d , and a subsequent integration-by-parts on the right-hand side, also using the definition of .div, proves (6.16). 

Remark 6.1 If we define .π h := ( ◦ e2 )(u(t0 ) − u) ∈ Cω0 (I , W 1,2 () ∩ L20 ()) for arbitrary .t0 ∈ I , then a simple adaptation of the proof of Proposition 6.3 shows that the assertion of Proposition 6.3, where we substitute .u(0) by .u(t0 ) in the estimate .(6.17)3 , is still valid, but with .π h (t0 ) = 0 instead of .π h (0) = 0 +   q  (·) in .W 1,2 (). In addition, note that .π 1 ∈ L(q ) (I, W 1,q (·) () ∩ L0 ()) and + 

p (·)

π 2 ∈ L(p ) (I, L0 contrast to .π h .

.

()), by definition (cf. (6.18)), do not depend on .t0 ∈ I , in

Admittedly, Proposition 6.3 is to some extent rather unsatisfactory, in the sense that we are still not able to recover the pressures, in particular, the pressure  associated to .H ∈ Lp (·) (QT )d×d , in full regularity, i.e., we cannot guarantee that  p (·) .π 2 ∈ L 0, (QT ), even if the variable exponent is smooth and does not depend on time. Actually, we wish the following pressure reconstruction theorem to be true (CAUTION: The following theorem is a lie!). Untruth (Pressure Reconstruction in Full Regularity) Let . ⊆ Rd , .d ≥ 2, be an arbitrary bounded domain, .I := (0, T ), .T < ∞, .QT := I × , and .p ∈ Plog (QT ) with .p− > 1. Furthermore, let .u ∈ Cω0 (I , Y ), with .div(u(t)) = 0 in .(W01,2 ())∗ for

234

6 Pressure Reconstruction

∞ (Q )d , it holds every .t ∈ I , and .H ∈ Lp(·,·) (QT )d×d such that for every .φ ∈ C0,div T

 .



−

u(t, x) · ∂t φ(t, x) dtdx =

H (t, x) : ∇φ(t, x) dtdx .

QT

(6.20)

QT p(·,·)

Then, there exist .π ∈ L0, (QT ) and .π h ∈ Cω0 (I , W 1,2 () ∩ L20 ()), satisfying 1,2 1,2 () ∩ L2 (), ∗ . (π h (t)) = 0 in .(W 0 0 ()) for every .t ∈ I and .π h (0) = 0 in .W ∞ d such that for every .φ ∈ C0 (QT ) , it holds  −

[u(t, x) + ∇π h (t, x)] · ∂t φ(t, x) dtdx QT



.

(6.21) [H (t, x) + π(t, x)Id ] : ∇φ(t, x) dtdx ,

= QT

and there exist constants .cp = c(d, , p), ch = c(d, ) > 0 such that π Lp(·,·) (QT ) ≤ cp H Lp(·,·) (QT )d×d , .

π h (t)W 1,2 () ≤ ch u(t) − u(0)Y

for all t ∈ I .

(6.22)

Unfortunately, as already announced, Untruth 6.1 cannot be valid, even if the variable exponent is smooth and does not depend on time, as can be observed in the following three remarks. Remark 6.2 (Failure of Pressure Reconstruction in Variable Bochner–Lebesgue Spaces) Let us assume that the Untruth 6.1 tells the truth after all. Then, since for each .H ∈ C0∞ (QT )d×d , the function .u := −PH Y div(H ) ∈ C ∞ (I , H ), where .PH : Y → H denotes the orthogonal projection from Y into H and .Y the Volterra operator with respect to Y defined in Proposition 2.24 (ii), satisfies for every ∞ d .φ ∈ C 0,div (QT )  .

−

 u(t, x) · ∂t φ(t, x) dtdx =

QT

H (t, x) : ∇φ(t, x) dtdx ,

(6.23)

QT p(·,·)

Untruth 6.1 would provide .π ∈ L0, (QT ) and .π h ∈ Cω0 (I , W 1,2 () ∩ L20 ()), satisfying both . (π h (t)) = 0 in .(W01,2 ())∗ for every .t ∈ I and .π h (0) = 0 in 1,2 () ∩ L2 (), such that (6.21) as well as (6.22) are valid. As a consequence, .W 0 p(·,·) p(·,·) if we denote by .F (H ) ⊆ L0, (QT ) the set of all .π ∈ L0, (QT ) that satisfy (6.21) and (6.22) with respect to .π h ∈ Cω0 (I , W 1,2 () ∩ L20 ()), satisfying both 1,2 ∗ 1,2 () ∩ L2 (), . (π h (t)) = 0 in .(W 0 0 ()) for every .t ∈ I and .π h (0) = 0 in .W

6.1 Pressure Reconstruction

235

then the resulting mapping3 p(·,·)

F : C0∞ (QT )d×d ⊆ Lp(·,·) (QT )d×d → 2L0,

.

(QT )

(6.24)

is well-defined, i.e., we have that .F (H ) = ∅ for all .H ∈ C0∞ (QT )d×d , and for every .H ∈ C0∞ (QT )d×d , owing to (6.22), it holds for every .π ∈ F (H ) π Lp(·,·) (QT ) ≤ cp H Lp(·,·) (QT )d×d .

(6.25)

.

If we choose .φ := ∇ϕ ∈ C0∞ (QT )d for arbitrary .ϕ ∈ C0∞ (QT ) in (6.21), then we observe, using that .div(u(t) + ∇(π h (t))) = div(u(t)) + (π h (t)) = 0 in .(W01,2 ())∗ for every .t ∈ I , for every .ϕ ∈ C0∞ (QT ), .H ∈ C0∞ (QT )d×d and .π ∈ F (H ) that   2 . H (t, x) : ∇ ϕ(t, x) dtdx = − π (t, x) ϕ(t, x) dtdx . (6.26) QT

QT p(·,·)

Although the mapping .F : C0∞ (QT )d×d ⊆ Lp(·,·) (QT )d×d → 2L0, (QT ) might be well-defined, it cannot satisfy (6.25), even if .p ∈ P∞ () ∩ C ∞ (), i.e., if the variable exponent is smooth and does not depend on time, as can be observed on the following page. The following remark shows that the well-definedness of (6.24) and the validity of the inequality (6.25), immediately also implies a variable Bochner–Lebesgue version of the well-known Hessian–Laplacian inequality. The latter follows from a result on the existence of strong solutions to the Poisson problem (cf. [49, Sec. 14.1]) and states the following in its variable exponent form. Proposition 6.4 (Hessian–Laplacian Inequality) Let . ⊆ Rd , .d ≥ 2, be a domain and .p ∈ Plog () with .p− > 1. Then, there exists a constant .c = c(p, ) > 0 such that for every .ϕ ∈ C0∞ () it holds ∇ 2 ϕLp(·) ()d×d ≤ c ϕLp(·) () .

.

Proof A direct consequence of [49, Theorem 14.1.2.] or [50, Theorem 31].

(6.27) 

Remark 6.3 Let . ⊆ ≥ 2, be an arbitrary bounded domain, .I := (0, T ), .T < ∞, .QT := I ×, and .p ∈ Plog (QT ) with .p− > 1. Suppose that the mapping (6.24) is well-defined and satisfies the inequality (6.25). Then, according to Remark 6.2, the mapping (6.24) likewise satisfies (6.26). Thus, using the inequality (6.25), (6.26) and the norm-conjugate formula (cf. Proposition 2.7), we conclude for every .ϕ ∈ Rd , .d

3 For

a set S, we denote by the set .2S := {M | M ⊆ S} its power set.

236

6 Pressure Reconstruction

C0∞ (QT ) that  ϕLp (·,·) (QT ) ≥

1 2

sup π∈Lp(·,·) (QT ) πLp(·,·) (Q ) ≤1

( ϕ, π)Lp(·,·) (QT )

T

≥

1 2

T)

.

( ϕ, π )Lp(·,·) (QT )

sup H ∈C0∞ (QT )d×d π∈F (H );π Lp(·,·) (Q

≤1

≥

1 2cp

≥

1 ∇ 2 ϕLp (·,·) (QT )d×d , 4cp

sup H ∈C0∞ (QT )d×d H Lp(·,·) (Q )d×d ≤1 T

(6.28)

−(∇ 2 ϕ, H )Lp(·,·) (QT )d×d

i.e., a variable Bochner–Lebesgue space version of the Hessian–Laplacian inequality (6.27), where we combined Proposition 2.7 with the density of .C0∞ (QT )d×d in p(·,·) (Q )d×d for .(6.28) .4 .L T 4 The following remark makes clear that the Hessian–Laplacian inequality (cf. Proposition 6.4) does not admit a congruent extension to the context of variable Bochner–Lebesgue spaces, i.e., that inequality (6.28) cannot be valid in general, which confirms Remark 6.2. Remark 6.4 (Invalidity of Hessian–Laplacian Inequality) Let . ⊆ Rd , .d ≥ 2, be an arbitrary bounded domain, .I := (0, T ), .T < ∞, and .QT := I × . Moreover, let   FH–L := (∇ 2 ϕ, ϕ) | ϕ ∈ C0∞ () ⊆ C0∞ ()d×d × C0∞ () .

.

Let .η ∈ C0∞ () with .η ≡ 1 in G, where .G ⊂⊂  is a domain, and .A ∈ Md×d sym \ {0} with .tr(A) = 0. Thus, if we set .ϕ(x) := η(x)x  Ax for every .x ∈ , then it holds ∞ 2 .ϕ ∈ C () with .∇ ϕ = 2A and . ϕ = 2tr(A) = 0 in G. In other words, 0 2  .(∇ ϕ, ϕ) ∈ FH–L with .int(supp(∇ 2 ϕ)) \ supp( ϕ) = ∅ and . ϕ ≡ 0. In consequence, according to Proposition 3.1, there exists an exponent .p ∈ C ∞ (Rd ) with .p− > 1 that does not admit a constant .c > 0 such that for every .ϕ ∈ C0∞ (QT ), it holds ∇ 2 ϕLp(·) (QT )d×d ≤ c ϕLp(·) (QT ) .

.

norm-conjugate formula with respect to dense sets of .Lp(·) (G), such as .C0∞ (G), can be found in [49, Cor. 3.4.13].

4A

6.2 Application to Model Problems

237

Recapitulatory, Untruth 6.1 is indeed a lie. As a consequence, we cannot expect  to reconstruct the pressure associated to .H ∈ Lp (·) (QT )d×d in full regularity, i.e.,  p (·) we are incapable of showing that .π 2 ∈ L0, (QT ). This leads to the breakdown of a series of normally quite robust tools from the mathematical analysis of fluid mechanics in the context of variable Bochner–Lebesgue spaces, such as the wellknown parabolic .L∞ - and Lipschitz truncation techniques (cf. Sect. 6.3).

6.2 Application to Model Problems Finally, this section is devoted to the extension of the existence results from Chap. 5, Sect. 5.5, for the unsteady .p(·, ·)-Stokes equations with an optional, lower-order variable exponent non-linearity (cf. Theorem 5.2) and the unsteady .p(·, ·)-Navier– Stokes equations (cf. Theorem 5.3) by the reconstruction of the kinematic pressure. Again, we will first consider the .p(·, ·)-Stokes–like system. Theorem 6.1 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .p, q := p∗ − ε ∈ P (QT ) with .p− ≥ 2 and  2 − d×d d×d . Moreover, let .S : QT × Msym → Msym be a mapping satisfying .ε ∈ 0, p d (S.1)–(S.4) with respect to p and .d : QT × Rd → Rd a mapping satisfying (D.1)– −  (D.3) with respect to .r = q.5 Then, for arbitrary .f ∈ L(p ) (QT )d , .F ∈  (·,·) d×d p (QT , Msym ) and .u0 ∈ H , the following statements apply: L (p+ )

q,p

(i) There exist .u ∈ Wε,σ (QT ) and .π ∈ Cω0 (I , L0 in H and for every .φ ∈ C0∞ (QT )d , it holds

()) such that .uc (0) = u0

 −

u(t, x) · ∂t φ(t, x) + π(t, x)∂t div(φ)(t, x) dtdx

.

QT

 d(t, x, u(t, x)) · φ(t, x) + S(t, x, ε(u)(t, x)) : ε(φ)(t, x) dtdx

+ QT



f (t, x) · φ(t, x) + F (t, x) : ε(φ)(t, x) dtdx ,

= QT

i.e., .∂t u + d(·, ·, u) − div(S(·, ·, ε(u))) − ∇∂t π = f − div(F ) in .D (QT )d . In p (·) addition, if .p ∈ Plog (), then we have that .π ∈ Cω0 (I , L0 ()). q,p (ii) If .∂ ∈ C 1,1 , then there exist .u ∈ Wε,σ (QT ), .π h ∈ Cω0 (I , W 1,2 () ∩ L20 ()), π 1 ∈ L(q

.

+ )

(I, W 1,(q

+ )

(q + )

() .∩L0

()) and .π 2 ∈ L(p

+ )

(p+ )

(I, L0

()) such

that for .p − ≥ 2 and .ε ∈ (0, d2 p − ], we have .p∗ − ε ≥ p ≥ 2 (cf. Remark 5.7) and, thus, .r = q in Proposition 5.6. 5 Note

238

6 Pressure Reconstruction

that .uc (0) = u0 in H and for every .φ ∈ C0∞ (QT )d , it holds  .

−

[u(t, x) + ∇π h (t, x)] · ∂t φ(t, x) dtdx QT

 [d(t, x, u(t, x)) + ∇π 1 (t, x)] · φ(t, x) dtdx

+ 

QT

[S(t, x, ε(u)(t, x)) + π 2 (t, x)Id ] : ε(φ)(t, x) dtdx

+ QT

 f (t, x) · φ(t, x) + F (t, x) : ε(φ)(t, x) dtdx ,

= QT

i.e., .∂t u + d(·, ·, u) − div(S(·, ·, ε(u))) + ∇[π 1 − π 2 + ∂t π h ] = f − div(F ) in .D (QT )d . In addition, if .p ∈ Plog (), then we have that   (q + ) (I, W 1,q  (·) () ∩ Lq (·) ()) and .π ∈ L(p+ ) (I, Lp (·) ()). .π 1 ∈ L 2 0 0 q,p

Proof Theorem 5.2 provides the existence of .u ∈ Wε,σ (QT ) with a representation ∞ 0 d .uc ∈ H (QT ) such that .uc (0) = u0 in H and for every .φ ∈ C 0,div (QT ) , it holds  .

−

u(t, x) · ∂t φ(t, x) dtdx QT



+

d(t, x, u(t, x)) · φ(t, x) + S(t, x, ε(u)(t, x)) : ε(φ)(t, x) dtdx QT

 f (t, x) · φ(t, x) + F (t, x) : ε(φ)(t, x) dtdx .

= QT



Therefore, if we define the functions .h := f − d(·, ·, u) ∈ Lq (·,·) (QT )d ⊆ +   (p+ ) (Q , Md×d ), L(q ) (QT )d and .H := F −S(·, ·, ε(u)) ∈ Lp (·,·) (QT , Md×d T sym ) ⊆ L sym  2d qd  6 observing that .p ≥ max d+2 , d+q , i.e., .s = p in Proposition 6.1, for assertion (i), first with constant exponents and then with .p ∈ Plog (), we resort to Proposition 6.1 and for assertion (ii), we similarly resort to Proposition 6.3. 

Finally, we turn to the reconstruction of the pressure for the unsteady .p(·, ·)Navier–Stokes equations in a bounded Lipschitz domain . ⊆ Rd , .d ≥ 2, with log .p ∈ P (QT ) satisfying .p− ≥ pC = 3d+2 d+2 .

6 Due .(t .

dq d+q

→ ≤

to .p − ≥ 2, we have .p ≥

2d d+2

in .QT . Since

.



d dt dt d+t

dt d+t ) : R≥0 → R≥0 is non-decreasing, and .q dp∗ d(d+2)p d(d+2) d+p∗ = d 2 +(d+2)p ≤ d 2 +(d+2)2 p ≤ p in .QT , where we



=

d2 (d+t)2

≥ 0, i.e.,

≤ p∗ in .QT , we find that again used that .p − ≥ 2.

6.2 Application to Model Problems

239

Theorem 6.2 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), log .T < ∞, .QT := I × , and .p ∈ P (QT ) with .p− ≥ pC . Moreover, let .S : QT × d×d d×d Msym → Msym be a mapping satisfying (S.1)–(S.4) with respect to p. Then, for arbitrary .f ∈ L(p statements apply:

− )



(QT )d , .F ∈ Lp (·,·) (QT , Md×d sym ) and .u0 ∈ H , the following (p+ )

p,p

(i) There exist .u ∈ Wε,σ (QT ) and .π ∈ Cω0 (I , L0 in H and for every .φ ∈ C0∞ (QT )d , it holds

()) such that .uc (0) = u0

 .

−

u(t, x) · ∂t φ(t, x) dtdx QT

 [S(t, x, ε(u)(t, x)) − u(t, x) ⊗ u(t, x)] : ε(φ)(t, x) dtdx

+ QT

 =

π (t, x)∂t div(φ)(t, x) dtdx QT



f (t, x) · φ(t, x) + F (t, x) : ε(φ)(t, x) dtdx ,

+ QT

i.e., .∂t u − div(S(·, ·, ε(u))) + div(u ⊗ u) − ∇∂t π = f − div(F ) in .D (QT )d . p (·) In addition, if .p ∈ Plog (), then we have that .π ∈ Cω0 (I , L0 ()). p,p

(ii) If .∂ ∈ C 1,1 , then there exist .u ∈ Wε,σ (QT ), .π h ∈ Cω0 (I , W 1,2 () ∩ L20 ()), + 

+ 

(p+ )

+ 

(p+ )

π 1 ∈ L(p ) (I, W 1,(p ) () .∩L0 ()) and .π 2 ∈ L(p ) (I, L0 that .uc (0) = u0 in H and for every .φ ∈ C0∞ (QT )d , it holds

.

 .

()) such



−

[u(t, x) + ∇π h (t, x)] · ∂t φ(t, x) dtdx + QT

∇π 1 (t, x) · φ(t, x) dtdx QT



[S(t, x, ε(u)(t, x))−u(t, x) ⊗ u(t, x)+π 2 (t, x)Id ] : ε(φ)(t, x) dtdx

+ QT

 f (t, x) · φ(t, x) + F (t, x) : ε(φ)(t, x) dtdx ,

= QT

i.e., .∂t u − div(S(·, ·, ε(u))) + div(u ⊗ u) + ∇[π 1 − π 2 + ∂t π h ] = f − div(F ) in .D (QT )d . In addition, if .p ∈ Plog (), then we have that .π 1 ∈ +   +  p (·) p (·) L(p ) (I, W 1,p (·) () ∩ L0 ()) and .π 2 ∈ L(p ) (I, L0 ()).

240

6 Pressure Reconstruction p,p

Proof Theorem 5.3 provides the existence of .u ∈ Wε,σ (QT ) with a representation ∞ 0 d .uc ∈ H (QT ) such that .uc (0) = u0 in H and for every .φ ∈ C 0,div (QT ) , it holds  .

−

u(t, x) · ∂t φ(t, x) dtdx QT

 [S(t, x, ε(u)(t, x)) − u(t, x) ⊗ u(t, x)] : ε(φ)(t, x) dtdx

+ QT

 f (t, x) · φ(t, x) + F (t, x) : ε(φ)(t, x) dtdx .

= QT

− 

Therefore, if we define the functions .h := f ∈ L(p ) (QT )d and .H := F −  (p+ ) (Q , Md×d ), then, observing S(·, ·, ε(u)) + u ⊗ u ∈ Lp (·,·) (QT , Md×d T sym ) .⊆ L sym  2d dp−  that .p ≥ max d+2 , d+p− in .QT , i.e., .s = p in Proposition 6.1, for assertion (i), again first with constant exponents and then with .p ∈ Plog (), we resort to Proposition 6.1 and for assertion (ii), we similarly resort to Proposition 6.3. 

6.3 Applicability of Parabolic L∞ - and Lipschitz Truncation In this section, we examine the parabolic .L∞ - and (solenoidal) Lipschitz truncation techniques for their potential to admit extensions to the framework of variable exponent Bochner–Lebesgue spaces. In doing so, we will avoid delving too deeply into the technical sophistications of these techniques. Nevertheless, we will point out the most essential tools used in these methods, which now fail in the context of variable Bochner–Lebesgue spaces. For detailed accounts of these techniques, we recommend the seminal contributions [25, 48, 69, 164].

6.3.1 Parabolic L∞ - and Lipschitz Truncation Reviewing the pioneering work of J. Wolf [164] addressing the parabolic .L∞ truncation technique, which was used to prove the existence of weak solutions of the unsteady p-Navier–Stokes equations for constant exponents .p > 2d+2 d+2 , one easily becomes aware of the dependence of this method on the reconstruction of the pressure terms in appropriate function spaces, such as, for example, described –at least for the particular case of constant exponents– in Proposition 6.3, see also, e.g., [48, 165]. To extend the parabolic .L∞ -truncation technique to the framework of variable Bochner–Lebesgue spaces, the validity of Untruth 6.1 becomes indispensable. Unfortunately, as we already noted in Remark 6.2, Untruth 6.1 cannot be valid, even if the variable exponent is smooth and does not depend on time. On these grounds, there is just a little hope that the parabolic .L∞ -truncation technique will find use in the context of variable Bochner–Lebesgue spaces.

6.3 Applicability of Parabolic L∞ - and Lipschitz Truncation

241

The parabolic Lipschitz truncation technique was developed by L. Diening, M. R˚užiˇcka and J. Wolf in [48] as an improvement of the .L∞ -truncation technique, which provided the existence of weak solutions of the unsteady p-Navier–Stokes 2d equations for constant exponents .p > d+2 . Regrettably, also this method makes elementary use of the pressure reconstruction theorem (cf. Proposition 6.3), suggesting that even this tool fails in the context of variable Bochner–Lebesgue spaces.

6.3.2 Parabolic Solenoidal Lipschitz Truncation The dependence of both the parabolic .L∞ - and the parabolic Lipschitz truncation techniques on the reconstruction of the pressure in full regularity, inter alia, motivated L. Diening et. al in [25] to develop the so-called parabolic solenoidal Lipschitz truncation technique, which does not require the local reconstruction of the pressure terms and, therefore, appeared to be promising with regard to a potential variable exponent version of the parabolic solenoidal Lipschitz truncation technique. However, a more in-depth review of their procedure reveals that they merely postponed the problem because their construction depends decisively on the validity of the following result concerning the solvability and stability for the biLaplace equation with double divergence right-hand side. Proposition 6.5 (Bi-Laplace Equation) Let .B ⊆ Rd , .d ≥ 2, a ball and .p ∈ 2,p (1, ∞). Then, for every .H ∈ Lp (B)d×d , there exists a unique function .f ∈ W0 (B) such that for every .φ ∈ C0∞ (B), it holds ( φ, f )Lp (B) = (∇ 2 φ, H)Lp (B)d×d .

(6.29)

.

In particular, there exists a constant .c > 0, which is independent of B, such that ∇ 2 f Lp (B)d×d ≤ cHLp (B)d×d .

(6.30)

.

2 7 p d×d → W In other words, the operator . −2 B ◦ div : L (B) 0 (B) is well 2,p defined and continuous, where . 2B : W0 (B) → (W02,p (B))∗ is given by 2,p 2,p  2 . f, gW 2,p (B) := ( g, f )Lp (B) for all .f ∈ W B 0 (B) and .g ∈ W0 (B), and 0  2 p d×d → (W 2,p (B))∗ by . div2 (H), g 2,p 2 .div : L (B) W0 (B) := (∇ g, H)Lp (B)d×d for 0  2,p all .H ∈ Lp (B)d×d and .g ∈ W0 (B). 2,p



Proof See [25, Corollary 2.5].

−2 . B



to [164, Lem. 2.4], . 2B : W0 (B) → (W02,p (B))∗ 2,p 2,p ∗ : (W0 (B)) → W0 (B) exists.

7 Due

2,p

is

an

isomorphism,

i.e.,

242

6 Pressure Reconstruction

More precisely, the parabolic solenoidal Lipschitz truncation technique exploits for .p ∈ (1, ∞) that the linear-induced operator 2,p G : Lp (I × B)d×d ∼ = Lp (I, Lp (B)d×d ) → Lp (I, W0 (B)) ,

.

(6.31)

2 for every .H ∈ Lp (I × B)d×d defined by .G(H )(t) := ( −2 B ◦ div )(H (t)) in 2,p p d×d , .W 0 (B) for almost every .t ∈ I , is well-defined and for every .H ∈ L (I × B) it holds the inequality

 G(H )Lp (I ×B) ≤ cH Lp (I ×B)d×d ,

(6.32)

.

where the constant .c > 0 is still independent of B. Unfortunately, the operator (6.31) does not admit an extension to variable exponent spaces, as can be seen in the following remark. Remark 6.5 Let .B ⊆ Rd , .d ≥ 2, be a ball, .I := (0, T ), .T < ∞, .QT := I × B, and log .p ∈ P (QT ) with .p− > 1. Suppose that the mapping +

2,p+

G : C0∞ (QT )d×d ⊆ Lp(·,·) (QT )d×d → Lp (I, W0

.

(B))

admits a constant .c > 0 such that for every .H ∈ C0∞ (QT )d×d , it holds  G(H )Lp(·,·) (QT ) ≤ cH Lp(·,·) (QT )d×d .

.

(6.33)

Then, using the inequality (6.33) and (6.29) and the norm-conjugate formula (cf. Proposition 2.7) together with the density of .C0∞ (QT )d×d in .Lp(·,·) (QT )d×d , we conclude for every .φ ∈ C0∞ (QT ) that  φLp (·,·) (QT ) ≥

.

1 2

sup π ∈Lp(·,·) (QT ) π Lp(·,·) (Q ) ≤1

( φ, π )Lp(·,·) (QT )

T

≥

1 2

sup H ∈C0∞ (QT )d×d  G (H )Lp(·,·) (Q ) ≤1

( φ, G(H ))Lp(·,·) (QT )

T

1 ≥ 2c

≥

sup H ∈C0∞ (QT )d×d H Lp(·,·) (Q )d×d ≤1 T

(∇ 2 φ, H )Lp(·,·) (QT )d×d

1 ∇ 2 φLp (·,·) (QT )d×d , 4c

which, according to Remark 6.3, cannot be valid in general.

6.3 Applicability of Parabolic L∞ - and Lipschitz Truncation

243

Remark 6.6 (Conclusion) We emphasize that the foregoing observations do not prove that the parabolic .L∞ - or (solenoidal) Lipschitz truncation techniques do not admit appropriate extensions to the framework of variable Bochner–Lebesgue spaces, but, rather, that their methods of proof require fundamental modifications at the indicated sections, see Untruth 6.1 or Remark 6.5. Unfortunately, the reparation of these failing tools, or even just determining whether these tools are substitutable, lies far beyond the scope of this book. Nevertheless, lifting these techniques to the framework of variable Bochner–Lebesgue spaces undoubtedly represents an attractive research direction, in particular, with regard to a potential existence proof 2d for the unsteady .p(·, ·)-Navier–Stokes equations up to the lower bound .p− > d+2 , 2d which has so far only been accomplished for constant exponents .p > d+2 (cf. [25, 48]). In fact, without these techniques, we are only able to establish the existence of weak solutions of the unsteady .p(·, ·)-Navier–Stokes equations up to 3d , which surprisingly corresponds to the lower bound the lower bound .p− > d+2 that is typically imposed when one tries to prove the existence of weak solutions of the steady .p(·)-Navier–Stokes equations without the steady analogues of the ∞ .L - and (solenoidal) Lipschitz truncation techniques, but using standard pseudomonotonicity methods, see, e.g., [147, Theorem 2.4 & Remark 2.6, p. 63].

Chapter 7

Existence Theory for Irregular Domains

This chapter addresses the extension of the existence results of Chap. 5 to the case of irregular, i.e., non-Lipschitz, domains, in which the formula of integration-by-parts q,p for .Wε,σ (QT ) (cf. Proposition 4.23), which unequivocally formed a cornerstone of the analysis therein, no longer needs to be available. ˚εq,p (QT ) ∩c H0 (QT ) → V ˚εq,p (QT )∗ is bounded, .C 0 More precisely, if .A : V 0 Bochner coercive and satisfies the .C -Bochner condition (M) –in accordance with ˚εq,p (QT ) ∩c H0 (QT ) is a the assumptions of Theorem 5.1– and if .(un )n∈N ⊆ V sequence satisfying .

˚εq,p (QT ) ∩c H0 (QT ) in V

un  u Aun  ξ

in

˚εq,p (QT )∗ V

(n → ∞) ,

q,p

where .u ∈ Wε,σ (QT ) with a representation .uc ∈ H0 (QT ),1 ˚εq,p (QT )∗ , and .V .

lim sup Aun , un V ˚q,p (Q n→∞

ε

T)

(7.1)

(n → ∞) , .

dσ u dt

= −ξ in

1 1 ≤ − uc (T )2H + uc (0)2H , 2 2

(7.2) q,p

then we are not able to apply the formula of integration-by-parts for .Wε,σ (QT ) ˚εq,p (QT ) and, in (7.2) in order to identify the right-hand side in (7.2) with .ξ , u V therefore, to be in the position to refer to the .C 0 -Bochner condition (M) of .A. As a consequence, we now need to find a way to perform the passage to the limit without q,p the usage of the formula of integration-by-parts for .Wε,σ (QT ).

H0 (Q

q,p

we explicitly have to assume that .u ∈ Wε,σ (QT ) possesses a representation .uc ∈ T ) because we cannot refer to Proposition 4.23 (i).

1 Here,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaltenbach, Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents, Lecture Notes in Mathematics 2329, https://doi.org/10.1007/978-3-031-29670-3_7

245

246

7 Existence Theory for Irregular Domains

For this, we choose a method of solution whose basic idea is fairly intuitive. We switch from the (.C 0 -)Bochner condition (M) to a notion of continuity that allows us to perform the passage to the limit without the help of the formula of q,p integration-by-parts for .Wε,σ (QT ). This will be obtained by means of the socalled Bochner–Sobolev condition (M), where the prefix Sobolev is intended to indicate that we immediately incorporate information from the generalized time derivative . ddtσ , whereas the .C 0 -Bochner condition (M) takes the indirect route via the weak sequential topology of .H0 (QT ). The Bochner–Sobolev condition (M) is simply formulated in the manner to guarantee the identification of weak limits in the passage to the limit and, thus, provides in a straightforward way an abstract existence result for solenoidal generalized evolution equations in irregular domains. To justify the formulation of an abstract existence result involving the Bochner–obolev condition (M), the main task of this chapter is to characterize operators that actually meet the notion of Bochner–Sobolev condition (M), i.e., to clarify that this abstract notion is indeed meaningful, especially with regard to the unsteady .p(·, ·)-Stokes and .p(·, ·)-Navier–Stokes equations. Here, we will at an early stage become aware of that Bochner strongly continuous operators satisfy the Bochner–Sobolev condition (M), such as, e.g., the unsteady con˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ (cf. Proposition 5.3). In vective term .C : V order to demonstrate that operators such as, e.g., the unsteady extra stress tensor ˚εq,p (QT ) → V ˚εq,p (QT )∗ (cf. Proposition 5.2) satisfy the Bochner–Sobolev .S : V condition (M), a whole machinery must be set in motion. The central idea is to apply a Minty–Trick like argument, which merges the abstract theory of monotone operators with the theory of finite Radon measures. Furthermore, in order to be able to apply this Minty–Trick like argument, a parabolic compensated compactness ˚εq,p (QT ) will be indispensable. principle, or briefly PCCP, in the framework of .V Here, we will follow ideas from [140, 173, 175]. More precisely, we will improve these results, in the sense that we will overcome the necessity of the artificial relation .p+ < (p− )• 2 for the variable exponent .p ∈ Plog (QT ). This will be accomplished on the basis of a special formula of integration-by-parts for so-called q,p anisotropic variable Bochner–Lebesgue spaces, i.e., subspaces of .Xε (QT ) that take additional integrability of the divergence into account.

7.1 Bochner–Sobolev Condition (M) In this section, we introduce the Bochner–Sobolev condition (M), which is perfectly adapted to our approximation scheme, which is derived from the standard approach, see, for example, [164], to approximate an irregular domain . ⊆ Rd , .d ≥ 2, from the inside by a non-decreasing sequence of potentially more regular domains .(n )n∈N , e.g., by a sequence of Lipschitz domains.

2 Recall

that .(·)• ∈ W 1,∞ (1, ∞) is defined by .s• := min{s + 2, s∗ } for every .s ∈ (1, ∞).

7.1 Bochner–Sobolev Condition (M)

247

Throughout the entire chapter, unless otherwise specified, let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, .QT := I × , and .q, p ∈ P∞ (QT ) with − − > 1, such that .V ˚−q,p → H . We call .(n ) .q , p n∈ N a non-decreasing exhaustion of ., if .n ⊆ n+1 ⊆  for all .n ∈ N and . = n∈N n . Moreover, for every n ˚εq,p (Qn ) → V ˚εq,p (QT ) the .n ∈ N, we define .Q := I × n and denote by .n : V T T ˚εq,p (Qn ) defined by .n u := u in .Qn and zero extension operator, for every .u ∈ V T T q,p ∗ n ˚ε (QT )∗ → V ˚εq,p (Qn )∗ its adjoint operator. .n u := 0 in .QT \ Q , and by .n : V T T ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is said to Definition 7.1 An operator .A : V satisfy the (i) Bochner–Sobolev condition (M) with respect to a non-decreasing exhaustion q,p n n ∞ .(n )n∈N of ., if for sequences .un ∈ Wε,σ (Q ) ∩ H (Q ), .n ∈ N, and T T q,p ∗ ∗ ˚ε (QT ) from .(v n )n∈N ⊆ V .

dσ un + ∗n An un = ∗n v ∗n dt

˚εq,p (Qn )∗ in V T

for all n ∈ N , .

(7.3)

in H

(n → ∞) , .

(7.4)

˚ε (QT ) in V

q,p

(n → ∞) , .

(7.5)

in H∞ (QT )

(n → ∞) , .

(7.6)

An un  u∗

˚εq,p (QT )∗ in V

(n → ∞) , .

(7.7)

v ∗n → v ∗

˚εq,p (QT )∗ in V

(n → ∞) ,

(7.8)

(n un )ω (0) → u0  n un  u ∗

 n un  u

˚εq,p (QT )∗ . it follows that .Au = u∗ in .V (ii) Bochner–Sobolev condition (M), if it satisfies the Bochner–Sobolev condition (M) with respect to every non-decreasing exhaustion .(n )n∈N of .. Here, .(n un )ω := n (un )ω ∈ Cω0 (I , H ), .n ∈ N, denote the zero extensions of the weakly continuous representations .(un )ω ∈ Cω0 (I , L20,σ (n )), .n ∈ N, of q,p n n ∞ .un ∈ Wε,σ (Q ) ∩ H (Q ), .n ∈ N. T T Remark 7.1 Without additionally assuming that the non-decreasing exhaustion (n )n∈N consists of bounded Lipschitz domains and that .q, p ∈ Plog (QT ) satisfy .p− ≥ 2 and .q ≥ p in .QT , we cannot guarantee that the mappings q,p n .(·)c : Wε,σ (Q ) → H0 (QnT ), .n ∈ N, in the sense of Proposition 4.23 T (i), exist. Nonetheless, Proposition 4.19 (i) always provides the mappings q,p n n 2 ( )), .n ∈ N, where we exploit, ∞ 0 .(·)ω : Wε,σ (Q ) ∩ H (Q ) → Cω (I , L n T T 0,σ q,p q,p in particular, that the inclusions .Wε,σ (QnT ) ⊆ Wσ,− (QnT ), .n ∈ N, (cf. Proposition 4.20) hold. Even though we are actually interested in approximating . by a non-decreasing exhaustion .(n )n∈N consisting of Lipschitz domains, we do not impose this regularity assumption in Definition 7.1 to cover also the case of a constant exhaustion, i.e., when .n =  for every .n ∈ N. .

248

7 Existence Theory for Irregular Domains

Admittedly, at first glance, the Bochner–Sobolev condition (M) may seem to be an artificial property. In fact, it merely contains the approximation scheme we have chosen and all the convergences one can expect, such as in Theorem 5.1. However, if one recalls the definition of the (.C 0 -)Bochner condition (M) and its usage in Theorem 5.1, it should become apparent that this notion was also solely formulated in the spirit of guaranteeing the identification of weak limits of the Galerkin approximation implemented in it. The main difference of Bochner– Sobolev condition (M) in direct comparison with the (.C 0 -)Bochner condition (M) is that the usual limit superior inequality (5.44) is replaced by a sequence of solenoidal generalized evolution equations coupled to a non-decreasing exhaustion .(n )n∈N of ., i.e., by (7.3). This has the advantage of making the notion of the Bochner– Sobolev condition (M) much more amenable to localization arguments, which will prove crucial later in this chapter when we try to identify operators such as the ˚εq,p (QT ) → V ˚εq,p (QT )∗ as operators satisfying unsteady extra stress tensor .S : V the Bochner–Sobolev condition (M) by falling back on parabolic compensated compactness methods. ˚εq,p (QT )∩H∞ (QT ) of The following lemma illustrates that the weak limit .u ∈ V q,p n n ∞ a sequence .un ∈ Wε,σ (QT ) ∩ H (QT ), .n ∈ N, satisfying (7.3)–(7.8) inevitably possesses certain regularity properties and that there is an additional convergence property hidden in (7.3)–(7.8). Lemma 7.1 Let .un ∈ Wε,σ (QnT ) ∩ H∞ (QnT ), .n ∈ N, be a sequence, where .(n )n∈N is a non-decreasing exhaustion of ., such that q,p

.

dσ un dt

= ∗n u∗n

(n un )ω (0)  u0 ∗

 n un  u u∗n  u∗

˚εq,p (Qn )∗ in V T

for all n ∈ N , .

in H

(n → ∞) , .

(7.10)

in H∞ (QT )

(n → ∞) , .

(7.11)

˚εq,p (QT )∗ in V

(n → ∞) .

(7.12)

(7.9)

Then, it holds: q,p σu ˚εq,p (QT )∗ and .uω (0) = u0 in (i) .u ∈ Wε,σ (QT ) ∩ H∞ (QT ) with . ddt = u∗ in .V H , where the initial condition has to be understood in the sense of the unique weakly continuous representation .uω ∈ Cω0 (I , H ) (cf. Proposition 4.19). (ii) .(n un )ω (t)  uω (t) in H .(n → ∞) for every .t ∈ I .

Proof In principle, we proceed as in the proof of Theorem 5.1, step 2 and 3 up to modifications. ad (i). Let first .φ ∈ C ∞ (QT )d with .div(φ) = 0 and .supp(φ) ⊆ [0, T ) × . ∞ ( )) for Then, there exists some integer .n0 ∈ N such that .φ ∈ C ∞ (I , C0,σ n every .n ∈ N with .n ≥ n0 . Therefore, testing the equations (7.9) for any ∞ ∞ ˚q,p n .n ∈ N, where .n ≥ n0 , with .φ ∈ C (I , C 0,σ (n )) ⊆ Vε (QT ) and a subsequent application of the non-symmetric formulas of integration-by-parts

7.1 Bochner–Sobolev Condition (M)

249

for .Wσ,− (QnT ) ∩ H∞ (QnT ), .n ∈ N, (cf. Proposition 4.19 (ii)), since .un ∈ q,p Wσ,− (QnT ) ∩ H∞ (QnT ), .n ∈ N, (cf. Proposition 4.20), yields q,p

 −

(un (s), ∂t φ(s))L2

0,σ (n )

I

.

ds = ∗n u∗n , φ V ˚q,p (Qn ) ε

T

(7.13)

+ ((un )ω (0), φ(0))L2

0,σ

(n ) .

In particular, (7.13) also reads  − I

.

((n un )(s), ∂t φ(s))H ds = u∗n , φ V ˚q,p (Q

T)

ε

(7.14)

+ ((n un )ω (0), φ(0))H . for every .n ∈ N with .n ≥ n0 . Then, by passing for .n → ∞ in (7.14), using (7.10)–(7.12) in doing so, for every .φ ∈ C ∞ (QT )d with .div(φ) = 0 and .supp(φ) ⊆ [0, T ) × , we observe that  .

− I

(u(s), ∂t φ(s))H ds = u∗ , φ V ˚q,p (Q ε

T)

+ (u0 , φ(0))H .

(7.15)

∞ (Q )d in (7.15), according to Definition 4.6, we obtain .u ∈ Choosing .φ ∈ C0,div T q,p ∞ σu ˚εq,p (QT )∗ . In particular, it holds .u ∈ = u∗ in .V Wε,σ (QT )∩H (QT ) with . ddt q,p Wσ,− (QT ) ∩ H∞ (QT ) (cf. Proposition 4.20). Therefore, Proposition 4.19 (i) provides a weakly continuous representation .uω ∈ Cω0 (I , H ) and we are allowed q,p to apply the non-symmetric formula of integration-by-parts for .Wσ,− (QT ) ∩ H∞ (QT ) (cf. Proposition 4.19 (ii)) in (7.15) with .φ = vϕ ∈ C ∞ (QT )d , where ∞ .v ∈ V is chosen arbitrarily and .ϕ ∈ C (I ) with .supp(ϕ) ⊆ [0, T ) and .ϕ(0) = 1. In this way, for every .v ∈ V, we arrive at

(uω (0) − u0 , v)H = 0 .

.

(7.16)

Due to the density of .V in H , we conclude from (7.16) that .uω (0) = u0 in H . ad (ii). We fix an arbitrary time slice .t ∈ (0, T ]. Since .((n un )ω (t))n∈N ⊆ H is bounded (cf. (7.11)), there exists a subsequence .((n un )ω (t))n∈ t , with . t ⊆ N, and an element .u t ∈ H such that (n un )ω (t)  u t

.

in H

( t n → ∞) .

(7.17)

Let anew .φ ∈ C ∞ (QT )d with .div(φ) = 0 and .supp(φ) ⊆ (0, T ]× be arbitrary. ∞ ( )) for Again, there exists some integer .n0 ∈ N such that .φ ∈ C ∞ (I , C0,σ n every .n ∈ N with .n ≥ n0 . Then, we test (7.9) for any .n ∈ t , where .n ≥ n0 , ˚εq,p (Qn ) and use the non-symmetric formulas of integration-bywith .φχ[0,t] ∈ V T

250

7 Existence Theory for Irregular Domains

parts for .Wσ,− (QnT ) ∩ H∞ (QnT ), .n ∈ t , to find for every .n ∈ t with .n ≥ n0 that  t  t − ((n un )(s), ∂t φ(s))H ds = u∗n (s), φ(s) V˚εq,p (s) ds 0 0 . (7.18) q,p

− ((n un )ω (t), φ(t))H , . By passing for . t n → ∞ in (7.18), using anew (7.11) and (7.12) and additionally (7.17) in doing so, we obtain for every .φ ∈ C ∞ (QT )d with .div(φ) = 0 and .supp(φ) ⊆ (0, T ] ×  that 

t

− .

 (u(s), ∂t φ(s))H ds =

0

0

t

u∗ (s), φ(s) V˚εq,p (s) ds

(7.19)

− (u t , φ(t))H . The non-symmetric formula of integration-by-parts for .Wσ,− (QT ) ∩ H∞ (QT ) applied to (7.19) with .φ = vϕ ∈ C ∞ (QT )d , for arbitrary .v ∈ V and .ϕ ∈ C ∞ (I ) with .supp(ϕ) ⊆ (0, T ] and .ϕ(t) = 1, then for every .v ∈ V results in q,p

(uω (t) − u t , v)H = 0 .

.

(7.20)

From (7.20) we conclude that .uω (t) = u t in H , i.e., due to (7.17), it holds (n un )ω (t)  uω (t)

.

in H

( t n → ∞) .

(7.21)

As this argumentation remains valid for each subsequence of ((n un )ω (t))n∈N ⊆ H , .uω (t) ∈ H is a weak accumulation point of each subsequence of .((n un )ω (t))n∈N ⊆H . Hence, the standard convergence principle (cf. [166, Prop. 10.13 (4)]) yields that (7.21) holds even if . t = N.  

.

The next proposition states that under the assumptions of Proposition 4.23, i.e., q,p the assumptions under which the formula of integration-by-parts for .Wε,σ (QT ) applies, the Bochner–Sobolev condition (M), in fact, is a generalization of the .C 0 Bochner condition (M) (cf. Definition 5.3). Proposition 7.1 (.C 0 -Bochner Condition (M) .⇒ Bochner–Sobolev Condition (M)) Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain and .q, p ∈ Plog (QT ) with .p− ≥ 2 and .q ≥ p in .QT . Furthermore, let q,p q,p .A : Vε (QT ) ∩ H∞ (QT ) → Vε (QT )∗ satisfy the .C 0 -Bochner condition (M). q,p q,p ∞ Then, .A : Vε (QT ) ∩ H (QT ) → Vε (QT )∗ satisfies the Bochner–Sobolev condition (M) with respect to every non-decreasing exhaustion .(n )n∈N of . consisting of Lipschitz domains.

7.1 Bochner–Sobolev Condition (M)

251 q,p

Proof Let us assume that .un ∈ Wε,σ (QnT ), .n ∈ N, satisfies (7.3)–(7.8). Then, q,p Proposition 4.23 and Lemma 7.1 imply that .u ∈ Wε,σ (QT ) with a representation 0 .uc ∈ H (QT ) and

.

dσ u = v ∗ − u∗ dt

in Vε (QT )∗ , q,p

uc (0) = u0

(7.22)

in H ,

as well as (n un )c (t)  uc (t)

.

in H

(n → ∞)

for all t ∈ I .

(7.23)

Moreover, we infer from (7.5), (7.6), and (7.23), and the characterization of weak q,p convergence in .Vε (QT ) ∩c H0 (QT ) (cf. Chap. 5, Sect. 5.3, or by analogy with [92, Proposition 2.6]) that  n un  u

.

q,p

in Vε (QT ) ∩c H0 (QT )

(n → ∞) .

(7.24) q,p

Eventually, making use of the formulas of integration-by-parts for .Wε,σ (QnT ), .n ∈ q,p N, and .Wε,σ (QT ) (cf. Proposition 4.23 (ii)), also exploiting (7.3)–(7.8), (7.22) and (7.23), we deduce that .

  dσ un , un lim sup A(n un ), n un Vεq,p (QT ) = lim sup ∗n v ∗n − q,p dt n→∞ n→∞ Vε (Qn ) T

≤

lim v ∗ , n un Vεq,p (QT ) n→∞ n

 1 + lim sup − (n un )c (T )2H 2 n→∞  1 + (n un )c (0)2H 2 ≤ v ∗ , u Vεq,p (QT ) 1 1 − uc (T )2H + uc (0)2H 2 2   dσ u ∗ ,u = v − q,p dt Vε (QT ) = u∗ , u Vεq,p (QT ) .

(7.25)

Based on (7.24), (7.7) and (7.25), we conclude that .Au = u∗ in .Vε (QT )∗ using q,p q,p the .C 0 -Bochner condition (M) of .A : Vε (QT ) ∩ H∞ (QT ) → Vε (QT )∗ .   q,p

252

7 Existence Theory for Irregular Domains

As a result, if . ⊆ Rd , .d ≥ 2, is a bounded Lipschitz domain and .q, p ∈ P (QT ) satisfy .p− ≥ 2 and .q ≥ p in .QT , then the Bochner–Sobolev condition (M) is indeed a meaningful property because all operators satisfying the .C 0 -Bochner condition (M), e.g., all operators appearing so far in this book, meet the Bochner–Sobolev condition (M), at least with respect to non-decreasing exhaustions .(n )n∈N consisting of Lipschitz domains. Nevertheless, for the case of non-Lipschitz domains, which are actually the objects of interest in this chapter, we cannot make any statement yet. Here, the following proposition is intended to partially remedy the situation. It states, apart from that, that the Bochner–Sobolev condition (M), completely analogous to the .C 0 -Bochner condition (M), is stable under Bochner strongly continuous perturbations. log

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ satisfy the Proposition 7.2 Let .A : V q,p ˚ε (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ Bochner–Sobolev condition (M) and let .B : V be Bochner strongly continuous. Then, it holds: ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ satisfies the Bochner–Sobolev (i) .B : V condition (M). ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ satisfies the Bochner–Sobolev (ii) .A + B : V condition (M). Proof Since .A ≡ 0 certainly satisfies the Bochner–Sobolev condition (M), the q,p proof of (ii) suffices. So, let .un ∈ Wε,σ (QnT )∩H∞ (QnT ), .n ∈ N, where .(n )n∈N is a non-decreasing exhaustion of ., be a sequence satisfying (7.3)–(7.8) with respect ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ , i.e., to .A + B : V dσ un + ∗n (A + B)(n un ) = ∗n v ∗n . dt (n un )ω (0) → u0  n un  u ∗

 n un  u (A + B)(n un )  u∗ v ∗n → v ∗

˚εq,p (Qn )∗ in V T

for all n ∈ N , .

(7.26)

in H

(n → ∞) , .

(7.27)

˚ε (QT ) in V

q,p

(n → ∞) , .

(7.28)

in H∞ (QT )

(n → ∞) , .

(7.29)

in

˚εq,p (QT )∗ V

(n → ∞) , .

(7.30)

in

˚εq,p (QT )∗ V

(n → ∞) .

(7.31)

7.2 L1 -Monotonicity

253 q,p

Apart from that, by resorting to Lemma 7.1, we further find that .u ∈ Wε,σ (QT )∩ ∞ 0 .H (QT ) has a weakly continuous representation .uω ∈ Cω (I , H ) and (n un )ω (t)  uω (t)

.

in H

(n → ∞)

for all t ∈ I .

(7.32)

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is Bochner strongly Therefore, since .B : V continuous, we infer from (7.28), (7.29) and (7.32) that .B(n un ) → Bu ˚εq,p (QT )∗ .(n → ∞). In consequence, if we introduce the sequence in .V ∗ ∗ ˚εq,p (QT )∗ , .n ∈ N, and .w ∗ := v ∗ − Bu ∈ V ˚εq,p (QT )∗ , .w n := v n − B(n un ) ∈ V then (7.26), (7.30) and (7.31) can be rewritten as .

dσ un + ∗n A(n un ) = ∗n w ∗n dt

˚εq,p (Qn )∗ in V T

˚εq,p (QT )∗ A(n un )  u∗ − Bu in V w∗n → w∗

˚εq,p (QT )∗ in V

for all n ∈ N , (n → ∞) ,

(7.33)

(n → ∞) .

˚εq,p (QT ) ∩ H∞ (QT ) → Eventually, the Bochner–Sobolev condition (M) of .A : V q,p ∗ ˚ε (QT ) , (7.33) and (7.27)–(7.29) prove that .Au = u∗ − Bu in .V ˚εq,p (QT )∗ , i.e., V q,p ∗ ∗ ˚ε (QT ) . .(A + B)u = u in .V   ˚εp,p (QT ) ∩ H∞ (QT ) → Since for .p− > pC , the unsteady convective term .C : V is Bochner strongly continuous (cf. Proposition 5.3), we conclude from Proposition 7.2 (i) to the following first non-trivial example of an operator satisfying the Bochner–Sobolev condition (M) even in irregular domains. ˚εp,p (QT )∗ V

˚εp,p (QT ) ∩ Corollary 7.1 If .p− > pC , then the unsteady convective term .C : V p,p ∞ ∗ ˚ε (QT ) satisfies the Bochner–Sobolev condition (M). H (QT ) → V

7.2 L1 -Monotonicity We are already aware of that Bochner strongly continuous operators satisfy the Bochner–Sobolev condition (M). But there is another class of operators satisfies the Bochner–Sobolev condition (M) without being Bochner strongly continuous. One might expect all monotone and demi-continuous operators to form a subclass of operators satisfying the Bochner–Sobolev condition (M). However, monotonicity and demi-continuity do not suffice, but the following stronger monotonicity concept.

254

7 Existence Theory for Irregular Domains

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is said Definition 7.2 An operator .A : V q,p ˚ε (QT ) ∩ H∞ (QT ) → to be .L1 -monotone, if there exists a mapping .A : V   ) with following properties: Lq (·,·) (QT )d × Lp (·,·) (QT , Md×d sym ˚εq,p (QT ) ∩ H∞ (QT ) and ˚εq,p (QT )∗ for every .u ∈ V (L.1) .Au = Jεσ (A u) in .V  .A 0 = (0, 0) . ˚εq,p (QT ) ∩ H∞ (QT ), .f , g ∈ Lq  (·,·) (QT )d and .F , G ∈ (L.2) For every .u, v ∈ V    Lp (·,·) (QT , Md×d sym ), such that .A u = (f , F ) and .A v = (g, G) , it holds (f − g) · (u − v) + (F − G) : (ε(u) − ε(v)) ∈ L1 (QT , R≥0 ) .

.

q  (·,·) (Q )d ˚εq,p (QT ) ∩ H∞ (QT ), .(f n ) (L.3) For sequences .(un )n∈N ⊆ V T n∈N ⊆ L   and .(F n )n∈N ⊆ Lp (·,·) (QT , Md×d sym ), such that .A un = (f n , F n ) for every .n ∈ N, and .(a n )n∈N := (f n · un + F n : ε(un ))n∈N ⊆ L1 (QT , R≥0 ), from

un  u

.

∗

un  u fn  f

˚εq,p (QT ) in V

(n → ∞) ,

in H∞ (QT )

(n → ∞) ,

q  (·,·)

(n → ∞) ,

in L

Fn  F

in L

(QT )d

p (·,·)

(QT , Md×d sym )

(n → ∞) ,

and for a measurable set .S ⊆ QT with .|S| = 0 and .a := f · u + F : ε(u) ∈ L1 (QT , R≥0 )   . lim sup a n (t, x) dtdx ≤ a(t, x) dtdx for all closed K ⊆ QT \ S , n→∞

K

K

it follows that A u = (f , F ) .

.

(L.4) From A uLq  (·,·) (Q

.

T)

d×d → ∞, Msym )

d ×Lp (·,·) (Q , T

it follows that Au, u V ˚q,p (Q

.

ε

T)

→ ∞.

At first glance, .L1 -monotonicity may seem to be a somewhat artificial property. In actual fact, this monotonicity concept is simply an abstraction of a class of oper˚εq,p (QT ) → ators under which, among others, the unsteady extra stress tensor .S : V q,p ∗ ˚ Vε (QT ) (cf. Proposition 5.2) falls.

7.2 L1 -Monotonicity

255

d×d Proposition 7.3 Let .S : QT × Md×d sym → Msym be a mapping satisfying (S.1)– (S.4) and, in addition, (S.5), i.e., .S(·, ·, 0) = 0 almost everywhere in .QT . Then, ˚εq,p (QT ) → V ˚εq,p (QT )∗ , for every .u ∈ V ˚εq,p (QT ) defined by .Su := .S : V q,p σ ∗ 1 ˚ Jε (0, S(·, ·, ε(u))) in .Vε (QT ) , is .L -monotone. 



Proof We set .S u := (0, S(·, ·, ε(u))) in .Lq (·,·) (QT )d × Lp (·,·) (QT , Md×d sym ) ˚εq,p (QT ). for all .u ∈ V ad (L.1). ad (L.2). ad (L.3). ad (L.4).

Follows from the definition, where we make use of the condition (S.5) for .S 0 = (0, 0) . Immediate consequence of the condition (S.4). Follows by a similar, but slightly simplified, procedure as can be found [175, Prop. 3.1]. −  − + We use .ρp (·,·) (S(·, ·, ε(u))) ≤ 2(p ) [α p 2p (ρp(·,·) (δ)+ρp(·,·) (ε(u))) ˚εq,p (QT ) (cf. (3.103)), (3.111), and + ρp (·,·) (β)] for every .u ∈ V Lemma 2.1 (iii).  

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ , Remark 7.2 For .L1 -monotone .A : V  (·,·)  (·,·) q,p d×d ∞ q d p ˚ .A : Vε (QT ) ∩ H (QT ) → L (QT ) × L (QT , Msym ) is always demi-continuous. The underlying idea of .L1 -monotonicity is to create a special monotonicity concept that is amenable to the theory of Radon measures. More precisely, if ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is .L1 -monotone, then for every .u ∈ .A : V  q,p ˚ε (QT ) .∩H∞ (QT ), .f ∈ Lq (·,·) (QT )d and .F ∈ Lp (·,·) (QT , Md×d ) such that V sym  1 .A u = (f , F ) , it holds .f · u + F : ε(u) ∈ L (QT , R≥0 ), since we have that  .A 0 = (0, 0) , i.e., the mapping .μ := f · u + F : ε(u) dtdx : B(QT ) → R≥0 , where .B(QT ) is the Borel .σ -algebra of .QT , defined by  .μ(E) := f (t, x) · u(t, x) + F (t, x) : ε(u)(t, x) dtdx (7.34) E

for every .E ∈ B(QT ), is a finite Radon measure. In this connection, before delving too deeply into a thorough examination of the general concept of .L1 -monotonicity, it may be prudent to begin with a brief overview of important definitions and facts about finite Radon measures first.

7.2.1 Finite Radon Measures The purpose of the present section is to briefly recall definitions and basic facts from the theory of finite Radon measures. For proofs and extensive presentations, we refer, e.g., to [7, 55, 58, 91, 150]. Throughout this section, let X be a locally compact, separable metric space such as .Rn , .n ∈ N, or its open or closed subsets. Moreover, we denote by .B(X) the Borel .σ -algebra of X, i.e., the .σ -algebra generated by open subsets of X.

256

7 Existence Theory for Irregular Domains

Definition 7.3 A measure .μ : B(X) → [0, +∞] is called a Radon measure, if it is locally finite, i.e., for every .x ∈ X, there exists an open set .O x with .μ(O) < ∞, and it is inner regular, i.e., .μ(E) = sup μ(K) | K ⊆ E, K is compact for every 3 .E ∈ B(X). We denote by .M(X) the set of all finite Radon measures on .B(X). Then, for .μ ∈ M(X), we define .μM(X) := μ(X).  We define .C0 (X) to be the closure of .C00 (X) := f ∈ C 0 (X) | supp(f ) ⊂⊂ X with respect to the norm .ϕC0 (X) := supx∈X |ϕ(x)|. Then, the following proposition allows for an identification of .M(X) with the set of all positive functionals in ∗ .C0 (X) , see [58, Satz 2.10, p. 338]) for a proof. Proposition 7.4 The Riesz representation operator .JM : M(X) → C0 (X)∗ , defined by  := .J μ, f f dμ C0 (X) M X

for every .μ ∈ M(X) and .f ∈ C0 (X), is well-defined, linear and satisfies JM μC0 (X) ≤ μM(X) for every .μ ∈ M(X). In addition, for every positive ∗ ∈ C (X)∗ , i.e., .f ∗ , f .f 0 C0 (X) ≥ 0 for all .f ∈ C0 (X) with .f ≥ 0 in X, there exists a unique .μ ∈ M(X) such that .JM μ = f ∗ in .C0 (X)∗ . .

This identification, in turn, allows for an introduction of a notion of weak-* convergence in .M(X), as we have done before in a similar way for .Y ∞ (QT ) and ∞ .H (QT ) (cf. Definition 2.18). Definition 7.4 (Weak-* Convergence in .M(X)) A sequence of Radon measures (μn )n∈N ⊆ M(X) is said to converge weakly-*, or vaguely, to a Radon measure .μ ∈ M(X) as .n → ∞, if .

∗

JM μn  JM μ

.

in C0 (X)∗

(n → ∞) .

(7.35)

∗

With the common abuse of notation, we always write .μn  μ in .M(X) .(n → ∞) instead of (7.35). Proposition 7.5 (Weak-* Compactness) Let .(μn )n∈N ⊆ M(X) be a bounded sequence, i.e., .supn∈N μn M(X) < ∞. Then, there exists a subsequence .(μn )n∈ ⊆ M(X), with a cofinal subset . ⊆ N, and a finite Radon measure ∗ .μ ∈ M(X) such that .μn  μ in .M(X) .( n → ∞). Proof See [150, Theorem 21.18].

 

Proposition 7.6 (Portemanteau for Weak-* Convergence) For a sequence (μn )n∈N ⊆ M(X) and a finite Radon measure .μ ∈ M(X), the following statements are equivalent:

.

∗

(i) .μn  μ in .M(X) .(n → ∞). (ii) .μ(int(E)) ≤ lim infn→∞ μn (E) ≤ lim supn→∞ μn (E) ≤ μ(E) for every bounded set .E ∈ B(X). 3A

measure .μ : A → [0, +∞] on a measurable space .(X, A) is called finite, if .μ(X) < ∞.

7.2 L1 -Monotonicity

(iii)

.

257





dμ ≤ lim infn→∞ E f dμn ≤ lim supn→∞ E f dμn ≤ E f dμ for every bounded set .E ∈ B(X) and .f ∈ C 0 (X) with .f ≥ 0 in X. int(E) f

 

Proof The proof is postponed to the appendix (cf. Proposition 9.2).

7.2.2 Minty–Trick Like Argument for L1 -Monotone Operators This section illustrates how .L1 -monotonicity and the theory of finite Radon measures interact synergistically in the identification of weak limits in the absence of suitable sources of compactness. More precisely, we will observe that the hardto-obtain limit superior inequality in Definition 7.2, (L.3), can be reduced to a point-wise inequality that, in turn, can be obtained by falling back on parabolic compensated compactness methods. ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ be .L1 -monotone. Proposition 7.7 Let .A : V q,p ∞ q  (·,·) (Q )d and ˚ε (QT ) ∩ H (QT ), .(f n ) Then, for .(un )n∈N ⊆ V T n∈N ⊆ L  p (·,·) (Q , Md×d ) such that . u = (f , F ) for all .n ∈ N, .(F n )n∈N ⊆ L T n A n n sym and .(μn )n∈N := (f n · un + F n : ε(un ) dtdx)n∈N ⊆ M(QT ), defined according to (7.34), from ˚εq,p (QT ) in V

(n → ∞) , .

(7.36)

un  u

in H∞ (QT )

(n → ∞) , .

(7.37)

fn  f

in Lq (·,·) (QT )d

(n → ∞) , .

(7.38)

Fn  F

in Lp (·,·) (QT , Md×d sym )

(n → ∞) , .

(7.39)

in M(QT )

(n → ∞) ,

(7.40)

un  u

.

∗

∗

μn  μ





where .μ = a 0 dtdx + μs in .M(QT ) for .a 0 ∈ L1 (QT , R≥0 ) and .μs ∈ M(QT ) with .μs ⊥ dtdx,4 provided by Lebesgue’s measure decomposition theorem (cf. [58, p. 285]), and a 0 ≤ a := f · u + F : ε(u)

.

a.e. in QT ,

(7.41)

it follows that: ˚εq,p (QT )∗ . (i) .A u = (f , F ) . In particular, it holds .Au = Jεσ (f , F ) in .V (ii) .a 0 = a almost everywhere in .QT . ˚εq,p (QT ) ≥ Au, uχ(t  ,t) V ˚εq,p (QT ) (iii) .lim infn→∞ Aun , un χ(t  ,t) V for every   .t , t ∈ I with .t < t. measures .μ, ν : A → [0, +∞] on a measurable space .(X, A) are called singular, written ⊥ ν, if there exists a set .A ∈ A such that .μ(A) = ν(X \ A) = 0.

4 Two .μ

258

7 Existence Theory for Irregular Domains

Proof ad (i). Owing to .μs ⊥ dtdx, there exists .S ∈ B(QT ) such that .|S| = 0 and s .μ (QT \ S) = 0. Then, by combining Proposition 7.6 (ii), (7.40) and (7.41), we find for every closed set .K ⊆ QT \ S, taking into account that .μs (K) = 0, that  lim sup n→∞

K

a n (s, y) dsdy = lim sup μn (K) n→∞

≤ μ(K)  = a 0 (s, y) dsdy

.



(7.42)

K

≤

a(s, y) dsdy . K

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ , i.e., Therefore, the .L1 -monotonicity of .A : V  Definition 7.2, (L.3), yields that .A u = (f , F ) . In particular, it holds .Au = ˚εq,p (QT )∗ , i.e., (i), where we used Definition 7.2, Jεσ (A u) = Jεσ (f , F ) in .V (L.1). ad (ii). Based on (i), i.e., we have that .A u = (f , F ) , it holds, due to Definition 7.2, (L.2), for every measurable set .K ⊆ QT and .n ∈ N that5  .

K

(f n − f ) · (un − u) + (F n − F ) : (ε(un ) − ε(u)) dsdy ≥ 0 .

(7.43)

By rearranging (7.43), we arrive for every measurable set .K ⊆ QT and .n ∈ N at  K .

a n (s, y) dsdy ≥ Jεσ ((f n − f )χK , (F n − F )χK ), u V ˚q,p (Q ε

T)

(7.44)

+ Jεσ (f χK , F χK ), un V ˚εq,p (QT ) . Taking the limit inferior with respect to .n → ∞ on both sides in (7.44), in particular,  using (7.36), (7.38), (7.39), as well as the weak continuity of .Jεσ : Lq (·,·) (QT )d ×  ∗ ˚q,p Lp (·,·) (QT , Md×d sym ) → Vε (QT ) (cf. Proposition 4.17 (ii)) in doing so, provides further for every measurable set .K ⊆ QT that  .

lim inf n→∞

K

a n (s, y) dsdy ≥ Jεσ (f χK , F χK ), u V ˚q,p (Q ε



=

(7.45)

f (s, y) · u(s, y) + F (s, y) : ε(u)(s, y) dsdy K

 =

a(s, y) dsdy . K

5 Here,

T)

in favor of readability, we suppress the .(s, y) -dependency.

7.2 L1 -Monotonicity

259

Then, combining (7.45) and (7.42), for every closed set .K ⊆ QT \ S, we find that 

 a(s, y) dsdy ≤ lim inf n→∞

K

a n (s, y) dsdy K



≤ lim sup n→∞



.

≤

a n (s, y) dsdy K

(7.46)

a 0 (s, y) dsdy K

 ≤

a(s, y) dsdy . K

Owing to .|S| = 0, we conclude from (7.46) to .a = a 0 almost everywhere in .QT , i.e., (ii).  ad (iii). Corresponds to (7.45), if we choose .K = t  , t × for arbitrary .t  , t ∈ I with .t  < t and exploit that .a = f · u + F : ε(u) in .L1 (QT ) and .Au = Jεσ (f , F ) ˚εq,p (QT )∗ (cf. (i)), as well as .a n = f n · un + F n : ε(un ) in .L1 (QT ) and in .V σ ˚εq,p (QT )∗ for every .n ∈ N. .Aun = Jε (f n , F n ) in .V   By the comparison of Proposition 7.7 and the definition of the Bochner–Sobolev condition (M) (cf. Definition 7.1), it should become apparent that both contain (7.5) and (7.6), or (7.36) and (7.37), respectively, that (7.7), by Definition 4.3, (L.4), implies (7.38) and (7.39), and that (7.40) follows from (7.38) and (7.39) with the aid of Proposition 7.5. Hence, the only point that still needs some additional attention is the inequality (7.41). Here, we intend to apply a parabolic compensated compactness principle, or briefly PCCP, see, e.g., [175] for an overview. Regrettably, conventional results concerning PCCPs mostly cover only the non-solenoidal case, in which (7.3) is a sequence of generalized evolution equations (cf. Definition 3.9) rather than a sequence of solenoidal generalized evolution equations (cf. Definition 4.9). The latter is a drastic simplification to our situation since the solenoidal case is significantly less amenable to the localization argument typically applied in a PCCP. Nevertheless, there are some contributions worth mentioning by V.V. Zhikov [173], who considers exclusively time-independent exponents .p ∈ P∞ (), and by S.E. Pastukhova [140], who additionally covers time-dependent exponents ∞ + < (p − ) . Inasmuch as the .p ∈ P (QT ) but imposes the rather artificial relation .p • basic principle of electrorheological fluids lies in the dependence it of the extra stress tensor .S on a time-varying electro-magnetic field .E (cf. (1.7)), admitting time- and space-dependent variable exponents is of absolute necessity (cf. Introduction 1.2). Therefore, Pastukhova’s PCCP may appear to be the more attractive result. On the other hand, the imposed relation .p+ < (p− )• in Pastukhova’s PCCP is a rather restrictive assumption. To give an example, in three dimensions, exponents ∞ − = 2 are covered by Pastukhova’s PCCP only if they addition.p ∈ P (QT ) with .p ally satisfy .p+ < 10 . 3 Pastukhova’s artificial upper bound can mainly be attributed , apart from other causes, to the need for a local pressure reconstruction, which is necessary to locally transform the sequence of solenoidal generalized evolution

260

7 Existence Theory for Irregular Domains

equations into a sequence of generalized evolution equations and, therefore, to make the localization argument applied in a PCCP accessible. Recall that we have already established that the parabolic pressure reconstruction in full regularity, i.e., with p (·,·) .π 2 ∈ L ,0 (QT ), does not hold in the case of variable exponents (cf. Remark 6.2). Thus, we+are only able to recover the pressure with lower integrability, i.e., with (p ) .π 2 ∈ L (Q T ), which is then, in turn, solely controllable if, in addition, the ,0 artificial relation .p+ < (p− )• is satisfied, i.e., if the integrability of our solution is − at least .u ∈ L(p )• (QT )d . From a methodological point of view, the artificial upper bound in [140] comes from not exploiting the given variable exponent structure since Pastukhova’s PCCP is completely formulated in terms of the constant limit exponents .p− , p+ . On the other hand, V.V. Zhikov and S.E. Pastukhova together in [175, Lemma 3”] outlined a PCCP addressing the non-solenoidal case involving a given .log-Hölder continuous variable exponent structure. This has the advantage that the assumption .p+ < (p− )• , which is also necessary in the non-solenoidal case for other technical reasons, can be fulfilled locally and, thus, is omitted globally. An essential component of their argumentation, even though they have left out all the details, is the application of an appropriate formula of integration-by-parts, which is not specified there, but, e.g., Proposition 3.22 or [51, Theorem 7.1] would undoubtedly remedy the situation. We follow this approach and integrate a given .log-Hölder continuous variable exponent structure in our PCCP for the solenoidal case. Inasmuch as we also intend to treat time-dependent variable exponents, we are also forced to reconstruct the pressure locally in order to transfer the sequence of solenoidal generalized evolution equations locally into a sequence of generalized evolution equations. Then, we need to apply the correct formula of integrationby-parts. Since we will switch to a sequence of generalized evolution equations, q,p the formula of integration-by-parts for .Wε,σ (QT ) is not suitable. The formula of q,p integration-by-parts for .Wε (QT ) is not applicable in this very situation either because the pressure terms cannot be reconstructed in the correct function spaces (cf. Remark 6.2). The next section, therefore, addresses the introduction of so-called q,p anisotropic variable exponent Bochner – Lebesgue spaces, subspaces of .Xε (QT ) that take additional integrability of the divergence into account and, thus, can be q,p q,p considered as hybrid spaces between .Vε (QT ) and .Xε (QT ).

7.3 Anisotropic Variable Exponent Bochner–Lebesgue Spaces In this section, we introduce anisotropic variable Bochner–Lebesgue spaces. Since anisotropic variable Bochner–Lebesgue spaces can be considered as hybrid spaces, q,p q,p inheriting properties of both .Vε (QT ) and .Xε (QT ), the majority of results q,p q,p already obtained for .Vε (QT ) and .Xε (QT ) admit congruent adaptations to anisotropic variable exponent Bochner–Lebesgue spaces. We will, therefore, rarely give detailed proofs, but rather outline how the assertions follow similar to the proofs in Chaps. 3 and 4.

7.3 Anisotropic Variable Exponent Bochner–Lebesgue Spaces

261

Throughout the entire section, unless otherwise specified, let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, .I := (0, T ), .T < ∞, .QT := I × , and .q, p, s ∈ Plog (QT ) with .q − , p− , s − > 1.

q,p,s

7.3.1 The Space Xε,div (QT ) Before introducing the anisotropic variable Bochner–Lebesgue space, we first introduce its steady analogue, i.e., for time-independent exponents .q, ˜ p˜ ∈ P∞ (), q(·), ˜ p(·) ˜ q(·), ˜ p(·) ˜ ˚ε ˚ε () and .X () that incorporates additional the hybrid space between .V integrability of the divergence. Therefore, for .q, ˜ p, ˜ s˜ ∈ P∞ (), we define the anisotropic variable Sobolev space  ˜ p(·),˜ ˜ s (·) ˜ p(·) ˜ ˚εq(·), ˚q(·), () := x ∈ X () | div(x) ∈ Ls˜(·) () . X ε,div

.

The same argumentation as in the proof of Proposition 2.9 demonstrates that ˜ p(·),˜ ˜ s (·) ˚q(·), () is a Banach space, if equipped with the canonical norm X ε,div

.

:=  · X  · X + div(·)Ls˜(·) () . ˜ p(·),˜ ˜ s (·) ˜ p(·) ˜ ˚q(·), ˚q(·), () ()

.

ε

ε,div

Definition 7.5 We define for every .t ∈ I , the anisotropic time slice space ˚q(t,·),p(t,·),s(t,·) () . ˚q,p,s (t) := X X ε,div ε,div

.

Furthermore, we define the anisotropic limiting time slice spaces +

˚q,p,s := X ˚q ,p X ε,div div,+

.

+ ,s +

−

˚q,p,s := X ˚q ,p X ε,div div,−

() ,

− ,s −

() .

˚q,p,s (t), .t ∈ I , we next introduce By means of the time slice spaces .X ε,div anisotropic variable Bochner–Lebesgue spaces, by analogy with Definition 3.2 and Definition 4.2, respectively. Definition 7.6 We define the anisotropic variable Bochner–Lebesgue space  q,p,s q,p s(·,·) .X (QT ), ε,div (QT ) := x ∈ Xε (QT ) | div(x) ∈ L ˚q,p,s (t) for a.e. t ∈ I . x(t) ∈ X ε,div Furthermore, we define the limiting anisotropic Bochner–Lebesgue spaces q,p,s

+ ,p + ,s + }

q,p,s

− ,p − ,s − }

Xdiv,+ (QT ) := Lmax{q

.

Xdiv,− (QT ) := Lmin{q

q,p,s

˚ (I, X div,+ ), q,p,s

˚ (I, X div,− ) .

262

7 Existence Theory for Irregular Domains q,p,s

Proposition 7.8 The space .Xε,div (QT ) forms a Banach space, if equipped with the norm q,p,s q,p . ·  X (QT ) :=  · Xε (QT ) + div(·)Ls(·,·) (QT ) . ε,div

In addition, q,p,s

q,p,s

q,p,s

Xdiv,+ (QT ) → Xε,div (QT ) → Xdiv,− (QT )

.

and q,p

q,p,s

q,p

Vε (QT ) → Xε,div (QT ) → Xε

.

(QT ).  

Proof Follows along the lines of the proof of Proposition 3.2.

q,p,s q(·,·) (Q )d × Lp(·,·) Proposition 7.9 The operator .div T ε : Xε,div (QT ) → L div d×d s(·,·) (QT , Msym ) × L (QT ), defined by .ε x := (ε x, div(x)) in .Lq(·,·) (QT )d × s(·,·) (Q ) for every .x ∈ X q,p,s (Q ), where . denotes Lp(·,·) (QT , Md×d T T ε sym ) × L ε,div the operator from Proposition 3.3, is an isometric isomorphism into .R(εdiv ). In q,p,s particular, .Xε,div (QT ) is separable and reflexive.

 

Proof Follows along the lines of the proof of Proposition 3.3. q,p,s

7.3.2 Duality in Xε,div (QT ) q,p,s

q,p

Because .Xε,div (QT ), in contrast to .Vε (QT ), is not equipped with an incomq,p,s pressibility constraint, a thorough description of .Xε,div (QT )∗ is available, such as q,p it is for .Xε (QT )∗ (cf. Proposition 3.4). 



Proposition 7.10 The operator .Jεdiv : Lq (·,·) (QT )d × Lp (·,·) (QT , Md×d sym ) ×    q,p,s Ls (·,·) (QT ) → Xε,div (QT )∗ , for every .f ∈ Lq (·,·) (QT )d , .F ∈ Lp (·,·) (QT , Md×d sym ),  (·,·) q,p,s s .π ∈ L (QT ) and .x ∈ Xε,div (QT ) defined by Jεdiv (f , F , π ), x X q,p,s (QT ) := Jε (f , F ), x Xεq,p (QT ) + (π, div(x))Ls(·,·) (QT ) ,

.

ε,div

where .Jε denotes the operator from Proposition 3.4, is well-defined, linear and q,p,s Lipschitz continuous with constant 2. In addition, for every .x ∗ ∈ Xε,div (QT )∗ ,   s  (·,·) (Q ) such there exist .f ∈ Lq (·,·) (QT )d , .F ∈ Lp (·,·) (QT , Md×d T sym ) and .π ∈ L q,p,s that .x ∗ = Jεdiv (f , F , π ) in .Xε,div (QT )∗ and .

1 ∗ x X q,p,s (QT )∗ ≤ f Lq  (·,·) (QT )d + F Lp (·,·) (QT )d×d + π Ls  (·,·) (QT ) ε,div 2 ≤ 2x ∗ X q,p,s (QT )∗ . ε,div

Proof Follows along the lines of the proof of Proposition 3.4.

 

7.3 Anisotropic Variable Exponent Bochner–Lebesgue Spaces

263

Corollary 7.2 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain and log ˚q,p,s .q, p, s ∈ P () with .q − , p− , s − > 1. Then, the mapping .div ε : Xε,div → s(·) (), for every .x ∈ X ˚q,p,s defined by Lq(·) ()d × Lp(·) (, Md×d sym ) × L ε,div div  in .Lq(·) ()d × Lp(·) (, Md×d ) × Ls(·) (), where .ε x := (ε x, div(x)) sym .ε denotes the operator from Corollary 3.2, is an isometric isomorphism into div ˚q,p,s is separable and reflexive. Furthermore, the .R(ε ). In particular, .X ε,div q  (·) ()d × Lp (·) (, Md×d ) × Ls  (·) () → (X ˚q,p,s )∗ , for every : L operator .Jdiv sym ε ε,div    q (·) ()d , .F ∈ Lp (·) (, Md×d ), .π ∈ Ls (·) () and .x ∈ X ˚q,p,s defined by .f ∈ L sym

ε,div

Jdiv ˚q,p,s := Jε (f, F), x X ˚εq,p + (π, div(x))Ls(·) () , ε (f, F, π ), x X

.

ε,div

where .Jε denotes the operator from Corollary 3.2, is well-defined, linear and ˚q,p,s )∗ , there Lipschitz continuous with constant 2. In addition, for every .x∗ ∈ (X ε,div    s (·) () such that .x∗ = ) and . π ∈ L exist .f ∈ Lq (·) ()d , .F ∈ Lp (·) (, Md×d sym ˚q,p,s ∗ Jdiv ε (f, F, π ) in .(Xε,div ) and .

1 ∗ ∗ x (X ˚q,p,s )∗ ≤ fLq  (·) ()d + FLp (·) ()d×d + π Ls  (·) () ≤ 2x (X ˚q,p,s )∗ . ε,div ε,div 2

Remark 7.3 (Time slices of functionals .x ∗ ∈ Xε,div (QT )∗ ) q,p,s







s (·,·) (Q ) (i) For all functions .f ∈ Lq (·,·) (QT )d , .F ∈ Lp (·,·) (QT , Md×d T sym ), .π ∈ L q,p,s and .x ∈ Xε,div (QT ), Fubini’s theorem, Proposition 7.10 and Corollary 7.2 1 imply that .(t → Jdiv ˚q,p,s (t) ) ∈ L (I ) with ε (f (t), F (t), π (t)), x(t) X ε,div

 Jεdiv (f , F , π ), x X q,p,s (QT ) =

.

ε,div

I

Jdiv ˚q,p,s (t) dt . ε (f (t), F (t), π (t)), x(t) X ε,div

q,p,s ˚q,p,s ∗ (ii) For .x ∗ ∈Xε,div (QT )∗ , we define .x ∗ (t) := Jdiv ε (f (t), F (t), π(t)) in .Xε,div (t)   for a.e. .t ∈ I for functions .f ∈ Lq (·,·) (QT )d , .F ∈ Lp (·,·) (QT , Md×d sym ) and  q,p,s s (·,·) ∗ div ∗ .π ∈ L (QT ) that satisfy .x = Jε (f , F , π ) in .Xε,div (QT ) .

q,p,s

7.3.3 Smoothing in Xε,div (QT ) In Proposition 4.14 (i), we noticed that the smoothing method via transversal expansion of bounded Lipschitz domains and a subsequent modification by a contravariant Piola transform preserves the incompressibility constraint. Moreover, if we apply the smoothing operator from Proposition 4.14 to non-solenoidal functions, such as, q,p,s e.g., to .x ∈ Xε,div (QT ), the divergence .div(x) ∈ Ls(·,·) (QT ) will be preserved up to multiplicative factors that are given by the family .(det(Jh ))h∈(0,h2 ) ⊆ C ∞ () (cf. Proposition 7.11 (i) below). We are, therefore, able to generalize the smoothing

264

7 Existence Theory for Irregular Domains q,p,s

operator from Proposition 4.14 to a smoothing operator for the space .Xε,div (QT ), whereas the space truncation smoothing operator (cf. Proposition 3.14) covers only the particular case of .s = p in .QT . Proposition 7.11 Let .q, p, s ∈ Plog (QT ) with .p− , s − > 1 and .q ≥ p in .QT . For q,p,s .x ∈ X ε,div (QT ) and .h ∈ (0, h2 ), we define, in accordance with Proposition 4.14, the smoothing operator h h SQ x := Ph SQ x ∈ C0∞ (QR )d , T ,σ T

.

h : Lq(·,·) (Q )d →C ∞ (Q )d , .h ∈ where .SQ (0, h2 ), are the transversal expansion T R 0 T smoothing operators from Proposition 4.10 and .(Ph )h∈(0,h2 ) ⊆ C ∞ ()d×d were q,p,s introduced in Lemma 4.2. Then, for every .x ∈ Xε,div (QT ), it holds: h h (i) . (SQ x)h∈(0,h2 ) ⊆ C0∞ (QR )d with .supp(SQ x) ⊆ [−ζ h, T + ζ h] × h T ,σ T ,σ h h (div(x)) for every .h ∈ (0, h2 ). In addition, it holds .div(SQT ,σ x) = det(Jh )SQ T in .QR for every .h ∈ (0, h2 ). (ii) There exists a constant .c > 06 (not depending on .q, p, s ∈ Plog (QT )) such that h sup |SQ x| ≤ c Md+1 (QT x) T ,σ

a.e. in QR ,

h sup |div(SQ x)| ≤ c Md+1 (QT div(x)) T ,σ

a.e. in QR ,

.

h∈(0,h2 )

h∈(0,h2 )

 h sup |ε(SQ x)| ≤ c Md+1 (QT x) + Md+1 (QT ε(x)) a.e. in QR . T ,σ

h∈(0,h2 )

(iii) For extensions .q, p, s ∈ Plog (Rd+1 ) of .q, p, s ∈ Plog (QT ), respectively, with − ≤ q ≤ q + , .p − ≤ p ≤ p + and .s − ≤ s ≤ s + in .Rd+1 , there exist constants .q log d+1 .cq , cp , cs > 0 (depending on .q, p, s ∈ P (R ) and .c > 0, respectively) such that .

h sup SQ xLq(·,·) (Q )d ≤ cq xLq(·,·) (QT )d , T ,σ R

h∈(0,h2 )

h sup div(SQ x)Ls(·,·) (Q ) ≤ cs div(x)Ls(·,·) (QT ) , T ,σ R

 h sup ε(SQ x)Lp(·,·) (Q )d×d ≤ cp xLp(·,·) (QT )d +ε(x)Lp(·,·) (QT )d×d . T ,σ R h∈(0,h2 ) h∈(0,h2 )

h (iv) .SQ x → x in Xε,div (QT ) .(h → 0), i.e., .C ∞ (I , C0∞ ()d ) lies densely in T ,σ q,p,s .X ε,div (QT ). q,p,s

6 We

assume for .c > 0 the same dependencies as described in Remark 4.9.

7.3 Anisotropic Variable Exponent Bochner–Lebesgue Spaces

265

Proof ad (i). Follows as in the proof of Proposition 4.14 (i) (cf. (4.59)) by means of Lemma 4.2 (i). ad (ii). For the first and third inequality we proceed exactly as in the proof of Proposition 4.14 (ii). The second inequality follows from (i), Proposition 4.10 (ii) and the estimate .det(Jh )L∞ () ≤ 2 for every .h ∈ (0, h2 ) (cf. Proposition 4.9 (ii)). ad (iii). Follows as in the proof of Proposition 4.10 (iii) from (ii) by means of Proposition 2.12. ad (iv). By proceeding as in the proof of Proposition 4.14 (iv), we obtain h SQ x→x T ,σ

.

q,p

in Xε

(QT )

(h → 0) .

(7.47)

Combining (i), Proposition 4.10 (iv) and the estimate . det(Jh ) − 1L∞ () ≤ κ0 h for all .h ∈ (0, h2 ) (cf. Proposition 4.9 (iii)), we observe that h div(SQ x) − div(x)Ls(·,·) (QT ) ≤  det(Jh ) − 1L∞ () T ,σ

.

(7.48)

h × SQ (div(x))Ls(·,·) (QT ) T h + SQ (div(x)) − div(x)Ls(·,·) (QT ) T

→0

(h → 0) .

h Putting everything together, (7.47) and (7.48) prove that .SQ x → x in T ,σ q,p,s .X   ε,div (QT ) .(h → 0).

The same method as in Proposition 7.11 can be used to construct a smoothing ˚q,p,s :=X ˚q(·),p(·),s(·) (), provided that . ⊆ Rd , .d ≥ 2, is a bounded operator for .X ε,div ε,div Lipschitz domain and .q, p, s ∈ Plog () with .q − , p− , s − > 1. In addition, ˚εq,p → Lp(·) (), i.e., ˚q,p,s → X similar to Corollary 4.3, we observe, due to .X ε,div ˚q,p,s (cf. Propositions 3.5 and .xLp(·) ()d ≤ cp ε(x)Lp(·) ()d×d for every .x ∈ X ε,div 2.13), that the assumption .q ≥ p in . can be omitted. Corollary 7.3 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain, let ∞ d d .(ν h )h∈[0,1] ⊆ C (R ) be the transversal expansion mappings provided by Propo˚q,p,s := X ˚q(·),p(·),s(·) (). sition 4.9, .q, p, s ∈ Plog () with .q − , p− , s − > 1 and .X ε,div ε,div q,p,s ˚ For .x ∈ X ε,div and .h ∈ (0, h2 ), we define the smoothing operators Sh,σ x := Ph Sh x,

.

Sh x := (ωζdh ∗ (ℰ x)) ◦ ν h ∈ C ∞ (Rd )d ,

where .(ωhd )h>0 ⊆ C0∞ (Rd ) are the scaled standard mollifiers from Remark 2.6 and 1 d d  .ℰ x ∈ L (R ) is defined by .ℰ x := x in . and .ℰ x := 0 in . . Then, for every q,p,s ˚ .x ∈ X ε,div , it holds: (i) .(Sh,σ x)h∈(0,h2 ) ⊆ C0∞ ()d with .supp(Sh,σ x) ⊆ h for every .h ∈ (0, h2 ). In addition, it holds .div(Sh,σ x) = det(Jh )Sh (div(x)) in . for every .h ∈ (0, h2 ).

266

7 Existence Theory for Irregular Domains

(ii) There exists a constant .c > 0 (not depending on .q, p, s ∈ Plog ()) such that sup |Sh,σ x| ≤ c Md (ℰ x)

a.e. in  ,

sup |div(Sh,σ x)| ≤ c Md (ℰ div(x))

a.e. in  ,

.

h∈(0,h2 )

h∈(0,h2 )

 sup |ε(Sh,σ x)| ≤ c Md (ℰ x) + Md (ℰ ε(x))

a.e. in  .

h∈(0,h2 )

(iii) For extensions .q, p, s ∈ Plog (Rd ) of .q, p, s ∈ Plog (), respectively, with − ≤ q ≤ q + , .p − ≤ p ≤ p + and .s − ≤ s ≤ s + in .Rd , there exist constants .q log d .cq , cp , cs > 0 (depending on .q, p, s ∈ P (R ) and .c > 0, respectively) such that .

sup Sh,σ xLq(·) ()d ≤ cq xLq(·) ()d ,

h∈(0,h2 )

sup div(Sh,σ x)Ls(·) () ≤ cs div(x)Ls(·) () ,

h∈(0,h2 )

 sup ε(Sh,σ x)Lp(·) ()d×d ≤ cp xLp(·) ()d + ε(x)Lp(·) ()d×d .

h∈(0,h2 )

˚q,p,s .(h → 0), i.e., .C ∞ ()d lies densely in .X ˚q,p,s . (iv) .Sh,σ x → x in .X ε,div ε,div 0 In view of the introduction of a further notion of a generalized time derivative, q,p,s which is now supposed to be a functional in .Xε,div (QT )∗ , the density of .C0∞ (QT )d q,p,s in .Xε,div (QT ) again becomes indispensable. The latter, by analogy with Proposition 3.18, Proposition 4.13, and Proposition 4.16, will be obtained by a simple truncation argument with respect to the time variable. Proposition 7.12 Let .q, p, s ∈ Plog (QT ) with .p− , s − > 1 and .q ≥ p in .QT and let .(ϕh )h∈I◦ ⊆ C0∞ (I ) be the family of cut-off functions from Proposition 3.18.  T q,p,s For .x ∈ Xε,div (QT ) and .h ∈ (0, h2 ) ∩ I◦ζ , where .I◦ζ := 0, 4ζ , we define, in accordance with Proposition 4.16, the smoothing operator ˚h x := S h (ϕζ h x) ∈ C ∞ (Q )d . S 0 QT ,σ QT ,σ R

.

q,p,s

Then, for every .x ∈ Xε,div (QT ), it holds: ˚h x)h∈(0,h )∩I ζ ⊆ C ∞ (QT )d with .supp(S ˚h x) ⊆ Iζ h ×  for every (i) .(S h 2 QT ,σ QT ,σ 0 ◦ ζ ˚h x) = det(Jh )S ˚h (div(x)) in .h ∈ (0, h2 ) ∩ I◦ . In addition, it holds .div(S QT ,σ QT ζ .QT for every .h ∈ (0, h2 ) ∩ I◦ .

7.3 Anisotropic Variable Exponent Bochner–Lebesgue Spaces

267

(ii) There exists a constant .c > 0 (not depending on .q, p, s ∈ Plog (QT )) such that ˚h x| ≤ c Md+1 (Q x) |S T QT ,σ

a.e. in QT ,

˚h x)| ≤ c Md+1 (Q div(x)) |div(S T QT ,σ

a.e. in QT ,

sup

.

h∈(0,h2 )∩I◦ζ

sup h∈(0,h2 )∩I◦ζ

sup h∈(0,h2 )∩I◦ζ



˚h x)| ≤ c Md+1 (Q x) + Md+1 (Q ε(x)) a.e. in QT . |ε(S T T QT ,σ

(iii) For extensions .q, p, s ∈ Plog (Rd+1 ) of .q, p, s ∈ Plog (QT ), respectively, with − ≤ q ≤ q + , .p − ≤ p ≤ p + and .s − ≤ s ≤ s + in .Rd+1 , there exist constants .q log d+1 .cq , cp , cs > 0 (depending on .q, p, s ∈ P (R ) and .c > 0, respectively) such that .

sup

h∈(0,h2 )∩I◦ζ

sup h∈(0,h2 )∩I◦ζ

sup h∈(0,h2 )∩I◦ζ

˚h x q(·,·) S L (QT )d ≤ cq xLq(·,·) (QT )d , QT ,σ

˚h x) s(·,·) div(S L (QT ) ≤ cs div(x)Ls(·,·) (QT ) , QT ,σ

 ˚h x) p(·,·) ε(S L (QT )d×d ≤ cp xLp(·,·) (QT )d +ε(x)Lp(·,·) (QT )d×d . QT ,σ

˚h x → x in X q,p,s (QT ) .(h → 0), i.e., .C ∞ (QT )d lies densely in (iv) .S QT ,σ ε,div 0 q,p,s .X (Q ). T ε,div Proof ad (i). We simply combine Proposition 4.13 (i) and Proposition 7.11 (i). ad (ii) and (iii). We fall back on Proposition 7.11 (ii) and (iii) and exploit that .suph∈I ϕh L∞ (R) ≤ 1. ◦ ad (iv). Applying Proposition 7.11 (iii) and (iv) and Lebesgue’s theorem on dominated convergence (cf. Proposition 2.4), we obtain h ˚h x − x q,p,s q,p,s S QT ,σ X (QT ) ≤ SQT ,σ [(1 − ϕζ h )x]X (QT ) ε,div

ε,div

h + SQ x T ,σ .

− xX q,p,s (QT ) ε,div

≤ max{cq , 2(1 + |QT |)cp , cs }(1 − ϕζ h )xX q,p,s (QT ) ε,div

h + SQ x T ,σ

→0

− xX q,p,s (QT )

(h → 0) .

ε,div

 

By analogy with Chap. 4, Sect. 4.3, Proposition 7.11 (iii) ensures for an in time T := I˜ × , where .I˜ ⊇ I is a bounded interval, and in time extended cylinder .Q T ), with .q − ≤ q˜ ≤ q + , .p− ≤ p˜ ≤ p+ and .s − ≤ ˜ p, ˜ s˜ ∈ Plog (Q extended exponents .q,

268

7 Existence Theory for Irregular Domains

q,p,s q, ˜ p,˜ ˜s  T , that the smoothing operators .S h s˜ ≤ s + in .Q QT ,σ : Xε,div (QT ) → Xε,div (QT ), .h ∈ (0, h2 ), remain well-defined, linear and bounded. Hence, their adjoint operators q, ˜ p,˜ ˜s  ∗ q,p,s h ∗ ∗ .(S QT ,σ ) : Xε,div (QT ) → Xε,div (QT ) , .h ∈ (0, h2 ), defined by

   h h (SQ )∗ x˜ ∗ , x X q,p,s (Q ) := x˜ ∗ , SQ x X q,˜ p,˜ ˜s  T ,σ T ,σ T ε,div ε,div (QT )

 .

q, ˜ p,˜ ˜s  ∗ q,p,s for every .x˜ ∗ ∈ Xε,div (Q T ) , .x ∈ Xε,div (QT ) and .h ∈ (0, h2 ), are also well-defined, linear and bounded. In addition, these adjoint operators inherit the following approximative properties.

Proposition 7.13 Let .q, p, s ∈ Plog (QT ) with .p− , s − > 1 and .q ≥ p in .QT . q, ˜ p,˜ ˜s  ∗ Then, for every .x˜ ∗ ∈ Xε,div (Q T ) , it holds: q, ˜ p,˜ ˜s

∗ T )∗ → X (i) If .∗QT : Xε,div (Q ε,div (QT ) denotes the adjoint operator of the q,p,s q, ˜ p,˜ ˜s  zero extension operator .QT : .Xε,div (QT ) → Xε,div (Q T ), for every .x ∈ q,p,s T \ QT , then (QT ) defined by .QT x := x in .QT and .QT x := 0 in .Q X q,p,s

ε,div

h (SQ )∗ x˜ ∗  ∗QT x˜ ∗ T ,σ

.

in Xε,div (QT )∗ q,p,s

(h → 0) .

T ), respectively, with (ii) For extensions .q, p, s ∈ Plog (Rd+1 ) of .q, ˜ p, ˜ s˜ ∈ Plog (Q − + − + − + .q ≤ q ≤ q , .p ≤ p ≤ p and .s ≤ s ≤ s in .Rd+1 , there exists a constant .cq,p,s > 0 (depending on .q, p, s ∈ Plog (Rd+1 )) such that .

h sup (SQ )∗ x˜ ∗ X q,p,s (QT )∗ ≤ cq,p,s x˜ ∗ X q,˜ p,˜ ˜s ˜ T ,σ (Q

h∈(0,h2 )

ε,div

ε,div

T)

∗

.

Proof Follows analogously to the proof of Proposition 4.15, but now by using Proposition 7.11.  

7.3.4 Generalized Time Derivative in Xε,div (QT )∗ and q,p,s Formula of Integration-by-Parts for Wε,div (QT ) q,p,s

In this subsection, we will introduce a further notion of a generalized time derivative, q,p,s which is now supposed to be a functional in .Xε,div (QT )∗ . We follow the basic procedure in Chap. 3, Sect. 3.6. For the entire section, we further assume that .q, p, s ∈ Plog (QT ), with − − > 1 and .q ≥ p in .Q , are chosen so that .X ˚q,p,s → Y . .p , s T div,−

7.3 Anisotropic Variable Exponent Bochner–Lebesgue Spaces

269

q,p,s

Definition 7.7 A function .x ∈ Xε,div (QT ) possesses a generalized time derivative q,p,s q,p,s in .Xε,div (QT )∗ , if there exists a functional .x ∗ ∈ Xε,div (QT )∗ such that for every ∞ d .φ ∈ C (QT ) , it holds 0  .

− I

(x(t), ∂t φ(t))Y dt = x ∗ , φ X q,p,s (QT ) .

(7.49)

ε,div

q,p,s ∗ ∗ In this case, we define . dx dt := x in .Xε,div (QT ) .

Remark 7.4 The notion of a generalized time derivative from Definition 7.7, similar to the notion of a generalized time derivative from Definition 3.6, is consistent with the standard notion of the distributional time derivative living in .D (QT )d . In fact, q,p,s q,p,s ∗ if .x ∈ Xε,div (QT ) has a generalized time derivative . dx dt ∈ Xε,div (QT ) , then   Proposition 7.10 provides functions .f ∈ Lq (·,·) (QT )d , .F ∈ Lp (·,·) (QT , Md×d sym )  div (f , F , π ) in .X q,p,s (Q )∗ . Therefore, and .π ∈ Ls (·,·) (QT ) such that . dx = J T ε ε,div dt we deduce from (7.49) that .∂t x = f − div(F ) − ∇π in .D (QT )d . It is precisely for d this reason that we employ the same notation . dt in both Definitions 3.6 and 7.7. Lemma 7.2 The generalized time derivative in the sense of Definition 7.7 is unique. Proof Results as in the proof of Lemma 3.7 by referring to the density of .C0∞ (QT )d q,p,s in .Xε,div (QT ) (cf. Proposition 7.12 (iv)).   Definition 7.8 We define the anisotropic variable Bochner–Sobolev space q,p,s .W ε,div (QT )

   dx  q,p,s q,p,s ∗ := x ∈ Xε,div (QT )  ∃ ∈ Xε,div (QT ) . dt q,p,s

Proposition 7.14 The space .Wε,div (QT ) forms a separable, reflexive Banach space, if equipped with the norm  · W q,p,s (QT )

.

ε,div

  d ·  :=  · X q,p,s (QT ) +  .  dt  q,p,s ε,div X (QT )∗ ε,div

 

Proof Follows along the lines of the proof of Proposition 3.21. ˚q,p,s X div,+

˚q,p,s X div,−

˚q,p,s : ˚q,p,s : Because the identity mappings .idX → Y and .idX →Y div,+ div,− are embeddings, which are, based on Corollary 7.3 (iv), also dense, the adjoint oper˚q,p,s )∗ and .(idX ˚q,p,s )∗ are embeddings ˚q,p,s )∗ : Y ∗ → (X ˚q,p,s )∗ : Y ∗ → (X ators .(idX div,+ div,− div,+ div,− as well. Consequently, also the mappings

∗  div := idX ˚q,p,s RY idX ˚q,p,s : e− div,+ div,− . ∗  div q,p,s ˚ ˚q,p,s : e+ := idX RY idX div,− div,+

˚q,p,s → (X ˚q,p,s )∗ , X div,− div,+ ˚q,p,s → (X ˚q,p,s )∗ , X div,+ div,−

(7.50)

270

7 Existence Theory for Irregular Domains

where .RY : Y → Y ∗ again denotes the Riesz isomorphism with respect to Y , are embeddings. This ensures the well-posedness of the following Bochner–Sobolev spaces (cf. Definition 2.19). Definition 7.9 We define the limiting anisotropic Bochner–Sobolev spaces 1,max{q + ,p+ ,s + },max{(q − ) ,(p− ) ,(s − ) }

q,p,s

Wdiv,+ (QT ) := Wediv

.

+

1,min{q − ,p− ,s − },min{(q + ) ,(p+ ) ,(s + ) }

q,p,s

Wdiv,− (QT ) := Wediv −

˚q,p,s , (X ˚q,p,s )∗ ) , (I, X div,+ div,−

∗ ˚ ˚ (I, X div,− , (Xdiv,+ ) ) . q,p,s

q,p,s

q,p,s

Proposition 7.15 (Alternative Characterization of .Wε,div (QT )) For a function q,p,s q,p,s ∗ ∗ .x ∈ X ε,div (QT ) and a functional .x ∈ Xε,div (QT ) , the following statements are equivalent: ∗ ∗ (i) .x ∈ Wε,div (QT ) with . dx dt = x in .Xε,div (QT ) . q,p,s

q,p,s

∈

(ii) .x

q,p,s

Wdiv,− (QT )

with

.

div x d e− dt

=

 ∗ ∗ J˚−1 q,p,s (idX q,p,s (QT ) ) x

Xdiv,+ div,+ + ) ,(p + ) ,(s + ) } q,p,s ∗ min{(q ˚ in .L (I, (Xdiv,+ ) ), where .(idX q,p,s (QT ) )∗ is the adjoint div,+ q,p,s q,p,s operator of .idX q,p,s (QT ) : Xdiv,+ (QT ) → Xε,div (QT ) and .JX ˚q,p,s : div,+ div,+ +  +  +  q,p,s q,p,s min{(q ) ,(p ) ,(s ) } (I, (X ∗ ∗ ˚ .L div,+ ) ) → Xdiv,+ (QT ) is from Proposition 2.20.

Proof Is based on the same argumentation as in the proof of Proposition 3.23.

 

Proposition 7.16 We denote again by  .ET : → the in time extension via reflection operator from Proposition 3.24, where .qT :=  ET p, sT :=  ET s ∈ Plog (Q3T ), .3I := (−T , 2T ), and .Q3T := 3I × . ET q, pT :=  Then, it holds: q,p Xε (QT )

q ,p Xε T T (Q3T )

q,p,s qT ,pT ,sT (i)  .ET : X (Q3T ) is well-defined and Lipschitz continuε,div (QT ) → Xε,div ous with constant 3. q,p,s qT ,pT ,sT (ii)  .ET : W (Q3T ) is well-defined and Lipschitz continε,div (QT ) → Wε,div q,p,s uous with constant 12. To be more precise, if we have that .x ∈ Wε,div (QT )  q,p,s div ∗ q (·,·) (Q )d , .F ∈ with . dx T dt = Jε (f , F , π ) in .Xε,div (QT ) for .f ∈ L   q T ,pT ,sT d×d p (·,·) s (·,·)  (QT , Msym ) and .π ∈ L (QT ), then .ET x ∈ Wε,div (Q3T ) with L

.

d ET x − − − = Jεdiv ( ET f ,  ET F ,  ET π) dt

q ,p ,pT

T T in Xε,div

(Q3T )∗ ,

− − − q  (·,·) (Q )d ,  p (·,·) (Q , Md×d ) and  where  .ET f ∈ L T .ET F ∈ L T .ET π ∈ 3T 3T sym 

LsT (·,·) (Q3T ) are defined as in Proposition 3.24. Proof Follows analogously to the proof of Proposition 3.24 from Proposition 2.25, but now using Propositions 7.10 and 7.15.  

7.3 Anisotropic Variable Exponent Bochner–Lebesgue Spaces

271

log Proposition 7.17 ) with .p− ≥ 2, .s − > 1 and .q ≥ p in .QT .q, p, s ∈ P (QT q,p,s  TLet and let .I∧ := 0, 2ζ . For every .x ∈ Wε,div (QT ) and .h ∈ (0, h2 ), we define the smoothing operator ∞ ∞ d  h x := S h S QT ,σ Q3T ,σ (ET x)|QT ∈ C (I , C0 () ) .

.

q,p,s

Then, for every .x ∈ Wε,div (QT ), it holds: h x)h∈(0,h2 ) ⊆ W (i) .(S QT ,σ ε,div (QT ) with error functionals .(Eh (x))h∈(0,h2 )∩I∧ ⊆ q,p,s Xε,div (QT )∗ such that for every .h ∈ (0, h2 ) ∩ I∧ , it holds q,p,s

   d h h ∗ dET x S + Eh (x) x = (SQT ,σ ) . dt QT ,σ dt

in Xε,div (QT )∗ , q,p,s

T T T h where .(SQ )∗ : Xε,div (Q3T )∗ → Xε,div (QT )∗ denotes the adjoint T ,σ q,p,s qT ,pT ,sT h operator of the smoothing operator .SQ : Xε,div (QT ) → Xε,div (Q3T ) T ,σ (cf. Proposition 7.13). (ii) For extensions .q T , p T , s T ∈ Plog (Rd+1 ) of .qT , pT , sT ∈ Plog (Q3T ), resp., with .q − ≤ q T ≤ q + , .p− ≤ pT ≤ p+ and .s − ≤ s T ≤ s + in .Rd+1 , there exists a constant .cq T ,pT ,s T > 0 such that

q ,p ,s

 h S 

sup

.

 

q,p,s

q,p,s QT ,σ x Wε,div (QT )

h∈(0,h2 )∩I∧

≤ cq T ,pT ,s T xW q,p,s (QT ) , . ε,div

Eh (x)X q,p,s (QT )∗ ≤ cq T ,pT ,s T xX q,p,s (QT ) .

sup

ε,div

h∈(0,h2 )∩I∧

ε,div

(7.51) (7.52)

(iii) It holds

.

h x → x S QT ,σ

in Xε,div (QT )

(h → 0) ,

d h dx SQT ,σ x  dt dt

in Xε,div (QT )∗

(h → 0) ,

in Xε,div (QT )∗

(h → 0) .

Eh (x)  0

q,p,s q,p,s q,p,s

Proof ad (i). Analogous to the argumentation in the proof of Proposition 4.22, we h x)h∈(0,h2 ) ⊆ W q,p,s (QT ) and that for every .φ ∈ C ∞ (QT )d observe that .(S QT ,σ ε,div 0 and .h ∈ (0, h2 ) ∩ I∧ , it holds   d h SQT ,σ x, φ q,p,s dt Xε,div (QT ) (7.53) .  

h  ( ET x)(t), ∂t (SQ3T ,σ ) (QT φ) (t) Y dt . =− 3I

272

7 Existence Theory for Irregular Domains

h Unfortunately, we have that .((SQ ) (QT φ))h∈(0,h2 )∩I∧ ⊆ C0∞ (Q3T )d (cf. 3T ,σ h φ)h∈(0,h2 )∩I∧ ⊆ C0∞ (Q3T )d because Remark 4.11). However, we have that .(SQ T ,σ h h ∞ d .(S QT ,σ φ)h∈(0,h2 )∩I∧ ⊆ C0 (QR ) with .supp(SQT ,σ φ) ⊆ [−ζ h, T + ζ h] × h ⊆ Q3T for all .h ∈ I∧ (cf. Proposition 7.11

h (i)). Hence, we proceed in (7.53) with  h  ( replacing .∂t (SQ ) φ) by .∂t S Q T QT ,σ φ and paying for this with the errors 3T ,σ  ∞ d d .(Eh (x))h∈(0,h2 )∩I∧ ⊆ D (QT ) , for every .h ∈ (0, h2 ) ∩ I∧ and .φ ∈ C (QT ) 0 defined by

h   h .Eh (x), φ C ∞ (Q )d :=  φ − (SQ ) (QT φ) L2 (Q )d . ET x, ∂t SQ T T ,σ 3T ,σ 3T

0

In this way, also exploiting that  .ET x and Proposition 7.13, from (7.53) for we deduce that    

h  d h SQT ,σ x, φ ( ET x)(t), ∂t SQ =− φ (t) Y dt . ,σ T q,p,s dt 3I X (QT )

qT ,pT ,sT ∈ Wε,div (Q3T ) (cf. Propositions every .φ ∈ C0∞ (QT )d and .h ∈ (0, h2 )

7.16) ∩ I∧ ,

(7.54)

ε,div

+ Eh (x), φ C ∞ (QT )d 0    dET x h , SQT ,σ φ +Eh (x), φ C ∞ (QT )d = 0 q ,p ,s dt X T T T (Q3T ) ε,div

     h ∗ dET x ,φ = (SQ ) +Eh (x), φ C ∞ (QT )d . T ,σ 0 q,p,s dt X (QT ) ε,div

By proceeding as for the estimates (4.91)–(4.99) in the proof of Proposition 4.22, also making use of .p− ≥ 2, i.e., employing Corollary 2.1, we then arrive for every ∞ d .φ ∈ C (QT ) and .h ∈ (0, h2 ) ∩ I∧ at 0  

 Eh (x), φ ∞  C (QT )d ≤ c xL2 (QT )d φL2 (QT )d + ∇φL2 (QT )d×d 0

.

≤ 4c (1 + |QT |)2 xX q,p,s (QT ) φX q,p,s (QT ) . ε,div

(7.55)

ε,div

Therefore, on the basis of (7.55) and of the density of .C0∞ (QT )d in .Xε,div (QT ) (cf. Proposition 7.12 (iv)), the functionals .(Eh (x))h∈(0,h2 )∩I∧ ⊆ D (QT )d admit unique and, therefore, not relabeled, continuous extensions to functionals q,p,s ∗ .(Eh (x))h∈(0,h2 )∩I∧ ⊆ X ε,div (QT ) satisfying q,p,s

.

sup h∈(0,h2 )∩I∧

Eh (x)X q,p,s (QT )∗ ≤ 4c (1 + |QT |)2 xX q,p,s (QT ) . ε,div

ε,div

(7.56)

With the renewed exploitation of the density of .C0∞ (QT )d in .Xε,div (QT ), (7.54) then extends to (i). q,p,s

7.3 Anisotropic Variable Exponent Bochner–Lebesgue Spaces

273

ad (ii). The inequality (7.52) is just (7.56). By applying Proposition 7.11 (iii), we further find that sup h∈(0,h2 )∩I∧

≤

.

h x q,p,s S QT ,σ X (QT ) ε,div

sup h∈(0,h2 )∩I∧

h SQ ( ET x)X qT ,pT ,sT (Q3T ) 3T ,σ

(7.57)

ε,div

≤ max{cq , 2(1 + |QT |)cp , cs } ET xX qT ,pT ,sT (Q3T ) . ε,div

By resorting to Proposition 7.13 (ii), we observe, apart from that, that sup h∈(0,h2 )∩I∧ .

      h ∗ dET x  (S )  QT ,σ dt 

q,p,s

Xε,div (QT )∗

   d ET x   . ≤ cq T ,pT ,s T   dt  qT ,pT ,sT X (Q3T )∗

(7.58)

ε,div

Eventually, combining (7.56)–(7.58) and (i), also using the Lipschitz continuity q,p,s qT ,pT ,sT of the extension operator  .ET : W (Q3T ) (cf. Proposiε,div (QT ) → Wε,div tion 7.16), we conclude that (7.51) holds. ad (iii). Follows on the basis of the same argumentation as in the proof of Proposition 4.22 (iii), but now by making use of Propositions 7.13 and 7.14.   The usage of Proposition 7.17 allows us to derive the validity of a formula q,p,s of integration-by-parts for .Wε,div (QT ), which in retrospect was our underlying motivation for the introduction of anisotropic variable Bochner–Lebesgue spaces. Proposition 7.18 Let .q, p, s ∈ Plog (QT ) with .p− ≥ 2, .s − > 1 and .q ≥ p in .QT . Then, it holds: q,p,s

(i) Each function .x ∈ Wε,div (QT ) possesses a unique continuous representation 0 .x c ∈ Y (QT ). q,p,s (ii) For every .x, y ∈ Wε,div (QT ) and .t, t  ∈ I with .t  ≤ t, it holds  t .

t

 s=t

dx (s), y(s) ds = (x c (s), y c (s))Y s=t  dt ˚q,p,s (s) X ε,div   t dy (s), x(s) ds . − dt ˚q,p,s (s) t X ε,div

274

7 Existence Theory for Irregular Domains q,p,s

(iii) The from (i) resulting mapping .(·)c : Wε,div (QT ) → Y 0 (QT ) is a strongq,p,s weak-embedding, i.e., for a sequence .(x n )n∈N ⊆ Wε,div (QT ) from q,p,s

xn → x

in Xε,div (QT )

(n → ∞) ,

dx n dx  dt dt

in Xε,div (QT )∗

(n → ∞) ,

.

q,p,s

it follows that .(x n )c → x c in .Y 0 (QT ) .(n → ∞). In particular, q,p,s 0 .(·)c : W ε,div (QT ) → Y (QT ) is an embedding. Proof Follows along the lines of the proof of Proposition 4.23, but now using Proposition 7.17.  

7.4 First Parabolic Compensated Compactness Principle q,p,s

Aided by the formula of integration-by-parts for .Wε,div (QT ) (cf. Proposition 7.18), we are, in the end, in a position to give a proof of a PCCP for solenoidal variable Bochner–Lebesgue spaces without the variable exponent .p ∈ Plog (QT ) being subject to the artificial relation .p+ < (p− )• . We emphasize that the following proposition is the first PCCP for solenoidal generalized evolution equations that dispenses with such a relation but, nevertheless, still needs to assume that .p− ≥ 2. This lower bound will be relaxed in a second PCCP (cf. Proposition 8.4), later in Chap. 8. Proposition 7.19 (First Parabolic Compensated Compactness Principle) Let  ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, .QT :=  I × , and log .p, q := p∗ − ε, r := max{p◦ , p} ∈ P (QT ), with .p− ≥ 2, .ε ∈ 0, d2 p− and q,r n n − ∞ .p◦ ∈ [2, (p )∗ ). Furthermore, let .un ∈ Wε,σ (Q ) ∩ H (Q ), .n ∈ N, where T T ∗ σ .(n )n∈N is a non-decreasing exhaustion of ., .(un )n∈N = (Jε (f n , F n ))n∈N ⊆ ∗ be sequences with the following ˚q,p ˚εq,p (QT )∗ , and .(v ∗ ) V n n∈N ⊆ Vε (QT ) properties: .

(i)

.

dσ u n dt

˚εq,r (Qn )∗ for every .n ∈ N.7 + ∗n u∗n = ∗n v ∗n in .V T

∗ ˚εq,p (QT ) .(n → ∞), .n un  (ii) .n un  u in .V u in .H∞ (QT ) .(n → ∞) and .(n un )(t)  u(t) in H .(n → ∞) for almost every .t ∈ I , where .u ∈ ˚εq,p (QT ) ∩ H∞ (QT ) has a representation .uω ∈ C 0 (I , H ). V ω

7 Here, .n

∈ N.

∗ ˚q,r n ∗ ˚εq,r (Qn ) )∗ n v ∗ we are a little imprecise and identify .∗n v ∗n := (idV n ∈ Vε (QT ) for every T

7.4 First Parabolic Compensated Compactness Principle

275





(iii) .f n  f in .Lq (·,·) (QT )d .(n → ∞), .F n  F in .Lr (·,·) (QT , Md×d sym )  (·,·) q,p d×d p ∗ ∗ ˚ε (QT )∗ .(n → ∞), where .F ∈ L (QT , Msym ), and .v n → v in .V .(n → ∞). ∗

(iv) .μn := a n dtdx  μ = a 0 dtdx + μs in .M(QT ) .(n → ∞), where 1 s s ⊥ dtdx, and .(a ) .a 0 ∈ L (QT , R≥0 ) and .μ ∈ M(QT ), with .μ n n∈N := (f n · n un + F n : ε(n un ))n∈N ⊆ L1 (QT , R≥0 ) is bounded. Then, it holds .a 0 ≤ a := f · u + F : ε(u) almost everywhere in .QT . Remark 7.5 The treatment of the variable exponent .r := max{p◦ , p} ∈ Plog (QT ) with a parameter .p◦ ∈ [2, (p− )∗ ) is merely preparatory work for Chap. 8, in which we will examine the solvability of solenoidal generalized evolution equations (cf. Definition 4.9) for variable exponents .p ∈ Plog (QT ) with a lower bound .p− < 2. For the scope of this chapter, the simplified case .p◦ = 2, i.e., the case in which log .r = p ∈ P (QT ), is already perfectly adequate. Proof of Proposition 7.19 The main components of the proof are the pressure reconstruction theorem (cf. Proposition 6.3) and the formula of integration-by-parts for anisotropic variable Bochner–Lebesgue spaces (cf. Proposition 7.18). We divide the proof of into three main parts: 1. Localization of assumptions: To begin with, we infer from (ii) by the use of p∗ (·,·)−δ (Q )d and Proposition 3.8 and Proposition 3.11 T  that .(n un )n∈N ⊆ L p (·,·)−δ (Q )d for every .δ ∈ 0, (p − ) − 1 , and for all .δ ∈ (0, (p − ) − 1] .u ∈ L ∗ T ∗ ∗  n un → u

.

(n un )(t) → u(t)

in Lp∗ (·,·)−δ (QT )d

(n → ∞) , .

in H

(n → ∞)

(7.59) for a.e. t ∈ I .

(7.60)

Note also that (7.60) is a consequence of (7.59)8 since .(p− )∗ > 2 for .p− ≥ 2.  On the other hand, Lemma 4.4 provides sequences .(g n )n∈N ⊆ Lq (·,·) (QT )d and  p (·,·) (Q , Md×d ), satisfying .v ∗ = J σ (g , G ) in .V ˚εq,p (QT )∗ .(Gn )n∈N ⊆ L T n n sym n ε   for all .n ∈ N, and functions .g ∈ Lq (·,·) (QT )d and .G ∈ Lp (·,·) (QT , Md×d sym ), ˚εq,p (QT )∗ , such that satisfying .v ∗ = Jεσ (g, G) in .V gn → g .

Gn → G



in Lq (·,·) (QT )d in L

p (·,·)

(QT , Md×d sym )

(n → ∞) , (n → ∞) .

(7.61)

We fix an arbitrary point .(t0 , x0 ) ∈ QT . Then, based on the uniform continuity of .p : QT → [2, ∞)9 and .(·)∗ : [1, ∞) → (1, ∞) (cf. Definition 3.3), there exists a cylinder .Q := J × G satisfying .(t0 , x0 ) ∈ Q, where .G ⊂⊂ 

8 After

possibly switching to a subsequence.

276

7 Existence Theory for Irregular Domains

− := is a domain with .∂G ∈ C 1,1 and .J ⊂⊂ I an interval, such that .pQ + + inf(t,x) ∈Q p(t, x), .pQ := sup(t,x) ∈Q p(t, x) and .qQ := sup(t,x) ∈Q q(t, x) satisfy + + + − pQ ≤ qQ = (pQ )∗ − ε < (pQ )∗ ,

.

(7.62)

where the first inequality comes from .q ≥ p in .QT , which is based on ε ∈ (0, d2 p− ] (cf. Remark 5.7). Without loss of generality, we may assume, on the basis of (7.60), for .tinf := inf(J ), that

.

(n un )ω (tinf ) → uω (tinf )

.

in H

(n → ∞) ,

(7.63)

where .((n un )ω )n∈N ⊆ Cω0 (I , H ) denote weakly continuous representations of q,r the functions .un ∈ Wε,σ (QnT ) ∩ H∞ (QnT ), .n ∈ N, whose existence, thanks to Proposition 4.19 (i), is guaranteed. Otherwise, we substitute J by a smaller interval .J˜ ⊆ J with .t0 ∈ J˜ such that (7.63) is satisfied. Moreover, we define the + + + − := max{p◦ , pQ exponent .rQ }, i.e., .rQ < (pQ )∗ (cf. (7.62)), and the functions . + 

hn := g n − f n ∈ L(qQ ) (Q)d , + 

H n := Gn −F n ∈ L(rQ ) (Q, Md×d sym ) , n ∈ N , + 

(7.64)

h := g − f ∈ L(qQ ) (Q)d , + 

H := G − F ∈ L(pQ ) (Q, Md×d sym ) . Apparently, there exists some integer .n0 ∈ N such that .G ⊂⊂ n for every n ∈ N with .n ≥ n0 . Then, we can test (i) for every .n ∈ N, where .n ≥ n0 , with ∞ (Q)d . In doing so, for every .n ∈ N with .n ≥ n and an arbitrary .φ ∈ C0,div 0 ∞ d .φ ∈ C (Q) , we observe that 0,div .

 −

un (s, y) · ∂t φ(s, y) dsdy Q



.

=

(7.65) hn (s, y) · φ(s, y)+H n (s, y) : ∇φ(s, y) dsdy .

Q

we exploit that .p ∈ Plog (QT ) admits an extension .p ∈ Plog (Rd+1 ) (cf. Proposition 2.10), i.e., .p|QT = p in .QT , which consequently also satisfies .p ∈ C 0 (QT ) and, therefore, implies the uniform continuity of .p : QT → [2, ∞).

9 Here,

7.4 First Parabolic Compensated Compactness Principle

277

+  +  On the other hand, we infer from (7.61), (iii) and .(pQ ) ≥ (rQ ) , i.e., employing Corollary 2.1, that + 

in L(qQ ) (Q)d

hn  h .

Hn  H

in L

+  (rQ )

(n → ∞) ,

(Q, Md×d sym )

(7.66)

(n → ∞) .

As a result of (7.66) and (ii), we can pass for .n → ∞ in (7.65) and conclude for ∞ (Q)d every .φ ∈ C0,div  −

u(s, y) · ∂t φ(s, y) dsdy Q



.

(7.67) h(s, y) · φ(s, y) + H (s, y) : ∇φ(s, y) dsdy .

= Q

2. Local pressure reconstruction: In this step, we apply the pressure reconstruction theorem (cf. Proposition 6.3) to (7.65) and (7.67). To this end, we identify for the rest of the proof .(un |J ×G )n∈N ∈ L∞ (J, L2 (G)d ) and .u|J ×G ∈ L∞ (J, L2 (G)d ) with the representations .((n un )ω |J ×G )n∈N 0 2 d 2 d 0 .⊆ Cω (J , L (G) ) and .uω | J ×G ∈ Cω (J , L (G) ), respectively. Then, for any .n ∈ N with .n ≥ n0 , we obtain, by applying Proposition 6.3 with constant exponents, functions + 

+ 

(q + )

π 1,n ∈ L(qQ ) (J, W 1,(qQ ) (G) ∩ L0 Q (G)) ,

.

(r + )

+ 

π 2,n ∈ L(rQ ) (J, L0 Q (G)) , π h,n ∈ Cω0 (J , W 1,2 (G) ∩ L20 (G)) , + 

+ 

(q + )

π 1 ∈ L(qQ ) (J, W 1,(qQ ) (G) ∩ L0 Q (G)) , (pQ+ )

+ 

π 2 ∈ L(pQ ) (J, L0

(G)) , π h ∈ Cω0 (J , W 1,2 (G) ∩ L20 (G)) ,

with .(π h,n (t)) = (π h (t)) = 0 in .(W01,2 ())∗ for all .t ∈ J and 1,2 (G), such that for every .n ∈ N with .n ≥ n .π h,n (tinf ) = π h (tinf ) = 0 in .W 0 and .φ ∈ C0∞ (Q)d , it holds  .

−

[un (s, y) + ∇π h,n (s, y)] · ∂t φ(s, y) dsdy Q

 [hn (s, y) + ∇π 1,n (s, y)] · φ(s, y) dsdy

= Q



+

[H n (s, y) + π 2,n (s, y)Id ] : ∇φ(s, y) dsdy , Q

(7.68)

278

7 Existence Theory for Irregular Domains

and likewise for every .φ ∈ C0∞ (Q)d , it holds  .

−

[u(s, y) + ∇π h (s, y)] · ∂t φ(s, y) dsdy Q



=

[h(s, y) + ∇π 1 (s, y)] · φ(s, y) dsdy

(7.69)

Q



+

[H (s, y) + π 2 (s, y)Id ] : ∇φ(s, y) dsdy . Q

In addition, for every .n ∈ N with .n ≥ n0 , it holds (cf. Proposition 6.3) π 1,n 

.

+ 

+ 

L(qQ ) (J,W 1,(qQ ) (G))

π 2,n 

+ 

L(rQ ) (Q)

≤ c1,Q hn 

+ 

L(qQ ) (Q)d

≤ c2,Q H n 

+ 

(7.70)

,

L(rQ ) (Q)d×d

,

π h,n (t)W 1,2 (G) ≤ ch,G un (t) − un (tinf )L2 (G)d

for all t ∈ J ,

where .c1,Q , c2,Q , ch,G > 0 are independent of .n ∈ N. Next, we recall the construction in the proof of Proposition 6.3 and realize that we can assume that the above functions have the forms  π 1,n =  ◦ eqQ+ (hn ) ,

.

π 2,n = ( ◦ div)(H n ) ,  π 1 =  ◦ eqQ+ (h) , π 2 = ( ◦ div)(H ) ,

π h,n = ( ◦ e2 )(un (tinf ) − un ) ,

n ∈ N,

π h = ( ◦ e2 )(u(tinf ) − u) ,

where the operators10 +

+

+

+ 

+

 ˚rQ ,rQ )∗ ) → L(rQ ) (J, L(rQ ) (G)) ,  ◦ div : L(rQ ) (J, (X 0 .

+ 

+ 

+ 

(q + )

(7.71)

 ◦ eqQ+ : L(qQ ) (Q)d → L(qQ ) (J, W 1,(qQ ) (G) ∩ L0 Q (G)) ,  ◦ e2 : Cω0 (J , L2 (G)d ) → Cω0 (J , W 1,2 (G) ∩ L20 (G)) ,

are defined as in Corollary 6.1 and Lemma 6.2. Due to the linearity of the operators in (7.71), the boundedness of the operators .(7.71)1,2 (cf. Corollary 6.1),

10 For .π 2 + 

+ 

+ ) (pQ

∈ L(pQ ) (J, L0

+ ) (pQ

L(pQ ) (J, L0

+ 

+

+

˚pQ ,pQ )∗ ) → (G)), we exploit that . ◦ div : L(pQ ) (J, (X

(G)) is well-defined.

7.4 First Parabolic Compensated Compactness Principle

279

and the boundedness of . ◦ e2 : L∞ (J, L2 (G)d ) → L∞ (J, W 1,2 (G) ∩ L20 (G)) (cf. Corollary 6.1 (ii)), we conclude from (ii), (7.63) and (7.66) to the convergences .

π 1,n  π 1 π 2,n  π 2 ∗

π h,n  π h

+ 

(q + )

+ 

in L(qQ ) (J, W 1,(qQ ) (G) ∩ L0 Q (G)) + 

(r + )

(n → ∞) ,

in L(rQ ) (J, L0 Q (G))

(n → ∞) ,

in L∞ (J, W 1,2 (G) ∩ L20 ())

(n → ∞) .

(7.72)

The linearity and boundedness of .𝒫 ◦ e2 : L2 (G)d → W 1,2 (G) ∩ L20 (G) (cf. Proposition 6.2 (ii)) yields a constant .ch,G > 0 (not depending on .n ∈ N or .t ∈ J ) such that for every .t ∈ J and .n ∈ N with .n ≥ n0 , it holds π h,n (t) − π h (t)W 1,2 (G) = (𝒫◦ e2 )(un (tinf ) − un (t)) − (𝒫◦ e2 )(u(tinf ) − u(t))W 1,2 (G) = (𝒫◦ e2 )(un (tinf ) − un (t) − (u(tinf ) − u(t)))W 1,2 (G)

.

(7.73)

≤ ch,G un (t) − u(t) − (un (tinf ) − u(tinf ))L2 (G)d ≤ ch,G un (t) − u(t)L2 (G)d + ch,G un (tinf ) − u(tinf )L2 (G)d . Next, we fix an open subset .G ⊂⊂ G, with .x0 ∈ G , and set .Q := J × G . Recalling that the functions .π h,n (t), π h (t) ∈ W 1,2 (G), .n ∈ N, are harmonic for every .t ∈ J , it follows by the well-known local regularity theory (cf. [77, Thm. 8.24]) that for every .n ∈ N with .n ≥ n0 , .s ∈ (1, ∞) and .t ∈ J .

π h,n (t) − π h (t)W 2,s (G ) ≤ c(s, d, G )π h,n (t) − π h (t)L2 (G) ,

(7.74)

where .c(s, d, G ) > 0 denotes a constant depending only on .d ∈ N, .s ∈ (1, ∞) and .dist(G , ∂G) > 0. Then, making use of (7.73) together with (7.60) and (7.63) in (7.74) shows that π h,n (t) → π h (t)

.

in W 2,s (G )

(n → ∞)

for a.e. t ∈ J .

(7.75)

On the other hand, we find, exploiting .supn∈N π h,n L∞ (J,L2 (G)) < ∞ (cf. (7.72)3 ), that the right-hand side in (7.74) is uniformly bounded with respect to .n ∈ N and .t ∈ J , i.e., for every .s ∈ (1, ∞), we have that

.

.

sup π h,n − π h L∞ (J,W 2,s (G ))

n∈N

≤ c(s, d, G ) sup π h,n − π h L∞ (J,L2 (G)) < ∞ . n∈N

280

7 Existence Theory for Irregular Domains

Therefore, for every .s ∈ (1, ∞), Lebesgue’s theorem on dominated convergence further gives us that π h,n → π h

.

in Ls (J, W 2,s (G ))

(n → ∞) .

(7.76)

3. Local limit superior inequality: We introduce .w := u + ∇π h as well as .w n := un + ∇π h,n for every .n ∈ N. Apart from that, we fix an arbitrary function .η ∈ C0∞ (Q ). Then, if test (7.68) for arbitrary .φ ∈ C0∞ (Q )d with ∞ ∞  d  d .φη ∈ C (Q ) , then for every .n ∈ N with .n ≥ n0 and .φ ∈ C (Q ) , we 0 0 observe that  − wn (s, y) · [∂t φ(s, y)η(s, y) + φ(s, y)∂t η(s, y)] dsdy Q



=

Q

.

[hn (s, y) + ∇π 1,n (s, y)] · φ(s, y)η(s, y) dsdy

(7.77)



+

Q

[H n (s, y) + π 2,n (s, y)Id ]

: [∇φ(s, y)η(s, y) + ∇η(s, y) ⊗ φ(s, y)] dsdy . + 

Rearranging (7.77) and using the definitions of .(hn )n∈N ⊆ L(qQ ) (Q)d and (r + ) d×d (cf. (7.64)) in doing so, we furthermore conclude for .(H n )n∈N ⊆ L Q (Q) every .n ∈ N with .n ≥ n0 and .φ ∈ C0∞ (Q )d that  .− η(s, y)w n (s, y) · ∂t φ(s, y) dsdy (7.78) Q



=

Q

∂t η(s, y)w n (s, y) · φ(s, y) dsdy



+  +  +

Q

Q

Q

η(s, y)[g n (s, y) − f n (s, y) + ∇π 1,n (s, y)] · φ(s, y) dsdy [(Gn (s, y) − F n (s, y) + π 2,n (s, y)Id )∇η(s, y)] · φ(s, y) dsdy η(s, y)[Gn (s, y) − F n (s, y) + π 2,n (s, y)Id ] : ∇φ(s, y) dsdy .

+ + + + − + − := max{qQ We introduce .σQ , rQ }, i.e., .σQ < (pQ )∗ because of .rQ < (pQ )∗ and p (·,·)−δ d (7.62). Hence, by the combination of (7.76) and that .(n un )n∈N ⊆ L ∗ (Q T)   p (·,·)−δ d − and .u ∈ L ∗ (QT ) for all .δ ∈ 0, (p )∗ − 1 with (7.59), it is readily seen +

+

that .(w n )n∈N ⊆ LσQ (Q )d and .w ∈ LσQ (Q )d with wn → w

.

+

in LσQ (Q )d

(n → ∞) .

(7.79)

7.4 First Parabolic Compensated Compactness Principle

281 +

Since for every .n ∈ N, we have that .div(ηw n ) = ∇η · wn ∈ LσQ (Q ),11 + +  + + σ+ (σ + )  d  d .∂t ηw n ∈ L Q (Q ) ⊆ L Q (Q ) , because .σ Q ≥ (σQ ) due to .σQ ≥ pQ ≥ 2, and sym ∈ Lr(·,·) (Q )d×d , which is based on .σ + ≥ r + , .ε(ηw n ) = ηε(w n ) + [∇η ⊗ w n ] Q Q σ + ,r,σQ+

Q it can easily be read from (7.78) that .(ηwn )n∈N ⊆ Wε,div

dηwn = Jεdiv (∂t ηwn + ηg n + Gn ∇η, ηGn , 0) dt + Jεdiv (π 2,n ∇η − F n ∇η + η∇π 1,n , 0, ηπ 2,n )

.

− Jεdiv (ηf n , ηF n , 0) =: Aηn + Bηn − Cηn

(Q ) with ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

σ + ,r,σQ+

Q in Xε,div

(Q )∗ .

(7.80) +

+

+ 

Similarly, based on .div(ηw) = ∇η · w ∈ LqQ (Q ), .∂t ηw ∈ LqQ (Q )d ⊆ L(qQ ) (Q )d , and .ε(ηw) = ηε(w) + [∇η ⊗ w]sym ∈ Lp(·,·) (Q )d×d , we derive from (7.69), or by q + ,p,qQ+

Q passing for .n → ∞ in (7.78), that .ηw ∈ Wε,div

(Q ) with ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

dηw = Jεdiv (∂t ηw + ηg + G∇η, ηG, 0) dt + Jεdiv (π 2 ∇η − F ∇η + η∇π 1 , 0, ηπ 2 )

.

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

− Jεdiv (ηf , ηF , 0) =: Aη + Bη − Cη

q + ,p,qQ+

Q in Xε,div

(Q )∗ .

(7.81) + 

+ 



(σQ ) (Q ) The (weak) continuity of .Jεdiv : L(σQ ) (Q )d × Lr (·,·) (Q , Md×d sym ) × L σ + ,r,σ +

Q Q (Q )∗ , (cf. Proposition 7.10), (7.79), (7.61), (iii), and that → Xε,div +  +  +  .(σ ) = min{(q ) , (r ) } yield Q Q Q

∗  + ,p,σ + σQ η Q  Aηn → idXε,div (Q ) A

Q in Xε,div

∗  + ,r,σ + σQ η Q Cηn  idXε,div (Q ) C

Q in Xε,div

.

σ + ,p,σQ+ σ + ,r,σQ+

(Q )∗

(n → ∞) , .

(7.82)

(Q )∗

(n → ∞) ,

(7.83)

σ + ,p,σ +

Q Q where we also have used for (7.82) that .(An )n∈N ⊆ Xε,div (Q )∗ because  p (·,·) (Q , Md×d ). In addition, by combining (ii), (7.76) and (7.79), .(ηGn )n∈N ⊆ L sym

η

σ + ,p,σQ+

Q we find that .(ηwn )n∈N ⊆ Xε,div

11 Here, .Q

.

(Q ) is bounded. Consequently, the reflexivity

for every .n ∈ N, we exploit that .div(wn ) = div(un ) + (π h,n ) = 0 almost everywhere in

282

7 Existence Theory for Irregular Domains σ + ,p,σQ+

Q of .Xε,div

(Q ) (cf. Proposition 7.9), (7.79) and (7.76) yield σ + ,p,σQ+

.

ηw n  ηw

Q in Xε,div

ηw n → ηw

in LσQ (Q )d

div(ηw n ) → div(ηw)

in LσQ (Q )

(Q )

+ +

+

+

σ ,r,σQ

η∇π h,n → η∇π h

Q in Xε,div

(Q )

(n → ∞) , .

(7.84)

(n → ∞) , .

(7.85)

(n → ∞) , .

(7.86)

(n → ∞) .

(7.87)

Then, by the combination of (7.82) with (7.84) and (7.83) with (7.87), respectively, one finds that Aηn , ηwn

.

+ σ + ,r,σQ

Q Xε,div

Cηn , η∇π h,n

(Q )

+ ,r,σ + σQ Q Xε,div (Q )

→ Aη , ηw

+ q + ,p,qQ

Q Xε,div

→ Cη , η∇π h

(Q )

+ q + ,p,qQ

Q Xε,div

(Q )

(n → ∞) , .

(7.88)

(n → ∞) .

(7.89)

Moreover, taking into account (7.85), (iii) and .(7.72)1,2 , in particular, that +  +  +  (σQ ) = min{(qQ ) , (rQ ) }, we obtain

.

Bηn , ηwn

+ σ + ,r,σQ

Q Xε,div

(Q )

= (2π 2,n ∇η + F n ∇η + η∇π 1,n , ηwn ) → (2π 2 ∇η + F ∇η + η∇π 1 , ηw)

.

+

LσQ (Q )d

+

(7.90)

LqQ (Q )d

= Bη , ηw

+ q + ,p,qQ

Q Xε,div

(Q )

(n → ∞) .

+ 

Using (7.89), .F n  F in .L(rQ ) (Q , Md×d sym ) .(n → ∞) (cf. (iii)) and .un → u in +

LrQ (Q )d .(n → ∞) (cf. (7.59)), we observe that   . lim sup a n (t, x)η(t, x)2 dtdx = lim sup Jεdiv (ηf n , ηF n , 0), ηun .

n→∞

Q

n→∞

− (F n , ∇η ⊗ (ηun )) 

 +

LrQ (Q )d×d

= lim sup Cηn , ηwn − η∇π h,n n→∞

− (F n ∇η, ηun ) n→∞

+

LrQ (Q )d + σ + ,r,σQ

Q Xε,div

+ σ + ,r,σQ

Q Xε,div



= lim sup Cηn , ηwn

+ σ + ,r,σQ

Q Xε,div

(Q )

(Q )

(Q )

7.4 First Parabolic Compensated Compactness Principle

283

− Cη , η∇π h

+ q + ,p,qQ

Q Xε,div

(Q )

− (F ∇η, ηu)Lp(·,·) (Q )d .

(7.91) σ + ,r,σQ+

Q Furthermore, by resorting to the formulas of integration by parts for .Wε,div

(Q )

qQ+ ,p,qQ+ and .Wε,div (Q ) (cf. Proposition 7.18 (ii)), respectively, which is allowed since + + + + − .σ Q = max{rQ , qQ } ≥ pQ and .pQ ≥ 2, defining .tsup := sup(J ) and recalling that .tinf := inf(J ), we obtain for every .n ∈ N with .n ≥ n0 continuous representations 2  d 0 .(ηw n )c , (ηw)c ∈ C (J , L (G ) ), such that for every .n ∈ N with .n ≥ n0 , it holds



 dηwn 1 1 2 2 , ηwn σ + ,r,σ + . = (ηw n )c (tsup ) 2  )d − (ηw n )c (tinf )L2 (G )d = 0 , L (G Q Q dt 2 2 Xε,div (Q )   dηw 1 1 , ηw q + ,p,q + = (ηw)c (tsup )2L2 (G )d − (ηw)c (tinf )2L2 (G )d = 0 , Q Q dt 2 2 X (Q ) ε,div

(7.92) where we have used a constant .τ > 0 such that .η(t) = 0  that there exists for every .t ∈ R \ tinf + τ, tsup − τ , from which it follows that .(ηwn )c (tsup ) = (ηw)c (tsup ) = (ηwn )c (tinf ) = (ηw)c (tinf ) = 0 in .L2 (G )d for every .n ∈ N with .n ≥ n0 . From (7.80), (7.81), (7.88), (7.90) and (7.92) we further deduce that  dηw n , ηw n σ + ,r,σ + = − Q Q dt Xε,div (Q )   dηw , ηw q + ,p,q + → Aη + Bη − Q dt X Q (Q ) 

Cηn , ηwn

+ σ + ,r,σQ

Q Xε,div

.

(Q )

Aηn

+ Bηn

(7.93)

ε,div

η

= C , ηw

+ q + ,p,qQ

Q Xε,div

(Q )

(n → ∞) .

Thus, if we insert (7.93) into (7.91), we find that the limit superior with respect to n → ∞ in (7.91) is actually a limit given by

.

 lim

n→∞ .

Q

a n (t, x)η(t, x)2 dtdx = Cη , ηw − η∇π h

+ q + ,p,qQ

Q Xε,div

− (F ∇η, ηu)Lp(·,·) (Q )d  = a(t, x)η(t, x)2 dtdx .

(Q )

(7.94)

Q

Taking into account (7.94), (iv) and .JM μs , η2 C0 (QT ) ≥ 0 since .μs ∈ M(QT ) is a positive measure, where .JM : M(QT ) → C0 (QT )∗ denotes the Riesz

284

7 Existence Theory for Irregular Domains

representation operator from Proposition 7.4, also recalling Definition 7.4, we obtain   . a 0 (t, x)η(t, x)2 dtdx ≤ a 0 (t, x)η(t, x)2 dtdx + JM μs , η2 C0 (QT ) Q

Q

= JM μ, η2 C0 (QT ) = lim JM μn , η2 C0 (QT ) n→∞  = lim a n (t, x)η(t, x)2 dtdx

(7.95)

n→∞ Q



=

Q

a(t, x)η(t, x)2 dtdx ,

Next, we choose .r > 0 such that .Brd+1 (t0 , x0 ) ⊂⊂ Q and .η = φ r (t0 − ·, x0 − ·) ∈ ffl C0∞ (Brd+1 (t0 , x0 )), where .φ ∈ C0∞ (B1d+1 (0)) ∩ SM(Rd+1 ) with . B d+1 (0) 1 φ(t, x)2 dtdx = 1 and .φ r ∈ C0∞ (Brd+1 (0)) for every .(t, x) ∈ Rd+1 is defined by .φ r (t, x) := φ( rt , xr ). Then, (7.95) for every sufficiently small .r > 0 reads

Brd+1 (t0 ,x0 )

a 0 (t, x)φ r (t0 −t, x0 −x)2 dtdx (7.96)

.

≤

a(t, x)φ r (t0 −t, x0 −x) dtdx . 2

Brd+1 (t0 ,x0 )

Proposition 9.1 (i) provides for a null sequence .(rn )n∈N ⊆ R>0 that the estimate (7.96) with .r = rn converges for almost every .(t0 , x0 ) ∈ QT to the estimate .a 0 (t0 , x0 ) ≤ a(t0 , x0 ) if we pass for .n → ∞. In other words, we have just proved   that .a 0 ≤ a almost everywhere in .QT . If we consider a constant exhaustion .(n )n∈N in Proposition 7.19, i.e., .n =  for all .n ∈ N, then we can omit the zero trace requirement on the sequence q,r n n ∞ .(un )n∈N ⊆ Wε,σ (Q ) ∩ H (Q ), .n ∈ N, which solely enabled us to extend this T T q,p n ˚ε (Q ) → V ˚εq,p (QT ), .n ∈ N, from .Qn , .n ∈ N, respectively, sequence via .n : V T T to the whole time-space cylinder .QT without suffering any losses, i.e., without generating constants that might depend critically on .n ∈ N. Corollary 7.4 Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, log .QT := I × , and .p, q := p∗ − ε, r := max{p◦ , p} ∈ P (QT ), with .p− ≥ 2,  2 − − and .p◦ ∈ [2, (p )∗ ). Furthermore, let .(un )n∈N ⊆ Y ∞ (QT ), with .ε ∈ 0, p d   p (·,·)−δ (Q )d for every .δ ∈ 0, (p − ) − 1 , .(ε(u )) r(·,·) .(un )n∈N ⊆ L ∗ T ∗ n n∈N ⊆ L  d×d q (·,·) (Q )d , 0 .(QT , Msym ) and representations .((un )ω )n∈N ⊆ Cω (I , Y ), .(fn )n∈N ⊆ L T   d×d r (·,·) q (·,·) d ⊆ L .(F n )n∈N (QT , Msym ), .(g n )n∈N ⊆ L (QT ) , and .(Gn )n∈N ⊆  ) be sequences with the following properties: Lp (·,·) (QT , Md×d sym

7.4 First Parabolic Compensated Compactness Principle

285

(i) For every .n ∈ N, it holds .div(un ) = 0 in .Lr(·,·) (QT ) and for every .φ ∈ ∞ (Q )d , it holds C0,div T  .

−

un (s, y) · ∂t φ(s, y) dsdy QT

 f n (s, y) · φ(s, y) + F n (s, y) : ε(φ)(s, y) dsdy

+ QT

 g n (s, y) · φ(s, y) + Gn (s, y) : ε(φ)(s, y) dsdy .

= QT

  (ii) .un → u in .Lp∗ (·,·)−δ (QT )d .(n → ∞) for every .δ ∈ 0, (p− )∗ − 1 , .ε(un )  ∗

∞ ε(u) in .Lp(·,·) (QT , Md×d sym ) .(n → ∞) and .un  u in .Y (QT ) .(n → ∞), where .u ∈ Y ∞ (QT ) has a representation .uω ∈ Cω0 (I , Y ). 



(iii) .f n  f in .Lq (·,·) (QT )d .(n → ∞), .F n  F in .Lr (·,·) (QT , Md×d sym )  (·,·)  (·,·) d×d p q .(n → ∞), where .F ∈ L (QT , Msym ), .g n → g in .L (QT )d  p (·,·) (Q , Md×d ) .(n → ∞). .(n → ∞), and .Gn → G in .L T sym ∗

(iv) .μn := a n dtdx  μ = a 0 dtdx + μs in .M(QT ) .(n → ∞), where 1 s s .a 0 ∈ L (QT , R≥0 ) and .μ ∈ M(QT ), with .μ ⊥ dtdx, and .(a n )n∈N := (f n · 1 un + F n : ε(un ))n∈N ⊆ L (QT , R≥0 ) is bounded. Then, it holds .a 0 ≤ a := f · u + F : ε(u) almost everywhere in .QT . Remark 7.6 Corollary 7.4 will be of crucial importance later in Chap. 8 when we unite Proposition 7.19 and Pastukhova’s hydro-mechanical parabolic compensated compactness principle [140, Lemma 3.4] to arrive at a second parabolic compensated compactness principle for solenoidal generalized evolution equations (cf. Definition 4.9), which makes redundant both the lower bound .p− ≥ 2 in Proposition 7.19 and the artificial relation .p+ < (p− )• in [140, Lemma 3.4]. Proof of Corollary 7.4 In principle, we would again divide the entire proof into three main parts. Since the entire argumentation of Proposition 7.19 carries over to Corollary 7.4 right after step 1, we solely check whether this step goes through except for some slight adjustments. 1. Localization of assumptions: We fix an arbitrary point .(t0 , x0 ) ∈ QT and choose a cylinder .Q = J × G ⊂⊂ QT satisfying .(t0 , x0 ) ∈ Q, where 1,1 and .J ⊂⊂ I an interval, such that .G ⊂⊂  is a domain with .∂G ∈ C − + := sup(t,x) ∈Q p(t, x) and the local exponents .pQ := inf(t,x) ∈Q p(t, x), .pQ + .q Q := sup(t,x) ∈Q q(t, x) are subject to the relation + + + − pQ ≤ qQ = (pQ )∗ − ε < (pQ )∗ ,

.

(7.97)

286

7 Existence Theory for Irregular Domains

where the first inequality again comes from .q ≥ p in .QT , based on .ε ∈ (0, d2 p− ]   (cf. Remark 5.7). Apparently, (ii) implies .for all δ ∈ 0, (p− )∗ − 1 un → u

.

un (t) → u(t)

in Lp∗ (·,·)−δ (Q)d

(n → ∞) , .

in L2 (G)d

(n → ∞)

(7.98) for a.e. t ∈ J ,

(7.99)

where (7.99) again follows from (7.98) after possibly switching to a subsequence since .(p− )∗ > 2 for .p− ≥ 2. Without loss of generality, we may assume, on the basis of (7.99), for .tinf := inf(J ), that (un )ω (tinf ) → uω (tinf )

.

in L2 (G)d

(n → ∞) .

(7.100)

Otherwise, we substitute J by a smaller interval .J˜ ⊆ J such that .t0 ∈ J˜ such + + := max{p◦ , pQ that (7.100) is satisfied. In addition, we introduce .rQ } > 0, i.e., + − .r Q < (pQ )∗ , and the functions + 

hn := g n − f n ∈ L(qQ ) (Q)d , + 

H n := Gn − F n ∈ L(rQ ) (Q, Md×d sym ) , n ∈ N , .

+ 

h := g − f ∈ L(qQ ) (Q)d , + 

H := G − F ∈ L(pQ ) (Q, Md×d sym ) . Consequently, if we test the weak formulation (i) for every .n ∈ N with an ∞ (Q)d , then for every .n ∈ N and .φ ∈ C ∞ (Q)d , we observe arbitrary .φ ∈ C0,div 0,div that  − un (s, y) · ∂t φ(s, y) dsdy Q



.

(7.101) hn (s, y) · φ(s, y) + H n (s, y) : ∇φ(s, y) dsdy .

= Q

+  +  On the other hand, we further infer from (iii) and .(pQ ) ≥ (rQ ) , i.e., employing Corollary 2.1, that

hn  h

+ 

in L(qQ ) (Q)d

(n → ∞) ,

(r + )

(n → ∞) .

.

Hn  H

in L

Q

(Q, Md×d sym )

(7.102)

7.4 First Parabolic Compensated Compactness Principle

287

As a result of (7.102) and (ii), we can next pass for .n → ∞ in (7.101) for ∞ (Q)d . In this way, for every .φ ∈ C ∞ (Q)d , we conclude arbitrary .φ ∈ C0,div 0,div that   .− u(s, y) · ∂t φ(s, y) dsdy = h(s, y) · φ(s, y) + H (s, y) : ∇φ(s, y) dsdy . Q

Q

Step 2 and step 3 now follow essentially the same argumentation as in the proof of Proposition 7.19.   Finally, let us combine Proposition 7.7 and Proposition 7.19, to identify the important class of .L1 -monotone operators as another subclass of operators that satisfies the Bochner–Sobolev condition (M), even on irregular domains. Proposition 7.20 (.L1 -Monotonicity .⇒ Bochner–Sobolev Condition (M)) Let  ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, .QT := I × , log .p, q := p∗ − ε ∈ P (QT ) with .p− ≥ 2 and .ε ∈ 0, d2 p− . Moreover, let ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ be .L1 -monotone. Then, the operator .A : V q,p n n ∞ for sequences .un ∈ Wε,σ (QT ) ∩ H (QT ), .n ∈ N, where .(n )n∈N is a non˚εq,p (QT )∗ from decreasing exhaustion of ., and .(v ∗n )n∈N ⊆ V .

.

dσ un + ∗n A(n un ) = ∗n v ∗n dt

˚ε (Qn )∗ in V T

for all n ∈ N , .

(7.103)

˚εq,p (QT ) in V

(n → ∞) , .

(7.104)

in H∞ (QT )

q,p

n un  u ∗

n un  u A(n un )  u∗ v ∗n → v ∗

(n → ∞) , .

(7.105)

in

˚εq,p (QT )∗ V

(n → ∞) , .

(7.106)

in

˚εq,p (QT )∗ V

(n → ∞) ,

(7.107)

˚εq,p (QT )∗ .(n → ∞) and for every .t  , t ∈ I it follows that .A(n un )  Au in .V  with .t ≤ t, it holds .

lim inf A(n un ), n un χ(t  ,t) V ˚q,p (Q n→∞

ε

T)

≥ Au, uχ(t  ,t) V ˚q,p (Q ) . ε

(7.108)

T

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ satisfies the Bochner– In particular, .A : V Sobolev condition (M). Proof To start with, given (7.104) and (7.106), we can choose for fixed but arbitrary t  , t ∈ I with .t  < t a cofinal subset . 0 ⊆ N, which possibly depends on these fixed time slices .t  , t ∈ I , such that

.

A(n un ), n un χ(t  ,t) V ˚q,p (Q

.

ε

T)

→ lim inf A(n un ), n un χ(t  ,t) V ˚q,p (Q n→∞

( 0 n → ∞) .

ε

T)

288

7 Existence Theory for Irregular Domains

Proceeding as in the proof of Lemma 7.1, we find that (7.103)–(7.107) imply both q,p that .u ∈ Wε,σ (QT ) ∩ H∞ (QT ) with a weakly continuous representation .uω ∈ 0 Cω (I , H ) and the convergence .(n un )ω (t)  uω (t) in H .(n → ∞) for every .t ∈ I . Furthermore, (7.104) and (7.106) together with Definition 7.2, (L.4), and   the reflexivity of both .Lq (·,·) (QT )d and .Lp (·,·) (QT , Md×d sym ) (cf. Proposition 2.7)  (·,·)  q d (QT ) and .(F n )n∈N ⊆ Lp (·,·) (QT , Md×d provide sequences .(f n )n∈N ⊆ L sym ), with .A (n un ) = (f n , F n ) for every .n ∈ N, as well as a cofinal set . ⊆ 0 , such that .

fn  f Fn  F



in Lq (·,·) (QT )d in L

p (·,·)

(QT , Md×d sym )

( n → ∞) , ( n → ∞) .

˚εq,p (QT )∗ . MoreBy the weak continuity of .Jεσ , we even have .u∗ = Jεσ (f , F ) in .V over, the sequence .(a n )n∈ := (f n · n un + F n : ε(n un ))n∈ ⊆ L1 (QT , R≥0 ) is bounded and, therefore, the sequence .(μn )n∈ := (a n dtdx)n∈ ⊆ M(QT ). It follows that Proposition 7.5 yields the existence of a not relabeled subsequence and ∗ a finite Radon measure .μ ∈ M(QT ) such that .μn  μ in .M(QT ) .( n → ∞). Thanks to Lebesgue’s decomposition theorem (cf. [58, p. 285]), there exist both a function .a 0 ∈ L1 (QT , R≥0 ) and a finite Radon measure .μs ∈ M(QT ) such that .μ = a 0 dtdx + μs in .M(QT ) and .μs ⊥ dtdx. Therefore, the first parabolic compensated compactness principle (cf. Proposition 7.19, .p◦ = 2) proves .a 0 ≤ a := f · u + F : ε(u) almost everywhere .QT . As a result, Proposition 7.7 implies σ ∗ ˚εq,p (QT )∗ , i.e., .a 0 = a almost everywhere in .QT and .Au = Jε (f , F ) = u in .V q,p ˚ε (QT )∗ .(n → ∞). on the basis of (7.106), we conclude that .A(n un )  Au in .V In addition, Proposition 7.7 provides for these fixed time slices .t  , t ∈ I with .t  < t that Au, uχ(t  ,t) V ˚q,p (Q ε

.

T)

≤ lim inf A(n un ), n un χ(t  ,t) V ˚q,p (Q

T)

ε

n→∞

= lim inf A(n un ), n un χ(t  ,t) V ˚q,p (Q

0 n→∞

T)

ε

(7.109)

= lim inf A(n un ), n un χ(t  ,t) V ˚q,p (Q ) . n→∞

ε

T

Because .t  , t ∈ I with .t  < t were chosen arbitrarily, (7.109) also proves (7.108).

 

7.5 Abstract Existence Result for Irregular Domains and p − ≥ 2

289

7.5 Abstract Existence Result for Irregular Domains and p− ≥ 2 After it has eventually been clarified that the Bochner–Sobolev condition (M) is indeed a meaningful condition because not only Bochner strongly continuous operators but also .L1 -monotone operators satisfy this condition, it is now justified to prove an existence result involving this abstract concept. Theorem 7.1 Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, .QT := I × , and .q, p ∈ Plog (QT ) with .p− ≥ 2 and .q ≥ ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ be bounded, p in .QT . Moreover, let .A : V ˚εq,p (QT )∗ and Bochner pseudo-monotone, Bochner coercive with respect to .u∗ ∈ V .u0 ∈ H , and satisfying the Bochner–Sobolev condition (M). Then, there exists q,p ∞ 0 .u ∈ Wε,σ (QT ) ∩ H (QT ) with a representation .uω ∈ Cω (I , H ) such that .

dσ u + Au = u∗ dt uω (0) = u0

˚εq,p (QT )∗ , in V in H .

Proof Similar to the proof of Theorem 5.1, it suffices to consider the case ˚εq,p (QT )∗ . Otherwise, we simply switch to the shifted operator u∗ = 0 in .V q,p ∗ ˚ ˚εq,p (QT )∗ , which again satisfies the .A − u : Vε (QT ) ∩ H∞ (QT ) → V Bochner–Sobolev condition (M) due to Proposition 7.2 (ii). Let .(n )n∈N be bounded Lipschitz domains with .n ⊂⊂ n+1 ⊂⊂  for every n n 2 .n ∈ N and . n∈N n = , .QT := I ×n , and .u0 := PHn (u0 |n ) ∈ Hn := L0,σ (n ) for all .n ∈ N, where .PHn : L2 (n )d → Hn for every .n ∈ N denotes the orthogonal projection from .L2 (n )d into .Hn . In particular, it is easy to observe that both .supn∈N un0 2Hn ≤ u0 2H and .ℰn un0 → u0 in H .(n → ∞), where .ℰn : Hn → H , .n ∈ N, denote the zero extension operators. As not only ˚εq,p (Qn ) → V ˚εq,p (QT ), .n ∈ N, but also .n : H∞ (Qn ) → H∞ (QT ), .n ∈ N, .n : V T T as well as .ℰn : Hn → H , .n ∈ N, are linear and continuous, it is readily seen that ˚εq,p (Qn ) ∩ H∞ (Qn ) → V ˚εq,p (Qn )∗ , .n ∈ N, the restrictions .An := ∗n An : V T T T remain bounded and Bochner pseudo-monotone. Apart from that, these restrictions ˚εq,p (Qn )∗ , .n ∈ N, and .un ∈ Hn , are also Bochner coercive with respect to .0 ∈ V T 0 .n ∈ N, respectively. To see this, assume for fixed but arbitrary .n ∈ N that a function ˚εq,p (Qn ) ∩ H∞ (Qn ) satisfies .un ∈ V T T .

.

1 1 n 2 un (t)2Hn + An un , un χ[0,t] V ˚εq,p (Qn ) ≤ u0 Hn T 2 2

for a.e. t ∈ I .

(7.110)

290

7 Existence Theory for Irregular Domains

Then, recalling the definitions of both .An and .un0 , (7.110) for almost every .t ∈ I can be rewritten as 1 1 n 2 (n un )(t)2H + A(n un ), n un χ[0,t] V ˚εq,p (QT ) ≤ u0 Hn 2 2 . 1 ≤ u0 2H . 2

(7.111)

˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ with respect The Bochner coercivity of .A : V q,p ∗ ˚ to .0 ∈ Vε (QT ) and .u0 ∈ H applied to (7.111), in turn, provides a constant .M > 0, which is independent of .n ∈ N, such that for every .n ∈ N un V ˚q,p (Qn )∩H∞ (Qn ) = n un V ˚q,p (Q

.

ε

T

T )∩H

ε

T

∞ (Q ) T

≤M.

(7.112)

˚εq,p (Qn )∩H∞ (Qn ) → V ˚εq,p (Qn )∗ Thus, for every fixed .n ∈ N, the operator .An : V T T T is bounded, Bochner pseudo-monotone and Bochner coercive with respect to ˚εq,p (Qn )∗ and .un ∈ Hn . Consequently, Theorem 5.1 yields a sequence .0 ∈ V T 0 q,p n n 0 .un ∈ Wε,σ (Q ), .n ∈ N, having representations .(un )c ∈ H (Q ), .n ∈ N, such T T that .

dσ un + A n un = 0 dt (un )c (0) =

un0

˚εq,p (Qn )∗ , in V T in Hn ,

(7.113)

n ∈ N.

˚εq,p (Qn ), where .t ∈ I By testing .(7.113)1 for every .n ∈ N with .un χ[0,t] ∈ V T is arbitrary, a subsequent application of the formula of integration-by-parts for q,p n .Wε,σ (Q ) (cf. Proposition 4.23 (ii)), and also exploiting .(7.113)2 for every .n ∈ N, T we arrive for every .n ∈ N at 1 1 n 2 2 . (u n )c (t)Hn + An un , un χ[0,t] V ˚εq,p (Qn ) ≤ u0 Hn T 2 2

for all t ∈ I .

(7.114)

˚εq,p (QT ) ∩ Because (7.114) also implies (7.110), the sequence .(n un )n∈N ⊆ V ∞ H (QT ) satisfies the a-priori estimate (7.112). From (7.112), the boundedness of ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ , the reflexivity of both .V ˚εq,p (QT ) and .A : V q,p ∗ ˚ .Vε (QT ) (cf. Proposition 4.17 (i)), as well as Corollary 2.2, we then conclude the ˚εq,p (QT ) ∩ H∞ (QT ) and existence of a not relabeled subsequence .(n un )n∈N ⊆ V q,p q,p ˚ε (QT ) ∩ H∞ (QT ) and .ξ ∈ V ˚ε (QT )∗ such that of elements .u ∈ V n un  u .

∗

n un  u A(n un )  ξ

˚εq,p (QT ) in V

(n → ∞) ,

in H∞ (QT )

(n → ∞) ,

˚εq,p (QT )∗ in V

(n → ∞) .

(7.115)

7.5 Abstract Existence Result for Irregular Domains and p − ≥ 2

291

On the basis of (7.113), (7.115) and .(n un )c (0) = ℰn un0 → u0 in H .(n → ∞), q,p Lemma 7.1 provides .u ∈ Wε,σ (QT ) ∩ H∞ (QT ) with a representation 0 .uω ∈ Cω (I , H ) and

.

dσ u = −ξ dt uω (0) = u0

˚ε (QT )∗ , in V q,p

(7.116)

in H ,

as well as (n un )c (t)  uω (t)

.

in H

(n → ∞)

for all t ∈ I .

(7.117)

Eventually, owing to .(7.113)1 , (7.115) and .(n un )c (0) → u0 in H .(n → ∞), ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ the Bochner–Sobolev condition (M) of .A : V q,p ˚ε (QT )∗ . proves that .Au = ξ in .V   Looking back to Propositions 7.2 (ii) and 7.20, we can prove another existence result that is less abstract and, hence, potentially more applicable, in the case that .A can be decomposed into an .L1 -monotone part .A0 and a Bochner strongly continuous part .B. Theorem 7.2 Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ),.T < ∞,  log .QT := I × , and .q, p := p∗ − ε ∈ P (QT ) with .p− ≥ 2 and .ε ∈ 0, d2 p− . Furthermore, we require the following: ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is bounded and .L1 -monotone. (i) .A0 : V ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is bounded and Bochner strongly (ii) .B : V continuous. ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ is Bochner pseudo(iii) .A = A0 + B : V ˚εq,p (QT )∗ and .u0 ∈ H . monotone and Bochner coercive with respect to .u∗ ∈ V Then, there exists .u ∈ Wε,σ (QT ) ∩ H∞ (QT ) with a representation .uω ∈ Cω0 (I , H ) such that q,p

dσ u + A0 u + Bu = u∗ . dt uω (0) = u0

˚εq,p (QT )∗ , in V

(7.118)

in H ,

which, in addition, satisfies the integration-by-parts inequality   1 1 dσ u 2  2 . uω (t)H − uω (t )H ≤ , uχ[t  ,t] 2 2 dt ˚εq,p (QT ) V for almost every .t  , t ∈ I with .t  < t, or for .t  = 0 and every .t ∈ (0, T ].

(7.119)

292

7 Existence Theory for Irregular Domains

Remark 7.7 The main advantage of Theorem 7.2 in direct comparison with Theorem 7.1 is its ability to prove the existence of solutions of solenoidal generalized evolution equations satisfying the integration-by-parts inequality (7.119). This feature of Theorem 7.2 will be of crucial importance later in Chap. 8 when we use Theorem 7.2 to establish the existence of approximative solutions that satisfy an energy inequality and are thus bounded in appropriate function spaces. ˚ε (QT ) ∩ H∞ (QT ) → V ˚ε (QT )∗ Proof Due to Proposition 7.20, .A0 : V q,p ˚ε (QT ) ∩ satisfies the Bochner–Sobolev condition (M). Thus, since .B : V ˚εq,p (QT )∗ is Bochner strongly continuous, Proposition 7.2 (ii) H∞ (QT ) → V ˚εq,p (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ satisfies the Bochner– proves that .A : V q,p ˚ε (QT ) ∩ H∞ (QT ) → V ˚εq,p (QT )∗ Sobolev condition (M). Inasmuch as .A : V is also bounded, Bochner pseudo-monotone and Bochner coercive with respect to ∗ ˚εq,p (QT )∗ and .u0 ∈ H , all assumptions of Theorem 7.1 are satisfied. The .u ∈ V existence of a solution of (7.118) would now immediately follow from Theorem 7.1. However, because we are actually interested the existence of a solution of (7.118) that in addition satisfies the integration-by-parts inequality (7.119), we need to go through the proof of Theorem 7.1 again superficially. ˚εq,p (QT )∗ . By analogy with the It suffices to consider the case .u∗ = 0 in .V proof of Theorem 7.1,  let .(n )n∈N be Lipschitz domains with .n ⊂⊂ n+1 ⊂⊂  for all .n ∈ N and . n∈N n = . For every .n ∈ N, we set .QnT := I × n , n 2 ( ), and .A := ∗ A : V ˚εq,p (Qn ) ∩ H∞ (Qn ) .u := PHn (u0 |n ) ∈ Hn := L n n n n T T 0,σ 0 ˚εq,p (Qn )∗ . Proceeding as in the proof of Theorem 7.1, we conclude the exis→V T q,p tence of .un ∈ Wε,σ (QnT ), .n ∈ N, with representations .(un )c ∈ H0 (QnT ), .n ∈ N, q,p satisfying (7.113)–(7.117) with respect to some .u ∈ Wε,σ (QT ) ∩ H∞ (QT ) that has a representation .uω ∈ Cω0 (I , H ) and even satisfies (7.118). If we apply Corollary 3.3 to .(7.115)1,2 and (7.117), also exploiting that .(p− )∗ > 2 for .p− ≥ 2, then, after possibly switching to a subsequence, we find that q,p

(n un )(t) → u(t)

.

in H

(n → ∞)

q,p

for a.e. t ∈ I .

(7.120)

˚εq,p (Qn ) for arbitrary .t  , t ∈ I By testing .(7.113)1 for every .n ∈ N with .un χ[t  ,t] ∈ V T  with .t < t and a subsequent application of the formula of integration-by-parts for q,p n  .Wε,σ (Q ) (cf. Proposition 4.23), we arrive for every .n ∈ N and .t , t ∈ I with T  .t < t at 1 (n un )c (t)2H + A(n un ), n un χ(t  ,t) V ˚εq,p (QT ) 2 . 1 = (n un )c (t  )2H . 2

(7.121)

˚εq,p (QT ) ∩ Due to .(7.115)1,2 and (7.117), the Bochner strong continuity of .B : V q,p ˚ε (QT )∗ further implies that H∞ (QT ) → V Bun → Bu

.

˚εq,p (QT )∗ in V

(n → ∞) .

(7.122)

7.6 Application to Model Problems

293

While on the other hand, by taking into account (7.113)–(7.117) and (7.122), ˚εq,p (QT )∗ for every .t  , t ∈ I Proposition 7.20 with .(v ∗n )n∈N := (−Bun )n∈N ⊆ V  with .t < t proves that .

lim inf A0 (n un ), n un χ(t  ,t) V ˚q,p (Q n→∞

T)

ε

≥ A0 u, uχ(t  ,t) V ˚q,p (Q ) . ε

T

(7.123)

σu ˚εq,p (QT ) in .V If we combine (7.122) and (7.123), using .(A0 + B)u = Au = − ddt   (cf. .(7.118)1 ), then for every .t , t ∈ I with .t < t, we observe that .

lim inf A(n un ), n un χ(t  ,t) V ˚q,p (Q n→∞

ε

T)

  dσ u ≥− . , uχ(t  ,t) dt ˚εq,p (QT ) V

(7.124)

Therefore, if take the limit inferior with respect to .n → ∞ on both sides in (7.121), using (7.120) and (7.124) in doing so, we conclude, in summary, for almost every   .t , t ∈ I with .t < t that   1 dσ u 1 2 . uω (t)H − ≤ uω (t  )2H . (7.125) , uχ(t  ,t) q,p 2 dt 2 ˚ Vε (QT ) Thanks to .(n un )c (0) = ℰn un0 → u0 in H .(n → ∞), taking again the limit inferior with respect to .n → ∞ in (7.123) in the case .t  = 0, proves that (7.125) holds also for .t  = 0 and almost every .t ∈ I . Then, taking into account .uω ∈ Cω0 (I , H ) and the weak lower semi-continuity of the . · H - norm, it is easy to see that (7.125) holds even for .t  = 0 and every .t ∈ I .  

7.6 Application to Model Problems As an application of Theorem 7.2, we extend the existence results concerning the unsteady .p(·, ·)-Stokes like system with an optional, lower-order variable exponent non-linearity, i.e., Theorem 5.2, and the unsteady .p(·, ·)-Navier–Stokes equations, i.e., Theorem 5.3, from the required Lipschitz regularity of the domain to arbitrary bounded domains. In agreement with Chap. 5, Sect. 5.5, we will start with the unsteady .p(·, ·)-Stokes like system with .p ∈ Plog (QT ) satisfying .p− ≥ 2.

7.6.1 Unsteady p(·, ·)-Stokes Equations in an Irregular Domain with p − ≥ 2 Theorem 7.3 Let  ⊆ Rd , d ≥ 2, be a bounded domain, I := (0, T ), T < ∞, QT := I × , and p, q := p∗ − ε ∈ Plog (QT ) with p− ≥ 2 and ε ∈ (0, d2 p− ]. d×d Furthermore, let S : QT × Md×d sym → Msym be a mapping satisfying (S.1)–(S.5)

294

7 Existence Theory for Irregular Domains

with respect to p and d : QT × Rd → Rd a mapping satisfying (D.1)–(D.3) with −   respect to r = q.12 Then, for arbitrary f ∈ L(p ) (QT )d , F ∈ Lp (·,·) (QT , Md×d sym ) q,p ∞ and u0 ∈ H , there exists u ∈ Wε,σ (QT ) ∩ H (QT ) with a representation uω ∈ Cω0 (I , H ) such that .

dσ u + Su + Du = Jεσ (f , F ) dt uω (0) = u0

˚εq,p (QT )∗ , in V in H ,

i.e., for every φ ∈ C ∞ (QT )d with div(φ) = 0 in QT and supp(φ) ⊆ [0, T ) × , it holds  .− u(t, x) · ∂t φ(t, x) dtdx QT

 d(t, x, u(t, x)) · φ(t, x) + S(t, x, ε(u)(t, x)) : ε(φ)(t, x) dtdx

+ QT



= (u0 , φ(0))H +

f (t, x) · φ(t, x) + F (t, x) : ε(φ)(t, x) dtdx , QT

which, in addition, for every t ∈ I satisfies the energy inequality 1 uω (t)2H . + 2

 t

≤

0

d(s, y, u(s, y)) · u(s, y)+S(s, y, ε(u)(s, y)) : ε(u)(s, y) dsdy 

1 u0 2H + 2

 t 0

f (s, y) · u(s, y) + F (s, y) : ε(u)(s, y) dsdy .



(7.126) ˚εq,p (QT )∩H∞ (QT ) → V ˚εq,p (QT )∗ Proof We have already proved that S +D : V is bounded, Bochner pseudo-monotone and Bochner coercive with respect to all ˚εq,p (QT )∗ , where f ∈ L(p− ) (QT )d and F ∈ Lp (·,·) (QT , Md×d ), Jεσ (f , F ) ∈ V sym and u0 ∈ H (cf. Proposition 5.14). Apart from that, Proposition 3.31 and Propo˚εq,p (QT ) → V ˚εq,p (QT )∗ is bounded and L1 -monotone, sition 7.3 proved that S : V q,p ˚εq,p (QT )∗ is bounded and Bochner ˚ε (QT ) → V and in Proposition 5.6 that D : V strongly continuous. Therefore, the assertion is a direct application of Theorem 7.2, where the energy inequality (7.126) follows from the integration-by-parts inequality (7.119).  

that for p − ≥ 2 and ε ∈ (0, d2 p − ], we have p∗ − ε ≥ p ≥ 2 (cf. Remark 5.7) and, thus, r = q in Proposition 5.6.

12 Recall

7.6 Application to Model Problems

295

Remark 7.8 (i) We cannot make any statement about the uniqueness of a solution provided by Theorem 7.3. Even under the assumptions of Proposition 5.16, i.e., if d : QT × Rd → Rd also satisfies (5.70), it is still unclear whether the solution provided by Theorem 7.3 is unique. This can mainly be attributed to the fact that we are not aware of whether the integration-by-parts inequality (7.119) applies to all solutions or only to solutions provided by Theorem 7.2. (ii) In the case d ≡ 0, Theorem 7.3 simply treats the unsteady p(·, ·)-Stokes equations. Therefore, Theorem 7.3 proves the –not necessarily unique– weak solvability of the unsteady p(·, ·)–Stokes equations in a bounded domain  ⊆ Rd , d×d d ≥ 2, with a variable exponent non-linearity S : QT × Md×d sym → Msym that satisfies (S.1)–(S.5) with respect to p ∈ Plog (QT ) with p− ≥ 2.

7.6.2 Unsteady p(·, ·)-Navier–Stokes Equations in an Irregular Domain with p − > 3d+2 d+2 Last but not least, we establish the weak solvability of the unsteady .p(·, ·)-Navier– Stokes equations in a bounded domain . ⊆ Rd , .d ≥ 2, with .p ∈ Plog (QT ) satisfying .p− > pC = 3d+2 d+2 . Theorem 7.4 Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, log .QT := I × , and .p ∈ P (QT ) with .p− > pC . Furthermore, let .S : QT × d×d d×d Msym → Msym be a mapping satisfying (S.1)–(S.5) with respect to p. Then, for − 



arbitrary .f ∈ L(p ) (QT )d , .F ∈ Lp (·,·) (QT , Md×d sym ) and .u0 ∈ H , there exists p,p ∞ 0 .u ∈ Wε,σ (QT ) ∩ H (QT ) with a representation .uω ∈ Cω (I , H ) such that .

dσ u + Su + Cu = Jεσ (f , F ) dt uω (0) = u0

˚εp,p (QT )∗ , in V in H ,

i.e., for every .φ ∈ C ∞ (QT )d with .div(φ) = 0 in .QT and .supp(φ) ⊆ [0, T ) × , it holds  .− u(t, x) · ∂t φ(t, x) dtdx QT



 S(t, x, ε(u)(t, x)) − u(t, x) ⊗ u(t, x) : ε(φ)(t, x) dtdx

+ QT



= (u0 , φ(0))H +

f (t, x) · φ(t, x) + F (t, x) : ε(φ)(t, x) dtdx , QT

296

7 Existence Theory for Irregular Domains

which, in addition, for every .t ∈ I satisfies the energy inequality 1 2 . uω (t)H + 2 ≤

 t 0

S(s, y, ε(u)(s, y)) : ε(u)(s, y) dsdy



1 u0 2H + 2

 t 0

(7.127)

f (s, y) · u(s, y) + F (s, y) : ε(u)(s, y) dsdy . 

˚εp,p (QT ) ∩ H∞ (QT ) → V ˚εp,p (QT )∗ Proof We have already proved that .S + C : V is bounded, Bochner pseudo-monotone and Bochner coercive with respect to all σ ˚εq,p (QT )∗ , where .f ∈ L(p− ) (QT )d and .F ∈ Lp (·,·) (QT , Md×d ), .Jε (f , F ) ∈ V sym and .u0 ∈ H (cf. Proposition 5.12). On the other hand, we know that ˚εp,p (QT ) → V ˚εq,p (QT )∗ is bounded and .L1 -monotone (cf. Propositions 3.31 .S : V ˚εp,p (QT ) ∩ H∞ (QT ) → V ˚εp,p (QT )∗ is bounded and and 7.3) and that .C : V Bochner strongly continuous (cf. Proposition 5.3). Therefore, the assertion is a direct application of Theorem 7.2, where we use for the energy inequality (7.127) ˚εp,p (QT ) = 0 for every .t ∈ I .   that .Cu, uχ(0,t) V Remark 7.9 For exactly the same reasons as in Remark 7.8, it is entirely unclear whether a solution provided by Theorem 7.4 is unique, even under the assumptions d×d of Proposition 5.17, i.e., if .S : QT ×Md×d sym → Msym additionally satisfies the strong monotonicity property (5.72).

Chapter 8

Existence Theory for p− < 2 .

In this chapter, we establish the weak solvability of the unsteady .p(·, ·)-Navier– Stokes equations in a bounded, possibly irregular domain with a variable exponent 3d log . At the same time, we will observe that .p ∈ P (QT ) satisfying .p− > d+2 the existence result Theorem 7.3, and the closely connected weak solvability of the unsteady .p(·, ·)-Stokes equations in a bounded, possibly irregular domain, 2d . can be generalized to variable exponents .p ∈ Plog (QT ) satisfying .p− > d+2 Nevertheless, the focus still lies on the unsteady .p(·, ·)-Navier–Stokes equations, which are from a physical point of view of greater importance than the unsteady .p(·, ·)-Stokes equations. Let us begin with a brief overview of historical milestones in the weak solvability of the unsteady .p(·, ·)-Navier–Stokes equations. In this context, it should be recalled once again that the unsteady .p(·, ·)-Navier–Stokes equations in a bounded domain . ⊆ Rd , .d ≥ 2, and over a finite time interval d .I := (0, T ), .T < ∞, search for a velocity vector field .u : QT → R and a scalar kinematic pressure .π : QT → R that solve the system ∂t u − div(S(·, ·, ε(u))) + div(u ⊗ u) + ∇π = f − div(F ) div(u) = 0 .

u=0 u(0) = u0

in QT , in QT , on I × ∂ ,

(8.1)

in  ,

where .f : QT → Rd is a given vector field, .F : QT → Md×d sym is a given symmetric tensor field, jointly describing external body forces, and .u0 :  → Rd is the velocity vector field at time .t = 0. Moreover, we assume that the extra stress tensor d×d d×d .S : QT × Msym → Msym satisfies the conditions (S.1)–(S.5) with respect to a .logHölder continuous exponent .p : QT → (1, +∞). We have already elaborated the weak solvability of the system (8.1) in a bounded, possibly irregular domain with the admissible lower bound .p− > pC = 3d+2 d+2 ≥ 2 (cf. Theorem 7.4). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaltenbach, Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents, Lecture Notes in Mathematics 2329, https://doi.org/10.1007/978-3-031-29670-3_8

297

298

8 Existence Theory for .p − < 2

This lower bound corresponds to the lower bound of O.A. Ladyžhenskaya in [104], who initiated the mathematical analysis of the system (8.1) (cf. [105, 106]), at least for constant exponents. More precisely, she established the existence of weak solutions to the system (8.1) for constant exponents .p ∈ [pC , ∞) using the theory of monotone operators in combinations with compactness arguments. The same results have been obtained by J.–L. Lions in [116] for the extra stress tensor governed by the full gradient rather than by the symmetric part of the gradient. Due to the linearity of the extra stress tensor .S(·, ·, A) := A in .Md×d sym for every d×d .A ∈ Msym , the solvability of (8.1) for the particular case .p = 2 stands apart and was already obtained in 1951 by E. Hopf [87] and for the Cauchy problem even earlier in 1934 by J. Leray [113]. Note that for .d ≥ 3, the exponent .p = 2 does not belong to the set of admissible exponents, i.e., .[pC , ∞), established by O.A. Ladyžhenskaya and J.–L. Lions. For this reason, during the last 50 years, there have been numerous attempts to lower the bound on the admissible exponents, at least up to .p ≥ 2. The existence of measure-valued weak solutions for the system (8.1) was proved 2d for constant exponents .p > d+2 in [77, 134], see also, e.g., [119, 120], using Ball’s theorem on Young measures. The existence of weak solutions that satisfy a variational inequality instead of a weak formulation was proved for the system (8.1) for constant exponents .p ≥ 2 and dimensions .d ≥ 2 in [41]. Important existence results addressing the existence of weak solutions in three dimensions, mainly related to the dilatant flow zone, i.e., if .p > 2, were foremost obtained by J. Málek, J. Neˇcas and M. R˚užiˇcka in [17, 118–120] under the assumption that . ⊆ R3 is a bounded domain with .C 3 -boundary, that the extra stress tensor has a potential, i.e., there is a function . ∈ C 1 (Rd×d ) such that for d×d every .A ∈ Md×d sym , it holds .S(·, ·, A) = ∂A (A) in .Msym , and that the data are p,p 2 ˚ sufficiently smooth, i.e., that .u0 ∈ V and .f ∈ L (QT )3 , which is related to the method of proof. We emphasize, though, that especially the pseudo-plastic flow zone, i.e., the case in which .p < 2, is of particular interest in applications. A first step towards an existence result for .p < 2 goes back to R. Landes and V. Mustonen [112], where an .L∞ -approximation of the corresponding sequence of Galerkin solutions is constructed that converges strongly to a potential weak solution. Further improvements have been achieved by J. Wolf in [164], and even earlier by J. Frehse, J. Málek and M. Steinhauer in [69] for space-periodic boundary conditions, which, with the aid of the parabolic .L∞ -truncation technique and a deep understanding of the pressure, established the existence of weak solutions to the system (8.1) in a possibly irregular and unbounded domain for constant exponents 2d+2 .p > d+2 . Eventually, Wolf’s result was surpassed by himself, L. Diening and M. R˚užiˇcka in [48] by using the parabolic Lipschitz truncation technique, which enabled them to prove the existence of weak solutions to the system (8.1) in a potentially irregular 2d domain for constant exponents .p > d+2 . Their result is to some extent optimal, in 2d the sense that for .p < d+2 one can still give a sense to the convective term due to the information provided by the time derivative, i.e., that .u ∈ H∞ (QT ), as then at least .u ⊗ u ∈ L∞ (I, L1 ()d×d ). Nevertheless, because one loses the compact

8 Existence Theory for .p − < 2

299

˚p,p →→ H , any parabolic compactness result, e.g., Landes’ and embedding .V Mustonen’s compactness principle (cf. Proposition 3.10), the Aubin–Lions’ lemma (cf. [12, 116]), or Simon’s compactness theorem (cf. [156, Thm. 3, p. 80]), is no longer at disposal. In consequence, one may well encounter problems identifying the weak limit of the convective term after passing to the limit. The existence theory for the unsteady .p(·, ·)-Navier–Stokes equations is not developed to this extent. First systematic results addressing the existence of both strong and weak solutions are due to M. R˚užiˇcka [147], who similar to the works [17, 119, 120] assumed that . ⊆ R3 is a domain having .C 3,1 -boundary, that the ˚p,p (0) and .f ∈ L2 (QT )3 , and data are sufficiently smooth, i.e., at least .u0 ∈ V d×d d×d that the extra stress tensor .S : QT × Msym → Msym assumes a form similar to (1.4) governed by a variable exponent .p ∈ C 1 (QT ) that is subject to the relation (Fig. 8.1) 

 8 3 + p− .2 ≤ p ≤ p < min , . 3 2(3 − p − ) −

+

(8.2)

For periodic boundary conditions, this result extends to . 95 < p− ≤ p+ < p− + 1 (cf. [146]). For an extensive examination addressing the existence of strong solutions, please also refer to [43, 45]. Returning to the theory of weak solutions, recent progress has been made by V.V. Zhikov [173], who, by using PCCP methods, proved the existence of weak solutions to the system (8.1) for a time-independent, not necessarily .log-Hölder continuous exponent .p ∈ P∞ () satisfying (Fig. 8.1) √  3d d + 3d 2 + 4d . max , ≤ p− ≤ p+ < ∞ . d +2 d +2 

(8.3)

By adapting Zhikov’s methods to both time- and space-dependent exponents, S.E. Pastukhova [140] proved the existence of weak solutions to the system (8.1) for a variable exponent .p ∈ P∞ (QT ) satisfying (Fig. 8.1)   3d − + − − −d +2 . < p ≤ p < (p )• = min p + 2, p . d +2 d

(8.4)

To the best of the author’s knowledge, these results concerning the existence of weak solutions for the unsteady .p(·, ·)-Navier–Stokes equations have not been improved ever since. In summary, Zhikov’s result was the first to prove the existence of weak solutions of the system (8.1) for variable exponents without imposing an artificial upper bound but applied exclusively to time-independent variable exponents. In contrast to that, R˚užiˇcka’s and Pastukhova’s argumentations also include timedependent variable exponents but are fundamentally based on the required artificial

300

8 Existence Theory for .p − < 2

upper bound for this variable exponent. The improvements of this book concerning the existence of weak solutions of the unsteady .p(·, ·)-Navier–Stokes equations are in a sense twofold: First, we will make no assumptions on the regularity of the domain under consideration, other than its boundedness. Second, we will consider 3d time-dependent variable exponents .p ∈ Plog (QT ) with lower bound .p− > d+2 , i.e., we do not impose any artificial upper bounds (Fig. 8.1).

Fig. 8.1 Comparison of admissible variable exponents in the existence results for the unsteady equations of M. R˚užiˇcka [147], V.V. Zhikov [173], S.E. Pastukhova [140] and Theorem 8.1 (below) in the case .d = 3

.p(·, ·)-Navier–Stokes

8.1 Second Parabolic Compensated Compactness Principle

301

8.1 Second Parabolic Compensated Compactness Principle In this section, we generalize the first PCCP (cf. Proposition 7.19) to the case .p−< 2, q,p,s i.e., the case in which the special formula of integration-by-parts for .Wε,div (QT ) is no longer available. More precisely, we will relax the required lower bound .p− ≥ 2 2d in Proposition 7.19 to .p− > d+2 . Because for .p− < pC the unsteady convective term .C (cf. Proposition 5.3) behaves too badly, in the sense that we cannot any longer ˚εp,p (QT ) ∩ H∞ (QT ) into .V ˚εp,p (QT )∗ , the assumption determine that .C maps .V 2d − > .p d+2 is too weak for the treatment of the unsteady .p(·, ·)-Navier–Stokes 3d equations. We will note, however, that under the assumption .p− > d+2 , which is p,p ˚ε (QT ) ∩ H∞ (QT ) → L3 (QT )d (cf. just sufficient to guarantee the embedding .V Proposition 3.8), we can prove a second PCCP that is exactly adapted to peculiarities of the unsteady .p(·, ·)-Navier–Stokes equations, i.e., to the specific properties of the convective term .C. We adopt several steps of the proof of Proposition 7.19, such as, e.g., step 1 and step 2. Nonetheless, because this time we cannot rely q,p,s on the formula of integration-by-parts for .Wε,div (QT ), as we have done before in Proposition 7.19, step 3, the basic idea now is that we immediately pass for .r → 0 with the localization parameter right after we passed for .n → ∞ with the approximation parameter. This method has already been used by S.E. Pastukhova [140], see also V.V. Zhikov [173], in an existence proof for the unsteady .p(·, ·)Navier–Stokes equations, where .p ∈ P∞ (QT ) satisfied (8.4). We will follow their argumentation, but we will generalize their result to some extent. By combining their methods and the first PCCP, more precisely Corollary 7.4, we find ourselves in the position to omit the condition .p+ < (p− )• . Their argumentation, especially the ability to pass for .r → 0 with the localization parameter, essentially relies on the following local Poincaré–Wirtinger type inequalities. Proposition 8.1 Let .x0 ∈ Rd , .d ≥ 2, .r > 0, .p ∈ (1, ∞), .ϕ ∈ L∞ (B1d (0), R≥0 ), ffl d 0 satisfying . B d (0) ϕ(x) dx = 1, and .ϕr (x) := ϕ( x−x r ) for every .x ∈ Br (x0 ). Then, 1 for each .q ∈ (1, p∗ ), there exists a constant .c = c(q, p, d,  ϕ) > 0 (not depending on .r > 0) such that for all .u ∈ W 1,p (Brd (x0 ))d , satisfying . B d (x0 ) u(x)ϕr (x) dx = 0 r

in .Rd , it holds  .

Brd (x0 )

|u(x)|q ϕr (x) dx rq

1 q

 ≤c

1 |∇u(x)| dx p

Brd (x0 )

p

. 

Proof See [175, Proposition 8.3]. )

Proposition ∈ .d ≥ 2, .r > 0, .p ∈ (1, d),  8.2 Let .z0 := (t0 , x0 Ir (t0 ) := t0 , t0 + r 2 , .Qr (z0 ) := Ir (t0 ) × Brd (x0 ), .ϕ ∈ L∞ (B1d (0), R≥0 ), satisfyffl d 0 ing . B d (0) ϕ(x) dx = 1, and .ϕr (x) := ϕ( x−x r ) for every .x ∈ Br (x0 ). Then, there 1 exists a constant .c = c(p, d, ϕ) > 0 (not depending on .r > 0) such that for all p 1,p (B d (x ))d ) ∩ L∞ (I (t ), L2 (B d (x ))d ), satisfying for almost .u ∈ L (Ir (t0 ), W r 0 r 0 r 0 .

Rd+1 ,

8 Existence Theory for .p − < 2

302

 every .t ∈ Ir (t0 ) that . B d (x0 ) u(t, x)ϕr (x) dx = 0 in .Rd , it holds r

|u(t, x)|p∗ ϕr (x) dtdx ≤ c 1 + ess sup r p∗ t∈Ir (t0 )

.

Qr (z0 )

×

Brd (x0 )

|u(t, x)|2 ϕr (x) dx r2



|∇u(t, x)|p dtdx . Qr (z0 )

Proof See [175, Proposition 8.4].



Next, we will prove a generalized version of Zhikov’s and Pastukhova’s hydromechanical parabolic compensated compactness principle (cf. [140, Lemma 3.4]). Proposition 8.3 (Zhikov–Pastukhova, Generalized) Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞,.QT := I × , and let .p, q := max{2, p∗ − ε} ∈ Plog (QT ) for some .ε ∈ 0, d2 p− be such that for fixed .α ∈ {0, 1}, it holds − ≤ p α := αp + (1 − α)2 and .p C 3d 2d + (1 − α) < p− ≤ p+ ≤ q + < (p− )∗ . d +2 d +2

α

.

(8.5)

let .(un )n∈N ⊆ Lp◦ (I, W 1,p◦ ()d ) ∩ Y ∞ (QT ), where .p◦ ∈ Moreover, + α − . max{p , p }, (p )∗ , be a sequence such that there exist representa tions .((un )ω )n∈N ⊆ Cω0 (I , Y ) and let .(f n )n∈N ⊆ Lq (·,·) (QT )d, .(F n )n∈N ⊆   q (·,·) (Q )d and .(G ) p (·,·) (Q , Md×d ) Lp◦ (QT , Md×d T n n∈N ⊆ L T sym ), .(g n )n∈N ⊆ L sym be sequences with the following properties: (i) For every .n ∈ N, it holds .div(un ) = 0 in .Lp◦ (QT ) and for every .φ ∈ ∞ (Q )d , it holds C0,div T

.

−

un (s, y) · ∂t φ(s, y) dsdy QT

+

f n (s, y) · φ(s, y) + F n (s, y) : ε(φ)(s, y) dsdy

QT

= QT

g n (s, y) · φ(s, y)

+ [Gn (s, y) + αun (s, y) ⊗ un (s, y)] : ε(φ)(s, y) dsdy . −

(ii) .un u in .Lp (QT )d .(n → ∞), .ε(un ) ε(u) in .Lp(·,·) (QT , Md×d sym ) ∗

(n → ∞), .un u in .Y ∞ (QT ) .(n → ∞) and .un (t) u(t) in Y .(n → ∞) for a.e. .t ∈ I , where .u ∈ Y ∞ (QT ) has a representation .uω ∈ Cω0 (I , Y ).

.

8.1 Second Parabolic Compensated Compactness Principle

303





(iii) .f n f in .Lq (·,·) (QT )d .(n → ∞), .F n F in .Lp◦ (QT , Md×d sym ) .(n → ∞),  (·,·)  (·,·) d×d p q d (QT , Msym ), .g n → g in .L (QT ) .(n → ∞) and where .F ∈ L  (·,·) d×d p .Gn → G in .L (QT , Msym ) .(n → ∞). ∗

(iv) .μn :=a n dtdx μ=a 0 dtdx + μs in .M(QT ) .(n → ∞), where .a 0 ∈ L1 (QT , R≥0 ) and .μs ∈ M(QT ), with .μs ⊥ dtdx, and .(a n )n∈N := (f n · un + 1 .F n : ε(un ))n∈N ⊆ L (QT , R≥0 ) is bounded. Then, it holds .a 0 ≤ a := f · u + F : ε(u) almost everywhere in .QT . Remark 8.1 The control parameter .α ∈ {0, 1} shall help us to distinguish between the presence of the convective term in (i), i.e., when .α = 1, and its absence, i.e., when .α = 0. To be more precise, in the case of the absence of the convective term in (i), the assertion of Proposition 8.3 improves, in the sense that the imposed lower 2d 3d bound reduces from .p− > d+2 to .p− > d+2 . This circumstance will later allow us 2d − . p ≥ 2 in Theorem 7.3 to .p− > d+2 . to relax the required lower bound Proof of Proposition 8.3 We distinguish between the cases .α = 1 and .α = 0. Nevertheless, since the procedure is more or less the same, we give a comprehensive proof only for the case .α = 1 and highlight merely those sections of the proof for the case .α = 0 that deviate from the proof of the case .α = 1. The case .α = 1: We divide the proof into five main parts: 1. Localization of assumptions: Let .G ⊆  be a domain with .∂G ∈ C 1,1 and set .Q := I × G. Apparently, due to (ii), the sequence .(un |Q )n∈N is bounded in p− (I, W 1,p− (G)d ) ∩ L∞ (I, L2 (G)d ) and satisfies .u (t) u(t) in .L2 (G)d for .L n almost every .t ∈ I . Thus, we infer, using Proposition 3.7 and Corollary 3.3, that −)

un u

in L(p

un → u

in Ls (Q)d

(n → ∞)

for all s < (p− )∗ , .

(8.7)

un (t) → u(t)

in L2 (G)d

(n → ∞)

for a.e. t ∈ I ,

(8.8)

∗

(Q)d

(n → ∞) , .

.

(8.6)

where (8.8) follows again from (8.7) after possibly switching to a subsequence 2d (cf. (8.5)). In particular, it is easy to see that because .(p− )∗ > 2 for .p− > d+2 the convergences (8.6) and (8.7) imply that .un

⊗ un u ⊗ u

un ⊗ un → u ⊗ u

in L

(p− )∗ 2

(Q, Md×d sym )

in Ls (Q, Md×d sym )

(n → ∞) , . (n → ∞)

(8.9) for all s
1, then we conclude from (iii) and (8.9),    (p− )∗  −   ≤ min (p+ ) , (p2 )∗ due to .p◦ ≥ p+ , where we exploit that .r◦ = min p◦ , 2 the convergences in L(q

hn h .

+ )

(Q)d



in Lr◦ (Q, Md×d sym )

Hn H

(n → ∞) ,

(8.12)

(n → ∞) ,

+ 



where .h := g − f ∈ L(q ) (Q)d and .H := G − F + u ⊗ u ∈ Lr◦ (Q, Md×d sym ). If ∞ (Q)d , then, also taking we employ (i) for every .n ∈ N with an arbitrary .φ ∈ C0,div ∞ (Q)d , we into account the above definitions (8.11), for every .n ∈ N and .φ ∈ C0,div observe that − un (s, y) · ∂t φ(s, y) dsdy Q

.

(8.13)

=

hn (s, y) · φ(s, y)+H n (s, y) : ∇φ(s, y) dsdy . Q

Therefore, if we pass for .n → ∞ in (8.13), using (8.12) and (8.7) in doing so, then ∞ (Q)d we conclude that for every .φ ∈ C0,div

− u(s, y) · ∂t φ(s, y) dsdy Q

.

=

(8.14) h(s, y) · φ(s, y) + H (s, y) : ∇φ(s, y) dsdy .

Q

2. Local pressure reconstruction: From now until the end of the proof, we always identify the functions .(un )n∈N ⊆ Y ∞ (QT ) and .u ∈ Y ∞ (QT ) with the representations .((un )ω )n∈N ⊆ Cω0 (I , Y ) and .uω ∈ Cω0 (I , Y ), respectively. Then, on the basis of (8.8), for fixed time slice .t0 ∈ I , for which un (t0 ) → u(t0 )

.

in L2 (G)d

(n → ∞)

(8.15)

8.1 Second Parabolic Compensated Compactness Principle

305

holds, we apply the second pressure reconstruction theorem (cf. Proposition 6.3 and Remark 6.1). In this way, for every .n ∈ N, we obtain functions π 1,n ∈ L(q

.

+ )

(I, W 1,(q

+ )

(q + )

(G) ∩ L0

(G)) ,

p



π 2,n ∈ Lp◦ (I, L0 ◦ (G)) , 0 π th,n ∈ Cω0 (I , W 1,2 (G) ∩ L20 (G)) ,

π 1 ∈ L(q r◦

+ )

(I, W 1,(q

+ )

(q + )

(G) ∩ L0

(G)) ,

r◦

π 2 ∈ L (I, L0 (G)), π th0 ∈ Cω0 (I , W 1,2 (G) ∩ L20 (G)) , 0 0 with .(π th,n (t)) = (π th0 (t)) = 0 in .(W01,2 (G))∗ for all .t ∈ I and .π th,n (t0 ) = t0 ∞ 1,2 d π h (t0 ) = 0 in .W (G), such that for every .n ∈ N and .φ ∈ C0 (Q) , it holds

.

− Q

0 [un (s, y) + ∇π th,n (s, y)] · ∂t φ(s, y) dsdy

[hn (s, y) + ∇π 1,n (s, y)] · φ(s, y) dsdy

=

(8.16)

Q

+

[H n (s, y) + π 2,n (s, y)Id ] : ∇φ(s, y) dsdy , Q

and for every .φ ∈ C0∞ (Q)d , it holds

.

− Q

[u(s, y) + ∇π th0 (s, y)] · ∂t φ(s, y) dsdy

[h(s, y) + ∇π 1 (s, y)] · φ(s, y) dsdy

=

(8.17)

Q

+

[H (s, y) + π 2 (s, y)Id ] : ∇φ(s, y) dsdy . Q

0 As we already noted in Remark 6.1, the functions .(π th,n )n∈N and .π th0 depend on the choice of .t0 ∈ I with (8.15), indicated by the superscript .t0 , but the remaining functions .(π 1,n )n∈N , .π 1 , .(π 2,n )n∈N and .π 2 are completely independent of .t0 ∈ I . This circumstance will be crucial later in step 5. In addition, for every .n ∈ N, there

8 Existence Theory for .p − < 2

306

hold the estimates (cf. Proposition 6.3) π 1,n L(q + ) (I,W 1,(q + ) (G)) ≤ cq + ,G hn L(q + ) (Q)d ,

(8.18)

.

π 2,n Lr◦ (Q) ≤ cr◦ ,G H n Lr◦ (Q)d×d , 0 (t)W 1,2 (G) ≤ ch,G un (t) − un (t0 )L2 (G)d π th,n

for all t ∈ I ,

where the constants .cq + ,G , cr◦ ,G , ch,G > 0 do not depend on .n ∈ N. Based on (8.12), (8.15) and (ii), following the proof of Proposition 7.19, step 2, we deduce that

.

+ )

(I, W 1,(q

+ )

(q + )

(G) ∩ L0

(n → ∞) ,

π 1,n π 1

in L(q

π 2,n π 2

in Lr◦ (I, L0◦ (G))

(n → ∞) ,

in L∞ (I, W 1,2 (G) ∩ L20 (G))

(n → ∞) ,

r



∗

0

π th0 π th,n

(G))

(8.19)

and for every .G ⊂⊂ G and .s ∈ (1, ∞), in addition, the local strong convergence in Ls (I, W 2,s (G )) (n → ∞) .

0 π th,n → π th0

.

(8.20)

Apart from that, since the function .π th0 (t) ∈ W 1,2 (G) is a harmonic function for every .t ∈ I , the local higher regularity theory (cf. [67, 6.3.1., Thm. 2]) proves that .π th0 (t) ∈ W 2+d,2 (G ) for all .t ∈ I and .G ⊂⊂ G. Moreover, [67, 6.3.1., Thm. 2] yields .π th0 (t)W 2+d,2 (G ) ≤ c(d, G , G)π th0 (t)L2 (G) for every   .t ∈ I , where .c(d, G , G) > 0 depends only on .d ∈ N and .dist(G , ∂G) > 0, i.e., t0 t0 ∞ 2+d,2  0 2 .π (G )). Thus, since also .π h ∈ Cω (I , L (G)), Proposition 2.19 h ∈ L (I, W yields that .π th0 ∈ Cω0 (I , W 2+d,2 (G )) for all .G ⊂⊂ G, which, due to the compact embedding .W 2+d,2 (G ) →→ C 2 (G ) (cf. [5, Thm. 6.3]) for all .G ⊂⊂ G with  0,1 , shows that .π t0 ∈ C 0 (I , C 2 (G )) for all .G ⊂⊂ G with .∂G ∈ C 0,1 . .∂G ∈ C h 3. Preparatory work for Proposition 8.1/8.2: In this step, we make all preparations necessary to apply Proposition 8.1 and Proposition 8.2 right after passing for .n → ∞. We fix an arbitrary domain .G ⊂⊂ G with .∂G ∈ C ∞ and, on the basis of (8.8), an interval .I  := (t0 , t1 ) ⊆ I such that un (t1 ) → u(t1 )

.

in L2 (G )d

(n → ∞) ,

(8.21)

and define .Q := I  × G . For .A ∈ Rd×d with .tr(A) = 0, .ξ ∈ (C0∞ (G )) \ {0} and ∞ 2   ∈ I × G and .n ∈ N .ϕ := ξ ∈ C (G ), we introduce for every .(t, x) 0 mnϕ,A (t) :=  .

unϕ,A (t, x)

:=

1 ϕ(y) dy  G

G

0 [un (t, y) + ∇π th,n (t, y) − Ay]ϕ(y) dy ,

0 un (t, x) + ∇π th,n (t, x) − Ax

− mnϕ,A (t) ,

(8.22)

8.1 Second Parabolic Compensated Compactness Principle

307

i.e., we have that .(unϕ,A )n∈N ⊆ Lp◦ (I, W 1,p◦ (G )d ) ∩ L∞ (I, L2 (G )d ) as well as n ) ∞ d   .(m ϕ,A n∈N ⊆ L (I ) , and analogously for every .(t, x) ∈ I × G 1 .mϕ,A (t) :=  G ϕ(y) dy

G

[u(t, y) + ∇π th0 (t, x) − Ay]ϕ(y) dy ,

uϕ,A (t, x) := u(t, x) + ∇π th0 (t, x) − Ax − mϕ,A (t) , −

−

i.e., we have that .uϕ,A ∈ Lp (I, W 1,p (G )d ) ∩ L∞ (I, L2 (G )d ) as well as .mϕ,A ∈ L∞ (I )d . We introduced these functions because, by construction, they satisfy for every .n ∈ N

.

G

unϕ,A (t, x)ϕ(x) dx =

G

uϕ,A (t, x)ϕ(x) dx = 0

for a.e. t ∈ I ,

(8.23)

which is essential for the applicability of both Propositions 8.1 and 8.2. Since 0 tr(A) = 0 and .(π th,n ) = (π th0 ) = div(un ) = div(u) = 0 almost everywhere in Q for every .n ∈ N, we have that

.

div(unϕ,A ) = div(uϕ,A ) = 0

.

a.e. in I × G

(8.24)

for every .n ∈ N. In addition, (8.7) and (8.20) also imply .mnϕ,A → mϕ,A in .Ls (I )d − .(n → ∞) for every .s < (p )∗ , from which we, in turn, again by resorting to (8.7) and (8.20), conclude that unϕ,A → uϕ,A

.

in Ls (I × G )d

Due to .(p− )∗ > 2 for .p− > unϕ,A (t) → uϕ,A (t)

.

2d d+2 ,

(n → ∞)

for all s < (p− )∗ .

(8.25)

we may assume, without loss of generality, that

in L2 (G )d

(n → ∞)

for t ∈ {t0 , t1 } .

(8.26)

For .t = t0 , (8.26) results from (8.15) and .mnϕ,A (t0 ) → mϕ,A (t0 ) .(n → ∞), as

0 π th,n (t0 ) = π th0 (t0 ) = 0 for every .n ∈ N. For .t = t1 , we may switch to a smaller interval .I˜ = (t0 , t˜1 ) ⊆ I  that satisfies both (8.21) and (8.26). On the other hand, similar to Proposition 7.19, if we test (8.16) for all .n ∈ N with .φξ ∈ C0∞ (Q )d , where .φ ∈ C0∞ (Q )d , then for every .φ ∈ C0∞ (Q )d and .n ∈ N, we observe that

0 .− ξ(y)[un (s, y) + ∇π th,n (s, y)] · ∂t φ(s, y) dsdy (8.27)

.

Q

=

Q

ξ(y)[hn (s, y) + ∇π 1,n (s, y)] · φ(s, y) dsdy

+

Q

[H n (s, y) + π 2,n (s, y)Id ] : [∇φ(s, y)ξ(y) + φ(s, y) ⊗ ∇ξ(y)] dsdy .

8 Existence Theory for .p − < 2

308

If we introduce .σ◦ := max{q + , r◦ }, i.e., it holds .σ◦ < (p− )∗ due to (8.5) 3d and .((p− )∗ /2) < (p− )∗ for .p− > d+2 , then it can be read from (8.27) that σ◦ ,p◦ t0  .(ξ [un + ∇π (Q ) (cf. Definition 3.6) and for every .n ∈ N h,n ])n∈N ⊆ Wε

.

⎫ ⎪ ⎬

 d 0 ] = Jε (ξ [hn +∇π 1,n ], 0) ξ [un +∇π th,n dt

⎪ +Jε (H n ∇ξ +π 2,n ∇ξ, ξ [H n +π 2,n Id ])⎭

σ ,p◦

in Xε ◦

(Q )∗ . (8.28)



As a consequence, on the basis of .(mnϕ,A )n∈N ⊆ W 1,σ◦ (I  , Rd )1 , we even have that σ◦ ,p◦ n ) .(ξ u (Q ) with representations .((ξ unϕ,A )c )n∈N ⊆ Y 0 (Q ) (cf. ϕ,A n∈N ⊆ Wε Proposition 3.26 (i)) such that for every .n ∈ N dξ unϕ,A dt

=

.

 d 0 ] ξ [un + ∇π th,n dt   d d mnϕ,A ,0 − Jε ξ R dt

(8.29) in

σ ,p Xε ◦ ◦ (Q )∗ .



be more precise, we have that .mnϕ,A ∈ W 1,σ◦ (I  , Rd ) (cf. Definition 2.19 and Proposition 2.24 (i)) for every .n ∈ N with

1 To

dRd mnϕ,A dt .

1 [hn (·, y) + ∇π 1,n (·, y)]ϕ(y) dy G G ϕ(y) dy

[H n (·, y) + π 2,n (·, y)Id ]∇ϕ(y) dy +

= 

G

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭



in Lσ◦ (I  , Rd ) .

In fact, let .η ∈ C0∞ (I  , Rd ) be arbitrary. Then, it holds .φ := ηϕ ∈ C0∞ (Q )d with .∂t φ = ϕ∂t η ∈ C0∞ (Q )d . Therefore, by taking into account (8.16), for every .η ∈ C0∞ (I  , Rd ) and for every .n ∈ N, we observe that

1 0 .− mnϕ,A (s) · ∂t η(s) ds = −  [un (s, y) + ∇π th,n (s, y) − Ay] · ∂t φ(s, y) dsdy  ϕ(y) dy  I Q G

1 0 = − [un (s, y) + ∇π th,n (s, y)] · ∂t φ(s, y) dsdy  ϕ(y) dy  Q G

1  [hn (s, y) + ∇π 1,n (s, y)]ϕ(y) dy = I  G ϕ(y) dy G

[H n (s, y) + π 2,n (s, y)Id ]∇ϕ(y) dy · η(s) ds . + G

8.1 Second Parabolic Compensated Compactness Principle

309

Thus, testing (8.29) for every .n ∈ N with .ξ unϕ,A ∈ Wε ◦ ◦ (Q ) and a subsequent σ ,p application of the formula of integration-by-parts for .Wε ◦ ◦ (Q ) (cf. Proposition 3.26 (ii)) in conjunction with (8.23), also making use of that .ϕ = ξ 2 , yields for every .n ∈ N that σ ,p



.

  d 0 ξ [un + ∇π th,n ] , ξ unϕ,A σ ,p dt Xε ◦ ◦ (Q )     dRd mnϕ,A dξ unϕ,A n + Jε ξ , 0 , ξ uϕ,A = σ ,p dt dt Xε ◦ ◦ (Q ) s=t1 1 (ξ unϕ,A )c (s)2L2 (G )d = s=t0 2

d mn d ϕ,A R n (s) · uϕ,A (s, y)ϕ(y) dy ds + dt G I s=t1 1 (ξ unϕ,A )c (s)2L2 (G )d = . s=t0 2

(8.30)

4. Local passage to the limit with respect to .n → ∞: If we test the identity σ ,p (8.28) for every .n ∈ N with .ξ unϕ,A ∈ Xε ◦ ◦ (Q ), also using that .ϕ = ξ 2 , then we get for every .n ∈ N 

 d t0  n ξ [un + ∇π h,n ] , ξ uϕ,A σ ,p dt Xε ◦ ◦ (Q ) = Jε (ξ [hn + ∇π 1,n ], 0), ξ unϕ,A Xεσ◦ ,p◦ (Q )

.

+ Jε (H n ∇ξ + π 2,n ∇ξ, ξ [H n + π 2,n Id ]), ξ unϕ,A Xεσ◦ ,p◦ (Q )

= [hn (s, y) + ∇π 1,n (s, y)] · unϕ,A (s, y)ϕ(y) dsdy

(8.31)

Q

+

Q

[H n (s, y) + π 2,n (s, y)Id ] : ε(unϕ,A ϕ)(s, y) dsdy .

If we recall the definitions of the functions .(unϕ,A )n∈N ⊆ Lp◦ (I  , W 1,p◦ (G )d ) ∩ 

L∞ (I  , L2 (G )d ) (cf. (8.22)) and .(H n )n∈N ∈ Lp◦ (Q , Md×d sym ) (cf. (8.11)), using

8 Existence Theory for .p − < 2

310

(8.24), then for every .n ∈ N, the last integral in (8.31) can be decomposed into

.

Q

[H n (s, y) + π 2,n (s, y)Id ] : ε(unϕ,A ϕ)(s, y) dsdy

(8.32)

=

[Gn (s, y) − F n (s, y)] : ε(unϕ,A ϕ)(s, y) dsdy

Q

+

Q

π 2,n (s, y)Id : ε(unϕ,A ϕ)(s, y) dsdy

+

Q

=

[Gn (s, y) − F n (s, y)] : ε(un )(s, y)ϕ(y) dsdy

Q

+

un (s, y) ⊗ un (s, y) : ε(unϕ,A ϕ)(s, y) dsdy

Q

0 [Gn (s, y) − F n (s, y)] : [∇ 2 π th,n (s, y) − A]ϕ(y) dsdy

+

Q

[Gn (s, y) − F n (s, y) + π 2,n (s, y)Id ] : ∇ϕ(y) ⊗ unϕ,A (s, y) dsdy

+

Q

un (s, y) ⊗ un (s, y) : ε(unϕ,A ϕ)(s, y) dsdy .

Moreover, by successively integrating by parts, exploiting that .div(un ) = 0 in Lp◦ (Q ) for all .n ∈ N, and by applying the product rule .div(u⊗v) = Duv+udiv(v) in .L1 (G )d for every .u, v ∈ W 1,2 (G )d , for every .n ∈ N, the last integral in (8.32) can be rewritten as

. un (s, y) ⊗ un (s, y) : ε(unϕ,A ϕ)(s, y) dsdy (8.33)

.

Q

=−

Q

=−

Q

div(un ⊗ un )(s, y) · unϕ,A (s, y)ϕ(y) dsdy 

 Dun (s, y)un (s, y) · unϕ,A (s, y)ϕ(y) dsdy

=−

Q

unϕ,A (s, y) ⊗ un (s, y) : Dun (s, y)ϕ(y) dsdy

=−

−

Q

Q

unϕ,A (s, y) ⊗ un (s, y) : Dunϕ,A (s, y)ϕ(y) dsdy 0 unϕ,A (s, y) ⊗ un (s, y) : [A − ∇ 2 π th,n (s, y)]ϕ(y) dsdy

8.1 Second Parabolic Compensated Compactness Principle

=−

−

Q

=

Q

|unϕ,A |2 2

(s, y) · un (s, y)ϕ(y) dsdy

0 unϕ,A (s, y) ⊗ un (s, y) : [A − ∇ 2 π th,n (s, y)]ϕ(y) dsdy

|unϕ,A (s, y)|2 2

Q

−

∇

Q

311

un (s, y) · ∇ϕ(y) dsdy

0 unϕ,A (s, y) ⊗ un (s, y) : [A − ∇ 2 π th,n (s, y)]ϕ(y) dsdy .

We recall that .a n = f n · un + F n : ε(un ) in .L1 (Q , R≥0 ) for all .n ∈ N. Then, for every .n ∈ N, we get

.

Q

a n (s, y)ϕ(y) dsdy = [f n (s, y) · un (s, y)+Gn (s, y) : ε(un )(s, y)]ϕ(y) dsdy Q

+

Q

[hn (s, y) + ∇π 1,n (s, y)] · unϕ,A (s, y)ϕ(y) dsdy

s=t1 1 (ξ unϕ,A )c (s)2L2 (G )d s=t0 2

0 + [Gn (s, y) − F n (s, y)] : [∇ 2 π th,n (s, y) − A]ϕ(y) dsdy −

Q

+

Q

+

Q

+

Q

[Gn (s, y) − F n (s, y) + π 2,n (s, y)Id ] : ∇ϕ(y) ⊗ unϕ,A (s, y) dsdy 0 unϕ,A (s, y) ⊗ un (s, y) : [∇ 2 π th,n (s, y) − A]ϕ(y) dsdy

|unϕ,A (s, y)|2 2

un (s, y) · ∇ϕ(y) dsdy ,

(8.34)

where the right-hand side results from side if we just add the sum the left-hand t0  d n 1 σ ,p ξ [u +  + ∇π ] − 12 [(ξ unϕ,A )c (s)2L2 (G )d ]s=t n s=t0 h,n , ξ uϕ,A Xε ◦ ◦ (Q ) = 0 (cf. dt (8.30)) and use (8.31)–(8.33). In particular, the first term on the right-hand side in (8.34) results from the left-hand side if we add the fifth line in (8.32). Taking the limit with respect to .n → ∞ on either side in (8.34), also taking into account (ii), (iii), (iv) in conjunction with Proposition 7.6 (iii), (8.7), (8.19), (8.20), (8.25) and

.

8 Existence Theory for .p − < 2

312

(8.26), we arrive at

. a 0 (s, y)ϕ(y) dsdy ≤ Q

≤

Q

ϕ dμ

(8.35)

[f (s, y) · u(s, y) + F (s, y) : ε(u)(s, y)]ϕ(y) dsdy

Q

+

Q

[h(s, y) + ∇π 1 (s, y)] · uϕ,A (s, y)ϕ(y) dsdy

s=t1 1 uϕ,A (s)ξ 2L2 (G )d s=t0 2

+ [G(s, y) − F (s, y)] : [∇ 2 π th0 (s, y) + ε(u)(s, y) − A]ϕ(y) dsdy −

+

Q

Q

+

Q

+

Q

[G(s, y) − F (s, y) + π 2 (s, y)Id ] : ∇ϕ(y) ⊗ uϕ,A (s, y) dsdy uϕ,A (s, y) ⊗ u(s, y) : [∇ 2 π th0 (s, y) − A]ϕ(y) dsdy |uϕ,A (s, y)|2 u(s, y) · ∇ϕ(y) dsdy , 2 



p (·,·) (Q , Md×d ) in the where we replaced .G ∈ Lp (·,·) (QT , Md×d T sym ) by .F ∈ L sym first line in (8.35) after passing for .n → ∞ in (8.34), which resulted in the added summand .ε(u) ∈ Lp(·,·) (QT , Md×d sym ) in the fourth line in (8.35), and where also  used that . Q ϕ dμs ≥ 0 since .μs is a positive measure and .ϕ ≥ 0. It is important to 3d note that the ability to pass for .n → ∞ in (8.34) is essentially based on .p− > d+2 . In fact, the convergences of the fifth and the seventh line in (8.34) depend on this assumption since it guarantees that .(p− )∗ > r◦ ≥ 3 and, therefore, that .unϕ,A → uϕ,A in .Lr◦ (Q )d .(n → ∞) (cf. (8.25)), .un → u in .Lr◦ (Q )d .(n → ∞) (cf. (8.7))  and .π 2,n π 2,n in .Lr◦ (Q ) .(n → ∞). However, we will observe in the proof of the case .α = 0, i.e., the case in which .r◦ = p◦ and the seventh line in (8.34) does 3d not occur, that the requirement .p− > d+2 becomes redundant. Note also that the convergences of the first summand in the first integral on the right-hand side in (8.34) and of the integral in the second line in (8.34) are based on the assumption .q + < (p− )∗ . 5. Passage to the limit with the localization: Let .z0 := (t0 , x0 ) ∈ Q be a fixed point for which (8.15) holds and all following mean value integrals with respect to radii .(rn )n∈N ⊆ R>0 , where .(rn )n∈N is a suitable null sequence, containing .f , .g, .F , .G, .a 0 , .a, .u, .ε(u), .∇π 1 , .π 2 , and .∇u(t0 ) converge for .z0 ∈ Q in terms of Proposition 9.1 (ii) by passing for .rn → 0 .(n → ∞). Here, we have to assume that − the evaluations .∇u(z0 ) ∈ Rd×d and .∇u(t0 ) ∈ Lp ()d×d are well-defined. In fact, thanks to (8.8) and Proposition 9.1 (ii), after switching finitely many times to a subsequence of .(rn )n∈N ⊆ R>0 , the above assumptions are satisfied for almost

8.1 Second Parabolic Compensated Compactness Principle

313

every .z0 = (t0 , x0 ) ∈ Q. In particular, we take advantage of the fact that .π 1 and .π 2 are independent of .t0 ∈ I (cf. Remark 6.1). Since we are already aware of that .π th0 ∈ C 0 (I , C 2 (G )) for all .G ⊂⊂ G with .∂G ∈ C 0,1 , that is, .∇ 2 π th0 ∈ C 0 (I , C 0 (G )d×d ) ⊆ C 0 (I × G )d×d for all subsets .G ⊂⊂ G with .∂G ∈ C 0,1 , we can, furthermore, guarantee that all mean value integrals with respect to the radii 2 t0 converge for .z = (t , x ) ∈ Q in terms of .(rn )n∈N ⊆ R>0 containing .∇ π 0 0 0 h Proposition 9.1 (ii) if we pass for .rn → 0 .(n → ∞).   Moreover, we define .Qr (z0 ) := Ir (t0 ) × Brd (x0 ), where .Ir (t0 ) := t0 , t0 + r 2 , for .r > 0. If we choose .A := ∇u(z0 ), i.e., .tr(∇u(z0 )) = div(u)(z0 ) = 0, .G := Brd (x0 ), .I  := (t0 , t1 ), for .t1 ∈ Ir (t0 ) such that .unϕ,A (t1 ) → uϕ,A (t1 ) in .L2 (G )d  · −x0 2 2  d .(n → ∞) and .un (t1 ) → u(t1 ) in .L (G ) .(n → ∞), and .ϕ := ϕr := ξ r ) ,   3d (γ∗ ) d ∞ d − with .γ∗ > max{p◦ , q + },2 .ξ := η 2 , .η ∈ C (B (0)) ∩ SM(R ), .γ ∈ 0 1 d+2 , p ffl 2 p− 1,p− (B d (x ))d ) ∩ . r 0 B d (0) ξ(x) dx = 1, in (8.35), set .ur := uϕr ,∇u(z0 ) ∈ L (I, W 1

L∞ (I, L2 (Brd (x0 ))d ) (cf. (8.22)) and divide by .|Qr (z0 )| = r 2 |Brd (x0 )| > 0, we get for every .r > 0 and almost every .t1 ∈ Ir (t0 )

1 t1 0≤ . a 0 (s, y)ϕr (y) dsdy (8.36) r 2 t0 Brd (x0 )

1 t1 a(s, y)ϕr (y) dsdy ≤ 2 r t0 Brd (x0 )

s=t1 1 |ur (s, y)|2 − ϕr (y) dy 2 Brd (x0 ) r2 s=t0  |F (s, y)| + |G(s, y)|

+ Qr (z0 )

 × |∇ 2 π th0 (s, y)| + |ε(u)(z0 ) − ε(u)(s, y)| ϕr (y) dsdy  |F (s, y)| + |G(s, y)| + |π 2 (s, y)| |ur (s, y)||∇ϕr (y)| dsdy

+ Qr (z0 )

+ Qr (z0 )

+ Qr (z0 )

 |∇u(z0 )| + |∇ 2 π th0 (s, y)| |ur (s, y)||u(s, y)|ϕr (y) dsdy |ur (s, y)|2 |u(s, y)||∇ϕr (y)| dsdy 2  |h(s, y)| + |∇π 1 (s, y)| |ur (s, y)|ϕr (y) dsdy

+ Qr (z0 )

to .p − > guaranteed.

2 Thanks

3d d+2

and .(p − )∗ > max{p◦ , q + } (cf. (8.5)), the existence of such a constant is

8 Existence Theory for .p − < 2

314

1 =: 2 r

t1 Brd (x0 )

t0



1 − 2

Brd (x0 )

a(s, y)ϕr (y) dsdy

|ur (s, y)|2 ϕr (y) dy r2

s=t1

+

s=t0

5 

Iir (z0 ) ,

i=1

where we used .[G − F ] : ∇u(z0 ) = [G − F ] : ε(u)(z0 ) in .L1 (QT ), since .G, F ∈  Lp (·,·) (QT , Md×d sym ), to obtain the third line in (8.36) from the fourth line in (8.35). Moreover, for every .r > 0, we define Ar (z0 ) := ess sup

.

t∈Ir (t0 )

1 2

Brd (x0 )

|ur (t, y)|2 ϕr (y) dy . r2

Then, observing that .t1 ∈ Ir (t0 ) was chosen arbitrarily in (8.36), we infer from (8.36) for every .r > 0 Ar (z0 ) ≤

a(s, y)ϕr (y) dsdy Qr (z0 )

.

1 + 2

(8.37)

 |ur (t0 , y)|2 ϕr (y) dy + Iir (z0 ) . 2 r 5

Brd (x0 )

i=1

2d it holds By resorting to Proposition 8.1, which is allowed since for .p− > d+2 t0 − ∗ 2 .(p ) > 2, also using that we have .∇ur (t0 , ·) = ∇u(t0 , ·) + ∇ π (t0 , ·) − ∇u(z0 ) h − = ∇u(t0 , ·) − ∇u(z0 ) in .Lp (Brd (x0 ))d×d , owing to .π th0 (t0 ) = 0 (cf. Remark 6.1), for every .r > 0, we observe that

1 2

Brd (x0 )

|ur (t0 , y)|2 ϕr (y) dy. ≤ c r2

 Brd (x0 )

|∇u(t0 , y) − ∇u(z0 )|

=: Br (z0 ) .

p−

 ϕr (y) dy

2 p−

(8.38)

Then, by the choice of .z0 = (t0 , x0 ) ∈ Q and the sequence .(rn )n∈N ⊆ R>0 , we find that Brn (z0 ) → |∇u(z0 ) − ∇u(z0 )|2 = 0

.

(n → ∞) .

(8.39)

Thus, if .Iirn (z0 ) → 0 .(n → ∞) for every .i = 1, . . . , 5, then we conclude from (8.36) to .a 0 (z0 ) ≤ a(z0 ). So, let us verify that .Iirn (z0 ) → 0 .(n → ∞) holds for every .i = 1, . . . , 5. Making again use of .π th0 (t0 ) = 0 in .W 1,2 (G) ∩ L20 (G), i.e., d×d 2 t0 .∇ π (z0 ) = 0 in .Msym , we get that h I1rn (z0 )→ 0

.

(n → ∞) .

(8.40)

8.1 Second Parabolic Compensated Compactness Principle

For .(I2rn (z0 ))n∈N , recalling that .γ ∈ note that for any .r > 0



3d − d+2 , p



315

with .γ∗ > max{p◦ , q + }, we first

 x − x (γ∗ ) −1  x − x  1 0 0 ∇η (γ∗ ) η , r r r cη ≤ γ ϕr (x) , for all x ∈ Brd (x0 ) , r ∗

∇ϕr (x) = .

i.e., |∇ϕr (x)|γ∗

(8.41)

where .cη > 0 is independent of .r > 0. By applying Hölder’s inequality with respect to . r1◦ + γ1∗ < 1, where we exploit, in particular, that .r◦ ≤ p◦ ≤ p◦ < γ∗ and also .(8.41)2 , we obtain for every .n ∈ N rn 2 (z0 ) ≤ c





.I

Qrn (z0 )





|G(s, y)|r◦ + |F (s, y)|r◦ + |π 2 (s, y)|r◦ dsdy

≤ c(z0 )Crn (z0 ) ,

 1

r◦

Crn (z0 )

(8.42)

where for every .r > 0  Cr (z0 ) :=

.

Qr (z0 )

|ur (s, y)|γ∗ ϕr (y) dsdy r γ∗



1 γ∗

,

and where we have, apart from that, exploited the fact that the first integral on the right-hand side in .(8.42)1 converges, by the choice of the point .z0 ∈ Q and the null sequence .(rn )n∈N ⊆ R>0 , if we pass for .n → ∞ and, hence, is bounded by a constant .c(z0 ) > 0, which depends on the fixed point .z0 ∈ Q but not on .n ∈ N. On the other hand, based on .γ < p− ≤ pα = 3d+2 d+2 ≤ d, Proposition 8.2 provides a constant .c > 0, which does not depend on .n ∈ N, such that for every .n ∈ N  Crn (z0 )γ∗ ≤ c 1 + Arn (z0 ) Drn (z0 ) ,

(8.43)

.

−

where, due to the identity .∇urn = ∇u + ∇ 2 π th0 − ∇u(z0 ) in .Lp (I × Brdn (x0 ))d×d for every .n ∈ N, .π th0 (t0 ) = 0 in .W 1,2 (G) ∩ L20 (G) and by the choice of .z0 ∈ Q and .(rn )n∈N ⊆ R>0 , we have that .

Drn (z0 ) :=

Qrn (z0 )

  ∇ur (s, y)γ dsdy → 0 n

(n → ∞) .

(8.44)

Hence, if .(Arn (z0 ))n∈N ⊆ R≥0 is bounded, then we infer .Crn (z0 ) → 0 .(n → ∞) from (8.44) in (8.43). By applying Hölder’s inequality with . γ1∗ + γ1∗ + γ1∗ < 1

8 Existence Theory for .p − < 2

316

because .γ >

we obtain for every .n ∈ N

3d d+2 ,





I3rn (z0 ) ≤ crn

|∇u(z0 )| + |∇ 2 π th0 (s, y)|

Qrn (z0 )



.



×

|u(s, y)|γ∗ ϕr (y) dsdy

1 γ∗

Qrn (z0 )

γ∗

 ϕr (y) dsdy

1 γ∗

(8.45) Crn (z0 )

≤ c(z0 )Crn (z0 ) . Similarly, by applying Hölder’s inequality with . γ1∗ + advantage of .(8.41)2 , for every .n ∈ N, we find that  I4rn (z0 )



≤c Qrn (z0 )

.

|urn (s, y)| dsdy γ∗





×

|u(s, y)|γ∗ dsdy Qrn (z0 )

1 γ∗

+

1 γ∗

< 1, taking again

1 γ∗

1 γ∗

(8.46) Crn (z0 ) .

Once again, on the basis of the choice of .z0 ∈ Q and .(rn )n∈N ⊆ R>0 , there exists a constant .c(z0 ) > 0, which depends on the fixed point .z0 ∈ Q but not on .n ∈ N, such that for every .n ∈ N 

 Qrn (z0 )

|urn (s, y)|γ∗ dsdy

1 γ∗





≤

|u(s, y)|γ∗ dsdy

1 γ∗

Qrn (z0 )





+ Qrn (z0 ) .

|∇π th0 (s, y)|γ∗

dsdy





+

|∇u(z0 )y| dsdy γ∗

1 γ∗

1 γ∗

(8.47)

Qrn (z0 )





+

|mϕrn ,∇u(z0 ) (s)| ds γ∗

Irn (t0 )

1 γ∗





≤ c(z0 ) + Qrn (z0 )

≤ c(z0 ) ,

|u(s, y) + ∇π th0 (s, y) + ∇u(z0 )y|γ∗ dsdy

1 γ∗

8.1 Second Parabolic Compensated Compactness Principle

317

where we have made use of Minkowski’s and Jensen’s inequality, and the equality  d d Br (x0 ) ϕrn (y) dy = |Br (x0 )|. Hence, we conclude from (8.47) in (8.46) that for every .n ∈ N

.

I4rn (z0 ) ≤ c(z0 )Crn (z0 ) .

(8.48)

.

Apart from that, due to . (q1+ ) + γ1∗ < 1, because .q + < γ∗ , Hölder’s inequality yields for every .n ∈ N  I5rn (z0 ) ≤ crn



.

|h(s, y)| + |∇π 1 (s, y)|

(q + )

 ϕr (y) dsdy

Qrn (z0 )

≤ c(z0 )Crn (z0 ) .

1 (q + )

Crn (z0 ) (8.49)

To summarize, if .(Arn (z0 ))n∈N ⊆ R is bounded, then we can conclude that Iirn (z0 ) → 0 .(n → ∞) for .i = 2, . . . , 5. By combining (8.37), (8.42), (8.45), (8.48), (8.49), and (8.43), for every .n ∈ N, we get

.

Arn (z0 ) ≤

.

Qrn (z0 )

≤ Qrn (z0 )

a(s, y)ϕrn (y) dsdy + Brn (z0 ) + I1rn (z0 ) + c(z0 )Crn (z0 ) a(s, y)ϕrn (y) dsdy + Brn (z0 ) + I1rn (z0 ) 1

(8.50)

1

+ c(z0 )[1 + Arn (z0 )] γ∗ Drn (z0 ) γ∗ . 1

1 Consequently, for .n0 ∈ N such that .Drn (z0 ) γ∗ ≤ 2c(z for every .n ∈ N with 0) .n ≥ n0 , which exists on the basis of (8.44), also using that .Brn (z0 ) → 0 .(n → ∞) 1 (cf. (8.39)), .I1rn (z0 ) → 0 .(n → ∞) (cf. (8.40)) and .[1 + Arn (z0 )] γ∗ ≤ 1 + Arn (z0 ) for every .n ∈ N, for every .n ∈ N with .n ≥ n0 , we observe that

1 Ar (z0 ) ≤ 2 n .

Qrn (z0 )

a(s, y)ϕrn (y) dsdy

+ Brn (z0 ) + I1rn (z0 ) +

(8.51) 1 ≤ c(z0 ) , 2

i.e., .(Arn (z0 ))n∈N is bounded. Hence, we infer .Crn (z0 ) → 0 .(n → ∞) from (8.51) and (8.44) in (8.43), which together with (8.40), (8.42), (8.45), (8.48), and (8.49) proves that .Iirn (z0 ) → 0 .(n → ∞) for .i = 1, . . . , 5. The latter and (8.39) in (8.36) imply, by the choice of .z0 ∈ Q and .(rn )n∈N ⊆ R>0 , that .a 0 (z0 ) ≤ a(z0 ). Because the assumptions on .z0 ∈ Q are satisfied almost everywhere in Q, thanks to (8.8) and Proposition 9.1 (ii), we further deduce that .a 0 ≤ a holds almost everywhere in Q. Finally, since .QT can be covered by countably many cylinders .Q = I × G, where 1,1 , we conclude that .a ≤ a almost everywhere in .Q . This .G ⊆  with .∂G ∈ C 0 T completes the proof of Proposition 8.3 in the case .α = 1.

8 Existence Theory for .p − < 2

318

The case .α = 0: In the case .α = 0, the convective term in (i) does not occur. This allows us to consider variable exponents .q, p ∈ Plog (QT ) satisfying 2d . p− ≤ p+ ≤ q + < (p− )∗ . Moreover, we have that .p− ≤ pα = 2 ≤ d and d+2 <  + − .p◦ ∈ max{2, p }, (p )∗ in this case. In principle, we proceed as in the proof of the case .α = 1 and divide the proof into five main steps. However, since the argumentation primarily stays the same except for individual arguments, we highlight merely these changing arguments instead of repeating the entire proof. To be more precise, step 1 simplifies since we are not forced to deal with the convective term. In particular, we now need to set .r◦ := p◦ . The rest of step 1 remains unchanged. Step 2 and step 3 likewise remain essentially unchanged. In step 4, the calculation (8.33) is omitted, thanks to the absence of the convective term. Moreover, the last two integrals in (8.34), which necessitated the assumption − > 3d in the case .α = 1, do not occur. Because of .r = p , we have that .p ◦ ◦ d+2 n p .π 2,n π 2 in .L ◦ (Q) .(n → ∞) in this case, which on the basis of .u ϕ,A → uϕ,A p d − ◦ in .L (Q) .(n → ∞), since .p◦ < (p )∗ , allows us to pass for .n → ∞ in the third last term in (8.34). The rest of step 4 remains unchanged. Step 5 also simplifies because the terms .I3r (z0 ) and .I4r (z0 ) are dropped. In particular, these are the only 3d terms that made use of the assumption .p− > d+2 in the case .α = 1. All told, the 2d − case .α = 0 gets along with .p > d+2 . 

Eventually, by combining Proposition 8.3 and Corollary 7.4, we now arrive at a second parabolic compensated compactness principle adjusted to the peculiarities of both the convective term and variable Bochner–Lebesgue spaces, which, thus, allows us to prove the existence of weak solutions of the unsteady .p(·, ·)-Navier– 3d Stokes equations for .p ∈ Plog (QT ) satisfying .p− > d+2 . Proposition 8.4 (Second Parabolic Compensated Compactness Principle) Let  ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0,  T ), .T < ∞, .QT := I × , log .p, q := max{2, p∗ − ε} ∈ P (QT ) for some .ε ∈ 0, d2 p− such that for fixed .α ∈ {0, 1}, it holds .

3d 2d + (1 − α) , d +2 d +2   and .r := max{p◦ , p} ∈ Plog (QT ), where .p◦ ∈ pα , (p− )∗ and .pα = αpC + q,r (1 − α)2. Furthermore, let .(un )n∈N ⊆ Wε,σ (QT ) ∩ H∞ (QT ), .(u∗n )n∈N = ∗ ˚εq,r (QT )∗ , and .(v ∗ ) ˚q,p (Jεσ (f n , F n ))n∈N ⊆ V n n∈N ⊆ Vε (QT ) be sequences with the following properties: p− > α

.

(i)

.

dσ u n dt

˚εq,r (QT )∗ for every .n ∈ N. + u∗n + αCun = v ∗n in .V ∗

˚ε (QT ) .(n → ∞), .un u in .H∞ (QT ) .(n → ∞) and .un (t) (ii) .un u in .V ˚εq,p (QT ) ∩ H∞ (QT ) u(t) in H .(n → ∞) for almost every .t ∈ I , where .u ∈ V 0 has a representation .uω ∈ Cω (I , H ). q,p

8.1 Second Parabolic Compensated Compactness Principle

319





(iii) .f n f in .Lq (·,·) (QT )d .(n → ∞), .F n F in .Lr (·,·) (QT , Md×d sym )  (·,·) q,p d×d p ∗ ∗ ˚ε (QT )∗ .(n → ∞), where .F ∈ L (QT , Msym ), and .v n → v in .V .(n → ∞). ∗

(iv) .μn := a n dtdx μ = a 0 dtdx + μs in .M(QT ) .(n → ∞), where .a 0 ∈ L1 (QT , R≥0 ) and .μs ∈ M(QT ), with .μs ⊥ dtdx, and .(a n )n∈N := (f n · un + F n : ε(un ))n∈N ⊆ L1 (QT , R≥0 ) is bounded. Then, it holds .a 0 ≤ a := f · u + F : ε(u) almost everywhere in .QT . 

Proof We apply Lemma 4.4, to obtain .(g n )n∈N ⊆ Lq (·,·) (QT )d and .(Gn )n∈N ⊆  ∗ σ ∗ ˚q,p Lp (·,·) (QT , Md×d sym ), satisfying .v n = Jε (g n , Gn ) in .Vε (QT ) for every .n ∈ N,  (·,·)  (·,·) q d p (QT ) and .G ∈ L (QT , Md×d as well as functions .g ∈ L sym ), satisfying q,p ∗ σ ∗ ˚ .v = Jε (g, G) in .Vε (QT ) , such that

.



gn → g

in Lq (·,·) (QT )d

(n → ∞) ,

Gn → G

in Lp (·,·) (QT , Md×d sym )



(n → ∞) .

Moreover, there exist representations .((un )ω )n∈N ⊆ Cω0 (I , H ) of .(un )n∈N ⊆ q,r Wε,σ (QT ) ∩ H∞ (QT ) (cf. Propositions 4.19 and 4.20). On  the basis of (ii), Propositions 3.8 and 3.11, for every .δ ∈ 0, (p− )∗ − 1 , we have that p (·,·)−δ (Q )d and .u ∈ Lp∗ (·,·)−δ (Q )d with .(un )n∈N ⊆ L ∗ T T un → u

.

in Lp∗ (·,·)−δ (QT )d

(n → ∞) .

(8.52)

Next, we distinguish between the cases  3d.α = 1and .α = 0: The case .α = 1: Since .p : QT → d+2 , ∞ and .(·)∗ : [1, ∞) → (1, ∞) are uniformly continuous and, by assumption, .p◦ > pα = pC holds, .QT can be covered by finitely many cylinders of type .Q := J × G ⊆ QT , where .G ⊆  is a domain − := inf(t,x) ∈Q p(t, x), and .J ⊆ I an interval, such that the local exponents .pQ + + := := .p p(t, x) and . q q(t, x) either satisfy sup sup   (t,x) ∈Q (t,x) ∈Q Q Q (I)

.

or

− pQ ≤ pC ,

− > pC (II) pQ

+ pQ < p◦

and

+ − qQ < (pQ )∗ ,

and

+ − qQ < (pQ )∗ .

Next, we distinguish between the cases (I) and (II): − + + − 3d < pQ ≤ pQ ≤ qQ < (pQ )∗ , .r|Q = p◦ in Q, i.e., .r  |Q = p◦ in (I) It holds . d+2 + − Q, because .pQ < p◦ , and .pQ ≤ pα . Moreover, .(un )n∈N ⊆ Lp◦ (J, W 1,p◦ (G)d ) ∩ + − Y ∞ (Q), where .p◦ ∈ (max{pQ , pC }, (pQ )∗ ) (cf. .p◦ in Proposition 8.3), with 2 0 representations .((un )ω )n∈N ⊆ Cω (J , L (G)d ), satisfies:

8 Existence Theory for .p − < 2

320

(i) For every .n ∈ N, it holds .div(un ) = 0 in .Lp◦ (Q) and for every ∞ d .φ ∈ C 0,div (QT ) , it holds

.

−

un (s, y) · ∂t φ(s, y) dsdy Q

+

f n (s, y) · φ(s, y) + F n (s, y) : ε(φ)(s, y) dsdy

Q

=

Q

g n (s, y) · φ(s, y)

+ [Gn (s, y) + un (s, y) ⊗ un (s, y)] : ε(φ)(s, y) dsdy . −

(ii) .un u in .LpQ (Q)d .(n → ∞), .ε(un ) ε(u) in .Lp(·,·) (Q, Md×d sym ) ∗

(n → ∞), .un u in .Y ∞ (Q) .(n → ∞) and .un (t) u(t) in .L2 (G)d ∞ (Q) has a representation .(n → ∞) for almost every .t ∈ J , where .u ∈ Y 0 2 d .uω ∈ Cω (J , L (G) ). .





(iii) .f n f in .Lq (·,·) (Q)d .(n → ∞), .F n F in .Lp◦ (Q, Md×d sym ) .(n → ∞),  (·,·)  (·,·) d×d p q d where .F ∈ L (Q, Msym ), .g n → g in .L (Q) , and .Gn → G in  (·,·) d×d p .L (Q, Msym ) .(n → ∞). ∗

(iv) .μn |Q = a n dtdx|Q μ|Q = a 0 dtdx|Q + μs |Q 3 in .M(Q) .(n → ∞), where .a 0 |Q ∈ L1 (Q, R≥0 ) and .μs |Q ∈ M(Q), with .μs |Q ⊥ dtdx|Q , and 1 .(a n |Q )n∈N ⊆ L (Q, R≥0 ) is bounded. Thus, it follows from Proposition 8.3 that .a 0 ≤ a holds almost everywhere in Q. − −  − (II) We have that .p◦ ∈ (pC , (pQ )∗ ) (cf. .p◦ in Corollary 7.4) and .2(pQ ) < (pQ )∗ − 

− since .pQ > pC . Therefore, (8.52) yields .un ⊗ un → u ⊗ u in .L(pQ ) (Q, Md×d sym ) →  d×d p (·,·) (Q, Msym ) .(n → ∞) (cf. Corollary 2.1). Moreover, we have that L ∞ (Q), with .(u ) p∗ (·,·)−δ (Q)d for all .δ ∈ (0, (p − ) − 1], .(un )n∈N ⊆ Y n n∈N ⊆ L Q ∗ d×d r(·,·) .(ε(un ))n∈N ⊆ L (Q, Msym ) and representations .((un )ω )n∈N ⊆ Cω0 (J , L2 (G)d ).  n := Gn + un ⊗ un ∈ Lp (·,·) (Q, Md×d Hence, defining .G sym ), .n ∈ N, and  (·,·) d×d p  .G := G + u ⊗ u ∈ L (Q, Msym ), it holds:

3 Let .μ :

A → [0, +∞] be a measure on a measurable space .(X, A) and .B ∈ A. Then, : AB → [0, +∞], with .AB := {A ∩ B | A ∈ A}, given by .(μ|B )(A) := μ(A ∩ B) for all .A ∈ A, is the to B restricted measure on .(X ∩ B, AB ).

.μ|B

8.1 Second Parabolic Compensated Compactness Principle

321

(i) For every .n ∈ N, it holds .div(un ) = 0 in .Lr(·,·) (Q) and for every ∞ d .φ ∈ C 0,div (Q) , it holds

.

−

un (s, y) · ∂t φ(s, y) dsdy Q

+ Q

=

Q

f n (s, y) · φ(s, y) + F n (s, y) : ε(φ)(s, y) dsdy  n (s, y) : ε(φ)(s, y) dsdy . g n (s, y) · φ(s, y) + G

− (ii) .un → u in .Lp∗ (·,·)−δ (Q)d .(n → ∞) for all .δ ∈ (0, (pQ )∗ − 1], .ε(un ) ε(u) d×d p(·,·) ∞ in .L (Q, Msym ) .(n → ∞) and .un u in .Y (Q) .(n → ∞), where ∞ .u ∈ Y (Q) has a representation .uω ∈ Cω0 (J , L2 (G)d ). 



(iii) .f n f in .Lq (·,·) (Q)d .(n → ∞), .F n F in .Lr (·,·) (Q, Md×d sym ) .(n → ∞),  q  (·,·) (Q)d .(n → ∞), and .G n → G  ), . g → g in . L where .F ∈ Lp (·,·) (Q, Md×d n sym  (·,·) d×d p in .L (Q, Msym ) .(n → ∞). ∗

(iv) .μn |Q = a n dtdx|Q μ|Q = a 0 dtdx|Q + μs |Q in .M(Q) .(n → ∞), where .a 0 |Q ∈ L1 (Q, R≥0 ) and .μs |Q ∈ M(Q), with .μs |Q ⊥ dtdx|Q and 1 .(a n |Q )n∈N ⊆ L (Q, R≥0 ) is bounded. Thus, it follows from Corollary 7.4 that .a 0 ≤ a holds almost everywhere in Q. Altogether, because .QT can be covered by finitely many cylinders Q such that (I) or (II) applies, we conclude that .a 0 ≤ a holds almost everywhere in .QT .  2d , ∞ and .(·)∗ : [1, ∞) → (1, ∞) are The case .α = 0: Since .p : QT → d+2 uniformly continuous and, by assumption, .p◦ > pα = 2 holds, .QT can be covered by finitely many cylinders of type .Q := J × G ⊆ QT , where .G ⊆  is a domain − := inf(t,x) ∈Q p(t, x), and .J ⊆ I an interval, such that the local exponents .pQ + + := := .p p(t, x) and . q q(t, x) either satisfy sup sup   (t,x) ∈Q (t,x) ∈Q Q Q (I)

.

or

− pQ ≤ 2,

− (II) pQ >2

+ pQ < p◦

and

+ − qQ < (pQ )∗ ,

and

+ − qQ < (pQ )∗ .

Next, we distinguish between the cases (I) and (II): − + + − 2d (I) It holds . d+2 < pQ ≤ pQ ≤ qQ < (pQ )∗ , .r|Q = p◦ in Q, i.e., .r  |Q = p◦ in + − α Q, because .pQ < p◦ , and .pQ ≤ p . Moreover, .(un )n∈N ⊆ Lp◦ (J, W 1,p◦ (G)d ) ∩ + − Y ∞ (Q), where .p◦ ∈ (max{pQ , 2}, (pQ )∗ ) (cf. .p◦ in Proposition 8.3), with representations .((un )ω )n∈N ⊆ Cω0 (J , L2 (G)d ), satisfies:

8 Existence Theory for .p − < 2

322

∞ (Q )d , (i) For every .n ∈ N, it holds .div(un ) = 0 in .Lp◦ (Q) and for every .φ ∈ C0,div T it holds

.− un (s, y) · ∂t φ(s, y) dsdy Q

+ Q

f n (s, y) · φ(s, y) + F n (s, y) : ε(φ)(s, y) dsdy

=

Q

g n (s, y) · φ(s, y) + Gn (s, y) : ε(φ)(s, y) dsdy .

−

(ii) .un u in .LpQ (Q)d .(n → ∞), .ε(un ) ε(u) in .Lp(·,·) (Q, Md×d sym ) ∗

(n → ∞), .un u in .Y ∞ (Q) .(n → ∞) and .un (t) u(t) in .L2 (G)d ∞ (Q) has a representation .(n → ∞) for almost every .t ∈ J , where .u ∈ Y 0 2 d .uω ∈ Cω (J , L (G) ). .





(iii) .f n f in .Lq (·,·) (Q)d .(n → ∞), .F n F in .Lp◦ (Q, Md×d sym ) .(n → ∞),  (·,·)  (·,·) d×d p q d where .F ∈ L (Q, Msym ), .g n → g in .L (Q) .(n → ∞) and .Gn → G  in .Lp (·,·) (Q, Md×d ) . (n → ∞). sym ∗

(iv) .μn |Q = a n dtdx|Q μ|Q = a 0 dtdx|Q + μs |Q in .M(Q) .(n → ∞), where .a 0 |Q ∈ L1 (Q, R≥0 ) and .μs |Q ∈ M(Q), with .μs |Q ⊥ dtdx|Q , and 1 .(a n |Q )n∈N ⊆ L (Q, R≥0 ) is bounded. Hence, it follows from Proposition 8.3 that .a 0 ≤ a holds almost everywhere in Q. − − (II) We have that .p◦ ∈ (2, (pQ )∗ ) (cf. .p◦ in Corollary 7.4) and .pQ > 2. Moreover, for .(un )n∈N ⊆ Y ∞ (Q), with .(un )n∈N ⊆ Lp∗ (·,·)−δ (Q)d for every .δ ∈ − (0, (pQ )∗ − 1], .(ε(un ))n∈N ⊆ Lr(·,·) (Q, Md×d sym ) and representations .((un )ω )n∈N ⊆ 0 2 d Cω (J , L (G) ), it holds:

(i) For every .n ∈ N, it holds .div(un ) = 0 in .Lr(·,·) (Q) and for every ∞ d .φ ∈ C 0,div (Q) , it holds

.

−

un (s, y) · ∂t φ(s, y) dsdy Q

+ Q

f n (s, y) · φ(s, y) + F n (s, y) : ε(φ)(s, y) dsdy

=

Q

g n (s, y) · φ(s, y) + Gn (s, y) : ε(φ)(s, y) dsdy .

− (ii) .un → u in .Lp∗ (·,·)−δ (Q)d .(n → ∞) for all .δ ∈ (0, (pQ )∗ − 1], .ε(un ) ε(u) ∗

∞ in .Lp(·,·) (Q, Md×d sym ) .(n → ∞) and .un u in .Y (Q) .(n → ∞), where ∞ (Q) has a representation .u ∈ C 0 (J , L2 (G)d ). .u ∈ Y ω ω

8.2 Unsteady p(·, ·)-Navier–Stokes Equations in an Irregular Domain with. . . 

323



(iii) .f n f in .Lq (·,·) (Q)d .(n → ∞), .F n F in .Lr (·,·) (Q, Md×d sym ) .(n → ∞),  (·,·)  (·,·) d×d p q d (Q, Msym ), .g n → g in .L (Q) .(n → ∞) and .Gn → G where .F ∈ L  (·,·) d×d p in .L (Q, Msym ) .(n → ∞). ∗

(iv) .μn |Q = a n dtdx|Q μ|Q = a 0 dtdx|Q + μs |Q in .M(Q) .(n → ∞), where .a 0 |Q ∈ L1 (Q, R≥0 ) and .μs |Q ∈ M(Q), with .μs |Q ⊥ dtdx|Q , and 1 .(a n |Q )n∈N ⊆ L (Q, R≥0 ) is bounded. Therefore, it follows from Corollary 7.4 that .a 0 ≤ a holds almost everywhere in Q. Taken together, we conclude that .a 0 ≤ a holds almost everywhere in .QT . 

8.2 Unsteady p(·, ·)-Navier–Stokes Equations in an Irregular 3d Domain with p − > d+2 Eventually, we are in the position to establish the weak solvability of the unsteady p(·, ·)-Navier–Stokes equations in an arbitrary bounded domain . ⊆ Rd , .d ≥ 2, 3d with .p ∈ Plog (QT ) satisfying .p− > d+2 .

.

Theorem 8.1 Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), log .T < ∞, .QT := I × , and .p, q := max{2, p∗ − ε} ∈ P (QT ) with .p− >  2 − 3d d×d d×d d+2 and .ε ∈ 0, d p . Moreover, let .S : QT × Msym → Msym be a mapping − 

satisfying (S.1)–(S.5) with respect to p. Then, for arbitrary .g ∈ Lmin{2,(p ) } (QT )d , p (·,·) (Q , Md×d ) and .u ∈ H , there exists .u ∈ V ˚εq,p (QT ) ∩ H∞ (QT ) .G ∈ L T 0 sym with a representation .uω ∈ Cω0 (I , H ) such that .uω (0) = u0 in H and for every ∞ d .φ ∈ C 0,div (QT ) , it holds

−

u(t, x) · ∂t φ(t, x) dtdx QT

.

[S(t, x, ε(u)(t, x)) − u(t, x) ⊗ u(t, x)] : ε(φ)(t, x) dtdx

+

(8.53)

QT

g(t, x) · φ(t, x) + G(t, x) : ε(φ)(t, x) dtdx ,

= QT

which, in addition, satisfies for every .t ∈ I the energy inequality (7.127). Proof 1. Existence of solutions: 1.1 Approximative solutions: For .r := max{p◦ , p} ∈ Plog (QT ), where .p◦ ∈ (pC , (p− )∗ ), let ˚εq,r (QT ) → V ˚εq,r (QT )∗ , Sn : V

.

n ∈ N,

8 Existence Theory for .p − < 2

324

˚εq,p (QT ) → V ˚εq,p (QT )∗ , for every .n ∈ N and be an approximation of .S : V q,r ˚ε (QT ) defined by .v ∈ V Sn v := Jεσ (0, Sn (·, ·, ε(v)))

.

˚ε (QT )∗ , in V q,r

d×d where the tensor-valued mappings .Sn : QT × Md×d sym → Msym , .n ∈ N, for d×d  every .n ∈ N, .A ∈ Msym and almost every .(t, x) ∈ QT are defined by

Sn (t, x, A) := S(t, x, A) +

.

1 p◦ −2 |A| A n

in Md×d sym .

˚εq,r (QT ) → V ˚εq,r (QT )∗ , .n ∈ N, are bounded, continThe operators .Sn : V uous and .L1 -monotone (cf. Propositions 3.31 and 7.3) since the mappings d×d d×d .Sn : QT × Msym → Msym , .n ∈ N, satisfy (S.1)–(S.5) with respect to r. ˚εq,r (QT ) ∩ H∞ (QT ) → V ˚εq,r (QT )∗ is bounded Due to .r − > pC , .C : V and Bochner strongly continuous (cf. Proposition 5.3).4 Propositions 5.12 ˚εq,r (QT ) ∩ H∞ (QT ) →V ˚εq,r (QT )∗ , implies that the operators .Sn + C : V .n ∈ N, are Bochner pseudo-monotone and Bochner coercive with respect to ˚εq,r (QT )∗ , where .v ∗ := J σ (g, G) ∈ V ˚εq,p (QT )∗ and ˚εq,r (QT ) )∗ v ∗ ∈ V .(idV ε q,r ˚ε (QT ) → V ˚εq,p (QT ), ˚εq,r (QT ) )∗ is the adjoint operator of .idV ˚εq,r (QT ) : V .(idV and .u0 ∈ H . Thus, Theorem 7.2, or Theorem 7.4, yields the existence q,r of a sequence .(un )n∈N ⊆ Wε,σ (QT ) ∩ H∞ (QT ) having representations 0 .((un )ω )n∈N ⊆Cω (I , H ) such that dσ un ˚εq,r (QT ) )∗ v ∗ + Sn un + Cun = (idV dt . (un )ω (0) = u0

˚ε (QT )∗ , in V q,r

in H,

(8.54)

n ∈ N,

which, in addition, for every .t ∈ I and .n ∈ N satisfies   1 1 dσ un 2 2 (un )ω (0)H ≤ , un χ(0,t) . (un )ω (t)H − . 2 2 dt ˚εq,r (QT ) V

4 Since

.C :

− − ˚εr ,r (QT ) ∩ H∞ (QT ) → V ˚εr ,r (QT )∗ V

(8.55)

is bounded and Bochner strongly continuous (cf. Proposition 5.3), the same is true for its not relabeled restriction ˚εq,r (QT ) ∩ H∞ (QT ) → V ˚εq,r (QT )∗ . .C : V

8.2 Unsteady p(·, ·)-Navier–Stokes Equations in an Irregular Domain with. . .

325

1.2 A priori estimates: Taking into account (8.54) in (8.55) and that ˚εq,r (QT ) = 0 for every .n ∈ N and .t ∈ I , we observe for .Cun , un χ(0,t) V every .n ∈ N and .t ∈ I that 1 (un )ω (t)2H + Sun − v ∗ , un χ(0,t) V ˚εq,p (QT ) 2 .

t 1 1 + |ε(un )(s, y)|p◦ dy ds ≤ u0 2H . 2 0  n

(8.56)

˚εq,p (QT ) → V ˚εq,p (QT )∗ is Bochner coercive with respect to Since .S : V q,p ∗ ˚ε (QT )∗ and .u0 ∈ H (cf. Proposition 5.10), (8.56) yields a constant .v ∈ V .M > 0 (independent of .n ∈ N) such that for every .n ∈ N, it holds un V ˚q,p (Q

.

ε

T )∩H

∞ (Q ) T

≤M.

(8.57)

On the basis of the parabolic interpolation inequality (cf. Proposition 3.7) and Corollary 2.1, there exists a constant .cp− > 0 such that, by recourse to (8.57), we, in turn, obtain for every .n ∈ N un ⊗ un L(p− )∗ /2 (QT )d×d ≤ un 2 (p− )∗ L

(QT )d

≤ cp− un  ˚q,p 2

.

Vε (QT )∩H∞ (QT )

(8.58)

≤ cp− M 2 . If we now exploit (8.57) in (8.56) in the special case .t = T , then we further get for every .n ∈ N that 1 (un )ω (T )2H + 2

F n (s, y) : ε(un )(s, y) dsdy QT

.

(8.59)

1  ≤ u0 2H + v ∗ V ˚εq,p (QT )∗ M =: M , 2 

where .(F n )n∈N := (Sn (·, ·, ε(un )))n∈N ⊆ Lr (·,·) (QT , Md×d sym ). Hence, if we introduce the sequence .(a n )n∈N := (F n : ε(un ))n∈N ⊆ L1 (QT , R≥0 ), then it can be read from (8.59) that for every .n ∈ N a n L1 (QT ) ≤ M  .

.

(8.60)

Moreover, if we introduce .(μn )n∈N := (a n dtdx)n∈N ⊆ M(QT ), then (8.60) also reads for every .n ∈ N μn M(QT ) ≤ M  .

.

(8.61)

8 Existence Theory for .p − < 2

326

d×d Using (8.57) and the growth conditions of .S : QT × Md×d sym →Msym (cf. (S.2) or (3.103)), also taking advantage of Lemma 2.1 (iii), for every .n ∈ N, we conclude that .S(·, ·,

 1/(p+ ) ε(un )Lp (·,·) (QT )d×d ≤ 1 + ρp (·,·) (S(·, ·, ε(un )))

(8.62)

 1/(p+ ) −  −  + ≤ 1 + 2(p ) α (p ) 2p ρp(·,·) (δ) + ρp(·,·) (ε(un )) + ρp (·,·) (β)   −  −  + p+ ≤ 1 + 2(p ) α (p ) 2p ρp(·,·) (δ) + ε(un )Lp(·,·) (Q )d×d + 1 

T

1/(p+ ) + ρp (·,·) (β)   1/(p+ ) −  −  + + ≤ 1 + 2(p ) α (p ) 2p ρp(·,·) (δ) + M p + 1 + ρp (·,·) (β) .

Due to .(S(·, ·, ε(un )) : ε(un ))n∈N ⊆ L1 (QT , R≥0 ), we can also infer from (8.56) that for every .n ∈ N .

1 p ε(un )L◦p◦ (Q )d×d ≤ M  . T n

(8.63)

From (8.63) we, in turn, deduce that for every .n ∈ N p◦   1  |ε(un )|p◦ −2 ε(un )  p n L

.

◦ (QT

= )d×d

≤

1 p◦  ε(un )Lp◦ (Q )d×d T np◦ 1 

np◦ −1

(8.64)



M .

We define .s := max{p◦ , p+ , ((p− )∗ /2) } and the embedding .es := (idV˚s,s )∗ ˚q,p → (V ˚s,s )∗ , where .(idV˚s,s )∗ is the adjoint operator of RH idV˚−q,p : V − s,s ˚ ˚s,s : V .idV → H . Then, taking into account (8.58), (8.62) and (8.64) in 1,p− ,s  ˚q,p , (V ˚s,s )∗ ) (cf. .(8.54)1 , from (8.54) we derive that .(un )n∈N ⊆ Wes (I, V −  Definition 2.19) as well as the existence a constant .M > 0 (independent of .n ∈ N) such that for every .n ∈ N, it holds    des un    dt 

.



˚s,s )∗ ) Ls (I,(V

≤ M  .

(8.65)

8.2 Unsteady p(·, ·)-Navier–Stokes Equations in an Irregular Domain with. . .

327 q,p

˚ε (QT ) 1.3 Convergence of approximative solutions: By the reflexivity of .V  (cf. Proposition 4.17 (i)), .Lp (·,·) (QT , Md×d ) (cf. Proposition 2.7), and sym 1,p− ,s 

.Wes

˚ , (V ˚s,s )∗ ) (cf. Proposition 2.23), Corollary 2.2, and Propo(I, V − q,p

sition 7.5, we obtain from (8.57), (8.61), (8.62) and (8.65) a not relabeled −  ˚q,p , (V ˚s,s )∗ ) ˚εq,p (QT ) ∩ H∞ (QT ) ∩ We1,p ,s (I, V subsequence .(un )n∈N ⊆ V − s −  q,p 1,p ,s q,p ∞ ˚ , (V ˚s,s )∗ ), ˚ε (QT ) ∩ H (QT ) ∩ We as well as elements .u ∈ V (I, V − s  p (·,·) (Q , Md×d ) and .μ ∈ M(Q ) such that .F ∈ L T T sym ˚ε (QT ) in V

q,p

(n → ∞) ,

un u

in H∞ (QT )

(n → ∞) ,

un u

in Wes

un u ∗

.

1,p− ,s 

˚ , (V ˚s,s )∗ ) (I, V − q,p

(n → ∞) ,



S(·, ·, ε(un )) F ∗

μn μ

in Lp (·,·) (QT , Md×d sym )

(n → ∞) ,

in M(QT )

(n → ∞) ,





(8.66)



p (·,·) (Q , Md×d ) → Lr (·,·) (Q , Md×d ), (8.64) Given .Lp◦ (QT , Md×d T T sym ), L sym sym and .(8.66)4 together imply .F n

F



in Lr (·,·) (QT , Md×d sym )

(n → ∞) .

(8.67)

Lebesgue’s decomposition theorem (cf. [58, p. 285]) yields .a 0 ∈ L1 (QT , R≥0 ) and .μs ∈ M(QT ) such that .μ = a 0 dtdx + μs in s ⊥ dtdx . On the other hand, .(8.66) together with the .M(QT ) and .μ 3 fundamental theorem for Bochner–Sobolev functions (cf. Proposition 2.24) ˚s,s )∗ ) (cf. and the characterization of weak convergence in .C 0 (I , (V ˚s,s )∗ ) Proposition 2.16) yields representations .((es un )c )n∈N ⊆ C 0 (I , (V  1,s s,s ∗ ˚ of .(es un )n∈N ⊆ W (I, (V ) ), defined by .(es un )(t) := es (un (t)) in ˚s,s )∗ for almost every .t ∈ I and every .n ∈ N, and analogously a .(V ˚s,s )∗ ) of .es u ∈ W 1,s  (I, (V ˚s,s )∗ ), defined by representation .(es u)c ∈ C 0 (I , (V s,s ∗ ˚ .(e s u)(t) := es (u(t)) in .(V ) for almost every .t ∈ I , such that (es. un )c (t) (es u)c (t)

˚s,s )∗ in (V

(n → ∞)

for all t ∈ I .

(8.68)

Moreover, with recourse to Proposition 2.19, since .u ∈ H∞ (QT ) satisfies 0 ˚s,s )∗ ), we can establish the existence of a weakly con.(e s u)c ∈ C (I , (V tinuous representation .uω ∈ Cω0 (I , H ) of .u ∈ H∞ (QT ), whereby we have ˚s,s )∗ is an embedding, since taken into account that .(idV˚s,s )∗ RH : H → (V s,s ˚ ˚s,s : V .idV → H is a dense embedding. Because .(un )n∈N ⊆ H∞ (QT ) is bounded (cf. (8.57)), i.e., .supn∈N (un )ω (t)H ≤ M for every .t ∈ I , Proposition 2.17 applied to (8.68) yields the point-wise weak convergence .(un )ω (t)

uω (t)

in H

(n → ∞)

for all t ∈ I .

(8.69)

8 Existence Theory for .p − < 2

328

Eventually, owing to .(8.66)1,2 and (8.69), Corollary 3.3 proves, in addition, that .un

⊗ un → u ⊗ u

in Lσ (QT , Md×d sym )

(n → ∞)

for all σ
2 for .p− > d+2 switched to a subsequence. 1.4 Passage to the limit: Because of .(un )ω (0) = u0 in H for every .n ∈ N, we conclude using (8.69) that .uω (0) = u0 in H . By applying the second parabolic compensated compactness principle (cf. Proposition 8.4) for ∗ ˚εq,r (QT )∗ and .α = 1 with respect to .(8.54)1 with .(un )n∈N := (Sn un )n∈N ⊆ V q,p ∗ ∗ ˚ε (QT )∗ for every .n ∈ N, .(8.66)1,2,5 , (8.67), .f = f = 0 ∈ .v n := v ∈ V n  (·,·) q d (QT ) for every .n ∈ N, and (8.69), we conclude that L .a 0

≤ a := F : ε(u)

(8.72)

a.e. in QT .

Next, let .S ∈ B(QT ) be such that .|S| = 0 and .μs (QT \ S) = 0. Then, due to .S(·, ·, ε(un )) : ε(un ) ≤ a n almost everywhere in .QT for every .n ∈ N and (8.72), by applying Proposition 7.6 (ii), we find that5

n→∞ .

S(·, ·, ε(un )) : ε(un ) dtdx ≤ lim sup

lim sup

n→∞

K

≤

a n dtdx K

a 0 dtdx

(8.73)

K

≤

a dtdx . K

˚εq,p (QT ) → V ˚εq,p (QT )∗ is for any closed .K ⊆ QT \ S . Inasmuch as .S : V 1 .L -monotone (cf. Proposition 7.3), we conclude that .F = S(·, ·, ε(u)) in p (·,·) (Q , Md×d ) from (L.3) applied to .(8.66) .L T 1,2,4 and (8.73). Therefore, if sym ∞ (Q )d and subsequently pass for we test .(8.54)1 with an arbitrary .φ ∈ C0,div T .n → ∞, using .(8.66)1 , (8.67) and (8.70) in doing so, then we conclude the weak formulation (8.53).

5 Here,

we abbreviate a bit, since we would proceed in the same way as for the inequality (7.42) in Proposition 7.7.

8.3 Unsteady p(·, ·)-Stokes Equations in an Irregular Domain with p − >

2d d+2

329

2. Energy inequality: The inequality (8.56) implies for every .t ∈ I that 1 (un )ω (t)2H + Sun , un χ(0,t) V ˚εq,p (QT ) 2 . 1 ≤ u0 2H + v ∗ , un χ(0,t) V ˚εq,p (QT ) . 2

(8.74)

˚ε (QT )∗ , we conclude from .(8.66)4 with the Since .Su = Jεσ (0, F ) in .V   σ q ∗ ˚q,p weak continuity of .Jε : L (·,·) (QT )d × Lp (·,·) (QT , Md×d sym ) → Vε (QT ) (cf. Proposition 4.17 (ii)) to q,p

˚εq,p (QT )∗ in V

Sun Su

.

(n → ∞) .

(8.75)

˚εq,p (QT ) → V ˚εq,p (QT )∗ , for every .t ∈ I Owing to the .L1 -monotonicity of .S : V and .n ∈ N, it holds Sun , un χ(0,t) V ˚q,p (Q ε

T)

.

≥ Sun − Su, uχ(0,t) V ˚q,p (Q

T)

ε

+ Su, un χ(0,t) V ˚q,p (Q ) .

(8.76)

T

ε

Therefore, by taking into account .(8.66)1 and (8.75) in (8.76), we deduce that for every .t ∈ (0, T ] lim inf Sun , un χ(0,t) V ˚q,p (Q

.

n→∞

ε

T)

≥ Su, uχ(0,t) V ˚q,p (Q ) . ε

T

(8.77)

Eventually, taking the limit inferior with respect to .n → ∞ on either side in (8.74), using .(8.66)1 , (8.69), the weak lower semi-continuity of the . · H -norm, and (8.77), we conclude for every .t ∈ I .

1 1 2 ∗ uω (t)2H + Su, uχ(0,t) V ˚εq,p (QT ) ≤ uω (0)H + v , uχ(0,t) V ˚εq,p (QT ) , 2 2

which precisely corresponds to the desired energy inequality (7.127).



8.3 Unsteady p(·, ·)-Stokes Equations in Irregular Domain 2d with p − > d+2 The second parabolic compensated compactness principle (cf. Proposition 8.3) in the case .α = 0 allows us to generalize the obtained existence result for the unsteady .p(·, ·)-Stokes equations with an optional, lower-order variable exponent non-linearity in an irregular domain, i.e., Theorem 7.3, from required the lower 2d bound .p− ≥ 2 to the more natural lower bound .p− > d+2 .

8 Existence Theory for .p − < 2

330

Theorem 8.2 Let . ⊆ Rd , .d ≥ 2, be a bounded domain, .I := (0, T ), .T < ∞, log .QT := I × , and .p, q := max{2, p∗ − ε}, r := max{2, p∗ } − ε ∈ P (QT )  2 − 2d d×d d×d − with .p > d+2 and .ε ∈ 0, d p . Moreover, let .S : QT × Msym → Msym be a mapping satisfying (S.1)–(S.5) with respect to p and .d : QT × Rd → Rd a mapping −  satisfying (D.1)–(D.3) with respect to r. Then, for arbitrary .f ∈ Lmin{2,(p ) } (QT )d ,  q,p p (·,·) (Q , Md×d ) and .u ∈ H , there exists .u ∈ W ∞ .F ∈ L T 0 ε,σ (QT ) ∩ H (QT ) with sym 0 a representation .uω ∈ Cω (I , H ) such that .

dσ u + Su + Du = Jεσ (f , F ) dt uω (0) = u0

˚εq,p (QT )∗ , in V in H ,

which, in addition, satisfies for every .t ∈ I the energy inequality (7.126). Proof We refrain from giving a detailed proof since the procedure, except for obvious adjustments, where we, above all, now need to apply Proposition 8.3 in the 2d case .α = 0 to get along with .p− > d+2 , is very similar to the proof of Theorem 8.1. 

Chapter 9

Appendix

In the Appendix, we address three results whose proofs have been postponed until now: We prove the point-wise Poincaré inequality for the symmetric gradient, a generalized Lebesgue differentiation theorem and a Portemanteau theorem for weak-* convergence.

9.1 Point-Wise Poincaré Inequality for the Symmetric Gradient We have yet to give a proof for the point-wise Poincaré inequality near the boundary of a bounded Lipschitz domain involving only the symmetric part of the gradient. Theorem 9.1 Let . ⊆ Rd , .d ≥ 2, be a bounded Lipschitz domain and .r(x) := dist(x, ∂) for .x ∈ . Then, there exist global constants .c0 = c0 (d, ), h0 = h0 () > 0, such that for every .x ∈ E01,1 () (cf. Remark 2.3), it holds  |x(x)| ≤ c0

.

d B2r(x) (x)∩

|ε(x)(y)| dy |x − y|d−1

for a.e. x ∈  with r(x) ≤ h0 .

(9.1)

Proof The approach of this proof is oriented on [37, 40]. We divide the proof into three main steps: Step 1: To begin with, we will prove that for a bounded Lipschitz domain in .Rd , there exist global constants .h0 := h0 (), c1 := c1 (d, ) > 0 such that for every d  .x ∈  with .r(x) ≤ h0 , the intersection .∂B 2r(x) (x) ∩  contains a hyperspherical d (x) with surface area greater or equal than the quantity .c1 (2r(x))d−1 . cap of .∂B2r(x)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaltenbach, Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents, Lecture Notes in Mathematics 2329, https://doi.org/10.1007/978-3-031-29670-3_9

331

332

9 Appendix

Fig. 9.1 Sketch of the construction of Step 1 in two dimensions

To this end, we exploit that the bounded Lipschitz domain . satisfies the uniform exterior cone property (cf. [78, Theorem 1.2.2.2]),  i.e., that there exist a global  height .h > 0 and a global opening angle .θ ∈ 0, π2 such that for every boundary point .x ∈ ∂, there exists an axis .ξx ∈ Sd−1 with .x + C(ξx , θ, h) ⊆  , where d .C(ξx , θ, h) := {y ∈ R | |y| cos(θ ) < y · ξx < h} denotes an open cone (Fig. 9.1). We introduce the global constant .h0 := h4 and fix an arbitrary point .x ∈  with .r(x) ≤ h0 . On the basis of the compactness of .∂, we can find a boundary point .x ∈ ∂ having the property |x − x| = r(x) .

.

(9.2)

Then, the uniform exterior cone property yields a unit vector .ξx ∈ Sd−1 such that   .x + C(ξx , θ, h) ⊆  . Inasmuch as . is closed, we even have the inclusion x + C(ξx , θ, h) ⊆  .

.

(9.3)

Because of .|x−x| = r(x) ≤ h0 = h4 , also resorting to the reverse triangle inequality, we observe that |(x + hξx ) − x| = |hξx − (x − x)| ≥ h − |x − x| ≥ h − h0 =

.

3h ≥ 3h0 ≥ 3r(x) , 4

Point-Wise Poincaré Inequality for the Symmetric Gradient

333

d d / B3r(x) (x). But owing to .x ∈ B2r(x) (x) (cf. (9.2)), the i.e., there holds .x + hξx ∈ d sphere .∂B2r(x) (x) intersects the axis .x + (0, h) ξx non-trivially, i.e., there exists d (x) ∩ (x + (0, h) ξx ) . sx,x ∈ ∂B2r(x)

.

(9.4)

d Our next objective is to establish that .∂B2r(x) (x) ∩ (x + C(ξx , θ, h)) contains a d (x) (cf. (9.4)) and hyperspherical cap of the desired size. Based on .sx,x ∈ ∂B2r(x) d .x ∈ ∂B (x) (cf. (9.2)), we have that r(x)

r(x) ≤ |sx,x − x| ≤ |sx,x − x| + |x − x| = 3r(x) .

.

(9.5)

Denote by .L(ξx , θ, h) := {y ∈ Rd | |y| cos(θ ) = y · ξx ≤ h} the lateral surface of the cone .C(ξx , θ, h). From the compactness of .x + L(ξx , θ, h), we obtain a point .s ˜x,x ∈ x + L(ξx , θ, h) such that |sx,x − s˜x,x | = dist(sx,x , x + L(ξx , θ, h)) .

.

(9.6)

Then, by considering the right-angled triangle .(sx,x , x, s˜x,x ), i.e., we have that (sx,x − s˜x,x , s˜x,x − x) = π2 and . (sx,x − x, s˜x,x − x) = θ , also making use   of (9.5), (9.6) and .sin(θ ) > 0 for every .θ ∈ 0, π2 , it can readily be seen that

.

dist(sx,x , x + L(ξx , θ, h)) = |sx,x − s˜x,x | = sin( (sx,x − x, s˜x,x − x))|sx,x − x|

.

(9.7)

= sin(θ )|sx,x − x| ≥ sin(θ )r(x) . Next, denote by .B(ξx , θ, h) := {y ∈ Rd | |y| cos(θ ) ≤ y · ξx = h} the basis of the cone .C(ξx , θ, h). On the grounds of (9.5), in particular, with renewed recourse to the reverse triangle inequality, from which it is easy to derive for every .y ∈ B(ξx , θ, h) that |sx,x − (x + y)| ≥ |y| − |sx,x − x| ≥ y · ξx − |sx,x − x| = h − |sx,x − x| ,

.

i.e., .dist(sx,x , x + B(ξx , θ, h)) ≥ h − |sx,x − x|, we conclude that dist(sx,x , x + B(ξx , θ, h)) ≥ h − |sx,x − x| ≥ h − 3r(x)

.

≥h−

h 3h = = h0 ≥ r(x) . 4 4

(9.8)

334

9 Appendix

Taking into account  .∂C(ξx , θ, h) = L(ξx , θ, h) ∪ B(ξx , θ, h) and .sin(θ ) ∈ (0, 1) for all .θ ∈ 0, π2 , (9.7) and (9.8) further imply dist(sx,x , x + ∂C(ξx , θ, h)) ≥ sin(θ )r(x) .

(9.9)

.

  d (x) |  (y−x, sx,x −x) ≤ γθ the hyperspherical Denote by .K(x, x) := y ∈ ∂B2r(x) d cap of .∂B2r(x) (x) with center .sx,x and radius .νθ 2r(x) > 0, where .νθ := sin(γθ ) > 0 and .γθ = 2 arcsin( sin(θ) 4 ) > 0 is the opening angle. Then, there exists a constant .c1 := c1 (d, γθ ) > 0, depending only .d ∈ N and .γθ > 0, such that ℋd−1 (K(x, x)) = c1 (2r(x))d−1 .

(9.10)

.

d It is left to show that .K(x, x) ⊆ ∂B2r(x) (x) ∩  . Since for every d .y ∈ K(x, x) ⊆ ∂B 2r(x) (x) the triangle .(y, sx,x , x) is isosceles, owing to .|sx,x − x| = |y − x| = 2r(x), also making use of (9.9), we infer that

 |sx,x − y| = 4r(x) sin



.

(y − x, sx,x − x) 2

 ≤ 4r(x) sin

γ

θ

2

= r(x) sin(θ ) ≤ dist(sx,x , x + ∂C(ξx , θ, h)) , d (sx,x ) ⊆ x + C(ξx , θ, h) ⊆  (cf. (9.2)), and, i.e., .K(x, x) ⊆ Bdist(s x,x ,x+∂ C(ξx ,θ,h)) therefore, d K(x, x) ⊆ B2r(x) (x) ∩  .

.

(9.11)

Step 2: This step shows that it suffices to treat the special case .x ∈ C0∞ ()d . In fact, suppose that there exist global constants .c0 = c0 (d, ), h0 = h0 () > 0 such that for every .x ∈ C0∞ ()d and .x ∈  with .r(x) ≤ h0 , it holds  |x(x)| ≤ c0

.

d B2r(x) (x)∩

|ε(x)(y)| dy . |x − y|d−1

(9.12)

Next, let .x ∈ E01,1 (). By definition (cf. Definition 2.8), there exists a sequence ∞ d such that .x → x in .E 1,1 () .(n → ∞). Without loss of .(xn )n∈N ⊆ C () n 0 generality, we may assume that .xn → x .(n → ∞) almost everywhere in .. Otherwise, we simply switch, with the aid of Proposition 2.2, to a subsequence ∞ d .(xnk )k∈N ⊆ C () for which .xnk → x .(k → ∞) holds almost everywhere in . 0 and replace .(xn )n∈N ⊆ C0∞ ()d by this subsequence .(xnk )k∈N ⊆ C0∞ ()d . On the

Point-Wise Poincaré Inequality for the Symmetric Gradient

335

other hand, if we define the measurable functions .(fn )n∈N ⊆ L0 () and .f ∈ L0 () for almost every .x ∈  by  fn (x) :=

d B2r(x) (x)∩

|ε(xn )(y)| dy , |x − y|d−1

d B2r(x) (x)∩

|ε(x)(y)| dy , |x − y|d−1

.

 f (x) :=

n ∈ N,

we deduce by the weak-type .L1 -estimate for the Riesz potential operator (cf. [157, Theorem 5.1.1]), that there exists a constant .c > 0, not depending on .n ∈ N, such that for every .λ > 0, it holds    |fn − f | > λ  ≤ c ε(xn ) − ε(x) 1 d×d → 0 L () λ

.

(n → ∞) ,

i.e., .fn → f .(n → ∞) in measure in .. In consequence, on the basis of .|| < ∞, there exists a not relabeled subsequence of .(fn )n∈N ⊆ L0 () such that .fn → f .(n → ∞) almost everywhere in .. Putting everything together, in particular, by falling back on (9.12), we conclude for almost every .x ∈  with .r(x) ≤ h0 that |x(x)| = lim |xn (x)| n→∞  ≤ lim c0

.

n→∞

d B2r(x) (x)∩

|ε(xn )(y)| dy |x − y|d−1

= c0 lim fn (x) n→∞

= c0 f (x)  = c0

d B2r(x) (x)∩

|ε(x)(y)| dy , |x − y|d−1

which precisely corresponds to the desired point-wise inequality (9.1)/(9.12) for any x ∈ E01,1 (). Step 3: Next, we prove (9.1). Based on Step 2, it suffices to treat the case ∞ d .x ∈ C () . We fix .x ∈  with .r(x) ≤ h0 and consider the mappings 0 d 1 d−1 → Sd−1 ⊆ Rd , .i = 1, . . . , d, for every .η = .i = ( , . . . ,  ) : R i i (η1 , . . . , ηd−1 ) ∈ Rd−1 and .j = 1, . . . , d defined by .

j

d (η) :=

.

⎧ ⎨√

if j = d

⎩√

else

1 |η|2 +1 ηj |η|2 +1

and

⎧ ⎪ √−ηj ⎪ ⎪ 2 ⎪ ⎨ |η| +1 1 j i (η) := √ 2 |η| +1 ⎪ ⎪ ⎪ ηj ⎪ ⎩√ |η|2 +1

if j = i if j = d . else

336

9 Appendix

Fig. 9.2 Limit behavior of the parametrizations .i : R → S1 ⊆ R2 , .i = 1, 2, for .α → 0 and .α → ∞

It is not difficult to see that the functions .i : Rd−1 → Rd , .i = 1, . . . , d, are smooth and injective (Fig. 9.2). In the particular case .i = d, the Jacobian for every .η ∈ Rd−1 is given by (Dd )(η) =

.

1 3

+ 1) 2 ⎞ ⎛d−1 2 −η1 η2 ... −η1 ηd−2 −η1 ηd−1 j =2 ηj + 1 d−1 2 ⎜ −η η ... −η2 ηd−1 ⎟ 2 1 ⎟ ⎜ j =1 ηj + 1 −η2 η3 ⎟ ⎜ j =2 ⎟ ⎜ .. ×⎜ ⎟, .. ⎟ ⎜ . . ⎟ ⎜  2 + 1⎠ ⎝ −ηd−1 η1 −ηd−1 η2 . . . −ηd−1 ηd−2 d−2 η j =1 j −η1 −η2 ... −ηd−2 −ηd−1 (|η|2

i.e., .(Dd )(0) = (Id−1 , 0) ∈ Rd×(d−1) , and, thus, .det((Dd )(0) (Dd )(0)) = 1. Because for every .η ∈ Rd−1 and .i = 1, . . . , d −1, it holds .(Di )(η) = Ei (Dd )(η) in .Rd×(d−1) , where .Ei := (e1 , . . . , ei−1 , −ei , ei+1 , . . . , ed ) ∈ Md×d for sym .i = 1, . . . , d − 1, we have that

.

det((Di )(0) (Di )(0)) = det((Dd )(0) Ei Ei (Dd )(0)) = det((Dd )(0) (Dd )(0)) = 1

Point-Wise Poincaré Inequality for the Symmetric Gradient

337

d−1 →R(d−1)×(d−1) , for all .i = 1, . . . , d. As .det : R(d−1)×(d−1) →R and .D i Di : R .i = 1, . . . , d, are continuous, there exists a constant .α0 > 0 such that for every d−1 (0), it holds .η ∈ Bα 0 .

det((Di )(η) (Di )(η)) ≥

1 . 2

(9.13)

Therefore, .rk((Di )(η)) = d − 11 for all .η ∈ Bαd−1 (0) and .i = 1, . . . , d, i.e., 0 d−1 (0) →  (B d−1 (0)), .i = 1, . . . , d, are immersions into .Sd−1 . Moreover, .i : Bα i α0 0 if we set . := (1 , . . . , d ) : Rd−1 → Rd×d , it is easy to see that .det((η)) = 0 for every .η ∈ (R \ {0})d−1 , see, e.g., Fig. 9.2. Note also that the author checked this manually by solving the corresponding homogeneous linear problem. Hence, 2 d−1 . If we define for every .α ∈ (0, α ) .(η) ∈ GL(d, R) for every .η ∈ (R \ {0}) 0    α  for i = 1, . . . , d − 1 , .Qα := η = (η1 , . . . , ηd−1 ) ∈ Bαd−1 (0)  |ηi | > 2d then, since .| det | : GL(d, R) → R>0 and . : Qα ⊆ (R \ {0})d−1 → GL(d, R) are continuous and for every .α ∈ (0, α0 ), the image .(| det | ◦ )(Qα ) is a compact subset of .R>0 , there exists a constant .μα > 0, depending on .α ∈ (0, α0 ), such that for every .α ∈ (0, α0 ) and .η ∈ Qα , there holds | det((η))| ≥ μα .

.

(9.14)

On the other hand, for every .η ∈ Qα , .α ∈ (0, α0 ) and .i = 1, . . . , d, there holds .

    1 1 . < arccos √ (i (η), ed ) = arccos  α2 + 1 |η|2 + 1

Therefore, the sets .i (Qα ), .i = 1, . . . , d, are all contained in the hyperspherical cap .Kα of .Sd−1 with center .ed and opening angle .arccos((α 2 + 1)−1/2 ). Thus, since 2 −1/2 ) → 0 as .α → 0, for .α ∈ (0, α ) sufficiently small, the sets .arccos((α + 1) 1 0 d−1 .i (Qα1 ), .i = 1, . . . , d, are contained in the hyperspherical cap .Kα1 of .S with center .ed and surface area .c1 , where .c1 is the global constant from Step 1. Rescaling and translating .K(x, x), it follows from (9.10) and (9.11) that 1 (K(x, x) − x) ⊆ Sd−1 , . 2r(x)

1 For

 ℋ

d−1

 1 (K(x, x) − x) = c1 . 2r(x)

a tensor .A ∈ Rm×n , .m, n ∈ N, we denote by .rk(A) ∈ {0, . . . , min{m, n}} its rank. ∈ N, we denote by .GL(n, R) := {A ∈ Rn×n | det(A) = 0} the general linear group.

2 For .n

338

9 Appendix

1 Thus, there is an orthogonal matrix .Sx ∈ Rd×d with .Sx (Kα1 ) = 2r(x) (K(x, x) − x). Consequently, we have that d  .

1 (K(x, x) − x) . 2r(x)

Sx (i (Qα1 )) ⊆ Sx (Kα1 ) =

i=1

(9.15)

But (9.15) in conjunction with (9.11) implies for .xi := Sx ◦ i : Qα1 → Sd−1 , .i = 1, . . . , d, that   d x d .x + 2r(x) i (Qα0 ) ⊆ K(x, x) ⊆ ∂B2r(x) (x) ∩  .

(9.16)

i=1

Furthermore, for every .η ∈ Qα1 , .i = 1, . . . , d and .t ∈ [0, 2r(x)], we have that

.

d dt [x(x

+ txi (η)) · xi (η)] = (∇x)(x + txi (η)) : xi (η) ⊗ xi (η) = ε(x)(x + txi (η)) : xi (η) ⊗ xi (η) ,

(9.17)

where we made use of .xi (η)⊗xi (η) ∈ Md×d sym for every .η ∈ Qα1 and .i = 1, . . . , d. We integrate (9.17) with respect to .t ∈ [0, 2r(x)] and subsequently use the Newton– Leibniz formula, to obtain for every .η ∈ Qα1 and .i = 1, . . . , d that x(x) · xi (η) = x(x + 2r(x)xi (η)) · xi (η)  2r(x) . − ε(x)(x + txi (η)) : xi (η) ⊗ xi (η) dt .

(9.18)

0

If we use (9.16), i.e., that .x(x + 2r(x)xi (η)) = 0 for every .η ∈ Qα1 and .i = 1, . . . , d, in (9.18), then we further observe for every .η ∈ Qα1 and .i = 1, . . . , d that  |x(x) · xi (η)| ≤

.

0

2r(x)

|ε(x)(x + txi (η))| dt .

Furthermore, if we define .cα1 := d−1 L∞ (Q

α1 )

d×d

> 0, then for every .η ∈ Qα1 3

| det((η))|2 = | det((η) (η))| ≤ cα1 λmin ((η) (η)) ,

.

3 Here,

(9.19)

(9.20)

it is essential that for a tensor .A ∈ Rn×n , .n ∈ N, we have that .|A| ≥ λmax (A), where ∈ R denotes the largest eigenvalue of .A.

.λmax (A)

Point-Wise Poincaré Inequality for the Symmetric Gradient

339

where .λmin (A) ∈ R denotes the smallest eigenvalue of a tensor .A ∈ Rd×d . Using the relation (9.20), we derive for every .η ∈ Qα1 that d 

|x(x) · xi (η)|2 =

i=1

d 

2 |(S x x(x)) · i (η)|

i=1

=

d 

2 |((η)S x x(x))i |

i=1 = ((η)S x x(x)) · ((η)Sx x(x)) .

2 ≥ λmin ((η) (η))|S x x(x)|

(9.21)

= λmin ((η) (η))|x(x)|2 ≥ ≥

1 |det((η))|2 |x(x)|2 cα1 μ2α1 cα1

|x(x)|2 ,

where we have exploited the min-max theorem of R. Courant and E. Fischer in the first inequality, (9.20) in the second inequality and (9.14) in the last inequality. Consequently, also using (9.19), we infer from (9.21) for every .η ∈ Qα1 that 1  d 2 μα1 x 2 |x(x)| ≤ |x(x) ·  (η)| √ i cα1 i=1

.

≤

d 

|x(x) · xi (η)|

(9.22)

i=1

≤

d   i=1

2r(x) 0

|ε(x)(x + txi (η))| dt .

We integrate (9.22) with respect to .η ∈ Qα1 , divide by .|Qα1 | > 0, use the transformation theorem, which is allowed since .xi : Qα1 → xi (Qα1 ) ⊆ Sd−1 ,

340

9 Appendix

i = 1, . . . , d, are immersions, and employ the so-called ”onion formula” (cf. [56, Hilfsatz 1.8]) to establish that

.

.

μα1 √ |x(x)| cα1 ≤



d  i=1

≤2

(9.23)

0

Qα1



d  i=1

2  |Qα1 | i=1

|ε(x)(x + txi (η))|

2r(x) 0

Qα1 d

=

2r(x)



| det((Dxi )(η) (Dxi )(η))| dt dη | det((Dxi )(η) (Dxi )(η))|

|ε(x)(x + txi (η))|| det((Dxi )(η) (Dxi )(η))| dt dη 

xi (Qα1 ) 0

2r(x)

|ε(x)(x + tξ )| dt do(ξ )

 2r(x)  2d |ε(x)(x + tξ )| t d−1 dt do(ξ ) d−1 |Qα1 | S t d−1 0  |ε(x)(y)| 2d dy , = d |Qα1 | B2r(x) (x)∩ |x − y|d−1 ≤

where we used that .| det((Dxi )(η) (Dxi )(η))|=| det((Di )(η) (Di )(η))| ≥ 12 for every .η ∈ Qα1 ⊆ Bαd−1 (0) (cf. (9.13)) in the second line. Finally, with 0 the observation that all constants in (9.23) exclusively depend on the Lipschitz characteristics of ., we conclude the assertion.  

9.2 Generalized Lebesgue Differentiation Theorem We prove a Lebesgue differentiation theorem like result stated in a similar form in [140, 173, 175], without giving proofs. For the sake of completeness, we will now catch up on that. ffl Proposition 9.1 Let .d ∈ N, .ϕ ∈ C0∞ (B1d (0)) ∩ SM(Rd ), . B d (0) ϕ(y) dy = 1, and 1

ϕr ∈ C0∞ (Brd (0)), .r > 0, defined by .ϕr (x) := ϕ( xr ) for every .x ∈ Rd and .r > 0. Furthermore, let .(rn )n∈N ⊆ R>0 be a sequence such that .rn → 0 .(n → ∞). Then, the following statements apply:

.

(i) If .f ∈ L1 (Rd ), then .

Brdn (x0 )

ϕrn (x0 − y)f (y) dy → f (x0 )

(n → ∞)

for a.e. x0 ∈ Rd .

Generalized Lebesgue Differentiation Theorem

ffl If .f ∈ C 0 (Rd ), then . B d

rn (x0 )

ϕrn (x0 − y)f (y) dy → f (x0 ) .(n → ∞) for every

x0 ∈ (ii) If .f ∈ L1 (Rd+1 ), then there exists a cofinal subset . ⊆ N such that for almost   every .z0 := (t0 , x0 ) ∈ Rd+1 and .Qr (z0 ) := t0 , t0 + r 2 × Brd (x0 ), .r > 0, it holds .

Rd .

341

.

Qrn (z0 )

ϕrn (x0 − y)f (s, y) dsdy → f (z0 )

(  n → ∞) .

ffl If .f ∈ C 0 (Rd+1 ), then . Qr (z0 ) ϕrn (x0 − y)f (s, y) dsdy → f (z0 ) .(n → ∞) n for every .z0 := (t0 , x0 ) ∈ Rd+1 . Proof ad (i). First, we define .ωd := |B1d (0)|, .ϕ ω := ωϕd ∈ C0∞ (B1d (0)) ∩ SM(Rd ) and .ϕrω ∈ C0∞ (Brd (0)), .r > 0, by .ϕrω (x) := r1d ϕ ω ( xr ) for every .x ∈ Rd and .r > 0. Then, we have that .ϕ ω L1 (Rd ) = 1 and for every .n ∈ N and .x0 ∈ Rd , there holds .

Brdn (x)

ϕrn (x0 − y)f (y) dy = (ϕrωn ∗ f )(x0 ) .

As a consequence, by resorting to Proposition 2.14 (iii), we establish that (ϕrωn ∗ f )(x0 ) → f (x0 ) .(n → ∞) for almost every .x0 ∈ Rd . If .f ∈ C 0 (Rd ), then .ϕrωn ∗ f → f in .C 0 (K) .(n → ∞) holds for any compact set .K ⊆ Rd (cf. [31, Proposition 4.21]), i.e., we have .(ϕrωn ∗ f )(x0 ) → f (x0 ) .(n → ∞) for any .x0 ∈ Rd . ad (ii). Let .ϕ ω ∈ L1 (Rd+1 ) be defined by .ϕ ω (t, x) := χ(−1,0) (t)ϕ ω (x) for every .(t, x) ∈ Rd+1 . Then, we define .(ϕ ωr )r>0 ⊆ L1 (Rd+1 ) by .ϕ ωr (t, x) := 1 ϕ ω ( rt2 , xr ) for every .r > 0 and .(t, x) ∈ Rd+1. It is easy to check that for r d+2 every .n ∈ N and .z0 = (t0 , x0 ) ∈ Rd+1 , there holds4

.

.

Qrn (z0 )

ϕrn (x0 − y)f (s, y) dsdy = (ϕ ωrn ∗ f )(z0 ) .

(9.24)

Inasmuch as .C0∞ (Rd+1 ) lies densely in .L1 (Rd+1 ), there exists a sequence ∞ d+1 ) such that .f 1 d+1 ) .(m → ∞). Therefore, .(f m )m∈N ⊆ C (R m → f in .L (R 0 for every .ε > 0, there exists an integer .m0 (ε) ∈ N such that .f − f m L1 (Rd+1 ) < ε for all .m ∈ N with .m ≥ m0 (ε). Then, using Young’s inequality (cf. [31, Theorem 4.15]), in particular, exploiting that .ϕ ωrn L1 (Rd+1 ) = ϕ ω L1 (Rd+1 ) = 1 for every

4 Here,

the crucial ingredient is .χ(−1,0)

 t0 −s  rn2

= χ(t0 ,t0 +rn2 ) (s) for every .s ∈ R and .n ∈ N.

342

9 Appendix

n ∈ N, we find that .(ϕ ωrn ∗ f )n∈N ⊆ L1 (Rd+1 ) and for every .m, n ∈ N with .m ≥ m0 (ε) .

ϕ ωrn ∗ f − f L1 (Rd+1 ) ≤ ϕ ωrn ∗ (f − f m )L1 (Rd+1 ) + ϕ ωrn ∗ f m − f m L1 (Rd+1 ) + f m − f L1 (Rd+1 )

.

≤ (ϕ ωrn L1 (Rd+1 ) + 1)f m − f L1 (Rd+1 )

(9.25)

+ ϕ ωrn ∗ f m − f m L1 (Rd+1 ) ≤ 2ε + ϕ ωrn ∗ f m − f m L1 (Rd+1 ) . A simple application of the transformation formula shows that for every .m, n ∈ N with .m ≥ m0 (ε) and .z0 = (t0 , x0 ) ∈ Rd+1 , there holds |(ϕ ωrn ∗ f m )(z0 ) − f m (z0 )|     =  ϕrn (x0 − y)f m (s, y) dsdy − f m (t0 , x0 )

.

(9.26)

Qrn (z0 )

=

(−1,0)×B1d (0)

≤ Lip(f m )

ϕ(y)|f m (t0 − rn2 s, x0 − rn y) − f m (t0 , x0 )| dsdy

(−1,0)×B1d (0)

ϕ(y)|(rn2 s, rn y) | dsdy

≤ Lip(f m )(rn2 + rn ) . Appealing to [31, Proposition 4.18], for every .m, n ∈ N with .m ≥ m0 (ε), there holds the inclusion supp(ϕ ωrn ∗ f m ) ⊆ supp(ϕ ωrn ) + supp(f m ) .

⊆ Qrn (0) + supp(f m ) ⊂⊂ Rd+1 .

(9.27)

As a result, if we combine (9.26) and (9.27), then we get for every fixed .m ∈ N with m ≥ m0 (ε)

.

ϕ ωrn ∗ f m − f m L1 (Rd+1 ) ≤ |Qrn (0) + supp(f m )|Lip(f m )(rn2 + rn )

.

→ 0 (n → ∞) .

(9.28)

Generalized Lebesgue Differentiation Theorem

343

If we take the limit superior with respect to .n → ∞ in (9.25) for some fixed .m ∈ N with .m ≥ m0 (ε), taking into account (9.28) in doing so, then we conclude that .

lim sup ϕ ωrn ∗ f − f L1 (Rd+1 ) ≤ 2ε . n→∞

Inasmuch as .ε > 0 was chosen arbitrarily, we find that .ϕ ωrn ∗ f → f in .L1 (Rd+1 ) .(n → ∞). Hence, Proposition 2.2, in turn, provides a cofinal subset . ⊆ N such that .ϕ ωrn ∗ f → f in .(  n → ∞) almost everywhere in .Rd+1 , which, owing to (9.24), proves the first claim in (ii). To prove the second claim in (ii), consider next .f ∈ C 0 (Rd+1 ). According to [31, Prop. 4.21], there exists a sequence .(f m )m∈N ⊆ C ∞ (Rd+1 ) such that fm → f

.

in C 0 (K) (m → ∞)

(9.29)

for every compact set .K ⊆ Rd+1 . We fix an arbitrary .z0 := (t0 , x0 ) ∈ Rd+1 and set .K := B1d+1 (z0 ). By proceeding as for (9.26), for every .n ∈ N such that .Qrn (z0 ) ⊆ K, we observe that    

Qrn (z0 )

  ϕrn (x0 − y)f (s, y) dsdy − f (z0 ) ≤ Qrn (z0 )

  + 

.

ϕrn (x0 − y)|f (s, y) − f m (s, y)| dsdy

Qrn (z0 )

  ϕrn (x0 − y)f m (s, y) dsdy − f m (z0 )

(9.30)

+ |f m (z0 ) − f (z0 )| ≤ Lip(f m )(rn2 + rn ) + 2f − f m C 0 (K) . Thus, if take the limit superior with respect to .n → ∞ in (9.30) and subsequently pass for .m → ∞, using (9.29) in doing so, we eventually conclude that

.

  lim sup  n→∞

Qrn (z0 )

  ϕrn (x0 − y)f (s, y) dsdy − f (z0 )

(9.31)

≤ 2f − f m C 0 (K) → 0 (m → ∞) . Since .z0 := (t0 , x0 ) ∈ Rd+1 was chosen arbitrarily, (9.31) eventually proves the second claim in (ii).  

344

9 Appendix

9.3 Portemanteau Theorem for Weak-* Convergence Finally, we give a proof for the Portemanteau theorem for weak-* convergence of sequences of finite Radon measures, which was used crucially in Chaps. 7 and 8. For a detailed explanation of the notation employed in the following proposition, we refer back to Chap. 7, Sect. 7.2.1. Proposition 9.2 (Portemanteau for Weak-* Convergence) Let X be a locally compact, separable metric space and .B(X) the Borel .σ -algebra of X. Then, for a sequence of finite Radon measures .(μn )n∈N ⊆ M(X) and a finite Radon measure .μ ∈ M(X), the following statements are equivalent: ∗

(i) .μn  μ in .M(X) .(n → ∞). (ii) .μ(int(E)) ≤ lim infn→∞ μn (E) ≤ lim supn→∞ μn (E) ≤ μ(E) for every bounded set .E ∈ B(X).     (iii) . int(E) f dμ ≤ lim infn→∞ E f dμn ≤ lim supn→∞ E f dμn ≤ E f dμ for every bounded set .E ∈ B(X) and every .f ∈ C 0 (X) with .f ≥ 0 in X. Proof The equivalence of (i) and (ii) has been proved in [150, Thm. 21.15, p. 250]. In addition, for .f = 1 ∈ C 0 (X), (iii) implies (ii). Thus, it is still pending to establish that (i) and/or (ii) imply (iii). To this end, consider a continuous function .f ∈ C 0 (X) with .f ≥ 0 in X and a bounded set .E ∈ B(X). Appealing to Urysohn’s lemma (cf. [150, Lem. B.2]), there exist sequences .(gk )k∈N , (hk )k∈N ⊆ C00 (X)5 such that 0 ≤. gk ≤ gk+1 ≤ χint(E) ≤ χE ≤ χE ≤ hk+1 ≤ hk

for all k ∈ N ,

(9.32)

and .

gk → χint(E)

(k → ∞)

a.e. in X ,

h k → χE

(k → ∞)

a.e. in X .

(9.33)

Consequently, for every fixed .k ∈ N, observing that .fgk , f hk ∈ C00 (X) with .fgk ≤ f χE ≤ f hk in X, based on (9.32) and .f ∈ C 0 (X) with .f ≥ 0 in X, we find that

5 Recall

that .C00 (X) := {f ∈ C 0 (X) | supp(f ) ⊂⊂ X}.

Portemanteau Theorem for Weak-* Convergence

345



 f gk dμ = lim

n→∞ X

X

f gk dμn



≤ lim inf n→∞

f dμn E



≤ lim sup

.

f dμn

n→∞

(9.34)

E



≤ lim

n→∞ X

f hk dμn



=

f hk dμ . X

Applying Beppo Levi’s theorem on monotone convergence to either side in (9.34), which is allowed due to (9.32) and (9.33), we conclude that   f dμ = lim fgk dμ k→∞ X

int(E)



≤ lim inf n→∞

.

f dμn E



≤ lim sup n→∞



≤ lim

k→∞ X

f dμn

(9.35)

E

f hk dμ



=

f dμ . E

Inasmuch as the continuous function .f ∈ C 0 (X) with .f ≥ 0 in X and the bounded set .E ∈ B(X) were chosen arbitrarily in (9.35), we have just proved the implication from (i) and/or (ii) to (iii).  

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