Table Of ContentM o n o g r a f i e M a t e m a t y c z n e
Instytut Matematyczny Polskiej Akademii Nauk (IMPAN)
Volume 73
(New Series)
Founded in 1932 by
S. Banach, B. Knaster, K. Kuratowski,
S. Mazurkiewicz, W. Sierpinski, H. Steinhaus
Managing Editor:
Marek Bozejko (Wroclaw University, Poland)
Editorial Board:
Jean Bourgain (IAS, Princeton, USA)
Joachim Cuntz (University of Münster, Germany)
Ursula Hamenstädt (University of Bonn, Germany)
Gilles Pisier (Texas A&M University, USA)
Piotr Pragacz (IMPAN, Poland)
Andrew Ranicki (University of Edinburgh, UK)
Slawomir Solecki (University of Illinois, Urbana-Champaign, USA)
Przemyslaw Wojtaszczyk (IMPAN and Warsaw University, Poland)
Jerzy Zabczyk (IMPAN, Poland)
Henryk Zoladek (Warsaw University, Poland)
Volumes 31–62 of the series
Monografie Matematyczne were published by
PWN – Polish Scientific Publishers, Warsaw
Pavel Plotnikov • Jan Sokołowski
Compressible Navier–Stokes
Equations
Theory and Shape Optimization
Pavel Plotnikov Jan Sokołowski
Siberian Branch of RAS Institut Élie Cartan de Nancy, UMR 7502
Lavrentyev Institute of Hydrodynamics Université de Lorraine, CNRS
Novosibirsk Vandœuvre-lès-Nancy
Russia France
ISBN 978-3-0348-0366-3 ISBN 978-3-0348-0367-0 (eBook)
DOI 10.1007/978-3-0348-0367-0
Springer Basel Heidelberg New York Dordrecht London
Library of Congress Control Number: 2012945409
Mathematics Subject Classification (2010 ): 35Q30, 35Q35, 35Q93, 49Q10, 49Q12, 49K20, 49K40, 76N10, 76N25
© Springer Basel 2012
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Contents
Preface ix
Introduction xi
1 Preliminaries 1
1.1 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Interpolation of Banach spaces . . . . . . . . . . . . . . . . 6
1.1.3 Invertibility of linear operators . . . . . . . . . . . . . . . . 8
1.1.4 Fixed point theorems . . . . . . . . . . . . . . . . . . . . . 8
1.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Hölder spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Measure and integral . . . . . . . . . . . . . . . . . . . . . . 12
1.2.3 Lebesgue measure in Rd. Lebesgue spaces . . . . . . . . . . 16
1.3 Compact sets in Lp spaces . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.1 Functions of bounded variation . . . . . . . . . . . . . . . . 20
1.3.2 Lp spaces of Banach space valued functions . . . . . . . . . 23
1.4 Young measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6 Mollifiers and DiPerna & Lions lemma . . . . . . . . . . . . . . . . 35
1.7 Partial differential equations . . . . . . . . . . . . . . . . . . . . . . 39
1.7.1 Elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . 39
1.7.2 Stokes problem . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.7.3 Parabolic equations . . . . . . . . . . . . . . . . . . . . . . 43
2 Physical background 45
2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1.1 Isentropic flows. Compressible Navier-Stokes equations . . . 47
2.2 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . 48
2.3 Power and work of hydrodynamic forces . . . . . . . . . . . . . . . 50
2.4 Navier-Stokes equations in a moving frame . . . . . . . . . . . . . 51
2.5 Flow around a moving body . . . . . . . . . . . . . . . . . . . . . . 53
v
vi Contents
3 Problem formulation 57
3.1 Weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Renormalized solutions . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Work functional. Optimization problem . . . . . . . . . . . . . . . 61
4 Basic statements 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Bounded energy functions . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Functions of bounded mass dissipation rate . . . . . . . . . . . . . 68
4.4 Compactness properties . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4.1 Two lemmas on compensated compactness . . . . . . . . . . 71
4.4.2 Proof of Theorem 4.4.2 . . . . . . . . . . . . . . . . . . . . 76
4.5 Basic integral identity . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.6 Continuity of viscous flux . . . . . . . . . . . . . . . . . . . . . . . 87
4.7 Viscous flux. Localization . . . . . . . . . . . . . . . . . . . . . . . 95
5 Nonstationary case. Existence theory 99
5.1 Problem formulation. Results . . . . . . . . . . . . . . . . . . . . . 99
5.2 Regularized equations . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3 Passage to the limit. The first level . . . . . . . . . . . . . . . . . . 118
5.3.1 Step 1. Convergence of densities and momenta . . . . . . . 121
5.3.2 Step 2. Weak convergence of the kinetic energy tensor . . . 122
5.3.3 Step 3. Weak limits of ε∇(cid:3) ⊗u . . . . . . . . . . . . . . 124
n n
5.3.4 Step 4. Proof of estimates (5.3.8–5.3.9) . . . . . . . . . . . . 125
5.3.5 Step 5. Proof of Theorem 5.3.1 . . . . . . . . . . . . . . . . 129
5.3.6 Proof of Theorem 5.3.2 . . . . . . . . . . . . . . . . . . . . 130
5.3.7 Local pressure estimate . . . . . . . . . . . . . . . . . . . . 132
5.3.8 Normal derivative of the density . . . . . . . . . . . . . . . 136
5.4 Passage to the limit. The second level . . . . . . . . . . . . . . . . 139
5.5 Passage to the limit. The third level . . . . . . . . . . . . . . . . . 157
6 Pressure estimate 167
7 Kinetic theory. Fast density oscillations 175
7.1 Problem formulation. Main results . . . . . . . . . . . . . . . . . . 175
7.2 Proof of Theorem 7.1.9. . . . . . . . . . . . . . . . . . . . . . . . . 185
7.3 Proof of Theorem 7.1.12 . . . . . . . . . . . . . . . . . . . . . . . . 204
7.4 Adiabatic exponent γ >3/2 . . . . . . . . . . . . . . . . . . . . . . 223
8 Domain convergence 225
8.1 Hausdorff and Kuratowski-Mosco convergences . . . . . . . . . . . 225
8.2 Capacity, quasicontinuity and fine topology . . . . . . . . . . . . . 229
8.3 Applications to flow around an obstacle . . . . . . . . . . . . . . . 235
8.4 S-compact classes of admissible obstacles . . . . . . . . . . . . . . 244
Contents vii
9 Flow around an obstacle. Domain dependence 249
9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
9.2 Existence theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.3 Main stability theorem . . . . . . . . . . . . . . . . . . . . . . . . . 255
10 Existence theory in nonsmooth domains 281
10.1 Existence theory in nonsmooth domains . . . . . . . . . . . . . . . 281
10.2 Continuity of the work functional . . . . . . . . . . . . . . . . . . . 282
10.3 Applications of the stability theorem to optimization problems . . 296
11 Sensitivity analysis. Shape gradient of the drag functional 299
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
11.1.1 Change of variables in Navier-Stokes equations . . . . . . . 300
11.1.2 Extended problem. Perturbation theory . . . . . . . . . . . 304
11.2 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 308
11.2.1 Function spaces. Results . . . . . . . . . . . . . . . . . . . . 308
11.2.2 Step 1. Estimates of composite functions . . . . . . . . . . . 312
11.2.3 Step 2. Stokes equations . . . . . . . . . . . . . . . . . . . . 314
11.2.4 Step 3. Transport equations . . . . . . . . . . . . . . . . . . 315
11.2.5 Step 4. Fixed point scheme. Estimates . . . . . . . . . . . . 319
11.2.6 Step 5. Proof of Theorem 11.2.6 . . . . . . . . . . . . . . . 328
11.3 Dependence of solutions on N . . . . . . . . . . . . . . . . . . . . . 330
11.3.1 Problem formulation. Basic equations. Transposed problem 330
11.3.2 Transposed problem . . . . . . . . . . . . . . . . . . . . . . 333
11.3.3 Special dual space . . . . . . . . . . . . . . . . . . . . . . . 347
11.3.4 Very weak solutions . . . . . . . . . . . . . . . . . . . . . . 348
11.3.5 Uniqueness and existence. Main theorem. . . . . . . . . . . 354
11.4 Shape derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
11.4.1 Preliminaries. Results . . . . . . . . . . . . . . . . . . . . . 355
11.4.2 Proof of Theorem 11.4.6 . . . . . . . . . . . . . . . . . . . . 359
11.4.3 Conclusion. Material and shape derivatives . . . . . . . . . 370
11.5 Shape derivative of the drag functional. Adjoint state. . . . . . . . 371
12 Transport equations 377
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
12.2 Existence theory in Sobolev spaces . . . . . . . . . . . . . . . . . . 380
12.3 Proof of Theorem 12.2.3 . . . . . . . . . . . . . . . . . . . . . . . . 382
12.3.1 Normal coordinates . . . . . . . . . . . . . . . . . . . . . . 383
12.3.2 Model equation . . . . . . . . . . . . . . . . . . . . . . . . . 384
12.3.3 Local estimates . . . . . . . . . . . . . . . . . . . . . . . . . 385
12.3.4 Estimates near the inlet . . . . . . . . . . . . . . . . . . . . 390
12.3.5 Partition of unity . . . . . . . . . . . . . . . . . . . . . . . . 392
12.3.6 Proofs of Lemmas 12.3.2 and 12.3.3 . . . . . . . . . . . . . 395
12.3.7 Proof of Lemma 12.3.4 . . . . . . . . . . . . . . . . . . . . . 398
viii Contents
13 Appendix 407
13.1 Proof of Lemma 2.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 407
13.2 Normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
13.3 Geometric results. Approximation of unity . . . . . . . . . . . . . . 413
13.4 Singular limits of normal derivatives . . . . . . . . . . . . . . . . . 418
Bibliography 441
Notation 451
Index 453
Preface
This book is a result of scientific collaboration, for more than ten years, between
threecountries,France,RussiaandPoland.Themainsupportforthejointresearch
comesfromUniversitédeLorraine,InstitutÉlieCartanandfromCentreNationale
de la Recherche Scientifique (CNRS) for the visits of Pavel I. Plotnikov in France.
Also the support of the Systems Research Institute of the Polish Academy of Sci-
ences for the visits in Poland is acknowledged. The research of Pavel I. Plotnikov
was partially supported by Contract 02.740.11.0617 with the Ministry of Educa-
tion and Science of the Russian Federation and by grant 10-01-00447-a from the
Russian Foundation of Basic Research, while the research of Jan Sokołowski was
partiallysupportedbytheprojectANR-09-BLAN-0037Geometric analysis of op-
timal shapes (GAOS) financed by the French Agence Nationale de la Recherche
(ANR) and by grant N51402132/3135 of the Polish Ministerstwo Nauki i Szkol-
nictwa Wyższego “Optymalizacja z wykorzystaniem pochodnej topologicznej dla
przepływówwośrodkachściśliwych” intheSystemsResearchInstituteofthePol-
ish Academy of Sciences. The monograph is published in the Polish mathematical
seriesbyBirkhäuserBasel.TheauthorsaregreatlyindebtedtoJerzyTrzeciakfor
his careful work on editing the monograph, and his patience.
Pavel Plotnikov
Lavrentyev Institute of Hydrodynamics and
Novosibirsk State University
Jan Sokołowski
Université de Lorraine, Institut Élie Cartan and
Systems Research Institute of the Polish Academy of Sciences
Nancy, Novosibirsk and Warsaw, March 2012
ix
