Navier—Stokes Equations in Irregular Domains

582 Pages·1995·14.277 MB·English

by  Liudas Stupelis (auth.)

Navier-Stokes Equations in Irregular Domains
Mathematics and Its Applications
Managing Editor:
M.HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 326
Navier-Stokes Equations
in Irregular Domains
by
L. Stupelis
Institute ofM athematics and Informatics,
Vilnius, Lithuania
'~·'
SPRINGER-SCIENCE+BUSINESS MEDIA. B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4562-1 ISBN 978-94-015-8525-5 (eBook)
DOI 10.1007/978-94-015-8525-5
This is an updated and revised translation of the original Russian work
The Boundary Value Problems for the System of
Navier-Stokes Equations in Piecewise Smooth Domains,
Vilnius, Mokslas, © 1992
Printed on acid-free paper
All Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint of the hardcover 1st edition 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To my Teacher
Olga Alexandrovna Ladyzhenskaya
Contents
Preface xiii
Chapter 1. Preliminaries 1
1.1. Normed Linear Spaces and Operators . . . . . . . . . . . . . . . . . . 1
1.1.1. Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2. Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3. Regularization of Operators . . . . . . . . . . . . . . . . . . . . 10
1.2. Some Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1. Spaces of Generalized Functions . . . . . . . . . . . . . . . . 11
1.2.2. Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.3. Function Spaces. Classes of Domains . . . . . . . . . . . . 17
1.2.4. Imbedding Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.2.5. Spaces with Negative Norms . . . . . . . . . . . . . . . . . . . 42
1.3. Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.3.1. Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.3.2. Singular Integral Operators in Weighted Spaces . . . . 45
1.3.3. Generalized Hardy-Littlewood Inequality . . . . . . . . . 50
Chapter 2. Boundary Value Problems for the Stokes System . . . . . . . . . . . . 55
2.1. Statement of the Boundary Value Problems and the Green
Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.1.1. Statement of the Boundary Value Problems . . . . . . . . 55
2.1.2. The Green Formulas and the Problems Formally
Adjoint with Respect to Them . . . . . . . . . . . . . . . . . . 56
2.2. The Green Matrices on the Half-space and the Half-plane 60
2.2.1. Green Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.2.2. The Main Properties of the Green Matrices . . . . . . . . 64
2.3. Weighted Spaces of Functions on JR+ . . . . . . . . . . . . . . . . . . 66
2.3.1. Weighted Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . 66
vii
viii
2.3.2. Spaces of Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.3.3. Spaces with Negative Norms . . . . . . . . . . . . . . . . . . . 74
2.3.4. Weighted HOlder Spaces . . . . . . . . . . . . . . . . . . . . . . . 76
2.4. Strong Solvability of the Boundary Value Problems in the
Half-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.4.1. The First Boundary Value Problem in Weighted
Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.4.2. Solvability of the Second Boundary Value Problem . 91
2.4.3. Solvability of the Boundary Value Problems in
Weighted Holder Classes . . . . . . . . . . . . . . . . . . . . . . . 92
2.5. Generalized Solutions of Boundary Value Problems . . . . . . 101
2.5.1. Auxiliary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.5.2. Generalized Solutions of the Boundary Value
Problems for the Stokes System . . . . . . . . . . . . . . . . . 105
2.5.3. Solvability of the Boundary Value Problems in
Weighted Holder Spaces . . . . . . . . . . . . . . . . . . . . . . . 114
2.6. Weak Solutions of the Boundary Value Problems in the
Half-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.6.1. The First Boundary Value Problem . . . . . . . . . . . . . . 124
2.6.2. The Second Boundary Value Problem . . . . . . . . . . . . 127
2.7. 'Perturbed' Boundary Value Problems in the Half-space . . . 132
2.7.1. The First Boundary Value Problem . . . . . . . . . . . . . . 132
2. 7 .2. The First Boundary Value Problem in Weighted
Holder Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.7.3. The Second Boundary Value Problem . . . . . . . . . . . . 141
2.8. Local Solutions in Half-space . . . . . . . . . . . . . . . . . . . . . . . . 148
2.8.1. The First Boundary Value Problem . . . . . . . . . . . . . . 148
2.8.2. The Second Boundary Value Problem . . . . . . . . . . . . 157
2.9. Local Properties of Solutions near Boundary . . . . . . . . . . . . 166
2.9.1. Local Estimates of Solutions near Boundary . . . . . . . 166
2.9.2. The Case of Two Independent Variables . . . . . . . . . . 174
Chapter 3. Boundary Value Problems in Plane and Bihedral Angles . . . . . . 176
3.1. Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
3 .1.1. Weighted Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . 177
3.1.2. Weighted HOlder Spaces . . . . . . . . . . . . . . . . . . . . . . . 179
CONTENTS IX
3.2. Model Problems in Plane Angle . . . . . . . . . . . . . . . . . . . . . . 182
3.2.1. Auxiliary Problem in Arc . . . . . . . . . . . . . . . . . . . . . . 183
3.2.2. Finding of the Maximal Strip on the Complex Plane
without Spectra of the Boundary Value Problems 189
3.2.3. Solvability in Weighted Hilbert Spaces of Model
Problems in Plane Angle . . . . . . . . . . . . . . . . . . . . . . . 193
3.3. Boundary Value Problems Depending on a Parameter . . . . . 209
3.3.1. Boundary Value Problems Depending on Parameter in
Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
3.3.2. Auxiliary Boundary Value Problems in a Plane
Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
3.3.3. The Noether Property of Boundary Value Problems
and Coercive Estimates . . . . . . . . . . . . . . . . . . . . . . . . 225
3.3.4. Unique Solvability of the Boundary Value Problems in
a Plane Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
3.4. Solvability in Weighted Hilbert Spaces of the Boundary
Value Problems in a Bihedral Angle . . . . . . . . . . . . . . . . . . . 242
3.4.1. Solvability of the Boundary Value Problems in
Weighted Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . 243
3.4.2. The Boundary Value Problems in Weighted Hilbert
Spaces with Negative Norms . . . . . . . . . . . . . . . . . . . 245
3.4.3. Local Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
3.5. The Green Matrices of the Boundary Value Problems in a
Bihedral Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
3.5.1. Existence of the Green Matrix . . . . . . . . . . . . . . . . . . 262
3.5.2. The Main Properties of the Green Matrix and
Pointwise Estimates of its Elements . . . . . . . . . . . . . . 266
3.6. L -Estimates of Solutions to the Boundary Value Problems
8
in Bihedral Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
3.6.1. Estimates of Solutions in Weighted Sobolev Spaces 277
3.6.2. Unique Solvability of the Boundary Value Problems in
Weighted Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . 282
3.7. Estimates of Solutions to the Boundary Value Problems in a
Variational Form by Weighted L -Norms . . . . . . . . . . . . . . . 283
8
3.7.1. A Priori Estimates of Solutions . . . . . . . . . . . . . . . . . 283
3.7.2. Unique Solvability of the Boundary Value Problems in
Generalized Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
X
3.8. Solvability of the Boundary Value Problems in Weighted
Holder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
3.8.1. Estimates of Solutions of the Boundary Value
Problems in Weighted Holder Spaces . . . . . . . . . . . . . 292
3.8.2. Solvability of the Problem (3.4.1), (3.4.2) in Weighted
Holder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
Chapter 4. The First Boundary Value Problem in a Given Domain . . . . . . . 295
4.1. Classes C1·a . . . . . . • . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . 295
4.1.1. Holder Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
4.1.2. Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . 300
4.2. Classes N1·a . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . 302
4.2.1. Weighted HOlder Classes . . . . . . . . . . . . . . . . . . . . . . . 302
4.2.2. Continuous Differential Operators in Weighted HOlder
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
4.2.3. Diffeomorphisms Preserving Spaces . . . . . . . . . . . . . 308
4.3. Spaces Vi,v(Q, M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
4.3.1. Weighted Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . 309
4.3.2. Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . 310
4.4. The First Boundary Value Problem in a Bounded Angular
Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
4.4.1. Simple Connected Domain . . . . . . . . . . . . . . . . . . . . . 311
4.4.2. Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
4.4.3. Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 318
4.4.4. The Boundary Value Problem in Weighted Sobolev
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
4.5. The First Boundary Value Problem for the Navier-Stokes
System in Domains with Noncompact Boundaries . . . . . . . . 333
4.5.1. Spaces of Solenoidal Vector-Functions on Domains
whose 'Exits' to Infinity Contain a Cone . . . . . . . . . . 333
4.5.2. Spaces of Solenoidal Vector-Functions on Domains
with Noncompact Boundaries . . . . . . . . . . . . . . . . . . . 343
4.5.3. The First Boundary Value Problem for the
Navier-Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . 356
Chapter 5. Steady Motion of a Fluid with a Free Surface . . . . . . . . . . . . . . . 363
5.1. Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
CONTENTS xi
5.2. The Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
5.2.1. Notation and Auxiliary Propositions . . . . . . . . . . . . . 366
5.2.2. The Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . 369
5.3. Auxiliary Linear Problem in a Given Bounded Domain . . . 375
5.3.1. Properties of the Domain and Diffeomorphisms of the
Class NI+J,o: n c1..x . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
5.3.2. Auxiliary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . 379
5.3.3. The Noether Property of the Boundary Value Problem
and a priori Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 390
5.3.4. Solvability of the Boundary Value Problem in Hilbert
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
5.4. Solvability of the Boundary Value Problem in Weighted
HOlder and Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 406
5.4.1. Solvability in N~·o:(G) x Ni·o:(G) . . . . . . . . . . . . . . . . 406
5.4.2. Solvability in N~"t~·o:(G) x N~!:·o:(G) and V~"t~-t<G) x
v:.t~t(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
5.5. Auxiliary Nonlinear Problem . . . . . . . . . . . . . . . . . . . . . . . . . 412
5.6. Variation of the Solution to the Auxiliary Problem in a
Varying Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
5.7. Solvability of the Free Boundary Problem . . . . . . . . . . . . . . 423
5.8. The Two-Dimensional Free Boundary Problem . . . . . . . . . . 434
5.9. The Case of Complete Wetting . . . . . . . . . . . . . . . . . . . . . . . 442
5.9.1. Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . 442
5.9.2. Auxiliary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . 444
5.9.3. Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 455
5.9.4. Weighted Holder Spaces . . . . . . . . . . . . . . . . . . . . . . . 466
5.9.5. A Classical Solution of an Auxiliary Linearized
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
5.9.6. A Nonlinear Auxiliary Problem . . . . . . . . . . . . . . . . . 498
5.9.7. Variation of the Solution to the Auxiliary Problem in
a Varying Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
5.9.8. Solvability of the Free Boundary Problem . . . . . . . . . 507
Appendix 1. The Green Matrices on the Half-Space and Half-Plane . . . . . . . 518
1. The Green Matrices on the Half-Space 518
2. The Green Matrices on the Half-Plane 528