This volume and its companion, both written by a winner of the 1994 Fields Medal, provide a unique and rigorous treatise on mathematical aspects of fluid mechanics models. These models consist of systems of nonlinear partial differential equations for which, despite a long history of important mathematical contributions, no complete mathematical understanding is available. This second volume focuses on compressible Navier-Stokes equations. It is probably the first reference covering the issue of global solutions in the large. It includes entirely new material on compactness properties of solutions for the Cauchy problem, the existence and regularity of stationary solutions, and the existence of global weak solutions. Written by one of the world's leading researchers in nonlinear partial differential equations, Mathematical Topics in Fluid Mechanics will be an indispensable reference for every serious researcher in the field. Its topicality and the clear, concise, and deep presentation by the author make it an outstanding contribution to one of the most important branches of science, the rigorous mathematical modeling of physical phenomena.
Author(s): Pierre-Louis Lions
Series: Oxford Lecture Series in Mathematics and Its Applications 10
Publisher: Oxford University Press, USA
Year: 1998
Language: English
Pages: 368
Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Preface......Page 8
CONTENTS......Page 12
CONTENTS LIST FOR VOLUME 1......Page 14
5.1 Preliminaries......Page 16
5.2 Compactness results and propagation of oscillations......Page 22
5.3 Proofs of compactness results in the whole space case......Page 30
5.4 Proofs of compactness results in the other cases......Page 45
5.5 General pressure laws......Page 51
5.6 Other boundary value problems......Page 54
6.1 Preliminaries......Page 64
6.2 Existence and regularity results for time-discretized problems......Page 66
6.3 A priori estimates......Page 72
6.4 Compactness......Page 95
6.5 Existence proofs......Page 99
6.6 The isothermal case in two dimensions......Page 112
6.7 Stationary problems......Page 127
6.8 Exterior problems and related questions......Page 143
6.9 Regularity of solutions......Page 159
6.10 Related problems......Page 173
6.11 General compressible models......Page 177
7.1 A priori bounds......Page 187
7.2 Existence results......Page 195
7.3 Existence proofs via regularization......Page 197
7.4 Existence proofs via time discretization......Page 212
7.5 General pressure laws......Page 220
7.6 Other boundary-value problems......Page 224
8.1 Pure transport of entropy......Page 228
8.2 A semi-stationary model......Page 239
8.3 A Stokes-like model......Page 251
8.4 On some shallow water models......Page 266
8.5 Compactness properties for compressible models with temperature......Page 269
8.6 Global existence results for some compressible models with temperature......Page 277
8.7 On compressible Euler equations......Page 286
8.8 On a low Mach number model......Page 296
Appendix A: A few facts about some function spaces......Page 303
Appendix B: On a weakly continuous product......Page 305
Appendix C: A remark on the limiting case of Sobolev inequalities......Page 307
Appendix D: Continua and limits......Page 310
Appendix E: On sums of L^P spaces......Page 312
Appendix F: A remark on parabolic equations......Page 315
Bibliography......Page 322
Errata (Volume 1)......Page 352
Index......Page 356
