Mathematical Topics in Fluid Mechanics Volume 2: Compressible Models

by Lions, Pierre-Louis

• ISBN13: 9780198514886
• ISBN10: 0198514883
• Format: Hardcover
• Copyright: 1998-06-04
• Publisher: Clarendon Press

Summary

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.

Table of Contents

5 Compactness results for compressible isentropic Navier-Stokes equations	1	(48)
5.1 Preliminaries	1	(6)
5.2 Compactness results and propagation of oscillations	7	(8)
5.3 Proofs of compactness results in the whole space case	15	(15)
5.4 Proofs of compactness results in the other cases	30	(6)
5.5 General pressure laws	36	(3)
5.6 Other boundary value problems	39	(10)
6 Stationary problems	49	(123)
6.1 Preliminaries	49	(2)
6.2 Existence and regularity results for time-discretized problems	51	(6)
6.3 A priori estimates	57	(23)
6.4 Compactness	80	(4)
6.5 Existence proofs	84	(13)
6.6 The isothermal case in two dimensions	97	(15)
6.7 Stationary problems	112	(16)
6.8 Exterior problems and related questions	128	(16)
6.9 Regularity of solutions	144	(14)
6.10 Related problems	158	(4)
6.11 General compressible models	162	(10)
7 Existence results for Cauchy problems	172	(41)
7.1 A priori bounds	172	(8)
7.2 Existence results	180	(2)
7.3 Existence proofs via regularization	182	(15)
7.4 Existence proofs via time discretization	197	(8)
7.5 General pressure laws	205	(4)
7.6 Other boundary-value problems	209	(4)
8 Related problems	213	(75)
8.1 Pure transport of entropy	213	(11)
8.2 A semi-stationary model	224	(12)
8.3 A Stokes-like model	236	(15)
8.4 On some shallow water models	251	(3)
8.5 Compactness properties for compressible models with temperature	254	(8)
8.6 Global existence results for some compressible models with temperature	262	(9)
8.7 On compressible Euler equations	271	(10)
8.8 On a low Mach number model	281	(7)
Appendix A: A few facts about some function spaces	288	(2)
Appendix B: On a weakly continuous product	290	(2)
Appendix C: A remark on the limiting case of Sobolev inequalities	292	(3)
Appendix D: Continua and limits	295	(2)
Appendix E: On sums of L(p) spaces	297	(3)
Appendix F: A remark on parabolic equations	300	(7)
Bibliography	307	(30)
Errata (Volume 1)	337	(4)
Index	341