Advances in the Theory of Shock Waves

by Liu, Tai-Ping; Szepessy, Anders; Metivier, Guy; Smoller, Joel; Temple, Blake; Yong, Wen-An; Zumbrun, Kevin

• ISBN13: 9780817641870
• ISBN10: 0817641874
• Format: Hardcover
• Copyright: 2001-06-01
• Publisher: Birkhauser

Summary

This volume provides a comprehensive treatment of central themes in the modern mathematical theory of shock waves. Authored by leading scientists, the work covers: * the uniqueness of weak solutions to hyperbolic systems of conservation laws in one space variable (Tai-Ping Liu) * the multidimensional stability problem for shock fronts (Guy Métivier)* shock wave solutions of the Einstein-Euler equations of general relativity (Joel Smoller and Blake Temple)* fundamental properties of hyperbolic systems with relaxation (Wen-An Yong)* the multidimensional stability problem for planar viscous shock waves (Kevin Zumbrun)The five articles, each self-contained and interrelated, combine the rigor of mathematical analysis with careful attention to the physical origins and applications of the field. A timely reference text for professional researchers in shock wave theory, the book also provides a basis for graduate seminars and courses for students of mathematics, physics, and theoretical engineering.

Table of Contents

Preface	vii
Well-Posedness Theory for Hyperbolic Systems of Conservation Laws	1	(24)
Tai-Ping Liu
Scalar conservation law	1	(6)
Glimm scheme	7	(8)
Wave tracing	15	(3)
Nonlinear functional	18	(7)
Stability of Multidimensional Shocks	25	(80)
Guy Metivier
Introduction	25	(1)
The uniform stability condition	26	(13)
The uniform stability estimates	39	(20)
Well posedness of the linearized shock front equations	59	(10)
The existence of multidimensional shocks	69	(17)
Stability of weak shocks	86	(19)
Shock Wave Solutions of the Einstein Equations: A General Theory with Examples	105	(154)
Joel Smoller
Blake Temple
Introduction	105	(12)
Solutions of the Einstein equations when the metric is only Lipschitz continuous across an interface	117	(22)
Matching an FRW to a TOV metric across a shock wave	139	(21)
A class of exact shock wave solutions of the Einstein equations --- blast waves in GR	160	(16)
Cosmology with a shock wave	176	(39)
General comments on FRW-TOV shock waves	215	(28)
The Oppenheimer-Snyder limit and the solution for k = 0	243	(16)
Basic Aspects of Hyperbolic Relaxation Systems	259	(48)
Wen-An Yong
Introduction	259	(3)
Relaxation criterion	262	(6)
The Chapman-Enskog expansion	268	(4)
Admissible boundary conditions	272	(4)
Stability conditions	276	(5)
Typical examples	281	(3)
Moment closure systems	284	(5)
Discrete velocity kinetic models	289	(5)
Relaxation limits for smooth solutions	294	(5)
Shock structure problems	299	(8)
Multidimensional Stability of Planar Viscous Shock Waves	307	(1)
Kevin Zumbrun
Introduction	307	(15)
The Evans function and its low frequency limit	322	(12)
Necessary conditions for stability	334	(8)
Sufficient conditions for stability	342	(60)
Pointwise bounds for scalar equations	402	(33)
One-dimensional stability: the stability index	435	(16)
Discussion and open problems	451	(3)
Appendices: extensions and auxiliary calculations	454