Advances in the Theory of Shock Waves

Gathers recent insights in well posedness for conservation laws, multidimensional shock stability, dissipation / relaxation phenomena, and applications of shock wave theory to general relativity. Five self -contained and interrelated articles, written by recognized experts in the field, provide a survey of central themes in the modern mathematical theory of shock waves. Combines the rigor of mathematical analysis with attention to the physical origins and applications of the field. Useful for graduate seminars or courses, or as a reference text for mathematicians, physicists, and theoretically motivated engineers.

Klappentext

In the field known as "the mathematical theory of shock waves," very exciting and unexpected developments have occurred in the last few years. Joel Smoller and Blake Temple have established classes of shock wave solutions to the Einstein Euler equations of general relativity; indeed, the mathematical and physical con sequences of these examples constitute a whole new area of research. The stability theory of "viscous" shock waves has received a new, geometric perspective due to the work of Kevin Zumbrun and collaborators, which offers a spectral approach to systems. Due to the intersection of point and essential spectrum, such an ap proach had for a long time seemed out of reach. The stability problem for "in viscid" shock waves has been given a novel, clear and concise treatment by Guy Metivier and coworkers through the use of paradifferential calculus. The L 1 semi group theory for systems of conservation laws, itself still a recent development, has been considerably condensed by the introduction of new distance functionals through Tai-Ping Liu and collaborators; these functionals compare solutions to different data by direct reference to their wave structure. The fundamental prop erties of systems with relaxation have found a systematic description through the papers of Wen-An Yong; for shock waves, this means a first general theorem on the existence of corresponding profiles. The five articles of this book reflect the above developments.

Inhalt
I. Well-Posedness Theory for Hyperbolic Systemsof Conservation Laws.- 1. Scalar conservation law.- 2. Glimm scheme.- 3. Wave tracing.- 4. Nonlinear functional.- II. Stability of Multidimensional Shocks.- 1. The uniform stability condition.- 2. The uniform stability estimates.- 3. Well posedness of the linearized shock front equations.- 4. The existence of multidimensional shocks.- 5. Stability of weak shocks.- III. Shock Wave Solutions of the Einstein Equations:A General Theory with Examples.- 1. Introduction.- 2. Solutions of the Einstein equations when the metricis only Lipschitz continuous across an interface.- 3. Matching an FRW to a TOV metric across a shock wave.- 4. A class of exact shock wave solutions of the Einstein equations blast waves in GR.- 5. Cosmology with a shock wave.- 6. General comments on FRWCTOV shock waves.- 7. The OppenheimerCSnyder limit and the solution for k = 0.- IV. Basic Aspects of Hyperbolic Relaxation Systems.- 1. Introduction.- 2. Relaxation criterion.- 3. The ChapmanCEnskog expansion.- 4. Admissible boundary conditions.- 5. Stability conditions.- 6. Typical examples.- 7. Moment closure systems.- 8. Discrete velocity kinetic models.- 9. Relaxation limits for smooth solutions.- 10. Shock structure problems.- V. Multidimensional Stability of Planar Viscous Shock Waves.- 1. Introduction.- 2. The Evans function and its low frequency limit.- 3. Necessary conditions for stability.- 4. Sufficient conditions for stability.- 5. Pointwise bounds for scalar equations.- 6. One-dimensional stability: the stability index.- 7. Discussion and open problems.- 8. Appendices: extensions and auxiliary calculations.