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AuthorLiu, Tai-Ping. author
TitleAdvances in the Theory of Shock Waves [electronic resource] / by Tai-Ping Liu, Guy Mรฉtivier, Joel Smoller, Blake Temple, Wen-An Yong, Kevin Zumbrun ; edited by Heinrich Freistรผhler, Anders Szepessy
ImprintBoston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2001
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Descript VIII, 520 p. online resource


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 Einsteinยญ Euler equations of general relativity; indeed, the mathematical and physical conยญ 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 apยญ proach had for a long time seemed out of reach. The stability problem for "inยญ 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 semiยญ 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 propยญ 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


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 โ{128}{148} blast waves in GR -- 5. Cosmology with a shock wave -- 6. General comments on FRWโ{128}{147}CTOV shock waves -- 7. The Oppenheimerโ{128}{147}CSnyder limit and the solution for k = 0 -- IV. Basic Aspects of Hyperbolic Relaxation Systems -- 1. Introduction -- 2. Relaxation criterion -- 3. The Chapmanโ{128}{147}CEnskog 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

Mathematics Partial differential equations Applied mathematics Engineering mathematics Physics Gravitation Mathematics Partial Differential Equations Applications of Mathematics Mathematical Methods in Physics Classical and Quantum Gravitation Relativity Theory


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