Advances in the Theory of Shock Waves: 47 (Progress in Nonlinear Differential Equations and Their Applications) - Tapa dura

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.

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 M,tivier)

* 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.