Shock-Waves and Geometry

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Mathematical Sciences: Shock Waves and Geometry0102493TempleThis project is focused on the interplay between principles of geometry and the principles of the mathematical theory of shock waves. There are three main areas of research. The first is analysis of regularity and shock formation in initial value problems for weak solutions of the Einstein equations. The second area is development of shock-matching methods with application to blast waves in general relativity. The third area is investigation of oscillations in systems of conservation laws, with particular emphasis on analyzing oscillations that are generated in transonic flow. As part of the project, we combine Glimm's theory of wave interactions with the theory of general relativity to analyze shock wave interactions for Einstein's equations. Since the Einstein equations contain the compressible Euler equations as a subsystem, results on shock wave propagation in general relativity are expected to have new implications for analysis of shock waves in classical fluids.We are developing a theory of shock wave propagation in Einstein's theory of general relativity. A shock wave is best known as the blast wave that marks the leading edge of an explosion. In that case, the shock wave is the surface between the rapidly expanding material and the ambient, undisturbed air into which the shock front propagates. In general relativity, shock waves are waves in the curvature of spacetime itself. There are thus new and interesting fundamental issues involved in extending Einstein's equations to a setting that admits shock wave propagation. As part of the project, we are constructing new exact solutions of Einstein's equations that incorporate a blast wave into the Big Bang -- the explosion that first set the universe into expansion. The mathematical results will also have application to shock waves in ordinary fluids.