The Geometry of Shock Waves

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DMS-9500694 Temple This proposal addresses the problem of constructing, analyzing and numerically computing shock-wave solutions of systems of nonlinear, hyperbolic conservation laws, with emphasis on the compressible Euler equations of gas dynamics. Shock-waves are modeled by discontinuous solutions in inviscid equations, and shock-waves are what introduce time-irreversibility, decay, loss of information, dissipation and increase of entropy into the dynamics of solutions. The author's recent construction of a shock wave solution of the Einstein equations takes advantage of the fact that the gravitational metric is continuous across a shock, even though the fluid variables are discontinuous. This allows for solutions that are weaker than shock wave, and the proposal explores the organizing effect that the gravitational metric has on the geometry of shock wave propagation. In the nonlinear theory of sound waves, the authors address the open problem of explaining the stability of periodic solutions. Their idea is to treat Lie bracket errors in wave interactions as virtual waves, and thereby they can identify and study a nonlinear cancellation mechanism that explains why the dissipation due to increasing entropy defeats the amplification of waves due to nonlinear interaction in the long time. The proposal involves the analysis and interpretation of the author's recent construction of a new shock wave solution of the Einstein equations of general relativity. This exact solution resolves a problem that was first posed in a famous 1939 paper by J. Robert Oppenheimer and his student H. Snyder. The new shock wave solution of the Einstein equations of general relativity provides a model in which the universe begins with a shock wave explosion instead of the well established ``Big Bang'' scenario (in which the entire universe burst from a single point). The proposal explores the possible application of this construction to modeling the dynamics of stars.