The Geometry of Shock Waves

DMS-9500694 Temple该提案解决了非线性双曲守恒律系统的冲击波解的构造、分析和数值计算问题，重点是气体动力学的可压缩欧拉方程。冲击波是由非粘性方程中的不连续解模拟的，冲击波是将时间不可逆性、衰减、信息损失、耗散和熵增加引入到解的动力学中的东西。作者最近构造的爱因斯坦方程的激波解利用了这样一个事实，即引力度规在激波上是连续的，即使流体变量是不连续的。这允许比冲击波更弱的解决方案，并且该提案探索了引力度量对冲击波传播几何形状的组织效应。在声波的非线性理论中，作者解决了解释周期解的稳定性的公开问题。他们的想法是将波相互作用中的李括号误差视为虚波，从而他们可以识别和研究一种非线性抵消机制，该机制解释了为什么由于熵增加而导致的耗散会在很长一段时间内击败由于非线性相互作用而导致的波的放大。 该提案涉及分析和解释作者最近构建的广义相对论爱因斯坦方程的新冲击波解。这个精确解解决了一个问题，这个问题最早是在1939年由J·罗伯特·奥本海默（J. Robert Oppenheimer）和他的学生H.斯奈德爱因斯坦广义相对论方程的新的冲击波解提供了一个模型，在这个模型中，宇宙从冲击波爆炸开始，而不是公认的“大爆炸”情景（整个宇宙从一个点爆炸）。该提案探讨了这种结构在模拟恒星动力学方面的可能应用。

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.