Shock-Waves and Geometry

数学科学：冲击波和几何0102493圣殿这个项目的重点是几何原理和冲击波的数学理论的原则之间的相互作用。 主要有三个研究领域。 第一部分是对Einstein方程初值问题弱解的正则性和激波形成的分析。 第二个领域是发展冲击波匹配方法，并将其应用于广义相对论中的爆炸波。 第三个领域是研究守恒律系统中的振荡，特别强调分析跨音速流中产生的振荡。 作为该项目的一部分，我们结合联合收割机格里姆的波相互作用理论与广义相对论来分析爱因斯坦方程的冲击波相互作用。 由于爱因斯坦方程包含可压缩的欧拉方程作为子系统，因此，关于广义相对论中冲击波传播的结果有望对经典流体中冲击波的分析产生新的影响。我们正在发展爱因斯坦广义相对论中的冲击波传播理论。冲击波最为人所知的是标志着爆炸前沿的冲击波。 在这种情况下，冲击波是快速膨胀的材料和周围未受干扰的空气之间的表面，冲击波阵面传播到其中。在广义相对论中，冲击波是时空本身曲率中的波。因此，有新的和有趣的基本问题涉及扩展爱因斯坦方程的设置，承认冲击波传播。 作为该项目的一部分，我们正在构建爱因斯坦方程的新的精确解，这些解将爆炸波纳入大爆炸--第一次使宇宙膨胀的爆炸。 数学结果也将应用于普通流体中的冲击波。

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.