Strong Shock Waves in Cosmology, General Relativity, and Classical Fluids

这个项目研究非线性偏微分方程组理论中的基本开放问题，涉及产生冲击波的大振幅或高振荡解。该项目的第一部分解决了与爱因斯坦广义相对论方程中的冲击波传播有关的数学和宇宙学问题。该项目的第二部分解决了关于经典流体中强波的振荡和共振相互作用的未解决问题，特别是可压缩流体的欧拉方程。冲击波是在流体中发展和传播的陡峭锋面。这种流体流动的数学模型涉及到具有非常奇异结构的解的高度非线性的微分方程组，这是很难分析的。对于产生冲击波的大振幅或高度振荡的解，没有适用的通用数学理论，许多基本问题仍未解决。这些问题是本研究项目的主题，该项目发展了重力冲击波和可压缩流体中的冲击波的数学理论。这项工作对宇宙的宇宙学模型和飞机设计的跨音速流动建模具有潜在的应用价值。

This project investigates fundamental open problems in the theory of nonlinear partial differential equations concerning large amplitude or highly oscillatory solutions that generate shock waves. The first part of the project addresses both mathematical and cosmological issues related to shock wave propagation in the Einstein equations of general relativity. The second part of the project addresses unsolved problems regarding oscillatory and resonant interactions of strong waves in classical fluids, particularly the Euler equations for a compressible fluid.Shock waves are steep fronts that develop and propagate in fluids. Mathematical models of such fluid flow involve highly nonlinear differential equations that have solutions with very singular structure, which is difficult to analyze. For large amplitude or highly oscillatory solutions that generate shock waves, there is no general mathematical theory that applies, and many fundamental problems remain unsolved. Such problems are the topics of this research project, which develops the mathematical theory of gravitational shock waves and of shock waves in compressible fluids. The work has potential application to cosmological models of the universe and to modeling of transonic flow for aircraft design.