Systems of Hyperbolic Conservation Laws and Nonlinear Wave Equations

非线性双曲偏微分方程组(PDE)被用来描述波浪运动，包括气体动力学、水波和交通流。例如，气体动力学的可压缩欧拉方程(从恒星形成到飞机设计有许多应用)是以双曲偏微分方程组的形式表示物理守恒定律的。可压缩欧拉方程的解通常会产生不连续性，这就是所谓的冲击波；当飞机的移动速度超过音速时，后者表现为音爆。一般而言，拟线性双曲型偏微分方程组的解可以产生有限时间奇性。在这项研究中，研究者重点研究了可压缩欧拉方程和非线性波动方程，它们的解在有限时间内形成了激波和尖点奇点(如液晶方程中出现的那些奇点)。该项目使用分析和数值技术来加强对这些方程的基本理解。通过对波的奇异行为的研究，可以更好地理解双曲偏微分方程组的基本特征。这一研究项目寻求加深对包括冲击波在内的大量数据解决方案的结构的理解。将研究解的变化及其传播、解的一般规律性以及对真空附近解的行为的理解等问题。该项目的第二部分集中于模拟向列相液晶的准线性波动系统，其中的解发展为尖点奇异性。这项研究包含了许多新的方法。在Lipschitz连续相依性的研究中，由于标准的Soblev范数不能产生有用的信息，因此将使用Finsler类型的最优传输度量。为了研究一般的规律性，将使用Thom横截性定理。

Nonlinear hyperbolic partial differential equations (PDEs) are used for the mathematical description of wave-like motion, including gas dynamics, water waves, and traffic flow. For example, the compressible Euler equations of gas dynamics (which have numerous applications ranging from star formation to aircraft design) are in the form of a system of hyperbolic PDEs expressing physical conservation laws. Solutions of compressible Euler equations often develop discontinuities, which are known as shock waves; the latter are manifested as a sonic boom when an aircraft moves faster than the speed of sound. In general, solutions of quasi-linear hyperbolic PDEs can develop finite-time singularities. In this research, the investigator focuses on the compressible Euler equations and the nonlinear wave equations whose solutions form shock waves and cusp singularities (such as those that occur in liquid crystal equations) in finite time. The project uses both analytical and numerical techniques to enhance basic understanding of these equations. This research on singular behavior of waves is expected to lead to better understanding of fundamental features of hyperbolic PDEs. This research project seeks to deepen understanding of the structure of solutions with large data including shock waves. The issues of the variation of solutions and their propagation, the generic regularity of solutions, and understanding of behavior of solution near vacuum will be investigated. The second part of the project focuses on the quasi-linear wave system modeling nematic liquid crystals, where the solution develops a cusp singularity. The research contains a number of new approaches. In the study of Lipschitz continuous dependence, a Finsler type optimal transport metric will be utilized, since the standard Sobolev norms do not yield useful information. To study the generic regularity, the Thom transversality theorem will be used.