Systems of Hyperbolic Conservation Laws and Nonlinear Wave Equations

非线性双曲型偏微分方程（PDEs）用于波浪运动的数学描述，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）以双曲偏微分方程系统的形式表示物理守恒定律。可压缩欧拉方程的解经常出现不连续点，这就是冲击波；后者表现为当飞机运动速度超过音速时的音爆。一般来说，拟线性双曲偏微分方程的解具有有限时间奇异性。在这项研究中，研究者着重于可压缩欧拉方程和非线性波动方程，其解在有限时间内形成激波和尖点奇点（如液晶方程中出现的那些）。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。对波的奇异行为的研究有望使人们更好地理解双曲偏微分方程的基本特征。该研究项目旨在加深对包括冲击波在内的大数据解决方案结构的理解。研究了解的变异及其传播、解的一般规则性以及解在真空附近的行为。该项目的第二部分侧重于模拟向列液晶的准线性波系统，其中解决方案发展为尖端奇点。这项研究包含了许多新方法。在Lipschitz连续依赖的研究中，由于标准Sobolev规范不能产生有用的信息，因此将使用Finsler型最优传输度量。为了研究泛正则性，将使用托姆横截定理。

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.