Systems of Hyperbolic Conservation Laws and Nonlinear Wave Equations

双曲守恒定律和非线性波动方程组

• 批准号: 1715012
• 负责人: Geng Chen
• 金额: $ 14.5万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1715012&HistoricalAwards=false
• 关键词: Systems; Hyperbolic; Conservation; Laws; Nonlinear

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.

非线性双曲型偏微分方程（PDE）用于波动运动的数学描述，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（从星星形成到飞机设计都有许多应用）是以双曲偏微分方程组的形式表达物理守恒定律的。可压缩欧拉方程的解经常会产生不连续性，这被称为激波;后者表现为当飞机移动速度超过音速时的音爆。一般来说，拟线性双曲型偏微分方程的解可以发展出有限时间奇点。在这项研究中，研究者的重点是可压缩的欧拉方程和非线性波动方程，其解决方案形成冲击波和尖点奇点（如那些发生在液晶方程）在有限的时间。该项目使用分析和数值技术来提高对这些方程的基本理解。本文对波动奇异性的研究有助于更好地理解双曲型偏微分方程的基本特征。 该研究项目旨在加深对包含冲击波在内的大数据的解决方案结构的理解。本课程将探讨解的变化及其传播，解的一般规律性，以及对真空附近解行为的理解。该项目的第二部分侧重于准线性波系统建模的液晶，其中的解决方案开发一个尖点奇点。这项研究包含了一些新的方法。在Lipschitz连续依赖的研究中，Finsler型最优传输度量将被利用，因为标准的Sobolev范数不产生有用的信息。为了研究通有正则性，将使用Thom横截性定理。