Systems of Hyperbolic Conservation Laws and Nonlinear Wave Equations

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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.