The aim of this research program is to develop new methods of mathematical analysis and techniques for studying some nonlinear partial differential equations governing the motion of compressible flows and related applications. Compressible fluids such as gases are ubiquitous in nature. Understanding of dynamics of compressible fluids is a crucial ingredient to prediction and control of important physical processes arising in aerodynamics, atmospheric science, astrophysics, plasma physics, biology and medicine, material science, and others. While the highly idealized one-dimensional problems are rather well understood, the general theory for the real-life multi-dimensional case is mathematically underdeveloped. The project will advance the mathematical understanding of the multi-dimensional equations of compressible flows and related problems in emerging applications. The research program will advance knowledge of the fundamental areas of mathematics and mechanics as well as applications. Graduate students will be involved in this research and trained on the outstanding problems in the related research fields.
This project is devoted to a mathematical study of some nonlinear partial differential equations in multi-dimensional conservation laws and related applications. In particular, the study focuses on the following topics from the theory of inviscid and viscous compressible flows and related applications: (a) the existence and stability of vortex sheets in multi-dimensional compressible elastodynamics, (b) the global smooth isometric embedding of surfaces of negative curvature, (c) the stochastic partial differential equations of compressible flows, and (d) the active liquid crystal systems in biology/biophysics. The goal of the research is to develop novel analytic methods and efficient techniques for solving some important problems in multi-dimensional inviscid and viscous conservation laws and applications, to explore new phenomena of the motion of compressible flows, and to gain insights into the general multi-dimensional problems and emerging real-world applications.
