Partial Differential Equations in Conservation Laws and Applications

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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) Mixed-type PDE problems for transonic flows past an obstacle. (b) Mixed-type PDE problems for isometric embedding. (c) Existence and regularity of global solutions to the compressible multi-dimensional Navier-Stokes equations. (d) Global solutions to the viscous flows of related applications in liquid crystals. The goals of the research are: (a) To develop novel analytic methods and efficient techniques for solving some important problems in multi-dimensional inviscid and viscous conservation laws and applications. (b) To explore the qualitative behavior of flow motion. (c) To establish new connections of the isometric embedding problem with elastodynamics. (d) To gain insights into other multi-dimensional problems of conservation laws and emerging applications.The aim of this research program is to develop new methods of analysis and techniques for studying some nonlinear partial differential equations governing the motion of compressible fluid flows and related applications. Compressible fluids such as gases are important in nature. Their study is crucial for understanding aerodynamics, atmospheric science, astrophysics, plasma physics, biology, elastodynamics, etc. While the one-dimensional problems are rather well understood, the general theory for the 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, and will provide education and training to students in this important field.