Analysis of Nonlinear Partial Differential Equations in Conservation Laws and Related Applications

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This project is devoted to the study of multi-dimensional conservation laws arising in fluid dynamics and related applications. In particular, the study focuses on the following topics from the theory of two-dimensional compressible isentropic Euler equations and related partial differential equations: (a) the steady and unsteady transonic flows past an obstacle, (b) a fluid dynamic approach for the Gauss-Codazzi equations of isometric embedding/immersion, (c) two-dimensional solutions with special data of the Euler equations, and (d) the two-dimensional problems arising in magnetohydrodynamics, elastodynamics, and radiation hydrodynamics. The goals of the research are: (a) developing novel analytic methods and numerical schemes to construct multi-dimensional solutions, (b) exploring global structure of solutions, (c) understanding long-time behavior, for example, evolution of singularities and stability, and (d) gaining insights into the multi-dimensional problems, in particular identifying the functional spaces of multi-dimensional solutions.The project is devoted to a mathematical study of some nonlinear partial differential equations governing the motion of compressible fluid flows and related applications. Compressible fluids are ubiquitous in nature, the most common examples being gases. Their study is crucial for understanding aerodynamics, atmospheric science, astrophysics, plasma physics, etc. While the one-dimensional problems are rather well understood, the general theory for the multi-dimensional case is mathematically underdeveloped. The project is to advance the mathematical understanding of the multi-dimensional equations of compressible flows. The aim of the project is to advance knowledge in this fundamental area of mathematics and mechanics, and to provide education and training to students in this important field.