Analysis and Applications of Nonlinear Partial Differential Equations in Conservation Laws

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The research program is for the study of multi-dimensional conservation laws arising in fluid dynamics and magnetohydrodynamics. In particular, the investigator studies some nonlinear problems on the multi-dimensional Euler equations for inviscid compressible flow, the magnetohydrodynamics (MHD) equations for viscous compressible flow in an electromagnetic field, and related applications. The objectives of this research are (1) to develop analytic techniques and numerical schemes to construct global solutions of the multi-dimensional Euler equations with certain symmetries and study the qualitative behavior of the solution near the origin; (2) to construct some special two-dimensional global solutions of Euler equations, obtain global structure of solutions, and study evolution of discontinuities and stability; (3) to explore the wave interactions and construct two-dimensional Riemann solutions of the pressure gradient system and the Euler equations; and (4) to study the formation of singularity, long-time behavior, global existence, and stability of solutions to both the one-dimensional and multi-dimensional problems of the viscous MHD equations when initial data are large. New ideas and techniques need to be developed to solve these problems. The study will provide deep insight into the structure and qualitative behavior of solutions, and will shed light on the general multi-dimensional problems for the compressible Euler and MHD equations.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 occur all around us in nature, e.g. gases and plasmas, whose study is crucial to understanding aerodyanmics, environmental science, astrophysics, and plasma physics, etc. While the one-dimensional Euler equations for inviscid compressible fluid flow are rather well understood, the general theory for the multi-dimensional case is comparatively mathematically underdeveloped. The mathematical theory of magnetohydrodynamic equations for viscous compressible conducting fluid flow in an electromagnetic field in any space dimensions is even less developed. The purpose of this research program is to investigate some important problems to advance the mathematical understanding of the multi-dimensional compressible fluid flows and magnetohydrodynamics. Success in this project will advance knowledge of this fundamental area of mathematics and mechanics, and will provide education and training to students on the outstanding problems in the field.