Analysis and Applications of Nonlinear Partial Differential Equations in Conservation Laws

本研究计划是研究流体力学和磁流体力学中出现的多维守恒定律。重点研究了无粘可压缩流动的多维欧拉方程、电磁场中粘性可压缩流动的磁流体力学方程及其相关应用。本研究的目标是：(1)发展具有一定对称性的多维欧拉方程全局解的解析技术和数值格式，并研究解在原点附近的定性行为；(2)构造Euler方程的二维特殊整体解，得到解的整体结构，研究不连续结构的演化和稳定性；(3)探索波浪相互作用，构建压力梯度系统的二维Riemann解和欧拉方程；(4)研究初始数据大时粘性MHD方程一维和多维问题解的奇点形成、长时间行为、全局存在性和稳定性。要解决这些问题，需要发展新的思想和技术。该研究将深入了解解的结构和定性行为，并将为可压缩欧拉方程和MHD方程的一般多维问题提供启示。该项目致力于对控制可压缩流体运动的非线性偏微分方程及其相关应用进行数学研究。可压缩流体在自然界中无处不在，如气体、等离子体等，对其研究对于理解空气动力学、环境科学、天体物理学、等离子体物理学等至关重要。虽然一维无粘可压缩流体流动的欧拉方程已经被很好地理解，但多维情况下的一般理论在数学上相对欠发达。粘性可压缩导电流体在任何空间尺度的电磁场中流动的磁流体动力学方程的数学理论更是不发达。本研究计划的目的是探讨一些重要的问题，以促进对多维可压缩流体流动和磁流体力学的数学理解。这个项目的成功将推进数学和力学这一基础领域的知识，并将为学生提供有关该领域突出问题的教育和培训。

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.