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

本项目致力于研究流体力学中出现的多维守恒定律及其相关应用。特别是从二维可压缩等熵欧拉方程及相关偏微分方程的理论出发，重点研究以下主题：(a)通过障碍物的定常和非定常跨声速流动；(b)等距嵌入/浸没的高斯-科达齐方程的流体动力学方法；(c)欧拉方程特殊数据的二维解；(d)磁流体动力学、弹性动力学和辐射流体动力学中出现的二维问题。研究的目标是：(a)开发新的解析方法和数值格式来构建多维解；(b)探索解的全局结构；(c)理解长期行为，例如奇点和稳定性的演化；(d)获得对多维问题的见解，特别是识别多维解的功能空间。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，最常见的例子是气体。他们的研究对于理解空气动力学、大气科学、天体物理学、等离子体物理学等至关重要。虽然一维问题很好理解，但多维情况的一般理论在数学上还不发达。该项目旨在促进对可压缩流动多维方程的数学理解。该项目的目的是促进数学和力学这一基础领域的知识，并为这一重要领域的学生提供教育和培训。

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.