Partial Differential Equations in Conservation Laws and Applications

本课题致力于多维守恒律中一些非线性偏微分方程解的数学研究及其应用。具体而言，本研究从无粘性和粘性可压缩流动理论及其应用出发，重点研究了以下问题：(A)跨声速跨声速绕流的混合型偏微分方程组问题。(B)等距嵌入的混合型偏微分方程组问题。(C)可压缩多维Navier-Stokes方程整体解的存在性和正则性。(D)液晶中相关应用的粘性流动的全球解决方案。研究的目标是：(A)发展新的分析方法和有效的技术来解决多维无粘和粘性守恒定律及其应用中的一些重要问题。(B)探索水流运动的定性行为。(C)建立等距嵌入问题与弹性动力学的新联系。(D)洞察守恒定律和新兴应用的其他多维问题。这项研究计划的目的是发展新的分析方法和技术，研究一些控制可压缩流体运动的非线性偏微分方程组及其相关应用。可压缩流体，如气体，在自然界中很重要。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、生物学、弹性动力学等是至关重要的。虽然一维问题已经被很好地理解了，但对于多维情况的一般理论在数学上还不够发达。该项目将促进对可压缩流动的多维方程和新兴应用中相关问题的数学理解，并将为学生提供这一重要领域的教育和培训。

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.