Hyperbolic Conservation Laws and Applications

本研究计划的目的是发展新的数学分析方法和技术，以研究控制可压缩流动运动的非线性偏微分方程及其相关应用。可压缩流体，如气体，在自然界中无处不在。了解可压缩流体的动力学是预测和控制空气动力学、大气科学、天体物理学、等离子体物理学、生物和医学、材料科学等领域重要物理过程的关键因素。虽然高度理想化的一维问题相当容易理解，但现实生活中的多维情况的一般理论在数学上是不发达的。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。该研究计划将推进数学和力学基础领域的知识以及应用。研究生将参与本研究，并对相关研究领域的突出问题进行培训。本课题致力于多维守恒律中某些非线性偏微分方程的数学研究及其应用。主要研究方向包括：无粘和粘性可压缩流动理论及相关应用：(a)多维可压缩弹性动力学中涡旋片的存在性和稳定性；(b)负曲率曲面的全局光滑等长嵌入；(c)可压缩流动的随机偏微分方程；(d)生物/生物物理学中的主动液晶系统。研究的目标是开发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和应用中的一些重要问题，探索可压缩流体运动的新现象，并深入了解一般的多维问题和新兴的现实世界应用。

The aim of this research program is to develop new methods of mathematical analysis and techniques for studying some nonlinear partial differential equations governing the motion of compressible flows and related applications. Compressible fluids such as gases are ubiquitous in nature. Understanding of dynamics of compressible fluids is a crucial ingredient to prediction and control of important physical processes arising in aerodynamics, atmospheric science, astrophysics, plasma physics, biology and medicine, material science, and others. While the highly idealized one-dimensional problems are rather well understood, the general theory for the real-life 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. The research program will advance knowledge of the fundamental areas of mathematics and mechanics as well as applications. Graduate students will be involved in this research and trained on the outstanding problems in the related research fields.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) the existence and stability of vortex sheets in multi-dimensional compressible elastodynamics,(b) the global smooth isometric embedding of surfaces of negative curvature,(c) the stochastic partial differential equations of compressible flows, and(d) the active liquid crystal systems in biology/biophysics. The goal of the research is to develop novel analytic methods and efficient techniques for solving some important problems in multi-dimensional inviscid and viscous conservation laws and applications, to explore new phenomena of the motion of compressible flows, and to gain insights into the general multi-dimensional problems and emerging real-world applications.