Hyperbolic Systems of Conservation Laws and Applications

这个研究项目涉及守恒定律双曲系统理论的几个方面，涉及的项目可分为两类。第一类是将消失黏度方法应用于(a)具有弱耗散机制的双曲系统，如摩擦、衰退记忆和松弛，以及(b)具有大初始数据的双曲守恒律的特殊系统。这些项目将建立具有重要应用的系统解决方案的存在性、稳定性和定性行为；例如，多孔介质的流动和具有衰退记忆的粘弹性材料的运动。第二类是守恒定律双曲系统的熵弱解对物理参数（如绝热指数和光速）的连续依赖。这些项目的目的是在气体动力学和狭义相对论中可压缩流体中对非线性通量函数的依赖问题进行研究。本研究项目涉及连续介质物理与守恒定律双曲系统理论之间的界面问题。在弹性、塑性、流体力学、半导体、气体动力学和燃烧研究中出现的大多数偏微分方程都可以表述为守恒定律。这个项目研究了这种守恒定律系统的数学性质，利用潜在的物理结构来指导分析，同时反过来阐明方程的性质，以帮助更好地理解连续体物理学。该计划的结果也有望提出将推进系统数值模拟的计算算法。

This research program deals with several aspects of the theory of hyperbolic systems of conservation laws and involves projects that can be divided into two categories. The first category is the application of the method of vanishing viscosity to (a) hyperbolic systems with weakly dissipative mechanisms such as friction, fading memory, and relaxation and (b) special systems of hyperbolic conservation laws with large initial data. These projects will establish the existence, stability, and qualitative behavior of solutions to systems with important applications; for example, porous media flows and the motion of viscoelastic materials with fading memory. The second category is the continuous dependence of entropy weak solutions to hyperbolic systems of conservation laws on physical parameters such as the adiabatic exponent and speed of light. The aim of these projects is to initiate an investigation on this issue of dependence on nonlinear flux functions with applications to compressible fluids in gas dynamics and special relativity.This research project concerns problems lying on the interface between continuum physics and the theory of hyperbolic systems of conservation laws. Most of the partial differential equations arising in the study of elasticity, plasticity, fluid mechanics, semiconductors, gas dynamics, and combustion can be formulated as conservation laws. This project investigates mathematical properties of systems of such conservation laws, making use of the underlying physical structure to direct the analysis, while conversely elucidating properties of the equations to help better understand continuum physics. The results of this program are also expected to suggest computational algorithms that will advance numerical simulation of the systems.