Entropies, geometric structures, and interactions for systems of conservation laws

首席研究员和他的合作者研究双曲守恒定律，这是一类偏微分方程，涵盖了应用数学中的许多基本方程，特别是在流体动力学中。当前研究的主要方向是：(A)基于一维守恒律系统本征框架几何性质的系统研究方法；(B)使用消失粘度近似来获得守恒定律在几个空间维度上的相互作用估计；(C)守恒律系统的径向对称解，特别是可压缩欧拉系统的解。所有项目的动机都是需要提供科学家和工程师使用的关于物理模型属性的严格陈述。结果评估了各种物理模型的有效性范围，并将用于划定和改进现有模型。特别强调的是理解在高速飞行、交通流、燃烧和爆炸等模拟中使用的复杂非线性方程的解决方案。

The Principal Investigator and his collaborators study hyperbolic conservation laws, a class of partial differential equations that covers many of the fundamental equations in applied math, and in particular in fluid dynamics. The main lines of current research are: (A) a systematic approach to the study of one-dimensional systems of conservation laws based on geometric properties of their eigen-frames; (B) employing vanishing viscosity approximations to obtain interaction estimates for conservation laws in several space dimensions; (C) radially symmetric solutions to systems of conservation laws, and in particular solutions to the compressible Euler system.All of the projects are motivated by the need to provide rigorous statements about the properties of physical models that are used by scientists and engineers. The results assess the range of validity of various physical models and will be of use in delimiting and refining existing models. Particular emphasis is put on understanding solutions of complex, nonlinear equations that are used in simulations of e.g. high speed flight, traffic flow, combustion and detonations.