Mathematical Sciences: Mathematical Problems in CompressibleFluid Flow

调查员将研究与以下相关的各种问题 的存在性、稳定性和数值逼近 某些可压缩流体流动模型的解。 先前 关于一维球对称流动的工作 将扩展到一般多维流和模型 用于化学反应流和涉及相的流 过渡。 该项目的两个长期目标是： 证明了Navier-Stokes方程解的整体存在性 具有大的不连续柯西的可压缩流方程 数据;第二，了解这些方程的解决方案， 一维的情况下，与黎曼初始数据，在条款 相应的无粘欧拉方程的解 流体. 本项目研究的数学模型试图 预测可能发生化学反应的流体的行为， 以及液体或气体状态的流体 可以同时存在。 的设计与开发 数值求解模型方程的计算机方法是 通过严格的数学理解， 为什么存在解决方案，在什么意义上，以及这些解决方案的敏感性如何， 解决方案是针对系统的初始状态。 主 这个项目的目标是提供这种严格的数学 理解数学模型;次要目标 是将这些数学结果应用于设计、测试 和分析计算机算法产生近似 解决方案

The investigator will study various questions relating to the existence, stability, and numerical approximation of solutions of certain models of compressible fluid flow. Previous work concerning one-dimensional and spherically symmetric flows will be extended to general multidimensional flows and to models for chemically reacting flows and flows involving phase transitions. Two long-range goals of the project are: first, to prove the global existence of solutions of the Navier-Stokes equations for compressible flow with large, discontinuous Cauchy data; second, to understand solutions of these equations in the one-dimensional case, with Riemann initial data, in terms of solutions of the corresponding Euler equations for inviscid fluids. The mathematical models studied in this project attempt to predict the behavior of fluids which may react chemically so as to ignite, and fluids in which both the liquid or gaseous states may be present simultaneously. The design and development of computer methods for numerically solving the model equations is greatly facilitated by a rigorous mathematical understanding of why solutions do exist and in what sense, and how sensitive these solutions are to the initial state of the system. The primary goal of this project is to provide this rigorous mathematical understanding of the mathematical models; a secondary objective is to apply these mathematical results to the design, testing, and analysis of computer algorithms for generating approximate solutions.