Mathematical Sciences: Mathematical Problems in CompressibleFluid Flow

研究人员将研究与某些可压缩流体流动模型解的存在性、稳定性和数值逼近有关的各种问题。以前关于一维和球对称流动的工作将扩展到一般的多维流动，以及化学反应流动和涉及相变的流动的模型。该项目的两个长期目标是：第一，证明具有大的、不连续的柯西数据的可压缩流动的Navier-Stokes方程解的整体存在性；第二，了解这些方程在具有Riemann初始数据的一维情况下的解，即相应的无粘流体的Euler方程的解。这个项目中研究的数学模型试图预测可能发生化学反应从而点燃的流体的行为，以及可能同时存在液态或气态的流体的行为。通过对为什么存在解、在什么意义上以及这些解对系统初始状态的敏感性的严格的数学理解，极大地促进了用于数值求解模型方程的计算机方法的设计和开发。这个项目的主要目标是提供对数学模型的这种严格的数学理解；次要目标是将这些数学结果应用于计算机算法的设计、测试和分析，以生成近似解。

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.