Mathematical Problems in Compressible Fluid Flow

本课题研究与某些偏微分方程组的存在性、稳定性、正则性和定性特征有关的问题。这些系统在数学上密切相关，它们都与可压缩流体流动的Navier-Stokes方程具有共同特征，并且都用作具体应用中出现的物理现象的模型，包括可压缩流体和气体的流动、浅水理论、磁流体动力学、燃烧理论和半导体建模。主要目标是通过应用分析和偏微分方程领域的技术，开发关于这些模型的精确的、数学上确定的陈述。这项研究在几个方面与基础物理科学有关：第一，它可以验证、确定适用范围的限制，或者在某些情况下使模型无效；第二，对模型进行严格的数学分析可以加深对物理现象的理解；最后，通过确定模型适合的拓扑结构并阐明这种结构的潜在原因，这项工作可以为开发有效计算近似解的计算机程序奠定基础。提出者将研究有关可压缩流体和材料的重要模型的数学问题。这些模型出现在广泛的应用中，包括超音速飞行、动态气象学、半导体理论和燃烧过程。虽然构建这些模型的主要目标是实现预测能力，但它们太复杂了，无法在任何明确的意义上解决。另一方面，通过计算方法经常可以产生足够的近似解。然而，这些方法的智能设计关键取决于对解决方案为什么存在、在什么意义上以及它们以什么方式对数据中的噪声敏感的理解。该项目将为这些模型提供如此严格的数学分析，并确定其适用范围的可能限制，从而为开发有效计算近似解的计算机程序奠定基础。

This project investigates questions relating to the existence, stability, regularity, and qualitative features of certain systems of partial differential equations. These systems are closely related mathematically, all sharing common features with the Navier-Stokes equations of compressible fluid flow, and all serving as models for physical phenomena arising in concrete applications, including the flow of compressible fluids and gases, shallow water theory, magnetohydrodynamics, combustion theory, and semiconductor modeling. The primary goal is to develop precise, mathematically certain statements about these models by applying techniques from the fields of analysis and partial differential equations. This research is related to the underlying physical science in several ways: first, it can validate, determine limits on the range of applicability, or in some cases invalidate the models; second, rigorous mathematical analysis of the models can lead to deeper understanding of the physical phenomena; finally, by determining topologies in which the models are well-posed and elucidating underlying reasons for this structure, the work can set the stage for the development of computer procedures for the effective computation of approximate solutions.The proposer will study mathematical questions concerning important models of compressible fluids and materials. These models arise in a broad range of applications, including supersonic flight, dynamic meteorology, semiconductor theory, and combustion processes. While the main goal in constructing these models is to achieve a predictive capability, they are far too complicated to be solved in any explicit sense. On the other hand, adequate approximate solutions can frequently be generated by computational methods. However, the intelligent design of such methods depends crucially on understanding of why solutions do exist, in what sense, and in what ways they are sensitive to noise in the data. The project will provide such a rigorous mathematical analysis for these models and determine possible limits on their range of applicability, thereby setting the stage for the development of computer procedures for the effective computation of approximate solutions.