Mathematical Sciences: Mathematical Problems in Compressible Fluid Flow

-申请者将研究粘性可压缩流体流动理论中出现的某些偏微分方程系统的存在性、稳定性、规律性、大时间行为和解的数值近似等各种问题。不连续的解决方案特别有趣，并且在项目中起统一的作用。在以往的工作中，一维流动和球对称流动已经得到了一个相当完善的理论。最近的研究将其中的一些结果扩展到一般的多维流。然而，对于多维流还有大量的工作要做，这将占据项目的大部分。还建议继续研究一维流动的各种问题，特别是粘性接触不连续面的严格数学分析。申请者将研究有关可压缩流体流动重要模型的各种数学问题。这些模型在广泛的应用中出现，从超音速飞行到动态气象学，等等。虽然构建这些模型的主要目的是实现预测能力，但它们太复杂了，无法以任何明确的方式“解决”。另一方面，通常可以用计算机方法生成适当的近似解。然而，这些方法的智能设计关键取决于对解决方案存在的原因、意义以及它们对数据噪声敏感的方式的严格理解。因此，该项目的主要目标是为这些模型提供如此严格的数学分析；第二个目标是将这些数学见解应用于生成近似解的算法的智能设计和分析。

- The proposer will investigate various questions relating to the existence, stability, regularity, large-time behavior, and numerical approximation of solutions of systems of certain partial differential equations arising in the theory of viscous, compressible fluid flow. Discontinuous solutions are of particular interest, and play a unifying role in the project. A fairly well-developed theory has been attained in previous work for flows in one space dimension and for flows which are spherically symmetric. More recent research has extended some of these results to general multidimensional flows. A great deal of work remains to be done for multidimensional flows, however, and this will occupy the greater part of the project. It is also proposed to continue work on various issues remaining for one-dimensional flows, especially the rigorous mathematical analysis of viscous contact discontinuities. The proposer will study various mathematical questions concerning important models of compressible fluid flow. These models arise in a broad range of applications, from supersonic flight to dynamic meteorology, among many others. While the main purpose for 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 often be generated by computer methods. However, the intelligent design of such methods depends crucially on a rigorous understanding of why solutions do exist, in what sense, and in what ways they are sensitive to noise in the data. The primary goal of this project is therefore to provide such a rigorous mathematical analysis for these models; a secondary objective is to apply these mathematical insights to the intelligent design and analysis of algorithms for generating approximate solutions.