Mathematical Sciences: Mathematical Problems in Compressible Fluid Flow

-提出者将研究与粘性、可压缩流体流动理论中出现的某些偏微分方程组的解的存在性、稳定性、正则性、大时间行为和数值逼近有关的各种问题。不连续的解决方案特别令人感兴趣，并在项目中起到统一的作用。在以前的工作中，对于一维空间的流动和球对称的流动，已经获得了相当完善的理论。最近的研究已经将其中一些结果扩展到一般的多维流动。然而，对于多维流动，仍有大量工作要做，这将占据项目的更大部分。还建议继续研究一维流动的各种遗留问题，特别是粘性接触不连续性的严格数学分析。提出者将研究与可压缩流体流动的重要模型有关的各种数学问题。这些模型出现在广泛的应用中，从超音速飞行到动态气象学，以及许多其他应用。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，无法在任何明确的意义上“解决”。另一方面，计算机方法通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；次要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.