Mathematical Sciences: Mathematical Problems in Compressible Fluid Flow

9703703霍夫这是一项继续研究各种问题的建议，这些问题与在连续介质力学的各个领域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。不连续的解决方案特别令人感兴趣，并将在项目中起到统一的作用。对于整个空间中不受外力作用的小范围流动，已经得到了相当完善的理论。这些结果将推广到有边界的区域，以及具有大初始数据和强迫项的流动。将发展一个连续依赖理论，足以提供一个框架，在其中可以研究近似这些解的数值程序。整个分析将扩展到在相关物理问题中产生的微分方程组，如磁流体动力学和粘弹性。最后，将继续最近关于Navier-Stokes扩散波逐点行为的工作：将对相关物理系统进行扩散波的推导和逐点分析，并将给出平面粘性激波相对于多维摄动的稳定性分析。提出者将研究与可压缩流体和材料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，包括超音速飞行、动态气象学、半导体理论以及粘弹性材料的设计和使用。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，计算机方法常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在哪些方面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。

9703703 Hoff This is a proposal to continue the investigation of 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 various areas of continuum mechanics. Discontinuous solutions are of particular interest, and will play a unifying role in the project. A fairly well-developed theory has been attained for flows in the whole space which are not subject to external forces and which are small. These results will be extended to regions with boundaries and to flows with large initial data and with forcing terms. A continuous-dependence theory will be developed, sufficient to provide a framework in which numerical procedures for approximating these solutions can be studied. The entire analysis will be extended to systems of differential equations arising in related physical problems, such as magnetohydrodynamics and viscoelasticity. Finally, recent work concerning the pointwise behavior of Navier-Stokes diffusion waves will be continued: the derivation and pointwise analysis of diffusion waves will be carried out for related systems of physical interest, and a stability analysis of planar viscous shock waves with respect to multidimensional perturbations will be given. The proposer will study various 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 the design and use of viscoelastic materials. 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 computer methods. The intelligent design of such methods depends crucially, however, on a rigorous understanding of why solutions do e xist, 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.