Mathematical Sciences: Mathematical Problems in Compressible Fluid Flow

小霍夫9703703 这是一个建议，继续调查有关的存在性，稳定性，规律性，大时间的行为，和数值逼近的某些偏微分方程系统的解决方案，在连续介质力学的各个领域的各种问题。 不连续的解决方案是特别感兴趣的，并将在项目中发挥统一的作用。对于不受外力作用的小空间中的流动，已经有了一个相当完善的理论。 这些结果将被扩展到具有边界的区域和具有大初始数据和强迫项的流动。一个连续的依赖理论将被开发，足以提供一个框架，在该框架中，可以研究这些解决方案的近似数值程序。整个分析将扩展到微分方程系统所产生的相关物理问题，如磁流体力学和粘弹性。最后，最近的工作有关的逐点行为的Navier-Stokes扩散波将继续：推导和逐点分析的扩散波将进行相关的系统的物理利益，和一个稳定性分析的平面粘性冲击波相对于多维扰动将给出。 该提议者将研究各种数学问题有关的重要模型的可压缩流体和材料。 这些模型出现在广泛的应用中，包括超音速飞行、动力气象学、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，适当的近似解决方案，可以经常产生的计算机方法。 然而，这种方法的智能设计关键取决于对解决方案为什么存在、在什么意义上存在以及它们对数据中的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严格的数学分析;次要目标是将这些数学见解应用于智能设计和算法分析，以生成近似解。

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.