Mathematical Problems in Compressible Fluid Flow

9986658Hoff这是一项关于粘性、可压缩流体流动理论中出现的某些偏微分方程解的存在性、稳定性、正则性和大时间行为的各种问题的研究。以前关于三维空间中Navier-Stokes方程间断解存在性的工作将扩展到有限区域，这就需要对边界上可压缩涡量的产生有新的认识。我们将对这些方程的解中奇点的传播和不连续曲面的正则性进行严格的分析。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解的数值程序。一维空间中的Navier-Stokes方程的吸引子的维度将根据所施加的力的大小和系统的尺度不变量来估计。最后，将前人关于标量守恒律下大振幅粘性激波多维稳定性的工作推广到更适用的系统情形。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以什么方式对数据中的噪音敏感的严格理解。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。

9986658HoffThis is a proposal to study various questions relating to the existence, stability, regularity, and large-time behavior of solutions of certain partial differential equations arising in the theory of viscous, compressible fluid flow. Previous work on the existence of discontinuous solutions of the Navier-Stokes equations in three space dimensions will be extended to finite regions, requiring new insight into the production of compressible vorticity at boundaries. A rigorous analysis of the propagation of singularities and the regularity of discontinuity surfaces in solutions of these equations will be given. A continuous-dependence theory will be developed, sufficient to provide a framework in which numerical procedures for approximating solutions can be studied. The dimension of the attractor for the Navier-Stokes equations in one space dimension will be estimated in terms of the size of the applied force and scale-invariants of the system. Finally, previous work on the multidimensional stability of large-amplitude viscous shocks for scalar conservation laws will be extended to the more applicable case of systems.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 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.