Mathematical Problems in Compressible Fluid Flow

9986658 hoff这是一个关于粘性可压缩流体流动理论中某些偏微分方程解的存在性、稳定性、规律性和大时间行为的各种问题的研究方案。先前关于三维空间中Navier-Stokes方程不连续解的存在性的工作将扩展到有限区域，这需要对边界处可压缩涡量的产生有新的见解。本文将对这些方程的解中的奇点传播和不连续曲面的正则性进行严格的分析。一个连续相关理论将被发展，足以提供一个框架，其中数值程序近似解可以研究。一维空间中Navier-Stokes方程的吸引子的维数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，先前关于标量守恒定律下的大振幅粘性冲击的多维稳定性的工作将扩展到更适用的系统情况。申请者将研究关于可压缩流体和材料的重要模型的各种数学问题。这些模型出现在广泛的应用，包括超音速飞行，动态气象学，半导体理论，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，无法以任何明确的方式“解决”。另一方面，计算机方法可以经常生成适当的近似解。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、意义和对数据噪声敏感的方式的严格理解。因此，这个项目的主要目标是为这些模型提供这样一个严格的数学分析。

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.