Mathematical Problems in Compressible Fluid Flow

可压缩流体流动的数学问题

• 批准号: 9986658
• 负责人: David Hoff
• 金额: $ 15.9万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9986658&HistoricalAwards=false
• 关键词: Mathematical; Problems; Compressible; Fluid; Flow

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.

9986658 Hoff这是一个研究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建议。以前的工作在三维空间中的Navier-Stokes方程的不连续解的存在将被扩展到有限区域，需要新的洞察力在边界处的可压缩涡的生产。严格的分析传播的奇异性和规则性的不连续表面的解决方案，这些方程。一个连续的依赖理论将被开发，足以提供一个框架，其中可以研究近似解的数值程序。在一维空间中的Navier-Stokes方程的吸引子的尺寸将估计的大小施加的力和系统的标度不变量。最后，将把以前关于标量守恒律的大振幅粘性激波的多维稳定性的工作推广到更适用的系统情况，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。