Mathematical Problems in Compressible Fluid Flow

这个项目将解决与某些偏微分方程系统的解的存在性，规律性，定性特征和大时间行为有关的各种数学问题。这些系统在数学上密切相关，都与多维可压缩流体流动的Navier-Stokes方程共享共同特征，并出现在各种应用中，包括可压缩流体和气体的流动，浅水理论，磁流体力学和自引力恒星系统的建模。该项目的主要目标是通过应用分析和偏微分方程领域的技术来实现这些模型的精确，数学上的某些陈述。这项研究试图通过验证模型，对模型的适用范围设定限制，或者在某些情况下甚至使正在考虑的模型无效，来影响基础物理科学。在智力水平上，它采用的推理过程与将实验室数据与基于数值模拟的预测进行比较的推理过程截然不同。事实上，确保现实模型适定性的数学机制几乎总是有物理对应物;这意味着用严格的数学严谨性分析这些模型的挑战也是在更新，更深层次上理解物理现象的挑战。此外，通过确定正确的框架，其中的数学模型是适定性和数学机制，使它如此，这项研究可以指出的方法，有效的设计方法，通过计算机模拟生成近似解。这个项目将解决与可压缩流体和材料的重要物理模型相关的各种数学问题。这些模型的应用范围很广，包括超音速飞行、动力气象学、燃烧过程和恒星动力学。虽然构建这些模型的主要目标是实现预测能力，但组成这些模型的方程组太复杂，无法在任何明确意义上“求解”。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，这些方法的智能和有效设计关键取决于对解决方案存在的原因、它们存在的意义以及它们对数据中的噪声敏感的方式的严格理解。因此，该项目将设法为这些模型提供严格的数学框架，并确定其适用范围的可能限制。该项目的第二个目标是吸引和培养年轻数学家到这一研究领域的发展。具体而言，PI将继续与现在和未来的研究生和博士后合作研究，将在补助金期间的第二年教授关于可压缩流数学问题的高级研究生课程，并将继续监督印第安纳州大学REU计划的本科生研究，大约每三个夏天。

This project will address various mathematical questions relating to the existence, regularity, qualitative features, and large-time behavior of solutions of certain systems of partial differential equations. These systems are closely related mathematically, all sharing common features with the Navier-Stokes equations of multidimensional, compressible fluid flow and arise in a variety of applications, including the flow of compressible fluids and gases, shallow-water theory, magnetohydrodynamics, and the modeling of self-gravitating stellar systems. The primary goal of the project is to achieve precise, mathematically certain statements about these models by applying techniques from the fields of analysis and partial differential equations. This research seeks to impact the underlying physical science by validating the models, setting limits on the range of applicability of the models, or in some cases even invalidating the models under consideration. At the intellectual level it applies a reasoning process quite distinct from that in which laboratory data is compared with predictions based on numerical simulations. Indeed, the mathematical mechanisms which insure well-posedness of realistic models nearly always have physical counterparts; this means that the challenge to analyze these models with strict mathematical rigor is also a challenge to understand the physical phenomena at a newer, deeper level. Additionally, by identifying the correct framework in which a mathematical model is well-posed and the mathematical mechanisms which make it so, this research can point the way to the effective design of methods for generating approximate solutions by computer simulation. This project will address various mathematical questions relating to important physical models of compressible fluids and materials. These models arise in a broad range of applications, including supersonic flight, dynamic meteorology, combustion processes, and stellar dynamics. While the main goal in constructing these models is to achieve a predictive capability, the systems of equations which comprise them 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 and effective design of such methods depends crucially, however, on a rigorous understanding of why solutions do exist, in what sense they exist, and in what ways they are sensitive to noise in the data. The project will therefore seek to provide this rigorous mathematical framework for these models and to determine possible limits on the range of their applicability. A second goal of the project will be to attract and nurture the development of younger mathematicians to this area of research. Specifically, the PI will continue collaborative research with present and future graduate students and postdocs, will teach an advanced graduate-level course on mathematical problems in compressible flow in the second year of the grant period, and will continue supervising undergraduate research in Indiana University's REU program approximately every third summer.