Nonlinear Stability of Multidimensional Structures in Fluid Dynamics

在这个项目中，主要研究者打算对可压缩流体的数学研究中出现的某些重要多维结构的存在性和非线性稳定性（或不稳定性）进行严格的调查。这些结构包括边界层、激波、爆轰波阵面和涡面。关于这些主题有大量的应用文献，但直到最近，它们还抵制仔细的数学分析，特别是在多维情况下。在主要研究者，他的合作者和其他人最近的工作中，新的工具已经变得可用，来自PDE，谐波分析和动力系统，这使得对在研究这种结构的稳定性时出现的高度奇异摄动问题进行严格的研究成为可能。物理学家和工程师对这个项目中考虑的这类问题所做的许多早期工作都是形式上的或启发式性质。从严格的数学角度来处理同样的问题，不仅可以为以前的工作打下坚实的基础，而且可以加深理解，发现新的现象，并揭示错误。所建议的边界层工作涉及到与高速飞机稳定性有关的工程问题，并且可能对研究这些问题的工程师感兴趣。研究结果将通过在高质量期刊上发表以及在会议和其他大学举办讲座的方式传播。该赠款还将促进首席研究员与法国和中国同事的国际研究合作。 根据这项赠款进行的研究预计将对其他数学家的工作，研究生课程的准备和研讨会，以及对研究生和博士后的培训产生影响。

In this project the principal investigator intends to carry out a rigorous investigation of the existence and nonlinear stability (or instability) of certain important multidimensional structures arising in the mathematical study of compressible fluids. These structures include boundary layers, shock waves, detonation fronts, and vortex sheets. There is a vast applied literature on these topics, but until relatively recently they have resisted careful mathematical analysis, particularly in the multidimensional case. In recent work by the principal investigator, his collaborators, and others, new tools have become available, coming from PDE, harmonic analysis, and dynamical systems, that permit a rigorous study of the highly singular perturbation problems that arise in investigating the stability of such structures.Much of the earlier work done by physicists and engineers on problems of the type under consideration in this project has been of a formal or heuristic nature. Approaching the same problems from a mathematically rigorous point of view not only puts previous work on a firm foundation but can also deepen understanding, uncover new phenomena, and bring errors to light. The proposed work on boundary layers is related to problems in engineering concerned with the stabilization of high speed aircraft and may be of interest to engineers working on such problems. The results of the research would be disseminated by publication in high quality journals and by lectures given at conferences and other universities. The grant will also foster international research collaboration of the principal investigator with colleagues in France and China. The research carried out under this grant is expected to have an impact on the work of other mathematicians, on graduate course preparation and seminars, and on the training of graduate students and postdocs.