Stability of shocks and layers in Fluid Mechanics and related problems

流体力学中冲击和层的稳定性及相关问题

基本信息

批准号：
1614918
负责人：
Alexis Vasseur
金额：
$ 40.73万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1614918&HistoricalAwards=false
关键词：
Stability shocks layers Fluid Mechanics

项目摘要

The main part of the project concerns the study of dramatic behaviors in fluid mechanics known as shocks. A typical example, in oceanography, is the propagation of a tsunami. Understanding flows with shocks is of particular importance for engineering and industry. The investigator studies the stability of such structures using state of the art mathematical tools. The project focuses on these specific goals: (1) Advance the knowledge of physical systems that are of fundamental importance in engineering, meteorology, medicine, or natural disasters, such as flooding and tsunamis; (2) Develop powerful mathematical tools, based on the theory of nonlinear partial differential equations, that can be used to study these systems. Graduate students are involved in the work of the project, which provides good opportunities for training junior researchers in mathematical analysis and physical modeling.The investigator studies the stability of discontinuous solutions in compressible fluid mechanics, known as shocks. They are linked to intriguing behaviors of fluids, especially in supersonic flows. This project advances the understanding of these phenomena. A special emphasis is on the study of asymptotic limits from viscous models such as compressible Navier-Stokes equations, or from kinetic models. These problems involve the study of layers subject to large perturbations. The investigator also studies the existence and properties of solutions to related models, such as compressible Navier-Stokes equations with degenerated viscosities, or nonlinear nonhomogeneous kinetic equations. For this purpose, the investigator uses and develops deep analysis tools, such as the De Giorgi method and the relative entropy method.

该项目的主要部分涉及流体力学中称为冲击的戏剧性行为的研究。在海洋学中，一个典型的例子是海啸的传播。了解冲击流对于工程和工业尤其重要。研究人员使用最先进的数学工具研究此类结构的稳定性。该项目侧重于以下具体目标：（1）增进对工程、气象学、医学或自然灾害（例如洪水和海啸）具有根本重要性的物理系统的了解；（2）开发基于非线性偏微分方程理论的强大数学工具，可用于研究这些系统。研究生参与该项目的工作，这为培训初级研究人员进行数学分析和物理建模提供了良好的机会。研究人员研究可压缩流体力学中不连续解的稳定性，即冲击。它们与流体的有趣行为有关，特别是在超音速流动中。该项目增进了对这些现象的理解。特别强调的是对粘性模型（例如可压缩纳维-斯托克斯方程）或动力学模型的渐近极限的研究。这些问题涉及对受到大扰动的层的研究。研究人员还研究相关模型解的存在性和性质，例如具有退化粘度的可压缩纳维-斯托克斯方程或非线性非齐次动力学方程。为此，研究者使用并开发了深度分析工具，例如 De Giorgi 方法和相对熵方法。