Stability of shocks and layers in Fluid Mechanics and related problems

该项目的主要部分涉及流体力学中被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。理解带有冲击的流动对工程和工业特别重要。研究者使用最先进的数学工具研究这种结构的稳定性。该项目侧重于以下具体目标：(1)推进在工程、气象、医学或自然灾害（如洪水和海啸）中具有重要基础意义的物理系统的知识；(2)基于非线性偏微分方程理论，开发强大的数学工具，可用于研究这些系统。研究生参与该项目的工作，为培养初级研究人员在数学分析和物理建模方面提供了良好的机会。研究者研究可压缩流体力学中不连续解的稳定性，即冲击波。它们与流体的有趣行为有关，特别是在超音速流动中。这个项目促进了对这些现象的理解。特别强调的是对粘性模型（如可压缩的Navier-Stokes方程）或动力学模型的渐近极限的研究。这些问题涉及受大扰动影响的层的研究。研究者还研究了相关模型的解的存在性和性质，例如具有退化粘度的可压缩Navier-Stokes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，如De Giorgi方法和相对熵法。

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.