Stability of shocks and layers in Fluid Mechanics and related problems

该项目的主要部分涉及对被称为冲击的流体力学中戏剧性行为的研究。在海洋学中，一个典型的例子是海啸的传播。理解有冲击的流动对工程和工业特别重要。研究人员使用最先进的数学工具研究这种结构的稳定性。该项目侧重于这些具体目标：(1)增进对工程、气象学、医学或自然灾害(如洪水和海啸)至关重要的物理系统的知识；(2)基于非线性偏微分方程组理论，开发可用于研究这些系统的强大的数学工具。研究生参与了该项目的工作，这为初级研究人员提供了在数学分析和物理模型方面进行培训的良好机会。研究人员研究可压缩流体力学中不连续解的稳定性，即所谓的冲击。它们与流体的有趣行为有关，特别是在超音速流动中。这个项目推进了对这些现象的理解。一个特别的重点是研究来自粘性模型的渐近极限，例如可压缩的Navier-Stokes方程或来自动力学模型。这些问题涉及到对受大扰动的层的研究。研究人员还研究了相关模型解的存在性和性质，如具有简并粘性的可压缩Navier-Stokes方程或非线性非齐次动力学方程。为此，研究人员使用并开发了深度分析工具，如德乔尔基方法和相对熵方法。

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.