Stability of shocks and layers in Fluid Mechanics and related problems

University Of Texas At Austin, Austin TX

Investigators

Abstract

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.