Topics in Fluid dynamics with free boundaries, and Kinetic theory

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This research proposes to develop new methods to advance the current level of scientific knowledge on a diverse collection of recognized questions in a few different areas of the mathematical analysis of non-linear partial differential equations. The first part of this project involves questions regarding the dynamics of fluids, and solutions to these questions are expected to increase our understanding of water waves, tsunamis, hurricanes, and other fluid phenomena. The second part of this project studies the dynamics of plasmas from a mathematical point of view, and we anticipate that these studies will increase our understanding of the physical phenomena such as the solar wind, galactic nebulae, and the Van Allen radiations belts. The third part of this project focuses on the study the relativistic Kinetic theory and it is expected that the research will increase our physical understanding in a wide variety of places in astrophysics, for instance in high atmosphere aerodynamics where the air is a very rarefied gas and fluid equations are probably not sufficient. This project will involve training in research of postdoctoral researchers, graduate students and undergraduate students from the University of Pennsylvania and beyond, with participation of under-represented groups. The PI is working to develop innovative Active Learning Calculus courses in order to further the goal of developing a diverse and globally competitive STEM workforce and to improve STEM education at the collegiate level. The objective of the proposed research is to fully understand both global in time existence of solutions and singularity formation of solutions when this occurs for several different fundamental physical models in non-linear partial differential equations. One part of this research is to study fluid dynamics problems with free boundaries such as the Muskat problem and the Surface Quasi-Geostrophic equations. Another part of this work looks at problems related to the relativistic Vlasov-Maxwell system which is a fundamental model of plasma physics. And a third part of this proposal will study problems on the the relativistic Boltzmann equation which is the central model in relativistic Kinetic theory. The PI proposes to develop several new methods in the Analysis of partial differential equations in the course of developing deeper understanding of these different equations. It is expected that the techniques developed to be useful for future mathematical and physical developments.