Analysis of Non-Linear Partial Differential Equations in Free Boundary Fluid Dynamics and Kinetic Theory

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This project aims to advance scientific understanding and develop new methods in the study of non-linear partial differential equations that predict the future behavior of fluid flow with moving boundaries, and particle dynamics in Kinetic Theory. The first part of the project considers a basic mathematical model in petroleum engineering, which was formulated by the petroleum engineer M. Muskat in 1934 to describes the mixture of water into an oil sand. The second part of this project considers the study of a highly accurate mathematical model for a dilute hot plasma. These types of plasmas appear regularly in fundamental physical problems from astrophysics, nuclear fusion, and tokamaks. In the third part of this project the PI will study fundamental partial differential equations from relativistic particle dynamics and it is expected that this research will increase our physical understanding in a variety of places in astrophysics, for instance in high atmosphere aerodynamics where the air is a very rarefied gas and fluid equations are no longer a suitable mathematical model. This project will involve training in research and teaching of postdoctoral researchers, graduate students and undergraduate students from the University of Pennsylvania and other universities. The project will also involve outreach to undergraduate students through the UPenn Center for Undergraduate Research & Fellowships program. The PI is fully committed to facilitate the training and education of these students through teaching courses, regular direct mentoring, and running regular research seminars. The PI is actively working to develop new innovative mathematics courses at the University of Pennsylvania 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 PI is engaging in outreach activities to groups that are traditionally under-represented in mathematics, and these activities will continue over the course of this project. The objective of this research is to fully understand both global existence and singularity formation for several different fundamental physical models in non-linear partial differential equations. The first part of this work studies fluid dynamics problems with free boundaries, such as the Surface Quasi-Geostrophic equations and the Muskat problem. Another part of this work looks at existence and uniqueness problems for the relativistic Landau equation from plasma physics. And a third part of this project considers mathematical problems in the the relativistic Boltzmann equation from Kinetic theory with physically relevant non-integrable particle interactions. The PI proposes to develop new methods to advance the current level of scientific knowledge on a diverse collection of recognized questions in these different areas in the mathematical analysis of non-linear partial differential equations. It is expected that the techniques developed will be useful for future mathematical and physical developments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.