Analysis of non-linear partial differential equations in Kinetic theory and related fields

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This project proposes to develop new methods in the mathematical analysis of nonlinear partial differential equations. The objective is to gain a deeper understanding of global-in-time existence and uniqueness theorems for kinetic equations with initial data from three different categories. The first category is nearby vacuum initial data. The second category encompasses spatially homogenous equations with initial data that do not have a size restriction. Building on that, the third category of initial data will be those that are large but weakly inhomogeneous (i.e., the solution starts out sufficiently close to some large homogeneous solution in a suitable Banach space). Another goal of the project is to determine the large-time convergence to equilibrium of these solutions. This research will lead to a deeper understanding of partial differential equations, which in turn introduce novel techniques that are expected to be useful for future mathematical and physical developments.This project pursues a multifaceted approach to physical questions arising in nonlinear partial differential equations from kinetic theory and related fields. The research vision includes developing new tools to study analytical problems about fundamental mathematical models; these tools are expected to be generalizable to a wide array of problems in applied mathematics. The results of the project will be disseminated through publication in journal articles, will be posted on the principal investigator's web site, and will be presented at international conferences. The principal investigator plans to bring together experts in areas related to this proposal. The project will involve graduate and undergraduate students from the University of Pennsylvania and other universities. The principal investigator is committed to help foster the training and education of these students through a variety of means, including directed studies and seminars related to the project. He will actively seek out students from diverse backgrounds to work on the project.