Boundary Problems of Kinetic Theory

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The kinetic theory on rarefied particles have played an important role in the understanding of problems in gas dynamics, plasma physics, and fluid equations. In many physical situations (for example, plasma in tokamak fusion reactor), the particles interact with boundaries and their interaction affects global dynamics significantly. Singular behavior of particle distribution near the boundary is major difficulty in both mathematical understanding and the experiment. This project aims to develop new mathematical tools to handle these problems and answer several fundamental questions. This research project studies several important boundary problems arising in kinetic models and fluid equations. In particular, the PI plans to study regularity property of the Vlasov-Maxwell-Boltzmann system with boundary conditions. The equation is nonlinear, nonlocal, and degenerated dissipative coupled with hyperbolic equation (wave equation). Due to the characteristics boundary condition, the solutions are singular in general. Successful investigations for the proposal would mark a significant development in the PDE theory with singularity. Results of the project are expected to improve modeling capabilities for a wide range of problems of interest to other scientists and engineers.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.