Boundary Problems in the Boltzmann theory

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The kinetic theory and its models such as the Boltzmann equation and Vlasov equations have played an important role in the understanding of problems in gas dynamics, plasma physics, and fluid equations. In many physical situations, the particles in various models interact with boundaries, and this interaction creates several interesting phenomena such as the formation of singularities. In general, boundary effects may not stay only near the boundary but also impact on the whole interior dynamics. Therefore the boundary effects are important and have rich application in many cases. However, boundary problems in kinetic models are mathematically challenging due to their singular nature. This project aims to develop new mathematical tools to handle these problems.This research project studies several important boundary problems arising in kinetic models and fluid equations. The first topic regards optimal regularity of Boltzmann solutions in bounded domains with several physical boundary conditions for both dynamical and stationary problems. The second topic concerns the boundary-field interaction in kinetic models such as Vlasov-Poisson-Boltzmann systems. The third topic is to understand the relation between steady Boltzmann solutions and the incompressible Navier-Stokes-Fourier systems in the presence of a boundary when the mean free path is sufficiently small. The last topic concerns fluid-material interaction such as surfactant dynamics on viscous surface waves. Results of the project are expected to improve modeling capabilities for a wide range of problems of interest to other scientists and engineers.