Theory and Application of Kinetic Equations

点击翻译按钮获取中文摘要

My proposed research program are focused on analysis, application, and numerical computation of kinetic equations. ***In the analysis and computation part, I propose to conduct research in three main subjects:***1. Well-posedness of nonlinear Boltzmann equations: Together with R. Alonso, Y. Morimoto, and T. Yang, we will investigate the well-posedness of nonlinear Boltzmann equations with non-cutoff kernels. Our main goal is to relax the smoothness and fast-decay assumptions for solutions in the current literature and look for spaces that are as large as possible where uniqueness can hold. We will also investigate regularization of solutions for non-cutoff Boltzmann equations starting from rough data. ***2. Nonlinear boundary layer kinetic equation: Boundary layer kinetic equations play a crucial role in determining the proper boundary conditions for fluid equations (such as Navier-Stokes and Euler) derived as the hydrodynamic limits of kinetic equations. Most of the current theoretical work regarding these equations in gas dynamics are restricted to the linear case. Together with Q. Li and J. Lu, we aim at rigorously justifying the existence of the solutions for nonlinear boundary layer Boltzmann equations for various parameter regions. We will also design efficient numerical methods for solving nonlinear equations and couple them with fluid equations in the interior. ***3. Motion of a solid in a gas: This research is to study the interaction of a moving solid with surrounding gases. I will study the setting where there is an external force acting on the solid and the gases are modelled by various kinetic equations. Current theoretical works are mostly restricted to free streaming gases. My goal is to study collisional gases and rigorously characterize the long-time asymptotic behaviour of the solid and compare with existing numerical results. ***Simultaneously, I propose to work on kinetic models in math biology, focusing on two major applications:***5. Aggregation behaviour: Kinetic and fluid types of equations have been widely applied in modelling aggregation behaviour of agents like birds. Together with R. Fetecau and H. Huang, we propose to study mean field limits of stochastic aggregation models. Current results in this direction are either over the whole space or with relatively regular interaction potentials. Our goal is to extend to models with singular potentials posed on domains with boundaries.***4. Chemotaxis: Chemotaxis studies the motion of bacteria under the influence of exterior environment. Together with B. Perthame and M. Tang, we will study non-classical kinetic transport equations modelling the evolution of the density of bacteria with internal states and noise. Our main goal is to understand ***these equations in different regions of the biological parameters involved. We will derive limiting equations and measurable quantities in different regions and compare them with experimental data.