Exterior Problems for the Vlasov-Maxwell-Boltzmann System

Description

The project aims to investigate the behavior of rarefied colliding plasma outsidean obstacle under the influence of the self-induced electro-magnetic field governed bythe Maxwell system. The problem has strong physical background and is challengingin mathematics. The feasibility of the project is based on our previous studies onthe exterior problem for the classical Boltzmann equation, and the recent work on thespectrum analysis for the Vlasov-Maxwell-Boltzmann system.Even though there are satisfactory mathematical theories for fluid dynamics in themacroscopic level, there are very few results on the exterior problem for the kineticequations in the mesoscopic level. For example in the macroscopic level, there areimportant works on the subsonic, supersonic and transonic flows by Bers, Courant-Friedrichs,Mowawetz, Finn-Gilbarg, Gilbarg-Shiffman, Dong, Chen, Chen-Feldman,Liu-Xin-Yin, Elling-Liu, etc. On the other hand, in the mesoscopic level, with the pioneercontributions made by Hilbert and the Girttingen school, Chapman and Enskog,Grad, Kogan and Maslova, in the context of DiPerna-Lions renormalized solutions,there are some studies by Hamdache and Arkeryd-Cercignani. For the pertubation ofan equilibrium, Ukai-Asano gave a rigorous analysis on the well-posedness of the exteriorproblem when the far field Mach number is sufficiently small. This approach waslater improved in our recent joint works to include more general boundary condition.Based on our recent study on the precise spectrum structure of the linearized Valsov-Maxwell-Boltzmann system around a global equilibrium for both the one-species andthe two-species cases that are very different from the classical Boltzmann equation, inthis project, we aim to further develop the analytic approach initiated by Ukai-Asano tostudy the exterior problems on the Vasov-Maxwell-Boltzmann equations. For this, wewill apply the analytic techniques such as those in the spirit of the principle of limitingabsorption used in the scattering theory, decomposition of the solution operators, andthe velocity averaging lemma introduced by Golse-Lions-Perthame-Sentis. New ideasare also needed to decompose the solution operator in order to show its boundednessand compactness in some suitable functions spaces.The study in this project can be applied to other complicated systems of kineticequations, therefore, it will enrich the mathematical theories in this aspect.