Some mathematical theories on the Vlasov-Maxwell-Boltzmann system
關於 Vlasov-Maxwell-Boltzmann 系統的一些數學理論
Student thesis: Doctoral Thesis
Related Research Unit(s)
The Vlasov-Maxwell-Boltzmann system is a system used to describe the dynamics of charged particles in the presence of an electromagnetic field. There have been extensive studies on the global existence of solutions to the one and two-species Vlasov-Maxwell- Boltzmann system with uniform background density. In this project, we first deal with the energy estimates of the one-species Vlasov-Maxwell-Boltzmann system when the background density is space-dependent and assume that the norm of the perturbation of the background density from the stationary state in some Sobolev spaces is small enough. To derive the energy estimates, we need to consider microscopic, macroscopic and electromagnetic dissipations separately. Next, we combine these three types of dissipations to get energy estimates for full dissipation. Then, we will handle the time decay estimates for this system for both the linearized and non-linear systems. For the non-linear system, we will only handle the estimates with constant background density. There have been studies on the time decay estimates for the model with uniform background densities, the convergence rates obtained are of algebraic type, being 3/8 and 3/4 for the linearized one-species and the two-species model, respectively. In our project, we will study the time decay rate for the linearized one-species VMB model. Furthermore, the dissipation rate for the non-linear one-species VMB model with uniform ionic background will also be studied. The tools we use include Sobolev inequalities and spectral analysis.
- Mathematical models, Electrodynamics