Some Mathematical Theories of Kinetic Related Models


Student thesis: Doctoral Thesis

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Award date24 Jul 2017


This thesis is concerned with some mathematical studies of the Vlasov-Maxwell-Boltzmann (VMB) system, which describes the time evolution of dilute charged particles under the influence of binary collisions and the self-induced Lorentz force and the compressible magnetohydrodynamics (MHD) system describing the mutual interaction of the fluid flow and the magnetic field.
In the first part, two mathematical problems for the VMB system are studied. One is the spectral analysis of the VMB system without the angular cutoff assumption. The spectrum structure reveals the influence of the angular singularity in the cross section. The large time behavior and the optimal convergence rates of solutions to the Cauchy problem for the non-cutoff VMB system are obtained by the combination of the spectral analysis with some refined weighted energy estimates. The other one is the nonlinear stability of the rarefaction wave for a macroscopic model derived from the one-dimensional VMB system. By studying the large time behavior of solutions to the corresponding Cauchy problem, the rarefaction wave for the system is proved time-asymptotically stable under the smallness assumption on the perturbation while additional smallness assumption on the initial boundary data and the wave strength are imposed because of the influence of the magnetic field.
In the second part, the well-posedness theory of the compressible MHD boundary layer is investigated. The local-in-time existence and uniqueness of solution to the MHD boundary layer equations, which are derived from the compressible MHD system in the small viscosity, heat conductivity and magnetic diffusivity limit with non-slip condition for the velocity field and perfectly conducting condition for the magnetic field on the boundary, are established provided that the initial tangential magnetic field is nonzero.