Some problems on conservation laws and Vlasov-Poisson-Boltzmann equation

Abstract

In this thesis, we studied a mathematical study of conservation laws and gas motion under the influence of self-induced forcing. The models considered are the 2 × 2 system of hyperbolic conservation laws with artificial viscosity and the Vlasov-Poisson- Boltzmann system in kinetic theory. First, the existence of strong travelling wave profiles for a class of 2 × 2 viscous conservation laws is considered when the corresponding inviscid systems are hyperbolic. Apart from some technical assumptions, the only main assumption is hyperbolicity in accordance with which existence theory can be applied to systems which are not strictly hyperbolic. Characteristic fields can be neither genuinely nonlinear nor linearly degenerate. The Vlasov-Poisson-Boltzmann system, meanwhile, is a classical physical model for the time evolution of charged particles. Second, the two-species Vlasov-Poisson- Boltzmann system with a non-constant background density in the whole space is investigated. There is a stationary solution when the background density goes to zero. The global-in-time classical solutions and the nonlinear stability of solutions to the Cauchy problem near the stationary state in some Sobolev space without any time derivatives are constructed. The convergence rate in time to the global Maxwellian and the uniform-in-time stability of solutions are also obtained using the energy method. The macroscopic conservation laws are essentially used to deal with the a priori estimates on both the microscopic and macroscopic parts of the solution in the proof. Additionally, some interactive energy functionals are introduced to overcome the difficulty that stems from no-time derivatives in the energy functional.