| In this thesis,we will study some mathematical theories of some fundamental equations arising from the kinetic theory of diluted gases.The Vlasov-Poisson-Boltzmann system is a physical model describing mutual interactions of diluted charged particles through collisions in the self-consistent electronic field.The results obtained in the paper include the global well-posedness of the Cauchy problems of the scaled Vlasov-PossionBoltzmann system and hydrodynamic limits of the scaled Vlasov-Possion-Boltzmann system to the impressible Navier-Stokes-Fourier-Poisson system.This thesis is divided into the following five chapters:The first chapter is an introduction,which focuses on the background and its theoretical and practical significance on the mathematical theories.And literature reviews,research results in related fields are reviewed.Then,notations which will be used later are given.At last,we gives the main problems to be studied in this thesis and results obtained.The second chapter establishes the local existence of the scaled Vlasov-PoissonBoltzmann equations through the Hahn-Banach extension theorem and proves the local well-posedness of the classical solution of the scaled Vlasov-Poisson-Boltzmann equations with small initial data.The third part chapter establishes closed energy estimates in the case of hard spheres by making full use of macro-microscopic decomposition energy method and the basic properties of Sobolev space,and then we give the global classical solutions of the scaled Vlasov-Poisson-Boltzmann equations near Maxwellians.This result,together with the second chapter,is devoted to build the classical solution theory in the perturbation framework.The fourth chapter mainly consider the incompressible Navier-Stokes-Fourier-Poisson fluid dynamics limit problems.The fifth chapter establishes the global smooth solution of the incompressible Navier-Stokes-Fourier-Poisson equation through a high-order priori estimates. |
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