Some Mathematical Theories On Certain Kinetic Equations

This thesis is concerned with some mathematical theories on certain kinetic equations arising from the kinetic theory of dilute gases. It contains three parts:The first part is an introduction to some complex but elementary equations in the kinetic theory of dilute gases and some recent progress on the mathematical theories related to these equations are reviewed;In the second part, we devote ourselves to the construction of globally smooth so-lutions near a given global Maxwellian to the Cauchy problem of the one-species Vlasov-Poisson-Boltzmann system. A satisfactory wellposedness theory in the per-turbative framework is established through introducing on a new nonlinear time and velocity weighted energy method. In our results, we do not ask the initial perturba-tion to satisfy the "neutral condition" and our analysis is based on some optimal or almost optimal temporal decay estimates on the solution itself and on its derivatives with respect to the space variable and the velocity variable;Finally, we justify the global-in-time diffusive limit of the Boltzmann equation inside a periodic domain T3. We only assume that in the initial expansion the kinetic parts are well-prepared, while the fluid parts could be general, i.e. the fluids parts are not required to satisfy the incompressibility and Boussinesq relations. For this case, the fluid initial layers are created and preserved in the periodic domain. Employing the method of time-averaging, we analyze the propagation of the initial layers in the spaces Ker(A) and Ker(A)⊥.