Some mathematical theories on the Cauchy problem and boundary layer problem for the Boltzmann equation

Abstract

This thesis focuses on the mathematical theories on the Cauchy problem and the boundary layer problem of the Boltzmann equation. The Cauchy problem is well-known. The boundary layer problem arises in the physical consideration of the condensation-evaporation problem, and is the first order approximation of the Boltzmann equation for small Knudsen number near a plane. The thesis is mainly divided into two parts. In the first part, the Cauchy problems of the Boltzmann equation with potential force in the whole space and in torus are investigated. In the whole space, we consider the Cauchy problem with potential force with some less restrictive assumptions compared to the previous works. We obtain the well-posedness theory and the optimal convergence rate of the solution to the Boltzmann equation even for the hard potential case by energy method, when the initial data is sufficiently close to a steady state. In torus, global existence and stability of solutions to the Cauchy problem of the Boltzmann equation with potential forces for hard potentials are considered. We prove the stationary state is asymptotically stable with exponential rate in time for any initially smooth, periodic, origin symmetric small perturbation which preserves the same total mass, momentum and mechanical energy as the natural steady state and any origin symmetric small potential force. In the second part, the boundary layer solutions to the Boltzmann equation with mixed boundary condition for the inverse power law are discussed. The existence of boundary layer solutions to the Boltzmann equation with mixed boundary condition, that is Dirichlet boundary condition weakly perturbed by diffuse reflection boundary condition at the wall, is considered. The boundary condition is imposed on the incoming particles, and the solution is supposed to approach to a global Maxwellian in the far field. Like the problem with Dirichlet boundary condition, the existence of a solution depends on the Mach number of the far field Maxwellian. Furthermore, an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of positive characteristic speeds is also given.