L2 solution of linear non-cutoff Boltzmann equation with boundary conditions

Abstract

In this paper, we are concerned about the existence of solution to linear non-cutoff Boltzmann equation with boundary conditions. In [9] and [18], the existence of the solutions to the kinetic equation and linear Boltzmann equation were obtained by characteristic line and iteration methods. However, their methods are not applicable in the context of non-cutoff linear operators. To overcome this difficulty, we cut off the non-cutoff linear operator L to LR,δ, the latter is compatible with the method in [9]. Via this process, we are able to use the sequence of solutions associated to LR,δ to approximate the solution associated to L, which is corresponding to the original equation, in turn, obtain the weak solutions of linear non-cutoff Boltzmann equation with specular and diffusion reflection boundary conditions.