Diffusive approximation of the Liouville equation

Abstract

In statistical physics phase space behavior of an ensemble of non interacting particles can be described by the Liouville equation. In the stationary case with inflow boundary conditions on a (finite) slab the method of characteristics provides solutions with jump type discontinuities. The goal of this work was to overcome the uniqueness issues using a vanishing viscosity method. Since existing results cannot handle problems with non symmetric, parameter dependent collision operators, these approaches are extended. In particular existence of an unique solution to the parabolic-elliptic degenerated diffusive version of the stationary Liouville equation is proven. Furthermore some basic properties such as smoothness and a bound by the posed boundary conditions were established. Hereby the intrinsic Krein space structure of this problem was pointed out.