Boltzmann Equation and Knudsen Group - Boundary Shape and Boundary Conditions

Abstract

A Boltzmann type equation is considered where both, the physical and the velocity space, are bounded. It is assumed that the boundary conditions consist of a reflective as well as a diffusive component. Existence of positive bounds from below and above has been proved. It has been demonstrated that under conditions on the shape of the boundary, the underlying Knudsen type transport semigroup can be embedded in a not necessarily positive group for time t ∈ ℝ. As a consequence, the Boltzmann type equation considered has a unique global solution for time t ∈ [τ 0, ∞) for some τ 0 < 0. The time τ 0 does not depend on the initial value at time 0. It can be arbitrarily small depending on the intensity of the collisions.