Knudsen Type Group for Time in $\mathbb{R}$ and Related Boltzmann Type Equations

Abstract

We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both, reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time $t\in\mathbb{R}$. Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time $t\ge 0$ or for time $t\in [\tau_0,\infty)$ for some $\tau_0<0$ which is independent of the initial value at time 0. Depending on the collision kernel, $\tau_0$ can be arbitrarily small.