On eventual compactness of collisionless kinetic semigroups with velocities bounded away from zero

TL;DR

This paper studies the long-term behavior of collisionless kinetic equations with bounded velocities, showing that under certain conditions, the associated semigroup becomes compact and converges exponentially to equilibrium.

Contribution

It demonstrates that the semigroup is eventually compact and converges exponentially without irreducibility, contrasting with cases with slow velocities.

Findings

Semigroup is eventually compact under certain conditions.

Exponential convergence to equilibrium without irreducibility.

Contrast with algebraic convergence when velocities are unbounded.

Abstract

In this paper, we consider the long time behaviour of collisionless kinetic equation with stochastic diffuse boundary operators for velocities bounded away from zero. We show that under suitable reasonable conditions, the semigroup is eventually compact. In particular, without any irreducibility assumption, the semigroup converges exponentially to the spectral projection associated to the zero eigenvalue as $t \to \infty.$ This contrasts drastically to the case allowing arbitrarily slow velocities for which the absence of a spectral gap yields at most algebraic rate of convergence to equilibrium. Some open questions are also mentioned.