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

Bertrand Lods (1) , Mustapha Mokhtar-Kharroubi (2)

1 Department of Statistics and Economics
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

A. 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 → ∞. This contrasts drastically to the case allowing arbitrarily slow velocities for which the absence of a spectral gap yields to at most algebraic rate of convergence to equilibrium. Some open questions are also mentioned.

Mots clés: Kinetic equation; Boundary operators; Non-zero velocities; Convergence to equilibrium

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Bertrand Lods, Mustapha Mokhtar-Kharroubi. On eventual compactness of collisionless kinetic semigroups with velocities bounded away from zero. 2021. ⟨hal-03229699v1⟩