# Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators

### B. Lods

Università degli Studi di Torino & Collegio Carlo Alberto, Department of Economics and Statistics, Corso Unione Sovietica, 218/bis, 10134 Torino, Italy### M. Mokhtar-Kharroubi

Université de Bourgogne Franche-Comté, Laboratoire de Mathématiques, CNRS UMR 6623, 16, route de Gray, 25030 Besançon Cedex, France### R. Rudnicki

Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland

## Abstract

This paper deals with collisionless transport equations in bounded open domains $Ω⊂R_{d}$ $(d⩾2)$ with $C_{1}$ boundary $∂Ω$, orthogonally invariant velocity measure $m(dv)$ with support $V⊂R_{d}$ and stochastic partly diffuse boundary operators $H$ relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic $C_{0}$-semigroups $(U_{H}(t))_{t⩾0}$ on $L_{1}(Ω×V,dx⊗m(dv))$. We give a general criterion of irreducibility of $(U_{H}(t))_{t⩾0}$ and we show that, under very natural assumptions, if an invariant density exists then $(U_{H}(t))_{t⩾0}$ converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then $(U_{H}(t))_{t⩾0}$ is *sweeping* in the sense that, for any density $φ$, the total mass of $U_{H}(t)φ$ concentrates near suitable sets of zero measure as $t→+∞$. We show also a general weak compactness theorem of interest for the existence of invariant densities. This theorem is based on several results on smoothness and transversality of the dynamical flow associated to $(U_{H}(t))_{t⩾0}$.

## Cite this article

B. Lods, M. Mokhtar-Kharroubi, R. Rudnicki, Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 4, pp. 877–923

DOI 10.1016/J.ANIHPC.2020.02.004