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

Lods, B.; Mokhtar-Kharroubi, M.; Rudnicki, R.

doi:10.1016/j.anihpc.2020.02.004

Lods, B.¹ ; Mokhtar-Kharroubi, M.² ; Rudnicki, R.³

¹ Università degli Studi di Torino & Collegio Carlo Alberto, Department of Economics and Statistics, Corso Unione Sovietica, 218/bis, 10134 Torino, Italy
² Université de Bourgogne Franche-Comté, Laboratoire de Mathématiques, CNRS UMR 6623, 16, route de Gray, 25030 Besançon Cedex, France
³ Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 877-923.

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Résumé

This paper deals with collisionless transport equations in bounded open domains $Ω \subset R^{d}$ $(d ⩾ 2)$ with $C^{1}$ boundary ∂Ω, orthogonally invariant velocity measure $m (d v)$ with support $V \subset R^{d}$ 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 $L^{1} (Ω \times V, d x \otimes m (d v))$ . 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 $t \to + \infty$ . 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 ${(U_{H} (t))}_{t ⩾ 0}$ .

MR Zbl

DOI : 10.1016/j.anihpc.2020.02.004

Classification : 82C40, 35F15, 47D06
Mots-clés : Kinetic equation, Stochastic semigroup, Convergence to equilibrium

Affiliations des auteurs :

Lods, B. ¹ ; Mokhtar-Kharroubi, M. ² ; Rudnicki, R. ³

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     title = {Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {877--923},
     publisher = {Elsevier},
     volume = {37},
     number = {4},
     year = {2020},
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     mrnumber = {4104829},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.02.004/}
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Lods, B.; Mokhtar-Kharroubi, M.; Rudnicki, R. Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 877-923. doi : 10.1016/j.anihpc.2020.02.004. http://www.numdam.org/articles/10.1016/j.anihpc.2020.02.004/

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