Geodesic
Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators
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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This paper deals with collisionless transport equations in bounded open domains Ω⊂Rd (d⩾2) with C1 boundary ∂Ω, orthogonally invariant velocity measure m(dv) with support V⊂Rd and stochastic partly diffuse boundary operators H relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic C0-semigroups (UH(t))t⩾0 on L1(Ω×V,dx⊗m(dv)). We give a general criterion of irreducibility of (UH(t))t⩾0 and we show that, under very natural assumptions, if an invariant density exists then (UH(t))t⩾0 converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then (UH(t))t⩾0 is sweeping in the sense that, for any density φ, the total mass of UH(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 (UH(t))t⩾0.

DOI : 10.1016/j.anihpc.2020.02.004
Classification : 82C40, 35F15, 47D06
Keywords: Kinetic equation, Stochastic semigroup, Convergence to equilibrium

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 
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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},
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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

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