JournalsaihpcVol. 37, No. 4pp. 877–923

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

A subscription is required to access this article.

Abstract

This paper deals with collisionless transport equations in bounded open domains \mathrm{\Omega } \subset \mathbb{R}^{d}$$(d⩾2) with C1\mathscr{C}^{1} boundary ∂Ω, orthogonally invariant velocity measure m(dv)\boldsymbol{m}(\mathrm{d}v) with support VRdV \subset \mathbb{R}^{d} and stochastic partly diffuse boundary operators H\mathsf{H} relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic C0C_{0}-semigroups (UH(t))t0\left(U_{\mathsf{H}}(t)\right)_{t⩾0} on L1(Ω×V,dxm(dv))L^{1}(\mathrm{\Omega } \times V,\mathrm{d}x \otimes \boldsymbol{m}(\mathrm{d}v)). We give a general criterion of irreducibility of (UH(t))t0\left(U_{\mathsf{H}}(t)\right)_{t⩾0} and we show that, under very natural assumptions, if an invariant density exists then (UH(t))t0\left(U_{\mathsf{H}}(t)\right)_{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))t0\left(U_{\mathsf{H}}(t)\right)_{t⩾0} is sweeping in the sense that, for any density φ, the total mass of UH(t)φU_{\mathsf{H}}(t)\varphi concentrates near suitable sets of zero measure as t+t\rightarrow + \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 (UH(t))t0\left(U_{\mathsf{H}}(t)\right)_{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