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Article Dans Une Revue Annales de l'Institut Henri Poincaré C, Analyse non linéaire Année : 2020

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

Résumé

This paper deals with collisionless transport equations in bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega $, orthogonally invariant velocity measure $\bm{m}(\d v)$ with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $% L^{1}(\Omega \times V,\d x \otimes \bm{m}(\d v)).$ We give a general criterion of irreducibility of $% \left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ and we show that, under very natural assumptions, if an invariant density exists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ converges strongly (not simply in Cesar\`o means) to its ergodic projection. We show also that if no invariant density exists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ is \emph{sweeping} in the sense that, for any density $\varphi $, the total mass of $% U_{\mathsf{H}}(t)\varphi $ concentrates near suitable sets of zero measure as $% t\rightarrow +\infty .$ We show also a general weak compactness theorem which provides a basis for a general theory on existence of invariant densities. This theorem is based on a series of results on smoothness and transversality of the dynamical flow associated to $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}.$
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Dates et versions

hal-01940537 , version 1 (30-11-2018)

Identifiants

  • HAL Id : hal-01940537 , version 1

Citer

Bertrand Lods, Mustapha Mokhtar-Kharroubi, Ryszard Rudnicki. Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2020, 37 (4), pp.877-923. ⟨hal-01940537⟩
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