Article Dans Une Revue Annales de l'Institut Henri Poincaré (C), Analyse non linéaire (Nonlinear Analysis) 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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hal-01940537 , version 1 (30-11-2018)

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  • HAL Id : hal-01940537 , version 1

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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 (Nonlinear Analysis), 2020, 37 (4), pp.877-923. ⟨hal-01940537⟩
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