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

Abstract

This paper deals with collisionless transport equations in bounded open domains $\Omega \subset {\mathbb{R}}^d$ $(d\ge 2)$ with ${\mathcal{C}}^1$ boundary $\partial \Omega$, orthogonally invariant velocity measure $\bm{m}(\d v)$ with support $V\subset {\mathbb{R}}^d$ 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 ${\left(U_{\mathsf{H}}(t)\right)}_{t\ge 0}$ on We give a general criterion of irreducibility of and we show that, under very natural assumptions, if an invariant density exists then ${\left(U_{\mathsf{H}}(t)\right)}_{t\ge 0}$ converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then ${\left(U_{\mathsf{H}}(t)\right)}_{t\ge 0}$ is \emph{sweeping} in the sense that, for any density $\phi$, the total mass of $U_{\mathsf{H}}(t)\phi$ 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 ${\left(U_{\mathsf{H}}(t)\right)}_{t\ge 0}.$

This preprint supersedes the previous version. In version1, a gap was contained in Lemma A.11. We corrected Lemma A.11 which results now in a new and different kind of result for Theorem 5.1 covering the diffuse case. The main existence result (Theorem 5.6) has been corrected under some additional condition on the accomodation coefficient