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

Abstract

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