6533b7dcfe1ef96bd127341e

RESEARCH PRODUCT

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

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This paper deals with collisionless transport equationsin bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega $, orthogonallyinvariant velocity measure $\bm{m}(\d v)$ with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing andincoming fluxes. Under very general conditions, such equations are governedby 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 densityexists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ converges strongly (notsimply in Cesar\`o means) to its ergodic projection. We show also that if noinvariant 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 theoremwhich provides a basis for a general theory on existence of invariantdensities. This theorem is based on a series of results on smoothness andtransversality of the dynamical flow associated to $\left( U_{\mathsf{H}}(t)\right) _{t\geq0}.$