A semigroup approach to the convergence rate of a collisionless gas

Abstract

We study the rate of convergence to equilibrium for a collisionless (Knudsen)gas enclosed in a vessel in dimension $n \in \{2,3\}$. By semigroup arguments,we prove that in the $L^1$ norm, the polynomial rate of convergence$\frac{1}{(t+1)^{n-}}$ given by Tsuji et al [2010] and Kuo et al[2013,2014,2015] can be extended to any $C^2$ domain, with standard assumptionson the initial data. This is to our knowledge, the first quantitative result incollisionless kinetic theory in dimension equal to or larger than 2 relying ondeterministic arguments that does not require any symmetry of the domain, nor amonokinetic regime. The dependency of the rate with respect to the initialdistribution is detailed. Our study includes the case where the temperature atthe boundary varies. The demonstrations are adapted from a deterministicversion of a subgeometric Harris' theorem recently established by Ca\~nizo andMischler. We also compare our model with a free-transport equation withabsorbing boundary.