A semigroup approach to the convergence rate of a collisionless gas

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 in [25], [17] and [18] can be extended to any $ C^2 $ domain, with standard assumptions on the initial data. This is to our knowledge, the first quantitative result in collisionless kinetic theory in dimension equal to or larger than 2 relying on deterministic arguments that does not require any symmetry of the domain, nor a monokinetic regime. The dependency of the rate with respect to the initial distribution is detailed. Our study includes the case where the temperature at the boundary varies. The demonstrations are adapted from a deterministic version of a subgeometric Harris' theorem recently established by Cañizo and Mischler [7]. We also compare our model with a free-transport equation with absorbing boundary.