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 ∈ {2, 3}. By semigroup arguments, we prove that in the L 1 norm, the polynomial rate of convergence 1 (t+1) n− given in [TAG10] and [KLT13] can be extended to any C 2 domain, with standard assumptions on the inital 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 dependancy of the rate with respect to the initial distribution is detailed. We adress for the first time the case where the temperature at the boundary varies and the equilibrium distribution is no longer explicit. The demonstrations are adapted from a deterministic version of a subgeometric Harris' theorem recently established by Cañizo and Mischler [CnM19]. We also compare our model with a free-transport equation with absorbing boundary.Acknowledgements: I would like to acknowledge my supervisor, Stéphane Mischler, for suggesting me this problem and for all the fruitful discussions and advices he offered me. This work was supported by grants from Région Île de France.

Existence of a Non-Equilibrium Steady State for the Non-linear BGK equation on an interval

“…There have also been works in a spatially homogeneous setting in the presence of scatterers [CLM15] for the Boltzmann equation and [Eva16] for Kac's toy model for the Boltzmann equation. We also mention the preprint [Ber19] which shows exponential convergence towards non-equilibrium steady states for the free transport equation in a domain with Maxwell boundary conditions.…”