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 ∂ − G := {(x, v) ∈ ∂D × R n , −(v • n x) < 0}. Given a function φ on (0, ∞) ×D × R n , γ ± φ denotes its trace on (0, ∞) × ∂ ± G, provided this object is well-defined. The boundary operator K is defined, for all (t, x, v) ∈ R + ×∂ − G and for φ supported on (0, ∞)×∂ + G such that φ(t, x, •) belongs to L 1 ({v : v • n x < 0}), by Kφ(t, x, v) = α(x)M (x, v) {v ∈R n :v •nx<0} φ(t, x, v)|v • n x |dv + (1 − α(x))φ(t, x, v − 2(v • n x)n x).
We study the asymptotic behavior of the kinetic free-transport equation enclosed in a regular domain, on which no symmetry assumption is made, with Cercignani-Lampis boundary condition. We give the first proof of existence of a steady state in the case where the temperature at the wall varies, and derive the optimal rate of convergence towards it, in the L 1 norm. The strategy is an application of a deterministic version of Harris' subgeometric theorem, in the spirit of [10] and [4]. We also investigate rigorously the velocity flow of a model mixing pure diffuse and Cercignani-Lampis boundary conditions with variable temperature, for which we derive an explicit form for the steady state, providing new insights on the role of the Cercignani-Lampis boundary condition in this problem.
Acknowledgements:The author acknowledges financial support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement number 864066).
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