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 ∂ − 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).

Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups

“…Subgeometric convergence rates have also been studied for classical models as the Fokker-Planck equation, and the Boltzmann equation and its relatives; for this we refer to Kavian et al (2021); Carrapatoso and Mischler (2017) and the references therein, and the classical papers by Caflisch (1980b,a). We also mention the recent works by Bernou and Fournier (2019); Bernou (2020a), where convergence to equilibrium for a collisionless model of a gas is investigated, the last one using techniques related to the present paper.…”

Quantitative tauberian approach to collisionless transport equations with diffuse boundary operators

“…We mention also the important contributions [16,17] which obtain optimal rate of convergence when the spatial domain is a ball. The two very recent works [5,6] provide (optimal) convergence rate for general domains Ω. All these works are dealing with partial diffuse boundary operator of Maxwell-type for which…”

“…where, as above M θ(x) is a Maxwellian distribution given by (1.11) for which the temperature θ(x) depends (continuously) on x ∈ ∂Ω and γ(x) is a normalization factor ensuring H to be stochastic. Optimal rate of convergence for the boundary condition (1.12) in dimension d = 2, 3 has been obtained recently in [5] thanks to a clever use of Harris's subgeometrical convergence theorem for Markov processes. A related probabilistic approach, based on coupling, has been addressed in [6] in dimension d 2 whenever θ(x) = θ is constant.…”

“…Optimal rate of convergence for the boundary condition (1.12) in dimension d = 2, 3 has been obtained recently in [5] thanks to a clever use of Harris's subgeometrical convergence theorem for Markov processes. A related probabilistic approach, based on coupling, has been addressed in [6] in dimension d 2 whenever θ(x) = θ is constant. As mentioned, in [5], the obtained rate of convergence is optimal and given by…”

“…A related probabilistic approach, based on coupling, has been addressed in [6] in dimension d 2 whenever θ(x) = θ is constant. As mentioned, in [5], the obtained rate of convergence is optimal and given by…”