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Article Dans Une Revue Archive for Rational Mechanics and Analysis Année : 2016

Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation

Résumé

The aim of the present paper is twofold: (1) We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators writing as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another—smaller or larger—Banach space under the condition that a certain iterate of the “mild perturba- tion” part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series. (2) We apply this method to the kinetic Fokker-Planck equation when the spatial domain is either the torus with periodic boundary conditions, or the whole space with a confinement potential. We then obtain spectral gap es- timates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W_1.
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Dates et versions

hal-01098081 , version 1 (22-12-2014)
hal-01098081 , version 2 (25-01-2015)
hal-01098081 , version 3 (27-10-2015)

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Paternité - Pas d'utilisation commerciale

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Stéphane Mischler, C Mouhot. Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation. Archive for Rational Mechanics and Analysis, 2016, 221 (2), pp.677-723. ⟨hal-01098081v3⟩
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