Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation
Mischler, Stéphane; Mouhot, Clément (2016), Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation, Archive for Rational Mechanics and Analysis, 221, 2, p. 677-723. 10.1007/s00205-016-0972-4
TypeArticle accepté pour publication ou publié
Journal nameArchive for Rational Mechanics and Analysis
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)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 . 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.
Subjects / KeywordsFokker-Planck equation; Kolmogorov-Fokker-Planck equation; hypocoercivity; hypodissipativity; spectral mapping theorem; semigroup; enlargement; spectral gap
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