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

S Mischler; C Mouhot

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arXiv · 1412.7487

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

S Mischler·C Mouhot

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Abstract

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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S Mischler, C Mouhot. 2014-12-23. Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation. https://arxiv.org/abs/1412.7487

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