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Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation

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Abstract

The aim of the present paper is twofold:

  1. 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 written 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 perturbation” 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. 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 estimates 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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Authors and Affiliations

  1. Université Paris-Dauphine and IUF CEREMADE, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    S. Mischler

  2. Centre for Mathematical Sciences, University of Cambridge, DPMMS, Wilberforce road, Cambridge, CB3 0WA, UK

    C. Mouhot

Authors
  1. S. Mischler
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  2. C. Mouhot
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Correspondence to C. Mouhot.

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Communicated by L. Saint-Raymond

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Mischler, S., Mouhot, C. Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation. Arch Rational Mech Anal 221, 677–723 (2016). https://doi.org/10.1007/s00205-016-0972-4

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  • DOI: https://doi.org/10.1007/s00205-016-0972-4

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