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Recent Advances in Kinetic Equations and Applications pp 1–50Cite as

Sharpening of Decay Rates in Fourier Based Hypocoercivity Methods

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Part of the Springer INdAM Series book series (SINDAMS,volume 48)

Abstract

This paper is dealing with two L2 hypocoercivity methods based on Fourier decomposition and mode-by-mode estimates, with applications to rates of convergence or decay in kinetic equations on the torus and on the whole Euclidean space. The main idea is to perturb the standard L2 norm by a twist obtained either by a nonlocal perturbation build upon diffusive macroscopic dynamics, or by a change of the scalar product based on Lyapunov matrix inequalities. We explore various estimates for equations involving a Fokker–Planck and a linear relaxation operator. We review existing results in simple cases and focus on the accuracy of the estimates of the rates. The two methods are compared in the case of the Goldstein–Taylor model in one-dimension.

Keywords

  • Hypocoercivity
  • Linear kinetic equations
  • Entropy–entropy production inequalities
  • Goldstein–Taylor model
  • Fokker–Planck operator
  • Linear relaxation operator
  • Linear BGK operator
  • Transport operator
  • Fourier modes decomposition
  • Pseudo-differential operators
  • Nash’s inequality

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Acknowledgements

This work has been partially supported by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR) and the Amadeus project Hypocoercivity no. 39453PH. J.D. and C.S. thank E. Bouin, S. Mischler and C. Mouhot for stimulating discussions that took place during the preparation of [15]: some questions raised at this occasion are the origin for this contribution. A.A., C.S., and T.W. were partially supported by the FWF (Austrian Science Fund) funded SFB F65.

Ⓒ 2020 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

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Authors and Affiliations

  1. Technische Universität Wien, Institut für Analysis und Scientific Computing, Wien, Austria

    Anton Arnold & Tobias Wöhrer

  2. CEREMADE (CNRS UMR n∘7534), PSL University, Université Paris-Dauphine, Place de Lattre de Tassigny, Paris, France

    Jean Dolbeault

  3. Fakultät für Mathematik, Universität Wien, Wien, Austria

    Christian Schmeiser

Authors
  1. Anton Arnold
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  2. Jean Dolbeault
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  4. Tobias Wöhrer
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Correspondence to Jean Dolbeault .

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Editors and Affiliations

  1. De Vinci Pôle Universitaire Research Center, Courbevoie, France

    Prof. Francesco Salvarani

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Arnold, A., Dolbeault, J., Schmeiser, C., Wöhrer, T. (2021). Sharpening of Decay Rates in Fourier Based Hypocoercivity Methods. In: Salvarani, F. (eds) Recent Advances in Kinetic Equations and Applications. Springer INdAM Series, vol 48. Springer, Cham. https://doi.org/10.1007/978-3-030-82946-9_1

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