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\(\text {L}^2\)-Hypocoercivity and Large Time Asymptotics of the Linearized Vlasov–Poisson–Fokker–Planck System

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Abstract

This paper is devoted to the linearized Vlasov–Poisson–Fokker–Planck system in presence of an external potential of confinement. We investigate the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar product adapted to the presence of a Poisson coupling. Our framework provides estimates which are uniform in the diffusion limit. As an application in a simple case, we study the one-dimensional case and prove the exponential convergence of the nonlinear Vlasov–Poisson–Fokker–Planck system without any small mass assumption.

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Acknowledgements

The Authors would like to thank two referees for very helpful comments and suggestions. This work has been partially supported by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR) and by the research unit Dynamical systems and their applications (UR17ES21), Ministry of Higher Education and Scientific Research, Tunisia.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, 7021, Zarzouna, Banzart, Tunisia

    Lanoir Addala

  2. CEREMADE (CNRS UMR n 7534), PSL Research University, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775, Paris 16, France

    Jean Dolbeault & Xingyu Li

  3. Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, 2092, El-Manar, Tunisia

    M. Lazhar Tayeb

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  1. Lanoir Addala
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  2. Jean Dolbeault
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  4. M. Lazhar Tayeb
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Correspondence to Jean Dolbeault.

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Addala, L., Dolbeault, J., Li, X. et al. \(\text {L}^2\)-Hypocoercivity and Large Time Asymptotics of the Linearized Vlasov–Poisson–Fokker–Planck System. J Stat Phys 184, 4 (2021). https://doi.org/10.1007/s10955-021-02784-4

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