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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Communicated by Stefano Olla.
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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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DOI: https://doi.org/10.1007/s10955-021-02784-4
Keywords
- Confinement
- Vlasov–Poisson–Fokker–Planck system
- Convergence
- Large-time behavior
- Rate of convergence
- Hypocoercivity
- Diffusion limit