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Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

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Abstract

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted \(L^2\) space, and where dependencies on the mean-free path \(\tau \) and the Debye length \(\delta \) are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions \(\tau \rightarrow \infty \) to the strongly collisional regime \(\tau \rightarrow 0\). As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the \(\tau \)-dependent constraint on \(\delta \) ensuring exponential decay with explicit \(\tau \)-dependent rates towards the stationary solution. In the strongly collisional limit \(\tau \rightarrow 0\), we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a \(L^2\) space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.

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Notes

  1. In contrast we use the term for any limit corresponding to sending the diffusive parameter to infinity.

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Acknowledgements

Research of L. Miguel Rodrigues was partially supported by the ANR Project BoND ANR-13-BS01-0009-01.

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Authors and Affiliations

  1. CNRS UMR 5208, Institut Camille Jordan, Université de Lyon 1, 43 bd 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Maxime Herda

  2. IRMAR, UMR CNRS 6625, Université de Rennes 1, 263 avenue du General Leclerc, 35042, Rennes Cedex, France

    L. Miguel Rodrigues

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  1. Maxime Herda
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Herda, M., Rodrigues, L.M. Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit. J Stat Phys 170, 895–931 (2018). https://doi.org/10.1007/s10955-018-1963-7

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