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Pré-Publication, Document De Travail Année : 2017

Large-time behavior of solutions to Vlasov-Poisson-Fokker-Planck equations: from evanescent collisions to diffusive limit

Maxime Herda

Résumé

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. Our goal is to 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 $\tau\to\infty$ to the strongly collisional regime $\tau\to0$. 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 $\tau\to0$, 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 thorough preliminary 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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Dates et versions

hal-01541854 , version 1 (19-06-2017)
hal-01541854 , version 2 (07-11-2017)

Identifiants

  • HAL Id : hal-01541854 , version 1

Citer

Maxime Herda, Luis Miguel Miguel Rodrigues. Large-time behavior of solutions to Vlasov-Poisson-Fokker-Planck equations: from evanescent collisions to diffusive limit. 2017. ⟨hal-01541854v1⟩
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