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

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 $\tau \to \infty$ to the strongly collisional regime $\tau \to 0$. 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 $\tau \to 0$, 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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Cited by in corpus (3)

• L2-Hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker-Planck system
• Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit
• On a Vlasov-Fokker-Planck equation for stored electron beams