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

## TL;DR

This paper studies the long-term behavior of solutions to the 2D Vlasov-Poisson-Fokker-Planck equations, analyzing various regimes from evanescent collisions to diffusive limits with explicit convergence rates.

## Contribution

It provides uniform global-in-time estimates across all collision regimes, explicitly tracking dependencies on mean-free path and Debye length, and explores asymptotic regimes with strong convergence results.

## Key findings

- Uniform decay rates towards stationary solutions.
- Analysis of regimes from evanescent to strongly collisional.
- Explicit dependence of estimates on physical parameters.

## 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 $\tau\to0$. 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 $\tau\to0$, 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 detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.