Title: 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
$\tau\to0$. 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.