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The present contribution investigates the dynamics generated by thetwo-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles ina steady inhomogeneous background of opposite charges. We provide global intime estimates that are uniform with respect to initial data taken in a boundedset 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 themean free path covers the full range of possible values: from the regime ofevanescent collisions $\tau\to\infty$ to the strongly collisional regime$\tau\to0$. As a counterpart, the largeness of the Debye length, that enforcesa 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 withexplicit $\tau$-dependent rates towards the stationary solution. In thestrongly collisional limit $\tau\to0$, we also examine all possible asymptoticregimes selected by a choice of observation time scale. Here also, our emphasisis on strong convergence, uniformity with respect to time and to initial datain bounded sets of a $L^2$ space. Our proofs rely on a detailed study of thenonlinear elliptic equation defining stationary solutions and a carefultracking and optimization of parameter dependencies ofhypocoercive/hypoelliptic estimates.

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