LAUSR

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 $$L^2$$L2 space, and where dependencies on the mean-free path $$\tau $$τ and the Debye length $$\delta $$δ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions $$\tau \rightarrow \infty $$τ→∞ to the strongly collisional regime $$\tau \rightarrow 0$$τ→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 $$\tau $$τ-dependent constraint on $$\delta $$δ ensuring exponential decay with explicit $$\tau $$τ-dependent rates towards the stationary solution. In the strongly collisional limit $$\tau \rightarrow 0$$τ→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 $$L^2$$L2 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.