Paper detail

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

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 $τ$ and the Debye length $δ$ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions $τ\to\infty$ to the strongly collisional regime $τ\to0$. 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 $τ$-dependent constraint on $δ$ ensuring exponential decay with explicit $τ$-dependent rates towards the stationary solution. In the strongly collisional limit $τ\to0$, 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.