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 $$L^2$$ L 2 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 $$L^2$$ 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.
Journal of Statistical Physics – Springer Journals
Published: Feb 1, 2018
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