# Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Statistical Physics Springer Journals

# Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

, Volume 170 (5) – Feb 1, 2018
37 pages

/lp/springer_journal/large-time-behavior-of-solutions-to-vlasov-poisson-fokker-planck-O6QdV2xoFc
Publisher
Springer US
Subject
Physics; Statistical Physics and Dynamical Systems; Theoretical, Mathematical and Computational Physics; Physical Chemistry; Quantum Physics
ISSN
0022-4715
eISSN
1572-9613
D.O.I.
10.1007/s10955-018-1963-7
Publisher site
See Article on Publisher Site

### 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 $$L^2$$ 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

Journal of Statistical PhysicsSpringer Journals

Published: Feb 1, 2018

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