Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a VlasovFokkerPlanck equation
Abstract
In this paper, we study Landau damping in the weakly collisional limit of a VlasovFokkerPlanck equation with nonlinear collisions in the phasespace $(x,v) \in \mathbb T_x^n \times \mathbb R^n_v$. The goal is fourfold: (A) to understand how collisions suppress plasma echoes and enable Landau damping in agreement with linearized theory in Sobolev spaces, (B) to understand how phase mixing accelerates collisional relaxation, (C) to understand better how the plasma returns to global equilibrium during Landau damping, and (D) to rule out that collisiondriven nonlinear instabilities dominate. We give an estimate for the scaling law between Knudsen number and the maximal size of the perturbation necessary for linear theory to be accurate in Sobolev regularity. We conjecture this scaling to be sharp (up to logarithmic corrections) due to potential nonlinear echoes in the collisionless model.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 DOI:
 10.48550/arXiv.1704.00425
 arXiv:
 arXiv:1704.00425
 Bibcode:
 2017arXiv170400425B
 Keywords:

 Mathematics  Analysis of PDEs;
 Physics  Plasma Physics
 EPrint:
 Corrected minor error in proof of multiplier properties (corrections essentially confined to appendix)