Title: Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation,
Abstract: In this paper, we study Landau damping in the weakly collisional limit of a
Vlasov-Fokker-Planck equation with nonlinear collisions in the phase-space
$(x,v) \in \mathbb T_x^n \times \mathbb R^n_v$. The goal is four-fold: (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 collision-driven 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.