Landau damping for the linearized Vlasov Poisson equation in a weakly collisional regime

TL;DR

This paper proves Landau damping for the linearized Vlasov-Poisson equation with weak collisions, showing uniform results as the collision parameter approaches zero, for both Boltzmann and Fokker-Planck operators.

Contribution

It establishes uniform Landau damping results in Sobolev spaces for the linearized Vlasov-Poisson equation with weak collisional effects, covering two types of collision operators.

Findings

Landau damping holds uniformly as collision parameter tends to zero.

Results apply to both linear Boltzmann and Fokker-Planck collision operators.

Damping is proven in Sobolev spaces for the weakly collisional regime.

Abstract

In this paper, we consider the linearized Vlasov-Poisson equation around an homogeneous Maxwellian equilibrium in a weakly collisional regime: there is a parameter $\eps$ in front of the collision operator which will tend to $0$. Moreover, we study two cases of collision operators, linear Boltzmann and Fokker-Planck. We prove a result of Landau damping for those equations in Sobolev spaces uniformly with respect to the collision parameter $\eps$ as it goes to $0$.