Linear Landau Damping for a Two Species Vlasov-Poisson System for Electrons and Ions

This paper concerns the linear Landau damping for the two species Vlasov-Poisson system for ions and electrons near Penrose stable equilibria. The result is an extension of the result on the one species Vlasov-Poisson equation by Mouhout and Villani (Acta Math. 207(1):29–201, 2011, [12]), Villani (Landau Damping. Notes de cours, CEMRACS, 2010, [15]). Different from Mouhout and Villani (Acta Math. 207(1):29–201, 2011, [12]), Villani (Landau Damping. Notes de cours, CEMRACS, 2010, [15]), we do not describe the ions as a background species but as a species which is also described by a separate Vlasov equation. We show an exponential decay of the electric energy for the linearised system near Penrose stable equilibria.