On the Existence of Exponentially Decreasing Solutions of the Nonlinear Landau Damping Problem

TL;DR

This paper proves the existence of a broad class of periodic solutions to the nonlinear Vlasov-Poisson system in one dimension that decay exponentially over time, extending previous results by relaxing the flatness condition.

Contribution

It introduces a new stability condition for the linearized problem, enlarging the class of solutions with exponential decay in the nonlinear Landau damping problem.

Findings

Existence of periodic solutions with exponential decay.

Extension of decay results to broader classes of solutions.

Replacement of flatness condition with a stability criterion.

Abstract

In this paper we prove the existence of a large class of periodic solutions of the Vlasov-Poisson in one space dimension that decay exponentially as t goes to infinity. The exponential decay is well known for the linearized version of the Landau damping problem and it has been proved in [4] for a class of solutions of the Vlasov-Poisson system that behaves asymptotically as free streaming solutions and are sufficiently flat in the space of velocities. The results in this paper enlarge the class of possible asymptotic limits, replacing the fatness condition in [4] by a stability condition for the linearized problem.