Linearized wave-damping structure of Vlasov-Poisson in $\mathbb R^3$

TL;DR

This paper analyzes the linearized Vlasov-Poisson system in three-dimensional unbounded space, revealing a mixed wave-damping behavior with weakly-damped Klein-Gordon waves and Landau damping, contrasting with confined or screened cases.

Contribution

It demonstrates the coexistence of Klein-Gordon-type waves and Landau damping in the unconfined Vlasov-Poisson system, with dispersive estimates and novel long-time behavior analysis.

Findings

Electric field decomposes into weakly-damped Klein-Gordon and Landau-damped parts.

Klein-Gordon waves approximate linearized Euler-Poisson equations.

Landau damping occurs at rates comparable to confined or screened cases.

Abstract

In this paper we study the linearized Vlasov-Poisson equation for localized disturbances of an infinite, homogeneous Maxwellian background distribution in $\mathbb R^3_x \times \mathbb R^3_v$. In contrast with the confined case $\mathbb T^d _x \times \mathbb R_v ^d$, or the unconfined case $\mathbb R^d_x \times \mathbb R^d_v$ with screening, the dynamics of the disturbance are not scattering towards free transport as $t \to \pm \infty$: we show that the electric field decomposes into a very weakly-damped Klein-Gordon-type evolution for long waves and a Landau-damped evolution. The Klein-Gordon-type waves solve, to leading order, the compressible Euler-Poisson equations linearized about a constant density state, despite the fact that our model is collisionless, i.e. there is no trend to local or global thermalization of the distribution function in strong topologies. We prove dispersive estimates on the Klein-Gordon part of the dynamics. The Landau damping part of the electric field decays faster than free transport at low frequencies and damps as in the confined case at high frequencies; in fact, it decays at the same rate as in the screened case. As such, neither contribution to the electric field behaves as in the vacuum case.