Linearized Wave-Damping Structure of Vlasov--Poisson in $\mathbb{R}^3$

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