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In this paper we study the linearized Vlasov-Poisson equation for localizeddisturbances 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 notscattering towards free transport as $t \to \pm \infty$: we show that theelectric field decomposes into a very weakly-damped Klein-Gordon-type evolutionfor long waves and a Landau-damped evolution. The Klein-Gordon-type wavessolve, to leading order, the compressible Euler-Poisson equations linearizedabout a constant density state, despite the fact that our model iscollisionless, i.e. there is no trend to local or global thermalization of thedistribution function in strong topologies. We prove dispersive estimates onthe Klein-Gordon part of the dynamics. The Landau damping part of the electricfield decays faster than free transport at low frequencies and damps as in theconfined case at high frequencies; in fact, it decays at the same rate as inthe screened case. As such, neither contribution to the electric field behavesas in the vacuum case.

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