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

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

We present results on 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 $\mathbb{T}^3_x \times \mathbb{R}^3_v$, or the unconfined case $\mathbb{R}^3_x \times \mathbb{R}^3 _v$ with screening, the dynamics of the disturbance are not scattering towards free transport as $t \to \pm \infty$: instead 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 lacks collisions that normally are the basis of hydrodynamic behaviours. The Landau-damped part shows a specific structure reminiscent of the screened unconfined case. This is a joint work with Jacob Bedrossian (University of Maryland) and Nader Masmoudi (NYU).