On the stability of homogeneous equilibria in the Vlasov-Poisson system on $\mathbb{R}^3$

Abstract

The goal of this article is twofold. First, we investigate the linearized Vlasov-Poisson system around a family of spatially homogeneous equilibria in $\mathbb{R}^3$ (the unconfined setting). Our analysis follows classical strategies from physics and their subsequent mathematical extensions. The main novelties are a unified treatment of a broad class of analytic equilibria and the study of a class of generalized Poisson equilibria. For the former, this provides a detailed description of the associated Green's functions, including in particular precise dissipation rates (which appear to be new), whereas for the latter we exhibit explicit formulas. Second, we review the main result and ideas in our recent work on the full global nonlinear asymptotic stability of the Poisson equilibrium in $\mathbb{R}^3$.