On the stability of homogeneous equilibria in the Vlasov–Poisson system on

Abstract

The goal of this article is twofold. First, we investigate the linearized Vlasov–Poisson system around a family of spatially homogeneous equilibria in (the unconfined setting). Our analysis follows classical strategies from physics (Binney and Tremaine 2008 Galactic Dynamics (Princeton University Press); Landau 1946 Acad. Sci. USSR. J. Phys. 10 25–34; Penrose 1960 Phys. Fluids 3 258–65) and their subsequent mathematical extensions (Bedrossian et al 2022 SIAM J. Math. Anal. 54 4379–406; Degond 1986 Trans. Am. Math. Soc. 294 435–53; Glassey and Schaeffer 1994 Transp. Theory Stat. Phys. 23 411–53; Grenier et al 2021 Math. Res. Lett. 28 1679–702; Han-Kwan et al 2021 Commun. Math. Phys. 387 1405–40; Mouhot and Villani 2011 Acta Math. 207 29–201). 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 (Ionescu et al 2022 (arXiv:2205.04540)) on the full global nonlinear asymptotic stability of the Poisson equilibrium in .