On the stability of homogeneous equilibria in the Vlasov–Poisson system on R3

Abstract

AbstractThe goal of this article is twofold. First, we investigate the linearized Vlasov–Poisson system around a family of spatially homogeneous equilibria inR3(the unconfined setting). Our analysis follows classical strategies from physics (Binney and Tremaine 2008Galactic Dynamics(Princeton University Press); Landau 1946Acad. Sci. USSR. J. Phys.1025–34; Penrose 1960Phys. Fluids3258–65) and their subsequent mathematical extensions (Bedrossianet al2022SIAM J. Math. Anal.544379–406; Degond 1986Trans. Am. Math. Soc.294435–53; Glassey and Schaeffer 1994Transp. Theory Stat. Phys.23411–53; Grenieret al2021Math. Res. Lett.281679–702; Han-Kwanet al2021Commun. Math. Phys.3871405–40; Mouhot and Villani 2011Acta Math.20729–201). 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 (Ionescuet al2022 (arXiv:2205.04540)) on the full global nonlinear asymptotic stability of the Poisson equilibrium inR3.