Nonlinear Landau Damping for the Vlasov–Poisson System in $\mathbb {R}^3$: The Poisson Equilibrium

Abstract

We prove asymptotic stability of the Poisson homogeneous equilibrium among solutions of the Vlasov–Poisson system in the Euclidean space $\mathbb {R}^3$. More precisely, we show that small, smooth, and localized perturbations of the Poisson equilibrium lead to global solutions of the Vlasov–Poisson system, which scatter to linear solutions at a polynomial rate as $t\rightarrow \infty $. The Euclidean problem we consider here differs significantly from the classical work on Landau damping in the periodic setting, in several ways. Most importantly, the linearized problem cannot satisfy a “Penrose condition”. As a result, our system contains resonances (small divisors) and the electric field is a superposition of an electrostatic component and a larger oscillatory component, both with polynomially decaying rates.