Stability of a point charge for the repulsive Vlasov–Poisson system

Abstract

We consider solutions of the repulsive Vlasov–Poisson system which are a combination of a point charge and a small gas, i.e., measures of the form ${\delta }_{(\mathcal{X}(t),\mathcal{V}(t))}+{\mu }^{2}d\mathbf{x}d\mathbf{v}$ for some $(\mathcal{X},\mathcal{V}):\mathbb{R}\to {\mathbb{R}}^6$ and a small gas distribution $\mu :\mathbb{R}\to L_{\mathbf{x},\mathbf{v}}^2$, and study asymptotic dynamics in the associated initial value problem. If initially suitable moments on ${\mu }_0=\mu (t=0)$ are small, we obtain a global solution of the above form, and the electric field generated by the gas distribution $\mu$ decays at an almost optimal rate. Assuming in addition boundedness of suitable derivatives of ${\mu }_0$, the electric field decays at an optimal rate, and we derive modified scattering dynamics for the motion of the point charge and the gas distribution. Our proof makes crucial use of the Hamiltonian structure. The linearized system is transport by the Kepler ODE, which we integrate exactly through an asymptotic action-angle transformation. Thanks to a precise understanding of the associated kinematics, moment and derivative control is achieved via a bootstrap analysis that relies on the decay of the electric field associated to $\mu$. The asymptotic behavior can then be deduced from the properties of Poisson brackets in asymptotic action coordinates.