Stability of a point charge for the Vlasov-Poisson system: the radial case

TL;DR

This paper proves that small, radial perturbations of a point charge in the Vlasov-Poisson system lead to global solutions that disperse over time, using linearized analysis and action-angle coordinates.

Contribution

It provides a rigorous analysis of the stability and long-term behavior of the Vlasov-Poisson system near a point charge with radial symmetry, including exact linearized solutions.

Findings

Solutions are global and disperse to infinity.

Dispersal occurs via modified scattering along linearized trajectories.

The analysis employs exact integration and action-angle coordinates.

Abstract

We consider the Vlasov-Poisson system with initial data a small, radial, absolutely continuous perturbation of a point charge. We show that the solution is global and disperses to infinity via a modified scattering along trajectories of the linearized flow. This is done by an exact integration of the linearized equation, followed by the analysis of the perturbed Hamiltonian equation in action-angle coordinates.