Existence and non-uniqueness of stationary states for the Vlasov-Poisson equation on $\mathbb{R}^3$ subject to attractive background charges

TL;DR

This paper proves the existence of stationary plasma solutions with background charges and demonstrates their non-uniqueness when the background charge is attractive, due to trapped particles orbiting the charge.

Contribution

It establishes the existence of stationary solutions for the Vlasov-Poisson equation with arbitrary background charges and shows non-uniqueness in the attractive case, highlighting trapped particle effects.

Findings

Existence of stationary solutions for plasma with background charges.

Non-uniqueness of solutions when the background charge is attractive.

Presence of trapped particles explains non-uniqueness.

Abstract

We prove the existence of stationary solutions for the density of an infinitely extended plasma interacting with an arbitrary configuration of background charges. Furthermore, we show that the solution cannot be unique if the total charge of the background is attractive. In this case, infinitely many different stationary solutions exist. The non-uniqueness can be explained by the presence of trapped particles orbiting the attractive background charge.