Steady states of the spherically symmetric Vlasov-Poisson system as fixed points of a mass-preserving algorithm

TL;DR

This paper provides a new proof for the existence of spherically symmetric steady states in the Vlasov-Poisson system using a mass-preserving algorithm, linking numerical approximation with mathematical analysis.

Contribution

It introduces a novel proof technique for steady states, connecting numerical schemes with rigorous existence results in the Vlasov-Poisson system.

Findings

Existence of spherically symmetric steady states established.

The proof strategy parallels numerical approximation methods.

Potential extension to axially symmetric solutions and relativistic cases.

Abstract

We give a new proof for the existence of spherically symmetric steady states to the Vlasov-Poisson system, following a strategy that has been used successfully to approximate axially symmetric solutions numerically, both to the Vlasov-Poisson system and to the Einstein-Vlasov system. There are several reasons why a mathematical analysis of this numerical scheme is important. A generalization of the present result to the case of flat axially symmetric solutions would prove that the steady states obtained numerically in \cite{AR3} do exist. Moreover, in the relativistic case the question whether a steady state can be obtained by this scheme seems to be related to its dynamical stability. This motivates the desire for a deeper understanding of this strategy.