Stability of isotropic steady states for the relativistic Vlasov-Poisson system

TL;DR

This paper proves the orbital stability of all isotropic steady states in the relativistic Vlasov-Poisson system using a non-variational approach based on Hamiltonian monotonicity and symmetry rearrangements, without subcritical assumptions.

Contribution

It introduces a new method for establishing stability of isotropic models in the relativistic Vlasov-Poisson system without subcritical conditions, extending previous results.

Findings

Stability of all isotropic steady states is established.

A new non-variational method is developed for stability analysis.

Overcomes difficulties related to potential continuity and homogeneity breaking.

Abstract

?In this work, we study the orbital stability of stationary solutions to the relativistic Vlasov-Manev system. This system is a kinetic model describing the evolution of a stellar system subject to its own gravity with some relativistic corrections. For this system, the orbital stability was proved for isotropic models constructed as minimizers of the Hamiltonian under a subcritical condition. We obtain here this stability for all isotropic models by a non-variationnal approach. We use here a new method developed in [23] for the classical Vlasov-Poisson system. We derive the stability from the monotonicity of the Hamiltonian under suitable generalized symmetric rearrangements and from a Antonov type coer- civity property. We overcome here two new difficulties : the first one is the a priori non-continuity of the potentials, from which a greater control of the re- arrangements is necessary. The second difficulty is related to the homogeneity breaking which does not give the boundedness of the kinetic energy. Indeed, in this paper, we does not suppose any subcritical condition satisfied by the steady states.