Global existence and nonlinear stability for the relativistic Vlasov-Poisson system in the gravitational case

TL;DR

This paper proves the global existence and nonlinear stability of certain steady states in the relativistic Vlasov-Poisson system under spherical symmetry, overcoming limitations of previous energy and variational methods.

Contribution

It introduces a direct, non-variational approach to establish nonlinear stability for the relativistic Vlasov-Poisson system, which was previously inaccessible.

Findings

Stable steady states are nonlinearly stable under spherically symmetric perturbations.

Smooth solutions near these steady states exist globally in time.

Previous energy and variational techniques do not apply to the relativistic case.

Abstract

As is well known from the work of R. Glassey} and J. Schaeffer, the main energy estimates which are used in global existence results for the gravitational Vlasov-Poisson system do not apply to the relativistic version of this system, and smooth solutions to the initial value problem with spherically symmetric initial data of negative energy blow up in finite time. For similar reasons the variational techniques by which Y. Guo and G. Rein obtained nonlinear stability results for the Vlasov-Poisson system do not apply in the relativistic situation. In the present paper a direct, non-variational approach is used to prove nonlinear stability of certain steady states of the relativistic Vlasov-Poisson system against spherically symmetric, dynamically accessible perturbations. The resulting stability estimates imply that smooth solutions with spherically symmetric initial data which are sufficiently close to the stable steady states exist globally in time.