Stable ground states for the relativistic gravitational Vlasov-Poisson system

TL;DR

This paper demonstrates the existence of stable ground states in the relativistic gravitational Vlasov-Poisson system by leveraging concentration compactness and rigidity properties, despite the system's critical nature and potential for blow-up.

Contribution

It introduces a novel approach to establish stability of ground states in the relativistic case without relying on minimizer uniqueness, expanding the understanding of stability in such systems.

Findings

Existence of stable relativistic ground states due to broken scaling symmetry

Orbital stability proven without requiring minimizer uniqueness

Application of concentration compactness and rigidity techniques

Abstract

We consider the three dimensional gravitational Vlasov-Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the stability of a large class of ground states has been derived by various authors. The relativistic problem is critical and displays finite time blow up solutions. Using standard concentration compactness techniques, we however show that the breaking of the scaling symmetry allows the existence of stable relativistic ground states. A new feature in our analysis which applies both to the classical and relativistic problem is that the orbital stability of the ground states does not rely as usual on an argument of uniqueness of suitable minimizers --which is mostly unknown-- but on strong rigidity properties of the transport flow, and this extends the class of minimizers for which orbital stability is now proved.