Stable Ground States for the Relativistic Gravitational Vlasov–Poisson System

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.

Keywords:

• Gravitational systems
• Stability
• Variational approaches
• Vlasov–Poisson system

Mathematics Subject Classification:

• 35A15
• 37C75
• 35Q75
• 74A25

Acknowledgments

Mohammed Lemou was supported by the Agence Nationale de la Recherche, ANR Jeunes Chercheurs MNEC. Pierre Raphaël was supported by the Agence Nationale de la Recherche, ANR ONDENONLIN. Florian Méhats was supported by the ACI Nouvelles Interfaces des Mathématiques ACINIM 176-2004 funded by the French ministry of research as well as the Agence Nationale de la Recherche, ANR project QUATRAIN.