Stable ground states for the gravitational Vlasov-Manev system

Mohammed Lemou (1) , Florian Méhats (1) , Cyril Rigault (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In this work, we prove the orbital stability of ground state stationary solutions to the so-called Vlasov-Manev (VM) system. This system is a kinetic model which has a similar Vlasov structure as the classical Vlasov-Poisson system, but is coupled to a potential in $-1/r- 1/r^2$ (Manev potential) instead of the usual gravitational potential in $-1/r$, and in particular the potential field does not satisfy a Poisson equation but a fractional-Laplacian equation. The ground states are constructed as minimizers of the Hamiltonian, and the orbital stability is deduced both from the compactness of minimizing sequences and the rigidity of the flow. In driving this analysis, there are two mathematical obstacles: the first one is related to the possible blow-up of solutions to the VM system, which we overcome by imposing a sub-critical condition on the constraints of the variational problem. The second difficulty (and the most important) is related to the nature of the Euler-Lagrange equations (fractional-Laplacian equations) to which classical results for the Poisson equation do not extend. In this paper we prove the uniqueness of the minimizer under equimeasurabilty constraints, without using Euler-Lagrange equations.

Domaines: Equations aux dérivées partielles [math.AP]

Mohammed Lemou, Florian Méhats, Cyril Rigault. Stable ground states for the gravitational Vlasov-Manev system. 2010. ⟨hal-00487932v1⟩