Asymptotic Behaviour for the Vlasov-Poisson System in the Stellar-Dynamics Case

Abstract

We study an optimal inequality which relates potential and kinetic energies in an appropriate framework for bounded solutions of the Vlasov-Poisson (VP) system. Optimal distribution functions, which are completely characterized, minimize the total energy. From this variational approach, we deduce bounds for the kinetic and potential energies in terms of conserved quantities (mass and total energy) of the solutions of the VP system and a nonlinear stability result. Then we apply our estimates to the study of the large time asymptotics and observe two different regimes.

The Orbital Stability of the Ground States and the Singularity Formation for the Gravitational Vlasov Poisson System

“…In dimension N = 3 where the problem is subcritical, we prove using concentration compactness techniques that every minimizing sequence to a large class of minimization problems attained on steady states solutions are up to a translation shift relatively compact in the energy space. This implies in particular the orbital stability in the energy space of the spherically symmetric polytropes what improves the nonlinear stability results obtained for this class in [16,19,11]. In dimension N = 4 where the problem is L 1 critical, we obtain the polytropic steady states as best constant minimizers of a suitable Sobolev type inequality relating the kinetic and the potential energy.…”

“…To show the compactness at the infinity, it suffices to estimate the decay of E fn . The following elementary computation is adapted from [11].…”

“…(ii) The nonlinear stability statement of the obtained steady states in [16,19,11,37] is measured in terms of a distance which under some very specific constraints on the perturbation is proved to control the L 2 norm. One of the difficulties the authors are confronted with is the weak L p convergence of the minimizing sequences of (1.14) which is a consequence of their strategy ie their choice of minimization problem.…”

Boundary Value Problems for the Stationary Nordström-Vlasov System

“…There are also some results for initial-boundary value problems [1,23] and boundary value problems [7,22,30,31]. A lot of studies concerns the stationary solutions and their stability [5,6,18,32,34].…”

A new variational approach to the stability of gravitational systems

“…However, the nonlinear stability of a large class of stationary solutions of so-called ground state type has been obtained in the framework of the concentration compactness techniques as introduced by Lions in [29,30], see [44], [11,13,14,15,9,38,20,21,22]. In particular, in [21], we showed that for a large class of convex functions j, the two parameters according to the scaling symmetry of (1.1) minimization problem…”