Isotropic Steady States in Galactic Dynamics

Abstract

The present paper completes our earlier results on nonlinear stability of stationary solutions of the Vlasov-Poisson system in the stellar dynamics case. By minimizing the energy under a mass-Casimir constraint we construct a large class of isotropic, spherically symmetric steady states and prove their nonlinear stability against general, i. e., not necessarily symmetric perturbations. The class is optimal in a certain sense, in particular, it includes all polytropes of finite mass with decreasing dependence on the particle energy. 1

Reduction and a Concentration-Compactness Principle for Energy-Casimir Functionals

“…Although the main purpose of the present paper is to get a more general understanding of the techniques developed in [1,2,3,4,5,10,11] we want to at least indicate some possible applications of these techniques. First we should mention that [5] differs from the other papers in so far as there the Casimir functional is used as part of the constraint under which then the total energy is minimized.…”

“…First we should mention that [5] differs from the other papers in so far as there the Casimir functional is used as part of the constraint under which then the total energy is minimized. This made it possible to relax the growth conditions on Q-0 < k ≤ 7/2 is covered in [5]-,but since in the reduction process we turn C(f ) + E kin (f ) into a new functional of ρ, [5] seems to be outside the present framework.…”

“…Given a large ensemble of particles which interact by gravitational attraction we consider energy-Casimir functionals which are defined on the space of phase space density functions, and certain reduced versions of these which are defined on the space of spatial density functions. This reduction procedure should put the techniques developed in [1,2,3,4,5,10,11] into a more general framework and make them applicable to problems outside kinetic theory. However, to be specific we start by recalling the Vlasov-Poisson system which describes the time evolution of a large ensemble of particles interacting by the gravitational field which they create collectively:…”

“…Obviously, the potential energy is not a functional of f itself, but of the induced spatial density ρ, and the crucial question is how along a minimizing sequence the spatial density can or cannot split into parts or spread uniformly in space. This was analyzed in the context of the Vlasov-Poisson system and with various variations in [1,2,3,4,5,10,11].…”

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…”