A new variational approach to the stability of gravitational systems

Abstract

Abstract. We consider the three dimensional gravitational Vlasov Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A well-known conjecture in astrophysics is that the steady state solutions which are nonincreasing functions of their microscopic energy are nonlinearly stable by the ow. This was proved at the linear level by Antonov in 1961. Since then, standard variational techniques based on concentration compactness methods as introduced by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of stationary solutions of ground state type.In this paper, we propose a new variational approach based on the minimization of the Hamiltonian under equimeasurable constraints which are conserved by the nonlinear transport ow, and recognize any anisotropic steady state solution which is a decreasing function of its microscopic energy as a local minimizer. The outcome is the proof of its nonlinear stability of under radially symmetric perturbations.

Scattering Structure and Landau Damping for Linearized Vlasov Equations with Inhomogeneous Boltzmannian States

“…This work contributes to the solution of two problems addressed in [7] which are, for non homogeneous profiles, the construction are quasi-modes and the long time behavior of the Vlasov equation. Other works on the stability of the Vlasov equation for non homogeneous states are in [50,6,39,43], with particular mention to [7,3] on the dynamics of the HMF equation around inhomogeneous backgrounds and to the recent exposure [8]. The stability of the linear and non linear Vlasov equation is studied by numerical methods in many works, let us mention [40,7] where trapped particles are discussed and [17,23,24] for various tests cases.…”

“…The method and tools are strongly related to the physical context: they are an alternative to Hamiltonian theories [45,46,3] with action-angle variables popular in plasma physics, and we use a good deal of Koopmanism [35,53,54] to construct the eigenstructure of the different operators; the HMF model [7,3,22] is one example. The extension to non linear equations is an open problem, the techniques seem so far very different [48,47,30,32,50,39,41,43,10,9,22].…”

Non linear stability of spherical gravitational systems described by the Vlasov-Poisson equation

“…This approach avoids the delicate step of linearization of the Hamiltonian and reduces the stability problem for the full distribution function f to a minimization problem for a generalized energy involving the Poisson field φ f only. The main outcome is the radial stability of nonincreasing anisotropic models, solving in this way the first stability conjecture (see [30] for the proof): Theorem 1.1 (Radial stability of nonincreasing anisotropic models, [30]). Let Q(x, v) = F (e, ) be a continuous, nonnegative compactly supported steady state solution to (1.1).…”

“…For the full non radial problem, (1.14) is lost. However, we claim that the strategy developed in [30] coupled with a new generalized Antonov coercivity property allows us to derive the classical conjecture (conjecture 2) of orbital stability of nonincreasing isotropic models. Theorem 1.2 (Orbital stability of nonincreasing isotropic models).…”

“…To completely solve conjecture 1, we proposed in [30] a different approach based on fine monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements as first observed in pioneering breakthrough works in the physics literature, see in particular Lynden-Bell [36], Gardner [12], Wiechen, Ziegler, Schindler [50], Aly [1]. This approach avoids the delicate step of linearization of the Hamiltonian and reduces the stability problem for the full distribution function f to a minimization problem for a generalized energy involving the Poisson field φ f only.…”