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

In this work, we prove the nonlinear stability of galaxy models derived from the three dimensional gravitational Vlasov Poisson system, which is a canonical model in astrophysics to describe the dynamics of galactic clusters. The stability of the so-called spherical models (which are radially symmetric steady states solutions) is a major question in astrophysics and, for decades, this problem has been the subject of a considerable amount of works in both mathematical and physics communities. A well known conjecture [6] is the stability of spherical models which are nonincreasing functions of their microscopic energy. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov [2] in 1961. In a previous work [30], we proved the stability of anisotropic spherical models under radially symmetric perturbations using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics literature [36, 12, 50, 1]. In a more recent work [31], we show how this approach combined with a new generalized Antonov type coercivity property implies the orbital stability of spherical isotropic models under general perturbations. In this paper, we summarize the results obtained in this work and give the main lines of the proofs. Joint work with Florian Méhats and Pierre Raphaël