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Non linear stability of spherical gravitational systems described by the Vlasov-Poisson equation. (English) Zbl 1319.35266

Summary: 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 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 a previous work [Commun. Math. Phys. 302, No. 1, 161–224 (2011; Zbl 1339.35320)], we proved the stability of anisotropic spherical models under radially symmetric perturbations using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements. In a more recent work [Invent. Math. 187, No. 1, 145–194 (2012; Zbl 1232.35170)], 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.

MSC:

35Q83 Vlasov equations
85A05 Galactic and stellar dynamics
35Q85 PDEs in connection with astronomy and astrophysics
70M20 Orbital mechanics
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