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A new nonlinear stability result for self-gravitating systems

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 any steady state solutions which is a nonincreasing functions of its microscopic energy is nonlinearly stable by the flow. We recently proved this conjecture for anisotropic, spherically symmetric steady states. The aim of this talk is to present the main lines of this mathematical proof. The starting point of the strategy is a new variational approach based on the minimization of the Hamiltonian under equimeasurable constraints which are conserved by the nonlinear transport flow. Then we 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 under radially symmetric perturbations.

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