Mohammed Lemou

The orbital nonlinear stability of steady states solutions to the gravitational Vlasov-Poisson system, which are decreasing functions of the energy, has been proved in 2012. However, this result is partially based on compactness arguments and does not provide a complete quantitative information on the perturbation. In this talk, I will start by presenting a quantitative version of this nonlinear stability result. In particular a refined functional inequality on extended rearrangements of functions is proved, which is then combined with a Poincaré-like inequality. Another advantage of this approach is its applicability to other systems like the so-called Hamiltonian Mean Field (HMF), where the space domain is bounded and where the decreasing property of the steady states is no more sufficient to guarantee their stability. In fact, an additional explicit criteria is needed for HMF, under which the non-linear stability is proved. In a last part of this talk, we show that this criteria is sharp by proving a nonlinear instability result for HMF when the criteria is not satisfied. To this aim, we use an iterative procedure that was introduced by Grenier in a different context.