Nonlinear instability of inhomogeneous steady states solutions to the HMF Model

In this work we prove the nonlinear instability of inhomogeneous steady states solutions to the Hamiltonian Mean Field (HMF) model. We first study the linear instability of this model under a simple criterion by adapting the techniques developed in [19]. In a second part, we extend to the inhomogeneous case some techniques developed in [14, 17, 18] and prove a nonlinear instability result under the same criterion.