In this talk, we will give new optimal decay rates for the Nesterov acceleration scheme of classical gradient descent depending on the local geometry of the function to minimize. Only bounds on the rates are known for convex or strongly convex functions. We will give a more complete description of this rates using Lojasievicz and flatness conditions and explain how these decays can be obtained studying an ODE.This is a joint work with V. Apidopoulos, J.-F. Aujol and Ch. Dossal.;;