Quadratic lifespan for the sublinear α-SQG sharp front problem

In this paper we consider the generalized surface quasi-geostrophic α-SQG equations, in the "sublinear regime" α ∈ (0, 1) and we study the stability of vortex patches close to vortex discs. We shall prove that for regular, Sobolev initial vortex patches ε-close to a vortex disc, the solutions stay ε-close to a vortex disc for a time interval of order O(ε- 2). The proof is based on a paradifferential Birkhoff normal form reduction, implemented in the case where the dispersion relation is sublinear.