Quadratic lifespan for the sublinear $\alpha$-SQG sharp front problem

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