Propagation of Sobolev regularity for the critical dissipative quasi-geostrophic equation

We show that under a smallness assumption on the L^∞ norm of the initial data, there is a propagation of any Sobolev regularity greater than −1/2 for some weak solutions of the critical dissipative quasi-geostrophic equation. We also prove the existence of solutions in a space close to the homogeneous Sobolev space $\dot{H}$ ¹, theses solutions are given by a fixed point theorem.