On the sharp time decay rates for the 2D generalized quasi-geostrophic equation and the Boussinesq system

We compute the sharp time decay rates of the solutions of the IVP for quasi-geostrophic equation and the Boussinesq model, subject to fractional dissipation. Moreover, we explicitly identify the asymptotic profiles, the kernel of the $\alpha$ stable processes, which are analogues of the Oseen vortices.