Global existence of the two-dimensional QGE with sub-critical dissipation

In this paper, we study the sub-critical dissipative quasi-geostrophic equations $({\bf S}_\alpha)$. We prove that there exists a unique local-in-time solution for any large initial data $\theta_0$ in the space ${\bf{\mathcal X}}^{1-2\alpha}(\mathbb R^2)$ defined by (\ref{ec}). Moreover we show that $({\bf S}_\alpha)$ has a global solution in time if the norms of the initial data in ${\bf{\mathcal X}}^{1-2\alpha}(\mathbb R^2)$ are bounded by $1/4$. Also, we prove a blow-up criterion of the non global solution of $({\bf S}_\alpha)$.