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

In this paper, we study the sub-critical dissipative quasi-geostrophic equations $({\mathbf{S}}_{\alpha })$. We prove that there exists a unique local-in-time solution for any large initial data ${\theta }_0$ in the space ${\bf{\mathcal X}}^{1-2\alpha}(\mathbb R^2)$definedby(\text{ref}ec)\mathrm{.}Moreoverweshowthat$({\bf S}\alpha)$ has a global solution in time if the norms of the initial data in ${\mathcal{X}}^{1-2\alpha }({\mathbb{R}}^2)$ are bounded by $1\mathrm{/}4$. Also, we prove a blow-up criterion of the non global solution of $({\bf S}\alpha)$.