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

In this paper, we study the sub-critical dissipative quasi-geostrophic equations (Sα). We prove that there exists a unique local-in-time solution for any large initial data θ0 in the space X1−2α(R2) defined by (1). Moreover, we show that (Sα) has a global solution in time if the norms of the initial data in X1−2α(R2) are bounded by 1/4. Also, we prove a blow-up criterion of the local-in-time solution of (Sα).