In this paper, we study the sub-critical dissipative quasi-geostrophic equations (S α ). We prove that there exists a unique local-in-time solution for any large initial data θ 0 in the space X 1−2α (R 2 ) defined by (1). Moreover, we show that (S α ) has a global solution in time if the norms of the initial data in X 1−2α (R 2 ) are bounded by 1/4. Also, we prove a blow-up criterion of the local-in-time solution of (S α ).

In this paper we consider a periodic 2-dimensional quasi-geostrophic equations with subcritical dissipation. We show the global existence and uniqueness of the solution ∈ ([0, T],  1−2 a, (T 2 )) for small initial data in the Lei-Lin-Gevrey spaces  1−2 a, (T 2 ). Moreover, we establish an exponential type explosion in finite time of this solution.

In this paper, we study the asymptotic behavior of the two-dimensional quasi-geostrophic equations with subcritical dissipation. More precisely, we establish that θtX1−2α vanishes at infinity.