The 2D quasi-geostrophic equations in the Sobolev space

First of all, a brief review is given for some recent results on the well-posedness theory for the two dimensional Quasi-Geostrophic Equations (2D QGE) in the Sobolev Space H-s. The focus here is on the cases with critical and super-critical dissipation. Next, some new results on the continuity properties of the solutions are derived for the 2D QGE with critical or super-critical dissipation, which are important part of the well-posedness theory. Then, a new result for the criterion on parallel to del theta parallel to(BMO) for the global regularity of the solutions of 2D QCE in H-1 is proved for both inviscid and dissipative cases. With this new result, an improved global regularity result for 2D QGE in H-s (s >= 1) is obtained for the case with critical dissipation. Finally, we show that this criterion is not only sufficient for existence of global strong solutions, but also sufficient for uniqueness of the weak solution of 2D QG equations.