On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces

We prove the local smoothing effect of the 2D critical and supercritical dissipative quasi-geostrophic equations in critical Besov spaces. As an application, a global well-posedness result is established by adapting a method in Kiselev, Nazarov, and Volberg (2007) [16], and an idea in Dong and Du (2008) [15], with suitable modifications. Moreover, we show that the unique solution obtained in Chen, Miao, and Zhang (2007) [11], is a classical solution. These generalize some previous results in Dong (2010) [13], , Dong and Du (2008) [15]. The main ingredients of the proofs are two commutator estimates and the preservation of suitable modulus of continuity of the solutions.