On regularity criterion for the dissipative quasi-geostrophic equations

In this paper we consider the 2D dissipative quasi-geostrophic equations and study the regularity criterion of the solutions. By means of a commutator estimate based on frequency localization and Bony's paraproduct decomposition, we obtain a regularity criterion∇⊥θ∈Lr(0,T;B˙p,∞0)for2p+αr⩽α,2α⩽p⩽∞, which is an obvious extension of the result in [B. Dong, Z. Chen, A remark on regularity criterion for the dissipative quasi-geostrophic equations, J. Math. Anal. Appl. 329 (2007) 1212–1217] where their restrictive condition on p   is 4α⩽p⩽∞.