A logarithmically improved regularity criterion for the surface quasi-geostrophic equation

In this paper, we consider the two-dimensional dissipative surface quasi-geostrophic equation, and establish a logarithmically improved regularity criterion. Consequently, our result extends the regularity criterion result of Dong and Pavlovié (2009). In addition, a logarithmically improved regularity criterion to the three-dimensional Navier-Stokes equations is also derived by the same arguments. Therefore, this result extends and improves many previous works.