Remarks on the logarithmical regularity criterion of the supercritical surface quasi-geostrophic equation in Morrey spaces

Abstract

This paper is concerned with the regularity criterion of the weak solutions of the two-dimensional supercritical surface quasi-geostrophic equation in a critical Morrey space. It is proved that if the weak solution θ(x,t) of the supercritical quasi-geostrophic equation satisfies the growth condition∫0T‖∇θ(t)‖Ṁp,qr1+ln(e+‖∇θ‖Lp)dt<∞with2p+αr=α,2α<p<∞, then the solution θ(x,t) is regular on R2×(0,T]. This improves the earlier results by Xiang (2010).