A Logarithmically Improved Regularity Criterion for the Supercritical Quasi-geostrophic Equations in Besov Space

In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov spaceB,(R~2). The result shows that if θ is a weak solutions satisfiesα∫~T ‖▽θ(·, s) ‖α/α-r B/1+㏑(e+‖▽θ(·, s) ‖~2,ds < ∞ for some 0 < r < α and 0 < α < 1, then θ is regular at t = T. In view of the embedding L~2M~p  B,∞with 2 ≤ p <2 rand 0 ≤ r < 1, we r see that our result extends the results due to [20] and [31].