Logarithmically Improved Regularity Criteria for Supercritical Quasi-geostrophic Equations in Orlicz-morrey Spaces

This article provides a regularity criterion for the surface quasigeostrophic equation with supercritical dissipation. This criterion is in terms of the norm of the solution in a Orlicz-Morrey space. The result shows that, if a weak solutions θ satisfies Z T 0 ‖∇θ(·, s)‖ α α−r M L2 logP L 1 + ln(e+ ‖∇⊥θ(·, s)‖L2/r ) ds <∞, for some 0 < r < α and 0 < α < 1, then θ is regular at t = T . In view of the embedding L2/r ⊂ Mp ⊂ M L2 logP L with 2 < p < 2/r and P > 1, our result extends the results due to Xiang [29] and Jia-Dong [15].