Logarithmically improved regularity criteria for supercritical quasi-geostrophic equations in Orlicz-Morrey spaces

Abstract

This article provides a regularity criterion for the surface quasi-geostrophic 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 $\theta $ satisfies $$ \int_0^T\frac{\| \nabla \theta (\cdot,s)\| _{\mathcal{M}_{L^2\log^P L} ^{2/r}} ^{\frac{\alpha }{\alpha -r}}} {1+\ln (e+\| \nabla ^{\bot }\theta (\cdot,s)\| _{L^{2/r}})}ds1, our result extends the results due to Xiang [29] and Jia-Dong [15].

Keywords

• Quasi-geostrophic equations
• logarithmical regularity criterion
• Orlicz-Morrey space