Electron. J. Diff. Equ., Vol. 2016 (2016), No. 137, pp. 1-7.

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

Sadek Gala, Maria Alessandra Ragusa

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
$$ \int_0^T\frac{\| \nabla \theta (\cdot,s)\| _{\mathcal{M}_{L^2\log^P L} ^{2/r}} ^{\frac{\alpha }{\alpha -r}}} {1+\ln (e+\| \nabla ^{\bot }\theta (\cdot,s)\| _{L^{2/r}})}ds<\infty , $$
for some $0<r<\alpha $ and $0<\alpha <1$, then $\theta $ is regular at t=T. In view of the embedding $L^{2/r}\subset {\mathcal{M}_p}^{2/r} \subset \mathcal{M}_{L^2\log^PL}^{2/r}$ with $2<p<2/r$ and P>1, our result extends the results due to Xiang [29] and Jia-Dong [15].

Submitted September 28, 2015. Published June 8, 2016.
Math Subject Classifications: 35Q35, 76D03.
Key Words: Quasi-geostrophic equations; logarithmical regularity criterion; Orlicz-Morrey space.

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Sadek Gala
Department of Mathematics
University of Mostaganem, Box 227
Mostaganem 27000, Algeria
email: sadek.gala@gmail.com
Maria Alessandra Ragusa
Dipartimento di Matematica e Informatica
Università di Catania
Viale Andrea Doria, 6 95125 Catania, Italy
email: maragusa@dmi.unict.it

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