# A regularity criterion for the dissipative quasi-geostrophic equations

### Hongjie Dong

The Division of Applied Mathematics, Brown University, 182 George Street, Box F, Providence, RI 02912, USA### Nataša Pavlović

Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712, USA

## Abstract

We establish a regularity criterion for weak solutions of the dissipative quasi-geostrophic equations (with dissipation $(−\mathrm{\Delta })^{\gamma / 2}$, $0 < \gamma ⩽1$). More precisely, we show that if $\theta ∊L_{t}^{r_{0}}((0,T);B_{p,\infty }^{\alpha }(\mathbb{R}^{2}))$ with $\alpha = \frac{2}{p} + 1−\gamma + \frac{\gamma }{r_{0}}$ is a weak solution of the 2D quasi-geostrophic equation, then *θ* is a classical solution in (0,T \times \mathbb{R}^{2}\right.. This result extends the regularity result of Constantin and Wu [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. I. H. Poincaré – AN (2007), doi:10.1016/j.anihpc.2007.10.001] to scaling invariant spaces.

## Cite this article

Hongjie Dong, Nataša Pavlović, A regularity criterion for the dissipative quasi-geostrophic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, pp. 1607–1619

DOI 10.1016/J.ANIHPC.2008.08.001