Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation

In this article we apply the method used in the recent elegant proof by Kiselev, Nazarov and Volberg of the well-posedness of critically dissipative 2D quasi-geostrophic equation to the super-critical case. We prove that if the initial value satisfies ‖∇θ0‖1−2s L∞ ‖θ0‖ L∞ < cs for some small number cs > 0, where s is the power of the fractional Laplacian, then no finite time singularity will occur for the super-critically dissipative 2D quasi-geostrophic equation. © 2007 Elsevier Inc. All rights reserved.