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

Abstract

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 ‖ ∇ θ 0 ‖ L ∞ 1 − 2 s ‖ θ 0 ‖ L ∞ 2 s < c s for some small number c s > 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.