Local Regularity for the Modified SQG Patch Equation

We study the patch dynamics on the whole plane and on the half-plane for a family of active scalars called modified surface quasi-geostrophic (SQG) equations. These involve a parameter α that appears in the power of the kernel in their Biot-Savart laws and describes the degree of regularity of the equation. The values α=0 and α=½ correspond to the two-dimensional Euler and SQG equations, respectively. We establish here local-in-time regularity for these models, for all α ∊ (0,½) on the whole plane and for all small α > 0 on the half-plane. We use the latter result in [16], where we show existence of regular initial data on the half-plane that lead to a finite-time singularity.© 2016 Wiley Periodicals, Inc.

Duke Scholars

Author Alexander A. Kiselev Mathematics