%0 Journal Article
%T Local regularity for the modified SQG patch equation
%A Alexander Kiselev
%A Yao Yao
%A Andrej Zlatos
%J Mathematics
%D 2015
%I arXiv
%X We study the patch dynamics on the whole plane and on the half-plane for a family of active scalars called modified SQG equations. These involve a parameter $\alpha$ which appears in the power of the kernel in their Biot-Savart laws and describes the degree of regularity of the equation. The values $\alpha=0$ and $\alpha=\frac 12$ correspond to the 2D Euler and SQG equations, respectively. We establish here local-in-time regularity for these models, for all $\alpha\in(0,\frac 12)$ on the whole plane and for all small $\alpha>0$ on the half-plane. We use the latter result in [16], where we show existence of regular initial data on the half-plane which lead to a finite time singularity.
%U http://arxiv.org/abs/1508.07611v1