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 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 $\alpha =\frac{1}{2}$ correspond to the 2D Euler and SQG equations, respectively. We establish here local-in-time regularity for these models, for all $\alpha \in (0,\frac{1}{2})$ 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.