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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 $α$ which 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 $α=\frac 12$ correspond to the 2D Euler and SQG equations, respectively. We establish here local-in-time regularity for these models, for all $α\in(0,\frac 12)$ 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 which lead to a finite time singularity.