Local regularity for the modified SQG patch equation

We study the patch dynamics for a family of active scalars called modified SQG equations, on the whole plane and on the half-plane. 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 = 1 2 correspond to the 2D Euler and SQG equations, respectively. We establish here local-in-time existence and uniqueness results for these models, for all α ∈ (0, 12) on the whole plane and for all small α > 0 on the half-plane. The main novelty of this work is both in showing existence of local patch solutions on the half-plane and in proving their uniqueness on both domains.