Eulerian uniqueness of the α-SQG patch problem

We consider the patch problem of the α-SQG equation with α=0 being the 2D Euler and α= 1/2 the SQG equations respectively. In the Eulerian setting, we prove the uniqueness of patch solutions of regularity W^2, 1/1-2α + when 0<α< 1/2 and C^1, 4α+ when 0<α< 1/4. The proof is intrinsic to the modified Biot-Savart law and independent of the local existence of patch solutions.