Eulerian uniqueness of the $\alpha$-SQG patch problem

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