Remark on the uniqueness of the mild solution of SQG equation

We study the two-dimensional surface quasi-geostrophic equation. Motivated by the uniqueness for the three-dimensional incompressible Navier-Stokes equations, we demonstrate that the uniqueness of the mild solution of the two-dimensional surface quasi-geostrophic equation holds in the scaling critical Lebesgue space with a unique structure of the non-linear term.