On the uniqueness of the quasi-geostrophic equation with the fractional Laplacian

We consider the uniqueness of the solution of the surface quasi-geostrophic equation with fractional Laplacian. We show that the uniqueness holds in non-homogeneous Besov spaces without any additional assumption which is supposed to constract solutions. When the power of the fractional Laplacian is close to 2, we prove that the uniqueness with the regularity index $s=-1\mathrm{/}2$. We extract the least regularity $s=-1\mathrm{/}2$ for the well-definedness of the nonlinear term of the equation.