On wellposedness and limiting behavior of generalized SQG equations

We consider the two-dimensional generalized surface quasi-geostrophic equations in $\mathbb{R}^2$, given by \[\partial_t θ+u\cdot \nabla θ=0,\quad u=-\nabla^{\perp}Λ^{β-2}θ,\,β\in [1,2).\] When $β=1$, the equation defines the SQG equation and for $β>1$, it defines a family of more singular active