On a multi-dimensional transport equation with nonlocal velocity

We study a multi-dimensional nonlocal active scalar equation of the form _ut+v⋅∇u=0$u_t+v\cdot \mathrm{\nabla }u=0$ in R⁺×R^d${\mathbb{R}}^{+}\times {\mathbb{R}}^d$, where v=Λ^−2+α∇u$v={\Lambda }^{-2+\alpha }\mathrm{\nabla }u$ with Λ=(−Δ)^1/2$\Lambda ={(-\mathrm{\Delta })}^{1/2}$. We show that when α∈(0,2]$\alpha \in (0,2]$ certain radial solutions develop gradient blowup in finite time. In the case when α=0$\alpha =0$, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces.