On a multi-dimensional transport equation with nonlocal velocity

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