On a multi-dimensional transport equation with nonlocal velocity

Abstract

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

13 pages, minor revision, to appear in Adv. Math