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 $v={\Lambda }^{-2+\alpha }\nabla u$ with $\Lambda =(-\Delta {)}^{1/2}$. We show that when $\alpha \in (0,2]$ 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.

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

References in corpus (1)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

Cited by in corpus (1)

• Remarks on a nonlocal transport