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arXiv · 1404.6665

On a multi-dimensional transport equation with nonlocal velocity

Hongjie Dong
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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.

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Hongjie Dong. 2014-07-25. On a multi-dimensional transport equation with nonlocal velocity. https://doi.org/10.1016/j.aim.2014.07.028

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