On a multi-dimensional transport equation with nonlocal velocity

TL;DR

This paper investigates a multi-dimensional nonlocal scalar transport equation, demonstrating finite-time gradient blowup for certain parameters and global well-posedness for others, revealing complex behaviors depending on the nonlocal term.

Contribution

It introduces a detailed analysis of a multi-dimensional nonlocal transport equation, identifying conditions for blowup and global existence based on the parameter lpha.

Findings

Gradient blowup occurs for lpha in (0,2]

Global well-posedness for lpha=0

Behavior depends critically on the nonlocal term parameter lpha

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.