Global well-posedness in the critical Besov space of the skew mean curvature flow in $\mathbb{R}^d: d\ge 5$

TL;DR

This paper proves global regularity for the skew mean curvature flow in high-dimensional Euclidean spaces with small initial data, using advanced analytical techniques to improve upon previous results.

Contribution

It introduces a new div-curl lemma and interaction Morawetz estimate to establish global well-posedness in the critical Besov space, significantly enhancing prior work.

Findings

Global well-posedness in critical Besov space for d≥5

Use of a novel div-curl lemma for bilinear estimates

Improved results over previous studies in high dimensions

Abstract

In this paper, we are devoted to studying the global regularity for the skew mean curvature flow with small initial data in $\mathbb{R}^d\, (d\ge 5)$. By using a new div-curl lemma which was first introduced by the third author to establish a bilinear estimate, and also the interaction Morawetz estimate, the global well-posedness for the skew mean curvature flow in the critical Besov space is established, and hence the corresponding result obtained by Huang, Li and Tataru (Int. Math. Res. Not. 2024, no. 5, 3748-3798) in $\mathbb{R}^d\, (d\ge 5)$ is substantially improved.