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

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.