Global existence of the $s$-Schr\"odinger maps in $n \geq 5$

Abstract

We study the $s$-Schr\"odinger map equation \[ \begin{cases} \partial_t u = -u \wedge (-\Delta)^s u \quad \text{on } \mathbb{R}^n \times \mathbb{R}\\ u: \mathbb{R}^n \times \mathbb{R} \to \mathbb{S}^2\\ u(x,0) = u_0. \end{cases} \] For $s \in (1/2,1)$ and $n \geq 5$ we prove that for small initial data $u_0$ in the critical Besov space one can obtain a global solution.