Global existence of the $s$-Schrödinger maps in $n\ge 5$

Abstract

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