Global existence and uniqueness of Schrödinger maps in dimensions \(d\geq 4\) (Q2381963)

For \(\sigma\geq 0\) and \(n\in \{1,2,\dots \}\) let \(H^\sigma=H^\sigma({\mathbb R}^d, {\mathbb C}^n)\) denote the Banach space of \({\mathbb C}^n\)-valued Sobolev functions on \({\mathbb R}^d\). For \(\sigma\geq 0\) and \(Q=(Q_1,Q_2,Q_3)\in {\mathbb S}^2\) define complete metric space \(H_Q^\sigma = H_Q^\sigma({\mathbb R}^d;{\mathbb S}^2\hookrightarrow {\mathbb R}^3) = \{f:{\mathbb R}^d\to{\mathbb R}^3; | f(x)| \equiv 1, f-Q\in H^\sigma \}\) with induced distance \(d_Q^\sigma(f,g) =\| f-g\| _{H^\sigma}\), and \(H^\infty_Q=\bigcap_{\sigma\in{\mathbb Z}_+} H_Q^\sigma\). Let \(s:{\mathbb R}^d\times{\mathbb R}\to {\mathbb S}^2\hookrightarrow {\mathbb R}^3\) is a continuous function. The authors consider the Schrödinger map initial-value problem \[ \begin{cases} \partial s = s\times \Delta_x s,\quad \text{ on}\quad {\mathbb R}^d \times {\mathbb R} \cr s(0)=s_0.\end{cases} \] It is proved that in dimensions \(d\geq 4\) this problem admits a unique global (in time) solution \(s\in C({\mathbb R}:H_Q^\infty)\), provided that \(s_0\in H_Q^\infty\) and \(\| s_0-Q\| _{H^{d/2}}\ll 1\), where \(Q\in {\mathbb S}^2\).