Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces

Abstract

We consider the Schrödinger map initial-value problem $$\cases{ \partial_t\phi=\phi\times\Delta \phi &\text{on } \mathbb{R}^d\times\mathbb{R},\cr
\phi(0)=\phi_0, &{} }$$ where $\phi\colon\mathbb{R}^d\times\mathbb{R}\to\mathbb{S}^2\hookrightarrow \mathbb{R}^3$ is a smooth function. In all dimensions $d\geq 2$, we prove that the Schrödinger map initial-value problem admits a unique global smooth solution $\phi\in C(\mathbb{R}:H^\infty_Q)$, $Q\in\mathbb{S}^2$, provided that the data $\phi_0\in H^\infty_Q$ is smooth and satisfies the smallness condition $\|\phi_0-Q\|_{\dot{H}^{d/2}}\ll 1$. We prove also that the solution operator extends continuously to the space of data in $\dot H^{d/2}\cap \dot H^{d/2-1}_Q$ with small $\dot{H}^{d/2}$ norm.