Anal. PDE 6 (3), 601-686, (2013) DOI: 10.2140/apde.2013.6.601
KEYWORDS: Schrödinger maps, global regularity, energy-critical, critical Besov spaces, subthreshold, 35Q55, 35B33

We consider the Schrödinger map initial value problem

$$\left\{\begin{array}{c}\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ {\partial}_{t}\phi =\phi \times \Delta \phi ,\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phi \left(x,0\right)={\phi}_{0}\left(x\right),\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \end{array}\right.$$

with ${\phi}_{0}:{\mathbb{R}}^{2}\to {\S}^{2}\hookrightarrow {\mathbb{R}}^{3}$ a smooth ${H}_{Q}^{\infty}$ map from the Euclidean space ${\mathbb{R}}^{2}$ to the sphere ${\S}^{2}$ with subthreshold ($<4\pi $) energy. Assuming an a priori ${L}^{4}$ boundedness condition on the solution $\phi $, we prove that the Schrödinger map system admits a unique global smooth solution $\phi \in C\left(\mathbb{R}\to {H}_{Q}^{\infty}\right)$ provided that the initial data ${\phi}_{0}$ is sufficiently energy-dispersed, i.e., sufficiently small in the critical Besov space ${\u1e02}_{2,\infty}^{1}$. Also shown are global-in-time bounds on certain Sobolev norms of $\phi $. Toward these ends we establish improved local smoothing and bilinear Strichartz estimates, adapting the Planchon–Vega approach to such estimates to the nonlinear setting of Schrödinger maps.