#### Vol. 6, No. 3, 2013

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Conditional global regularity of Schrödinger maps: Subthreshold dispersed energy

### Paul Smith

Vol. 6 (2013), No. 3, 601–686
##### Abstract

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 ×\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 ${\phi }_{0}:{ℝ}^{2}\to {§}^{2}↪{ℝ}^{3}$ a smooth ${H}_{Q}^{\infty }$ map from the Euclidean space ${ℝ}^{2}$ to the sphere ${§}^{2}$ 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 $\phi \in C\left(ℝ\to {H}_{Q}^{\infty }\right)$ provided that the initial data ${\phi }_{0}$ is sufficiently energy-dispersed, i.e., sufficiently small in the critical Besov space ${Ḃ}_{2,\infty }^{1}$. 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.

##### Keywords
Schrödinger maps, global regularity, energy-critical, critical Besov spaces, subthreshold
Primary: 35Q55
Secondary: 35B33