Conditional global regularity of Schrödinger maps: Subthreshold dispersed energy

with $φ_{0} : ℝ^{2} \to §^{2} ↪ ℝ^{3}$ a smooth $H_{Q}^{\infty}$ map from the Euclidean space $ℝ^{2}$ to the sphere $§^{2}$ with subthreshold ( $< 4 π$ ) energy. Assuming an a priori $L^{4}$ boundedness condition on the solution $φ$ , we prove that the Schrödinger map system admits a unique global smooth solution $φ \in C (ℝ \to H_{Q}^{\infty})$ provided that the initial data $φ_{0}$ 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 $φ$ . 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

Mathematical Subject Classification 2010

Primary: 35Q55

Secondary: 35B33

Milestones

Received: 19 April 2011

Revised: 8 June 2012

Accepted: 6 July 2012

Published: 11 July 2013

Authors

Paul Smith
Department of Mathematics University of California, Berkeley 970 Evans Hall Berkeley, CA 94720-3840 United States

Vol. 6, No. 3, 2013

Paul Smith

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors