Vol. 6, No. 3, 2013

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Conditional global regularity of Schrödinger maps: Subthreshold dispersed energy

Paul Smith

Vol. 6 (2013), No. 3, 601–686

We consider the Schrödinger map initial value problem

tφ = φ × Δφ, φ(x,0) = φ0(x),

with φ0 : 2 §23 a smooth HQ map from the Euclidean space 2 to the sphere §2 with subthreshold ( < 4π) energy. Assuming an a priori L4 boundedness condition on the solution φ, we prove that the Schrödinger map system admits a unique global smooth solution φ C( HQ) provided that the initial data φ0 is sufficiently energy-dispersed, i.e., sufficiently small in the critical Besov space 2,1. 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.

Schrödinger maps, global regularity, energy-critical, critical Besov spaces, subthreshold
Mathematical Subject Classification 2010
Primary: 35Q55
Secondary: 35B33
Received: 19 April 2011
Revised: 8 June 2012
Accepted: 6 July 2012
Published: 11 July 2013
Paul Smith
Department of Mathematics
University of California, Berkeley
970 Evans Hall
Berkeley, CA 94720-3840
United States