TY - JOUR

T1 - Asymptotic stability of harmonic maps under the schrödinger flow

AU - Gustafson, Stephen

AU - Kang, Kyungkeun

AU - Tsai, Tai Peng

N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.

PY - 2008/12/1

Y1 - 2008/12/1

N2 - For Schrödinger maps from ℝ2 × ℝ+ to the 2-sphere &Sdbl;2, it is not known if finite energy solutions can form singularities (blow up) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispersive sense, i.e., scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the generalized Hasimoto transform, and Strichartz (dispersive) estimates for a certain two space-dimensional linear Schrödinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length scale of a nearby harmonic map.

AB - For Schrödinger maps from ℝ2 × ℝ+ to the 2-sphere &Sdbl;2, it is not known if finite energy solutions can form singularities (blow up) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispersive sense, i.e., scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the generalized Hasimoto transform, and Strichartz (dispersive) estimates for a certain two space-dimensional linear Schrödinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length scale of a nearby harmonic map.

UR - http://www.scopus.com/inward/record.url?scp=57349108766&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57349108766&partnerID=8YFLogxK

U2 - 10.1215/00127094-2008-058

DO - 10.1215/00127094-2008-058

M3 - Article

AN - SCOPUS:57349108766

VL - 145

SP - 537

EP - 583

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 3

ER -