Asymptotic stability of harmonic maps under the schrödinger flow

Stephen Gustafson, Kyungkeun Kang, Tai Peng Tsai

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)537-583
Number of pages47
JournalDuke Mathematical Journal
Volume145
Issue number3
DOIs
Publication statusPublished - 2008 Dec 1

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Harmonic Maps
Asymptotic Stability
Energy
Blow-up
Singularity
Dispersive Estimates
Equivariant Map
Strichartz Estimates
Topological Degree
Local Well-posedness
Scatter
Global Solution
Equivariant
Length Scale
Thing
Two Parameters
Linear equation
Infinity
Decay
Tend

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Gustafson, Stephen ; Kang, Kyungkeun ; Tsai, Tai Peng. / Asymptotic stability of harmonic maps under the schrödinger flow. In: Duke Mathematical Journal. 2008 ; Vol. 145, No. 3. pp. 537-583.
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abstract = "For Schr{\"o}dinger 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 Schr{\"o}dinger 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.",
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Asymptotic stability of harmonic maps under the schrödinger flow. / Gustafson, Stephen; Kang, Kyungkeun; Tsai, Tai Peng.

In: Duke Mathematical Journal, Vol. 145, No. 3, 01.12.2008, p. 537-583.

Research output: Contribution to journalArticle

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