Duke Mathematical Journal

Asymptotic stability of harmonic maps under the Schrödinger flow

Stephen Gustafson,Kyungkeun Kang, and Tai-Peng Tsai

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Abstract

For Schrödinger maps from $\mathbb{R}^2\times\mathbb{R}^+$ to the $2$-sphere $\mathbb{S}^2$, 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

Article information

Source
Duke Math. J. Volume 145, Number 3 (2008), 537-583.

Dates
First available: 15 December 2008

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1229349904

Digital Object Identifier
doi:10.1215/00127094-2008-058

Mathematical Reviews number (MathSciNet)
MR2462113

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35B40: Asymptotic behavior of solutions

Citation

Gustafson, Stephen; Kang, Kyungkeun; Tsai, Tai-Peng. Asymptotic stability of harmonic maps under the Schrödinger flow. Duke Mathematical Journal 145 (2008), no. 3, 537--583. doi:10.1215/00127094-2008-058. http://projecteuclid.org/euclid.dmj/1229349904.


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