### Abstract

For Schrödinger maps from ℝ^{2} × 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 language | English |
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Pages (from-to) | 537-583 |

Number of pages | 47 |

Journal | Duke Mathematical Journal |

Volume | 145 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2008 Dec 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*145*(3), 537-583. https://doi.org/10.1215/00127094-2008-058