### Abstract

We consider the Schrödinger map initial-value problem, where φR{double-struck}^{d} × R{double-struck} → S{double-struck}^{2} → R{double-struck}^{3} is a smooth function. In all dimensions d ≥ 2, we prove that the Schrödinger map initial-value problem admits a unique global smooth solution φ ε C(R{double-struck} : H_{Q}^{∞}), Q ε S{double-struck}^{2}, provided that the data φo ε H_{Q}^{∞} is smooth and satisfies the smallness condition ||φ0 - Q||_{H}^{d/2} << 1. We prove also that the solution operator extends continuously to the space of data in H.^{d/2} ∩ H_{Q} ^{d/2-1} with small H ^{d/2}. norm.

Original language | English (US) |
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Pages (from-to) | 1443-1506 |

Number of pages | 64 |

Journal | Annals of Mathematics |

Volume | 173 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2011 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Bejenaru, I., Ionescu, A. D., Kenig, C. E., & Tataru, D. (2011). Global Schrödinger maps in dimensions d ≥ 2: Small data in the critical sobolev spaces.

*Annals of Mathematics*,*173*(3), 1443-1506. https://doi.org/10.4007/annals.2011.173.3.5