## Abstract

In this paper we prove that the one-dimensional Schrödinger equation with derivative in the nonlinear term is globally well-posed in H^{s} for s > 1/2 for data small in L^{2}. To understand the strength of this result one should recall that for s 1/2 the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the "I-method" used by the same authors to obtain global well-posedness for s > 2/3. The same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in H^{s} for s > 1/2.

Original language | English (US) |
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Pages (from-to) | 64-86 |

Number of pages | 23 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 34 |

Issue number | 1 |

DOIs | |

State | Published - 2003 |

## Keywords

- Almost conserved energies
- Global well-posedness
- Schrödinger equation with derivative