A refined global well-posedness result for schrödinger equations with derivative

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao

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109 Scopus citations

Abstract

In this paper we prove that the one-dimensional Schrödinger equation with derivative in the nonlinear term is globally well-posed in Hs for s > 1/2 for data small in L2. 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 Hs for s > 1/2.

Original languageEnglish (US)
Pages (from-to)64-86
Number of pages23
JournalSIAM Journal on Mathematical Analysis
Volume34
Issue number1
DOIs
StatePublished - 2003

Keywords

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

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