Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis

Research output: Contribution to journalArticlepeer-review

Abstract

The initial value problem for the L2 critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well-posed in Hs(double-struck T signd), for s > 4/9 and s > 2/3 in 1D and 2D respectively, confirming in 2D a statement of Bourgain in [4]. We use the "I- method". This method allows one to introduce a modification of the energy functional that is well defined for initial data below the H1(double- struck T signd) threshold. The main ingredient in the proof is a "refinement" of the Strichartz's estimates that hold true for solutions defined on the rescaled space, double-struck T sign λd = ℝd/λℤd, d = 1, 2.

Original languageEnglish (US)
Pages (from-to)37-65
Number of pages29
JournalDiscrete and Continuous Dynamical Systems
Volume19
Issue number1
DOIs
StatePublished - Sep 2007
Externally publishedYes

Keywords

  • Global well-posedness
  • Nonlinear Schrödinger equation
  • Nonlinear dispersive equations

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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