Global Well-Posedness for the $L^2$-critical nonlinear Schr\"odinger equation in higher dimensions

Daniela De Silva; Natasa Pavlovic; Gigliola Staffilani; Nikolaos Tzirakis

Global Well-Posedness for the $L^2$-critical nonlinear Schr\"odinger equation in higher dimensions

Analysis of PDEs 2007-05-23 v1

Authors: Daniela De Silva , Natasa Pavlovic , Gigliola Staffilani , Nikolaos Tzirakis

Abstract

The initial value problem for the $L^{2}$ critical semilinear Schr\"odinger equation in $\R^n, n \geq 3$ is considered. We show that the problem is globally well posed in $H^{s}({\Bbb R^{n}})$ when $1>s>\frac{\sqrt{7}-1}{3}$ for $n=3$ , and when $1>s> \frac{-(n-2)+\sqrt{(n-2)^2+8(n-2)}}{4}$ for $n \geq 4$ . We use the `` $I$ -method'' combined with a local in time Morawetz estimate.

Keywords

nonlinear schrödinger equation global well-posedness l2-critical i-method morawetz estimate

Cite

@article{arxiv.math/0607632,
  title  = {Global Well-Posedness for the $L^2$-critical nonlinear Schr\"odinger equation in higher dimensions},
  author = {Daniela De Silva and Natasa Pavlovic and Gigliola Staffilani and Nikolaos Tzirakis},
  journal= {arXiv preprint arXiv:math/0607632},
  year   = {2007}
}

Comments

18 pages, no figures

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