Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions

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

doi:10.3934/cpaa.2007.6.1023

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2007, Volume 6, Issue 4: 1023-1041. Doi: 10.3934/cpaa.2007.6.1023

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Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions

1.
Department of Mathematics, Barnard College, Columbia University, 2990 Broadway, New York, NY 10027
2.
Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712
3.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307
4.
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801

Received: October 2006

Revised: June 2007

Published: December 2007

Abstract / Introduction Related Papers Cited by

Abstract

The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation in $\mathbb R^n, n \geq 3$ is considered. We show that the problem is globally well posed in $H^s(\mathbb R^n )$ 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.

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Mathematics Subject Classification: 35Q55.

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