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2007, Volume 6, Issue 4: 1023-1041. Doi: 10.3934/cpaa.2007.6.1023
This issue Previous Article Well-posedness for one-dimensional derivative nonlinear Schrödinger equations Next Article Orbital stability of solitary waves of the Schrödinger-Boussinesq equation

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:  September 30, 2006
Revised:  May 31, 2007
Published:  November 2007
Abstract 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.
    Mathematics Subject Classification: 35Q55.

    Citation:
    shu

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