Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS

P. Blue; J. Colliander

doi:10.3934/cpaa.2006.5.691

Article Contents

2006, Volume 5, Issue 4: 691-708. Doi: 10.3934/cpaa.2006.5.691

This issue Previous Article An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential Next Article On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations

Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS

P. Blue^1, and
J. Colliander^1,

1.
University of Toronto, Toronto, Ontario, M5S 2E4

Received: January 2006

Revised: May 2006

Published: December 2006

Abstract Related Papers Cited by

Abstract

The $L^2$-critical defocusing nonlinear Schrödinger initial value problem on $\mathbb R^d$ is known to be locally well-posed for initial data in $L^2$. Hamiltonian conservation and the pseudoconformal transformation show that global well-posedness holds for initial data $u_0$ in Sobolev $H^1$ and for data in the weighted space $(1+|x|) u_0 \in L^2$. For the $d=2$ problem, it is known that global existence holds for data in $H^s$ and also for data in the weighted space $(1+|x|)^\sigma u_0 \in L^2$ for certain $s, \sigma < 1$. We prove: If global well-posedness holds in $H^s$ then global existence and scattering holds for initial data in the weighted space with $\sigma = s$.

Keywords:

Mathematics Subject Classification: 35Q55.

Citation:

Article Metrics

HTML views() PDF downloads(116) Cited by(0)

Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content