December  2006, 5(4): 691-708. doi: 10.3934/cpaa.2006.5.691

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

1. 

University of Toronto, Toronto, Ontario, M5S 2E4, Canada

Received  January 2006 Revised  May 2006 Published  September 2006

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 $\sigma = s$.
Citation: P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691
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