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

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

2005-07-29

arxiv.org/abs/math/0508001v2

Mathematics

Analysis of PDEs

21 pages, 1 figure. v2: A correction to proof of lemma 2.1 has been added following the paper

Scientific paper

The L^2 -critical defocusing nonlinear Schrodinger initial value problem on 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.

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