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

P. Blue; J. Colliander

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

P. Blue·J. Colliander

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Abstract

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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P. Blue, J. Colliander. 2005-07-29. Global well-posedness in Sobolev space implies global existence for weighted L^2 initial data for L^2 -critical NLS. https://arxiv.org/abs/math/0508001

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