English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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.

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