In this article we prove that the defocusing, cubic nonlinear Schrödinger initial value problem is globally well posed and scattering for . The proof uses the bilinear estimates of Planchon and Vega and a frequency-localized interaction Morawetz estimate similar to the high-frequency estimate of Colliander, Keel, Staffilani, Takaoka, and Tao and especially the low-frequency estimate of Dodson.
"Global well-posedness and scattering for the defocusing, -critical, nonlinear Schrödinger equation when ." Duke Math. J. 165 (18) 3435 - 3516, 1 December 2016. https://doi.org/10.1215/00127094-3673888