Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d=2

Abstract

In this article we prove that the defocusing, cubic nonlinear Schrödinger initial value problem is globally well posed and scattering for $u_0\in L^2\left({\mathbf{R}}^2\right)$. 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.