Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schr{ö}dinger equation when $d=2$

Abstract

In this paper we prove that the defocusing, cubic nonlinear Schr{ö}dinger initial value problem is globally well-posed and scattering for $u_0\in L^2({\mathbf{R}}^2)$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in \cite{CKSTT4}. Since we are considering an $L^2$ - critical initial value problem we will localize to low frequencies.

75 pages