Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schrödinger equation when $d=1$

Abstract

In this paper we prove that the defocusing, quintic nonlinear Schrödinger initial value problem is globally well-posed and scattering for $u_0\in L^2(\mathbf{R})$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [11]. Since we are considering an $L^2$ - critical initial value problem we will localize to low frequencies.

48 pages