Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schr{ö}dinger equation when $d\ge 3$

Abstract

In this paper we prove that the defocusing, $d$-dimensional mass critical nonlinear Schr{ö}dinger initial value problem is globally well-posed and scattering for $u_0\in L^2({\mathbf{R}}^d)$ and $d\ge 3$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [10]. Since we are considering an $L^2$ - critical initial value problem we will localize to low frequencies.

53 pages