Global well-posedness and scattering for the three-dimensional defocusing cubic Schrödinger equation in $H^s$, $s>\frac{1}{2}$

Abstract

We prove global well-posedness and scattering for the three-dimensional defocusing cubic nonlinear Schrödinger equation with arbitrary initial data in H^s(R^3), $s>\frac{1}{2}$. The proof combines the $I$-method with improved long-time bilinear $L^2_{t,x}$ estimates for frequency-localized components of the solution. The key high--low frequency estimate follows from a directional interaction identity and an induction on frequency.