Global well-posedness and scattering for defocusing, cubic NLS in $\mathbb{R}^3$

TL;DR

This paper establishes the global existence and scattering behavior of solutions to the defocusing cubic nonlinear Schrödinger equation in three-dimensional space for initial data with regularity above a certain threshold, using advanced decomposition and iteration techniques.

Contribution

It extends the well-posedness and scattering results to initial data in $H^s$ for $s>49/74$, combining resonance and linear-nonlinear decompositions with large time iteration methods.

Findings

Proves global well-posedness for $s>49/74$

Demonstrates scattering for the same class of initial data

Integrates resonance decomposition with large time iteration techniques

Abstract

We prove global well-posedness and scattering for the defocusing, cubic NLS on $\mathbb{R}^3$ with initial data in $H^s(\mathbb{R}^3)$ for $s>49/74$. The proof combines the ideas of resonance decomposition in \cite{CKSTT4} and linear-nonlinear decomposition in \cite{ben1}\cite{roy} together with the idea of large time iteration.