Global well-posedness for the defocusing, quintic nonlinear Schr\"odinger equation in one dimension

TL;DR

This paper proves global well-posedness for the one-dimensional quintic defocusing nonlinear Schrödinger equation with initial data of low regularity, improving previous regularity thresholds by developing refined Morawetz estimates.

Contribution

It introduces an improved method for obtaining almost Morawetz estimates, lowering the regularity requirement for global solutions in the 1D quintic NLS.

Findings

Global well-posedness established for s > 8/29

Improved error estimates in Morawetz inequalities

Extension of well-posedness results to lower regularity data

Abstract

In this paper, we prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schr\"odinger equation. We show that a unique solution exists for $u_{0} \in H^{s}(\mathbf{R})$, $s > {8/29}$. This improves the result in [13], which proved global well-posedness for $s > {1/3}$. The main new argument is that we obtain almost Morawetz estimates with improved error.