Almost conservation laws and global rough solutions to a Nonlinear Schr\"odinger equation

TL;DR

This paper establishes an almost conservation law to prove global well-posedness for the cubic defocusing nonlinear Schrödinger equation in certain Sobolev spaces, extending understanding of solution behavior over time.

Contribution

It introduces an almost conservation law approach to achieve global solutions for NLS in H^s spaces with s > 4/7 for n=2 and s > 5/6 for n=3, advancing previous results.

Findings

Global well-posedness in H^s for n=2 and s > 4/7

Global well-posedness in H^s for n=3 and s > 5/6

Development of an almost conservation law technique

Abstract

We prove an "almost conservation law" to obtain global-in-time well-posedness for the cubic, defocussing nonlinear Schr\"odinger equation in H^s(R^n) when n = 2, 3 and s > 4/7, 5/6, respectively.