Almost Morawetz estimates and global well-posedness for the defocusing $L^2$-critical nonlinear Schr{\"o}dinger equation in higher dimensions

TL;DR

This paper establishes global well-posedness for the defocusing $L^2$-critical nonlinear Schrödinger equation in higher dimensions using advanced analytical techniques, including the I-method and almost Morawetz estimates.

Contribution

It introduces a novel combination of energy increment methods, interaction Morawetz, and almost Morawetz estimates to extend well-posedness results to higher dimensions.

Findings

Global well-posedness in 3D for s > 2/5

Global well-posedness in n ≥ 4 for s > (n - 2)/n

Effective use of combined Morawetz estimates

Abstract

In this paper, we consider the global well-posedness of the defocusing, $L^{2}$ - critical nonlinear Schr{\"o}dinger equation in dimensions $n \geq 3$. Using the I-method, we show the problem is globally well-posed in $n = 3$ when $s > {2/5}$, and when $n \geq 4$, for $s > \frac{n - 2}{n}$. We combine energy increments for the I-method, interaction Morawetz estimates, and almost Morawetz estimates to prove the result.