Global Well-Posedness for the $L^2$-critical nonlinear Schr\"odinger equation in higher dimensions

TL;DR

This paper proves global well-posedness for the $L^2$-critical nonlinear Schrödinger equation in higher dimensions using the $I$-method and Morawetz estimates, extending the range of initial data regularity for which solutions exist.

Contribution

It establishes new global well-posedness results for the $L^2$-critical NLS in dimensions three and higher, employing the $I$-method combined with Morawetz estimates.

Findings

Global well-posedness in $H^s$ for specific $s$ ranges

Extension of well-posedness to higher dimensions $n \\geq 3$

Application of the $I$-method with Morawetz estimates

Abstract

The initial value problem for the $L^{2}$ critical semilinear Schr\"odinger equation in $\R^n, n \geq 3$ is considered. We show that the problem is globally well posed in $H^{s}({\Bbb R^{n}})$ when $1>s>\frac{\sqrt{7}-1}{3}$ for $n=3$, and when $1>s> \frac{-(n-2)+\sqrt{(n-2)^2+8(n-2)}}{4}$ for $n \geq 4$. We use the ``$I$-method'' combined with a local in time Morawetz estimate.