Global Well-Posedness for the $L^2$-critical nonlinear Schrödinger equation in higher dimensions

Abstract

The initial value problem for the $L^2$ critical semilinear Schrödinger equation in ${\mathbb{R}}^n,n\ge 3$ is considered. We show that the problem is globally well posed in $H^s({\mathbb{R}}^n)$ 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 $n\ge 4$. We use the ``$I$-method'' combined with a local in time Morawetz estimate.

18 pages, no figures