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Article Contents

Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions

• The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation in $\mathbb R^n, n \geq 3$ 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 $n \geq 4$. We use the "$I$-method" combined with a local in time Morawetz estimate.
Mathematics Subject Classification: 35Q55.

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