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

Daniela De Silva·Natasa Pavlovic·Gigliola Staffilani·Nikolaos Tzirakis
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Abstract

The initial value problem for the $L^{2}$ critical semilinear Schrödinger 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.

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Daniela De Silva, Natasa Pavlovic, Gigliola Staffilani, Nikolaos Tzirakis. 2006-07-25. Global Well-Posedness for the $L^2$-critical nonlinear Schrödinger equation in higher dimensions. https://arxiv.org/abs/math/0607632

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