On the 1D Cubic Nonlinear Schrödinger Equation in an Almost Critical Space

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• 作者：Shaoming Guo
• 关键词：Cubic nonlinear Schrödinger equation ; Almost global well ; posedness ; Modulation spaces ; Restriction estimates
• 刊名：Journal of Fourier Analysis and Applications
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：23
• 期：1
• 页码：91-124
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Fourier Analysis; Signal,Image and Speech Processing; Abstract Harmonic Analysis; Approximations and Expansions; Partial Differential Equations; Mathematical Methods in Physics;
• 出版者：Springer US
• ISSN：1531-5851
• 卷排序：23

文摘

We consider the Cauchy problem for the one dimensional cubic nonlinear Schrödinger equation \(iu_t+u_{xx}-|u|^2u=0\). As the first step local well-posedness in the modulation space \(M_{2,p}\) (\(2\le p<\infty \)) is derived (see Theorem 1.4), which covers all the subcritical cases. Afterwards in order to approach the endpoint case, we will prove the almost global well-posedness in some Orlicz type space (see Theorem 1.8), which is a natural generalization of \(M_{2,p}\), and is almost critical from the viewpoint of scaling. The new ingredient is an endpoint version of the two dimensional restriction estimate (see Lemma 3.7).