Globally well and ill posedness for non-elliptic derivative Schr?dinger equations with small rough data

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• 作者：Baoxiang Wang
• 关键词：Non-elliptic derivative Schrö ; dinger equation ; Gabor frame ; Modulation spaces ; Well posedness ; Ill posedness
• 刊名：Journal of Functional Analysis
• 出版年：2013
• 出版时间：15 December, 2013
• 年：2013
• 卷：265
• 期：12
• 页码：3009-3052
• 全文大小：431 K

文摘

We show that there exists such that the cubic (quartic) non-elliptic derivative Schr?dinger equations with small data in modulation spaces for () are globally well-posed if , and ill-posed if . In 2D cubic case, using the Gabor frame, we get some time-global dispersive estimates for the Schr?dinger semi-group in anisotropic Lebesgue spaces, which include a time-global maximal function estimate in the space . By resorting to the smooth effect estimate together with the dispersive estimates in anisotropic Lebesgue spaces, we show that the cubic hyperbolic derivative NLS in 2D has a unique global solution if the initial data in Feichtinger-Segal algebra or in weighted Sobolev spaces are sufficiently small.