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Remark on the periodic mass critical nonlinear Schrödinger equation


Author: Nobu Kishimoto
Journal: Proc. Amer. Math. Soc. 142 (2014), 2649-2660
MSC (2010): Primary 35Q55; Secondary 46E35, 11P21
DOI: https://doi.org/10.1090/S0002-9939-2014-12024-7
Published electronically: May 2, 2014
MathSciNet review: 3209321
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Abstract: We consider the mass critical NLS on $ \mathbb{T}$ and $ \mathbb{T} ^2$. In the $ \mathbb{R}^d$ case the Strichartz estimates enable us to show well-posedness of the IVP in $ L^2$ (at least for small data) via the Picard iteration method. However, counterexamples to the $ L^6$ Strichartz on $ \mathbb{T}$ and the $ L^4$ Strichartz on $ \mathbb{T}^2$ were given by Bourgain (1993) and Takaoka-Tzvetkov (2001), respectively, which means that the Strichartz spaces are not suitable for iteration in these problems. In this note, we show a slightly stronger result, namely, that the IVP on $ \mathbb{T}$ and $ \mathbb{T}^2$ cannot have a smooth data-to-solution map in $ L^2$ even for small initial data. The same results are also obtained for most of the two dimensional irrational tori.


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Additional Information

Nobu Kishimoto
Affiliation: Department of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo, Kyoto 606-8502, Japan
Address at time of publication: Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo, Kyoto 606-8502, Japan
Email: nobu@kurims.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2014-12024-7
Keywords: Mass critical nonlinear Schr\"odinger equation, periodic boundary condition, Strichartz estimates, Picard iteration method.
Received by editor(s): March 30, 2012
Received by editor(s) in revised form: July 24, 2012
Published electronically: May 2, 2014
Additional Notes: This work was partially supported by Grant-in-Aid for Scientific Research 23840022.
Communicated by: James E. Colliander
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.