Remark on the periodic mass critical nonlinear Schrödinger equation

Kishimoto, Nobu

doi:10.1090/S0002-9939-2014-12024-7

Remark on the periodic mass critical nonlinear Schrödinger equation
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Proc. Amer. Math. Soc. 142 (2014), 2649-2660 Request permission

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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