Remark on the periodic mass critical nonlinear Schrödinger equation

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.

12 pages. The main result in 2d was extended to some of irrational tori

Cited by in corpus (1)

• Periodic Cubic Hyperbolic Schrödinger equation on $\T^2$