Remark on the periodic mass critical nonlinear Schrödinger equation

Abstract

We consider the mass critical NLS onT\mathbb {T}andT2\mathbb {T} ^2. In theRd\mathbb {R}^dcase the Strichartz estimates enable us to show well-posedness of the IVP inL2L^2(at least for small data) via the Picard iteration method. However, counterexamples to theL6L^6Strichartz onT\mathbb {T}and theL4L^4Strichartz onT2\mathbb {T}^2were 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 onT\mathbb {T}andT2\mathbb {T}^2cannot have a smooth data-to-solution map inL2L^2even for small initial data. The same results are also obtained for most of the two dimensional irrational tori.