Global well-posedness of the cubic nonlinear Schrödinger equation on ${\mathbb{T}}^2$

Abstract

We prove global well-posedness for the cubic nonlinear Schrödinger equation for periodic initial data in the mass-critical dimension $d=2$ for initial data of arbitrary size in the defocusing case and data below the ground state threshold in the focusing case. The result is based on a new inverse Strichartz inequality, which is proved by using incidence geometry and additive combinatorics, in particular, the inverse theorems for Gowers uniformity norms by Green-Tao-Ziegler. This allows to transfer the analogous results of Dodson for the non-periodic mass-critical NLS to the periodic setting. In addition, we construct an approximate periodic solution which implies sharpness of the results.

94 p. v2: Paper reorganized. Several corrections, in particular in Sect. 6 (new numbering), 97 p. v3: further minor corrections and simplifications, 94 p. v4: This version of the article has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements or corrections. The Version of Record is available online at the DOI below