Global well-posedness of the cubic nonlinear Schrödinger equation on $\mathbb{T} ^{2}$
Abstract
Abstract We prove global well-posedness for the cubic nonlinear Schrödinger equation for periodic initial data in the mass-critical dimension $d=2$ 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 that implies the sharpness of the results.
Fields of Study & Subjects
Advanced Mathematical Physics Problems
Publication details
- Journal / source
- Inventiones mathematicae
- Publisher
- Springer Science and Business Media LLC
- Volume
- 245
- Issue
- 1
- Pages
- 513–629
- Publication date
- 31 March 2026
- Language
- EN