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

TL;DR

This paper establishes global well-posedness for the cubic nonlinear Schrödinger equation on the two-dimensional torus, using novel inverse Strichartz inequalities derived from incidence geometry and additive combinatorics.

Contribution

It introduces a new inverse Strichartz inequality based on incidence geometry and additive combinatorics, extending Dodson's non-periodic results to the periodic setting.

Findings

Proves global well-posedness for all initial data in the defocusing case.

Achieves well-posedness for data below the ground state in the focusing case.

Constructs an approximate solution demonstrating the sharpness of the results.

Abstract

We prove global well-posedness for the cubic nonlinear Schr\"odinger 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.