What question did this study set out to answer?

To establish global well-posedness for the cubic nonlinear Schrödinger equation in a mass-critical dimension of 2, for periodic initial data.

April 1, 2026Open Access

Global well-posedness of the cubic nonlinear Schrödinger equation on T ^2

SHSebastian Herr BKBeomjong Kwak

Key Points

To establish global well-posedness for the cubic nonlinear Schrödinger equation in a mass-critical dimension of 2, for periodic initial data.
Utilized a new inverse Strichartz inequality
Applied incidence geometry techniques
Used results from additive combinatorics
Leveraged inverse theorems for Gowers uniformity norms
Extended non-periodic results to a periodic framework
Global well-posedness is proven for arbitrary size initial data in the defocusing case
Results hold for data below the ground state threshold in the focusing case
Constructed an approximate periodic solution to demonstrate result sharpness

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.

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