Work

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

Year: 2026

Type: article

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.

Source: Inventiones mathematicae

DOI: 10.1007/s00222-026-01418-4

Authors: Sebastian Herr, Beomjong Kwak

Institutions: Bielefeld University, Korea Advanced Institute of Science and Technology

Language: English

Topic: Advanced Mathematical Physics Problems

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: National Research Foundation, Deutsche Forschungsgemeinschaft, National Research Foundation of Korea, Universität Bielefeld