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arXiv · 2502.17073

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

Sebastian Herr·Beomjong Kwak
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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.

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Sebastian Herr, Beomjong Kwak. 2025-02-24. Global well-posedness of the cubic nonlinear Schr\"odinger equation on $\mathbb{T}^{2}$. https://doi.org/10.1007/s00222-026-01418-4

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