Global well-posedness of the cubic nonlinear Schrödinger equation on $\mathbb{T}^{2}$
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911624052342784 |
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| author | Herr, Sebastian Kwak, Beomjong |
| author_facet | Herr, Sebastian Kwak, Beomjong |
| contents | We prove global well-posedness for the cubic nonlinear Schrödinger 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global well-posedness of the cubic nonlinear Schrödinger equation on $\mathbb{T}^{2}$ Herr, Sebastian Kwak, Beomjong Analysis of PDEs We prove global well-posedness for the cubic nonlinear Schrödinger 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. |
| title | Global well-posedness of the cubic nonlinear Schrödinger equation on $\mathbb{T}^{2}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.17073 |