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

Sebastian Herr, Beomjong Kwak

TL;DR

This work addresses the global well-posedness of the cubic nonlinear Schrödinger equation on the two-dimensional torus in the mass-critical regime. It introduces a sharp inverse Strichartz inequality on $\mathbb{T}^2$ built from incidence geometry and additive combinatorics, and couples it with a degree-lowering framework for inverse Gowers norms to transfer Dodson’s nonperiodic results to the periodic setting. The authors also develop a theory of rectangular resonances via Bohr sets and multiprogressions, along with extinction and profile-structure arguments, to control large-data dynamics and obtain sharpness results through approximate periodic solutions. Collectively, these techniques yield global well-posedness results for defocusing data of arbitrary size in $H^s(\mathbb{T}^2)$ and for focusing data below the ground-state threshold, marking the first mass-critical periodic GWP results in this setting and providing a robust bridge from Euclidean to periodic dispersive theory.

Abstract

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.

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

TL;DR

This work addresses the global well-posedness of the cubic nonlinear Schrödinger equation on the two-dimensional torus in the mass-critical regime. It introduces a sharp inverse Strichartz inequality on built from incidence geometry and additive combinatorics, and couples it with a degree-lowering framework for inverse Gowers norms to transfer Dodson’s nonperiodic results to the periodic setting. The authors also develop a theory of rectangular resonances via Bohr sets and multiprogressions, along with extinction and profile-structure arguments, to control large-data dynamics and obtain sharpness results through approximate periodic solutions. Collectively, these techniques yield global well-posedness results for defocusing data of arbitrary size in and for focusing data below the ground-state threshold, marking the first mass-critical periodic GWP results in this setting and providing a robust bridge from Euclidean to periodic dispersive theory.

Abstract

We prove global well-posedness for the cubic nonlinear Schrödinger equation for periodic initial data in the mass-critical dimension 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.

Paper Structure

This paper contains 21 sections, 106 theorems, 705 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Degree-lowering on inverse Gowers inequalities
  4. Norms and inverse theorems concerning rectangular resonances
  5. Norms concerning tensor products
  6. An inverse theorem on the rectangle resonance
  7. Limit properties of profiles
  8. Proof of Theorem \ref{['thm:inverse L4 Stri']}
  9. Proof of Proposition \ref{['prop:modulus conc']}
  10. Proof of Proposition \ref{['prop:conc to P']} and Proposition \ref{['prop:conc to [N]^2']}
  11. Proof of Proposition \ref{['prop:conc to Q']}
  12. Inverse inequalities on sparse sublattices
  13. Inverse property of the $\mathcal{\mathcal{N}}$-norm for degree $2$ nilsequences
  14. $(3,2)$-*-reducibility of the $\mathcal{\mathcal{N}}$-norm
  15. Proof of Proposition \ref{['prop:conc to Q']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $s>0$. The defocusing eq:NLS is globally well-posed for initial data $u_0 \in H^s(\mathbb{T}^2)$. Moreover, we have the following quantitative bound: Let $M>0$ and $T>0$. For $u_0\in H^s(\mathbb{T}^2)$ such that $\|u_0\|_{L^2(\mathbb{T}^2)}\le M$, the solution $u$ to the defocusing eq:NLS with t

Theorems & Definitions (222)

  • Theorem 1.1: GWP for defocusing NLS
  • Theorem 1.2: GWP for focusing NLS
  • Proposition 1.3: Theorem 1.2 in herr2024strichartz
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Proposition 2.1: Szemeredi-trotter
  • Proposition 2.2: tao2006additive
  • Lemma 2.3
  • ...and 212 more