Global Well-Posedness of the Energy-Critical Nonlinear Schr\"odinger Equation on $\mathbb{T}^4$

TL;DR

This paper establishes global well-posedness for the energy-critical cubic nonlinear Schrödinger equation on 4-dimensional tori, including both defocusing and focusing cases, with initial data in the Sobolev space H^1.

Contribution

It proves global well-posedness on tori and extends the understanding of focusing solutions below the ground state threshold in four dimensions.

Findings

Global well-posedness for defocusing cubic NLS on 4D tori.

Focusing solutions with energy below the ground state are global.

Results hold for both rational and irrational tori.

Abstract

In this paper, we first prove global well-posedness for the defocusing cubic nonlinear Schr\"odinger equation (NLS) on 4-dimensional tori - either rational or irrational - and with initial data in $H^1$. Furthermore, we prove that if a maximal-lifespan solution of the focusing cubic NLS $u: I\times\mathbb{T}^4\to \mathbb{C}$ satisfies $\sup_{t\in I}\|u(t)\|_{\dot{H}^1(\mathbb{T}^4)}<\|W\|_{\dot{H}^1(\mathbb{R}^4)}$, then it is a global solution. $W$ denotes the ground state on Euclidean space, which is a stationary solution of the corresponding focusing equation in $\mathbb{R}^4$.