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

Abstract

In this paper, we first prove global well-posedness for the defocusing cubic nonlinear Schrödinger 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 ${\sup }_{t\in I}\Vert u(t){\Vert }_{{\dot{H}}^1({\mathbb{T}}^4)}<\Vert W{\Vert }_{{\dot{H}}^1({\mathbb{R}}^4)}$, 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 ${\mathbb{R}}^4$.