Global well-posedness for the energy-critical focusing nonlinear Schrödinger equation on [formula omitted

Abstract

In this paper, we consider the global (in time) well-posedness for the focusing cubic nonlinear Schrödinger equation (NLS) on 4-dimensional tori –either rational or irrational– and with initial data in H1. We prove that if a maximal-lifespan solution of the focusing cubic NLS u:I×T4→C satisfies supt∈I⁡‖u(t)‖H˙1(T4)<‖W‖H˙1(R4), 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 R4. As a consequence, we also construct the global solution with some threshold conditions related to the modified energy of the initial data which is the energy modified by the mass of the initial data and the best constants of Sobolev embedding on T4.