Global well-posedness of the energy critical Nonlinear Schr\"odinger equation with small initial data in H^1(T^3)

TL;DR

This paper proves global well-posedness for the energy-critical quintic nonlinear Schrödinger equation on the three-dimensional torus with small initial data in H^1, using refined estimates and modified function space techniques.

Contribution

It introduces a refined trilinear Strichartz estimate and adapts critical function space theory to establish the first energy-critical global well-posedness on compact manifolds.

Findings

Global well-posedness for small data in H^1 on T^3

First energy-critical result on compact manifolds

Refined trilinear Strichartz estimates for Schrödinger solutions

Abstract

A refined trilinear Strichartz estimate for solutions to the Schr\"odinger equation on the flat rational torus T^3 is derived. By a suitable modification of critical function space theory this is applied to prove a small data global well-posedness result for the quintic Nonlinear Schr\"odinger Equation in H^s(T^3) for all s \geq 1. This is the first energy-critical global well-posedness result in the setting of compact manifolds.