Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr\"odinger equation in $\R^{1+4}$

TL;DR

This paper proves global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in four spatial dimensions, providing improved bounds and a simplified approach compared to previous methods.

Contribution

It introduces a simpler derivation of the frequency-localized interaction Morawetz estimate, leading to better bounds on spacetime norms for solutions.

Findings

Established global well-posedness and scattering.

Derived improved bounds on the $L^6_{t,x}$-norm.

Simplified the derivation of key estimates.

Abstract

We obtain global well-posedness, scattering, uniform regularity, and global $L^6_{t,x}$ spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schr\"odinger equation in $\R\times\R^4$. Our arguments closely follow those of Colliander-Keel-Staffilani-Takaoka-Tao, though our derivation of the frequency-localized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the $L^6_{t,x}$-norm.