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

Abstract

We obtain global well-posedness, scattering, uniform regularity, and global $L_{t,x}^6$ spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}\times {\mathbb{R}}^4$. 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 $L_{t,x}^6$-norm.