Global well - posedness and scattering for the focusing, energy - critical nonlinear Schr\"odinger problem in dimension $d = 4$ for initial data below a ground state threshold

TL;DR

This paper proves global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equation in four dimensions for initial data below a ground state, overcoming previous dimensional limitations.

Contribution

It introduces long time Strichartz estimates to handle the four-dimensional case, extending results previously known only in five dimensions and higher.

Findings

Established global well-posedness in 4D for sub-ground state data.

Achieved scattering results in 4D using novel long time estimates.

Overcame the logarithmic blowup obstacle in four dimensions.

Abstract

In this paper we prove global well - posedness and scattering for the focusing, energy - critical nonlinear Schr\"odinger initial value problem in four dimensions. Previous work proved this in five dimensions and higher using the double Duhamel trick. In this paper, using long time Strichartz estimates we are able to overcome the logarithmic blowup in four dimensions.