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

Abstract

In this paper we prove global well - posedness and scattering for the focusing, energy - critical nonlinear Schrödinger 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.

39 pages