The defocusing energy-supercritical NLS in four space dimensions

Abstract

We consider a class of defocusing energy-supercritical nonlinear Schrödinger equations in four space dimensions. Following a concentration-compactness approach, we show that for $1<s_c<3/2$, any solution that remains bounded in the critical Sobolev space ${\dot{H}}_x^{s_c}({\mathbb{R}}^4)$ must be global and scatter. Key ingredients in the proof include a long-time Strichartz estimate and a frequency-localized interaction Morawetz inequality.

52 pages