The defocusing $\dot{H}^{1/2}$-critical NLS in high dimensions

TL;DR

This paper proves that solutions to the defocusing ot;H^{1/2}-critical nonlinear Schrf6dinger equation in dimensions five and higher are global and scatter if they remain bounded in ot;H^{1/2}, combining concentration-compactness and Morawetz inequalities.

Contribution

It establishes global existence and scattering for ot;H^{1/2}-critical NLS in high dimensions using a novel combination of analytical techniques.

Findings

Solutions bounded in ot;H^{1/2} are global and scatter.

The proof extends techniques to higher dimensions.

Combines concentration-compactness with Morawetz inequality.

Abstract

We consider the defocusing $\dot{H}^{1/2}$-critical nonlinear Schr\"odinger equation in dimensions $d\geq 5$. In the spirit of Kenig and Merle [Trans. Amer. Math. Soc. 362 (2010), 1937--1962], we combine a concentration-compactness approach with the Lin--Strauss Morawetz inequality to prove that if a solution $u$ is bounded in $\dot{H}^{1/2}$ throughout its lifespan, then $u$ is global and scatters.