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

Abstract

We consider the defocusing ${\dot{H}}^{1/2}$-critical nonlinear Schrödinger equation in dimensions $d\ge 5$. 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.

17 pages