On scattering for the defocusing high dimensional inter-critical NLS

TL;DR

This paper proves global well-posedness and scattering for the inter-critical defocusing high-dimensional nonlinear Schrödinger equation, assuming boundedness of the critical norm, using concentration compactness, Duhamel, and Morawetz techniques.

Contribution

It establishes the scattering result for the inter-critical NLS in high dimensions under the critical norm boundedness assumption, extending previous methods to this regime.

Findings

Proves global well-posedness under critical norm boundedness.

Establishes scattering for the inter-critical NLS in high dimensions.

Uses concentration compactness and interaction Morawetz estimates to exclude critical elements.

Abstract

In this paper, we study the critical norm conjecture for the inter-critical nonlinear Schr{\"o}dinger equation with critical index $s_c$ satisfying $\frac{1}{2}<s_c<1$ when $d\geq 5$. Under the assumption of uniform boundedness of the critical norm, we prove the global well-posedness and scattering for the Cauchy problem. We follow the standard `Concentration compactness/Rigidity method' established in \cite{KenigMerle1,KenigMerle2}, and treat three scenarios for the critical element respectively. Moreover, double Duhamel method and interaction Morawetz estimate are applied to exclude the critical element.