The quintic NLS on perturbations of $\mathbf{R}^3$

TL;DR

This paper proves that solutions to the energy-critical defocusing quintic nonlinear Schrödinger equation on R^3 continue to scatter to linear solutions even when the Euclidean metric is slightly deformed, using advanced microlocal analysis.

Contribution

It demonstrates scattering stability under small metric perturbations for the energy-critical NLS on R^3, introducing a novel long-time microlocal weak dispersive estimate.

Findings

Finite-energy solutions still scatter under small metric deformations.

Developed a microlocal weak dispersive estimate for perturbed geometries.

Extended scattering results to non-Euclidean, perturbed settings.

Abstract

Consider the defocusing quintic nonlinear Schr\"{o}dinger equation on $\mathbf{R}^3$ with initial data in the energy space. This problem is "energy-critical" in view of a certain scale-invariance, which is a main source of difficulty in the analysis of this equation. It is a nontrivial fact that all finite-energy solutions scatter to linear solutions. We show that this remains true under small compact deformations of the Euclidean metric. Our main new ingredient is a long-time microlocal weak dispersive estimate that accounts for the refocusing of geodesics.