Well-posedness for energy-critical nonlinear Schr\"odinger equation on waveguide manifold

TL;DR

This paper establishes well-posedness for the energy-critical nonlinear Schrödinger equation on waveguide manifolds in dimensions three and four, using recent scale-invariant Strichartz estimates and decoupling techniques.

Contribution

It provides a unified, simplified approach to well-posedness results leveraging recent advances in Strichartz estimates on waveguides.

Findings

Well-posedness results for 3D and 4D energy-critical NLS on waveguides.

Application of Bourgain-Demeter decoupling in this context.

Simplified proof framework for existing results.

Abstract

In this article, we utilize the scale-invariant Strichartz estimate on waveguide which is developed recently by Barron \cite{Barron} based on Bourgain-Demeter $l^2$ decoupling method \cite{BD} to give a unified and simpler treatment of well-posedness results for energy critical nonlinear Schr\"odinger equation on waveguide when the whole dimension is three and four.