On the cubic NLS on 3D compact domains

TL;DR

This paper establishes bilinear estimates for the 3D cubic nonlinear Schrödinger equation on compact domains, leading to global well-posedness results for initial data in certain Sobolev spaces, extending known results to bounded domains.

Contribution

It provides bilinear estimates on 3D domains with boundary conditions, matching Euclidean and boundaryless cases, and proves global well-posedness for the cubic NLS on bounded domains.

Findings

Bilinear estimates match Euclidean case on non-trapping domains.

Bilinear estimates match boundaryless case on bounded domains.

Global well-posedness for cubic NLS in $H^s_0(\Omega)$ for $1<s\leq 3$.

Abstract

We prove bilinear estimates for the Schr\"odinger equation on 3D domains, with Dirichlet boundary conditions. On non-trapping domains, they match the $\mathbb{R}^3$ case, while on bounded domains they match the generic boundary less manifold case. As an application, we obtain global well-posedness for the defocusing cubic NLS for data in $H^s_0(\Omega)$, $1<s\leq 3$, with $\Omega$ any bounded domain with smooth boundary.