On the cubic NLS on 3D compact domains

Fabrice Planchon

Published 2012-10-16, updated 2013-01-04Version 2

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.