On the cubic NLS on 3D compact domains

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 $H_0^s(\mathrm{\Omega })$, $1<s\le 3$, with $\mathrm{\Omega }$ any bounded domain with smooth boundary.