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 $H^s_0(\Omega)$, $1<s\leq 3$, with $\Omega$ any bounded domain with smooth boundary. 
Fabrice Planchon
math.AP