  In this article, we study long time dynamics for defocusing cubic NLS on
three dimensional product space. First, we apply the decoupling method in
Bourgain-Demeter \cite{BD} to establish a bilinear Strichartz estimate.
Moreover, we prove global well-posedness for defocusing, cubic NLS on three
dimensional product space with rough initial data ($H^s$, $s>\frac{5}{6}$)
based on I-method and the bilinear estimate. At last, we discuss the growth of
higher Sobolev norm problem which is tightly linked to the weak turbulence
phenomenon.
