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In this article, we study long time dynamics for defocusing cubic NLS onthree dimensional product space. First, we apply the decoupling method inBourgain-Demeter \cite{BD} to establish a bilinear Strichartz estimate.Moreover, we prove global well-posedness for defocusing, cubic NLS on threedimensional 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 ofhigher Sobolev norm problem which is tightly linked to the weak turbulencephenomenon.

Long time dynamics for defocusing cubic NLS on three dimensional product space