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

Abstract

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 [6] 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 > 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.

Global Well-posedness for the Focusing Cubic NLS on the Product Space $\mathbb{R} \times \mathbb{T}^3$

“…The techniques used in Euclidean and tori settings are frequently combined and applied to the waveguides problems. We refer to [8,9,10,16,19,20,21,22,33,35,34,36] for some NLS results in the waveguide setting.…”