SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 3644-3660, January 2021.
In this article, we study long time dynamics for defocusing cubic nonlinear Schrödinger equations (NLS) on three dimensional product space. First, we apply the decoupling method in Bourgain and Demeter [Ann. of Math. (2), 182 (2015), pp. 351--389] to establish a bilinear Strichartz estimate. Moreover, we prove global well-posedness for defocusing, cubic NLS on a three dimensional product space with rough initial data ($H^s$, $s>\frac{5}{6}$) based on the I-method and the bilinear estimate. At last, we discuss the growth of the higher Sobolev norm problem which is tightly linked to the weak turbulence phenomenon.

中文翻译：

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三维积空间上三次非线性薛定谔方程散焦的长时间动力学

SIAM 数学分析杂志，第 53 卷，第 3 期，第 3644-3660 页，2021
年1 月。在本文中，我们研究了在三维积空间上散焦三次非线性薛定谔方程 (NLS) 的长时间动力学。首先，我们在 Bourgain 和 Demeter [Ann. 的数学。(2), 182 (2015), pp. 351--389] 建立双线性 Strichartz 估计。此外，我们证明了基于 I 方法和粗略初始数据 ($H^s$, $s>\frac{5}{6}$) 在三维乘积空间上散焦三次 NLS 的全局适定性和双线性估计。最后，我们讨论了与弱湍流现象密切相关的更高 Sobolev 范数问题的增长。