Improved global well-posedness for the cubic NLS on two-dimensional waveguide $ \mathbb{R}\times \mathbb{T} $

In this article, we show that the solution to the defocusing cubic nonlinear Schrödinger equation (NLS) posed on the two-dimensional waveguide $ \mathbb{R}\times \mathbb{T} $, $ i\partial_tu+\Delta_{ \mathbb{R}\times \mathbb{T}}u = |u|^2u $, is globally well-posed in $ H^s( \mathbb{R}\times \mathbb{T}) $ with $ s>\frac{1}{2} $. The proof is based on the $ I $-method. Inspired by Colliander-Keel-Staffilani-Takaoka-Tao [Discrete Contin. Dyn. Syst. 21 (2008), 665-686], we construct the modified energy to improve the energy increment. The main difficulty lies in controlling the resonant interactions caused by the modified energy. To this end, we establish refined bilinear Strichartz estimates with angular truncation on the rescaled waveguide, thereby generalizing results previously obtained by Takaoka [J. Differential Equations 394 (2024), 296-319]. Furthermore, we demonstrate polynomial growth of $ H^s $ norms with $ \frac{1}{2}<s<1 $. Our result extends the recent work of Deng-Fan-Yang-Zhao-Zheng [J. Funct. Anal. 287 (2024), 110595].