Global wellposedness of NLS in H$^1$ ($\mathbb{R}$) + H$^s$ ($\mathbb{T}$)

We show global wellposedness for the defocusing cubic nonlinear Schrödinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) +
H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS and global results for the quadratic NLS in this hybrid setting.