#### Vol. 14, No. 4, 2021

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On global-in-time Strichartz estimates for the semiperiodic Schrödinger equation

### Alex Barron

Vol. 14 (2021), No. 4, 1125–1152
##### Abstract

We prove global-in-time Strichartz-type estimates for the Schrödinger equation on manifolds of the form ${ℝ}^{n}×{\mathbb{𝕋}}^{d}$, where ${\mathbb{𝕋}}^{d}$ is a $d$-dimensional torus. Our results generalize and improve a global space-time estimate for the Schrödinger equation on $ℝ×{\mathbb{𝕋}}^{2}$ due to Z. Hani and B. Pausader. As a consequence we prove global existence and scattering in ${H}^{1∕2}$ for small initial data for the quintic NLS on $ℝ×\mathbb{𝕋}$ and the cubic NLS on ${ℝ}^{2}×\mathbb{𝕋}$.

##### Keywords
Strichartz estimates on product space, Schrödinger equation on waveguide, decoupling
Primary: 35Q55
Secondary: 42B37