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

Barron, Alex

doi:10.2140/apde.2021.14.1125

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

Alex Barron

Vol. 14 (2021), No. 4, 1125–1152

DOI: 10.2140/apde.2021.14.1125

Abstract

We prove global-in-time Strichartz-type estimates for the Schrödinger equation on manifolds of the form $ℝ^{n} \times 𝕋^{d}$ , where $𝕋^{d}$ is a $d$ -dimensional torus. Our results generalize and improve a global space-time estimate for the Schrödinger equation on $ℝ \times 𝕋^{2}$ 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 $ℝ \times 𝕋$ and the cubic NLS on $ℝ^{2} \times 𝕋$ .

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Keywords

Strichartz estimates on product space, Schrödinger equation on waveguide, decoupling

Mathematical Subject Classification 2010

Primary: 35Q55

Secondary: 42B37

Milestones

Received: 22 February 2019

Revised: 5 September 2019

Accepted: 2 December 2019

Published: 6 July 2021

Authors

Alex Barron
Department of Mathematics University of Illinois Urbana-Champaign Urbana, IL United States

Vol. 14, No. 4, 2021