Long-time Strichartz estimates on 3D waveguide with applications

We study long-time Strichartz estimates for the Schrödinger equation on waveguide manifolds, and use them to establish upper bounds on the growth of Sobolev norms for the nonlinear Schrödinger equation on three-dimensional waveguides.

💡 Research Summary

The paper studies long‑time Strichartz estimates for the linear Schrödinger equation on three‑dimensional waveguide manifolds of the form (\mathbb{R}^{m}\times\mathbb{T}^{n}) (with (m+n=3)) and uses these estimates to control the growth of Sobolev norms for the nonlinear Schrödinger equation (NLS) on such geometries.

Background. Strichartz estimates are a fundamental tool for dispersive PDEs, providing space‑time integrability of solutions. On Euclidean space (\mathbb{R}^{d}) the classical estimates hold globally in time without loss, while on the torus (\mathbb{T}^{d}) one typically needs a frequency cut‑off or incurs an (\varepsilon)‑derivative loss. Waveguides combine both structures, so the dispersive effect from the Euclidean factor and the compactness of the torus factor interact in a non‑trivial way. Prior work (Barron 2020) gave global‑in‑time Strichartz estimates on waveguides but with an (\varepsilon)‑loss at the endpoint.

Main Linear Results.

1. Theorem 1.1 derives a family of long‑time estimates from the global ones by Hölder interpolation and dyadic frequency localisation. For any (p\ge2), any finite time interval (