Long-time Strichartz estimates on 3D waveguide with applications

TL;DR

This work advances the analysis of dispersive equations on waveguides by deriving long-time Strichartz estimates for the Schrödinger equation on manifolds of the form $\mathbb{R}^m\times\mathbb{T}^n$, with precise bounds that depend on the Euclidean and torus components. Building on global-in-time estimates, the authors establish sharp $L^2\to L^4$ bounds for $p=4$ and develop improved long-time estimates for key 3D cases, enabling robust control of nonlinear interactions. These estimates are then leveraged to study the nonlinear Schrödinger equation on the 3D waveguide $\mathbb{R}\times\mathbb{T}^2$ with nonlinearity $|u|^{\mu-1}u$ ($3<\mu\le5$), yielding polynomial bounds on the growth of high Sobolev norms $\|u(t)\|_{H^s}$ for $s>1$ and establishing almost conservation laws in the energy-critical/ subcritical regimes. The results bridge Euclidean and compact geometry effects, with implications for long-time dynamics, scattering tendencies in mixed geometries, and the stability analysis of nonlinear waves in waveguide-like media.

Abstract

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.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6: Growth of higher Sobolev norms to NLS on 3D waveguide
• Remark 1.7
• proof : Proof of Theorem \ref{['thm:BCP RmTn']}
• proof : Proof of Theorem \ref{['thm:example']}
• proof : Proof of Theorem \ref{['thm:long-time Strichartz for p=4']}