Bilinear Strichartz estimates on rescaled waveguide manifolds with applications

TL;DR

This paper establishes new bilinear Strichartz estimates for Schrödinger equations on rescaled waveguide manifolds, leading to improved well-posedness and scattering results for nonlinear Schrödinger equations in mixed Euclidean-torus spaces.

Contribution

It introduces global and local bilinear Strichartz estimates on rescaled waveguides, filling gaps in existing theory and enabling lower regularity well-posedness results.

Findings

Global-in-time bilinear Strichartz estimates with $N_2^ ext{epsilon}$ loss

Local angularly refined bilinear estimates on 2D waveguides

Well-posedness and scattering results for nonlinear Schrödinger equations

Abstract

We focus on the bilinear Strichartz estimates for free solutions to the Schr\"odinger equation on rescaled waveguide manifolds $\mathbb{R} \times \mathbb{T}_\lambda^n$, $\mathbb{T}_\lambda^n=(\lambda\mathbb{T})^n$ with $n\geq 1$ and their applications. First, we utilize a decoupling-type estimate originally from Fan-Staffilani-Wang-Wilson [Anal. PDE 11 (2018)] to establish a global-in-time bilinear Strichartz estimate with a `$N_2^\epsilon$' loss on $\mathbb{R} \times \mathbb{T}^n_\lambda$ when $n\geq1$, which generalize the local-in time estimate in Zhao-Zheng [SIAM J. Math. Anal. (2021)] and fills a gap left by the unresolved case in Deng et al. [J. Func. Anal. 287 (2024)]. Second, we prove the local-in-time angularly refined bilinear Strichartz estimates on the 2d rescaled waveguide $\mathbb{R} \times \mathbb{T}_\lambda$. As applications, we show the local well-posedness and small data scattering for nonlinear Schr\"odinger equations with algebraic nonlinearities in the critical space on $\mathbb R^m\times\mathbb{T}^n$ and the global well-posedness for cubic NLS on $\mathbb{R} \times \mathbb{T}$ in the lower regularity space $H^s$ with $s>\frac{1}{2}$.