Uniform Strichartz estimates on the lattice

TL;DR

This paper establishes uniform Strichartz estimates for discrete Schr"odinger and Klein-Gordon equations on a lattice, with implications for the continuum limit and nonlinear equations.

Contribution

It proves uniform Strichartz estimates on the lattice with fractional derivatives, advancing the understanding of discrete models and their continuum limits.

Findings

Uniform Strichartz estimates hold for all h in (0,1]

Application to local well-posedness of discrete nonlinear Schr"odinger equations

Framework useful for studying continuum limits of discrete models

Abstract

In this paper, we investigate Strichartz estimates for discrete linear Schr\"odinger and discrete linear Klein-Gordon equations on a lattice $h\mathbb{Z}^d$ with $h>0$, where $h$ is the distance between two adjacent lattice points. As for fixed $h>0$, Strichartz estimates for discrete Schr\"odinger and one-dimensional discrete Klein-Gordon equations are established by Stefanov-Kevrekidis \cite{SK2005}. Our main result shows that such inequalities hold uniformly in $h\in(0,1]$ with additional fractional derivatives on the right hand side. As an application, we obtain local well-posedness of a discrete nonlinear Schr\"odinger equation with a priori bounds independent of $h$. The theorems and the harmonic analysis tools developed in this paper would be useful in the study of the continuum limit $h\to 0$ for discrete models, including our forthcoming work \cite{HY} where strong convergence for a discrete nonlinear Schr\"odinger equation is addressed.