# American Institute of Mathematical Sciences

June  2019, 39(6): 3239-3264. doi: 10.3934/dcds.2019134

## Uniform Strichartz estimates on the lattice

 1 Department of Mathematics, Chung-Ang University, Seoul 06974, Republic of Korea 2 Korea Institute for Advanced Study, Seoul 20455, Korea 3 Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 54896, Korea

* Corresponding author: Changhun Yang

Received  June 2018 Published  February 2019

Fund Project: This Research of the first author was supported by the Chung-Ang University Research Grants in 2018. The second author was supported in part by Samsung Science and Technology Foundation under Project Number SSTF-BA1702-02

In this paper, we investigate Strichartz estimates for discrete linear Schrödinger 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 Schrödinger and one-dimensional discrete Klein-Gordon equations are established by Stefanov-Kevrekidis [21]. 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 Schrödinger 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 [7] where strong convergence for a discrete nonlinear Schrödinger equation is addressed.

Citation: Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134
##### References:

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##### References:
Strichartz estimates $(q,r)$ pair for $d = 3$
Degenerate point in Fourier side for $d = 2$
Domain in lattice and Fourier side for $d = 2$
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