Uniform Strichartz estimates on the lattice

Younghun Hong; Changhun Yang

doi:10.3934/dcds.2019134

Article Contents

2019, Volume 39, Issue 6: 3239-3264. Doi: 10.3934/dcds.2019134

This issue Previous Article On the well-posedness of the inviscid multi-layer quasi-geostrophic equations Next Article Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains

Uniform Strichartz estimates on the lattice

Younghun Hong^1, and
Changhun Yang^2,3, ,

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
* Corresponding author: Changhun Yang

Received: May 31, 2018

Published: February 2019

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 37K60.

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Figure 1. Strichartz estimates $(q,r)$ pair for $d = 3$

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Figure 2. Degenerate point in Fourier side for $ d = 2 $

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Figure 3. Domain in lattice and Fourier side for $ d = 2 $

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