Uniform-in-time asymptotic limits of generalized Kuramoto models

TL;DR

This paper establishes uniform-in-time asymptotic limits for generalized Kuramoto models, including continuum and mean-field limits, with stability estimates and convergence results that extend previous findings.

Contribution

It introduces uniform stability estimates and proves global-in-time convergence of solutions in both continuum and mean-field limits for generalized Kuramoto models.

Findings

Proved uniform-in-time stability estimates.

Established global-in-time existence of measure-valued solutions.

Demonstrated convergence of lattice solutions to continuum models.

Abstract

We study two uniform-in-time asymptotic limits for generalized Kuramoto (GK) models. For these GK type models, we first derive the uniform stability estimates with respect to initial data, natural frequency and communication network under a suitable framework, and then as direct applications of this uniform stability estimate, we establish two asymptotic limits which are valid in the whole time interval, namely uniform-in-time continuum and mean-filed limit to the continuum and kinetic GK models, respectively. In the mean-field limit setting (the number of particles tends to infinity), we show global-in-time existence of measure-valued solutions to the corresponding kinetic equation. On the other hand, in a continuum limit setting (the lattice size tends to zero), we show that the lattice GKM solutions converge to a classical solution to the continuum GK model in supremum norm. Two asymptotic limits improve earlier results for the generalized GK type models.