Contractivity of the Wasserstein metric for the kinetic Kuramoto equation

TL;DR

This paper analyzes the kinetic Kuramoto model, demonstrating synchronization and exponential contraction in Wasserstein distance for identical oscillators, extending previous L1-contraction results.

Contribution

It provides new synchronization and contractivity estimates for the kinetic Kuramoto model, including a general exponential contraction principle in Wasserstein distance.

Findings

Complete synchronization for certain initial data

Exponential decay of Wasserstein p-distance between solutions

Extension beyond previous L1-contraction results

Abstract

We present synchronization and contractivity estimates for the kinetic Kuramoto model obtained from the Kuramoto phase model in the mean-field limit. For identical Kuramoto oscillators, we present an admissible class of initial data leading to time-asymptotic complete synchronization, that is, all measure valued solutions converge to the traveling Dirac measure concentrated on the initial averaged phase. If two initial Radon measures have the same natural frequency density function and strength of coupling, we show that the Wasserstein p-distance between corresponding measure valued solutions is exponentially decreasing in time. This contraction principle is more general than previous L1-contraction properties of the Kuramoto phase model.