Relaxation dynamics of the continuum Kuramoto model with non-integrable kernels

TL;DR

This paper rigorously analyzes the long-term behavior of the continuum Kuramoto model with singular kernels, establishing existence of solutions and exponential relaxation under certain conditions.

Contribution

It introduces a novel regularization approach to construct global weak solutions for the singular continuum Kuramoto model with fractional dissipation.

Findings

Existence of global weak solutions via regularization and compactness.

Exponential relaxation toward the initial phase average in $L^2$-norm.

Applicability to physically relevant singular kernels like power-law and Coulomb types.

Abstract

We study the asymptotic behavior of the continuum Kuramoto model with a fractional Laplacian-type kernel. For this, we construct global weak solutions via a two-parameter regularization procedure using a kernel truncation with fractional dissipation. Using a priori uniform estimates derived in fractional Sobolev spaces, we employ compactness arguments to construct global weak solutions to the singular continuum Kuramoto model. Furthermore, we also establish an exponential relaxation toward the initial phase average in $L^2$-norm under suitable assumptions on initial data and system parameters. These findings provide a rigorous characterization of the existence of solutions and the emergent dynamics of Kuramoto ensembles under physically important strongly singular interactions, including power-law singular kernels and Coulomb-type kernels.