Stability of partially locked states in the Kuramoto model through Landau damping with Sobolev regularity

TL;DR

This paper proves nonlinear Landau damping for the Kuramoto model's partially locked states assuming Sobolev regularity, extending previous results that required analytic regularity, through a novel bootstrap approach.

Contribution

It introduces a new bootstrap method to establish nonlinear Landau damping under Sobolev regularity, broadening the class of initial conditions for stability analysis.

Findings

Nonlinear Landau damping is valid under Sobolev regularity.

A robust bootstrap argument is developed for the Volterra equation.

Stability results extend beyond analytic regularity assumptions.

Abstract

The Kuramoto model is a mean-field model for the synchronisation behaviour of oscillators, which exhibits Landau damping. In a recent work, the nonlinear stability of a class of spatially inhomogeneous stationary states was shown under the assumption of analytic regularity. This paper proves the nonlinear Landau damping under the assumption of Sobolev regularity. The weaker regularity required the construction of a different more robust bootstrap argument, which focuses on the nonlinear Volterra equation of the order parameter.