Hydrodynamic limit of the Kuramoto-Sakaguchi equation with inertia and noise effects

TL;DR

This paper investigates the hydrodynamic limit of a nonlinear parabolic integro-differential equation modeling coupled oscillators with inertia and noise, and proves a Hardy-type inequality as part of the analysis.

Contribution

It establishes the hydrodynamic limit for the Kuramoto-Sakaguchi equation with inertia and noise, and proves a Hardy-type inequality on the real line.

Findings

Hydrodynamic limit derived for the Kuramoto-Sakaguchi equation with inertia and noise

Proved a Hardy-type inequality over the whole real line

Provides mathematical foundation for understanding collective synchronization phenomena

Abstract

We consider the Kuramoto-Sakaguchi-Fokker-Planck equation (namely, parabolic Kuramoto-Sakaguchi, or Kuramoto-Sakaguchi equation, which is a nonlinear parabolic integro-differential equation) with inertia and white noise effects. We study the hydrodynamic limit of this Kuramoto-Sakaguchi equation. During showing this main result, as a support, we also prove a Hardy-type inequality over the whole real line.