A Diffusion Limit for the Parabolic Kuramoto--Sakaguchi Equation with Inertia

Abstract

In this paper, we study a macroscopic description on the ensemble of Kuramoto oscillators with finite inertia in a random media characterized by a white noise. In a mesoscopic regime, it is well known that the dynamics of a large Kuramoto ensemble in a random media is governed by the Kuramoto--Sakaguchi--Fokker--Planck (in short, parabolic Kuramoto--Sakaguchi) equation for one-oscillator distribution function. For this parabolic Kuramoto--Sakaguchi equation, we present a global existence of weak solutions in any finite-time interval. Furthermore, we rescale the kinetic equation using the diffusion scaling, and formally derive a drift-diffusion equation by using Hilbert-like expansion in a small parameter $\varepsilon$. For the rigorous justification of this asymptotic limit, we introduce a new free energy functional ${\mathcal E}$ consisting of total mass, kinetic energy, entropy functional, and interaction potential and show the uniform boundedness of this free energy with respect to the small parameter $\varepsilon$. This uniform boundedness of ${\mathcal E}$ combined with $L^1$-compactness argument enables us to derive the drift-diffusion equation. We also classified all ${\mathcal C}^2$-stationary solutions to the drift-diffusion equation in terms of synchronization parameters $\kappa$ and $\sigma$.