A diffusion limit for the parabolic kuramoto-sakaguchi equation with inertia

In this paper, we study a macroscopic description on the ensemble of Kuramoto os- cillators 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 ϵ. For the rigorous justification of this asymptotic limit, we introduce a new free energy functional ϵ 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 ϵ. This uniform boundedness of E combined with L¹-compactness argument en- ables us to derive the drift-diffusion equation. We also classified all C²-stationary solutions to the drift-diffusion equation in terms of synchronization parameters κ and σ.