Trend to equilibrium and diffusion limit for the inertial Kuramoto-Sakaguchi equation

Francis Filbet (1) , Myeongju Kang (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 KIAS - Korea Institute for Advanced Study

In this paper, we study the inertial Kuramoto-Sakaguchi equation for interacting oscillatory systems. On the one hand, we prove the convergence toward corresponding phase-homogeneous stationary states in weighted Lebesgue norm sense when the coupling strength is small enough. In [10], it is proved that when the noise intensity is sufficiently large, equilibrium of the inertial Kuramoto-Sakaguchi equation is asymptotically stable. For generic initial data, every solutions converges to equilibrium in weighted Sobolev norm sense. We improve this previous result by showing the convergence for a larger class of functions and by providing a simpler proof. On the other hand, we investigate the diffusion limit when all oscillators are identical. In [19], authors studied the same problem using an energy estimate on renormalized solutions and a compactness method, through which error estimates could not be discussed. Here we provide error estimates for the diffusion limit with respect to the mass m ≪ 1 using a simple proof by imposing slightly more regularity on the solution.

Domaines: Equations aux dérivées partielles [math.AP]

Francis Filbet, Myeongju Kang. Trend to equilibrium and diffusion limit for the inertial Kuramoto-Sakaguchi equation. 2023. ⟨hal-04263367v2⟩