Trend to Equilibrium and Diffusion Limit for the Inertial Kuramoto–Sakaguchi Equation

Abstract. 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 [Choi et al., SIAM J. Math. Anal., 53 (2021), pp. 3188–3235], 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 [ Ha, Shim, and Zhang, SIAM J. Math. Anal., 52 (2020), pp. 1591–1638 ], 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 [Formula: see text] using a simple proof by imposing slightly more regularity on the solution.