2023/10/30 by Francis Filbet, Myeongju Kang, Filbet, Francis +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2310.19344
openalex publication_date 2023/10/30 · openalex created_date 2023/11/01 · openalex updated_date 2026/07/28
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.