On the rigorous derivation of hydrodynamics of the Kuramoto model for synchronization phenomena

Young-Pil Choi ()
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Young-Pil Choi: Yonsei University

Partial Differential Equations and Applications, 2023, vol. 4, issue 1, 1-20

Abstract: Abstract We study the rigorous derivation of hydrodynamics of the Kuramoto model for synchronization phenomena, introduced by Choi and Lee (Math Models Methods Appl Sci 30: 2175–2227, 2020), which is pressureless Euler equations with nonlocal interaction forces. We present two different ways of deriving that hydrodynamic model. We first discuss the asymptotic analysis for the inertial kinetic Kuramoto equation with a strong local frequency alignment force. We show that a weak solution to the kinetic equation converges to the classical solution of that hydrodynamic synchronization model under certain assumptions on the initial data. We also provide the derivation from the particle Kuramoto model with inertia as the number of oscillators goes to infinity in the mono-kinetic case. Our proofs are based on a modulated energy-type estimate combined with the bounded Lipschitz distance between local densities.

Keywords: Hydrodynamics of the Kuramoto model; Inertia; Synchronization; Mean-field limit; Hydrodynamic limit; 35Q70; 34C15; 35Q83; 35Q35 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-022-00219-7

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