Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia

Young-Pil Choi; Seung-Yeal Ha; Seok-Bae Yun; Young-Pil Choi; Seung-Yeal Ha; Seok-Bae Yun

doi:10.3934/nhm.2013.8.943

Networks and Heterogeneous Media

2013, Volume 8, Issue 4: 943-968. doi: 10.3934/nhm.2013.8.943

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Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia

1.
Department of Mathematics, Imperial College London, London SW7 2AZ
2.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747
3.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747

Received: 01 December 2012 Revised: 01 June 2013
Primary: 92D25, 76N10; Secondary: 74A25.

We present the global existence and long-time behavior of measure-valued solutions to the kinetic Kuramoto--Daido model with inertia. For the global existence of measure-valued solutions, we employ a Neunzert's mean-field approach for the Vlasov equation to construct approximate solutions. The approximate solutions are empirical measures generated by the solution to the Kuramoto--Daido model with inertia, and we also provide an a priori local-in-time stability estimate for measure-valued solutions in terms of a bounded Lipschitz distance. For the asymptotic frequency synchronization, we adopt two frameworks depending on the relative strength of inertia and show that the diameter of the projected frequency support of the measure-valued solutions exponentially converge to zero.
- Kuramoto-Dadio model,
- well-posedness,
- measure-valued solutions,
- Neunzert's mean-field approach,
- Vlasov equation,
- large time behavior
Citation: Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia[J]. Networks and Heterogeneous Media, 2013, 8(4): 943-968. doi: 10.3934/nhm.2013.8.943

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Abstract

We present the global existence and long-time behavior of measure-valued solutions to the kinetic Kuramoto--Daido model with inertia. For the global existence of measure-valued solutions, we employ a Neunzert's mean-field approach for the Vlasov equation to construct approximate solutions. The approximate solutions are empirical measures generated by the solution to the Kuramoto--Daido model with inertia, and we also provide an a priori local-in-time stability estimate for measure-valued solutions in terms of a bounded Lipschitz distance. For the asymptotic frequency synchronization, we adopt two frameworks depending on the relative strength of inertia and show that the diameter of the projected frequency support of the measure-valued solutions exponentially converge to zero.

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