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

doi:10.3934/nhm.2013.8.943

Article Contents

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

This issue Previous Article Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures Next Article Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes

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: November 30, 2012

Revised: May 31, 2013

Published: November 2013

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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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Mathematics Subject Classification: Primary: 92D25, 76N10; Secondary: 74A25.

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