A local sensitivity analysis for the kinetic Kuramoto equation with random inputs

Seung-Yeal Ha; Shi Jin; Jinwook Jung

doi:10.3934/nhm.2019013

Article Contents

2019, Volume 14, Issue 2: 317-340. Doi: 10.3934/nhm.2019013

This issue Previous Article Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions Next Article Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

A local sensitivity analysis for the kinetic Kuramoto equation with random inputs

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
2.
Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Republic of Korea
3.
School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
4.
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

^* Corresponding author: Jinwook Jung
^* Corresponding author: Jinwook Jung

Received: April 30, 2018

Revised: December 31, 2018

Published: April 2019

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Abstract

We present a local sensivity analysis for the kinetic Kuramoto equation with random inputs in a large coupling regime. In our proposed random kinetic Kuramoto equation (in short, RKKE), the random inputs are encoded in the coupling strength. For the deterministic case, it is well known that the kinetic Kuramoto equation exhibits asymptotic phase concentration for well-prepared initial data in the large coupling regime. To see a response of the system to the random inputs, we provide propagation of regularity, local-in-time stability estimates for the variations of the random kinetic density function in random parameter space. For identical oscillators with the same natural frequencies, we introduce a Lyapunov functional measuring the phase concentration, and provide a local sensitivity analysis for the functional.

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Mathematics Subject Classification: Primary: 35Q82, 35Q92, 37H99.

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