Asymptotic stability of the phase-homogeneous solution to the Kuramoto-Sakaguchi equation with inertia

Young Pil Choi; Seung Yeal Ha; Qinghua Xiao; Yinglong Zhang

doi:10.1137/20M1368719

Asymptotic stability of the phase-homogeneous solution to the Kuramoto-Sakaguchi equation with inertia

Young Pil Choi, Seung Yeal Ha, Qinghua Xiao, Yinglong Zhang

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We present global-in-time existence and uniqueness of strong solutions around a phase-homogeneous solution, and its large-time behavior for the Kuramoto-Sakaguchi equation with inertia. Our governing equation describes the evolution of the probability density function for a large ensemble of Kuramoto oscillators under the effects of inertia and stochastic noises. In this paper, we take a perturbative framework around the Maxwellian type equilibrium and use the classical energy method together with careful analysis based on the decomposition of the perturbation. We establish the global-in-time existence and uniqueness of strong solutions with large initial data when the noise strength is large enough. For the large-time behavior, we show the exponential decay of solutions toward the equilibrium under the same assumptions as those for the global solutions.

Original language	English
Pages (from-to)	3188-3235
Number of pages	48
Journal	SIAM Journal on Mathematical Analysis
Volume	53
Issue number	3
DOIs	https://doi.org/10.1137/20M1368719
Publication status	Published - 2021

Bibliographical note

Funding Information:
\ast Received by the editors September 22, 2020; accepted for publication (in revised form) March 16, 2021; published electronically June 3, 2021. https://doi.org/10.1137/20M1368719 Funding: The work of the first author was supported by National Research Foundation of Korea grants 2017R1C1B2012918 and 2017R1A4A1014735, the POSCO Science Fellowship of the POSCO TJ Park Foundation, and the Yonsei University Research Fund of 2019-22-0212. The work of the second author was supported by National Research Foundation of Korea grant 2020R1A2C3A01003881. The work of the third author was supported by Youth Innovation Promotion Association CAS grant (2017379) and National Natural Science Foundation of China grant 11871469. The work of the fourth author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1A5A1028324).

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics

All Science Journal Classification (ASJC) codes

Analysis
Computational Mathematics
Applied Mathematics

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10.1137/20M1368719

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title = "Asymptotic stability of the phase-homogeneous solution to the Kuramoto-Sakaguchi equation with inertia",

abstract = "We present global-in-time existence and uniqueness of strong solutions around a phase-homogeneous solution, and its large-time behavior for the Kuramoto-Sakaguchi equation with inertia. Our governing equation describes the evolution of the probability density function for a large ensemble of Kuramoto oscillators under the effects of inertia and stochastic noises. In this paper, we take a perturbative framework around the Maxwellian type equilibrium and use the classical energy method together with careful analysis based on the decomposition of the perturbation. We establish the global-in-time existence and uniqueness of strong solutions with large initial data when the noise strength is large enough. For the large-time behavior, we show the exponential decay of solutions toward the equilibrium under the same assumptions as those for the global solutions.",

author = "Choi, {Young Pil} and Ha, {Seung Yeal} and Qinghua Xiao and Yinglong Zhang",

note = "Funding Information: \ast Received by the editors September 22, 2020; accepted for publication (in revised form) March 16, 2021; published electronically June 3, 2021. https://doi.org/10.1137/20M1368719 Funding: The work of the first author was supported by National Research Foundation of Korea grants 2017R1C1B2012918 and 2017R1A4A1014735, the POSCO Science Fellowship of the POSCO TJ Park Foundation, and the Yonsei University Research Fund of 2019-22-0212. The work of the second author was supported by National Research Foundation of Korea grant 2020R1A2C3A01003881. The work of the third author was supported by Youth Innovation Promotion Association CAS grant (2017379) and National Natural Science Foundation of China grant 11871469. The work of the fourth author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1A5A1028324). Publisher Copyright: {\textcopyright} 2021 Society for Industrial and Applied Mathematics",

year = "2021",

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AU - Choi, Young Pil

AU - Ha, Seung Yeal

AU - Xiao, Qinghua

AU - Zhang, Yinglong

N1 - Funding Information: \ast Received by the editors September 22, 2020; accepted for publication (in revised form) March 16, 2021; published electronically June 3, 2021. https://doi.org/10.1137/20M1368719 Funding: The work of the first author was supported by National Research Foundation of Korea grants 2017R1C1B2012918 and 2017R1A4A1014735, the POSCO Science Fellowship of the POSCO TJ Park Foundation, and the Yonsei University Research Fund of 2019-22-0212. The work of the second author was supported by National Research Foundation of Korea grant 2020R1A2C3A01003881. The work of the third author was supported by Youth Innovation Promotion Association CAS grant (2017379) and National Natural Science Foundation of China grant 11871469. The work of the fourth author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1A5A1028324). Publisher Copyright: © 2021 Society for Industrial and Applied Mathematics

PY - 2021

Y1 - 2021

N2 - We present global-in-time existence and uniqueness of strong solutions around a phase-homogeneous solution, and its large-time behavior for the Kuramoto-Sakaguchi equation with inertia. Our governing equation describes the evolution of the probability density function for a large ensemble of Kuramoto oscillators under the effects of inertia and stochastic noises. In this paper, we take a perturbative framework around the Maxwellian type equilibrium and use the classical energy method together with careful analysis based on the decomposition of the perturbation. We establish the global-in-time existence and uniqueness of strong solutions with large initial data when the noise strength is large enough. For the large-time behavior, we show the exponential decay of solutions toward the equilibrium under the same assumptions as those for the global solutions.

AB - We present global-in-time existence and uniqueness of strong solutions around a phase-homogeneous solution, and its large-time behavior for the Kuramoto-Sakaguchi equation with inertia. Our governing equation describes the evolution of the probability density function for a large ensemble of Kuramoto oscillators under the effects of inertia and stochastic noises. In this paper, we take a perturbative framework around the Maxwellian type equilibrium and use the classical energy method together with careful analysis based on the decomposition of the perturbation. We establish the global-in-time existence and uniqueness of strong solutions with large initial data when the noise strength is large enough. For the large-time behavior, we show the exponential decay of solutions toward the equilibrium under the same assumptions as those for the global solutions.

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Asymptotic stability of the phase-homogeneous solution to the Kuramoto-Sakaguchi equation with inertia

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