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.
|Number of pages||48|
|Journal||SIAM Journal on Mathematical Analysis|
|Publication status||Published - 2021|
Bibliographical noteFunding 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).
© 2021 Society for Industrial and Applied Mathematics
All Science Journal Classification (ASJC) codes
- Computational Mathematics
- Applied Mathematics