On the global well-posedness of BV weak solutions to the Kuramoto-Sakaguchi equation

Title
On the global well-posedness of BV weak solutions to the Kuramoto-Sakaguchi equation
Author
박진영
Keywords
BV weak solution; Continuous dependence; The Kuramoto model; The Kuramoto-Sakaguchi equation; Synchronization
Issue Date
2017-01
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Citation
JOURNAL OF DIFFERENTIAL EQUATIONS, v. 262, no. 2, page. 978-1022
Abstract
The Kuramoto model is a prototype phase model describing the synchronous behavior of weakly coupled limit-cycle oscillators. When the number of oscillators is sufficiently large, the dynamics of Kuramoto ensemble can be effectively approximated by the corresponding mean-field equation, namely "the Kuramoto-Sakaguchi (KS) equation". This KS equation is a kind of scalar conservation law with a nonlocal flux function due to the mean-field interactions among oscillators. In this paper, we provide a unique global solvability of bounded variation (BV) weak solutions to the kinetic KS equation for identical oscillators using the method of front-tracking in hyperbolic conservation laws. Moreover, we also show that our BV weak solutions satisfy local-in-time L-1-stability with respect to BV-initial data. For the ensemble of identical Kuramoto oscillators, we explicitly construct an exponentially growing BV weak solution generated from BV perturbation of incoherent state for any positive coupling strength. This implies the nonlinear instability of incoherent state in a positive coupling strength regime. We provide several numerical examples and compare them with our analytical results. (C) 2016 Elsevier Inc. All rights reserved.
URI
https://www.sciencedirect.com/science/article/pii/S0022039616303291?via%3Dihubhttp://repository.hanyang.ac.kr/handle/20.500.11754/106086
ISSN
0022-0396; 1090-2732
DOI
10.1016/j.jde.2016.10.004
Appears in Collections:
COLLEGE OF NATURAL SCIENCES[S](자연과학대학) > MATHEMATICS(수학과) > Articles
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