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

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• 作者：Debora Amadori^a ; ^{amadori@univaq.it" class="auth_mail" title="E-mail the corresponding author} ; Seung-Yeal Ha^b ; ^c ; ^{syha@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Jinyeong Park^d ; ^{pjy40@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：BV weak solution ; Continuous dependence ; The Kuramoto model ; The Kuramoto&ndash ; Sakaguchi equation ; Synchronization
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：262
• 期：2
• 页码：978-1022
• 全文大小：717 K

文摘

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 hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303291&_mathId=si1.gif&_user=111111111&_pii=S0022039616303291&_rdoc=1&_issn=00220396&md5=281448a2a9914e2b387fd85a7d374eb2" title="Click to view the MathML source">L¹hContainer hidden">hCode">h altimg="si1.gif" overflow="scroll">L1h>-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.