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A Diffusion Limit for the Parabolic Kuramoto--Sakaguchi Equation with Inertia
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2020-04-02 , DOI: 10.1137/19m1237454
Seung-Yeal Ha , Woojoo Shim , Yinglong Zhang

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1591-1638, January 2020.
In this paper, we study a macroscopic description on the ensemble of Kuramoto oscillators with finite inertia in a random media characterized by a white noise. In a mesoscopic regime, it is well known that the dynamics of a large Kuramoto ensemble in a random media is governed by the Kuramoto--Sakaguchi--Fokker--Planck (in short, parabolic Kuramoto--Sakaguchi) equation for one-oscillator distribution function. For this parabolic Kuramoto--Sakaguchi equation, we present a global existence of weak solutions in any finite-time interval. Furthermore, we rescale the kinetic equation using the diffusion scaling, and formally derive a drift-diffusion equation by using Hilbert-like expansion in a small parameter $\varepsilon$. For the rigorous justification of this asymptotic limit, we introduce a new free energy functional ${\mathcal E}$ consisting of total mass, kinetic energy, entropy functional, and interaction potential and show the uniform boundedness of this free energy with respect to the small parameter $\varepsilon$. This uniform boundedness of ${\mathcal E}$ combined with $L^1$-compactness argument enables us to derive the drift-diffusion equation. We also classified all ${\mathcal C}^2$-stationary solutions to the drift-diffusion equation in terms of synchronization parameters $\kappa$ and $\sigma$.


中文翻译:

具有惯性的抛物型Kuramoto-Sakaguchi方程的扩散极限。

SIAM数学分析杂志,第52卷,第2期,第1591-1638页,2020年1月。
在本文中,我们研究了在具有白噪声特征的随机介质中具有有限惯性的Kuramoto振荡器的整体的宏观描述。在介观状态下,众所周知,在一个随机介质中,一个大型仓本集合的动力学是由一元振荡器的仓本-坂口-福克-普朗克(简而言之,抛物线式仓本-坂口)方程控制的。分配功能。对于此抛物线型Kuramoto-Sakaguchi方程,我们提出了在任何有限时间间隔内弱解的整体存在。此外,我们使用扩散比例缩放对动力学方程式进行缩放,并使用小参数$ \ varepsilon $中的希尔伯特式展开形式来正式推导漂移扩散方程式。为了严格证明渐近极限,我们引入了一个新的自由能函数$ {\ mathcal E} $,该函数由总质量,动能,熵函数和相互作用势组成,并显示了该自由能相对于小参数$ \ varepsilon $的均匀有界性。$ {\ mathcal E} $的统一有界性与$ L ^ 1 $ -compactness自变量相结合使我们能够导出漂移扩散方程。我们还根据同步参数$ \ kappa $和$ \ sigma $对漂移扩散方程的所有$ {\ mathcal C} ^ 2 $平稳解进行了分类。
更新日期:2020-04-02
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