SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 3188-3235, January 2021.
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.

中文翻译：

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Kuramoto-Sakaguchi方程相齐解的渐近稳定性与惯性

SIAM 数学分析杂志，第 53 卷，第 3 期，第 3188-3235 页，2021 年 1 月。
我们提出了围绕相位齐次解的强解在时间上的全局存在性和唯一性，以及它对具有惯性的仓本-坂口方程的大时间行为。我们的控制方程描述了在惯性和随机噪声的影响下，大量 Kuramoto 振荡器的概率密度函数的演变。在本文中，我们围绕麦克斯韦型平衡采用微扰框架，并使用经典能量方法并基于微扰分解进行仔细分析。当噪声强度足够大时，我们建立了具有大量初始数据的强解的全局时间存在性和唯一性。对于大时间行为，