Particle and Kinetic Models for Swarming Particles on a Sphere and Stability Properties
Particle and Kinetic Models for Swarming Particles on a Sphere and Stability Properties
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DOI：
10.1007/s10955-018-2169-8
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发表时间：
2018-11
影响因子：
1.6
通讯作者：
Seung‐Yeal Ha;Dohyun Kim;Jaeseung Lee;S. E. Noh
中科院分区：
文献类型：
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作者：
Seung‐Yeal Ha;Dohyun Kim;Jaeseung Lee;S. E. Noh
我们提出了粒子和动力学模型的描述在随机噪声的存在下，在一个球体上的群集粒子，并研究其稳定性。在没有噪声的情况下，所提出的粒子模型可以从用于量子同步的洛赫矩阵模型简化，并且动力学模型可以使用BBGKY层次从粒子群集模型形式地导出。对于不含噪声的粒子模型，我们证明了它在Lebesgue范数下关于初值是一致稳定的。这种一致稳定性和粒子在细胞的方法产生的整体存在的测量值的解决方案，相应的无粘动力学模型。我们还表明，非相干态的动力学模型是非线性稳定的，只要噪声强度和耦合强度之间的比率是足够大的。
We present particle and kinetic models for the description of swarming particles on a sphere in the presence of random noises, and study their stability properties. In the absence of noises, the proposed particle model can be reduced from the Lohe matrix model for quantum synchronization, and the kinetic model can be formally derived from the particle swarming model using the BBGKY hierarchy. For the particle model without noises, we show that it is uniformly stable with respect to initial data in a Lebesgue norm. This uniform stability and particle-in-cell method yield a global existence of a measure-valued solution to the corresponding inviscid kinetic model. We also show that the incoherent state for the kinetic model is nonlinearly stable, as long as the ratio between noise strength and coupling strength is sufficiently large.