集体动力学中一般Vicsek模型的Cauchy理论和平均场极限近似
Cauchy theory for general Vicsek models in collective dynamics and mean-field limit approximations
摘要：
在本文中，我们提供了一个局部柯西理论的环面和在整个空间的一般Vicsek动力学的动力学水平。我们考虑相当一般的相互作用核，非线性粘性和非线性摩擦。特别是，我们包括正常化的内核，显示一个奇点时，粒子的通量消失。因此，在柯西理论的动力学方程，我们扩展到更一般的相互作用和完成的程序开始在Gamba等。（2016）（作者假设奇点不会发生）和Figalli et.（2017）（作者证明了奇点不会发生在空间齐次的情况下）。此外，我们推导出一个明确的较低的存在时间，以及一个全球存在的标准，适用于，除其他情况下，获得一个长期的理论，非重正化内核和原始Vicsek问题，没有任何先验假设。在本文的第二部分，我们还建立了一个近似（正则化）系统，与原来的一个重合时，通量不消失的大粒子极限的平均场极限。基于对极限动力学方程的证明，我们证明了在短时间内，近似粒子系统的动力学与原奇异动力学重合的概率趋于多粒子极限中的一个.
英文摘要：
In this paper we provide a local Cauchy theory both on the torus and in the whole space for general Vicsek dynamics at the kinetic level. We consider rather general interaction kernels, nonlinear viscosity and nonlinear friction. Particularly, we include normalised kernels which display a singularity when the flux of particles vanishes. Thus, in terms of the Cauchy theory for the kinetic equation, we extend to more general interactions and complete the program initiated in Gamba et. al. (2016) (where the authors assume that the singularity does not take place) and in Figalli et. al. (2017) (where the authors prove that the singularity does not happen in the space homogeneous case). Moreover, we derive an explicit lower time of existence as well as a global existence criterion that is applicable, among other cases, to obtain a long time theory for non-renormalised kernels and for the original Vicsek problem without any a priori assumptions. On the second part of the paper, we also establish the mean-field limit in the large particle limit for an approximated (regularized) system that coincides with the original one whenever the flux does not vanish. Based on the results proved for the limit kinetic equation, we prove that for short times, the probability that the dynamics of this approximated particle system coincides with the original singular dynamics tends to one in the many particle limit.
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