Cauchy theory and mean-field limit for general Vicsek models in collective dynamics
Cauchy theory and mean-field limit for general Vicsek models in collective dynamics
复制标题
DOI：
--
复制
发表时间：
2020-04
期刊：
影响因子：
--
通讯作者：
M. Briant;S. Merino-Aceituno
中科院分区：
文献类型：
--
作者：
M. Briant;S. Merino-Aceituno
本文在粒子和动力学水平上给出了环面和整个空间的局部柯西理论。我们考虑一般的相互作用核，非线性黏度和非线性摩擦。我们还在大粒子极限中建立了平均场极限。特别地，我们包括了在粒子通量消失时显示奇点的归一化核。因此，就动力学方程的柯西理论而言，我们将扩展到更一般的相互作用，并完成Gamba等人（2016年）（作者假设奇点不发生）和Figalli等人（2017年）（作者证明奇点在空间齐次情况下不发生）中启动的程序。此外，我们推导了一个明确的较低存在时间以及一个适用于其他情况的全局存在准则，该准则在没有任何先验假设的情况下获得了非重整核和原始Vicsek问题的长时间理论。
In this paper we provide a local Cauchy theory both on the torus and in the whole space for general Vicsek dynamics at the particle and at the kinetic level. We consider rather general interaction kernels, nonlinear viscosity and nonlinear friction. We also establish the mean-field limit in the large particle limit. 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.