CAUCHY THEORY FOR GENERAL KINETIC VICSEK MODELS INCOLLECTIVE DYNAMICS AND MEAN-FIELD LIMITAPPROXIMATIONS

In this paper we provide a local Cauchy theory both on the torus and in the wholespace for general Vicsek dynamics at the kinetic level. We consider rather general interaction kernels,nonlinear viscosity, and nonlinear friction. Particularly, we include normalized kernels which displaya singularity when the flux of particles vanishes. Thus, in terms of the Cauchy theory for the kineticequation, we extend to more general interactions and complete the program initiated in [I. M. Gambaand M.-J. Kang,Arch. Ration. Mech. Anal., 222 (2016), pp. 317--342] (where the authors assumethat the singularity does not take place) and in [A. Figalli, M.-J. Kang, and J. Morales,Arch. Ration.Mech. Anal., 227 (2018), pp. 869--896] (where the authors prove that the singularity does not happenin the spatially homogeneous case). Moreover, we derive an explicit lower time of existence as wellas a global existence criterion that is applicable, among other cases, to obtain a long time theory fornonrenormalized kernels and for the original Vicsek problem without any a priori assumptions. Onthe second part of the paper, we also establish the mean-field limit in the large particle limit for anapproximated (regularized) system that coincides with the original one whenever the flux does notvanish. 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 originalsingular dynamics tends to one in the many particle limit.