Global weak solutions for Kolmogorov-Vicsek type equations with orientational interactions

Abstract

We study the global existence and uniqueness of weak solutions to kinetic Kolmogorov-Vicsek models that can be considered a non-local non-linear Fokker-Planck type equation describing the dynamics of individuals with orientational interactions. This model is derived from the discrete Couzin-Vicsek algorithm as mean-field limit \cite{B-C-C,D-M}, which governs the interactions of stochastic agents moving with a velocity of constant magnitude, i.e. the the corresponding velocity space for these type of Kolmogorov-Vicsek models are the unit sphere. Our analysis for $L^p$ estimates and compactness properties take advantage of the orientational interaction property meaning that the velocity space is a compact manifold.

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• Phase transitions and macroscopic limits in a BGK model of body-attitude coordination
• Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles
• Derivation of the viscoelastic stress in Stokes flows induced by non-spherical Brownian rigid particles through homogenization
• Bulk topological states in a new collective dynamics model
• Dynamics of a spatially homogeneous Vicsek model for oriented particles on the plane
• Model reduction of a kinetic swarming model by operator projection
• Body-attitude coordination in arbitrary dimension
• Nematic alignment of self-propelled particles in the macroscopic regime
• Macroscopic limit of a Fokker-Planck model of swarming rigid bodies
• Cauchy theory for general Vicsek models in collective dynamics and mean-field limit approximations