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

TL;DR

This paper proves the global existence and uniqueness of weak solutions for a class of kinetic equations modeling collective behavior with orientational interactions, derived from agent-based models on the sphere.

Contribution

It establishes the first rigorous analysis of weak solutions for Kolmogorov-Vicsek type equations with non-local interactions on the sphere.

Findings

Proved global existence of weak solutions.

Established uniqueness of solutions.

Analyzed $L^p$ estimates and compactness properties.

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.