Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flow

TL;DR

This paper proves the global existence, uniqueness, and exponential convergence to steady state of solutions for a space-homogeneous nonlinear Fokker-Planck equation modeling self-driven particles with orientation interaction.

Contribution

It establishes the well-posedness and long-term behavior of solutions for the spatially homogeneous Kolmogorov-Vicsek model, a significant step in understanding its dynamics.

Findings

Global existence and uniqueness of weak solutions

Exponential convergence to Fisher-von Mises distribution

Steady state characterized by a specific distribution

Abstract

We consider the so-called spatially homogenous Kolmogorov-Vicsek model, a non-linear Fokker-Planck equation of self-driven stochastic particles with orientation interaction under the space-homogeneity. We prove the global existence and uniqueness of weak solutions to the equation. We also show that weak solutions exponentially converge to a steady state, which has the form of the Fisher-von Mises distribution.