Global Weak Solutions for Kolmogorov–Vicsek Type Equations with Orientational Interactions

We study the global existence and uniqueness of weak solutions to kinetic Kolmogorov–Vicsekmodels that canbe considered as non-local, non-linear, Fokker– Planck type equations describing the dynamics of individuals with orientational interactions. This model is derived from the discrete Couzin–Vicsek algorithm as mean-field limit (Bolley et al., Appl Math Lett, 25:339–343, 2012; Degond et al., Math Models Methods Appl Sci 18:1193–1215, 2008), which governs the interactions of stochastic agents moving with a velocity of constant magnitude, that is, the corresponding velocity space for these types of Kolmogorov–Vicsek models is 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.