The Cucker–Smale Equation: Singular Communication Weight, Measure-Valued Solutions and Weak-Atomic Uniqueness

Abstract

The Cucker–Smale flocking model belongs to a wide class of kinetic models that describe a collective motion of interacting particles that exhibit some specific tendency, e.g. to aggregate, flock or disperse. This paper examines the kinetic Cucker–Smale equation with a singular communication weight. Given a compactly supported measure as an initial datum we construct a global in time weak measure-valued solution in the space Cweak(0 , ∞; M). The solution is defined as a mean-field limit of the empirical distributions of particles, the dynamics of which is governed by the Cucker–Smale particle system. The studied communication weight is ψ(s) = | s| -α with α∈(0,12). This range of singularity admits the sticking of characteristics/trajectories. The second result concerns the weak-atomic uniqueness property stating that a weak solution initiated by a finite sum of atoms, i.e. Dirac deltas in the form miδxi⊗δvi, preserves its atomic structure. Hence these coincide with unique solutions to the system of ODEs associated with the Cucker–Smale particle system.