The Cucker-Smale equation: singular communication weight, measure-valued solutions and weak-atomic uniqueness

TL;DR

This paper studies the kinetic Cucker-Smale model with a singular communication weight, constructing global measure-valued solutions and proving the uniqueness of solutions initiated by finite atomic measures, linking them to particle system dynamics.

Contribution

It introduces a framework for global weak measure solutions with singular weights and establishes atomic solution uniqueness, connecting measure solutions to particle system ODEs.

Findings

Constructed global weak measure solutions for singular weights.

Proved weak-atomic solutions preserve their atomic structure.

Linked measure solutions to particle system ODEs.

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. The 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 $C_{weak}(0,\infty;\mathcal{M})$. The solution is defined as a mean-field limit of the empirical distributions of particles, which dynamics is governed by the Cucker-Smale particle system. The studied communication weight is $\psi(s)=|s|^{-\alpha}$ with $\alpha \in (0,\frac 12)$. This range of singularity admits 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 $m_i \delta_{x_i} \otimes \delta_{v_i}$, preserves its atomic structure. Hence they coincide with unique solutions to the system of ODEs associated with the Cucker-Smale particle system.