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. 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 $\alpha \in (0,\frac{1}{2})$. 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.