On weak solutions to the kinetic Cucker–Smale model with singular communication weights

Abstract

We establish the local-in-time existence of weak solutions to the kinetic Cucker–Smale model with singular communication weights ϕ ( x ) = | x | − α \phi (x) = |x|^{-\alpha } with α ∈ ( 0 , d ) \alpha \in (0,d) . In the case α ∈ ( 0 , d − 1 ] \alpha \in (0, d-1] , we also provide the uniqueness of weak solutions extending the work of Carrillo et al [MMCS, ESAIM Proc. Surveys, vol. 47, EDP Sci., Les Ulis, 2014, pp. 17–35] where the existence and uniqueness of weak solutions are studied for α ∈ ( 0 , d − 1 ) \alpha \in (0,d-1) .