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We introduce a multi-dimensional variant of the kinetic Cucker-Smale modelwith singular and matrix-valued communication weight, which reduces to thesingular kinetic Cucker-Smale equation in the one-dimensional case. We proposean appropriate notion of weak measure-valued solution to this second-ordersystem and a suitable first-order reduction, which persist beyond the blow-uptime in classical norms. The core of the paper is to show that bothformulations are equivalent in this singular regime, thus extending theprevious results from regular to weakly singular communication weights. As aconsequence, we obtain: (i) global-in-time well-posedness of weakmeasure-valued solutions, (ii) quantitative convergence rates to equilibrium,and (iii) uniform-in-time mean-field limits thanks to the above-mentionedequivalence and our recent work on fibered gradient flows.

Measure solutions to a kinetic Cucker-Smale model with singular and matrix-valued communication