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We analyze the one dimensional Cucker-Smale (in short CS) model with a weaksingular communication weight $\psi(x) = |x|^{-\beta}$ with $\beta \in (0,1)$.We first establish a global-in-time existence of measure-valued solutions tothe kinetic CS equation. For this, we use a proper change of variable toreformulate the particle CS model as a first-order particle system and providethe uniform-in-time stability for that particle system. We then extend thisstability estimate for the singular CS particle system. By using that stabilityestimate, we construct the measure-valued solutions to the kinetic CS equationglobally in time. Moreover, as a direct application of the uniform-in-timestability estimate, we show the quantitative uniform-in-time mean-field limitfrom the particle system to that kinetic CS equation in $p$-Wassersteindistance with $p \in [1,\infty]$. Our result gives the uniqueness ofmeasure-valued solution in the sense of mean-field limits, i.e., themeasure-valued solutions, approximated by the empirical measures associated tothe particle system, uniquely exist. Similar results for the first-order modelalso follow as a by-product. We also reformulate the continuity-type equation,which is derived from the first-order model, as an integro-differentialequation by employing the pseudo-inverse of the accumulative particledistribution. By making use of a modified $p$-Wasserstein distance, we providethe contractivity estimate for absolutely continuous solutions of the continuumequation.

One dimensional singular Cucker-Smale model: uniform-in-time mean-field limit and contractivity