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

Analysis of PDEs