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

We analyze the one dimensional Cucker–Smale (in short CS) model with a weak singular communication weight ψ(x)=|x|^−β with β∈(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∈[1,∞]. 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.