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 β∈(0,1). first establish global-in-time existence of measure-valued solutions to kinetic CS equation. For this, we use proper change variable reformulate particle as first-order system and provide uniform-in-time stability for that system. then extend this estimate By using estimate, construct equation globally in time. Moreover, direct application show quantitative mean-field limit from p-Wasserstein distance p∈[1,∞]. Our result gives uniqueness solution sense limits, i.e., solutions, approximated by empirical measures associated system, uniquely exist. Similar results also follow by-product. continuity-type equation, which is derived model, an integro-differential employing pseudo-inverse accumulative distribution. making modified distance, contractivity absolutely continuous continuum