Wook Yoon | Searcharxiv

arXiv2026

Uniform-in-time stability and mean-field limit of the Cucker-Smale-type model with noncompact support

We study the uniform-in-time stability and mean-field limit of an infinite Cucker--Smale-type (ICS-type) model in a fully noncompact setting. First, we clarify the main difficulties arising from the original ICS model and its corresponding kinetic Cucker--Smale (KCS) model, when the spatial support is noncompact. In particular, we show that the velocity diameter may remain constant in time, which invalidates the classical approach based on position and velocity diameters. Moreover, we construct a counterexample showing that the original KCS model does not satisfy uniform-in-time stability in the noncompact spatial setting. To overcome these obstacles, we introduce a new ICS-type model whose communication weight depends on a pairwise spatial moment. Under suitable assumptions, we also establish the uniform-in-time stability of the ICS-type model with respect to initial data in the noncompact setting. Finally, we derive the uniform-in-time mean-field limit for the ICS-type model, together with uniform-in-time stability for the corresponding KCS-type model.

math.AP↗

arXiv2024

Mono-cluster flocking and uniform-in-time stability of the discrete Motsch-Tadmor model

The Motsch-Tadmor (MT) model is a variant of the Cucker-Smale model with a normalized communication weight function. The normalization poses technical challenges in analyzing the collective behavior due to the absence of conservation of momentum. We study three quantitative estimates for the discrete-time MT model considering the first-order Euler discretization. First, we provide a sufficient framework leading to the asymptotic mono-cluster flocking. The proposed framework is given in terms of coupling strength, communication weight function, and initial data. Second, we show that the continuous transition from the discrete MT model to the continuous MT model can be made uniformly in time using the finite-time convergence result and asymptotic flocking estimate. Third, we present uniform-in-time stability estimates for the discrete MT model. We also provide several numerical examples and compare them with analytical results.

math.NA↗