Local well-posedness of the generalized Cucker-Smale model

2014/06/09 by José A. Carrillo, Carrillo, J. A., Young-Pil Choi +3
Computer Science · Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Biology Tumor Growth #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1406.1792

openalex publication_date 2014/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

In this paper, we study the local well-posedness of two types of generalized Cucker-Smale (in short C-S) flocking models. We consider two different communication weights, singular and regular ones, with nonlinear coupling velocities v|v|^β-2 for β> (3-d)/(2). For the singular communication weight, we choose ψ¹(x) = 1/|x|^α with α∈ (0,d-1) and β≥ 2 in dimension d > 1. For the regular case, we select ψ²(x) ≥ 0 belonging to (L_loc^∞ ∩ Lip_loc)(ℝ^d) and β∈ ((3-d)/(2),2). We also remark the various dynamics of C-S particle system for these communication weights when β∈ (0,3).

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