Local well-posedness of the generalized Cucker-Smale model with singular kernels

• Main Authors: Carrillo José A., Choi Young-Pil, Hauray Maxime
• Format: Article
• Language: English
• Published: EDP Sciences 2014-12-01
• Series: ESAIM: Proceedings and Surveys
• Online Access: http://dx.doi.org/10.1051/proc/201447002

In this paper, we study the local well-posedness of two types of generalized kinetic Cucker-Smale (in short C-S) equations. We consider two different communication weights in space with nonlinear coupling of the velocities, v | v | β − 2 for β > 3-d/2, where singularities are present either in space or in velocity. For the singular communication weight in space, ψ1(x) = 1 / | x | α with α ∈ (0,d − 1), d ≥ 1, we consider smooth velocity coupling, β ≥ 2. For the regular one, we assume ψ2(x) ∈ (Lloc∞ ∩ Liploc) (Rd) but with a singular velocity coupling β ∈ (3-d/2, 2). We also present the various dynamics of the generalized C-S particle system with the communication weights ψi,i = 1,2 when β ∈ (0,3). We provide sufficient conditions of the initial data depending on the exponent β leading to finite-time alignment or to no collisions between particles in finite time.