Geodesic
Local well-posedness of the generalized Cucker-Smale model with singular kernels
José A. Carrillo1 ; Young-Pil Choi1 ; Maxime Hauray2
1 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom,
2 Centre de Mathématiques et Informatique (CMI), Université de Provence, Technopôle Château-Gombert, Marseille, France,
ESAIM. Proceedings, Tome 47 (2014), pp. 17-35.

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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.
DOI : 10.1051/proc/201447002

José A. Carrillo 1 ; Young-Pil Choi 1 ; Maxime Hauray 2

1 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom,
2 Centre de Mathématiques et Informatique (CMI), Université de Provence, Technopôle Château-Gombert, Marseille, France, 
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     title = {Local well-posedness of the generalized {Cucker-Smale} model with singular kernels},
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José A. Carrillo; Young-Pil Choi; Maxime Hauray. Local well-posedness of the generalized Cucker-Smale model with singular kernels. ESAIM. Proceedings, Tome 47 (2014), pp. 17-35. doi : 10.1051/proc/201447002. http://geodesic.mathdoc.fr/articles/10.1051/proc/201447002/

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