Volume 47, December 2014MMCS, Mathematical Modelling of Complex Systems
|Page(s)||17 - 35|
|Published online||18 December 2014|
Local well-posedness of the generalized Cucker-Smale model with singular kernels
1 Department of Mathematics, Imperial
College London, London
SW7 2AZ, United
2 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom,
3 Centre de Mathématiques et Informatique (CMI), Université de Provence, Technopôle Château-Gombert, Marseille, France,
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.
© EDP Sciences, SMAI 2014
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