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

¹ Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom,
carrillo@imperial.ac.uk
² Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom,
young-pil.choi@imperial.ac.uk
³ Centre de Mathématiques et Informatique (CMI), Université de Provence, Technopôle Château-Gombert, Marseille, France,
maxime.hauray@univ-amu.fr

Abstract

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) ∈ (L_loc^∞ ∩ Lip_loc) (R^d) 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.