| Summary: | 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.
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