Local well-posedness of the generalized Cucker-Smale model

In this paper, we study the local well-posedness of two types of generalized Cucker-Smale (in short C-S) flocking models. We consider two different communication weights, singular and regular ones, with nonlinear coupling velocities $v|v|^{β-2}$ for $β> \frac{3-d}{2}$. For the singular communication weight, we choose $ψ^1(x) = 1/|x|^α$ with $α\in (0,d-1)$ and $β\geq 2$ in dimension $d > 1$. For the regular case, we select $ψ^2(x) \geq 0$ belonging to $(L_{loc}^\infty \cap \mbox{Lip}_{loc})(\mathbb{R}^d)$ and $β\in (\frac{3-d}{2},2)$. We also remark the various dynamics of C-S particle system for these communication weights when $β\in (0,3)$.

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