Local well-posedness of the generalized Cucker-Smale model

Abstract

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 $v|v{|}^{\beta -2}$ for $\beta >\frac{3-d}{2}$. For the singular communication weight, we choose ${\psi }^1(x)=1/|x{|}^{\alpha }$ with $\alpha \in (0,d-1)$ and $\beta \ge 2$ in dimension $d>1$. For the regular case, we select ${\psi }^2(x)\ge 0$ belonging to $(L_{loc}^\infty \cap \mbox{Lip}_{loc})(\mathbb{R}^d)$ and $\beta \in (\frac{3-d}{2},2)$. We also remark the various dynamics of C-S particle system for these communication weights when $\beta \in (0,3)$.

References in corpus (2)

• Novel type of phase transition in a system of self-driven particles
• Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight

Cited by in corpus (1)

• Discrete Cucker-Smale's flocking model with a weakly singular weight