Stability of Collision-Avoiding Initial Configurations for Cucker-Smale Model

Abstract

In this thesis, we present the explicit calculation of collision-avoiding initial configurations for multi-particle Cucker-Smale system when the communi- cation weight ψ equals to 1. Between the two types of ψ: ψ_1(r) = r^(−β) and ψ_2(r) = (1 + r^2)^(−β/2) for r ≥ 0, we consider the latter. It is known that the particles form a flocking behavior when β ∈ [0, 1]. We extend the result to the case when β ∈ (0, 1), i.e., ψ ̸= 1 . We show that ψ is stable against the perturbation of β, i.e. the collision-avoiding initial positions and velocities surprisingly do not change much.