On Cucker-Smale dynamical systems with degenerate communication

Abstract

This note introduces a new method for establishing alignment in systems of collective behavior with degenerate communication protocol. The communication protocol consists of a kernel defining interaction between pairs of agents. Degeneracy presumes that the kernel vanishes in a region, which creates a zone of indifference. A motivating example is the case of a local kernel. Lapses in communication create a lack of coercivity in the energy estimates. Our approach is the construction of a corrector functional that compensates for this lack of coercivity. We obtain a series of new results: unconditional alignment for systems on $\mathbb{R}^d$ with degeneracy at close range and fat tail in the long range, and for systems on the circle with purely local kernels. The results are proved in the context of both the agent based model and its hydrodynamic counterpart (Euler alignment model). The method covers bounded and singular communication kernels.