On Cucker-Smale dynamical systems with degenerate communication
14 Apr 2019
•
Dietert Helge
•
Shvydkoy Roman
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.(read more)