Flocking With Short-Range Interactions

Abstract

We study the large-time behavior of continuum alignment dynamics based on Cucker-Smale (CS)-type interactions which involve short-range kernels, that is, communication kernels with support much smaller than the diameter of the crowd. We show that if the amplitude of the interactions is larger than a finite threshold, then unconditional hydrodynamic flocking follows. Since we do not impose any regularity nor do we require the kernels to be bounded, the result covers both regular and singular interaction kernels. Moreover, we treat initial densities in the general class of compactly supported measures which are required to have positive mass on average (over balls at small enough scale), but otherwise vacuum is allowed at smaller scales. Consequently, our arguments of hydrodynamic flocking apply, mutatis mutandis, to the agent-based CS model with finitely many Dirac masses. In particular, discrete flocking threshold is shown to depend on the number of dense clusters of communication but otherwise does not grow with the number of agents.Date: May 2, 2019. 1991 Mathematics Subject Classification. 92D25, 35Q35, 76N10.

On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity

“…This gives the conclusion that diam Ω(t ) remains uniformly bounded for all time (by N R 0 /2), and that the velocities align exponentially fast to a constant (following an argument of Morales, Peszek, and Tadmor [16], but with simplifications due to the a priori knowledge of the upper bound in Proposition 8). As happy a situation as this might seem on its face, the result is actually rather weak.…”

“…would be for Ω to be chain connected at scale R 0 /2 (in the sense described by Morales, Peszek, and Tadmor in [16]) and that some chain…”

Cucker–Smale Type Dynamics of Infinitely Many Individuals with Repulsive Forces

“…This is consistent with a sharp distinction between the macroscopic size of the system and the typical interaction length among the individuals. In this direction, we mention the recent paper [26], where, in the framework of mean-field approximation, the large time behavior of a continuum alignment dynamics based on CS-type interactions with finite-range kernel is studied.…”

Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

“…Both counterexamples can be ruled out assuming graph-connectivity of the flock at scale r 0 . If the flock is nearly aligned and initially connected such an assumption will propagate in time resulting in exponential alignment, see [25,30] and references therein.…”

“…Note that if a flock is dense at scale r ′ , it is dense at any larger scale r ′′ > r ′ , and every flock is trivially dense at the global scale r = diam Ω n . It is also clear that every part of a dense flock can be connected by a graph with legs of size < r. In fact, dense flocks are also automatically chain connected at scale r in the sense defined in [25]. If r = r 0 , where r 0 is the communication range (11), then clearly there exists a c > 0 depending only on δ, c 0 , r 0 such that…”