# Flocking with short-range interactions

## TL;DR

This paper analyzes the long-term behavior of continuum flocking models with short-range, possibly singular interactions, establishing conditions for unconditional flocking that apply to both continuum and agent-based models.

## Contribution

It proves unconditional hydrodynamic flocking for short-range kernels without regularity assumptions, extending results to singular kernels and general initial densities.

## Key findings

- Unconditional flocking occurs if interaction amplitude exceeds a finite threshold.
- Results apply to both continuum and agent-based models with finitely many agents.
- Flocking threshold depends on the number of dense communication clusters, not on the total number of agents.

## 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.