Flocking With Short-Range Interactions
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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.
KeywordsAlignment Cucker–Smale Agent-based system Large-crowd hydrodynamics Interaction kernels Short-range Chain connectivity Flocking
Mathematics Subject Classification92D25 35Q35 76N10
Research was supported by NSF grants DMS16-13911, RNMS11-07444 (KI-Net) and ONR Grant N00014-1812465. JP was also supported by the Polish MNiSW grant Mobilność Plus no. 1617/MOB/V/2017/0.
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