Discrete Cucker--Smale Flocking Model with a Weakly Singular Weight

Peszek, Jan

doi:10.1137/15m1009299

Cited by 66 publications

(67 citation statements)

References 30 publications

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“…Note that it is well-posed for β < 3 (in view also of the fact that the density is bounded for regular solutions). Computation similar to the discrete case proves (15), and from this point on the proof is exactly the same.…”

Section: Hydrodynamic Systemsmentioning

confidence: 60%

On Cucker–Smale dynamical systems with degenerate communication

Dietert

Shvydkoy

2020

Anal. Appl.

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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 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.

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Section: Hydrodynamic Systemsmentioning

confidence: 60%

On Cucker–Smale dynamical systems with degenerate communication

Dietert

Shvydkoy

2020

Anal. Appl.

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“…Lemma 5.2 Let ρ ε , u ε be a regular solution of (55), satisfying (56) and (57). Then there exists constants C and C(ε) such that…”

Section: Comparison With the Pm Equation -Theoretical Resultsmentioning

confidence: 99%

“…For n ≥ −1, on each interval [t n ,t n+1 ] (we assume that t −1 = 0) we consider the problem Idea of the proof. The proof of existence can be found in [56], while the proof of uniqueness can be found in [57]. Existence outside of times of collision is straightforward.…”

Section: Collision-avoidancementioning

confidence: 99%

Singular Cucker–Smale Dynamics

Minakowski

Mucha

Peszek

et al. 2019

Active Particles, Volume 2

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The existing state of the art for singular models of flocking is overviewed, starting from microscopic model of Cucker and Smale with singular communication weight, through its mesoscopic mean-filed limit, up to the corresponding macroscopic regime. For the microscopic Cucker-Smale (CS) model, the collision-avoidance phenomenon is discussed, also in the presence of bonding forces and the decentralized control. For the kinetic mean-field model, the existence of global-in-time measure-valued solutions, with a special emphasis on a weak atomic uniqueness of solutions is sketched. Ultimately, for the macroscopic singular model, the summary of the existence results for the Euler-type alignment system is provided, including existence of strong solutions on one-dimensional torus, and the extension of this result to higher dimensions upon restriction on the smallness of initial data. Additionally, the pressureless Navier-Stokes-type system corresponding to particular choice of alignment kernel is presented, and compared -analytically and numerically -to the porous medium equation.

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“…We specifically address the case of short-range interactions D φ ≪ D S 0 . Moreover, since we do not impose any boundedness of φ, (1.8) includes both -bounded communication kernels, [4,5,12,11,2,17], and singular ones [19,20,21,23,24,25,7].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Flocking With Short-Range Interactions

2019

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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.Date: May 2, 2019. 1991 Mathematics Subject Classification. 92D25, 35Q35, 76N10.

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Discrete Cucker--Smale Flocking Model with a Weakly Singular Weight

Cited by 66 publications

References 30 publications

On Cucker–Smale dynamical systems with degenerate communication

On Cucker–Smale dynamical systems with degenerate communication

Singular Cucker–Smale Dynamics

Flocking With Short-Range Interactions

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