Eulerian dynamics with a commutator forcing

Shvydkoy, Roman; Tadmor, Eitan

doi:10.1093/imatrm/tnx001

Cited by 64 publications

(142 citation statements)

References 19 publications

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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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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Flocking With Short-Range Interactions

2019

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“…It has been studied recently that the so called strongly singular interaction has a regularization effect, which prevents the solution from finite time singularity formations. In 1D, global regularity is obtained in [9] for s ∈ (1, 2), and in [17] for s ∈ [2, 3) through a different approach. 9,17]).…”

Section: • If Infmentioning

confidence: 99%

“…In 1D, global regularity is obtained in [9] for s ∈ (1, 2), and in [17] for s ∈ [2, 3) through a different approach. 9,17]). Consider the 1D Euler-Alignment system (5)-(6) with smooth periodic initial data (ρ 0 , G 0 ).…”

Section: • If Infmentioning

confidence: 99%

On the Euler-alignment system with weakly singular communication weights

Tan

2020

Nonlinearity

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We study the pressureless Euler equations with nonlocal alignment interactions, which arises as a macroscopic representation of complex biological systems modeling animal flocks. For such Euler-Alignment system with bounded interactions, a critical threshold phenomenon is proved in [18], where global regularity depends on initial data. With strongly singular interactions, global regularity is obtained in [9], for all initial data. We consider the remaining case when the interaction is weakly singular. We show a critical threshold, similar as the system with bounded interaction. However, different global behaviors may happen for critical initial data, which reveals the unique structure of the weakly singular alignment operator.

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“…The significance of condition (3) lies in the fact that such kernels prevent collisions between agents and consequently, the discrete system (1) is well-posed even though the right hand side is not Lipschitz. This issue has received extensive treatment in works of Peszek et al [2,13,14,15], and [21,22,20,6] for the Euler-alignment system. A quantitative expression of non-collision is an integral part of our approach, so we will revisit the question in Section 2.1 below.…”

Section: Introductionmentioning

confidence: 99%

“…All our results pertaining to (2) hold for a given strong solution. We note, however, that in a variety of situations, both for smooth and singular kernels, such solutions have been constructed, see [1,6,21,22,20,23,24]. In particular, the 1D case is completely understood, and small initial data results are known in multi-D case, [5,7,9,23,19].…”

Section: Introductionmentioning

confidence: 99%

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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Eulerian dynamics with a commutator forcing

Cited by 64 publications

References 19 publications

Flocking With Short-Range Interactions

Flocking With Short-Range Interactions

On the Euler-alignment system with weakly singular communication weights

On Cucker–Smale dynamical systems with degenerate communication

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