Singularity formation for the fractional Euler-alignment system in 1D

Arnaiz, Victor; Castro, Ángel

doi:10.1090/tran/8228

Cited by 11 publications

(10 citation statements)

References 20 publications

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“…The alignment term on the right hand side of the velocity equation in (1) is then given by the following singular integral operator:…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…In view of no-vacuum condition (ρ 0 > 0) necessary to develop a well-posedness theory (c.f. [1], [28]), we consider the periodic domain T n , where a uniform lower bound on the density is compatible with finite mass. When dealing with the n-dimensional torus, the term (2) can be expressed in terms of the periodized kernel φ α (z) :=…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Global existence and limiting behavior of unidirectional flocks for the fractional Euler Alignment system

Lear¹

2021

Preprint

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In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels φ(x) := |x| −(n+α) for α ∈ (0, 2). Here, we consider the critical case α = 1 and establish a couple of global existence results of smooth solutions, together with a full description of their long time dynamics. The first one is obtained via Schauder-type estimates under a null initial entropy condition and the other is a small data result. In fact, using Duhamel's approach we get that any solution is almost Lipschitz-continuous in space. We extend the notion of weak solution for α ∈ [1, 2) and prove the existence of global Leray-Hopf solutions. Furthermore, we give an anisotropic Onsager-type criteria for the validity of the natural energy law for weak solutions of the system. Finally, we provide a series of quantitative estimates that show how far the density of the limiting flock is from a uniform distribution depending solely on the size of the initial entropy.This system represents a multi-dimensional hydrodynamic version of the celebrated Cucker-Smale flocking model introduced in [7, 8], with a huge number of applications ranging from biology or robotics to social sciences, see [4,29,21] for recent surveys and references therein. The system (1) is designed to describe the interaction between agents governed by laws of self-organization with communication protocol encoded into the kernel φ.In many realistic situations and applications, the communication among agents takes place in local neighborhoods induced by short-range kernels. Particularly interesting is the case of singular communication kernels. Clearly, singularity at the origin strongly emphasizes local communication.For models with singular kernels given by φ(x) := |x| −(n+α) for 0 < α < 2 the operator L φ ≡ L α becomes the (negative of) classical fractional Laplacian:R n f (y) − f (x) |x − y| n+α dy Λ α := (−∆) α/2 , 0 < α < 2.

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“…The alignment term on the right hand side of the velocity equation in (1) is then given by the following singular integral operator:…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Global existence and limiting behavior of unidirectional flocks for the fractional Euler Alignment system

Lear¹

2021

Preprint

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“…The corresponding alignment term on the right hand side of the momentum equation in (1) is then given by the following singular integral:…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…In [16,17,18] Tadmor and the second author proved global existence of smooth solutions for the one-dimensional system (1) with alignment term given by (3) in the full range 0 < α < 2, with focus on the most difficult critical case α = 1. In addition, the authors proved in [17] that all regular solutions converge exponentially fast to a so called flocking state, consisting of a traveling wave, ρ(x, t) = ρ ∞ (x − tū), with a fixed speed ū,…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Unidirectional flocks in hydrodynamic Euler Alignment system II: Singular models

Lear¹,

Shvydkoy²

2020

Preprint

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In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels φ(x) := |x| −(n+α) for α ∈ (0, 2). The solutions describe unidirectional parallel motion of agents governing multi-dimensional collective behavior of flocks. Here, we consider the range 1 < α < 2 and establish the global regularity of smooth solutions, together with a full description of their long time dynamics. Specifically, we develop the flocking theory of these solutions and show long time convergence to traveling wave with rapidly aligned velocity field.

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Finite- and infinite-time cluster formation for alignment dynamics on the real line

Leslie,

Tan

2024

J. Evol. Equ.

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Singularity formation for the fractional Euler-alignment system in 1D

Abstract: We study the formation of singularities for the Euler-alignment system with influence function ψ = k α | x | 1 + α \psi =\frac {k_\alpha }{|x|^{1+\alpha }} in 1D. As in [Commun. Math. Sci. 17 (2019), pp. 1779–1794] the problem is reduc… Show more

Cited by 11 publications

References 20 publications

Global existence and limiting behavior of unidirectional flocks for the fractional Euler Alignment system

Global existence and limiting behavior of unidirectional flocks for the fractional Euler Alignment system

Unidirectional flocks in hydrodynamic Euler Alignment system II: Singular models

Finite- and infinite-time cluster formation for alignment dynamics on the real line

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