Critical thresholds in 1D Euler equations with non-local forces

Carrillo, José A.; Choi, Young Pil; Tadmor, Eitan; Tan, Changhui

doi:10.1142/s0218202516500068

Cited by 118 publications

(119 citation statements)

References 20 publications

(27 reference statements)

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“…We refer to the recent reviews [7,14] for the detailed descriptions of the modeling and related literature. There are several works on the pressureless Euler alignment system; the global regularity of classical solutions is obtained in the Eulerian formulation [22] and in the Lagrangian formulation [15,23], and critical thresholds between the supercritical regions with finite-time breakdown and the subcritical region with global-in-time regularity of classical solutions are investigated in one dimension [3,8,33]. More recently, the global regularity for the pressureless fractional Euler alignment system is also established in [19,28,30,31].…”

Section: Introductionmentioning

confidence: 99%

The global Cauchy problem for compressible Euler equations with a nonlocal dissipation

Choi

2019

Math. Models Methods Appl. Sci.

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This paper studies the global existence and uniqueness of strong solutions and its largetime behavior for the compressible isothermal Euler equations with a nonlocal dissipation. The system is rigorously derived from the kinetic Cucker-Smale flocking equation with strong local alignment forces and diffusions through the hydrodynamic limit based on the relative entropy argument. In a perturbation framework, we establish the global existence of a unique strong solution for the system under suitable smallness and regularity assumptions on the initial data. We also provide the large-time behavior of solutions showing the fluid density and the velocity converge to its averages exponentially fast as time goes to infinity.

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Section: Introductionmentioning

confidence: 99%

The global Cauchy problem for compressible Euler equations with a nonlocal dissipation

Choi

2019

Math. Models Methods Appl. Sci.

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“…Further previous results related to systems of the form (1.1) include long-time asymptotics and critical thresholds without pressure terms in one dimension [13,14] and weak-strong uniqueness and nonuniqueness results [16] for the corresponding Euler equations. As far as the Navier-Stokes-Poisson (NSP) system is concerned, various questions like existence, long-time behavior, and stability of solutions have been studied during the last 20 years.…”

Section: Introductionmentioning

confidence: 93%

On long-time asymptotics for viscous hydrodynamic models of collective behavior with damping and nonlocal interactions

Carrillo

Wróblewska-Kamińska

Zatorska

2019

Math. Models Methods Appl. Sci.

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Hydrodynamic systems arising in swarming modelling include nonlocal forces in the form of attractive-repulsive potentials as well as pressure terms modelling strong local repulsion. We focus on the case where there is a balance between nonlocal attraction and local pressure in presence of confinement in the whole space. Under suitable assumptions on the potentials and the pressure functions, we show the global existence of weak solutions for the hydrodynamic model with viscosity and linear damping. By introducing linear damping in the system, we ensure the existence and uniqueness of stationary solutions with compactly supported density, fixed mass and center of mass. The associated velocity field is zero in the support of the density. Moreover, we show that global weak solutions converge for large times to the set of these stationary solutions in a suitable sense. In particular cases, we can identify the limiting density uniquely as the global minimizer of the free energy with the right mass and center of mass.

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“…This is due to the fact that η t ceases to be a diffeomorphism. The critical threshold phenomena for flocking models are studied in [2,14]. In fact, in that case, the constant C is given by…”

Section: )mentioning

confidence: 99%

Hydrodynamic Cucker--Smale Model with Normalized Communication Weights and Time Delay

Choi¹,

Haškovec

2019

SIAM J. Math. Anal.

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We study a hydrodynamic Cucker-Smale-type model with time delay in communication and information processing, in which agents interact with each other through normalized communication weights. The model consists of a pressureless Euler system with time delayed non-local alignment forces. We resort to its Lagrangian formulation and prove the existence of its global in time classical solutions. Moreover, we derive a sufficient condition for the asymptotic flocking behavior of the solutions. Finally, we show the presence of a critical phenomenon for the Eulerian system posed in the spatially onedimensional setting.for t ≥ 0 and x ∈ R d with d ∈ N the space dimension. The constant τ ≥ 0 denotes the fixed delay in communication and information processing. Here and in the sequel we denote by the subscript {·} t the time-dependence of the respective variable. The influence function ψ : R d → R + satisfies the following set of assumptions:

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Critical thresholds in 1D Euler equations with non-local forces

Cited by 118 publications

References 20 publications

The global Cauchy problem for compressible Euler equations with a nonlocal dissipation

The global Cauchy problem for compressible Euler equations with a nonlocal dissipation

On long-time asymptotics for viscous hydrodynamic models of collective behavior with damping and nonlocal interactions

Hydrodynamic Cucker--Smale Model with Normalized Communication Weights and Time Delay

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