Emergent Behaviors of Thermodynamic Cucker--Smale Particles

Ha, Seung‐Yeal; Kim, Jeong-ho; Ruggeri, Tommaso

doi:10.1137/17m111064x

Cited by 51 publications

(20 citation statements)

References 31 publications

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“…In their work, they derived an exponential flocking estimate for (1.1) as long as initial states satisfy a kind of small assumptions. Their work has been extended to several directions, e.g., nonexistence of mono-cluster flocking [31], time-delay effect [22], uniform stability and its kinetic limit [30], global well-posedness of the hydrodynamic TCS model in [29], coupling with fluids [12,13], disctete TCS model [28], although they are all dealing with TCS ensemble over the complete graph. As far as the authors know, this is the first work dealing with the emergent dynamics of TCS ensemble other than the complete graph.…”

Section: A Review Of Previous Resultsmentioning

confidence: 99%

Emergent behaviors of continuous and discrete thermomechanical Cucker–Smale models on general digraphs

Dong

Kim

2019

Math. Models Methods Appl. Sci.

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We present emergent dynamics of continuous and discrete thermomechanical Cucker-Smale(TCS) models equipped with temperature as an extra observable on general digraph. In previous literature, the emergent behaviors of the TCS models were mainly studied on a complete graph, or symmetric connected graphs. Under this symmetric setting, the total momentum is a conserved quantity. This determines the asymptotic velocity and temperature a priori using the initial data only. Moreover, this conservation law plays a crucial role in the flocking analysis based on the elementary ℓ 2 energy estimates. In this paper, we consider a more general connection topology which is registered by a general digraph, and the weights between particles are given to be inversely proportional to the metric distance between them. Due to this possible symmetry breaking in communication, the total momentum is not a conserved quantity, and this lack of conservation law makes the asymptotic velocity and temperature depend on the whole history of solutions. To circumvent this lack of conservation laws, we instead employ some tools from matrix theory on the scrambling matrices and some detailed analysis on the state-transition matrices. We present two sufficient frameworks for the emergence of mono-cluster flockings on a digraph for the continuous and discrete models. Our sufficient frameworks are given in terms of system parameters and initial data.Definition 1.1. [31, 35] Let C := (X, V, Θ) be a time-dependent configuration on the extended state space R N d × R N d × R d . Then the configuration C exhibits asymptotic mono-cluster flocking, if and only if the following conditions hold: sup 0≤t<∞ D(X(t)) < ∞, lim t→∞ D(V (t)) = 0, lim t→∞ D(Θ(t)) = 0. December 10, 2018 1:46 WSPC/INSTRUCTION FILE

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Section: A Review Of Previous Resultsmentioning

confidence: 99%

Emergent behaviors of continuous and discrete thermomechanical Cucker–Smale models on general digraphs

Dong

Kim

2019

Math. Models Methods Appl. Sci.

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“…In [12], the TCS model was derived based on a heuristic argument from the multitemperature fluid model via the space homogeneity assumption, Galilean invariance and entropy principle. In a series of papers by the first author and his collaborators, the TCS model has been extensively studied from various point of view, e.g., its emergent collective behavior [7,9,12], kinetic description and mean-field limit [7], hydrodynamic description [8], singular interaction kernels [1], coupled with other various kinds of fluids [2,3]. For notational simplicity, we set z := (x, v, θ), dz := dxdvdθ, ω := T d…”

Section: Moon-jin Kang Seung-yeal Ha Jeongho Kim and Woojoo Shimmentioning

confidence: 99%

“…Moreover, we are interested in the setting where particle's temperature fields are close to equilibrium temperature θ e pointwise. Note that the convergence rates of velocity and temperature fields toward equilibria are proportional to the coupling strengths (see [9,10,11,13] for TCS and CS models). In this situation, under the limit process (ε → 0), we can expect that f ε converges, in some weak sense, to a mono-kinetic distribution:…”

Section: Moon-jin Kang Seung-yeal Ha Jeongho Kim and Woojoo Shimmentioning

confidence: 99%

Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime

Kang¹,

Ha²,

Kim³

et al. 2020

Communications on Pure &Amp; Applied Analysis

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We present a hydrodynamic limit from the kinetic thermomechanical Cucker-Smale (TCS) model to the hydrodynamic Cucker-Smale (CS) model in a strong local alignment regime. For this, we first provide a global existence of weak solution, and flocking dynamics for classical solution to the kinetic TCS model with local alignment force. Then we consider one-parameter family of well-prepared initial data to the kinetic TCS model in which the temperature tends to common constant value determined by initial datum, as singular parameter ε tends to zero. In a strong local alignment regime, the limit model is the hydrodynamic CS model in [8]. To verify this hydrodynamic limit rigorously, we adopt the technique introduced in [5] which combines the relative entropy method together with the 2-Wasserstein metric.

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“…The TCS model was first introduced in [20] as a generalization of the CS flocking model in a self-consistent temperature field, since the TCS model reduces to the CS model when temperature field is constant. Until now, various topics on the TCS model have been studied, including the emergent behavior of microscopic TCS ensemble [18,19], the mean-field limit [18], well-posedness and emergent dynamics of hydrodynamic model [17], discrete time model [15], etc.…”

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confidence: 99%

Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain

Kim¹,

Zou²

2020

Kinetic &Amp; Related Models

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We study local existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale (in short TCS) model coupled with incompressible Navier-Stokes (NS) equations in the whole space. The coupled system consists of the kinetic TCS equation for particle ensemble and the incompressible NS equations for a fluid via a drag force. For the strong solution, we investigate the blow-up mechanism for the coupled system, and we also study the global existence of a weak solution in the whole space.

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Emergent Behaviors of Thermodynamic Cucker--Smale Particles

Cited by 51 publications

References 31 publications

Emergent behaviors of continuous and discrete thermomechanical Cucker–Smale models on general digraphs

Emergent behaviors of continuous and discrete thermomechanical Cucker–Smale models on general digraphs

Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime

Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain

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