On the weak flocking of the kinetic Cucker-Smale model in a fully non-compact support setting

Seung-Yeal Ha; Xinyu Wang

arxiv: 2507.07493 · v3 · submitted 2025-07-10 · 🧮 math.AP

On the weak flocking of the kinetic Cucker-Smale model in a fully non-compact support setting

Seung-Yeal Ha , Xinyu Wang This is my paper

Pith reviewed 2026-05-19 06:06 UTC · model grok-4.3

classification 🧮 math.AP

keywords Cucker-Smale modelkinetic equationweak flockingnon-compact supportmoment estimatesasymptotic behavioremergent dynamics

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The pith

The kinetic Cucker-Smale model exhibits asymptotic flocking for weak solutions with non-compact support when distributions decay exponentially or polynomially in phase space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the challenge of proving flocking in the kinetic Cucker-Smale model when the support of the distribution is not compact in space and velocity. In such cases, the communication weight may lose its positive lower bound and spatial differences may not grow linearly, breaking standard diameter-based arguments. The authors overcome this by establishing refined upper bounds on second-order moments of spatial and velocity deviations from the center of mass and mean velocity. They also prove uniqueness of weak solutions by controlling the deviation of particle trajectories from the mean flow. For initial data with exponential or polynomial decay, these bounds imply that velocity deviations vanish asymptotically while spatial deviations remain bounded, showing flocking is robust beyond compact support.

Core claim

For distribution functions with exponential or polynomial decay in phase space, the second moment for the velocity deviation from an average velocity tends to zero asymptotically, while the second moment for spatial deviation from the center of mass remains bounded uniformly in time. This holds for weak solutions of the kinetic Cucker-Smale model in a fully non-compact support setting.

What carries the argument

Refined estimates on the upper bounds for the second-order spatial-velocity moments, together with the estimate on the deviation of particle trajectories.

If this is right

Mono-cluster flocking dynamics persist in non-compact settings.
Uniqueness of weak solutions follows from the trajectory deviation estimates.
The result generalizes earlier flocking theorems that required compact support.
Both exponential and polynomial decay classes yield the same qualitative behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Decay conditions appear sufficient to control the long-time behavior without needing bounded support.
Similar moment methods might extend to other nonlocal alignment models with singular interactions.
Bounded spatial moments suggest the center of mass moves in a controlled way over infinite time.

Load-bearing premise

The initial distribution functions must possess exponential or polynomial decay in phase space for the refined moment estimates to close and yield the asymptotic flocking conclusion.

What would settle it

A counterexample distribution with exponential decay where the velocity second moment fails to approach zero or where the spatial second moment grows without bound would disprove the emergent flocking claim.

read the original abstract

We study the emergent behaviors of the weak solutions to the kinetic Cucker-Smale (in short, KCS) model in a non-compact spatial-velocity support setting. Unlike the compact support situation, non-compact support of a weak solution can cause a communication weight to have zero lower bounds, and position difference does not have a uniformly linear growth bound. These cause the previous approach based on the nonlinear functional approach for spatial and velocity diameters to break down. To overcome these difficulties, we derive refined estimates on the upper bounds for the second-order spatial-velocity moments and show the uniqueness of the weak solution using the estimate on the deviation of particle trajectories. For the estimate of emergent dynamics, we consider two classes of distribution functions with decaying properties (an exponential decay or polynomial decay) in phase space, and then verify that the second moment for the velocity deviation from an average velocity tends to zero asymptotically, while the second moment for spatial deviation from the center of mass remains bounded uniformly in time. This illustrates the robustness of the mono-cluster flocking dynamics of the KCS model even for fully non-compact support settings in phase space and generalizes earlier results on flocking dynamics in a compact support setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They get flocking for non-compact supports in the kinetic Cucker-Smale model via new second-moment bounds and trajectory uniqueness, but the decay classes need to be shown to persist in time.

read the letter

The main thing to know is that this paper pushes flocking analysis for the kinetic Cucker-Smale equation into the fully non-compact support regime. Previous methods using nonlinear functionals for diameters break down because communication weights can hit zero and positions lack uniform linear growth. The authors replace those with direct upper bounds on the second spatial and velocity moments, then add a uniqueness result from controlling how particle trajectories deviate. That combination lets them treat general weak solutions first before specializing to initial data with exponential or polynomial decay in phase space, where they conclude that velocity second moment goes to zero while spatial second moment stays bounded. The estimates come from the equation itself rather than assumed quantities, so there is no obvious circularity. The logic is straightforward: moment control first, then asymptotic behavior for the decaying cases. This is a clean technical step that generalizes the compact-support theorems without forcing extra structure. A soft spot worth checking is whether the assumed decay rates survive for positive times. The transport and nonlocal alignment terms can spread mass, and the abstract does not spell out an a-priori argument that the exponential or polynomial tails remain controlled along the flow. If the moment bounds close this loop, the conclusion holds; if they only work under an extra standing assumption, the result is narrower than it first appears. The gap is not fatal but it is the part that needs the clearest write-up. This is aimed at people already working on kinetic models for alignment and swarming. Anyone following the Cucker-Smale literature or related multi-agent kinetic equations will find the new estimates useful. It is solid enough on the technical side to go to a serious referee, with the decay propagation question as the main item for revision.

Referee Report

1 major / 1 minor

Summary. The manuscript studies weak solutions of the kinetic Cucker-Smale equation in the fully non-compact phase-space support regime. It derives refined a-priori upper bounds on the second moments of spatial and velocity deviations from the center of mass and mean velocity, establishes uniqueness of weak solutions via estimates on particle-trajectory deviations, and, for initial data belonging to exponential-decay or polynomial-decay classes, proves that the velocity-deviation second moment tends to zero while the spatial-deviation second moment remains uniformly bounded, thereby obtaining asymptotic mono-cluster flocking.

Significance. If the decay-class invariance is established, the work provides a meaningful extension of flocking theory beyond compact-support assumptions, where diameter functionals cease to be useful. The moment-based approach is a clear technical strength and avoids circularity by deriving estimates directly from the kinetic equation.

major comments (1)

[Asymptotic analysis for decaying distributions] Asymptotic flocking section (following the moment estimates): the restriction to initial data with exponential or polynomial decay is invoked to conclude that ∫|v−v̄|² f dx dv → 0 as t→∞. However, the transport and nonlocal alignment terms can spread tails; the manuscript does not supply an a-priori proof that the assumed decay rate persists uniformly in time or that an integrable tail bound follows from the derived moment estimates. This leaves a gap between the general moment bounds and the time-asymptotic conclusion for the stated classes.

minor comments (1)

[Introduction] The abstract states that the communication weight may have zero lower bounds; a brief remark on how the moment estimates circumvent this without invoking a positive lower bound would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment point by point below.

read point-by-point responses

Referee: Asymptotic flocking section (following the moment estimates): the restriction to initial data with exponential or polynomial decay is invoked to conclude that ∫|v−v̄|² f dx dv → 0 as t→∞. However, the transport and nonlocal alignment terms can spread tails; the manuscript does not supply an a-priori proof that the assumed decay rate persists uniformly in time or that an integrable tail bound follows from the derived moment estimates. This leaves a gap between the general moment bounds and the time-asymptotic conclusion for the stated classes.

Authors: We thank the referee for this observation. The second-moment bounds derived directly from the kinetic equation already control the spreading induced by transport and alignment, and for the exponential/polynomial classes these bounds suffice to obtain the required tail integrability that closes the asymptotic argument. Nevertheless, to make the invariance of the decay class fully explicit and remove any ambiguity, we will insert a short additional lemma (or appendix paragraph) proving uniform persistence of the decay rates from the moment estimates. This constitutes a clarification rather than a change in the main results. revision: yes

Circularity Check

0 steps flagged

No circularity: moment bounds derived directly from the kinetic equation

full rationale

The paper derives refined upper bounds on second-order spatial-velocity moments directly from the weak form of the kinetic Cucker-Smale equation for general non-compact weak solutions. These bounds are then applied to initial data belonging to exponential or polynomial decay classes in phase space to obtain the asymptotic velocity flocking and uniform spatial moment bound. No step reduces the target flocking statement to a fitted parameter, a self-definition, or a load-bearing self-citation; the estimates are obtained from the transport-alignment structure of the PDE itself and remain independent of the final conclusion. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard existence theory for weak solutions of kinetic equations plus the additional structural assumption of decay at infinity; no free parameters or new postulated entities are introduced.

axioms (2)

domain assumption Weak solutions to the kinetic Cucker-Smale equation exist and satisfy the integral form of the continuity and momentum equations.
The paper studies properties of weak solutions, presupposing their existence as a starting point for the estimates.
domain assumption The initial distribution functions decay exponentially or polynomially in phase space.
This decay is invoked to close the moment estimates and obtain the asymptotic velocity alignment.

pith-pipeline@v0.9.0 · 5743 in / 1431 out tokens · 52644 ms · 2026-05-19T06:06:12.993427+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We derive refined estimates on the upper bounds for the second-order spatial-velocity moments and show the uniqueness of the weak solution... for two classes of distribution functions with decaying properties (an exponential decay or polynomial decay) in phase space, and then verify that the second moment for the velocity deviation... tends to zero asymptotically, while the second moment for spatial deviation... remains bounded uniformly in time.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

[1]

S. Ahn, H. Choi, S.-Y. Ha, and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), pp. 625–643

work page 2012
[2]

Butta and C

P. Butta and C. Marchioro, Cucker-Smale type dynamics of infinitely many individuals with repulsive forces, J. Stat. Phys., 181 (2020), pp. 2094–2108

work page 2020
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J. A. Canizo, J. A. Carrillo, and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), pp. 515–539

work page 2011
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J. A. Carrillo, Y. P. Choi, and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, ESAIM Proceedings and Surveys., 47 (2014), pp. 17–35

work page 2014
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J. A. Carrillo, M. R. D’ Orsogna, and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models., 2 (2009), pp. 363–378

work page 2009
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J. A. Carrillo, M. Fornasier, J. Rosado, and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), pp. 218–236

work page 2010
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Chen and X

Z. Chen and X. Yin, The kinetic Cucker-Smale model: well-posedness and asymptotic behavior, SIAM J. Math. Anal., 51 (2019), pp. 3819–3853

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Y. P. Choi and S. Salem, Cucker-Smale flocking particles with multiplicative noises: stochastic mean- field limit and phase transition, Kinet. Relat. Models., 12 (2019), pp. 573–592

work page 2019
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Cucker and J.-G

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control., 55 (2010), pp. 1238–1243

work page 2010
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Cucker and S

F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math., 2 (2007), pp. 197–227

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Cucker and S

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control., 52 (2007), pp. 852–862

work page 2007
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S.-Y. Ha, J. Kim, J. Park, and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), pp. 319–365

work page 2019
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S.-Y. Ha, J. Jeong, S. E. Noh, Q. Xiao, and X. Zhang, Emergent dynamics of Cucker-Smale flocking particles in a random environment, J. Differ. Equ., 262 (2017), pp. 2554–2591

work page 2017
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S.-Y. Ha, J. Kim, P. Pickl and X. Zhang, A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication, Kinet. Relat. Models., 12 (2019), pp. 1045–1067

work page 2019
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S.-Y. Ha, J. Kim, and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models., 11 (2018), pp. 1157–1181

work page 2018
[16]

Ha and J.-G

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), pp. 297–325

work page 2009
[17]

Ha and E

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Model., 1 (2008), pp. 415–435

work page 2008
[18]

S.-Y. Ha, X. Wang, and X. Xue. On the exponential weak flocking behavior of the kinetic Cucker-Smale model with non-compact support. Math. Models Methods Appl. Sci., 35 (2025), pp. 781–824

work page 2025
[19]

T. K. Karper, A. Mellet, and K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2013), pp. 215–243

work page 2013
[20]

T. K. Karper, A. Mellet, and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, in Hyperbolic Conservation Laws and Related Analysis with Applications. Springer Proc. Math. Stat. 49, Springer, 2014, pp. 227–242

work page 2014
[21]

Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst., 34 (2014), pp. 3683–3702

work page 2014
[22]

Li and S.-Y

Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), pp. 693–709

work page 2015
[23]

Li and X

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), pp. 3156–3174

work page 2010
[24]

P. B. Mucha and J. Peszek, The Cucker-Smale equation: singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), pp. 273–308

work page 2018
[25]

Natalini and T

R. Natalini and T. Paul, On the mean field limit for Cucker–Smale model, Discrete Contin. Dyn. Syst. B, 27 (2022), pp. 2873–2889

work page 2022
[26]

Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale’s flocking model with a singular communication weight, J

J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale’s flocking model with a singular communication weight, J. Differ. Equ., 257 (2014), pp. 2900–2925. WEAK MONO-CLUSTER FLOCKING OF KINETIC CS MODEL 33

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Peszek, Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J

J. Peszek, Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), pp. 3671–3686

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Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), pp. 694–719

work page 2007
[29]

Villani, Optimal Transport Old and New, Springer-verlag, 2009

C. Villani, Optimal Transport Old and New, Springer-verlag, 2009

work page 2009
[30]

Wang and X

X. Wang and X. Xue, The flocking behavior of the infinite-particle Cucker-Smale model, Proc. Amer. Math. Soc., 150 (2022), pp. 2165–2179

work page 2022
[31]

Wang and X

X. Wang and X. Xue, The collective behavior of the Cucker-Smale model on infinite graphs, Sci SinMath. (in Chinese), 53 (2023), pp. 1799–1826

work page 2023
[32]

Wang and X

X. Wang and X. Xue, Formation behavior of the kinetic Cucker-Smale model with non-compact support, Proc. Roy. Soc. Edinburgh Sect. A., 153 (2023), pp. 1315–1346

work page 2023
[33]

Wang and X

X. Wang and X. Xue, Pattern formation of the Cucker-Smale type kinetic models based on gradient flow, ESAIM Math. Model. Numer. Anal., 57 (2023), pp. 3165–3200

work page 2023
[34]

Wang and X

X. Wang and X. Xue, Flocking behavior of the Cucker-Smale model on infinite graphs with a central vertex group, J. Stat. Phys., 2024, 191: 47. (Seung-Yeal Ha) Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Republic of Korea Email address: syha@snu.ac.kr (Xinyu Wang) Department of Mathema...

work page 2024

[1] [1]

S. Ahn, H. Choi, S.-Y. Ha, and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), pp. 625–643

work page 2012

[2] [2]

Butta and C

P. Butta and C. Marchioro, Cucker-Smale type dynamics of infinitely many individuals with repulsive forces, J. Stat. Phys., 181 (2020), pp. 2094–2108

work page 2020

[3] [3]

J. A. Canizo, J. A. Carrillo, and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), pp. 515–539

work page 2011

[4] [4]

J. A. Carrillo, Y. P. Choi, and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, ESAIM Proceedings and Surveys., 47 (2014), pp. 17–35

work page 2014

[5] [5]

J. A. Carrillo, M. R. D’ Orsogna, and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models., 2 (2009), pp. 363–378

work page 2009

[6] [6]

J. A. Carrillo, M. Fornasier, J. Rosado, and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), pp. 218–236

work page 2010

[7] [7]

Chen and X

Z. Chen and X. Yin, The kinetic Cucker-Smale model: well-posedness and asymptotic behavior, SIAM J. Math. Anal., 51 (2019), pp. 3819–3853

work page 2019

[8] [8]

Y. P. Choi and S. Salem, Cucker-Smale flocking particles with multiplicative noises: stochastic mean- field limit and phase transition, Kinet. Relat. Models., 12 (2019), pp. 573–592

work page 2019

[9] [9]

Cucker and J.-G

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control., 55 (2010), pp. 1238–1243

work page 2010

[10] [10]

Cucker and S

F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math., 2 (2007), pp. 197–227

work page 2007

[11] [11]

Cucker and S

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control., 52 (2007), pp. 852–862

work page 2007

[12] [12]

S.-Y. Ha, J. Kim, J. Park, and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), pp. 319–365

work page 2019

[13] [13]

S.-Y. Ha, J. Jeong, S. E. Noh, Q. Xiao, and X. Zhang, Emergent dynamics of Cucker-Smale flocking particles in a random environment, J. Differ. Equ., 262 (2017), pp. 2554–2591

work page 2017

[14] [14]

S.-Y. Ha, J. Kim, P. Pickl and X. Zhang, A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication, Kinet. Relat. Models., 12 (2019), pp. 1045–1067

work page 2019

[15] [15]

S.-Y. Ha, J. Kim, and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models., 11 (2018), pp. 1157–1181

work page 2018

[16] [16]

Ha and J.-G

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), pp. 297–325

work page 2009

[17] [17]

Ha and E

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Model., 1 (2008), pp. 415–435

work page 2008

[18] [18]

S.-Y. Ha, X. Wang, and X. Xue. On the exponential weak flocking behavior of the kinetic Cucker-Smale model with non-compact support. Math. Models Methods Appl. Sci., 35 (2025), pp. 781–824

work page 2025

[19] [19]

T. K. Karper, A. Mellet, and K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2013), pp. 215–243

work page 2013

[20] [20]

T. K. Karper, A. Mellet, and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, in Hyperbolic Conservation Laws and Related Analysis with Applications. Springer Proc. Math. Stat. 49, Springer, 2014, pp. 227–242

work page 2014

[21] [21]

Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst., 34 (2014), pp. 3683–3702

work page 2014

[22] [22]

Li and S.-Y

Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), pp. 693–709

work page 2015

[23] [23]

Li and X

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), pp. 3156–3174

work page 2010

[24] [24]

P. B. Mucha and J. Peszek, The Cucker-Smale equation: singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), pp. 273–308

work page 2018

[25] [25]

Natalini and T

R. Natalini and T. Paul, On the mean field limit for Cucker–Smale model, Discrete Contin. Dyn. Syst. B, 27 (2022), pp. 2873–2889

work page 2022

[26] [26]

Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale’s flocking model with a singular communication weight, J

J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale’s flocking model with a singular communication weight, J. Differ. Equ., 257 (2014), pp. 2900–2925. WEAK MONO-CLUSTER FLOCKING OF KINETIC CS MODEL 33

work page 2014

[27] [27]

Peszek, Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J

J. Peszek, Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), pp. 3671–3686

work page 2015

[28] [28]

Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), pp. 694–719

work page 2007

[29] [29]

Villani, Optimal Transport Old and New, Springer-verlag, 2009

C. Villani, Optimal Transport Old and New, Springer-verlag, 2009

work page 2009

[30] [30]

Wang and X

X. Wang and X. Xue, The flocking behavior of the infinite-particle Cucker-Smale model, Proc. Amer. Math. Soc., 150 (2022), pp. 2165–2179

work page 2022

[31] [31]

Wang and X

X. Wang and X. Xue, The collective behavior of the Cucker-Smale model on infinite graphs, Sci SinMath. (in Chinese), 53 (2023), pp. 1799–1826

work page 2023

[32] [32]

Wang and X

X. Wang and X. Xue, Formation behavior of the kinetic Cucker-Smale model with non-compact support, Proc. Roy. Soc. Edinburgh Sect. A., 153 (2023), pp. 1315–1346

work page 2023

[33] [33]

Wang and X

X. Wang and X. Xue, Pattern formation of the Cucker-Smale type kinetic models based on gradient flow, ESAIM Math. Model. Numer. Anal., 57 (2023), pp. 3165–3200

work page 2023

[34] [34]

Wang and X

X. Wang and X. Xue, Flocking behavior of the Cucker-Smale model on infinite graphs with a central vertex group, J. Stat. Phys., 2024, 191: 47. (Seung-Yeal Ha) Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Republic of Korea Email address: syha@snu.ac.kr (Xinyu Wang) Department of Mathema...

work page 2024