A local existence of viscous self-organized hydrodynamic model

Zhang, Teng-Fei; Jiang, Ning

doi:10.1016/j.nonrwa.2016.09.016

Cited by 14 publications

(7 citation statements)

References 10 publications

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“…Corresponding works for the body alignment model are only formal [35,37,38]. Existence theory for the hydrodynamic models derived from the Vicsek model can be found in [40,90] and numerical methods in [44,73]. Both questions are still open for the body orientation model.…”

Section: Body−orientation Not Aligned Alignedmentioning

confidence: 99%

Bulk topological states in a new collective dynamics model

Degond,

Diez,

2021

Preprint

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In this paper, we demonstrate the existence of topological states in a new collective dynamics model. This individual-based model (IBM) describes self-propelled rigid bodies moving with constant speed and adjusting their rigid-body attitude to that of their neighbors. In previous works, a macroscopic model has been derived from this IBM in a suitable scaling limit. In the present work, we exhibit explicit solutions of the macroscopic model characterized by a non-trivial topology. We show that these solutions are well approximated by the IBM during a certain time but then the IBM transitions towards topologically trivial states. Using a set of appropriately defined topological indicators, we reveal that the breakage of the non-trivial topology requires the system to go through a phase of maximal disorder. We also show that similar but topologically trivial initial conditions result in markedly different dynamics, suggesting that topology plays a key role in the dynamics of this system.

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Section: Body−orientation Not Aligned Alignedmentioning

confidence: 99%

Bulk topological states in a new collective dynamics model

Degond,

Diez,

2021

Preprint

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“…In Degond and Motsch (2008), a methodological breakthrough has been achieved by introducing the so-called generalised collisional invariants (GCI) to rigorously link kinetic and macroscopic equations in the context of collective behaviour of self-propelled particles. This technique is now rigorously justified (Jiang et al 2016) and has already been successfully applied to a wide range of problems (Jiang et al 2017;Zhang and Jiang 2017). It will be the key here to derive the macroscopic model in Sect.…”

Section: Introductionmentioning

confidence: 99%

Phase Transitions and Macroscopic Limits in a BGK Model of Body-Attitude Coordination

et al. 2020

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In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body attitude of an agent is modelled by a rotation matrix in $${\mathbb {R}}^3$$ R 3 as in Degond et al. (Math Models Methods Appl Sci 27(6):1005–1049, 2017). The starting point of this study is a BGK equation modelling the evolution of the distribution function of the system at a kinetic level. The main novelty of this work is to show that in the spatially homogeneous case, self-organisation may appear or not depending on the local density of agents involved. We first exhibit a connection between body-orientation models and models of nematic alignment of polymers in higher-dimensional space from which we deduce the complete description of the possible equilibria. Then, thanks to a gradient-flow structure specific to this BGK model, we are able to prove the stability and the convergence towards the equilibria in the different regimes. We then derive the macroscopic models associated with the stable equilibria in the spirit of Degond et al. (Arch Ration Mech Anal 216(1):63–115, 2015, Math Models Methods Appl Sci 27(6):1005–1049, 2017).

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“…Here we omit the details of the derivation. Actually, one can refer to an analogous derivation in [13] (see also in [31]) for the SOH models in three dimensions. However, the system (1.4) has a coefficient ½ × Ò ¾ which is singular near ¼.…”

Section: 2mentioning

confidence: 99%

Coupled Self-organized Hydrodynamics and Navier–Stokes Models: Local Well-posedness and the Limit from the Self-organized Kinetic-Fluid Models

Jiang

Luo

Zhang

2019

Arch Rational Mech Anal

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A coupled system of self-organized hydrodynamics and Navier-Stokes equations (SOH-NS), which models self-propelled particles in a viscous fluid, was recently derived by Degond et al. [14], starting from a micro-macro particle system of Vicsek-Navier-Stokes model, through an intermediate step of a self-organized kinetic-kinetic model by multiple coarse-graining processes. We first transfer SOH-NS into a non-singular system by stereographic projection, then prove the local in time well-posedness of classical solutions by energy method. Furthermore, employing the Hilbert expansion approach, we justify the hydrodynamic limit from the self-organized kinetic-fluid model to macroscopic dynamics. This provides the first analytically rigorous justification of the modeling and asymptotic analysis in [14].

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A local existence of viscous self-organized hydrodynamic model

Cited by 14 publications

References 10 publications

Bulk topological states in a new collective dynamics model

Bulk topological states in a new collective dynamics model

Phase Transitions and Macroscopic Limits in a BGK Model of Body-Attitude Coordination

Coupled Self-organized Hydrodynamics and Navier–Stokes Models: Local Well-posedness and the Limit from the Self-organized Kinetic-Fluid Models

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