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

Jiang, Ning; Luo, Yi-Long; Zhang, Teng-Fei

doi:10.1007/s00205-019-01470-w

Cited by 10 publications

(3 citation statements)

References 30 publications

(39 reference statements)

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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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Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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“…Starting with the model (1.1) (actually in the second case, see below), they proposed the continuum model through considering some parameter tending to zero [1]. Moreover, describing the agents by some statistical distribution of their speed and positions is also referred as the kinetic model or mean-field model, and some results of well-posedness in this model have been obtained [19][20][21][22][23][24].…”

Section: Brief Reviewmentioning

confidence: 99%

Local well‐posedness of classical solutions for the kinetic self‐organized model of Cucker–Smale type

Zhou,

Zhang,

2024

Math Methods in App Sciences

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We study in this paper the local well‐posedness of classical solutions to the Cauchy problem for a kinetic self‐organized model of Cucker–Smale type for collective motions, which includes two cases of velocity alignment mechanisms: normalized and nonnormalized reorientation cases. The main concern is the a priori estimates for both cases. Our treatments rely on two key ingredients: One point is to deal with the possible singularity and high nonlinearity arising from the normalized/nonnormalized mean velocity, and the other is to control the growth with respect to the microscopic velocity variable in the whole space by employing weighted estimates.

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“…Finally, we mention two works which do incorporate swimming. Jiang et al [29] provide a proof of local well-posedness for a microscopic "self-organized kinetic" model coupled with Navier-Stokes and rigorously justify the hydrodynamic limit to a macroscopic closure model. Further related work on swimmers includes Kanzler and Schmeiser [30], who consider a kinetic transport model for myxobacteria in which the particles interact via collisions rather than through a surrounding fluid medium.…”

Section: Introductionmentioning

confidence: 99%

On the stabilizing effect of swimming in an active suspension

Albritton¹,

Ohm²

2022

Preprint

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We consider a kinetic model of an active suspension of rod-like microswimmers. In certain regimes, swimming has a stabilizing effect on the suspension. We quantify this effect near homogeneous isotropic equilibria ψ = const. Notably, in the absence of particle (translational and orientational) diffusion, swimming is the only stabilizing mechanism. On the torus, in the non-diffusive regime, we demonstrate linear Landau damping up to the stability threshold predicted in the applied literature. With small diffusion, we demonstrate nonlinear stability of arbitrary equilibrium values for pullers (rear-actuated swimmers) and enhanced dissipation for both pullers and pushers (front-actuated swimmers) at small concentrations. On the whole space, we prove nonlinear stability of the vacuum equilibrium due to generalized Taylor dispersion. Contents 1. Introduction 1 2. Landau damping 9 3. Taylor dispersion 15 4. Enhanced dissipation 23 Appendix A. Nondimensionalization 32 Appendix B. Strong solution theory 33 Appendix C. Proof of Lemma 4.3 36 References 37

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

Cited by 10 publications

References 30 publications

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

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

Local well‐posedness of classical solutions for the kinetic self‐organized model of Cucker–Smale type

On the stabilizing effect of swimming in an active suspension

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