### Abstract

This paper studies global existence, hydrodynamic limit, and large-time behavior of weak solutions to a kinetic flocking model coupled to the incompressible Navier-Stokes equations. The model describes the motion of particles immersed in a Navier-Stokes fluid interacting through local alignment. We first prove the existence of weak solutions using energy and L^{p} estimates together with the velocity averaging lemma. We also rigorously establish a hydrodynamic limit corresponding to strong noise and local alignment. In this limit, the dynamics can be totally described by a coupled compressible Euler-incompressible Navier-Stokes system. The proof is via relative entropy techniques. Finally, we show a conditional result on the large-time behavior of classical solutions. Specifically, if the mass-density satisfies a uniform in time integrability estimate, then particles align with the fluid velocity exponentially fast without any further assumption on the viscosity of the fluid.

Original language | English |
---|---|

Pages (from-to) | 273-307 |

Number of pages | 35 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 33 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2016 Mar 1 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematical Physics
- Applied Mathematics

### Cite this

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*,

*33*(2), 273-307. https://doi.org/10.1016/j.anihpc.2014.10.002

}

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*, vol. 33, no. 2, pp. 273-307. https://doi.org/10.1016/j.anihpc.2014.10.002

**On the analysis of a coupled kinetic-fluid model with local alignment forces.** / Carrillo, Jose A.; Choi, Young Pil; Karper, Trygve K.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the analysis of a coupled kinetic-fluid model with local alignment forces

AU - Carrillo, Jose A.

AU - Choi, Young Pil

AU - Karper, Trygve K.

PY - 2016/3/1

Y1 - 2016/3/1

N2 - This paper studies global existence, hydrodynamic limit, and large-time behavior of weak solutions to a kinetic flocking model coupled to the incompressible Navier-Stokes equations. The model describes the motion of particles immersed in a Navier-Stokes fluid interacting through local alignment. We first prove the existence of weak solutions using energy and Lp estimates together with the velocity averaging lemma. We also rigorously establish a hydrodynamic limit corresponding to strong noise and local alignment. In this limit, the dynamics can be totally described by a coupled compressible Euler-incompressible Navier-Stokes system. The proof is via relative entropy techniques. Finally, we show a conditional result on the large-time behavior of classical solutions. Specifically, if the mass-density satisfies a uniform in time integrability estimate, then particles align with the fluid velocity exponentially fast without any further assumption on the viscosity of the fluid.

AB - This paper studies global existence, hydrodynamic limit, and large-time behavior of weak solutions to a kinetic flocking model coupled to the incompressible Navier-Stokes equations. The model describes the motion of particles immersed in a Navier-Stokes fluid interacting through local alignment. We first prove the existence of weak solutions using energy and Lp estimates together with the velocity averaging lemma. We also rigorously establish a hydrodynamic limit corresponding to strong noise and local alignment. In this limit, the dynamics can be totally described by a coupled compressible Euler-incompressible Navier-Stokes system. The proof is via relative entropy techniques. Finally, we show a conditional result on the large-time behavior of classical solutions. Specifically, if the mass-density satisfies a uniform in time integrability estimate, then particles align with the fluid velocity exponentially fast without any further assumption on the viscosity of the fluid.

UR - http://www.scopus.com/inward/record.url?scp=84959295241&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84959295241&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2014.10.002

DO - 10.1016/j.anihpc.2014.10.002

M3 - Article

AN - SCOPUS:84959295241

VL - 33

SP - 273

EP - 307

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

SN - 0294-1449

IS - 2

ER -