### 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 |
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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