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 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.
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José A. Carrillo, Young-Pil Choi, Trygve K. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 2, pp. 273–307DOI 10.1016/J.ANIHPC.2014.10.002