JournalsaihpcVol. 33, No. 2pp. 273–307

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

  • José A. Carrillo

    Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
  • Young-Pil Choi

    Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
  • Trygve K. Karper

    Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, N-7491, Norway
On the analysis of a coupled kinetic-fluid model with local alignment forces cover
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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 LpL^{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.

Cite this article

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. 273–307

DOI 10.1016/J.ANIHPC.2014.10.002