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.
|Number of pages||35|
|Journal||Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire|
|Publication status||Published - 2016 Mar 1|
Bibliographical noteFunding Information:
JAC was partially supported by the project MTM2011-27739-C04-02 DGI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya . JAC acknowledges support from the Royal Society by a Wolfson Research Merit Award. YPC was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Ref. 2012R1A6A3A03039496 ). JAC and YPC were supported by Engineering and Physical Sciences Research Council grants with references EP/K008404/1 (individual grant) and EP/I019111/1 (platform grant). The work of TK was supported by the Norwegian Research Council (proj. 205738 ).
© 2015 Elsevier Masson SAS.
All Science Journal Classification (ASJC) codes
- Mathematical Physics
- Applied Mathematics