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.
@article{AIHPC_2016__33_2_273_0, author = {Carrillo, Jos\'e A. and Choi, Young-Pil and Karper, Trygve K.}, title = {On the analysis of a coupled kinetic-fluid model with local alignment forces}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {273--307}, publisher = {Elsevier}, volume = {33}, number = {2}, year = {2016}, doi = {10.1016/j.anihpc.2014.10.002}, zbl = {1339.35233}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.10.002/} }
TY - JOUR AU - Carrillo, José A. AU - Choi, Young-Pil AU - Karper, Trygve K. TI - On the analysis of a coupled kinetic-fluid model with local alignment forces JO - Annales de l'I.H.P. Analyse non linéaire PY - 2016 SP - 273 EP - 307 VL - 33 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2014.10.002/ DO - 10.1016/j.anihpc.2014.10.002 LA - en ID - AIHPC_2016__33_2_273_0 ER -
%0 Journal Article %A Carrillo, José A. %A Choi, Young-Pil %A Karper, Trygve K. %T On the analysis of a coupled kinetic-fluid model with local alignment forces %J Annales de l'I.H.P. Analyse non linéaire %D 2016 %P 273-307 %V 33 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2014.10.002/ %R 10.1016/j.anihpc.2014.10.002 %G en %F AIHPC_2016__33_2_273_0
Carrillo, José A.; Choi, Young-Pil; Karper, Trygve K. On the analysis of a coupled kinetic-fluid model with local alignment forces. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 2, pp. 273-307. doi : 10.1016/j.anihpc.2014.10.002. http://www.numdam.org/articles/10.1016/j.anihpc.2014.10.002/
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