We present a global existence theory for strong solution to the Cucker-Smale-Navier-Stokes system in a periodic domain, when initial data is sufficiently small. To model interactions between flocking particles and an incompressible viscous fluid, we couple the kinetic Cucker-Smale model and the incompressible Navier-Stokes system using a drag force mechanism that is responsible for the global flocking between particles and fluids. We also revisit the emergence of time-asymptotic flocking via new functionals measuring local variances of particles and fluid around their local averages and the distance between local averages velocities. We show that the particle and fluid velocities are asymptotically aligned to the common velocity, when the viscosity of the incompressible fluid is sufficiently large compared to the sup-norm of the particles' local mass density.
Bibliographical noteFunding Information:
H.-O. Bae is supported by Basic Science Research Program (BSRP) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MOEST) ( 2013R1A1A2004601 ). Y.-P. Choi is supported by BSRP through NRF funded by MOEST ( 2012R1A6A3A03039496 ). S.-Y. Ha and M.-J. Kang is partially supported by NRF-SRC( PARC ) grant ( NRF-2009-0083521 ).
All Science Journal Classification (ASJC) codes
- Applied Mathematics