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In this paper, we study global existence and large time behaviors of strongsolutions to the kinetic Cucker--Smale model coupled with the three dimensionalincompressible Navier--Stokes equations in the whole space. Using the maximalregularity estimate on the Stokes equations, global-in-time strong solutions tothe Cauchy problem of the coupled system are obtained under small initial dataregime. The optimal decay rate of the system, in the sense that the energy ofthe system decays at a rate coinciding with the one corresponding to theunderling incompressible Navier--Stokes equations without the coupling term, isalso analyzed. Compared with previous results set in spatial-periodic orbounded domains, we circumvent the difficulty caused by unboundedness of thedomain using the Fourier splitting method.

Global Existence and Large Time Behaviors of Strong Solutions to the  Kinetic Cucker--Smale Model Coupled with the Three Dimensional Incompressible  Navier--Stokes Equations