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

In this paper, we study global existence and large-time behaviors of strong solutions to a kinetic Cucker–Smale-type model coupled with the three-dimensional incompressible Navier–Stokes equations in the whole space. By virtue of the maximal regularity estimate on the Stokes equations, global-in-time strong solutions to the Cauchy problem of the coupled system are obtained under a small initial data regime. The optimal energy decay rate of the system is also analyzed, which is in the sense that the energy of the system decays at a rate coinciding with the one corresponding to the underlying linear parabolic equations. Comparing with previous results in a spatial-periodic or bounded domain, we circumvent the difficulty caused by unboundedness of the domain using the Fourier splitting method.