H-boundedness of the pullback attractor of the micropolar fluid flows with infinite delays

where u(x, t) = (u,u,u) represents the velocity, ω(x, t) = (ω,ω,ω) stands for the angular velocity field of rotation of particles, p is the pressure, f and f̃ represent the external force and moment, respectively. The positive parameters ν , νr , c, ca, cd are the viscous coefficients. In fact, ν is the usual Newtonian kinetic viscosity, and νr is the dynamics microrotation viscosity, and c, ca, cd denote the angular viscosity (see []). From [, ] we see that these equations express the balance ofmomentum,mass, andmoment ofmomentum, accordingly. When microrotation effects are neglected (i.e., ω = ), the equations reduce to the incompressible Navier-Stokes equations. Therefore, the equations of micropolar fluid flows can be regarded as a generalization of the Navier-Stokes equations in the sense that they take into account the microstructure of the fluid. For physical background, we refer, for example, to [, ]. Due to their wide applications, the micropolar fluid flows have drawn much attention from mathematicians and physicists and have been well studied. For the theories on the existence and uniqueness of solutions of the micropolar fluid flows, we refer to [–].