  We study a three-dimensional, incompressible, viscous, micropolar fluid with
anisotropic microstructure on a periodic domain. Subject to a uniform
microtorque, this system admits a unique nontrivial equilibrium. We prove that
when the microstructure is inertially oblate (i.e. pancake-like) this
equilibrium is nonlinearly asymptotically stable.
  Our proof employs a nonlinear energy method built from the natural energy
dissipation structure of the problem. Numerous difficulties arise due to the
dissipative-conservative structure of the problem. Indeed, the dissipation
fails to be coercive over the energy, which itself is weakly coupled in the
sense that, while it provides estimates for the fluid velocity and
microstructure angular velocity, it only provides control of two of the six
components of the microinertia tensor. To overcome these problems, our method
relies on a delicate combination of two distinct tiers of energy-dissipation
estimates, together with transport-like advection-rotation estimates for the
microinertia. When combined with a quantitative rigidity result for the
microinertia, these allow us to deduce the existence of global-in-time decaying
solutions near equilibrium.
