  In this paper, we provide a much simplified proof of the main result in [Lin
and Zhang, Comm. Pure Appl. Math.,67(2014), 531--580] concerning the global
existence and uniqueness of smooth solutions to the Cauchy problem for a 3D
incompressible complex fluid model under the assumption that the initial data
are close to some equilibrium states. Beside the classical energy method, the
interpolating inequalities and the algebraic structure of the equations coming
from the incompressibility of the fluid are crucial in our arguments. We
combine the energy estimates with the $L^\infty$ estimates for time slices to
deduce the key $L^1$ in time estimates. The latter is responsible for the
global in time existence.
