  In this paper we consider a coupled hydrodynamical system which involves the
Navier-Stokes equations for the velocity field and kinematic transport
equations for the molecular orientation field. By applying the Chemin-Lerner's
time-space estimates for the heat equation and the Fourier localization
technique, we prove that when initial data belongs to the critical Besov spaces
with negative-order, there exists a unique local solution, and this solution is
global when initial data is small enough. As a corollary, we obtain existence
of global self-similar solutions. In order to figure out the relation between
the solution obtained here and weak solution of standard sense, we establish a
stability result, which yields in a direct way that all global weak solutions
associated with the same initial data must coincide with the solution obtained
here, namely, weak-strong uniqueness holds.
