  Paicu and Zarnescu have studied an order tensor system which describes the
flow of a liquid crystal. They have proven the existence of weak solutions, the
propagation of higher regularities, namely $H^s$ with $s>1$ and the weak-strong
uniqueness in dimension two. This paper is devoted to the propagation of lower
regularities, namely $H^s$ for $0<s\leq 1$ and to prove the uniqueness of the
weak solutions. For the completeness of this research, we also propose an
alternative approach in order to prove the existence of weak solutions.
