A Global 2D Well-Posedness Result on the Order Tensor Liquid Crystal Theory

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 $0<s\le 1$ 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.