Continuation Criterion for the 2D Liquid Crystal Flows

We consider the 2D liquid crystal systems, which consists of Navier-Stokes system coupled with wave maps or biharmonic wave maps, respectively. By logarithmic Sobolev inequalities, we obtain a blow-up criterion ∇ , ∈ 1 ( 0 , ; 0 ∞ , ∞ ( ℝ 2 ) ) for the case with wave maps, and we prove the existence of a global-in-time strong solutions for the case with biharmonic wave maps.