Blow up Criterion for Nematic Liquid Crystal Flows

In this paper, we establish a blow up criterion for the short time classical solution of the nematic liquid crystal flow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals, in dimensions two and three. More precisely, 0 < T-* < +infinity is the maximal time interval iff (i) for n = 3, vertical bar omega vertical bar + vertical bar del d vertical bar(2) is not an element of (LtLx infinity)-L-1(R-3 x [0, T-*]); and (ii) for n = 2, vertical bar del d vertical bar(2) is not an element of (LtLx infinity)-L-1(R-2 x [0, T-*]).