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 ow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals, in dimensions two and three. More precisely, $0<T_*<+\infty$ is the maximal time interval iff (i) for $n=3$, $|ω|+|\nabla d|^2\notin L^1_tL^\infty_x(\mathbb R^3\times [0,T_*])$, and (ii) for $n=2$, $|\nabla|^2\notin L^1_tL^\infty_x(\mathbb R^2\times [0,T_*])$.

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