LPS's Criterion for Incompressible Nematic Liquid Crystal Flows
  In this paper we derive LPS's criterion for the breakdown of classical solutions to the incompressible nematic liquid crystal flow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals in $\mathbb R^3$. We show that if $0<T<+\infty$} is the maximal time interval for the unique smooth solution $u\in   C^\infty([0,T),\mathbb   R^3)$, then $|u|+|\nabla d|\notin L^q([0,T],L^p(\mathbb   R^3))$, where $p$ and $q$ safisfy the Ladyzhenskaya-Prodi-Serrin's condition: $\frac{3}{p}+\frac{2}{q}=1$ and $p\in(3,+\infty]$ 
Qing Chen, Zhong Tan, Guochun Wu
math.AP