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 ℝ3. We show that if 0 < T < + ∞ is the maximal time interval for the unique smooth solution u ∈ C∞([0,T],ℝ3), then |u|+|∇d|∉Lq([0,T],Lp(ℝ3)) where p and q safisfy the Ladyzhenskaya-Prodi-Serrin's condition: