Blow-up criterion of classical solutions for the incompressible nematic liquid crystal flows

In this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in R3. We establish a criterion for possible breakdown of such solutions at a finite time. More precisely, if (u,d) is smooth up to time T provided that ∫0T||∇×u(t,⋅)||BMO(R3)+||∇d(t,⋅)||L4(R3)8dt<∞.