An Osgood type regularity criterion for the liquid crystal flows

In this paper, we prove an Osgood type regularity criterion for the model of liquid crystals, which says that the condition $$sup_{2 leq q< infty} int nolimits_0^T frac{| bar{S}_{q} nabla {bf u}(t)|_{L^infty}}{q , {rm ln} , q} {rm d} t implies the smoothness of the solution. Here, ${{bar S_q=sumnolimits_{k=-q}^q dot {triangle}_k}}$ with ${dot{triangle}_k}$ being the frequency localization operator.