A Serrin type criterion for incompressible hydrodynamic flow of liquid crystals in dimension three

In the paper, we establish a Serrin type criterion for strong solutions to a simplified density-dependent Ericksen-Leslie system modeling incompressible, nematic liquid crystal materials in dimension three. The density may vanish in an open subset of $\Omega$. As a byproduct, we establish the Serrin type criterion for heat flow of harmonic map whose gradients belong to $L^r_xL^s_t$, where $\frac{2}{s}+\frac{3}{r}\le1$, for $3< r\le\infty$.