Serrin blow-up criterion for strong solutions to the 3-D compressible nematic liquid crystal flows with vacuum

In this article, we extend the well-known Serrin's blow-up criterion for solutions of the 3-D incompressible Navier-Stokes equations to the 3-D compressible nematic liquid crystal flows where the initial vacuum is allowed. It is proved that for the initial-boundary value problem of the 3-D compressible nematic liquid crystal flows in a bounded domain, the strong solution exists globally if the velocity satisfies the Serrin's condition and $L^1(0,T;L^{infty})$-norm of the gradient of the velocity is bounded