  We extend the well-known Serrin's blowup criterion for the three-dimensional
(3D) incompressible Navier-Stokes equations to the 3D viscous compressible
cases. It is shown that for the Cauchy problem of the 3D compressible
Navier-Stokes system in the whole space, the strong or smooth solution exists
globally if the velocity satisfies the Serrin's condition and either the
supernorm of the density or the $L^1(0,T;L^\infty)$-norm of the divergence of
the velocity is bounded. Furthermore, in the case that either the shear
viscosity coefficient is suitably large or there is no vacuum, the Serrin's
condition on the velocity can be removed in this criteria.
