The blow-up criterion for the compressible Navier-Stokes equations

We prove a blow-up criterion in terms of the upper bound ofthe density for the strong solution to the 3-D compressible Navier-Stokesequations. The initial vacuum is allowed. The main ingredient of theproof is a priori estimate for an important quantity under the assumptionthat the density is upper bounded, whose divergence can be viewed asthe effective viscous flux.