Blow-up criterions of strong solutions to 3D compressible Navier-Stokes   equations with vacuum
  In the paper, we establish a blow-up criterion in terms of the integrability of the density for strong solutions to the Cauchy problem of compressible isentropic Navier-Stokes equations in \mathbb{R}^3 with vacuum, under the assumptions on the coefficients of viscosity: \frac{29\mu}{3}>\lambda. This extends the corresponding results in [20, 36] where a blow-up criterion in terms of the upper bound of the density was obtained under the condition 7\mu>\lambda. As a byproduct, the restriction 7\mu>\lambda in [12, 37] is relaxed to \frac{29\mu}{3}>\lambda for the full compressible Navier-Stokes equations by giving a new proof of Lemma 3.1. Besides, we get a blow-up criterion in terms of the upper bound of the density and the temperature for strong solutions to the Cauchy problem of the full compressible Navier-Stokes equations in \mathbb{R}^3. The appearance of vacuum could be allowed. This extends the corresponding results in [37] where a blow-up criterion in terms of the upper bound of (\rho,\frac{1}{\rho}, \theta) was obtained without vacuum. The effective viscous flux plays a very important role in the proofs. 
Huanyao Wen and Changjiang Zhu
math.AP