Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum

摘要

In the paper, we establish a blow-up criterion in terms of the integrabilityof the density for strong solutions to the Cauchy problem of compressibleisentropic Navier-Stokes equations in \mathbb{R}^3 with vacuum, under theassumptions on the coefficients of viscosity: \frac{29\mu}{3}>\lambda. Thisextends the corresponding results in [20, 36] where a blow-up criterion interms of the upper bound of the density was obtained under the condition7\mu>\lambda. As a byproduct, the restriction 7\mu>\lambda in [12, 37] isrelaxed to \frac{29\mu}{3}>\lambda for the full compressible Navier-Stokesequations by giving a new proof of Lemma 3.1. Besides, we get a blow-upcriterion in terms of the upper bound of the density and the temperature forstrong solutions to the Cauchy problem of the full compressible Navier-Stokesequations in \mathbb{R}^3. The appearance of vacuum could be allowed. Thisextends the corresponding results in [37] where a blow-up criterion in terms ofthe upper bound of (\rho,\frac{1}{\rho}, \theta) was obtained without vacuum.The effective viscous flux plays a very important role in the proofs.