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 integrability of the density for strong solutions to the Cauchy problem of compressible isentropic Navier-Stokes equations in R3 with vacuum, under the assumptions on the coefficients of viscosity: 29μ3>λ. This extends the corresponding results in Huang et al. (2011), Sun et al. (2011) [20,36] where a blow-up criterion in terms of the upper bound of the density was obtained under the condition 7μ > λ. As a byproduct, the restriction 7μ > λ in Fan et al. (2010), Sun et al. (2011) [12,37] is relaxed to 29μ3>λ 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 R3. The appearance of vacuum could be allowed. This extends the corresponding results in Sun et al. (2011) [37] where a blow-up criterion in terms of the upper bound of (ρ,1ρ,θ) was obtained without vacuum. The effective viscous flux plays a very important role in the proofs.