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

Abstract

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.