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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{29μ}{3}>λ. 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 7μ>λ. As a byproduct, the restriction 7μ>λin [12, 37] is relaxed to \frac{29μ}{3}>λfor 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 (ρ,\frac{1}ρ, θ) was obtained without vacuum. The effective viscous flux plays a very important role in the proofs.