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# Communications in Mathematical Sciences

## Volume 12 (2014)

### Number 8

### Blowup criterion for 3-dimensional compressible Navier-Stokes equations involving velocity divergence

Pages: 1427 – 1435

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n8.a3

#### Authors

#### Abstract

In this paper, we provide a sufficient condition, in terms of only velocity divergence, for global regularity of strong solutions to the three-dimensional Navier-Stokes equations with vacuum in the whole space, as well as for the case of a bounded domain with Dirichlet boundary conditions. More precisely, we show that the weak solutions of the Cauchy problem or the Dirichlet initial-boundary-value problem of the 3D compressible Navier-Stokes equations are indeed regular provided that the $L^2(0, T; L^{\infty})$-norm of the divergence of the velocity is bounded. Additionally, initial vacuum states are allowed and the viscosity coefficients are only restricted by the physical conditions.

#### Keywords

compressible Navier-Stokes equations, blowup criterion, vacuum, velocity divergence

#### 2010 Mathematics Subject Classification

35B65, 35Q30, 76N10