Communications in Mathematical Sciences

Volume 12 (2014)

Number 8

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

Pages: 1427 – 1435



Lili Du (Department of Mathematics, Sichuan University, Chengdu, China)

Yongfu Wang (Department of Mathematics, Sichuan University, Chengdu, China)


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.


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

2010 Mathematics Subject Classification

35B65, 35Q30, 76N10

Full Text (PDF format)