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

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

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Published 14 May 2014