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Blowup criterion for viscous baratropic flows with vacuum states. (English) Zbl 1213.35135
The paper is concerning barotropic and not baratropic flow of a viscous compressible Navier-Stokes fluid. In some previous papers, the authors proved a blow-up criteria in terms of the norm of the velocity gradient of the velocity, but with a strong restriction involving shear and bulk viscosity coefficients. In the present paper this result is improved, by removing this restriction. A new blow-up criterion is given in terms of the deformation tensor. The key steps are some new estimates in \(L_2\) of both density and velocity gradients. The result is obtained by a new energy estimate which uses the effective stress tensor. The case of zero density (vacuum state) is also studied. A very interesting estimate of the velocity gradient, in terms of divergence and vorticity, is given in Lemma 2.3, by using Poincaré and Ehrling type inequalities.

MSC:
35B44 Blow-up in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35Q30 Navier-Stokes equations
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