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Huang, Xiangdi; Li, Jing; Xin, Zhouping

Blowup criterion for viscous baratropic flows with vacuum states. (English) Zbl 1213.35135

Commun. Math. Phys. 301, No. 1, 23-35 (2011).
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.
Reviewer: Gelu Paşa (Bucureşti)

Cited in 6 Reviews
Cited in 74 Documents

MSC:

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

Keywords:

unsteady compressible Navier-Stokes fluids; regularity and uniqueness; zero density; vacuum states; new energy estimate; Poincaré and Ehrling type inequalities
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\textit{X. Huang} et al., Commun. Math. Phys. 301, No. 1, 23--35 (2011; Zbl 1213.35135)
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