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Journal of Zhejiang University SCIENCE A 2007 Vol.8 No.10 P.1681~1690

http://doi.org/10.1631/jzus.2007.A1681


Non-formation of vacuum states for Navier-Stokes equations with density-dependent viscosity


Author(s):  ZHANG Ting, FANG Dao-yuan

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Corresponding email(s):   zhangting79@hotmail.com, dyf@zju.edu.cn

Key Words:  Compressible Navier-Stokes equations, Vacuum states, Density-dependent viscosity


ZHANG Ting, FANG Dao-yuan. Non-formation of vacuum states for Navier-Stokes equations with density-dependent viscosity[J]. Journal of Zhejiang University Science A, 2007, 8(10): 1681~1690.

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author="ZHANG Ting, FANG Dao-yuan",
journal="Journal of Zhejiang University Science A",
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pages="1681~1690",
year="2007",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2007.A1681"
}

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%A FANG Dao-yuan
%J Journal of Zhejiang University SCIENCE A
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%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.2007.A1681

TY - JOUR
T1 - Non-formation of vacuum states for Navier-Stokes equations with density-dependent viscosity
A1 - ZHANG Ting
A1 - FANG Dao-yuan
J0 - Journal of Zhejiang University Science A
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SP - 1681
EP - 1690
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PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2007.A1681


Abstract: 
We consider the Cauchy problem, free boundary problem and piston problem for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Using the reduction to absurdity method, we prove that the weak solutions to these systems do not exhibit vacuum states, provided that no vacuum states are present initially. The essential requirements on the solutions are that the mass and energy of the fluid are locally integrable at each time, and the Lloc1-norm of the velocity gradient is locally integrable in time.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

[1] Duan, R., Zhao, Y.C., 2005. A note on the non-formation of vacuum states for compressible Navier-Stokes equations. J. Math. Anal. Appl., 311:744-754.

[2] Hoff, D., Smoller, J., 2001. Non-formation of vacuum states for compressible Navier-Stokes equations. Commun. Math. Phys., 216(2):255-276.

[3] Jiang, S., 1994. On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas. Math. Z., 216:317-336.

[4] Xin, Z.P., Yuan, H.J., 2006. Vacuum state for spherically symmetric solutions of the compressible Navier-Stokes equations. J. Hyperbolic Differ. Equ., 3(3):403-442.

[5] Zhang, T., 2006. Compressible Navier-Stokes equations with density-dependent viscosity. Appl. Math. J. Chin. Univ. Ser. B, 21(2):165-178.

[6] Zhang, T., Fang, D.Y., 2006. Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient. Arch. Rational Mech. Anal., 182(2):223-253.

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