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BLOW-UP OF SMOOTH SOLUTIONS TO THE NAVIER–STOKES EQUATIONS OF COMPRESSIBLE VISCOUS HEAT-CONDUCTING FLUIDS

Part of: Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  01 April 2010

ZHONG TAN  and
YANJIN WANG
ZHONG TAN
Affiliation:
School of Mathematical Sciences, Xiamen University, Fujian 361005, PR China (email: [email protected])
YANJIN WANG*
Affiliation:
School of Mathematical Sciences, Xiamen University, Fujian 361005, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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We give a simpler and refined proof of some blow-up results of smooth solutions to the Cauchy problem for the Navier–Stokes equations of compressible, viscous and heat-conducting fluids in arbitrary space dimensions. Our main results reveal that smooth solutions with compactly supported initial density will blow up in finite time, and that if the initial density decays at infinity in space, then there is no global solution for which the velocity decays as the reciprocal of the elapsed time.

Keywords

blow-upNavier–Stokes equationscompressibleheat-conductinglife span

MSC classification

Secondary: 35Q30: Navier-Stokes equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 2 , April 2010 , pp. 239 - 246
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

Supported by National Science Foundation of China (Grant No. 10531020) and National Natural Science Foundation of China-NSAF (Grant No. 10976026).

References

[1]Cho, Y. and Jin, B. J., ‘Blow-up of viscous heat-conducting compressible flows’, J. Math. Anal. Appl. 320 (2006), 819–826.CrossRefGoogle Scholar
[2]Matsumura, A. and Nishida, T., ‘The initial value problem for the equations of motion of viscous and heat-conductive gases’, J. Math. Kyoto Univ. 20 (1980), 67–104.Google Scholar
[3]Sideris, T. C., ‘Formation of singularity in three dimensional compressible fluids’, Comm. Math. Phys. 101 (1985), 475–487.CrossRefGoogle Scholar
[4]Xin, Z., ‘Blow-up of smooth solutions to compressible Navier–Stokes equations with compact density’, Comm. Pure. Appl. Math. 51 (1998), 229–240.3.0.CO;2-C>CrossRefGoogle Scholar
[5]Ziemer, W., Weakly Differential Functions (Springer, Berlin, 1989).CrossRefGoogle Scholar