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On Blowup of Classical Solutions to the Compressible Navier-Stokes Equations

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Abstract

In this paper, we study the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums. We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group (see Definition 2.2). The results hold regardless of either the size of the initial data or the far fields being vacuum or not. This improves the blowup results of Xin (Comm Pure Appl Math 51:229–240, 1998) by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy. Furthermore, the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time, if the initial data have an isolated mass group satisfying some suitable conditions.

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References

  1. Cho Y., Kim H.: Existence results for viscous polytropic fluids with vacuum. J. Diff. Eqs. 228, 377–411 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cho Y., Kim H.: On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manus Math. 120, 91–129 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cho Y., Jin B.J.: Blow-up of viscous heat-conducting compressible flows. J. Math. Anal. Appl. 320, 819–826 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Du, D., Li, J., Zhang, K.: it Blowup of Smooth Solutions to the Navier-Stokes Equations for Compressible Isothermal Fluids. http://arxiv.org/abs/1108.1613v1 [math.Ap], 2011

  5. Feireisl, E.: it Dynamics of viscous compressible fluids. Oxford: Oxford University Press, 2004

  6. Hoff D.: Discontinuous solutions of the Navier-Stokes equations for multi-dimensional heat-conducting fluids. Arch. Rat. Mech. Anal. 193, 303–354 (1997)

    Article  MathSciNet  Google Scholar 

  7. Hoff D., Serre D.: The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51, 887–898 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Huang, X.D., Li, J.: it Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier-Stokes System with Vacuum and Large Oscillations. http://arxiv.org/abs/1107.4655v3 [math-ph], 2011

  10. Huang X.D., Li J., Xin Z.P.: Blow-up criterion for viscous barotropic flows with vacuum states. Commun. Math. Phys. 301, 23–35 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Huang X.D., Li J., Xin Z.P.: Serrin type criterion for the three-dimensional viscous compressible flows. SIAM J. MATH. Anal. 43, 1872–1886 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang X.D., Li J., Xin Z.P.: Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Comm. Pure Appl. Math. 65, 549–585 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang X.D., Xin Z.P.: A blow-up criterion for classical solutions to the compressible Navier-Stkoes equations. Sci. in China 53(3), 671–686 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kazhikhov A.V., Shelukhin V.V.: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. Prikl. Mat. Meh. 41, 282–291 (1977)

    MathSciNet  Google Scholar 

  15. Lions, P.-L.: it Mathematical topics in fluid mechanics. Vol. 2. Oxford: The Clarendon Press, Oxford University Press, 1998

  16. Luo, Z., Xin, Z.P.: it Global Well-Posedness and Blowup Behavior of Classical Solutions with Large Oscillations and Vacuum to the Two-Dimensional Isentropic Compressible Navier-Stokes Equations. Preprint, 2011

  17. Matsumura A., Nishida T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980)

    MathSciNet  MATH  Google Scholar 

  18. Matsumura A., Nishida T.: Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89, 445–464 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Nash J.: Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962)

    MathSciNet  MATH  Google Scholar 

  20. Serrin, J.: it Mathematical principles of classical fluid-mechanics. Handbuch der Physik, Vol. 811. Berlin-Heidelberg-New York: Spring-Verlag, 1959, pp. 125–265

  21. Rozanova O.: Blow up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity. J. Diff. Eqs. 245, 1762–1774 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xin Z.: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51, 229–240 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematical Science, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

    Zhouping Xin

  2. Laboratory of Science and Technology on Computational Physics, Institute of Applied Mathematics and Computational Physics, Beijing, P.R.China

    Wei Yan

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  1. Zhouping Xin
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  2. Wei Yan
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Correspondence to Zhouping Xin.

Additional information

Communicated by P. Constantin

This research is supported in parts by Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants CUHK 4042/08P and CUHK 4041/11P.

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Xin, Z., Yan, W. On Blowup of Classical Solutions to the Compressible Navier-Stokes Equations. Commun. Math. Phys. 321, 529–541 (2013). https://doi.org/10.1007/s00220-012-1610-0

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