Skip to main content

Serrin-Type Blowup Criterion for Full Compressible Navier–Stokes System

Abstract

The authors establish a Serrin-type blowup criterion for the Cauchy problem of the three-dimensional full compressible Navier–Stokes system, which states that a strong or smooth solution exists globally, provided that the velocity satisfies Serrin’s condition and that the temporal integral of the maximum norm of the divergence of the velocity is bounded. In particular, this criterion extends the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier–Stokes equations to the three-dimensional full compressible system and is just the same as that of the barotropic case.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  2. 2.

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

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Fan J., Jiang S., Ou Y.: A blow-up criterion for compressible viscous heat-conductive flows. Annales de l’Institut Henri Poincare (C) Analyse non lineaire 27, 337–350 (2010)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  4. 4.

    Feireisl E.: Dynamics of Viscous Compressible Fluids. Oxford Science Publication, Oxford (2004)

    MATH  Google Scholar 

  5. 5.

    Feireisl E., Novotny A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3, 358–392 (2001)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  6. 6.

    Hoff D.: Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120(1), 215–254 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Hoff D.: Discontinuous solutions of the Navier–Stokes equations for multidimensional flows of heat-conducting fluids. Arch. Rational Mech. Anal. 139, 303–354 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Huang, X.D.: Some results on blowup of solutions to the compressible Navier–Stokes equations. PhD thesis, The Chinese University of Hong Kong (2009)

  9. 9.

    Huang X.D., Li J.: On breakdown of solutions to the full compressible Navier–Stokes equations. Methods Appl. Anal. 16, 479–490 (2009)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Huang, X.D., Li, J.: Global classical and weak solutions to the three-dimensional full compressible Navier–Stokes system with vacuum and large oscillations. http://arxiv.org/abs/1107.4655

  11. 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)

    MathSciNet  MATH  Google Scholar 

  12. 12.

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

    MathSciNet  ADS  MATH  Article  Google Scholar 

  13. 13.

    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. Commun. Pure Appl. Math. 65, 549–585 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

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

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Kazhikhov A.V.: Cauchy problem for viscous gas equations. Sib. Math. J. 23, 44–49 (1982)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    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. J. Appl. Math. Mech. 41, 273–282 (1977)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Lions P.L.: Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  18. 18.

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

    MathSciNet  MATH  Google Scholar 

  19. 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. 20.

    Nirenberg L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13(3), 115–162 (1959)

    MathSciNet  MATH  Google Scholar 

  21. 21.

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

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Serrin J.: On the uniqueness of compressible fluid motion. Arch. Rational Mech. Anal. 3, 271–288 (1959)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  23. 23.

    Serrin J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 9, 187–195 (1962)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  24. 24.

    Sun Y.Z., Wang C., Zhang Z.F.: A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier–Stokes equations. J. Math. Pures Appl. 95, 36–47 (2011)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Sun Y.Z., Wang C., Zhang Z.F.: A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows. Arch. Rational Mech. Anal. 201, 727–742 (2011)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  26. 26.

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

    MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jing Li.

Additional information

Communicated by P.-L. Lions

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Huang, X., Li, J. & Wang, Y. Serrin-Type Blowup Criterion for Full Compressible Navier–Stokes System. Arch Rational Mech Anal 207, 303–316 (2013). https://doi.org/10.1007/s00205-012-0577-5

Download citation

Keywords

  • Weak Solution
  • Cauchy Problem
  • Stokes Equation
  • Strong Solution
  • Smooth Solution