Skip to main content

Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows

Abstract

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.

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

References

  1. 1

    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959); II, Comm. Pure Appl. Math. 17, 35–92 (1964)

  2. 2

    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  Article  MATH  Google Scholar 

  3. 3

    Chen M., Liu S.: Blow-up criterion for 3D viscous-resistive compressible magnetohydrodynamic equations. Math. Method. Appl. Sci. 36(9), 1145–1156 (2013)

    Article  MATH  Google Scholar 

  4. 4

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

    MathSciNet  Article  ADS  MATH  Google Scholar 

  5. 5

    Ducomet B., Feireisl E.: The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars. Commun. Math. Phys. 226, 595–629 (2006)

    MathSciNet  ADS  Article  Google Scholar 

  6. 6

    Fan J., Jiang S., Ou Y.: A blow-up criterion for compressible viscous heat-conductive flows. Ann. Inst. H. Poincaré (C) Analyse non-lineaire 27, 337–350 (2010)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  7. 7

    Fan J., Yu W.: Strong solution to the compressible magnetohydrodynamic equations with vacuum. Nonlin Anal. Real World Appl. 10(1), 392–409 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8

    Feireisl E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  9. 9

    Haspot, B.: Regularity of weak solutions of the compressible barotropic Navier-Stokes equations. Available at http://arxiv.org/abs/1001.1581v1 [math.AP], 2010

  10. 10

    He C., Xin Z.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Diff. Eq. 213, 235–254 (2005)

    MathSciNet  Article  ADS  MATH  Google Scholar 

  11. 11

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

    MathSciNet  Article  MATH  Google Scholar 

  12. 12

    Hu X., Wang D.: Global solutions to the three-dimensional full compressible magnetohydrodynamic flows. Commun. Math. Phys. 283(1), 255–284 (2008)

    ADS  Article  MATH  Google Scholar 

  13. 13

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

  14. 14

    Huang X.D., Li J.: On breakdown of solutions to the full compressible Navier-Stokes equations. Meth. Appl. Anal. 16(4), 479–490 (2009)

    MATH  Google Scholar 

  15. 15

    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/1107.4655v3 [math-ph], 2011

  16. 16

    Huang X.D., Li J., Wang Y.: Serrin-type blowup criterion for full compressible Navier-Stokes system. Arch. Rat. Mech. Anal. 207, 303–316 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17

    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 

  18. 18

    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  Article  MATH  Google Scholar 

  19. 19

    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)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20

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

    MathSciNet  Article  MATH  Google Scholar 

  21. 21

    Laudau, L.D., Lifshitz, E.M.: Electrodynamics of continuous media. 2nd ed., New York: Pergamon, 1984

  22. 22

    Lions P. L.: Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford University Press, New York (1998)

    Google Scholar 

  23. 23

    Lu M., Du Y., Yao Z.A. et al.: A blow-up criterion for compressible MHD equations. Commun. Pure Appl. Anal. 11, 1167–1183 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    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 

  25. 25

    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 

  26. 26

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

    MathSciNet  MATH  Google Scholar 

  27. 27

    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  Article  ADS  MATH  Google Scholar 

  28. 28

    Rozanova, O.: Blow up of smooth solutions to the barotropic compressible magnetohydrodynamic equations with finite mass and energy. In: Proceedings of Symposia in Applied Mathematics 2009, Hyperbolic Problems: Theory, Numerics and Applications, Vol. 67, Providence, RI: Amer. Math. Soc., 2009, pp. 911–918

  29. 29

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

    MathSciNet  Article  MATH  Google Scholar 

  30. 30

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

    MathSciNet  ADS  Article  MATH  Google Scholar 

  31. 31

    Solonnikov, V. A.: General boundary value problems for systems elliptic in the sense of A. Douglis and L. Nirenberg. I, Izv. Akad. Nauk SSSR Ser. Mat. 28(3), 665–706 (1964), English transl., Amer. Math. Soc. Transl. Ser. II 56, 193–232 (1964)

  32. 32

    Solonnikov V.A.: General boundary value problems for Douglis-Nirenberg elliptic systems. II. Proc. Steklov Inst. Math. 92, 269–339 (1968)

    Google Scholar 

  33. 33

    Solonnikov V.A.: On Green’s matrices for elliptic boundary problem. I. Proc. Steklov Inst. Math. 110, 123–170 (1970)

    MathSciNet  Google Scholar 

  34. 34

    Solonnikov V.A.: On Green’s matrices for elliptic boundary problem. II. Proc. Steklov Inst. Math. 116, 187–226 (1971)

    MathSciNet  Google Scholar 

  35. 35

    Suen A.: A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density. Dis. Cont. Dyn. Sys. 33, 3791–3805 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36

    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 

  37. 37

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

    MathSciNet  Article  MATH  Google Scholar 

  38. 38

    Volpert A.I., Khudiaev S.I.: On the Cauchy problem for composite systems of nonlinear equations. Mat. Sb. 87, 504–528 (1972)

    MathSciNet  Google Scholar 

  39. 39

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

    MathSciNet  Article  MATH  Google Scholar 

  40. 40

    Xu X.Y., Zhang J.W.: A blow-up criterion for 3D compressible magnetohydrodynamic equations with vaccum. Math. Models Method. Appl. Sci. 22(2), 1150010 (2012)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jing Li.

Additional information

Communicated by L. Caffarelli

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Huang, X., Li, J. Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows. Commun. Math. Phys. 324, 147–171 (2013). https://doi.org/10.1007/s00220-013-1791-1

Download citation

Keywords

  • Weak Solution
  • Strong Solution
  • Smooth Solution
  • Bounded Smooth Domain
  • Blowup Criterion