Skip to main content

Blowup Criterion for Viscous Baratropic Flows with Vacuum States

Abstract

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional (3D) barotropic compressible Navier-Stokes equations. More precisely, if a solution of the 3D barotropic compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce’s criterion for 3-dimensional incompressible Euler equations (Ponce in Commun Math Phys 98:349–353, 1985). In addition, initial vacuum states are allowed in our cases.

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

References

  1. 1

    Beal 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)

    Article  ADS  Google Scholar 

  2. 2

    Bendali A., Domíguez J.M., Gallic S.: A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three-dimensional domains. J. Math. Anal. Appl. 107(2), 537–560 (1985)

    MATH  Article  MathSciNet  Google Scholar 

  3. 3

    Bourguignon J.P., Brezis H.: Remarks on the Euler equation. J. Funct. Analysis 15, 341–363 (1974)

    MATH  Article  MathSciNet  Google Scholar 

  4. 4

    Cho Y., Choe H.J., Kim H.: Unique solvability of the initial boundary value problems for compressible viscous fluid. J.Math. Pures Appl. 83, 243–275 (2004)

    MATH  MathSciNet  Google Scholar 

  5. 5

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

    MATH  Article  MathSciNet  Google Scholar 

  6. 6

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

    MATH  Article  MathSciNet  Google Scholar 

  7. 7

    Choe H.J., Kim H.: Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Diff. Eqs. 190, 504–523 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  8. 8

    Choe H.J., Bum J.: Regularity of weak solutions of the compressible Navier-Stokes equations. J. Korean Math. Soc. 40(6), 1031–1050 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  9. 9

    Constantin P.: Nonlinear inviscid incompressible dynamics. Phys. D. 86, 212–219 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  10. 10

    Desjardins B.: Regularity of weak solutions of the compressible isentropic Navier-Stokes equations. Comm. Part. Diff. Eqs. 22(5-6), 977–1008 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  11. 11

    Fan J.S., Jiang S.: Blow-Up criteria for the navier-stokes equations of compressible fluids. J.Hyper. Diff. Eqs. 5(1), 167–185 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  12. 12

    Feireisl E., Novotny A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  13. 13

    Feireisl E.: On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J. 53(6), 1705–1738 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  14. 14

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

    MATH  Google Scholar 

  15. 15

    Hoff D.: Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Amer. Math. Soc. 303(1), 169–181 (1987)

    MATH  MathSciNet  Google Scholar 

  16. 16

    Hoff D.: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rat. Mech. Anal. 132, 1–14 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  17. 17

    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)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  18. 18

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

  19. 19

    Huang, X.D., Xin, Z.P.: A Blow-up criterion for the compressible Navier-Stokes equations. http://arxiv.org/abs/0902.2606v1 [math-ph], 2009

  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)

    MATH  Article  MathSciNet  Google Scholar 

  21. 21

    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 

  22. 22

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

  23. 23

    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)

    MATH  MathSciNet  Google Scholar 

  24. 24

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

    MATH  MathSciNet  Google Scholar 

  25. 25

    Ponce G.: Remarks on a paper: “Remarks on the breakdown of smooth solutions for the 3-D Euler equations”. Commun. Math. Phys. 98(3), 349–353 (1985)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  26. 26

    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)

    MATH  Article  MathSciNet  Google Scholar 

  27. 27

    Salvi R., Straskraba I.: Global existence for viscous compressible fluids and their behavior as t → ∞. J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 40, 17–51 (1993)

    MATH  MathSciNet  Google Scholar 

  28. 28

    Serre D.: Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Sér. I Math. 303, 639–642 (1986)

    MATH  MathSciNet  Google Scholar 

  29. 29

    Serre D.: Sur l’équation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur. C. R. Acad. Sci. Paris Sér. I Math. 303, 703–706 (1986)

    MATH  MathSciNet  Google Scholar 

  30. 30

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

    MATH  Article  MathSciNet  Google Scholar 

  31. 31

    Vaigant V.A., Kazhikhov A.V.: On the existence of global solutions to two-dimensional Navier-Stokes equations of a compressible viscous fluid. Siberian Math. J. 36(6), 1108–1141 (1995)

    Article  MathSciNet  Google Scholar 

  32. 32

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

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xiangdi Huang.

Additional information

Communicated by P. Constantin

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Huang, X., Li, J. & Xin, Z. Blowup Criterion for Viscous Baratropic Flows with Vacuum States. Commun. Math. Phys. 301, 23–35 (2011). https://doi.org/10.1007/s00220-010-1148-y

Download citation

Keywords

  • Weak Solution
  • Global Existence
  • Strong Solution
  • Smooth Solution
  • Local Existence