Skip to main content
SpringerLink
Log in

On the blow up criterion for the 2-D compressible Navier-Stokes equations

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

Motivated by [10], we prove that the upper bound of the density function j9 controls the finite time blow up of the classical solutions to the 2-D compressible isentropic Navier-Stokes equations. This result generalizes the corresponding result in [3] concerning the regularities to the weak solutions of the 2-D compressible Navier-Stokes equations in the periodic domain.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  3. B. Desjardins: Regularity of weak solutions of the compressible isentropic Navier-Stokes equations. Comm. P. D. E. 22 (1997), 977–1008.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. N. Itaya: On the Cauchy problem for the system of fundamental equations describing movement of compressible viscous fluids. Kōdai Math. Sem. Rep. 23 (1971), 60–120.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Kozono and Y. Taniuchi: Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Comm. Math. Phys. 214 (2000), 191–200.

    Article  MATH  MathSciNet  Google Scholar 

  7. P. L. Lions: Mathematical Topics in Fluid Mechanics, Vol 2. Compressible Models. Oxford lecture series in Mathematics and its Applications, 10, Oxford Sciences Publications. The Clarendon Press, Oxford University Press, New York, 1998.

    Google Scholar 

  8. A. Tani: On the first initial-boundary value problem of compressible viscous fluid motion. Publ. RIMS. Kyoto Univ. 13 (1977), 193–253.

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. V. A. Vaigant and A. V. Kazhikhov: On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid. Sibirsk. Mat. Zh. 36 (1995), 1283–1316 translation in it Siberian Math. J. 36 (1995). (In Russian.)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Central University of Finance and Economics, Beijing, 100081, China

    Lingyu Jiang & Yidong Wang

Authors
  1. Lingyu Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yidong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lingyu Jiang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jiang, L., Wang, Y. On the blow up criterion for the 2-D compressible Navier-Stokes equations. Czech Math J 60, 195–209 (2010). https://doi.org/10.1007/s10587-010-0009-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-010-0009-3

Keywords

MSC 2010

Search

Navigation