Skip to main content
SpringerLink
Log in

A Blowup Criterion for Ideal Viscoelastic Flow

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

We establish an analog of the Beale–Kato–Majda criterion for singularities of smooth solutions of the system of PDE arising in the Oldroyd model for ideal viscoelastic flow.

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Caflisch R.E., Klapper I., Steele G.: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Commun. Math. Phys. 184(2), 443–455 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Kato, T.: Remarks on the Euler and Navier–Stokes equations in \({\mathbb{R}^2}\) . Nonlinear functional analysis and its applications, Part 2 (Berkeley, California, 1983), 1–7, Proceedings of Symposium in Pure Mathematics, 45, Part 2, American Mathematical Society, Providence, RI, 1986

  4. Kupferman R., Mangoubi C., Titi E.S.: A Beale–Kato–Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime. Commun. Math. Sci 6(1), 235–256 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Lei Z., Liu C., Zhou Y.: Global solutions for incompressible viscoelastic fluids. Arch. Ration. Mech. ~Anal. 188(3), 371–398 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lei Z., Masmoudi N., Zhou Y.: Remarks on the blowup criteria for Oldroyd models. J. Differ. Equ. 248(2), 328–341 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lin F.-H., Liu C., Zhang P.: On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 58(11), 1437–1471 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lin F., Zhang P.: On the initial-boundary value problem of the incompressible viscoelastic fluid system. Commun. Pure Appl. Math. 61(4), 539–558 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Majda, A.J., Bertozzi, A.L.: Vorticity and incompressible flow. In: Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)

  10. Sideris T.C., Thomases B.: Global existence for three-dimensional incompressible isotropic elastodynamics via the incompressible limit. Commun. Pure Appl. Math. 58(6), 750–788 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012-1185, USA

    Xianpeng Hu & Ryan Hynd

Authors
  1. Xianpeng Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ryan Hynd
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ryan Hynd.

Additional information

Communicated by G. P. Galdi

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1004733.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hu, X., Hynd, R. A Blowup Criterion for Ideal Viscoelastic Flow. J. Math. Fluid Mech. 15, 431–437 (2013). https://doi.org/10.1007/s00021-012-0124-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00021-012-0124-z

Keywords

Search

Navigation