Journal of Mathematical Fluid Mechanics

, Volume 15, Issue 3, pp 431–437

A Blowup Criterion for Ideal Viscoelastic Flow



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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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(1), 61–66 (1984)MathSciNetADSMATHCrossRefGoogle Scholar
  2. 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)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 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, 1986Google Scholar
  4. 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)MathSciNetMATHGoogle Scholar
  5. 5.
    Lei Z., Liu C., Zhou Y.: Global solutions for incompressible viscoelastic fluids. Arch. Ration. Mech. ~Anal. 188(3), 371–398 (2008)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Lei Z., Masmoudi N., Zhou Y.: Remarks on the blowup criteria for Oldroyd models. J. Differ. Equ. 248(2), 328–341 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Lin F.-H., Liu C., Zhang P.: On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 58(11), 1437–1471 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 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)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Majda, A.J., Bertozzi, A.L.: Vorticity and incompressible flow. In: Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)Google Scholar
  10. 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)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations