Mathematische Annalen

, Volume 357, Issue 2, pp 687–709 | Cite as

Global existence results for Oldroyd-B fluids in exterior domains: the case of non-small coupling parameters

  • Daoyuang Fang
  • Matthias HieberEmail author
  • Ruizhao Zi


Consider the set of equations describing Oldroyd-B fluids in an exterior domain. It is shown that this set of equations admits a unique, global solution in a certain function space provided the initial data, but not necessarily the coupling constant, is small enough.

Mathematics Subject Classification (2000)

35Q35 76D03 76D05 



This work was carried out while the first and the third authors are visiting the Department of Mathematics at the Technical University of Darmstadt. They would express their gratitude to Prof. Matthias Hieber for his kind hospitality and the Deutsche Forschungsgemeinschaft (DFG) for financial support. We would like thank Paolo Galdi for stimulating discussion concerning Oldroyd-B fluids and the third author also would like to thank Tobias Hansel for his sincere help. Daoyuan Fang and Ruizhao Zi were partially supported by NSFC 11271322, 10931007 and ZNSFC Z6100217.


  1. 1.
    Chemin, J.Y., Masmoudi, N.: About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33, 84–112 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Fernández-Cara, E., Guillén, F., Ortega, R.: Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind. Ann. Scuola Norm. Sup. Pisa 26, 1–29 (1998)zbMATHGoogle Scholar
  3. 3.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. I. Springer, New York (1994)Google Scholar
  4. 4.
    Girault, V., Raviart, P.A.: Finite Elements Methods for Navier-Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)CrossRefGoogle Scholar
  5. 5.
    Guillopé, C., Saut, J.C.: Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. 15, 849–869 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hieber, M., Naito, Y., Shibata, Y.: Global existence results for Oldroyd-B fluids in exterior domains. J. Diff. Equ. 252, 2617–2629 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kreml, O., Pokorny M.: On the local strong solutions for a system describing the flow of a viscoelastic fluid. In: Nonlocal and Abstract Parabolic Equations and their Applications, Banach Center Publ., vol. 86, Polish Acad. Sci. Inst. Math., Warsaw, pp. 195–206 (2009)Google Scholar
  8. 8.
    Lei, Z.: Global existence of cloassical solutions for some Oldroyd-B model via the incompressible limit. Chin. Ann. Math. 27, 565–580 (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Lei, Z.: On 2D viscoelasticity with small strain. Arch. Ration. Mech. Anal. 198, 13–37 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lei, Z., Liu, C., Zhou, Y.: Global solutions for incompressible viscoelastic fluids. Arch. Ration. Mech. Anal. 188, 371–398 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lei, Z., Masmoudi, N., Zhou, Y.: Remarks on the blowup criteria for Oldroyd models. J. Diff. Equ. 248, 328–341 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lei, Z., Zhou, Y.: Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit. SIAM J. Math. Anal. 37, 797–814 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lin, F.H., Liu, C., Zhang, P.: On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math. 58, 1437–1471 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lin, F.H., Zhang, P.: On the initial-boundary value problem of the incompressible viscoelastic fluid system. Comm. Pure Appl. Math. 61, 539–558 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lions, P.L., Masmoudi, N.: Global solutions for some Oldroyd models of non-Newtonian flows. Chin. Ann. Math. Ser. B 21, 131–146 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Molinet, L., Talhouk, R.: On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law. Nonlinear Diff. Equ. Appl. 11, 349–359 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oldroyd, J.G.: Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. Roy. Soc. Lond. 245, 278–297 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Talhouk, R.: Existence locale et unicité d’écoulements de fluides viscoélastiques dans des domaines non bornés (Local existence and uniqueness of viscoelastic fluid flows in unbounded domains). C. R. Acad. Sci. Paris 328, 87–92 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouChina
  2. 2.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Center of Smart InterfacesDarmstadtGermany

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