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

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Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  3. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. I. Springer, New York (1994)

    Google Scholar 

  4. Girault, V., Raviart, P.A.: Finite Elements Methods for Navier-Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)

    Book  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hieber, M., Naito, Y., Shibata, Y.: Global existence results for Oldroyd-B fluids in exterior domains. J. Diff. Equ. 252, 2617–2629 (2012)

    MathSciNet  MATH  Google Scholar 

  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)

  8. Lei, Z.: Global existence of cloassical solutions for some Oldroyd-B model via the incompressible limit. Chin. Ann. Math. 27, 565–580 (2006)

    Article  MATH  Google Scholar 

  9. Lei, Z.: On 2D viscoelasticity with small strain. Arch. Ration. Mech. Anal. 198, 13–37 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Oldroyd, J.G.: Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. Roy. Soc. Lond. 245, 278–297 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

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.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou,  310027, China

    Daoyuang Fang & Ruizhao Zi

  2. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, Darmstadt,  64289, Germany

    Matthias Hieber

  3. Center of Smart Interfaces, Petersenstr. 32, Darmstadt,  64287, Germany

    Matthias Hieber

Authors
  1. Daoyuang Fang
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  2. Matthias Hieber
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  3. Ruizhao Zi
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Correspondence to Matthias Hieber.

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Fang, D., Hieber, M. & Zi, R. Global existence results for Oldroyd-B fluids in exterior domains: the case of non-small coupling parameters. Math. Ann. 357, 687–709 (2013). https://doi.org/10.1007/s00208-013-0914-5

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  • DOI: https://doi.org/10.1007/s00208-013-0914-5

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