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Global Solution to the Incompressible Oldroyd-B Model in the Critical L p Framework: the Case of the Non-Small Coupling Parameter

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Abstract

This paper is dedicated to the global well-posedness issue of the incompressible Oldroyd-B model in the whole space \({\mathbb{R}^d}\) with \({d \geqq 2}\) . It is shown that this set of equations admits a unique global solution in a certain critical L p-type Besov space provided that the initial data, but not necessarily the coupling parameter, is small enough. As a consequence, even through the coupling effect between the equations of velocity u and the symmetric tensor of constrains τ is not small, one may construct the unique global solution to the Oldroyd-B model for a class of large highly oscillating initial velocity. The proof relies on the estimates of the linearized systems of (u, τ) and \({(u, \mathbb{P}{\rm div}\tau)}\) which may be of interest for future works. This result extends the work by Chemin and Masmoudi (SIAM J Math Anal 33:84–112, 2001) to the non-small coupling parameter case.

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References

  1. Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 343. Springer, Heidelberg, (2011)

  2. Bony J.M.: Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires. Annales Scinentifiques de l’école Normale Supérieure 14, 209–246 (1981)

    MATH  MathSciNet  Google Scholar 

  3. Cannone M.: A generalization of a theorem by Kato on Navier–Stokes equations. Rev. Mat. Iberoamericana 13, 515–541 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cannone, M., Meyer, Y., Planchon, F.: Solutions autosimilaires des équations de Navier–Stokes. In: S éminaire “Équations aux Dérivées Partielles” de l’École Polytechnique, Exposé VIII, (1993–1994)

  5. Charve F., Danchin R.: A global existence result for the compressible Navier–Stokes Navier–Stokes equations in the critical L p framwork. Arch. Rational Mech. Anal. 198, 233–271 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Chemin, J.Y.: Localization in Fourier space and Navier–Stokes system. Phase Space Analysis of Partial Differential Equations. Proceedings 2004, CRM series, Pisa, 53–136, (2004)

  7. Chemin J.Y., Lerner N.: Flot de champs de vecteurs non Lipschitziens et équations de Navier–Stokes. J. Differential Equations 121, 314–228 (1995)

    Google Scholar 

  8. Chemin J.Y.: Théorèmes d’ unicité pour le système de Navier–Stokes tridimensionnel. Journal d’Analyse Mathématique 77, 25–50 (1999)

    Article  MathSciNet  Google Scholar 

  9. 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  MATH  MathSciNet  Google Scholar 

  10. Chen Q.L., Miao C.X.: Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces. Nonlinear Anal., 68, 1928–1939 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen Q.L., Miao C.X., Zhang Z.F.: Global well-posedness for compressible Navier–Stokes equations with highly oscillating initial velocity. Comm. Pure Appl. Math. 63, 1173–1224 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Constantin P., Kliegl M.: Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress. Arch. Rational Mech. Anal. 206, 725–740 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Danchin R.: Density-dependent incompressible viscous fluids in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A 133, 1311–1334 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Danchin R.: A Lagrangian approach for the incompressible Navier–Stokes equations with varibable density. Comm. Pure Appl. Math. 65, 1458–1480 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fang D.Y., Hieber M., Zi R.Z.: Global existence results for Oldroyd-B Fluids in exterior domains: The case of non-small coupling parameters. Math. Ann. 357, 687–709 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fang, D.Y., Zi, IR.Z.: Incompressible limit of Oldroyd-B fluids in the whole space, Preprint, (2013)

  17. 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 

  18. Fujita H., Kato T.: On the Navier–Stokes initial value problem I. Arch. Rational Mech. Anal. 16, 269–315 (1964)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. 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  MATH  MathSciNet  Google Scholar 

  20. Guillopé C., Salloum Z., Talhouk R.: Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete Contin. Dyn. Syst. Ser. B 14, 1001–1028 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    ADS  MATH  MathSciNet  Google Scholar 

  22. Hmidi T.: Régularité höldérienne des poches de tourbillon visqueuses. J. Math. Pures Appl. 84, 1455–1495 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hoff D., Zumbrun K.: Multi-dimensional diffusion waves for the Navier–Stokes equations of compressible flow. Indiana Univ. Math. J. 44, 603–676 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Koch H., Tataru D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157, 22–35 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. 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, 235–256 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  28. Lei Z., Liu C., Zhou Y.: Global existence for a 2D incompressible viscoelastic model with small strain. Commun. Math. Sci. 5, 595–616 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  31. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  33. 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  MATH  MathSciNet  Google Scholar 

  34. Lions P.L., Masmoudi N.: Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. Ser. B 21, 131–146 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  35. Molinet L., Talhouk R.: On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law. Nonlinear Differ. Equ. Appl. 11, 349–359 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Qian J.Z.: Well-posedness in critical spaces for incompressible viscoelastic fluid system. Nonlinear Anal. 72, 3222–3234 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Talhouk, R.: Analyse Mathématique de Quelques Écoulements de Fluides Viscoélastiques, Thèse, Université Paris-Sud, (1994)

  39. Vishik M.: Hydrodynamics in Besov spaces. Arch. Rational Mech. Anal. 145, 197–214 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  40. Weissler F.: The Navier–Stokes initial value problem in L p. Arch. Rational Mech. Anal. 74, 219–230 (1980)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  41. Zhang T., Fang D.Y.: Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical L p framework. SIAM J. Math. Anal. 44, 2266–2288 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  42. Zi, R.Z.: Global solutions to Oldroyd-B model in hybrid Besov spaces. Preprint, (2013)

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Ruizhao Zi

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

    Daoyuan Fang & Ting Zhang

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  1. Ruizhao Zi
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  2. Daoyuan Fang
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  3. Ting Zhang
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Correspondence to Ting Zhang.

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Communicated by F. Lin

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Zi, R., Fang, D. & Zhang, T. Global Solution to the Incompressible Oldroyd-B Model in the Critical L p Framework: the Case of the Non-Small Coupling Parameter. Arch Rational Mech Anal 213, 651–687 (2014). https://doi.org/10.1007/s00205-014-0732-2

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  • DOI: https://doi.org/10.1007/s00205-014-0732-2

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