Skip to main content
SpringerLink
Log in

On the Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In two space dimension, we prove the global existence of smooth solutions to a coupled microscopic-macroscopic co-rotational FENE dumbbell model which arises from the kinetic theory of diluted solutions of polymeric liquids with noninteracting polymer chains.

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. (1984). Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94: 61–66

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Bahouri H., Chemin J.Y. (1994). Équations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluids. Arch. Ration. Mech. Anal. 127: 159–181

    Article  MATH  MathSciNet  Google Scholar 

  3. Barrett J.W., Schwab C., Süli E. (2005). Existence of global weak solutions for some polymeric flow models. Math. Models Methods Appl. Sci. 15: 939–983

    Article  MATH  MathSciNet  Google Scholar 

  4. Bird, R.B., Curtis, C.F., Armstrong, R.C., Hassager, O.: Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory. New York: Weiley Interscience, 1987

  5. Bony J.M. (1981). Calcul symbolique et propagation des singularités pour les quations aux drivées partielles non linéaires. Ann. Sci. École Norm. Sup., (4), 14: 209–246

    MATH  MathSciNet  Google Scholar 

  6. Chemin, J.Y.: Perfect Incompressibe Fluids. New York: Oxford University Press, 1998

  7. Chemin J.Y. (1999). Théormes d’unicité pour le systéme de Navier-Stokes tridimensionnel. J. Anal. Math. 77: 27–50

    Article  MATH  MathSciNet  Google Scholar 

  8. Chemin J.Y., Masmoudi N. (2001). About lifespan of regular solutions of equations related to viscoelastic fluid. SIAM. J. Math. Anal. 33: 84–112

    Article  MATH  MathSciNet  Google Scholar 

  9. Constantin P., Fefferman C., Titi E., Zarnescu A. (2007). Regularity for coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systems. Commun. Math. Phys. 270: 789–811

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Constantin, P., Masmoudi, N.: Global well-posdness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D. Commun. Math. Phys., DOI:10.1007/s00220-007-0384-2

  11. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford: Oxford Science Publication, 1986

  12. Du Q., Liu C., Yu P. (2005). FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul. 4: 709–731

    Article  MATH  MathSciNet  Google Scholar 

  13. Weinan E., Li T.J., Zhang P.W. (2004). Well-posedness for the dumbbell model of polymeric fluids. Commun. Math. Phys. 248(2): 409–427

    MATH  ADS  Google Scholar 

  14. Jourdain B., Leliévre T., Le Bris C. (2004). Existence of solution for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal. 209: 162–193

    Article  MATH  MathSciNet  Google Scholar 

  15. Jourdain B., Leliévre T., Le Bris C., Otto F. (2006). Long-time asymptotics of a multiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal. 181: 97–148

    Article  MATH  MathSciNet  Google Scholar 

  16. Lin, F., Zhang, P.: The FENE dumbell model near equilibrium. Acta Math. Sin. (2007, in press)

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Lin F., Liu C., Zhang P. (2007). On a Micro-Macro Model for Polymeric Fluids near Equilibrium. Comm. Pure Appl. Math. 60(6): 838–866

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  20. Matskewich T., Sobolevskii P.E. (1998). The sharp constant in Hardy’s inequality for complement of bounded domain. Nonlinear Anal. 33: 105–120

    Article  MATH  MathSciNet  Google Scholar 

  21. Planchon F. (2003). An extension of the Beale-Kato-Majda criterion for the Euler equations. Commun. Math. Phys. 232: 319–326

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. Renardy M. (1991). An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal. 22: 313–327

    Article  MATH  MathSciNet  Google Scholar 

  23. Triebel, H.: Theory of Function Spaces. Monograph in mathematics, Vol. 78, Basel: Birkhauser Verlag, 1983

  24. Zhang H., Zhang P.W. (2006). Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal. 181: 373–400

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Academy of Mathematics & Systems Science, The Chinese Academy of Sciences, 100080, Beijing, China

    Ping Zhang

  2. Courant Institute, New York University, 10012, New York, NY, USA

    Fanghua Lin & Ping Zhang

  3. School of Mathematical Science, Peking University, 100871, Beijing, China

    Zhifei Zhang

Authors
  1. Fanghua Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ping Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhifei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ping Zhang.

Additional information

Communicated by P. Constantin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lin, F., Zhang, P. & Zhang, Z. On the Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model. Commun. Math. Phys. 277, 531–553 (2008). https://doi.org/10.1007/s00220-007-0385-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0385-1

Keywords

Search

Navigation