Skip to main content
SpringerLink
Log in

Well-Posedness for the Dumbbell Model of Polymeric Fluids

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

Abstract

The dumbbell model is a coupled hydrodynamic-kinetic model for polymeric fluids in which the configurations of the dumbbells are described by stochastic differential equations. We prove well-posedness of this model by deriving directly a priori estimates on the stochastic model. Our results can be used to analyze stochastic simulation methods such as the ones that are based on Brownian configuration fields.

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. Bird, R.B., Curtiss, C.F., Armstrong, R.C., Hassager, O.: Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory. New York: John Wiley, 1987

  2. Doi, M., Edwards, S.F.: The theory of polymer dynamics. Oxford: Oxford University Press, 1986

  3. E, W., Li, T., Zhang, P.: Convergence of a stochastic method for the modeling of polymeric fluids. Acta Mathematicae Applicatae Sinica, English Series, 18(4), 529–536 (2002)

    Google Scholar 

  4. E, W., Li, T., Zhang, P.: Convergence analysis of BCF methods. In preparation

  5. Guillope, C., Saut, J.C.: Existence results for the flow of viscoelastic fluids with a differnetial costitutive law. Nonlinear Anal. 15, 849–869 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hulsen, M.A., van Heel, A.P.G., van den Brule, B.H.A.A.: Simulation of viscoelastic flows using Brownian configuration fields. J. Non-Newtonian Fluid Mech. 70, 79–101 (1997)

    Article  Google Scholar 

  7. Jourdain, B., Lelièvre, T., Le Bris.: C.L Numerical analysis of micro-macro simulations of polymeric fluid flows: A simple case. Math. Models and Methods in Appl.Sci. 12(9), 1205–1243 (2002)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  9. Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus (2nd edition), New York: Springer-Verlag, 1991

  10. Laso, M., Öttinger, H.C.: Calculation of viscoelastic flow using molecular models: The CONNFFESSIT approach. J. Non-Newtonian Fluid Mech. 47, 1–20 (1993)

    Article  MATH  Google Scholar 

  11. Li, T., Zhang, H., Zhang, P.: Local existence for the dumbbell model of polymeric fluids. Comm. PDE 29, 903–923 (2004)

    Google Scholar 

  12. Li, T., Zhang, P., Zhou, X.: Well-posedness and numerical analysis of stochastic models for liquid crystal polymers. Submitted to Comm. Math. Sci.

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Majda, A.J.: Compressible fluid flow and systems of conservation laws in several space variables. New York: Springer-Verlag, 1984

  15. Öttinger, H.C.: Stochastic processes in polymeric liquids. Berlin-Heidelberg-New York: Springer-Verlag, 1996

  16. Renardy, M.: Local existence of solutions of the dirichlet initial-boundary value problem for incompressible hypoelastic materials. SIAM J. Math. Anal. 21, 1369–1385 (1990)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544, USA

    Weinan E

  2. School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China

    Weinan E

  3. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China

    Tiejun Li & Pingwen Zhang

Authors
  1. Weinan E
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tiejun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pingwen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by P. Constantin

Supported by ONR grant N00014-01-1-0674 and National Science Foundation of China through a Class B Award for Distinguished Young Scholars 10128102.

Partially supported by the special funds for Major State Research Projects G1999032804 and National Science Foundation of China for Distinguished Young Scholars 10225103.

Rights and permissions

Reprints and permissions

About this article

Cite this article

E, W., Li, T. & Zhang, P. Well-Posedness for the Dumbbell Model of Polymeric Fluids. Commun. Math. Phys. 248, 409–427 (2004). https://doi.org/10.1007/s00220-004-1102-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1102-y

Keywords

Search

Navigation