Skip to main content
SpringerLink
Log in

Global Solutions for Incompressible Viscoelastic Fluids

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We prove the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near- equilibrium initial data. The results hold in both two- and three-dimensional spaces. The results and methods presented in this paper are also valid for a wide range of elastic complex fluids, such as magnetohydrodynamics, liquid crystals, and mixture problems.

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. Agemi R. (2000) Global existence of nonlinear elastic waves. Invent. Math. 142(2): 225–250

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Alinhac S. (1995) Blowup for Nonlinear Hyperbolic Equations. Birkhäuser Boston, Boston

    MATH  Google Scholar 

  3. Alinhac S. (2001) The null condition for quasilinear wave equations in two space dimensions. I. Invent. Math. 145(3): 597–618

    MathSciNet  MATH  Google Scholar 

  4. Alinhac S. (2001) The null condition for quasilinear wave equations in two space dimensions. II. Am. J. Math. 123(6): 1071–1101

    Article  MathSciNet  MATH  Google Scholar 

  5. Byron Bird, R., Curtiss, C.F., Armstrong, R.C., Hassager, O.: Dynamics of polymeric liquids. In: Kinetic Theory, vol. 2, 2nd edn. Wiley Interscience, New York, 1987

  6. Chen Y., Zhang P. (2006) The global Existence of Small Solutions to the Incompressible Viscoelastic Fluid System in 2 and 3 space dimensions. Comm. Partial Differ. Equ. 31(10–12): 1743–1810

    Google Scholar 

  7. Christodoulou D. (1986) Global existence of nonlinear hyperbolic equations for small data. Commun. Pure Appl. Math. 39: 267–286

    Article  MathSciNet  MATH  Google Scholar 

  8. Dafermos C. (2000) Hyperbolic Conservation Laws in Continuum Physics. Springer, Heidelberg

    MATH  Google Scholar 

  9. Davidson, P.A.: An Introduction to Magnetohydrodynamics. Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001

  10. de Gennes P. (1976) Physics of Liquid Crystals. Oxford University Press, London

    Google Scholar 

  11. Gurtin M.E. (1981) An Introduction to Continuum Mechanics. Academic, New York

    MATH  Google Scholar 

  12. Joseph, D.: Instability of the rest state of fluids of arbitrary grade greater than one. Arch. Ration. Mech. Anal. 75(3), 251–256 (1980/1981)

    Google Scholar 

  13. Kawashima S., Shibata Y. (1992) Global existence and exponential stability of small solutions to nonlinear viscoelasticity. Commun. Math. Phys. 148: 189–208

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Klainerman S. (1985) Uniform decay estimates and the Lorentz invariance of the classical wave equation. Commun. Pure Appl. Math. 38: 321–332

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Klainerman S. (1986) The null condition and global existence to nonlinear wave equations. Lect. Appl. Math. 23: 293–326

    MathSciNet  Google Scholar 

  16. Klainerman S., Majda A. (1981) Singular limits of quasilinear hyperbolic system with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34: 481–524

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Klainerman S., Sideris T.C. (1996) On almost global existence for nonrelativistic wave equations in 3D. Commun. Pure Appl. Math. 49: 307–322

    Article  MathSciNet  MATH  Google Scholar 

  18. Ladyzhenskaya O.A., Seregin G.A. (1999) On the regularity of solutions of two- dimensional equations of the dynamics of fluids with nonlinear viscosity. Zapiski Nauchn. Semin. POMI 259: 145–166

    Google Scholar 

  19. Larson R.G. (1995) The Structure and Rheology of Complex Fluids. Oxford University Press, New York

    Google Scholar 

  20. Lei Z. (2006) Global existence of classical solutions for some Oldroyd-B model via the incompressible limit. Chin. Ann. Math. Ser. B 27(5): 565–580

    Article  MATH  Google Scholar 

  21. Lei Z., Liu C., Zhou Y. (2007) Global Existence for a 2D Incompressible Viscoelastic Model with Small Strain. Comm. Math. Sci. 5(3): 545–561

    MathSciNet  Google Scholar 

  22. Lei Z., Zhou Y. (2005) Global existence of classical solutions for 2D Oldroyd model via the incompressible limit. SIAM J. Math. Anal. 37(3): 797–814

    MathSciNet  MATH  Google Scholar 

  23. Lin F.H., Liu C. (1995) Nonparabolic dissipative systems modelling the flow of liquid crystals. Commun. Pure Appl. Math. 48(5): 501–537

    Article  MathSciNet  MATH  Google Scholar 

  24. Lin F.H., Liu C. (2000) Existence of solutions for Erichsen-Leslie system. Arch. Ration. Mech. Anal. 154(2): 135–156

    Article  MathSciNet  MATH  Google Scholar 

  25. Lin F.H., Liu C., Zhang P. (2005) On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 58(11): 1437–1471

    Article  MathSciNet  MATH  Google Scholar 

  26. Lions J.L. (1978) On some questions in boundary value problems of mathematical physics. In: Contemporary Development in Continuum Mechanics and PDEs. North-Holland, Amsterdam

  27. Liu C., Walkington N.J. (2001) An Eulerian description of fluids containing visco- hyperelastic particles. Arch. Ration. Mech. Anal. 159: 229–252

    Article  MathSciNet  MATH  Google Scholar 

  28. Málek J., Necas J., Rajagopal K.R. (2002) Global analysis of solutions of the flows of fluids with pressure-dependent viscosities. Arch. Ration. M ech. Anal. 165(3): 243–269

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Renardy M., Hrusa W.J., Nohel J.A. (1987) Mathematical Problems in Viscoelasticity. Longman Scientific and Technical, Wiley, New York

    MATH  Google Scholar 

  31. Schowalter W.R. (1978) Mechanics of Non-Newtonian fluids. Pergamon, New York

    Google Scholar 

  32. Sideris T.C. (2000) Nonresonance and global existence of prestressed nonlinear elastic waves. Ann. Math. 151: 849–874

    Article  MathSciNet  MATH  Google Scholar 

  33. Sideris T.C., Tu S.Y. (2001) Global existence for system of nonlinear wave equations in 3D with multiple speeds. SIAM J. Math. Anal. 33: 477–488

    Article  MathSciNet  MATH  Google Scholar 

  34. Sideris T.C., Thomases B. (2004) Global existence for 3D incompressible isotropic elastodynamics via the incompressible limit. Commun. Pure Appl. Math. 57: 1–39

    Article  Google Scholar 

  35. Sideris, T.C., Thomases, B.: Global Existence for 3D incompressible isotropic elastodynamics. Commun. Pure Appl. Math. (2007, in press)

  36. Slemrod M. (1999) Constitutive relations for Rivlin-Erichsen fluids bases on generalized rational approximation. Arch. Ration. Mech. Anal. 146(1): 73–93

    Article  MathSciNet  MATH  Google Scholar 

  37. Teman R. (1977) Navier–Stokes Equations. North-Holland, Amsterdam

    Google Scholar 

  38. Yue P., Feng J., Liu C., Shen J. (2004) A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515: 293–317

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Zhen lei

  2. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, People’s Republic of China

    Zhen lei

  3. Department of Mathematics, Pennsylvania State University, State college, PA, 16802, USA

    Chun Liu

  4. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Yi Zhou

  5. Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Shanghai, People’s Republic of China

    Yi Zhou

Authors
  1. Zhen lei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhen lei.

Additional information

Communicated by A. Bressan

Rights and permissions

Reprints and permissions

About this article

Cite this article

lei, Z., Liu, C. & Zhou, Y. Global Solutions for Incompressible Viscoelastic Fluids. Arch Rational Mech Anal 188, 371–398 (2008). https://doi.org/10.1007/s00205-007-0089-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-007-0089-x

Keywords

Search

Navigation