Archive for Rational Mechanics and Analysis

, Volume 188, Issue 3, pp 371–398 | Cite as

Global Solutions for Incompressible Viscoelastic Fluids

  • Zhen leiEmail author
  • Chun Liu
  • Yi Zhou


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.


Global Solution Global Existence Nonlinear Wave Equation Deformation Tensor Strain Energy Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Agemi R. (2000) Global existence of nonlinear elastic waves. Invent. Math. 142(2): 225–250CrossRefADSMathSciNetzbMATHGoogle Scholar
  2. 2.
    Alinhac S. (1995) Blowup for Nonlinear Hyperbolic Equations. Birkhäuser Boston, BostonzbMATHGoogle Scholar
  3. 3.
    Alinhac S. (2001) The null condition for quasilinear wave equations in two space dimensions. I. Invent. Math. 145(3): 597–618MathSciNetzbMATHGoogle Scholar
  4. 4.
    Alinhac S. (2001) The null condition for quasilinear wave equations in two space dimensions. II. Am. J. Math. 123(6): 1071–1101CrossRefMathSciNetzbMATHGoogle Scholar
  5. 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, 1987Google Scholar
  6. 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): 1743–1810Google Scholar
  7. 7.
    Christodoulou D. (1986) Global existence of nonlinear hyperbolic equations for small data. Commun. Pure Appl. Math. 39: 267–286CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Dafermos C. (2000) Hyperbolic Conservation Laws in Continuum Physics. Springer, HeidelbergzbMATHGoogle Scholar
  9. 9.
    Davidson, P.A.: An Introduction to Magnetohydrodynamics. Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001Google Scholar
  10. 10.
    de Gennes P. (1976) Physics of Liquid Crystals. Oxford University Press, LondonGoogle Scholar
  11. 11.
    Gurtin M.E. (1981) An Introduction to Continuum Mechanics. Academic, New YorkzbMATHGoogle Scholar
  12. 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. 13.
    Kawashima S., Shibata Y. (1992) Global existence and exponential stability of small solutions to nonlinear viscoelasticity. Commun. Math. Phys. 148: 189–208CrossRefADSMathSciNetzbMATHGoogle Scholar
  14. 14.
    Klainerman S. (1985) Uniform decay estimates and the Lorentz invariance of the classical wave equation. Commun. Pure Appl. Math. 38: 321–332CrossRefADSMathSciNetzbMATHGoogle Scholar
  15. 15.
    Klainerman S. (1986) The null condition and global existence to nonlinear wave equations. Lect. Appl. Math. 23: 293–326MathSciNetGoogle Scholar
  16. 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: 481–524CrossRefADSMathSciNetzbMATHGoogle Scholar
  17. 17.
    Klainerman S., Sideris T.C. (1996) On almost global existence for nonrelativistic wave equations in 3D. Commun. Pure Appl. Math. 49: 307–322CrossRefMathSciNetzbMATHGoogle Scholar
  18. 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: 145–166Google Scholar
  19. 19.
    Larson R.G. (1995) The Structure and Rheology of Complex Fluids. Oxford University Press, New YorkGoogle Scholar
  20. 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): 565–580CrossRefzbMATHGoogle Scholar
  21. 21.
    Lei Z., Liu C., Zhou Y. (2007) Global Existence for a 2D Incompressible Viscoelastic Model with Small Strain. Comm. Math. Sci. 5(3): 545–561MathSciNetGoogle Scholar
  22. 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): 797–814MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lin F.H., Liu C. (1995) Nonparabolic dissipative systems modelling the flow of liquid crystals. Commun. Pure Appl. Math. 48(5): 501–537CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Lin F.H., Liu C. (2000) Existence of solutions for Erichsen-Leslie system. Arch. Ration. Mech. Anal. 154(2): 135–156CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Lin F.H., Liu C., Zhang P. (2005) On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 58(11): 1437–1471CrossRefMathSciNetzbMATHGoogle Scholar
  26. 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, AmsterdamGoogle Scholar
  27. 27.
    Liu C., Walkington N.J. (2001) An Eulerian description of fluids containing visco- hyperelastic particles. Arch. Ration. Mech. Anal. 159: 229–252CrossRefMathSciNetzbMATHGoogle Scholar
  28. 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): 243–269CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Renardy M. (1991) An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal. 22: 313–327CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Renardy M., Hrusa W.J., Nohel J.A. (1987) Mathematical Problems in Viscoelasticity. Longman Scientific and Technical, Wiley, New YorkzbMATHGoogle Scholar
  31. 31.
    Schowalter W.R. (1978) Mechanics of Non-Newtonian fluids. Pergamon, New YorkGoogle Scholar
  32. 32.
    Sideris T.C. (2000) Nonresonance and global existence of prestressed nonlinear elastic waves. Ann. Math. 151: 849–874CrossRefMathSciNetzbMATHGoogle Scholar
  33. 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: 477–488CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Sideris T.C., Thomases B. (2004) Global existence for 3D incompressible isotropic elastodynamics via the incompressible limit. Commun. Pure Appl. Math. 57: 1–39CrossRefGoogle Scholar
  35. 35.
    Sideris, T.C., Thomases, B.: Global Existence for 3D incompressible isotropic elastodynamics. Commun. Pure Appl. Math. (2007, in press)Google Scholar
  36. 36.
    Slemrod M. (1999) Constitutive relations for Rivlin-Erichsen fluids bases on generalized rational approximation. Arch. Ration. Mech. Anal. 146(1): 73–93CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    Teman R. (1977) Navier–Stokes Equations. North-Holland, AmsterdamGoogle Scholar
  38. 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: 293–317CrossRefADSMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsNortheast Normal UniversityChangchunPeople’s Republic of China
  3. 3.Department of MathematicsPennsylvania State UniversityState collegeUSA
  4. 4.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China
  5. 5.Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of EducationShanghaiPeople’s Republic of China

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