Initial value problems for viscoelastic liquids

  • M. Renardy
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 195)


Cauchy problems for equations modelling non-Newtonian fluids are discussed and recent existence theorems for classical solutions, based on semigroup methods, are presented. Such existence results depend in a crucial manner on the symbol of the leading differential operator. Both “parabolic” and “hyperbolic” cases are discussed. In general, however, the leading differential operator may be of non-integral order, arising from convolution with a singular kernel. This has interesting implications concerning the propagation of singularities. In particular, there are cases where C-smoothing coexists with finite wave speeds.


Evolution Problem Stokes Operator Quasilinear Parabolic Equation Fading Memory Viscoelastic Liquid 
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  1. [1]
    S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 12 (1959), 623–727 and 17 (1964), 35–92.CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    B. Bernstein, E. A. Kearsley and L. J. Zapas, A study of stress relaxation with finite strain. Trans. Soc. Rheology 7 (1963), 391–410.ADSCrossRefMATHGoogle Scholar
  3. [3]
    L. Boltzmann, Zur Theorie der elastischen Nachwirkung, Ann. Phys. 7 (1876), Ergänzungsband, 624–654.Google Scholar
  4. [4]
    B. D. Coleman and M. E. Gurtin, Waves in materials with memory II, Arch. Rat. Mech. Anal. 19 (1965), 239–265.CrossRefMATHGoogle Scholar
  5. [5]
    C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. PDE 4 (1979), 219–278.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    C. M. Dafermos and J. A. Nohel, A nonlinear hyperbolic Volterra equation in viscoelasticity, Amer. J. Math., Supplement (1981), 87–116.Google Scholar
  7. [7]
    M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems, J. Chem. Soc. Faraday 74 (1978), 1789–1832 and 75 (1979), 38–54.CrossRefGoogle Scholar
  8. [8]
    Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Math. Z. 178 (1981), 297–329.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    A. E. Green and R. S. Rivlin, Nonlinear materials with memory, Arch. Rat. Mech. Anal. 1 (1957), 1–21.CrossRefMATHGoogle Scholar
  10. [10]
    W. J. Hrusa, A nonlinear functional differential equation in Banach space with applications to materials with fading memory, Arch. Rat. Mech. Anal.Google Scholar
  11. [11]
    T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasilinear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rat. Mech. Anal. 63 (1976), 273–294.MATHGoogle Scholar
  12. [12]
    T. Kato, Linear equations of “hyperbolic” type I, J. Fac. Sci. Univ. Tokyo 17 (1970), 241–258 and II, J. Math. Soc. Japan 25 (1973), 648–666.MATHMathSciNetGoogle Scholar
  13. [13]
    T. Kato, Quasi-linear equations of evolution with application to partial differential equations, in: W. N. Everitt (ed.), Spectral Theory of Differential Equations, Springer Lecture Notes in Mathematics 448, 1975, 25–70.Google Scholar
  14. [14]
    A. Kaye, Cc A Note 134, The College of Aeronautics, Cranfield, Bletchley, England 1962.Google Scholar
  15. [15]
    J. U. Kim, Global smooth solutions of the equations of motion of a nonlinear fluid with fading memory, Arch. Rat. Mech. Anal. 79 (1982), 97–130.CrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    A Narain and D. D. Joseph, Linearized dynamics for step jumps of velocity and displacement of shearing flows of a simple fluid, Rheol. Acta 21 (1982), 228–250.CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    W. Noll, A mathematical theory of the mechanical behavior of continuous media, Arch. Rat. Mech. Anal. 2 (1958), 197–226.CrossRefMATHMathSciNetGoogle Scholar
  18. [18]
    J. G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. London A 200 (1950), 523–541.ADSCrossRefMATHMathSciNetGoogle Scholar
  19. [19]
    M. Renardy, A quasilinear parabolic equation describing the elongation of thin filaments of polymeric liquids, SIAM J. Math. Anal. 13 (1982), 226–238.CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    M. Renardy, A class of quasilinear parabolic equations with infinite delay and application to a problem of viscoelasticity, J. Diff. Eq. 48 (1983), 280–292.ADSCrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    M. Renardy, Local existence theorems for the first and second initial-boundary value problems for a weakly non-Newtonian fluid, Arch. Rat. Mech. Anal. 83 (1983), 229–244.CrossRefMATHMathSciNetGoogle Scholar
  22. [22]
    M. Renardy, Singularly perturbed hyperbolic evolution problems with infinite delay and an application to polymer rheology, SIAM J. Math. Anal. 15 (1984).Google Scholar
  23. [23]
    M. Renardy, A local existence and uniqueness theorem for a K-BKZ fluid, submitted to Arch. Rat. Mech. Anal.Google Scholar
  24. [24]
    M. Renardy, Some remarks on the propagation and non-propagation of discontinuities in linearly viscoelastic liquids, Rheol. Acta 21 (1982), 251–254.CrossRefMATHMathSciNetGoogle Scholar
  25. [25]
    M. Renardy, On the domain space for constitutive laws in linear viscoelasticity, Arch. Rat. Mech. Anal.Google Scholar
  26. [26]
    P. E. Rouse, A theory of the linear viscoelastic properties of dilute solutions of coiling polymers, J. Chem. Phys. 21 (1953), 1271–1280.ADSCrossRefGoogle Scholar
  27. [27]
    I. M. Rutkevich, The propagation of small perturbations in a viscoelastic fluid, J. Appl. Math. Mech. (1970), 35–50.Google Scholar
  28. [28]
    P. E. Sobolevskii, Equations of parabolic type in a Banach space, AMS Transl. 49 (1966), 1–62.Google Scholar
  29. [29]
    V. A. Solonnikov, General boundary value problems for DouglisNirenberg elliptic systems, Proc. Steklov Inst. 92 (1967), 269–339.Google Scholar
  30. [30]
    V. A. Solonnikov, Estimates of the solutions of the nonstationary linearized system of Navier-Stokes equations, Proc. Steklov Inst. 70 (1964), 213–317.Google Scholar
  31. [31]
    B. H. Zimm, Dynamics of polymer molecules in dilute solution: viscoelasticity, flow birefringence and dielectric loss, J. Chem. Phys. 24 (1956), 269–278.ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • M. Renardy
    • 1
  1. 1.Mathematics Research Center and Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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