Analysis of Generalized Newtonian Fluids

  • Michael Růžička
Part of the Lecture Notes in Mathematics book series (LNM, volume 2073)


In this paper we want to present an optimal existence result for the steady motion of generalized Newtonian fluids. Moreover, we present an optimal error estimate for a FEM approximation of the corresponding steady p-Stokes system. The presented results are based on long lasting cooperations with L. Berselli, L. Diening, J. Málek, A. Prohl and J. Wolf.



The author have been partially supported by the SFB/TR 71 “Geometric Partial Differential Equations”.


  1. 1.
    E. Acerbi, N. Fusco, Regularity for minimizers of nonquadratic functionals: the case 1 < p < 2. J. Math. Anal. Appl. 140(1), 115–135 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    E. Acerbi, N. Fusco, On the finite element approximation of p-Stokes systems. SIAM J. Numer. Anal. 50(2), 373–397 (2012). doi:10.1137/10080436XMathSciNetCrossRefGoogle Scholar
  3. 3.
    L. Belenki, L.C. Berselli, L. Diening, M. Růžička, On the finite element approximation of p-Stokes systems, preprint SFB/TR 71 (2010)Google Scholar
  4. 4.
    H. Bellout, F. Bloom, J. Nečas, Solutions for incompressible non–Newtonian fluids. C. R. Acad. Sci. Paris 317 795–800 (1993)zbMATHGoogle Scholar
  5. 5.
    L.C. Berselli, L. Diening, M. Růžička, Existence of strong solutions for incompressible fluids with shear dependent viscosities. J. Math. Fluid Mech. 12(1), 101–132 (2010). Online First, doi:10.1007/s00021-008-0277-yGoogle Scholar
  6. 6.
    R.B. Bird, R.C. Armstrong, O. Hassager, Dynamic of Polymer Liquids, 2nd edn. (Wiley, New York, 1987)Google Scholar
  7. 7.
    F. Brezzi, M. Fortin, in Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15 (Springer, New York, 1991)Google Scholar
  8. 8.
    M. Bulíček, L. Consiglieri, J. Málek, On solvability of a non-linear heat equation with a non-integrable convective term and data involving measures. Nonlinear Anal. Real World Appl. 12(1), 571–591 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Buliček, P. Gwiazda, J. Málek, A. Świerczewska, On steady flows of an incompressible fluids with implicit power-law-like rheology. Adv. Calc. Var. 2(2), 109–136 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ph. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numérique 9(R-2), 77–84 (1975)Google Scholar
  11. 11.
    L. Diening, M. Růžička, K. Schumacher, A decomposition technique for John domains. Ann. Acad. Sci. Fenn. Ser. A. I. Math. 35(1), 87–114 (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    L. Diening, C. Ebmeyer, M. Růžička, Optimal convergence for the implicit space-time discretization of parabolic systems with p-structure. SIAM J. Numer. Anal. 45, 457–472 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    L. Diening, F. Ettwein, Fractional estimates for non–differentiable elliptic systems with general growth. Forum Mathematicum 20(3), 523–556 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    L. Diening, Ch. Kreuzer, Linear convergence of an adaptive finite element method for the p-Laplacian equation. SIAM J. Numer. Anal. 46, 614–638 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    L. Diening, J. Málek, M. Steinhauer, On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM Control Optim. Calc. Var. 14(2), 211–232 (2008). doi:10.1051/cocv:2007049MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    L. Diening, M. Růžička, Interpolation operators in Orlicz–Sobolev spaces. Num. Math. 107, 107–129 (2007). doi:10.1007/s00211-007-0079-9CrossRefzbMATHGoogle Scholar
  17. 17.
    L. Diening, M. Růžička, An existence result for non-Newtonian fluids in non-regular domains. Discrete Contin. Dyn. Syst. Ser. S 3(2), 255–268 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    L. Diening, M. Růžička, J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Scuola Norm. Sup. Pisa Cl. Sci. V IX, 1–46 (2010)Google Scholar
  19. 19.
    H. Eberlein, M. Růžička, Existence of weak solutions for unsteady motions of Herschel–Bulkley fluids. J. Math. Fluid Mech. (2011). doi:10.1007/s00021-011-0080-zGoogle Scholar
  20. 20.
    J. Frehse, J. Málek, M. Steinhauer, An existence result for fluids with shear dependent viscosity – steady flows. Nonlinear Anal. Theory Methods Appl. 30, 3041–3049 (1997)CrossRefzbMATHGoogle Scholar
  21. 21.
    H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974)zbMATHGoogle Scholar
  22. 22.
    V. Girault, J.L. Lions, Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. Port. Math. (N.S.) 58(1), 25–57 (2001)Google Scholar
  23. 23.
    V. Girault, L.R. Scott, A quasi-local interpolation operator preserving the discrete divergence. Calcolo 40(1), 1–19 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    E. Giusti, Metodi Diretti nel Calcolo delle Variazioni (Unione Matematica Italiana, Bologna, 1994)zbMATHGoogle Scholar
  25. 25.
    M. Gutmann, An Existence Result for Herschel–Bulkley Fluids Based on the Lipschitz Truncation Method (Diplomarbeit, Universität Freiburg, Freiburg, 2009)Google Scholar
  26. 26.
    A. Kufner, O. John, S. Fučík, Function Spaces (Academia, Praha, 1977)zbMATHGoogle Scholar
  27. 27.
    R. Landes, Quasimonotone versus pseudomonotone. Proc. R. Soc. Edinb. Sect. A 126(4), 705–717 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (Dunod, Paris, 1969)zbMATHGoogle Scholar
  29. 29.
    J. Málek, J. Nečas, M. Růžička, On the non-Newtonian incompressible fluids. Math. Models Methods Appl. Sci. 3, 35–63 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    J. Málek, K.R. Rajagopal, M. Růžička, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models Methods Appl. Sci. 5, 789–812 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    J. Málek, K.R. Rajagopal, in Mathematical Issues Concerning the Navier-Stokes Equations and Some of Its Generalizations. Handb. Differ. Equ. in Evolutionary equations, Vol. II, (Elsevier/North-Holland, Amsterdam, 2005), pp. 371–459Google Scholar
  32. 32.
    J. Málek, M. Růžička, V.V. Shelukhin, Herschel–Bulkley fluids: existence and regularity of steady flows. M3AS 15(12), 1845–1861 (2005)Google Scholar
  33. 33.
    J. Malý, W.P. Ziemer, in Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs, vol. 51 (American Mathematical Society, Providence, 1997)Google Scholar
  34. 34.
    M.M. Rao, Z.D. Ren, in Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146 (Marcel Dekker Inc., New York, 1991)Google Scholar
  35. 35.
    S. Rauscher, Existenz schwacher Lösungen stationärer Bewegungen von Fluiden mit scherspannungsabhängiger Viskoität im unbeschränkten Gebiet (Diplomarbeit, Universität Freiburg, Freiburg, 2011)Google Scholar
  36. 36.
    M. Růžička, L. Diening, Non–Newtonian Fluids and Function Spaces. Nonlinear Analysis, Function Spaces and Applications, Proceedings of NAFSA 2006 Prague, vol. 8, 2007, pp. 95–144Google Scholar
  37. 37.
    L.R. Scott, S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483–493 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970)zbMATHGoogle Scholar
  39. 39.
    B. Weber, Existenz sehr schwacher Lösungen für mikropolare elektrorheologische Flüssigkeiten (Diplomarbeit, Universität Freiburg, Freiburg, 2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut of Applied MathematicsFreiburg UniversityFreiburgGermany

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