Annali di Matematica Pura ed Applicata (1923 -)

, Volume 196, Issue 3, pp 1185–1202 | Cite as

Homogenization of an incompressible stationary flow of an electrorheological fluid

  • Miroslav BulíčekEmail author
  • Martin Kalousek
  • Petr Kaplický


We combine two-scale convergence, theory of monotone operators and results on approximation of Sobolev functions by Lipschitz functions to prove a homogenization process for an incompressible flow of a generalized Newtonian fluid. We avoid the necessity of testing the weak formulation of the initial and homogenized systems by corresponding weak solutions, which allows optimal assumptions on lower bound for a growth of the elliptic term. We show that the stress tensor for homogenized problem depends on the symmetric part of the velocity gradient involving the limit of a sequence selected from a family of solutions of initial problems.


Non-Newtonian incompressible fluids Two-scale convergence Homogenization Lipschitz approximation 

Mathematics Subject Classification

35Q35 36B27 76M50 



We thank the anonymous referee for relevant remarks that helped to improve our article.


  1. 1.
    Acerbi, E., Fusco, N.: An approximation lemma for \(W^{1,p}\) functions. In: Material Instabilities in Continuum Mechanics (Edinburgh, 1985–1986), Oxford Sci. Publ., pp. 1–5. Oxford Univ. Press, New York (1988)Google Scholar
  2. 2.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992). doi: 10.1137/0523084 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ball, J.M., Murat, F.: Remarks on Chacon’s biting lemma. Proc. Am. Math. Soc. 107(3), 655–663 (1989). doi: 10.2307/2048162 MathSciNetzbMATHGoogle Scholar
  4. 4.
    Breit, D., Diening, L., Schwarzacher, S.: Solenoidal Lipschitz truncation for parabolic PDEs. Math. Models Methods Appl. Sci. 23(14), 2671–2700 (2013). doi: 10.1142/S0218202513500437 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bulíček, M., Gwiazda, P., Málek, J., Świerczewska-Gwiazda, A.: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44(4), 2756–2801 (2012). doi: 10.1137/110830289 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)Google Scholar
  7. 7.
    Diening, L., Málek, J., Steinhauer, M.: 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:2007049 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Diening, L., Růžička, M., Wolf, J.: Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 9(1), 1–46 (2010)Google Scholar
  9. 9.
    Diestel, J., Uhl, J.: Vector Measures. American Mathematical Society, Mathematical Surveys and Monographs (1977)Google Scholar
  10. 10.
    Feireisl, E., Novotný, A.: Singular limits in thermodynamics of viscous fluids. Advances in Mathematical Fluid Mechanics. Birkhäuser Verlag, Basel (2009). doi: 10.1007/978-3-7643-8843-0
  11. 11.
    Frehse, J., Málek, J., Steinhauer, M.: On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34(5), 1064–1083 (electronic) (2003). doi: 10.1137/S0036141002410988
  12. 12.
    Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(3), 489–507 (1978)Google Scholar
  13. 13.
    Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989). doi: 10.1137/0520043 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)Google Scholar
  15. 15.
    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, NJ (1970)Google Scholar
  16. 16.
    Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, pp. 136–212. Pitman, Boston, MA (1979)Google Scholar
  17. 17.
    Visintin, A.: Towards a two-scale calculus. ESAIM Control Optim. Calc. Var. 12(3), 371–397 (electronic) (2006). doi: 10.1051/cocv:2006012
  18. 18.
    Zhikov, V.V.: Homogenization of a Navier–Stokes-type system for electrorheological fluid. Complex Var. Elliptic Equ. 56(7–9), 545–558 (2011). doi: 10.1080/17476933.2010.487214 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Miroslav Bulíček
    • 1
    Email author
  • Martin Kalousek
    • 2
  • Petr Kaplický
    • 2
  1. 1.Mathematical InstituteCharles UniversityPraha 8, KarlínCzech Republic
  2. 2.Department of Mathematical AnalysisCharles UniversityPraha 8, KarlínCzech Republic

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