Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 1317–1333 | Cite as

Evolutionary Oseen Model for Generalized Newtonian Fluid with Multivalued Nonmonotone Friction Law

  • Stanisław Migórski
  • Sylwia DudekEmail author
Open Access


The paper deals with the non-stationary Oseen system of equations for the generalized Newtonian incompressible fluid with multivalued and nonmonotone frictional slip boundary conditions. First, we provide a result on existence of a unique solution to an abstract evolutionary inclusion involving the Clarke subdifferential term for a nonconvex function. We employ a method based on a surjectivity theorem for multivalued L-pseudomonotone operators. Then, we exploit the abstract result to prove the weak unique solvability of the Oseen system.


Oseen model generalized Newtonian fluid hemivariational inequality L-pseudomonotonicity Clarke subdifferential friction-type law slip boundary condition 

Mathematics Subject Classification

47J20 47J22 49J40 49J45 74G25 74G30 74M15 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.


  1. 1.
    Aubin, J.P., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Springer-Verlag, Berlin (1984)CrossRefzbMATHGoogle Scholar
  2. 2.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  3. 3.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    Djoko, J.K., Lubuma, J.M.: Analysis of a time implicit scheme for the Oseen model driven by nonlinear slip boundary conditions. J. Math. Fluid Mech. 18, 717–730 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dudek, S., Kalita, P., Migórski, S.: Stationary flow of non-Newtonian fluid with nonmonotone frictional boundary conditions. Zeitschrift für angewandte Mathematik und Physik 66, 2625–2646 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dudek, S., Kalita, P., Migórski, S.: Stationary Oberbeck–Boussinesq model of generalized Newtonian fluid governed by multivalued partial differential equations. Appl. Anal. 96, 2192–2217 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dziurda, J.: A generalized solution of Oseen’s equations. Universitatis Iagellonicae Acta Mathematica 23, 131–149 (1982)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fujita, H.: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. In: Mathematical Fluid Mechanics and Modeling (Kyoto, 1994), RIMS Kokyuroko, vol. 888, Kyoto University, Kyoto (1994), 199–216Google Scholar
  10. 10.
    Fujita, H.: Non-stationary Stokes flows under leak boundary conditions of friction type. J. Comput. Appl. Math. 19, 1–8 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Goeleven, D., Motreanu, D., Dumont, Y., Rochdi, M.: Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics. Kluwer Academic Publishers, Boston (2003)zbMATHGoogle Scholar
  12. 12.
    Gross, S., Reusken, A.: Numerical Methods for Two-Phase Incompressible Flows. Springer-Verlag, Berlin (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Han, W., Migórski, S., Sofonea, M.: A class of variational–hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 3891–3912 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kashiwabara, T.: On a strong solution of the non-stationary Navier–Stokes equations under slip or leak boundary conditions of friction type. J. Differ. Equ. 254, 756–778 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kulig, A., Migórski, S.: Solvability and continuous dependence results for second order nonlinear inclusion with Volterra-type operator. Nonlinear Anal. 75, 4729–4746 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  17. 17.
    Málek, J., Rajagopal, K.R.: Mathematical issues concerning the Navier–Stokes equations and some of their generalizations. In: Dafermos, C., Feireisl, E. (eds.) Handbook of Evolutionary Equations, vol. II. Elsevier, Amsterdam (2005)Google Scholar
  18. 18.
    Migórski, S.: Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems. Comput. Math. Appl. 52, 677–698 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Migórski, S., Ochal, A.: Hemivariational inequalities for stationary Navier–Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Migórski, S., Ochal, A.: Navier–Stokes problems modelled by evolution hemivariational inequalities, Discrete Contin. Dyn. Syst. Suppl. 731–740 (2007)Google Scholar
  21. 21.
    Migórski, S., Ochal, A., Sofonea, M.: Nonlinear inclusions and hemivariational inequalities. Models and analysis of contact problems. In: Advances in Mechanics and Mathematics, vol. 26, Springer, New York (2013)Google Scholar
  22. 22.
    Migórski, S., Ochal, A., Sofonea, M.: Evolutionary inclusions and hemivariational inequalities, chapter 2 in advances in variational and hemivariational inequalities: theory, numerical analysis, and applications. In: Han, W., Migórski, S., Sofonea, M. (eds.) Advances in Mechanics and Mathematics Series, vol. 33, pp. 39–64. Springer, New York (2015)Google Scholar
  23. 23.
    Migórski, S., Paczka, D.: On steady flow of non-Newtonian fluids with frictional boundary conditions in reflexive Orlicz spaces. Nonlinear Anal. Series B: Real World Appl. 39, 337–361 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker Inc, New York (1995)zbMATHGoogle Scholar
  25. 25.
    Panagiotopoulos, P.D.: Nonconvex energy functions, hemivariational inequalities and substationary principles. Acta Mech. 42, 160–183 (1983)Google Scholar
  26. 26.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985)CrossRefzbMATHGoogle Scholar
  27. 27.
    Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer-Verlag, Berlin (1993)CrossRefzbMATHGoogle Scholar
  28. 28.
    Růžička, M., Diening, L.: Non-Newtonian fluids and function spaces, In: J. Rákosnik (ed.): Nonlinear Analysis, Function Spaces and Applications, Proceedings of the Spring School, Prague, May 30–June 6, 2006, vol. 8, Czech Academy of Sciences, Mathematical Institute, Praha, pp. 95–143 (2007)Google Scholar
  29. 29.
    Sofonea, M., Migórski, S.: Variational-Hemivariational Inequalities with Applications. Monographs and Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton (2017)zbMATHGoogle Scholar
  30. 30.
    Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1979)zbMATHGoogle Scholar
  31. 31.
    Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis. SIAM, Philadelphia (1997)zbMATHGoogle Scholar
  32. 32.
    Zeidler, E.: Nonlinear Functional Analysis and Applications II A/B. Springer, New York (1990)zbMATHGoogle Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.College of Applied MathematicsChengdu University of Information TechnologyChengduPeople’s Republic of China
  2. 2.Chair of Optimization and ControlJagiellonian University in KrakowKrakowPoland
  3. 3.Institute of Mathematics Faculty of Physics, Mathematics and Computer ScienceKrakow University of TechnologyKrakowPoland

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