Visco-plastic, newtonian, and dilatant fluids: Stokes equations with variable exponent of nonlinearity

1. Ladyzhenskaya O.A. New equations for description of motion of viscous incompressible fluids and global solvability of boundary value problems for them, Proc. Steklov Inst. Math., 102 (1967), 85-104.

2. Fernandez-Cara E., Guillen F., Ortega R. Some theoretical results for visco-plastic and dilatant fluids with variable density. Nonlinear Analysis, 28 (1997), №6, 1079-1100.

3. Solonnikov V.A. Weighted Schauder estimates for evolution Stokes problem, Annali Univ. Ferrara, 52 (2006), 137-172.

4. Temam R., Navier-Stokes equations: theory and numerical analysis, North-Holland Publ., Amsterdam, New York, Oxford, 1979.

5. Ruzicka M., Electrorheological fluids: Modeling and mathematical theory, in: Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000.

6. Simon J. Nonhomogeneous viscous incompressible fluids: existence of velocity, density and preassure, SIAM J. Math. Anal., 21 (1990), №5, 1093-1117.

7. Langa J.A., Real J., Simon J. Existence and regularity of the pressure for the stochastic Navier-Stokes equations, Applied Mathematics and Optimization, 48 (2003), №3, 195-210.

8. Lavrenjuk S.P., Onishkevich G.M. Stabilization of solutions of the problem without initial data for system of Navier-Stocks type, Lviv, 1996, Ya.S. Pidstrygach IPPMM, Preprint №7.

9. Antontsev S.N., Daz J.I., de Oliveira H.B., Mathematical models in dynamics of non-Newtonian fluids and in glaciology, Proceedings of the CMNE/CILAMCE Congress. APMTAC, SEMNI and ABMEC, Universidade do Porto, Porto, 2007, 20 p.

10. Wrbylewska A. Existence results for unsteady flows of nonhomogeneous non-Newtonian incompressible fluids, Monotonicity methods in generalized Orlicz spaces, 2011, Preprints of PhD Programme, №2011-015.

11. Zhikov V.V., Pastukhova S.E. On the Navier-Stokes equations: existence theorems and energy equalities, Proceed. Steklov Inst. Math., 278 (2012), 67-87.

12. Fukao T. Variational inequality for the Stokes equations with constraint. Descrete Contin. Dyn. Syst., 2011, Dynamical Systems, Differential Equations and Applications, 8th AIMS Conference, Suppl., V.I, 437-446.

13. Serrin J. Mathematical principles of classical fluid mechanics. Moscow, 1963, translated from: Berlin, GЃNottingen, Heidelberg, 1959.

14. Adams R.A. Sobolev spaces, Academic Press, New York, San Francisco, London, 1975.

15. Gajewski H., Groger K., Zacharias K., Nonlinear operator equations and operator differential equations. Mir, Moscow, 1978, translated from: Akademie-Verlag, Berlin, 1974.

16. Evans L.C. Partial differential equations, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, RI, 1998.

17. Bokalo T., Buhrii O. Some integrating by parts formulas in variable indices of nonlinearity function spaces, Visnyk (Herald) of Lviv Univ. Series Mech.-Math., 71 (2009), 13-26.

18. Diening L., Harjulehto P., Hasto P., R.u.zi.cka M., Lebesgue and Sobolev spaces with variable exponents, Springer, Heidelberg, 2011.

19. Kovacik O., Rakosnik J., On spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math. J., 41 (1991), №116, 592-618.

20. Fan X.-L., Zhao D., On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.

21. Antontsev S., Shmarev S. Evolution PDEs with nonstandard growth conditions. Existence, uniqueness, localization, blow-up. Atlantis Studies in Diff. Eq., V.4, Paris: Atlantis Press, 2015.

22. Radulescu V., Repov.s D., Partial differential equations with variable exponents: variational methods and qualitative analysis, CRC Press, Boca Raton, London, New York, 2015.

23. Buhrii O.M. Finiteness of time vanishing of the solution of a nonlinear parabolic variational inequality with variable exponent of nonlinearity, Mat. Stud., 24 (2005) №2, 167-172.

24. Buhrii O., Domanska G., Protsakh N. Initial boundary value problem for nonlinear differential equation of the third order in generalized Sobolev spaces, Visnyk (Herald) of Lviv Univ. Series Mech.-Math., 64 (2005), 44-61.

25. Buhrii O., Buhrii N. Integro-differential systems with variable exponents of nonlinearity, Open Math., 15 (2017), 859-883.

26. Lions J.-L. Some methods of solving of nonlinear boundary value problems, Mir, Moscow, 1972, translated from: Dunod, Gauthier-Villars, Paris, 1969.

27. Aubin J.-P. Un theoreme de compacite, Comptes rendus hebdomadaires des seances de lЃfacademie des sciences. 256 (1963) №24, 5042-5044.

28. Bernis F. Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann., 279 (1988), 373-394.

29. Simon J. Compact sets in the space Lp(0; T;B), Annali di Mat. Pura ed Appl., 146 (1987), №4, 65-96.

30. Bokalo M.M., Tsebenko A.M. Existence of optimal control in the coefficients for problem without initial condition for strongly nonlinear parabolic equations, Mat. Stud., 45 (2016), №1, 40-56.

31. Brezis H. Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, Dordrecht, Heidelberg, London, 2011.

32. Panat O.T., Problems for hyperbolic equations and hyperbolic-parabolic systems in generalized Sobolev spaces, Ph.D. thesis, Lviv, Ukraine, 2010.

33. Buhrii O., Buhrii M., On existence in generalized Sobolev spaces solutions of the initial-boundary value problems for nonlinear integro-differential equations arising from theory of European option, Visn. Lviv Univ., Herald of Lviv University, Ser. Mech.-Math., 81 (2016), 61-84.