Some nonlinear Stokes system is considered.  The initial-boundary value problem for the system is investigated  and the existence and uniqueness of the weak solution for the problem is proved.

[1] R. Temam. Navier-Stokes equations: theory and numerical analysis, Mir, Moscow, 1981 (translated
from: North-Holland Publ., Amsterdam, New York, Oxford, 1979).
[2] M. Rúžička. Electrorheological fluids: Modeling and mathematical theory, in: Lecture Notes in Mathematics,
1748, Springer-Verlag, Berlin, 2000.
[3] V.A. Solonnikov. Estimates of solutions to the linearized system of the Navier-Stokes equations, Trudy
of the Steklov Math. Inst. 70 (1964) 213-317.
[4] V.A. Solonnikov. On estimates of solutions of the non-stationary Stokes problem in anisotropic Sobolev
spaces and on estimates for the resolvent of the Stokes operator, Russian Math. Surveys. 58, №2 (2003)
331-365.
[5] V.A. Solonnikov. Weighted Schauder estimates for evolution Stokes problem, Annali Univ. Ferrara. 52
(2006) 137-172.
[6] I.S. Mogilevskii. On a boundary value problem for the time-dependent Stokes system with general boundary
conditions, Mathematics of the USSR-Izvestiya. 28, №1 (1987) 37-66.
[7] G.P. Galdi, C.G. Simader, H. Sohr. On Stokes problem in Lipschitz domain, Annali di Matematica
pura ed applicata. CLXVII (IV) (1994) 147-163.

[8] G.P. Galdi, C.G. Simader, H. Sohr. A class of solution to stationary Stokes and Navier-Stokes equations
with boundary data in $W^{-\frac{1}{q},q}$, Math. Ann. 331 (2005) 41-74.
[9] O.M. Buhrii. Visco-plastic, Newtonian, and dilatant fluids: Stokes equations with variable exponent of
nonlinearity, Mat. Stud. 49 (2018) 165-180.
[10] O. Buhrii, M. Khoma On initial-boundary value problem for nonlinear integro-differential Stokes system,
Visnyk (Herald) of Lviv Univ. Series Mech.-Math. 85 (2018) 107-119.
[11] H. Gajewski, K. Groger, K. Zacharias. Nonlinear operator equations and operator differential equations,
Mir, Moscow, 1978 (translated from: Akademie-Verlag, Berlin, 1974).
[12] J.-L. Lions. Quelques méthodes de résolution des problémes aux limites non linéaires, Mir, Moscow,
1972 (translated from: Dunod Gauthier-Villars, Paris, 1969).
[13] E. Suhubi. Functional analysis, Kluwer Acad. Publ., Dordrecht, Boston, London, 2003.
[14] O. Buhrii, N. Buhrii. Integro-differential systems with variable exponents of nonlinearity, Open Math.
15 (2017) 859-883.
[15] Brezis H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer, New York,
Dordrecht, Heidelberg, London, 2011)
[16] Bystr¨om J., Sharp constants for some inequalities connected to the p-Laplace operator, J. of Ineq. in
Pure and Appl. Math., 2005, 6 (2): Article 56
[17] J.A. Langa, J. Real, J. Simon. Existence and regularity of the pressure for the stochastic Navier-Stokes
equations, Applied Mathematics and Optimization. 48, №3 (2003) 195-210.
[18] J. Simon. Nonhomogeneous viscous incompressible fluids: existence of velocity, density and preassure,
SIAM J. Math. Anal. 21, №5 (1990) 1093-1117.
[19] J.-P. Aubin. Un theoreme de compacite, Comptes rendus hebdomadaires des seances de l’academie des
sciences. 256 (24) (1963) 5042-5044.
[20] F. Bernis. Existence results for doubly nonlinear higher order parabolic equations on unbounded domains,
Math. Ann. 279 (1988) 373-394.
[21] O. Buhrii, M. Khoma Integration by parts formulas for functions from generalized Sobolev spaces.
International Scient. Conf. “Applied Mathematics and Information Technology” dedicated to the 60th
anniversary of the Department of Applied Mathematics and Information Technology (September 22-24,
2022, Chernivtsi): Book of Materials. – Chernivtsi, 2022. – P. 107-110.
[22] J. Droniou Intégration et espaces de Sobolev á valeurs vectorielles. Lecture notes, Universite de
Provence, Marseille, 2001.