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On steady inner flows of an incompressible fluid with the viscosity depending on the pressure and the shear rate. (English) Zbl 1163.76335
Summary: We consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate. The existence of weak solutions for steady flows of such fluids subject to homogeneous Dirichlet boundary conditions is established by M. Franta, J. Málek and K.R. Rajagopal, Proc. Roy. Soc. A Math. Phys. Eng. Sci. 461, No. 2055, 651–670 (2005; Zbl 1145.76311)]. In this paper we treat non-homogeneous Dirichlet boundary conditions, assuming either that the normal part of velocity on the boundary is equal to zero or that the boundary data are small. We also relax the requirement concerning how to fix the pressure. Such a model has relevance to some important engineering applications.

##### MSC:
 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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##### References:
 [1] Málek, J.; Rajagopal, K.R., Mathematical issues concerning the navier – stokes equations and some of its generalizations, (), 371-459 · Zbl 1095.35027 [2] Szeri, A.Z., Fluid film lubrication: theory and design, (1998), Cambridge University Press · Zbl 1001.76001 [3] Barus, C., Isothermals, isopiestics and isometrics relative to viscosity, Amer. J. sci., 45, 87-96, (1893) [4] Bridgman, P.W., The physics of high pressure, (1931), The MacMillan Company New York · JFM 57.0068.01 [5] Bair, S.; Kottke, P., Pressure – viscosity relationship for elastohydrodynamics, Tribology trans., 46, 289-295, (2003) [6] Málek, J.; Rajagopal, K.R., Mathematical properties of the solutions to the equations governing the flow of fluids with pressure and shear rate dependent viscosities, () [7] Franta, M.; Málek, J.; Rajagopal, K.R., On steady flows of fluids with pressure- and shear-dependent viscosities, Proc. roy. soc. A math. phys. eng. sci., 461, 2055, 651-670, (2005) · Zbl 1145.76311 [8] M. Bulíček, J. Málek, K.R. Rajagopal, Mathematical analysis of unsteady flows of fluids with pressure, shear-rate and temperature dependent material moduli, that slip at solid boundaries, SIAM J. Math. Anal. (in press) [9] Bulíček, M.; Málek, J.; Rajagopal, K.R., Navier’s slip and evolutionary navier – stokes-like systems with pressure and shear-rate dependent viscosity, Indiana univ. math. J., 56, 51-85, (2007) · Zbl 1129.35055 [10] Bogovskii, M.E., Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. akad. nauk SSSR, 5, 1037-1040, (1979) [11] Amrouche, C.; Girault, V., Decomposition of vector spaces and applications to the Stokes problems in arbitrary dimension, Czech. math. J., 44, 109-141, (1994) · Zbl 0823.35140 [12] Novotný, A.; Straškraba, I., () [13] Málek, J.; Nečas, J.; Rokyta, M.; Růžička, M., Weak and measure-valued solutions to evolutionary pdes, (1996), CRC Press London · Zbl 0851.35002 [14] Málek, J.; Nečas, J.; Rajagopal, K.R., Global analysis of the flows of fluids with pressure-dependent viscosities, Arch. ration. mech. anal., 165, 3, 243-269, (2002) · Zbl 1022.76011 [15] M. Lanzendörfer, Numerical simulations of the flow in the journal bearing, Master’s Thesis, Charles University in Prague, Faculty of Mathematics and Physics, 2003 [16] Renardy, M., Some remarks on the navier – stokes equations with a pressure-dependent viscosity, Comm. partial differential equations, 11, 779-793, (1986) · Zbl 0597.35097 [17] Gazzola, F., A note on the evolution of navier – stokes equations with a pressure-dependent viscosity, Z. angew. math. phys., 48, 760-773, (1997) · Zbl 0895.76018 [18] Gazzola, F.; Secchi, P., Some results about stationary navier – stokes equations with a pressure-dependent viscosity, (), 31-37 · Zbl 0940.35156 [19] Hron, J.; Málek, J.; Nečas, J.; Rajagopal, K.R., Numerical simulations and global existence of solutions of two dimensional flows of fluids with pressure and shear dependent viscosities, Math. comput. simul., 61, 3-6, 297-315, (2003) · Zbl 1205.76159 [20] Málek, J.; Nečas, J.; Rajagopal, K.R., Global existence of solutions for flows of fluids with pressure and shear dependent viscosities, Appl. math. lett., 15, 8, 961-967, (2002) · Zbl 1062.76011 [21] Kaplický, P.; Málek, J.; Stará, J., On global existence of smooth two-dimensional steady flows for a class of non-Newtonian fluids under various boundary conditions, (), 213-229 · Zbl 0953.35120 [22] Miranville, A.; Wang, X., Upper bound on the dimension of the attractor for nonhomogeneous navier – stokes equations, Discrete contin. dynam. systems, 2, 95-110, (1996) · Zbl 0949.35112 [23] Lions, J.L., Quelques méthodes de résolution des problèmes aux limites non linéares, (1969), Dunod Paris · Zbl 0189.40603 [24] Dunford, N.; Schwartz, J.T., () [25] Galdi, G.P., () [26] E. Hopf, On nonlinear partial differential equations, in: Lecture series of the Symposium on Partial Differential Equations, Berkley, 1955, The Univ. of Kansas, 1957, pp. 1-29 [27] Temam, R., Navier – stokes equations: theory and numerical analysis, (2001), AMS Chelsea Publishing Providence, RI, Reprint of the 1984 edition · Zbl 0981.35001
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