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Existence theory for steady flows of fluids with pressure and shear rate dependent viscosity, for low values of the power-law index. (English) Zbl 1198.35174
The existence of weak solutions of a steady flow of an incompressible, homogeneous, non-Newtonian fluid is established on an open, bounded domain \(\Omega \subset {\mathbb R}^d, d \geq 2\) that is supplemented by no-slip boundary conditions. The Cauchy stress is defined with a pressure and shear-rate dependent viscosity such that non-Newtonian features are incorporated. Under certain assumptions on the constitutively determined part of the Cauchy stress the result is obtained by passing two times to the limit in an \((\epsilon, \eta)\)-approximate system. First the existence of the approximations is sketched. It can be shown with help of a Galerkin approximation; the bounds for the pressure and velocity approximations are calculated. Then the \(\epsilon \rightarrow 0\) limit is presented, where the pressure is decomposed into a strongly convergent and a weakly convergent part to show pointwise convergence of the pressure and the velocity derivatives. Here also Lipschitz approximations of Sobolev functions are considered to obtain these results. Similarly such findings are obtained in the \(\eta \rightarrow 0\) limit, proving the overall statement.

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984), 125 - 145. · Zbl 0565.49010
[2] Amrouche, Ch. and Girault, V., Decomposition of vector spaces and applica- tion to the Stokes problem in arbitrary dimension. Czechoslovak Math. J. 44 (1994), 109 - 140. · Zbl 0823.35140
[3] Boccardo, L. and Murat, F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19 (1992)(6), 581 - 597. · Zbl 0783.35020
[4] Bogovski\?ı, M. E., Solutions of some problems of vector analysis, associated with the operators div and grad (in Russian). In: Trudy Sem. S. L. Soboleva 2. Novosibirsk: Akad. Nauk SSSR Sibirsk. Otdel. 1980, pp. 5 - 40.
[5] Bridgman, P. W., The Physics of High Pressure. New York: MacMillan 1931.
[6] Bulí\check cek, M., Málek, J. and Rajagopal, K. R., Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity. Indiana Univ. Math. J. 56 (2007), 51 - 85. · Zbl 1129.35055
[7] Bulí\check cek, M., Málek, J. and Rajagopal, K. R., Mathematical analysis of un- steady flows of fluids with pressure, shear-rate and temperature dependent material moduli, that slip at solid boundaries. To appear in: SIAM J. Math. Anal. (2009).
[8] Diening, L., Málek, J. and Steinhauer, M., On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM: Control Optim. Calc. Var. 14 (2008)(2), 211 - 232. · Zbl 1143.35037
[9] Franta, M., Málek, J. and Rajagopal, K. R., On steady flows of fluids with pressure- and shear-dependent viscosities. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005)(2055), 651 - 670. · Zbl 1145.76311
[10] Frehse, J., Málek, J. and Steinhauer, M., An existence result for fluids with shear dependent viscosity-steady flows. Nonlin. Analysis TMA 30 (1997), 3041 - 3049. · Zbl 0902.35089
[11] Frehse, J., Málek, J. and Steinhauer, M., On analysis of steady flows of flu- ids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34 (2003)(5), 1064 - 1083. · Zbl 1050.35080
[12] Gazzola, F., A note on the evolution of Navier-Stokes equations with a pressure-dependent viscosity. Z. Angew. Math. Phys. 48 (1997)(5), 760 - 773. · Zbl 0895.76018
[13] Gazzola, F. and Secchi, P., Some results about stationary Navier-Stokes equa- tions with a pressure-dependent viscosity. In: Navier-Stokes Equations: The- ory and Numerical Methods (Varenna 1997; ed.: R. Salvi). Pitman Res. Notes Math. Ser. 388. Harlow: Longman 1998, pp. 31 - 37. · Zbl 0940.35156
[14] Hron, J., Málek, J. and Rajagopal, K. R., Simple flows of fluids with pressure dependent viscosities. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 477 (2001), 277 - 302. 371 · Zbl 1052.76017
[15] Hron, J., Málek, J., Ne\check cas, J. and Rajagopal, K. R., Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear-dependent viscosities.Math.Comput.Simulation 61 (2003), 297 - 315. · Zbl 1205.76159
[16] Lady\check zenskaja, O. A., New equations for the description of the motions of vis- cous incompressible fluids, and global solvability for their boundary value prob- lems (in Russian). Trudy Steklov’s Math. Institute 102 (1967), 85 - 104. · Zbl 0202.37802
[17] Lady\check zenskaja, O. A., Modifications of the Navier-Stokes equations for large gradients of the velocities (in Russian). Zap. Nau\check cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 126 - 154.
[18] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon & Breach 1969. · Zbl 0184.52603
[19] Lanzendörfer, M., On steady inner flows of an incompressible fluid with the viscosity depending on the pressure and the shear rate. Nonlinear Anal. Real World Appl. 10 (2009)(2), 1943 - 1954. · Zbl 1163.76335
[20] Lions, J.-L., Quelques Méthodes de Résolution des Probl‘emes aux Limites non Linéaires (in French). Paris: Dunod, Gauthier-Villars 1969. · Zbl 0189.40603
[21] Málek, J. and Rajagopal, K. R., Mathematical issues concerning the Navier- Stokes equations and some of its generalizations. In: Evolutionary Equations II. Handb. Diff. Equ. Amsterdam: Elsevier/North-Holland 2005, pp. 371 - 459. · Zbl 1095.35027
[22] Málek, J. and Rajagopal, K. R., Mathematical properties of the solutions to the equations governing the flow of fluids with pressure and shear rate depen- dent viscosities. In: Mathematical Theory in Fluid Dynamics IV. Amsterdam: Elsevier/North-Holland 2007, pp. 407 - 444.
[23] Málek, J., Ne\check cas, J., Rokyta, M. and R\ring u\check zi\check cka, M., Weak and Measure-Valued Solutions to Evolutionary PDEs. London: Chapman & Hall 1996.
[24] Málek, J., Ne\check cas, J. and Rajagopal, K. R., Global analysis of the flows of fluids with pressure-dependent viscosities. Arch. Ration. Mech. Anal. 165 (2002)(3), 243 - 269. · Zbl 1022.76011
[25] Novotný, A. and Stra\check skraba, I., Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Ser. Math. Appl. 27. Oxford: Oxford Univ. Press 2004.
[26] Rajagopal, K. R., On implicit constitutive theories. Appl. Math. 48 (2003)(4), 279 - 319. · Zbl 1099.74009
[27] Rajagopal, K. R. and Srinivasa, A. R., On thermomechanical restrictions of continua. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004)(2042), 631 - 651. · Zbl 1041.74002
[28] Renardy, M., Some remarks on the Navier-Stokes equations with a pressure- dependent viscosity. Comm. Part. Diff. Equ. 11 (1986)(7), 779 - 793. · Zbl 0597.35097
[29] R\ring u\check zi\check cka, M., A note on steady flow of fluids with shear dependent viscosity. Nonlin. Anal. 30 (1997), 3029 - 3039. · Zbl 0906.35076
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