On steady flows of an incompressible fluids with implicit power-law-like rheology ∗ MIROSLAV BUL ´ I ˇ CEK 1 PIOTR GWIAZDA 2 JOSEF M ´ ALEK 1 AGNIESZKA ´ SWIERCZEWSKA-GWIAZDA 2 November 6, 2007 1 Mathematical Institute, Charles University, Sokolovsk´a 83, 186 75 Prague, Czech Republic 2 Institute of Applied Mathematics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw Banacha 2, 02-097 Warszawa, Poland Abstract We consider steady flows of incompressible fluids with power-law-like rhe- ology given by an implicit constitutive equation relating the Cauchy stress and the symmetric part of the velocity gradient in such a way that it leads to a maximal monotone (possibly multivalued) graph. Such a framework includes Bingham fluids, Herschel-Bulkley fluids, and shear-rate dependent fluids with discontinuous viscosities as special cases. We assume that the fluid adheres to the boundary. Using tools such as the Young measures, properties of spatially depen- dent maximal monotone operators and Lipschitz approximations of Sobolev functions, we are able to extend the results concerning large data existence of weak solutions to those values of the power-law index that are of importance from the point of view of engineering and physical applications. 1 Problem formulation We consider the following problem associated to a fixed, yet arbitrary parameter q ∈ (1, ∞) that is connected to its dual exponent q ′ through the relation q ′ = * M. Bul´ ıˇ cek’s research is supported by the Jindˇ rich Neˇ cas Center for Mathematical Modeling, the project LC06052 financed by M ˇ SMT. P. Gwiazda and A. Swierczewska-Gwiazda were supported by the Grant of Ministry of Science and Higher Education, Nr N201 033 32/2269. J. M´ alek’s contribution is a part of the research project MSM 0021620839 financed by M ˇ SMT; the support of GA ˇ CR 201/06/0352 is also acknowledged. 1