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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.

##### MSC:
 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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