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Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. (English) Zbl 1151.76426
Summary: In the present paper we prove the existence of weak solutions \(u:Q \to \mathbb{R}^{n}\) to the equations of non-stationary motion of an incompressible fluid with shear rate dependent viscosity in a cylinder \(Q = \Omega \times (0, T)\), where \( \Omega \subset \mathbb{R}^{n} \) denotes an open set. For the power-low model with \(q > 2\frac{n+1}{n+2}\) we are able to construct a weak solution \(u\in L^q (0, T; W_0^{1,q}(\Omega)^n)\cap C_w([0,T];L^2(\Omega)^n)\) with \(\nabla \cdot u = 0\).

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35D05 Existence of generalized solutions of PDE (MSC2000)
54B15 Quotient spaces, decompositions in general topology
34A34 Nonlinear ordinary differential equations and systems, general theory
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