Existence of Weak Solutions to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with Shear Rate Dependent Viscosity

Abstract.

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 = Ω × (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 ∇ · u = 0.