Abstract
We investigate in this paper existence of a weak solution for a stationary incompressible Navier‐Stokes system with non‐linear viscosity and with non‐homogeneous boundary conditions for velocity on the boundary. Our concern is with the viscosity obeying the power‐law dependence ν(ξ) = ∣Tr(ξξ*)∣^r/2−1^, r < 2, on shear stress ξ. It is corresponding to most quasi‐Newtonian flows with injection on the boundary. Since for r ⩽ 2 the inertial term precludes any a priori estimate in general, we suppose the Reynolds number is not too large. Using the specific algebraic structure of the Navier‐Stokes system we prove existence of at least one approximate solution. The constructed approximate solution turns out to be uniformly bounded in W^1,r^ (Omega;)^n^ and using monotonicity and compactness we successfully pass to the limit for r ≥ 3__n__/(n + 2). For 3__n__/(n + 2) > r > 2__n__/(n + 2) our construction gives existence of at least one very weak solution. Furthermore, for r ≥ 3__n__/(n + 2) we prove that all weak solutions lying in the ball in W of radius smaller than critical are equal. Finally, we obtain an existence result for the flow through a thin slab.