ON THE STATIONARY QUASI-NEWTONIAN FLOW OBEYING A POWER-LAW

We investigate in this paper existence of a weak solution for a statio nary 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 v (xi)= \Tr(xi xi)\(r/2-1) r < 2, on shear stress xi. It is correspondi ng to most quasi-Newtonian hows with injection on the boundary. Since for r less than or equal to 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,W-r(Ome ga)(n) and using monotonicity and compactness we successfully pass to the limit for r greater than or equal to 3n/(n + 2). For 3n/(n + 2) > r > 2n/(n + 2) our construction gives existence of at least one very w eak solution. Furthermore, for r greater than or equal to 3n/(n + 2) w e prove that all weak solutions lying in the ball in W-0(1,r) of radiu s smaller than critical are equal. Finally, we obtain an existence res ult for the flow through a thin slab.