The paper is concerned with the equations of motion for incompressible fluids that slip at the wall. Particular interest is in the domain dependence of weak solutions. We prove that the solutions depend continuously on the perturbation of the boundary provided that the latter remains in the class C^1,1. The result is applicable to a wide class of shape optimization problems and is optimal in terms of boundary regularity.

We consider the steady motion of an incompressible fluid whose viscosity depends on the pressure and the shear rate. The system is completed by suitable boundary conditions involving non-homogeneous Dirichlet, Navier's slip and inflow/outflow parts. We prove the existence of weak solutions and show that the resulting level of the pressure is fixed by the boundary conditions. The problem is motivated by particular applications from tribology.

We consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate. Suitable boundary conditions on the surface force at the inflow/outflow part of boundary are given. As an advantage of this, the mean value of the pressure over the domain is no more a free parameter which would have to be prescribed otherwise. We prove the existence and the uniqueness of weak solutions (the later for small data) and discuss particular applications of the results.