On the Navier problem for the stationary Navier–Stokes equations

The Navier problem is to find a solution of the steady-state Navier–Stokes equations such that the normal component of the velocity and a linear combination of the tangential components of the velocity and the traction assume prescribed value a and s at the boundary. If Ω is exterior it is required that the velocity converges to an assigned constant vector u0 at infinity. We prove that a solution exists in a bounded domain provided ‖a‖L2(∂Ω) is less than a computable positive constant and is unique if ‖a‖W1/2,2(∂Ω)+‖s‖L2(∂Ω) is suitably small. As far as exterior domains are concerned, we show that a solution exists if ‖a‖L2(∂Ω)+‖a−u0⋅n‖L2(∂Ω) is small.