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 Omega is exterior it is required that the velocity converges to an assigned constant vector u(0) at infinity. We prove that a solution exists in a bounded domain provided parallel to a parallel to(L2(theta Omega)) is less than a computable positive constant and is unique if parallel to a parallel to(w)1/2,2((partial derivative Omega)) + parallel to S parallel to(L2(partial derivative Omega)) is suitably small. As far as exterior domains are concerned, we show that a solution exists if parallel to a parallel to(L2(partial derivative Omega)) +parallel to a -u(0) . n parallel to(L2(partial derivative Omega)) is small. (C) 2011 Elsevier Inc. All rights reserved.