Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions

We establish the wellposedness of the time-independent Navier-Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navierʹs slip condition and a restricted Coulomb-type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary-value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data.