A weak solvability of a steady variational inequality of the Navier-Stokes type with mixed boundary conditions

We study a steady flow of a viscous incompressible fluid in a channel with a non-Dirichlet boundary condition on the output. In order to control the kinetic energy of the fluid, in the channel, we assume that possible backward flows on the output are in some sense bounded. Flow fields which satisfy this assumption fill up a convex subset of a certain function space. We formulate a variational inequality of the Navier-Stokes type on this convex set and we prove the existence of its weak solution. Moreover, we also study the relation of the weak solution to the Navier-Stokes equation.