  We apply a phase field approach for a general shape optimization problem of a
stationary Navier-Stokes flow. To be precise we add a multiple of the
Ginzburg--Landau energy as a regularization to the objective functional and
relax the non-permeability of the medium outside the fluid region. The
resulting diffuse interface problem can be shown to be well-posed and
optimality conditions are derived. We state suitable assumptions on the problem
in order to derive a sharp interface limit for the minimizers and the
optimality conditions. Additionally, we can derive a necessary optimality
system for the sharp interface problem by geometric variations without stating
additional regularity assumptions on the minimizing set.

Optimization and Control