  In this paper we consider the three dimensional exterior Stokes problem and
study the solvability of the corresponding continuous and discrete formulations
that arise from the coupling of a dual-mixed variational formulation with the
boundary integral equation method. More precisely, after employing the
incompressibility condition to eliminate the pressure, we consider the
resulting velocity-stress-vorticity approach with different kind of boundary
conditions on an annular bounded domain, and couple the underlying equations
with either one or two boundary integral equations arising from the application
of the usual and normal traces to the Green representation formula in the
exterior unbounded region. As a result, we obtain saddle point operator
equations, which are then analyzed by the well-known Babuska-Brezzi theory. We
prove the well-posedness of the continuous formulations, identifying previously
the space of solutions of the associated homogeneous problem, and specify
explicit hypotheses to be satisfied by the finite element and boundary element
subspaces in order to guarantee the stability of the respective Galerkin
schemes. In particular, following a similar analysis given recently for the
Laplacian, we are able to extend the classical Johnson & N\'ed\'elec procedure
to the present case, without assuming any restrictive smoothness requirement on
the coupling boundary, but only Lipschitz-continuity. In addition, and
differently from known approaches for the elasticity problem, we are also able
to extend the Costabel & Han coupling procedure to the 3D Stokes problem by
providing a direct proof of the required coerciveness property, that is without
argueing by contradiction, and by using the natural norm of each space instead
of mesh-dependent norms. Finally, we briefly describe concrete examples of
discrete spaces satisfying the aforementioned hypotheses.
