  We consider the compressible Navier--Stokes equation in a perturbed
half-space with an outflow boundary condition as well as the supersonic
condition. For a half-space, it has been known that a certain planar stationary
solution exist and it is time-asymptotically stable. The planar stationary
solution is independent of the tangential directions and its velocities of the
tangential directions are zero. In this paper, we show the unique existence of
stationary solutions for the perturbed half-space. The feature of our work is
that our stationary solution depends on all directions and has multidirectional
flow. Furthermore, we also prove the asymptotic stability of this stationary
solution.
