  We study the well-posedness of compressible vortex sheets and entropy waves
in two-dimensional steady supersonic Euler flows over Lipschitz walls with $BV$
incoming flows. Both the Lipschitz wall of $BV$ tangential angle function and
the $BV$ incoming flow perturb a background strong vortex sheet/entropy wave.
In particular, when the total variation of the incoming flow perturbation
around the background strong vortex sheet/entropy wave is small, we prove that
the two-dimensional steady supersonic Euler flows containing a strong vortex
sheet/entropy wave past the Lipschitz wall are $L^{1}$--stable. The weak waves
are reflected after the nonlinear waves interact with the strong vortex
sheet/entropy wave and the wall boundary. Using the wave-front tracking method,
the existence of solutions in $BV$ over the Lipschitz walls is first shown,
when the total variation of the incoming flow perturbation around the
background strong vortex sheet/entropy wave is suitably small. Then we
establish the $L^{1}$--stability of the solutions with respect to the incoming
flows. To achieve this, a Lyapunov functional, equivalent to the
$L^{1}$--distance between two solutions containing the strong vortex
sheets/entropy waves, is carefully constructed to include the nonlinear waves
generated by both the wall boundary and the incoming flow. This Lyapunov
functional is then proved to decrease in the flow direction, leading to the
$L^{1}$--stability of the solutions. Furthermore, the uniqueness of these
solutions extends to a larger class of viscosity solutions.
