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We study the well-posedness of compressible vortex sheets and entropy wavesin two-dimensional steady supersonic Euler flows over Lipschitz walls with $BV$incoming flows. Both the Lipschitz wall of $BV$ tangential angle function andthe $BV$ incoming flow perturb a background strong vortex sheet/entropy wave.In particular, when the total variation of the incoming flow perturbationaround the background strong vortex sheet/entropy wave is small, we prove thatthe two-dimensional steady supersonic Euler flows containing a strong vortexsheet/entropy wave past the Lipschitz wall are $L^{1}$--stable. The weak wavesare reflected after the nonlinear waves interact with the strong vortexsheet/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 thebackground strong vortex sheet/entropy wave is suitably small. Then weestablish the $L^{1}$--stability of the solutions with respect to the incomingflows. To achieve this, a Lyapunov functional, equivalent to the$L^{1}$--distance between two solutions containing the strong vortexsheets/entropy waves, is carefully constructed to include the nonlinear wavesgenerated by both the wall boundary and the incoming flow. This Lyapunovfunctional is then proved to decrease in the flow direction, leading to the$L^{1}$--stability of the solutions. Furthermore, the uniqueness of thesesolutions extends to a larger class of viscosity solutions.

$L^1$-Stability of Vortex Sheets and Entropy Waves in Steady Compressible Supersonic Euler Flows over Lipschitz Walls