$ L^1 $–stability of vortex sheets and entropy waves in steady supersonic Euler flows over Lipschitz walls

Abstract

<p style='text-indent:20px;'>We study the well-posedness of compressible vortex sheets and entropy waves in two-dimensional steady supersonic Euler flows over Lipschitz walls with <inline-formula><tex-math id="M2">\begin{document}$ BV $\end{document}</tex-math></inline-formula> incoming flows. Both the Lipschitz wall of <inline-formula><tex-math id="M3">\begin{document}$ BV $\end{document}</tex-math></inline-formula> tangential angle function and the <inline-formula><tex-math id="M4">\begin{document}$ BV $\end{document}</tex-math></inline-formula> incoming flow perturb a background strong vortex sheet/entropy wave. In particular, when the total variation of the incoming flow perturbation around the background strong vortex sheet/entropy wave is small, we prove that the two-dimensional steady supersonic Euler flows containing a strong vortex sheet/entropy wave past the Lipschitz wall are <inline-formula><tex-math id="M5">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula>–stable. The weak waves are reflected after the nonlinear waves interact with the strong vortex sheet/entropy wave and the wall boundary. The existence of solutions in <inline-formula><tex-math id="M6">\begin{document}$ BV $\end{document}</tex-math></inline-formula> over the Lipschitz walls is first shown, when the total variation of the incoming flow perturbation around the background strong vortex sheet/entropy wave is suitably small, by using the wave-front tracking method. Then we establish the <inline-formula><tex-math id="M7">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula>–stability of the solutions with respect to the incoming flows. To achieve this, a Lyapunov functional, equivalent to the <inline-formula><tex-math id="M8">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula>–distance between two solutions containing the strong vortex sheets/entropy waves, is carefully constructed to include the nonlinear waves generated by both the wall boundary and the incoming flow. This Lyapunov functional is then proved to decrease in the flow direction, leading to the <inline-formula><tex-math id="M9">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula>–stability of the solutions. Furthermore, the uniqueness of these solutions extends to a larger class of viscosity solutions.</p>