Stability of Compressible Vortex Sheets in Steady Supersonic Euler Flows over Lipschitz Walls

Abstract

We are concerned with the stability of compressible vortex sheets in two‐dimensional steady supersonic Euler flows over Lipschitz walls under a $BV$ boundary perturbation, since steady supersonic Euler flows are important in many physical situations. It is proved that steady compressible vortex sheets in supersonic flow are stable in structure globally, even under the $BV$ perturbation of the Lipschitz walls. In order to achieve this, we develop a modified Glimm difference scheme and identify a Glimm‐type functional to obtain the required $BV$ estimates by incorporating the Lipschitz boundary and the strong vortex sheets naturally and by tracing the interaction not only between the boundary and weak waves but also between the strong vortex sheets and weak waves. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution and the corresponding approximate strong vortex sheets to a strong compressible vortex sheet of the entropy solution. The asymptotic stability of entropy solutions in the flow direction is also established.