Local stability of direct Mach configurations

Abstract

We investigate the local stability of direct Mach configurations for the two-dimensional steady compressible Euler equations. Given a piecewise constant Mach configuration that consists of an incident shock, a reflected shock, a Mach stem and a slip line, with subsonic downstream states, we prove the following: under a transversality condition on the shock polar loops and a suitable condition on the truncation segment, every sufficiently small perturbation of the incoming flow gives rise to a Mach configuration that is a small perturbation of the background one, in the sense that the reflected shock, the Mach stem and the slip line, together with the downstream subsonic flow, stay close to their background counterparts; moreover, the solution is unique in the class of solutions satisfying the a priori estimate (for incoming flows that are small in a slightly stronger norm). Lagrangian coordinates are employed to transform the unknown contact discontinuity curve into a fixed boundary and to reduce the full Euler system to a first-order elliptic system for the flow direction and the pressure. The key idea in dealing with the contact discontinuity is to solve a mixed boundary value problem, in divergence form with discontinuous coefficients, for a single elliptic equation, so that the contact discontinuity conditions are naturally preserved as compatibility conditions for the solutions of the elliptic problem.