Well-Posedness of Transonic Characteristic Discontinuities in   Two-Dimensional Steady Compressible Euler Flows
  In our previous work, we have established the existence of transonic characteristic discontinuities separating supersonic flows from a static gas in two-dimensional steady compressible Euler flows under a perturbation with small total variation of the incoming supersonic flow over a solid right-wedge. It is a free boundary problem in Eulerian coordinates and, across the free boundary (characteristic discontinuity), the Euler equations are of elliptic-hyperbolic composite-mixed type. In this paper, we further prove that such a transonic characteristic discontinuity solution is unique and $L^1$--stable with respect to the small perturbation of the incoming supersonic flow in Lagrangian coordinates. 
Gui-Qiang Chen, Vaibhav Kukreja, Hairong Yuan
math.AP math-ph math.MP