Stability of Transonic Contact Discontinuity for Two-Dimensional Steady Compressible Euler Flows in a Finitely Long Nozzle

We consider the stability of transonic contact discontinuity for two-dimensional steady compressible Euler flows in a finitely long nozzle. This is first work on mixed-type problem across as free boundary nozzles. start with Euler-Lagrangian transformation to straighten new coordinates. However, upper nozzle wall subsonic region depending mass flux becomes after transformation. Then we develop ideas and techniques solve free-boundary three steps: (1) fix generate iteration scheme corresponding fixed value hyperbolic-elliptic mixed type by building some powerful estimates both first-order hyperbolic equation second-order nonlinear elliptic Lipschitz domain; (2) update constructing mapping that has point; (3) establish via inverse Lagrangian coordinate original interface admits unique piecewise smooth solution near background state, which consists flow supersonic discontinuity.