Stabilization Effect Of Frictions For Transonic Shocks In Two-Dimensional Steady Compressible Euler Flows

In this paper we show existence and stability of transonic shocks in a two dimen-sional straight duct for the complete compressible steady Euler system with frictions, under perturbations of the upcoming supersonic flows and the pressure at the exit. It is a nonlinear free boundary problem of a elliptic-hyperbolic composite-mixed system, with the shock-front being the free boundary. The essential part is to obtain estimates and existence for nonclassical nonlocal boundary value problems of linear first-order elliptic-hyperbolic coupled system after introducing Lagrangian transform and charac-teristic decomposition. This result demonstrates the stabilization effect of frictions for transonic shocks in ducts for given back pressure, which was previously proved to be unstable for Euler equations without frictions.