On stability of transonic shocks for stationary rayleigh flows in two-dimensional ducts

Transonic shocks play a pivotal role in design of supersonic nozzles in gas dynamics. Previous studies have shown that for stationary compressible Euler flows in rectilinear ducts with constant cross-sections, they are not stable under perturbations of upcoming supersonic flows and back pressures at the exits of the ducts, while expanding the nozzle and frictions have stabilization effects. In this paper we study the Rayleigh flows, namely the effects of heat-exchange, on stabilization of transonic shocks, in two-dimensional rectilinear ducts. We proved that for given heat-exchange per unit mass of gas, almost all the associated unidimensional transonic shocks are stable, provided that the perturbations of the upstream supersonic flows and downstream back pressures satisfy some symmetry conditions, while for given heat-exchange per unit volume of gas, the resultant unidimensional transonic shocks are not stable. Mathematically we study a nonlinear free-boundary problem of a system of conservation laws of hyperbolic-elliptic composite-mixed type. The proof depends on decoupling the elliptic and hyperbolic part of the subsonic Euler system in Lagrangian coordinates by characteristic decomposition. Since heat-exchange introduces complicated interactions in the Euler system, we need to study a general linear variable-coefficient first-order elliptic-hyperbolic strongly coupled system with nonlocal boundary conditions, by Fourier analysis and careful investigation of reduced boundary value problems of ordinary differential equations.