Stability of Cylindrical Transonic Shocks for the Two-Dimensional Steady Compressible Euler System

We establish the stability of a class of cylindrical symmetric transonic shocks for two-dimensional complete compressible steady Euler system. This result partly explains the effectiveness of the popular quasi-one-dimensional model of nozzle flows used in aerodynamics. Mathematically, we solve a nonlinear free boundary problem of an elliptic-hyperbolic composite system, with the circular transonic shock front as the free boundary. We accomplish this by finding a (locally) unique fixed point of an appropriately defined boundary profile updating mapping. To define this mapping, we encounter a series of nonclassical boundary value problems in an annulus, which involve a new type of nonlocal elliptic problem, and integral-like solvability conditions to determine the position of the free boundary. This reflects an interesting new feature of boundary value problems of elliptic-hyperbolic composite systems.