A free boundary value problem for the full Euler system and 2-d transonic shock in a large variable nozzle

In this paper, we establish the existence and uniqueness of a transonic shock solution to the full steady compressible Euler system in a class of de Laval nozzles with a large straight divergent part when a given variable exit pressure lies in a suitable range.Thus, for this class of nozzles, we have solved the transonic shock problem posed by Courant-Friedrichs in Section 147 of [5].By introducing a new elaborate iteration scheme, we are able to solve this boundary value problem for a coupled elliptic-hyperbolic system with a free boundary without some stringent requirements in the previous studies.One of the key ingredients in this approach is to solve a boundary value problem for a first order linear system with nonlocal terms and a free parameter.