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This paper concerns the existence of transonic shock solutions to the 2-Dsteady compressible Euler system in an almost flat finite nozzle ( in the sensethat it is a generic small perturbation of a flat one ), under physicalboundary conditions proposed by Courant-Friedrichs in\cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed atthe exit of the nozzle. In the resulting free boundary problem, the location ofthe shock-front is one of the most desirable information one would like todetermine. However, the location of the normal shock-front in a flat nozzle canbe anywhere in the nozzle so that it provides little information on thepossible location of the shock-front when the nozzle's boundary is perturbed.So one of the key difficulties in looking for transonic shock solutions is todetermine the shock-front. To this end, a free boundary problem for thelinearized Euler system will be proposed, whose solution will be taken as aninitial approximation for the transonic shock solution. In this paper, asufficient condition in terms of the geometry of the nozzle and the given exitpressure is derived which yields the existence of the solutions to the proposedfree boundary problem. Once an initial approximation is obtained, a furthernonlinear iteration could be constructed and proved to lead to a transonicshock solution.

On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure