| We establish the existence and uniqueness of transonic shocks in the steady flows through a two-dimensional nozzle with varying cross-sections. The flow is governed by the steady full Euler equations. The problem is approached by a one-phase free boundary problem where the shock front is a free boundary. We show that the solutions behind the shock front remain subsonic in a downstream region and the shock front is smooth for the given supersonic flows perturbed from the uniform supersonic flows. The steady full Euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem. To solve the free boundary problem, the Schauder fixed point theorem is used. |
