Transonic shocks for 2-D steady Euler flows with large gravity in a nozzle for polytropic gases

In this paper, we are concerned with the existence of transonic shock solutions for two-dimensional (2-d) steady Euler flows of polytropic gases with the vertical gravity in a horizontal nozzle under a pressure condition imposed at the exit of the nozzle. The acceleration of the gravity g is assumed to take a generic value. We first show that the existence of special transonic shock solutions with the flow states depending only on the variable in the gravity direction can be established if and only if the Mach number of the incoming flow satisfies certain conditions. However, the shock position of the special solutions is arbitrary in the nozzle. We determine the shock position and establish the existence of transonic shock solution when the boundary data are small perturbations of the special shock solutions under certain conditions. Mathematically, the perturbation problem can be formulated as a free boundary problem of a nonlinear system of hyperbolic-elliptic mixed type and composite. Key difficulties in the analysis mainly comes from the vertical gravity. Methods and techniques are developed in this paper to deal with these key difficulties. Finally, it turns out that the vertical gravity plays a dominant role in the mechanism determining the shock position. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2026.