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In this paper, we prove the existence of two-dimensional solutions to thesteady Euler-Poisson system with continuous transonic transitions across sonicinterfaces of codimension 1. First, we establish the well-posedness of aboundary value problem for a linear second order system that consists of anelliptic-hyperbolic mixed type equation with a degeneracy occurring on aninterface of codimension 1, and an elliptic equation weakly coupled together.Then we apply the Schauder fixed point theorem to prove the existence oftwo-dimensional solutions to the potential flow model of the steadyEuler-Poisson system with continuous transonic transitions across sonicinterfaces. With the aid of Helmholtz decomposition, established in [6], weextend the existence result to the full Euler-Poisson system for the case ofnonzero vorticity. Most importantly, the solutions constructed in this paperare classical solutions to Euler-Poisson system, thus their sonic interfacesare not weak discontinuities in the sense that all the flow variables are $C^1$across the interfaces.

The steady Euler-Poisson system and accelerating flows with transonic $C^1$-transitions