2023/08/09 by Myoungjean Bae, Bae, Myoungjean, Ben Duan +3
Computer Science · Mathematics · #35J47 #35J57 #35J66 #35M10 #76N10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2308.04694
openalex publication_date 2023/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper, we prove the existence of two-dimensional solutions to the steady Euler-Poisson system with continuous transonic transitions across sonic interfaces of codimension 1. First, we establish the well-posedness of a boundary value problem for a linear second order system that consists of an elliptic-hyperbolic mixed type equation with a degeneracy occurring on an interface of codimension 1, and an elliptic equation weakly coupled together. Then we apply the Schauder fixed point theorem to prove the existence of two-dimensional solutions to the potential flow model of the steady Euler-Poisson system with continuous transonic transitions across sonic interfaces. With the aid of Helmholtz decomposition, established in [6], we extend the existence result to the full Euler-Poisson system for the case of nonzero vorticity. Most importantly, the solutions constructed in this paper are classical solutions to Euler-Poisson system, thus their sonic interfaces are not weak discontinuities in the sense that all the flow variables are C1 across the interfaces.