Structural Stability of Supersonic Solutions to the Euler–Poisson System

Three-dimensional supersonic flows of Euler-Poisson system for potential flow

“…In this paper, we are aimed to construct a three dimensional supersonic solution to (1.4). Even though the main framework would be similar to [1], there are several technical differences to be taken care of, and it is our main goal to treat all the technical differences occurring when one considers a three dimensional supersonic solution of the steady Euler-Poisson system. This work may seem as a simple extension of [1] by adding a few technical changes.…”

“…Even though the main framework would be similar to [1], there are several technical differences to be taken care of, and it is our main goal to treat all the technical differences occurring when one considers a three dimensional supersonic solution of the steady Euler-Poisson system. This work may seem as a simple extension of [1] by adding a few technical changes. But we must emphasize that this work is a keystone to investigate a three dimensional supersonic solution to (1.1) with a nonzero vorticity in the regime of Helmholtz decomposition.…”

“…Furthermore, the system (1.4) is a natural example of a quasi-linear hyperbolic-elliptic mixed system of second order arising from the study of physical phenomena, therefore the main result obtained from this paper can provide a footstone in the study of a nonlinear hyperbolic-elliptic mixed type system of second order partial differential equations in a multidimensional domain. Moreover, we remove the extra compatibility conditions given in [1] in this paper. For a fixed constant L > 0, we define a nozzle Ω L of length L with a rectangular cross section D := (−1, 1) × (−1, 1) ⊂ R 2 by Ω L := (x 1 , x 2 , x 3 ) ∈ R 3 : 0 < x 1 < L, (x 2 , x 3 ) ∈ D .…”

“…Lemma 1.2 (One dimensional supersonic solutions [1,12]). For any small constant δ > 0, there exists a constant L 1 > 0 depending on (γ, J 0 , b 0 , u 0 , E 0 ) so that the initial value problem (1.9) has a unique smooth solution (u, E) on [0, L 1 ] with satisfying that inf…”

“…A proof of Lemma 1.2 can be easily given from [1,Lemma 1.1] or [12] so we skip it in this paper. Instead, we add a brief comment on how L 1 varies depending on (u 0 , E 0 ).…”