Subsonic Solutions for Steady Euler--Poisson System in Two-Dimensional Nozzles

Abstract

In this paper, we prove the existence and stability of subsonic flows for a steady full Euler-Poisson system in a two-dimensional nozzle of finite length when imposing the electric potential difference on a noninsulated boundary from a fixed point at the entrance, and prescribing pressure at the exit of the nozzle. The Euler-Poisson system for subsonic flow is a hyperbolicelliptic coupled nonlinear system. One of the crucial ingredients of this work is the combination of Helmholtz decomposition for the velocity field and stream function formulation. In terms of the Helmholtz decomposition, the Euler-Poisson system is rewritten as a second order nonlinear elliptic system of three equations and transport equations for entropy and pseudo-Bernoulli's invariant. The associated elliptic system in a Lipschitz domain with nonlinear boundary conditions is solved with the help of the estimates developed in [M. Bae, B. Duan, and C. J. Xie, Existence and Stability of Multidimensional Steady Potential Flows for Euler-Poisson Equations, preprint, arXiv:1211.5234, 2012 based on its nice structure. The transport equations are resolved via the flow map induced by the stream function formulation. Furthermore, the delicate estimates for the flow map give the uniqueness of the solutions.

Contact discontinuities for 3-D axisymmetric inviscid compressible flows in infinitely long cylinders

“…with σ * 4 defined in (4.74), so that we obtain from (4.76) that f (1) = f (2) for σ ≤ σ 4 . Then, by (4.75), we have W (1) = W (2) .…”

“…with σ * 4 defined in (4.74), so that we obtain from (4.76) that f (1) = f (2) for σ ≤ σ 4 . Then, by (4.75), we have W (1) = W (2) . Therefore (f (1) , W (1) , ϕ (1) , ψ (1) ) = (f (2) , W (2) , ϕ (2) , ψ Then, by the definition of the Bernoulli invariant (1.4), we have…”

“…where ǫ 1 is given in (4.41), so that (4.43) implies that (φ, ψ) ∈ K f * For each j = 1, 2, let (1) , t(r, ψ (1) , Dψ (1) , Λ * )) − F(S * − S − 0 , Dφ (2) , t(r, ψ (2) , Dψ (2) , Λ * )), G * := G(W * , ∂ r W * , t(r,ψ (1) , Dψ (1) , Λ * ), Dφ (1)…”

“…To complete the estimate of f 1,α,(0,L) = f 0,(0,L) + ( f ) ′ α,(0,L) , we now estimate f 0,(0,L) . Define ρ (1) , u (1) x , ρ (2) , and u (2) x by…”

“…Finally, it remains to prove the uniqueness of a solution to Problem 4.2. For a fixed W * ∈ P(M 1 ), let (f (1) , ϕ (1) , ψ (1) ) and (f (2) , ϕ (2) , ψ (2) ) be two solutions to Problem 4.2, and suppose that each solution satisfies the estimate given in (4.6) of Lemma 4.2. Define a transformation T : N − L,f (1) → N − L,f (2) by…”