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Supersonic flows of the Euler-Poisson system in three-dimensional cylinders

Myoungjean Bae, Hyangdong Park

Abstract

In this paper, we prove the unique existence of three-dimensional supersonic solutions to the steady Euler-Poisson system in cylindrical nozzles when prescribing the velocity, entropy, and the strength of electric field at the entrance. We first establish the unique existence of irrotational supersonic solutions in a cylindrical nozzle with an arbitrary cross section by extending the results of \cite{bae2021three} with an aid of weighted Sobolev norms. Then, we establish the unique existence of three-dimensional axisymmetric supersonic solutions to the Euler-Poisson system with nonzero vorticity in a circular cylinder. In particular, we construct a three-dimensional solution with a nonzero angular momentum density (or equivalently a nonzero swirl). Therefore this is truly a three dimensional flow in the sense that the Euler-Poisson system cannot be reduced to a two dimensional system via a stream function formulation. The main idea is to reformulate the system into a second order hyperbolic-elliptic coupled system and two transport equations via the method of Helmholtz decomposition, and to employ the method of iterations. Several technical issues, including the issue of singularities on the axis of symmetry and the issue of corner singularities in a Lipschitz domain, are carefully addressed.

Supersonic flows of the Euler-Poisson system in three-dimensional cylinders

Abstract

In this paper, we prove the unique existence of three-dimensional supersonic solutions to the steady Euler-Poisson system in cylindrical nozzles when prescribing the velocity, entropy, and the strength of electric field at the entrance. We first establish the unique existence of irrotational supersonic solutions in a cylindrical nozzle with an arbitrary cross section by extending the results of \cite{bae2021three} with an aid of weighted Sobolev norms. Then, we establish the unique existence of three-dimensional axisymmetric supersonic solutions to the Euler-Poisson system with nonzero vorticity in a circular cylinder. In particular, we construct a three-dimensional solution with a nonzero angular momentum density (or equivalently a nonzero swirl). Therefore this is truly a three dimensional flow in the sense that the Euler-Poisson system cannot be reduced to a two dimensional system via a stream function formulation. The main idea is to reformulate the system into a second order hyperbolic-elliptic coupled system and two transport equations via the method of Helmholtz decomposition, and to employ the method of iterations. Several technical issues, including the issue of singularities on the axis of symmetry and the issue of corner singularities in a Lipschitz domain, are carefully addressed.
Paper Structure (24 sections, 32 theorems, 422 equations, 1 figure)

This paper contains 24 sections, 32 theorems, 422 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Theorems
  3. Potential flows
  4. Nonzero vorticity flows
  5. Potential flows (Proof of Theorem \ref{['Irr-MainThm']})
  6. Outline of the proof
  7. A priori estimates of a smooth solution
  8. A priori estimates of the first order derivatives
  9. Second order derivative estimates
  10. A priori estimates of $D^3(V, W)$
  11. A priori estimates of the fourth order derivatives
  12. Proof of Proposition \ref{['proposition-apriori estimate of smooth solution']}
  13. Proof of Proposition \ref{['lemma-wp of lbvp for potential']}
  14. Proof of Theorem \ref{['Irr-MainThm']}
  15. Nonzero vorticity flows (Proof of Theorem \ref{['MainThm']})
  16. ...and 9 more sections

Key Result

Lemma 2.1

For any given constant $\bar{\delta}>0$ sufficiently small, there exists a constant $\bar{L}>0$ depending on $(\gamma, J_0, S_0, b_0, \rho_0, E_0,\bar{\delta})$ so that the initial value problem one-re-uE has a unique smooth solution $(\bar{\rho},\bar{E})(x_1)$ on $[0,\bar{L}]$ with satisfying that Note that the above inequality is equivalent to for some constant $\hat{\delta}>0$.

Figures (1)

  • Figure 1: The $\bar{u}$-$\bar{E}$ phase plane: In (i), the background solution is period, therefore we have $L^*<\bar{L}$. In (ii) and (iii), we can fix $L^*$ as $L^*=\bar{L}$.

Theorems & Definitions (55)

  • Lemma 2.1: One-dimensional supersonic solutions (bae2021structural, luo2012transonic)
  • Definition 2.2: Background solutions
  • Definition 2.4: A weighted Sobolev norm
  • Definition 2.5: A weighted Sobolev norm with involving a time-like variable
  • Theorem 2.6: Potential flows
  • Remark 2.7
  • Definition 2.8
  • Remark 2.10
  • Theorem 2.11: Nonzero vorticity flows
  • Remark 2.12
  • ...and 45 more