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Subsonic Euler flows in a three-dimensional finitely long cylinder with arbitrary cross section

Shangkun Weng, Changkui Yao

TL;DR

The paper addresses the existence and uniqueness of subsonic steady Euler flows in a 3D finitely long cylinder with arbitrary cross-section. It employs a deformation-curl decomposition to decouple the hyperbolic and elliptic modes of the steady Euler system and augments this with a separation-of-variables technique to improve regularity near corner interactions between entrance, exit, and wall. Boundary data include the normal momentum, vorticity, Bernoulli constant, and entropy at the entrance, plus normal momentum at the exit, all subject to compatibility; solutions are sought in the Sobolev space $H^3(\Omega)$. A nonlinear fixed-point scheme on a ball in $H^3(\Omega)$ is used to prove existence and uniqueness for small boundary perturbations, yielding the stability estimate $\| (\rho,u_1,u_2,u_3,K)-(\bar{\rho},\bar{u},0,0,\bar{K})\|_{3,\Omega} \le \mathcal{C}\sigma$. The approach extends prior results to cylinders with arbitrary cross-sections, providing a robust structural stability framework for 3D subsonic flows in more general geometries.

Abstract

This paper concerns the well-posedness of subsonic flows in a three-dimensional finitely long cylinder with arbitrary cross section. We establish the existence and uniqueness of subsonic flows in the Sobolev space by prescribing the normal component of the momentum, the vorticity, the entropy, the Bernoulli's quantity at the entrance and the normal component of the momentum at the exit. One of the key points in the analysis is to utilize the deformation-curl decomposition for the steady Euler system introduced in \cite{WX19} to deal with the hyperbolic and elliptic modes. Another one is to employ the separation of variables to improve the regularity of solutions to a deformation-curl system near the intersection between the entrance and exit with the cylinder wall.

Subsonic Euler flows in a three-dimensional finitely long cylinder with arbitrary cross section

TL;DR

The paper addresses the existence and uniqueness of subsonic steady Euler flows in a 3D finitely long cylinder with arbitrary cross-section. It employs a deformation-curl decomposition to decouple the hyperbolic and elliptic modes of the steady Euler system and augments this with a separation-of-variables technique to improve regularity near corner interactions between entrance, exit, and wall. Boundary data include the normal momentum, vorticity, Bernoulli constant, and entropy at the entrance, plus normal momentum at the exit, all subject to compatibility; solutions are sought in the Sobolev space . A nonlinear fixed-point scheme on a ball in is used to prove existence and uniqueness for small boundary perturbations, yielding the stability estimate . The approach extends prior results to cylinders with arbitrary cross-sections, providing a robust structural stability framework for 3D subsonic flows in more general geometries.

Abstract

This paper concerns the well-posedness of subsonic flows in a three-dimensional finitely long cylinder with arbitrary cross section. We establish the existence and uniqueness of subsonic flows in the Sobolev space by prescribing the normal component of the momentum, the vorticity, the entropy, the Bernoulli's quantity at the entrance and the normal component of the momentum at the exit. One of the key points in the analysis is to utilize the deformation-curl decomposition for the steady Euler system introduced in \cite{WX19} to deal with the hyperbolic and elliptic modes. Another one is to employ the separation of variables to improve the regularity of solutions to a deformation-curl system near the intersection between the entrance and exit with the cylinder wall.
Paper Structure (6 sections, 12 theorems, 113 equations)

This paper contains 6 sections, 12 theorems, 113 equations.

Table of Contents

  1. Introduction and main results
  2. Some preliminaries
  3. A deformation-curl reformulation to the steady Euler system
  4. Some results for Sobolev spaces on $\Omega=(0,L)\times\Sigma$
  5. Existence and uniqueness of steady subsonic flows
  6. Proof of Theorem 1.1

Key Result

Theorem 1.1

Given $(B_{0},K_{0}) \in (H_0^3(\Sigma))^2$, $J_{0} \in H^2_0(\Sigma)$ and $(m_{0},m_{L}) \in (H^{\frac{5}{2}}(\Sigma))^2$ satisfying the compatibility conditions compm, there exists a positive small constant $\sigma_0$ depending on the background subsonic state $(\bar{\rho},\bar{u},0,0,\bar{K})$ an where $\mathcal{C}>0$ depends on $(\bar{\rho}, \bar{u},\bar{K})$ and $(m_{0},B_{0},K_{0},J_{0},m_{L

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 11 more