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Global three-dimensional subsonic Euler flows past an axisymmetric obstacle with large vorticity

Dehua Wang, Tian-Yi Wang, Weiqiang Wang

Abstract

In this paper, we prove the existence and uniqueness of subsonic solutions to the steady Euler flows past a smooth, axisymmetric obstacle. Specifically, for a broad class of prescribed positive axial velocities in the upstream, the subsonic Euler flow exists provided that the upstream density exceeds a critical threshold. The non-degeneracy of the axial velocity is rigorously established by combining the strong maximum principle with a refined continuity argument. The asymptotic behavior of the flow is obtained from uniform integral estimates for the difference between the flow and the upstream state. In addition, this result accommodates flows with large vorticity under a structural condition, thereby differing from previous results in the two-dimensional case.

Global three-dimensional subsonic Euler flows past an axisymmetric obstacle with large vorticity

Abstract

In this paper, we prove the existence and uniqueness of subsonic solutions to the steady Euler flows past a smooth, axisymmetric obstacle. Specifically, for a broad class of prescribed positive axial velocities in the upstream, the subsonic Euler flow exists provided that the upstream density exceeds a critical threshold. The non-degeneracy of the axial velocity is rigorously established by combining the strong maximum principle with a refined continuity argument. The asymptotic behavior of the flow is obtained from uniform integral estimates for the difference between the flow and the upstream state. In addition, this result accommodates flows with large vorticity under a structural condition, thereby differing from previous results in the two-dimensional case.
Paper Structure (11 sections, 15 theorems, 261 equations, 3 figures)

This paper contains 11 sections, 15 theorems, 261 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Stream Function Formulation
  3. Existence of Subsonic Solution of \ref{['1.13']} in a Nozzle
  4. Uniform Estimates for Subsonic Flows in Nozzles
  5. Fine Properties of Subsonic Solutions
  6. Asymptotic behavior at far fields
  7. Positivity of axial velocity away from axis
  8. The uniqueness of subsonic Euler flows past an axisymmetric obstacle
  9. Existence of the critical density in the upstream
  10. Proof of Theorem \ref{['thm1']}
  11. Limit of Subsonic Flows: Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1.1

Suppose that the upstream axial velocity $u_{\infty}(r)$ in 1.3-2 satisfies and the following structural condition: for $r\ge 0$. There exists a critical value $\rho_{cr}>\rho_{\infty}^{*}>0$, such that if the upstream density $\rho_{\infty}$ in 1.3-2 is larger than $\rho_{cr}$, the Euler system 1.3 with the boundary conditions 1.3-1--1.3-2 admits a subsonic solution $(\rho,u,v)\in (C^{1,\alpha}

Figures (3)

  • Figure 1: Subsonic Euler flows past an axisymmetric obstacle
  • Figure 2: The schematic of topological structure of streamline
  • Figure 3: The relationship between $\mathcal{M}$ and $\rho$

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • Lemma 3.1: Existence of \ref{['2.9-1']}
  • Lemma 3.2
  • ...and 10 more