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On two-dimensional steady compactly supported Euler flows with constant vorticity

Changfeng Gui, Jun Wang, Wen Yang, Yong Zhang

TL;DR

This work analyzes two-dimensional steady compactly supported Euler flows with constant vorticity in bounded ring and disk-like domains, focusing on closed streamline configurations and three overdetermined boundary-value problems. It develops a shape-derivative framework and employs Crandall–Rabinowitz bifurcation to construct local solution curves that bifurcate from radial annuli or disks, yielding nontrivial admissible domains. The paper establishes flexibility, rigidity, and stability results for each problem class, including explicit bifurcation points such as $\gamma^* = \frac{4}{\lambda^2-2\lambda^2\ln\lambda-1}$ and related higher-mode generalizations, and extends the analysis to perturbed Neumann data. These findings provide new insights into overdetermined elliptic problems in fluid mechanics and suggest avenues for higher-dimensional and more general vorticity settings.

Abstract

In this paper, we mainly construct local solution curves for the two-dimensional steady compactly supported incompressible Euler equations with free boundaries and constant vorticity. Our work is distinguished from most existing studies on two-dimensional steady water waves by its focus on perturbations near annular flows, rather than laminar flows. More precisely, we consider three classes of steady Euler flows with compact support, corresponding to partially overdetermined, two-phase overdetermined, and overdetermined elliptic problems. The primary contribution of our work is threefold. For each class, we first establish the flexibility result (i.e., the existence of nontrivial admissible domains) via shape derivatives and local bifurcation theory. Second, we give and discuss the corresponding rigidity result respectively. Third, we apply the implicit function theorem to demonstrate the stability of standard annular flows under perturbations of the Neumann boundary condition. Our results also offer novel insights into the theory of elliptic overdetermined problems.

On two-dimensional steady compactly supported Euler flows with constant vorticity

TL;DR

This work analyzes two-dimensional steady compactly supported Euler flows with constant vorticity in bounded ring and disk-like domains, focusing on closed streamline configurations and three overdetermined boundary-value problems. It develops a shape-derivative framework and employs Crandall–Rabinowitz bifurcation to construct local solution curves that bifurcate from radial annuli or disks, yielding nontrivial admissible domains. The paper establishes flexibility, rigidity, and stability results for each problem class, including explicit bifurcation points such as and related higher-mode generalizations, and extends the analysis to perturbed Neumann data. These findings provide new insights into overdetermined elliptic problems in fluid mechanics and suggest avenues for higher-dimensional and more general vorticity settings.

Abstract

In this paper, we mainly construct local solution curves for the two-dimensional steady compactly supported incompressible Euler equations with free boundaries and constant vorticity. Our work is distinguished from most existing studies on two-dimensional steady water waves by its focus on perturbations near annular flows, rather than laminar flows. More precisely, we consider three classes of steady Euler flows with compact support, corresponding to partially overdetermined, two-phase overdetermined, and overdetermined elliptic problems. The primary contribution of our work is threefold. For each class, we first establish the flexibility result (i.e., the existence of nontrivial admissible domains) via shape derivatives and local bifurcation theory. Second, we give and discuss the corresponding rigidity result respectively. Third, we apply the implicit function theorem to demonstrate the stability of standard annular flows under perturbations of the Neumann boundary condition. Our results also offer novel insights into the theory of elliptic overdetermined problems.
Paper Structure (18 sections, 22 theorems, 192 equations, 7 figures)

This paper contains 18 sections, 22 theorems, 192 equations, 7 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Function spaces and abstract operators
  4. Shape derivatives
  5. Spherical harmonics
  6. The partially overdetermined elliptic problem (\ref{['e1.2']})
  7. The trivial solution of (\ref{['e1.2']})
  8. The spectrum of the linearized operator $\partial_{\eta}G(\gamma,0)$
  9. Proof of Theorem \ref{['thm1.1']}
  10. Proof of Corollary \ref{['co']}
  11. The two-phase overdetermined elliptic problem (\ref{['e1.3']})
  12. The spectrum of the linearized operator $\partial_{\eta}H(\gamma_2,0)$
  13. Proof of Theorem \ref{['thm1.2']}
  14. The overdetermined elliptic problem (\ref{['e1.4']})
  15. The spectrum of the linearized operator $\partial_{(\eta,\xi)}\mathcal{G}(\gamma,(0,0))$
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any $\lambda\in (0,1)$, there exists $s_0>0$ and a curve of solutions $(\gamma(s),\eta(s))$ to (e1.2), parametrized by $s\in(-s_0,s_0)$, with the following properties:

Figures (7)

  • Figure 1: The nontrivial domains $\Omega_{\eta(s)}\setminus B_{\lambda}$ for Theorem \ref{['thm1.1']} bifurcating from $B_{1}\setminus B_{\lambda}$.
  • Figure 2: The nontrivial domains $B_{\lambda}\cup\left(\Omega_{\eta(s)}\setminus B_{\lambda}\right)$ for Theorem \ref{['thm1.2']} bifurcating from $B_1$
  • Figure 4: The nontrivial domains $\Omega_{\eta(s)}\setminus D_{\xi(s)}$ for Theorem \ref{['thm1.3']} bifurcating from $B_{1}\setminus B_{\lambda}$.
  • Figure 5: Sketches of $\gamma_k^*$ as a function of $\lambda$ for $k=1,2,3,5,10,20,100$.
  • Figure 6: Sketches of $\gamma_{2k}^*/\gamma_1$ as a function of $\lambda$ for $k=1,2,3,5,10,20,100$.
  • ...and 2 more figures

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.1
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.2
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.3
  • Lemma 3.1
  • ...and 31 more