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

1. Problem Statement

The paper investigates two-dimensional steady, incompressible Euler flows with constant vorticity in compactly supported domains (i.e., fluid regions with closed free boundaries). Unlike the classical "water waves" literature which focuses on strip-shaped domains with open streamlines, this work addresses the "closed-streamline" configuration relevant to physical phenomena like liquid drops or rotating fluid columns.

The authors analyze three distinct classes of overdetermined elliptic free-boundary problems, where the domain shape is unknown and must be determined alongside the stream function $\psi$:

1. Partially Overdetermined Problem (Single Free Boundary):

   • Domain: An annulus $\Omega \setminus D_0$ where $D_0$ is a fixed inner disk and $\partial \Omega$ is a free boundary.
   • Conditions: $\Delta \psi = \gamma$ in the domain, $\psi=0$ on $\partial \Omega$, $\psi=1$ on $\partial D_0$, and $|\nabla \psi|^2 = Q$ (Bernoulli constant) on $\partial \Omega$.
   • Context: Models a fluid ring with a fixed inner core.

2. Two-Phase Overdetermined Problem:

   • Domain: A disk $\Omega$ containing a core region $D$ (both boundaries are free).
   • Conditions: Two distinct constant vorticities $\gamma_1$ (in $D$) and $\gamma_2$ (in $\Omega \setminus D$). Continuity of $\psi$ and flux across the interface $\partial D$ is required. The outer boundary $\partial \Omega$ satisfies $\psi=0$ and $|\nabla \psi|^2 = Q$.
   • Context: Models a fluid drop with a distinct inner core of different vorticity.

3. Fully Overdetermined Problem (Two Free Boundaries):

   • Domain: An annulus $\Omega \setminus D$ where both the outer boundary $\partial \Omega$ and inner boundary $\partial D$ are free.
   • Conditions: Constant vorticity $\gamma$. Dirichlet conditions $\psi=0$ on $\partial \Omega$ and $\psi=1$ on $\partial D$. Neumann conditions $\partial_\nu \psi = q_{out}$ on $\partial \Omega$ and $\partial_\nu \psi = q_{in}$ on $\partial D$.
   • Context: A fully free-floating fluid ring where both boundaries adjust to satisfy the Bernoulli condition.

2. Methodology

The authors employ a combination of Shape Derivatives, Local Bifurcation Theory (specifically the Crandall–Rabinowitz theorem), and the Implicit Function Theorem.

• Reformulation: The free-boundary problems are reformulated as abstract operator equations $G(\gamma, \eta) = 0$ (or similar for the other cases), where:

  • $\gamma$ (or $\gamma_2$) acts as the bifurcation parameter.
  • $\eta$ (and $\xi$ for the inner boundary) represents the shape perturbation of the boundary from a reference circle (or annulus).
  • The operators map the vorticity and shape perturbation to the deviation of the Neumann data from the required constant.

• Linearization and Spectrum Analysis:

  • The authors compute the Fréchet derivative of the operator at the trivial solution (standard circular/annular domains).
  • Using Spherical Harmonics (Fourier cosine series in 2D), they derive the dispersion relation for the linearized operator.
  • They identify bifurcation points $\gamma^*$ where the linearized operator has a non-trivial kernel (i.e., the determinant of the associated matrix vanishes).

• Bifurcation and Stability:

  • Flexibility (Existence): By verifying the transversality condition of the Crandall–Rabinowitz theorem, they prove that non-trivial branches of solutions (non-circular domains) bifurcate from the trivial radial solutions at specific critical vorticities.
  • Rigidity: They establish conditions under which the only solutions are the trivial radial ones (using Reichel's theorem and Aleksandrov's theorem).
  • Stability: Using the Implicit Function Theorem, they prove that the trivial radial solutions are locally unique and stable under small perturbations of the Neumann boundary condition (Bernoulli constant), provided the vorticity is not at a bifurcation point.

3. Key Contributions and Results

A. Flexibility (Existence of Non-Trivial Domains)

The paper proves that for specific values of constant vorticity, the standard circular/annular domains are not the only solutions.

• Theorem 1.1 (Partially Overdetermined): For any inner radius $\lambda \in (0,1)$, there exists a critical vorticity $\gamma^*$ (and a sequence $\gamma^*_k$) such that non-trivial, non-annular domains bifurcate.
  • Notably, while $\gamma^*_1$ is always negative, the paper shows that for small $\lambda$, higher modes $k \ge 2$ yield positive critical vorticities, allowing non-trivial domains even with positive vorticity.
• Theorem 1.3 (Two-Phase): Bifurcation occurs when the outer vorticity $\gamma_2$ equals the inner vorticity $\gamma_1$ (and other specific ratios). This leads to non-concentric disk configurations.
• Theorem 1.5 (Fully Overdetermined): Two distinct critical vorticities ($\gamma^*_1$ and $\gamma^{**}_1$) are found where the annulus bifurcates into non-annular shapes with perturbations on both inner and outer boundaries.

B. Rigidity Results

• The authors establish that if the vorticity is non-negative and the solution is positive (or under specific geometric constraints), the domain must be a standard annulus (or disk) and the solution must be radially symmetric.
• This relies on Reichel's Theorem, which states that for certain overdetermined problems, radial symmetry is forced unless the domain is a specific shape.
• A key insight is that non-trivial bifurcating solutions often require sign-changing stream functions, whereas positive solutions enforce symmetry.

C. Stability Results

• Corollaries 1.2, 1.4, and 1.6: The authors prove that the trivial (radial) solutions are locally unique and stable under small perturbations of the Neumann boundary condition (i.e., if the Bernoulli constant $Q$ is slightly perturbed by a function $\rho(\theta)$, a unique deformed domain exists).
• This stability holds provided the vorticity $\gamma$ is not one of the bifurcation values where the linearized operator becomes singular.

4. Significance and Novelty

• Shift from Laminar to Vortical Flows: Most existing literature on steady water waves focuses on irrotational (laminar) flows or small-amplitude perturbations. This paper rigorously constructs large-amplitude, compactly supported solutions with constant vorticity.
• New Perspectives on Overdetermined Problems: The work connects fluid dynamics with the theory of overdetermined elliptic problems. It demonstrates that "flexibility" (existence of non-symmetric domains) and "rigidity" (uniqueness of symmetric domains) coexist depending on the vorticity parameter.
• Methodological Advancement:
  • The authors utilize shape derivatives to simplify the linearization process, avoiding the cumbersome Hanzawa transformations used in previous works.
  • They treat vorticity as the bifurcation parameter rather than the domain radius, allowing for a more comprehensive analysis of the solution space for any fixed geometry.
• Physical Relevance: The results provide a theoretical foundation for understanding the stability and shape of rotating fluid drops, liquid bridges, and vortex rings where surface tension or pressure balance creates closed boundaries.

In summary, the paper successfully bridges the gap between classical free-boundary fluid dynamics and modern nonlinear analysis, providing a complete picture of existence, uniqueness, and stability for steady Euler flows with constant vorticity in compact domains.