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

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.

💡 Research Summary

The paper investigates steady two‑dimensional incompressible Euler flows with constant vorticity in domains that are compactly supported and possess free boundaries. Three distinct classes of problems are considered: (i) a partially overdetermined elliptic problem (equation (1.2)) where a fixed inner circle Bλ is surrounded by an unknown outer free boundary; (ii) a two‑phase overdetermined problem (equation (1.3)) with a core D of one vorticity γ₁ and an outer region of another vorticity γ₂, both bounded by free interfaces; and (iii) a truly overdetermined problem (equation (1.4)) in which both inner and outer boundaries satisfy Neumann (Bernoulli) conditions.

The authors treat the constant vorticity γ as a bifurcation parameter and the boundary perturbations η, ξ as unknown functions in suitable Hölder spaces. By computing shape derivatives of the associated nonlinear operators, they linearize the problems and obtain eigenvalue problems for the Laplacian with mixed Dirichlet–Neumann conditions. Critical values γ* emerge as bifurcation points. For (1.2) they show explicitly that γ* = 4λ² – 2λ² ln λ – 1 (which is negative for all λ∈(0,1)). Near this value a smooth curve of solutions (γ(s), η(s)) exists for s∈(−s₀,s₀). At s=0 the domain is the standard annulus; for s≠0 the free boundary is perturbed as η(s)=s α₁ cos θ+o(s), producing non‑radial admissible domains. Higher Fourier modes (k≥2) generate additional bifurcation points γ*ₖ, allowing non‑trivial domains even for positive vorticity provided the stream function changes sign.

Rigidity results are proved using the maximum principle and Serrin‑type symmetry theorems. When γ≥0 and the stream function ψ is positive, ψ must be monotone in the radial variable, forcing the admissible domain to be a concentric annulus (or a disk when the inner core is absent). Thus, non‑radial solutions can only occur if ψ changes sign, a phenomenon reminiscent of sign‑changing solutions in overdetermined problems.

Stability under perturbations of the Bernoulli constant is addressed by considering a perturbed problem (1.6) where |∇ψ|² = Q + ρ(θ) on the outer free boundary. Applying the implicit function theorem, the authors prove that for sufficiently small ρ there exists a unique perturbation η(ρ) in the same function space, and the solution ψ remains close to the unperturbed radial profile. Moreover, η(ρ) admits an explicit Fourier expansion η(ρ)=∑ₖ (τₖ/σₖ) cos(kθ)+o(‖ρ‖), where τₖ are the Fourier coefficients of ρ and σₖ are the eigenvalues arising from the linearized operator. This yields quantitative stability estimates.

For the two‑phase problem (1.3) the analysis is analogous. Fixing γ₁≠0 and treating γ₂ as a parameter, a bifurcation occurs at γ₂* = γ₁. Near this point a curve of non‑concentric configurations Bλ∪Ω_η(s)\Bλ exists, with the same cosine perturbation structure. If an additional Neumann condition ∂_νψ₂ = m is imposed on the outer boundary, the authors prove a rigidity theorem: the only admissible configuration is a concentric disk.

The truly overdetermined problem (1.4) is treated in the same framework. The authors identify bifurcation points where the inner and outer Neumann data match, leading to non‑radial domains, and they establish corresponding rigidity and stability results under suitable spectral conditions.

Overall, the paper combines shape‑derivative calculus, spectral analysis of mixed boundary eigenvalue problems, and the implicit function theorem to obtain three complementary results for each class of Euler flows: (1) Flexibility – existence of non‑trivial free‑boundary domains via local bifurcation; (2) Rigidity – conditions under which the domain must be a concentric annulus or disk; (3) Stability – continuous dependence of the solution and the domain on small perturbations of the Bernoulli (Neumann) data. By focusing on compactly supported, closed‑streamline configurations rather than the more commonly studied open‑wave (strip‑like) settings, the work opens a new avenue in the study of overdetermined elliptic problems and steady fluid flows with constant vorticity. The results have potential implications for the design of fluid devices with prescribed vortex structures and for the mathematical theory of free‑boundary problems in fluid dynamics.