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

TL;DR

This paper investigates steady, compactly supported 2D Euler flows with constant vorticity, revealing existence, rigidity, and stability results for various overdetermined free-boundary problems.

Contribution

It introduces new existence and stability results for nontrivial Euler flows with constant vorticity, expanding the understanding of overdetermined elliptic problems.

Findings

Existence of nontrivial admissible domains for three classes of flows.

Rigidity results confirming uniqueness under certain conditions.

Standard annular flows are stable under small boundary perturbations.

Abstract

In this paper, we study the two-dimensional steady compactly supported incompressible Euler equations with free boundaries. We consider flows with constant vorticity that are perturbations of annular equilibria, in contrast to the laminar flows that predominate in the existing literature on steady water waves. More precisely, we analyze three distinct classes of steady Euler flows with compact support, which correspond, respectively, to partially overdetermined, two-phase overdetermined, and (fully) overdetermined elliptic free-boundary problems. Our main contributions are threefold. For each class, we first prove a flexibility result-the existence of nontrivial admissible domains-by combining shape derivatives with local bifurcation theory. Second, we establish the corresponding rigidity results. Third, we apply the implicit function theorem to show that the standard annular flows are stable under small perturbations of the Neumann boundary condition. These results provide new perspectives on the theory of overdetermined elliptic problems.