Free boundary problems for the two-dimensional Euler equations in exterior domains

TL;DR

This paper classifies steady Euler flows in two-dimensional exterior domains with free boundaries, showing under certain conditions that the flow must be circular and the domain a disk complement, using a refined moving planes method.

Contribution

It provides new classification results for steady Euler flows with free boundaries in exterior domains, extending understanding of flow symmetry and domain shape.

Findings

Flow without interior stagnation points and constant boundary norm implies circular flow and disk complement domain.

If stagnation points form a disk with certain conditions, the flow is necessarily circular.

Refined moving planes method is used to establish symmetry and domain shape results.

Abstract

In this paper we present some classification results for the steady Euler equations in two-dimensional exterior domains with free boundaries. We prove that, in an exterior domain, if a steady Euler flow devoid of interior stagnation points adheres to slip boundary conditions and maintains a constant norm on the boundary, along with certain additional conditions at infinity, then the domain is the complement of a disk, and the flow is circular, namely the streamlines are concentric circles. Additionally, we establish that in the entire plane, if all the stagnation points of a steady Euler flow coincidentally form a disk, then, under certain additional reasonable conditions near the stagnation points and at infinity, the flow must be circular. The proof is based on a refinement of the method of moving planes.