Circular flows for the Euler equations in two-dimensional annular domains

TL;DR

This paper proves that steady Euler flows in two-dimensional annular and exterior domains without stagnation points are necessarily circular, inheriting the domain's radial symmetry, using flow trajectory analysis and symmetry methods.

Contribution

It establishes the radial symmetry of steady Euler flows in various 2D domains under specific boundary and behavior conditions, extending symmetry results to new settings.

Findings

Flow without stagnation points is circular in annular domains.

Flow symmetry is proven using the method of moving planes and elliptic equation analysis.

Domains with constant boundary flow norm are disks or annuli.

Abstract

In this paper, we consider steady Euler flows in two-dimensional bounded annuli, as well as in exterior circular domains, in punctured disks and in the punctured plane. We always assume rigid wall boundary conditions. We prove that, if the flow does not have any stagnation point, and if it satisfies further conditions at infinity in the case of an exterior domain or at the center in the case of a punctured disk or the punctured plane, then the flow is circular, namely the streamlines are concentric circles. In other words, the flow then inherits the radial symmetry of the domain. The proofs are based on the study of the trajectories of the flow and the orthogonal trajectories of the gradient of the stream function, which is shown to satisfy a semilinear elliptic equation in the whole domain. In exterior or punctured domains, the method of moving planes is applied to some almost circular domains located between some streamlines of the flow, and the radial symmetry of the stream function is shown by a limiting argument. The paper also contains two Serrin-type results in simply or doubly connected bounded domains with free boundaries. Here, the flows are further assumed to have constant norm on each connected component of the boundary and the domains are then proved to be disks or annuli.