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

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.

Symmetry results for compactly supported steady solutions of the 2D Euler equations

“…This is based on a topological observation that makes use of the Brouwer degree and the lack of stagnation points of u in Ω. We would like to point out that this observation works also in the framework of [15,Theorem 1.13]. As a consequence, the result there holds without assuming a priori that Ω is an annular domain.…”

“…There are some related results available, though. In [15] radial symmetry is proved for solutions in bounded domains under constant tangential velocity at the boundary: however this constant is not allowed to be 0. Another symmetry result is [10] for nonnegative and compactly supported vorticity, and here the velocity field need not have compact support (it is an immediate consequence of the divergence theorem that the unique compactly supported velocity field with nonnegative vorticity is 0).…”

“…Let us emphasize that the regularity condition in assumption (A1) prevents the appearance of isolated stagnation points. This is a rather typical assumption in many rigidity results in this fashion, see [12][13][14][15]. Moreover, the set Ω is assumed to be of annular type.…”

“…Then we are led to Theorem A. In its proof we are largely inspired by the works of Hamel and Nadirashvili, [12][13][14][15]. In particular, Theorem A can be seen as a version of [15,Theorem 1.13] with a degenerate boundary condition.…”

“…In its proof we are largely inspired by the works of Hamel and Nadirashvili, [12][13][14][15]. In particular, Theorem A can be seen as a version of [15,Theorem 1.13] with a degenerate boundary condition. The main idea is to show that the stream function φ solves a semilinear equation, and then to use symmetry results for this kind of problems to conclude.…”