[Paper Review] Circular flows for the Euler equations in two-dimensional annular domains

François Hamel, Nikolaï Nadirashvili|arXiv (Cornell University)|Sep 4, 2019

Nonlinear Partial Differential Equations21 references4 citations

TL;DR

This paper establishes radial symmetry results for steady Euler flows in two-dimensional annular domains, proving that under rigid wall boundary conditions and absence of stagnation points, the flow must be circular (streamlines are concentric circles). It further classifies free boundary problems where constant-speed boundary conditions force the domain to be a disk or annulus, using novel comparison principles and the method of moving planes for semilinear elliptic equations derived from the stream function.

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.

Motivation & Objective

• To determine under what geometric and boundary conditions steady Euler flows in annular domains must be radially symmetric (circular).
• To classify free boundary problems where constant-speed boundary conditions force the domain to be a disk or annulus.
• To develop new comparison principles for semilinear elliptic equations in doubly connected domains to prove symmetry results.
• To extend Liouville-type and Serrin-type theorems to unbounded and punctured domains with appropriate decay or regularity conditions.

Proposed method

• Uses the method of moving planes adapted to almost circular domains between streamlines to prove radial symmetry of the stream function.
• Analyzes the stream function as a solution to a semilinear elliptic equation derived from the Euler equations.
• Applies ODE and PDE techniques to study trajectories and orthogonal trajectories of the stream function gradient.
• Employs strong maximum principle and weak maximum principle in small domains to derive contradiction in the moving plane argument.
• Imposes boundary conditions: tangential flow on boundaries (rigid walls), constant speed on boundary components, and decay/infinity conditions at infinity or at the origin.
• Uses limiting arguments in the moving plane method to show that symmetry holds up to the limit, proving full radial symmetry.

Experimental results

Research questions

• RQ1 Under what conditions does a steady Euler flow in a two-dimensional annular domain become circular (i.e., streamlines are concentric circles)?
• RQ2 Can a free boundary problem for the Euler equations be solved such that constant-speed boundary conditions force the domain to be a disk or annulus?
• RQ3 What comparison principles for semilinear elliptic equations in doubly connected domains can be established to prove symmetry results?
• RQ4 How does the method of moving planes apply to domains that are not exactly symmetric, but nearly circular between streamlines?

Key findings

• In bounded annuli, punctured disks, unbounded exterior domains, and the punctured plane, a steady Euler flow with no stagnation points and tangential boundary conditions is necessarily circular.
• For free boundary problems in simply or doubly connected domains, if the flow has constant speed on each boundary component, then the domain must be a disk or an annulus.
• The stream function and its level sets (streamlines) are radially symmetric under the stated conditions, implying the flow inherits the domain's radial symmetry.
• A new comparison principle for semilinear elliptic equations is established in doubly connected domains, which is crucial for proving symmetry via the method of moving planes.
• The moving plane method is successfully adapted to domains between streamlines, even when the domain is not exactly symmetric, by using limiting arguments and small domain estimates.