On the rigidity of the 2D incompressible Euler equations

TL;DR

This paper proves that steady incompressible Euler flows in 2D bounded domains are necessarily circular under certain conditions, confirming conjectures and solving open problems about flow rigidity and symmetry.

Contribution

It establishes the rigidity of steady Euler flows in disks and annuli, confirming conjectures and characterizing flows with specific boundary and stagnation point conditions.

Findings

Steady flows in a disk with one interior stagnation point are circular.

Steady flows in an annulus without interior stagnation points are circular.

No-slip boundary conditions lead to flow rigidity, allowing only circular flows in certain domains.

Abstract

We consider rigidity properties of steady Euler flows in two-dimensional bounded domains. We prove that steady Euler flows in a disk with exactly one interior stagnation point and tangential boundary conditions must be circular flows, which confirms a conjecture proposed by F. Hamel and N. Nadirashvili in [J. Eur. Math. Soc., 25 (2023), no. 1, 323-368]. Moreover, for steady Euler flows on annuli with tangential boundary conditions, we prove that they must be circular flows provided there is no stagnation point inside, which answers another open problem proposed by F. Hamel and N. Nadirashvili in the same paper. We secondly show that the no-slip boundary conditions would result in absolute rigidity in the sense that except for the disks (\emph{resp}. annuli), there is no other smooth simply (\emph{resp}. doubly) connected bounded domain on which there exists a steady flow with only one (\emph{resp}. no) interior stagnation point and no-slip boundary conditions, and if present on the other hand, the flow must be circular. The arguments are based on the geometry of streamlines and 'local' symmetry properties for the non-negative solutions of semi-linear elliptic problems.