Flexibility and rigidity in steady fluid motion

TL;DR

This paper investigates the structural properties and stability of steady solutions to fluid equations, demonstrating symmetry results and stability under domain perturbations, with implications for fluid dynamics theory.

Contribution

It establishes Liouville-type theorems for steady solutions, showing they must have certain symmetries, and proves their structural stability under domain deformations.

Findings

Steady solutions in specific domains must have structural symmetries.

Solutions can be deformed within small domain perturbations.

Arnol'd stable solutions are shown to be structurally stable.

Abstract

Flexibility and rigidity properties of steady (time-independent) solutions of the Euler, Boussinesq and Magnetohydrostatic equations are investigated. Specifically, certain Liouville-type theorems are established which show that suitable steady solutions with no stagnation points occupying a two-dimensional periodic channel, or axisymmetric solutions in (hollowed out) cylinder, must have certain structural symmetries. It is additionally shown that such solutions can be deformed to occupy domains which are themselves small perturbations of the base domain. As application of the general scheme, Arnol'd stable solutions are shown to be structurally stable.