Isolated steady solutions of the 3D Euler equations

TL;DR

This paper demonstrates the existence of isolated smooth steady solutions to the 3D Euler equations on certain Riemannian manifolds, combining dynamical systems, spectral geometry, and contact topology techniques.

Contribution

It introduces new methods to identify isolated steady solutions of the Euler equations on Riemannian manifolds, extending to Euclidean space with partial uniqueness results.

Findings

Existence of isolated steady solutions on specific Riemannian manifolds.

Analytic steady solutions on ^3 with local uniqueness properties.

Contrast with shear flows having infinitely many nearby solutions.

Abstract

We show that there exist closed three-dimensional Riemannian manifolds where the incompressible Euler equations exhibit smooth steady solutions that are isolated in the $C^1$-topology. The proof of this fact combines ideas from dynamical systems, which appear naturally because these isolated states have strongly chaotic dynamics, with techniques from spectral geometry and contact topology, which can be effectively used to analyze the steady Euler equations on carefully chosen Riemannian manifolds. Interestingly, much of this strategy carries over to the Euler equations in Euclidean space, leading to the weaker result that there exist analytic steady solutions on $\mathbf{T}^3$ such that the only analytic steady Euler flows in a $C^1$-neighborhood must belong to a certain linear space of dimension six. For comparison, note that in any $C^k$-neighborhood of a shear flow there are infinitely many linearly independent analytic shears.