# Isolated steady solutions of the 3D Euler equations

## TL;DR

This paper shows that incompressible fluid flows on certain 3D manifolds can have isolated steady solutions, which is a novel finding in fluid dynamics.

## Contribution

The paper proves the existence of isolated smooth steady states for the 3D Euler equations on specific Riemannian manifolds.

## Key findings

- Isolated steady solutions of the 3D Euler equations exist on certain compact Riemannian manifolds.
- These solutions are isolated in the C1-topology and exhibit strongly chaotic dynamics.
- A weaker result is shown for analytic steady solutions on T3 with constraints on nearby solutions.

## Abstract

In the study of the stationary incompressible fluid flows, one finds a subtle interplay between flexibility and rigidity properties, that is, between the existence of a wealth of solutions and the significant constraints that they must satisfy. A recent survey by Drivas and Elgindi draws attention to the fact that none of the known steady states are isolated, and they conjecture that no smooth enough steady state is isolated. In this article, we show that on a broad range of 3-dimensional compact Riemannian manifolds the incompressible fluid flows admit an isolated steady state.

We show that there exist closed three-dimensional Riemannian manifolds where the incompressible Euler equations exhibit smooth steady solutions that are isolated in the C1-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 T3 such that the only analytic steady Euler flows in a C1-neighborhood must belong to a certain linear space of dimension six. For comparison, note that in any Ck-neighborhood of a shear flow, there are infinitely many linearly independent analytic shears.