# Stability Theory of the 3-Dimensional Euler Equations

## TL;DR

This paper analyzes the stability of shear flow steady states in 3D Euler equations, revealing spectral stability, instability, and parametric instability, and connecting these to turbulence transition.

## Contribution

It extends stability analysis of shear flows from 2D to 3D Euler equations, deriving a simplified linearised system and identifying conditions for stability and instability.

## Key findings

- Some shear flows are spectrally stable.
- Other shear flows are spectrally unstable.
- All shear flows are parametrically unstable.

## Abstract

The Euler equations on a three-dimensional periodic domain have a family of shear flow steady states. We show that the linearised system around these steady states decomposes into subsystems equivalent to the linearisation of shear flows in a two-dimensional periodic domain. To do so, we derive a formulation of the dynamics of the vorticity Fourier modes on a periodic domain and linearise around the shear flows. The linearised system has a decomposition analogous to the two-dimensional problem, which can be significantly simplified. By appealing to previous results it is shown that some subset of the shear flows are spectrally stable, and another subset are spectrally unstable. For a dense set of parameter values the linearised operator has a nilpotent part, leading to linear instability. This is connected to the nonnormality of the linearised dynamics and the transition to turbulence. Finally we show that all shear flows in the family considered (even the linearly stable ones) are parametrically unstable.