# On stability of Euler flows on closed surfaces of positive genus

## TL;DR

This paper investigates the stability of ideal 2D Euler flows on closed surfaces of positive genus, showing harmonic solutions are linearly stable and that all flows are stable against harmonic perturbations.

## Contribution

It provides a rigorous proof of linear stability for harmonic Euler flows on closed surfaces of positive genus, extending stability analysis to these complex geometries.

## Key findings

- Harmonic solutions are linearly stable on such surfaces.
- All Euler flows are stable against harmonic velocity perturbations.
- The stability analysis uses the Hodge-Helmholtz decomposition.

## Abstract

Incompressible flows of an ideal two-dimensional fluid on a closed orientable surface of positive genus are considered. Linear stability of harmonic, i.e. irrotational and incompressible, solutions to the Euler equations is shown using the Hodge-Helmholtz decomposition. We also demonstrate that any surface Euler flow is stable with respect to harmonic velocity perturbations.