Arnold stability and rigidity in Zeitlin's model of hydrodynamics

TL;DR

This paper proves Lyapunov stability of steady states in Zeitlin's discretisation of 2-D Euler equations using Arnold's geometric approach, revealing a rigidity condition on the state matrix and supporting the model's reliability.

Contribution

It introduces a matrix-theoretic proof of Arnold stability and rigidity in Zeitlin's model, connecting geometric stability with matrix structure analysis.

Findings

Lyapunov stability of steady states established

Stable solutions satisfy a specific matrix rigidity condition

Results align with known properties of 2-D Euler equations

Abstract

Zeitlin's model is a discretisation of the 2-D Euler equations that preserves the underlying geometric structure. This feature makes it suitable for studying the qualitative behaviour of the dynamics. Here, we utilise Arnold's geometric approach to prove Lyapunov stability of steady states in Zeitlin's model. Furthermore, we show that such Arnold stable stationary solutions are subject to a rigidity condition that enforces a specific form of the matrix describing the state. Our argument relies on matrix theory and is therefore detached, and conceptually different, from the nonlinear stability analysis as developed for the 2-D Euler equations. Nevertheless, our results concur with those known for the 2-D Euler equations, which hints at links between matrix theory and nonlinear PDE techniques. Furthermore, our results show that the Zeitlin's model, as a numerical discretisation, is reliable for studying stationary solutions.