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Arnold stability and rigidity in Zeitlin's model of hydrodynamics

Luca Melzi, Klas Modin

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.

Arnold stability and rigidity in Zeitlin's model of hydrodynamics

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.
Paper Structure (7 sections, 10 theorems, 59 equations, 1 figure)

This paper contains 7 sections, 10 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on nonlinear stability of Lie--Poisson systems
  3. Euler's equations on the sphere
  4. Lyapunov stability for the Zeitlin model
  5. Rigidity results
  6. Proof of Proposition \ref{['prop:rigidity_-6']}
  7. Proof of Proposition \ref{['prop:rigidity_-2']}

Key Result

Theorem 1.1

If $L>-6$, then $W_0,P_0$ is Lyapunov stable in the Frobenius norm. In particular, if $W_0,P_0$ is a steady state of type eq:Zeitlin_steady_f with $f'>-6$ everywhere, then it is Lyapunov stable in the Frobenius norm.

Figures (1)

  • Figure 1: Snapshots of a typical time evolution of vorticity in Euler's equations on the sphere. A smooth initial vortex configuration undergoes intermediate chaotic stretch-and-fold motion, where vorticity region of equal sign (red or blue) tend to mix into larger vortex condensates. At some point, the mixing stops, with a few remaining and interacting coherent vortex structures. These numerical results are consistent with the conjectured long-term behaviour Sverak2012Shnirelman2013. For details of the numerical simulations, see the study by Modin and Viviani MoVi2026.

Theorems & Definitions (15)

  • Theorem 1.1: Stability
  • Theorem 1.2: Rigidity
  • Proposition 2.1
  • Remark 2.2
  • Corollary 2.3: Euler--Arnold equations
  • proof : Proof of Proposition \ref{['prop:Arnold']}
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 5 more