Instability of Equilibria for the 2D Euler Equations on the torus

TL;DR

This paper investigates the linear and nonlinear stability of specific stationary solutions to the 2D Euler equations on a torus, demonstrating that most are nonlinearly unstable through analytical and numerical methods.

Contribution

It provides a detailed analysis of the stability of cosine-sine vorticity solutions using truncation methods, deriving explicit instability conditions and eigenvalue bounds.

Findings

Most subsystems are linearly stable.

Explicit lower bounds for eigenvalues are derived.

Most stationary solutions are nonlinearly unstable.

Abstract

We consider the hydrodynamics of an incompressible fluid on a 2D periodic domain. There exists a family of stationary solutions with vorticity given by $\Omega^*=\alpha\cos (\mathbf{p} \cdot \mathbf{x} )+\beta \sin (\mathbf{p} \cdot \mathbf{x} )$. This situation can be approximated as a structure preserving finite dimensional Hamiltonian system by a truncation introduced by Zeitlin (1990,2005) or by the more standard Galerkin style finite element method. We use these two truncations to analyse the linear stability of these solutions and analytical and numerical results are compared. Following the methods used by Li (2000) the problem is divided into subsystems and we prove that most subsystems are linearly stable. We derive a sufficient condition for a subsystem to be linearly unstable and derive an explicit lower bound for the associated real eigenvalues independent of the truncation size $N$. Then we show that the corresponding eigenvectors are in $\ell^2$. This together with known stability results for the 2D periodic Euler equations allows us to conclude that most of these stationary solutions are nonlinearly unstable. We confirm our results with a numerical computation of the spectrum for a large, finite truncation. Finally we discuss the essential spectrum of the full problem as the limit of the truncated problem.