Instability of two-dimensional Taylor-Green Vortices

For a wide class of linear Hamiltonian operators we develop a general criterion that characterizes the unstable eigenvalues as the zeros of a holomorphic function given by the determinant of a finite-dimensional matrix. We apply the latter result to prove the spectral instability of the Taylor-Green vortex in two-dimensional ideal fluids. The linearized Euler operator at this steady state possesses different invariant subspaces, within which we apply our criterion to rule out or detect instabilities. We show linear stability of odd perturbations, for which the unstable spectrum can appear only on the real axis. We exclude this possibility by applying our stability criterion. Real instabilities, instead, exist and can be detected with the same criterion if we consider suitable rescalings of the Taylor-Green vortex. In the subspace of functions even in both variables, the problem is reduced to finding a single complex root of our stability function. We successfully locate this value by combining our general criterion with a rigorous computer-assisted argument. As a consequence, we fully characterize the unstable spectrum of the Taylor-Green vortex.

💡 Research Summary

The paper introduces a novel, fully constructive instability criterion for linear Hamiltonian operators and applies it to the two‑dimensional Taylor‑Green vortex, a classical steady solution of the incompressible Euler equations on the periodic torus.

1. General instability criterion (Theorem 1.5).
Let (L=JH) be a Hamiltonian operator on a Hilbert space (X), where (J) is densely defined, closed and skew‑adjoint, and (H) is bounded self‑adjoint. Decompose the self‑adjoint part as (H=H_{s}+H_{u}) with the following properties:

• (H_{s}) is strictly positive (its quadratic form is bounded below by a positive constant).
• (H_{u}) has finite rank, its range being a finite‑dimensional subspace (V_{u}) of dimension (n).
• (H_{s}) and (H_{u}) commute.

Define the skew‑adjoint operator (T:=H_{s}^{1/2}JH_{s}^{1/2}). For any (\lambda\in\mathbb{C}\setminus i\mathbb{R}) set
\