An Evans function for the linearised 2D Euler equations using Hill's determinant

TL;DR

This paper develops an Evans function based on Hill's determinant to analyze the spectrum of the linearized 2D Euler equations around shear flows, providing a complete spectral characterization and eigenvalue counting method.

Contribution

It introduces a novel Evans function approach for the linearized 2D Euler equations using Hill's determinant, enabling precise spectral analysis and eigenvalue enumeration.

Findings

The Evans function characterizes the point spectrum of the linearized Euler operator.

The number of discrete eigenvalues equals twice the count of non-zero lattice points inside the unstable disk.

The method allows counting isolated eigenvalues with non-zero real part.

Abstract

We study the point spectrum of the linearisation of Euler's equation for the ideal fluid on the torus about a shear flow. By separation of variables the problem is reduced to the spectral theory of a complex Hill's equation. Using Hill's determinant an Evans function of the original Euler equation is constructed. The Evans function allows us to completely characterise the point spectrum of the linearisation, and to count the isolated eigenvalues with non-zero real part. We prove that the number of discrete eigenvalues of he linearised operator for a specific shear flow is exactly twice the number of non-zero integer lattice points inside the so-called unstable disk.