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

Date: Thursday 4th April 2024

Abstract

In this talk, I will discuss the point spectrum of the linearisation of Euler’s equation for an ideal fluid on the torus about a so-called `shear flow'. Using 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 one to completely characterise the point spectrum of the linearisation, and to count the isolated eigenvalues with non-zero real part. I will show that the number of discrete eigenvalues of the linearised operator for a specific shear flow is exactly twice the number of non-zero integer lattice points inside the so-called unstable disc.