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

2021/01/15 by Holger R. Dullin, Dullin, Holger R., Robert Marangell +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2101.05920

openalex publication_date 2021/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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