An Evans function for the linearised 2D Euler equations using Hill's determinant
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 nonzero real part. In particular this approach also works in the case where complex eigenvalues appear.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.05920
 Bibcode:
 2021arXiv210105920D
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematical Physics
 EPrint:
 29 pages, 10 figures