Eigenvalues of the Linearized 2D Euler Equations via Birman–Schwinger and Lin’s Operators

Abstract

We study spectral instability of steady states to the linearized 2D Euler equations on the torus written in vorticity form via certain Birman–Schwinger type operators $K_{\lambda }(\mu )$ and their associated 2-modified perturbation determinants $\mathcal D(\lambda ,\mu )$. Our main result characterizes the existence of an unstable eigenvalue to the linearized vorticity operator $L_\mathrm{vor}$ in terms of zeros of the 2-modified Fredholm determinant $\mathcal D(\lambda ,0)={\text {det}}_{2}(I-K_{\lambda }(0))$ associated with the Hilbert Schmidt operator $K_{\lambda }(\mu )$ for $\mu =0$. As a consequence, we are also able to provide an alternative proof to an instability theorem first proved by Zhiwu Lin which relates existence of an unstable eigenvalue for $L_\mathrm{vor}$ to the number of negative eigenvalues of a limiting elliptic dispersion operator $A_{0}$.