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 $\mathcal{D}(\lambda ,0)={\det }_2(I-K_{\lambda }(0))$ 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 $A_0$.