2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/57411The linear stability of a steady state solution of 2D Euler equations of an ideal fluid is being studied. We give an explicit geometric construction of approximate eigenfunctions for the linearized Euler operator $L$ in vorticity form acting on Sobolev spaces on two dimensional torus. We show that each nonzero Lyapunov-Oseledets exponent for the flow induced by the steady state contributes a vertical line to the essential spectrum of $L$. Also, we compute the spectral and growth bounds for the group generated by $L$ via the maximal Lyapunov-Oseledets exponent. When the flow has arbitrarily long orbits, we show that the essential spectrum of $L$ on $L_2$ is the imaginary17 pagesMathematical PhysicsSpectral Theory76E99; 37D25; 47B33; 47D99text