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The linear stability of a steady state solution of 2D Euler equations of an ideal fluid is being studied. We give an ex- plicit 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 essen- tial spectrum of L on L2 is the imaginary axis.

Essential Spectrum of the Linearized 2D Euler Equation and Lyapunov–Oseledets Exponents