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

The 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 L2 is the imaginary axis. Mathematics Subject Classification (1991). 76, 35.