Essential Spectrum of the Linearized

In this note we continue the work in [SL], and give a full description of the essential spectrum for the linearized Euler operator L in dimension two. We prove that the essential spectrum of the operator is one solid vertical strip symmetric with respect to the imaginary axis. The width of the strip is determined by the maximal Lyapunov exponent Λ for the flow induced by the steady state. For classical results concerning linearized Euler equations see, e.g., [C, DR, L, Y]. Recent advances concerning the essential spectrum of the linearized Euler operator can be found in [FSV, FSV2, FV, FV2, V, VF]. In particular, it was proved in [V, VF] that the essential growth bound for the group generated by L is equal to Λ. Using this result, it was proved in [LV] that the essential spectral bound for L is equal to Λ. We study the linearized Euler operator L in vorticity form, Lw = −〈u,∇〉w − 〈curl w,∇〉 curlu, on the Sobolev spaceH 1 = H 0 1 (T ) of scalar functions w having zero means ∫