Abstract.
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.
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Shvydkoy, R., Latushkin, Y. Essential Spectrum of the Linearized 2D Euler Equation and Lyapunov–Oseledets Exponents. J. math. fluid mech. 7, 164–178 (2005). https://doi.org/10.1007/s00021-004-0114-x
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DOI: https://doi.org/10.1007/s00021-004-0114-x