Essential spectrum of the linearized 2D Euler equation and Lyapunov-Oseledets exponents

Details Essential spectrum of the linearized 2D Euler equation and Lyapunov-Oseledets exponents Essential spectrum of the linearized 2D Euler equation and Lyapunov-Oseledets exponents

2003-06-10

arxiv.org/abs/math-ph/0306026v1

Physics

Mathematical Physics

17 pages

Scientific paper

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 $L_2$ is the imaginary

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