The Essential Spectrum of the Linearized 2D Euler Operator is a Vertical Band.pdf

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The Essential Spectrum of the Linearized 2D Euler Operator is a Vertical Band.pdf

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The Essential Spectrum of the Linearized 2D Euler Operator is a Vertical Band

a r X i v : m a t h - p h / 0 3 0 6 0 2 7 v 1 1 0 J u n 2 0 0 3 Contemporary Mathematics The Essential Spectrum of the Linearized 2D Euler Operator is a Vertical Band Roman Shvidkoy and Yuri Latushkin Abstract. We prove that the essential spectrum of the operator obtained by linearization about a steady state of the Euler equations governing the motion of inviscid ideal fluid in dimension two is a vertical strip whose width is determined by the maximal Lyapunov exponent of the flow induced by the steady state. 1. Introduction 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?1 w,?〉 curlu, on the Sobolev spaceH01 = H 0 1 (T 2) of scalar functions w having zero means ∫ wdx = 0 on the 2-torus T2 = R2/2πZ2. Here u = (u1, u2) ? is a steady state (velocity) solution of the Euler equations 〈u,?〉u + ?p = 0, div u = 0, p is the pressure, 〈·, ·〉 denotes the scalar product, ? – transposition, curlu = ??2u1 + ?1u2 is the scalar curl of the two-dimensional vector field, and v = curl?1 w denotes the unique solution of curl v = w, div v = 0. 1991 Mathematics Subject Classification. 76,35. Key words and phrases. Inviscid fluids, Euler equations