In this note we give a description of the continuous spectrum of the
linearized Euler equations in three dimensions. Namely, for all but countably
many times $t\in \R$, the continuous spectrum of the evolution operator $G_t$
is given by a solid annulus with radii $e^{t\mu}$ and $e^{t M}$, where $\mu$
and $M$ are the smallest and largest, respectively, Lyapunov exponents of the
corresponding bicharacteristic-amplitude system of ODEs

Shvydkoy, Roman

English

arXiv

Roman Shvydkoy

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 897–900Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comPartial Differential EquationsContinuous spectrum of the 3D Euler equation is a solid annulusLe spectre continu de l’equation d’Euler 3D est un anneauRoman Shvydkoy 1Department of Mathematics, Statistics and Computer Science, 851 S. Morgan St., M/C 249, University of Illinois at Chicago, Chicago, IL 60607, United Statesa r t i c l e i n f o a b s t r a c tArticle history:Received 30 October 2009Accepted 15 July 2010Available online 27 July 2010Presented by Pierre-Louis LionsIn this Note we give a description of the continuous spectrum of the linearized Eulerequations in three dimensions. Namely, for all but countably many times t ∈ R, thecontinuous spectrum of the evolution operator Gt is given by a solid annulus with radii etμand etM , where μ and M are the smallest and largest, respectively, Lyapunov exponents ofthe corresponding bicharacteristic-amplitude system of ODEs.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éOn donne dans cette Note une description du spectre continu de l’équation d’Eulerlinearisée en dimension 3. Précisément, pour presque tout t ∈ R, le spectre continu del’opérateur d’évolution Gt est constitué d’un anneau de rayons etμ et etM , où μ et M sont,respectivement, le plus petit et le plus grand exposant de Lyapunov du système d’EDObicaractéristique-amplitude associé.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. BackgroundThis Note addresses the geometric structure of the continuous spectrum (in Browder or Fredholm sense) of the linearizedthree-dimensional Euler equation. The question has its origin in the so-called elliptic instabilities of ideal fluids foundimplicated in transition to turbulence of some lamina flows in a subcritical range of Reynolds numbers (see Bayly [1], Pateraand Orszag [7], Pierrehumbert [8]). The works of Lifschitz and Hameiri [5], and Friedlander and Vishik [3] laid the basisfor a general approach to shortwave instabilities which incorporated elliptic and a broad range of other equilibria, includingthose possessing exponential stretching of trajectories. In [14] Vishik draws a connection between shortwave instabilitiesand the essential spectrum of the linearized Euler equation, as opposed to the point spectrum, and describes the essentialspectral radius in terms of the maximal Lyapunov–Oseledets exponent of a bicharacteristic-amplitude system of ODEs, BASfor short (see below). Further investigation on the structure of the spectrum continued in a series of papers [4,11,12,10],where the Sacker–Sell theory of linear skew-product flows was introduced into the subject. It was shown that in generalthe shortwave instabilities and the corresponding points of the essential spectrum of the Euler equations are linked to thedynamical spectrum of the BAS (in the Sacker–Sell sense). At the same time the spectrum in two dimensions was completelydescribed in [11] by a direct construction of approximate eigenfunctions. Thus, the spectrum of the Euler semigroup is asolid annulus centered at the origin, while the essential spectrum of the generator is the corresponding vertical band, soE-mail address: shvydkoy@math.uic.edu.1 The work is partially supported by the US National Science Foundation Grant DMS-0907812.1631-073X/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2010.07.009898 R. Shvydkoy / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 897–900that the two are related by the Spectral Mapping Theorem. Over the past several years attempts were made to find a similardescription in three dimensions. In this Note we offer a complete answer to this question on the level of the semigroupproving that, just as in two dimensions, its essential spectrum is given by a solid annulus. A partial description of thespectrum for the generator was also provided in [10]. However, the spectral mapping theorem in this case remains open,leaving a possibility for further investigation.2. Technical description of the resultLet u0 ∈ (C2(T3))3 be a solution to the stationary Euler equations(u0 · ∇)u0 + ∇p = 0, ∇ · u0 = 0, (1)on the three-dimensional torus T3. Evolution of small perturbations to u0 is described up to the linear approximation bythe systemvt = −(u0 · ∇)v − (v · ∇)u0 − ∇q, ∇ · v = 0. (2)The solution map for (2) is given by a C0-group {Gt}t∈R of operators acting on the energy space L2div = (L2div(T3))3 ofdivergence-free vector fields. The method of geometric optics projects evolution of oscillating localized perturbations of theform v(x, t) = b(x, t)eiS(x,t)/δ + O (δ), 0 < δ  1, where the initial amplitude b(x,0) = b0(x) and phase S(x,0) = ξ0 · x aregiven, and ξ0 · b0(x) = 0 for all x ∈ T3. If written in the Lagrangian coordinates of the flow u0, x0 → ϕt(x0), the evolutionof the vectors b(t) = b(ϕt(x0), t) and ξ(t) = ∇ S(ϕt(x0), t) is governed by the bicharacteristic-amplitude system (BAS) ofODEs:xt = u0(x), x(0) = x0, (3a)ξt = −∂u0(x)ξ, ξ(0) = ξ0 ⊥ b0, (3b)bt = −∂u0(x)b + 2(∂u0(x)b, ξ)ξ |ξ |−2, b(0) = b0. (3c)One can see that b · ξ = 0 is a conservation law of (3)which is consistent with the incompressibility of the flow. We viewEqs. (3a)–(3b) as a Hamiltonian system over the cotangent bundle T ∗T3 = T3 ×R3, with the Hamiltonian H(x, ξ) = u0(x) · ξ .The corresponding flow projected onto the compact space Ω = Tn × Sn−1 is given byχt(x0, ξ0) =(ϕt(x0),∂ϕ−t (x0)ξ0|∂ϕ−t (x0)ξ0|). (4)As the amplitude equation (3c) is homogeneous in ξ it can be written in the form bt = a0(χt(x0, ξ0))b. Therefore, thefundamental matrix solution of (3c), Bt(x, ξ) defines a cocycle over the flow {χt} on the fibre bundle F with base Ω andfibers given by F (x, ξ) = F (ξ) = {b ∈ C3: b · ξ = 0}. We call it the b-cocycle. Let ΣB denote the dynamical spectrum ofthe b-cocycle, that is, the set of λ ∈ R such that the rescaled cocycle Bλt = e−λt Bt does not have exponential dichotomy(uniform exponential hyperbolicity) on the bundle, see [9]. In the resent paper [12], we showed a key connection betweenthe Fredholm spectrum of Gt and the dynamical spectrum ΣB , expressed by the following two identities:σΦ(Gt; L2div) = {z ∈ C: |z| ∈ etΣB }, (5)for all but countably many t ∈ R, and∣∣σΦ(Gt; L2div)∣∣ = etΣB , (6)for all t without exceptions. Here σΦ stands for the Fredholm spectrum over L2div. The classical results of Sacker and Sell onthe structure of a dynamical spectrum, see [9], in our case reveal that ΣB is either a single interval [μ, M] or the unionof at most two disjoint intervals, as limited by the dimension of the fibers F (ξ). The main contribution of this Note is toadd one last technical piece to (5)–(6), which is to show that ΣB is in fact a single interval. We thus obtain the followingtheorem:Theorem 2.1. Let μ and M be the minimal and maximal Lyapunov–Oseledets exponents of the b-cocycle, respectively. Then the fol-lowing identity:σess(Gt; L2div) = σΦ(Gt; L2div) = {z ∈ C: etμ  |z|  etM} (7)holds for all but countably many t ∈ R, while the identity∣∣σess(Gt; L2div)∣∣ = ∣∣σΦ(Gt; L2div)∣∣ = [etμ, etM] (8)holds for all t ∈ R without exception.Here σess stands for the essential spectrum in the Browder sense [2]. We added it to the statement as this is the type ofspectrum that was examined in all the previous works. As we see from (7) it is in fact the same as the Fredholm spectrumfor almost all t . The identities (7)–(8) for σess follow immediately from the identities for σΦ due to the general inclusionσΦ⊂σess and the Nussbaum Theorem [6] on the equality between the spectral radii (applied also to the inverse of Gt ).R. Shvydkoy / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 897–900 8993. Connectedness of the dynamics spectrumLet us suppose, on the contrary, that Σ = [μ,γ1] ∪ [γ2, M], where γ1 < γ2. Then there exists a continuous projection-valued functionP (x, ξ) : F (ξ) → F (ξ) (9)corresponding to the exponential splitting of the cocycle Bt . So, for a small ε > 0 and λ ∈ (γ1, γ2) there are constantsc, C > 0 such that∥∥Bλt (x, ξ)b∥∥  Ce−εt‖b‖, t > 0, for all b ∈ Rg P (x, ξ); (10)∥∥Bλt (x, ξ)b∥∥  ceεt‖b‖, t > 0, for all b ∈ Ker P (x, ξ). (11)Moreover, relations (10) and (11) characterize the range and kernel of P (x, ξ), respectively.Let us fix an arbitrary x0 ∈ T3, and consider the splittingF (ξ) = Rg P (x0, ξ) ⊕ Ker P (x0, ξ). (12)Since the dimension of F (ξ) over C is two, and (12) is non-trivial, the dimension of Rg P (x0, ξ) is one. Let us now fix afinite local coordinate chart {U j} j∈ J of S2. For each j ∈ J we can find a continuous mappingξ → b jC(ξ) ∈ F (ξ), ξ ∈ U j,with ‖b jC(ξ)‖ = 1, such thatRg P (x0, ξ) =[b jC(ξ)]C, (13)where [·]C denotes the linear span over C.We will now de-complexify the spaces given by (13). To this end, let us writeb jC(ξ) = b jre(ξ) + ib jim(ξ),where b jre(ξ) and bjim(ξ) are real vectors. Since bjC(ξ) · ξ = 0, and ξ is real, we also haveb jre(ξ),bjim(ξ) ∈ F (ξ).Furthermore, the cocycle Bt , being the fundamental matrix of solutions of a system with real coefficients, maps real vectorsto real vectors. Thus, we obtain∥∥Bλt (x0, ξ)b jre(ξ)∥∥ + ∥∥Bλt (x0, ξ)b jim(ξ)∥∥ ∼ ∥∥Bλt (x0, ξ)b jC(ξ)∥∥  Ce−εt,for all t > 0. This implies that both vectors b jre(ξ) and bjim(ξ) belong to Rg P (x0, ξ), span it over C, and are linearly depen-dent over R. Let us define the mapL j(ξ) =[b jre(ξ),bjim(ξ)]R, ξ ∈ U j . (14)We see that each L j(ξ) is a one-dimensional subspace of the real tangent plane to the surface of the sphere S2 at ξ . It isnot hard to see that L j ’s agree on intersections of the charts. Indeed, let ξ ∈ U j ∩ Uk , for some j = k. According to (13),vectors b jC(ξ) and bkC(ξ) are linearly dependent over C. Thus, for some z = x + iy, b jC(ξ) = zbkC(ξ), which implies lineardependenciesb jre(ξ) = xbkre(ξ) − ybkim(ξ), (15)b jim(ξ) = ybkre(ξ) + xbkim(ξ). (16)It is then clear that L j(ξ) = Lk(ξ). As a result, (14) defines a continuous one-dimensional subbundle of the tangent bundleof S2. According to [13, Theorem 27.16], one can then select a continuous non-vanishing tangent field on the sphere, whichcontradicts the classical Hairy Ball Theorem.Remark 3.1. A simple example disclosed in [12] shows that the “all but countable many” condition in Theorem 2.1 isessential. Consider a two-dimensional parallel shear flow with constant profile, u0 = 〈U ,0〉. The linearized equation (2)takes the form vt = −U∂x1 v . We then obtain Gt v0(x1, x2) = v0(x1 − Ut mod 2π, x2). So, the spectrum of Gt is the unitcircle for all t /∈ πU−1Q, and a finite set of the circle otherwise.900 R. Shvydkoy / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 897–900Remark 3.2. It is desired to obtain a similar description of the essential spectrum for the generator L, which would expo-nentially correspond to the solid annulus. So far partial results has been given in [10]. Let us consider the closed subsetΩ⊥ = {(x, ξ) ∈ Ω: u0(x) · ξ = 0}. The flow χt leaves Ω⊥ invariant. We can therefore consider the restrictions of the flow,χ⊥t , and the cocycle B⊥t on the fiber bundle F ⊥ on Ω⊥ with the same fibers. Let ΣB⊥ be the corresponding dynamicalspectrum of the cocycle {B⊥t }. It was shown in [10] that ΣB⊥⊂Re(σess(L)). We can see now that if there is a stagnationpoint x0 ∈ T3 of the flow, i.e. u0(x0) = 0, then the corresponding x0-slice of Ω⊥ is (again) the sphere S2. So, the argumentabove applies with this specific choice of x0, which implies that ΣB⊥ = [μ⊥, M⊥], for some μ⊥  μ and M⊥  M . It remainto be seen, however, whether ΣB⊥ is in fact the same as ΣB . So far, it has been established only in the two-dimensionalcase, see [4,11].References[1] B.J. Bayly, Three-dimensional instability of elliptical flow, Phys. Rev. Lett. 57 (17) (1986) 2160–2163.[2] F.E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1960/1961) 22–130.[3] S. Friedlander, M.M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett. 66 (17) (1991) 2204–2206.[4] Y. Latushkin, M. Vishik, Linear stability in an ideal incompressible fluid, Comm. Math. Phys. 233 (3) (2003) 439–461.[5] A. Lifschitz, E. Hameiri, Local stability conditions in fluid dynamics, Phys. Fluids A 3 (11) (1991) 2644–2651.[6] R.D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970) 473–478.[7] S.A. Orszag, A.T. Patera, Subcritical transition to turbulence in plane channel flows, Phys. Rev. Lett. 45 (1980) 989–993.[8] R.T. Pierrehumbert, Universal short-wave instability of two-dimensional eddies in an inviscid fluid, Phys. Rev. Lett. 57 (1986) 2157–2159.[9] R.J. Sacker, G.R. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (3) (1978) 320–358.[10] R. Shvydkoy, The essential spectrum of advective equations, Comm. Math. Phys. 265 (2) (2006) 507–545.[11] R. Shvidkoy, Y. Latushkin, The essential spectrum of the linearized 2D Euler operator is a vertical band, in: Advances in Differential Equations andMathematical Physics, Birmingham, AL, 2002, in: Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 299–304.[12] R. Shvidkoy, Y. Latushkin, Operator algebras and the Fredholm spectrum of advective equations of linear hydrodynamics, J. Funct. Anal. 257 (2009)3309–3328.[13] N. Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, NJ, 1951.[14] M. Vishik, Spectrum of small oscillations of an ideal fluid and Lyapunov exponents, J. Math. Pures Appl. (9) 75 (6) (1996) 531–557.

Continuous spectrum of the 3D Euler equation is a solid annulus

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Continuous spectrum of the 3D Euler equation is a solid annulus

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