On 2D Euler equations. I. On the energy–Casimir ...faculty.missouri.edu/~liyan/Euler-L.pdf · On 2D Euler equations. I. On the energy–Casimir stabilities and the spectra for linearized

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JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 2 FEBRUARY 2000

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On 2D Euler equations. I. On the energy–Casimirstabilities and the spectra for linearized 2D Euler equations

Yanguang (Charles) Lia)

Department of Mathematics, University of Missouri, Columbia, Missouri 65211

~Received 10 July 1998; accepted for publication 30 September 1999!

In this paper, we study a linearized two-dimensional Euler equation. This equationdecouples into infinitely many invariant subsystems. Each invariant subsystem isshown to be a linear Hamiltonian system of infinite dimensions. Another importantinvariant besides the Hamiltonian for each invariant subsystem is found and isutilized to prove an ‘‘unstable disk theorem’’ through a simple energy–Casimirargument@Holm et al., Phys. Rep.123, 1–116 ~1985!#. The eigenvalues of thelinear Hamiltonian system are of four types: real pairs (c,2c), purely imaginarypairs (id,2 id), quadruples (6c6 id), and zero eigenvalues. The eigenvalues arecomputed through continued fractions. The spectral equation for each invariantsubsystem is a Poincare´-type difference equation, i.e., it can be represented as thespectral equation of an infinite matrix operator, and the infinite matrix operator is asum of a constant-coefficient infinite matrix operator and a compact infinite matrixoperator. We have obtained a complete spectral theory. ©2000 American Insti-tute of Physics.@S0022-2488~00!00702-7#

I. INTRODUCTION

In Ref. 1, Henshaw and Kreiss numerically studied the propagation of perturbationssolution of the two-dimensional incompressible Navier–Stokes~2D N-S! equations with no bodyforce, and at high Reynolds numbers. They numerically solved the equations starting withsmooth initial data for which the Fourier modes have random phases. This flow evolved oveinitially through a complicated state containing many shear layers and then into a state ovortex structures~with shear layers! which persists for a long time.

~1! Changing the viscosity, the large vortex structures do not change much.~2! Adding high mode perturbation to the initial data, the large vortex structures do not ch

much.~3! Changing the LaplacianD to D2, the large vortex structures do not change much.~4! Changing the LaplacianD to an operator which isD for low modes andD2 for high modes,

the large vortex structures do not change much.

In Ref. 2, Matthaeuset al. studied the same problem, and found similar results. Matthaeuset al.run the numerics for a much longer time. These numerics indicate that for relatively long tim~notinfinite, long time!, the solution to the 2D N-S equation~without body force, i.e., decayingturbulence! has the large vortex structures. Such structures persist in the solution to the 2Dequation by the claims 1, 3, 4 above.~Cf. Under decay boundary conditions, the Kato theorstates that for finite time the solution to the 2D N-S equation converges to the solution to tEuler equation in norm as viscosity approaches zero, see Ref. 3.! Moreover, the large vortexstructures are stable with respect to the change in initial data.

In Ref. 4, Robert and Sommeria studied the organized structures in two-dimensionalfluid flows by a theory of equilibrium statistical mechanics. The theory takes into account aknown constants of motion for the two-dimensional Euler equations. The microscopic states

a!Electronic mail: [email protected]

7280022-2488/2000/41(2)/728/31/$17.00 © 2000 American Institute of Physics

03 Jan 2006 to 128.206.49.84. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp

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729J. Math. Phys., Vol. 41, No. 2, February 2000 On the energy–Casimir stabilities and the . . .

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the possible vorticity fields, while a macroscopic state is defined as a probability distributivorticity at each point of the domain, which describes in a statistical sense the fine-scale vofluctuations. The organized structure appears as a state of maximal entropy, with the constrall the constants of motion. The vorticity field obtained as the local average of this opmacrostate is asteady solutionof the Euler equations.

The above numerical results show that certain relatively long-time large vortex structurthe 2D Euler equation persist for 2D N-S equation at high Reynolds numbers. Such structustable with respect to the change in initial data. We believe such structures are the exhibicertain unstable manifolds. The above theoretical results show that the probability mean ostructures for 2D Euler flows are steady solutions to the 2D Euler equations. Thus, we beliecertain unstable manifolds, if there are any, of steady solutions to 2D Euler equations are rsible for such large vortex structures. Therefore, a dynamical system study on certain unmanifolds of certain steady solutions to 2D Euler equations, and their persistence for 2Dequations at high Reynolds numbers, is crucial for studying the large vortex structures. Indehave built achaos-molecules-modelon 2D turbulence based upon the above motivations.5 Webelieve that our chaos-molecules-model captures the qualititive frames of the hyperbolictures, and the energy inverse cascade and the enstrophy cascade nature for 2D turbulenc

In this paper, we study the linearized 2D Euler equation at a fixed point. This study is thefor future analytical studies on the unstable manifolds for the 2D Euler equation. In Ref. 6have begun numerical studies on the unstable manifolds for the 2D Euler equation.

The current study is also important in the linear hydrodynamic stability theory. By utilithe Energy–Casimir method,7 we obtain anunstable disk theoremwhich is not in the category othe classical Rayleigh theorem.8

Next we discuss the approaches used in this study. Through the energy–Casimir mnonlinear stabilities of various types of two-dimensional ideal fluid flows have bestablished.7,9–11 Below we give a brief description on the energy-Casimir method. LetD be aregion on the~x,y!-plane bounded by the curvesG i( i 51,2). An ideal fluid flow inD is governedby the 2D Euler equation written in the stream-function form:

]

]tDc5@¹c, ¹Dc#, ~I.1!

where

@¹c, ¹Dc#5]c

]x

]Dc

]y2

]c

]y

]Dc

]x,

with the boundary conditions

cuG i5ci~ t !, c1[0,

d

dt RG i

]c

]nds50.

For every functionf (z), the functional

F5E ED

f ~Dc! dx dy ~I.2!

is a constant of motion~a Casimir! for ~I.1!. The conditional extremum of the kinetic energy

E51

2 E ED

¹c•¹c dx dy ~I.3!

for fixed F is given by the Lagrange’s formula,9


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etherdisk

stem.re of

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730 J. Math. Phys., Vol. 41, No. 2, February 2000 Yanguang Li

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dH5d~E1lF !50, ⇒c05l f 8~Dc0!, ~I.4!

wherel is the Lagrange multiplier. Thus,c0 is the stream function of a stationary flow, whicsatisfies

c05F~Dc0!, ~I.5!

whereF5l f 8. The second variation is given by9

d2H51

2 E ED$¹f•¹f1F8~Dc0!~Df!2% dx dy. ~I.6!

Let c5c01w be a solution to the 2D Euler equation~I.1!. Arnold proved the followingestimates.10

~a! Whenc<F8(Dc0)<C, 0,c<C,`,

E ED$¹w~ t !•¹w~ t !1c@Dw~ t !#2% dx dy<E E

D$¹w~0!•¹w~0!1C@Dw~0!#2% dx dy,

for all tP(2`,1`).~b! Whenc<2F8(Dc0)<C, 0,c,C,`,

E ED$c@Dw~ t !#22¹w~ t !•¹w~ t !% dx dy<E E

D$C@Dw~0!#22¹w~0!•¹w~0!% dx dy,

for all tP(2`,1`). Therefore, when the second variation~I.6! is positive definite, or when

E ED$¹f•¹f1@maxF8~Dc0!#~Df!2% dx dy

is negative definite, the stationary flow~I.5! is nonlinearly stable~Liapunov stable!. In this paper,we have found an invariant for the linearized 2D Euler equation, and use this invariant togwith an energy–Casimir-type argument to study linear stability, and to prove an unstabletheorem. The linearized 2D Euler equation is an infinite-dimensional linear Hamiltonian syFor finite-dimensional linear Hamiltonian systems, it is well known that the eigenvalues afour types: real pairs (c,2c), purely imaginary pairs (id,2 id), quadruples (6c6 id), and zeroeigenvalues.12–14 The same is true for the linearized 2D Euler equation. The eigenvaluescomputed through continued fractions following the work of Meshalkin and Sinai.15 The linear-ized 2D Euler equation can also be written in an infinite-matrix form. The spectral equationinfinite-matrix operator defines a Poincare´-type difference equation.16,17That is, the infinite-matrixoperator can be written as the sum of a constant-coefficient infinite matrix operator and a coinfinite matrix operator. In this paper, we follow a spectral theory developed by Duren18 to studythe spectra of the constant-coefficient infinite matrix operator through characteristic polynoThen we apply the Weyl’s essential spectra theorem to the perturbation of the constant-coeinfinite matrix operator by the compact infinite matrix operator,19 to achieve a complete spectrtheory.

Finally, we discuss some preliminaries on the 2D Euler equation. Consider thedimensional incompressible Euler equation written in vorticity form,

]V

]t52u

]V

]x2v

]V

]y,

~1.7!]u

]x1

]u

]y50;


h


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under periodic boundary conditions in bothx andy directions with period 2p, whereV is vorticity,u andv are respectively velocity components alongx andy directions. We also require that botu andv have means zero,

E0

2pE0

2p

u dx dy5E0

2pE0

2p

v dx dy50.

ExpandV into Fourier series,

V5 (kPZ2/$0%

vkeik•X,

wherev2k5vk, k5(k1 ,k2)T, andX5(x,y)T. In this paper, we use 0 for (0,0)T; the context willalways make it clear. By the relation between vorticityV and stream functionC,

V5]v]x

2]u

]y5DC,

where the stream functionC is defined by

u52]C

]y, v5

]C

]x;

the system~I.7! can be rewritten as the following kinetic system,

vk5 (k5p1q

A~p,q!vpvq , ~I.8!

whereA(p,q) is given by,

A~p,q!51

2@ uqu222upu22#~p1q22p2q1!5

1

2@ uqu222upu22#Up1 q1

p2 q2U, ~I.9!

FIG. 1. An illustration on the locations of the modes (k85rk) and (uk8u5uku) in the definitions ofEk1 andEk

2 ~Proposition1!.


is

tion

ctionf theinuedf the


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whereuqu25q121q2

2 for q5(q1 ,q2)T, similarly for p.Remark I.1: Notice that direct calculation shows that the nonlinear term in (I.8)

Sk5p1qA(p,q)vpvq , where A(p,q)5upu22(q1p22p1q2). Then, A(q,p)5uqu22(p1q22q1p2).The A(p,q) in (I.8) is the average of A˜ (p,q) and A(q,p), A(p,q)5 1

2@A(p,q)1A(q,p)#.For any two functionalsF1 andF2 of $vk%, define their Lie–Poisson bracket:

$F1 ,F2%5 (k1p1q50

Uq1 p1

q2 p2Uvk

]F1

]vp

]F2

]vq

. ~I.10!

Then the 2D Euler equation~I.8! is a Hamiltonian system,20

vk5$vk ,H%, ~I.11!

where the HamiltonianH is the kinetic energy,

H51

2 (kPZ2/$0%

uku22uvku2. ~I.12!

Following are Casimirs~i.e., invariants that Poisson commute with any functional! of the Hamil-tonian system~I.11!:

Jn5 (k11¯1kn50

vk1¯vkn

. ~I.13!

The following proposition is concerned with the equilibrium manifolds of the 2D Euler equa~I.8!.

Proposition 1: For any kPZ2/$0%, the infinite-dimensional space

Ek1[$$vk8%uvk850, i f k8Þrk,;r PR%,

and the finite-dimensional space

Ek2[$$vk8%uvk850, i f uk8uÞuku%,

entirely consist of fixed points of the system (I.8).Proof: Let k0PZ2/$0%, $vk

0%PEk01 . For any p,qPZ2/$0%, vp

0vq0Þ0 implies that p

5r 1k0, q5r 2k0 for somer 1 ,r 2PR. Then,A(p,q)50; thus we always haveA(p,q)vp0vq

050,and $vk

0% is a fixed point of~I.8!. Let k0PZ2/$0%, $vk0%PEk0

2 . For anyp,qPZ2/$0%,vp0vq

0Þ0implies thatupu5uk0u, uqu5uk0u. Then,A(p,q)50; thus we always haveA(p,q)vp

0vq050, and

$vk0% is a fixed point of~I.8!. h

Figure 1 shows an example on the locations of the modes (k85rk) and (uk8u5uku) in thedefinitions ofEk

1 andEk2 ~Proposition 1!.

The paper is organized as follows: Sec. II contains the formulations of the problem. SeIII contains the Liapunov stability. Section IV contains the properties of the eigenvalues olinearized 2D Euler equation as a linear Hamiltonian system. Section V holds the contfraction study of the eigenvalues. Section VI has the infinite-matrix study of the spectra olinearized 2D Euler equation. Section VII is the conclusion.

II. THE FORMULATIONS OF THE PROBLEM

Denote$vk%kPZ2/$0% by v. Consider the simple fixed pointv* :

vp* 5G, vk* 50, if kÞp or 2p, ~II.1!


r

per.

ion 1,-

w.


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of the 2D Euler equation~I.8!, which belongs to the two-dimensional intersection spaceEp1ùEp

2

~Proposition 1!, whereG is an arbitrary complex constant. Thelinearized two-dimensional Euleequationat v* is given by

vk5A~p,k2p!Gvk2p1A~2p,k1p!Gvk1p . ~II.2!

This is the linearized two-dimensional Euler equation that we are going to study in this paDefinition 1 (Classes): For any kˆ PZ2/$0%, we define the classS k to be the subset of Z2/$0%:

S k5$k1npPZ2/$0%unPZ, p is specified in (II.1)%.

See Fig. 2 for an illustration of the classes. According to the classification defined in Definitthe linearized two-dimensional Euler equation~II.2! decouples into infinitely many invariant subsystems:

v k1np5A„p,k1~n21!p…Gv k1~n21!p1A„2p,k1~n11!p…Gv k1~n11!p . ~II.3!

Each invariant subsystem can be rewritten as a linear Hamiltonian system as shown beloDefinition 2 (the quadratic Hamiltonian): The quadratic HamiltonianHk is defined as

Hk522 Im H (nPZ

rnGA„p,k1(n21)p…v k1~n21!pv k1npJ52Up1 k1

p2 k2U Im H (

nPZGrnrn21v k1~n21!pv k1npJ , ~II.4!

wherern5@ uk1npu222upu22# and ‘‘Im’’ denotes ‘‘imaginary part’’.Then the invariant subsystem~II.3! can be rewritten as a linear Hamiltonian system,

i v k1np5rn21 ]Hk

]v k1np. ~II.5!

Let

v k1np5an1 ibn , nPZ,

FIG. 2. An illustration of the classesS k and the diskD upu .


ic


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i.e., an and bn are the real and imaginary parts ofv k1np . Then the linear Hamiltonian system~II.5! can be rewritten in the form

an51

2rn

21 ]Hk

]bn,

~II.6!

bn521

2rn

21 ]Hk

]an,

where

Hk5U k1 p1

k2 p2U(

nPZrnrn21@G r~anbn212an21bn!1G i~an21an1bn21bn!#,

whereG5G r1 iG i , i.e., G r andG i are the real and imaginary parts ofG. Explicitly,

an51

2 U k1 p1

k2 p2U@rn11G ran112rn21G ran211rn11G ibn111rn21G ibn21#,

~II.7!

bn521

2 U k1 p1

k2 p2U@rn21G rbn212rn11G rbn111rn11G ian111rn21G ian21#.

If we rescale the variables as follows,

t51

2 U k1 p1

k2 p2U t, an5rnan , bn5rnbn ,

the linear Hamiltonian system~II.7! can be rewritten in the simpler form

dan /dt5rn@G r~ an112an21!1G i~ bn111bn21!#,

dbn /dt52rn@G r~ bn212bn11!1G i~ an111an21!#.

Next we discuss the fixed pointv* from a variational-principle point of view. Consider the kinetenergy and the enstrophy of the 2D Euler equation~I.8!,

E51

2 (kPZ2/$0%

uku22uvku2, J5 (kPZ2/$0%

uvku2.

If we extremize the kinetic energyE for fixed enstrophyJ5c ~a constant!, we have the criticalstates by the Lagrange formula,

]L

]l5J2c50,

]L

]vk5~ uku221l!vk50, ;kPZ2/$0%,

whereL52E1l(J2c). Thus, the critical states satisfy the relations,

l52uqu22, for some qPZ2/$0%,


y

the

to

ergy–

)


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vk50, if ukuÞuqu, ~II.8!

(uku5uqu

uvku25c.

Thus, we have thecritical manifold for extremizing the kinetic energy for fixed enstrophy,

M uquc 5H vUvk50, if ukuÞuqu, (

uku5uquuvku25cJ ,

which is a submanifold of the equilibrium manifoldEq2 ~Proposition 1!. Denote byI the linear

combination,

I 52E2upu22J. ~II.9!

Then, we have the following variational principle for the fixed pointv* ~II.1!.Variational principle:The fixed pointv* is a conditionally critical state of the kinetic energ

E for fixed enstrophyJ5uGu2, and is an absolute critical state ofI.

III. LIAPUNOV STABILITY

Definition 3 (an important functional): For each invariant subsystem (II.3), we definefunctional Ik which is the restriction of the functional I (II.9) to the classS k ,

I k5I ~restricted to ( k!5 (nPZ

$uk1npu222upu22%uv k1npu2. ~III.1!

Lemma III.1: Ik is a constant of motion for the system (II.5).Proof: Differentiating I k , we have

I k5 (nPZ

rn@v k1npv k1np1v k1npv k1np#

52 i (nPZ

F v k1np

]Hk

]v k1np2v k1np

]Hk

]v k1npG

51

2 Up1 k1

p2 k2U(

nPZ@v k1np~Grnrn21v k1~n21!p2Grn11rnv k1~n11!p!

2v k1np~Grn11rnv k1~n11!p2Grnrn21v k1~n21!p!#50.

This completes the proof of the lemma. h

Next we define the concept of disk inZ2/$0% which is needed in the unstable disk theorembe proved below.

Definition 4 (the disk): The disk of radiusupu in Z2/$0%, denoted by Dupu , is defined as

D upu5$kPZ2/$0%uuku,upu%.

The closure of Dupu , denoted by Dupu , is defined as

D upu5$kPZ2/$0%uuku<upu%.

See Fig. 2 for an illustration. Next we prove the unstable disk theorem using a simple enCasimir-type argument,7,20

Theorem III.1 „unstable disk theorem…: If S kùD upu5B, then the invariant subsystem (II.3is Liapunov stable for all tPR; in fact,


)

-


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(nPZ

uv k1np~ t !u2<s (nPZ

uv k1np~0!u2, ;tPR,

where

s5@maxnPZ

$2rn%#@minnPZ

$2rn%#21, 0,s,`.

Proof: By Lemma~III.1!, I k is a constant of motion for the invariant subsystem~II.3!. Then,

(nPZ

rnuv k1np~ t !u25 (nPZ

rnuv k1np~0!u2, ;tPR. ~III.2!

If S kùD upu5B, then

uk1npu.upu, ;nPZ.

Thus, there exists a constantd.0 such that

d,2rn,2. ~III.3!

By ~III.2!,

minnPZ

$2rn% (nPZ

uv k1np~ t !u2<2 (nPZ

rnuv k1np~ t !u2

52 (nPZ

rnuv k1np~0!u2

<maxnPZ

$2rn% (nPZ

uv k1np~0!u2,

that is,

(nPZ

uv k1np~ t !u2<s (nPZ

uv k1np~0!u2,

where

s5@maxnPZ

$2rn%#@minnPZ

$2rn%#21.

By relation ~III.3!,

12d,s,2d21.

This completes the proof of the theorem. h

Remark III.1: If kip, i.e., ' real scalara such that kˆ 5ap, then the invariant subsystem (II.3reduces to

v k1np50, ;nPZ;

thus, it is obviously Liapunov stable for all tPR. In fact, this is a linearization inside the equilibrium space Ep

1 of the 2D Euler equation (cf. Definition 1).


ms

l t


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If uku5upu, then the invariant subsystem~II.3! decomposes into two decoupled systems,

v k1np5A„p,k1~n21!p…Gv k1~n21!p1A„2p,k1~n11!p…Gv k1~n11!p ~n>1!,~III.4!

v k1np5A„p,k1~n21!p…Gv k1~n21!p1A„2p,k1~n11!p…Gv k1~n11!p ~n<21!,~III.5!

whereA(p,k)50. The equation forv k is

v k5A~p,k2p!Gv k2p1A~2p,k1p!Gv k1p . ~III.6!

Each of~III.4! and ~III.5! is a Hamiltonian system with the Hamiltonian

Hk

152Up1 k1

p2 k2U Im H (

n51

`

Grnrn21v k1~n21!pv k1npJ ,

Hk

252Up1 k1

p2 k2U Im H (

n521

2`

Grnrn21v k1~n21!pv k1npJ ,

which has the same representation as~II.5!. Denote byIk

1and I

k

2the restrictions ofI k to the

systems~III.4! and ~III.5!:

Ik

15 (

n51

`

rnuv k1npu2, ~III.7!

Ik

25 (

n521

2`

rnuv k1npu2. ~III.8!

Lemma III.2: If uku5upu, then Ik

1and I

k

2are respectively constants of motion for the syste

(III.4) and (III.5).Proof: The same as that for Lemma III.1. h

Theorem III.2 „half class stability theorem…: If uku5upu, k1p¹D upu , then the linearHamiltonian system (III.4) is Liapunov stable for all tPR. In fact,

(n51

`

uv k1np~ t !u2<s (n51

`

uv k1np~0!u2, ;tPR,

where

s5@maxn>1

$2rn%#@minn>1

$2rn%#21, 0,s,`.

If uku5upu, k2p¹D upu , then the linear Hamiltonian system (III.5) is Liapunov stable for alPR. In fact,

(n521

2`

uv k1np~ t !u2<s (n521

2`

uv k1np~0!u2, ;tPR,

where


ystem


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s5@ maxn<21

$2rn%#@ minn<21

$2rn%#21, 0,s,`.

Proof: The same argument on Ik

1and I

k

2as that on Ik in the proof of Theorem III.1. h

Remark III.2: If k5(p2 ,2p1)T or k5(2p2 ,p1)T, then k1p¹D upu and k2p¹D upu . There-fore, by Theorem III.2, both systems (III.4) and (III.5) are Liapunov stable for all tPR. Points in

S k are on the tangent lines to the circle of radiusupu at k ~cf. Fig. 2!.

IV. PROPERTIES OF THE POINT SPECTRUM FOR THE LINEARIZEDTWO-DIMENSIONAL EULER EQUATION AS A LINEAR HAMILTONIAN SYSTEM

In this section, we study the properties of the eigenvalues for the linear Hamiltonian s~II.6!. The right-hand side of~II.6! defines a linear operator denoted byL, i.e.,

LS ab D5S j

h D , ~IV.1!

where

a5~¯ a21 a0 a1 ¯ !T, b5~¯ b21 b0 b1 ¯ !T;

j5~¯ j21 j0 j1 ¯ !T , h5~¯ h21 h0 h1 ¯ !T;

jn51

2rn

21 ]Hk

]bn, hn52

1

2rn

21 ]Hk

]an, nPZ.

HereL has the infinite matrix representation:

whereun5G rrn , vn5G irn ,

k51

2U k1 p1

k2 p2U .

We define the enstrophy norm which is thel 2 norm:

i~a,b!Ti25 (nPZ

~an21bn

2!. ~IV.2!

Lemma IV.1: The linear operatorL maps l23 l 2 into l23 l 2 :


s the


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L: l 23 l 2° l 23 l 2 .

Proof: Notice that

rn→upu22 as unu→`.

Let

r* 5maxnPZ

urnu

be the maximum ofurnu. Thenr* ,`, and from~IV.1! we have

ujnu<c$uan11u1uan21u1ubn11u1ubn21u%,

uhnu<c$uan11u1uan21u1ubn11u1ubn21u%,

wherec5 12r* uGuuk1p22 k2p1u. Thus

I S jh D I 2

<8c2I S ab D I 2

,

which proves the lemma. h

A complex numberl is an eigenvalue ofL, if there exists (a,b)TP l 23 l 2 such that

LS ab D5lS a

b D . ~IV.3!

Lemma IV.2: The eigenvalues of the linear operatorL have the following properties:

~1! If lPC is an eigenvalue ofL, then bothl ~the complex conjugate ofl! and 2l are alsoeigenvalues.

~2! If l is a real eigenvalue, thenl is a multiple eigenvalue.~3! If l is a simple eigenvalue which is not real, then its corresponding eigenvector satisfie

relation b56 ia.

Proof: Since L is a real linear operator, ifl is an eigenvalue ofL, then l is also aneigenvalue. Next we show that ifl is an eigenvalue ofL, then2l is also an eigenvalue. Letl andv be an eigenvalue and a corresponding eigenvector. Starting from the equation~II.3!, we have

lv k1np5A„p,k1~n21!p…Gv k1~n21!p1A„2p,k1~n11!p…Gv k1~n11!p .

Then,

v k1np5~21!nv k1np

satisfies

~2l!v k1np5A„p,k1~n21!p…Gv k1~n21!p1A„2p,k1~n11!p…Gv k1~n11!p .

Therefore,2l is also an eigenvalue. To prove claims 2 and 3, notice that the HamiltonianHk

~II.6! is invariant under the transformation

an52bn ,

bn5an .


,

x

or

in


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Therefore, ifl is an eigenvalue and

eltS ab D

solves the linear system~II.6!, then

eltS 2ba D

also solves the system~II.6!. If l is real, then (a,b)T can be chosen to be real. Assume that

S ab D5hS 2b

a D . ~IV.4!

Thenh2521, which leads to a contradiction. Thus (a,b)T and (2b,a)T are linearly indepen-dent, andl is a multiple eigenvalue which proves claim 2. Ifl is simple and not real, then (a,b)T

and (2b,a)T are linearly dependent and~IV.4! implies thatb56 ia, which proves claim 3.hIn fact, we have the following theorem.Theorem IV.1: The eigenvalues of the linear operatorL defined in (IV.1) have the following

properties:

~1! The eigenvalues ofL are of four types: real pairs(c,2c), purely imaginary pairs( id,2 id), quadruples(6c6 id), and zero eigenvalues.

~2! If l is a real eigenvalue, thenl is a multiple eigenvalue. Ifl is a simple nonreal eigenvaluethen its corresponding eigenvector satisfies the relationb56 ia.

~3! If S kùD upu5B, then all the eigenvalues ofL are either purely imaginary and in compleconjugate pairs (l5 ic,2 ic; c is real and not zero) or zeros.

~4! If uku5upu, then zero is a multiple eigenvalue ofL.~5! If uku5upu and k1p¹D upu (or k2p¹D upu), then all the eigenvalues for the system (III.4) [

(III.5)] are either purely imaginary and in complex conjugate pairs or zeros.

Proof: Claims 1 and 2 are proved in Lemma IV.2. Next we prove claim 3. IfS kùD upu5B, then I k is negative definite for (a,b)TP l 23 l 2 . In fact, there exists a constantd.0, suchthat

d,2rn,2, ;nPZ. ~IV.5!

Thus,

2I k.di~a,b!Ti2. ~IV.6!

Let l be an eigenvalue ofL and (a,b)T be its corresponding eigenvector, which are writtenterms of real and imaginary parts,

l5l r1 il i , ~ a,b !T5~ a~1!,b~1!!T1 i ~ a~2!,b~2!!T. ~IV.7!

Then both the real and the imaginary parts ofelt(a,b)T, denoted by (a (1),b (1)) and (a (2),b (2)),are real solutions to system~II.6!, where

a~1!5elr t@a~1! cosl i t2a~2! sinl i t#,

b~1!5elr t@b~1! cosl i t2b~2! sinl i t#;

a~2!5elr t@a~1! sinl i t1a~2! cosl i t#,


m

the

of


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b~2!5elr t@b~1! sinl i t1b~2! cosl i t#.

Denote byIk

( j )( j 51,2) the values ofI k evaluated at (a ( j ),b ( j ))T. Then

Ik

~1!~ t !1I

k

~2!~ t !5e2lr t (

nPZrn[( a~1!)21~ a~2!!21~ b~1!!21~ b~2!!2]

5Ik

~1!~0!1I

k

~2!~0!5 (

nPZrn@~ a~1!!21~ a~2!!21~ b~1!!21~ b~2!!2#Þ0

by Lemma III.1 and relations~IV.5! and ~IV.6!. Thus,

e2lr t51, i.e., l r50.

This proves claim 3. To prove claim 4, notice that ifuku5upu, then system~II.3! decomposes intosystems~III.4!–~III.6!. Thus the vectors (a,b)T defined as

a051, an50 ~nÞ0!, bn50 ~;nPZ!,

and

b051, bn50 ~nÞ0!, an50 ~;nPZ!,

give two linearly independent eigenvectors ofL with eigenvalue zero. This proves claim 4. Clai5 follows from the proof for claim 3 when restricted to system~III.4! or ~III.5!. h

Remark IV.1: For a finite-dimensional linear Hamiltonian system, it is well known thateigenvalues are of four types: real pairs(c,2c), purely imaginary pairs( id,2 id), quadruples(6c6 id), and zero eigenvalues.12–14 There is also a complete theorem on the normal formssuch Hamiltonians.14

V. THE POINT SPECTRUM OF THE LINEARIZED TWO-DIMENSIONAL EULEREQUATION: A CONTINUED FRACTION STUDY

Rewrite the equation~II.3! as follows,

rn21zn5a@ zn112 zn21#, ~V.1!

where zn5rnein(u1p/2)v k1np , u1g5p/2, G5uGueig, a5 12 uGu up2

p1

k2

k1u, and rn5uk1npu22

2upu22. Let zn5eltzn , wherelPC. Thenzn satisfies

anzn1zn212zn1150, ~V.2!

wherean5l(arn)21. Let wn5zn /zn21 .15 Thenwn satisfies

an11

wn5wn11 . ~V.3!

Iteration of ~V.3! leads to the continued fraction solution,15

wn~1!5an211

1

an2211

an231�

. ~V.4!

Rewrite ~V.3! as follows:


d

e two


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wn51

2an1wn11. ~V.5!

Iteration of ~V.5! leads to the continued fraction solution,15

wn~2!52

1

an11

an1111

an121�

. ~V.6!

Before we study the two continued fraction solutions~V.4! and~V.6!, we quote two theorems onthe convergence of continued fractions.21

Theorem V.1 „Sleszynski–Pringsheim’s theorem…: The continued fraction

K ~an /bn!5a1

b11a2

b21a3

b31�

,

where$an% and $bn% are complex numbers and all anÞ0, converges if for all n

ubnu>uanu11.

Under the same condition

uK ~an /bn!u<1.

Theorem V.2 „Van Vleck’s theorem…: Let 0,e,p/2, and let bn satisfy

2p/21e,arg$bn%,p/22e

for all n. Then the continued fractionK (1/bn) converges if and only if

(n51

`

ubnu5`.

Notice that asn→6`,

an→a52la21upu2. ~V.7!

Then we have the corollary.Corollary 1: If Re$a%Þ0, or Re$a%50 (uau.2), then the two continued fractions (V.4) an

(V.6) converge.Proof: If Re $a%Þ0, then there exists an positive integerN and a positive constante such that

2p/21e,arg$an%,p/22e

for all unu>N, or

2p/21e,arg$2an%,p/22e

for all unu>N. In either case, applying Van Vleck’s theorem, we have the convergence of thcontinued fractions~V.4! and ~V.6!. If Re$a%50 (uau.2), then there exists a positive integerNsuch that


the


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uanu.2

for all unu>N. Then, applying S´ leszynski–Pringsheim’s theorem, we have the convergence oftwo continued fractions~V.4! and ~V.6!. h

Remark V.1: In fact, as proved in Theorem VI.5, Re$a%50 and uau<2 correspond to thecontinuous spectrum (5essential spectrum) of the system.

When Re$a%Þ0, or Re$a%50 (uau.2), asn→2`,

wn~1!→w~1!5a1

1

a11

a1�

5a1K~1/a!, ~V.8!

and asn→1`,

wn~2!→w~2!52

1

a11

a11

a1�

52K~1/a!. ~V.9!

Both w(1) andw(2) satisfy the equation

w22aw2150. ~V.10!

~i! When Re$a%Þ0, the solutions of~V.10! can be written as

w65 12@ a6dAa214#, ~V.11!

whered5sign (Re$a%) sign (Re$Aa214%). ~Note that if Re$a%Þ0, then Re$Aa214%Þ0.!~ii ! When Re$a%50 (uau.2), let a5 i j, wherej is a real number. The solutions of~V.10! can

be written as:

w65i

2@j6dAj224#, ~V.12!

whered5sign (j) sign (Aj224).Lemma V.1: The solutions (V.11) and (V.12) satisfy the inequality

uw1u.1.uw2u,

and the continued fractions (V.8) and (V.9) have the values

w~1!5v1 , w~2!5w2 .

Proof: First we show thatuw1u.1 when Re$a%Þ0:

w1w15 14~ uau21ua 214u1 adAa2141adAa214!. ~V.13!

Let a5a11 ia2 andAa2145b11 ib2 . Then

b1b25a1a2 , ~V.14!

b122b2

25a122a2

214. ~V.15!

From ~V.14!, we have


n

real


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a12~a2b2!5b2

2~a1b1!.

Thus,a2b2 is either zero or of the same sign asa1b1 . Therefore,

adAa2141adAa21452d@a1b11a2b2#.0. ~V.16!

Together with

ua214u>42uau2,

we have uw1u.1. When Re$a%50(uau.2), it is obvious thatuw1u.1. Notice that w1w2

521, we haveuw2u,1 in both cases. Thus, we have

uw1u.1.uw2u.

From the relationw1w2521, when Re$a%Þ0, Re$w1% and Re$w2% are of opposite signs. WheRe$a%Þ0, Re$w(1)% and Re$w(2)% are of opposite signs, and Re$w(1)% and Re$w1% are of the samesign; thus,w(1)5w1 andw(2)5w2 . When Re$a%50(uau.2), by Sleszynski–Pringsheim’s theo-rem,

uK ~1/a!u<1.

Then,

uw~1!u5ua1K ~1/a!u>uau2uK ~1/a!u.1.

Thus,w(1)5w1 andw(2)5w2 . h

Definition 5: Define w(* ) as follows:

wn~* !5wn

~1! for n<1, wn~* !5wn

~2! for n.1.

Thenwn(* ) solves~V.3!, provided thatw1

(1)5w1(2) , i.e.,

f 5a01S 1

a2111

a2211

a231�

D 1S 1

a111

a211

a31�

D 50, ~V.17!

where f 5 f (l,k,p), l5l/a. Let z(* ) satisfy

wn~* !5zn

~* !/zn21~* ! .

Then asn→1`,

zn~* !;~w2!n,

and asn→2`,

zn~* !;~w1!n.

Thus by Lemma V.1,z(* )P l 2 . Therefore Eq.~V.17! determines eigenvalues.If S kùD upu5B, thenrn,0 for anynPZ. If Re$l%Þ0, then Re$an%Þ0 and are of a fixed sign

for any nPZ. Then Re$f%Þ0. Therefore, in such cases, there is no eigenvalue with nonzeropart. This fact is already obtained in Theorem IV.1. Moreover, we have the following fact.

Lemma V.2: IfS kùD upu5B, then Eq. (V.17) determines no eigenvalue.


er the

e

l


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Proof: As discussed above, ifS kùD upu5B, then the possible solutionl to ~V.17! has to beimaginary. Therefore,a has to be imaginary. Rewrite Eq.~V.2! as follows:

Lzn[rn

r@zn112zn21#5azn . ~V.18!

If S kùD upu5B, then 0,rn /r,1 for all nPZ. Thus, i Li<2. Then if l is an eigenvalue,uau<i Li<2. Notice that Eq.~V.17! should be solved under the condition Re$a%Þ0 or Re$a%50(uau.2). Thus, in this case, Eq.~V.17! determines no eigenvalue. h

Remark V.2: In fact, as proved in Theorem VI.5, Re$a%50 and uau<2 correspond to the

continuous spectrum (5 essential spectrum) of the system. Thus, ifS kùD upu5B, the point spec-trum is empty (cf. Theorem VI.5). Then the problem is reduced to solving Eq. (V.17) und

conditionsS kùD upuÞB, Re$a%Þ0 or Re$a%50(uau.2).Example:Let p5(1,1)T. In this case, only one classS k labeled byk5(1,0)T has non-empty

intersection withD upu . @The other class labeled byk5(0,1)T gives the complex conjugate of thsystem led by the class labeled byk5(1,0)T.# For this class,urn /ru<1 for all nPZ. Thus, thelinear operatorL defined in~V.18! has normi Li<2. Therefore, Eq.~V.17! determines no reaeigenvalue. Numerical calculation on Eq.~V.17! gives the eigenvalue

l50.248 223 024 782 551 i0.351 720 765 265 20.

By Theorem IV.1, Eq.~V.17! determines a quadruple of eigenvalues~see Fig. 3 for an illustra-tion!.

VI. THE SPECTRA OF THE LINEARIZED TWO-DIMENSIONAL EULER EQUATION: ANINFINITE MATRIX STUDY

A. The general setup

Rewrite ~II.3! as follows:

zn5 ia@rn21zn211rn11zn11# ~nPZ!, ~VI.1!

FIG. 3. The quadruple of eigenvalues determined by Eq.~V.17! for the system led by the classS k labeled by k5(1,0)T, whenp5(1,1)T.


n

-


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where

zn5einuv k1np , G5uGueig, u1g5p/2, a51

2uGuUp1 k1

p2 k2U .

Relabel$zn% as follows:

zn5z2n , n>1,

z2n5z2n11 , n>0.

Then

z2n5 ia@rn21z2~n21!1rn11z2~n11!# ~n>2!, ~VI.2!

z2n115 ia@r2n11z2~n21!111r2n21z2~n11!11# ~n>1!, ~VI.3!

z25 ia@r0z11r2z4#, ~VI.4!

z15 ia@r1z21r21z3#, ~VI.5!

for z5(z1 ,z2 ,...)T. Notice that Eqs.~VI.2! and ~VI.3! are decoupled, the coupling betweecomponents ofz with even and odd indices is through Eqs.~VI.4! and~VI.5!. The right-hand sidesof ~VI.2!–~VI.5! define a bounded linear operatorLA : l 2° l 2 , with the infinite matrix representation

A5 iaS 0 r1 r21 0 0 0 0

r0 0 0 r2 0 0 0

r0 0 0 0 r22 0 0 s

0 r1 0 0 0 r3 0

0 0 r21 0 0 0 r23

s � s �

D . ~VI.6!

More importantly,

rn→r52upu22 as unu→`. ~VI.7!

Define the infinite matrix

B5 ibS 0 1 1 0 0 0 0

1 0 0 1 0 0 0

1 0 0 0 1 0 0 s

0 1 0 0 0 1 0

0 0 1 0 0 0 1

s � s �

D , ~VI.8!

whereb5ar52aupu22. Define the infinite matrixC as

C5A2B, ~VI.9!

that is,


rix

t, i.e.,

ist


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C5 iaS 0 r1 r21 0 0 0 0

r0 0 0 r2 0 0 0

r0 0 0 0 r22 0 0 s

0 r1 0 0 0 r3 0

0 0 r21 0 0 0 r23

s � s �

D , ~VI.10!

where rn5rn2r. Denote byLB and LC the bounded linear operators with the infinite matrepresentations byB and C. According to Duren,18 LA , LB , and LC are called(23211)-operators, since their entriescn,n1m satisfy the conditioncn,n1m50 if umu.2. LB is a(23211)-operator with constant coefficients, since its entriescn,n1m are independent ofn whenn.2 andiLB is self-adjoint.

Theorem VI.1: The bounded linear operatorLC : l 2° l 2 is a compact operator.Proof: Denote byLC

(N) the linear operator represented through the matrixCN3N obtained fromC by replacing its entriescm,n by 0, whenm.N. Let $z( j )% be a bounded sequence inl 2 . Then$LC

(1)z( j )% is a bounded sequence in which each element has only one nonzero componen

(LC~1!z~ j !)n50, when n.1.

Thus, there exists a subsequence$z(1 j )%, such that$LC(1)z(1 j )% converges inl 2 . Similarly, we can

get a subsequence of$z(1 j )%, denoted as$z(2 j )%, such that$LC(2)z(2 j )% converges inl 2 , and so on.

Therefore, we have a nested list of subsequences:

z~11! z~12!¯

z~21! z~22!¯

] ] ]

] ] ]

] ] ]

We choose the subsequence$z(nn)% of $z( j )%, which is the diagonal of the above list. There exconstantsz andN0 , such that

iLC~ n!z~nn!2LCz~nn!i<

z

n2 , for all n.N0 and all n. ~VI.11!

For anye.0, chooseN large enough, such that

z

N2,

1

3e. ~VI.12!

Since the subsequence$LC(N)z(Nj )% converges, there existsN, such that

iLC~N!z~Nj 1

!2LC~N!z~Nj 2

!i, 13e, ; j 1 , j 2.N. ~VI.13!

Let N15max$N,N%. Then

iLC~N!z~nn!2LC

~N!z~ nn!i, 13e, ;n,n.N1 . ~VI.14!

Thus


i.e.

nt

-


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iLCz~nn!2LCz~ nn!i<iLCz~nn!2LC~N!z~nn!i1iLC

~N!z~nn!2LC~N!z~ nn!i1iLC

~N!z~ nn!2LCz~ nn!i

, 13e1 1

3e1 13e5e, ;n,n.N1 .

Therefore,$LCz(nn)% is a Cauchy sequence inl 2 and thus converges. This proves thatLC is acompact operator. h

Remark VI.1: In fact, a theorem of Achieser and Glasmann18,22 states that a(2M11)-operator is compact if and only if its diagonal sequence entries tend to zeros,,cn,n1m→0, as n→` for each fixed m, umu<M . Here we give the proof for self-containedness.

B. The spectra of the linear operator LB

Next we will follow a theory of Duren18 to study the spectra of the constant-coefficieinfinite-matrix bounded self-adjoint operatoriLB .

The characteristic polynomialfor the difference equation

~B2lI !z50, ~VI.15!

whereI is the identity matrix, is defined as

f B~w,l!5 ib2lw21 ibw4. ~VI.16!

Define the rescaled characteristic polynomial as follows:

f B~w,l !512lw21w4, ~VI.17!

wherel5 ibl. In fact, f B(w,l) is the characteristic polynomial for the difference equation

~B2lI !z50, ~VI.18!

whereB52 ib21B. The roots off B(w,l) are

w* , 2w* ,1

w*, 2

1

w*, ~VI.19!

where

w* 5F l1Al224

2G1/2

.

Definition 6: Thespectral curveof the linear operatorLB (with the infinite matrix represen

tation by B), denoted by CB , is defined to be the set of alllPC for which the characteristic

polynomial f B(w,l) has a root of modulus one. Thespectral point-set of the operatorLB ,

FIG. 4. The spectral curveCB .


l

se is

ts


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denoted by PB , is defined to be the set of alllPC for which the characteristic polynomia

f B(w,l) has a multiple root. Denote by SB(l) the number of roots of f˜B(w,l), of modulus less

than 1 (counted with multiplicity).Notice that

l5w*2 1w

*22. ~VI.20!

Let w* be a root of modulus 1~then all the four roots are of modulus 1!, w* 5eiu,uP@0,2p).Thus the spectral curveCB is the segment of the real axis,

CB :l52 cos 2u, uP@0,2p!. ~VI.21!

See Fig. 4. The spectral point-setPB consists of two points,

PB :l562, ~VI.22!

which are the boundary points of the spectral curveCB . At l562, the four roots off B(w,l) are

at l52: 1,21,1,21;

at l522: i ,2 i ,2 i ,i .

The functionSB(l) is~VI.23!

SB~ l !5H 0, if lPCB ,

2, if l¹CB .

The general solution to~VI.18! is

zn5c1w*n 1c2~2w* !n1c3w

*2n1c4~2w* !2n, if l¹PB , ~VI.24!

zn5c1w*n 1c2nw

*n 1c3~2w* !n1c4n~2w* !n, if lPPB ~ thenw* 51,i !; ~VI.25!

under the restrictions

2lz11z21z350, ~VI.26!

z12lz21z450. ~VI.27!

Lemma VI.1: The general solution (VI.24) and (VI.25) is in l2 if and only if uw* u,1 andc35c450 in (VI.24) or uw* u.1 and c15c250 in (VI.24).

Proof: See Ref. 18, p. 24, Lemma 5. h

Definition 7: Let L: l 2° l 2 be a linear operator. The set of pointssp(L) in the complexl-plane C such that(L2lI ) has no inverse (i.e., L2lI is not 1-1) is called the point spectrumof L. The set of pointss r(L) in C such that(L2lI )21 exists and is a linear operator withdomain not everywhere dense is called the residual spectrum ofL. The set of pointssc(L) in Csuch that(L2lI )21 exists and is an unbounded linear operator with domain everywhere dencalled the continuous spectrum ofL. The set of pointsr(L) in C such that(L2lI )21 exists andis a bounded linear operator with domain everywhere dense is called the resolvent set ofL. Thesets(L)5sp(L)øs r(L)øsc(L) is called the spectrum ofL.

Without loss of generality, assumeuw* u,1. Thenl is an eigenvalue if and only if there exisa nontrivial solution (c1 ,c2) to the following system:


e

rom


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S 2lw* 1w*2 1w

*3 lw* 1w

*2 2w

*3

w* 2lw*2 1w

*4 2w* 2lw

*2 1w

*4 D S c1

c2D50. ~VI.28!

Theorem VI.2: The l2 point spectrumsp(B) of the linear operatorLB is empty.Proof: The determinant

detS 2lw* 1w*2 1w

*3 lw* 1w

*2 2w

*3

w* 2lw*2 1w

*4 2w* 2lw

*2 1w

*4 D 52w

*3 @w

*4 22lw

*2 1~ l221!#50

~VI.29!

implies that

w*4 22lw

*2 1l22150, ~VI.30!

sincew* Þ0. Herew* is a root of f B(w,l) ~VI.17!,

w*4 2lw

*2 1150. ~VI.31!

From ~VI.30! and ~VI.31!, we have

w*2 5

l222

l. ~VI.32!

Notice also that

l5w*2 1w

*225

l222

l1

l

l222,

which implies thatl562. Thenuw* u51. Thus, if uw* u,1, then Eq.~VI.28! has only a trivialsolution. Therefore, the point spectrum ofLB is empty; equivalently, the point spectrum ofLB isempty. h

Theorem VI.3: The l2 residual spectrums r(B) of the linear operatorLB is empty.Proof: lPs r(B) if and only if the dimension of the orthocomplement of (LB2lI )+ l 2 is

positive and (LB2lI )21 exists. From the inner product relation

^~LB2lI !z~1!,z~2!&5^z~1!,~LB* 2 lI !z~2!&5^z~1!,~LB2 lI !z~2!&,

sinceLB is self-adjoint,LB5LB* ~the adjoint ofLB!, where^,& denotes the inner product over th

complex field. We have that iflPs r(B), then lPsp(B). By Theorem VI.2,sp(B) is empty;thus,s r(B) is empty and, equivalently,s r(B) is empty. h

SinceiLB is self-adjoint, this theorem is well known, but we furnish a short proof here. F~VI.15!, ~VI.18!, ~VI.21!, and ~VI.22!, the spectral curveCB for the linear operatorLB is thesegment of the imaginary axis,

CB :l5 i2b cos 2u, uP@0,2p!; ~VI.33!

the spectral point-setPB for the linear operatorLB is

PB :l56 i2b, ~VI.34!

which are the boundary points of the spectral curveCB .


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Theorem VI.4: The l2 continuous spectrumsc(B) of the linear operatorLB is the spectralcurve, sc(B)5CB . The l2 resolvent setr(B) of the linear operatorLB is the complement of CBin the finite complex plane C, r(B)5(CB)8.

Proof: First we show that iflPCB , thenlPsc(B). Since bothsp(B) ands r(B) are empty,by Theorems VI.2 and VI.3, for anylPCB ,(LB2lI )21 exists and is everywhere densely dfined. We need to show that (LB2lI )21 is unbounded. For anylPCB , there exists a root off B(w,l) of modulus one:

w* 5eiu, uP@0,2p!.

Define the elements

zn~N!5H einu, n<N,

0, n.N.

Thenz(N)P l 2 for each finiteN, andiz(N)i→`, asN→`. There exists a constantd independentof N, such that

i~B2lI !z~N!i<d, ;N.

Thus

iz~N!i

i~B2lI !z~N!i→`, as N→`.

Therefore, (LB2lI )21 is unbounded, andlPsc(B). Next we show that ifl¹CB , then l

Pr(B). For anyl¹CB , the corresponding roots off B(w,l) are ~VI.19!, such that

uw* u5u2w* u,1,uw*21u5u~2w* !21u. ~VI.35!

For anyyP l 2 , we want to construct a solution to

~B2lI !z5y, ~VI.36!

using the method of variation of coefficients. Explicitly, we need to solve

zn2lzn121zn145yn12 ~n>1! ~VI.37!

under the constraints

2lz11z21z35y1 ,~VI.38!

z12lz21z45y2 .

Assume a solution to~VI.37! has the form

zn5cn~1!w

*n 1cn

~2!~2w* !n1cn~3!w

*2n1cn

~4!~2w* !2n. ~VI.39!

If

Dcn~1!w

*n111Dcn

~2!~2w* !n111Dcn~3!w

*2~n11!1Dcn

~4!~2w* !2~n11!50, ~VI.40!

Dcn~1!w

*n121Dcn

~2!~2w* !n121Dcn~3!w

*2~n12!1Dcn

~4!~2w* !2~n12!50, ~VI.41!



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Dcn~1!w

*n131Dcn

~2!~2w* !n131Dcn~3!w

*2~n13!1Dcn

~4!~2w* !2~n13!50, ~VI.42!

Dcn~1!w

*n141Dcn

~2!~2w* !n141Dcn~3!w

*2~n14!1Dcn

~4!~2w* !2~n14!5yn12 , ~VI.43!

whereDcn( l )5cn11

( l ) 2cn( l ) ( l 51,2,3,4), thenzn given in ~VI.39! solves~VI.37!. Solving ~VI.40!–

~VI.43!, we have

Dcn~ l !5~21! l yn12

Dn~ l !

Wn~ l 51,2,3,4!, ~VI.44!

where

Wn5Uw*n11 ~2w* !n11 w

*2~n11! ~2w* !2~n11!

w*n12 ~2w* !n12 w

*2~n12! ~2w* !2~n12!

w*n13 ~2w* !n13 w

*2~n13! ~2w* !2~n13!

w*n14 ~2w* !n14 w

*2~n14! ~2w* !2~n14!

U , ~VI.45!

Dn~1!5U ~2w* !n11 w

*2~n11! ~2w* !2~n11!

~2w* !n12 w*2~n12! ~2w* !2~n12!

~2w* !n13 w*2~n13! ~2w* !2~n13!

U52w*2n@w

*2421#, ~VI.46!

Dn~2!5Uw

*n11 w

*2~n11! ~2w* !2~n11!

w*n12 w

*2~n12! ~2w* !2~n12!

w*n13 w

*2~n13! ~2w* !2~n13!

U52~2w* !2n@12w*24#, ~VI.47!

Dn~3!5Uw

*n11 ~2w* !n11 ~2w* !2~n11!

w*n12 ~2w* !n12 ~2w* !2~n12!

w*n13 ~2w* !n13 ~2w* !2~n13!

U52w*n @w

*4 21#, ~VI.48!

Dn~4!5Uw

*n11 ~2w* !n11 w

*2~n11!

w*n12 ~2w* !n12 w

*2~n12!

w*n13 ~2w* !n13 w

*2~n13!

U52~2w* !n@12w*4 #. ~VI.49!

The Wn defined in~VI.45! satisfies the Wronskian relation

Wn115Wn . ~VI.50!

The representation~VI.44! can be extended ton>0. Choosec0( l )50. We have

cn~ l !5 (

j 50

n21

Dcj~ l !5

1

W0(j 50

n21

D j~ l !~21! l y j 12 ~n>1!. ~VI.51!

The expressions~VI.46!–~VI.49! lead to

cn~1!5

2@12w*24#

W0(j 50

n21

w*2 j y j 12 , ~VI.52!



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cn~2!5

2@12w*24#

W0(j 50

n21

~2w* !2 j y j 12 , ~VI.53!

cn~3!5

2@12w*4 #

W0(j 50

n21

w*j y j 12 , ~VI.54!

cn~4!5

2@12w*4 #

W0(j 50

n21

~2w* ! j y j 12 . ~VI.55!

With these representations ofcn( l ) , zn given by ~VI.39! is a special solution to~VI.37!. The

general solution to~VI.37! is

zn5~cn~1!1a~1!!w

*n 1~cn

~2!1a~2!!~2w* !n1~cn~3!1a~3!!w

*2n1~cn

~4!1a~4!!~2w* !2n,~VI.56!

wherea( l )( l 51,2,3,4) are arbitrary constants. Set

a~3!522@12w

*4 #

W0(j 50

`

w*j y j 12 , ~VI.57!

a~4!522@12w

*4 #

W0(j 50

`

~2w* ! j y j 12 . ~VI.58!

Let

f n5cn~1!w

*n 1cn

~2!~2w* !n1~cn~3!1a~3!!w

*2n1~cn

~4!1a~4!!~2w* !2n. ~VI.59!

Then, from expressions~VI.52!–~VI.55! and ~VI.57!–~VI.59!, we have

f n52@12w

*24#

W0(j 50

n21

@w*n2 j1~2w* !n2 j #yj 122

2@12w*4 #

W0(j 5n

`

@w*j 2n1~2w* ! j 2n#yj 12 .

~VI.60!

Finally,

zn5a~1!w*n 1a~2!~2w* !n1 f n . ~VI.61!

Next we choosea(1) anda(2) to satisfy the constraints~VI.38!:

M S a~1!

a~2!D5F y11l f 12 f 22 f 3

y22 f 11l f 22 f 4G , ~VI.62!

where

M5F2lw* 1w*2 1w

*3 lw* 1w

*2 2w

*3

w* 2lw*2 1w

*4 2w* 2lw

*2 1w

*4 G .

As shown in the proof of Theorem~VI.2!, M is nonsingular. Then,

S a~1!

a~2!D5M 21F y11l f 12 f 22 f 3

y22 f 11l f 22 f 4G , ~VI.63!



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where

M 215k21F2w* 2lw*2 1w

*4 2lw* 2w

*2 1w

*3

2w* 1lw*2 2w

*4 2lw* 1w

*2 1w

*3 G ,

wherek52w*3 (w

*4 22lw

*2 1l221). Thus

zn5„w*n ,~2w* !n

…M 21F y11l f 12 f 22 f 3

y22 f 11l f 22 f 4G1 f n ~VI.64!

solves Eq.~VI.36!. Rewrite f n given in ~VI.60! as follows:

f n5(j 51

`

g~n, j !yj , ~VI.65!

where

g~n, j !550, j51;

2@12w*24#

W0@w

*n2j121~2w* !n2j12#, 2<j<n11;

22@12w

*4 #

W0@w

*j2n221~2w* !j2n22#, j>n12 .

Rewritezn given in ~VI.64! as follows:

zn5(j 51

`

G~n, j !yj , ~VI.66!

where

G~n, j !5„w*n ,~2w* !n

…M 21

S d1,j1lg~1,j !2g~2,j !2g~3,j !

d2,j2g~1,j !1lg~2,j !2g~4,j !D 1g~n, j !, ~VI.67!

whered l , j is the Kronecker delta:d l , j51(l 5 j ) andd l , j50(lÞ j ). From the expression~VI.67!,we see that there exists a constantK independent ofn,j such that

(j 51

`

uG~n, j !u<K, ;n51,2,... ; ~VI.68!

(n51

`

uG~n, j !u<K, ; j 51,2,... . ~VI.69!

Then, we have thel ` norm relation,

izi`5supn

uznu<supn

(j 51

`

uG~n, j !uuyj u<F supn

(j 51

`

uG~n, j !uG iyi`<Kiyi` , ~VI.70!

and thel 1 norm relation,


nd

C

al


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izi15 limN→`

F (n51

N

uznuG< limN→`

F (n51

N

(j 51

`

uG~n, j !uuyj uG5 limN→`

F (j 51

`

uyj u (n51

N

uG~n, j !uG<K(

j 51

`

uyj u5Kiyi1 .

~VI.71!

Thus the linear operator defined in~VI.66!, which mapsy into z, is bounded inl ` and l 1 .Therefore, by Riesz convexity theorem,18,23,24(LB2lI )21 defined in~VI.66! is bounded inl 2 .Since, by Theorems~VI.2! and~VI.3!, (LB2lI )21 exists and is everywhere densely defined ais bounded, we havelPr(B). In summary, we have shown that iflPCB , thenlPsc(B); andif l¹CB , then lPr(B). Thus, sc(B)5CB and r(B)5(CB)8. Equivalently,sc(B)5CB andr(B)5(CB)8. h

In summary, the spectrum ofLB is as depicted in Fig. 5.

C. The spectra of the linear operator LA

Now we apply Weyl’s essential spectrum theorem19 to obtain the spectral theorem forLA .Theorem VI.5 „The spectral Theorem ofLA…: (1) If S kùD upu5B, then the entire l2

spectrum of the linear operatorLA is its continuous spectrum which is the spectral curveBdefined in (VI.33), i.e., s(LA)5sc(LA)5CB . See Fig. 5.

(2) If S kùD upuÞB, then the entire essential l2 spectrum of the linear operatorLA is itscontinuous spectrum which is the spectral curve CB defined in (VI.33), i.e., sess(LA)5sc(LA)5CB . That is, the residual spectrum ofLA is empty, s r(LA)5B. The point spectrum ofLA issymmetric with respect to both real and imaginary axes. See Fig. 6.

Proof: First, we want to show that, in both cases, the residual spectrum ofLA is empty. ByWeyl’s essential spectrum theorem,19 the essential spectrum ofiLA is the same with the essentispectrum ofiLB , sess( iLA)5sess( iLB)5 iCB . Let il rPs r( iLA), then il rP iCB . By the argu-ment in the proof of Theorem VI.3,il rPsp„( iLA)* …, where (iLA)* is the adjoint ofiLA ,

~ iLA!* 5 iLB1~ iLC!* .

By Weyl’s essential spectrum theorem,19 the essential spectrum of (iLA)* is the same with theessential spectrum ofiLB , sess„( iLA)* …5sess( iLB)5 iCB . Thus, il rPsess„( iLA)* …. Sincesess„( iLA)* … andsp„( iLA)* … are disjoint,s r( iLA)5B. The claimsp(LA)5B in case 1 follows

FIG. 5. The continuous spectrum ofLB andLA .


ntial

e case

d

a

useful

),


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from the proof of Lemma V.2 and the fact that the spectral curveCB corresponds to Re$a%50 anduau<2. The property ofsp(LA) in case 2 has been proved in Theorem IV.1. Then Weyl’s essespectrum theorem implies the rest of the claims. h

Remark VI.2: By the above theorem, the computation of eigenvalues is reduced to th

that S kùD upuÞB. By Corollary 1, if l¹CB5sess(LA), the two continued fractions (V.4) an(V.6) converge, and solutions of Eq. (V.17) lead to eigenvalues.

Remark VI.3: The width of the continuous spectrumsc(LA) is 4ubu, where b52aupu22 and

a5 12uGuu

p2 k2

p1 k1u. Although uau can increase to infinity asuku increases to infinity, a is essentially

scaling factor forLA as can be seen in the expression for the infinite-matrix A.Next we discuss an alternative way of representing eigenvalues. This approach is not

for practical computation. Consider the linear difference equation,

~A2lI !z50, ~VI.72!

whereA defined in~VI.6! is the representation matrix ofLA . Explicitly,

rn21z2~n21!2lz2n1rn11z2~n11!50 ~n>2!,~VI.73!

r2n11z2~n21!112lz2n111r2n21z2~n11!1150 ~n>1!,

under the constraints

2lz11r1z21r21z350,~VI.74!

r0z12lz21r2z450,

where l5( ia)21l. By Theorem VI.1, the linear operatorLA is a compact perturbation ofLB .Thus, the difference equation~VI.73! is of Poincare´–Perron type. The Poincare´–Perron theoremstated specifically for the difference equation~VI.72! is as follows:16–18

Theorem VI.6 „Poincare–Perron theorem…: For any lPC, let w* ,2w* ,1/w* ,21/w* bethe roots of the characteristic polynomial fB(w,l) defined in (VI.16), which are given in (VI.19whereuw* u<1. Then there exists a fundamental set of solutions zn

( j )( j 51,2,3,4)to the differenceequation (VI.73), such that

FIG. 6. The spectrum ofLA in case 2.


state.tem isriant‘‘un-linear

ctions.tmatrixatrixsystemcontin-

ionaryndingt the


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lim supn→`

uzn~ j !u1/n5uw* u ~ j 51,2!,

~VI.75!

lim supn→`

uzn~ j !u1/n5

1

uw* u ~ j 53,4!.

It is easy to see that

z~ j !P l 2 ~ j 51,2!; z~ j !¹ l 2 ~ j 53,4!;

if uw* u,1. By definition, whenl¹CB @defined in~VI.33!#, uw* u,1. Next we study the condi-tions for the point spectrum ofLA . Let

zn5c1zn~1!1c2zn

~2! . ~VI.76!

Substitutezn into the constraints~VI.74!, we have

M S c1

c2D50, ~VI.77!

where

M5S 2lz1~1!1r1z2

~1!1r21z3~1! 2lz1

~2!1r1z2~2!1r21z3

~2!

r0z1~1!2lz2

~1!1r2z4~1! r0z1

~2!2lz2~2!1r2z4

~2! D . ~VI.78!

Theorem VI.7: If l¹CB [the spectral curve forLB , defined in (VI.33)], and detM50[where M is defined in (VI.78)], thenlPsp(A) (the point spectrum ofLA).

Proof: If detM50, then there is a nontrivial solution to~VI.77!. Thus there is a nonzerosolution to~VI.73!, which satisfies the constraints~VI.74!. Therefore,l is an eigenvalue. h

VII. CONCLUSION

In this paper, we study the linearized two-dimensional Euler equation at a stationaryThis equation decouples into infinitely many invariant subsystems. Each invariant subsysshown to be a linear Hamiltonian system of infinite dimensions. Another important invabesides the Hamiltonian for each invariant subsystem is found and is utilized to prove anstable disk theorem’’ through a simple energy–Casimir argument. The eigenvalues of theHamiltonian system are of four types: real pairs (c,2c), purely imaginary pairs (id,2 id), qua-druples (6c6 id), and zero eigenvalues. The eigenvalues are studied through continued fraThe spectral equation for each invariant subsystem is a Poincare´-type difference equation, i.e., ican be represented as the spectral equation of an infinite matrix operator, and the infiniteoperator is a sum of a constant-coefficient infinite matrix operator and a compact infinite moperator. We have a complete spectral theory. The essential spectrum of each invariant subis a bounded band of continuous spectrum. The point spectrum can be computed throughued fractions.

This study is the first step toward understanding the unstable manifold structures of statstates of the two-dimensional Euler equation, which we believe to be the key for understatwo-dimensional turbulence. In particular, we will be interested in investigating whether or nounstable manifolds of 2D Euler equations are degenerate~i.e., figure eight structures!. Degeneracywill imply that the dynamics of 2D Euler equations is not turbulent.


allywithship

e 2D

vier-

Rep.

kl.

r the

ns a

, No.


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ACKNOWLEDGMENTS

This work was started at MIT, continued at the Institute for Advanced Study, and fincompleted at the University of Missouri. The author benefited greatly from discussionsProfessor Thomas Witelski at MIT. This work was supported by the AMS Centennial Fellowand the Guggenheim Fellowship.

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On 2D Euler equations. I. On the energy–Casimir ...faculty.missouri.edu/~liyan/Euler-L.pdf · On 2D Euler equations. I. On the energy–Casimir stabilities and the spectra for linearized