Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

Подождите немного. Документ загружается.

spinor field o

is geodesic and shear free iff it is a

repeated principal spinor of the Weyl spinor.

In the spin-coefficient formalism, o

is geodesic

and shear-free iff  and  vanish, and, from [16],isa

repeated principal spinor of the Weyl spinor

provided 

=

= 0. It will be repeated three

times if also 

= 0 and four times if 

= 0, but

one must have 

6¼ 0 for some k if the spacetime is

not to be flat.

Suppose that o

is a (twice) repeated principal

spinor of the Weyl spinor, then at once from the first

two expressions in [18] both  and  vanish. If it is

repeated three times, one gets the same result from

the third and fourth expressions in [18], while if o

is repeat ed four times then the fifth and sixth

expressions of [18] should be used.

For the converse, suppose that  =  = 0. Then, by

the first equation in [14], o

canberescaledtoensure

that  = 0andaspinorfield

can be chosen which is

normalized against o

and parallelly propagated along

‘

, so that, by the fifth equation in [14],  = 0. From

the second expression in [17], one can see at once that



= 0, so that the first two equations in [18] simplify

to give expressions for D

and 

.BycommutingD

and  on 

and using the second expression of [15]

with the relevant parts of [17], it can be concluded that



= 0, as required.

Another application which is easy to describe is

the solution of the type-D vacuum equations. A

type-D solution is one for which the Weyl spinor has

two (linearly independent) repeated principal spi-

nors. If these are taken as the normalized dyad, then

from [16] only 

is nonzero among the 

. By the

Goldberg–Sachs theorem, both spinors are geodesic

and shear free, so that the spin coefficients , , ,

and  all vanish. With these conditions, the spin-

coefficient equations simplify to the point that

careful choices of coordinates and the remaining

freedom in the dyad enable the equations to be

solved explicitly. One obtains metrics that depend

only on a few parameters. Analogous methods

reduce the Einstein equations to simpler systems

for the other vacuum algebraically special metrics,

that is, the other vacuum metrics for which the Weyl

spinor does not have four distinct principal null

directions (Mason 1998).

The spin-coefficient formalism has also been

extensively used in the study of asymptotically flat

spacetimes and gravitat ional radiation (Penrose and

Rindler 1984, 1986, Stewart 1990).

The Positive-Mass Theorem

A very important application of spinor calculus in

recent years was the proof by Witten (1981) of the

positive-mass (or positive-energy) theorem. The

proof was motivated by ideas from supergravity

and gave rise to an increased interest in spinors in

general relativity.

The positive-mass theorem is the following asser-

tion: given an asymptotically flat spacetime M with

a spacelike hypersurface , which is topologically

and in which the dominant energy condition

holds, the total (or Arnowitt–Deser–Misner (ADM))

momentum is timelike and future-pointing. (The

dominant-energy condition is the requirement that

is non-negative for every pair of future-

pointing timelike or null vectors U

and V

We follow the notation of Penrose and Rindler

(1984, 1986), where the proof begins by considering

the 2-form  defined in terms of a spinor field 

 by

 ¼i





^ dx

If 

tends to a constant spinor at spatial infinity on

, then

4G

 ! p







½19

as the spacelike spherical surface S tends to spatial

infinity, where p

is the ADM momentum. Suppose

 has unit normal t

, intrinsic metric h

= g

 t

and the dual-volume 3-form is d

= t

d. Then

Stokes’ theorem states that

 ¼



d

We calculate

d ¼  þ 

where

 ¼ 4GT

‘

d

 ¼i 







d

where ‘

= 





and we have used the Einstein field

equations to replace curvature terms in  by the

energy–momentum tensor T

. Provided the matter

satisfies the dominant-energy condition,  is every-

where a positive multiple of the volume form on 

and its integral is positive (it can vanish only in

vacuum). To make the integral of  positive, 

required to satisfy



¼ 0 ½20

where D

= h

, which is the projection of the four-

dimensional covariant derivative rather than the

intrinsic covariant derivative of . Equation [20] is

the Sen–Witten equation; it is elliptic and reduces to

672 Spinors and Spin Coefficients

the Dirac equation on a maximal surface; furthermore,

given an asymptotically constant value for 

on an

asymptotically flat 3-surface  with the topology of

, it has a unique solution. Equation [20] removes

part of the derivative of 

from  to leave

 ¼h







d

Now h

is negative definite and  has timelike

normal so that  is a positive multiple of the v olume

form on  (unless 

is covariantly constant, a case

which is dealt with separately). Thus, the integral of

d is non-negative and therefore, by [19], so is the

inner product of the ADM momentum p

with any

null vector constructed from asymptotically constant

spinors. Furthermore, this inner product is strictly

positive, except in a vacuum spacetime admitting a

constant spinor. Such spacetimes can be found

explicitly and cannot be asymptotically flat, so that

the ADM momentum is always timelike and future

pointing, and vanishes only in flat spacetime.

The basic positive-energy theorem outlined above

can be extended in several directions:



to prove that the total momentum at future null

infinity is also timelike and future pointing;



to deal with surfaces  which have inner

boundaries, for example, at black holes;



to prove inequalities between charge and mass; and



to deal with spacetimes which are asymptotically

anti-de Sitter rather than flat.

Further Applications of Spinors

Supersymmetry is a symmetry in quantum field

theory relating bosons and fermions. In the languag e

of spinors, bosons are represented by fields with an

even number of spinor indices and fermions by fields

with an odd number of indices. Thus, the gauge

transformations of supersymmetry are generated by

spinors with a single index.

Supergravity is supersymmetry in the case that one of

the fields is the graviton. A supergravity theory is

labeled by an integer N for the number of independent

supersymmetries and much of the numerology of these

theories follows from properties of spinors. N = 1

supergravity contains a graviton and a spin-3/2 field

coupled together, and the presence of the super-

symmetry allows the Buchdahl condition to

be evaded. Supergravity theory with one supersymme-

try in 11 spacetime dimensions depends on one spinor,

which, in 11 dimensions, has 32 components. This is as

many components as eight Dirac spinors in a four-

dimensional spacetime, and, by a process of dimen-

sional reduction, N = 1 supergravity in 11 dimensions

is related to N = 8 supergravity in four dimensions. For

reasons related to the Buchdahl conditions, 8 is the

largest N that is considered in four dimensions.

In superstring theory and in some supergravity

theories, one often wishes to consider spaces

with ‘‘residual supersymmetry,’’ by which is meant

that there is a spinor field satisfying a condition of

covariant constancy in some connection (Candelas et

al. 1985). The existence of such constant spinors, as a

result of spinor Ricci identities analogous to those

given above, typically imposes strong restrictions on

the curvature. Riemannian manifolds admitting con-

stant spinors for the Levi-Civita connection are Ricci-

flat (Hitchin 1974); Lorentzian ones can often be

found in terms of a few functions. Manifolds of

special holomorphy, which are of interest in super-

string theory, can usually be characterized as admit-

ting special spinors (Wang 1989).

See also: Clifford Algebras and Their Representations;

Dirac Operator and Dirac Field; Einstein Equations: Exact

Solutions; Einstein’s Equations with Matter; General

Relativity: Overview; Geometric Flows and the Penrose

Inequality; Index Theorems; Relativistic Wave Equations

Including Higher Spin Fields; Spacetime Topology,

Causal Structure and Singularities; Supergravity; Twistor

Theory: Some Applications [in Integrable Systems,

Complex Geometry and String Theory]; Twistors.

Further Reading

Benn IM and Tucker RW (1987) An Introduction to Spinors and

Geometry with Applications in Physics. Bristol: Adam Hilger.

Budinich P and Trautman A (1988) The Spinorial Chessboard.

Berlin: Springer.

Candelas P, Horowitz G, Strominger A, and Witten E (1985)

Vacuum configurations for superstrings. Nuclear Physics

B 258: 46–74.

Cartan E (1981) The Theory of Spinors. New York: Dover.

Chevalley CC (1954) The Algebraic Theory of Spinors. New

York: Columbia University Press.

Dirac PAM (1928) The quantum theory of the electron.

Proceedings of the Royal Society of London A 117: 610–624.

Harvey FR (1990) Spinors and Calibrations. Boston: Academic

Press.

Hitchin NJ (1974) Harmonic spinors. Advances in Mathematics.

14: 1–55.

Mason LJ (1998) The asymptotic structure of algebraically special

spacetimes. Classical Quantum Gravity 15: 1019–1030.

Penrose R and Rindler W (1984, 1986) Spinors and Space–Time.

vol. 1 and 2. Cambridge: Cambridge University Press.

Stewart J (1990) Advanced General Relativity. Cambridge:

Cambridge University Press.

van der Waerden BL (1960) Exclusion principle and spin. In:

Fierz M and Weisskopf VF (eds.) Theoretical Physics in

the Twentieth Century: A Memorial Volume to Wolfgang Pauli,

pp. 199–244. New York: Interscience.

Wang MY (1989) Parallel spinors and parallel forms. Annals of

Global Analysis and Geometry 7: 59–68.

Witten E (1981) A new proof of the positive energy theorem.

Communications in Mathematical Physics 80: 381–402.

Spinors and Spin Coefficients 673

Stability of Flows

S Friedlander, University of Illinois-Chicago,

Chicago, IL, USA

Introduction

This article gives a brief disc ussion of a topic with

an enormous literature, namely the stability/instabil-

ity of fluid flows. Following the seminal observa-

tions and experiments of Reynolds in 1883, the issue

of stability of a fluid flow became one of the central

problems in fluid dynamics: stable flows are robust

under inevitable disturbances in the environment,

while unstable flows may break up, sometimes

rapidly. These possibilities were demonstrated in a

relatively simple experiment where flow in a pipe is

examined at increasing speeds. As a dimensionless

parameter (now known as the Reynolds number)

increases, the flow completely changes its nature

from a stable flow to a completely different regime

that is irregular in space and time. Reynolds called

this ‘‘turbulence’’ and observed that the transition

from the simple flow to the chaotic flow was caused

by the phenomenon of instability.

Even though the topic has been the subject of

intense study over more than a century, Reynolds

experiment is still not fully explained by current

theory. Although there is no rigorous proof of

stability of the simple flow (known as Poiseuille

flow in a circular pipe), analytical an d numerical

investigations of the equations suggest theoretical

stability for all Reynolds numbers. However, experi-

ments show instability for sufficiently large

Reynolds numbers. A plausible explanation for this

phenomenon is the instability of such flows with

respect to small but finite disturbances combined

with their stability to infinitesimal disturbances.

The issue of fluid stability, in contexts much

more complex than the fundamental experiment of

Reynolds, arises in a multitude of branches of

science, including engineeering, physics, astrophy-

sics, oceanography, and meteorology. It is far

beyond the scope of this short article to even

touch upon most of the extensive literature. In the

bibliography we list just a few of the substantive

books where classical results can be found

(Chandrasekhar 1961, Drazin and Reid 1981,

Gershuni and Zhukovitiskii 1976, Joseph 1976,

Lin 1967, Swinney and Gollub 1985). Recent

extensive bibliographies on mathematical aspects

of fluid instability are given in several articles in the

Handbook of Mathematical Fluid Dynamics

(Friedlander and Serre 2003) and the compendium

of articles on hydrodynamics and nonlinear

instabilities in Godreche and Maneville (1998).

The Equations of Motion

The Navier–Stokes equations for the motion of an

incompressible, constant density, viscous fluid are

þðq rÞq ¼



rP þ  r

q ½1a

div q ¼ 0 ½1b

where q(x, t) denotes the velocity vector, P(x, t) the

pressure, and the constants  and  are the density

and kinematic viscosity, respectively. This system is

considered in three (or sometimes two) spatial

dimensions with a specified initial velocity field

qðx; 0 Þ¼q

ðxÞ½1c

and physic ally appropriate boundary conditions : for

example, zero velocity on a rigid boundary, or

periodicity conditions for flow on a torus. This

nonlinear system of partial differential equations

(PDEs) has proved to be remarkably challenging,

and in three dimensions the fund amental issues of

existence and uniqueness of physically reasonable

solutions are still open problems.

It is often useful to consider the Navier–Stokes

equations in nondimensional form by scaling the

velocity and length by some intrinsic scale in the

problem, for example, in Reynolds’ experiment by

the mean speed U and the diameter of the pipe d.

This leads to the nondimensional equations

þðq rÞq ¼rP þ

q ½2a 

div q ¼ 0 ½2b

where the Reynolds number R is

R ¼ Ud= ½3

In many situations, the siz e of R has a crucial

influence on stability. Roughly speaking, when R is

small the flow is very sluggish and likely to be

stable. However, the effects of viscosity are actually

very complicated and not only is viscosity able to

smooth and stabilize fluid motions, sometimes it

actually also destroys and destabilizes flows.

The Euler equations, which predate the Navier–

Stokes equations by many decades , neglect the

effects of viscosity and are obtained from [1a] by

setting the viscosity parameter  to zero. Since this

removes the highest-derivative term from the equa-

tions, the nature of the Euler equations is funda-

mentally differe nt from that of the Navier–Stokes

equations and the limit of vanishing viscosity (or

infinite Reynolds number) is a very singular limit.

Since all real fluids are at least very weakly viscous,

it could be argu ed that only the the Navier–Stokes

equations are physically relevant. However, many

important physical phenomena, such as turbulence,

involve flows at very high Reyn olds numbers (10

higher). Hence, an understanding of turbulence is

likely to involve the asymptotics of the Navier–

Stokes equations as R !1. The first step towards

the construction of such asymptotics is the study of

inviscid fluids governed by the Euler equations:

þðq rÞq ¼rP ½4a

div q ¼ 0 ½4b

Stability issues for the Euler equations are in many

respects distinct from those of the Navier–Stokes

equations and in this article we will briefly touch

upon stability results for both systems.

Comments on Some ‘‘Classical’’

Instabilities

To illustrate the complexity of the structure of

instabilities that can arise in the Navier–Stokes

equations, we mention one classical example,

namely the centrifugal instabilities called Taylor–

Couette instabilities. Consider a fluid between two

concentric cylinders rotating with different angular

velocities. If the inner cylinder rotates sufficiently

faster than the outer one, the centrifugal force is

stronger on inside particles than outside particles

and a disturbance which exchanges the radial

position of particles is enhanced, that is, the

configuration is unstable. As the angular velocity

of the inner cylinder is increased above a certain

critical rate, the instability is manifested in a series

of small toroidal (Taylor) vortices that fill the space

between the cylinders. There follows a hierarchy of

successive instabilities: azimuthal traveling waves,

twisting regimes, and qua siperiodic regimes until

chaotic solutions appear. Such a sequence of

bifurcations is a scenario for a transition to

turbulence postulated by Ruelle–Takens. Details

concerning bifurcation theory and fluid behavior

can be found in the book of Chossat and Iooss

(1994).

We note that phenomena of successive bifurca-

tions connected with loss of stability, such as

regimes of Taylor–Couette instabilities, occur at

moderately large Reynolds numbers. Fully devel-

oped turbulence is a phenomenon associated with

very high Reynolds numbers. These are parameter

regimes basically inaccessible in current numerical

investigations of the Navier–Stokes equations and

turbulent models. The Euler equations lie at the

limit as R !1. It is an interesting observation that

results at the limit of infinite Reynolds number are

sometimes also applicable and consistent with

experiments for flows with only moderate Reynolds

number.

There is a huge diversity of forces that couple

with fluid motion to produce instability. We will

merely mention a few of these which an interested

reader could pursue in consultation with texts listed

in the ‘‘Further readi ng’’ sect ion and referenc es

therein.

1. The so-called Be´nard problem of convective

instability concerns a horizontal layer of fluid

between parallel plates and subject to a tempera-

ture gradient. The governing equations are the

Navier–Stokes equation for a nonconstant den-

sity fluid and the heat equation. In this problem,

the critical parameter governing the onset of

instability is called the Rayleigh number. The

patterns that can develop as a result of instability

are strongly influenced by the boundary condi-

tions in the horizontal coordinates. With lattice

type conditions, bifurcating solutions include

rolls, rectangles, and hexagons. Convection rolls

are themselves subject to secondary instabilities

that may break the trans lation symmetry and

deform the rolls into meandering shapes. Further

refinements of convective instabilities include

doubly diffusive convection, where the density

depends on concentration as well as temperature.

Competition between stabilizing diffusivity and

destabilizing diffusivity can lead to the so-called

‘‘salt-finger’’ instabilities.

2. Of considerable interest in astrophysics and

plasma physics are the instabilities that occur in

electrically conducting fluilds. Here the fluid

equations are coupled with Maxwell’s equations.

Much work has been done on the topic of

magnetohydrodynamical (MHD) stability, which

was developed to address various important

physical issues such as thermonuclear fusion,

stellar and planetary interiors, and dynamo

theory. For example, dynamo theory addresses

the issue of how a magnetic field can be

generated and sustained by the motion of an

electrically conducting fluid. In the simplest

scenario, the fluid motion is assumed to be a

given divergence-free vector field and the study of

2 Stability of Flows

the instabilities that may occur in the evolution

of the magnetic field is called the kinematic

dynamo problem. This gives rise to interesting

problems in dynamical systems and actually is

closely analogous to the topic of vorticity

generation in the three-dimensional (3D) fluid

equations in the absence of MHD effects.

In the next section we discuss certain mathema-

tical results that have been rigorously proved for

particular problems in the stability of fluid flows.

We restrict our attention to the ‘‘basic’’ equations,

that is, [2a] and [2b], [4a] and [4b], observing that

even in rather simple configurations there are still

more open problems than precise rigorous results.

The Navier–Stokes Equations:

Mathematical Definitions of

Stability/Instability

Instability occurs when there is some disturbance of

the internal or external forces acting on the fluid

and, loosely speaking, the question of stability or

instability considers whether there exist disturbances

that grow with time. There are many mat hematical

definitions of stability of a solution to a PDE. Most

of these definitions are closely related but they may

not be equivalent. Because of the distinctly different

nature of the Navier–Stokes equations for a viscous

fluid and the Euler equations for an inviscid fluid,

we will adopt somewhat different precise definitions

of stability for the two systems of PDEs. Both

definitions are related to the concept known as

Lyapunov stability. A steady state described by a

velocity field U

(x) is called Lyapunov stable if

every state q(x, t) ‘‘close’’ to U

(x)att = 0 stays

close for all t > 0. In mathematical terms, ‘‘close-

ness’’ is defined by considering metrics in a normed

space X. While in finite-dimensional systems the

choice of norm is not significant because all Banach

norms are equivalent, in infinite-dimensi onal sys-

tems, such as a fluid configuration, this choice is

crucial. The point was emphasized by Yudovich

(1989) and it is a version of the definit ion of

stability given in this book that we will adopt in

connection with the parabolic Navier–Stokes

equations.

Definitions for a General Nonlinear

Evolution Equation

Consider an evolution equation for u(x, t) whose

phase space is a Banach space X:

¼ Lu þ Nðu; uÞ

We assume that if the initial value u(x,0)2 X is

given, the future evolution u(x, t), t > 0, of the

equation is uniquely defined (at least for sufficiently

small initial data). Without loss of generality, we

can assume that zero is a steady state.

We define a version of Lyapunov (nonlinear)

stability and its converse instability.

Definition 1 Let (X, Z) be a pair of Banach spaces.

The zero steady state is called (X, Z) nonlinearly

stable if, no matter how small >0, there exists

>0 so that u(x,0)2 X and

kuðx; 0Þk

<

imply the following two assertions:

(i) there exists a global in time solution such that

u(x, t) 2 (([0, 1); X);

(ii) ku(x, t)k

<for a.e. t 2 [0, 1).

The zero state is called nonlinearly unstable if either

of the above assertions is violated. We note that

under this strong definition of stability, loss of

existence of a solution is a particular case of

instability. The concept of existence that we will

invoke in considering the Navier–Stokes equations is

the existence of ‘‘mild’’ solutions introduced by Kato

and Fujita (1962). Local-in-ti me existence of mild

solutions is known in X = L

for q  n, where n

denotes the space dimension. (L

denotes the usual

Lebesque space).

We now state two theorems for the Navier–Stokes

equations [2a] and [2b]. The theorems are valid in any

space dimension n and in finite or infinite domains. Of

course, the most physically relevant cases are n = 3or

2. Both theorems relate properties of the spectrum of

the linearized Navier–Stokes equations to stability or

instability of the full nonlinear system. Let

(x), P

(x) be a steady state flow:

ðU

rÞU

¼rP

F ½5a

rU

¼ 0 ½5b

where U

2 C

vanishes on the boundary of the

domain D and F is a suitable external force. We

write [2a] and [2b] in perturbation form as

qðx; tÞ¼U

ðxÞþuðx; tÞ½6

where

¼ L

u þ Nðu; u Þ½7a

ru ¼ 0 ½7b

Stability of Flows 3

with

u ðU

rÞu ðu rÞU

u rP

½8

Nðu; uÞðu rÞu rP

½9

Here P

and P

are, respectively, the portions of the

pressure required to ensure that L

u and N(u, u)

remain divergence free. The operators L

and N act

on the space of divergence-free vector-valued func-

tions in the closure of the Sobolev space W

s, p

that

vanish on the boundary of D.

We note that the spectrum of the elliptic linear

operator L

with appropriate bounda ry conditions

in a bounded domain is purely discrete: that is, it

consists of a countable number of eigenvalues of

finite multiplicity with the sole limit point being at

infinity.

Theorem 2 (Nonlinear instability). Let 1 < p < 1

be arbitrary. Suppose that the operator L

over L

has spectrum in the right half of the complex plane.

Then the flow U

(x) is (L

, L

) nonli nearly unstable

for any q > max(p, n).

Theorem 3 (Asymptotic Lyapunov stability). Let

q > n be arbitrary. Assume that the operator L

over L

has spectrum confined to the left half of the

complex plane. Then the flow U

(x) is (L

, L

)

nonlinearly stable.

A recent proof of these theorems is given in

Friedlander et al. (2006) using a bootstrap type

argument. In Theorem 2, the space L

, q > n, is used

as an auxiliary space inwhich the norm of the

nonlinear term is control led, while the final instabil-

ity result is proved in L

for p 2 (1, 1). We note

that this includes the most physically relevant case

of instability in the L

energy norm. An earlier proof

of the theorems under the restriction p  n was

given by Yudovich (1989).

To apply Theorem 2 or 3 to conclude nonlinear

instability or stability of a given flow U

,itis

necessary to have information concerning the spec-

trum of the linear operator L

. Obtaining such

information has been the goal of much of the

literature concerning fluid stability (see the biblio-

graphy and the references therein). However, except

in the case of some relatively simple flows, the

eigenvalues of L

have not yet been calcu lated

explicitly. Perhaps the example that is the most

tractable is plane parallel shear flows. Here the

eigenvalue problem is governed by an ordinary

differential equation (ODE) known as the Orr–

Sommerfeld equation, which has been the subject of

extensive analytical and numerical investigations.

Consider the parallel flow U

= (U(z), 0, 0) in the

strip 1  z  1. For disturbances of the form

ðzÞe

iðk

xþk

yÞ

t

½10

the eigenvalue  is determined by the following

equation with k

= k

þ k

U  i





 k

  U



ikR

 k

 ½11

with boundary conditions  = 0atz = 1. We note

that the discreteness of the spectrum is preserved if

periodicity conditions are imposed in the (x, y)

plane.

The complexity of the spectral problem [11] is

apparent even for the simple case U(z) = 1  z

(known as plane Poiseuille flow). Unstable eigenva-

lues exist but only in certain regions of (k, R)

parameter space. There is a critical Reynolds number,

= 5772, below which Re <0 for all wave

numbers k.ForR > R

, instability occurs in a band

of wave numbers and the thickness of this band

shrinks to zero as R !1 (i.e., the inviscid limit).

Hence, Poiseuille flow with R < R

can be considered

as an example where the stability Theorem 3 can be

applied, that is, the flow is nonlinearly stable to

infinitesimal disturbances. However, extremely care-

ful experiments are needed to obtain agreement with

the theoretical value of R

= 5772. Rather it is more

usual in an experiment with R  2000 that the flow

exhibits instability in the form of streamwise streaks

that appear near the walls. These structures do not

look like traveling waves of the form given by

expression [10], rather they are finite-amplitude

effects of nonmodal growth. Such linear growth of

disturbances, along with energy growth and pseudos-

pectra have recently been investigated extensively.

An example where Theorem 3, proving nonlinear

instability, can be applied is the so-called

Kolmogorov flow. This is also a shear flow with the

spectral problem for the linearized operator given by

eqn [11]. In this example, the profile is oscillatory in z

with U(z) = sin mz. In an elegant paper, Meshalkin

and Sinai (1961) used continued fractions to prove

the existence of a real uns table positive eigenvalue. It

is interesting, and in some sense surprising, that the

particular case of sinusoidal profiles leads to a

nonconstant-coefficient eigenvalue problem, where

it is possible to construct in explicit form the

transcendental characteristic equation that relates

the eigenvalues  and the wave numbers. Usually,

4 Stability of Flows

this can be done only for constant-coefficient equa-

tions. In the case U(z) = sin mz, a Fourier series

representation for the eigenfunctions leads to a

tridiagonal infinite matrix for the algebraic system

satisfied by the Fourier coefficients. This is amenable

to examination using continued fractions. Analysis of

the characteristic equation shows that there exist real

eigenvalues >0 provided R is larger than some

critical value for each wave number k with k

< m

The Euler Equation: Linear and

Nonlinear Stability/Instability

We conclude this brief article with some discussion

of instabilities in the inviscid Euler equations whose

existence is likely to be important as a ‘‘trigger’’ for

the development of instabilities in high-Reynolds-

number viscous flows. As we mentioned, the Euler

equations are very different from the Navier–Stokes

equations in their mathematical structure. The

Euler equations are degenerate and nonelliptic. As

such, the spectrum of the linearized operator L

not amenable to standard spectral theory of elliptic

operators. For example, unlike the Navier–Stokes

operator, the spectrum of L

is not purely discrete

even in bounded domains. To define L

we consider

a steady Euler flow {U

(x), P

(x)}, where

rU

¼rP

½12a

rU

¼ 0 ½12b

We assume that U

2 C

. For the Euler equations,

appropriate boundary conditions include zero nor-

mal component of U

on a rigid boundary, or

periodicity conditions (i.e., flow on a torus) or

suitable decay at infinity in an unbounded domain.

The theorems that we will be describing have been

proved mainly in the cases of the second and third

conditions stated above. There are many classes of

vector fields U

(x), in two and three dimensions,

that satisfy [12a] and [12b]. We write [4a] and [4b]

in perturbation form as

qðx; tÞ¼U

ðxÞþuðx; tÞ½13

with

¼ L

u þ Nðu; u Þ½14a

ru ¼ 0 ½14b

Here

u ðU

rÞu ðu rÞU

rP

½15

Nðu; uÞðu rÞu rP

½16

Linear (spectral) instability of a steady Euler flow

(x) concerns the structure of the spectrum of L

Assuming U

2 C

), the linear equation

¼ L

u; ru ¼ 0 ½17

defines a strongly continuous group in every Sobolev

space W

s, p

with generator L

. We denote this group

by exp {L

t}. For the issue of spectral instability of

the Euler equation it proves useful to study not only

the spectrum of L

but also the spectrum of the

evolution operator exp {L

t}. This permits the

development of an explicit formula for the growth

rate of a small perturbation due to the essential (or

continuous) spectrum. It was proved by Vishik

(1996) that a quantity , refered to as a ‘‘fluid

Lyapunov exponent’’ gives the maximum growth

rate of the essential spectrum of exp{L

t}. This

quantity is obtained by computing the exponential

growth rate of a cert ain vector that satisfies a

specific system of ODEs over the trajectories of the

flow U

(x). This proves to be an effective mechan-

ism for detecting instabilities in the essential

spectrum which result due to high-spatial-frequency

perturbations. For example, for this reason any flow

(x) with a hyperbolic fixed point is linearly

unstable with growth in the sense of the L

-norm.

In two dimensions,  is equal to the maximal

classical Lyapunov exponent (i.e., the exponential

growth of a tangent vector over the ODE

x = U

(x)).

In three dimensions, the exis tence of a nonzero

classical Lyapunov exponent implies that  > 0.

However, in three dimensions there are also exam-

ples where the classical Lyapunov exponent is zero

and yet  > 0. We note that the delicate issue of the

unstable essential spectrum is strongly dependent on

the function space for the perturbations and that ,

for a given U

, will vary with this function space.

More details and examples of instabilities in the

essential spectrum can be found in references in the

bibliography.

In contrast with instabilities in the essential

spectrum, the existence of discrete unstable eigenva-

lues is independent of the norm in which growth is

measured. From this point of view, such instabilities

can be considered as ‘‘strong.’’ However, for most

flows U

(x) we do not know the existence of such

unstable eigenvalues. For fully 3D flows there are no

examples, to our knowledge, where such unstable

eigenvalues have been proved to exist for flows with

standard metrics. The case that has received the

most attention in the literature is the ‘‘relatively

simple’’ case of plane parallel shear flow. The

eigenvalue problem is governe d by the Rayleigh

Stability of Flows 5

equation (which is the inviscid version of the Orr–

Sommerfeld equation [11]):

U 

i



 k

  U

 ¼ 0

 ¼ 0atz ¼1 ½18

The celebrated Rayleigh stability criterion says that

a sufficient condition for the eigenvalues  to be

pure imaginary is the absence of an inflection point

in the shear profile U(z). It is more difficult to prove

the converse; however, there have been several

recent results that show that oscillating profiles

indeed produce unstable eigenvalues. For example, if

U(z) = sin mz the continued fraction proof of

Meshalkin and Sinai can be adapted to exhibit the

full unstable spectrum for [18]. We note the ‘‘fluid

Lyapunov exponent’’  is zero for all shear flows;

thus the only way the unstable spectrum can be

nonempty for shear flows is via discrete unstable

eigenvalues.

As we have discussed, it is possible to show that

many classes of steady Euler flows are linearly

unstable, either due to a nonempty unstable essential

spectrum (i.e., cases where  > 0) or due to unstable

eigenvalues or possibly for both reasons. It is natural

to ask what this means abo ut the stability/instability

of the full nonli near Euler equations [14]–[16]. The

issue of nonlinear stability is complex and there are

several natural precise definitions of nonlinear

stability and its converse instability.

One definition is to consider nonlinear stability

in the energy norm L

and the enstrophy norm H

which are natural function spaces to measure

growth of disturbances but are not ‘‘correct’’ spaces

for the Euler equations in terms of proven proper-

ties of existence and uniqu eness of solutions to the

nonlinear equation. Fa lling under this definition is

the most frequently employed method to prove

nonlinear stability, which is an elegant technique

developed by Arnol’d (cf. Arnol’d and Khesin

(1998) and references therein). This is based on

the existence of the so-called energy-Casimirs. The

vorticity curl q is transported by the motion of

the fluid so that at time t it is obtained from the

vorticity at time t = 0 by a volume-preserving

diffeomorphism. In the terminology of Arnol’d,

the velocity fields obtained in this manner at any

two times are called isovortical. For a given field

(x), the class of isovortical fields is an infinite-

dimensional manifold M, which is the orbit of the

group of volume-preserving diffeomorphisms in the

space of divergence-free vector fields. The steady

flows are exactly the critical points of the energy

functional E restricted to M. If a critical point is a

strict local maximum or minim um of E, then the

steady flow is nonlinearly stable in the space J

divergence-free vectors u(x, t) (satisfying the bound-

ary conditions) that have finite norm,

kuk

kuk

þkcurl uk

½19

This theory can be applied, for example, to show

that any shear flow with no inflection points in the

profile U(z) is nonlinearly unstable in the function

space J

, that is, the classical Rayleigh criterion

implies not only spectral stability but also nonlinear

stability.

We note that Arnol’d’s stability method cannot be

applied to the Euler equations in three dimensi ons

because the second vari ation of the energy defined

on the tangent space to M is never definite at a

critical point U

(x). This result is suggestive, but

does not prove, that most Euler flows in three

dimensions are nonlinearly unstable in the Arnol’d

sense. To quote Arnol’d, in the context of the Euler

equations ‘‘there appear to be an infinitely great

number of unstable configurations.’’

In recent years, there have been a number of

results concerning nonlinear instability for the

Euler e quation. Most of these results prove non-

linear instability under certain assumptions on the

structure of the spectrum of the linearized Euler

operator. To date, none of the approaches prove

the definitive result that in general linear instability

implies nonlinear instability. As we have remarked,

this is a much more delicate issue for Euler than for

Navier–Stokes because of the existence, for a

generic Euler flow, of a nonempty essential

unstable spectrum. To give a flavor of the mathe-

matical treatment of nonlinear instability for the

Euler equations, we present one recent result and

refer the interested reader to articles listed in the

‘‘Further reading’’ section for f urther results and

discussions.

In the context of Euler equations in two dimen-

sions, we adopt the following definition of Lyapu-

nov stability.

Definition 4 An equilibrium solution U

(x)is

called Lyapunov stable if for every ">0 there exists

>0 so that for any divergence-free vector u(x,0)2

1þs, p

, s > 2=p, such that ku(x,0)k

<the unique

solution u(x, t)to[14]–[16] satisfies

kuðx; tÞk

<" for t 2½0; 1Þ

We note that we require the initial value u(x,0) to

be in the Sobolev space W

1þs, p

, s > p=2, since it is

known that the 2D Euler equations are globally in

time well posed in this function space.

6 Stability of Flows

Definition 5 Any steady flow U

(x) for which the

conditions of Definition 4 are violated is called

nonlinearly unstable in L

Observe that the open issues (in three dimensions)

of nonuniqueness or nonexistence of solutions to

[14]–[16] would, under Definition 5, be scenarios

for instability.

Theorem 6 (Nonlinear instability for 2D Euler

flows). Let U

(x) 2 C

) be satisfy [12]. Let 

be the maximal Lyapunov exponent to the ODE

x = U

(x). Assume that there exists an eigenvalue 

in the L

spectrum of the linear operator L

given

by [15] with Re >. Then in the sense of

Definition 5, U

(x) is Lyapunov unstable with

respect to growth in the L

-norm.

The proof of this result is given in Vishik and

Friedlander (2003) and uses a so-called ‘‘bootstrap’’

argument whose origins can be found in referenc es

in that article. We remark that the above result gives

nonlinear instability with respect to growth of the

energy of a perturbation which seems to be a

physically reasonable measure of instability.

In order to apply Theorem 6 to a specific 2D flow

it is necessary to know that the linear operator L

has an eigenvalue with Re >. As we have

discussed, such knowledge is lacking for a generic

flow U

(x). Once again, we turn to shear flows. As

we noted =0 for shear flows, any shear profile for

which unstable eigenvalues have been proved to

exist provides an example of nonlinear instability

with respect to growth in the energy.

We conclude with the observation that it is

tempting to speculate that, given the complexity

of flows in three dimensions, most, if not all, such

inviscid flows are nonlinearly unstable. It is clear

from the concept of the fluid Lyapunov exponent

that stret ching in a flow is associated with

instabilities and there are more m echanisms for

stretching in three, as opposed to two, dimensions.

However, to date there are virtually no mathema-

tical results for the nonlinear stability problem for

fully 3D flows and many challenging issues rem ain

entirely open.

Acknowledgments

The author is very grateful to IHES and ENS-

Cachan for their hospitality during the writing of

this paper. She thanks Misha Vishik for much

helpful advice.

The work is partially supported by NSF grant

DMS-0202767.

See also: Compressible Flows: Mathematical Theory;

Incompressible Euler Equations: Mathematical Theory;

Magnetohydrodynamics; Newtonian Fluids and

Thermohydraulics; Non-Newtonian Fluids; Topological

Knot Theory and Macroscopic Physics.