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

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Julia sets are ‘‘hairy’’ at the origin, that is, their

blow-ups fill in densely the whole plane (this

phenomenon is related to the universal geometry of

the Feigenbaum attractors; McMullen (1996)).

However, some of them have zero Lebesgue measure

(Yarrington, thesis 1995) and Hausdorff dimension

smaller than 2 (Avila–Lyubich, preprint 2004). It is

unknown whether this happens for all of them or

not (in particular, the answer is unknown for the

Feigenbaum map born in the cascade of doubling

bifurcations).

Regular or Stochastic Dichotomy

Stochastic Maps

An S-unimodal map f is called stochastic if it has an

absolutely continuous invariant measure . In this

case, f is topologically chaotic (see the section

‘‘Topo logical dyn amics’’) and  is supporte d on the

transitive cycle of intervals [J

. Moreover,  has a

positive characteristic exponent,

 ¼

log jDf jd>0

and Lebesgue almost all orbits are equidistributed

with respect to , that is, for Lebesgue a.e. x 2 I,

ðf

xÞ!

 d

for any continuous function . The map f

jJ is mixing

with respect to , and in fact, is weakly Bernoulli.

Here are two important criteria for stochasticity:



Collet–Eckmann condition (see Holomorphic

Dynamics). These maps have extra strong sto-

chastic properties, notably, the exponential decay

of correlations.



Martens–Nowicki condition.Tostateit,weneedto

define the principal nest of intervals, I

I

3

0. Here I

= [, ], where  is the fixed point with

negative multiplier, and I

nþ1

is inductively defined

as the component of f

l

) containing 0, where l

is the moment of first return of the orbit of 0 to I

Let us consider the scaling factors 

= jI

j=jI

n1

j.If

ﬃﬃﬃﬃﬃ



< 1 then f is stochastic.

Let N[2, 1=4] be the set of parameters c for

which the quadratic map P

is topologically chaotic.

Not every such map is stochastic. However, the set

of stochastic parameters has positive Lebesgue

measure (Jakobson 1981), and in fact,

Theorem 5 (Lyubich 2000). For a.e. c 2N, the

map P

satisfies the Martens–Nowicki condition,

and thus, is stochastic.

Avila and Moreira (2005) went on to prove that

for a.e. c 2N, the map P

is Collet–Eckmann.

Renormalization Horseshoe

Let us consider the complexification of the renor-

malization operator [2],

R :

[

2T



!Q ½3

acting in the space of quadratic-like maps.

Theorem 6 (Lyubich 2002). The ‘‘Strong Renor-

malization Conjecture’’ is valid for the operator [3].

Let I[2, 1=4] be the set of parameters for

which the quadratic map P

is infinitely renormaliz-

able. The above theorem implies that this set has

zero Lebesgue measure. (Avila and Moreira went on

to prove that HD(I) < 1.)

Regular or Stochastic Dichotomy

Putting together Theorems 5 and 6, we obtain:

Theorem 7 For a.e. c 2 [2, 1=4], the quadratic

map P

is either regular or stochastic.

This result gives a complete probabilistic picture

of dynamics in the real quadratic family. It has been

later transferred to any nondegenerate real analytic

family of S-unimodal maps (Avila–Lyubich–de

Melo), and further to a generic smooth family of S-

unimodal maps (Avila–Moreira).

Palis has formulated a strong general conjecture

(in all dimensions) asserting that a typical (from

the probabilistic point of view) smooth dynamical

system f has finitely many attractors supporting

SRB measures (see Lyapunov Exponents and

Strange Attractors) that govern the behavior of

Lebesgue a.e. trajectories of f. The above results

confirm the Palis Conjecture in the s etting of S-

unimodal maps.

Other Universality Classes

From a more general point of view, renormalization

is an appropriately rescaled return map to a relevant

piece of the phase space, viewed as an operator in

some class of dynamical systems. From this point of

view, most dynamical systems are ‘‘renormalizable,’’

and the renormalization approach often provides a

deep insight into the nature of the systems in

question.

Here is a partial list of classes of nonlinear

systems that exhibit universality with an underlying

renormalization mechanism (we provide a few

348 Universality and Renormalization

relevant names, but there are many more people

who contributed to the corresponding theories):



Holomorphic germs near indifferent equilibria

(Yoccoz, Shishikura, McMullen);



critical circle maps (Kadanoff, Feigenbaum, Rand,

Lanford, Swiatek, de Faria, Yampolsky);



non-renormalizable quadratic-like maps of

Fibonacci type (Lyubich–Milnor);



conservative two-dimensional diffeomorphisms

near the point of breaking of KAM tori (MacKay,

Koch); and



dissipative He´non-like maps (Collet–Eckmann–

Koch, de Carvalho–Lyubich–M artens).

See also: Fractal Dimensions in Dynamics; Holomorphic

Dynamics; Lyapunov Exponents and Strange Attractors;

Multiscale Approaches.

Further Reading

Collet P and Eckmann J-P (1980) Iterated Maps of the Interval as

Dynamical Systems. Boston: Birkha¨user.

Cvitanovic

P (1984) Universality in Chaos. Bristol: Adam Hilger.

Douady A and Hubbard JH (198 5) On the dynamics of

polynomial-like maps. Annales Scientifiques de l’E

cole

Normale Supe´rieure 18: 287–343.

Lyubich M (1999) Feigenbaum–Coullet–Tresser universality and

Milnor’s hairiness conjecture. Annals of Mathematics 149:

319–420.

Lyubich M (2000) Quadratic family as a qualitatively solvable

model of dynamics. Notices of the American Mathematical

Society 47(9): 1042–1052.

McMullen C (1996) Renormalization and 3-Manifolds Which

Fiber Over the Circle, Annals of Math. Studies, vol. 135.

Princeton: Princeton University Press.

de Melo W and van Strien S (1993) One-Dimensional Dynamics.,

Berlin: Springer.

Sullivan D (1993) Linking the universalities of Milnor–Thurston,

Feigenbaum and Ahlfors–Bers. In: Topological Methods

in Modern Mathematics, The Proceedings of Symposium

held in honor of John Milnor’s 60th Birthday, SUNY at

Stony Brook, 1991, pp. 543–564. Houston, TX: Publish or

Perish.

Vul EB, Sinai YaG, and Khanin KM (1984) Feigenbaum

universality and the thermodynamical formalism. Russian

Mathematical Surveys 39: 1–40.

Universality and Renormalization 349

Variational Methods in Turbulence

F H Busse, Universita

t Bayreuth, Bayreuth, Germany

Introduction

The problem of fluid turbulence is commonly

regarded as one of the most challenging problems of

theoretical physics and mathematics. There is general

agreement that the Navier–Stokes equations (NSEs)

provide a satisfactory basis for the description of

turbulent motions of homogeneous Newtonian fluids

such as gases and most liquids. But the difficulty of

generating solutions of these equations for high-

Reynolds-number flows has prevented accurate

answers to simple questions such as the question of

the discharge of turbulent pipe flow as a function of

the pressure head or the question of the heat transport

by turbulent convection in a fluid layer heated from

below. In view of this difficulty, it has become an

attractive idea to obtain rigorous bounds on turbulent

transports. Variational methods have played an

important role in the derivation of such bounds.

There is another motivation for the use of varia-

tional methods for the understanding of turbulent

fluid systems. Experimenters have sometimes noted

the tendency of turbulent flows to maximize trans-

ports under given external conditions. In his pioneer-

ing paper, Howard (1963) mentions that the Malkus

hypothesis of a maximum heat transport by thermal

convection had motivated him to derive upper bounds

through the use of variational methods. The techni-

ques developed by Howard have later been applied to

other kinds of turbulent transports by Busse. While

relatively simple ordinary differential equations are

obtained when the equation of continuity is not

imposed as a constraint, the Euler–Lagrange equa-

tions for a stationary value of the variational

functional lead to nonlinear partial differential equa-

tions when solenoidal extremalizing vector fields are

required. Nevertheless, using boundary layer methods

one can derive approximate analytical solutions even

in the limit of asymptotically large Rayleigh and

Reynolds numbers (Busse 1969, 1978).

In the following, we shall first discuss the energy

method which provides necessary conditions for the

existence of turbulent solutions of the underlying

equations and then turn to the problem of upper

bounds for the turbulent momentum transport in the

plane Couette flow configuration as a particular

example. The properties and physical relevance of

the extremalizing vector fields will be discussed in a

final section.

Energy Method

For simplicity, we consider the NSEs for a homo-

geneous incompressible fluid with a constant kine-

matic viscosity  in an arbitrary fixed domain D.

Using the diameter d of the domain as length scale

and d

= as timescale, we can write the NSEs of

motion in dimensionless form,

v þ v rv ¼rp þ f þr

v ½ 1a 

rv ¼ 0 ½1b

where f denotes some given steady distribution of a

force density. On the boundary @D of the domain D,

steady velocities parallel to the boundary may be

specified. We assume that the basic steady solution

of the problem is given by v

= Re

v where the

average of (

=2 over the domain D (indicated by

angular brackets) is unity, hj

i= 2. Any velocity

field v

different from v

, that is, with u v



60, must obey the equations

u þv

ru þurv

þuru ¼r

p þr

½2a

ru ¼ 0 ½2b

together with the homogeneous boundary conditions

for u on @D. By multiplying eqn [2a] by u and

averaging the result over the domain D we obtain

the relationship

hu ui ¼ hjruj

iRehu ðu rÞ

vi½3

where the vanishing of u on @D and equations such

hu ðv

rÞu i¼

ru ui

hrðv

u uÞi ¼ 0

have been used to prove that the terms

ru, u ru and r

p do not enter the balance [3].

This balance is called the Reynolds–Orr energy

equation and is the basis for the appli cation of the

energy method. The lowest value Re for which the

right-hand side of [3] is non-negative is called the

energy Reynolds number Re

. For Re < Re

the

steady solution v

is absolutely stable and the energy

of any disturbance u must decay exponentially in

time. Re > Re

is a necessary condition for the

existence of a persistent turbulent state of fluid flow.

is determined as the solution of the variational

problem:

For a given flow

v in D find the minimum Re

of the

functional



hjr



h



u ð



u rÞ

½4

among all vector fields



u which satisfy the conditions

r



u = 0 in D,



u = 0 on @D, and h



u (



u r)

vi < 0.

For Re Re

there will exist at least one vector

field u, namely the minimizing solution



u of the

variational problem [4], the energy of which does

not decay, at least not initially. In the derivation of

the Euler–Lagrange equations as necessary condi-

tions for stationary values of the variational func-

tional [4],

Gð









Þ¼ @

 þ @





½5a







¼ 0 ½5b

the constraint r



u = 0 has been taken into account

through the Lagrange multiplying function

. G is a

stationary value of the functional [4] andingeneral

there exist many of those which are determined as

eigenvalues of the linear boundary value problem [5]

together with its boundary condition



= 0on@D.

Only the infinum of all G provides the energy

Reynolds number Re

. Many details on the energy

method can be found in Joseph’s book (1976). Here

we just wish to remark that the Reynolds–Orr balance

[3] remains valid when the problem is considered in a

system rotating with a constant angular velocity 

since the Coriolis force does not contribute to the

energy balance [3]. The values of Re

are usually

much smaller than the critical values Re

for the onset

of infinitesimal disturbances as can be seen from

Table 1. Here the experimentally determined values

for the instability of the basic flow state have also

been listed. A unique situation occurs in the small gap

limit of the Taylor–Couette system where Re

and Re

coincide for a special value of the dimensionless mean

rotation rate 

(Busse 2002).

Variational Problem for Turbulent

Momentum Transport

In order to introduce the variational method for

bounds on turbulent transports we consider the

simplest configu ration for which a nontrivial solu-

tion of the NSEs of motion exists: the configuration

of plane Couette flow (Figure 1). The Reynolds

number is defined in this case in terms of the

constant relative motion U

i between the plat es,

Re = U

d=, where i is the unit vector parallel to the

plates and  is the kinematic viscosity of the fluid.

Using the distance d between the plates as length

scale and d

= as timescale, the basic equations can

be written in the form

v þ v rv ¼rp þr

v ½6

rv ¼ 0 ½7

We use a Cartesian system of coordinates with the

x, z-coordinates in the directions of i and k,

Table 1 Reynolds numbers for shear flows

(from exp.)

Plane Couette flow 82.6 1300 1

Poiseuille flow (channel flow) 99.2

2000

5772

Hagen–Poiseuille flow

(pipe flow)

81.5

2100

Circular Couette flow with



= Re

82.6 82:6 82.6

The maximum velocity and the channel width d (radius d in the

case of pipe flow) have been used in definition of Re.

–

z =

Figure 1 Geometrical configuration of the plane Couette flow

problem.

352 Variational Methods in Turbulence

respectively, where k is the unit vector normal to the

plates such that the boundary conditions are given by

v ¼

Re i at z ¼

½8

After separating the velocity field v into its mean

and fluctuating parts, v = U þ



v with



v = U,





v = 0,

where the bar denotes the average over planes

z = const., we obtain by multiplying eqn [6] by



and averaging it over the entire fluid layer (indicated

by angular brackets)



i¼



uw 



hjr



i½9

Here u denotes the component of



v perpendicular to

k and w is its z-component. We define fluid

turbulence under stationary conditions by the prop-

erty that quantities averaged over planes z = const.

are time independent. Accordingly, the equation for

the mean flow U can be integrated to yield

U ¼

wu hwuiRe i ½10

where the boundary condition [8] has been

employed. With this relationship, U can be elimi-

nated from the problem and the energy ba lance

hjruj

iþhju w huwij

i¼Rehu

wi½11

is obtained where the identity h

ihuwi

hjuw huwij

i has been used.

Since the momentum transport in the x-direction

between the moving rigid plates is described by

M = dU

=dz j

z = 1 =2

= hu

wiþRe, we can con-

clude immediately that the momentum transport

by turbulent flow always exceeds the corresponding

laminar value because h u

wi is positive according

to the relationship [11]. Since a lower bound on M

thus exists, an upper bound  on hu

wi as a

function of Re is of primary interest. Following

Howard (1963), it c an be shown that (Re)isa

monotonous function and it is therefor equivalent

to ask for a lower bound R of Re at a given value 

of hu

wi. We are thus led to the following

formulation of the variational problem:

Find the minimum R() of the functional

Rð

v;Þ

hjr



þ

w h

wij

½12

among all solenoidal vector fields

v 

u þ k

w (with

u k = 0) that satisfy the boundary condition

v = 0at

z = 1=2 and the condition h

wi > 0.

The Euler–Lagrange equations as necessary con-

ditions for an extremal value of the functional are

given by



þ k

u 



¼r þr

v ½13

r

v ¼ 0 ½14

where dU



=dz is defined by



w h

wii



R 

hjr



½15

and where  = h

wi has been set. When eqns

[13]–[15] are compared with the equations for



and for U, a strong similarity can be noticed. The

variational problem does not exhibit any time

dependence, but the Euler–Lagrange equations may

still be regarded as the symmetric analogue of the

NSEs for steady flow.

Upper Bounds on the Turbulent

Momentum Transport

A simple analytical solution of the variational

problem can be obtained when the constraint

r

v = 0 is dropped. In that case it is evident that

the minimum of the functional [12] is reached

when

v is independent of x , y,andwhen

w = f (z) holds. The Euler–Lagrange equations

then assume the form of an ordinary differential

equation,

¼½ðf

=hf

i1ÞR þhf

i=hf

if ½16

Since the variational functional [12] is homogeneous

v, we are free to use a normalization condition for

which we choose max [f (z)] = 1. Multiplication of

eqn [16] by f

and integration yield



ð1  k

Þð1  f

with k

¼ =½2ð R þ Þhf

i2hf

i½17

This equation can be solved in terms of elliptical

integrals. The minimum R() is determined by the

relationships

R ¼

½K

ð1 þ k

ÞþK

=D  3k

KDÞ

 ¼ 8k

½18

where D(k )andK(k) are the complete elliptical

integrals usually labeled by these letters. For

details, see the analysis by Howard (1963) of an

analogous problem. In the asymptotic case of large

Variational Methods in Turbulence 353

Reynolds numbers, relationships [18] yield the

upper bound

ðReÞ¼

128

½19

In solving the full eqns [13]–[15], it is convenient

to eliminate eqn [14] throug h the general represen-

tation of the solenoidal vector field

v ¼rðr kÞþr k ½20

We assume that the minimizing vector field

v does

not depend on x, although a rigorous proof for this

property can be given only for small values of .

Introducing the notations  @ =@y and w 

@

=@y

we are thus led to the general ansatz

w ¼ w

ðNÞ



n¼1



2

ðzÞ

ðyÞ½21a

 ¼ 

ðNÞ



n¼1



ðzÞ

ðyÞ½21b

where N may tend to infinity and the functions 

(y)

satisfy the equation



¼



½22

In the following, it will be assumed that the positive

wavenumbers 

are ordered according to their siz e,



n1

<

nþ1

. The solutions of the form [21] of

the Euler–Lagrange equations exhibit a boundary

layer structure for large  as sketched in Figure 2.

Accordingly, the N– solutions are characterized by a

hierarchy of N boundary layers at each plate and

provide the upper bound sequentially with increasing

 starting with N = 1. The extremalizing vector fields

thus exhibit a bifurcation structure similar to that

found in many cases of the transition to turbulence.

The thicknesses of the boundary layers decrease with

increasing  and their ratio from one layer to the next

approaches the factor 4 as indicated in Figure 3. The

typical scale of motion increases linearly with

distance from the wall as assumed in Prandtl’s

mixing-length theory. But the disc reteness of the

scales reflects the fact that effective transports require

preferred scales. Asymptotically, the upper bound for

the momentum transport approaches

ðReÞ¼0:010 Re

½23

which represent s a significant improvement over the

relationship [19]. Nevertheless, the upper bound still

exceeds the measured values of the momentum

transport by more than a factor 10.

Discussion

Bounds like those for the momentum transport have

been obtained for many other kinds of turbulent

transports. For details we refer to the review articles

listed below. Usually, the form ulation of the upper

bound problem requires that the external conditions

are homogeneous in two spatial dimensions such

that a separa tion of the turbulent velocity, tempera-

ture, or magnetic fields into mean and fluctuating

parts is possible. In this respect, the variational

methods for upper bounds are more restricted than

those used for determination of the energy Reynolds

number Re

. The latter problem, incidentally,

corresponds to the limit  !0 of variational

problems of the type [12] as can be seen from a

comparison with expression [4].

In recent years, the background field method has

been introduced by Doering and Constantin (1994) as

an alternative way for obtaining bounds on properties

of turbulent flows. When optimized, it becomes

equivalent to the variational method discussed in this

article as has been demonstrated by Kerswell (1998).

The fact that not optimized bounds can be obtained

–1/2

w θ

N – 1

N – 3

N – 2

Figure 2 Qualitative sketch of the boundary layer structure of

the extremalizing N–  solution.

Figure 3 Qualitative sketch of the nested boundary layers that

characterize the vector field of maximum transport. The profile of

the mean shear is shown on the right side.

354 Variational Methods in Turbulence

relatively easily emphasizes the point that the extre-

malizing vector fields are the most interesting aspect of

the variational problems. They often exhibit simila-

rities with the observed turbulent velocity fields, in

particular as far as the mean flows are concerned. In

the case of convection in a layer heated from below,

the transition of the bound from the 1 –  solution to

the 2 –  solution corresponds closely to the experi-

mentally observed transition from convection rolls to

bimodal convection (Busse 1969).

The close similarities between variational functionals

for rather different physical systems suggest corre-

sponding similarities between the respective turbulent

fields. For example, the analogy between the fluctuat-

ing component of the temperature in turbulent convec-

tion and the streamwise component of the fluctuating

velocity field in shear flow turbulence has been

demonstratedandemployedinatheoryoftheatmo-

spheric boundary layer (Busse 1978). Better bounds

and more physically realistic properties of the extre-

malizing vector fields can be expected when additional

constraints are imposed. For example, the energy

balances for poloidal and toroidal components of the

velocity field can be applied separately. But these

developments are still in their initial stages.

See also: Bifurcations in Fluid Dynamics; Fluid

Mechanics: Numerical Methods; Turbulence Theories.

Further Reading

Busse FH (1969) On Howard’s upper bound for heat transport by

turbulent convection. Journal of Fluid Mechanics 37:

457–477.

Busse FH (1978) The optimum theory of turbulence. Advances in

Applied Mechanics 18: 77–121.

Busse FH (2002) The problem of turbulence and the manifold of

asymptotic solutions of the Navier–Stokes equations. In:

Oberlack M and Busse FH (eds.) Theories of Turbulence,

pp. 77–121. Wien: Springer.

Doering CR and Constantin P (1994) Variational bounds on

energy dissipation in incompressible flows: shear flow.

Physical Review E 49: 4087–4099.

Howard LN (1963) Heat transport by turbulent convection.

Journal of Fluid Mechanics 17: 405–432.

Howard LN (1972) Bounds on flow quantities. Annual Review of

Fluid Mechanics 4: 473–494.

Joseph DD (1976) Stability of fluid motions. vol. 1. Berlin:

Springer.

Kerswell RR (1998) Unification of variational principles for

turbulent shear flows: the background method of Doering–

Constantin and Howard–Busse’s mean-fluctuation formula-

tion. Physica D 121: 175–192.

Variational Techniques for Ginzburg–Landau Energies

S Serfaty, New York University, New York, NY, USA

Ginzburg–Landau-type problems are variational

problems which consider a Dirichlet-type energy

posed on complex-valued functions, penalized by a

potential term which has a well in the unit circle of

the complex plane. The denomination comes from

the physical model of superconductivity of Ginzburg

and Landau. They are phase-transition-type models

in the sense that they describe the state of the

material according to different ‘‘phases’’ which can

coexist in a sample and be separated by various

types of interfaces. We start by presenting the

physical model (readers familiar with it may wish

to skip the next two sections and go straight to the

section ‘‘The simplified model’’).

Introduction to the Ginzburg–Landau Model

The Ginzburg–Landau model was introduced by

Ginzburg and Landau in the 1950s as a pheno-

menological model to describe superconductivity,

and was later justified as a limit of the quantum

BCS theory of Bardeen–Cooper–Schrieffer. It is a

model of great importance and recognition in phy sics

(with several Nobel prizes awarded for it: Landau,

Ginzburg, Abrikosov). In addition to its importance

in the modeling of superconductivity, the Ginzburg–

Landau model turns out to be mathematically

extremely close to the Gross–Pitaevskii model for

superfluidity, and models for rotating Bose–Einstein

condensates, which all have in common the appear-

ance of topological defects called ‘‘vortices.’’

Superconductivity, which was discovered in 1911

by Kammerling Ohnes, consists in the complete loss

of resistivity of certain metals and alloys at very low

temperatures: the two most striking consequences of

it being the possibility of permanent superconduct-

ing currents and the particular behavior that an

external magnetic field applied to the sample gets

expelled from the material and can generate

vortices, through which it penetrates the sample.

The Energy Functional

After a series of dimension reductions, the Ginzburg–

Landau model describes the state of the

superconducting sample occupying a region 

and submi tted to the external magnetic field h

Variational Techniques for Ginzburg–Landau Energies 355

below the critical temperature, through its Gibbs

energy:

ð ; AÞ¼



ð1 j j

jcurl A  h

½1

In this expression, the first unknown is the

‘‘order parameter’’ in physics. It is a complex-valued

condensed wave function, indicating the local state

of the material, or the phase (in the Landau theory

approach of phase transitions): j j

is the density of

the ‘‘Cooper pairs’’ of superconducting electrons

explaining superc onductivity in the BCS approach.

With our normalization j j1 and where j j1

the material is in the superconducting phase, while

where j j0, it is in the normal phase (i.e., behaves

like a normal conductor), the two phases being able

to coexist in the sample.

The second unknown A is the electromagnetic

vector potential of the magnetic field, a function

from  to R

. The induced magnetic field in the

sample is deduced by h = curl A. The notation r

denotes the covariant derivative riA. The super-

conducting current is the vector j of components

¼hi ; ðr

i½2

where h.,.i denotes the scalar product in C

identified with R

Finally, the parameter " is the inverse of the

‘‘Ginzburg–Landau parameter’’ , a dimensionless

parameter (ratio of the penetration depth and

the coherence length) depending on the material only.

Most variational studies of Ginzburg–Landau

focus on the regime of large  or small ",

corresponding to ‘‘extreme type-II’’ superconduc-

tors, also called the London limit. In this limit, the

potential term acts as a singular perturbation, and

the characteristic size of the vortices is " ! 0;

vortices become line-like top ological singularities,

which makes it easier to extract and describe them.

This model is a U(1)-gauge theory, that is, it is

invariant under the gauge transformations:

7! e

i

A 7!A þr

½3

where  is a smooth real-valued function. The

physically relevant quantities are those that are

gauge invariant, such as the energy G

, j j, h, and

the superconducting current j.

For more on the model, we refer to the physics

literature (e.g., DeGennes (1966) and Tinkham

(1996)).

Reductions of the Model

The goal of variational studies of the Ginzburg–

Landau model is to relate the energy to the vortices

and the applied field. In three dimensions (3D),

vortices are filaments, or lines of zeros of the order

parameter , around which has a nonzero

winding number. These are quite delicate to describe

in 3D (we will mention some results below), so a

simplification that is commonly made consists in

reducing to a two-dimensional model.

When reducing to 2D, one assumes that every-

thing is independent of the vertical direction, and

that the applied magnetic field is also vertical. The

domain  is then a two-dimensional, bounded and

(for simplicity) s imply connected open set, which is

the horizontal section of an infinite vertical

cylinder. One can also imagine it represents a thin

film.

In 2D, the energy is written the same way:

ð ; AÞ¼



ð1 j j

þjcurl A  h

½4

where this time A is R

-valued, and the induced

magnetic field h = curl A = @

 @

is now a

real-valued function, which can be taken to be equal

to h

(now a real positive number) in R

n.

The stationary states of the system are the critical

points of G

, or the solutions of the Ginzburg–

Landau equations:

ðr

ð1 j j

Þ in 

r

h ¼hi ; r

i in 

h ¼ h

on @ 

  ¼ 0on@

½5

where r

denotes (@

, @

A common simplification consists in suppressing

the magnetic field, and thus in studying the

simplified energy

ðuÞ¼



jruj

ð1 juj

½6

where the order parameter is commonly denoted by

u, and is still complex valued. This energy, which

can be seen as a complex analog of the real-valued

Allen–Cahn model of phase transitions, has been

extensively studied, especially since the work of

Bethuel–Brezis–He´lein, where the domain  is

assumed to be two dimensional and simply con-

nected. The higher-dimensional case has also been

considered.

356 Variational Techniques for Ginzburg–Landau Energies

Vortices and Critical Fields

We now need to explain more precisely what a

vortex is. In two dimensions, a vortex is an object

centered at an isolated zero of u (or ), around

which the phase of u has a nonzero winding number

called the ‘‘degree of the vortex.’’ It is the simplest

example of a topological defect. If the zero is located

at x

, the winding number or degree is the integer

that can be computed by

2

@Bðx

;rÞ

@’

@

¼ d 2 Z ½7

where r is small enough, and ’ is the phase of u,that

is, u can be written u = juje

i’

. For example, the phase

’ = d,where is the polar angle centered at x

, yields

avortexofdegreed. Observe that the phase ’ is not a

well-defined function, it is multivalued (and defined up

to 2); however, we have the important relation

curl r’ ¼ 2



½8

where the a

’s are the zeros of u, d

’s the associated

degrees, and 

denotes the Dirac mass at x.

When " is small, it is clear from [4] or [6] that juj

prefers to be close to 1, and a scaling argument hints

that juj is different from 1 in regions of characteristic

size ". Of course this is an intuitive picture and several

mathematical notions are used to describe the vortices.

Vortices appear due to the applied field h

. For

type-II superconductors there are essentially three

critical fields, H

, H

, critical values of h

for

which phase transitions occur. For h



= O(jlog "j), there are no vortices and the

superconductor is in the superconducting phase

j j’1 everywhere. At H

the first vortices appear,

and their number increases as h

is raised. When

they become numerous they tend to arrange in

triangular lattices called Abrikosov lattices, as

observed in experiments and predicted by Abrikosov

from the Ginzburg–Landau model, in a very

influential work. At the second critical field

= O(1="

) bulk superconductivity is destroyed,

and surface superconductivity remains until

= O(1="

), the third critical field, above which

 0 and the material is normal.

Issues and Methods

The variational approach to Ginzburg–Landau con-

sists in expressing the energy in terms of reduced

quantities or objects, in particular in terms of the

vortices. This requires to develop mathematical tools

to describe and characterize the vortices (in particular

give some suitable definitions of a ‘‘vortex structure’ ’

for a given u or ), and estimate precisely the energetic

cost of each vortex and of their interaction. This

allows us to obtain results of variational convergence

of the energy G

, E

(or their variants), that is, to

derive -limits, or ‘‘reduced problems’ ’ posed in terms

of the vortices, which are easier to minimize than the

original ones. These limits depend on the regime of

applied field, and allow to characterization of, in turn,

the critical fields, and the optimal repartition and

number of the vortices, if any.

Variational methods also serve to solve some

inverse problems, that is, to prove the existence of

solutions of the equation which have some given

properties, such as a given repartition of vortices,

through local minimization procedures, or the use of

topological methods based on investigating the

topology of the energy levels.

Nonvariational approaches of Ginzburg–Landau

are also very useful, in particular to identify the

profiles of the solutions, to describe vortices of

nonminimizing critical points, or to perform a bifurca-

tion analysis around the normal solution at H

The Simplified Model

We first prese nt the variational study of E

[6] in

dimension 2, toget her with the mathematical tools

used for both [6] and [4]. We will restrict to the

asymptotics " ! 0, since this is the situation where

the most results are known.

Let us present informally the essential ingredients

of the analysis.

Tracing the Vortices

The easiest way to trace the vortices is to use the

current hiu, rui (or the ‘‘superconducting current’’

j = hi , r

i for the case with magnetic field). Here

we recall h.,.i denotes the scalar product in C as

identified with R

, that is, hiu, rui= (u  @

u, u 

u) with  the vector product in R

The curl of the current is the vorticity of the map u,

exactly like in fluid mechanics. Writing u = e

i’

have (at least formally) hiu, rui= 

r’ and since

 = juj is close to 1 (other than in the small vortex

regions), we have the approximation

curl hiu; rui¼curl ð

r’Þ’curl r’

¼2



½9

where the a

’s are the zeros of u (or its vortices) and

the d

’s their degrees, or

curl hi ; r

iþcurl A ’ curl r’

Variational Techniques for Ginzburg–Landau Energies 357