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

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where u

2 E

is given. The question of the well-

posedness of [2] is closely tied to the notion of a

-semigroup in E

. Let L(E

) denote the Banach

space of all bounded linear operators on E

endowed with the usual operator norm. A one-

parameter family T = {T(t) 2L(E

); t  0} is called

‘‘C

-semigroup’’ in L(E

) iff

1. T(0) = id

(normalization),

2. T(s þ t) = T(s)T(t) for all s, t  0 (semigroup

property), and

3. lim

t!0

T(t)x = x for all x 2 E

(strong continuity

at 0).

Given a C

-semigroup T , we define its (infinite-

simal) generator B by setting

domðBÞ :¼ x 2 E

; lim

t!0

TðtÞx  x

exists in E



and by defining

Bx :¼ lim

t!0

TðtÞx  x

for x 2 domðBÞ

This clearly defines a linear operator in E

and it is

well known that B is closed and densely defined.

Moreover, we have

Theorem 1 Assume that A : D(A)  E

! E

is the

generator of a C

-semigroup {T(t); t  0}. Then, given

2 D(A), problem [2] possesses a unique solution u

in C

([0, 1), E

), which is given by u(t) = T(t)u

Under suitable additional assumptions it can be

shown that the converse of Theorem 1 also holds

true. However, we shall not go into these details but

we prefer to present the following characterization

of generators of C

-semigroups:

Theorem 2 (Hille–Yosida). The operator A : D(A)

 E

! E

generates a C

-semigroup iff it is closed,

densely defined, and there exists !, M 2 R such that

the resolvent set (A) of A contains the ray (!, 1) and

such that k(  !)

(  A)

n

kMforall>! and

all n 2 N.

In applications, it is in general rather difficult to

derive a uniform estimate of powers of the resolvent

of an unbounded operator. Luckily, generators of

-semigroups of contractions (i.e., kT(t)k

L(E

)

 1

for all t  0) can be characterized in a rather useful

way. To formulate this result we call an operator

B : D(B)  E

! E

‘‘dissipative’’ iff for any x 2

D(B) there is an x

2 E

with hx

, xi= k xk

= kx

such that Rehx

, Bxi0. Here h, i denotes the

duality pairing between E

and E

. The operator B

is called ‘‘m-dissipative’’ if it is dissipative and

im(

 A) = E

for some 

> 0.

Theorem 3 (Lumer–Phillips). Let A : D(A)  E

be a closed and densely defined operator. Then

A generates a C

-semigroup of contractions in L(E

) iff

Aism-dissipative.

Before we shall discuss examples of C

-semigroups

and their infinitesimal generators, let us introduce the

following definition: given  2 (0,  ], let 



:= {z 2 C;

jarg (z)j <} denote the sector in C of angle 2.A

family of operators T = {T(z) 2L(E

); z 2 



}is

called a ‘‘holomorphic C

-semigroup’’ in L(E

)iff

1. [z 7!T(z)] : 



!L(E

) is holomorphic,

2. T(0) = id

and lim

z!0

T(z)x = x for all x 2 E

,and

3. T(w þ z) = T(w)T(z) for all w, z 2 



Generators of holomorphic C

-semigroups can be

characterized in the following way:

Theorem 4 A densely defined closed linear operator

A : D(A)  E

! E

generates a holomorphic

-semigroup iff there exist M > 0 and !

 0 such

that  2 (A) and k (  A)

1

kM for all  2 C

with Re >!

Examples 5

(i) Self-adjoint generators. Let E

be a Hilbert

space and assume that A is self-adjoint and that

there exists an 

2 R such that A  

. Then

A generates a holomorphic C

-semigroup {T(t); t 

0}. If {E

();  2 R} denotes the spectral resolution

of A, then T(t) =

exp (t)dE

() for t  0.

(ii) Dissipative operators in H ilbert spaces.

Assume again that E

is a Hilbert space. Then,

by Riesz’ representation formula, an operator A is

dissipative iff Re(ujAu)  0forallu 2 D(A).

(iii) The heat semigroup.LetM be either a

smooth compact closed Riemannian manifold or

with the Euclidean metric and write  for the

Laplace–Beltrami operator on M. Then it is known

that  2L(D

(M)), where D

(M) is the space of all

distributions on M. Given 1  p < 1, let

Dð

Þ :¼fu 2 L

ðMÞ;u 2 L

ðMÞg

and set 

u =u for u 2 D(

). Then 

generates a

holomorphic C

-semigroup on L

(M), the so-called

‘‘diffusion’’ or ‘‘heat semigroup’’ on M.If1< p < 1,

then it can be shown that D(

) = W

(M), where

(M) denotes the Sobolev space of order k 2 N, built

over L

(M).

If M = R

then the operators T(t) of the semigroup

generated by 

are given by

TðtÞuðxÞ¼

ð4tÞ

m=2

exp

jx  yj

uðyÞdy

for all t > 0 and almost all x 2 R

266 Evolution Equations: Linear and Nonlinear

Observe that the case L

(M) is excluded here. In

fact, it is known that if a linear operator A generates

a C

-semigroup on L

(M), then A must be

bounded. However, it can be shown that suitable

realizations of the Laplace–Beltrami operator on

spaces of continuous and Ho¨ lder continuous func-

tions generate holomorphic semigroups. For more

details on that topic the reader is referred to the

‘‘Furtherreading’’section.

(iv) Stone’s theorem and the Schro

dinger equa-

tion.LetE

be a Hilbert space and assume that A

is self-adjoint. Then Theorem 3 and Remark (ii)

imply that iA generates a C

-group {U(t); t 2 R}of

unitary operators. In fact, Stone’s theorem ensures

that every generator of a C

-group of unitary

operators is of the form iA with a self-adjoint

operator A. As an example of particular interest,

let us consider the Schro¨dinger equation

¼ u  Vu ½3

with a bounded potential V : R

! R.Letting

D(A):= H

)andAu :=u  Vu,itfollows

that A is self-adjoint in L

). Hence, the evolution

of [3] is governed by the group of unitary operators

generated by iA. Of course, the assumption that V be

bounded is rather restrictive. In fact, there are

numerous contributions which show that this assump-

tion can be weakened considerably. Again reader is

referredtothe‘‘Furtherreading’’sectionformore

details in this direction.

(v) The wave equation. Let us consider the

following initial-value problem

uðt; xÞ¼0; x 2 R

; t > 0

uð0; xÞ¼’

ðxÞ;@u=@tð0; xÞ

¼ ’

ðxÞ; x 2 R

½4

for the d’Alembert operator

= @

u=@t

 

m þ 1 dimensions. In order to associate with [4] a

semigroup, let us formally re-express [4] as the

following first-order system:

¼ AU; t > 0; Uð0Þ¼

where

U ¼ðu; u

Þ; A ¼

0id

 0



;  ¼ð’

;’

Letting now E

:= H

)  L

) and

D(A):= H

)  H

), it can be shown that A

generates a C

-group of linear operators in L(E

Hence, given any initial datum (’

, ’

) 2 H

)

), there exists a unique solution u 2 C

([0, 1), L

)) to the initial-value problem [4].It

can be shown that this solution possesses the

following additional regularity:

u 2 C

ð½0; 1Þ; L

ðR

ÞÞ \ Cð½0; 1Þ; H

ðR

ÞÞ

Hence, eqns [4] are satisfied for all t 2 [0, 1)

and for almost all x 2 R

(vi) Maxwell equations.LetE and H denote the

electric and magnetic field vector, respectively, " and 

the electrical permittivity and magnetic permeability,

respectively, and consider the initial-value problem for

Maxwell equations in vacuum and without charges

and currents: given sufficiently smooth vector fields

, H

) find a pair (E, H)suchthat

 rot H ¼ 0inð0; 1Þ  R



þ rot E ¼ 0inð0; 1Þ  R

Eð0; Þ ¼ E

; Hð0; Þ ¼ H

in R

½5

We assume that " and  belong to L

, L

sym

))

and are uniformly positive definite, that is, we

assume that there are "

> 0 and 

> 0 such that

ð"ðxÞyjyÞ"

jyj

; ððxÞyjyÞ

jyj

for all x, y 2 R

. Based on these assumptions we

endow the space L

)  L

) with the inner

product

ðu

; u

Þjðv

; v

ÞðÞ:¼ð"u

þðu

for (u

, u

), (v

, v

) 2 L

)  L

), and call this

Hilbert space E

. We further set

:¼fðu

; u

Þ2E

; ðrot u

; rot u

Þ2E

Finally, given u = (u

, u

) 2 E

, let

Au :¼ "

1

rot u

; 

1

rot u



It can be shown that iA is self-adjoint in E

Hence, Stone’s theorem ensures that A generates a

-group of unitary operators in L(E

). Therefore,

given (E

, H

) 2 E

, there exists a unique solution (E(),

H()) of [5]. For this solution, the energy functional

EðtÞ¼

ð"EðtÞjEðtÞÞ

3 þðHðtÞjHðtÞÞ



is constant on [0, 1).

Autonomous Inhomogeneous Equations

Next, we study problem [1] in the case A(t) = A for

all t 2 [0, T). Throughout this section we assume

that the following minimal hypotheses

1. A generates a C

-semigroup in L(E

2. f 2 L

((0, T), E

), and

3. u

2 E

Evolution Equations: Linear and Nonlinear 267

are satisfied. Later on we shall discuss several more

restrictive assumptions on (A, f , u

). A function

u : [0, T] ! E

is called a ‘‘(classical) solution’’ of

ðtÞ¼AuðtÞþf ðtÞ; t 2ð0; T; uð0Þ¼u

½6

iff u 2 C([0, T], E

) \ C

((0, T], E

), u(t) 2 D(A) for

all t 2 (0, T], and u satisfies [6] pointwise on [0, T].

It can be shown that [6] has at most one solution. If

it has a solution, this solution is represented by the

following variation-of-constant-formula:

uðtÞ¼TðtÞu

Tðt  sÞ f ðsÞds; t 2½0; T½7

where {T(t); t  0} denotes the semigroup generated

by A. Observe that the function u : [0, T] ! E

defined by [7], is continuous, but in general not

differentiable on (0, T]. For this reason one calls [7]

the ‘‘mild solution’’ of [6].

It is not difficult to see that if u

2 D(A)andf 2

([0, T], E

), then the mild solution is a classical

solution, that is, [6] is uniquely solvable in the

classical sense. In application to nonlinear problems,

the assumption f 2 C

([0, T], E

) is often too

restrictive. Fortunately, in the case of generators of

holomorphic semigroups, this assumption on f can

be weakened in two different directions. Let

kxk

:= kxk

þkAxk

denote the graph norm on

D(A). Then the closedness of A implies that

(D(A); kk

) is a Banach space. In the following,

we call this Banach space E

. Moreover, given  2

(0, 1), we write E



= (E

, E

)



for the complex

interpolation space between E

and E

. Then we

have the following result.

Theorem 6 Let A generate a holomorphic

-semigroup in L(E

) and assume that there is a

constant  2 (0, 1) such that

f 2 C



ð½0; T; E

ÞþCð½0; T; E



Then, given u

2 E

, the Cauchy problem [6]

possesses a unique classical solution. It is given by

uðtÞ¼TðtÞu

Tðt  sÞ f ðsÞds; t 2½0; T

where {T(t); t  0} stands for the semigroup gener-

ated by A.

In the following, we discuss an alternative

approach to the Cauchy problem [6], which is

based on the so-called theory of maximal regularity.

There are several different types of results on

maximal regularity, which we cannot discuss in full

detail here. We decided to give a brief introduction

to the theory of the so-called ‘‘maximal L

-regular-

ity.’’ For further results on maximal regularity, we

againdrawthereader’sattentiontothe‘‘Further

reading’’section.

The Banach space E

is called an unconditionality

of martingale ‘‘differences’’ (UMD) space if the

Hilbert transform is bounded on L

(R, E

) for

some q 2 (1, 1). It is known that Hilbert spaces,

the Lebesgue spaces L

(X,d) with 1 < p < 1 and

with a -finite measure space (X, ), and closed

subspaces of UMD spaces are UMD spaces.

Furthermore, UMD spaces are without exception

reflexive. Thus, the spaces L

(X,d), L

(X,d), and

spaces of continuous or Ho¨ lder continuous functions

are not UMD spaces.

Next, assume that A generates a holomorphic

-semigroup in L(E

) and that [0, 1)  (A).

Then, it is known that, given z 2 C, the fractional

power A

of A is a densely defined closed operator

in E

. We say that A has bounded imaginary powers

(BIP) of angle   0 if there exist positive constants

M and " such that

2LðE

Þ and kA

LðE

 M expðjtjÞ

t 2ð"; "Þ

½8

In order to have a neat notation, we write A 2

BIP()if[8] holds true.

Remarks 7 In the following, we assume that A

generates a holomorphic C

-semigroup in L(E

) and

that [0, 1)  (A).

(i) If Re z < 0, then A

is bounded on E

(ii) There are several representation formulas for

the fractional powers of A. Among them we

picked the following: if Re z 2 (1, 1) and x 2

D(A), then

x ¼

sinðzÞ

z

ðs þ AÞ

2

Ax ds

(iii) Assume that E

is a Hilbert space, that A is self-

adjoint, and that there is a positive constant 

such that A  . Further, let {E

() 2 R} be the

spectral resolution of A; then

:¼



ðÞ; z 2 C

Moreover, A 2 BIP(0).

(iv) Let again E

be a Hilbert space and assume that

A is m-dissipative and satisfies 0 2 (A). Then

A 2 BIP(=2).

Given p 2 (1, 1), Sobolev’s embedding theorem

ensures that W

((0, T), E

) is continuously injected

into C([0, T], E

). Consequently, given any function

u 2 W

((0, T), E

) and t 2 [0, T], the pointwise

268 Evolution Equations: Linear and Nonlinear

evaluation u(t) is well defined. In particular, the

trace at 0 with respect to time

tr : W

ðð0; TÞ; E

Þ!E

; u 7!uð0Þ

is a well-defined and bounded linear operator. In order

to formulate the next result, let E

s,p

= (E

, E

)

s,p

,with

p 2 (1, 1)ands 2 (0, 1), denote the real interpolation

space between the basic space E

and E

,thedomain

D(A)ofA, endowed with the graph norm. Further-

more, we set

:¼ L

ðð0; TÞ; E

:¼ L

ðð0; TÞ; E

Þ\W

ðð0; TÞ; E

and we write Isom(E, F) for the set of all topological

isomorphisms mapping the Banach space E onto the

Banach space F.

Theorem 8 (Dore and Venni). Suppose that E

is a

UMD space and that A 2 BIP() for some

 2 [0, =2). Then, given p 2 (1, 1), we have

ð@

þ A; trÞ2IsomðE

; E

 E

11=p;p

This means that, given (f , u

) 2 L

((0, T), E

) 

11=p,p

, there exists a unique solution u 2

((0, T), E

) \ W

((0, T), E

) of the Cauchy problem

[6]. Moreover, u depends continuously on (f , u

)

and fulfills the following aprioriestimate:

kuk

 cðkf k

þku

11=p;p

where c := k(@

þ A, tr)

1

L(E

E

11=p, p

, E

)

Nonautonomous Equations of Hyperbolic Type

According to Theorem 1 and the corresponding

remark, it is reasonable to impose in the study of the

Cauchy problem [1] the minimal hypothesis that,

given s 2 [0, T], each individual operator A(s) be the

generator of a C

-semigroup {T

(t); t  0} in L(E

If this semigroup is holomorphic, we call [1] of

‘‘parabolic type.’’ Otherwise the evolution equation [1]

is said to be of ‘‘hyperbolic type.’’

A family {A(t); t 2 [0, T]} of generators of

-semigroups in L(E

) is called ‘‘stable’’ iff there

exist positive constants M and ! such that (!, 1) 

(A(t)) for all 2 [0, T] and such that

j¼1

ð  Aðt

ÞÞ

1



 Mð  !Þ

k

for >!

and every finite sequence 0  t

 t

 t

 T

with k 2 N. Observe that the resolvent operators

(  A(t

))

1

do not commute in general. Therefore,

the order of the terms on the left-hand side of the above

estimate has to be obeyed. Assume that A= {A(t); t 2

[0, T]} is a family of m-dissipative operators. Then, A

is stable, since any m-dissipative operator B satisfies

the estimate k(  B)

1

k1= for all >0.

It turns out that the stability of a family of

generators is not sufficient to construct a solution of

[1] even in the case f  0. We also need a certain

time regularity of the mapping t 7!A(t). For this we

say that the family {A(t); t 2 [0, T]} has a common

domain D iff D is a dense subspace of E

such that

D(A(t)) = D for all t 2 [0, T]. The family {A(t); t 2

[0, T]} is called ‘‘strongly differentiable’’ iff it has a

common domain D and, given v 2 D, the function

t 7!A(t)v belongs to C

([0, T], E

We are now prepared to formulate the following

result.

Theorem 9 (Kato). Let {A(t); t 2 [0, T]} be a stable

and strongly differentiable family of generators of

-semigroups with common domain D. If f 2

([0, T], E

) and u

2 D then [1] possesses a

unique classical solution.

The above result is based on the construction of

an evolution operator U(t, s), which can be

considered as the generalization of the notion of a

-semigroup for autonomous equations to the case

evolution equations of the form

ðtÞ¼AðtÞuðtÞ; t 2ðs; T; uðsÞ¼v

for fixed s 2 [0, T). Once an evolution operator is

available, the solution of [1] is given by

uðtÞ¼Uðt; 0Þu

Uðt; sÞf ðsÞds; t 2½0; T

Of course, this generalizes [7] and if A(t)is

independent of t, then U(t, s) = T(t  s), where

{T(t); t  0} is the semigroup generated by A(0).

Furthermore, there are several extensions of the

Kato’s result. Among them the most interesting

contributions are concerned to weaken the time

regularity of f and to weaken the assumption that

{A(t); t 2 [0, T]} be strongly differentiable. In parti-

cular, it is possible to study [1] for families without

a common domain.

For the construction of evolution operators as

well as generalizations of Theorem 9, the reader is

againreferredtothe‘‘Furtherreading’’section.

Nonautonomous Equations of Parabolic Type

Throughout this section we assume that E

and E

are Banach spaces such that E

is dense and

continuously injected in E

. In the study of parabolic

evolution equations, the class of all operators in

L(E

, E

), considered as unbounded operators in E

with common domain E

, which generate holo-

morphic C

-semigroups in L(E

) has turned out to

Evolution Equations: Linear and Nonlinear 269

be very useful. In the following, we call this class

H(E

, E

). It is known that A 2H(E

, E

) iff there

exist constants !>0and  1 such that !  A 2

Isom(E

, E

) and such that



1



kð  AÞxk

jjkxk

þkxk

 

x 2 E

nf0g; Re   !

where kk

denotes the norm of E

. Using the above

characterization, it can be shown that H(E

, E

)is

an open subset of L(E

, E

). In the following, we

always endow H(E

, E

) with the topology induced

by the norm of L(E

, E

). As a consequence of this

convention it is meaningful to consider, for example,

continuous mappings from [0, T] into H(E

, E

Observe that if A 2 C([0, T], H(E

, E

)), then

A= {A(t); t 2 [0, T]} is a family of generators of

holomorphic semigroups with the common domain

. Then we have the following result.

Theorem 10 (Sobolevskii, Tanabe). Assume that

there is a  2 (0, 1) such that

ðA; f Þ2C



ð½0; T; HðE

; E

ÞE

Then, given u

2 E

, the Cauchy problem [1]

possesses a unique classical solution u. This solution

has the additional regularity

u 2 C



ðð0; T; E

Þ\C

1þ

ðð0; T; E

Finally, if u

2 E

, then u 2 C

([0, T], E

As in the hyperbolic case, the proof of Theorem 10

is based on the evolution operator U(t, s)forthe

homogeneous problem, although the constructions of

the corresponding evolution operators are completely

different.

In addition, there are several extensions and

generalizations of Theorem 10. In particular, the

assumption that the family {A(t); t 2 [0, T]} pos-

sesses a common domain can be weakened con-

siderably. Furthermore, it is possible to look at

parabolic evolution equations in the so-called inter-

polation and extrapolation scales. This offers a great

flexibility in the study of nonlinear problems.

Further details in this direction can be found in the

‘‘Furtherreading’’section.

Nonlinear Evolution Equations

Let E

, E

be Banach spaces such that E

dense and continuously embedded in E

.Assume

further that u

2 E

and that we are given a

nonlinear operator F 2 C([0, T]  V, E

), where V

is an open neighborhood of u

in E

.Inthis

section, we will discuss the well-posedness of the

Cauchy problem for the following nonlinear

evolution equation

ðtÞ¼Fðt; uðtÞÞ; t 2ð0; T; uð0Þ¼u

½9

in the Banach space E

.Wewillalwaysassume

that the nonlinear operator F either carries a quasi-

linear structure or is of fully nonlinear parabolic type.

By a ‘‘quasilinear structure,’’ we mean that there is

mapping A 2 C([0, T]  V, L(E

, E

)) and a suitable

‘‘lower-order term’’ f 2 C([0, T]  V, E

)suchthat

Fðt; vÞ¼Aðt; vÞv þ f ðt; vÞ

for all ðt; vÞ2½0; TV

Problem [9] is of fully nonlinear parabolic type if

F 2 C

([0, T]  V, E

) and if the Fre´chet derivative

F(0, u

)ofF with respect to v at (0, u

) belongs to

the class H(E

, E

Quasilinear Evolution Equations of Hyperbolic Type

Assume that E

is a reflexive Banach space and let

2 V  E

be chosen as above. We consider the

following abstract quasilinear evolution equation of

hyperbolic type:

ðtÞ¼Aðt; uðtÞÞuðtÞþf ðt; uðtÞÞ; t 2ð0; T

uð0Þ¼u

½10

and assume that the following hypotheses are

satisfied:

) A 2 C([0, T]  V, L(E

, E

)) is bounded on

bounded subsets of V and, given (t, v) 2

[0, T]  V, the operator A(t, v)ism-dissipative

and there is a constant 

such that

kAðt; vÞAðt; wÞk

LðE

 

kv  wk

for all t 2 [0, T] and all v, w 2 V.

) There is a Q 2 Isom(E

, E

) such that

QA(t, v)Q

1

= A(t, v) þ B(t, v), where B(t, v) 2

L(E

) is bounded, uniformly on bounded

subsets of V. Moreover,

kBðt; vÞBðt; wÞk

LðE

 

kv  wk

for all t 2 [0, T] and all v, w 2 V.

) f 2 C([0, T]  V, E

) is bounded on bounded

subsets of V and there are 

and 

such that

kf ðt; vÞf ðt; wÞk

 

kv  wk

for all v; w 2 V; j 2f0; 1g

Then we have the following result.

Theorem 11 (Kato). Assume that (H

), (H

and (H

) are satisfied. Then there is a maximal

270 Evolution Equations: Linear and Nonlinear

2 (0, T], depending only on ku

, and a unique

solution u to[10] such that

u ¼ uð; u

Þ2Cð½0; t

Þ; VÞ\C

ð½0; t

Þ; E

Moreover, the mapping u

7!u( , u

) is continuous

from V to C([0, t

), V) \ C

([0, t

), E

There are many applications of Theorem 11 to

different concrete partial differential equations

(PDEs), including symmetric hyperbolic first-

order systems, the Korteweg–de Vries equation,

nonlinear elastodynamics, quasilinear wave equa-

tions, Navier–Stokes and Euler equations, and

coupled Maxwell–Dirac equations. We decided

toexplaininsomedetailanapplicationtothe

so-called periodic Camassa–Holm equation:

 u

xxt

þ 3uu

¼ 2u

þ uu

xxx

t > 0; x 2 S

½11

where S

stands for the unit circle. In the above

model, the function u is the height of a unilinear

water wave over a flat bottom.

Set X := L

),V := H

), and Q := (I  @

)

1=2

With y := u  u

, eqn [11] can be re-expressed as

þðQ

2

Þy

¼2yðQ

2

yÞ

in L

ðS

which is of type [10] with

AðyÞ¼ðQ

2

yÞ@

; f ðyÞ¼2yðQ

2

yÞ

; y 2 V

where dom(A(y)) := {v 2 L

); (Q

2

y)v 2 H

)}.

Quasilinear Evolution Equations of Parabolic Type

Assume that E

and E

are Banach spaces such that

is dense and continuously injected in E

. More-

over, let ( , )

for each  2 (0, 1) be an admissible

interpolation functor (e.g., the real or complex

interpolation functor) and set E



:= (E

, E

)



for  2

(0, 1). Given a subset X  E



for some  2 (0, 1), we

set X



:= X \ E



for  2 [0, 1], equipped with the

topology induced by E



. Finally, we write C

1

(M, N)

for the class of all locally Lipschitz continuous

functions mapping the metric space M into the

metric space N.

Theorem 12 (Amann). Suppo se that 0 < <

<1, that X



is open in E



, and that

ðA; f Þ2C

1

ð½0; TX



; HðE

; E

ÞE



Then, given u

2 X



, there exists a unique maximal

2 (0, T], such that the quasilinear parabolic

Cauchy problem

ðtÞ¼Aðt;uðtÞÞuðtÞþf ðt;uðtÞÞ; t 2ð0;T; uð0Þ¼u

possesses a unique classical solution

u :¼ uð; u

Þ2Cð½0; t

Þ; X



Þ\C

ðð0; t

Þ; E

Assume A and f are independent of t and let u( , u

) be

the solution to corresponding autonomous problem

ðtÞ¼AðuðtÞÞuðtÞþf ðuðtÞÞ; t 2ð0; 1Þ; uð0Þ¼u

Then the mapping (t, u

) 7!u(t, u

) is a semiflow

on X



Due to its clarity and flexibility, Theorem 12 has

found a plethora of applications, which we cannot

discuss in detail here. Let us at least mention the

following: reaction–diffusion systems, population

dynamics, phase transition models, flows through

porous media, Stefan problems, and nonlinear and

dynamic boundary conditions in boundary-value

problems. In addition, many geometric evolution

equations fall into the scope of Theorem 12. Consider,

for example, the volume-preserving gradient flow of

the area functional of a compact hypersurface M in

mþ1

with respect to L

(M)andW

1

(M), respectively.

These flows are known as the averaged mean curvature

flow and the surface diffusion flow, respectively, and

have been investigated on the basis of Theorem 12.

Fully Nonlinear Evolution Equations

of Parabolic Type

Based on the theory of maximal regularity for linear

evolution equations, it is possible to investigate

abstract fully nonlinear parabolic problems of type

[9]. As there are different techniques of maximal

regularity, there are also different approaches to [9].

We present here a result which uses maximal

regularity properties in singular Ho¨ lder spaces C



Let E

and E

be Banach spaces such that E

continuously embedded into E

(density of E

in E

is not needed here). As before, V is an open subset of

and D

F stands for the Fre´chet derivative of

F(t, v ) with respect to the second variable.

Theorem 13 (Lunardi). Assume that F 2 C

([0, T] V, E

) such that D

F 2 C

([0, T] 

V, H(E

, E

)). Then, given u

2 V, there is a max-

imal t

2 (0, T] such that problem [9] has a solution

u 2 C([0, t

), E

) \ C

([0, t

), E

). This solution is

unique in the class

[

0<< 1



ðð0; t

 "; E

Þ\Cð½0; t

 "; E

for each " 2 (0, t

Theorem 13 has important applications to problems

for which the hypotheses of Theorem 12 (in particular

the assumption on the quasilinear structure) are not

Evolution Equations: Linear and Nonlinear 271

satisfied. We mention here fully nonlinear second-order

boundary-value problems, Hele–Shaw models, models

from combustion theory, and Bellman equations.

See also: Boltzmann Equation (Classical and Quantum);

Breaking Water Waves; Dissipative Dynamical Systems

of Infinite Dimension; Elliptic Differential Equations:

Linear Theory; Ginzburg–Landau Equation; Image

Processing: Mathematics; Incompressible Euler

Equations: Mathematical Theory; Nonlinear Schro

dinger

Equations; Partial Differential Equations: Some

Examples; Quantum Dynamical Semigroups; Relativistic

Wave Equations Including Higher Spin Fields; Semilinear

Wave Equations; Separation of Variables for Differential

Equations; Singularities of the Ricci Flow; Symmetric

Hyperbolic Systems and Shock Waves; Wave Equations

and Diffraction.

Further Reading

Amann H (1993) Nonhomogeneous linear and quasilinear elliptic

and parabolic boundary value problems. Teubner-Texte Math

133: 9–126.

Amann H (1995) Linear and Quasilinear Parabolic Problems,

vol. I. Basel: Birkha¨ user.

Arendt W, Bre´zis H, and Pierre M (eds.) (2004) Nonlinear

Evolution Equations and Related Topics. Basel: Birkha¨user.

Constantin A and Escher J (1998) Well-posedness, global existence,

and blowup phenomena for a periodic quasi-linear hyperbolic

equation. Communications on Pure and Applied Mathematics

LI: 443–472.

Da Prato G and Grisvard P (1979) Equations d’e´volution

abstraites nonline´aires de type parabolique. Annali di Mate-

matica Pura ed Applicata 120: 329–396.

Dore G and Venni A (1987) On the closedness of the sum of two

closed operators. Mathematische Zeitschrift 196: 189–201.

Escher J and Simonett G (1999) Moving Surfaces and Abstract

Evolution Equations, Progress in Nonlinear Differential

Equations and Their Application, vol. 35, pp. 183–212.

Basel: Birkha¨user.

Fattorini HO (1983) The Cauchy Problem. Reading: Addison-

Wesley.

Kato T (1993) Abstract Evolution Equations, Linear and Quasi-

linear, Revisited, Lecture Notes in Mathematics, vol. 1540,

pp. 103–125. Berlin: Springer.

Lunardi A (1991) Analytic Semigroups and Optimal Regularity in

Parabolic Equations. Basel: Birkha¨user.

Lunardi A (2004) Nonlinear Parabolic Equations and Systems,

Handbook of Differential Equations, vol. 1, pp. 385–436.

Amsterdam: North-Holland.

Pazy A (1983) Semigroups of Linear Operators and Applications

to Partial Differential Equations. Berlin: Springer.

Tanabe H (1979) Equations of Evolutions. London: Pitman.

Exact Renormalization Group

P K Mitter, Universite

de Montpellier 2, Montpellier,

France

Introduction

The renormalization group (RG) in its modern form

was invented by K G Wilson in the context of

statistical mechanics and Euclidean quantum field

theory (EQFT). It offers the deepest understanding of

renormalization in quantum field theory (QFT) by

connecting EQFT with the the theory of second-order

phase transition and associated critical phenomena.

Thermodynamic functions of many statistical mechan-

ical models (the prototype being the Ising model in two

or more dimensions) exhibit power-like singularities as

the temperature approaches a critical value. One of the

major triumphs of the Wilson RG was the prediction

of the exponents (known as critical exponents)

associated to these singularities. Wilson’s fundamental

contribution was to realize that many length scales

begin to cooperate as one approaches criticality and

that one should disentangle them and treat them one at

a time. This leads to an iterative procedure known

as the ‘‘renormalization group.’’ Singularities and

critical exponents then arise from a limiting process.

Ultraviolet singularities of field theory can also

be understood in the same way. Wilson reviews this

(Wilson and Kogut 1974) and gives the historical

genesis of his ideas (Wilson 1983).

The early work in the subject was heuristic, in the

sense that clever but uncontrolled approximations

were made to the exact equations often with much

success. Subsequently, authors with mathematical bent

began to use the underlying ideas to prove theorems.

Benfatto, Cassandro, Gallavotti, Nicolo, Olivieri et al.

pioneered the rigorous use of Wilson’s renormalization

group in the construction of super-renormalizable

QFTs, (see Benfatto and Gallavotti (1995) and

references therein). The subject saw further mathema-

tical development in the work of Gawedzki and

Kupiainen (1984, 1986) and that of Bałaban (1982),

and references therein. Bałaban in a series of papers

ending in Bałaban (1989) proved a basic result on the

continuum limit of Wilson’s lattice gauge theory.

Brydges and Yau (1990) simplified the mathematical

treatment of the renormalization group for a class of

models and this has led to further systemization and

simplification in the work of Brydges et al. (1998,

2003). Another method which has been intensely

developed during the same historical period is based on

phase cell expansions: Feldman, Magnen, Rivasseau,

and Se´ne´or developed the early phase cell ideas of

272 Exact Renormalization Group

Glimm and Jaffe and were able to prove independently

many of the results cited earlier (see Rivasseau (1991)

and references therein). Although these methods share

many features of the Wilson RG, they are different in

methodology and thus remain outside of the purview

of the present exposition.

A somewhat different line of development has been

the use of the RG to give simple proofs of perturbative

renormalizability of various QFTs: Gallavotti

and Nicolo, via iterative methods (see Benfatto and

Gallavotti (1995) and references therein), and

Polchinski (1984), who exploited a continuous version

of the RG for which Wilson (1974) had derived a

nonlinear differential equation. These early works

were devoted to the standard (

)

scalar field theory,

but subsequently Polchinski’s work has been extended

to a large class of models, including four-dimensional

nonabelian gauge theories (see Kopper and Muller

(2000) and references therein).

Finally, it should be mentioned that apart from

QFT and statistical mechanics, the RG method has

proved fruitful in other domains. An example is the

study of interacting fermion systems in condensed

matter physics (see Fermionic Systems and Renor-

malization: Statistical Mechanics and Condensed

Matter). In the rest of this article, our focus will be

on EQFT and statistical mechanics.

The RG as a Discrete Semigroup

We will first define a discrete version of the RG and

consider its continuous version later. As we will see,

the RG is really a semigroup, so calling it a group is

a misnomer.

Let  be a Gaussian random field (see, e.g., Gelfand

and Vilenkin (1964) for a discussion of random fields)

in R

. Associated to it there is a positive-definite

function which is identified as its covariance. In QFT

one is interested in the covariance

EððxÞðyÞÞ ¼ const: jx  yj

2½

dp e

ip:ðxyÞ

jpj

d2½

½1

Here [] > 0 is the (canonical) dimension of the

field, which for the standard massless free field is

[] = (d  2)=2. The latter is positive for d > 2.

However, other choices are possible but in EQFT

they are restricted by the Osterwalder–Schrader

positivity. It is assured if [] = (d  )=2, with

0 < 2. If <2, we get a generalized free field.

Observe that the covariance is singular for x = y and

this singularity is responsible for the ultraviolet

divergences of QFT. This singularity has to be initially

cut off and there are many ways to do this. A simple

way is as follows. Let u(x) be a smooth, rotationally

invariant, positive-definite function of fast decrease.

Examples of such functions are legion. Observe that

jx  yj

2

¼ const:

2½

x  y



½2

as can be seen by scaling in l. We define the unit

ultraviolet cutoff covariance C by cutting off at the

lower end point of the l integration (responsible for

the singularity at x = y)atl = 1,

Cðx  yÞ¼

2½

x  y



½3

C(x  y) is positive-definite and everywhere smooth.

Being positive-definite, it qualifies as the covariance

of a Gaussian probability measure denoted 

on a

function space  (which it is not necessary to specify

any further). The covariance C being smooth implies

that the sample fields of the measure are 

almost

everywhere sufficiently differentiable.

Remark Note that, more generally, we could have

cut off the lower end point singularity in [1] at any

>0. The -cutoff covariance is related to the unit

cutoff covariance by a scale transformation (defined

below) and we will exploit this relation later.

Let L > 1 be any real number. We define a scale

transformation S

on fields  by

ðxÞ¼L

½





½4

on covariances by

Cðx  yÞ¼L

2½

x  y



½5

and on functions of fields F()by

FðÞ¼FðS

Þ½6

The scale transformations form a multiplicative

group: S

= S

Now define a fluctuation covariance 



ðx  yÞ¼

2½

x  y



½7



(x  y) is smooth, positive-definite and of fast

decrease on scale L. It generates a key scaling

decomposition

Cðx  yÞ¼

ðx  yÞþS

Cðx  yÞ½8

Iterating this, we get

Cðx  yÞ¼

n¼0



ðx  yÞ½9

Exact Renormalization Group 273

where



ðx  yÞ¼S



ðx  yÞ¼L

2n½



x  y



½10

The functions 

(x  y) are of fast decrease on scale

nþ1

Thus, [9] achieves the decomposition into a sum

over increasing length scales as desired. Being

positive definite, 

qualify as covariances of

Gaussian probability measures, and therefore



n = 0





. Correspondingly introduce a family

of independent Gaussian random fields 

, called

fluctuation fields, distributed according to 



. Then

 ¼

n¼0



½11

Note that the fluctuation fields 

are slowly varying

over length scales L

. In fact, an easy estimate using

a Tchebycheff inequality shows that, for any >0,

jx  yjL

) 

j

ðxÞ

ðyÞj  ðÞconst:

2

½12

which reveals the slowly varying nature of 

scale L

. Equation [11] is an example of a multiscale

decomposition of a Gaussian random field.

The above implies that the 

integral of a function

can be written as a multiple integral over the fields 

We calculate it by integrating out the fluctuation fields



step by step, going from shorter to longer length

scales. This can be accomplished by the iteration of a

single transformation T

, a renormalization group trans-

formation, as follows. Let F() be a function of fields.

Then we define a RG transformation F ! T

F by

ðT

FÞðÞ¼S





 FðÞ

d



ðÞFð þ S

Þ½13

Thus the renormalization group transformation

consists of a convolution with the fluctuation

measure followed by a rescaling.

Semigroup Property

The discrete RG transformations form a semigroup:

¼ T

nþ1

for all n  0 ½14

To prove this, we must first see how scaling

commutes with convolution with a measure. We

have the property





 S

F ¼ S





F ½15

To see this, observe first that if  is a Gaussian

random field distributed with covariance 

then the

Gaussian field S

 is distributed according to S



This can be checked by computing the covariance of

. Now the left-hand side of [15] is just the

integral of F(S

 þ S

) with respect to d



().

By the previous observation, this is the integral of

F( þ S

) with respect to d



(), and the latter is

the right-hand side of [15]. Now we can check the

semigroup property trivially:

F ¼ S





 S





 F

¼ S





 



 F

¼ S

nþ1





þS



 F

¼ S

nþ1





nþ1

 F

¼ T

nþ1 F ½16

We have used the fact that 

þ S



=

nþ1 . This

is because S



has the representation [7] with

integration interval changed to [L

, L

nþ1

We note some properties of T

. T

has an unique

invariant measure, namely 

: for any bounded

function F,

d

F ¼

d

F ½17

To understand [17], recall the earlier observation

that if  is distributed according to the covariance C,

then S

 is distributed according to S

C.By[8],



þ S

C = C. Therefore,

d

F ¼

d





 F

d





 F

d

F ½18

The uniqueness of the invariant measure follows

from the fact that the semigroup T

is realized by a

convolution with a probability measure and, there-

fore, is positivity improving:

F  0;

a:e: ) T

F > 0;

a:e:

Finally, note that T

is a contraction semigroup

on L

(d

) for 1  p < 1. To see this, note that

since T

is a convolution with a probability measure

F = 

1



 S

F, we have, via Ho¨ lder’s inequal-

ity, jT

 T

jF j

. Then use the fact that 

is an

invariant measure.

Eigenfunctions

Let :p

n, m

:((x)) be a C Wick-ordered local mono-

mial of m fields with n derivatives. Define

n;m

ðXÞ¼

dx: p

n;m

ðxÞ

274 Exact Renormalization Group

The functions P

n, m

(X) play the role of eigenfunc-

tions of the RG transformation T

up to a scaling of

volume:

n;m

ðXÞ¼L

dm½n

n;m

ðL

1

XÞ½19

Because of the scaling in volume, P

n, m

(X) are not

true eigenfunctions. Nevertheless, they are very

useful because they play an important role in the

analysis of the evolution of the dynamical system

which we will later associate with T

. They are

classified as expanding (relevant), contracting

(irrelevant) or central (marginal), depending on

whether the exponent of L on the right-hand side

of [19] is positive, negative, or zero, respectively.

This depends, of course, on the space dimension

d and the field dimension [].

Gaussian measures are of limited interest. But we

can create new measures by perturbing the Gaus-

sian measure 

with local interactions. We cannot

study directly the situation where the interactions

are in infinite volume. Instead, we put them in a

very large volume which will eventually go to

infinity. We have a ratio of two length scales, one

from the size of the diameter of the volume and the

other from the ultraviolet cutoff in 

,andthis

ratio is enormous. The RG is useful whenever there

are two length scales whose ratio is very large. It

permits us to do a scale-by-scale analysis and at

each step the volume is reduced at the cost of

changing the interactions. The largeness of the ratio

is reflected in the large number of steps to be

accomplished, this number tending eventually to

infinity. This large number of steps has to be

controlled mathematically.

Perturbation of the Gaussian Measure

Let 

= [L

=2, L

=2]

 R

be a large cube in

. For any X  

,letV

(X, )bealocal

semibounded function where the fields are

restricted to the set X. Here ‘‘local’’ means that if

X, Y are sets with disjoint interiors then V

(X [

Y, ) = V

(X) þ V

(Y). Consider the integral

(known as the partition function in QFT and

statistical mechanics)

Zð

Þ¼

d

ðÞz

ð

;Þ½20

where

ðX;Þ¼e

V

ðX;Þ

½21

and

d

ð0Þ

ð

;Þ¼

Zð

d

ðÞe

V

ð

;Þ

½22

is the corresponding probability measure. V

typically not quadratic in the fields and therefore

leads to a non-Gaussian perturbation. For example,

ðX;Þ¼

dxðjr ðxÞj

þ g



ðxÞþ



ðxÞÞ ½23

where we take g

> 0. The integral [20] is well

defined because the sample fields are smooth.

We now proceed to the scale-by-scale analysis

mentioned earlier. Because 

is an invariant

measure of T

, we have the partition function

Z(

) in the volume 

Zð

Þ¼

d

ðÞz

ð

;Þ

d

ðÞT

ð

;Þ½24

The integrand on the right-hand side is a new

function of fields which, because of the final scaling,

live in the smaller volume 

N1

. This leads to the

following definition:

ð

N1

;Þ¼T

ð

;Þ½25

Because V

is local, z

has a factorization property for

unions of sets with disjoint interiors. This is no longer

the case for z

. Wilson noted that, nevertheless, the

integral is well approximated by an integrand which

does, but the approximator has new coupling con-

stants. The phrase ‘‘well approximated’’ is what all the

rigorous work is about and this was not evident in the

early Wilson era. The idea is to extract out a local part

and also consider the remainder. The local part leads

to a flow of coupling constants and the (unexponen-

tiated) remainder is an irrelevant term. This operation

and its mathematical control is an essential feature of

RG analysis.

Iterating the above transformation, we get, for all

0  n  N,

nþ1

ð

Nn1

;Þ¼T

ð

Nn

;Þ½26

After N iterations, we get

Zð

Þ¼

d

ðÞz

ð

;Þ½27

where 

is the unit cube. To take the limit as

N !1, we have to control the infinite sequence

of iterations. We cannot hope to control the

infinite sequence at the level of the entire partition

function. Instead, one chooses representative coor-

dinates for which the infinite sequence has a

chance of having a meaning. The coordinates are

provided by the coupling constants of the

extracted local part and the irrelevant terms (an

Exact Renormalization Group 275