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Then it can be proved that the scale space is the

unique solution of the heat equation :

 u ¼ 0

uð0; xÞ¼f ðxÞ

½12

Figure 4 is an example of [12] applied to a noisy

image at different scale, that is, at different time.

Note that noise is quickly removed but one has to

stop the evolution very early if we would like to

preserve some edges. In the nonlinear cases, several

operators have also been found based on curvature.

For instance, under suitable axioms (Alvarez et al.

1993), including contrast, scale, and affine invari-

ance, the associated scale space is

 signðÞðtÞ

1=3

jruj¼0

where  ¼ div

jruj



uð0; xÞ¼f ðxÞ

½13

This equation is called affine morphological scale

space (AMSS) and three restored images are shown

in Figure 5. Some qualitative differences are shown

in Figure 6.

Remark Scale space theory has shown the formal

link between some operators and PDEs. It has to be

noticed that one may propose some PDEs which do

not directly come from the scale space framework.

Starting from [12] which performs isotropic smoot h-

ing and smears edges, many nonlinear diffusion

models have been proposed to smooth images while

preserving edges (see e.g., Perona and Malik

(1990)).

To know more on scale space and PDEs, we refer

the reader to Weickert (1998) and Aubert and

Kornprobst (2002).

The Wavelet Approach

Before the 1980s , the Fourier transform played a

major role for analyzing osci llating signals. The

interest of such a transform for real application

increased after the discovery of the fast Four ier

transform. However, the Fourier transform has

some limit. The Fourier transform extracts from

the signal details of the frequency content but loses

all information on the location of particular fre-

quency. Moreover, for computing the Fourier trans-

form Ff (), we need to know f(t) for all the real

values of t. These difficulties can be overcome by

first windowing the signal, and then by taking its

Fourier transform:

win

f ð; tÞ¼

f ðsÞgðs  tÞe

is

where g is a window function. The parameter 

plays the role of a freque ncy localized around the

abscissa t of the temporal signal and F

win

f (, t) give

an information about what is happening around

s = t, for the frequency . The main drawback of

this method is that the window has a fixed length

which is a serious disadvantage when we want to

treat signals having variations of different orders of

magnitude. All these issues highlighted that a

mathematical theory of time–frequency representa-

tion was necessary. This was achieved with the

wavelet representation. In this section, we first recall

some elements of this theory (for 1D signal) and

then we show how it can be applied for restoring

noisy images.

The Wavelet Decomposition

The basic idea is to construct from a function ,

called mother wavelet, an orthonormal basis {

j, k

}of

(R) deduced from by translation and dilatation.

It is required that be regular, oscillating (but not

too much), that and F are well localized and that

has some null moments. Once this function is

Original image

40 iterations 90 iterations 150 iterations

Figure 4 Illustration of heat equation [12].

Original image 40 iterations 90 iterations 150 iterations

Figure 5 Illustration of the AMSS model [13].

Heat AMSS Heat AMSS

Figure 6 Some close-ups of Figures 4 and 5 showing

qualitative differences after 40 iterations.

Image Processing: Mathematics 7

chosen, we set

j, k

(x) = 2

j=2

t  k), j, k 2Z.An

elegant and practical way for obtaining such a basis is

to construct a multiresolution analysis of L

(R)

(Mallat 1989).

Definition 4 A multiresolution analysis of L

(R)is

a sequence V

, j 2Z of subspaces of L

(R), with the

following properties:

(i)

= {0},

(ii) V

 V

jþ1

(iii)

= L

(R),

(iv) f (t) 2V

if and only if f (2t) 2V

jþ1

, and

(v) There exists a regular function  with compact

support such that the family (t  k), k 2Z,is

an orthonormal basis of V

for the scalar

product of L

(R). Such a function  is called a

scaling function.

Then it is straightforward to check that the family



j, k

(t) defined by 

j, k

(t) = 2

j=2

(2

t  k) is an ortho-

normal basis of V

A basic example of multiresolution analysis of

(R) is to choose V

as the set of piecewise

constant functions on R and take  as the

characteristic function of the interval [0, 1):

(t) = 

[0, 1)

(t).

Let us now look at the link between wavelet basis

and multiresolution analysis. We just give main

ideas, all details can be found in the work of Mallat

(1989). Assume that we have a multire solution

analysis, and let us define W

as the orthogonal

complement of V

in V

.Webuildthemother

wavelet by imposing that the family (t  k),

k 2Z, is an orthonormal basis of W

. For example,

if (t) = 

[0, 1)

(t), it can be shown that (t) =



[0, 1=2)

(t)  

[1=2, 1)

(t) (called the Haar wavelet). B y

change of scal e, one get s that the family

j, k

(t) = 2

j=2

t  k), k 2Z,isanorthonormal

basis of W

, the orthogonal complement of V

jþ1

,thatis,

W

¼ V

jþ1

½14

Since the V

’s are a multiresolution analysis, we have

= 

J1

j = 1

and L

= 

j = þ1

j = 1

. It is then clear

that

j, k

(t) is an orthonormal basis of L

(R), that is,

for each function f 2L

(R), we get the following

decomposition:

f ðtÞ¼

þ1

1

j;k

ðtÞ with f

j;k

¼hf ;

j;k

Let us see now how in practice a multiresolution

analysis can be inte rpreted. Let f be a function in

(R). We denote A

j f (resp. D

j f ) the operator

which approximates f (resp. gives the details of f )at

resolution 2

. More precisely, A

j f (resp. D

j f ) is the

projection of f on V

(resp. on W

j f ðtÞ¼

k¼þ1

k¼1

hf ;

j;k

i

j;k

ðtÞ

j f is characterized by the sequence of scalar

products A

f = {hf , 

j, k

k 2Z.

We call A

f the

discrete approximation of f at resolution 2

In the same way, we have

j f ðtÞ¼

k¼þ1

k¼1

hf ;

j;k

ðtÞ

j f is characterized by the sequence of scalar

products D

f = {hf ,

j, k

k 2Z.

We call D

f the details of f at resolution 2

According to [14], approximation and detail are

linked by the relation

jþ1 f ¼ A

f þ D

This means that D

j f represents the details to be

added to obtain from one level of approximation to

the next level of approximation.

Finally, the decomposition of a signal f on a

wavelet basis is obtain ed as an accumulation of

details at scale 2

from 0 to þ1:

f ¼

j¼þ1

j¼1

j f ¼

j¼þ1

j¼1

k¼þ1

k¼1

hf ;

j;k

½15

Instead of considering the sum over all dyadic

levels j, one can sum over j  J for a fixed J; in this

case, we have

f ¼

k¼þ1

k¼1

jJ

hf ;

j;k

k¼þ1

k¼1

hf ;

J;k

i

J;k

We conclude this section by showing how we can

construct a 2D wavelet basis from the 1D case. We

can simply use a tensor product. Scaling function

and mother wavelet are given, respectively, as

follows:

ðx; yÞ¼ðxÞðyÞ;  ¼ð

;

with

ðx; yÞ¼ðxÞ ðyÞ

ðx; yÞ¼ðyÞ ðxÞ

ðx; yÞ¼ ðxÞ ðyÞ

As for the 1D case, A

j f denotes the projection of

f on V

, D

the horizontal details, D

the vertical

8 Image Processing: Mathematics

details, and D

the other details (the indice l in D

is the same as in

). For a 2D image f, we then have

the following decomposition (see Figure 7):

f ¼

2

k¼þ1

k¼1

jJ

hf ;

j;k

k¼þ1

k¼1

hf ;

J;k

i

J;k

Application to the Denoising Problem

We go back to the denoising problem. Our goal is to

solve this problem by using a variational approach

and wavelets. We recall that we have an ideal image

u that has been corrupted by a white Gaussian noise

 resul ting in an obs ervation f with f = u þ .Asit

has been seen in the sect ion ‘‘The vari ational

appro ach,’’ this question can be tackl ed by solving

the variational problem

min

ðjuj

Þþjf  uj



½16

for suitable choices of E, G,and. Here we propose to

choose G = L

()( is the domain image) and for E

the Besov space B

()) and  = Identity. Besov

spaces B



()) are used in many domains of

mathematics as harmonic analysis or approximation

theory. There exist different ways for defining them.

Roughly speaking, they consist of functions having 

derivatives in L

(); the third parameter q allows one

to make finer distinctions in smoothness. Here we are

only concerned with the Besov space B

()). One

important property needed here is that the norm of a

function in E = B

()) is equivalent to the l

-norm

of the wavelet coefficients, that is if {

j, k

}isan

orthonormal basis of L

()andifu

j, k,

are the wavelet

coefficients of u 2E,thenjuj

j, k,

Remark When one is concerned with a finite

domain, then some changes must be made with

respect to the construction given in [15] to obtain an

orthonormal basis of L

(). To avoid further

technical complications , we ignore this question.

Let us denote, respectively, by {u

j, k,

} and {f

j, k,

}

the wavelet coefficients of u and f, then solving [16]

amounts to finding the minimizer of the functional

FðuÞ¼

j;k;

jþ

j;k;

 f

j;k;

½17

One notes immediat ely that mi nimizing problem

[17] reduces to finding the minimizer s, given t,of

E(s) = js  tj

þ jsj and that the minimizer of E(s)is

given by s = t  (=2) if t >=2, s = 0ifjtj=2

and s = t þ (=2) if t < (=2).

Thus, we shrink the wavelet coefficients f

j, k,

toward zero by an amount of =2 to obtain the

minimizer. This is exactly the wavelet shrinkage

algorithm of Donoho and Johnstone (1994).Itis

remarkable that the wavelet shrinkage algorithm,

which has been found by using statistical tools, can

also be explained via a variational approach

(Chambolle et al. 1998). Figure 8 shows an example

of the result on a noisy image.

For more details, we refer the reader to Mallat

(1998).

Conclusion

Image processing is a challenging domain of applied

mathematics which has to deal with discrete and

continuous representations. In this article, we have

covered the core mathematical tools used in the

area. The example of gray-scale image restoration

allowed us to illustrate and compare the different

methodologies. Naturally, as mentioned in the

introduction, image processing refers to a wide

variety of applications and an intensive research

has been carri ed out on the different topics using the

methodologies described here. The reader will find

in the references (therein) several illustrations of

challenging problems.

See also: -Convergence and Homogenization; Convex

Analysis and Duality Methods; Elliptic Differential

–1

Figure 7 Illustration on the wavelets methodology.

Original noisy

ima

BV regularization Wavelet

shrinka

Figure 8 Illustration of two regularization methods.

Image Processing: Mathematics 9

Equations: Linear Theory; Evolution Equations: Linear

and Nonlinear; Fluid Mechanics: Numerical Methods;

Fractal Dimensions in Dynamics; Free Interfaces and

Free Discontinuities: Variational Problems; Geometric

Measure Theory; Ginzburg–Landau Equation;

Inequalities in Sobolev Spaces; Minimax Principle in the

Calculus of Variations; Optimal Transportation; Partial

Differential Equations: Some Examples; Stochastic

Differential Equations; Variational Techniques for

Ginzburg–Landau Energies; Wavelets: Applications;

Wavelets: Mathematical Theory.

Further Reading

Alvarez L, Guichard F, Lions P, and Morel J (1993) Axioms and

fundamental equations of image processing. Archive for

Rational Mechanics and Analysis 123(3): 199–257.

Ambrosio L, Fusco N, and Pallara D (2000) Functions of

Bounded Variation and Free Discontinuity Problems, Oxford

Mathematical Monographs. New York: Clarendon Press.

Aubert G and Aujol J (2005) Modeling very oscillating signals –

application to image processing. Applied Mathematics and

Optimization 51(2): 163–182.

Aubert G and Kornprobst P (2002) Mathematical Problems in

Image Processing: Partial Differential Equations and the

Calculus of Variations, Applied Mathematical Sciences,

vol. 147. New York: Springer.

Besag J (1974) Spatial interaction and the statistical analysis of

lattice systems (with discussion). Journal of Royal Statistical

Society 2: 192–236.

Chambolle A, DeVore R, Lee N, and Lucier B (1998) Non-linear

wavelet image processing: variational problems, compression,

and noise removal through wavelet shrinkage. IEEE Transac-

tions on Image Processing 7(3): 319–334.

Chambolle A and Lions P (1997) Image recovery via total

variation minimization and related problems. Nu¨ merische

Mathematik 76(2): 167–188.

Crandall M, Ishii H, and Lions P-L (1992) User’s guide to

viscosity solutions of second order partial differential equa-

tions. Bulletin of the American Society 27: 1–67.

Crandall M and Lions P (1981) Condition d’unicite´ pour les

solutions ge´ne´ralise´es des e´quations de Hamilton–Jacobi du

premier ordre. Comptes Rendus de l’Acade´mie des Sciences

292: 183–186.

Donoho D and Johnstone I (1994) Ideal spatial adaptation by

wavelet shrinkage. Biometrica 81: 425–455.

Geman S and Geman D (1984) Stochastic relaxation, Gibbs

distributions, and the Bayesian restoration of images. IEEE

Transactions on Pattern Analysis and Machine Intelligence

6(6): 721–741.

Gonzalez RC and Woods RE (1992) Digital Image Processing,

3rd edn. Addison-Wesley.

Kirkpatrick S, Gellat C, and Vecchi M (1983) Optimization by

simulated annealing. Science 220: 671–680.

Li S (1995) Markov Random Field Modeling in Computer Vision.

Tokyo: Springer.

Mallat S (1989) A theory for multiresolution signal decomposi-

tion: the wavelet representation. IEEE Transactions on

Pattern Analysis and Machine Intelligence 11(7): 674–693.

Mallat S (1998) A Wavelet Tour of Signal Processing. Cambridge:

Academic Press.

Meyer Y (2001) Oscillating Patterns in Image Processing and

Nonlinear Evolution Equations, University Lecture Series,

vol. 22. Providence, RI: American Mathematical Society.

Perona P and Malik J (1990) Scale-space and edge detection using

anisotropic diffusion. IEEE Transactions on Pattern Analysis

and Machine Intelligence 12(7): 629–639.

Rudin L, Osher S, and Fatemi E (1992) Nonlinear total variation

based noise removal algorithms. Physica D 60: 259–268.

Weickert J (1998) Anisotropic Diffusion in Image Processing.

Stuttgart: Teubner-Verlag.

Incompressible Euler Equations: Mathematical Theory

D Chae, Sungkyunkwan University, Suwon,

South Korea

Introduction

In this article we present comprehensive mathema-

tical results on the incompressible Euler equations.

Our presentation is focussed on the two aspects of

the equations. The first one is on the theories of

classical solutions and the problem of global in time

continuation/finite time blow-up of the local classi-

cal solutions. The second topic is concerned on the

weak solutions, mainly for the two-dimensional

(2D) Euler equations for existence and uniqueness

questions.

The motion of homogeneous incompressible ideal

fluid in a domain   R

is descr ibed by the

following system of Euler equati ons:

þðv rÞv ¼rp ½1

div v ¼ 0 ½2

vðx; 0 Þ¼v

ðxÞ½3

where v = (v

, v

, ..., v

), v

= v

(x, t), j = 1, 2, ..., n,

is the velocity of the fluid flows, p = p(x, t) is the

scalar pressure, and v

(x) is a given initial velocity

field satisfying div v

= 0. Here we use the standard

notion of vector calculus, denoting

10 Incompressible Euler Equations: Mathematical Theory

rp ¼

;

; ...;



ðv rÞv

k¼1

div v ¼

k¼1

Equation [1] represents the balance of momentum

for each portion of fluid, while eqn [2] represents

the conservation of mass of fluid during its motion,

combined with the homogeneity (constant density)

assumption on the fluid. Equations [1] and [2] are

first obtained by Euler in 1755. Although we could

consider, more generally, the inhomogeneous incom-

pressible Euler equations, in mathematical fluid

mechanics considerations the incompressible Euler

equations usually mean the above system [1]–[2].

For a bounded domain with fixed boundary @, the

natural boundary condition is

vðx; tÞðxÞ¼0 8ðx; tÞ2@ ½0; 1Þ ½4

where (x) is the unit normal vector at the boundary

point x 2 @. Several studies are concerned wi th the

Cauchy problem of the system [1]–[3], where we

consider the case

 ¼

ðwhole domain of R

Þ; or

ðperiodic domainÞ

(

½5

In this article for simplicity we suppose

=R

, n = 2, 3 unless otherwise stated. We note

that the Euler equation is obtained formally by

setting the viscosity = 0, or, equivalently, Reynolds

number = 1 in the Navier–Stokes equations. Thus,

we may view the Euler equations as the one

describing approximately the extremely high

Reynolds number turbulent flows. For detailed

mathematical studies on the finite Reynolds number

Navier–Stokes equations, see Temam (1984) and

Lions (1996). For much shorter and more compre-

hensive review see Constantin (1995). In the study of

the Euler equations the notion of vorticity, ! = curl v,

plays a very important role. In particular, we can

reformulate the system in terms of vorticity fields

only as follows. We first suppose we are working in

three-dimensional (3D) space, and rewrite [1] as

 v  curl v ¼r p þ

jvj



½6

Taking curl of [6], and using elementary vector

identities we obtain the following vorticity formulation:

þðv rÞ! ¼ ! rv ½7

div v ¼ 0; curl v ¼ ! ½8

!ðx; 0 Þ¼!

ðxÞ½9

The linear elliptic system [8] for v can be solved

explicitly in terms of ! to give the Biot–Savart law

vðx; tÞ¼

4

ðx  yÞ!ðy; tÞ

jx  yj

dy ½10

Substituting this v into [7] formally, we obtain a

integrodifferential system for !. The term in the

right-hand side of [7] is called the ‘‘vortex stretching

term,’’ and is regar ded as the main source of

difficulties in the mathematical theory of the 3D

Euler equations. In the 2D case we take the vorticity

as the scalar, ! = @v

=@x

 @v

=@x

, and the

evolution equation of ! becomes

þðv rÞ! ¼ 0 ½11

combined with the 2D Biot–Savart law,

vðx; tÞ¼

2

ðy

þ x

; y

 x

jx  yj

!ðy; tÞdy ½12

In many studies of the Euler equations it is

convenient to introduce the notion of ‘‘particle

trajectory mapping,’’ (, t) defined by

@ð; tÞ

¼ vðð; tÞ; tÞ

ð; 0Þ¼;  2 

½13

The mapping (, t) transforms from the location of

the initial fluid particles to the location at time t,

and the parameter  is called the Lagrangian particle

marker. If we denote the Jacobian of the transfor-

mation, det (r



(, t)) = J(, t), then we can show

easily that

¼ðdiv vÞJ

which implies the fact that the velocity field v

satisfies the incompressibility, div v = 0 if and only if

the mapping (, t) is volume preserving. At this

moment, we note that, although the Euler equations

are originally derived by applying the mass con-

servation and the momentum balance principles, we

could also derive them by applying the principle of

least action to the action defined by

AðÞ¼



@ðx; tÞ



dx dt

Here, (, t): ! is a parametrized family of

volume-preserving diffeomorphism. This variational

approach to the Euler equations implies that we can

Incompressible Euler Equations: Mathematical Theory 11

view solutions of the Euler equations as a geodesic

curve in the L

-metric on the infinite-dimensional

manifold of volume-preserving diffeomorphisms (see

for more details, e.g., Arnol’d and Khesin (1998)).

The 3D Euler equations have many conserved

quantities. We list some important ones below.

1. Energy

EðtÞ¼



jvðx; tÞj

dx ½14

2. Helicity

HðtÞ¼



vðx; tÞ!ðx; tÞdx ½15

3. Circulation



CðtÞ

v  dl ½16

where C(t) = {( , t)j 2 C} is the curve moving

along with the fluid.

4. Impulse

IðtÞ¼



x  ! dx ½17

5. Moment of impulse

Mðt Þ¼



x ðx  !Þdx ½18

The proof of conservations of the above quantities

can be carried out without difficulty by using

elementary vector calculus (for details see, e.g.,

Chorin and Marsden (1993), Majda and Bertoz zi

(2002), Marchioro and Pulvirenti (1994)). The

helicity above, in particular, represents the degree

of knotedness of the vortex lines in the fluid, where

the vortex lines are the integral curves of the

vorticity fields. Arnol’d and Khesin (1998) discuss

in detail aspects of helicity and other geometric

aspects of the Euler equations. For the 2D Euler

equations there is no analog of helicity, while the

circulation conservation is replaced by the vorticity

flux integral,

AðtÞ

!ðx; tÞdx ½19

where A(t) = {(, t)j 2 A} is a planar region

moving along the fluid. The impulse and the

moment of impulse integrals are replace by



ðx

; x

Þ! dx ½20a

and





jxj

! dx ½20b

respectively.

In the 2D ideal incompressible fluids we have

extra conserved quantities; namely for any p 2

[1, 1] the integral



j!ðx; tÞj

dx ½21

is conserved (as a matter of fact we can extend this

statement by replacing the integral by



f (!(x, t))dx

for any continuous function f ). There are many

known explicit solutions to the Euler equations (See

e.g., Lamb (1932) and Majda and Bertozzi (2002)).

Local Existence and the Blow-Up

Problem

The Classical Results

We first introd uce some notations of function

spaces. The Lebesgue space L

(), p 2 [1, 1], is the

Banach space defined by the norm

kf k

:¼



jf ðxÞj



1=p

; p 2½1; 1Þ

ess: sup

x2

jf ðxÞj; p ¼1

(

Let us set  := (

, 

, ..., 

) 2 (Z

[ {0})

with

jj= 

þ

þþ

. Then, D



:= D



D



where D

= @=@x

, j = 1, 2, ..., n. For given k 2 Z

and p 2 [1, 1) the Sobolev space, W

k, p

() is the

Banach space of functions consisting of functions

f 2 L

() such that

kf k

k;p

:¼





f ðxÞj



1=p

< 1

where the derivatives are in the sense of distribu-

tions. For p = 1 we replace the L

-norm by the L

norm. In order to cooperate with the fractional

derivatives of order s 2 R, we use the space L

()

defined by the Banach spaces norm,

kf k

s;p

:¼kð1  Þ

s=2

f k

where (1   )

s=2

f = F

1

[(1 þjj

)

s=2

F(f )()] with

F()andF

1

() denoting the Fourier transform

and its inverse. Below we outline the key ideas of

proving the local existenc e theorems for the Euler

equations. For more details we refer the reader to

Majda and Bertozzi (2002). For simplicity, we use

the function space H

) = W

m,2

), n = 2, 3.

Taking derivatives D



on [1], and then taking its

12 Incompressible Euler Equations: Mathematical Theory

inner product with D



v, and summing over the

multi-indices  with jjm, we obtain

kvk

¼

jjm

ðD



ðv rÞv ðv rÞD



v; D



vÞ



jjm

ððv rÞD



v; D



vÞ



jjm

ðD



rp; D



vÞ

I þII þIII

By integration by parts, we obtain

III ¼

jjm

ðD



p; D



div vÞ

¼ 0

Integrating by parts again, and using the fact that

div v = 0, we have

II ¼

jjm

ðv  rÞjD



jjm

div vjD



dx ¼ 0

We now use the so-called commutator type of

estimate,

jjm



ðfgÞfD



 Cðkrf k

kgk

m1

þkf k

kgk

and obtain

I 

jjm



ðv rÞv ðv rÞD



kvk

 Ckrvk

kvk

Summarizing the above estimates, I–III, we have

kvk

 Ckrvk

kvk

½22

Further estimate, using the Sobolev inequality, krvk

 Ckvk

for m > 5=2, gives

kvk

 Ckvk

Thanks to Gronwall’s lemma, we have the local-in-

time uniform estimate

kvðtÞk



1  Ctkv

 2kv

for all t 2 [0, 1=(2Ckv

)]. This is the key a priori

estimate for the construction of the local solutions.

The local-in-time solution of the Euler equations in

the Sobolev space H

) for m > n=2 þ 1, m 2 Z,

was obtained by Kato (1972). For the above-

constructed l ocal-in-time solutions, one of the

most outstanding open problems in mathematical

fluid m echanics is whether the solution can be

continued to any future time up to infinity, or the

solution will lose regul arity and blow up in finite

time. Even in terms of numerical experiments, the

answer is not yet settled down. In the direction of

solving t his problem there is a celebrated results,

called the Beale–Kato–Majda criterion (1984),

which states

lim sup

t%T



kvðtÞk

¼1 if and only if



k!ðsÞk

ds ¼1 ½23

We outline the proof of this result below (for more

details see Majda and Bertozzi (2002)). We first

recall the Beale–Kato–Majda’s version of the loga-

rithmic Sobolev inequality,

krvk

Ck!k

ð1 þlogð1 þkvk

ÞÞþCk!k

½24

for m > 5=2. Now suppose



k!(t)k

dt < 1.

Taking L

inner product of [7] with !, then after

integration by part we obtain

k!k

¼ðð! rÞv;!Þ

k!k

krvk

k!k

¼k!k

k!k

where we used the identity krvk

= k!k

. Apply-

ing the Gronwall lemma, we obtain

k!ðtÞk

k!

exp



k!ðsÞk



 Cð!

; T



Þ½25

for all t 2 [0, T



]. Substituting [24] into [22], and

combining this with [25], we have

kvk

 C 1 þk!k

½1 þ logð1 þkvk

Þ½kvk

Applying the Gronwall’s lemma, we obtain

kvðtÞk

kv

 exp C

exp C



k!ðÞk

d



 Cðv

; T



for all t 2 [0, T



] and for some constants C

, C

Thus, we proved the ‘‘necessity part’’ of [23], The

Incompressible Euler Equations: Mathematical Theory 13

‘‘sufficiency part’’ is an easy consequence of the

Sobolev inequality,



k!ðsÞk

ds  T



sup

0tT



krvðtÞk

 CT



sup

0tT



kvðtÞk

for m > 5=2.

Other Related Results

The previous local existence result in H

), m >

n=2 þ 1, is basically due to T Kato in 1972. He and

G Ponce extended this existence result using the

fractional Sobolev space, L

), s > n=2 þ 1, s 2 R

in 1986. These results were further extended, using

the Besov and the Triebel–Lizorkin spaces, by the

present author in 2001.

For bounded domain   R

, R Temam obtained

the local-existence result using the space W

k, p

()in

1975. On the other hand, in the setting of the

Ho¨ lder space, C

1, 

) L Lichtenstein (1925) and

W Wolibner (1933) obtained local existence of

solutions of the Euler equations. More recently,

J-Y Chemin considered the Zygmund C

), which

is identical to the Ho¨ lder space C

[s], s[s]

) for

noninteger s, where [s] means the largest integer not

greater than s, but is different from C

[s], 0

) for

integer s . He proved, in 1992, local existence of

solutions to the 3D Euler equations in this space in

1992. See Chemin (1998) for details of this proof.

The Beale–Kato–Majda criter ion for the finite-

time blow-up of the classical solutions of the 3D

Euler equations has been refined recently by many

authors; replacing the L

-norm of vorticity !(x, t)

by the weaker BMO (the space of functions with

bounded mean oscillations) norm (H Kozono and

Y Taniuchi, 2000), and by the even weaker Besov

space or Triebel–Lizorkin space norms by the

present author in 2001 (see Triebel (1983) for

more details on those spaces). Here we just note

that these spaces are refinements of the usual

Sobolev spaces. For a bounded domain case, there

is a result by A Ferrari in 1993. The blow-up

problem is still open even in the case of axisym-

metric 3D Euler equations if there is a nonzero swirl

(angular velocity). In this case, the blow-up is

controlled only by the angular component of the

vorticity as shown by the present author (1996). In

the region off the axis, in particular, the axisym-

metric 3D Euler equation has the same form as the

2D Boussinesq equations.

Some researchers also tried to approach to

regularity/singularity problem of the 3D Euler

equations by investigating the geometric structure

of the vortex stretching term, and obtained a

geometric type of blow-up criterion (P Constantin,

C Fefferman, and A Majda, 1996). For more

detailed review of studies in this directi on see

Constantin (1995).

Since the blow-up problem of the 3D Euler

equation itself looks too difficult to solve, it has

also been studied on the simplified model problems.

In 1985, P Constantin, P D Lax, and A Majda

considered the following 1D model problem of the

3D Euler equations:



þðHðÞÞ

¼ 0;ðx; 0Þ¼

ðxÞ

where H() is the Hilber t transform defined by

Hð!Þ¼



1

!ðyÞ

x  y

They proved finite-time blow-up of this model

problem by explicitly obtaining the solution. There

is another, 2D model problem of the 3D Euler

equations, the quasigeostro phic equations,



þðu rÞ ¼ 0

u ¼r

;¼ ð Þ

1=2

ðx; 0Þ¼

ðxÞ

½26

where r

= (@

, @

). Contrary to the above 1D

model equation, this 2D model has real physical

relevance in the atmospheric science, and (x, t)

represents the temperature of the air. The resem-

blance of this equation to the 3D Euler equation

was first observed by P Constantin, A Majda, and

E Tabak in 1994, and they derived the finite blow-

up criterion of the equations. In spite of many

interesting partial results, including the work by

D Cordoba (1998), the blow-up problem of [26] is

still open.

The 2D Euler Equations and the

Weak Solutions

The Case of W

1, p

Weak Solutions

In 2D Euler equations, the problem of global well-

posedness of the classical solutions is settled down.

This is an immediate consequence of the conserva-

tion of k !(t)k

as stated in [21] combined with the

Beale–Kato–Majda criterion [23]. On the other

hand, the notion of weak solutions is not well

understood. A weak solution of the Euler equations

is a singular (nondifferentiable) solution of the

equations. More precisely, by a weak solution of

14 Incompressible Euler Equations: Mathematical Theory

[1]–[2] in   (0, T) we mean a vector field v 2

C([0, T); L

loc

()) satisfying the integral identity:



vðx; tÞ

@ðx; tÞ

dxdt



vðx; 0Þðx; 0Þdx



vðx; tÞvðx;tÞ : rðx;tÞdx dt ¼ 0 ½27a

vðx; tÞr ðx; tÞdx dt ¼ 0 ½27b

for every vector test function  = (

, 

, ..., 

) 2

(  [0, T)) satisfying div  = 0, and for every

scalar test function 2 C

(  [0, T)). Here we

used the notation (u  v)

= u

,andA : B =

i, j = 1

for n  n matrices A and B.We

observe that [27a] and [27b] are obtained by

multiplying  and to [1] and [2], respe ctively,

and integrating by parts. Thus, even the locally

square-integrable vector fields, which are not differ-

entiable in the classical sense, could be solutions of

the Euler equations. For the general 3D Euler

equations, we do not yet have the global existence

theorems for the weak solutions. Actually, it is even

suggested that we need more weaker notion of

solution (the so-called ‘‘measure-valued solutions’’)

to describe generic global solutions for the 3D Euler

equations. For the 2D Euler equations, however, we

have global existence theorems for !

2 L

) \

) for p 2 [1, 1]. This better situation of 2D

Euler equations compared to the 3D case for the

weak solutions is mainly due to the conservation

law of L

-norm described in [21]. Here we present

briefly the existence proof of the weak solutions for

2D Euler equations in the simplest situation. We will

prove the global existence of weak solutions for

2 L

), 1 < p < 1. Let 

(x) = (1="

)(x="),

where  2 C

) is a standard mol lifier, satisfying

  0, supp   {x 2 R

jjxj < 1}, and

 dx = 1.

Let v

be the velocity associated with the initial

vorticity !

, given by the Biot–Savart law [12].

Define the sequence of initial data v

(x) = 



(x) =



(x  y)v

(y)dy. For each v

we have

global-in-time smooth solutions v

(x, t). Moreover,

thanks to [21], we have the following estimate of the

vorticity that is uniform in ":

k!ðtÞ

¼k!

k!

½28

where we used the property of the mollifier in the

second inequality. If we take the (distributional)

derivative of the Biot–Savart law [12], we find

rv = K  ! þ C!, where K(x) is a kernel funct ion

defining a singular integral operator of the convolu-

tion type, and C is a constant vector. The well-

known Calderon–Zygmund inequality implies that

krvk

 C

k!k

½29

Combining [28] and [29] we have

sup

0tT

krv

ðtÞk

 Cðv

Þ; 8T > 0 ½30

namely the sequence {v

} is uniformly bounde d in

(0, T; W

1, p

)). Next, we claim that {v

} satis-

fies the inequality

ðt

Þv

ðt

Þk

3

ðR

 Ckv

 t

j½31

for all t

, t

with 0 < t

 t

< T, where C is an

absolute constant. Here the negative-order Sobolev

space H

m

(), m > 0, is defined as the dual of

(), and can be identified with the space of

functions C

() completed with metric in H

().

Indeed, let  2 C

). Taking L

) inner pro-

duct of [1] with  we have the estimates

ðx; tÞ

 ðxÞdx





ð rÞp



 ðv

rÞv



r dx



ðv

rÞv



kp

ðtÞk

2

krk

þkv

ðtÞk

krk

 Cðkp

ðtÞk

2

þkv

Þkk

½32

where we used the Sobolev inequality krk



Ckk

and the energy equality in the last step.

Since [32] holds for all  2 C

), by taking the

closure of C

)inH

) we obtain

ðtÞ



2

 Ckp

ðtÞk

2

þkv

½33

We now estimate kp

(t)k

2

. Taking the divergence

operation on [1], we have the Poisson equation

p

¼divðv

rv

Let  2 C

), then

p

ðx; tÞðxÞdx



divðv

rv

Þ dx



ðv

rÞv

r dx



ðv

 rÞr  v



kv

ðtÞk

k

k

 Ckv

kk

½34

Incompressible Euler Equations: Mathematical Theory 15

where we used the energy equality [14] and the

Sobolev inequality in the last step. Since [34] holds

for all  2 C

), taking the closure of C

)in

), we obtain

p

ðx; tÞðxÞdx



 Ckv

kk

8 2 H

ðR

Þ½35

Thus,

kp

ðtÞk

4

 Ckv

8t 2½0; TÞ

This provides us with

ðtÞk

2

kD

ðtÞk

4

 Ckp

ðtÞk

4

 Ckv

Combining [33] with [36], we obtain

sup

0tT

ðtÞ



2

 Ckv

Thus, from

ðt

Þv

ðt

Þ¼

ðtÞ

we have

ðt

Þv

ðt

Þk

2

 sup

0tT

ðtÞ



2

 t

 Ckv

 t

Thus, [31] is proved as claimed. Thanks to the

Aubin–Nitsche compactness lemma together with

[30] and [31] we have a subsequence, denoted by the

same notation, {v

}andv in L

(0, T; W

1, p

)) such

that

! v weakly in L

ð0; T; W

1; p

ðR

ÞÞ ½36

and

! v in L

loc

ðR

ð0; TÞÞ ½37

as " !0. We know that as a classical solution each

and v

satisfies

ðx; 0Þ v

ðxÞdx

ð

 v

þr : v

 v

Þdx dt ¼ 0 ½38

for all  2 C

 [0, T)) with div  = 0and

r  v

dx dt ¼ 0 ½39

for all 2 C

 [0, T)). We can check easily that

the convergence [36] and [37] is enough to pass to the

limit " !0in[38] and [39] to obtain the correspond-

ing equations with v

and v

replaced by v and v

Thus, v is a weak solution of the Euler equations with

initial data v

. This completes the outline of the proof

of weak solutions to the 2D Euler equations.

Notes on Further Results

The study of weak solutions of the 2D Euler

equations was initiated by V Yudovich in 1963,

where he proved the existence of weak solutions for

initial data !

2 L

) \ L

). Subsequenthy,

theory of weak solutions has been developed by

studies of the vortex sheet problem due to DiPerna

and Majda in 1987. For the existence of weak

solutions t o the vortex sheet initial data, namely

the existence problem for initial vorticity !

1

) \M(R

), where M(R

)isthespaceof

Radon measures on R

, is still an outstanding open

problem. The main physical motivation of this

problem is to understand the dynamics of vortex

sheets in the 3D turbulence. For this problem

J M D elort proved existence assuming single-

signedness of the initial vortex sheet in 1991. The

proof is simplified by A Majda in 1993, using the

conservation of moment of impulse. The result is

also reproved by L C Evans and S Mu¨ ller in 1994,

using the weak compactness of the Hardy space.

Later in 2001, M C Lopes Filho, H J Nussenzveig

Lopes and Z Xin allowed the change of sign for

initial vortex sheet, but assumed special reflection

symmetry to prove existence of global weak solu-

tions. Related to this problem is the one of

characterizing the precise borderline function space

to which initial data belongs, and above which there

is no concentration phenomenon for weakly approx-

imating sequence of solutions; a recent analysis of

this problem was done by E Tadmor in 2001.

For the uniqueness problem of the weak solutions of

the 2D Euler equations, there are remarkable works by

V Scheffer (1993) and A Shnirelman (1997), where

they constructed explicitly an L

loc

 R) weak

solution starting from zero initial data. Also M Vishik

(1999) extended the uniqueness class of the weak

solutions of the 2D Euler equations, improving

previous work by V Yudovich (1995). The class

found by M Vishik, in particular, includes the BMO.

There is another problem closely related to the weak

solutions of the 2D Euler equations, called the vortex

patch problem. The main question was if there is any

singularity of the boundary of a patch

(t) = {X(, t) j 2 

}, where X(, t) is the particle

trajectory mapping generated by a weak solution

v(x, t), which is evolving from the initial data

(x) = 



(x), the characteristic function of set 

16 Incompressible Euler Equations: Mathematical Theory