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

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nonlinear elasticity for solid–solid phase transforma-

tions based on a huge body of work in the original

literature. The precise references can be found in the

extensive bibliographies of the books and review

articles that are cited in the subsequent section,

in particular in Ball (2004), Bhattacharya (2003),

Dolzmann (2003), James and Hane (2000),and

Mu¨ ller (1999). This article focuses on models for

single crystals; the behavior of polycrystals (which

strongly depends on the amount of symmetry

breaking in the transformation) was studied by

Bhattacharya and Kohn.

The formulation of solid–solid phase transforma-

tions via nonlinear continuum theory goes back to

Ericksen and the analysis via tools in the calculus of

variations was initiated by Ball and James, Chipot

and Kinderlehrer, and Fonseca. The Russian school

developed the theory in linear elasticity in the 1960s,

see Khachaturyan (1983) for a review. A detailed

discussion of the crystallographic and group-theo-

retical aspects is contained in Pitteri and Zanzotto

(2002).

Quasiconvexity was introduced by Morrey (1966)

and his results were extended to Carathe´odory

integrands by Acerbi and Fusco and Marcellini. A

modern treatment including Dacorogna’s relaxation

theorem and a summary of the various notions

of convexity and their properties can be found in

Dacorogna (1989).S

vera´k proved that rank-1

convexity does not imply quasiconvexity for m 3

and Milton modified his example to show that the

rank-1 convex hull of a set can be strictly smaller

than its quasiconvex hull. The explicit characteriza-

tions for nematic elastomers were obtained by

DeSimone and Dolzmann.

Lipschitz solutions to differential inclusions were

constructed by Mu¨ ller and S

vera´k based on Gro-

mov’s concept of convex integration, by Dacorogna

and Marcellini using Baire’s category argument, and

by Kirchheim in the framework of Banach Mazur

games. The structure of solutions of the two-well

problem with finite surface energy was analyzed by

Dolzmann and Mu¨ ller. Young measures (also called

parametrized measures or chattering controls) were

originally introduced as generalized solutions for

optimal control problems which do not admit

classical solutions (Young 1969). Tartar (1979)

introduced Young measures as a fundamental tool

for the analysis of oscillation effects in partial

differential equations and for the passage from

microscopic to macroscopic models. Gradient

Young measures were characterized by Kinder-

lehrer and Pedregal. The four-point configuration

was discovered independently in various contexts by

several authors including Scheffer, Aumann and

Hart, Casadio Tarabusi, Tartar, and Milton and

Nesi. The characterization of the quasiconvex hull

uses a quasiconvex function constructed by S

vera´k.

The quasiconvex hull of the two-well problem in 3D

was found by Ball and James, and the generalization

to n wells in 2D by Bhattacharya and Dolzmann.

Acknowledgments

The work of G Dolzmann was supported by the

NSF through grants DMS0405853 and

DMS0104118.

See also: Gamma-Convergence and Homogenization;

Variational Techniques for Ginzburg–Landau Energies.

Further Reading

Ball JM (2004) Mathematical models of martensitic microstruc-

ture. Materials Science and Engineering A 378(1–2): 61–69.

Bhattacharya K (2003) Microstructure of Martensite. Oxford:

Oxford University Press.

Dacorogna B (1989) Direct Methods in the Calculus of Varia-

tions. Berlin: Springer.

Dolzmann G (2003) Variational Methods for Crystalline Micro-

structure: Analysis and Computation, Lecture Notes in

Mathematics, vol. 1803. Berlin: Springer.

James RD and Hane KF (2000) Martensitic transformations and

shape-memory materials. Acta Materiali 48: 197–222.

Khachaturyan A (1983) Theory of Structural Transformations in

Solids. New York: Wiley.

Morrey CB (1966) Multiple Integrals in the Calculus of

Variations. Berlin: Springer.

Mu¨ ller S (1999) Variational methods for microstructure and

phase transitions. Proc. C.I.M.E. Summer School ‘‘Calculus of

Variations and Geometric Evolution Problems,’’ Cetraro,

1996, Lecture Notes in Mathematics, vol. 1713. Berlin–

Heidelberg: Springer.

Pitteri M and Zanzotto G (2002) Continuum Models for Phase

Transitions and Twinning in Crystals. London: Chapman and

Hall.

Tartar L (1979) Compensated compactess and partial differential

equations. In: Knops R (ed.) Nonlinear Analysis and

Mechanics: Heriot–Watt Symposion, vol. IV. London: Pitman.

Young LC (1969) Lectures on the Calculus of Variations and

Optimal Control Theory. Philadelphia–London–Toronto:

Saunders.

Vertex Operator Algebras see Two-Dimensional Conformal Field Theory and Vertex Operator Algebras

368 Variational Techniques for Microstructures

Viscous Incompressible Fluids: Mathematical Theory

J G Heywood, University of British Columbia,

Vancouver, BC, Canada

Introduction

The Navier–Stokes equations

ðu

þ u ruÞ¼rp þ u þ f ½1

ru ¼ 0 ½ 2

provide the simplest model for the motion of a

viscous incompressible fluid that is consistent with

the principles of mass and momentum conservation,

and with Stokes’ hypothesis that the internal forces

due to viscosity must be invariant with respect to

any superimposed rigid motion of the reference

frame. Despite their simplicity, they seem to govern

the motion of air, water, and many other fluids very

accurately over a wide range of conditions. Thus,

their mathematical theory is central to the rigorous

analysis of many experimental observations, from

the asymptotics of steady wakes and jets, to the

dynamics of convection cells, vortex shedding, and

turbulence. During the last 80 years, a great deal of

progress has been made on both the basic mathe-

matical theory of the equations and on its applica-

tion to the understanding of such phenomena. But

one of the most important matters, that of estimat-

ing the regularity of solutions over long periods of

time, remains a vexing and fascinating challenge.

Such an estimate will almost certainly be needed to

prove the ‘‘global’’ existence of smooth solutions. By

that we mean the existence of smooth solutions of

the initial-value problem over indefinitely long

periods of time without any restriction on the

‘‘size’’ of the data. To date we can prove the

‘‘local’’ existence of smooth solutions, but there

remains a concern that if the data are large,

solutions may develop singularities within a finite

period of time. In fact, there is a great deal more at

issue than this question of existence. A regularity

estimate is required to prove the reliability of the

equations as a predictive model. That is because any

estimate for the continuous dependence of solutions

on the prescribed data for a problem depends upon

a regularity estimate, as do error estimates for

numerical approximations. A global estimate for

the regularity of solutions is also required for a

mathematically rigorous theory of turbulence. In

fact, it may be hoped that the insight which

ultimately yields a global regularity estimate will

also be pivotal to our understanding of turbulence,

perhaps justifying Kolmogorff theory; see Heywood

(2003). In this article we aim to present a relatively

simple approach to the local existence, uniqueness,

and regularity theory for the initial boundary value

problem for the Navier–Stokes equations, and to

discuss some observations that bear on the question

of global regularity. A wider-ranging review of open

problems is given in Heywood (1990), and further

observations concerning the problem of global

regularity are given in Heywood (1994).

Setting the Problem

To focus on core issues, we shall make some

simplifying assumptions. The fluid under considera-

tion will be assumed to completely fill (without free

boundaries or vacuums) a bounded, connected,

time-independent domain   R

, n = 2 or 3, with

smooth boundary @. We are mainly interested in

the three-dimensional case, but comparisons with

the two-dimensional case are illuminating. The R

valued velocity u(x, t) = (u

(x, t), ..., u

(x, t)) and R-

valued pressure p(x, t) are functions of the position

x = (x

, ..., x

) 2  and time t  0. Equation [1] is

an expression of Newton’s second law of motion,

equating mass density times acceleration on the left

with several force densities on the right, due to

pressure and viscosity, and sometimes a prescribed

external force f. Written in full, using the summa-

tion convention over repeated indices, its ith

component is



þ u



¼

þ 

þ f

We will assume the density  and the coefficient of

viscosity  are positive constants.

In this article, we consider the initial boundary

value problem consisting of the equations [1], [2]

together with the initial and boundary conditions

t¼0

¼ u

; u

@

¼ 0 ½3

The initial velocity u

(x) is prescribed. It will be

assumed to possess whatever smoothness is con-

venient, and to satisfy ru

= 0 and u

@

= 0. The

boundary condition is a reasonable one, since fluids

adhere to rigid surfaces.

Notice that a further condition would be needed

to uniquely determine the pressure, since only its

derivatives appear in the problem as posed. We

prefer to do without auxiliary conditions for the

pressure, and to refer to u by itself as a solution of

Viscous Incompressible Fluids: Mathematical Theory 369

the problem provided there exists a scalar function p

which together with u satisfies [1]–[3]. The problem

is said to be uniquely solvable if there is a unique

solution u, in which case the gradient of the pressure

is also uniquely determined, along with the pressure

up to a constant. Notice also that under our

assumptions a potential force like gravity has no

effect on u.Ifu solves the problem in the absence of

such a force, then the inclusion of the force affects

only the pressure, from which the potential must be

subtracted. It turns out that the inclusion of a

prescribed nonpotential force, while complicating

many of the estimates below, does not affect in any

essential way those parts of the theory to be

presented here. Thus, for simplicity, we shall

henceforth assume that f  0.

Reynolds Number

We can make a slight further simplification of eqn [1]

by rescaling, with the objective of setting  = 1, or even

 = 1and = 1. This scaling is not required for the

existence theory we are presenting, but provides an

important insight for the study of stability, bifurcation,

and turbulence. The Reynolds number

R ¼

max jujjj



plays an important role in rescaling. It expresses the

ratio of the inertial to viscous effects. The notation

jj represents a characteristic length, such as the

minimum diameter of a bounded domain. Generally

speaking, a high Reynolds number corresponds to

what is meant by ‘‘large’’ data, and the higher the

Reynolds number the more inclined a flow is to

instability and turbulence, and perhaps to the

development of singularities. However, the size of

the Reynolds number has precise implications only

in comparing ‘‘dynamically similar’’ flows. We say

that two vector fields v(x, t) and u(x, t) are dynami-

cally similar if and only if v(x, t) = u(x=, t=) for

some , , >0. In such a case, if u is defined in

  [0, T), then v will be defined in   [0, T),

where ={x: x 2 }. Furthermore, if u satisfies

the Navier–Stokes equations, then v will satisfy



1

v

þ 

2

v rv

¼rpðx=; t=Þþ

1



v ½4

which has the form of the Navier–Stokes equations if

and only if the coefficients of the two inertial terms

on the left-hand side are equal. That is, if and only if

 ¼  ½5

in which case

þ v rv ¼rq þ v ½6

with

 ¼ = ½7

and q(x, t) = 



1

p(x=, t=). We refer to such u

and v as dynamically similar flows. The relation

[7], that follows from [5], is equivalent to the

equality of the Reynolds numbers for the two

flows,

RðuÞ¼

max jujjj



max jujjj1



¼ RðvÞ

The condition [5] can be satisfied simultaneously

with the condition  = 1. For example, one may

choose  = 1,  = =, and  = =. This achieves a

rescaling of the equation to

þ v rv ¼rq þ v ½8

without changing the domain. Different Reynolds

numbers result from varying the magnitude of the

velocity. In what follows, we will work with the

Navier–Stokes equation in this simplest possible

form.

Continuous Dependence on the Data

We begin our investigation of the initial boundary

value problem

þ u ru ¼rp þ u; ru ¼ 0

for x; tðÞ2  0; 1ðÞ,

t¼0

¼ u

; uj

@

¼ 0

½9

by considering two smooth solutions, say u and v,

taking possibly different initial values u

and v

. Let

their difference be w = v  u, with initial value w

and let q be the difference of the corresponding

pressures. Then, subtracting one equation from the

other, one obtains

þw rw þ u rw þ w ru ¼rq þ w ½10

Multiplying this by w, integrating over ,and

integrating by parts, one then obtains

þrw

¼w ru; wðÞ½11

370 Viscous Incompressible Fluids: Mathematical Theory

where



rwkk



w ru; wðÞ¼



since (and this should further explain our notation)

; wðÞ¼



 w dx



dx ¼

wkk

w; wðÞ¼



¼



dx ¼ rw

rq; wðÞ¼



dx ¼



dx ¼ 0

u rw; wðÞ¼



¼



dx ¼ 0

and similarly (w rw, w) = 0. In deriving these we

have used the fact that the vector fields are

divergence free and vanish on the boundary. In the

following, we will use such identities without further

mention.

We can estimate the nonlinear term on the right-

hand side of [11] by using the ‘‘Sobolev inequalities’’



 

r

; if n ¼ 2

kk

 kk

1=2

rkk

3=2

; if n ¼ 3

½12

proved by Ladyzhenskaya (1969), though with

larger constants. These are valid for any smooth

function  which vanishes on the boundary of .It

may be either scalar or vector valued. The norms on

the left are L

-norms; we use the notation



= (





dx)

1=p

for any p > 1, but usually

drop the subscript when p = 2. Using first Ho¨ lder’s

inequality and then [12], one obtains

w ru; wðÞ

 w



if n ¼ 2

1=2

3=2

if n ¼ 3

(

Young’s inequality

ab 

holds if a, b > 0, p, q > 1 and 1=p þ 1=q = 1. Taking

a =

ﬃﬃﬃ

, along with p = q = 2 in the two-

dimensional case, and a = (4=3)

3=4

rwkk, along

with p = 4=3, q = 4 in the three-dimensional case,

one obtains

w ru; wðÞ



; if n ¼ 2

256

; if n ¼ 3

(

½13

Using these estimates for the right-hand side of [11],

we obtain linear differential inequalities for w

that are easily integrated to give

wtðÞ



exp

d; if n ¼ 2

exp

128

d; if n ¼ 3

(

½14

It follows that if we can estimate the integrals on

the right, which concern only the solution u,andif

v is a second solution, perhaps differing only

slightly from u when t = 0, then we can estimate

the difference v(t)  u(t )

at later times. Moreover,

at any particular time this difference will be

bounded proportionally to v(0)  u(0)

.Theinte-

gral on the right-hand side of the two-dimensional

version of [14] is easily estimated using the energy

estimate [16] below. The estimation of the corre-

sponding integral in the three-dimensional case,

without a restriction on the size of the data,

remains an open problem. It can be regarded as

the most important open problem in the Navier–

Stokes theory. It would never be enough to some-

how prove that solutions are smooth without

estimating this integral, or something equivalent

to it. Of course, if solutions were known to be

smooth one could infer their uniqueness from [14],

since smoothness would imply that the integrals are

finite, which is enough to conclude that w(t)

zero if w

is zero.

Energy Estimate

If one multiplies the Navier–Stokes equation for u

by u, and proceeds as in deriving [11], one obtains

þru

¼ 0 ½15

and hence

utðÞ

d ¼

½16

Viscous Incompressible Fluids: Mathematical Theory 371

This settles the matter of continuous dependence in

the two-dimensional case. Together with [16], the

two-dimensional version of [14] implies

wtðÞ

 w

exp

; if n ¼ 2 ½17

We remark that the local rate of energy dissipa-

tion is 2 Du

rather than ru

,whereDu is the

stress tensor Du = (1=2)(r u þ (ru)

). However,

integrating over the domain, and integrating by

parts using the boundary condition u

@

= 0, one

may verify that the rate of total energy dissipation

2 Du

equals ru

. For the purpose of this

article, it is convenient to write the energy identity

as [15].

Estimates for ru(t)

Pointwise in Time

Of course, an estimate for ru(t)

pointwise in time

would imply an estimate for the integral of ru(t)

on the right-hand side of [14]. We can prove such an

estimate for at least a finite interval of time by an

argument due to Prodi (1962). It requires, in

preparation, some deep results concerning the

regularity of solutions of the steady Stokes equa-

tions. These cannot be proved here, but we can

briefly summarize what will be needed. Let

() = space of vector fields , with finite

-norms 

() = space of smooth vector fields with compact

support in ,

D() = { 2 C

(): r = 0},

J() = completion of D() in the L

-norm 

() = completion of D() in the norm r

G() = {rp: p 2 L

() with rp 2 L

()}, and

P : L

() !J() be the L

-projection of L

() onto

J(),

and define the Sobolev W

() norm by

ðÞ

¼ u

þru



=@x



Furthermore, observe that (rp, ) = 0forrp 2

G()and 2 J(), since it holds if p is smooth

and  2 D(). Therefore, Prp = 0, since

(Prp, ) = (rp, ) = 0, for all  2 J(). Later,

when we need it, we will also argue that

() = J()  G().

With these preparations, it is evident that every

smooth vector field u satisfying ru = 0and

@

= 0 can be regarded as a solution of the steady

Stokes problem

u þrp ¼ f and ru ¼ 0in u

@

¼0 ½18

with f = Pu. For such solutions, and hence for

all such u, we have the estimates

ðÞ

 cPu

½19

and

sup



ujj



Pu

; if n ¼ 2

c ru

Pu

; if n ¼ 3



½20

with constants independent of u. It can also be

shown that every such vector field u belongs to J

()

and hence to J(); see Heywood (1973).

Some history and remarks are in order. The

inequality [19] was proved independently by

Solonnikov (1964, 1966), and by Prodi’s student

Cattabriga (1961). In fact, they gave L

versions of

it for all orders of the derivatives. Several proofs

specific to the L

case needed here have been given

by Solonnikov and Sc

adilov (1973) and by Beira˜o da

Veiga (1997). The inequalities [20] can be proved by

combining [19] with appropriate Sobolev inequal-

ities, or better, by combining [19] with recent

inequalities of Xie (1991) which are of precisely

the form [20], but with 4u instead of P4u on the

right-hand side, and without the requirement that

ru = 0. The constant c in [19] depends upon the

regularity of the boundary, and tends to infinity

along with a bound for the boundary curvature.

Through the work of Xie (1992, 1997), there is

reason to believe that the inequalities [20] are

probably valid for arbitrary domains, with the

constant c = (2)

1

if n = 2, and c = (3)

1

if n = 3.

Xie’s efforts to prove this have been continued by

the author (Heywood 2001). If the inequalities

[20] can be proved for arbitrary domains (i.e.,

arbitrary open sets), with these fixed constants,

then the approach to Navier–Stokes theory pre-

sented in this article will extend immediately to

arbitrary domains, as explained in Heywood and

Xie (1997), with estimates independent of the

domain.

We go on now with an estimation of kru(t)k

based on [20]. Multiplying the Navier–Stokes

equation for u by P4u, and integrating over ,

one obtains

kruk

þkPuk

¼ u ru; PuðÞ

 sup



jujkrukkPuk½21

since (u

,Pu)=(Pu

,u)=(u

,u)=(ru

,ru)

and (rp,Pu)=0.

372 Viscous Incompressible Fluids: Mathematical Theory

The right-hand side of [21] can be estimated using

[20] and Young’s inequality:

sup



jujkrukkPuk



ckuk

1=2

krukkPuk

3=2

; if n ¼ 2

ckruk

3=2

kPuk

3=2

; if n ¼ 3

(



kPuk

þ ckuk

kruk

; if n ¼ 2

kPuk

þ ckruk

; if n ¼ 3

(

Thus,

kruk

þkPuk



ckuk

kruk

; if n ¼ 2

ckruk

; if n ¼ 3

(

½22

These differential inequalities are at the core of

present theory. Consider first the two-dimensional

case. It can be viewed as a linear differential

inequality

kruk

 ckuk

kruk



kruk

½23

with a coefficient ckuk

kruk

that is integrable, in

view of the energy estimate [16]. Integrating it yields

a ‘‘global’’ estimate; for all t  0,

krutðÞk

kru

exp

kuk

kruk

d

kru

exp

½24

However, if the three-dimensional version of [22]

is viewed as a linear differential inequality, the

coefficient to be integrated is kru(t)k

. Thus, the

same integral which is crucial to proving continuous

dependence on the data is also crucial to proving

regularity. What we can do in the three-dimensional

case, is view [22] as a nonlinear differential inequal-

ity of the form

’

 c’

or ’

 ckruk

’

½25

for ’(t) = kru(t)k

. Integrating the first of these, one

obtains a local estimate

krutðÞk



kru

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  2ckru

½26

for

0  t <

2ckru

without any restriction on the size of the data.

Integrating the second, one obtains a global estimate

krutðÞk



kru

1  ckru

kruk

d



kru

1 ðc= 2Þku

kru

½27

valid for all t  0, provided

kru

½28

This is a good interpretation of what we mean by

‘‘small data.’’ If Xie’s conjecture is correct, that the

constant in the three-dimensional version of [20] is

c = (3)

1

, then we obtain [25]–[28] with the

constant c = 3=(128

). Thus, 2=c ’ 842.

Further Regularity, Smoothing Estimates

Once one has an estimate of the form

krutðÞkMtðÞ; for 0  t < T ½29

as provided by [24], [26], or [27], one can estimate

the solution’s derivatives of all orders over the open

time interval (0, T). The initial time t = 0 must be

excluded from the interval, because the ‘‘imperfec-

tion’’ of prescribed data generally causes an impul-

sive acceleration along the boundary at time zero,

resulting in a thin boundary layer in which the

derivatives are so large that kru

(t)k and ku(t)k

()

tend to infinity as t !0

. But the solution quickly

smooths and remains smooth as long as [29]

remains in force. Thus, our working assumption up

to this point, that solutions are C

smooth in  

[0, 1) is not valid at t = 0. However, we will see

that they are smooth in

  (0, T) and continuous in

  [0, T). They are also continuous on [0, T) in the

() norm. This is sufficient regularity to justify

everything that we have done to this point.

In this section, we give estimates for the derivatives

of all orders with respect to time, of u and its first- and

second-order derivatives with respect to space. In the

next section, we will prove an existence theorem by

Galerkin approximation. It will be easily seen that all

of the estimates proved in this and previous sections,

for solutions that are assumed to be smooth, also hold

for the approximations, without any unproven

assumptions. Therefore, they will be inherited by the

solution that is obtained upon passing to the limit of

the approximations. At first, this solution will be

something of a generalized solution, not fully classical,

but one which is C

with respect to time over the

interval 0 < t < T,intheW

() norm. In a final step,

Viscous Incompressible Fluids: Mathematical Theory 373

viewing u at any fixed time as a solution of the steady

Stokes equations, we can apply regularity estimates for

the Stokes equations to infer that it is C

in all

variables throughout

  (0, T), with specific esti-

mates for each derivative.

The estimates of this section are obtained by

integrating an infinite sequence of differential inequal-

ities, for kuk, kruk, ku

k, kru

k, ku

k, kru

k, ....

The first two are [15] and [21], which have already

been dealt with. It turns out that after these first two,

each succeeding differential inequality is linearized by

the estimates obtained from its predecessor, which

explains why the time intervals for these additional

estimates do not become successively shorter. In fact,

in the two-dimensional case, the energy estimate

resulting from [15], which is valid for all time, already

gives the linearization [23] of [21], which then

provides an estimate valid for all time. Except for

noting such differences between the two- and three-

dimensional cases, we will henceforth deal with only

the three-dimensional case.

The differential inequalities just mentioned are

obtained by estimating the right-hand sides of two

sequences of differential identities, and ordering

them by an iteration between the two sequences.

The first sequence begins with and is patterned after

the energy identity,

kuk

þkruk

¼0

þkru

¼ u

ru; u

ðÞ

þkru

¼ u

ru; u

ðÞ

 2 u

ru

; u

ðÞ

etc:

½30

while the second begins with and is patterned after

Prodi’s identity,

kruk

þkPuk

¼ u ru; PuðÞ

kru

þkPu

¼ u

ru; Pu

ðÞ

þ u ru

; Pu

ðÞ

kru

þkPu

¼ u

ru; Pu

ð Þþ

etc:

½31

Before going on, notice that we can return to [22] and

use [29] to infer a more complete estimate of the form

krutðÞk

kPuk

d

 BM; tðÞ; for 0  t < T

½32

containing an integral of kPuk

on the left-hand side.

We will use the notation B(M, t) generically, for any

bound that depends only on the function M (t)andt.

We remark, that a term ku

can also be included

under the integral sign on the left-hand side of [32],

because ku

k and kPuk are of essentially the

same order, being the leading terms in the projection

þ P(u ru) = Pu of the Navier–Stokes equation.

Finally, one can also include kuk

()

under the

integral sign, in view of [19].

Going on, we obtain a third differential inequality

from the second identity of the sequence [30]. Its

right-hand side admits the estimate

 u

ru; u

ðÞku

kruk

 cku

1=2

kru

3=2

kruk



kru

þ ckruk

½33

which, in view of [29] or [32], produces a linear

differential inequality with integrable coefficients.

Its integration yields an estimate of the form

tðÞk

kru

d

 BM; t; ku

0ðÞkðÞ; for 0  t < T

½34

provided ku

(0)k is bounded. Since u

= P(u 

u ru), we have the estimate

0ðÞk¼kP u

 u

ru

ðÞk

ku

 u

ru

kB



ðÞ



½35

provided that u is smooth in

  [0, T). This is a

delicate point, having been forewarned of a regular-

ity breakdown at t = 0. But, we will be able to

replicate the estimate [35] for the Galerkin approx-

imations, ultimately validating [34] for the approx-

imations and the solution.

The integration of the next differential inequality,

which arises from the second of the identities [31],

requires that kru

(0)k < 1. Similarly to [35],we

have

0ðÞ

¼rP u

 u

ru

ðÞ

 B



ðÞ



½36

provided that u is smooth in

  [0, T). However,

there is a big difference between [35] and [36].Inthe

next section, we will not be able to obtain an analog of

[36] for the Galerkin approximations. Consequently,

the solution that is obtained will not be fully regular at

time t = 0. It will satisfy u 2 C(

  [0, T)) \ C

(

(0, T)), but not u 2 C

(  [0, T)). It will satisfy

374 Viscous Incompressible Fluids: Mathematical Theory

u(t)  u

()

!0 but not u(t)  u

()

!0,

as t !0

One may wonder whether this is a fault or

deficiency in the Galerkin method. It is not,

remembering what was said at the beginning of

this section. For most prescribed values of u

,no

matter how smooth, there is a breakdown in the

regularity of the solution as t !0

. In fact, it was

proved in Heywood and Rannacher (1982) that if

(t)

or any one of several other quantities,

including u(t)kk

()

, remains bounded as t !0

then there exists a solution p

of the overdetermined

Neumann problem

p

¼r u

ru

ðÞin 

@

¼ u

@

½37

Generically speaking, this problem is not solvable,

and therefore

lim sup

t!0

tðÞ

¼1

We mention that under our assumption that u

smooth, the correctly posed Neumann problem,

with boundary condition @p

=@n

@

=u

 n

@

,is

uniquely solvable for a solution p

2 W

()=R, and

rp(t ) rp

!0, as t !0

; see Heywood and

Rannacher (1982).

Since solutions are smooth for 0 < t < T, the

pressure in the Navier–Stokes equations satisfies the

overdetermined Neumann problem for all t 2 (0, T).

So it may seem appropriate to require that the

prescribed initial value u

be a function for which

problem [37] is solvable. We do not agree with that.

It is too difficult, if not impossible, to find such

functions, except by solving the Navier–Stokes

equations. For example, one might think that the

condition that [37] should be solvable might be

satisfied if u

2 D(), since such functions are zero

in a neighborhood of the boundary. In fact, K

Masuda has shown that if  is a three-dimensional

sphere, then the overdetermined Neumann problem

[37] is never solvable for nonzero u

2 D(). Hence,

the gradient of the initial pressure will have a

nonzero tangential component, causing an impulsive

tangential acceleration along the boundary.

If we are to use the Navier–Stokes equations to

make predictions of the future, we must solve the

initial boundary value problem for ‘‘man-made’’

initial values, and accept the fact that there is a

momentary breakdown in regularity along the

boundary, immediately following the initial time.

Thereafter, the solution smooths as ‘‘nature’’ takes

over. To prove the reliability of our predictions, we

need continuous dependence estimates and error

estimates for numerical methods that take into

account this initial breakdown in the regularity.

The continuous dependence estimate [14] meets this

requirement. So also do the error estimates given in

a series of four papers by Rannacher and the author,

beginning with Heywood and Rannacher (1982).

They were based on the ‘‘smoothing’’ regularity

estimates for solutions that are being presented here.

We go on with these now, as models for similar

estimates for the Galerkin approximations.

Estimating the right-hand side of the second of the

identities [31] using [20] and Young’s inequality,

and then multiplying through by t, we get the linear

differential inequality

t ru



þ tPu

ru

þc rukk

þrukk



þ Pu



t ru



½38

for t ru

, with coefficients that are integrable in

view of the previous estimates [32], [34], and [35].

Therefore, its integration yields an estimate analo-

gous to [32] of the form

t ru

ðtÞkk

tPu

d

 BM; t; u

ðÞ



; for 0 < t < T ½39

provided its ‘‘initial value’’ is finite. It is, due to the

time weight, in the sense that

lim sup

t!0

t ru

ðtÞ



¼ 0 ½40

This is proved by noting that if the lim sup were

positive, then the integral on the left-hand side of

[34] would be infinite. Finally, a term tu

can be

included under the integral sign on the left-hand side

of [39], because u

and Pu

are of essentially

the same order, being the leading terms in the

projection u

þ P(u

ru þ u ru

) = Pu

of the

time differentiated Navier–Stokes equation.

We continue inductively. Estimating the right-

hand side of the third of the identities [30] using

[12], [20], and Young’s inequality, and then multi-

plying through by t

, we get the linear differential

inequality



þ t

 2tu

þt

Pu

þ c ru

þru



½41

with coefficients that are integrable in view of

preceding estimates. In particular, the integrability

Viscous Incompressible Fluids: Mathematical Theory 375

of the first term on the right-hand side follows from

the boundedness of the integral

d ½42

which, we have pointed out, can be included on the

left-hand side of [39]. Finally, notice that the

boundedness of the integral [42] implies

lim sup

t!0

ðtÞ



¼ 0 ½43

Therefore, we can integrate [41] to get the estimate

ðtÞ

d

 BM; t; u

ðÞ



; for 0  t < T ½ 44

analogous to [34].

At this point, we have introduced every device

needed to proceed by induction to an infinite

sequence of time-weighted estimates, similar to

[39] and [44], but with successively higher orders

of time derivatives and weights. The dependence of

these estimates on u

()

was introduced through

[34] and [35]. It can be eliminated by beginning the

introduction of powers of t as weight functions one

step earlier, with the added advantage that the initial

velocity u

needs only belong to J

(). In the two-

dimensional case, the weight functions can be

introduced even another step earlier, with the

advantage that the initial velocity u

needs only

belong to J(). Each of these cases leads to an

existence theorem for solutions u 2 C

(  (0, T)),

with the initial values assumed in the norms of J

()

and J(), respectively.

Existence by Galerkin Approximation

Let {a

, a

, ...}and{

, 

, ...} denote the eigenfunc-

tions and eigenvalues of the Stokes equations,

a

þrp ¼ 

; ra

¼ 0in



@

¼0 ½45

chosen to be orthonormal in L

(). Clearly,

Pa

= 

, so they are also the eigenfunctions

and eigenvalues of the Stokes operator, P. Using

regularity estimates for the Stokes equations, each

eigenfunction is known to be C

smooth in .

The nth Galerkin approximation for problem [9]

is the solution

x; tðÞ¼

k¼1

tðÞa

xðÞ

of the system of ordinary differential equations

; a



þ u

ru

; a



¼ u

; a



for l ¼ 1; 2; ...; n ½46

satisfying the initial conditions (u

(0), a

) = (u

, a

for l = 1, 2, ..., n. Of course, since (u

, a

) = @c

=@t

and (u

, a

) = (Pu

, a

) = 

, the differential

equations can be written as

¼

i;j¼1

ra

; a



 

and the initial conditions as c

(0) = (u

(0), a

), for

l = 1, 2, ..., n.

The system [46] is at least locally solvable, on

some interval [0, T

), with each coefficient satisfying

2 C

[0, T

). Therefore, since the eigenfunctions

are also smooth, u

is C

smooth in   [0, T

). It

also satisfies all of the identities [30] and [31] on the

interval [0, T

). Indeed, multiplying [46] by c

and

summing over l from 1 to n has the effect of

converting a

into u

. The resulting identity for u

leads immediately to the energy identity

þru

¼ 0 ½47

The remaining identities in the sequence [30] are

obtained similarly. For example, the second is

obtained by taking the time derivative of [46],

multiplying through by dc

=dt and summing over l.

Prodi’s identity is obtained by multiplying [46] by



and summing, which has the effect of convert-

ing a

into Pu

. To obtain the second of the

identities [31] for u

, one differentiates [46], multi-

plies by 

=dt and sums. The remaining identities

in the sequence [31] are obtained similarly.

The initial conditions easily imply that u

(0)



, because u

2 J() and the eigenfunctions are

orthogonal and complete in J(). Therefore, inte-

gration of [47] yields the energy estimate

ðtÞk

kru

d 

½48

which is uniform in n.Sinceku

(t)k remains bounded,

the solution u

(t) can be continued for all time. Thus,

= 1 ,foralln. Hence, our early working assump-

tion about solutions, that they are smooth in  

[0, 1), is actually valid for the Galerkin approxima-

tions. The issue becomes one of obtaining estimates

for their derivatives that are uniform in n.Allofthe

estimates we have proved for solutions are proved in

exactly the same way for the approximations. The

only possible source of nonuniformity would arise

from the initial values of kru

k and ku

376 Viscous Incompressible Fluids: Mathematical Theory

The estimates [24], [26], and [27] are uniform in

n, since u

2 J

() and hence kru

(0)kkru

due to the orthogonality of the eigenfunctions in the

inner-product (ru, rv), and their completeness with

respect to functions in J

(). We also obtain a

uniform bound for ku

(0)k of the form [35],by

multiplying [46] by @c

=@t and summing over l.In

the last step, we also need the inequality

(0)k

()

ku

()

, which follows from the

orthogonality of the eigenfunctions in the inner

product (Pu, Pv), and their completeness with

respect to functions in J

() \ W

(); see

Ladyzhenskaya (1969, p. 46). Any attempt to find

a bound for kru

(0)k analogous to [36] is certain to

fail, as it would lead to a contradiction with afore-

mentioned results from Heywood and Rannacher

(1982).

Passage to the Limit

We now have L

-bounds for u

, ru

, u

, @

=@x

and ru

over any space-time region   (0, T

), with

0 < T

< T.WealsohaveL

-bounds for all orders of

the time derivatives of these quantities over any

subregion   (", T

), with 0 <"<T

< T.From

these L

-bounds, we may infer the existence of a

subsequence of the Galerkin approximations, again

denoted by {u

}, which converges, along with those of

its derivatives for which we have bounds, to a limit u

and its derivatives. The convergence u

!u and

!ru is strong in L

(  (0, T

)); the conver-

gence of u

is strong in L

(  (", T

)) and weak in

(  (0, T

)); the convergence of Pu

is weak in

(  (0, T

)); all time derivatives of u

, ru

con-

verge strongly in L

(  (", T

)).

Because of estimates for the time derivatives, trace

arguments give the strong convergence u

!u,

!ru, u

, and the weak convergence

Pu

!P u,inL

(), for every t > 0.

For any fixed time, u 2 W

(), and therefore u is

continuous in

 by a well known Sobolev inequal-

ity. Since u 2 J

(), it must equal zero along the

boundary. The estimates for the time derivatives of

, ru

, @

=@x

imply that u and its time

derivatives are time continuous in W

(). There-

fore, u, u

, u

, ... are classically continuous in

  (0, T).

Introduction of the Pressure

Because of the strong convergence u

!u, ru

ru, u

and the weak convergence

Pu

!P u,inL

(), for any t > 0, it is an easy

matter to let n !1 in [46], obtaining, for all t > 0,

; a



þðu ru; a

Þ¼ u; a



for l ¼ 1; 2; ... ½49

Since the eigenfunctions are complete in J(), and

D()  J(), this implies

ðu

þ u ru  u;Þ¼0; for all  2 Dð Þ½50

Therefore, there exists a vector field rp 2 G() such

that

þ u ru  u ¼rp ½ 51

Indeed, the usual test to determine whether a

smooth vector field w is conservative in some

domain , and therefore representable as a gradient,

is to check whether the curve integrals

w   ds ½52

vanish for every smooth closed curve C  . Here, 

is the unit tangent to the curve and ds is its arc

length. With a little reflection, one will realize that

these curve integrals can be approximated by

volume integrals of the form (w, ) with  2 D().

For this, one should choose  to have its support in

a small tubular neighborhood of the curve, and its

streamlines parallel to the curve, with unit net flux

through any section of the tube. If w is not smooth,

but only known to belong to L

(), one can

approximate it with its smooth mollifications. This

argument can be made rigorous. We previously

showed that J()andG() are orthogonal sub-

spaces of L

(). Now we have argued that

() = J()  G().

Classical C

Regularity

At any fixed time, we may regard u as a solution of the

steady Stokes problem [18] with f = u

 u ru.

Included in Cattabriga (1961) and Solonnikov (1964,

1966) are regularity estimates for all orders of

derivatives of the form

kuk

kþ2

ðÞ

 ckf k

ðÞ

From our estimates above, we easily conclude that

f u

 u ru 2 W

(). Hence, u 2 W

(). In

fact, in view of the regularity we have proven

with respect to time, f 2 C

(0, T; W

()) and u 2

(0, T; W

()). Thus begins a bootstrapping argu-

ment. In the next step, we observe that f 2

(0, T; W

()) and conclude that u 2 C

(0, T;

()). By induction, one obtains u 2 C

(0, T;

()) for every positive integer k. Then well-

known Sobolev inequalities imply that u 2 C

(  (0, T)).

Viscous Incompressible Fluids: Mathematical Theory 377