Galdi G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems

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16 I Steady-State Solutions: Formulation of the Problems and Open Questions

for one can exhibit examples of solutions that satisfy (I. 2.10) and decay more

slowly than any negative power of r (see Ham el 191 6 and cf. also Chapter

XII). It is interesting to remark that for these solutions the ﬂux Φ deﬁned

in (I.2. 7) is not zero. Nevertheless, it should be observed that if v(x) tends

uniform ly po intwise to

v 6= 0, and satisﬁes (I.2.10), Sazonov (1999) has shown,

with the help of the results of Smith (1965), that v admits such an expansion,

where the leading term is a sol ution to the (l inearized) Oseen equations; see

Chapter XII. This property can be considered the analogue of that established

by Babenko for the three-dimensional case.

In view of what we said, the fol lowing question deserves attention:

(viii) Is it possible to characterize the behavior, in a neighborhood of inﬁnity,

of a solution v satisfying (I.2.1 0) and tending uniformly to

v = 0?

The most important feature of Leray’s sol ution is that it is global in the

sense that it does not require restrictions on the size of the data. On the

other hand, by what we have seen, one does not know, to date, whether

such a solution satisﬁes condition (I.2.3).

Thus, we may wonder whether,

using a diﬀerent construction, we could prove existence in the full problem

(I.0. 4), (I.1.1), and (I.1.2) a t least in the small, that is, by im posing suitable

restrictions on the size of the data. Here again, we have to distinguish between

the cases v

∞

6= 0 and v

∞

= 0. In the form er case, the answer is positive and

is due to Finn and Smith (19 67b) and Galdi (1993); see also Chapter XI.

If, however, v

∞

= 0, no result is available. So we are led to formulate the

following questions:

(ix) Does existence in problem (I.0.4), (I.1.1), and (I.1.2) with v

∞

= 0 hold,

even for small data f and v

∗

(x) Can the solution of Finn and Smith be related to that of Leray with the

same data?

My conjecture is that for generic data, the answer to (ix) is negative. This

view is reinforced by the fact that the analogous problem for the linear case,

obtained by formally suppressing the term v · ∇v in (I. 0.4), admits an aﬃr-

mative answer if and only if the data satisfy a ﬁnite number of compatibility

conditions, o r in other words, the space of the data has ﬁnite codimension; see

Section 7 in Chapter V. (For an analogous situation in the three-simensional

case, see Galdi 2009.)

Question (x) can be framed within the more general problem of the unique-

ness of two-dimensional soluti ons. In this regard it should be pointed out that

such a subject is still essentially obscure and tha t no signiﬁcant result is avail-

able, with the exception of that of Finn and Smi th (1967b) and Galdi (1 993),

For this reason, in dimension two, the Leray solution strictly speaking should not

be called a “solution” to the steady-state exterior problem.

See the previous footnote. It should be observed that the solution of Finn and

Smith has a bounded Dirichlet integral.

I.3 Flow in Regions with Unbounded Boundaries 17

which concern a somewhat restricted class of sol utions; see Cha pter XII.

The main reason for the lack of results is that the traditional methods usually

employed to test the uni queness of solutions of Navier–Stokes equations abso-

lutely do not work in such a case. Perhaps the introduction of genuinely new

tools to attack uniqueness will open new avenues to a better understanding

of the entire problem of plane ﬂow.

All the above considerations raise the question of whether the classical

two-dimensional exterior problem with v

∞

6= 0 is indeed solvable for data of

“arbitrary size.” This question has been analyzed by Galdi (1999b). In that

paper it is assumed that v

∗

≡ f ≡ 0, v

∞

6= 0, and that R

− Ω 6= ∅ is

suﬃciently smooth, and symmetric around v

∞

, and it is shown that if there

is v such that (I.2.4), with the above data and v

≡ v

∞

, has no solution for

all |v

∞

| ≥ v, then the homogeneous problem

∆u = u · ∇u + ∇τ

∇ · u = 0

)

in Ω ,

∂Ω

= 0 ,

lim

|x|→∞

u(x) = 0 uniformly pointwise ,

(I.2. 11)

with u satisfying (I.2.9), must have at least one nonzero smooth solution

(u, τ) that is also symmetric (in a well deﬁned sense) around v

∞

. Although

at ﬁrst glance, this possibility seems to be easily ruled out, a mathemati-

cal proof showing tha t problem (I.2.11) admits only the zero solution is not

yet available, and it is probably far from being obvious. We thus have the

following:

(xi) Is u ≡ ∇τ ≡ 0 the only solution to (I.2.11) in the speciﬁed class?

It is worth emphasizing that the possibility that (I.2.11) may admit a

nonzero smooth solution is very q uestionable on physical grounds, and the

occurrence of such a situa tion would cast serious doubt on the meaning of the

two-dimensional assumption.

I.3 Flow in Regions with Unb ounded Boundaries

Even though some basic issues were formulated quite long ago, a systematic

study of ﬂow in region with unbounded boundaries began o nly more recently,

in the period 1976-1978, through the fundamental work of J.G. Heywood,

C.J. Amick, O.A. Ladyzhenskaya, and V.A. Solo nnikov. Therefore, it is not

In fact, the uniqueness results given by these authors are of such a “local” type

that it is not known whether the solution of Finn & Smith coincides with that of

Galdi.

18 I Steady-State Solutions: Formulation of the Problems and Open Questions

surprising that several basic questions are still far from being answered. Here,

we wish to mention a few of them.

It might be surprising that we don’t ﬁnd any direct contribution of Jean

Leray to the subject; however, one of the main questions still open seems to

be in fact due to him (Ladyzhenskaya 1959b, p. 77, 1959c, p. 551). Let us

describe this problem. Let Ω be a “distorted tube” of R

(n = 2, 3) with two

semi-inﬁnite cylindrical ends (strips, fo r n = 2),

i.e.,

Ω =

[

i=0

Ω

, (I.3.1)

where Ω

is a smooth bounded subset of Ω, whi le Ω

, i = 1, 2, are disjoint

regions that, in po ssibly diﬀerent coordinate systems (depending on Ω

, i =

1, 2), reduce to straight cylinders (strips, for n = 2), that is,

Ω

= {x ∈ R

: x

> 0, x

≡ (x

, . . . , x

n−1

) ∈ Σ

}, (I.3. 2)

with Σ

bounded and simply connected regions in R

n−1

. Denoting by Σ any

bounded intersection of Ω with a plane, which in Ω

reduces to Σ

, and by n

a unit vector orthogonal to Σ, oriented from Ω

toward Ω

(say) owing to the

incompressibility of the liquid and assuming that v vanishes at the boundary,

we at once deduce that the ﬂux Φ of v through Σ is a constant:

Φ ≡

v · n = const. (I.3. 3)

Therefore, a natural question that arises is that of establishing existence of

a ﬂow subject to a given ﬂux. This condition a lone, of course, may not be

enough to determine the ﬂow uniq uely, and similarly to what we do for ﬂows

in exterior regions, we must prescribe a velocity ﬁeld v

∞i

as |x| → ∞ in

the exits Ω

. However, in contrast to the case of ﬂows pa st bodies, v

∞i

need

not be uniform, and i n fact, if Φ 6= 0, it is easily seen that v

∞i

cannot be

uniform . Thus, one has to ﬁgure out how to prescribe v

∞i

. What is most

natural to expect is that the ﬂow corresponding to a given ﬂux Φ shoul d tend,

as |x| → ∞ in each Ω

, to the Poiseuille solution of the Navier–Stokes equation

in Ω

corresponding to the ﬂux Φ, that is, to a pair (v

(i)

, p

(i)

), where

(i)

≡ (0, . . ., v

(i)

)), ∇p

(i)

≡ (0, . . ., −C

) (I.3. 4)

for som e constants C

= C

(Φ), and

n−1

j=1

∂

(i)

∂x

= −C

in Σ

(i)

= 0 at ∂Σ

(I.3. 5)

Entirely analogous considerations could be performed for the case of more than

two “outlets” Ω

I.3 Flow in Regions with Unbounded Boundaries 19

If, for instance, n = 3 and the sections are circles of radius R

, the solution

to (I.3.4), (I.3.5) is the Hagen–Poiseuille ﬂow

(i)

) =

(1 − |x

The problem of determining a motion in a region Ω with cylindrical “exits,”

subject to a given ﬂux Φ and tending in each “exit” to the Poiseuille solution

corresponding to Φ, is known as Leray’s problem (Ladyzhenskaya 1959b, p.77,

1959c, p. 551). This problem has been the object of deep research by Amick

(1977, 1978), Horgan and Wheeler (1978), and Ladyzhenskaya and Solonnikov

(1980). However, its solvability has been shown only for Φ suﬃciently small.

We are therefore led to the following basic question:

(i) Is Leray’s problem solvable for any value of the ﬂux Φ?

Despite the seemingly diﬀerent natures of the two physical problems, due

to the quite diﬀerent shape of the regions of ﬂow, from the mathematical p oint

of view question (i) appears to be intimately related to the analogous problem

in a bounded regio n, which we discussed in Section 1.

Similar questions can b e formulated for regions having “outlets” to inﬁnity

whose cross sections are not necessarily bounded. So, assume that Ω is of the

type (I.3.1), (I.3.2), where now, the sections are allowed to vary with x

and become unbounded as x

tends to inﬁnity.

This time, the condition

that the limiting velocity ﬁelds v

∞i

are zero is no longer in conﬂict with the

conservation of ﬂux (I.3.3), and we may try to sol ve problem (I.0.4), (I.1.1)

(with v

∗

= 0) under the condition of prescribed ﬂux and vanishing velocity

as |x| tends to inﬁnity in each “outlet” Ω

Unlike ﬂow in exterior regions, here the case of two-dimensional solutions

presents results more complete than in the case of fully three-dimensional

motions, thanks to the thorough investigation of Amick and Fraenkel (19 80).

Speciﬁcally, these authors prove the existence of solutions and pointwise

asymptotic decay of the corresponding velo city ﬁelds under diﬀerent assump-

tions on the “growth” of Σ

and with a “small” ﬂux if the sections have a

certain rate of “growth.” However, two important issues are left out, that is,

uniqueness of solutio ns and their order of decay at large distances. These two

problems have been recently studied and solved for “small” ﬂux by K. Pileckas

in the particular case that each Ω

is a body of revolution of type



x ∈ R

: x

> 0, |x

| < g

)



provided the (smooth) positive functions g

) satisfy suitable “growth” con-

ditions as x

→ ∞. As expected, the decay rate of solutions is related to the

inverse power of the functions g

; see Pileckas (1996c, 1997); see also Nazarov

& Pileckas (1998).

We also assume that Σ cannot shrink to a point, that is, the measure |Σ| of Σ is

bounded below by a positive constant.

20 I Steady-State Solutions: Formulation of the Problems and Open Questions

On the other hand, several fundamental questions remain o pen fo r three-

dimensional ﬂows. Actually, one can prove that if the sections Σ

become

unbounded at a suﬃciently large rate, then a solution exists that, in the mean,

converges to zero a t large distances; otherwise, the problem admits only partial

answers, and in some cases, it i s completely unsolved. Let us brieﬂy explain

why. The leading idea is to try to obtain, as in previous instances, a bo und on

the Dirichlet integral D (say) of the velocity ﬁeld v. Now if the cross sections

become unbounded, we may distinguish the foll owing two possibilities:

(a)

∞

|Σ

−2

< ∞, i = 1, 2,

(b)

∞

|Σ

−2

= ∞, i = 1, 2.

In case (a), using inequality (I.1.6) one can show that the condition of constant

ﬂux is compatible with the ﬁniteness of D, and in fa ct, using more or less

standard, tools one proves an a priori bound that all ows us to obtain the

existence of a solution to the problem for arbitrary ﬂux Φ (Ladyzhenskaya and

Solonnikov 1980). However, in g eneral, one cannot prove a pointwise decay of

v at large distances; rather, only a weaker behavior in the mean is achieved.

We are thus led to formulate the following questions:

(ii) In case (a), is it possible to prove the pointwise decay of solutions whose

velocity ﬁeld has a bounded Dirichlet integral?

(iii) Is it possible to relate the asymptotic behavior of such solutions to the

rate of growth of cross sections Σ

There is one particular, though interesting, situa tion in which both ques-

tions (ii) and (iii) are positively answered, namely when each outlet Ω

“de-

generates” into a half-space (Borchers & Pileckas 1992, Chang 1992, 1993,

Coscia & Patria 1992 , Galdi & Sohr 1992); see also Chapter XIII. In such a

case, Ω becomes the so-called aperture domain (see Heywood,1976 ):

Ω = {x ∈ R

: x

6= 0 or x

∈ S},

with S a bounded region of the plane (the “aperture”). However, unlike the

results of Lady zhenskaya and Solonnikov, the results of all the preceding au-

thors require that the ﬂux Φ b e suﬃciently small. Therefore, we have the

following question:

(iv) Is it possible to show the known results for three-dimensional ﬂow in the

aperture domain for an arbitrary ﬂux Φ?

It should be remarked that the two-dimensional analogue of this problem

appears to be diﬃ cult to treat, and all methods employed by the above authors

Of course, we may assume that one section veriﬁes (a) and the other veriﬁes (b).

The considerations to follow should then be modiﬁed accordingly which, however,

would produce no conceptual diﬀerence.

I.3 Flow in Regions with Unbounded Boundaries 21

do not apply there. Nevertheless, by diﬀerent tools, one proves the existence of

a solution tha t tends to zero at large distances uniformly pointwise for any Φ

(Galdi, Padula, and Passerini 1995); however, the asymptotic structure of such

solutions is completely characterized only for small Φ and, more i mportantly,

when the aperture S is symmetric around x

(Galdi, Padula and Solonnikov

1996; cf. also Chapter XI II).

Therefore, we have the following question:

(v) Is it possible to characterize the asymptotic structure of plane ﬂow in an

aperture domain when the aperture is not symmetric, even for a small

ﬂux Φ?

Let us next consider case (b). Again using inequality (I.1 .6), one shows

that the nonzero ﬂux condition becomes incompatible with the existence of

solutions whose velo city ﬁeld has a bounded Dirichlet integral. However, if

≡

∞

|Σ

−3q/2+1

< ∞, i = 1, 2, some q > 2 , (I.3. 6)

one can show that solutions may still exist with corresponding velo city ﬁeld

v satisfying the condition

Ω

(∇v : ∇v)

q/2

< ∞, q > 2. (I.3. 7)

Therefore, a subspace S

(say) of functions satisfying (I.3.7) seems to be a

“most natural” space in which to set the exi stence problem. Whether this

conjecture is true is yet to be ascertained in the general case. However, if Ω

is a body of revolution deﬁned by a smooth positive function g

, K. Pileckas

(1996c, 1997 ) has proved the existence of solutions in the class (I.3.7) for

arbitrary ﬂux Φ, provided g

satisﬁes certain conditions at large distances more

restrictive than tho se merely required by (b) and (I.3.6).

Corresponding

decay estimates are also given.

A last p ossibility arises when the sections become unbo unded in such a

way that the integrals G

deﬁned in (I.3.6) are inﬁnite for every value o f

q > 1. For such regions of ﬂow it is not even clear i n which space the problem

has to be formulated. In this regard we should not overlook the approach of

Ladyzhenskaya and Solonnikov (1980), who, in a rather large class of regions

Ω with outlets Ω

of unbounded cross section, prove the existence of a solution

whose velocity ﬁeld has a ﬁnite Dirichlet integral on every bounded po rti on

of Ω. Growth estimates from above on such a quantity are then provided in

terms of the growth of the cross sections of Ω

. The question whether these

solutions tend to zero at inﬁnity i s, however, left open.

In f act, the asymptotes are given by suitable Jeﬀery–Hamel solutions; see Rosen-

head (1940).

Notice that since |Σ| ≥ Σ

> 0, in case (b) the integrals G

are inﬁnite for any

q ≤ 2.

In such a case, |Σ

| = πg

22 I Steady-State Solutions: Formulation of the Problems and Open Questions

Notes for Chapter I

Section I.1. The study of the properties of solutions to the Navier–Stokes

equations has received substantial attention also under boundary conditions

other than (I.1.1). A popular one is the following:

v + βn ·T (v, p) × n = b

∗

, at ∂Ω , (∗)

where β is a constant, n is the outer unit normal at ∂Ω, T (v, p) is the Cauchy

stress tensor (I.0.2), and b

∗

is a prescribed vector ﬁeld. Condition (∗) was

introduced for the ﬁrst time by Navier (1827).

If β = 0, (∗) reduces to

to (I.1.1), and v is totally prescribed (no sli p), while if 1/β → 0 , o nly v ·

n is prescribed, and we lose information on the tang ential component v

(pure slip). However, if β 6= 0 and ﬁnite, (∗) allows for v

to be nonzero,

by an amount that depends on the magnitude of the tangential stress at the

boundary (partial slip).

The Navier condition (∗), with 1/β 6= 0 or → 0, has been employed in a

wide range of problems. They include free surface problems (see, e.g., Solon-

nikov 1982, Maz’ja, Plamenevskii, & Stupyalis 1984), turbulence modeling

(see, e.g., Par´es 1992 , Galdi & Layton 200 0), and inviscid limits (see, e.g.,

Xiao & Xin 2007, Beir˜ao da Veiga 2010).

Concerning the use of (∗) in steady-state studies, after the pioneering work

of Solonnikov and

Sˇcadilov (1973), where pure slip boundary conditions are

used along wi th a linearized system of equations (Stokes equations), in the last

few years there has been considerably increasing interest. Besides the papers

of Beir˜ao da Veiga (2 004, 2005), which generalize and si mplify the proof of the

results of Solonnikov &

Sˇcadilov, we refer the interested reader, for exampl e,

to Ebemeyer & Frehse (2001) for ﬂow in bounded domains, Mucha (2003),

Konieczny (2006), and Beir˜ao da Veiga (2006) for ﬂow in inﬁnite channels and

pipes, to Konieczny (2009) fo r ﬂow in exterior domai ns, and to the literature

cited therein.

It is conceptually interesting to notice that some of the basic problems left

open for liquids m odeled by the Navier–Stokes equations may ﬁnd a complete

and posi tive answer for liquid models whose constitutive assumptions diﬀer

from tha t given in (I.0.2). Such models are generically referred to as non-

Newtonian.

Among the most successful and widely adopted models of non-

Newtonian liquids are those called generalized Newtonian, where the shear

viscosity coeﬃcient µ is no longer constant, but depends on the “amount of

shear,” namely, on |D(v)|, with D the stretching tensor deﬁned in (I.0.3).

Navier proposed (∗) (or better, an equation equivalent to (∗) with b

∗

= 0) in

alternative to the “adherence” condition (I.1.1), wit h the objective of explaining

the diﬀerence between the discharges in glass and copper tubes, as experimentally

observed by Girard (1816).

For this type of models and their range of applicability, we refer the interested

reader to the monograph of Bird, Armstrong, and Hassager (1987).

I.3 Flow in Regions with Unbounded Boundaries 23

For example, in an appropriate range of shear, for liquids such as latex paint,

styling gel, molasses, and blood, µ is found to be a decreasing function of

|D(v)| (shear-thinning liquids), whereas f or others, such as a mixture of corn

starch a nd water, and clay slurries, µ increases with |D(v)| (shear-t hinning

liquids). A prototypical example of the generalized Newtonian model is the

one with µ related to |D(v)| by the following formula (power law model)

µ = µ

+ ε

|D(v)|

, (∗∗)

where the (m aterial) constants µ

, ε

, and ε

satisfy µ

> 0, ε

≥ 0, and

∈ (−1, ∞). Thus, for ε

> 0, the model is shear-thinning if ε

∈ (−1, 0) and

shear-thickening if ε

∈ (0, ∞), whereas we recover the Newtonian (Navier–

Stokes) model if either ε

or ε

is zero. Now, for a given µ

, and ε

, ε

> 0 ar-

bitrarily small, one can show (Ladyzhenskaya 1967, §4, Galdi 2008, §2 .2.1(b))

that the b oundary value problem obtained using the constitutive assumption

(∗∗), namely,

∇ · [(µ

+ ε

|D(v)|

)D(v)] = ρv ·∇v + ∇p + f

∇ · v = 0

)

in Ω

v = v

∗

at ∂Ω ,

(∗ ∗ ∗)

where Ω is a bounded (suﬃciently smooth) domain with a possibly discon-

nected boundary, has a solutio n for any v

∗

, f in a suitable function class, and,

more importantly, with v

∗

satisfying the “natural” compatibility conditi on

(I.1. 2). As we have emphasized, unless ∂Ω is connected, this is an outstand-

ing open question for the Navier–Stokes model, obtained by setting ε

= 0

or ε

= 0.

The above result is conceptually very intriguing, in that it can

be rephrased, in common and provocative words, as follows: “what we do no t

know whether it is true for water, becomes certainly true if we add to water

a pinch of corn starch”!

Along the same lines, Galdi & Grisanti (2010) have considered (∗∗∗), with

Ω an exterior region of R

and v

∗

≡ 0, along with the asymptotic condition

(I.2. 4). They have shown that for arbitrary ε

> 0 and ε

∈ (−1, 0), the

resulting boundary value problem po ssesses a solution for any choice of f in

an appropriate function class, and for arbitrary v

∈ R

. As we pointed out in

Section I.2, this is another fundamental open question for the Navier–Stokes

model.

Section I.2. Among the many other signiﬁcant “exterior” problems that we

can f ormulate, and tha t we will not treat in this book , of particular interest is

the steady ﬂow of a Navier–Stokes liquid under the action of a given constant

shear. This problem, which has received great attention in the applied science

community thanks to the fundamental work of Saﬀman (196 5) on the eﬀect

This result can be extended to more general constitutive assumptions than (∗∗);

see Galdi, (2008).

24 I Steady-State Solutions: Formulation of the Problems and Open Questions

of shear o n the lift of a sphere, amounts to solving (I.2.4) with v

∗

≡ f ≡ 0,

along with the following condition at large distances:

lim

|x|→∞

(v(x) − v

∞

) = 0 , v

∞

:= κx

where κ is a nonzero constant and e

is the unit vector in the direction x

Unfortunately, the classical methods for existence of steady state solutions

in exterior domains, with which the reader will become familiar by ﬂi pping

through the pages of this book, all fail for the above problem. For instance, it

seems very unlikely that one could prove a bound of the type (I.2.9) for the

ﬁeld u := v − v

∞

, even for small data. I take this opportunity to bring this

intriguing open question to the attention of the interested mathematician.

Basic Function Spaces and Related Inequalities

Incipe parve puer risu cognoscere matrem.

VERGILIUS, Bucolica IV, v.60

Introduction

In this chapter we shall introduce some function spaces and enucleate certain

properties of fundamental importance for further developments. Particular

emphasis will b e given to what are called homogeneous Sobolev spaces, which

will play a fundamental role in the study of ﬂow in exterior domains. We shall

not a ttempt, however, to give an exhaustive treatment of the subject, since

this is beyond the scop e of the book. Therefore, the reader who wants more

details is referred to the specialized literature quoted throughout. As a rule,

we give proofs where they are elementary or relevant to the development of

the subject, or also when the result is new or does not seem to be widely

known.

II.1 Preliminaries

In this section we collect a number o f preparatory results. After i ntroducing

some basic notation, we shall recall the relevant properties of Banach spaces

and of certain classical spaces of smooth functions as well. We shall ﬁnally

deﬁne and analyze the properties of special subsets of the Euclidean space.

25G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations:

Steady-State Problems, Springer Monographs in Mathematics,