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

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366 VI Steady Stokes Flow in Domains with U nbounded Boundaries

|Σ

)| ≥ Σ

= const. > 0.

To ﬁx the ideas, we suppose that Ω has only two exits. Denote by Σ any

bounded intersection of Ω with an (n − 1)-dimensional plane, which in Ω

reduces to Σ and by n a unit vector orthogona l to Σ, oriented from Ω

toward

Ω

, say. Owing to the incompressibility of the liquid and assuming adherence

conditions at the boundary, we at once deduce that the ﬂux Φ through Σ of

the velocity ﬁeld v(x

, x

) associated with a given motion is a constant, that

is,

Φ ≡

v · n = const. (VI.0.1)

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

ﬂow subject to a given ﬂux. Clearly, this condition alone may not be enough

to determine the ﬂow uniquely and, sim ilarly to what we did for motio ns

in exterior domains, we must prescribe a velocity ﬁeld v

∞i

as |x| → ∞ in

the exits Ω

. However, unlike the case of ﬂows past a body, v

∞i

need not be

constant and, in fact, if Φ 6= 0, the corresponding v

∞i

can be a constant vector

only if

lim

|x|→∞

|Σ

)| = ∞. (VI.0.2)

To see this, we observe that if v

∞i

= const. and v(x) → v

∞i

as |x| → ∞ in

Ω

, uniformly (say), by the adherence conditions at the boundary it fol lows

that v

∞i

= 0 and so (VI.0.1) implies (VI.0.2) whenever Φ 6= 0. Thus, if |Σ

| is

uniform ly bounded, v

∞i

can not be a constant and one has to ﬁgure out how

to prescribe it. There are remarkable cases where v

∞i

is easily prescribed; this

happ ens when the exits Ω

, i = 1, 2, are cylindrical, nam ely,

) = Σ

= const.,

such as in tubes or pipes. In these situations it is reasonable to expect tha t

the ﬂow corresponding to a given ﬂux Φ should tend, as |x| → ∞, to the

Poiseuille solution of the Stokes equation in Ω

corresponding to the ﬂux Φ,

that is, to a pair (v

(i)

, p

(i)

) where

(i)

= v

(i)

, ∇p

(i)

= −C

(VI.0.3)

with C

= C

(Φ) (see Exercise VI.0.1), such that

n−1

j=1

∂

(i)

)

∂x

= −C

in Σ

= 0 at ∂Σ

(VI.0.4)

Thus, if n = 3 and the sections are circles of radius R

, the solution to (VI.0.3),

(VI.0.4) is the Hagen–Poiseuille ﬂow

VI Introduction 367

(i)

) = C

(1 − |x

Likewise, for n = 2 and Ω

a layer of depth d

, v

(i)

reduces to the Poiseuille

ﬂow

(i)

) = C

(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; see Ladyzhenskaya (1959b,

p. 175).

However, if it ha ppens that one of the sections Σ

is only uniformly

bounded but not constant, then, in general, one does not know the explicit

form of v

∞i

and, alternatively, one can prescribe at large distance in the exits

a “growth” condition (Ladyzhenskaya & Solonnikov 1980, Problem 1.1). Of

course, this condition must be such that, in the class of solutions verifying i t,

uniqueness is preserved. Moreover, once existence is established, one should

successively try to analyze the structure of solutions as |x| → ∞.

A further problem that arises when Σ is bounded is that the approach

of generalized solutions used for ﬂows in exterior domains (Theorem VI.2.1)

is not directly applicable and one has to modify it appropriately. This fact

is easily seen to be a consequence of (VI.0.1). In fact, using the Schwarz

inequality and inequality (II.5.5) we obtain

|Φ|

≤ C|Σ|

(n+1)/(n−1)

∇v : ∇v (VI.0.5)

which, for |Σ| uniformly bounded, implies an unbounded Dirichlet integral for

|v|

1,2

= ∞, unless Φ = 0.

Let us next suppose that Σ satisﬁes (VI.0.2). We may then prescribe a uniform

(zero) velocity ﬁeld v

∞i

at large distances in Ω

, i = 1, 2. We shall distinguish

the following two possibilities:

(i)

∞

|Σ

−(n+1)/(n−1)

< ∞, i = 1, 2,

(ii)

∞

|Σ

−(n+1)/(n−1)

= ∞, i = 1, 2.

In case (i) the condition of prescribed ﬂux is compatibl e with the approach of

generalized solutions, as a consequence of (VI.0.5). Nevertheless one must be

careful in choosi ng the function space where such sol utions are to be sought.

Actually, if one required v ∈ D

1,2

(Ω), by the results of Section III.5 one would

automatically impose zero ﬂux through and would therefore exclude a priori

all those solutions having Φ 6= 0. Instead, one should look for solutions in the

larger space

1,2

(Ω), where the condition Φ 6= 0 is allowed.

In case (ii) the non-zero ﬂux condition again becomes noncompatible with

the existence of generali zed solutions. However, if

368 VI Steady Stokes Flow in Domains with U nbounded Boundaries

≡

∞

|Σ|

[(1−n)(q−1)−q]/(n−1)

< ∞, i = 1, 2, some q > 2,

(VI.0.6)

since from (VI.0.1), the H¨older inequality and inequality (II.5.5)

|Φ|

≤ |v|

1,q

we deduce that now q-generalized solutions may still exist and that the “na t-

ural” space where they should be sought is

1,q

(Ω).

A last possibility arises when (VI.0.2) holds but the integrals G

are inﬁnite

for any value of q > 1. In this case it is not clear in which space the problem

has to be formulated.

Finally, we mention that, with the obvious modiﬁcations, all the above

reasonings apply to the circumstance when one section Σ

(say) is bounded

and the o ther is unbounded, as well as to the case where Ω has more than

two exits to inﬁnity.

The question of the unique solvability of the Stokes (and, more generally,

nonli near Navier–Stokes) problem in domains of the above types has been

investigated by several authors. In particular, Amick (1977, 1978) ﬁrst proved

solvability when the sections are constant (see Chapter XII), gi ving an aﬃr-

mative answer to Leray’s problem.

The case of an unbounded cross section

was ﬁrst posed and uniquely solved by Heywood (1976, Theorem 11) in the

special situation of the so-called aperture domain:

Ω = {x ∈ R

: x

6= 0 or x

∈ S} (VI.0.7)

with S a bounded domain of R

n−1

(see Section III.4.3, (III.4.4)). Successively,

under general assumptions on the “growth” o f Σ, the problem was thoroughly

investigated by Amick & Fraenkel (1980) (see also Amick (1979) and Remark

3.1) when Ω is a doma in in the plane having two exits to inﬁnity. In particular,

the authors show existence of solutions and p ointwise asymptotic decay of the

corresponding velocity ﬁelds.

. However, uniqueness is left out. It is interesting

to observe that, unlike the case of an exterior domain, for the general class of

regions o f ﬂow considered by Amick and Fraenkel, there is no Stokes paradox;

see also Section VI.2 and Section VI.4.

The entire question was independently reconsidered within a diﬀerent ap-

proach by Ladyzhenskaya & Solonnikov (1980), Solonnikov (1981,1 983) and

their associates; see Notes to this chapter. When Σ is uniform ly bounded,

these authors show, a mong other things, unique solvability in a class of solu-

tions having a Dirichlet integral that is ﬁnite on every bounded subset Ω

of Ω

and that may “grow” with a certain rate depending on Σ, as Ω

→ Ω; see also

Notice that since |Σ| ≥ Σ

> 0, in case (ii) the integrals G

are inﬁnite for any

q ≤ 2.

Under “small ” ﬂux condition in the nonlinear case; see Chapter XI.

Under “small” ﬂux condition in the nonlinear case, if Σ has a certain rate of

“growth.”

VI Introduction 369

Remark 1.1. If, in particular, the exits Ω

are cylindrical, the solution to the

Stokes problem corresponding to a given ﬂux tends in a well-deﬁned sense to

the corresponding Poiseuille solution in Ω

. Likewise, if Σ is unbounded and

satisﬁes condition (i), they prove existence of generalized solutions i n

1,2

(Ω)

corresponding to v

∞i

≡ 0 and to a prescribed ﬂux.

In case (ii), there is the remarkable contribution of Pileckas (1996a, 1996b,

1996c, 1997) who shows that in the particular case when each Ω

is a body of

revolution of the type



x ∈ R

: x

> 0, |x

| < f

)



, (VI.0.8)

the problem is uniquely solvable for any prescribed ﬂux, provided f

. satisﬁes

(VI.0.6)

and a “global” Lipschitz condition (see (ii) at the beginning of Sec-

tion 3). Furthermore, Pileckas shows that the decay rate of solutions is related

to the inverse power of the functions f

In the present chapter we prove existence and uniqueness of solutions to the

Stokes problem in a domain with exits, when these exits have either constant

sections Σ

or unbounded Σ

satisfying (i). Moreover, we shall p erform an

analysis of the poi ntwise asymptotic behavior either when Σ

is constant. We

also give some decay results when Σ

) becomes suitably unbounded as

|x| → ∞, and the exits are body of revolution as i n (VI.0.8). However, these

results are not sharp and we refer the reader to the cited papers of Pileckas

for more complete results, obtained by completely diﬀerent methods.

For simplicity, we shall describe the results in details only when the number

m of exits is two, leaving to the reader the (simple) task of generalizing them

to the case m > 2, and to the case when some of the exits are cylindrical,

while the others have an unbounded section verifying (i).

Finally, in the last section of the chapter, we shall furnish a full treatment

of the Stokes problem in the aperture domain (VI.0.7), which includes exis-

tence, uniqueness and L

-estimates of solutions together with their asymptoti c

behavior. Unlike the previously mentioned cases, for domain (VI.0.7) the sit-

uation is rendered easier by the fact that the problem can b e reduced to a

similar problem in a half space where explicit representations of solutions are

known; see Section IV.3 and Exercise IV.8.1.

Exercise VI.0.1 Show that for solutions to (VI.0.3), (VI.0.4) there is a one-to-one

correspondence between the pressure drop −C

and the ﬂux

(i)

)dx

In particular, show the existence of a positive constant c

= c

(Σ

, n) such that

= c

. The constant c

will be called the Poiseuille constant. Hint: Consider

the following problem

∆ψ

= −1 , in Σ

, ψ

∂Σ

= 0 ,

In this case, we have |Σ

| = c(n)f

n−1

370 VI Steady Stokes Flow in Domains with U nbounded Boundaries

and use the l inearity of problem (VI.0. 3) .

VI.1 Leray’s Problem: Existence, Uniqueness, and

Regularity

Let us consider a liq uid performing a steady slow motion in a domain Ω

(⊂ R

) of class C

∞ 1

with two cylindri cal ends, namely,

Ω =

[

i=0

Ω

with Ω

a compact subset of Ω and Ω

, i = 1, 2, disjoint domains, which in

possibly diﬀerent coordinate systems, are given by

Ω

= {x ∈ R

: x

< 0, x

∈ Σ

}

Ω

= {x ∈ R

: x

> 0, x

∈ Σ

Here, Σ

, i = 1, 2, are C

∞

-smooth, simpl y connected, bounded domains of

the plane, if n = 3 , while Σ

= (−d

, d

), d

> 0, if n = 2. We denote by Σ

a cross section of Ω, that is, any bounded intersection of Ω with an (n − 1)-

dimensional plane whi ch in Ω

reduces to Σ

. Moreover, n indicates a unit

vector orthogonal to Σ and oriented from Ω

toward Ω

(so that n = −e

Ω

and n = e

in Ω

The aim of this section is to solve the following Leray’s problem: Given

Φ ∈ R, to determine a solution v, p to the Stokes system

∆v = ∇p

∇· v = 0

)

in Ω (VI.1.1)

such that

v = 0 at ∂Ω

v ·n = Φ

(VI.1.2)

and

v → v

(i)

in Ω

as |x| → ∞, (VI.1.3)

Namely, for every x

∈ Ω there exists r = r(x

) such that ∂Ω ∩ B

) is a

boundary portion of cl ass C

∞

. This assumption will imply, in particular, that

the solutions we will determine are of class C

∞

. O f course, we may relax the

smoothness of Ω at the cost, however, of obtaining less regular solutions. Exten-

sion of results under weaker regularity assumptions on the boundary are left to

the reader as an exercise.

VI.1 Leray’s Problem: Existence, Uniqueness, and Regularity 371

where v

(i)

are the velocity ﬁelds (VI.0.3) and (VI.0.4) of the Poiseuille ﬂow in

Ω

corresponding to the ﬂux Φ.

We shall now give a generalized formulation of this problem, similar to

that furnished in Chapters IV and V for ﬂows in do mai ns with a compact

boundary. Multiplying (VI.1.1)

by ϕ ∈ D(Ω) and i ntegrating by parts we

deduce formally

(∇v, ∇ϕ) = 0, for all ϕ ∈ D(Ω). (VI.1.4)

We have

Deﬁnition VI.1.1. A vector ﬁeld v : Ω → R

is called a weak (or generalized)

solution to Leray’s problem (VI.1.1)–(VI.1.3) if and only if

(i) v ∈ W

1,2

loc

(Ω);

(ii) v veriﬁes (VI.1.4);

(iii) v is (weakly) divergence free in Ω;

(iv) v vanishes on ∂Ω (in the trace sense);

(v)

v · n = Φ (in the trace sense);

(vi) (v −v

(i)

) ∈ W

1,2

(Ω

), i = 1, 2.

Evidently, conditions (ii)-(v) translate in a generalized form the corre-

sponding properties (VI.1.1) and (VI.1.2), w hile (i) ensures a certain degree

of regularity. Also, it is easy to see that (vi) implies the validity of (VI.1.3) in

a well-deﬁned sense. Actually, from the trace inequality of Theorem II.4.1 we

deduce, with w ≡ v − v

(2)

|w(x

, x

≤ c

t>x



+ ∇w : ∇w)



where the constant c is independent of x

. So, by (vi),

|w(x

, x

→ 0, as |x| → ∞ in Ω

and similarly in Ω

Furthermore, it can be shown that to every weak solution we can associate

a correspo nding pressure ﬁeld p. Actually, directly from Lemma IV.1 .1 we ﬁnd

Lemma VI.1.1 Let v be a generalized solution to Leray’s problem. Then

there exists p ∈ L

loc

(Ω) such that

(∇v, ∇ψ) = (p, ∇·ψ), for all ψ ∈ C

∞

(Ω). (VI.1.5)

It is also simple to establish the smoothness of a weak solution v and the

corresponding pressure ﬁeld p. In fact, taking into account the regularity of

Ω, from Theorem IV.4.1 and Theorem IV. 5.1 we at once deduce the following

result.

372 VI Steady Stokes Flow in Domains with U nbounded Boundaries

Theorem VI.1.1 Let v be a weak solution to Leray’s problem (VI.1 .1),

(VI.1.3) and let p be the pressure associated to v by Lemma VI.1.1. Then

v, p ∈ C

∞

(Ω

), for any bounded domain Ω

⊂ Ω.

The objective of the remaining part of this section will be to prove exi stence

and uniqueness of a weak solution to Leray’s problem. To thi s end, we need

a suitable extension a (say) of the Poiseuille velocity ﬁelds v

(i)

which will

play the same role played by the ﬁeld (V.2.5 ) which, in the case of an exterior

domain, is used to extend the rigid body velocity ﬁeld V . Let us denote by

a(x) a vector ﬁeld enjoying the following properties:

(i) a ∈ W

2,2

loc

(Ω);

(ii) ∇ · a = 0 in Ω;

(iii) a = 0 at ∂Ω;

(iv) a = v

(1)

in Ω

, a = v

(2)

in Ω

, for some R > 0, where, for a > 0,

Ω

= {x ∈ Ω

: x

< −a}

Ω

= {x ∈ Ω

: x

> a}.

A way of constructing such a ﬁeld will be described. Let ζ

(x), i = 1, 2, be

functions from C

∞

) such that

(x) =







1 if x ∈ Ω

0 if x ∈ Ω − Ω

R/2

and set

V (x) =

i=1

(x)v

(i)

Clearly, V ∈ C

∞

) where

≡ Ω −

Ω

∪ Ω

Consider the problem

∇ · w = −∇ · V in A

w ∈ W

2,2

)

kwk

2,2,A

≤ ck∇ ·V k

1,2,A

Since ∇ ·V ∈ W

1,2

) and

∇· V = 0,

w exists, in view of Theorem III.3.3. Extend w to zero outside A

and denote

again by w such an extension. Evidently w ∈ W

2,2

(Ω) and so the ﬁeld

VI.1 Leray’s Problem: Existence, Uniqueness, and Regularity 373

a(x) = V (x) + w(x)

satisﬁes all requirements (i)-(iv) listed previously.

We look for a generalized solution to (VI.1.1)–(VI.1.3) of the form

v = u + a,

where

u ∈ D

1,2

(Ω).

Since, by inequality (II.5.5)

1,2

(Ω) ⊂ W

1,2

(Ω),

v satisﬁes (i) and (iii)-(vi) of Deﬁnition 1.1, while, f rom (VI.1.4), u must solve

the equation

(∇u, ∇ϕ) = (∆a, ϕ), for all ϕ ∈ D(Ω). (VI.1.6)

The existence of u is readily established by means of the Riesz representation

theorem. To this end, it suﬃces to show that the right-hand side of (VI. 1.6)

deﬁnes a linear functional in D

1,2

(Ω), i.e.,

|(∆a, ϕ)| ≤ c|ϕ|

1,2

, (VI.1.7 )

for some constant c (depending on a ) and for all ϕ ∈ D

1,2

(Ω). We split Ω as

follows:

Ω = Ω

∪



Ω −



Ω

∪ Ω



∪ Ω

≡ Ω

∪ Ω

, (VI.1.8)

and observe that in each Ω

the ﬁeld a coincides with the Poiseuille solution

(i)

satisfying (VI.0.4). Therefore, we have

Ω

∆a · ϕ =

Ω

∆v

(1)

· ϕ = −C

−∞



ϕ · ndΣ



= 0 (VI.1.9)

since ϕ carries no ﬂux. Likewise

Ω

∆a · ϕ = 0. (VI.1.10)

In Ω

, by the Schwarz inequality and inequality (II.5.5), we have



Ω

∆a · ϕ



≤ ck∆ak

2,Ω

|ϕ|

1,2,Ω

(VI.1.11)

and so (VI.1.7) follows from (VI.1.9 )–(VI.1.11) and property (i) of a . Exis-

tence is then acquired. To show uniqueness, let v

be another weak solution

corresponding to the same ﬂux Φ and Poiseuille velocity ﬁelds v

(i)

. Then, it

is readily shown that w = v −v

belongs to D

1,2

(Ω). In f act, we have in Ω

374 VI Steady Stokes Flow in Domains with U nbounded Boundaries

w = u − (v

−v

(1)

) + (a − v

(1)

) = u − (v

−v

(1)

)

and, likewise, in Ω

w = u − (v

− v

(2)

Therefore, taking into account condition (vi) of Deﬁnition VI.1.1 and that v

is i n C

∞

(Ω

), for any bounded Ω

⊂ Ω, we obtain

w ∈ D

1,2

(Ω).

Since w is zero at the bounda ry, from Exercise VI.1.1 we have

w ∈ D

1,2

(Ω)

and, w being solenoidal, we conclude

w ∈

1,2

(Ω).

However, by Exercise III.5.1,

1,2

(Ω) = D

1,2

(Ω)

so that

w ∈ D

1,2

(Ω).

This having been established, from (VI.1.4) it follows that

(∇w, ∇ϕ) = 0, for all ϕ ∈ D

1,2

(Ω),

implying w = 0 a.e. i n Ω, which is what we wanted to prove.

For a more general uniqueness result we refer the reader to Exercise VI.2.2.

Exercise VI.1.1 Let Ω be an inﬁnite “distorted pipe” of the type speciﬁed above

and let w ∈ D

1,2

(Ω). Assume that the t race of w at ∂Ω is zero. Show w ∈ D

1,2

(Ω).

Hint: Let ψ

be a C

∞

function in Ω that is one in Ω

(see (VI.1.8)) and vanishes

in Ω

, i = 1, 2. Then ψ

w → w in D

1,2

(Ω). Moreover, since w = 0 at ∂Ω, ψ

belongs to W

1,2

(Ω

02R

) and therefore it can be approximated there by functions from

∞

(Ω

02R

) ⊂ C

∞

(Ω).

In order to solve Leray’s problem completely, it remains to study the

asymptotic b ehavior of v. This will follow as a corollary to a general re-

sult concerning estimates of solutions to the Stokes problem in a semi-inﬁnite

channel which we are going to derive.

Lemma VI.1.2 Let

Ω = {x ∈ R

: x

> 0, x

∈ Σ}

VI.1 Leray’s Problem: Existence, Uniqueness, and Regularity 375

with Σ a C

∞

-smooth, bounded, and simply connected domain in R

n−1

. Given

f ∈ C

∞

(Ω

), Ω

any bounded subset of Ω, denote by u, τ ∈ C

∞

(Ω

) a solution

to the problem

∆u = ∇τ + f

∇ · u = 0

)

in Ω

u = 0 at ∂Ω − Σ

(VI.1.12)

with



x ∈ Ω : x

= 0



For s ≥ 1 and δ ∈ (0, s], set

= {x ∈ Ω : s < x

< s + 1},

s,δ

= {x ∈ Ω : s − δ < x

< s + δ + 1}.

Then, for all m ≥ 0 and q ≥ 1 the following estimate holds

kuk

m+2,q,ω

+ k∇τ k

m,q,ω

≤ c



kf k

m,q,ω

s,δ

+ kuk

1,q ,ω

s,δ



(VI.1.13)

where c = c(m, q, n, δ, Σ).

Proof. Evidently, it is enough to prove (VI.1.13) for s = 1, since the estimate

for arbitrary s ≥ 1 follows by making the change of variable x

→ x

− ξ,

ξ ≥ 0. This will autom atically imply that the constant c is independent of s.

Choose ψ ∈ C

∞

(R) such that ψ(t) = 0 fo r t ≤ 1, ψ(t) = 1 for t ≥ 2 and put

(x) = ψ



k(k + 1)x

−k

+ 2



[1 − ψ (k(k + 1)x

−k(2(k + 1 ) + 1) + 1]

with k a positive integer. Of course,

(x) =











0 if x

≤ 1 − 1/k

1 if 1 − 1/(1 + k) ≤ x

≤ 2 + 1/(1 + k)

0 if x

≥ 2 + 1/k.

We also put

= {x ∈ Ω : 1 − 1/(k + 1) ≤ x

≤ 2 + 1/(1 + k)}.

Let k

∈ N be such that

⊆ ω

1,δ

For k > k

, by setting

= ψ

u, q

= ψ

τ,

The assumptions of regularity on f , u, and τ are made f or the sake of simplicity.

Actually, for ﬁxed m ≥ 0, s ≥ 1, and δ ∈ (0, s], it would suﬃce to suppose

u ∈ W

1,q

(ω

s,δ

), τ ∈ L

(ω

s,δ

), and f ∈ W

m,q

(ω

s,δ