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

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XIII

Steady Navier–Stokes Flow in Domains with

Unbo unded B oundaries

Oν o´ι ϑεo`ι ϕιλoeυσιν, `απoϑν ´ησκει ν ´εoς.

MENANDROS

Introduction

Let us consider a steady Navier–Stokes ﬂow of a liquid ﬁlling a domain with

two unbounded “outlets.” The relevant region of ﬂow i s thus a domain Ω ⊂

, n = 2, 3 such that

Ω =

[

i=0

Ω

where Ω

is a compact set of R

while, in possibly diﬀerent coordinate systems,

Ω

= {x ∈ R

: x

> 0, x

≡ (x

, . . . , x

n−1

) ∈ Σ

)} i = 1, 2,

and Σ

= Σ

) are domains of R

n−1

smoothly varying with x

. The diﬃ-

culties one encounters in studying the mathematical properties of such a ﬂow

have already been discussed at some length in the Introduction to Chapter

VI and, therefore, they will not be repeated here. In this respect, we wish

only to recall that, even in the linearized Stokes approximation, many funda-

mental problems continue to be open when the cross sections Σ

remain either

bounded or unbounded. As expected, in the Navier–Stokes case, in additi on to

these unsolved questions, we have others that are merely due to the nonlinear

character of the equations.

897G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations:

Steady-State Problems, Springer Monographs in Mathematics,

DOI 10.1007/978-0-387-09620-9_13, © Springer Science+Business Media, LLC 2011

898 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

The objective of the present chapter is to study two characteristic prob-

lems (one with bounded cross section, the other with an unbounded one)

which, though completely solved in the linearized case, when analyzed in the

nonli near context present several basic aspects that are still far from clear.

The ﬁrst one is the so-called Leray’s problem; cf. also Section VI.1. In this

case Ω is a “di storted channel” of R

, n = 2, 3, that is, Σ

are bounded and

independent of x

, so that each outlet Ω

reduces to a semi-inﬁnite straight

cylinder fo r n = 3 a nd to a semi-inﬁnite strip for n = 2. One has to study

steady ﬂow tha t corresponds to a given velocity ﬂux Φ through the cross sec-

tion of Ω and whichin each Ω

tends to the Poiseuille ﬂow v

(i)

corresponding

to Φ. As we know, these ﬁelds satisfy

(i)

= v

(i)

n−1

j=1

∂

∂x

(i)

) = −C

in Σ

(i)

) = 0 at ∂Σ

(XIII.0.1)

with C

constants uniquely related to Φ; see Exercise VI.0.1.

The second is a problem introduced by Heywood (1976); cf. also Section

VI.5. Here Ω is an “aperture domain” of R

, namely,

Ω =



x ∈ R

: x

6= 0 or x

∈ S



with S a two-dimensional bounded domain. Therefore, the cross sections Σ

coincide with the whole of R

. The question is to determine a ﬂow correspond-

ing to a given velocity ﬂux Φ through S and whose veloci ty ﬁeld tends to zero

at l arge distances.

In both cases we wish to analyze existence, uniq ueness, and asymptotic

behavior of corresponding solutions. To solve these questions we shall use an

approach that is similar in principle to that employed i n the linear case. Of

course, we now have the further complication of the nonlinear term. This com-

plication manifests itself in several ways, whi ch we shall now brieﬂy describe.

As in the linear case, we look for a generalized solution v in the form

v = u + a.

In this relation a is a ﬂux carrier, i .e., a smooth solenoidal ﬁeld in Ω that

vanishes on ∂Ω, tends to the prescribed velocity ﬁeld at large distances, and

satisﬁes

a · n = Φ

with n unit normal to Σ, whereas

u ∈ D

1,2

(Ω).

XIII Introduction 899

As we have already seen in the case of a region of ﬂow with a compact bound-

ary, in the case at hand in order to guarantee the existence of a solution it is

enough to prove an a priori estimate for the Dirichlet integral of u. Moreover,

again as in the case of a compact boundary (cf. Sections VII.4 and IX.4), to

show existence without restrictions on |Φ|, we have to show that, for each

α > 0 there is a ﬂux carrier a = a(Φ; α) such that

−

Ω

u · ∇a · u ≤ α|u|

1,2

, for all u ∈ D(Ω). (XIII.0.2)

However, if the exits Ω

are cylindrical (or, more generally, have bounded cross

section) the existence of ﬂux carriers satisfying (XIII.0.2) is not yet known.

Nevertheless, one can construct, in such a case, ﬁelds a verifying the following

condition:



Ω

u · ∇a · u



≤ c|Φ||u|

1,2

, for all u ∈ D(Ω),

for some c = c(Ω, n). The consequence of this fact is that, unlike the linear

case, so far, one is able to produce existence of solutions to Leray’s problem

only for small values of |Φ| (compared to the coeﬃcient of kinematical viscosity

ν). The question of whether Leray’s problem is solvable for any value of the

ﬂux therefore remains open. On the other ha nd, if the outlets Ω

contain

a semi-inﬁnite cone, Ladyzhenskaya & Solonnikov (1 977) have shown that

there are vector ﬁelds a verifying condition (XIII.0.2) and, as a consequence,

for domains with this type of outl ets it is p ossible to show existence “in the

large,” that is, for arbitrary values of the ﬂux Φ. Thus, in particular, this kind

of existence holds for an “aperture domain. ”

Once a solution has been determined, the next task is to investigate its

asymptotic structure. In the case of Leray’s problem, one shows that, agai n if

|Φ| is suﬃciently small, all generali zed solutions (together with their deriva-

tives of arbitrary o rder) must tend to the corresponding Poiseuille velocity

ﬁeld exponentially fast. Similarly, for Heywood’s problem, one is able to give

a detailed asymptotic expansion, which resembles that given for the linear

case, provided |Φ| is suﬃciently small . If these problems can be solved for

arbitrary values of the ﬂux, it remai ns an open question.

Another point tha t we would like to emphasize is the two-dimensional

version of the ﬂow through an aperture. The situation is in a sense simil ar

to the plane exterior ﬂow which we have analyzed in the preceding chapter.

Speciﬁcally, in the case at hand we can prove, with no restriction on the ﬂux,

existence of vector ﬁelds v which solve the momentum equation, satisfy the

ﬂux and bo unda ry conditions and such that

Ω

∇v : ∇v < ∞. (XIII.0.3)

It is likely that in the case of the (three-dimensional) aperture ﬂow the restriction

on Φ can be removed, but no proof is known.

900 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

The diﬃculty is to show that (XIII.0.3) guarantees that v vanishes at large

distances. We shall not i nvestigate this question here, and refer the reader

to the paper by Galdi, Padula, & Passerini (1995), where it is shown that v

tends to zero unif ormly pointwise. Moreover, if the aperture S is symmetric

around the x

-axis,

and if |Φ| is suﬃciently small, it can be shown that the

solution behaves at large distances as a suitable Jeﬀery–Hamel solution, see

Galdi, Padula, & Sol onnikov (1996); see also Remark XIII.9. 5. If S i s no t

symmetric, the question of the asymptotic behavior is open.

Finally, we wish to remark that, even though obtained for pa rticular re-

gions of ﬂow, most of the results we ﬁnd could be extended without conceptual

diﬃculties to more general situations. For example, we could show existence of

generalized solutions (with no restriction on the ﬂux) in domains whose out-

lets contain and are contained in suitabl e semi-inﬁnite cones or for domains

of the type considered in Section VI.3. Likewise, we could furnish a complete

asymptotic description (for small values of the ﬂux) of generalized solutions in

domains whose outlets contain the body of revolution {|x

| < x

, α > 1}. For

other results concerning steady ﬂow in domains with unbounded b oundaries,

we refer the reader to the Notes for this Chapter.

XIII.1 Leray’s Problem: Generalized Solutions and

Rel ated Properties

Let us consider the steady ﬂow of a viscous liquid moving in a smo oth inﬁnite

“distorted channel.” We shall thus assume that the relevant region of ﬂow is

a domain Ω ⊂ R

, n = 2, 3, of class C

∞1

with two cylindri cal ends, namely,

Ω =

[

i=0

Ω

(XIII.1.1)

where Ω

is a compact subset of Ω and Ω

, i = 1, 2, a re 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, x

= (x

, . . . , x

n−1

) and Σ

, i = 1, 2, are C

∞

–smooth simply connected

domains of the plane if n = 3, while, if n = 2 (the case of plane ﬂow),

= (−d

, d

), for some d

> 0. Denote by Σ a cross section of Ω, that is,

any bounded intersection of Ω with an (n − 1)–dimensional plane tha t in Ω

Orthogonal to S.

Here applies t he same remark we made f or the linear case in Footnote 1 of Section

VI.1.

XIII.1 Leray’s Problem: Generalized Solutions and Related Properties 901

reduces to Σ

, and by n a unit vector orthogonal to Σ and oriented from Ω

toward Ω

(say).

The main objective of this and of the next three sections is to study the

solvability of the following Leray’s problem:

Given Φ ∈ IR, to ﬁnd a pair v, p

such that

ν∆v = v · ∇v + ∇p

∇ · v = 0

)

in Ω (XIII.1.2)

with

v = 0 at ∂Ω

v · n = Φ

(XIII.1.3)

and

v → v

(i)

as |x| → ∞ in Ω

(XIII.1.4)

where v

(i)

, i = 1, 2, are the veloci ty ﬁelds (XIII.0.1), of the Poiseuille ﬂow in

Ω

, corresponding to the ﬂux Φ.

We begin to give a generalized formulation of this problem, which paral lels

that furnished for the linearized case in Section VI.1. Speciﬁcally, multiplying

(XIII.1.2)

by ϕ ∈ D(Ω) and integrating by parts, we deduce

ν(∇v, ∇ϕ) = (v · ∇ϕ, v), for all ϕ ∈ D(Ω). (XIII.1.5)

We thus have the following deﬁnition

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

is called a weak (o r gener-

alized) solution to Leray’s problem (XIII.1.2)–(XIII.1.4) if and only if

(i) v ∈ W

1,2

loc

(Ω);

(ii) v satisﬁes (XIII.1.5);

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

(iv) v satisﬁes (XIII.1.3) in the trace sense;

(v) (v − v

(i)

) ∈ W

1,2

(Ω

), i = 1, 2.

Remark XI II.1.1 As shown in Section VI.1, for all v with v − v

(i)

) ∈

1,2

(Ω

) one has

|v(x

, x

) − v

(i)

→ 0 as |x

| → ∞ in Ω

Therefore, condition (v) of D eﬁnition XIII.1 .1 is the g eneralized version of

(XIII.1.4). 

For simplicity, we shall assume that no body forces are acti ng on the liquid.

Obvious generalizations are left to the reader as an exercise.

902 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

Remark XI II.1.2 Since

|(v · ∇ϕ, v)| ≤ sup

|∇ϕ|kvk

2,σ

with σ = supp ϕ, i dentity (XIII.1 .5) is well deﬁned for a generalized solution.



The following result shows that to every weak sol ution we can associate a

corresponding pressure ﬁeld.

Lemma XIII.1.1 Let v be a generalized solution to Leray’s problem (XIII.1.2)–

(XIII.1.4). Then there exi sts p ∈ L

loc

(Ω) such that

ν(∇v, ∇ψ) = (v · ∇ψ, v) + (p∇ · ψ) for all ψ ∈ C

∞

(Ω). (XIII.1.6)

Proof. In view of (i) of Deﬁnition XIII.1.1 and Corollary III.5.2 , it suﬃces to

show that

F(ψ) ≡ (v · ∇ψ, v) , ψ ∈ D

1,2

(Ω

)

deﬁnes a bo unded linear functional on D

1,2

(Ω

), with Ω

any bounded domain

of Ω. However, by the H¨older inequality and the embedding Theorem II.3.4,

we have the following.

|F(ψ)| ≤ |ψ|

1,2,Ω

kvk

4,Ω

≤ c|ψ|

1,2,Ω

kvk

1,2,Ω

with c = c(Ω

, n), and the l emma follows. ut

We end this section by establishing the diﬀerentiability properties of gen-

eralized solutions. Sp eciﬁcally, we have the following

Theorem XIII.1.1 Let v be a generalized solution to Leray’s problem

(XIII.1.2)– (XIII. 1.4) and let p be the corresponding pressure ﬁeld associated

to v by Lemma XIII.1.1. Then

v, p ∈ C

∞

(Ω

) (XIII.1.7)

for a ll bounded domains Ω

with Ω

⊂ Ω.

Proof. We show the proof f or n = 3; the case where n = 2 is treated analo-

gously. Since v ∈ W

1,2

loc

(Ω), we have

v · ∇v ∈ L

3/2

loc

(Ω) (XIII.1.8)

and, therefore, from Theorems IV.4.1 and IV.5.1 it follows that

v ∈ W

2,3/2

loc

(Ω), p ∈ W

1,3/2

loc

(Ω).

XIII.2 On the U niqueness of generalized Solutions to Leray’s Problem 903

Thus v and p have a b etter regularity than assumed at the outset and, with

the help of Theorem II.2.4 we infer

v · ∇v ∈ L

loc

(Ω) for all q < 3,

which improves on (XIII.1.8). Again using Theorem IV.4. 1 and Theorem

IV.5. 1, we ﬁnd

v ∈ W

2,q

loc

(Ω), p ∈ W

1,q

loc

(Ω), for all q < 3.

We then obtain a further improvement on the (local) summability of the

solution and iterating this procedure, we prove, by induction, the validity of

(XIII.1.7). ut

Remark XI II.1.3 If the domain Ω is not of class C

∞

, we can prove a weaker

version of (XIII.1.7 ), namely,

v, p ∈ C

∞

(Ω

)

only for every bounded Ω

with Ω

⊂ Ω. The regularity up to the boundary

will then depend on the assumed regularity of Ω, as speciﬁed by the assump-

tions of Theorem IV.5.1; cf. Footnote 1. 

XIII.2 On the Uniqueness of generalized Solutions to

Leray’s Problem

In this section we shall prove a general uniqueness result for g eneralized solu-

tions to Leray’s problem (XIII.1.2)–(XIII.1.4). To this end, we need to study

in some detail the continuity properties of the trilinear fo rm (v · ∇u, w).

We b egin to show some embedding inequaliti es in the domains Ω

, i = 1, 2.

For t ≥ 0, t

> t

> 0, we set

Ω

= {x ∈ Ω

: x

< −t}

Ω

= {x ∈ Ω

: x

> t}

Ω

0,t

= Ω − [Ω

∪ Ω

]

Ω

i,t

= Ω

∩Ω

, i = 1, 2.

(XIII.2.1)

Lemma XIII.2.1 Let Ω ≡ Ω

, and Σ ≡ Σ

, i = 1, 2, and let u ∈

m,q

(Ω

t,t+1

), m ≥ 0, q ≥ 1. Then the embedding inequalities (II.3.17)–

(II.3.18) hold in Ω

t,t+1

with constants c

, c

, and c

independent of t. More-

over, for all u ∈ D

1,2

(Ω) with u = 0 at ∂Ω − {x

= 0} we have

kuk

4,Ω

t,t+1

≤ κ|u|

1,2,Ω

t,t+1

kuk

6,Ω

t,t+1

≤ K|u|

1,2,Ω

t,t+1

(XIII.2.2)

904 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

where

κ =













|Σ| +

√

|Σ|

1/2



1/4

if n = 3





+ 1



1/4

if n = 2

and

K =







(1 +

√

|Σ|

1/2

)

1/3

if n = 3

(4κd

)

1/6

if n = 2.

Proof. The ﬁrst part of the lemma is an immediate consequence of Theorem

II.3.4. Actually, once we establish, from this theorem, inequalities (II. 3.17)–

(II.3.18) for t=0 with constants c

, i = 1, 2, 3, independent o f t, by virtue of

the translational i nvariance x

→ x

−t they remain established for all t > 0,

with the same constants c

. We shall now show the second part of the lemma.

From (II.3.9 ) (in the case n = 3) and the elementary inequality

|u(x

, x

≤

−d

|u(ξ

, x



∂u(ξ

, x

)

∂ξ



dξ

∈ (−d, d)

(in the case n = 2), we have

kuk

4,Σ

≤ λkuk

2,Σ

|u|

1,2,Σ

, (XII I .2.3)

where

λ =

(

if n = 3

2d if n = 2.

Furthermore, from (II. 5.3) we have

kuk

2,Σ

≤ µ|u|

1,2,Σ

, (XIII.2.4)

where µ is the Poincar´e constant for Σ. An upper bound for µ is obtained

from Exercise II.5.2 and (II.5.5), and we have

µ ≤











|Σ|

if n = 3

(2d)

if n = 2.

Integrating (XIII.2.3) over the variabl e x

, we derive

kuk

4,Ω

t,t+1

≤ λ

t+1

kuk

2,Σ

(XIII.2.5)

Recall that | Σ| = 2d if n = 2.

XIII.2 On the U niqueness of generalized Solutions to Leray’s Problem 905

On the other hand, for all x

∈ Σ, ξ

∈ (t, t + 1) and q ≥ 1,

|u(x

, t)|

≤ |u(x

, ξ

+ q

t+1

|u(x

, ξ)|

q−1

|∇u(x

, ξ)|dξ, (XIII.2.6)

and so, i ntegrating this latter inequality with q = 2 over x

and ξ

, with the

help of the Schwarz inequality we ﬁnd

kuk

2,Σ

≤ kuk

2,Ω

t,t+1

+ 2kuk

2,Ω

t,t+1

|u|

1,2,Ω

t,t+1

Using (XIII.2.4) in this relation f urnishes

kuk

2,Σ

≤ (µ + 2

√

µ)|u|

1,2,Ω

t,t+1

Inequality (XIII.2.2)

becomes then a consequence of the latter and of

(XIII.2.5). The case where n = 2 is treated in a completely si milar way.

Finally, we show (XIII.2 .2)

. We consider the case where n = 3, leaving to the

reader the two-dimensional case as an exercise. From (II.3.9) we have

kuk

6,Σ

≤

kuk

4,Σ

|u|

1,2,Σ

and so, integrating between t and t + 1 we ﬁnd

kuk

6,Ω

t,t+1

≤

t+1

kuk

4,Σ

|u|

1,2

Using (XIII.2.6) with q = 4 and integrating over x

∈ Σ and ξ

∈ [t, t + 1], it

follows that

kuk

4,Σ

≤ kuk

4,Ω

t,t+1

+ 4

Ω

t,t+1

|u|

|∇u|

and so, observing that from the H¨older inequality and (XIII.2.4),

kuk

4,Ω

t,t+1

≤ kuk

6,Ω

t,t+1

kuk

2,Ω

t,t+1

≤

√

µkuk

6,Ω

t,t+1

|u|

1,2,Ω

t,t+1

Ω

t,t+1

|u|

|∇u| ≤ kuk

6,Ω

t,t+1

|u|

1,2,Ω

t,t+1

from the last four displayed inequalities and recalling the value of µ, we con-

clude that

kuk

6,Ω

t,t+1

≤



1 +

|Σ|

1/2

√



kuk

6,Ω

t,t+1

|u|

1,2,Ω

t,t+1

and (XIII.2.2)

follows. The lemma is proved. ut

Next we shall investigate the continuity of the trili near form (v ·∇u, w)

in suita ble function spaces. To this end, set

= {v ∈ W

1,2

loc

(Ω) : |v − v

(i)

1,2,Ω

< ∞, i = 1, 2}

where, we recall, v

(i)

is the (uniquely determined) velocity ﬁeld of the

Poiseuille ﬂow in Ω

corresponding to the ﬂux Φ.