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

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

Taking into account that F is smooth and of compact support, reasoning as

in the proof of Lemma VI.5.1(ii) from (XIII.9.33)–(XIII.9.34) we then ﬁnd for

all x ∈ R

(x) = −

(x, y)F

(y)dy + 2ν

(y)

∂g

(x, y)

∂x

dσ

, (XIII.9.3 5)

where, for simplicity, we set g = g

. Our objective is to let ϕ

→ 1 into

(XIII.9.35) or, equivalently, R → ∞. To this end, we recall that g enjoys the

uniform estimate

g(x, y)| ≤

|x − y|

|α|+2

, |α| ≥ 0; (XIII. 9.36)

cf. (IV.3.50), (IV.3.51). Therefore, for all x ∈ R

we have

|g(x, y) ·v(y)∆ϕ

(y)|dy ≤ c

(x)

|v(y)∆ϕ

(y)|

|x − y|

−B

(x)

|v(y)∆ϕ

(y)|

|x −y|

where d < x

. The H¨older inequality, with r > 3 and s < 3/2, yields

(x)

|v(y)∆ϕ

(y)|

|x − y|

dy ≤

(x)

|x −y|

−2r

1/r

k∆ϕ

−B

(x)

|v(y)∆ϕ

(y)|

|x − y|

dy ≤

−B

(x)

|x − y|

−2s

1/s

k∆ϕ

(XIII.9.37)

and so, in view of (XIII.9.30) from (XIII.9.37) we conclude

lim

R→∞

g(x, y) · v(y)∆ϕ

(y)dy = 0. (XIII.9.38)

By similar reasonings, which we leave to the reader, we show

lim

R→∞

g(x, y) · ∇v(y) · ∇ϕ

(y)dy = 0

lim

R→∞

g(x, y) · ∇ϕ

(y)(p(y) − p

)(y)dy = 0

lim

R→∞

v(y) · ∇v(y) · g(x, y)ϕ

(y)dy =

v(y) · ∇v(y) ·g(x, y)dy.

(XIII.9.39)

XIII.9 Aperture Domain: Asymptotic Structure of Generalized Solutions 967

Collecting (XIII.9.27)

, (XIII.9.32), (XIII.9.3 4), (XIII.9.38), and (XIII.9.39)

we arrive at the following representation for p

p(x) = p

−

(x, y)v

(y)D

(y)dy+2ν

(y)

∂g

(x, y)

∂x

dσ

. (XIII.9.40)

Using the mean value theorem, together with (XIII.9.3 6), from (XIII.9.40) we

ﬁnd

p(x) = p

−

(x, y)v

(y)D

(y)dy + 2ν

∂g

(x, 0)

∂x

+ O(|x|

−4

(XIII.9.41)

Our next task is to furnish an estimate of the volume i ntegral I on the right-

hand side of (XIII.9.41). To this end, we wri te

I =

−B

(x)

(x, y)v

(y)D

(y)dy +

(x)

(x, y)v

(y)D

(y)dy.

Integrating by parts the ﬁrst integral, we ﬁnd

I = −

−B

(x)

(x, y)v

(y)v

(y)dy +

(x, y)v

(y)v

(y)n

(y)dσ

∂B

(x)

(x, y)v

(y)v

(y)n

(y)dσ

(x)

(x, y)v

(y)D

(y)dy

and so, employing the estimates of Theorem XIII.9.1 and Exercise XIII.9.2

we deduce tha t

I = −

−B

(x)

(x, y)v

(y)v

(y)dy +

(x, y)v

(y)v

(y)n

(y)dσ

+O(|x|

−4

), as |x| → ∞.

(XIII.9.42)

However, from (IV.3.46)

3,4

and (IV.3.48)

5,6

, it follows that g(x, y)|

= 0

for a ll x ∈ R

and from (XIII.9.42) we conclude that

I = −

−B

(x)

(x, y)v

(y)v

(y)dy+O(|x|

−4

), as |x| → ∞. (XIII.9.43)

Let us denote by I

the integral in (XIII.9.43). We have

= D

(x, 0)

−B

(x)

(x, y) − D

(x, 0)] v

(y)v

(y)dy

−D

(x, 0)

(x)

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

and so, by Theorem XIII.9.1 and (XIII.9.36) we ﬁnd

= D

(x, 0)A

−B

(x)

(x, y) − D

(x, 0)] v

(y)v

(y)dy

+ O(|x|

−4

)

(XIII.9.44)

where A

is deﬁned in Theorem XIII.9.1. By (XIII. 9.36) and the mean value

theorem we have

(x, y) − D

(x, 0)| ≤ c

|y|

|x −βy|

As a consequence, setting |x| = 2R, it follows that

Ω

(x, y) −D

(x, 0)|v

(y)dy ≤

|x|

Ω

(y)|y|dy.

Reasoning as in the proof of (XIII.9.21) we then show

Ω

(x, y) − D

(x, 0)|v

(y)dy = O(|x|

−4+δ

) (XIII.9.45)

where δ > 0 can be taken arbitrarily close to zero. Furthermore, aga in by

(XIII.9.36) and Theorem XIII.9.1, we have

(x, 0)|

Ω

≤

|x|

4−δ

Ω

3/2+δ

(y)dy = O(|x|

−4+δ

) (XIII.9.46)

and

Ω

−B

(x)

(x, y)|v

(y)dy ≤

|x|

4−δ

|x−y|≥1

|x − y|

|y|

= O(|x|

−4+δ

(XIII.9.47)

Collecting (XIII.9.41) and (XIII.9.43)–(XIII.9 .47) we then obtain the fol-

lowing result.

Theorem XIII.9.2 Let the ﬂux Φ satisfy the assumption of Theorem XIII.7.4.

Then the pressure ﬁeld associated to any generalized solution to (XIII.5.2)

that obeys the energy inequality (XIII.6.6) admi ts the following asymptotic

representation as |x| → ∞, x ∈ R

p(x) = p

+ D

(x, 0)A

+ 2νD

(x, 0)B

+ P (x) (XIII.9.48)

where p

and A

, B

are deﬁned in Lemma XI II.5. 1 and Theorem XIII.9.1,

respectively, while g

is the vector “pressure ﬁeld” associated to the Green’s

tensor for the Stokes problem in the R

; see (IV.3.46)–(IV.3.49). Final ly,

P (x) = O(|x|

−4+δ

with δ a positive number that can be taken arbitrarily close to zero.

XIII.9 Aperture Domain: Asymptotic Structure of Generalized Solutions 969

Remark XI II.9.3 From (XIII.9.36) and (XIII.9.48) we derive, in particular,

the following uniform asymptotic estimate

p(x) − p

= O(|x|

−3

), |x| → ∞, x ∈ R



Remark XI II.9.4 A consideration similar to that performed in Remark

XIII.9.1 applies to the representation formula (XIII.9 .48). 

Exercise XIII.9.1 Let v be any generalized solution to problem (XIII.5.2). Show

that v and the corresponding pressure ﬁeld p obey the asymptotic conditions (9 .1).

Hint: Consider (X III. 5.2) as a Stokes problem in R

with a body force −v ·∇v. Then

apply the estimate of Theorem IV.3.2, Theorem IV.3.3 together with the embedding

inequalities of Theorem II.3.4.

Exercise XIII.9.2 Let the assumptions of Theorem XIII.9.1 be fulﬁlled. Show that

the ﬁrst derivatives of the velocity ﬁeld v satisfy the following asymptotic estimate.

v(x) = O(|x|

−2

), |x| → ∞, k = 1, 2, 3.

Hint: From the representation formula (XIII.9.3)–(XIII.9.4) we have

(x) =

(x, y)v

(y)D

(y)dy + B

, g

)(x, 0) + ϕ

(x).

Diﬀerentiate both sides of this relation and argue as in the proof of (X.8.28). Finally,

use the asymptotic estimates given in (IV.3.50)

and in Lemma XIII.9.3.

Remark XI II.9.5 In the case of a two-dimensional aperture dom ain, the

asymptotic structure of any generalized solution v obeying the energy inequal-

ity and corresponding to small ﬂux Φ is known if the aperture S is symmetric

around the axi s orthogonal to S, see Galdi, Padula, & Solonnikov (1996); see

also Nazarov (1996). In such a case, v and the associated pressure p admit

the following representation for large |x|

v(x) = V

(x) + O(|x|

−1−α

)

p(x) = P

(x) + O(|x|

−2−α

)

x ∈ R

, α ∈ (0, 1),

where V

, P

is a suitable Jeﬀery–Hamel solution for the half-plane R

, cor-

responding to Φ (Rosenhead, 1940). Speciﬁcally, in a po lar coordinate system

(r, θ

) with the origin at x = 0 and θ counted from the positive [respectively,

negative] x

-axis, we have

(x) = ±Φ

(θ)

(x) = ±Φ

P(f

)

+ const.

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

Here, e

is the unit vector in the r direction, f

(θ) is a symmetric solution to

a suitable second-order nonlinear ordinary diﬀerential problem with

max

θ∈[0,2π]



(θ)|, |f

(θ)|



suﬃciently small, and P a known function of f

. 

XIII.10 N otes for the Chapter

Section XI II.1. The ﬁrst, signiﬁcant contribution to the solvabili ty of Leray’s

problem is due to Amick (1977, 1978)

who reduced the proof of existence to

the resolutio n of a well-known variational probl em relating to the stability

of Poiseuille ﬂow in a pip e; see Remark XIII.3 .5. Our deﬁnition of general-

ized solution is essentially taken from his work. However, Amick left out the

investigation of uniqueness while he examined the asymptotic structure of so-

lutions only in some particular cases. A rich and detailed analysis of Leray’s

problem and, more g enerally, of the problem of ﬂow in domains having outlets

to inﬁnity of bounded (not necessarily constant) cross sections is due to the St

Petersburg School. We refer, in particular, to the papers of Ladyzhenskaya &

Solonnikov (1980), Solonnikov (1983), Kapitanski

i (1982), Nazarov & Pileckas

(1983, 1990 ), Kapitanski

i & Pileckas (198 3) and the literature cited therein.

One of the main features of this approach is, as we already observed, that

condition (XIII.1.4 )

is replaced with suitable “growth” assumptions at large

distances.

Existence of solutions in two-dimensional, suitably symmetric channels

with “islands” has been investigated by Morimoto & Fujita (2002). In this

connection, we refer the reader to the other contributions by the same authors

provided in the bibl iography cited therein, and pertaining to more general ﬂow

regions, as well as to the review article of Morimoto (2007).

A diﬀerent approach to existence (and uniqueness) in a two-dimensional

inﬁnite channel, based on the theory of semi-Fredholm operators, has been

taken by Rabier (2002a, 2002b). More precisely, this author shows the exis-

tence of solutions “around” a Poiseuille ﬂow corresponding to any given ﬂux Φ

and to symmetric data, whereas for data that are not necessarily symmetric,

the same result holds provided 3|Φ|/(4ν) < 4647.

The problem o f existence and uniqueness of ﬂow in curved pipes has been

addressed by Galdi & Robertson (2 005).

Section XIII.2. The method used here has been inspired by the work of La-

dyzhenskaya & Solonnikov (1980). However, all main results, such as Theorem

XIII.2.1, are due to me.

A previous attempt of Ladyzhenskaya, unfortunately, failed, cf. Dokl. Akad. Na uk

SSSR, 124 (1959) 551-553.

XIII.10 Notes for the Chapter 971

Section XIII.3. The methods and results of this section are due to me.

Section XIII.4. The main results of this section, Theorem XIII.4.1 and

Corollary XIII.4.1, are due to me. They are obtained by coupling the ideas of

Horgan & Wheeler (1978) with those of Ladyzhenskaya & Solonnikov (1980).

In particular, Lemma XIII.4 .2 and Lemma XIII.4.3 are sp ecial cases of analo-

gous results of Ladyzhenskaya & Sol onnikov. Som ewhat weaker estimate can

be obtained from the work of Horgan (1978) and Ames & Payne (1989). Spa-

tial decay estimates in outlets with cross sections that reduce to a point at

large distances have been obtained by Iosif‘jan (1979).

In the case n = 2, the straight semi-inﬁnite cylinder reduces to the semi-

inﬁnite channel {x

∈ (0, d) , x

> 0}, where the decay of the solution (to the

appropriate data at inﬁnity) has been shown to be of exponential type, for

small data at l east. It is interesting to observe that if, instead, we consider the

three-dimensional semi-inﬁnite layer, {x

∈ (0, d) , x

∈ R , x

> 0}, solutions

only show a power-like decay (for small data), that turns o ut to be optimal;

see Pileckas (2002), Nazarov & Pileckas (1999b).

Section XIII.5. The aperture ﬂow problem was formulated by Heywood

(1976, §6). To him is due the generalized formulation given in Deﬁnition

XIII.5.1.

Properties of the pressure ﬁeld in weighted spaces, ana logous to those

furnished in Lemma XIII.5.1 in L

-spaces, are proved by Borchers, Galdi, &

Pileckas (1 993, Theorem 3.1).

Section XIII.6. Results of this section are due to me. The idea of exploiting

the energy inequality in the study of uniqueness is taken from Borchers, Galdi,

& Pileckas (1993).

Section XIII.7. Existence of generalized solutions for the ap erture ﬂow prob-

lem was ﬁrst proved by Heywood (1976, Theorem 17) under the assumption

of small ﬂux. This restriction was successively removed by Ladyzhenskaya &

Solonnikov (1977). Other signiﬁcant results in this direction, in classes of so-

lutions that are more regular than weak ones at large di stances, are due to

Borchers & Pileckas (1992) and Chang (1993); cf. a lso Coscia & Patria (1992),

Chang (1992). In particular, Borchers & Pi lecka s (1992) prove existence for

the formulation where the ﬂux condition is replaced by the prescription of the

pressure drop p

− p

−

Lemma XIII.7.1 and Theorem XIII.7.4 are due to me. Similar results can

be found in Borchers & Pileckas (1992) and Borchers, Galdi, & Pil eckas (1993).

There is much literature concerning steady ﬂow in domains with exits

having unbounded cross sections, more general tha n aperture domains. In

addition to the papers already quoted, we wi sh to mention the work of Pileckas

(1981, 1984) and Gal di & Sohr (1992). Special regards are deserved by the

contribution of Amick & Fraenkel (1980); cf. also Amick (1979). Here the

authors analyze existence and pointwise asymptotic decay of plane ﬂow in

domains with two outlets Ω

of unbounded cross section Σ

, i = 1, 2, under

very general assumptions on the way in which Σ

may g row at large distances.

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

For instance, Ω

can b e a symmetric channel of the type

Ω



x ∈ R

: x

∈ Σ

), x

> 0



(∗)

with

= (−x

, x

), k ≥ 0. (∗∗)

Existence is proved either with small or arbitrary ﬂux, depending on the rate

at which Σ

widens at inﬁnity (that i s, in case (∗), (∗∗), on the value of k).

It is interesting to observe that, as expected (cf. Remark VI.4.6), if this rate

is too slow, the pressure ﬁeld becomes unbounded at inﬁnity. Such a detailed

study is made possible since, in the two-dimensional case, it is possible to use

conformal-mapping techniques. Important questions, however, left out of this

work are uniq ueness and decay order of solutions.

The problem of existence, uniqueness and asymptotic behavior of solutions

in domains Ω with m ≥ 2 outlets Ω

, i = 1, . . .m, whose cross-section becomes

unbounded in such a way that the Dirichlet integral of the velo ci ty ﬁeld is

inﬁnite, has been recently studied and solved in a series of papers by Pileckas

(1996a , 1996b, 1996c, 1997) and Nazarov & Pileckas (1997), in the particular

case when each Ω

is a body of revolution of the type

{x ∈ R

: x

> 0, |x

| < g

)}, n = 2, 3.

Here g

) are smooth functions satisfyi ng the following “growth” conditions

(i) g

(t) ≥ g

= const. > 0 ; |g

) − g

)| ≤ M|t

− t

(ii) lim

t→∞

= 0 ;



≤ M

(iii)

∞

−(n−1)(q−1)−q

(t)dt < ∞, some q > 2;

∞

−(n+1)

(t)dt = ∞.

Notice that (iii) i mplies that the velocity ﬁeld has an inﬁnite Diri chlet in-

tegral, see the Introduction to Chapter VI. Moreover, requirements (i) and

(ii) exclude the case of aperture domains Then, for n = 2, if the total ﬂux

is suﬃciently small one shows the existence o f a unique solution and the va-

lidity of corresponding sharp asymptotic estimates. For n = 3 existence and

asymptotic decay are shown for arbi trary values of the total ﬂux, provided g

satisﬁes the further conditions

∞

4/3

(t)dt = ∞



1/3

(t)



≤ γ  1, for suﬃciently large t.

As expected, uniqueness only holds for small ﬂux. For detailed result and

further references, we refer the reader to the review article of Pileckas (1996c).

XIII.10 Notes for the Chapter 973

Section XIII.8. The results of this section are due to me. Results similar

to those proved in Lemma XIII.8.2 are given by Borchers & Pileckas (1992)

under the following assumption on the velocity ﬁeld v

Ω

|x|

1+ε

∇v : ∇v < ∞, fo r some ε > 0.

Section XIII.9. Asymptotic representation formulas sim ilar to (XIII.9.25)

and (XIII.9.48) are f urnished by Borchers & Pileckas (1992). Results of the

same type as those showed in Theorem XIII.9.1 and Theorem XIII.9.2 can be

derived from Borchers, Galdi, & Pileckas (1993, Theorem 4.1).

Existence, uni queness, and asymptotic behavior of solutions to the plane

aperture domain problem in the case of symmetric aperture have been given,

independently, by Nazarov (1996) and Galdi, Padula, & Solonnikov (1996).

Finally, existence, uniqueness, and asymptotic behavior of solutions in

two-dimensional domains constituted by a semi-inﬁnite channel merging into

a half-plane, have been addressed by Nazarov, Sequeira, & Videman (2001,

2002).

E ’a canzuncella

d’ ’aruta nuvella,

guardate, vedite . . . fenesce accussi’.

S. DI GIACOMO, A’ Te sta d’Aruta, vv. 42-44

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