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

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

(∇w, ∇ψ) = (U ·∇ψ, w) for all ψ ∈ D(Ω)

w ∈ D

1,2

(Ω).

(XIII.8.33)

Using the density property of D(Ω) into D

1,2

(Ω) along with Lemma X.2.1,

we may replace ψ with w in (XIII.8 .32) to derive

− |w|

1,2

= (U · ∇w, w). (XIII.8.34)

However, U is (weakly) divergence-free and, as a consequence,

(U · ∇w

, w

) = 0

along a sequence {w

} ⊂ D(Ω) approximating w in D

1,2

(Ω). Thus, again by

Lemma X.2.1, we obtain

(U · ∇w, w) = 0

and (XIII.8.32)

, (XIII.8.3 4) imply w ≡ 0 a.e. in Ω. From (XIII.8.2 4)

then ﬁnd

→ 0 strongly in L

4q/(4−q)

(Ω

which contradicts (XIII.8.23). The vali dity of (XIII.8.20) is therefore estab-

lished and (XIII.8.19) furnishes that, fo r all k ∈ N, there exists a generalized

solution w

, π

to (XIII.8.10) satisfying the estimate

1,2

+ kw

3q/(3−q)

+ |w

1,q

+ kπ

≤ c(hhv

∗

1/2,2

+ hhv

∗

1−1/q,q

)

(XIII.8.35)

with a constant c i ndependent of k. As a consequence, from {w

, π

} we can

select a subsequence, denoted a gain by {w

, π

}, and ﬁnd a pair {w, π} obey-

ing the property (XIII.8.24). Taking into account (XIII.8.25 )

and reasoning

as we did to recover (XIII.8.32) we show that {w, π} is a generalized solution

to the problem

∆w = v ·∇w + ∇π

∇ · w = 0

)

in Ω

w = v

∗

at ∂Ω

(XIII.8.36)

such that

w ∈ D

1,q

(Ω) ∩ L

3q/(3−q)

(Ω) ∩ D

1,2

(Ω)

π ∈ L

(Ω).

(XIII.8.37)

However, v, p is also a generalized solution to (XIII.8.36) and by assumption,

(XIII.8.36), and (XIII.8.37) we conclude that the diﬀerences

w ≡ v − w, π = p − π

XIII.8 Aperture Domain: Summability Properti es of Generalized Solutions 957

solve the generalized form of (XIII.8.36) with v

∗

≡ 0. Since w ∈ D

1,2

(Ω),

reasoning as we did to show that (XIII.8.33) admits only the i dentically van-

ishing generalized solution, we prove w ≡ 0, π ≡ const. a.e. in Ω. We then

conclude that

v ∈ D

1,q

(Ω) ∩ L

3q/(3−q)

(Ω), for all q ∈ (6/5, 3/2]. (XIII.8.38)

Since 3q/(3 − q) (> 2) can be chosen as close to 2 as we pl ease by picking

q (> 6/5) suitably clo se to 6/5, and since

|v · ∇v|

−1,r

≤ kvk

we may assert that

v · ∇v ∈ D

−1,q

(Ω) for all q ∈ (1, 3/2]

and, as a consequence, by Theorem V.3.3 it fo llows that

v ∈ D

1,q

(Ω) ∩ L

3q/(3−q)

(Ω) for all q ∈ (1, 3/2]. (XIII.8.39)

Since v ∈ D

1,q

(Ω), the lemma follows from (XIII.8.39) after application of

the interpol ation inequality (II.2.7). ut

Combining Lemma XIII.8.1, Theorem XIII.7.4, and Corollary XIII.7.1, we

obtain the (ﬁrst part of the) following result concerning (global) summabil-

ity properties of generalized sol utions to problem (XIII.5.2) in the aperture

domain Ω.

Theorem XIII.8.1 Let the ﬂux Φ satisfy the assumption of Theorem XIII.7.4.

Then any corresponding g eneralized solution v to (XIII.5.2) that obeys the

energy i nequality (XIII.6.6) enjoys the following summability properties:

v ∈ D

1,q

(Ω) ∩ L

3q/(3−q)

(Ω) for all q ∈ (1, 2].

Moreover, denoting by p the pressure ﬁeld associated to v by Lemma XIII.5.1

we have

p − p

∈ L

) for all s ∈ (1, 3],

where the constants p

are deﬁned in Lemma XIII.5.1.

Proof. We have to show only the summability of p. However, since

v · ∇v ∈ D

−1,s

(Ω) for all s ∈ (1, 3],

the stated property follows at once from Theorem VI.5.1. ut

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

XIII.9 Asymptotic Structure of Gener alize d Solutions

for Flow in an Aperture Domain

Using the results on the Stokes problem in the half-space as given in Theorem

IV.3. 2 and Theorem IV.3.3, it is a straightforward exercise to show that every

generalized solution to problem (XIII.5.2) satisﬁes the following asymptotic

properties

lim

|x|→∞

v(x) = 0

lim

|x|→∞,x∈R

(p(x) − p

) = 0,

(XIII.9.1)

for all |α| ≥ 0; cf. Exercise XIII.9.1. However, nothing can be said ab out the

order of decay, unless we put some suitable restriction on the magnitude of

the ﬂux Φ of the velocity ﬁeld through the aperture S. Actually, under the

hyp othesis of Theorem XIII.7.4, it is a relatively simple task to determine

the precise asymptotic structure of a ny corresponding generalized sol ution

v and of the associated pressure ﬁeld p. In showing this, we shall follow an

approach entirely analogous to that employed in similar circumstances for ﬂow

in exterior domains; cf. Section X.8 and X. 6.

First of all, we need a representation for v. By Theorem XI II.7.4, any

generalized solution corresponding to “small” Φ satisﬁes the condition

∈ L

(Ω), l, i = 1, 2, 3,

and so, by Lemma VI.5.1 , it follows that for all x ∈ R

v(x) admits the

representation

(x) = −

(x, y)v

(y)v

(y)dy

−

(y)T

, g

)(x, y)n

(y)dσ

(XIII.9.2)

where G

is the Green’s tensor for the Stokes problem in R

and g

is the

associated “pressure” ﬁeld. Relation (XI II.9.2) is the starting point of our

asymptotic analysis. Since what we shall say equally applies to R

and R

−

to ﬁx the i deas we shall deal with R

. Setting, for simplicity, G

= G and

= g, we begin to notice that from (XIII.9.2) we derive, in particular, for

all x ∈ R

(x) = −

(x, y)v

(y)v

(y)dy + T

, g

)(x, 0)

(y)dσ

−

, g

)(x, y) −T

, g

)(x, 0)] v

(y)n

(y)dσ

XIII.9 Aperture Domain: Asymptotic Structure of Generalized Solutions 959

Recalling the estimates (IV.3.50), from the latter relation, it follows for all

suﬃciently large x ∈ R

that

(x) = −

(x, y)v

(y)v

(y)dy + B

, g

)(x, 0) + ϕ

(x)

(XIII.9.3)

where

, ϕ

(x) = O(|x|

−3

). (XIII.9.4 )

Our next objective is to give an asymptotic estimate of the integral term in

(XIII.9.3). Speciﬁcally, we shall show that this term can be split as the sum of

two terms: the ﬁrst is proportional to D

(x, 0) and the second is O(|x|

−3+δ

for arbitrary positive δ. To reach this goal , we need some preliminary results.

For R > r > δ(S) and with the origin of coordinates in the interior of S, we

set

Ω

= R

∩ B

, a = r, R

Ω

r,R

= Ω

− Ω

Ω

= R

− Ω

a = r, R

and

G(R) =

Ω

∇v : ∇v. (XIII.9.5)

We have the foll owing.

Lemma XIII.9.1 Let the assumptions of Theorem XIII.7.4 be satisﬁed.

Then the generalized solution v veriﬁes

G(R) ≤ cR

−1

with c independent of R.

Proof. Without loss of generality, we set p

= 0. Multiply (XII I.5.2 )

by v,

integrate by parts over Ω

R,R

, R

> R, and use (XIII.5.2)

. We thus obtain

Ω

R,R

∇v : ∇v =

∂B

∪∂B



νv ·

∂v

∂n

− pv · n −

v ·n



. (XIII.9.6)

Since, by Theorem XIII.7.4,

v ·

∂v

∂n

, pv, v

∈ L

), (XIII.9.7)

letting R

→ ∞ along a suitable sequence into (XIII.9.6), we ﬁnd

νG(R) =

∂B



νv ·

∂v

∂n

−pv · n −

v · n



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

In view of (XIII.9.7) we have

G ∈ L

(R, ∞), for all R > δ(S),

and since G

(R) < 0, the result follows from Lemma X.8 .1. ut

Lemma XIII.9.2 Let the assumptions of Theorem XIII.7.4 be fulﬁlled. Then

the generali zed solution v satisﬁes

v(x) = O(|x|

−1

), as |x| → ∞ i n R

Proof. Set |x| = 2R, for R > δ(S). From (XIII.9.3)–(XIII.9.4) and (IV.3.50)

it follows that

|v(x)| ≤ c



Ω

(y)

|x − y|

dy +

Ω

(y)

|x −y|



+ O(|x|

−3

). (XIII.9.8)

Since, by Theorem XIII.7.4, v ∈ L

(Ω), we deduce

Ω

(y)

|x − y|

dy ≤ c

|x|

−2

. (XIII.9.9)

Furthermore, let δ(S) < ρ/2 < ρ < R and let ψ

be a smooth function in R

with

(x) =

(

0 if |x| ≤ ρ/2

1 if |x| ≥ ρ.

Extending v(x) by zero for x ∈ R

−

and applying inequali ty (II. 6.20) to the

function ψ

(x)v(x), we obta in

Ω

(y)

|x − y|

dy ≤ c

Ω

∇v : ∇v

with c

independent of R. Thus, from (XIII.9.5) and Lemma XIII.9.1 we

deduce

Ω

(y)

|x − y|

dy ≤ c

|x|

−1

and the lemma follows from this inequality, (XIII.9 .8), and (XIII.9.9). ut

Lemma XIII.9.3 Let the assumptions of Theorem XIII.7.4 be fulﬁlled. Then

the generali zed solution v satisﬁes

v(x) = O(|x|

−2

), as |x| → ∞ i n R

XIII.9 Aperture Domain: Asymptotic Structure of Generalized Solutions 961

Proof. From (XIII.9.3)–(XIII.9.4) and (IV.3.50) we see that it is enough to

show that for large |x| in R

(x) =

(x, y)v

(y)v

(y)dy = O(|x|

−2

) (XIII.9.10)

holds. From (IV.3.5 0) it follows, in parti cular, that

(x, y)| ≤

|x − y|

1+|α|

, 0 ≤ |α| ≤ 2, (XIII.9.11)

and so

|N(x)| ≤ c

(y)

|x − y|

. (XII I .9.12)

We split the regi on of integration R

into three subregions Ω

, Ω

R,3R

, and

Ω

, where R = |x|/2, and denote by I

, I

, and I

the three corresponding

integrals. Since

|x| ≤ 2 |x − y|, for y ∈ Ω

we ﬁnd

≤

|x|

, (XII I .9.13)

where, in the last step, we have used the property v ∈ L

(Ω); cf. T heorem

XIII.7.4. Also,

|y| ≤ 3|x − y|, for y ∈ Ω

and so

≤ 3c

Ω

(y)

|y|

Recalling that v ∈ L

(Ω), we conclude

≤

Ω

|x|

. (XIII.9.14)

Concerning the estimate for I

, we have

(y)

|x −y|

dy +

(x)

(y)

|x −y|

dy, (XIII.9.15)

where

= Ω

R,3R

− B

(x).

By Lemma XIII.9.2,

v(x) = O(|x|

−1

) (XIII.9.16)

and so

(x)

(y)

|x − y|

dy ≤

|x|

(x)

|x − y|

|x|

. (XIII.9. 17)

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

Furthermore, again by (XIII.9.16), for any positive η we have

(y)

|x −y|

dy ≤

|x|

3/2−η

1/2+η

(y)

|x − y|

and so, employing the H¨older inequality,

(y)

|x −y|

dy ≤

|x|

3/2− η



|x − y|



1/q



Ω

(

+η)q



1/q

By Theorem XIII.7.4, it follows that v ∈ L

(Ω) for al l r > 3/2 and arbitrarily

close to 3/2. Therefore, the preceding inequality yields

(y)

|x −y|

dy ≤

|x|

3/2−δ

(XIII.9.18)

where δ > 0 is arbitrarily close to zero. Collecting (XIII.9.10), (XIII.9.12)–

(XIII.9.15), (XIII.9.17), and (XIII.9.18), with the help of (XIII.9.3)–(XIII.9.3)

we derive

v(x) = O(|x|

−3/2+δ

). (XIII.9.19)

With this improved estimate on v we can give an i mproved estimate of the

ﬁrst integral on the right-hand side of (XIII.9.15). Actually, from (XIII.9.19)

we ﬁnd

(y)

|x −y|

dy ≤

|x|

(2−4δ)/ ( 3−2δ)

(y)

|x −y|

dy.

Thus, applying the H¨older inequal ity and recalling that v ∈ L

(Ω) for all

r ∈ (3/2, 6] we a rrive at

(y)

|x −y|

≤

|x|

and the proof of the lemma is complete. ut

Lemma XIII.9.4 Let the assumptions of Theorem XIII.7.4 be fulﬁlled. Then

the generali zed solution v satisﬁes as |x| → ∞ in R

(x, y) − D

(x, 0)] v

(y)v

(y)dy = O(|x|

−3+δ

with δ an arbitrary small positive number.

Proof. We denote by I the integral we want to estimate and write

I =

Ω

(x, y) − D

(x, 0)]v

(y)v

(y)dy

Ω

(x, y) − D

(x, 0)] v

(y)v

(y)dy

≡ I

+ I

(XIII.9.20)

XIII.9 Aperture Domain: Asymptotic Structure of Generalized Solutions 963

where |x| = 2R. From the mean value theorem and (XIII.9.11) we deduce

(x, y) − D

(x, 0)| = |y

(x, βy)| ≤

c|y|

|x − βy|

, β ∈ (0, 1).

We have

|x| ≤ 2|x −βy|, y ∈ Ω

and so

| ≤

|x|

Ω

(y)|y|dy.

From Lemma XIII.9.3 and the H¨older inequality we ﬁnd

| ≤

|x|

|Ω

1/q

kvk

3/2

3q/2,Ω

and since, by Theorem XIII.7.4, we can take q arbitrarily close to 1, we derive

| ≤

|x|

3−δ

(XIII.9.21)

with δ > 0 arbitrarily close to zero. Moreover, from (XIII.9.1 1) it follows that

| ≤

Ω

(x, y)v

(y)v

(y)|dy + |D

(x, 0)|

Ω

(y)dy

≤ c



Ω

(y)

|x −y|

dy +

|x|

Ω



(XIII.9.22)

Therefore, using the estimate on v g iven in Lemma XIII.9.3 and then applying

Lemma I I.9.2, we infer that

Ω

(y)

|x − y|

dy ≤

|x|

|x − y|

|y|

≤

|x|

. (XIII.9.23)

Likewise, again from Lemma XIII.9.3 and the property v ∈ L

(Ω) for all

r ∈ (3/2, 6], we ﬁnd for any positive δ suﬃciently close to zero

Ω

(y)dy ≤

|x|

1−δ

Ω

3/2+δ

|x|

1−δ

. (XIII.9.24)

The lemma then fol lows from (XIII.9.20)– (XIII. 9.24). ut

The representation (XIII.9.3)–(XIII.9.4), together with Lemma XIII.9.4,

allows us to show the fol lowing result concerning the asymptotic structure of

a generalized solution corresponding to a ﬂux of suitably restricted size.

Theorem XIII.9.1 Let the ﬂux Φ satisfy the assumption of Theorem XIII.7.4.

Then any corresponding generalized solution v to problem (XIII.5.2) that

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

obeys the energy inequality (XIII.6.6) admits the foll owing asymptotic repre-

sentation as |x| → ∞, x ∈ R

(x) = A

(x, 0) + B

, g

)(x, 0) + V

(x), j = 1, 2, 3.

(XIII.9.25)

Here G

≡ {G

}, g

≡ {g

} is the Green’s tensor-solution for the Stokes

problem in R

(see (IV.3.46)-(IV.3.49)), G

≡ (G

, G

), and

, B

:= ±

, V

(x) = O(|x|

−3+δ

with δ an arbitrary positive number.

Remark XI II.9.1 From (XIII.9.25) and (IV.3 .50) we obtain the uni form

asymptotic estimate f or v (see also Lemma XIII.9.3):

v(x) = O(|x|

−2

Moreover, comparing (XIII.9.25) with the analogous formula (VI. 5.23) ob-

tained in the linear case, we recognize in (XIII.9.25) the presence of the extra

terms involving the qua ntities A

. Therefore, unlike the case of a n exterior

domain (when v

∞

6= 0), the nonlinearities give an explicit contribution to the

asymptotic expansion. 

Remark XI II.9.2 Following the methods of Coscia & Patria (1992), one can

show that the ﬁrst derivatives o f v obey the asymptotic estimate

v(x) = O(|x|

−3

), as |x| → ∞.

We shall not perform this study here; we refer the reader to Theorem 12 of

the paper by Coscia & Patria; cf. also Exercise XIII.9.2. 

The ﬁnal part of this section is devoted to the proo f of an asymptotic

representation for the pressure ﬁeld p. As before, we shall show thi s in R

, an

analogous formula holding in R

−

. Let ϕ

(x) be a smooth function that equals

one for |x| ≥ 2R and zero for |x| ≤ R with |∇ϕ

| ≤ MR

−1

, M indep endent

of R. Setting

w = ϕ

v, τ = ϕ

(p − p

with p, p

introduced in Lemma XIII.5.1, we easily establish that w, τ satisfy

the following problem

ν∆w = ∇τ + F

∇ · w = g

)

in R

w = v at x

= 0

(XIII.9.26)

with

XIII.9 Aperture Domain: Asymptotic Structure of Generalized Solutions 965

F = 2 ∇ϕ

·∇v + ∆ϕ

v + ϕ

v · ∇v + (p −p

) · ∇ϕ

g = ∇ϕ

· v.

(XIII.9.27)

Denote, further, by h a solution to the non-homogeneous Stokes problem

ν∆h = ∇σ

∇ · h = −∇ϕ

· v

)

in R

h = 0 at x

= 0.

(XIII.9.28)

As we know from Theorem IV.3.2, this probl em admits a (unique) solution

satisfying the estimate

|h|

2,q

+ |σ|

1,q

≤ ck ∇ϕ

·vk

1,q

for a ll q > 1, (XIII.9.29)

with a constant c i ndependent of R. Using Theorem XIII.7 .4, together with

the properties of ϕ

, we easily show

lim

R→∞

(∇ϕ

= 0 for all q > 1, and |β|, | χ|=0,1. (XIII.9.30)

Choosing q = q

< 3 and modifying σ by the addition of an appropriate

constant, from Theorem II.6.3 we ﬁnd

kσk

/(3−q

)

≤ γ

|σ|

. (XIII.9.31)

Thus, from (XI II.9.29) and (XIII.9.31 ), with the help of Theorem II.9.1, we

ﬁnd for all x ∈ R

|σ(x)| ≤ γ

(|σ|

1,q

+ |σ|

1,q

) q

> 3,

with γ

independent of R. From this relation, (XIII.9.29), and (XIII.9.30) we

then conclude

lim

R→∞

σ(x) = 0, for all x ∈ R

. (XIII.9.32)

Setting

u = w + h,

from (XIII.9.2 6)–(XIII.9.28) we ﬁnd

ν∆u = ∇τ

+ F

∇ · u = 0

)

in R

u = v at x

= 0

(XIII.9.33)

with

= τ + σ. (XIII.9.34)