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

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706 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

where

c = c (Ω, B) for all R ∈ [0, B]

and

= min{1, R

1/2

Then a ny generalized soluti on corresponding to f, v

∗

, and v

∞

satisﬁes (v +

∞

) ∈ L

(Ω)

along with the inequality

kv + v

∞

√

. (X.7.3)

Proof. We begin to show tha t under hypothesis (X.7.2) there exi sts a gener-

alized solution w

, say, verifying the condition

+ v

∞

√

(X.7.4)

To thi s end, we may employ, for example, the method of successive approxima-

tions. We introduce a sequence of approximating solutions {w

, π

}, deﬁned

by w

≡ π

≡ 0 and, for k ≥ 1,

∆w

+ R

∂w

∂x

= Rw

k−1

· ∇w

k−1

+ ∇π

+ Rf

∇ · w

= 0











in Ω

lim

|x|→∞

(x) = 0,

= v

∗

+ v

∞

at ∂Ω.

(X.7.5)

By Theorem VII.7.1 with m = 0 and q = 6/5, we know that, for k = 1, there

is a solution w

, π

such that

∈ L

(Ω) ∩ D

1,12/7

(Ω) ∩ D

2,6/5

(Ω),

∈ D

1,6/5

(Ω),

and obeying the estimate

+ a

1,12/7

+ |w

2,6/5

+ |π

1,6/5

≤ c

(Rkfk

6/5

+ kv

∗

−v

∞

7/6,6/5(∂Ω)

)

≡ c

(X.7.6)

where c

= c

(q, Ω) is independent of R for R ∈ (0, B] and

Observe that this property fol lows from the assumptions on the data and Lemma

X.6.1.

X.7 Energy Equation and Uniqueness when v

∞

6= 0 707

= min{1, R

1/2

}, a

= min{1, R

1/4

Since, by Theorem II.6.1,

1,2

≤ c

2,6/5

, (X.7.7)

(X.7.6) furnishes, in particular,

+ |w

1,2

+ |w

2,6/5

+ |π

1,6/5

≤ cD, (X.7.8)

with c = c(Ω, B) independent of R ∈ (0, B]. We next show, by induction, that

the following inequality is veriﬁed for all k ∈ N

+ |w

1,2

+ |w

2,6/5

+ |π

1,6/5

≤ 2cD, (X.7.9)

provided D is “small enough.” Thus, assuming w

, π

obey (X.7.9), by The-

orem VII.7.1 we recover

k+1

+ |w

k+1

1,2

+ |w

k+1

2,6/5

+ |π

k+1

1,6/5

≤ c



D + Rkw

· ∇w

6/5



(X.7.10)

Now we have

· ∇w

6/5

≤ kw

1,2

and, by i nduction hypothesis,

· ∇w

6/5

≤ 4c

so that (X.7.10) shows that if

D < a

/4c

R, (X.7 .11)

then inequality (X.7.9) i s satisﬁed for all k ∈ N. It is now easy to prove that

, π

} is a Cauchy sequence in the space

S ≡



(Ω) ∩

1,2

(Ω) ∩

2,6/5

(Ω)



1,6/5

(Ω).

In fact, from (X. 7.5) we deduce that

k+1

−w

+ |w

k+1

− w

1,2

+ |w

k+1

− w

2,6/5

+ |π

k+1

−π

1,6/5

≤ c Rkw

·∇w

− w

k−1

·∇w

k−1

6/5

and since

· ∇w

−w

k−1

· ∇w

k−1

6/5

≤ kw

− w

k−1

1,2

+kw

−w

k−1

1,2

in view of (X.7.9) we conclude, for al l k ≥ 1,

708 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

k+1

− w

+ |w

k+1

− w

1,2

+ |w

k+1

− w

2,6/5

+ |π

k+1

−π

1,6/5

≤ (4c

RD/a

)(a

−w

k−1

+ |w

−w

k−1

1,2

From this inequality we receive, for all k ≥ 1,

k+1

−w

+ |w

k+1

− w

1,2

+|w

k+1

− w

2,6/5

+ |π

k+1

− π

1,6/5

≤ (4c

RD/a

)

k+1

which, by (X. 7.11) and virtue of a standard argument, implies that the se-

quence {w

, π

} is a Cauchy sequence in the space S. Denoting by w, π the

limiting ﬁeld, we then have

w ∈ L

(Ω) ∩ D

1,2

(Ω) ∩ D

2,6/5

(Ω)

π ∈ D

1,6/5

(Ω)

and, moreover, w, π obey the estimate (X.7.9). In addition, setting

= w + v

∞

, (X.7.1 2)

and recalling that, in particular, w is a generalized solution, from Lemma

X.6. 1 we ﬁnd

+ v

∞

∈ L

(Ω) , v

∞

· ∇w

∈ L

4/3

(Ω) (X.7.13)

and, if the data satisfy (X.7.2)

, we also have that w

obeys (X.7.4). Now

let v denote any generalized solution corresponding to f , v

∗

, and v

∞

. From

Lemma X.6.1 we ﬁnd

v + v

∞

∈ L

(Ω)

∞

·∇v ∈ L

4/3

(Ω)

(X.7.14)

and, by Theorem X.7.1, we obtain that

v veriﬁes the generalized energy equality (X.2.28) (X.7.15)

for any extension A of v

∗

and v

∞

Observing that f ∈ L

6/5

(Ω) implies f ∈

−1,2

(Ω) (as a consequence of Sobolev inequality (II.3.11)), the uniqueness

Theorem X.3.1, together with (X.7.12)– (X.7.1 5)

and (X.7.4), furnishes w = v

a.e. in Ω. The lemma is proved. ut

From Lemma X.7.1 and Theorem X.7.1 we immediately obtain the follow-

ing uniqueness theorem for generalized s olutions corresponding to v

∞

6= 0.

Actually, v satisﬁes the energy equation (X.2.29) in its classical form.

See footnote 1 in this section.

X.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 709

Theorem X.7.3 Let Ω be a three-dimensional exterior domain of class C

Further, let

f ∈ L

6/5

(Ω) ∩ L

4/3

(Ω), v

∗

∈ W

5/4,4/3

(∂Ω), v

∞

6= 0.

Then if (X.7.2)

is satisﬁed, v is the only generalized solution achieving these

data.

X.8 The Asymptotic Structure of Generalized Solutions

when v

∞

6= 0

In this section we conclude the study of the asymptotic behavior of generalized

solutions with v

∞

6= 0, by showing that they have the same structure as the

corresponding solutions to the Oseen problem. In pa rticular, they exhibit a

“downstream” paraboloi dal wake region and the rate of convergence of the

velocity ﬁeld v to −v

∞

is more rapid outside the wake than inside.

For the reader’s convenience, we collect the estimates on the tensor ﬁeld E

derived in (VII.3.24), (VII.3.32), and Exercise VII.3.1, which will be frequently

used in the sequel:

(x − y)| ≤ c |x − y|

−1

|∇E

(x − y)| ≤ c |x − y|

−3/2

∂B

(x)

|∇E

(x − y)| ≤ c R

−1/2

(X.8.1)

Our ﬁrst objective is to recover an appropriate uniform estimate for u(x) =

v(x)+v

∞

, for large values of |x|, where, as usual and wi tho ut loss of g enerality,

we take v

∞

= e

. For simplicity, we shall assume that the body force f is of

bounded support. Let Ω denote, temporarily, the exterior of B

, where ρ is

taken large enough to satisfy B

⊃ supp (f). By Theorem X.1.1 we then have

that v, p ∈ C

∞

(Ω) and that v, p enjoy the summability properties proved in

Theorem X.6.4. From the asymptotic formulas (X.5.50)–(X.5.52) we have

u(x) = N[u(x)] + O(1/|x|) as |x| → ∞ (X.8.2)

where

= (N [u(x)])

Ω

(x − y)u

(y)D

(y)dy.

We wish to g ive a unifo rm estimate for N . To this end, setting |x| = 2R, we

may write

Ω

(x − y)u

(y)D

(y)dy +

Ω

(x − y)u

(y)D

(y)dy

≡ N

+ N

(X.8.3)

710 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

Recalling (X.8.1)

, it follows that

| ≤

kuk

3,Ω

|u|

1,3/2,Ω

≤

|x|

kuk

|u|

1,3/2

This inequality and Theorem X.6.4 yield

| ≤ c

|x|

−1

. (X.8.4)

Furthermore, from Theorem II.6.1 (cf. (II. 6.20)) and (X.8.1)

we also have

| ≤



Ω

|x − y|



1/2



Ω

∇u : ∇u



1/2

≤ c

Ω

∇u : ∇u (X.8.5)

with c

independent of u and R. Setting

G(R) ≡

Ω

∇u : ∇u (X. 8.6)

from (X.8.2)–(X.8.5) we deduce that estimating the nonlinear term N[u(x)]

is reduced to estimating the functions G(R), R = |x|/2. This latter question

will be analyzed in the next two lemm as. We begin with a simple but very

useful result.

Lemma X.8.1 Suppose that for all t > a ≥ 0,

(i) y ∈ C

([a, t]),

(ii) y(t) ≥ 0,

(iii) y

(t) ≤ 0,

and that, for some β ∈ [0, 1),

∞

y(s)s

−β

ds < ∞.

Then, for all t ≥ a,

y(t)t

1−β

≤ (1 − β)

∞

y(s)s

−β

ds + y(a)a

1−β

Proof. The assertion is an immediate consequence of the identity

y(t)t

1−β

[y(s)s

1−β

]ds + y(a)a

1−β

and of the assumptions made on y. ut

Lemma X.8.1 allows us to prove the following estimate for G(R).

X.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 711

Lemma X.8.2 Let v be a generali zed solution to the Navier–Stokes problem

in Ω = R

−B

corresponding to f = 0 and v

∞

6= 0. Then, setting u = v+v

∞

for a ll R > ρ, it holds that

G(R) ≤ cR

−1+ε

(X.8.7)

where G is deﬁned in (X.8.6), ε is an arbitrary po sitive number and c is

independent of R.

Proof. As usual, we assume v

∞

= e

. We recall that v and the corresponding

pressure p are in C

∞

(Ω). Multiplying (X.0.8 ) by u and i ntegrating over Ω

R,R

∗

> R), it follows that

Ω

R,R

∗

∇u : ∇u =

∂B

∪∂B

∗



u ·

∂u

∂n

−

v · n − p( u · n)



. (X.8.8)

From Theorem X.6.4 we derive, in particular,

Ψ ≡ |u|

+ |∇u|

3/2

+ |u|

5/2

+ |p|

5/3

∈ L

(Ω). (X.8.9)

Therefore, there exists a sequence {R

} tending to inﬁnity as k tends to

inﬁnity, such that

∂B

Ψ(R

, ω)dω = o(R

−1

). (X.8.10)

Using Young’s inequality several times and the H¨older inequali ty, we deduce

(with r denoting either R or R

∗

)

F (r) ≡

∂B



u ·

∂u

∂n

−

v · n − p(u · n)



≤ c

(

∂B



+ |∇u|

3/2

+ u

5/2

+ p

5/3



+ r

2/q



∂B



1/q

)

(X.8.11)

where q is an arbitrary number greater than one. Taking in (X.8.11) r ≡ R

∗

, q = 3/2 and using (X.8.9), (X.8.10), it foll ows that

lim

k→∞

F (R

) = 0,

and(X.8.8) furnishes

G(R) = F (R). (X.8.12)

For any ε ∈ (0, 1), by Young’s inequality,

H(R) ≡ R

−ε

2/q



∂B



1/q

≤ c



−εq

∂B



712 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

and so, taking q < 3/(3−ε), and recalling that, by Theorem X.6.4, u ∈ L

(Ω)

for a ll s > 2, we have

H ∈ L

(ρ, ∞). (X.8.13)

Collecting (X.8.9), (X.8.11), and (X.8.13) yields

−ε

F (R) ∈ L

(ρ, ∞) (X.8.14)

and, since

(R) = −

∂B

∇u : ∇u < 0, (X.8.15)

from (X.8.12), (X.8.14), and (X.8.15), with the help of Lemma X.8.1, we

obtain (X.8.7), and the proof is compl ete. ut

Using (X.8.3)–(X.8.5), together with the results of Lemma X.8.2, we then

have the fol lowing uniform estimate for the nonli near term:

|N [u(x)]| ≤ c|x|

−1+ε

, for any ε ∈ (0, 1], (X.8.16)

for some c = c(ε) (c → ∞ as ε → 0). From (X.8.2) and (X.8.16 ) we thus

obtain the following.

Lemma X.8.3 Let v be a generali zed solution to the Navier–Stokes problem

in a three-dimensional exterior domain Ω with f of bounded support and

∞

6= 0. Then, for all suﬃciently large |x|, the following uniform estima te

holds:

v(x) + v

∞

= O(1/|x|

1−ε

), for any ε ∈ (0, 1].

With this result in hand, we can now show the following theorem which

furnishes the asymptotic structure of any generalized solution corresponding

to a body force of bounded support.

Theorem X.8.1 Let v be a generalized solution to the Navier–Stokes prob-

lem (X.0.8)–(X.0.4) in a three-dimensional exterior domain of class C

, cor-

responding to v

∞

6= 0, f of bounded support in Ω, and, moreover, with

f ∈ L

(Ω), v

∗

∈ W

2−1/q

,1/q

(∂Ω),

for some q

> 3 and all q ∈ (1, q

]. Then for all suﬃciently large |x|, v admits

the following representation:

v(x) + v

∞

= m · E(x) + V(x) (X.8.17)

where E is the Oseen fundamental tensor,

= −

∂Ω

(u, p) + R(δ

−u

)]n

+ R

Ω

, (X.8.18)

with u := v + v

∞

, and V(x) satisﬁes the estimate

V(x) = O(|x|

−3/2+δ

) for any δ > 0. (X.8.19)

X.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 713

Proof. In view o f Theorem X. 5.3, formula (X.5.50)

, it suﬃces to show the

uniform estimate



Ω

(y)u

(y)D

(x − y)dy



= O(|x|

−3/2+δ

). (X.8.20)

Setting |x| = R suﬃciently large, we divide the region Ω into three parts:

Ω

R/2

, Ω

R/2,3R

, Ω

and denote the corresponding integrals over these regions by I

, I

, and I

respectively. Using (X.8.1)

and the H¨older inequality, we ﬁnd

| ≤

3/2

Ω

R/2

≤ c

3(1 /q

−1/2)

kuk

and so, choosing q = 3/(3 − δ), from Theorem X.6.4 i t follows tha t

| ≤ c

|x|

−3/2+δ

. (X.8.21)

Furthermore, recalling Lemma X.8.3 , we have

| ≤ c

−2+2ε

Ω

R/2,3R

|E(x − y)|dy ≤ c

−2+2ε

|E|

1,1,B

(x)

and so, in view of (X.8.1)

, choosi ng ε = δ/2, we recover

| ≤ c

|x|

−3/2+δ

. (X.8.22)

Finally, from Lemma X.8.3 and an obvious majo ration it foll ows that

| ≤ c

Ω

|y|

−2+2ε

|∇E(x − y)|dy ≤ c

(x)

|y|

−2+2ε

|∇E(x − y)|dy ,

where we have used the fact that | x| = R. Now, for any y ∈ B

(x) we ﬁnd

|x − y| ≤ |y| + R ≤ 2 |y|, so that the above inequality leads to the following

one

| ≤ c

(x)

|x − y|

−2+2ε

|∇E(x −y)|dy ,

which, in turn, by (X.8.1)

furnishes

| ≤ c

∞

−2+2ε

−1/2

dr ≤ c

−3/2+2ε

This latter, with the choice ε = δ/2, implies

| ≤ c

|x|

−3/2+δ

. (X.8.23)

The validity of (X.8.20) is a consequence of (X.8.21)–(X.8.23) and the theorem

is proved. ut

714 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

Remark X.8.1 From (X.8. 17) and the properties of the tensor ﬁeld E

(cf. (VII.3 .24)–(VII. 3.26)), we deduce, in particular, that any generalized so-

lution of Theorem X.8.1 exhibi ts a paraboloidal wake region in the direction

of v

∞

(= e

). Speciﬁcally, for all suﬃciently large |x| we have the uniform

estimate

v(x) + v

∞

= O(1/|x|). (X.8.24)

On the other hand, denoting by ϕ the angle m ade by a ray starting from the

origin (in Ω

, say) with the negatively directed x

-axis, for all x satisfying

|x|(1 + cos ϕ) ≥ |x|

2σ

, σ ∈ (0, 1/2], (X.8.25)

we have

v(x) + v

∞

= O(1/|x|

1+α

) (X.8.26)

where

α = min(2σ, 1/2 − δ). (X.8.27)

Relations (X.8.24)–(X.8.27) then show the m entioned behavior of v. Compar-

ing (X.8.26) with the analo gous estimate for E given in (VII.3 .26) or, what

amounts to the same thing, with the estimate for the velocity ﬁeld of the cor-

responding Oseen linearized problem given by (VII.6.18), we recognize that

(X.8.26) is apparently weaker, because we cannot take α = 2σ, for σ ≥ 1/4.

This latter circumstance is due to the fact that, in the proof of Theorem X.8.1 ,

we have given a uniform bo und for the nonlinear term

N(x) ≡

Ω

(y)u

(y)D

(x − y)dy.

However, a lso for N , we can prove estimates whose decay order is faster tha n

(X.8.21) if x belong s to the region R

described by (X.8.25). This will, in turn,

furnish more accurate estimates for the remainder V(x), deﬁned in (X.8.17),

for x ∈ R

. For example, Finn (1959b, Theorem 8) has shown the following

asymptotic bound for V(x):

V(x) = O



|x|

−3/2+δ−2σ



, for all x ∈ R

Sharper estimates for V(x) can be found i n the work of Finn (1965a, Theo-

rem 5.1) and Vasil’ev (1973) and, under sui table restriction on m, in that of

Babenko & Vasil’ev (1973). 

An immediate consequence of Theorem X.8.1 is that the uniform estimate

(X.8.24) is sharp in the sense speciﬁed by the following result, whose ﬁrst

formulation traces back to the work of Udeschini (1941), Berker (1952), and

Finn (1959b).

See Remark IX.3.1.