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

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

Again by (i) of Lemma X.4.1, for a given ε > 0 we may choose R so that

|V |

1,3/2,Ω

R < ε,

while, by (X.4.30) (possibly along a new subsequence),

lim

m→∞

−uk

4,Ω

= 0

and so from (X.4.43) and (X.4.41), it follows that

lim sup

m→∞

|(u

· ∇V , u

) −(u · ∇ V , u)| ≤ 4M

which, in turn, by the arbitrariness of ε furnishes

lim

m→∞

· ∇V , u

) = (u · ∇V , u). (X.4.44)

Since, clearly,

lim

m→∞

(∇V , ∇u

) = (∇V , ∇u) (X.4.45)

from (X.4.22), (X.4.23), (X.4.42), (X.4.44), (X.4.4 5), and Theorem II.1.3 we

conclude

|u|

1,2

≤

lim

m→∞

1,2

= −(u · ∇V , u) −

(∇V , ∇u) − (V · ∇V , u) − [f , u] .

Recalling that v = u + V , we then recover (X.4.19) and the proof of part

(i) is complete. In order to show part (ii), we notice that from (X.4.27) for

Φ ≤ 1/(2R) and η = 1/(4R), we deduce

1,2

≤ 4R|f |

−1,2

+ 4|V |

1,2

+ 4Rc

kV k

1,2,Ω

√



kV + v

∞

3,Ω

R |V |

1,2

+ |v

∞

||V |

1,6/5



(X.4.46)

Passing to the limit m → ∞ in (X.4.46) and employing T heorem I I.1.3 and

the estimates (X.4.6) for V , we infer the validity o f (X.4.20), which concludes

the proof of the theorem. ut

Remark X.4.3 If the number s of compact regions Ω

is one, existence is

proved under the following condition on Φ:

Φ ≡

|Φ

4πr

;

cf. Remark X. 4.1. 

X.4 Existence of Generalized Solutions 687

Remark X.4.4 We would like to make some comments about the extension

of Theorem X.4.1 to the two-dimensional case. Taking into account Remark

X.4. 2, we verify at once that, for Ω ⊂ R

, with Ω 6= R

, under the assumption

that the ﬂux of v

∗

through ∂Ω satisﬁes the following restriction

i=1

|Φ

| <

2π

, (X.4.47)

there is at least one ﬁeld v satisfying conditions (i)–(iii) and (v) of Deﬁnition

X.1. 1.

Nevertheless, with the informati on derived so far, we are not able to

draw any conclusion ab out the behavior at inﬁnity of v. This is because, unlike

the three-dimensional case, we have no Sobolev-like (or weighted) inequality

which ensures some type of decay for v + v

∞

as |x| → ∞. A fundamental

problem is then to analyze what is the behavior at inﬁnity of vector ﬁelds

satisfying merely (i)-(iii) and (v) of D eﬁnition X.1.1, when Ω is a planar

domain. This will b e the object of Section XII.3 . Let us now consider the

case Ω = R

. In such a circumstance the procedure adopted in Theorem

X.4. 1 does not produce any ki nd of existence. Actually, we can still construct

an approximating sequence {u

} and show that it satisﬁes estimate (X.4.28).

Therefore, we can ﬁnd a ﬁeld u ∈ D

1,2

) for which (X.4.29) holds. However,

we can not establish (X.4.30) and, as a consequence, (X.4.32). In fact, in view

of the example given in Exercise II.7.3, we know that condition (X.4.28) alone

is not suﬃcient to ensure any kind of convergence of {u

} to u in any space

), q ≥ 1, R > 0. Because of this, we are not able to show (X.4.32) and,

as a consequence, we can not prove tha t the ﬁeld u (≡ v) satisﬁes the identity

(X.1.2) 

Remark X.4.5 In dimension n ≥ 4, existence of generalized solutions to

(X.0.8), (X.0.4) can be proved along the same lines of Theorem X.4.1 provided

∗

∈ W

1−1/s,s

(∂Ω), Φ

(n)

< 1/R,

where Φ

(n)

is deﬁned in Remark X.4. 2. In such a case, the asymptotic estimate

(X.4.18)

becomes

n−1

|v(x) + v

∞

| = O(1/|x|

n/2−1

) as |x| → ∞.



Remark X.4.6 If v

∗

≡ v

∞

≡ 0, Theorem X.4.1, with the exception of

(X.4.18)

, holds without requiring any regularity of the domain. This is be-

cause, the only point in the proof where regulari ty is needed is in the con-

struction of the extension V which, in this case, can be taken as identically

zero. 

The ﬁrst proof of this result is due to Russo (2009).

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

X.5 On the Asym pt otic Behavior of Generalized

Solutions: Prelimi nary Results and Representation

Formulas

In the present section we begin to study the asymptotic behavior o f a gener-

alized solution v to the Navier–Stokes equation in a three-dimensional exte-

rior domain (we postpone the analogous two-dimensional case until Chapter

XII). Speciﬁcally, we shall prove that if the body force has a certain degree

of summability at inﬁnity, then v b ehaves essentially as the corresponding

solution of the linearized approximations (cf. Theorem V.3.1 and Theorem

VII.6.1), that is, we have

lim

|x|→∞

|v(x) + v

∞

| = 0

lim

|x|→∞

v(x)| = 0, 1 ≤ |α| ≤ s,

uniform ly, where s is related to the degree of summability of the derivatives

of f. An analo gous property is shown for the pressure ﬁeld p associated to v

by Lemma X.1.1. Successively, we prove that, as in the linear problem, v and

p admit an integral representation valid for almost all points in Ω. We have

the following theorem.

Theorem X.5.1 Let v be a generalized solution to the Navier–Stokes prob-

lem in an exterior three-dimensional domain Ω. Assume that for some R >

δ(Ω

) and some q ∈ (3/2, ∞),

f ∈ L

(Ω

). (X.5.1)

Then

lim

|x|→∞

|v(x) + v

∞

| = 0 (X.5.2)

uniform ly. Furthermore, if for m ≥ 0 and some r ∈ (3, ∞), q ∈ (3/2 , 2]

f ∈ W

m,r

(Ω

) ∩ L

(Ω

), (X. 5.3)

then

lim

|x|→∞

v(x)| = 0, 1 ≤ |α| ≤ m + 1, (X.5.4)

uniform ly. Finally, under assumption (X.5.3) there is p

∈ R such that

lim

|x|→∞

(p(x) −p

)| = 0, 0 ≤ |α| ≤ m, (X.5.5)

uniform ly, where p is the pressure ﬁeld associated to v by Lemma X.1.1.

Proof. We shall consider throughout the case v

∞

= 0, since the case v

∞

6= 0

is treated in a completely analog ous way. Moreover, to simplify notation, we

X.5 Asymptotic Behavior: Preliminary Results and Representation Formulas 689

shall put R = 1 throughout the proof. From Lemma V.3.1 we have for a.a.

x ∈ Ω

with di st (x, ∂Ω

) > d

(x) =

(x)

(d)

(x − y)[f

(y) + v

(y)]dy

−

β(x)

(d)

(x − y)v

(y)dy

≡ I

(x) + I

(x).

(X.5.6)

As in the proo f of Theorem V.3.1 we show, for q > 3/2,

(x)| ≤ c

kfk

q,B

(x)

. (X.5.7)

Furthermore, recalling that

(d)

(x − y)| ≤ c

|x −y|

−1

, y ∈ B

(x),

we ﬁnd

(x)| ≤ c

kv/|x −y|k

2,B

(x)

|v|

1,2,B

(x)

On the other hand, by Theorem II.6.1,

kv/|x − y|k

2,B

(x)

≤ c

|v|

1,2

and therefore

(x)| ≤ c

|v|

1,2,B

(x)

. (X.5.8)

Finally, as in the proo f of Theorem V.3.1,

(x)| ≤ c

kvk

6,B

(x)

(X.5.9)

and so, recalling that, by Theorem II.6.1,

v ∈ L

(Ω

) (X.5.10)

the property (X.5.2) follows from (X.5.1) and (X.5.6)–(X.5.10). Let us now

show (X.5.4). We begin to notice that, by the regularity Theorem X.1.1, we

have

v ∈ L

∞

(Ω

R,R

) f or all R

> R,

which, because of (X.5.1), implies

v ∈ L

∞

(Ω

). (X.5.11)

As a consequence, from the inequality

kv · ∇vk

≤ kvk

2t/(2−t)

|v|

1,2

, 1 ≤ t ≤ 2,

with a view at (X.5.10), (X.5.1 1) we obtain

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

v · ∇v ∈ L

(Ω

), for all s ∈ [3/2, 2]. (X.5.12)

The assumption on f and (X.5.12) lead to

(f + v ·∇v) ∈ L

(Ω

), for some q ∈ [3/2, 2],

and thus, by Theorem V.5.2, it follows that

v ∈ D

2,q

(Ω

), p ∈ D

1,q

(Ω

), R

> R. (X.5.13)

Using (X.5.13) and Theorem II.6.1 yields

v ∈ D

1,3q/(3−q)

(Ω

) (X.5.14 )

and from (X.5.11), (X.5.14) we conclude tha t

v · ∇v ∈ L

3q/(3−q)

(Ω

). (X.5.15)

Next, recalling that I

∈ C

∞

), from (X.5.6) with F = f + v ·∇v we ﬁnd

(x) = D

(x)

(d)

(x − y)F

(y)dy −

β(x)

(d)

(x − y)v

(y)dy.

(X.5.16)

Using the deﬁnition of generalized diﬀerentiation we easily show that

(x)

(d)

(x − y)F

(y)dy =

(x)

(d)

(x −y))F

(y)dy (X.5.17)

and so, bearing in mind that

(d)

(x − y)| ≤ c

|x − y|

−2

, y ∈ B

(x),

we recover, by (X.5.17) and the H¨older inequality,

I(x) ≡



(x)

(d)

(x − y)F

(y)dy



≤ c

(k|x −y|k

(x)

kf k

r,B

(x)

k|x −y|

−2

3q/(4q−3),B

(x)

kv · ∇vk

3q/(3−q),B

(x)

Since both r

and 3q/(4q − 3) are strictly l ess than 3/2, from (X.5.15) and

(X.5.2) we obtain

I(x) → 0 as |x| → ∞. (X.5.18)

Since



β(x)

(d)

(x − y)v

(y)dy



≤ c

kvk

6,B

X.5 Asymptotic Behavior: Preliminary Results and Representation Formulas 691

from this i nequality, and from (X.5.16) and (X.5.18) we infer (X.5.4) for |α| =

1. Let us now pass to higher-order derivatives. To this end, we begin to show

v · ∇v ∈ W

1,r

(Ω

) (X.5.19)

for all suﬃciently large R.

By (X.5.11) and the fact that v is a generalized

solution, we ﬁnd

v · ∇v ∈ L

(Ω

) f or all s ≥ 2 (X.5.20)

and therefore

f + v · ∇v ∈ L

(Ω

From Theorem V.5.3 we then deduce

v ∈ D

2,r

(Ω

) p ∈ D

1,r

(Ω

) (X.5.21)

and so, since

(v ·∇v) = (D

v) · ∇v + v · D

∇v (X.5.22)

from (X.5.20), (X.5.11), and (X.5.21) we prove, in pa rticular, (X.5.19). With

the help of Lemma V.3.1 we then have

(x) = D

(x)

(d)

(x − y)D

(y) + v

(y)]dy

−

β(x)

(d)

(x − y)D

(y)dy,

and since

[f + v · ∇v] ∈ L

(Ω

) r > 3,

we reason as in the case where |α| = 1 and conclude that

lim

|x|→∞

v(x)| = 0 |α| = 2 (X.5.23)

uniform ly. Relation (X.5.23) implies

v ∈ L

∞

(Ω

). (X.5.24)

On the other hand, from (X.5.19), (X.5.21), and Lemma V.4 .3, i t follows that

in particular

v ∈ D

3,r

(Ω

) (X.5.25)

and so, diﬀerentiating one mo re ti me (X.5.22) and employing (X.5.1 9), (X.5.2 1),

(X.5.24), and (X.5.25) we obtain

In the remaining part of the proof the number R need not be the same for all

formulas. However, it is understood that the indicated property holds for “suﬃ-

ciently large” R.

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

v · ∇v ∈ W

2,r

(Ω

), r > 3.

Consequently, by the same reasoning previously used, we prove (X.5 .2) with

|α| = 3. Iterating this procedure a s many times as needed, we then show

(X.5.4) in the general case. Let us now turn to the pressure p. Collecting

(X.5.13) and (X.5. 21) under assumption (X.5.3) we obtain

p ∈ D

1,r

(Ω

) ∩ D

1,q

(Ω

) (X.5.26)

where, we recall, r > 3 and 2 > q. Theorem II.9.1 then implies (X.5.5) with

α = 0. The proof of the general case (for m ≥ 1) is a direct consequence of

the momentum equation, that is,

∇p = −f + ∆v + v · ∇v. (X.5.27)

In fact, if (X.5.3) ho lds with some m ≥ 1, then from the embedding Theorem

II.3.4, it follows that

f (x) → 0 as |x| → ∞, for all |α| ∈ [0, m − 1]. (X.5.28)

By means of (X.5.2), (X.5.4), (X.5.27 ), and (X.5.28) we thus conclude that as

|x| → ∞

[−f(x) + ∆v(x) + v(x) · ∇v(x)] → 0

uniform ly for all multi-index α with |α| = m − 1. The theorem is therefore

proved. ut

Remark X.5.1 The arguments used in the proof of the preceding theorem

fail in dimension n ≥ 4. In fact, they do not ensure pointwise decay even for

v itself. The reason is that, in such a case, we cannot increase the term I

(x)

by a function vanishing at large distances. However, it is probably true that

results of the type presented in Theorem X.5.1 continue to hold, at least, for

n = 4. 

We shall now derive representation formulas for v and p analogous to those

derived in the linear case. Speciﬁcall y, we have the following theorem.

Theorem X.5.2 Let v be a generalized solution to the Navier–Stokes prob-

lem (X.0.8 ), (X.0.4 ) in an exterior three-dimensional domain Ω of class C

with

v ∈ W

2,r

loc

(Ω), r ∈ (1, ∞).

Then if for some R > δ(Ω

)

f ∈ L

(Ω

) ∩L

loc

(Ω), (X.5.29 )

with q ∈ (1 , 3/2) and s ∈ (1, ∞), the following representations hold for all

x ∈ Ω: if v

∞

= 0

X.5 Asymptotic Behavior: Preliminary Results and Representation Formulas 693

(x) = R

Ω

(x − y)f

(y)dy + R

Ω

(x − y)v

(y)D

(y)dy

∂Ω

(y)T

, q

)(x − y)

− U

(x − y)T

(v, p)(y)]n

dσ

(X.5.30)

and

(x) = R

Ω

(x − y)f

(y)dy − R

Ω

(y)v

(y)D

(x − y)dy

∂Ω

(y)T

, q

)(x − y) − U

(x − y)(T

(v, p)(y)

−Rv

(y)v

(y))]n

dσ

;

(X.5.31)

if v

∞

6= 0 (v

∞

= (1 , 0, 0)), setting u = v + v

∞

(x) = R

Ω

(x − y)f

(y)dy + R

Ω

(x − y)u

(y)D

(y)dy

∂Ω

(y)T

, e

)(x − y) − E

(x − y)T

(u, p)(y)

−Ru

(y)E

(x − y)δ

dσ

(X.5.32)

and

(x) = R

Ω

(x − y)f

(y)dy −R

Ω

(y)u

(y)D

(x − y)dy

∂Ω

(y)T

, e

)(x − y)

−E

(x − y)(T

(u, p)(y) −Ru

(y)u

(y))

− Ru

(y)E

(x − y)δ

dσ

;

(X.5.33)

where U, q and E, e are Stokes and Oseen fundamental solutions, respectively,

while u

and w

are deﬁned in (IV.8.1 1)

and (VII.6.3). All volume integrals

in (X.5.30)–(X.5.3 3) are absolutely convergent.

Furthermore, if

f ∈ L

(Ω

) ∩L

loc

(Ω),

for some r ∈ (1, 3/2) and t ∈ (3, ∞), then, denoting by p the pressure ﬁeld

associated to v by Lemma X.1.1, we have for a.a. x ∈ Ω: if v

∞

= 0

p(x) = p

− R

Ω

(x − y)f

(y)dy − R

Ω

(x − y)v

(y)D

(y)dy

∂Ω

(x − y)T

(v, p)(y) −2v

(y)

∂

∂x

(x − y)]n

dσ

;

(X.5.34)

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

if v

∞

6= 0

p(x) = p

− R

Ω

(x − y)f

(y)dy −R

Ω

(x − y)u

(y)D

(y)dy

∂Ω

(x − y)T

(u, p)(y) − 2u

(y)

∂

∂x

(x − y)

−R[e

(x − y)u

(y) − u

(y)e

(x − y)δ

dσ

(X.5.35)

where p

and p

are constants. All volume integrals in (X.5.34), (X.5.35) are

absolutely convergent.

Proof. We show (X.5.30), (X.5 .31) and (X.5.34), since the proof of (X.5.32),

(X.5.33) and (X.5.35) is somehow si milar and, therefore, left to the reader a s

an exercise. From the assumptions made and (V.3.6) with α = 0, we obtain

for R > δ(Ω

)

(x) = R

Ω

(R)

(x −y)F

(y)dy −

Ω

(R)

(x −y)v

(y)dy + s

(x), (X.5.36)

where

F = f + v · ∇v

and s(x) is the surface integral in (X.5.30). We recall that

|U(x − y)|, |U

(R)

(x − y)| ≤ c|x − y|

−1

, x, y ∈ R

, x 6= y

and so, taking into account (V.3.2), (V.3.3) we have



(R)

(x − y) − U

(x − y)]F

(y)dy



≤ c

Ω

R/2,R

(x)

|F (y)|

|x −y|

with

Ω

R/2,R

(x) = {y ∈ Ω : R/2 < |x − y| < R}.

Since

Ω

R/2,R

(x)

|F (y)|

|x − y|

dy ≤ k|x −y|

−1

,Ω

R/2,R

(x)

kfk

q,Ω

R/2,R

(x)

+ kv/|x − y|k

2,Ω

R/2,R

(x)

|v|

1,2,Ω

R/2,R

(x)

from Theorem II.6.1 and the assumption made on f we derive for all x ∈ Ω

lim

R→∞

Ω

(R)

(x − y)F

(y)dy =

Ω

(x − y)F

(y)dy (X.5. 37)

in the sense of absolute convergence. Furthermore, bearing in mind that

(R)

(x −y) is identically vanishing outside Ω

R/2,R

(x), from (V.3 .5) and The-

orem II.6 .1 we recover in the limit R → ∞

X.5 Asymptotic Behavior: Preliminary Results and Representation Formulas 695



Ω

(R)

(x − y)v

(y)dy



≤ ckvk

6,Ω

R/2,R

→ 0. (X.5.38)

Formula (X.5.30) then foll ows from (X.5.36 )–(X.5.38). To prove (X.5.31) we

notice that

Ω

(R)

(x − y)v

(y)D

(y)dy =

Ω

(y)v

(y)D

(R)

(x − y)dy

∂Ω

(x − y)v

(y)v

(y)n

dσ

(X.5.39)

Since

|∇U (x − y)|, |∇U

(R)

(x − y)| ≤ c|x − y|

−2

, x, y ∈ R,

we ﬁnd



Ω

(y)v

(y)(D

(R)

(x − y) − D

(x − y))dy



≤ ckv/|x − y|k

2,Ω

R,2R

and so

lim

R→∞

Ω

(y)v

(y)D

(R)

(x − y)dy =

Ω

(y)v

(y)D

(x − y)dy

in the sense of absolute convergence. This latter relation, together with

(X.5.39) and (X.5.30), proves (X.5.31). Let us now consider representation

(X.5.34) for the pressure ﬁeld. Setting ep = p − p

with p

given in Theorem

X.5. 1, from (IV.8.19) it follows, for all suﬃciently large R, that

ep(x) = p

−R

Ω

(x)

(x − y)F

(y)dy +

∂B

(x)

(x − y)T

(v, p)(y)

−2v

(y)

∂q

(x − y)

∂x

(y)dσ

+ σ(x),

(X.5.40)

where Ω

(x) = Ω ∩B

(x), σ(x) i s the surface integral in (X.5.3 4) and p

a constant, po ssibly depending on R. R ecalling that

(x − y)| ≤ |x − y|

−2

(x − y)| ≤ c|x −y|

−3

from Theorem X.5.1 and the assumption on f we immediately deduce for all

x ∈ Ω that

lim

R→∞

∂B

(x)

(x−y)T

(v, p)(y)−2v

(y)D

(x−y)]n

(y)dσ

= 0. (X.5.41)

Furthermore, for d < dist (x, ∂Ω), setting Ω

(x) = Ω − B

(x) we have