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

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

we recover

(v · ∇v, w

) = −(v · ∇w

, v) (X.3.30)

for a ll k ∈ N. Using (X.3.17) and Theorem II.6.1 it follows that

|(v · ∇v, w

−w)| ≤ M|v|

1,2

k(w

− w)/|x|k

≤ c|v|

1,2

−w|

1,2

(X.3.31)

while Lemma X.2.1 furnishes

|(v · ∇(w

−w), v)| ≤ kvk

−w|

1,2

. (X.3.32)

Recalling (X.3.26), we pass to the limit k → ∞ in (X.3.30) and use (X.3.31),

(X.3.32) to show the validity of (i). The proof of (ii) is analogous, and therefore

it will be omitted. In view o f (i ) and (ii ) from (X.3.29) we have

−1

|u|

1,2

≤ −(v · ∇w, v) + (w · ∇w, v) + (v · ∇v, v) − (w · ∇v, v)

= (u · ∇u, v).

(X.3.33)

However, by assumption and (II.6.10),

|(u · ∇u, v)| ≤ M ku/|x|k

|u|

1,2

≤ 2M |u|

1,2

which, once replaced into (X.3.33), furnishes

|u|

1,2

−1

− 2M) ≤ 0.

By (X.3.18) this inequali ty implies u(x) = 0 for a.a. x ∈ Ω, and the theorem

is proved. ut

Remark X.3.3 Theorem X.3.2 continues to hold in any dimension n ≥ 4,

provided we replace the bound on M given in (X.3.18) with a suitable one

depending on n. This generalization has been considered by Miyakawa (1995).



X.4 Existence of Generalize d Solutions

As in the case of a bounded domain, the fundamental contribution to the

existence of steady-state solutions to the Navier–Stokes problem in exterior

domains is due to J.Leray (1933, Chapitre III). Again, the leading idea is to

employ the following a prio ri estimate for solutions to (X.0.3), (X.0.4) with

∗

= 0:

Ω

∇v : ∇v ≤ M (X.4 .1)

where M depends only on f, Ω and ν. Actually, for a ny R > δ(Ω

), Leray

looked for a solution to (X.0.3), (X.0.4) in Ω

with v

∗

= 0 at ∂Ω and v = v

∞

X.4 Existence of Generalized Solutions 677

at ∂B

and he was able to prove the existence of a solution v

, p

such that

the Dirichlet integral admits a unifo rm bound

Ω

∇v

: ∇v

≤ M (X.4.2)

with M independent of R. Upon taking a suitable sequence, as R → ∞, a

solution v, p to (X.0.3), (X.0.4) was found. Because of (X.4.2), this solution

satisﬁes (X.4.1). In the case of three–dimensional ﬂow, the estimate (X.4.1)

is enough to ensure all requirements of a generalized solution, including the

behavior at inﬁnity. This latter property is in fact a consequence of Lemm a

II.6.2. However, for plane ﬂow the bound (X.4.1) is not enough to control

the behavior at inﬁnity of the solution. It is this circumstance that renders

the problem of existence of solutio ns in a two-dimensional exterior problem a

very diﬃcult one and, in several respects, it has to b e considered still an open

question; cf. Chapter XII.

The method we shall employ to show existence is, in fact, diﬀerent from

Leray’s and it is the same used for the case of a bounded domain, namely,

the Galerkin method. Such an approach is due to Fujita (1961). The proofs

given in Theorem IX.3.1 and Theorem IX.4.1 remain essentially unchanged to

cover the present case (cf. Remark I X.3.2). The only point that deserves some

attention is the construction of a suitable extension V of the velocity ﬁeld at

the boundary and at inﬁnity. Actually, as in the case where Ω is bounded,

we have to require that V satisﬁes an inequality of the type (IX.4.3) for

all u ∈ D

1,2

(Ω) and some α < ν. The ﬁrst part of this section is devoted

to the preceding question. In this regard, we observe that, for such a ﬁeld

V to exist, the vanishing of the overall ﬂux Φ of the velocity ﬁeld through

the bounding walls is not needed; rather, it is enough that Φ be “suﬃciently

small” in a way that will be made precise later. However, it is not known

whether an upper bound for Φ is in fact necessary and consequently, unlike

the corresponding linearized Stokes and Oseen approximations, the problem

of existence of steady Navier–Stokes ﬂow in exterior domains with arbitrary

ﬂux at the boundary remains open.

With a view to the rotational case, that will be treated in the following

chapter, we shall construct the extension ﬁeld V under more general assump-

tions on v

∞

than needed here.

To begin with, we need to introduce certain quantities. We recall that

Ω = R

− ∪

i=1

Ω

, s ≥ 1, where each Ω

is compact and, we assume, with a

non-empty interior. Furthermore, Ω

∩ Ω

= ∅, for i 6= j. We thus set

(x) =

4π

∇



|x − x



, x

∈

◦

Ω

, i = 1, . . . , s, (X.4.3 )

and observe that

∂Ω

· n = δ

, i, j = 1, . . . , s , (X.4.4)

where n denotes the outer normal to ∂Ω at ∂Ω

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

Lemma X.4.1 Let Ω be a locally Lipschitz domain of R

exterior to s ≥ 1

compact and disjoint sets Ω

, ..., Ω

, and let

∗

∈ W

1/2,2

(∂Ω), v

∞

:= a + A · x ,

where a ∈ R

and A is a second order tensor with trace ( A) = 0. Then, for

any η > 0, there exists ε = ε(η, v

∗

, Ω) > 0 and V = V (ε) : Ω → R

such

that, for some R > δ(Ω

) and all q ∈ (1, ∞), r ∈ (3/2, ∞):

(i) V + v

∞

∈ W

1,2

(Ω

) ∩D

1,q

(Ω

);

(ii) V + v

∞

∈ L

(Ω

);

(iii) V = v

∗

at ∂Ω;

(iv) ∇ · V = 0 in Ω.

Furthermore, for all u ∈ D

1,2

(Ω), it holds that

|(u · ∇(V + v

∞

), u)| ≤



η +

4π

i=1

|Φ

|max

x∈Ω

|x − x



|u|

1,2

. (X.4.5)

where

∂Ω

∗

· n, i = 1, . . . , s,

is the ﬂux of v

∗

through the boundary of Ω

. Finally, if kv

∗

1/2,2(∂Ω)

≤ M,

for som e M > 0, then

kV + v

∞

1,2

≤ C

∗

1/2,2(∂Ω)

kV + v

∞

r,Ω

R + |V + v

∞

1,q,Ω

R ≤ C

∗

1/2,2,(∂Ω)

(X.4.6)

where C

= C

(η, M, Ω), and C

= C

(r, q, R, Ω).

Proof. The procedure is simila r to, but si mpler than, that used in the proof

of Lemma IX.4. 2. We set

(x) = v

∗

(x) −

i=1

(x) + v

∞

, x ∈ ∂Ω .

Clearly, because of (X.4.4),

∂Ω

· n = 0, i = 1, . . ., s,

and so, according to Lemm a IX.4. 1, there is w ∈ W

2,2

(Ω) such that, setting

U := ∇ × w ∈ W

1,2

(Ω)

we have

X.4 Existence of Generalized Solutions 679

U = v

at ∂Ω

U(x) = 0, for all x ∈ Ω

, R > δ(Ω

Given ε > 0, we set

= ∇ ×(ψ

with ψ

the “cut-oﬀ” f unction deﬁned in Lemma II I.6.2. The desired extension

of v

∗

is then given by

V = V

i=1

−v

∞

≡ V

+ V

− v

∞

. (X.4.7 )

Taking into account the properties of the function w given in Lemma IX.4.1,

with the help of (X.4 .3) it i s immediate to show that V satisﬁes (i)-(iv). Let

us now estimate the trilinear form

a(u, V + v

∞

, u) ≡ (u ·∇(V + v

∞

), u) = a(u, V

+ V

, u), u ∈ D

1,2

(Ω).

Fix η > 0. By reasoning entirely anal ogous to that used in the proof of Lemma

IX.4. 2, and which we shall leave to the reader, we have

|a(u, V

, u)| ≤ η |u|

1,2

, (X.4.8)

provided ε ≤ η/



∗

1/2,2(∂Ω)



, for suitable c

= c

(Ω). Furthermore,

integrating by parts and recalling (X.4.3), we ﬁnd

|a(u, V

, u)| ≤

i=1

|Φ

Ω

|∇u|

|E(x − x

)| ≤

4π

i=1

|Φ

|max

x∈Ω

|x − x

|u|

1,2

(X.4.9)

Inequality (X.4.5) then follows from (X.4. 8) and (X.4.9). It remains to show

(X.4.6), under the further stated assumptions on v

∗

. By the same argument

used in the proof of Lemma IX.4.1 (speciﬁcally, the argument leading to

(IX.4.48)), we easily infer

1,2

≤ c

∗

1/2,2(∂Ω)

(X.4.10)

with c

= c

(η, M, Ω). Mo reover, from (X.4.3) we deduce

1,q

+ kV

≤ c

∗

1/2,2(∂Ω)

, for all q ∈ (1, ∞), r ∈ (3/2, ∞),

(X.4.11)

where c

= c

(Ω). Then, noticing that V (x) + v

∞

= V

for all |x| ≥ R, Ω

containing the support of ψ

we conclude that (X.4.6) follows from (X.4.10)

and (X.4 .11). ut

Notice that ∪

ε>0

supp (ψ

) is bounded.

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

Remark X.4.1 If s = 1, choosing x

= 0, we have

max

x∈Ω

|x|

≤

, r

= dist (0, ∂Ω),

condition (X.4.5) becomes

|(u · ∇(V + v

∞

), u)| ≤



η +

|Φ

4πr



|u|

1,2



Remark X.4.2 We would like to discuss the generali zation of Lemma X.4.1

to arbitrary di mension n ≥ 2, n 6= 3, in the case v

∞

≡ v

∞

∈ R

. If n ≥ 4, the

generalization is straightforward, provided we require a little more regularity

on v

∗

of the type

∗

∈ W

1−1/s,s

(∂Ω), s > n/2.

The extra regularity is needed to construct the extension ﬁeld V

, cf. Remark

IX.4. 7. If we assume this, we can prove the existence of a ﬁeld V satisfying

(i)-(iv) for all q ∈ (1, ∞) and all r ∈ (n/(n−1), ∞ ). Furthermore, the following

inequality holds

|(u ·∇V , u)| ≤ (η + Φ

(n)

)|u|

1,2

where

(n)

≡ c

i=1

|Φ

| max

x∈Ω

|x −x

n−2

and c = c(n). In the two-dimensional case, the starting point is, again, (X.4.7),

with

(x) =

2π

∇(log |x − x

|) . (X.4.12)

By following exactly the same arguments used for the proof of Lemma X.4.1,

we thus show that V satisﬁes (i)-(iv) with q ∈ (1, ∞) and r ∈ (2, ∞), as

well as the inequalities in (X.4.6), under the stated further assumption on v

∗

Moreover, we have

|(u · ∇V , u)| ≤ (η +

2π

i=1

|Φ

|)|u|

1,2

Given the properties of V

, the proof of this latter is, as in Lemma X.4. 1,

reduced to show the estimate

|(u · ∇V

, u)| ≤

2π

i=1

|Φ

||u|

1,2

. (X.4.13)

However, (X.4.13) can not be obtained by a procedure similar to that used

to obtain (X.4.9), and we have to argue diﬀerently. The following argument

X.4 Existence of Generalized Solutions 681

is due to A. Russo (2010b, Lemma 3). By integration by parts, the proof of

(X.4.13) reduces to show (i = 1, . . ., s)

:= 2π



Ω

u · ∇u · σ



≤ |u|

1,2

. (X.4.14)

We introduce a system of po lar coordinates (r, θ) with the origin at x

, and

set

f :=

2π

f(r, θ)dθ .

Extending u to 0 in Ω

, we ﬁnd

= 2π



Ω

u · ∇u ·

|x|



∞

2π



−u

)

∂u

∂θ

−(u

− u

)

∂u

∂θ



dθ



(X.4.15)

where we used

2π

∂u

∂θ

dθ =

2π

∂u

∂θ

dθ = 0 .

Employing the Schwarz inequality, along with the Wirtinger inequality (I I .5.17)

and Exercise II.5.1 2, we deduce



2π

− u

)

∂u

∂θ



≤

2π

−u

2π



∂u

∂θ



1/2

≤

2π



∂u

∂θ



2π



∂u

∂θ



1/2

(X.4.16)

and, likewise,



2π

− u

)

∂u

∂θ



≤

2π



∂u

∂θ



2π



∂u

∂θ



1/2

(X.4.17)

Therefore, o bserving that |

∂u

∂θ

| ≤ r|∇u|, we ﬁnd that (X.4.14) follows from

(X.4.15)–(X.4.17). 

Theorem X.4.1 Let Ω be a locally Lipschitz do mai n of R

, exterior to com-

pact and disjoint domains Ω

, ..., Ω

, s ≥ 1.

Moreover, let

f ∈ D

−1,2

(Ω), v

∗

∈ W

1/2

(∂Ω), v

∞

∈ R

and set

Φ :=

4π

i=1

max

x∈Ω

|x − x



∂Ω

∗

· n



The following prop erties hold.

The theorem continues to hold in the case where Ω = R

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

(i) Existence. If Φ < 1/R, there exists at l east one generalized solution v to

the Navier–Stokes problem (X.0.8), (X.0.4). Such a solution veriﬁes the

conditions:

|v(x) + v

∞

| = O(1/

|x|) as |x| → ∞

kpk

2,Ω

≤ c (|v|

1,2

+ Rkvk

1,2Ω

+ R|f|

1,2

)

(X.4.18)

for all R > δ(Ω

). In (X.4.18) p is the pressure ﬁeld associated to v by

Lemma X.1.1, whil e c = c( Ω, R) with c → ∞ as R → ∞. Furthermore,

v obeys the generalized energy inequality

|v|

1,2

+ R[f , v − V ] ≤ (∇v, ∇V ) − R(v · ∇V , v − V ) (X.4.19)

where V is the extension of v

∗

and v

∞

constructed in Lemma X.4.1.

(ii) Estimate by the data. If v

∗

∈ M

1/2,2

(∂Ω) (deﬁned in (IX.4.52)) and

Φ ≤ 1/(2R), then the generalized solution determined in (i) satisﬁes the

following estimate:

|v|

1,2

≤ 4R|f|

−1,2

+ C kv

∗

1/2,2(∂Ω)



1 + R



1 + kv

∗

1/2,2(∂Ω)

+ |v

∞



(X.4.20)

where C = C(Ω, R, M ) .

Proof. We shall employ the Galerkin method. We look for a solution of the

form

v = u + V ,

where V is the extension of v

∗

and v

∞

constructed in Lemma X.4 .1 and

corresponding to some η > 0 that will b e ﬁxed successively. Let {ψ

} ⊂

D(Ω) be the basis of D

1,2

(Ω), introduced in Lemma VII. 2.1. A sequence of

approximating solutions {u

} is then sought of the form

k=1

(∇u

, ∇ψ

) + (u

· ∇u

, ψ

) + (u

· ∇V , ψ

) + (V · ∇u

, ψ

)

= −[f, ψ

] −

(∇V , ∇ψ

) − (V · ∇V , ψ

), k = 1, 2, . . ., m.

(X.4.21)

For each m ∈ N we may establish exi stence to (X.4.21) in the same way as

in Theorem IX.3.1, provided we show a suitable uniform bound for |u

1,2

terms of the data. Thus, multiplying (X.4. 21)

by ξ

, summing over k from

1 to m and recalling that, for all m ∈ N,

(V · ∇u

, u

) = (u

· ∇u

, u

) = 0, (X.4.2 2)

(cf. Lemma IX.2.1), we obtain

X.4 Existence of Generalized Solutions 683

1,2

+ (u

·∇V , u

) = −[f, u

] −

(∇V , ∇u

) −(V · ∇V , u

(X.4.23)

Recalling the prop erties of V given in Lemma X.4.1, we have, by the Schwarz

inequality,

|(∇V , ∇u

)| ≤ |u

1,2

|V |

1,2

, (X.4.24)

and, by the H¨older and Sobolev inequalities

|(V · ∇V , u

)| ≤



Ω

V · ∇V · u



Ω

(V + v

∞

) ·∇V · u



Ω

∞

· ∇V · u



≤ [c

kV k

1,2,Ω

√

(kV + v

∞

3,Ω

R |V |

1,2

+ |v

∞

||V |

1,6/5

)]|u

1,2

(X.4.25)

Furthermore, again from Lemma X.4.1,

|(u

· ∇V , u

)| ≤ (η + Φ)|u

1,2

. (X.4.26)

Collecting (X.4.23)– (X.4.26), we ﬁnd

(1/R − η − Φ) |u

1,2

≤ |f |

−1,2

|V |

1,2

+ c

kV k

1,2,Ω

√



kV + v

∞

3,Ω

R |V |

1,2

+ |v

∞

||V |

1,6/5



(X.4.27)

Thus, if Φ < 1/R, we may choose η ∈ (0, 1/R − Φ) to obtain the following

uniform estimate

1,2

≤ C, (X.4.28)

with C = C(f , v

∗

, v

∞

, R). By (X.4.28) and Lemma IX.3.2 we may show exis-

tence of solutions to (X.4.21) for al l m. Moreover, we can select a subsequence,

denoted again by {u

}, such that as m → ∞

→ u in D

1,2

(Ω). (X.4.29)

Also, by Exercise II.5.8 a nd the Ca ntor diagonalization argument, we may

select another subsequence, that we still call {u

}, such that, as m → ∞,

→ u in L

(Ω

), q ∈ [1, 6) , (X.4.30)

for any R > δ(Ω

). By (X.4.28) and Theorem II.1.3 it foll ows that the limiting

ﬁeld u also obeys i nequality (X.4.28). Let us now pass to the lim it m → ∞

into (X.4.21)

. Observe that, setting

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

Ω

≡ supp (ψ

we have

)ψ

, V

∈ L

(Ω

and so, from (X.4.29) and (X.4.30) we obtain

(∇u

, ∇ψ

) → (∇u, ∇ψ

)

· ∇V , ψ

) → (u · ∇V , ψ

)

(V · ∇u

, ψ

) → (V · ∇u, ψ

(X.4.31)

Furthermore, reasoning as in the proof of Theorem IX.3.1

we have

· ∇u

, ψ

) → (u · ∇u, ψ

). (X.4.32)

Employing (X.4.31) and (X.4.32) into (X.4.21)

yields

(∇u, ∇ψ

) +(u · ∇u, ψ

) + (u · ∇V , ψ

) + (V · ∇u, ψ

)

= −[f, ψ

] −

(∇V , ∇ψ

) −(V · ∇V , ψ

(X.4.33)

It is now immediately veriﬁed that the ﬁeld

v ≡ u + V

is a generalized solution to the problem, in the sense of Deﬁnition X.1.1.

Actually, issues (i)-(iii) o f that deﬁnition are at once established. Also, from

Lemma X.4.1 and Lemma II.6.2 it f ollows, as |x| → ∞, that

|v(x) + v

∞

| ≤

|u(x)|+

|V (x) + v

∞

| = O(1/

|x|) (X.4.3 4)

and so (X. 4.18)

is proved. Finally, from (X.4.33) we obtain

(∇v, ∇ψ

) + (v · ∇v, ψ

) = −[f , ψ

] . (X. 4.35)

We have

v ·∇v = u · ∇v + (V + v

∞

) · ∇v − v

∞

·∇v .

Thus, recalling that u, (V + v

∞

) ∈ L

(Ω), by the H¨older inequality it follows

that

v · ∇v ∈ L

3/2

(Ω) + L

(Ω) .

Speciﬁcally, one has to follow the part of the proof that goes from (IX.3.15)

to (IX .3.20). In these formulas, for the case at hand, the integration has to be

performed over the bounded domain Ω

X.4 Existence of Generalized Solutions 685

Since any function ϕ ∈ D(Ω) can be approximated in the W

1,s

-norm by linear

combinations of ψ

for all s ≥ 2, the issue (v) follows from thi s latter property

and (X. 4.35). Let us next show the estima te for the pressure ﬁeld p, whose

existence is guaranteed by Lemma X.1.1. In particular, by this lemma,

p ∈ L

loc

(Ω). (X.4.36)

Fix R > δ(Ω

) and add to p the constant

C = C(R) = −

|Ω

Ω

so that

Ω

(p + C) = 0. (X.4. 37)

Successively, choose ψ such that

∇ · ψ = p + C in Ω

ψ ∈ W

1,2

(Ω

)

kψk

1,2

≤ c

kp + Ck

2,Ω

(X.4.38)

Because of (X.4.36) and (X.4.37), Theorem III.3.1 guarantees the existence of

ψ and it is clear that we can replace this function into identity (X.1.3). Thus,

from (X.1.3), and the Schwarz and H¨older inequalities, we ﬁnd

kp + Ck

2,Ω

≤ c

[(|v|

1,2

+ R|f|

−1,2

)|ψ|

1,2

+ Rkvk

4,Ω

|v|

1,2,Ω

kψk

4,Ω

(X.4.39)

Recalling that W

1,2

(Ω

) ,→ L

(Ω

), and using (X.4.38)

, from the ab ove

relation we prove (X.4.18)

. To show the theorem completely, we have to prove

the generalized energy inequality (X.4. 19). To this end, we observe that, by

(i) and (ii) of Lemma X.4.1, we have

V · ∇V ∈ L

6/5

(Ω), (X.4.40)

and since, by the Sobolev inequality (II.3.7),

≤ γ |u

1,2

≤ M (X.4.41)

we deduce (along a subsequence, at least)

lim

m→∞

(V · ∇V , u

) = (V · ∇V , u). (X.4.42)

Moreover, for ﬁxed R > δ(Ω

), by the H¨older inequali ty,

|(u

· ∇V , u

)−(u · ∇V , u)|

≤ |(u

−u) · ∇V , u

)| + |(u · ∇V , u

− u)|

≤ (ku

4,Ω

+ kuk

4,Ω

)ku

−uk

4,Ω

|V |

1,2

+(ku

+ kuk

)|V |

1,3/2,Ω

(X.4.43)