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

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

Proof. Setting u = v − e

, w = u − u

, and π = p −π

we deduce

∆w − R

∂w

∂x

= R(u · ∇v) + ∇π

∇ · w = 0











in Ω

lim

|x|→∞

w(x) = 0

w = 0 at ∂Ω.

Thus, (X.10.20)

follows directly from Theorem II.6.1, Theorem VII. 7.1, and

Lemma X.10.2. Since from Lemma X.10.2 we have

u · ∇v ∈ L

3/2

(Ω),

Theorem VII.7.1 in turn furnishes

|w|

1,12/5

≤ CR

3/4

Thus, by (X.10.20)

and the interpolation inequality (II. 2.7), we deduce with

θ = 4 (3 − t)/t

|w|

1,t

≤ |w|

12/5

|w|

(1−θ)

≤ CR

2−3/t

As a consequence, the estimate (X.10.20)

follows from this inequality and

Theorem II.6.1 . Moreover, (X.10.20)

is an immediate consequence of Theo-

rem II.6.1 and Theorem VII.7.1. In the rest of the proof the letter C wil l have

the same meaning as in Lemma X.10.2. Let us denote by E(x; R) the Oseen

tensor corresponding to the Reynolds number R. From Exercise VII.3.5 we

know that E ob eys the homogeneity condition

E(x; R) = RE(Rx; 1). (X.10. 21)

We next notice that, i n view of Theorem VII.6.2, we have

(x) =

∂Ω

∗i

, e

)(x − y) − E

(x − y)T

, π

)(y)

+ Ru

∗j

(y)E

(x −y)δ

(y)dσ

Therefore, assuming, without l oss, that Ω

⊂ B

1/2

, it follows that

(x)| ≤ CD{ sup

y∈Ω

1/2

|E(x − y; R)| + sup

y∈Ω

1/2

[|e(x −y)| + |∇

E(x − y; R)|]}.

(X.10.22)

where

D = ku

1,1,∂Ω

+ kπ

1,∂Ω

From (X.10.21) and (X.10.22) we deduce

X.10 Limit of Vanishing Reynolds Number and Stokes Problem 737

s,Ω

≤ CD

s−3

|y|≥R

sup

|z|≤R/ 2

|E(y − z; 1)|

+ R

[|e(y − z)|

+|∇

E(y − z; 1)|

]

dy.

(X.10.23)

To estimate the ﬁrst integral on the right-hand side of this inequality, we

observe that

|y|≥R

sup

|z|≤R/ 2

|E(y − z; 1)|

dy ≤

2≥|y|≥R

sup

|z|≤B/2

|E(y −z; 1)|

|y|≥2

sup

|z|≤B/2

|E(y −z; 1)|

dy .

(X.10.24)

From (VII. 3.21) we have

|E(y − z; 1)| ≤ c(|y −z|

−1

+ 1), |y|, |z| ≤ 2

and since

|z| ≤ R/2 , |y| ≥ R implies |y − z| ≥

|y|, (X.10.25)

recalling that s < 3, we ﬁnd

2≥|y|≥R

[ sup

|z|≤R/2

|E(y − z; 1)|]

dy ≤ C. (X.10.26)

Furthermore, by the mean value theorem,

|(E

(y − z) − E

(y))| = |z

∂

∂x

(y − βz)|, β ∈ (0, 1),

and so, from (VII.3.32) it follows for all |y| suﬃciently large (≥ |y

|, say) that

|y|≥|y

sup

|z|≤B/2

|E(y − z; 1)|

dy ≤ C

|y|≥|y



|E(y; 1)|

+ |y|

−3s/2



dy.

(X.10.27)

Taking into account that s > 2, from (X.10.27), the local regularity of E, and

(VII.3.2 8), we have

|y|≥2

sup

|z|≤B/2

|E(y − z; 1)|

dy ≤ C. (X.10.28)

In addition, from (VII.3.14) and (X.10.25) we inf er that

|y|≥R

sup

|z|≤R/ 2

|e(y −z)|

dy ≤ C R

{

2≥|y|≥R

|y|

−2s

|y|≥2

sup

|z|≤B/2

|y − z|

−2s

}

≤ C (1 + R

3−s

(X.10.29)

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

Likewise,

|y|≥R

sup

|z|≤R/ 2

|∇

E(y −z; 1)|

dy ≤

2≥|y|≥R

sup

|z|≤B/2

|∇

E(y − z; 1)|

|y|≥2

sup

|z|≤B/2

|∇

E(y − z; 1)|

dy .

(X.10.30)

Since, from (VII.3.21),

|∇E(y −z; 1)| ≤ C|y −z|

−2

, |y|, |z| ≤ 2,

by virtue of (X.10.25), it follows that

2≥|y|≥R

sup

|z|≤R/ 2

|∇

E(y − z; 1)|

dy ≤ C

2≥|y|≥R

|y|

−2s

dy ≤ C(1 + R

3−2s

(X.10.31)

Also, we use the asymptotic prop erties of ∇E(y; 1), cf. (VII.3.31), together

with the following ones on the second derivatives

E(y)| ≤ c|y|

−2

cf. (VII.3.35 ), to establish, as we did for (X.10.28 ), the following estimate:

|y|≥2

sup

|z|≤B/2

|∇

E(y − z; 1)|

dy ≤ C. (X.10. 32)

Thus, from (X.10.30)–(X.10.32) we recover

|y|≥R

sup

|z|≤R/2

|∇

E(y − z; 1)|

dy ≤ C(1 + R

3−s

). (X.10.33)

Collecting (X.10.23), (X.10.26), (X.10.28), (X.10.29), and (X.10.33) we ﬁnd

s,Ω

≤ DR

1−3/s

. (X.10.34)

On the other hand, by Theorem VII.2.1 and Theorem VII.6. 2, and with the

help of the embedding Theorem II.3.4, we ﬁnd

kπ

1,2,Ω

+ ku

s,Ω

+ |u

1,2

≤ C (X.10.35)

where π

has been modiﬁed by the addition of a suitable constant. Moreover,

from Theorem IV.4.1 and Theorem IV.5.1 we obtain, in particular,

2,2,Ω

≤ C (ku

1,2,Ω

+ kπ

1,2,Ω

) . (X.10.36)

Use of the trace T heorem II.4.4 yields

X.10 Limit of Vanishing Reynolds Number and Stokes Problem 739

D ≡ ku

1,1,∂Ω

+ kπ

1,∂Ω

≤ C (ku

2,2,Ω

+ kπ

1,2,Ω

)

and so this inequality, together with (X.10.35), (X.10.36), impl ies

D ≤ C.

Estimate (X.10.20)

then follows from this relation and (X.10.34), (X.10.35),

and the lemma is proved. ut

Lemma X.10.4 Let the assumptions of Lemma X.10.1 be satisﬁed. Denote

by v

, p

the solution to the f ollowing Stokes probl em

∆v

= ∇p

∇ · v

= 0

)

in Ω

= v

∗

at ∂Ω

lim

|x|→∞

(x) = e

(X.10.37)

and by u

, π

the solution to (X.10.19). Then there exists B > 0 such that

for a ny R ∈ (0, B], all q ∈ [3/2, 2] and all s ∈ (2, 3) we have

−v

+ e

3s/(3−s)

+ kπ

− p

+ |u

− v

1,3q/(3−q)

+kD

− v

2.q

+ k∇(π

−p

2,q

≤ CR

2−3/s

where C = C(Ω, v

∗

, B, q, s).

Proof. Setting

z = u

− v

+ e

, τ = π

− p

we ﬁnd

∆z = R

∂u

∂x

+ ∇τ

∇ · z = 0











in Ω

lim

|x|→∞

z(x) = 0

z = 0 at ∂Ω.

By Lemma X.10.3 we have, in parti cular, for all s ∈ (2, 3)



∂u

∂x



−1,s

≤ CR

1−3/s

Therefore, with the help of Theorem V.5.1, we obtain

kzk

3s/(3−s)

+ |z|

1,s

+ kτ k

≤ CR

2−3/s

. (X.10.38)

In addition, from Lemma V.4.3 and Lemm a X.10.3 , we have

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

|z|

1,3q/(3−q)

+ kD

2,q

+ k∇τk

2,q

≤ C (R + kzk

2,Ω

+ kτ k

2,Ω

) ,

and the proof becomes a consequence of this latter inequality and (X.10.38).

We are now in a position to prove the following main result.

Theorem X.10.1 Let Ω be an exterior domain of R

of class C

and let v

be a generalized solution to (X.0.8 ), (X.0.4) corresponding to v

∞

= e

and

to f ≡ 0. Moreover, l et v

, p

be the solution to the Stokes problem (X.10.1).

Then, if

∗

∈ W

11/2,2

(∂Ω),

there exist B > 0 and C = C(Ω, v

∗

, B, ε) > 0 such that

kv − v

(Ω)

+ kp − p

(Ω)

≤ CR

1−ε

where p is the pressure ﬁeld associated to v and ε is a positive number that

can b e ta ken arbitrarily close to zero.

Proof. Set

z = u

−v

+ e

, τ = π

− p

with u

, π

and v

, p

solving (X.10.19) and (X.10.37). Then, by Lemma

X.10.4 and a repeated use o f Theorem II.6. 1 we have

k∇zk

1,3q/(3−q)

≤ CR

2−3/s

and so, being 3q/(3 − q) > 3, by the embedding Theorem II.3.4 and taking s

arbitrarily close to 3, we deduce

k∇zk

(Ω)

≤ CR

1−ε

, (X.10.39)

where ε satisﬁes the property stated in the lemma. Likewise, again by L emma

X.10.4 and Theorem II.3.4, we have

k∇τ k

C(Ω)

≤ CR

1−ε

. (X.10.40)

Next, in view of (X.10.39) and Lemma X.10.4, it follows that

kzk

1,3s/(3−s)

≤ CR

1−ε

for a ll s arbitrarily close to 3. Therefore, with the help of Lemm a X.10.4 and

Theorem II.3.4, we ﬁnd

kzk

C(Ω)

≤ CR

1−ε

. (X.10.41)

Likewise, we prove

kτk

C(Ω)

≤ CR

1−ε

. (X.10.42)

C → ∞ as ε → 0.

X.10 Limit of Vanishing Reynolds Number and Stokes Problem 741

In a similar way, making use of the results of Lemma X.10. 3, we have

kv − u

−e

(Ω)

+ kp − π

(Ω)

≤ CR

1−ε

. (X.10. 43)

Since

kv − v

(Ω)

≤ kv − u

−e

(Ω)

+ ku

− v

+ e

(Ω)

kp − p

(Ω)

≤ kp − π

(Ω)

+ kπ

−p

(Ω)

the result follows from (X.10.39)–(X.10.43). ut

Remark X.10.1 If Ω is the complement of a bounded domain, Fischer,

Hsiao, & Wendland (1985, Theorem 1) have performed a more detailed anal-

ysis than that of Theorem X. 10.1 of the way in which v approaches v

. Such

an analysis is based on boundary integrals and pseudo-diﬀerential op erators

and leads to the following main result.

Theorem X.10.1

. Let Ω be a smooth exterior domain in R

bounded by a

simple closed surface and let v, p and v

, p

be solutions to (X.0.8), (X.0.4)

and (X.10 .2), respectively, corresponding to v

∗

≡ f ≡ 0. Then,

v(x) = v

(x) + q(x; R),

where

q(x; R) = O(R), as R → 0, uniformly in Ω.

Moreover, there exists w = w(x) deﬁned in Ω such that, for any given compact

subset K of Ω,

v(x) = v

(x) + Rw(x) + O(R

ln R

−1

), as R → 0,

holds uni formly on K. 

Theorem X.10.1 allows us to draw some interesting conclusions of the way

in which the force D exerted by the liquid on the boundary ∂Ω of the region

of ﬂow in the nonlinear motion described by (X.0.8), (X.0.4 ) approaches the

same quantity D

calculated in the Stokes approximation (X.10.1). To show

this, we observe that since

D = −

∂Ω

T ( v, p) · n, D

= −

∂Ω

T ( v

, p

) ·n,

with T the Cauchy stress tensor (IV.8.6) and n outer unit normal at ∂Ω,

under the assumptions of Theorem X.10.1, it follows at once that

|D − D

| ≤ CR

1−ε

, (X.10.44)

where C = C(Ω, v

∗

, B) and R ∈ (0, B].

Remark X.10.2 Under the same a ssumptions of Theorem X.10.1

, one can

show more detailed versions of (X.10.44), cf. Babenko (1976, eq. (6.7)), Fis-

cher, Hsiao, & Wendland (1985, Theorem 2). 

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

X.11 Notes for the Chapter

Section X.1. The variational formulation of the exterior problem in the way

used here has been introduced by Ladyzhenskaya (195 9b).

The i ntroduction o f the pressure ﬁeld p associa ted to a generalized solution

in the ﬁrst part of Lemma X.1.1, as a member of L

loc

, can be deduced from

the paper of Solonnikov &

Sˇcadilov (1 973). Nevertheless, from this paper

no summability properties for p in a neighborhood of inﬁnity are directly

obtainable. In fact, the second part of Lemma X.1. 1 is due to me.

The special case q = 2 and v

∞

= 0 of the results stated in Exercise X.1.1

can be found in Kozono & Sohr (1992a, Theorem 3(i)). Their proof, diﬀerent

than that indicated in Exercise X. 1.1, employs an i nterpolation theorem of

Aronszajn-Gagliardo.

Diﬀerentiability properties of generalized sol utions were ﬁrst determined

by Ladyzhenskaya (1959b, Chapter II).

Section X.2. The validity of the energy equation in exterior domains was

established by Finn (1959b, §III). However, the assumptions made there on the

body force and on the asymptotic behavior of solutions are a priori stronger

than those stated in Theorem X.2.1 and Theorem X.2.2. In particular, the

latter refer to the so-called physically reasonable solutions we mentioned i n

the Introduction to this chapter. For general properties of these solutio ns,

we refer the reader to the review articles of Finn (19 65b, 1973); cf. also Finn

(1961b, 1963, 1970).

Section X.3. The connection between generalized solutions satisfying the

energy inequality and their uniq ueness was ﬁrst pointed out by H. Kozono

and H. Sohr in a preprint in 1991, published later as their paper (1993).

However, Theorem X.3.1 and Theorem X.3.2 are placed in a diﬀerent context

and have been independently obtained by Galdi (1992a, 1992c).

Section X.4. The ﬁrst existence result for the Navier–Stokes problem in

exterior domains is due to Leray (1933).

Lemma X.4.1 is based on an idea of Finn (1961a, §2c) and it is due to me.

The proof of existence of generalized solutions satisfy ing the energy i nequality

(X.4.19) is also due to me; cf. Theorem X.4.1.

Existence in weighted Sobolev spaces has been investigated by Farwig

(1990, 1992 a, 1992b) and, successively, by Farwig and Sohr (1995, 1998), in

the case where v

∞

6= 0. A similar approach when v

∞

= 0 is not known.

An interesting question, that has been addresses by several authors, is

the existence of q-weak solutions for q 6= 2 and v

∞

= 0;

see Galdi & Padula

(1991), Kozono & Sohr (1993), Borchers & Miyakawa (1995), Miyakawa (1995,

1999), Kozono, Sohr, & Yamazaki (1997), Kozono & Yamazaki (1998). Now,

if q > 2, the answer is positive and quite trivial, in the lig ht of the asymp-

totic results of Theorem X.5.1, and provided f satisﬁes suitable summability

If v

∞

= 0, we know that generalized solutions are, in fact, q-solutions as well,

under suitable assumptions on the data; see Section X.5.

X.11 Notes for the Chapter 743

properties at large distances. On the other hand, if q ∈ (1, 2) the situation is

more involved, as we are going to explain. Take v

∗

≡ 0 (for simplicity) and

assume f ∈ D

−1,2

(Ω) ∩D

−1,q

(Ω). By a simple a rgument one shows that, due

to the structure of the nonlinear term, in order to prove exi stence by a ﬁxed

point approach we have to choose q = 3/2, in the three-dimensional case;

see

Kozono & Yama zaki (1998). However, existence of such a (3/2)-weak solution

means v ∈ D

1,3/2

(Ω) ∩ D

1,2

(Ω), and, therefore, p ∈ L

3/2

(Ω) ∩ L

(Ω).

turn, this implies, at once, tha t the solution (v, p), if exists, must obey the

following nonlocal compatibility condition

∂Ω

(T (v, p) + F ) · n = 0 , (∗)

where T is the Cauchy stress tensor, and, without l oss of generali ty, we have

written f = ∇·F in the weak sense, where F ∈ L

3/2

(Ω)∩L

(Ω); see Theorem

II.8.2.

Condition (∗) is immediately formally obtained by integrating (X.0.8)

(with f ≡ ∇ · F ) over Ω

, then letting R → ∞, and showing that, due to

the summability properties of v and p the surface integrals converge to zero,

along a sequence of surfaces, at least. From the physical viewpoint, (∗) means

that the total net force exerted on the “obstacle”, Ω

, must vanish. Clearly,

if Ω ≡ R

, (∗) is automatically satisﬁed and, in fact, one shows existence

(and uniqueness) for small data; Kozono & Nakao (1996) and also Maremonti

(1991). On the other hand, if Ω

6= ∅ and suﬃciently smooth, one can show

that such soluti ons can exist only for f in a subset of D

−1,2

(Ω) ∩D

−1,3/2

(Ω)

with empty i nterior; see Galdi (2009). In other words, for a generic F with

the speciﬁed summ abil ity properties, the above (3/2)-weak solution does not

exist. Finally, we wish to observe that, as shown by Kozono & Yama zaki

(1998), if the space D

1,3/2

(Ω) is restricted to a suitable larger homogeneous

Sobolev space o f Lorenz type, the compatibility condition (∗) is no longer

necessary, and existence of corresponding solutions, with velocity ﬁeld in this

latter space, can be still recovered.

Section X.5. In his paper of 1933, Leray left out the question of whether

a generalized solution tends, uniformly and pointwise, to the prescribed vec-

tor v

∞

, in the case when v

∞

6= 0; cf. Leray (1933, pp. 57-58). The general

case was successively proved by Finn (1959a, Theorem 2) and, independently,

by M. D. Faddeyev in his thesis published at Leningrad University in 1959;

cf. Ladyzhenskaya (1969, Chapter 5, Theorem 8). Convergence of higher-order

derivatives for the velocity ﬁeld and for the pressure ﬁeld was ﬁrst investigated

by Finn (1959b). In the same paper, Finn shows representation formulas of

In general, q = n/2 for n ≥ 3.

This property for p can be easily established by the same argument used in the

proof of Lemma X.1.1.

Of course, (∗) has to be understood in the trace sense, according to Theorem

III.2.2.

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

the type derived in Theorem X.5.2, f or solutions having suitable behavior at

large distances.

Section X.6. The ﬁrst study of the global summabili ty properties of a gener-

alized solution, v, when v

∞

6= 0, goes back to Babenko (1973). In this paper,

the crucial step is to show that

v + v

∞

∈ L

4−ε

(Ω) (∗∗)

for some small positive ε. Actually, once this condition is established, it is

relatively simple to prove that v enjoys the same summability properties (and

hence has the same asymptotic structure) of the Oseen fundamental solution;

see Babenko (1973, pp. 11- 21). I regret that some steps of Babenko’s proof

remain obscure to me.

The proo f of g lobal summability properties of generalized solutions devel-

oped in Lemma X.6.1 a nd Theorem X. 6.4 is due to me, and is inspired by the

paper o f Galdi & Sohr (1995).

Theorem X.6.5 with q = 2 is due to Finn (1960).

Section X.8. Empl oying the key prop erty (∗∗), Babenko (1973) showed that

every generalized solution corresponding to v

∞

6= 0 is “physically reason-

able” in the sense of Finn, that is, it behaves asymptotically as the Oseen

fundamental solutio n; cf. also the Introduction to this chapter. As remarked

previously, however, some steps of Babenko’s proof of (∗) remain unclear to

me. The proof given here, cf. Theorem X.8.1, is completely independent of

Babenko’s and is taken from the work of Galdi (1992b). Another proof has

been successively provided by Farwig & Sohr (1998)

The proofs of Theorem X.8.2 and T heorem X.8.3 rely on ideas of Finn

(1959b).

Section X.9. Lemma X.9.1 is due to Galdi & Si mader (1994). Theorem

X.9. 1, Theorem X.9.2(i), and Theorem X.9.3 are due to Gal di (1992a, 1992c).

Theorem X.9.5 is due to me. In this connection, see also Novotn´y & Padula

(1995)

Section X.10. The relation b etween Navier–Stokes and Stokes problems in

exterior domains, in the limit of vanishing Reynolds number, was ﬁrst inves-

tigated by Finn (1961a). The approach followed in this section is due to me.

For related questions, see also Arai (1995).

Steady Navier–Stokes Flow in

Three-Dimensional Exterior Domai ns.

Rotational Case

F.F. CHOPIN, Polonaise op. 53, bars 1–2.

Introduction

This chapter is devoted to the study of the m athematical properties of so-

lutions to the exterior boundary-value problem (X.0.1)–(X.0.2), in the case

ω 6= 0, so that in general, v

∞

= v

+ ω × x. As we observed in the In-

troduction to Chapter VIII, this study has an important bearing o n those

appli ed ﬁelds involving the free motion of rigid bodies in viscous liquids, such

as sedimentation and self-propulsio n phenomena.

Without loss of generality, we take ω = ω e

(ω > 0), whereas v

= v

with e a unit vector (unrelated, in principle, to e

) and v

≥ 0. Then, we

may use the Mozzi–Chasles transformation (VIII.0.5)–(VIII.0 .6) in (X.0.1)–

(X.0.2), so that our task reduces to the study of the following boundary-value

problem:

ν∆v = v · ∇v + 2ωe

×v + ∇p + f

∇· v = 0

)

in Ω

v = v

∗

at ∂Ω,

(XI.0.1)

745G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations:

Steady-State Problems, Springer Monographs in Mathematics,