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

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886 XII Two-Dimensional Flow in Exterior Domains

Stokes problem. However, similarly to what we proved for the linear case in

Section VII.8, this limiti ng process need not preserve the condition at inﬁnity

on the velocity ﬁeld and, i n fact, this condition i s preserved if and only if the

data are prescribed in a certain way.

For the sake of simplicity, we assume throughout that the body force f is

identicall y vanishing. Furthermore, as in Section XII.5, we shall denote by λ

the Reynolds number R.

The ﬁrst step is to show a uniform bound (independent of λ) for solutions

determined in Theorem XII.5.1. Thi s will be established in the following.

Lemma XII.9.1 Let the assumptions of Theorem XII.5.1 be satisﬁed and

let f ≡ 0 . Then, there is a c = c(Ω, q, v

∗

, λ

) > 0 such that

Ω

∇v : ∇v ≤ c.

Proof. We suppose, as usual, the origin of coordinates in Ω

. Set

σ = Φ∇log |x|

with m such that

∂Ω

∗

− σ) · n = 0.

In view of this latter property, by the results of Exercise III.3.5, there is a

ﬁeld V satisfying

(i) ∇ · V = 0 in Ω,

(ii) V = v

∗

−σ at ∂Ω,

(iii) V ∈ W

1,2

(Ω),

(iv) V ≡ 0 in Ω

, for some R > δ(Ω

Setting

v = w − e

+ σ + V , (XII.9.1)

from the property of v, σ, and V , and with the help of the embedding The-

orem II.3 .4, we deduce that the ﬁeld w veriﬁes the following properties

hwi

λ,q

< ∞

w = 0 at ∂Ω

lim

|x|→∞

w(x) = 0.

(XII.9.2 )

By Corollary XII.7.1, v satisﬁes the energy equality (X.2. 7) and so, by virtue

of (XII.9.1), (XII.9.2), and by an integration by parts that uses the relation

∂σ

∂x

∂σ

∂x

i, j = 1, 2,

XII.9 Limit of Vanishing Reynolds Number: Transition to the Stokes Problem 887

we ﬁnd

|w|

1,2

Ω

{∇w : ∇V −λ[w·∇w·(σ+V )+V ·∇w·V +V ·

∂w

∂x

]}. (XII.9.3)

By the properties of V and Theorem II.3.4, we have

kV k

< ∞ for all r ∈ [1, ∞).

Therefore, by using the Schwarz inequality along with inequality (II.2.5 ) in

(XII.9.3 ), we show

|w|

1,2

≤ λ



Ω

w · ∇w · (σ + V )



+ c

(XII.9.4 )

where c

depends on V , Ω, and λ

but not on λ. We now observe that, by

the H¨older inequality, it foll ows that



Ω

w · ∇w · (σ + V )



≤ kwk

3q/(3−2q)

|w|

1,3q/(3−q)

kσ + V k

q/(2q−2)

and so, since q/(2q − 2) > 2, from (XII. 9.4) and (XII.5.4) we recover

|w|

1,2

≤ c

hwi

λ,q

+ c

(XII.9.5 )

with c

independent of λ ∈ (0, λ

]. From the estimate (XII.5.47) and (XII.9.1)

we then obtain, at once,

hwi

λ,q

≤ c

with c

independent of λ ∈ (0, λ

]. As a consequence, the lemma follows f rom

this latter inequality, (XII.9 .5), and (XII.9.1). ut

The next lemm a shows other uniform bounds for solutio ns on every com-

pact set in Ω.

Lemma XII.9.2 Let the assumptions of Lemma XII.9.1 be satisﬁed. Then

there exists a constant c = c(Ω, R, q, λ

, v

∗

) such that for all R > δ(Ω)

kvk

2,q,Ω

+ kpk

1,q,Ω

≤ c

Proof. Using Theorem IV.4.1 and Theorem I V.5.1 in equation (XII.0.1), we

ﬁnd

kvk

2,q,Ω

+ kpk

1,q ,Ω

≤ c

(λkv · ∇vk

q,Ω

+ kvk

1,q ,Ω

+ kpk

q,Ω

+ kv

∗

2−1/q,q(∂Ω)

)

(XII.9.6 )

for all R

> R > δ(Ω

) and with c = c(Ω, q, R, R

). From Theorem II.3.4 and

(II.5.18) we ﬁnd for any s ∈ [1, ∞) and any R > δ(Ω

)

888 XII Two-Dimensional Flow in Exterior Domains

kvk

s,Ω

≤ c

(kvk

2,Ω

+ |v|

1,2,Ω

)

≤ c

(|v|

1,2

+ kv

∗

2−1/q,q(∂Ω)

)

and so, by Lemma XII.9.1,

kvk

s,Ω

≤ c

, s ≥ 1, R > δ(Ω

) (XII .9.7)

with c

= c

(Ω, R, s). Since

kv · ∇vk

q,Ω

≤ kvk

2q/(2−q),Ω

|vk

1,2,Ω

coupling (XII.9.6) and (XII.9.7 ) with the help of Lemma XII.9 .1,

kvk

2,q,Ω

+ kpk

1,q ,Ω

≤ c

(kpk

q,Ω

+ 1) (XII.9.8)

with c

independent of λ ∈ (0, λ

]. To estimate the pressure term on the

right-hand side of (XII.9.8 ), we use Lemma IV.1.1 to obtain (after the possible

modiﬁcation of p by adding a constant)

kpk

q,Ω

≤ c

(λkvk

2q,Ω

+ |v|

1,2

which, by (XII.9.7) and Lemma XII.9.1, in turn implies

kpk

q,Ω

≤ c

with c

independent of λ ∈ (0, λ

]. The lemma then follows from this latter

inequality and (XII.9.8). ut

The next result shows that solutions of Theorem XII.5.1 tend, in the limit

λ → 0, to solutions of a suitable Stokes problem.

Lemma XII.9.3 Let the assumptions of Lemma XII.9.1 be satisﬁed. Denote

by w , π the unique solution to the following Stokes problem

∆w = ∇π

∇ · w = 0

)

in Ω

w = v

∗

at ∂Ω

|w|

1,2

< ∞.

(XII.9.9 )

Then as λ → 0, the solutions v , p constructed i n Theorem XII.5.1 satisfy

∇v

→ ∇w in L

(Ω)

→ w in W

2,q

(Ω

)

→ π in W

1,q

(Ω

)

(XII.9.1 0)

for a ny R > δ(Ω

XII.9 Limit of Vanishing Reynolds Number: Transition to the Stokes Problem 889

Proof. Let {λ

}

n∈N

be any sequence converging to zero. From Lemm a XII.9.1

and Lemm a XII.9.2 it readily follows that, along a subsequence at least, con-

ditions (XII.9.10) hold. However, the ﬁeld w satisfying (XII.9.9) is uniquely

determined, whatever the sequence may be (cf. Theorem V.2.1) and, therefore,

the result is proved. ut

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

Theorem XII.9.1 Let the assumptions of Theorem XII.5.1 hol d and let

(v, p) be the solution constructed in that theorem corresponding to f ≡ 0.

Then, denoting by w, π the solution to the Stokes problem (XII.9 .9) we have

that, as λ → 0, v , p tend to w , π in the sense speciﬁed by Lemma XII.9.3.

Moreover, there is a w

∈ R

such that

lim

|x|→∞

w(x) = w

(XII.9.1 1)

and we have

+ e

4π

lim

λ→0

T (v)|log λ| (XII.9.12)

where

T (v) =

∂Ω

T ( v, p) · n.

Finally, the limit process preserves the prescription at inﬁnity, i.e., w

= −e

if and only if the data satisfy the conditions

∂Ω

∗

+ e

) · T (h

, π

) · n = 0 i=1,2 (XII.9.13)

where {h

, π

} are the “exceptional” solutions to the homogeneous Stokes

system constructed in Lemma V.5.1. In the particular case where Ω is exterior

to a unit circle, (XII.9.13 ) reduces to

∂Ω

∗i

+ δ

) = 0 i=1,2.

Proof. The ﬁrst part of the statement follows from Lemma XII.9.3. Moreover,

from Theorem V.3.2 we know that w sati sﬁes the representation

= w

∂Ω

∗i

, q

)(x −y) −U

(x − y)T

(w, π)(y)]n

(y)dσ

(XII.9.1 4)

for some w

∈ R

, which, by the regularity of v

∗

, proves (XII.9.11). On the

other hand, by Theorem XII.7.4, we have

(x) = λ

Ω

(x − y)u

(y)D

(y)dy +

∂Ω

(y)T

, e

)(x −y)

−E

(x − y)T

(u, p)(y) − λu

(y)E

(x − y)δ

dσ

(XII.9.1 5)

890 XII Two-Dimensional Flow in Exterior Domains

where u = v + e

. We wish to take λ → 0 into (XII.9.14). From the H¨older

inequality, (VII.3.41), (VII.3.43), (XII.5.4), and (XII.5.15) it follows for q ∈

(1, 6/5) that

Ω

(x −y)u

(y)dy| ≤ kuk

3q/(3−2q)

|u|

1,3q/(3−q)

kEk

q/(2q−2)

≤ λ

−1

−4(1−1/q)

hui

λ,q

where c

= c

(q, Ω). However, the solutions of Theorem XII.5.1 verify the

estimate (XII.5.47) and so, f rom the preceding inequality, we obtain



Ω

(x − y)u

(y)D

(y)dy



≤ c

|log λ|

−2

, (XII. 9.16)

where c

is independent of λ ∈ (0, λ

]. Now, integrating by parts over the

region Ω

, we ﬁnd

∂Ω

, e

(x − y)) −λE

(x − y)δ

] n

(y)dσ

= 0. (XII.9.17)

Moreover, by Lemma XII.9.2 and the compact embedding theorems o f trace

at the bounda ry (cf. Theorem II.4.1 and Theorem II.5.2; cf. also the Notes for

Chapter II), it follows that

kv − wk

1,q ,∂Ω

+ kp − πk

q,∂Ω

→ 0 as λ → 0. (XII.9.18)

With the aid of (XII.9.17), (XII.9.18), and the asymptotic formula (VII.3 .36)

we thus obtain

lim

λ→0

∂Ω

(y)T

, e

)(x − y)− λu

(y)E

(x − y)δ

dσ

∂Ω

∗i

(y)T

, q

)(x −y)dσ

(XII.9.1 9)

Finally, again from (VII.3. 36) and (XII.9. 18) we ﬁnd

−li m

λ→0

∂Ω

(x − y) T

(u, p)(y)n

(y)dσ

4π

lim

λ→0

[

∂Ω

(u, p)n

]|log λ|

−

∂Ω

(x − y)T

(w, π)(y)n

(y)dσ

(XII.9.2 0)

Collecting (XII.9.14)–(XII.9.16), (XII.9.19), and (XII.9.20) we obtain (XII.9.12).

The last part of the theorem is proved as in Section VII.8. Precisely, we have

= −e

if and only if u

≡ w + e

is a solution to the problem

XII.10 Notes for the Chapter 891

∆u

= ∇π

∇ · u

= 0

)

in Ω

= v

∗

+ e

at ∂Ω

lim

|x|→∞

(x) = 0.

However, as we know from the results of Section V.7, such a soluti on exists if

and only if (XII. 9.13) is sati sﬁed. The theorem is therefore proved.

Remark XI I.9.1 An interesting consequence of Theorem XII. 9.1 is the

derivation of an asymptotic formula (in the limit of vanishing Reynolds num-

ber) for the force, F(v) := −T (v) exerted by the liquid on a body moving in

it with constant velocity e

. Speciﬁcally, taking v

∗

≡ 0, from Theorem XII.9.1

we have that the limit solution w is identically zero so that from (XII. 9.12)

it easily foll ows in the limit λ → 0

F(v) = ( − 4 πe

+ o(1))|log λ|

−1

(XII.9.2 1)

where o(1) denotes a vector quantity tending to zero with λ. Thi s f ormula tells

us that, in the li mit of vanishingly small R eynolds number, the total force

exerted from the liquid on the body is determined entirely by the velocity

at inﬁnity e

and that it is directed along the line of this vector, namely, it

produces only “drag” and no “lift”. Surprisingl y enough, i t does not depend

on the shape of the body.

Equation (XII.9.21) was obtained for the ﬁrst time by Finn & Smi th

(1967b, Theorem 5.4 ).

However, due to the lack o f suitable uniq ueness theory,

the solutions v, p used by these authors diﬀer a priori from those in Theorem

XII.5.1, for which (XII.9.21) was derived. 

Remark XI I.9.2 Condition (XII.9.13) is satisﬁed if the body moves by self-

propulsion; see G aldi (1999a, 2004 §1.2.2). 

XII.10 Notes for the Chapter

Section XII.1. The ﬁrst systematic treatment of the two-dimensional ex-

terior problem for the Navier–Stokes equations traces back to the work of

Goldstein (1933a, 1933b) a nd Leray (1933). In particular, the latter author

proved existence of a solution to (XII.0. 1) with ﬁnite Dirichlet integral. Leray’s

solution is obtained as a suitable limit of a sequence of solutions {v

, p

} of

Due to a diﬀerent deﬁnition of T (v), the left-hand side of formula (XII.7.4) of

Finn and Smith (1967b) diﬀers from the left-hand side of (XII.9.21) by a factor

−1

892 XII Two-Dimensional Flow in Exterior Domains

suitable problems P

deﬁned on a family {Ω

} of bounded doma ins with

→ ∞ as k → ∞. For example, in the physically relevant case when

∗

≡ f ≡ 0 a nd v

∞

6= 0 (steady plane motion of a v iscous liquid past a

translating long cylinder), the generic P

is given by

∆v

= v

· ∇v

+ ∇p

∇ · v

= 0

)

in Ω

= 0 at ∂Ω

= −v

∞

at ∂B

(∗)

where, for simplicity, we have put R = 1. L eray proved the existence of a

uniform bound for the Dirichlet integral of v

, that is,

Ω

∇v

: ∇v

≤ M (∗∗)

with M independent of k. Passing to the limit k → ∞, he showed that (at

least al ong a subsequence) v

, p

converges to v

, p

with v

, p

solving

(XII.0.1 ). The limit solution v

still has a ﬁnite Dirichlet i ntegral. However,

Leray was not able to show that v

also veriﬁes (XII.0.2); cf. loc. cit. pp. 54-

55. The study of the asymptotic behavior of Leray’s solution was initiated by

Gilbarg & Weinberger (1974). These authors obtained, in particular, that v

converges (in the mean) to some vector, v

, which, as we have noticed in the

general context o f Section XII.3, need not coincide with −v

∞

However, a more fundamental question concerning Leray’s solution is the

following one (cf. Fi nn 1970 , p. 88): Is v

nontrivial? Actually, we are not

assured, a priori that v

is no nidentically zero. As a matter of fact, Leray’s

construction in the linear case would lead to an identically vanishing solution

(s)

as a consequence of the Stokes paradox.To see this, let us disregard in

(∗) the nonlinear term v

· ∇v

for each k ∈ N. Applying Leray’s procedure,

we would obtain that v

(s)

solves the Stokes problem w ith zero boundary data

and that, in view of (∗∗), v

(s)

has a ﬁnite Dirichlet integral. Therefore, by

Theorem V.2.2 we i nfer v

(s)

≡ 0.

It is worth emphasizing that the above question arises also if, instead of

a L eray’s solution, we consider a generalized solution to (XII.0. 1), (XII.0. 2)

constructed via Galerkin’s method. Actually, in such a case, we look for v =

u + V , where u ∈ D

1,2

(Ω) and V is a solenoidal extension of −v

∞

, such that

V (x) = −v

∞

, for all suﬃciently large |x|, and V = 0 at ∂Ω; see Remark

X.4. 2. Therefore, as we know from Theorem II.7.6(ii) and Theorem III.5.1,

V ∈ D

1,2

(Ω), for Ω locally Lipschitz (for exam ple). As a consequence, we can

not exclude u = −V , namely, we can not exclude v ≡ 0.

In the general nonlinear case, the answer to the question is still unknown.

However, for symmetric ﬂow (cf. Remark XII.3.2) Amick (1988, §4.2) has

shown that v

is nontrivial; see also Galdi (1999b, Theorem 3.1). On the

XII.10 Notes for the Chapter 893

other hand, it is very likely that if v

= 0, then v

≡ 0, but no proo f is

available yet; see Section XII.6. The validity of this latter condition has the

following important consequence. In fact, if it i s true, we would obtai n, in

particular, that, for symmetric ﬂow, v

must converge to a non-zero limit at

inﬁnity. Since, as observed in Remark XII.3.2, this convergence is uniformly

pointwise, from Theorem XII.8.1 we could then conclude that every symmet-

ric solution v

, p

has a t large distances the same asymptotic structure of the

Oseen fundamental solution. Moreover, one could prove v

= αv

∞

, for some

α ∈ (0, 1]; see Ga ldi (1998, Section 3).

Section XII.2. The counterexample given here is the same that appears in

Ladyzhenskaya’s book (1969, pp. xi-xii).

Section XII.3. Theorem XII.3.1 a nd Lemma XII.3.3 were obtained for the

ﬁrst time by Gilbarg & Weinberger (1978). Their proof of the theorem is

diﬀerent from ours since it relies on the maximum pri nciple for the vorticity

ﬁeld, which only holds in dimension two. Likewise, the proof of the lemm a

given by these authors is based on the Cauchy integral formula of complex

functions (cf. (XII.4 .29)) which, of course, is applicable only to plane ﬂow. On

the other ha nd, our proo f relies on a general theorem concerning pointwise

behavior of functions in spaces D

1,q

. Theorem XII.3.2 is due to me.

Section XII.4. The question of the poi ntwise rate of decay of higher order

derivatives for ∇v and p of the type considered in Exercise XII.4.1 is treated

in Russo (2010a). However this author’s estimates turn o ut to be more con-

servative tha n those given in (XII.4.6) and (XII.4.7).

Section XII.5. Existence with v

∞

6= 0 was ﬁrst shown by Finn & Smith

(1967b), and it relies on their work for the analogous linear problem, cf. Finn

& Smith (196 7a). However, this result is obtained under somewhat more re-

strictive assumptions on the body force and the smoothness of Ω and v

∗

than

those required in Theorem XII.5.1, which is taken from Galdi (1993). More-

over, the method of Finn and Smith is completely diﬀerent than that of Gal di.

Another approach to existence with v

∞

6= 0 is provided in Galdi (2004 , §2.1).

As we already mentioned, in the case v

∞

= 0, to date, no general existence

theory has been developed, and few results are available only under suitable

symmetry assumptions on the data. In particular, we refer the reader to the

work by Galdi (2004, §3.3) and to the more recent one by Yama zaki (2009).

More precisely, the former author assumes the domain

◦

Ω

to have two

orthogonal axes of symmetry, that we may take coinciding with the x

and x

directions. Moreover, denote by S the cla ss of vector functions w = (w

, w

)

such that

, x

) = −w

(−x

, x

) = w

, −x

) ,

, x

) = w

(−x

, x

) = −w

, −x

) .

Then, in Galdi (2004, Theorem 3.2) it is shown that for every b oundary data

∗

∈ W

1/2,2

(∂Ω) ∩ S , wi th ﬂux through ∂Ω suﬃciently small,

there exists

By a mere oversight, this latter assumption is not mentioned in Galdi, loc. cit.

894 XII Two-Dimensional Flow in Exterior Domains

at least one generalized solution to (XII.0.1)–(XII.0.2) with f = 0.

This

solution, which is, in fact, of class C

∞

(Ω) along with the associated pressure

ﬁeld p, is a lso in the class S . The simple (but crucial) observation for the

proof of this result consists in the fact that a vector ﬁeld w ∈ D

1,2

(Ω) ∩ S

obeys the following inequality

Ω

|w|

|x|

≤ c

Ω

|∇w|

and this, in turn, implies

lim

r→∞

2π

|w(r, θ)|

dθ = 0 ;

see Galdi, loc. cit., for details. As a matter of fact, this latter condition in

conjunction with T heorem XII.3.4 implies the stronger property f or the gen-

eralized solution:

lim

|x|→∞

v(x) = 0 , uniformly .

However, we are not expecting, in general, to give a speciﬁc order of decay for

such solutions in terms of negative powers of |x|. The reason is because, when

Ω is the exterior of a circle, they belong to the same class where solutions

(XII.2.7 ) belong, for which, as we know, no order of decay of the above type

can b e given.

In the paper of Yamazaki (2009) the case Ω = R

is considered, and it

is assumed that f =



∂F

∂x

, −

∂F

∂x



, where F belongs to the “antisymmetry

class” A deﬁned by the following conditions

w(x

, x

) = −w

(−x

, x

) , w

(

, −x

) = −w(x

, x

) ,

w(x

, x

) = −w(x

, x

) , w(−x

, −x

) = −w(x

, x

) .

Under the further assumptio n that F = O(| x|

−2

) as |x| → ∞, and its magni-

tude is suitably restricted, in Theorem 2.1 of Yamazaki, l oc. cit., it is shown

the existence of a corresponding generalized solution which satisﬁes further

summability properties. Moreover, the vorticity decays like |x|

−2

I conjecture that the two-dimensional, plane exterior problem correspond-

ing to v

∞

= 0 is, generically, not solvable. In the spirit of the approach

followed by Galdi (2009) where an analogous result is proved in the three-

dimensional case, a way of showing this conjecture could be by assessing that

the relevant nonlinear Navier–Stokes operator, properly deﬁned, is Fredholm

of negative index.

The case f 6= 0 can be easily handled, provided f ∈ S and decays suitably fast

at large distances.

XII.10 Notes for the Chapter 895

Section XII.7. The result of Galdi & Sohr (1995) on the summability prop-

erties of generalized solutions was later rediscovered by Sazonov (19 99), under

more stringent assumptions.

The “cut-oﬀ” technique employed in Theorem XII.7.2 has been extended

to dimension greater than two by Farwig and Sohr (1995, 1998).

Section XII.8. Though inspired by the work of Smith (1965), the method

presented here is due to me.

Employing the results of Smith (1965), Amick (1991) proved an analog of

Theorem XII.8.1 –Theorem XII.8.3 f or the case of symmetric ﬂows.

Section XII.9. The main result of this section i s from the work of Galdi

(1993).