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

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486 VII Steady Oseen Fl ow in Exterior Domains

|v|

1,q,Ω

+ kpk

q,Ω

≤ c



|f|

−1,q

+ kv

∗

1−1/q,q(∂Ω)

+ kvk

q,Ω

+ kpk

−1,q,Ω



with c = c(R, q, nΩ), which, in turn, implies

inf

(

h,eπ)∈

|v −

1,q ,Ω

+ kp − eπk

q,Ω

≤ c

|f|

−1,q

+ kv

∗

1−1/q,q(∂Ω)

+ inf

(

h,eπ)∈

kv −

q,Ω

+ kp − eπk

−1,q,Ω

With this inequality in hand, one can proceed exactly like in the proof of

Theorem V.5.1 and show the desired result. 

We end this section with a result similar to that shown in Theorem V.5.3

that will f urnish, in pa rticular, summability properties at large distances of

the pressure ﬁeld p associated to a q-weak solution, when q ≥ n + 1. To this

end we observe that if v is a q-weak solution corresponding to f ∈ D

−1,r

(Ω),

where a priori r 6= q, by Lemma VII.1.1 one can a lways associate to v a

pressure ﬁeld p satisfying (VII.1.2) with p ∈ L

loc

(Ω), µ = min(r, q).

Theorem VII.7.3 Let Ω be an exterior domain of R

and let v be a q-

generalized solution to (VII.0.2)–(VII.0.3) in Ω. Then, if f ∈ D

−1,r

(Ω), r >

n/(n − 1), fo r all R > δ(Ω

) it holds that

v ∈ D

1,r

(Ω

), p ∈ L

(Ω

where p is the pressure ﬁeld associated to v by Lemma VII.1.1.

Proof. By a reasoning analogous to that employed in Theorem V.5.3, one

obtains

v ∈ W

1,r

loc

(Ω), p ∈ L

loc

(Ω). (VII.7.4 3)

(We leave details to the reader.) Af terward, we recall that u = ϕv and π = ϕp

(with ϕ “cut-oﬀ” function deﬁned in Theorem V.5.3) obey problem (VII.7.1),

(VII.7.2 ) in R

. Using (VII.7.43) and proceeding again as in the proof of

Theorem V.5.2, one shows that if r > n/(n − 1),

F ∈ D

−1,r

), g ∈ L

). (VII.7.44)

Furthermore, denoting by φ an arbitrary element from C

∞

), we deduce

|(g, φ)| ≤ ck vk

r,Ω

kφk

,Ω

≤ c

kvk

r,Ω

kφk

/(n−r

),Ω

and therefore, by the Sobolev inequality,

g ∈ D

−1,r

). (VII.7 .45)

From (VII. 7.44), (VII.7.45 ), and Theorem VII.4.2 we i nfer the existence of a

solution w, τ to (VII.7.1), (VII.7.2) with

VII.8 Vanishing Reynolds Number. Transition to the Stokes Problem 487

w ∈ D

1,r

), τ ∈ L

The theorem will be therefore proved if we show ∇u ≡ ∇w. To this end, if

we set z = w −u and s = τ −π, it follows that z, s is a solution of class C

∞

to the homogeneous Oseen system in R

and, by Lemma VII.6.3, we have, for

all multi-index α, all x ∈ Ω and all d > 0

(x) = −

(x)

(d)

(x − y)D

(y)dy.

Choosing |α| = ` + 1, and using the asymptotic properties (VII.6.12) of the

function H

(d)

, from this identity, with the help of the H¨older inequality, we

derive

(x)| ≤ c



−(n+1+`)/2

n(1−1/q)

|u|

1,q

+ d

−(n+1+`)/2

n(1−1/r)

|w|

1,r



In this relati on we take

` > max{−1 + n(q − 2)/q, −1 + n(r −2)/2}

and then let d → ∞ to obtain

(x) = 0, for all x ∈ R

, | α| = ` + 1.

As a consequence, ∇z

(x) = ∇(w

− u

)(x) must be a polynom ial P(x) of

degree ` − 1. However, since

|∇w|

+ |∇u|

∈ L

there exists at least a sequence R

such that

lim

→∞

(|∇ w(R

, ω)| + |∇u(R

, ω)|) dω = 0,

implying P(x) ≡ 0, which completes the proof of the theorem. ut

VII.8 Limit of Vanishing Reynolds Number. Transition

to the Stokes Problem

In this last section we shall consider the behavior of solutions to the Oseen

problem in the limit R → 0. Although most of the results we ﬁnd apply

(even in a stronger form ) to three-dimensional ﬂow, here we shall be mainly

interested in plane mo tions. This is because, in such a case, the limiting process

is fairly more interesting, giving rise to a singular perturbation probl em, which

488 VII Steady Oseen Fl ow in Exterior Domains

throws additional light on the Stokes paradox. Concerning three-dimensional

ﬂow, we thus refer the reader to Chapter IX, directly in the nonlinear context.

Though diﬀering in the treatment, the basic ideas presented in this section

are due to Finn & Smith (1967a).

Let us consider the following Oseen problem

∆v + R

∂v

∂x

= ∇p

∇ · v = 0











in Ω

v = v

∗

at ∂Ω

lim

|x|→∞

v(x) = 0,

(VII.8.1 )

where Ω is a smooth exterior domain of R

and v

∗

is a prescribed regular

function on the boundary.

By virtue of Theorem VII.5.1, we know that there

is one and only one solution v, p to (VII.8.1) which, by Theorem VII.1.1, is o f

class C

∞

(Ω). Moreover, this solution satisﬁes the uniform bound

|v|

1,2

≤ c

(1 + B)kv

∗

1/2,2(∂Ω)

(VII.8.2 )

for all R ∈ (0, B] and with c

independent of R. Fix R

> δ(Ω

). From

Theorem IV.4.1 and Theorem IV.5.1 we obtain the foll owing estima tes for v:

kvk

2,2,Ω

+ kpk

1,2,Ω

≤ c



∂v

∂x



2,Ω

+ kvk

1,2,Ω

+ kpk

2,Ω

+ kv

∗

3/2,2(∂Ω)

(VII.8.3 )

where δ(Ω

) < R

< R

. Using (VII.7 .14), with q = 2, (VII.8.2), and (VII.8.3)

we thus recover the inequality

kvk

2,2,Ω

+ kpk

1,2,Ω

≤ c

(B, v

∗

), (VII.8.4)

where p has been possibly modiﬁed by adding a suitable constant depending

on Ω

. Again applying Theorem IV.4.1 and Theorem IV.5.1, for δ(Ω

) <

< R

we deduce

kvk

3,2,Ω

+ kpk

2,2,Ω

≤ c



∂v

∂x



1,2,Ω

+ kvk

1,2,Ω

+ kpk

2,Ω

+ kv

∗

5/2,3(∂Ω)

(VII.8.5 )

For simplicity, we shall restrict ourselves to the case of zero body force. Extension

of the results to nonzero f presents no conceptual diﬃculty and i s therefore left

to the reader as an exercise.

It will b ecome clear from t he context how smooth Ω and v must be.

VII.8 Vanishing Reynolds Number. Transition to the Stokes Problem 489

and so, using (VII.8.4), we obtain

kvk

3,2,Ω

+ kpk

2,2,Ω

≤ c

(B, v

∗

Iterating this procedure we therefore establi sh the following inequalities for

all m ≥ 0:

kvk

m+2,2,Ω

m+2

+ kpk

m+1,2,Ω

m+2

≤ C

(B, v

∗

), (VII.8.6)

with δ(Ω

) < R

m+2

< R

m+1

. We now let R → 0 alo ng a sequence {R

}, say,

and denote by {v

, p

} the corresponding solutio ns. In view of (VII.8.2), from

the weak compactness of the space

1,2

(see Exercise II.6 .2) it follows that,

at l east along a subsequence,

∇v

→ ∇w in L

, (VII.8.7 )

for some w ∈ D

1,2

(Ω). In addition, given a rbitrary ρ > δ(Ω

), from the

embedding Theorem II.3.4 and the compactness results of Exercise II.5.8, we

infer that w ∈ C

(Ω

) and that, for some π ∈ C

(Ω

), along a subsequence

of the previous one, it holds that

→ w in C

(Ω

)

→ π in C

(Ω

(VII.8.8 )

From (VII.8.1) (written for v = v

, p = p

), (VII.8.7), and (VII.8.8) we

conclude that the limit functions w, π obey the following Stokes system:

∆w = ∇π

∇ · w = 0

)

in Ω

w = v

∗

at ∂Ω

|w|

1,2

< ∞.

(VII.8.9 )

Because of Theorem V.2.2, w is uniquely determined since it is the only so-

lution to (VII.8.9)

1,2,3

verifying (VII.8.9)

. Therefore, (VII.8.7) and (VII.8.8)

are veriﬁed not only along a subsequence but as long as R → 0. We shall

next prove that in the lim iting process the continuity of the datum at inﬁnity

(VII.8.1 )

is generally lost. Actually, setting

I(a) ≡

∂Ω

T (a, s) · n

with s the pressure ﬁeld associated to the velocity ﬁeld a, by the results of

Theorem V.3.2 and Exercise V.3.2, we know that the solution w veriﬁes the

following conditions:

490 VII Steady Oseen Fl ow in Exterior Domains

I(w) = 0

(x) = w

∂Ω

∗

(y)T

, q

)(x − y)

−U

(x − y)T

(w, π)(y)]n

(y)dσ

(VII.8.1 0)

for all x ∈ Ω and for som e w

∈ R

that is in general not zero. The next step

is to investigate how relations (VII.8.10) come out from the limit process and,

in particular, the meaning of the vector w

. As we shall see, this vector which

within the Stokes approximation has apparently no clear meaning, is due to

the fact that, as R → 0, problem (VII.8.1) becomes singular in the sense that

the value at inﬁnity is in general not preserved. Actually, by Theorem VII.6.2

we have the representation

(x) =

∂Ω

∗

(y)T

, e

)(x −y)

−E

(x − y)T

(v, p)(y) −Rv

∗i

(x − y)δ

(y)dσ

(VII.8.1 1)

From (VII. 3.36) and (VII.8.11), it follows that

(x) =

4π

(v) log

∂Ω

∗i

(y)T

, q

)(x −y)

−U

(x − y)T

(v, p)(y)]n

(y)dσ

+ o(1) as R|x − y| → 0.

(VII.8.1 2)

Since, by (VII. 8.8), for any ﬁxed x ∈ Ω all terms in this relation tend to ﬁnite

limits as R → 0, this must be the case also for the ﬁrst term on the right-hand

side of (VII.8.12). Thus, in particular,

(v) → 0, as R → 0,

and we recover (VII.8.10 )

. By the same token, from (VII.8.12) we deduce

(VII.8.1 0)

, where

4π

lim

R→0

I(v) log

, (VII.8.13)

which furnishes the desired characterization of the ﬁeld w

. It is interesting

to observe that, according to the results of Section V.7, the vector w

is in

general not zero and that it is zero if and only if the restriction (V.7.2) on

∗

is satisﬁed. Therefore, in such a case and only in such a case the limiting

process preserves the condition a t inﬁnity.

In the ﬁnal part of this section we wish to derive a fundamental estimate

for the integral I(v) that will play an essential role in the existence of solutions

to the nonlinear exterior plane problem. Speciﬁcally, we have

Theorem VII.8.1 Let Ω be a two-dimensional exterior dom ain of class C

Assume fo r some q ∈ (1, 2]

VII.8 Vanishing Reynolds Number. Transition to the Stokes Problem 491

∗

∈ W

2−1/q,q

(∂Ω)

and denote by v, p the correspondi ng solution to (VII.8.1). Then, there exist

B > 0 and c = c(Ω, q, B) > 0 such that



∂Ω

T (v, p) ·n



≤ c|log R|

−1

∗

2−1/q,q(∂Ω)

for a ll R ∈ (0, B].

Proof. Fix R

> R

> δ(Ω

). Using Theorem IV.4.1 and Theorem IV.5.1 into

(VII.8.1 )

1,2

we ﬁnd

kvk

2,q,Ω

+ kpk

2,q,Ω

≤ c

(kvk

1,q ,Ω

+ kpk

q,Ω

+ kv

∗

2−1/q,q(∂Ω)

(VII.8.1 4)

By the trace Theorem II.4.4 it is

∗

1/2,2(∂Ω)

≤ c

∗

2−1/q,q(∂Ω)

and, therefore, (VII.8.2) yields

|v|

1,2

≤ c

(1 + B)kv

∗

2−1/q,q(∂Ω)

. (VII.8.15)

Moreover, from Theorem II.3.4 and inequality (II.5.18) we have

kvk

q,Ω

≤ c

(|v|

1,2

+ kv

∗

2−1/q,q(∂Ω)

)

which, in turn, with the help of (VII.8.2), furnishes

kvk

q,Ω

≤ c

∗

2−1/q,q(∂Ω)

. (VII.8.16)

Also, using the estimate (VII.7.14) for the pressure ﬁeld together with

(VII.8.1 5) and (VII.8.16), it foll ows that

kpk

q,Ω

≤ c

∗

2−1/q,q(∂Ω)

, (VII.8.17)

with c

= c

(Ω, q, B). Coll ecting (VII.8.14)–(VI I.8.17), we then conclude

kvk

2,q,Ω

+ kpk

2,q ,Ω

≤ c

∗

2−1/q,q(∂Ω)

(VII.8.1 8)

with c

= c

(Ω, q, B). With the help of (VII.8.18) we can now show the desired

estimate. Actually, from (VII.3.36) f ollows the existence of B

> 0 such tha t

|E(x − y)| ≤ |U(x − y)| +

4π

|log R| + c

E(x − y)| ≤ |D

U(x − y)| + c

(VII.8.1 9)

for all x ∈ Ω

, all y ∈ ∂Ω, and all R ∈ (0, B

], and with c

and c

depending o n Ω

, ∂Ω, and B

but otherwise independent of R. Since,

clearly,

492 VII Steady Oseen Fl ow in Exterior Domains

|U(x − y)| + |D

U(x − y)| ≤ c

(x − y)| ≤ c

x ∈ Ω

, y ∈ ∂Ω, (VII.8 .20)

from (VII.8.11), (VII.8.19), and (VII.8.20) we derive for all x ∈ Ω

and

all R ∈ (0, B

]

J(R) ≡ |log R|



∂Ω

T (v, p) ·n



≤ |v(x)|+ c

∂Ω

[|T (v, p) · n| + |v

∗

(VII.8.2 1)

with c

independent of R. Employing trace Theorem II.4.1 in the i ntegral on

the ri ght-hand side of (VII.8.21) in conjunction with (VII.8.18) we ﬁnd

J(R) ≤ |v(x)| + c

∗

2−1/q,q(∂Ω)

Integrating both sides of this relation over Ω

and using the H¨older in-

equality, we deduce

J(R) ≤ c



kvk

q,Ω

+ kv

∗

2−1/q,q(∂Ω)



. (VII. 8.22)

The desired estimate is then a consequence of (VII.8 .16) and (VII.8.22). ut

VII.9 Notes for the Chapter

Section VII.1. The ﬁrst complete treatment of exi stence and uniqueness

of the Oseen problem in exterior domains is due to Fax´en (1928/1929), who

generalized the method introduced by Odqvist in his thesis f or the Stokes

problem (Odqvist 1 930).

The variatio nal formulation (VII.1. 1) is taken from Finn (1965a). Theorem

VII.1.2 is due to me.

Section VII.2. Theorem VII.2.1 generalizes an analogous result of Finn

(1965a , Theorem 2.5).

Section VII.4. Theorem VII.4.1 is a detailed and expanded version of an

analogous one given by Galdi (1992). The case where m = 0, n = 3, q ∈ (1, 4),

and g ≡ 0 was ﬁrst proved by Babenko (1973). Other L

-estimates for n = 3

can b e found i n Salvi (1 991, Theorem 4).

Lemma VII.4.2 and Theorem VII.4.2 are due to me.

Section VII.5. Theorem VII.5.1 is due to me.

Section VII.6. Lemma VII.6.3 is essentially due to Fujita (1961), whi le The-

orem VII.6.2 is an extension of a classical result of Chang & Finn (1961).

Section VII.7. Most o f the results of this section are an expanded version

of those given by Galdi (1992). Theorem VII.7.3 was, however, proved by me

in the ﬁrst edition of this b ook.

VII.9 Notes for the Chapter 493

Existence and uniqueness results for three-dimensional ﬂows in weighted

anisotropic Sobolev spaces with weights reﬂecting the decay properties of the

fundamental solution have b een proved by Farwig (19 92a, 1992b) and Shi-

bata (1999). Generalization of results obtained by these authors are given by

Kraˇcmar, Novotn´y & Pokorn´y (2001). These authors also provide very detailed

estimates for the Oseen fundamental solution in dimensions 2 and 3. Estimates

for the Oseen volume potentials in weighted H¨older spaces have been stud-

ied by Solonnikov (1996), and in weighted anisotropic Lebesgue spaces by

Kraˇcmar, Novotn´y & Pokorn´y (2001); see also Kraˇcmar, Novotn´y & Pokorn´y

(1999).

A modiﬁed Oseen problem that contains an “anisotropic” second derivative

and that is releva nt in the study of certain non-Newtonian liquid models has

been investigated by Farwig, Novotn´y, & Pokorn´y (2000).

A boundary integral approach to the existence and uniqueness of solution

to the Oseen problem is addressed by Fischer, Hsiao & Wendland (19 85).

A detailed analysis of diﬀerent problems and results related to two-

dimensional ﬂows can be f ound in the review article of Olm stead & Gautesen

(1976) and in the references included therein.

VIII

Steady Generalized Oseen Flow in Exterior

Domains

. . . e quelle anime liete,

si fero spere sopra ﬁssi poli,

ﬁammando, volte, a guisa di comete.

DANTE, Paradiso XIV, vv. 10-12

Introduction

The Oseen approximation, which we analyzed to a large extent in the previous

chapter, a ims at describing the mo tion of a Navier–Stokes liquid around a

rigid body, B, that moves with a constant and “suﬃciently small” purely

translational velocity. However, assuming this ki nd of motion for B mig ht be

restrictive in several signiﬁcant applied problems, occurring on both large and

small scales, where B is allowed to translate and to rotate. Typical examples of

these problems are furnished by the ori entation of rigid bodies in the stream

of a viscous liquid, and by the self-propulsion of microorganisms in a viscous

liquid; see Galdi (2002) for a review of these and related questions.

In order to appropriately describe situations in which B moves by a generic,

but “small,” rigid motion, one needs the more general approximation that we

introduced in (VII.0.1), which we will call generalized Oseen approximation.

For the reader’s sake, we reproduce the relevant equations here:

ν∆v + v

· ∇v + ω × x · ∇v − ω ×v = ∇p + f

∇· v = 0

)

in Ω ,

v = v

∗

at ∂Ω,

(VIII.0.1)

along with the condition at inﬁnity

495G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations:

Steady-State Problems, Springer Monographs in Mathematics,