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

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356 V Steady Stokes Flow in Exterior Domains

where α is either 2 or n − 1. Then, the problem

(∇u, ∇ψ) − (π, ∇ ·ψ) = (G, ∇ψ) − (f

, ψ), for all ψ ∈ C

∞

(u, ∇χ) = −(g, χ), for all χ ∈ C

∞

(V.8.4)

admi ts at least one solution u, π such that

u ∈ D

1,q

), π ∈ L

), for all q > n/α,

(1 + |x|

α−1

)u ∈ L

∞

Moreover, this solution satisﬁes the estimate

k|x|

α−1

∞

+ |u|

1,q,R

+ kπk

q,R

≤ c (k(|x|

+ 1)Gk

∞

+ kf

−1,q,B

+ kgk

q,B

)

(V.8.5)

with c = c(n, q). Finally, if u

, π

is another pair satisfying (V.8.4) with the

same data as u, π a nd with

∈ W

1,r

loc

) ∩ L

), π

∈ L

loc

)

for som e r, s ∈ (1, ∞) and ρ > 0, then u ≡ u

, π ≡ π

+ const. a.e. in R

Proof. We approximate G with functions {G

} ⊂ C

∞

) of the typ e con-

structed in Lemma V.8.1 . In addition, by the elementary properties of mol li-

ﬁers, we see that the functions

= (f

)

1/h

, g

= (g)

1/h

, h ∈ N, h ≥ h

> 4/R

belong to C

∞

3R/4

) and sati sfy, as h → ∞,

−f

−1,q,B

+ kg

−gk

q,B

→ 0, for all q > n/α.

Let us consider the following problem for all h ≥ h

∆u

= ∇π

+ ∇· G

+ f

∇ · u

= g

)

in R

lim

|x|→∞

(x) = 0.

(V.8.6)

Proceeding as in Section IV.2, we look for a solution to (V.8.6) of the form

= w

+ h

, π

= τ

where w

and τ

are the volume potentials (IV.2.8) corresponding to the

body fo rce ∇·G

+ f

and h

is given in (IV.2.1 0) with g

in place of g. We

begin to furnish estimates for h

. From Calder´on–Zygmund Theorem II.11. 4

we immediately deduce

V.8 Further Existence and Uniqueness Results for q-generalized Solutions 357

1,q

≤ c

q,B

, (V.8.7 )

with c

= c

(n, q). Moreover, we have for |x| ≥ 2R,

(x)| ≤ c

R/2

(y)||x − y|

2−n

dy ≤ c

|x|

2−n

q,B

and so

k(|x|

α−1

+ 1)h

∞,B

≤ c

q,B

, (V.8.8)

with c

= c

(n, q, R). We shall next estimate w

. From (IV.2.8)

it follows

(omitting the index h)

(x) = −

(x − y)G

(y)dy +

(x − y)f

(y)dy = G

+ G

(V.8.9)

Clearly, agai n from the Calder´on–Zygmund theorem, we deduce for all q ∈

(1, ∞)

1,q

≤ c

kGk

. (V.8.10)

Moreover, denoting by ψ

a C

∞

-function which is one in B

3R/4

and zero

outside B

, we obtain, for all ϕ ∈ C

∞

)

|(f

, ϕ)| = |(f

, ψ

ϕ)| ≤ kfk

−1,q,B

kψ

ϕk

1,q

. (V.8.11)

Now, if q > n/(n − 1), by the Sobolev inequality (II.3.7) we easily show

kψ

ϕk

1,q

≤ c

|ϕ|

1,q

with c

= c

(r, q, n) and so (V. 8.11) yields

−1,q,R

≤ c

−1,q,B

Therefore, repeating the same argument employed in Section IV.2 (see (IV.2.27)–

(IV.2.28)) we recover for all q > n/(n − 1)

1,q

≤ c

−1,q,B

. (V.8.12)

Collecting (V.8.7), (V.8.9), (V.8.10), and (V.8.12) furnishes

1,q

≤ c

(kG

+ kf

−1,q,B

+ kg

q,B

). (V.8.13)

Moreover, recalling the expression (IV.2.8)

for τ

(= π

) and reasoning as in

(V.8.10) we readily prove

kπ

≤ c

(kG

+ kf

−1,q,B

This latter inequality and (V.8.13) then yield

1,q

+ kπ

≤ c

(kG

+ kf

−1,q,B

+ kg

q,B

), for all q > n/α.

(V.8.14)

358 V Steady Stokes Flow in Exterior Domains

We next show the pointwise estimate for w

. From (V.8.9) and from the

expression of the tensor U (omitting the index h),

(x)| ≤ c

k(|x|

+ 1)Gk

∞

|x −y|

1−n

|y|

−α

dy,

and so, Lemma II.9.2 implies

k(|x|

α−1

+ 1)G

∞,B

≤ c

k(|x|

+ 1)Gk

∞

. (V.8.15)

We have also

(x)|x|

α−1

| =



R/2

(y) · B(x, y)dy



, (V.8.16)

where, f or i = 1, . . . , n,

(x, y) = ψ

(y)U

(x − y)|x|

α−1

. (V.8.17)

Since for y ∈ B

and |x| ≥ 2R it is

|U(x − y)| + |∇U(x − y)| ≤ c| x|

1−α

with c = c(R, n), from (V.8.16) and (V.8.17) we ﬁnd

|x|

α−1

| ≤ kf

−1,q,B



(|B(x, y)|

+ |∇

B(x, y)|)dy



1/q

≤ c

−1,q,B

(V.8.18)

Thus, from (V.8.8), (V.8.9), (V.8.15), (V.8.18), and property (V.8.1)

of G

we recover

k(|x|

α−1

+ 1)u

∞,B

≤ c

(k(|x|

+ 1)Gk

∞

+ kf

−1,q,B

+ kg

q,B

(V.8.19)

We next pass to the limi t h → ∞. From the linearity and from the uniqueness

of problem (V.8.6), by virtue of (V.8.14) and by the properties of the approx-

imating functions G

, f

, and g

we obtain, in particular, that the sequence

, π

} is a Cauchy sequence in D

1,q

)×L

) for all q ∈ (n/α, ∞) and,

by the Sobolev inequality (II.3.7), it is also a Cauchy sequence in L

s = nq/(n − q), for all q ∈ (1, n). We may then assert the existence of two

ﬁelds u, π such that

u ∈ D

1,q

), π ∈ L

), for all q > n/α

with

lim

h→∞

−u|

1,q

= l im

h→∞

kπ

−πk

= 0 for all q > n/α

lim

h→∞

−u|

nq/(n−q)

= 0 for all q ∈ (1, n).

(V.8.20)

V.8 Further Existence and Uniqueness Results for q-generalized Solutions 359

After multi plying (V.8.6)

by ψ ∈ C

∞

) and (V.8.6)

by χ ∈ C

∞

integrating by parts and using (V.8.20) we deduce at once that u, π solves

(V.8.4). In addition, again by (V.8.20) a nd (V.8.14), it follows that u and π

satisfy the estimate

|u|

1,q

+kπk

≤ c

(kGk

+kf

−1,q,B

+kgk

q,B

), for all q > n/α. (V.8.21)

Furthermore, by (V.8.20 )

and Lemma II.2.2, we see that we can select a sub-

sequence {u

}, say, which converges pointwise to u, a.e. in R

. Consequently,

by passing to the lim it h

→ ∞ into (V.8.1 9) we conclude

k(|x|

α−1

+ 1)uk

∞,B

≤ c

(k(|x|

+ 1)Gk

∞

+ kf

−1,q,B

+ kgk

q,B

(V.8.22)

Finally, since by the embedding Theorem II.3.4,

kuk

∞,B

≤ ckuk

1,r,B

, r > n,

estimate (V. 8.5) becomes a consequence of this last inequality, of (V. 8.21)

and of (V. 8.22). Concerning uniqueness, let v = u − u

, p = π − π

. From

(V.8.4), the a ssumptions made on u

, π

and the regularity results of Theorem

IV.4. 2 we deduce that v, p is a C

∞

-smooth solution to the homogeneous Stokes

problem

∆v = ∇p

∇ · v = 0

)

in R

From Lemma V.3.1 we then have

(x) =

β(x)

(d)

(x − y)(u

(y) + u

(y))dy, (V.8.23)

where, we recall, β(x) = B

(x) − B

d/2

(x). Since u ∈ L

), for all q > n/α

and u

∈ L

), for some ρ > 0, using the H¨older inequality into (V.8.23)

and taking into account that kH

(d)

≤ M, independently of x and for all

t ≥ 1, we easily show that v(x) tends to zero pointwise as |x| tends to inﬁnity.

Theorem V.3.5 allows us to conclude v ≡ 0, p ≡ const. and the lemma is

completely proved. ut

The following result furnishes an extension o f the one just proved to the

case of an exterior domain and it represents the main contribution of this

section. For simplicity, we shall state it for homogeneous boundary data, i.e.,

∗

≡ 0, referring the reader to Exercise V.8.1 for the more general case v

∗

6= 0.

Theorem V.8.1 Let Ω ⊂ R

, n ≥ 3, be an exterior domain of class C

Suppose that the second-order tensor ﬁeld F in Ω satisﬁes

(1 + |x|

)F ∈ L

∞

(Ω),

360 V Steady Stokes Flow in Exterior Domains

with α either 2 or n −1. Then, the problem

(∇v, ∇ψ) − (p, ∇ ·ψ) = (F , ∇ψ) for all ψ ∈ C

∞

), (V.8.24)

admi ts one and only one solution v, p such that

v ∈ D

1,q

(Ω), p ∈ L

(Ω), for each q > n/α,

(1 + |x|

α−1

)v ∈ L

∞

(Ω).

Moreover, this solution satisﬁes the following estimate

k(|x|

α−1

+ 1)vk

∞

+ |v|

1,q

+ kpk

≤ ck(|x|

+ 1)F k

∞

, (V.8.25)

for each q > n/α and with c = c(n, q, Ω).

Proof. Since F ∈ L

(Ω) with arbi trary q > n/α ≥ n/(n − 1), from Theo-

rem V.5.1 we know that there exists a uniq ue (q-generalized) solution v, p to

(V.8.24) such that

v ∈ D

1,q

(Ω), p ∈ L

(Ω) for all q ∈ (n/α, n). (V. 8.26)

Moreover, by Sobo lev inequality (II.3.7), we have

v ∈ L

nq/(n−q)

(Ω), for all q ∈ (n/α, n). (V.8.27)

Finally, from the regularity results of Theorem IV.4.2 and Theorem IV.6.1,

we readily ﬁnd

v ∈ W

1,q

(Ω

), p ∈ L

(Ω

), for a ll q > n/α and all R > δ(Ω

). (V.8.28)

Let ϕ be the “cut-oﬀ” f unction of Lemma V.4.2. Problem (V.8.24) then goes

into problem (V.8.4) with u = ϕv, π = ϕp and

= ϕF

= T

(v, p)D

ϕ + D

ϕ + v

ϕ) − F

g = v · ∇ϕ;

see also (V.4.4). Thus, from (V.8.26)–(V.8.28), from Lemma V.8.2 and (V.5.24)

we obtain for all q > n/α

k(|x|

α−1

+ 1)uk

∞,R

+ |u|

1,q ,R

+ kπk

q,R

≤ c (k(|x|

+ 1)F k

∞,Ω

+ kvk

q,Ω

+ kpk

−1,q,Ω

) .

Recalling that ϕ is equal to one in Ω

R/2

, from this inequality it follows that

k(|x|

α−1

+ 1)vk

∞,Ω

R/2

+ |v|

1,q,Ω

R/2

+ kpk

q,Ω

R/2

≤ c (k(|x|

+ 1)F k

∞,Ω

+ kvk

q,Ω

+ kpk

−1,q,Ω

) .

(V.8.29)

V.8 Further Existence and Uniqueness Results for q-generalized Solutions 361

If we add (V.8.29) to (V.5.26 ), by reasoning exactly as we did to obtain

(V.5.28), we ﬁnd

kvk

1,q,Ω

+ k(|x|

α−1

+ 1)vk

∞,Ω

R/2

+ |v|

1,q,Ω

+ kpk

q,Ω

≤ c (k(|x|

+ 1)F k

∞,Ω

+ kvk

q,Ω

+ kpk

−1,q,Ω

) .

(V.8.30)

We now claim the existence of a constant κ = κ(n, q, Ω, R) such that

kvk

q,Ω

+ kpk

−1,q,Ω

≤ κk(|x|

+ 1)F k

∞,Ω

. (V.8.31)

To show the validity of (V.8.31), we use the usual contradiction argum ent.

Actually, the inval idity of (V.8.31) would imply the existence of a sequence

} verifying the assumptions of the theorem for each m ∈ N and of a

corresponding sequence of solutions {v

, p

} such that

q,Ω

+ kp

−1,q,Ω

= 1

k(|x|

+ 1)F

∞,Ω

≤ 1/m.

(V.8.32)

Since, clearly, for all s > n/(α −1)

s,Ω

R/2

≤ ck |x|

α−1

∞,Ω

R/2

from (V.8.32) and (V.8.30) we deduce, in particular,

1,q,Ω

+ kv

s,Ω

R/2

+ |v

1,q,Ω

+ kp

q,Ω

≤ M,

for a constant M independent of m. From the weak compactness of reﬂexive

Lebesgue spaces and the strong compactness results of Exercise II.5.8 and

Theorem II.5.3, it is easy to show the existence of a subsequence, denoted

again by {v

, p

}, and of two ﬁelds v, p such that

v ∈ L

(Ω

) ∩ D

1,q

(Ω) ∩ W

1,q

(Ω

)

p ∈ L

(Ω)

∇v

→ ∇v in L

(Ω)

→ p in L

(Ω)

→ v in L

(Ω

)

→ p in W

−1,q

(Ω

(V.8.33)

It is imm ediately seen that v is a q-generalized solution to the Sto kes sys-

tem (V.5.1) (see Deﬁnition V.5.1) corresponding to v

∗

≡ 0 and by vi rtue

of (V.8.32)

to F ≡ 0. Moreover, by (V.8.33)

, we deduce that, in the ex-

terior of a ball of suﬃciently large radius, v is in L

, for s > n/α and so

v is a q-generalized solution to the Stokes problem (V.0.1), (V.0.2) with

362 V Steady Stokes Flow in Exterior Domains

F ≡ v

∗

≡ v

∞

≡ 0. Thus, recalling that p ∈ L

(Ω), from Theorem V.3.4

we conclude v ≡ p ≡ 0 in Ω. However, by virtue of (V.8.33)

5,6

, this conclu-

sion contradicts (V.8.32)

and, therefore, (V.8.31) is proved. From (V.8.31)

and (V.8 .30) we then obtain, in particular,

k(|x|

α−1

+ 1)vk

∞,Ω

R/2

+ |v|

1,q

+ kpk

≤ ck(|x|

+ 1)F k

∞

, for all q > n/α.

(V.8.34)

Finally, since by the embedding Theorem II.3.4

kvk

∞,Ω

R/2

≤ ckvk

1,r,Ω

R/2

, r > n,

estimate (V.8.5) becomes a consequence of this last inequality and (V.8.34).

The proo f of the theorem is complete. ut

Exercise V.8.1 Let Ω and F be as in T heorem V.8.1. Show that, given

∗

∈ W

1−1/q,q

(∂Ω), g ∈ L

(Ω), q > n/α,

the problem

(∇v, ∇ψ) − (π, ∇ · ψ) = (F , ∇ψ) for all ψ ∈ C

∞

(v, ∇χ) = −(g, χ), for all χ ∈ C

∞

admits one and only one solution such that

v ∈ D

1,q

(Ω), p ∈ L

(Ω), (1 + |x|

)v ∈ L

∞

(Ω).

Moreover, show that for all R > δ(Ω

) this solution sati sﬁes the estimates

k(|x|

α−1

+ 1)vk

∞,Ω

+ |v|

1,q

+ kpk

≤ c(k(|x|

+ 1)F k

∞

+ kgk

+ kv

∗

1−1/q,q(∂Ω)

where we can take Ω

≡ Ω if q > n.

V.9 Notes for the Chapter

Section V.1. The ﬁrst existence and uniqueness theorems for the Stokes

problem in an exterior domain Ω is due to Boggi o (1910 ), for Ω

a closed ball.

In the same hypothesis on Ω, Oseen (1927, §§9.3,9.4) furnishes the expli ci t

form o f the Green’s tensor. For an arbitrary exterior domain, Lamb (1932) has

given a formal series development of a generic solution in terms of spherical

harmonics. The ﬁrst existence and uniqueness result in the general case can

be found in the work of Odqvist (1930, §4).

The variational formulation (V.1.1) has been introduced by Ladyzhenskaya

(1959b, §2). Lemma V.1.1 with q = 2 and Ω of class C

is due to Sol onnikov

Sˇcadilov (1973, §3).

Section V.2. A weaker version of Theorem V.2.1 is proved by Finn (1965a,

Theorem 2.5) and La dy zhenskaya (1969, Chapter 2, §2). Seemingly, Finn has

V.9 Notes for the Chapter 363

been the ﬁrst to recognize that, for existence, the condition of zero ﬂux of v

∗

through the boundary is not necessary (see Finn, loc. cit., Remark on p. 371).

The solenoidal extension of the boundary data , g iven in (V.2.5), in the

case ω = 0 is due to Ladyzhenskaya (1969, p. 41).

Section V.3. Lemma V.3.1 generalizes Lemma 4.2 of Fujita (1961). Theorem

V.3. 2, Theorem V.3.4, and Theorem V.3.5 are an extension of classical results

due to Chang & Finn (1961). A weaker version of the latter can be found in

Finn & Nol l (1957). Theorem V.3.3 is due to me; see also Galdi & Sima der

(1990).

Section V.4. All results and methods are originally due to me. Notwith-

standing, mainly in the literature of the early nineties, one can ﬁnd a number

of contributions by several authors, that cover, in part, some of these results.

However, their approach is diﬀerent than the one I introduced here.

Weaker versions of Lemma V.4.3 wi th m = 0, n = 3 and q = 2 were

originally given by Masuda (1975, Proposition 1 (iii)) and Heywood (1 980,

Lemma 1).

Theorem V.4.6, for m = 0, n = 3 and 1 < q < 3/2, was shown for the

ﬁrst time by Solonnikov (1973, Theorem 2.3). Generalizations of this result to

higher values of q were ﬁrst investigated by Maremonti & Solonnikov (1986);

see also Maremonti & Solonnikov (1 985). The extension of Solonnikov’s result

to arbitrary dimension n ≥ 3 can be deduced from the work of Borchers

& Sohr (198 7). Lemma V.4.3 and Lemm a V.4.4 and Theorem V.4.6 in the

particular case where m = 0 and n = 3 can be deduced from the work of

Maslennikova & Timoshin (1989, 1990). A way of avoiding quotient spaces in

Theorem V.4.6 is to modify suitably the conditions at inﬁnity. This view has

been considered by Maremonti & Solo nnikov (1990).

The validity o f (V.4.15) with m = 0 in a more restricted class of functions

has been disproved by Borchers & Miyakawa (199 2). The results contained

in Theorem V.4.8 have been the object of several researches. In this regard,

we refer the reader to the work of Sohr & Varnhorn (1990), Kozono & Sohr

(1991), Deuring (1990a, 1990b, 199 0c, 1991), and Deuring & von Wahl (1989).

Existence, uniqueness, and estimates for strong solutions in weighted

Sobolev spaces have been studied by Choquet-Bruhat & Christodoulou (1981),

Specovius-Neugebauer (1986), Farwig (1990), Girault & Sequeira (1991) and

Pulidori (1993).

Section V.5. Here we follow the ideas of Galdi & Simader (1990). Theorem

V.5. 1 in the case n ≥ 3, q ∈ (n/(n − 1), n) and Ω of class C

2,λ

, λ > 0, was

ﬁrst obtained by H. Kozono a nd H. Sohr in a preprint of 1989 and published

later in 1991. In particular, in this paper we ﬁnd a ﬁrst systematic study

of the Stokes problem in exterior domain in homogeneous Sobolev spaces.

The estimates contained in Theorem V.5.1 when q ∈ (1, n/(n −1)] were ﬁrst

derived by W. Borchers and T. Miyakawa in 1989 a nd published later in

1990. Generalizations of Theorem V.5.1 along the lines of Exercise V.5.1 are

considered by Kozono & Sohr (1992b) and Farwig, Sima der and Sohr (1993).

364 V Steady Stokes Flow in Exterior Domains

Most of the above results are reobtained, basically by the same metho ds,

in the paper by Maslennikova & Timoshin (1994)

Theorem V.5.3 is due to me.

Weak solutions in weighted Sobolev spaces have been analyzed by Gi rault

& Sequeira (1991), Pulidori (1993), Pulidori & Specovius-Neugebauer (1995)

and Specovius-Neugebauer (199 6).

Weak solutions in Lo rentz spaces have been studied by Kozono & Ya-

mazaki (1998).

Section V.7. Results of this section are essentially due to Galdi & Simader

(1990), or else can be obtained as corollary to their work. However, the Stokes

paradox, as presented here, was ﬁrst formulated in the particular case of a

domain exterior to a circle by Avudainayagam, Jothiram & Ramakrishna

(1986). For further results related to the plane, exterior Stokes problem, in

addition to the classical papers of Finn & Noll (1957) and Chang & Finn

(1961), we refer the reader to the work of Sequeira (1981, 1983, 1986) and of

Hsiao & McCamy (1981). Problem (V.7.1), (V.7.4) is related to the steady

motion of a viscous ﬂuid past a self-propell ed body that is moving at constant

small velocity. For this type of questions, see Pukhnacev (1990 a, 1990b) and

Galdi (1999a, 2002).

Section V.8. For results related to Theorem V.8.1, we refer to the paper o f

Novotn´y & Padula (1995).

Steady Stoke s Flow in Doma ins with

Unbo unded B oundaries

Nel dritto mezzo del campo maligno

vaneggia un pozzo assai largo e profondo

di cui suo loco dicer`o l’ordigno.

DANTE, Inferno XVIII, vv. 4-6

Introduction

So far, with the exception of the half-space, we have considered ﬂows occurring

in domains with a compact boundary. Nevertheless, from the point of view of

the applications it is very important to consider ﬂows in domains Ω having

an unbounded boundary, such as channels or pipes of possibly varying cross

section. In studying these problems, however, due to the particular geometry

of the region of ﬂow, completely new features, which we are goi ng to explain,

appear. To this end, assume Ω to be an unbounded domain of R

with m > 1

“exits” to inﬁnity, of the type (see Section III .4.3)

Ω =

[

i=0

Ω

where Ω

is a smooth compact subset of Ω whi le Ω

, i = 1, . . ., m, are disjoint

domains which, in possibly diﬀerent coo rdinate systems (depending on Ω

)

have the form

Ω

= {x ∈ IR

: x

> 0, x

≡ (x

, . . ., x

n−1

) ∈ Σ

)}.

Here Σ

= Σ

) are smoothly varying, sim ply connected domains in R

n−1

bounded for each x

> 0 with

365G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations:

Steady-State Problems, Springer Monographs in Mathematics,