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

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346 V Steady Stokes Flow in Exterior Domains

1,q

(Ω). The result stated in part (i) is then an obvious consequence of

Theorem V.5.1. We now pass to the proof of part (ii). Again by the Hahn-

Banach Theorem II.1.7(a), we extend f to some F ∈ D

−1,q

(Ω). Then, from

Lemma V.5.2, we ﬁnd

F = w +

i=1

[f , h

] `

, (V.5.36)

where [w, h

] = 0, i = 1, . . ., n, and where we employ the fact that, si nce

∈ D

1,q

(Ω), q

≥ n (q

> n, if n = 2), we have [f, h

] = [F , h

]. From

Theorem V.5.1 it then follows that there exists a unique v ∈ D

1,q

(Ω) such

that

[w, ϕ] = (∇v, ∇ϕ) , ϕ ∈ D

1,q

(Ω) . (V.5.37)

Furthermore, from Theorem II.1.6 and Theorem II.8.2 we may ﬁnd L

≡

{(L

)

} ∈ L

(Ω), i = 1, . . . , n, such tha t

, ϕ] = (L

, ∇ϕ) ≡ ((L

)

, D

) , ϕ ∈ D

1,q

(Ω) .

We further apply to each L

the Helmholtz decomposition Theorem III.1.2

to obtain L

= R

+ ∇r

, where R

≡ { (R

)

}, i = 1, . . ., n, satisfy

((R

)

, D

φ) = 0, fo r all φ ∈ D

1,q

(Ω), and all l = 1, . . . , n. The last dis-

played equation then becomes

, ϕ] = (∇r

, ∇ϕ) , ϕ ∈ D

1,q

(Ω) . (V.5.38)

The proo f then follows from (V.5. 36)–(V.5.38). ut

From the previous results we obtain the following one.

Corollary V.5.1 Let Ω be as in Theorem V.5.1 and let v ∈ D

1,q

(Ω). Then,

if 1 < q < n (1 < q ≤ n if n = 2)

|v|

1,q

≤ c sup

ϕ∈D

1,q

,ϕ 6= 0

|(∇v, ∇ϕ)|

|ϕ|

1,q

Proof. It f ollows at once from Theorem V.5.1 and Theorem V.5 .2. ut

Exercise V.5.1 Let Ω be as in Theorem V .5.1. Show that given f ∈ D

−1,q

(Ω),

∗

∈ W

1−1/q,q

(∂Ω), g ∈ L

(Ω), q > n/(n − 1) (q ≥ n/(n −1) if n = 2) there exists

v ∈ D

1,q

(Ω) solving (V.5.2), which equals v

∗

on ∂Ω in the trace sense and with

∇ · v = g in the generalized sense. Show, further, that existence of the above type

continues to hold if 1 < q ≤ n/(n − 1) (1 < q < n/(n − 1), if n = 2) provided the

following compatibility condition is satisﬁed:

[f , h] + (g, π) +

∂Ω

(n · ∇h · v

∗

−πv

∗

· n) = 0

V.5 Existence, Uniqueness, and L

-Estimates: q-generalized Solutions 347

for all (h, π) ∈ S

. Prove also that v and the corresponding pressure p (∈ L

(Ω))

verify the estimate

inf

(h,π)∈S

{|v − h|

1,q

+ kp − πk

} ≤ c(|f |

−1,q

+ kgk

+ kv

∗

1−1/q,q(∂Ω)

Finally, show that, if 1 < q < n, v tends to zero as |x| → ∞ in the following sense

n−1

|v(x)| = o(1/|x|

n/q−1

The last part of this section is devoted to the proof of a “regularization

at inﬁnity” for q-generalized solutions. In this respect, we recall that if v ∈

1,q

(Ω) satisﬁes (V.1.1) fo r all ϕ ∈ D(Ω), with f ∈ D

−1,r

(ω), for all bounded

subdomain ω wi th ω ⊂ Ω, and where a priori r 6= q, by Lemma IV.1.1 we

can associate to v a pressure ﬁeld p satisfying (V.1.2) with p ∈ L

loc

(Ω),

µ = min(r, q).

Theorem V.5.3 Let Ω be an exterior domain of R

, let v ∈ D

1,q

(Ω), 1 <

q < ∞, be weakly divergence-free satisfying (V.1.1) for a ll ϕ ∈ D(Ω), and let

ρ > δ(Ω

). Then, the following properties hold

(i) If

f ∈ D

−1,r

(Ω

), r > n/(n −1),

we have

v ∈ D

1,r

(Ω

), p ∈ L

(Ω

) (V.5.3 9)

for all R > ρ .

(ii) If

f ∈ L

(Ω

), 1 < s < ∞,

we have

v ∈ D

2,s

(Ω

), p ∈ D

1,s

(Ω

). (V.5 .40)

for all R > ρ.

In both cases, p is the pressure ﬁeld associated to v by Lemma IV.1.1.

Proof. The ﬁelds

u = ϕv, π = ϕp

solve the weak formulation of problem (V.4.3)–(V.4.4) in R

, namely,

(∇u, ∇ψ) −(π, ∇· ψ) = −[f

, ψ], for al l ψ ∈ C

∞

(u, ∇χ) = −(g, χ), for all χ ∈ C

∞

(V.5.41)

where (i = 1, . . . , n)

348 V Steady Stokes Flow in Exterior Domains

= (f

)

+ T

(v, p)D

ϕ + D

ϕ + v

ϕ),

, ψ] := [f , ϕψ]

g = v · ∇ϕ,

and T is deﬁned in (IV.8.6). By Theorem IV.4.5,

v ∈ W

1,r

loc

(Ω), p ∈ L

loc

(Ω)

and so

g ∈ L

). (V.5.42)

Furthermore, reasoning exactly as in the proof of Theorem V.5.1, one easily

shows that for r > n/(n − 1 )

∈ D

−1,r

). (V.5.43)

In view of Theorem IV.2.2, (V.5.42), and (V.5.43) we establish the existence

of a solution u

, π

to (V.5.41) such that

∈ D

1,r

), π

∈ L

), (V.5.44)

and, by the uniqueness pa rt of the same theorem we deduce

∇(u

− u) ≡ 0, π

− π ≡ const. (V.5.45)

Since ϕ = 1 in Ω

, conditions (V.5.4 4) and (V.5.45), after a possible mod-

iﬁcation of p by adding a constant (which causes no loss), prove (V.5.39).

Assume now f ∈ L

(Ω

). By Theorem IV.4.2 we deduce

v ∈ W

2,s

loc

(Ω), p ∈ W

1,s

loc

(Ω) (V.5.46)

and so u, π solve (V.4.3)–(V.4.4) a.e. in R

. (V. 5.46) implies

∈ L

), g ∈ W

1,s

We may then apply Theorem II.3.1 to (V.4.3)–(V.4.4) to establish the exis-

tence of a solution u

, π

such that

∈ D

2,s

), π

∈ D

1,s

). (V.5.47)

Setting w = u

−u, by Lemma V.3.1 we obta in

(x) =

β(x)



(d)

(x − y)D

(y)dy − D

(d)

(x −y)D

(y)



(V.5.48)

for a ll x ∈ R

and all d > 0. By properties (V.3.5) of H

(d)

, relation (V.5.48 ),

with the help of the H¨older inequality, implies for all suﬃciently large d,

V.6 Green’s Tensor and Some Related Properties 349

(x)| ≤ c log d



−n/r

s,β(x)

+ d

−n/q

k∇uk

q,β(x)



Letting d → ∞ into this inequality and recalling that ϕ = 1 in Ω

proves

(V.5.45)

. Consequently, (V.5.41)

yields

(π

− π, ∇ · ψ) = 0, for all ψ ∈ C

∞

which, by (V.5.46), in turn delivers (V.5.45)

. The theorem is completely

proved. ut

Remark V.5.3 For future reference, we wish to observe that results analo-

gous to those of Theorem V.5.3 are valid for the following Dirichlet problem

for the Poisson equation:

∆v = f in Ω, v = v

∗

at ∂Ω.

In particular, if v ∈ D

1,q

(Ω), for some q ∈ (1, ∞), and f ∈ L

(Ω

), 1 < s < ∞,

then v ∈ D

2,s

(Ω

) for all r > ρ. The proof of this assertion, completely

similar to (and simpler than) that of Theorem V.5. 3, is left to the reader as

an exercise. 

V.6 Green’s Te nsor and Some Related Properties

The results established in the previous two sections allow us to prove the exis-

tence of the Green’s tensor for the Stokes problem in a (suﬃciently smooth)

exterior domain. Actually, for ﬁxed y ∈ Ω, let us consider the functions

(x, y), a

(x, y) such that for all i, j = 1, . . ., n and a ll y ∈ Ω

∆

(x, y) +

∂a

(x, y)

∂x

= 0, x ∈ Ω,

∂A

(x, y)

∂x

= 0, x ∈ ∂Ω

(x, y) = U

(x − y), x ∈ ∂Ω

lim

|x|→∞

(x, y) = 0.

(V.6.1)

From Theorem V.4.8, we know that A

(x, y) and a

(x, y) exist and, from

Theorem V.1.1, that they are smooth i n Ω. Thus, i n analogy with the case of

a bounded domain, we have that the ﬁelds

(x, y) = U

(x − y) − A

(x, y)

(x, y) = q

(x − y) − a

(x, y)

deﬁne the Green’s tensor for the Stokes problem in the exterior domain Ω

(Finn 1965a, §2.6). It is not diﬃcult to show along the same lines of Odqvist

350 V Steady Stokes Flow in Exterior Domains

(1930, p. 358, see Finn 19 65, loc. cit.) that the tensor ﬁeld G satisﬁes the

following symmetry condition

(x, y) = G

(y, x).

In the rest of this book, we shall not make use of the Green’s tensor

solution, and, therefore, here we shall not provide a detailed study o f its

properties. Nevertheless, we would like to point out some features of G, that

do not appear to be widely known; see, e.g., Babenko (1980, Proposition I).

More speciﬁcally, from the result obtained i n Theorem V.5.1 we will show

that, if Ω

⊃ B

, some a > 0, then the tensor G does not satisfy certain

estimates which, on the other hand, are known to hold for the same quan-

tity in a bo unded domain (see (IV.8.4 )) and in a half-space (see (IV.3.3)).

For instance, the Green’s tensor for the Stokes problem i n an exterior three-

dimensional domain does not satisfy the following estimate:

|∇G

(x, y)| ≤ c|x − y|

−2

, (V.6.2)

for all x, y ∈ Ω, x 6= y, and with ∇ operating on either x or y. Actually, let

F be a second-order tensor ﬁeld with F

∈ C

∞

(Ω) and such that

(∇ · F , h) 6= 0 for all h ∈ S

, q > 3 . (V.6.3)

For example, we may take F = ψ∇h, where ψ = ψ(|x|) is a smooth, non

negative function such tha t ψ(|x|) = 0 if either |x| ≤ R or |x| ≥ 2R, R >

δ(Ω

). We then have

(∇ · F , h) = −

Ω

R,2R

ψ∇h : ∇h ,

which is, of course, non-zero. Now, in view of (V.6.1) and the properties of

G, g, it i s immediately recognized tha t the ﬁelds

v(x) =

Ω

G(x, y) · (∇ · F (y)) dy, p(x) = −

Ω

g(x, y) · ∇ · F (y)dy

deﬁne a solution to the Stokes system (V.5.1) with f = ∇ · F . Furthermore,

since

v(x) = −

Ω

∇G(x, y) · F (y)dy,

and F is of bo unded support, the validity of (V.6.2 ) would imply

v(x) = O(|x|

−2

). (V.6.4)

This property, with the aid of Theorem V.3.2, then furnishes that the vector

T deﬁned in (V.3.20) must be zero. Thus, from (V.3.19) and (V.3.21) we

obtain

Actually, if Ω = R

, then G ≡ U , and G obeys the same type of estimates

holding f or a bounded domain and a half-space; see ( IV.2.6).

V.7 Nonzero Boundary Data. Another form of the Stokes Paradox 351

∇v(x) = O(|x|

−3

Such a condition, along with (V.6.4) and the fact that v vanishes at the

boundary, leads, by Theorem II.7. 6, to the conclusio n that v ∈ D

1,q

(Ω), for

all q ∈ (1, ∞). By Theorem V.5.1, however, this is possible if and only if

(∇ · F , h) = 0 for all h ∈ S

, q > 3,

contradicting (V.6 .3). As a consequence, the invalidity of (V.6.2) is proved.

However, one can prove (in three dimensions) that the foll owing estimate,

weaker than (V.6.2), does hold:

Ω

|∇G

(x, y)||y|

−2

dy ≤ c |x|

−1

, x ∈ Ω (V.6.5)

and that

(x, y)| ≤ c |x − y|

−1

, x, y ∈ Ω , x 6= y;

see Finn (1965a) Theorem 3.1 and §2.6 .

V.7 A Characterization of Certain Flows with Nonzero

Boundary Data. Another Form of the Stokes Paradox

We wish to investigate the meaning of condition (V.5.4) in the context of slow

motions of a viscous ﬂow past a body, subject to zero body force and zero

velocity at inﬁnity. This last condition imposes, in fact, no serious restriction,

since the Stokes system is invariant if we change v into v +a, for any constant

vector a . In order to make the presentation clearer, we shall li mit ourselves

to considering smooth regions of motion and smooth velocity ﬁelds a t the

boundary as well, l eaving to the reader the (routine) task of extending the

results to less regular situations.

Consider the problem

∆v = ∇p

∇ · v = 0

)

in Ω

v = v

∗

at ∂Ω,

lim

|x|→∞

v(x) = 0.

(V.7.1)

Let us begin to show that (V.5.4) is equiva lent to the foll owing requirement

on v

∗

With ∇ operating on y. The symmetry prop erty of G allows us to draw the same

conclusion if ∇ operates on x.

Notice that, if a tensor function G satisﬁes (V.6.2), then (V.6.5) follows from

Lemma II.9.2.

352 V Steady Stokes Flow in Exterior Domains

∂Ω

∗

· T (h

, π

) ·n = 0, for all i = 1, 2, . . ., n, (V.7.2)

where {h

, π

} is a basis in S

constructed in the preceding section. In fact,

we write

v = w + V

+ σ,

where

σ(x) = Φ∇E(x)

Φ =

∂Ω

∗

·n,

and V

is a smooth solenoidal extension in Ω of compact support of the ﬁeld

∗

(x) −σ(x), x ∈ ∂Ω.

Thus, (V.7.1) can be equivalently rewritten a s

∆w = ∇p + f

∇· w = 0

)

in Ω

w = 0 at ∂Ω,

lim

|x|→∞

w(x) = 0,

(V.7.3)

where

f = −∆V

Clearly

f ∈ D

−1,q

(Ω), for all q ∈ (1, ∞)

and condition (V.5.4) furnishes

[f , h

] = −

Ω

∆V

· h

= −

Ω

∆(V

+ σ) · h

∂Ω

∗

·T (h

, π

) · n = 0

which proves (V.7.2). Suppo se now Ω ⊂ R

It is easy to show that a sol ution

to (V.7.1) veriﬁes (V.7.2) if and only if the following condition holds:

∂Ω

T ( v, p) · n = 0. (V.7.4)

From a physical point of view, this means that, within the approximation

we are considering, the net external force applied to the body is zero. This

happ ens, for example, if the body is self-propelled; Pukhnacev (1990a, 1990b),

Galdi (1999a). In fact, if (V.7.2 ) is sati sﬁed, then by Theorem V.5.1 there is a

solution w to (V.7.3) in the class D

1,q

(Ω), 1 < q ≤ 3/2. This in turn implies

We may take Ω ⊂ R

, n ≥ 3.

V.7 Nonzero Boundary Data. Another form of the Stokes Paradox 353

that a solution v to (V.7.1) exists in the class D

1,q

(Ω), 1 < q ≤ 3/2. In view

of Theorem V.3.2, however, such a circumstance is possible only if (V.7.4) is

satisﬁed (see Exercise V.3.3). Conversely, assume we have a solution to (V.7.1)

satisfying (V.7.4). Then, again by Theorem V.3.2, we have v ∈ D

1,q

(Ω),

1 < q ≤ 3/2, and so w ∈ D

1,q

(Ω) and, by Theorem V.5.1, (V.7.2) is sati sﬁed.

It is interesting to observe that i f R

− Ω = B

, from (V.0.4) we have (see

Remark V.5.2)

∂(h

)

∂x

= −

so that condition (V.7.2) becomes

∂Ω

∗

= 0. (V.7.5)

Let us next consider the case Ω ⊂ R

. We then show that a solution to

(V.7.1) exists if and only i f condition (V.7. 2) is satisﬁed. Since, as we shall see,

the vector ﬁeld v

∗

= const. does not verify (V.7.2) this latter statement can

be interpreted as another form of the Stokes paradox. Assume (V.7.2) holds.

Then, by T heorem V.5.1, there is a solution w to (V.7.3) and, consequently, a

solution v to (V.7.1). Conversely, assume that there is a solution v to (V.7.1);

then w = v − V

− σ is a solution to (V.7.3) which, by Theorem V.3.2,

belongs to the class D

1,q

(Ω), 1 < q < 2. As a consequence, by Theorem V.5.1,

condition (V.7.2) must be satisﬁed. We now show that v

∗

= v

does not verify

(V.7.2) for any nonzero choice of the constant vector v

. This is because from

(V.4.27) and Theorem V.3.2 it follows that

∂Ω

T (h

, π

) · n = −e

, i = 1, 2,

and therefore (V.7.2) would imply

· e

= 0, i = 1, 2;

that is,

= 0.

If Ω is the exterior of a unit circle, from (0.5) we deduce a gain that (V.7.2) is

equivalent to (V.7.5).

Exercise V.7.1 Prove that for Ω ⊂ R

, the ﬁeld v

∗

= const. does not verify

(V.7.2). Give a physical interpretation of this fact. Moreover, for Ω the exterior of

the closed unit ball centered at the origin, show that (V.7.2) is satisﬁed by

∗

= ω × x (V.7.6)

and that this ﬁnding is in agreement with (V.0.5). Finally, suppose that Ω is the

exterior of t he closed unit circle centered at t he origin and lying in the plane x

= 0.

354 V Steady Stokes Flow in Exterior Domains

Show that condition (V.7. 2) is again equivalent to (V.7.5), and that it is satisﬁed

by the ﬁeld (V.7.6), with ω directed along the x

-axis, and by the ﬁeld v

∗

= κx

κ 6= 0 (simple shear ﬂow).

V.8 Further Existence and Uni queness Result s for

q-generalized Solutions

One undesired feature concerning the q-generalized solutions constructed in

Section V.5 is the fact that their existence and uniqueness are recovered only

if we restrict suitably the range of values o f q, i.e., q ∈ (n/(n − 1), n). Now,

while the restriction from below (q > n/(n −1 )) is necessary unless f satisﬁes

the com patibility condition (V.5.4), the restriction from above (q < n) is due

to the circumstance that the estimates we a re able to derive for sol utions

under the sole assumption f ∈ D

−1,q

(Ω) are no t suﬃcient to guarantee their

uniqueness. However, we may wonder if, taking f from a suitable subclass

of the space D

−1,q

(Ω), we could remove the restriction q < n. Following the

work of Galdi & Simader (1994), in the present section we shall show that,

for n ≥ 3, this is indeed the case provided f is of the type ∇ · F , with F a

second-order tensor ﬁeld (i.e., f

= D

) such that either k(|x|

n−1

+ 1)F k

∞

or k(|x|

+ 1)F k

∞

is ﬁnite. Since the estimates we shall derive guarantee that

the solution tends to zero at large distances, we are not expecting tha t a

similar result holds also in the case of plane motions for, as we have learned

from the preceding section, a two-dimensional solution to the Stokes problem

tends to zero if and only i f the data satisfy compatibility condition (V.5.4).

We begin to show a simple approximation lemma.

Lemma V.8.1 Suppose

(1 + |x|

)F ∈ L

∞

), α > 0, n ≥ 2.

Then, there exists a sequence {F

} ⊂ C

∞

) such that

lim

h→∞

− Fk

= 0 for all s > n/α,

k(|x|

+ 1)F

∞

≤ 2(2

α−1

+ 1)k(|x|

+ 1)Fk

∞

(V.8.1)

Proof. Let ψ

, h ∈ N, be smooth functions in R

such that

|ψ

(x)| ≤ 1

(x) =

(

1 if |x| ≤ h

0 if |x| ≥ 2h.

(V.8.2)

Set

V.8 Further Existence and Uniqueness Results for q-generalized Solutions 355

(x) = ψ

(x)(F(x))

, ε = 1/h,

where, as usual, (·)

denotes molli ﬁcati on. Clearly, {F

} ⊂ C

∞

). Observ-

ing that F ∈ L

) for each s > n/α, we ﬁnd in the limit k → ∞

−Fk

≤ k(F)

1/h

− Fk

+ k(1 − ψ

)(F)

1/h

≤ 2k(F)

1/h

−Fk

+ k(1 − ψ

)Fk

→ 0

as a consequence of (V.8.2), of property (II.2.9)

of molliﬁers and of the

domi nated convergence theorem of Lebesgue given in Lemma II.2.1. Rela-

tion (V.8.1)

is then acquired. From the deﬁnition of molliﬁer, we obtain for

all ε ∈ (0, 1]

(|x|

+ 1)|(F(x))

| ≤ ε

−n



x − y



(|y|

+ 1)|F(y)|dy

+ ε

−n



x − y



||x|

− |y|

||F(y)|dy

≡ I

+ I

Now

≤ k( |x|

+ 1)Fk

∞

−n



x − y



dy = k(|x|

+ 1)Fk

∞

. (V.8.3)

Furthermore, for |x −y| ≤ ε ≤ 1

|x| ≤ |x − y| + |y| ≤ 1 + |y|

and so, in view of inequality (II.3.3) (with n ≡ 2 and q ≡ α) we derive

||x|

− |y|

| ≤ (2

+ 1)(1 + |y|

Therefore, recalling that j



x−y



= 0 for |x − y| ≥ ε, it follows tha t

≤ (2

+ 1)k(|x|

+ 1)Fk

∞

−n



x − y



dy = (2

+ 1)k(|x|

+ 1)Fk

∞

and the lemma is completely proved. ut

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

Lemma V.8.2 Assume G, f

, and g are a given second-order tensor, vector,

and scalar ﬁeld, respectively, in R

, n ≥ 3, satisfying

(1 + |x|

)G ∈ L

∞

, g ∈ L

), for each q > n/α,

supp (f

), supp (g) ⊂ B

R/2

, for some R > 0,