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

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306 V Steady Stokes Flow in Exterior Domains

Since p ∈ L

(Ω), by Theorem II.4.2, there exists at least one ψ satisfying

(V.1.4). Replacing such a ψ into (V.1.2) a nd using the H¨older inequality then

shows (V.1.3). ut

The next step is to investigate the regularity of q-generalized solutions.

Since regularity is a local property, such a study is most easily performed

by means of the results shown in Section IV.4 and Section IV.5. Speciﬁcally,

from Theorem IV.4.1 and Theorem IV.5.1 we have the following result, whose

proof is left to the reader as an exercise.

Theorem V.1.1 Let f ∈ W

m,q

loc

(Ω), m ≥ 0, 1 < q < ∞, and let

v ∈ W

1,q

loc

(Ω), p ∈ L

loc

(Ω)

with v weakly divergence-free, satisfy (V.1.2) for a ll ψ ∈ C

∞

(Ω). Then

v ∈ W

m+2,q

loc

(Ω), p ∈ W

m+1,q

loc

(Ω).

In particular, if f ∈ C

∞

(Ω), then

v, p ∈ C

∞

(Ω).

Also, if Ω is of class C

m+2

and f ∈ W

m,q

loc

(Ω), v

∗

∈ W

m+2−1/q,q

(∂Ω), then

v ∈ W

m+2,q

loc

(Ω), p ∈ W

m+1,q

loc

(Ω).

In particular, if Ω is of class C

∞

and f ∈ C

∞

(Ω), v

∗

∈ C

∞

(∂Ω) then v,

p ∈ C

∞

(Ω

), for all bounded Ω

⊂ Ω.

V.2 Existence and Uniqueness of Generalize d Solutions

for Three-Dimensional Flow

In this section we shall be concerned with the well-posedness of the Stokes

problem when the region of ﬂow is a three-dimensional exterior do mai n.

The two-dimensional case, being related to the Stokes paradox, i s in general

not solvable. Actually, as will be proved in Section V.7, it admits a solution

if and only if the data obey suitable restrictions; see also Theorem V.2.2.

Furthermore, the problem of existence of q-generalized solutions for any q > 1

will be treated in Section V.5.

As a matter of fact, both assumptions on v and p can be replaced by the following

one:

∇v ∈ L

loc

(Ω),

with v satisfying (V.1. 1) for all ϕ ∈ D(Ω). This is because, by Lemma II.6.1, v ∈

1,q

loc

(Ω) and, by Lemma IV.1.1, we infer the existence of p ∈ L

loc

(Ω) satisfying

(V.1.2).

Actually, n-dimensional with n ≥ 3; see Remark V.2.1.

V.2 Existence and Uniqueness for Three-Dimensional Flow 307

Theorem V.2.1 Let Ω be a locally Lipschitz exterior domain of R

. Given

f ∈ D

−1,2

(Ω), v

∗

∈ W

1/2,2

(∂Ω),

there exists one and only one generalized solution to the Stokes problem

(V.0.1), (V.0.2). This solution satisﬁes for all R > δ(Ω

) the f ollowing es-

timate

kvk

2,Ω

+ |v|

1,2

+ kpk

≤ c



|f |

−1,2

+ kv

∗

1/2,2(∂Ω)



(V.2.1)

where p is the pressure ﬁeld associated to v by L emma V.1.1 and c = c(Ω, R),

c → ∞ as R → ∞. Furthermore,

|v(x)| = o(1/

|x|) as |x| → ∞. (V.2.2)

Proof. The proof of existence and uniqueness goes exactly as in Theorem

IV.1. 1, provided we make a suitable extension of v

∗

. In this respect, it is

worth noticing that it is not required that the ﬂux of v

∗

on ∂Ω be zero. Set

Φ =

∂Ω

∗

· n, σ(x) = −Φ ∇E(x),

where E is the fundamental solution to the Laplace equation and with the

origin of coordinates taken in

Ω

. Clearly,

∆σ = 0 in Ω,

∂Ω

σ · n = Φ.

Putting w

∗

= v

∗

−σ, it follows that

∂Ω

∗

· n = 0

and we can apply the results in Exercise III.3.8 to construct a solenoida l ﬁeld

∈ W

1,2

(Ω), vanishing outside Ω

, for some ρ > δ(Ω

), that equals w

∗

∂Ω and, moreover,

1,2,Ω

≤ c

∗

1/2,2(∂Ω)

(V.2.3)

with c = c(Ω

). On the other hand we have, clearly,

∗

1/2,2(∂Ω)

≤ c

∗

1/2,2(∂Ω)

so that (V.2.3) implies

Recall that n is t he unit outer normal to ∂Ω.

308 V Steady Stokes Flow in Exterior Domains

1,2,Ω

≤ c

∗

1/2,2(∂Ω)

(V.2.4)

with c

= c

(Ω, ρ). A generalized solution to the exterior problem is then

sought in the form

v = w + V

+ σ,

where w ∈ D

1,2

(Ω) solves

(∇w, ∇ϕ) = −[f , ϕ] − (∇V , ∇ϕ),

with

V = V

+ σ.

Existence, uni queness, and estimate (V.2 .1) are proved along the same lines

of Lemm a IV.1.1 , provided we use Lemma V.1.1 instead of Lemma IV.1.1 and

note that, since ∆σ = 0 in Ω, we have

Ω

∇σ : ∇ ϕ = 0 ,

for any ϕ ∈ D

1,2

(Ω). To show estimate (V.2.2) we notice that for |x| suﬃ-

ciently large

|v(x)| ≤ c

(|w(x)| + |Φ||∇E(x)|)

= c

|w(x)| + O(1/|x|

and, since w ∈ D

1,2

(Ω), by Theorem II.7.6 and Lemma II.6 .3 it follows

|w(x)| = o(1/

|x|),

which furnishes (V.2.2). The proof of the theorem is then completed. ut

Exercise V.2.1 Show that Theorem V.2.1 holds also when ∇ · v = g 6≡ 0, where

g is a prescribed function of L

(Ω). In this case (V. 2.1) is modiﬁed accordingly, by

adding the term kgk

on its right-hand side. Notice that, unlike the case where Ω

is bounded (see Exercise IV.1.1), no relation between g and v

∗

is needed.

Remark V.2.1 If in Theorem V.2.1, v

∗

= v

+ ω × x ≡ V , the solenoidal

extension of the boundary velocity can be performed in an elementary way.

Speciﬁcally, we need a ﬁeld a that equals V near ∂Ω, equals 0 at large dis-

tances, and ha s ﬁrst derivatives in L

(Ω). Thus, assuming without loss of

generality v

= (v

, 0 , 0), we may take (Borchers 1992)

a = −

∇×



∇ × (ζv

) + ζx

ω)



, (V.2.5)

where ζ is an arbitrary function from C

∞

(Ω) that is zero near ∂Ω and one far

from ∂Ω. Consequently, in particular, if v

∗

= V , existence of a g eneralized

solution is proved without regularity assumptions on Ω. 

V.2 Existence and Uniqueness for Three-Dimensional Flow 309

Remark V.2.2 In spatial di mension n > 3 the results of Theorem V.2.1

continue to hold wi th estimate (V.2.2) replaced by

n−1

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

n/2−1

) as |x| → ∞.

In the case of plane motions, however, we have a diﬀerent situation that

resembles the Stokes paradox mentioned at the beginning of the chapter.

Actually, using the same method of proof, we can still construct a ﬁeld v

satisfying conditions (i)-(iii) and (v) of Deﬁnition V.1.1. However, we are

not able, for such a v, to check the validity of (iv), that is, to control the

behavior of the solution at large distances. This is because functions having

a ﬁnite Dirichlet integral in two space dimension need not tend to a ﬁnite

limit at inﬁnity, even i n a generalized sense; see Section II.6 and Section II.9.

Nevertheless, as wil l be shown in Theorem V.3.2 (see also Remark V.3.5),

every such solution does tend to a well-deﬁned vector, v

∞

, at inﬁnity, whenever

the body force is of compact support. However, we cannot conclude

∞

= 0 . (V.2.6)

Actually, (V.2.6) is in general not true, and in Section V.7 we shall prove

that (V.2.6) holds if and only if the data satisfy certain restrictions. The

meaning of the vector v

∞

will be clariﬁed in Section VII.8, within the context

of a singular perturbation theory based on the Oseen approximation. Here,

we end by pointing out the foll owing Stokes paradox for generalized solutions

(Heywood 1974).



Theorem V.2.2 Let v be a weak solution to the Stokes problem in an exte-

rior, locally Lipschitz two-dimensional domain correspo nding to f ≡ v

∗

≡ 0.

Then v = 0 a.e. in Ω.

Proof. By assumption,

(∇v, ∇ϕ) = 0, for all ϕ ∈ D

1,2

(Ω), (V.2.7)

where v ∈ D

1,2

(Ω) and v = 0 at ∂Ω (in the trace sense). By Theorem II.7.1

with q = n = 2, it follows that v ∈ D

1,2

(Ω) and, since v is solenoidal, this

implies v ∈

1,2

(Ω). On the other hand, since Ω is locally Lipschitz, by

Theorem III.5.1, we ﬁnd

1,2

(Ω) = D

1,2

(Ω)

and so v ∈ D

1,2

(Ω), whi ch together with (V.2.7) completes the proof of the

theorem. ut

In a private conversation in the summer of 2003, Olga Ladyzhenskaya pointed

out to me that a result entirely analogous to Theorem V.2.2 is stated, without

proof, at p. 43 of Ladyzhenskaya (1969).

310 V Steady Stokes Flow in Exterior Domains

V.3 Representation of Solutions. Behavior at Large

Distances and Related Resul ts

In order to perform an L

-theory in exterior domains of the type p erformed fo r

bounded domains in Section IV.6, we need to know more about the asymptotic

behavior of solutions at great distances. We shall see, in particular, tha t under

suitable conditi ons of “growth” at inﬁnity, they behave exactly as the Stokes

fundamental solution, provided the force f is of compact support. All this will

be proved as a consequence of some representation f ormulas we are about to

derive. In principle, thi s can be done qui te straightforwardly from the results

of Section IV.8. Actuall y we may write (IV.8.20) and (IV.8.21) on Ω ∩B

(x),

for suﬃciently large R, then let R → ∞ and require that the surface integrals

calculated at ∂B

(x) converge to zero. However, this method would imp ose

too severe restrictions a priori o n the behavior of v and p at large distances

and the results obtained under such assumptions would be of no use for further

purposes. We therefore employ another technique introduced by Fujita (1961)

in the nonlinear context, whi ch is based on a suitable “truncation” of the

fundamental solution.

Let ψ = ψ(t) be a C

∞

-function in R that equals one for |t| ≤ 1/2 and zero

for |t| ≥ 1. Setting

(x − y) = ψ



x − y



, R > 0,

there fol lows

(x − y) =

(

1 if |x − y| ≤ R/2

0 if |x − y| ≥ R

(x − y)| ≤ M R

−|α|

, |α| ≥ 0

(V.3.1)

where M is independent of x, y. The Stokes-Fujita tr uncated fundamental solu-

tion U

(R)

, q

(R)

is then deﬁned by (IV. 2.1) with Φ replaced by ψ

Φ. Evidently,

from (V.3.1)

we have

(R)

(x−y) = U

(x−y), q

(R)

(x−y) = q

(x−y), if |x − y| ≤ R/2, (V.3.2)

while

(R)

(x − y) ≡ q

(R)

(x − y) ≡ 0, if |x − y| ≥ R. (V.3.3)

Moreover, from (IV.2.2) it follows for x 6= y that

∆U

(R)

(x − y) +

∂

∂x

(R)

(x − y) = H

(R)

(x − y)

∂

∂x

(R)

(x − y) = 0

(V.3.4)

where H

(R)

is deﬁned by

V.3 Representation of Solutions and Behavior at Large Distances 311

(R)

(x − y) =

(

0 if x = y

∆

(ψ

Φ)(x − y) if x 6= y.

Since Φ(x −y) is biharmonic for x 6= y and all derivatives of ψ

(x −y) vanish

unless R/2 ≤ |x − y| ≤ R, we recover that H

(R)

is indeﬁnitely diﬀerentiable

and vanishes unl ess R/2 ≤ |x −y| ≤ R. Also, for u ∈ L

loc

(Ω) it is H

(R)

∗u ∈

∞

). Finally, from (V.3.1) and the properties of Φ we at once obtain the

following uniform bounds as R → ∞

(R)

(x − y)| =

(

O(log R/R

2+|α|

), if n = 2

O(R

−n−|α|

) if n > 2.

|α| ≥ 0 (V.3.5)

Consider now the Green’s f ormula (IV.8.10) in a domain Ω, not necessarily

bounded, with

u(y) = u

(R)

(x − y) ≡



(R)

, U

(R)

, . . . , U

(R)



π(y) = q

(R)

(x − y).

Such a procedure is meaningful, since, whatever the domain Ω may be, in view

of (V.3.3) the integration is al ways made on a bounded region (the support of

(x −y)). By repeating all the steps leading to (IV.8.20) and (IV.8.21) and

recalling (V.3.2) we thus readily obtain

(x) =

Ω

(R)

(x − y)D

(y)dy

−

∂Ω

(R)

(x − y)T

v, D

p)(y)

−D

(y)T

(R)

, q

(R)

)(x −y)]n

(y)dσ

−

Ω

(R)

(x − y)D

(y)dy.

(V.3.6)

Likewise, setting

(R)

(x) =

Ω

(R)

(x − y)D

(y)dy,

we have

∂(D

∂x

∂S

(R)

∂x

(x) +

∂Ω

∂q

(R)

(x −y)

∂x

v, D

p)(y)

−2D

(y)

∂

(x − y)

∂x



(y)dσ

−

Ω

∆H

(R)

(x − y)D

(y)dy;

(V.3.7)

312 V Steady Stokes Flow in Exterior Domains

see Exercise V. 3.1. Notice that if R < dist (x, ∂Ω), formulas (V.3.6) and

(V.3.7) do not require any regularity assumption on Ω. The following result

holds.

Lemma V.3.1 Let Ω be an arbitrary domain of R

. Let v ∈ W

1,r

loc

(Ω), 1 <

r < ∞, be weakly divergence-free and satisfy (V.1.1) for all ϕ ∈ D(Ω) and

some r ∈ (1, ∞). Then, if f ∈ W

m,q

loc

(Ω), 1 < q < ∞, it follows that v ∈

m+2,q

loc

(Ω) and, moreover, for all ﬁxed d > 0, for all |α| ∈ [0, m] and for

almost all x ∈ Ω with dist (x, ∂Ω) > d, v obeys the identity

(x) =

(x)

(d)

(x−y)D

(y)dy−

β(x)

(d)

(x−y)D

(y)dy (V.3.8)

where β(x) = B

(x) −B

d/2

(x).

Proof. The ﬁrst part of the lemma has already been proved in Theorem IV.4.2.

Concerning the validity of (V.3.8), we notice that if v a nd p are smooth, it

follows from (V.3.6), by taking R = d and recalling the properties o f H

(d)

The validity of (V.3.8) under the hypothesis of the lemma is recovered by

adopting exactly the same procedure used in the proo f of Theorem IV.8. 1,

and we leave it to the reader. ut

Remark V.3.1 The assumptions of Lemma V.3.1 do not require Ω to be

an exterior domain. Rather, Ω can be any domain of R

. Actually, Lemma

V.3. 1 will be applied in Chapter VI to the study of the asymptotic behavior

of Stokes ﬂow in doma ins with noncompact bounda ries. 

Remark V.3.2 For f uture purposes, we observe that a representation analo-

gous to that furnished in Lemma V.3.1 also holds fo r solutions to the Poisson

equation ∆u = f in an arbitrary domain Ω of R

. More precisely, set

(R)

(x − y) = ψ

(x − y)E(x −y), (V.3.9)

where E is the fundamental solution of Laplace’s equation (II.9.1) and ψ

given in (V.3.1). In a strict analogy with the Stokes-Fujita truncated funda-

mental solution, one has

(R)

(x − y) =

(

E(x − y) if |x − y| ≤ R/2

0 if |x − y| ≥ R.

(V.3.10)

Furthermore, for all x 6= y,

∆E

(R)

(x − y) = H

(R)

(x − y) (V.3.11)

with

(R)

(x − y) =

(

0 if x = y

∆ (ψ

(x − y)E(x −y)) if x 6= y.

(V.3.12)

V.3 Representation of Solutions and Behavior at Large Distances 313

Clearly, the function H

(R)

is inﬁnitely diﬀerentiable and vanishes unless R/2 ≤

|x − y| ≤ R, and by the properties of ψ

(R)

(x − y)| ≤ C

(

log R/R

2+|α|

, if n = 2

−n−|α|

if n > 2.

(V.3.13)

Repeati ng step by step the proof of Lemma II.9.1 with E

(R)

in place of E and

using (V.3.11) we deduce for almost all x ∈ Ω, with dist (x, ∂Ω) > R, the

following general representation formula

u(x) =

(x)

(R)

(x−y)D

f(y)dy−

(x)−B

R/2

(x)

(R)

(x−y)D

u(y)dy,

(V.3.14)

with | α| ≥ 0. 

Exercise V.3.1 Let Ω b e of cl ass C

m+2

, m ≥ 0, and let v ∈ W

m+2,q

(Ω

p ∈ W

m+1,q

(Ω

) solve a.e. the Stokes system (V.0.1)

1,2

, corresponding to f ∈

m,q

(Ω

), 1 < q < ∞, all ρ > δ(Ω

). Show the validity of (V.3.6) for al most all

x ∈ Ω and all R > δ(Ω

). Assuming, further, that the support S of f is bounded,

show the validity of (V.3.7) f or almost all x ∈ Ω and all R for which B

(x) ⊃ S.

Hint: ( V.3.6) is shown by the same technique of Theorem IV.8.1. For (V.3.7), the

only term that demands little care is that involving S

(R)

. However, for B

(x) ⊃ S

we have

(R)

(x) =

Ω

(x − y)D

(y)dy ≡ S(x)

and, since D

is a singular kernel, under the stated assumptions on f t he function

S belongs to L

(Ω

) for all r > δ(Ω

) and one has

q,Ω

≤ ckD

f k

q,Ω

;

see, e.g ., Mikhlin (1965, §29).

Lemma V.3.1 allows us to derive some informa tion concerning the point-

wise asymptotic behavior of q-weak solutions. For instance, we have

Theorem V.3.1 Let Ω be an a rbitrary exterior domain in R

, n ≥ 2, and

let v ∈ D

1,q

(Ω

) ∩ L

(Ω

), for some R > δ(Ω

) and some q, s ∈ (1, ∞).

Assume further that v is weakly divergence-free and satisﬁes (V.1.1 ), for all

ϕ ∈ D(Ω) with f ∈ W

m,r

(Ω

), m ≥ 0, n/2 < r < ∞. Then

lim

|x|→∞

v(x) = 0, for all |α| ∈ [0, m].

The hypotheses of the following theorem are not optimal on v. However, they

will suﬃce for further purposes. Furthermore, the theorem holds under alterna-

tive assumptions on f. We shall formulate these latter directly in the nonlinear

context; see Section X.5.

314 V Steady Stokes Flow in Exterior Domains

Proof. We show the result for n ≥ 3, the case n = 2 b eing treated analogously.

For ﬁxed d > 0 and all x ∈ Ω with dist (x, ∂Ω) > d, we have



(x)

(d)

(x − y)D

(y)dy



≤ ck|x − y|

2−n

q/(q−1),B

(x)

kfk

m,q,B

(x)

≤ c

kf k

m,q,B

(x)

(V.3.15)

and



β(x)

(d)

(x − y) D

(y)dy



β(x)

(d)

(x − y)v

(y)dy



≤ c max

i,j

(d)

(x − y)k

m,s

,β(x)

kvk

s,β(x)

≤ c

kvk

s,B

(x)

(V.3.16)

where we have exploited the properties of the function H

(d)

. The theorem is

then a consequence of this fa ct, (V.3.15 ), (V.3.16), and Lemma V.3.1. ut

Remark V.3.3 From Theorem V. 3.1 and the equation of motion (V.0.1) we

can immediately derive a pointwise behavior of the pressure ﬁeld p at large

distances. For example, if f satisﬁes the assumption of that theorem with

m ≥ 2 and, further, D

f(x) tends to zero as |x| → ∞, |α| ∈ [0, m − 2] we

have

lim

|x|→∞

∇p(x) = 0, 0 ≤ |α| ≤ m − 2.



Exercise V.3.2 Let v satisfy the hypotheses of Theorem V. 3.1, with the possible

exception of condition (iv) of Deﬁnition V.1.1. Assuming f ∈ W

m,r

(Ω

), r > n,

show

lim

|x|→∞

∇v(x) = 0, for all |α| ∈ [0, m].

Theorem V.3.1 is silent about the rate of decay of solutions. However, if

f is of compact support we can obtain sharp estimates for v, p and for their

derivatives of arbitrary order. In fact, we have

Theorem V.3.2 Let Ω be a C

-smooth, exterior domain and let v ∈

2,q

loc

(Ω), q ∈ (1, ∞), be weakly divergence-free and sati sfy (V.1.1) for all

ϕ ∈ D(Ω) wi th f ∈ L

(Ω). Assume, further, that the support of f is bounded.

Then, if at least one o f the following conditions is satisﬁed as |x| → ∞:

(i) |v(x)| = o (|x|)

(ii)

|x|≤r

|v(x)|

(1 + |x|)

n+t

dx = o (log r), some t ∈ (1, ∞),

V.3 Representation of Solutions and Behavior at Large Distances 315

there exist vector and scalar constants v

∞

, p

∞

such that for almost all x ∈ Ω

(x) = v

∞j

Ω

(x − y)f

(y)dy −

∂Ω

(x − y)T

(v, p)(y)

−v

(y)T

, q

)(x − y)]n

(y)dσ

≡ v

∞j

+ v

(1)

(x)

(V.3.17)

and

p(x) = p

∞

−

Ω

(x − y)f

(y)dy +

∂Ω

(x − y)T

(v, p)(y)

−2v

(y)

∂q

(x − y)

∂x

(y)]dσ

≡ p

∞

+ p

(1)

(x).

(V.3.18)

Moreover, as |x| → ∞, v

(1)

(x) and p

(1)

(x) are inﬁnitely diﬀerentiable and

there the following asymptotic representations hold:

(1)

(x) = T

(x) + σ

(x)

(1)

(x) = −T

(x) + η(x),

(V.3.19)

where

= −

∂Ω

(v, p)n

Ω

(V.3.20)

and, for all |α| ≥ 0,

σ(x) = O(|x|

1−n−|α|

)

η(x) = O(|x|

−n−|α|

(V.3.21)

Proof. Let us observe tha t, since the support S of f i s com pact, v and p

are inﬁni tely diﬀerentiable at each point of Ω − S, as follows from Theorem

IV.4. 1. We begin to show that (V.3.19)–(V.3 .21) are a consequence of (V.3.17),

(V.3.18). Actually we have

(1)

(x) = T

(x) +

∂Ω

(y)T

, q

)(x −y)n

dσ

Ω

(x − y) − U

(x)]f

(y)dy

−

∂Ω

(x − y) − U

(x)]T

(v, p)(y)n

(y)dσ

(V.3.22)