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

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606 IX Steady Navier–Stokes Flow in Bounded Domains

Remark IX.4.2 As already noticed, the above example only works if Φ < 0.

An example when Φ > 0 (outﬂow condition) has been recently f urnished by

Heywood (2010). 

Remark IX.4.3 By similar ideas and a slightly more complicated reasoning,

one is able to prove the invalidity of EC when Ω is the 3-dimensional spherical

shell



x ∈ R

: R

< |x| < R



; (IX.4.10)

see Takeshita (1989, Theorem 1) a nd Farwig, Kozono & Yanagisawa (2010,

Theorem 1). 

Remark IX.4.4 It is interesting to observe that the counterexample given

previously to the validity of EC implies, indirectly, that the problem

∇ · w = f in Ω,

Ω

f = 0,

w ∈ W

1,q

(Ω)

|w|

1,q

≤ ckfk

kwk

≤ ckfk

−1,q

(IX.4.11)

is, in general, not solvable even if f is in divergence form. Actually let Ω be

the annular region



x ∈ R

: R

< |x| < R



If (IX.4.11) were solvable for some q > 2, we could add to the extension

(IX.4.4) a ﬁeld w verifying (IX.4.11) with f = −∇ · (ψ

W ). The vector ﬁeld

U ≡ V + w is then solenoidal and assumes the value v

∗

at ∂Ω. Furthermore,

by the H¨older inequality, by the embedding Theorem II.3.4, and by (IX.4.11)

we have



Ω

u · ∇U · u



Ω

u · ∇u · U



≤ c

|u|

1,2

kUk

≤ c

|u|

1,2

(kψ

W k

+ k∇ · (ψ

W )k

−1,q

) ,

and being

k∇· (ψ

W )k

−1,q

≤ kψ

W k

→ 0 as ε → 0,

we obtain that, for all α > 0 there is an extension U = U (α) such that,



Ω

u · ∇U ·u



≤ α|u|

1,2

for any u ∈ H

(Ω), thus all owing EC for Ω. By the same token, one can show

that problem (IX.4.11)

with f ∈ C(Ω) does not admit a solution v ∈ C

(Ω)

such that

IX.4 Existence and Uniqueness with Nonhomogeneous Boundary Data 607

kvk

≤ ckfk

with c = c(n, Ω). 

The example shown previously rules out the general validity of EC, but, on

the other side, it also suggests that condition (IX.4.7) can possibl y b e replaced

by the weaker one:

m+1

i=1

|Φ

| < cν, (IX.4.12)

for some positive constant c. In fact, this is indeed the case. Speciﬁcally, follow-

ing the work of Galdi (1991), by suitably modifying the Leray-Hopf extension

(IX.4.5), we shall prove existence of solutio ns under the sole condition that the

ﬂuxes Φ

of v

∗

through each component Γ

of ∂Ω satisfy condition (IX.4. 12),

with a computable constant c that depends only on Ω and n.

However, if

∂Ω has more than one connected component, existence of solutions satisfying

merely (IX.4.6) with no restriction on the size of t he ﬂuxes Φ

remains open.

To show our ma in result we need some preparatory steps.

Lemma IX.4.1 Let Ω be a bounded locally Lipschitz domain in R

, n = 2, 3 .

Denote by ω

, i = 1, . . . , m, the (bounded) connected components of R

− Ω

and set

ω ≡

[

i=1

Then, given a ∈ W

1/2,2

(∂Ω) verifying the condition

a · n = 0, i = 1, 2, . . . , m + 1, (IX.4.13)

where n is the outer normal to ∂Ω and

≡ ∂ω

, for i = 1, . . ., m, Γ

m+1

≡ ∂(Ω ∪ ω),

We wish to emphasize that, clearly, the condition on the “smallness” of Φ

does

not imply a priori “smallness” of v

∗

. On the other hand, if we assume that the

trace norm of v

∗

at the boundary is small with respect to ν, then existence with

nonzero small ﬂuxes Φ

is proved in a direct elementary way.

For another approach t o existence with nonhomogeneous data, again due to Leray,

we refer the reader to the Notes for this Chapter. In that context, we shall also

give other existence results due to Amick (1984), Morimoto (1992) and Morimoto

and Ukai (1996), where the restriction (IX.4.7) can be removed.

Clearly, the number m is ﬁnite since ∂Ω i s compact and, furthermore,

min

i=1,...,m

dist (ω

, ∂(Ω ∪ ω)) > 0,

(cf. also Griesinger 1990a).

608 IX Steady Navier–Stokes Flow in Bounded Domains

there is w ∈ W

2,2

(Ω) if n = 3 [respectively w ∈ W

2,2

(Ω), if n = 2] such that

a = ∇×w [respectively a = ∇×w] in the trace sense at ∂Ω. Moreover, the

following inequality holds

kwk

2,2

≤ ckak

1/2,2(∂Ω)

[respectively kwk

2,2

≤ ckak

1/2,2(∂Ω)

] (IX.4.14)

where c = c(n, Ω).

Proof. Since

∂Ω =

m+1

[

i=1

from (IX.4.13) it follows that

∂Ω

a · n = 0

and so, by Exercise III.3.5, we may extend a to a ﬁeld v

∈ W

1,2

(Ω) with

∇· v

= 0, a nd verifying the inequality

1,2

≤ ckak

1/2,2(∂Ω)

. (IX.4.1 5)

If n = 2, for a ﬁxed x

∈ Ω we deﬁne a function w through the line integral

w(x) =

−v

), x ∈ Ω,

i.e., w is the stream function associated to v

. Since (IX.4.13) holds, w is

singlevalued. Furthermore,

∂w

∂x

= v

∂w

∂x

= −v

and so

|w|

1,2

+ |w|

2,2

≤ c

1,2

. (IX.4.16)

Also, we can modify w by an additive constant in such a way that

Ω

w = 0

so that by inequality (II.5.1 0), (IX.4.15), and (IX.4.16) we deduce (IX.4.14),

proving the lemma if n = 2. To prove it for n = 3, we notice that, again

by Exercise III.3.5, we can extend a at ∂ω

, i = 1, . . . , m into each ω

to a

solenoidal vector ﬁeld v

∈ W

1,2

(ω

) satisfying the estimate

1,2,ω

≤ c

kak

1/2,2(∂Ω)

, i = 1, . . ., m. (IX.4.17)

Moreover, denoting by B an open ball with B ⊃ Ω, since

IX.4 Existence and Uniqueness with Nonhomogeneous Boundary Data 609

m+1

a · n = 0,

by Corolla ry III.3.1 we can extend a at ∂(Ω ∪ω) to a solenoidal vector ﬁeld,

m+1

, in ω

m+1

≡ B − (Ω ∪ ω) such that

m+1

∈ W

1,2

(ω

m+1

) , v

m+1

(x) = 0 x ∈ ∂B

and, moreover,

m+1

1,2,ω

m+1

≤ c

kak

1/2,2(∂Ω)

. (IX.4.18)

It is then immediately veriﬁed that the vector ﬁeld:

v : x ∈ B →

(

if x ∈ Ω

if x ∈ ω

, i = 1, . . ., m + 1

(IX.4.19)

satisﬁes the foll owing properties

(i) v ∈ W

1,2

(B),

(ii) ∇ · v = 0 in B,

(iii) v = 0 at ∂B,

implying v ∈ H

(B). However, by means of an explicit representation fo rmula

it can be easily proved (see Exercise IX.4.1) that, given v ∈ H

(B) there is

w ∈ W

2,2

(B) such that

v = ∇ × w

kwk

2,2

≤ c

kvk

1,2

This last relation, together with (IX.4.17)–(IX.4.19) implies (IX.4.14) and the

restriction of w to Ω veriﬁes all requirements stated in the lemma. The proof

is therefore complete. ut

Exercise IX.4.1 Let Ω be a bounded domain in R

and let v ∈ H

(Ω). Show that

there exists w ∈ W

2,2

(Ω) such that v = ∇ × w. Hint: Take ﬁrst v ∈ D(Ω) and

consider the function U = E ∗ v, where E is the fundamental solution of Laplace’s

equation. Then v = ∇×w, where w = −∇×U . By the Calder´on-Zygmund Theorem

II.7.4 and Young’s inequality (II.5.3) it follows that

kwk

2,2,Ω

≤ Ckvk

1,2,Ω

where C = C(Ω). The result is then a consequence of this inequality and of the

density of D(Ω) into H

(Ω).

Remark IX.4.5 Using the same lines of proof, we at once recognize that

Lemma IX.4.1 is valid, more generally, with w ∈ W

2,q

(Ω), 1 < q < ∞, pro-

vided a ∈ W

1−1/q

(∂Ω). In particular, w obeys the following estimate

kwk

2,q

≤ ckak

1−1/q,q(∂Ω)



610 IX Steady Navier–Stokes Flow in Bounded Domains

Remark IX.4.6 Lemm a IX.4.1 admits of a suitable immediate extension

to arbitrary dimension n ≥ 4, which will be appropriate to our purposes.

Actually, if we set W = ∇U (i.e., W

= ∂U

/∂x

) with U deﬁned in Exercise

IX.4. 1, then it is easily seen that v = ∇ · W (i.e. v

= ∂W

/∂x

) and

that W

satisﬁes an estimate of the type (IX.4.14). Mo re generally, if a ∈

1−1/q,q

(∂Ω), 1 < q < ∞, then for all i, j = 1, . . ., n we have

2,q

≤ ckak

1−1/q,q(∂Ω)



The result just shown allows us to construct the desired extension of the

ﬁeld v

∗

. Let us introduce some notation ﬁrst. We denote by c = c(n, Ω) the

constant entering the problem:

∇· b = h in Ω

b ∈ W

1,2

(Ω)

|b|

1,2

≤ ckhk

(IX.4.20)

Moreover, if ∂Ω has m ore than one boundary, Γ

, .., Γ

m+1

, with Γ

, i =

1, . . ., m, the “interior” boundaries and Γ

m+1

the “outer” one, we set

d ≡ min

i,j

dist (Γ

, Γ

)

Ω

i,d

= {x ∈ Ω : dist (x, Γ

) < d/2}

(IX.4.21)

and, indicated by ω

, i = 1, ..., m, the (bounded) connected components o f

− Ω,

(x) = −∇E(x − x

), x

∈ ω

, i = 1, . . . , m

m+1

(x) = −σ

(x),

(IX.4.22)

where E(ξ) is the fundamental solution of Laplace’s equation deﬁned in

(II.9.1). Clearly, we have

· n = 1, i = 1 , . . ., m + 1, (IX.4.23)

where n denotes the outer normal to ∂Ω at Γ

. The following extension lemma

holds.

Lemma IX.4.2 Let Ω be a bounded locally Lipschitz domain in R

, n = 2, 3 ,

and let v

∗

∈ W

1/2,2

(∂Ω) satisfy

∂Ω

∗

· n = 0 . (IX.4.24)

Then, for any η > 0, there exist ε = ε(η, v

∗

, n, Ω) > 0, and a solenoidal vector

ﬁeld V = V (ε) such that

IX.4 Existence and Uniqueness with Nonhomogeneous Boundary Data 611

V ∈ W

1,2

(Ω), with V = v

∗

at ∂Ω

and verifying



Ω

u · ∇V · u



≤

(

η +

m+1

i=1



4cκ

kσ

2,Ω

i,d

+ κkσ

4,Ω

i,d



|Φ

)

|u|

1,2

(IX.4.25)

for all u ∈ H

(Ω). Here κ, κ

are constants depending on n and deﬁned in

(IX.4.34), (IX.4.35), a nd Lemma III.6 .1, respectively, c = c(n, Ω) is deﬁned

in (IX.4.20) and

∗

· n, i = 1, . . . , m + 1.

Furthermore, σ

and Ω

i,d

are given in (IX.4.21) and (IX.4.22). Finally, if v

∗

lies in a ball of W

1/2,2

(∂Ω), namely, kv

∗

1/2,2(∂Ω)

≤ M , for some M > 0,

then there is C = C(n, Ω, η, M ) > 0 such that

kV k

1,2

≤ C kv

∗

1/2,2(∂Ω)

. (IX.4.26)

Proof. We shall ﬁrst consider the case m > 0 . Let

(x) ≡ dist (x, Γ

), x ∈ Ω, i = 1, . . ., m + 1,

and denote by ρ

(x) the regularized distance of x from Γ

, in the sense of Stein

(cf. Lemma III. 6.1). Set

ψ(t) =







1 t ≤ 1

2 − t 1 ≤ t ≤ 2

0 t ≥ 2

and deﬁne

(x) ≡ ψ(4ρ

(x)/d), i = 1, . . . , m + 1. (IX.4.27)

Recalling the properties of ρ

(x), we have that the functions (IX.4.27 ) are

piecewise diﬀerentiable and that, moreover,

(x) = 1 if δ

(x) < d/4κ

(x) = 0 if δ

(x) ≥ d/2

|ψ

(x)| ≤ 1

|∇ψ

(x)| ≤ 4κ

supp (∇ψ

) ⊂ {x ∈ Ω : d/4κ

≤ δ

(x) ≤ d/2}

(IX.4.28)

where κ

and κ

are the constants introduced in Lemma III.6.1. In view of

(IX.4.23) and (IX.4.28) we recover that the ﬁeld

612 IX Steady Navier–Stokes Flow in Bounded Domains

(x) := v

∗

(x) −

m+1

i=1

(x)σ

(x), x ∈ ∂Ω (IX.4.29)

satisﬁes the m + 1 conditions

·n = 0 , i = 1, . . ., m + 1.

By Lemma IX.4.1 we then have that, if n = 3, there is a w ∈ W

2,2

(Ω)

[respectively w ∈ W

2,2

(Ω), if n = 2] such that v

(x) = ∇ × w(x), x ∈ ∂Ω

[respectively v

(x) = ∇ × w(x)]. For given ε > 0 we set

= ∇ ×(ψ

w) [respectively V

= ∇ × (ψ

w)] (IX.4.30)

where ψ

is the “cut-oﬀ” function deﬁned in Lemma III.6.2. From the prop-

erties of ψ

and w we easily realize that the ﬁeld

U (x) = V

(x) +

m+1

i=1

(x)σ

(x), x ∈ Ω,

is a W

1,2

(Ω)-extension of v

∗

. However, U is not solenoidal and, therefore, in

order to obtain the desired extension of v

∗

, we have to m odify U a ppropriately.

To this end, let us consider the ﬁeld b deﬁned by the following properties:

∇ · b = −

m+1

i=1

(x) ·∇(Φ

(x)) ≡ h(x)

b ∈ W

1,2

(Ω)

|b|

1,2

≤ c khk

(IX.4.31)

Since, by (IX.4.24) and (IX.4.28),

h ∈ L

(Ω), for all q ∈ (1, ∞)

Ω

h = 0,

Theorem III.3.1 ensures the existence of at least one vector b satisfying

(IX.4.31). Furthermore, using (IX.4.28)

3,4

, we obtain

|b|

1,2

≤

4cκ

m+1

i=1

kσ

2,Ω

i,d

|Φ

|. (IX.4.3 2)

The desired extension of v

∗

is then given by the ﬁeld:

V (x) := V

(x)+

m+1

i=1

(x)σ

(x)+b(x) ≡ V

(x)+V

(x)+b(x). (IX.4.33)

IX.4 Existence and Uniqueness with Nonhomogeneous Boundary Data 613

In fact, V is solenoidal, belongs to W

1,2

(Ω), and its trace at ∂Ω is v

∗

. Let us

now estimate the trilinear form:

a(u, V , u) ≡

Ω

u · ∇V ·u, u ∈ H

In this respect, we recall the inequality:

kuk

≤ κ|u|

1,2

(IX.4.34)

where

κ =

(

7/4

−13/ 8

|Ω|

1/12

if n = 3

|Ω|

1/4

√

2 if n = 2.

(IX.4.35)

Actually, (IX.4.34) and (IX.4.35) foll ow from (II.3.9), (II.3.10), and (II.5.5).

By the H¨older inequality, (IX.4.32), (IX.4.34), and (IX.4.35) we obtain

|a(u, b, u)| ≤ kuk

|b|

1,2

≤

m+1

i=1



4cκ

kσ

2,Ω

i,d

|Φ



|u|

1,2

. (IX. 4.36)

Furthermore, by an easily justiﬁed integration by parts we ﬁnd

|a(u, V

, u)| = |a(u, u, V

and so, again by the H¨older i nequality, (IX.4.28)

, (IX.4.33), and (IX.4.34),

it follows that

|a(u, V

, u)| ≤ kuk

|u|

1,2

≤ κ|u|

1,2

m+1

i=1

kσ

4,Ω

i,d

|Φ

|. (IX.4.3 7)

It remains to estimate the term

a(u, V

, u).

From the properties of the function ψ

established in Lemma III.6.2 we ﬁnd

that

(x)|







≤

εκ

δ(x)

|w(x)| + |∇w(x)|, if x ∈ Ω

= 0, if x 6∈ Ω

(IX.4.38)

where δ(x) is the distance from x ∈ Ω to ∂Ω,

Ω

≡ {x ∈ Ω : δ(x) ≤ 2γ(ε)};

and γ(ε) := exp(−1/ε). Observe that, for any k > 0,

|Ω

≤ c

ε , (IX.4. 39)

with c

= c

(n, k, Ω). Moreover, from the embedding Theorem II.3.4,

614 IX Steady Navier–Stokes Flow in Bounded Domains

|w(x)| ≤ c

k∇wk

2,2

k∇wk

≤ c

kwk

2,2

(IX.4.40)

which, by Lemma IX.4.1, in turn impl ies

k∇wk

+ |w(x)| ≤ c

1/2,2(∂Ω)

. (IX.4.41)

Thus, (IX.4.38), along with (IX.4.41), (IX.4.29) and (II.3.7), gives for a ll u ∈

(Ω)

k|u||V

≤

εkv

∗

1/2,2(∂Ω)

kuδ

−1



Ω

|∇w|



1/2

≤ c



εkv

∗

1/2,2(∂Ω)

kuδ

−1

+ |u|

1,2

k∇wk

3,Ω



(IX.4.42)

By Lemma III .6.3, we have

kuδ

−1

≤ c

|u|

1,2

while, by H¨older inequality and (IX.4.41),

k∇wk

3,Ω

≤ |Ω

k∇wk

4,Ω

≤ c

|Ω

∗

1/2,2(∂Ω)

Thus, from (IX.4.39) and (IX.4.42), we infer

k|u||V

≤ c

εkv

∗

1/2,2(∂Ω)

|u|

1,2

, (IX.4.4 3)

where c

= c

(n, Ω). Fix arbitrary η > 0 and choose

ε ≤

∗

1/2,2(∂Ω)

. (IX.4.44)

From (IX.4.43), (IX.4. 44), Lemma IX.2.1, and the Schwarz inequality we then

conclude that

|a(u, V

, u)| = |a(u, u, V

)| ≤ η |u|

1,2

. (IX.4.45)

Collecting (IX.4.33), (IX.4.36), (IX.4.37), and (IX.4.45) yields

|a(u, V , u)| ≤



η +

m+1

i=1



4cκ

kσ

2,Ω

i,d

+ κ

4κ

kσ

4,Ω

i,d



|Φ



|u|

1,2

(IX.4.46)

which coincides with (IX.4.25) if m > 0. If m = 0 the proof is sim pler,

since, then, one can take V = V

and proceed as before to arrive formally at

(IX.4.46) with identically vanishing Φ

. The ﬁrst part of the lemma is therefore

proved. In order to show (IX.4.26), we observe that, from (IX.4.32), (IX.4.33),

it o bviously follows that

IX.4 Existence and Uniqueness with Nonhomogeneous Boundary Data 615

1,2

+ kbk

1,2

≤ c

∗

1/2,2(∂Ω)

. (IX.4.47)

It remains to give an analogous estimate for V

. To this end, we notice that,

under the stated hypothesis on v

∗

, (IX.4.44) is certainly satisﬁed if we choose

ε = η/(c

M) ≡ ε

. Thus, from (IX.4.30) and the properties of the function

, we easily obta in that

1,2

≤ c



kwk

2,2

+ kw/(δ

+ δ)k

2,Ω

+ k∇w/δk

2,Ω



where

Ω



x ∈ Ω : γ

(ε)/(2κ

) ≤ δ(x) ≤ 2γ(ε)



and c

= c

(n, Ω, ε

). Consequently, from (IX.4.14 ) and the last two displayed

equations we deduce

1,2

≤ c

∗

1/2,2(∂Ω)

(IX.4.48)

where c

= c

(n, Ω, η, M). The proof of the lemma i s completed.

Remark IX.4.7 It is simple to generalize Lemma IX.4.2 to dimension n ≥ 4,

provided we make appropriate changes in the proof just given. Actually, it

suﬃces to use, instead of (IX.4.34), the Sobol ev inequality (II.3.7), to take

the ﬁeld b as solution to the following problem

∇ · b = h

b ∈ W

1,n/2

(Ω)

|b|

1,n/2

≤ ckhk

n/2

and, ﬁnally, to choose

= ∇ · (ψ

W )

as an extension of the ﬁeld v

, with W deﬁned in Remark IX.4.5. However, in

order that V

satisﬁes the estimate needed in the lemma, we should require

that v

∗

has slightly more regularity. Actually, in dimensions higher than three,

(IX.4.40) need not hold and we have, instead,

|W (x)| ≤ c

kW k

2,q

k∇W k

≤ c

kW k

2,q

)

q > n/2, s > n; (IX.4.49)

see Theorem II.3.4. On the other hand, taking into account Remark IX. 4.6, the

right-hand side of (IX.4.49) is ﬁnite provided v

∗

∈ W

1−1/q,q

(∂Ω), q > n/2.

Therefore, if n ≥ 4, under thi s additional condition on v

∗

the vector ﬁeld

(IX.4.33) belongs to W

1,q

(Ω),

and satisﬁes (IX.4 .45). One can then show

that the trilinear fo rm a(u, V , u) satisﬁes the estimate

Notice that, since h ∈ L

(Ω), for all r > 1, we may take b ∈ W

1,q

(Ω); see Remark

III.3.4.