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

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206 III The Function Spaces of Hydrodynamics

with Φ

independent of x

and so

|Φ

≡ |Φ

∞

(1−n)(q−1)

)dx

≤ Ckvk

. (III.4.19)

Therefore, if, for some ﬁxed q and n, there are m −1 integrals β

that diverge

we conclude Φ

= 0 for all i = 1, . . . , m and so,

(Ω) = H

(Ω). Actually,

if β

= ∞ for i = 1, . . . , m − 1 (say) we have Φ

= 0 for the corresponding

ﬂuxes, but since v is solenoidal in the whole of Ω we a lso have Φ

= 0. The

circumstance just described happens whenever 1 ≤ q ≤ n/(n − 1). In fact we

have the f ollowing result that represents, in the pa rti cular case of the domain

considered by us, a much more general one due to Maslennikova & Bogovski

(1983, Theorem 5) and that we recalled just before subsection (a).

Theorem III.4.5 Let Ω be as in Lemma III.4 .1 with m ≥ 2. Then

(Ω) =

(Ω) for all q ∈ [1, n/(n −1)].

Proof. By assumption (ii) made on f

), for x

suﬃciently large (> x

, say)

it holds that

)| ≤ Cx

and so

≥ C

∞

(1−n)(q−1)

)dx

= ∞, for all q ∈ [1, n/(n −1)],

and the proof i s achieved. ut

Suppose now that there are at least two of the quantities β

(β

and β

, say)

that are ﬁnite. The ﬂuxes Φ

and Φ

, then, need not be zero and so the spaces

(Ω) and H

(Ω) may not coincide. As a matter of fact, they are distinct

and dim[

(Ω)/H

(Ω)] is just the number of β

, which are ﬁnite minus one.

This result is due to Pileckas (1983, Theorem 7) for space dimension n = 2, 3.

Here, coupling the ideas of Maslennikova a nd Bogovski

i (1978) and those

of Ladyzhenskaya & Solonni kov (1976) we extend it to arbitrary dimension

n ≥ 2. We commence by introducing an auxiliary function.

Lemma III.4.2 Let f be a function satisfying properties (i) and (ii) of

Lemma III.4.1. Then, given σ < 1/2, there exists a function δ ∈ C

∞

)

such that, for all t > 0 and m ≥ 0 we have

(i) σf(t) ≤ δ(t) ≤ f(t),

(ii) |d

δ(t)/dt

| ≤ c/f

m−1

(t),

where c = (m, M ).

Proof. Let {t

} be a sequence of numbers such that

k+1

= t

+ αf(t

), t

= 0,

III.4 The Spaces H

207

with α a positive parameter to b e ﬁxed later. Let ϕ(ξ) denote a C

∞

function

on R with 0 ≤ ϕ ≤ 1 and

ϕ(ξ) =

(

1 if |ξ| ≤ 1/2

0 if |ξ| > ε + 1/2

for som e po sitive ε. Setting

(t) = ϕ((t − c

)/`

)

ε = η/`

, η > 0,

with

= (t

k+1

+ t

)/2

= t

k+1

− t

and η > 0 (to be ﬁxed later in the proof) we verify at once that

(t) =











0 if t < t

− η

1 if t

< t < t

k+1

0 if t > t

+ η

(III.4.20)

and

(t)/dt

| ≤ c/f

), (III.4.21)

where c = c(α, m). We now choose

η <

min

so that

+ η < t

k+1

− η,

and set

δ(t) =

∞

k=0

f(t

)ϕ

(t), (III.4.22)

with ϕ

(t) ≡ 0. Given t ≥ 0, we have three possibilities:

(a) t ∈ [t

− η, t

)

(b) t ∈ [t

, t

+ η)

+ η, t

k+1

−η),

for som e k ≥ 0. Correspondingly, (III.4.20) and (III.4.21) furnish

δ(t) = f(t

) + ϕ

k+1

(t)f(t

k+1

) in case (a)

δ(t) = f(t

)ϕ

(t) + f(t

k+1

) in case (b)

δ(t) = f(t

k+1

) in case (c).

(III.4.23)

208 III The Function Spaces of Hydrodynamics

By the properties of the function f, we obtain in case (a)

f(t) ≤ f(t

) + Mη, f(t

) ≤ f(t) + Mη (III.4 .24)

and

f(t

k+1

) ≤ f(t) + M|t

k+1

−t

|+ Mη = f(t) + αMf(t

) + Mη

and so, by (III.4.24),

f(t

k+1

) ≤ (1 + αM)f( t) + (αM

+ M)η. (III.4.25)

On the other hand,

f(t) ≤ f(t

k+1

) + αM f(t

) + Mη. (III.4.26)

Because

f(t

) ≤ f(t

k+1

) + αM f(t

by choosing

α = γ/M, γ < 1,

it follows that

f(t

) ≤ f(t

k+1

)/(1 − γ)

and (III.4.26) yields

f(t) ≤ f( t

k+1

)/(1 + γ) + M η. (III.4.27)

Selecting η = βf

/M, from (III. 4.24), (III.4.25), and (III.4.27) we derive

f(t)

1 + β

≤ f(t

) ≤ (1 + β)f(t)

(1 − γ)f(t)

1 + β(1 − γ)

≤ f(t

k+1

) ≤ (1 + γ)(1 + β)f(t)











(in case (a)). (III.4.28)

Therefore,

f(t)

1 + β

≤

δ(t) ≤ (1 + β)(2 + γ)f(t) (in case (a)). (II I .4.29)

Likewise, one shows

f(t)

1 + β

≤ f(t

) ≤ (1 + β)f(t)

(1 − γ)f(t) ≤ f(t

k+1

) ≤ [1 + γ(1 + β)]f(t)











(in case (b)) (III.4.30)

and so

III.4 The Spaces H

209

(1 − γ)f(t) ≤

δ(t) ≤ [2 + β + γ(1 + β)]f(t) (in case (b)). (III.4.31)

Finally, by (III.4.26) and by the properties of f we easily deduce

(1 −γ)f(t) ≤ f(t

k+1

) ≤ (1 − γ)f(t) (in case (c)), (III.4.32)

which gives

(1 − γ)f(t) ≤

δ(t) ≤ (1 − γ)f(t) (in case (c)). (III.4.33)

Collecting (I II.4. 23), (III.4.29), (II I .4.31), and (III.4.33) we ﬁnd

f(t) ≤

δ(t) ≤ σ

f(t) for all t ≥ 0 (III.4.34)

with

= min{1/(1 + β), 1 − γ}, σ

= (1 + β)(2 + γ).

Furthermore, from (III.4.21), (III.4.23), (III. 4.28), (III.4.30), and (III.4.32)

there fol lows

δ(t)/dt

| ≤

m−1

(t)

with c = c(m, M). Therefore, noting that σ ≡ σ

/σ

(< 1/2) can be chosen

as close to 1/2 a s we please by taking β and γ suﬃciently close to zero, we

may conclude tha t the function δ(t) =

δ(t)/σ

satisﬁes all requirements of the

lemma , which is therefore completely proved. ut

For β ∈ (0, 1), set

(β) = {x ∈ Ω

: βf

) < |x

| < f

)}.

We have

Lemma III.4.3 Let Ω

be a body of rotation of type (III.4.14 ). Then there

exists a vector b

∈ C

∞

(Ω

) such that for all x ∈ Ω

and all |α| ≥ 0,

(x)| ≤ cf

−n+|α|+1

)

∇ · b

(x) = 0

· n = 1,

(III.4.35)

where n is the unit no rmal to Σ

in the direction of increase of the coordinate

and c = c(n, α, M). Moreover, b

vanishes in ω

(β), for any ﬁxed β ∈ (0, 1).

Proof. Let ψ ∈ C

∞

(R), 0 ≤ ψ ≤ 1, with ψ(t) = 1 if t ≤ 1 and ψ(t) = 0 if

t ≥ 2, and set

Ψ(x) = ψ





)



210 III The Function Spaces of Hydrodynamics

where δ

is the function associated to f

by the preceding lemm a. We imme-

diately recognize that

Ψ(x) =

(

1 if |x

| ≤

βδ

)

0 if βδ

) ≤ |x

and therefore, by property (i) of the function δ

, Ψ identically vanishes in

(β). The ﬁeld b

is then deﬁned as

(x) =

)Ψ(x)

)

, k = 1, . . ., n − 1

(x) =

Ψ(x)

n−1

)

(III.4.36)

where

indicates diﬀerentiation with respect to x

. It is easy to check that

satisﬁes (III.4.35)

and vanishes in ω

(β) for any ﬁxed β ∈ (0, 1). More-

over, diﬀerentiating (III.4.36) and taking into account Lemma III.4.2, we also

deduce (III.4.35)

. Finally, setting

= x

/δ

), k = 1, . . ., n −1,

we obtain

)

(x) · n = 2ω

n−1

)

Ψ(ξ)|ξ|

n−2

d|ξ| > 0

with ω

n−1

the measure of the (n −1)-dimensional unit ball. Thus, (III.4.35)

also follows, after possible multiplication of b

by a sui table constant. The

lemma i s proved. ut

We are now in a position to prove the following results complementing

those of Theorem III.4.5.

Theorem III.4.6 Let Ω be as in Lemma III.4.1, m ≥ 2. Assume that the

integrals

∞

(1−n)(q−1)

(t)dt (III.4.37)

converge for i = 1, . . ., ` (≤ m), and diverge for i = `+1, . . . , m. Then, setting

(Ω) =

(Ω) / H

(Ω),

for q ∈ (n/( n −1), ∞) it holds that

dim K

(Ω) = ` − 1.

In addition, if ` ≥ 2, every v ∈

(Ω) can be uniquely represented as

III.4 The Spaces H

211

v = u +

`−1

i=1

where u ∈ H

(Ω). Here,

j=1

and Φ

, j = 1, . . . , `, with

j=1

= 0, are given by

v · n,

with n normal to Σ

in the direction o f increase of the coordinate x

in Ω

Moreover, {d

} is a basis in K

(Ω) constituted by C

∞

(Ω) vector ﬁelds that

vanish in a neighborhood of ∂Ω, in ω

(β), i = 1, . . ., `, for all suﬃciently

small β and in Ω

, i = ` + 1, . . ., m, for suﬃciently large |x|. Finally, for every

j = 1, . . ., ` and i = 1, . . . , ` − 1, the vectors d

satisfy a n estimate of type

(III.4.35)

along with the relations

· n = δ

−δ

i+1j

Proof. For i = 1, . . . , `, let S

be the i ntersection of Ω

with the pla ne x

= 1

(in the coordinate system to which Ω

is referred) and consider the open ball

), with x

intersection of S

with the axis x

= 0. We take

d < min



inf

t>0

(t), 1



so that B

) is contained in Ω

, while we m ay assume B

) strictly con-

taining S

∩ supp (b

) ≡ σ

, where b

are the vectors constructed in Lemma

III.4.2. This condition i s easily achieved by selecting β in such a way that

βf

(2) < (1/2)d,

which, in parti cular, imply

⊂ B

d/2

Let ψ

(x) be an inﬁnitely diﬀerentiable function that is one in B

3d/4

) and

is zero outside B

). Evidently, ψ

(x) is one near σ

. Setting

= {x ∈ Ω

: 0 < x

< 1}

and

212 III The Function Spaces of Hydrodynamics

(x) =

(

(x) if x ∈ Ω

−K

≡ C

(x)b

(x) if x ∈ Ω − C

(III.4.38)

we see that u

∈ C

∞

(Ω) and that supp (u

) ⊂ Ω

. Moreover, by the assump-

tion made on the integrals (III.4.37) and by (III.4.35)

we have u

∈ W

1,q

(Ω).

For i = 1, . . ., ` − 1 deﬁne

= u

− u

i+1

+ r

i,i+1

, (III.4.39)

where r

i,i+1

satisﬁes

∇ · r

i,i+1

= −∇ ψ

· u

+ ∇ψ

i+1

· u

i+1

in Γ ≡ ∪

i=1

. (III.4. 40)

Since Γ is locally Lipschitz and, moreover, by (III.4.3 5)

(−∇ψ

· u

+ ∇ψ

i+1

·u

i+1

) =

· n −

i+1

· n = 0

we may use Theorem III.3.3 to establish the existence of a solution r

i,i+1

(III.4.40). Actually, since the right-hand side of (III.4.4 0) belongs to C

∞

(Γ ),

we may take r

i,i+1

∈ C

∞

(Γ ). Therefore, the ﬁelds (III.4.39) are solenoidal,

belong to W

1,q

(Ω) ∩ C

∞

(Ω), vanish in a neighborhood of ∂Ω, and coincide

in Ω

∩ {x

≥ 1}, i = 1, . . ., `, with the ﬁelds b

. Furthermore, for every

j = 1, . . . , `

· n =

− b

i+1

) ·n = δ

− δ

i+1j

and Lemm a III.4.1 im plies, in particular,

∈

(Ω) and d

6∈ H

(Ω),

so that

(Ω) 6= H

(Ω).

Given now v ∈

(Ω), consider the vector

u = v −

`−1

i=1

where

j=1

Recalling that n denotes the normal to Σ

in the direction of the increase of

the coordinate x

in Ω

, we ﬁnd for all k = 1, . . . , `

u · n = Φ

−

`−1

i=1

(δ

−δ

i+1k

) = Φ

−

j=1

k−1

j=1

= 0.

III.4 The Spaces H

213

Since the ﬂux of u through the remaining exits Ω

, i = ` + 1, . . . , m, is zero

by assumption and so, from Lemma III.4.1 we deduce u ∈ H

(Ω). Moreover,

the vectors {d

} are linearly independent, which completes the proof of the

theorem. ut

Remark III.4.3 We wish to point out a noteworthy generalizati on of Theo-

rem III.4.6. In fact, it is not necessary to suppose the “exits” Ω

to be bodies

of rotati on but, rather, we can assume the foll owing conditions (Pileckas 1983,

p. 153; Sol onnikov 1981):

(a) D

⊂ Ω

⊂ D

, where, in possibly diﬀerent coordinate systems:

= {x ∈ R

: x

> 0, |x

| < f

)}

= {x ∈ R

: x

> 0, |x

| < a

), a

> 1};

(b) In the domains:

Ω

= Ω

∩{R < x

< R + f

(R)}

problem (III.4 .17) is sol vable with a constant c independent of R.



Remark III.4.4 Two f urther signi ﬁcant contributions to the problem of the

coincidence of the spaces

(Ω) and H

(Ω) are due to Solonnikov (1 983)

and to Maslennikova & Bogovski

i (1983). In the ﬁrst one (see Theorems 2.5

and 2.6) results are given avoiding assumptions on the shape of the “exits”

and imposing only some general restrictions. In the second one, the authors

provide necessary conditions and suﬃcient conditions for the above coinci-

dence to hold, in a class of domains with noncompact boundary which are

only requested to be strongly locally Lipschitz (see footnote 3 in thi s section).



Exercise III.4.4 Let Ω be a domain of the type considered in Exercise III. 4.1.

Assume Σ

) = Σ

, i = 1, . . . , N, where each section Σ

is a locall y Lipschitz

simply connected domain in R

n−1

independent of x

and

(Ω

) = H

(Ω

), 1 <

q < ∞, for all suﬃciently large R. Show

(Ω) = H

(Ω), 1 < q < ∞.

Exercise III.4.5 (Heywood 1976) Show that for any domain Ω ⊆ R

, n ≥ 2, we

have

(Ω) = H

(Ω) if and only if the only vector v ∈

(Ω) such that

Ω

(∇v : ∇ϕ + v · ϕ) = 0 for all ϕ ∈ D(Ω)

is the null vector.

214 III The Function Spaces of Hydrodynamics

III.5 The Spaces D

1,q

In this section we shall investigate the relevant properties of certain func-

tion spaces which, among other thing s, play a fundamental role in the study

of steady motions of a viscous incompressible ﬂuid in unbounded domains.

These spaces, denoted by D

1,q

(Ω), are subspaces of D

1,q

(Ω) deﬁned as the

completion of functions from D(Ω) in the norm of D

1,q

(Ω). If Ω is contained

in some ﬁnite layer, then, by inequality (II. 5.1) we have D

1,q

(Ω) = H

(Ω),

otherwise D

1,q

(Ω) ⊂ H

(Ω).

As in the case of spaces H

(Ω), it is of great interest to relate D

1,q

(Ω)

with the space

1,q

(Ω) =

v ∈ D

1,q

(Ω) : ∇ · v = 0 in Ω

and to p oint out domains Ω for whi ch

1,q

(Ω) 6= D

1,q

(Ω), for some q = q(n)

(in general, D

1,q

(Ω) ⊂

1,q

(Ω), for a ny domain Ω). This is because, as shown

for the ﬁrst time by Heywood (1976), whenever the coincidence does no t

hold, the “traditiona l” boundary-value problem for Stokes and Navier–Stokes

equations must be supplemented with appropriate extra conditions in order

to take into account physically important solutions that would otherwise be

excluded. Since for a bounded domain we have

1,q

(Ω) = H

(Ω),

1,q

(Ω) =

(Ω)

it follows tha t, in such a case, that result proved in Theorem III.4.1 also holds

for D

1,q

-spaces. Using this fact, we can then repeat verbatim the proofs of

Theorem III.4.1 and Theorem III.4.3 and show that they continue to be true

also for D

1,q

-spaces. We thus have the following theorem.

Theorem III.5.1 If Ω is a bounded domain satisfying the assumption of

Theorem III.4.1, then, for all q ∈ [1, ∞),

1,q

(Ω) =

1,q

(Ω) . (III.5.1)

If Ω is an exterior domain satisfying the assumption of Theorem III.4.2, then

(III.5.1) holds for all q ∈ (1, ∞). Finally, D

1,q

) =

1,q

), f or all q ∈

(1, ∞).

Assume now Ω with a noncompact boundary and having m “exits” Ω

to inﬁnity. Then, o ne can show results similar to those established in Sub-

section 4(c) for spaces H

(Ω), even though diﬀerent in some details. Pre-

cisely, one shows that if each Ω

contains a semi-inﬁnite cone, then

1,q

(Ω) 6=

1,q

(Ω) for all q > 1. Moreover, if Ω enjoys some further properties, then

dim



1,q

(Ω) / D

1,q

(Ω)



= m − 1 (Ladyzhenskaya & So lonnikov 1976, The-

orem 4.2).

Actually, the proof given by t hese authors is for q = 2. N evertheless, mutatis

mutandis it can be easily extended to all q > 1.

III.5 The Spaces D

1,q

215

In the special case when the domains Ω

are bodies of rotation deﬁned by

suitable functions f

, one can give a complete description of the coincidence of

the two spaces in a way similar to that used in Subsection 4(c) for the spaces

(Ω). In pa rti cular, the proof of Lemma III. 4.1 remains unchanged to show

the following result.

Lemma III.5.1 Let Ω be as in Lemma III.4.1. Then

1,q

(Ω) = D

1,q

(Ω),

1 < q < ∞, if and only if all vectors v ∈

1,q

(Ω) satisfy (III.4.15).

However, there is a modiﬁcation in the characterization f urnished by The-

orem III.4.6, in that the condition imposed on the functions f

must be appro-

priately changed. This is due to the fact that now v has only ﬁrst derivatives

in L

. Thus, using the notations of Subsection 4(c), with the aid of the H¨older

inequality and inequality (II.5.5), the ﬂux Φ

can be increased as follows:

|Φ

≡



)

v ·n



≤ C |Σ

q−1+q/(n−1)

)

|∇v|

≤ C

(n−1) (q−1)+q

)

|∇v|

and, therefore, we ﬁnd

|Φ

∞

(1−n)(q−1)−q

)dx

≤ C

|v|

1,q

So the vanishing of the ﬂuxes Φ

is this time related to the ﬁniteness of the

integrals

∞

(1−n)(q−1)−q

(t)dt

instead of integrals (III. 4.37). Consequently, Theorem III.4.6 is replaced by

the following one whose proof, which follows exactly the same lines as those

of Theorem III.4.6 , we leave to the reader as an exercise.

Theorem III.5.2 Let Ω be as in Lemma III.4.1, m ≥ 2. Assume that the

integrals

∞

(1−n)(q−1)−q

(t)dt (III.5.2)

converge for i = 1, . . . , ` (≤ m) and diverge for i = ` + 1, . . . , m. Then, setting

1,q

(Ω) =

1,q

(Ω) / D

1,q

(Ω),

for q ∈ (1, ∞),

dim F

1,q

(Ω) = ` − 1.

In addition, every v ∈

1,q

(Ω) can be uniquely represented as