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

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156 III The Function Spaces of Hydrodynamics

Therefore, i n view of Gagliardo’s Theorem II.4.3, we deduce u ·n = 0 at ∂Ω.

Conversely, assume ∇·u = 0 in Ω and u ·n = 0 at ∂Ω and take an arbitrary

h ≡ ∇φ ∈ G

(Ω). By L emma II.6.1, it follows that φ ∈ W

1,q

(Ω) (this is no

longer true if Ω is unbounded

) and from (III.2.2 ) we recover (III.2.1), which

implies u ∈ H

(Ω).

If Ω is a locally Lipschitz exterior domain, using (III.2.2) with ϕ ∈ C

∞

(Ω)

and (III.2.1) we can prove as in the previous case that u ∈ H

(Ω) impl ies

∇·u = 0 in Ω and u·n = 0 at ∂Ω. To prove the converse relation, however, we

should argue in a slightly more complicated way. Let u be a smooth solenoidal

function of L

(Ω), with u · n = 0 at ∂Ω a nd let ψ

be the “cut-oﬀ” function

(II.7.1). G iven an arbitrary φ ∈ D

1,q

(Ω), we can replace ϕ = ψ

φ into

(III.2.2) to ﬁnd

Ω

(u · ∇φ) = −

Ω

φ∇ψ

· u.

Since

lim

R→∞

Ω

(u ·∇φ) =

Ω

u · ∇φ,

in view of Lemma III.2.1 we w ill show u ∈ H

(Ω) if we prove

lim

R→∞

Ω

φ∇ψ

· u = 0. (III.2.3)

Now, by the H¨older inequality we ﬁnd



Ω

φ∇ψ

· u



≤ kuk

kφ∇ψ

Ω

where

Ω

is deﬁned in (II.7.3). The qua ntity kφ∇ψ

Ω

formally coincides

with (II.7.5) with the replacements u → φ, q → q

and so, proceeding as in the

part of the proof of Theorem II.7.1 that follows (III.6.5), we establish (III.2.3)

and the proof i s accomplished.

The above considerations are summari zed in the following.

Lemma III.2.2 Let Ω be a locally Li pschitz domain of R

, n ≥ 2, and let

u ∈ C

(Ω) ∩ L

(Ω), 1 < q < ∞ . Then u ∈ H

(Ω) if and o nly if ∇ · u = 0 in

Ω and u ·n = 0 at ∂Ω.

Remark III.2.1 By an argument entirely analogo us to that just shown, one

can prove that the result of Lemma III.2.2 continues to hold f or Ω a half-space.



If u is no longer assumed regular, we can nevertheless prove tha t the char-

acterization just described of the space H

(Ω) is still valid, provided we give

suitable generalizations of the deﬁnition of the trace of the normal comp onent

Take, for instance, Ω = {x ∈ R

: |x| > 1}, q = 2 and φ(x) = |x|

−1

. Then

∇φ ∈ L

(Ω) while φ 6∈ L

(Ω).

III.2 Relevant Properties of the Spaces H

and G

157

of u at the boundary and of identity (III.2.2), and provided, of course, that

the solenoidality conditio n is interpreted in the sense of weak derivatives. This

will be our next objective, that will be reached by arguments ba sically due to

Temam (1977, Chapter I, §1.3); see also Miyakawa (1982).

For q ∈ (1, ∞) let

(Ω) =

u ∈ L

loc

(Ω) : kuk

< ∞

, (III.2.4)

where

kuk

≡ kuk

+ k∇· uk

. (III.2.5)

Clearly, the functional (III.2.5) deﬁnes a norm in

and, by a simple rea-

soning, one shows that

is complete under this norm; see Exercise III.2.1.

The following result holds.

Theorem III.2.1 Let Ω be locally Li pschitz. Then, C

∞

(Ω) is dense in

(Ω), for all q ∈ [1, ∞).

Proof. Assume ﬁrst Ω bounded. From Lemma II.1.3 we ﬁnd a ﬁnite open cov-

ering of Ω, denoted by G = {G

, G

, . . ., G

}, with the following properties:

(i) G

⊂ Ω, (ii) ∂Ω ⊂ ∪

i=1

, and (iii) Ω

≡ Ω ∩ G

, i = 1, . . ., m, is a

star-shaped dom ain with respect to some interior point x

. We extend u|

Ω

to zero outside Ω

, continue to denote by u this extension. Let {ψ

}

i=0,1,...,m

be a partition of unity of Ω subordinate to G (see Lemma II.1.4), and set

= ψ

u. The result then follows if, for any ε > 0, we can ﬁnd ϕ

∈ C

∞

)

such that

− ϕ

(Ω)

< ε , for all i = 0, 1, . . ., m . (III.2.6)

In fact, setting Φ =

i=0

, and observing that

i=0

(x) = 1, x ∈ Ω,

from (III.2.6) we obtain

ku − Φk

(Ω)

≤

i=0

−ϕ

(Ω)

< (m + 1) ε .

Because of the properties of G

and of ψ

, the molliﬁer (u

)

of u

belongs to

∞

(Ω), for all suﬃciently smal l η > 0. Therefore, from (II.2.9 ) and Exercise

II.3.2, we at once obtain

− (u

)

(Ω)

→ 0 as η → 0 . (III.2.7)

We next pick i ∈ {1, . . ., m}. By means of a translation in R

, we may take

= 0 . Then, the domai n

Ω

(ρ)

= {x ∈ R

: ρ x ∈ Ω

} (III.2.8)

satisﬁes Ω

(ρ)

⊃ Ω

, for ρ ∈ (0, 1); see Exercise II.1.3. Setting

158 III The Function Spaces of Hydrodynamics

= u

(x) ≡ u(ρ x) , x ∈ Ω

(ρ)

, (III.2.9)

and recalling that ψ

∈ C

∞

), by the properties of the molliﬁer we deduce

)

∈ C

∞

), for all η > 0. Since |x| < δ(Ω

), for x ∈ Ω

, from the

continuity in the mean property (see Exercise II.2.8), given ε > 0, we can ﬁnd

ρ ∈ (0, 1) such that

ku − u

q,Ω

< ε .

Furthermore, from (II. 2.9) we also have, for some η = η(ρ) > 0,

− (u

)

q,Ω

< ε ,

so that we conclude

−ψ

)

q,Ω

≤ ku−(u

)

q,Ω

≤ ku −u

q,Ω

+ku

−(u

)

q,Ω

< 2 ε .

(III.2.10)

By the same token and by Exercise II.3.2,

k∇· (u

−ψ

)

) k

q,Ω

≤ C ku − (u

)

q,Ω

+ k∇ · u

− (∇· u

)

q,Ω

+k∇· u − ∇ · u

q,Ω

≤ C ε + ε + k∇ · u − ∇ · u

q,Ω

(III.2.11)

We now notice that, setting χ(x) = ∇ · u(x), x ∈ Ω

, by Exercise II.3.3 we

have ∇ · u

(x) = ρ χ(ρ x). As a consequence,

k∇ · u − ∇· u

q,Ω

≤ (1 −ρ)k∇ ·uk

q,Ω

+ ρ

Ω

|χ(x) − χ(ρ x)|

Thus, again by the continuity i n the mean property (see Exercise II.2.8), for

ρ suﬃciently close to 1, we deduce k∇ · u − ∇ · u

q,Ω

< ε, whi ch, along

with (III.2.7)–(III.2.11) allows us to conclude the validity of (III.2.6). This

concludes the proof when Ω is bounded. Next, assume Ω exterior, and, for

suﬃciently small η > 0, let ψ

be a “cut-oﬀ” function tha t is 1 in Ω

1/η

, 0

in Ω

2/η

and satisﬁes |∇ψ

| ≤ M η, with M i ndependent of η. It is at once

recognized that, for any u ∈

(Ω), we have u

≡ ψ

u ∈

(Ω). Given

ε > 0, we choose η such that

(1 + M η) kuk

q,Ω

1/η

+ k∇ · uk

q,Ω

1/η

< ε . (III.2.12)

Since supp (u

) ⊂ Ω

2/η

, following step by step the proof just given in the case

Ω bounded with, this time, {ψ

} partition of unity in Ω

2/η

, we show that,

corresponding to the given ε > 0, there is ϕ

∈ C

∞

) such that

−ϕ

(Ω)

< ε . (III.2.13)

Therefore, by the triangle inequality and the properties of ψ

, with the help

of (III.2.1 2) and (III.2.13), we conclude

III.2 Relevant Properties of the Spaces H

and G

159

ku −ϕ

(Ω)

≤ ku − u

(Ω)

+ ku

−ϕ

(Ω)

≤ (1 + Mη)kuk

q,Ω

1/η

+ k∇· uk

q,Ω

1/η

+ ε < 2ε

The proo f of the theorem is thus completed. ut

With the help of Theorem III.2.1 we are now able to deﬁne, suitably, the

trace of the normal component o f u ∈

(Ω) at ∂Ω, provided Ω is locally Lip-

schitz. Actually, ﬁx u ∈ C

∞

(Ω), and consider the following linear f unctional

on W

1−1/q

(∂Ω), 1 < q < ∞:

(ω) =

∂Ω

ω n ·u , ω ∈ W

1−1/q

(∂Ω).

Obviously, this functional is determined once the value of the normal compo-

nent of u at the boundary is speciﬁed. Let n· be the linear map that to each

u ∈ C

∞

(Ω) prescribes the corresponding functional F

deﬁned above; that

is,

n · u = F

Let us denote by W

−1/q,q

(∂Ω) the (strong) dual of W

1−1/q

(∂Ω), and by

h·, ·i

∂Ω

the corresponding duality pair. Using Gagliardo’s Theorem II.4.3 one

then proves that n·(·) |

∂Ω

can be extended to a bounded (linear) operator

from

into W

−1/q,q

(∂Ω). In fact, by that theorem we can extend ω to a

function ϕ ∈ W

1,q

(Ω) such that

kϕk

1,q

≤ c

kωk

1−1/q

(∂Ω)

Thus, by identity (III.2.2), the H¨older inequality, and (III.2.5) we obtain

|hF

, ωi

∂Ω

| =



Ω

(u ·∇ϕ + ϕ∇ · u)



≤ kuk

kϕk

1,q

≤ c

kuk

kωk

1−1/q

(∂Ω)

implying

kn · uk

−1/q,q

(∂Ω)

≤ c

kuk

, for all u ∈ C

∞

(Ω),

which is what we wanted to prove. Now, by the standard procedure used

to deﬁne generalized traces (see Theorem II.4.1), since, by Theorem III.2.1,

∞

(Ω) is dense in

(Ω), we m ay extend, by continuity, the map n· to

the whole of

(Ω). Moreover, the following generalization of (II.4.21) and

(III.2.2) ho lds:

hn · u, ωi

∂Ω

Ω

u · ∇ϕ +

Ω

ϕ∇· u, ϕ ∈ W

1,q

(Ω), (III.2.14)

where ω = γ(ϕ) is the trace of ϕ at ∂Ω.

The above results are summarized in the following

160 III The Function Spaces of Hydrodynamics

Theorem III.2.2 Assume Ω l ocally Lipschitz, and let u ∈

(Ω), 1 < q <

∞. Then n · u ∈ W

−1/q,q

(∂Ω) and the generalized Gauss identity (III.2.14)

holds.

After having obtained generalizations of the trace of the normal component

of a vector ﬁeld at the boundary and of (III.2.2), it is now straightforward

to obtain the desired characterization of any element u of the space H

(Ω).

In fact, by an argument completely analogous to that used in the pro of o f

Lemma I II.2.2 we prove the following.

Theorem III.2.3 Let

(Ω) = {u ∈ L

(Ω) : ∇ · u = 0 i n Ω, n · u = 0 at ∂Ω}.

Then, for any locally Lipschitz domain Ω of R

, n ≥ 2, we have

(Ω) = H

(Ω) .

Remark III.2.2 The coincidence of the spaces H

(Ω) and H

(Ω) can be

proved for any (suﬃciently smooth) domain for which identity (III.2.1 4) hol ds.

However, such a coincidence certainly does not hold for certain domains with

noncompact boundary; see Remark III.4.1. 

Exercise III.2.1 Show that the space

(Ω) endowed with the norm (III.2.5) is a

Banach space.

Exercise III.2.2 Prove the results of T heorem III.2.3 to the case where Ω = R

n ≥ 2.

Another question that will play an important role later is that of charac-

terizing the kernel of the map n·. In this regard, we have the following result,

of which Theorem III.2.3 is a special case.

Theorem III.2.4 Let Ω be a locally Lipschitz domain in R

, n ≥ 2, and let

0,q

(Ω) designate the completion of C

∞

(Ω) in the norm (III.2.5).

Then, for q ∈ (1, ∞) we have that

0,q

(Ω) =

u ∈

(Ω) : n · u = 0 at ∂Ω

. (III.2.15)

Proof. Denote by

0,q

(Ω) the space on the right-hand side of (III.2.15). It is

clear that

0,q

(Ω) is a closed subspace of

0,q

(Ω). Therefore, we only have

to show that every function from

0,q

(Ω) can be approximated by functions

from D(Ω) in the norm (III.2.5). To this end, we observe that the extension

of u ∈

0,q

(Ω) to the whole of R

, obtained by setting u = 0 outside Ω, is a n

III.3 The Problem ∇ · v = f 161

element o f

0,q

); see Exercise III. 2.3. Let us denote by u this extension.

Next, let Ω

, i = 0, . . ., m, and {ψ}

0,1,...,m

be the domains and the partition

of unity introduced in the proof of Theorem III.2 .1. Also, let Ω

(ρ)

be the

domain deﬁned in (III .2.8) but, this time, with ρ > 1, so that Ω

(ρ)

⊂ Ω

. It

is then clear that the function u

(x) = ψ

(x)u

(x), x ∈ Ω

(ρ)

, with u

deﬁned

in (I II.2.9), is of compact supp ort in Ω and belongs to

(Ω), and that its

molliﬁer, (ψ

, is in C

∞

(Ω), for all suﬃciently small η > 0. The result then

follows by using exactly, from now onward, the same procedure used in the

proof of Theorem I II.2.1. ut

Finally, it remains to investigate the prop erties of the function space

(Ω). However, we notice that m embers of G

(Ω) are gradients of func-

tions belonging to D

1,q

(Ω) and, in particular, it is easily shown that, in view

of Lemma II.6.2, G

(Ω) and

1,q

(Ω) are isomorphic via the mapping

i : u ∈ G

(Ω) → i(u) ∈

1,q

(Ω),

where i(u) is the class of functions p ∈ D

1,q

(Ω) such that u = ∇p. We may

then conclude that all relevant properties of G

(Ω) are immediately obtainable

from the analogous ones established f or the space

1,q

(Ω) in Section II.6.

Exercise III.2.3 Show that if u ∈

0,q

(Ω), with Ω locally Lipschitz, then its

extension to R

, obtained by setting u = 0 in R

− Ω, belongs to

0,q

). Hint:

Use ( III.2.14).

III.3 The Problem ∇ · v = f

In the proof of several results of this chapter we shall often consider an auxil-

iary problem whose interest goes well beyond this particular context. Actually,

we already encountered it in the proof of Theorem III.1.2, dealing with the

Helmholtz–Weyl decompositi on of the space L

(Ω).

The problem consists, essentially, in representing a scalar function as the

divergence of a vector ﬁeld in suitable function spaces and determining cor-

responding estimates. The resolution of such a problem is a fundamental tool

in several questions of mathematical ﬂuid mechanics and, therefore, we ﬁnd

it convenient to investigate it to some extent.

Let us begin to consider the case when Ω is a bounded domain in R

n ≥ 2. The problem is then formulated as follows: Given

f ∈ L

(Ω)

with

Ω

f = 0, (III.3.1)

to ﬁnd a vector ﬁeld v : Ω → R

such that

162 III The Function Spaces of Hydrodynamics

∇ · v = f

v ∈ W

1,q

(Ω)

|v|

1,q

≤ c kfk

(III.3.2)

where c = c(n, q, Ω).

Notice that (III.3.1 ) represents a compatibility condition, as a consequence

of (III.3.2)

and (I II.3.2)

. Also, since Ω is bounded, we may use the inequality

(II.5.1) into (III.3.2)

to deduce the stronger estimate

kvk

1,q

≤ c

kfk

. (III.3.3)

Problem (III.3.1), (I II.3.2) (which, of course, does not admit a unique

solution) has been studied by several authors and with diﬀerent methods (see

the Notes for this Cha pter). Here, we shall follow the approach of Bogovski

(1979, 1980) based on an explicit representation formula (see (III.3.8)), which

requires little regularity for Ω, e.g., Ω locally Lipschitz. In this latter respect

it should be emphasized that some regularity on Ω is in fact necessary for

the solvability of the problem; see Remark I II.3.9. We also point out that the

diﬃculty with (III.3.2) relies in the fact that we require that v vanishes (in a

suitable sense) at ∂Ω. If this condition is removed, resolution of the problem

is trivial; see Exercise III.3.1

To begin with, we assume that Ω is of a special shape. Speciﬁcally, we

have

Lemma III.3.1 Let Ω ⊂ R

, n ≥ 2, be star-like with respect to every point

of B

) with B

) ⊂ Ω. Then for any f ∈ L

(Ω), 1 < q < ∞, satisfying

(III.3.1), problem (III.3.2) has at least one solution v. Mo reover, the constant

c in (III.3.2)

admi ts the following estima te

c ≤ c

[δ(Ω)/R]

(1 + δ(Ω)/R), (III.3.4)

with c

= c

(n, q). Finally, if f ∈ C

∞

(Ω) then v ∈ C

∞

(Ω).

Proof. Let us assume ﬁrst f ∈ C

∞

(Ω). By the change of variables

x → x

= (x −x

)/R, (III.3.5)

we shift the origin of coordinates to the point x

and transform B

) into

(0) ≡ B. Moreover, Ω goes into a domai n Ω

that is star-like with respect

to every point o f B with

δ(Ω

) = δ(Ω)/R, (III.3.6)

while v goes into v

, f into f

and equation (III.3.2)

becomes

∇ · v

= Rf

≡ F

in Ω, (III.3.7)

where, of course, ∇ operates on the primed variables. Clearly, F

has mean

value zero in Ω

and F

∈ C

∞

(Ω

). Furthermore, if v

, F

satisfy (III.3.7),

the transformed functions v and f through the inverse of (III.3.5) satisfy

(III.3.2)

. Let now ω b e any function from C

∞

) such that

III.3 The Problem ∇ · v = f 163

(i) supp (ω) ⊂ B,

(ii)

ω = 1.

We wish to show that the vector ﬁeld

v(x) =

Ω

F (y)

x − y

|x − y|

∞

|x−y|



y + ξ

x − y

|x − y|



n−1

dξ

≡

Ω

F (y)N (x, y)dy

(III.3.8)

solves (III.3.7), where, for simplicity, we have omitted primes. By a straight-

forward calculation, we easily show that the ﬁeld v can be written in the

following equivalent useful forms

v(x) =

Ω

F (y)(x − y)



∞

ω(y + r(x −y))r

n−1



v(x) =

Ω

F (y)

x −y

|x −y|



∞



x + r

x − y

|x − y|



(|x − y| + r)

n−1



dy.

(III.3.9)

Making into the integral (III.3.9)

the change of variable z = x−y, we recover

at once v ∈ C

∞

). Moreover, from (III.3 .9)

, it follows that v is of compact

support in Ω. In fact, set

E =



z ∈ Ω : z = λz

+ (1 − λ)z

, z

∈ supp (F ), z

∈ B, λ ∈ [0, 1]



(III.3.10)

Since Ω is star-like with respect to every point of B, E is a compact subset

of Ω. Fix x ∈ Ω − E. For all y ∈ supp (F ) and all r ≥ 1,

y + r(x −y) 6∈ B

and, therefore, ω(y + r(x − y)) = 0, i.e., by (III.3.9)

, v(x) = 0. We thus

conclude

v ∈ C

∞

(Ω). (III.3.11)

Surrounding the point x ∈ Ω with a ball B

(x) of radi us ε suﬃciently small

and using integration by parts, from (III.3.8) one has

(x) = lim

ε→0

(

(x)

F (y)D

(x, y)dy+

∂B

(x)

F (y)

−y

|x − y|

(x, y)dσ

(III.3.12)

It is simple to show

The equation (III.3.8) is sometimes referred to as “Bogovski

i formula.” It is a

generalization of a similar representation due to Sobolev; see the Notes at the

end of this chapter.

164 III The Function Spaces of Hydrodynamics

lim

ε→0

|x−y|=ε

F (y)

− y

|x − y|

(x, y)dσ

= F (x)

Ω

−y

)(x

− y

)

|x −y|

ω(y)dy.

(III.3.13)

Actually, denoting by I

the integral on the left-hand side of (III.3.13),

∆

(x) ≡



(x) −F (x)

Ω

− y

)(x

−y

)

|x − y|

ω(y)dy



|z|=1



F (x −εz)

∞

ω(x + rz)(r + ε)

n−1



dσ

−F (x)

|z|=1



∞

ω(x + rz)r

n−1



dσ



and so, in the limit ε → 0 it follows

∆

(x) ≤

|z|=1

|F (x) − F (x − εz)|dσ

+ o(1),

which proves (III.3.1 3). On the other hand, the ﬁrst limit on the right-hand

side of (III.3.12) exists as a consequence of the Calder´on–Zygmund Theorem

II.11.4. To see this, we observe that from (III.3.9)

we have fo r ﬁxed y

(x, y) = D



−y

)

∞

ω(y + r(x − y))r

n−1



= δ

∞

ω(y + r(x −y))r

n−1

+(x

−y

)

∞

ω(y + r(x − y))r

|x −y|

∞



x + r

x − y

|x − y|



(|x − y| + r)

n−1

− y

|x − y|

n+1

∞



x + r

x − y

|x − y|



(|x − y| + r)

(III.3.14)

By expanding the powers of n in the last two integrals it easily follows that

(x, y) can be decomposed as

(x, y) = K

(x, x − y) + G

(x, y), (III.3.15)

where

III.3 The Problem ∇ · v = f 165

(x, x − y) =

|x − y|

∞



x + r

x − y

|x − y|



n−1

−y

|x −y|

n+1

∞



x + r

x − y

|x − y|



≡

(x, x − y)

|x − y|

(III.3.16)

while G

admi ts an estimate of the type

(x, y)| ≤ c

δ(Ω)

n−1

|x −y|

n−1

, x, y ∈ Ω, (III.3.17)

where c = c(ω, n). It is readily seen that, for each i and j, K

(x, z) is a

singular kernel, i .e., that k

(x, z) satisﬁes all conditions (II.11.15)–(II.11.17).

Actually, (II.11.15) is at o nce satisﬁed. Concerning (II.11.17), we notice that

(x, z)| ≤



∞

ω (x + rz) r

n−1



∞

ω (x + rz) r



≤ kωk

∞

δ(Ω)

+ kD

ωk

∞

δ(Ω)

n+1

n + 1

for |z| = 1 .

(III.3.18)

Therefore, also (II.1 1.17) is sati sﬁed. Furthermore,

|z|=1

(x, z) = δ

|z|=1

∞

ω(x + rz)r

n−1

|z|=1

∞

ω(x + rz)r

[δ

ω(x + y) + y

ω(x + y)] dy = 0

and so condition (II.11.16) is satisﬁed as well. Consequently, from (III.3.15)–

(III.3.17), the ﬁrst limit on the right-hand side of (III.3 .12) exists and (III. 3.12)

can b e rewritten as

(x) =

Ω

(x, x − y)F (y)dy +

Ω

(x, y)F (y)dy

+F (x)

Ω

−y

)(x

− y

)

|x − y|

ω(y)dy

≡ F

(x) + F

(x),

(III.3.19)

where the ﬁrst integral has to be understood in the Cauchy principal value

sense. We next show that (III.3.8) is a solution to (III.3.7). To this end, from

(III.3.12)–(III.3.14) and property (ii) of ω we have