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

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166 III The Function Spaces of Hydrodynamics

∇ · v =

Ω

F (y)



∞

ω(y + r(x −y))r

n−1

i=1

∞

−y

ω(y + r(x −y))r

i=1

F (x)

Ω

− y

)(x

− y

)

|x −y|

ω(y)dy

Ω

F (y)



∞

ω(y + r(x −y))r

n−1

∞



ω(y + r(x − y))





dy + F (x)

= −ω(x)

Ω

F (x) + F (x)

and so, since F has mean value zero over Ω,

∇ · v(x) = F (x), x ∈ Ω, (III.3.20)

which proves (III.3.7). It remains to show that v satisﬁes (III.3.2 )

. For 1 <

q < ∞, from (III.3.1 6) and (III.3.19), by the Calder´on–Zygmund Theorem

II.11.4 we obtain

≤ c

kF k

while Young’s inequality (II.11 .2) and (III.3.15)

furnish

≤ c

δ(Ω)

kF k

Finally, we obviously have

≤ c

δ(Ω)

kF k

We wish to emphasize that the constants c

and c

depend on ω, n, q but

not on Ω. As far as the constant c

is concerned, from (III.3.18) and Remark

II.11.2 we obtain

≤ c

δ(Ω)

(1 + δ(Ω)),

where c

= c

(n, q, ω). Restoring the primed notation, from the previous in-

equalities we recover

1,q ,Ω

≤ c

δ(Ω

)

(1 + δ(Ω

))kF

q,Ω

with c

= c

(n, q). Coming back to the original variables via the inverse

of transformation (III.3.4), recalling (III.3.6) and F

= Rf

, we obtain that

III.3 The Problem ∇ · v = f 167

the transformed solution v also satisﬁes (I II .3.2)

with a constant c obeying

(III.3.4). To complete the proof, we have to show solvability for arbitrary f

in L

(Ω) (obeying, of course, (III.3.1)). Thus, let f ∈ L

(Ω) satisfy (III.3.1)

and let {f

} ⊂ C

∞

(Ω) be a sequence approximating f in L

(Ω). Then, the

functions

∗

= f

− ϕ

Ω

, m ∈ N

with

ϕ ∈ C

∞

(Ω),

Ω

ϕ = 1

still approximate f in L

(Ω) and, at the same time, they obey (III.3.1) for

all m ∈ N. By what we have just shown, corresponding to each m ∈ N we

can ﬁnd a solution v

∈ C

∞

(Ω). By the estimate (III.3.3) and the linearity

of problem (III.3.2)

1,2

, as m → ∞ the sequence {v

} converges (strongly)

in W

1,q

(Ω) to a function v ∈ W

1,q

(Ω) that obeys (III.3.2)

1,3

in the sense of

generalized diﬀerentiation. The lemma i s therefore proved. ut

Remark III.3.1 The result just shown admits of a straightforward general-

ization to the case when f ∈ L

(Ω) ∩ L

(Ω), 1 < q, r < ∞. Speciﬁcally, one

easily shows that there exists a solution to (III.3.2)

, which further satisﬁes

v ∈ W

1,q

(Ω) ∩ W

1,r

(Ω)

|v|

1,q

≤ c kfk

|v|

1,r

≤ c kfk



Remark III.3.2 Formula (III.3.8) allows us to obtain solutions to (III.3.1)

and (III.3.2 ), in a dom ain Ω star-like with respect to a ball in the sense

speciﬁed in Lemma III.3.1, that satisfy estimates of the type (III.3.3) in

Sobolev spaces W

m,q

(Ω) of arbitrary order. To show this, for two multi-indices

α = (α

, . . . , α

), β = (β

, . . ., β

), we set β ≤ α to mean β

≤ α

for all

i = 1, . . . , n and, in such a case, we put

α−β

≡

∂

|α|−|β|

∂x

−β

. . .∂x

−β

≡

. . .

Applying the operator D

to both sides of (III.3.8), integrating by parts, and

using the Leibnitz rule we then ﬁnd for F ∈ C

∞

(Ω)

v(x) =

β≤α

Ω

(x, y)D

α−β

F (y)dy, (III.3.21)

where

168 III The Function Spaces of Hydrodynamics

(x, y) = (x − y)

∞

ω(y + r(x −y))r

n−1

dr. (III.3.22)

Taking into account that N

has the same properties as N, we apply the

same reasonings employed before to deduce the fol lowing inequality for all

f ∈ C

∞

(Ω)

k∇vk

`,q

≤ c kfk

`,q

(III.3.23)

for all ` ≥ 0 and q ∈ (1, ∞), where c satisﬁes an estimate of the type (II I.3.4).

Using (III.3.15) along with a density argument o f the type adopted in the last

part of the proof of Lemma III.3.1, we thus obtain, i n particular, a solution v

to (III.3.1) and (III.3.2) for any f in W

m,q

(Ω). Such a v belongs to W

m+1,q

(Ω)

and satisﬁes (III.3.23) for all ` = 0, . . ., m. 

Remark III.3.3 In several applications, the function f depends on a param-

eter t ∈ I, where I is an interval in R. In such a case, assuming that Ω is

star-like with respect to a ball, o ne immediately obtains from the representa-

tion (III.3.8) and the more general (III.3.21), (III.3.22), that if f(t) ∈ C

∞

(Ω),

t ∈ I, is continuous in t in the W

m,q

-norm, then the corresponding v = v(x, t)

given by (III.3.8) is continuous in the W

m+1,q

-norm and one has

kv(t

) −v(t

`+1,q

≤ c

kf(t

) −f(t

`,q

, t

∈ I , ` = 0, . . . , m .

Likewise, if f is diﬀerentiable in t, with the help of (III.3.2 1), (III.3.22), we

ﬁnd that the ﬁeld v(x, t) given by (III.3.8) is also diﬀerentiable in t and that



∇



∂v

∂t





`,q

≤ c



∂f

∂t



`,q

for all ` ≥ 0. Extension o f these results to m ore general domains will be given

in Exercise III.3.6 and Exercise III.3.7. 

Exercise III.3.1 Let Ω be an arbitrary domain in R

, n ≥ 2, and let f ∈ L

(Ω),

q ∈ (1, ∞). Show that there exists v ∈ D

1,q

(Ω) such that ∇ · v = f in Ω and

|v|

1,q

≤ c kfk

, c = c(n, q, Ω). Hint: Let {f

} ⊂ C

∞

(Ω) with f

→ f in L

(Ω).

Then, v

= (∇E ∗ f

) solves ∇ · v

= f

in Ω, and, by the Calder´on–Zygmund

Theorem II.11.4, satisﬁes |v

1,q

≤ c kf

Our next task is to extend the results of Lemma III.3.1 to the case of more

general domains. To this end, we propose

Lemma III.3.2 Let Ω ⊂ R

, n ≥ 2, be such that

Ω =

[

k=1

Ω

, N ≥ 2,

where each Ω

is a star-shaped domain with respect to some open ball B

with

⊂ Ω

, and let f ∈ L

(Ω) satisfy (III.3.1). Then, there exist N functions

such that for all k = 1, . . ., N:

III.3 The Problem ∇ · v = f 169

(i) f

∈ L

(Ω) ;

(ii) supp (f

) ⊂ Ω

;

(iii)

Ω

= 0 ;

(iv) f =

k=1

;

(v) kf

≤ C

kfk

, with

1 +

|Ω

1−1/q



1 +

|Ω

1−1/q



k−1

i=1

(1 + |F

1/q−1

−Ω

1−1/q

), k ≥ 2

and where F

= Ω

∩ D

and D

= ∪

s=i+1

Ω

Proof. Deﬁne

(x) =











f(x) −

(x)

Ω

f if x ∈ Ω

0 if x ∈ D

− Ω

(x) =











[1 − χ

]f(x) −

(x)

−Ω

f if x ∈ D

0 if x ∈ Ω

−D

(III.3.24)

with χ

characteristic function of the set F

. Clearly, it holds that

f = f

+ g

Ω

= 0

supp (f

) ⊂ Ω

, supp (g

) ⊂ D

∈ L

(Ω

) , g

∈ L

By the same token, we split g

= f

+ g

with f

and g

belonging to L

(Ω

) and L

), respectively, and satisfying

Observe that, since Ω is connected, we can always label the sets F

in such a

way that |F

| 6= 0, for all i = 1, ..., N − 1.

170 III The Function Spaces of Hydrodynamics

Ω

= 0

supp (f

) ⊂ Ω

, supp (g

) ⊂ D

This procedure gives rise to the following iteration scheme for the determina-

tion of the functions f

. We set g

= f and for k = 1, . . .N − 1

(x) =











[1 − χ

k−1

(x) −

(x)

−Ω

k−1

if x ∈ D

0 if x ∈ Ω

− D

(III.3.25)

with F

= Ω

∩D

and χ

characteristic function of F

; the functions f

are

then given by

(x) =











k−1

(x) −

(x)

Ω

k−1

if x ∈ Ω

0 if x ∈ D

−Ω

k = 1, . . .N − 1,

(x) = g

N−1

(x)

(III.3.26)

Relations (III.3.25) and (III.3.26) completely deﬁne the functions f

and prove

properties (i)-(iv). To show estimate (v), we o bserve that from (III.3.26), by

the H¨older inequality, for all k = 1, . . . , N

q,Ω

≤ kg

k−1

q,Ω



1 + |F

1/q−1

|Ω

1−1/q



Therefore, by estimating kg

k−1

q,Ω

from (III.3.25) in terms of kg

k−2

q,Ω

and

so on for k − 2 times, we arrive at (v). The lemma is proved. ut

Remark III.3.4 A noteworthy class of domains that satisfy the assumption

of Lemma III.3.2 is that constituted by domains Ω satisfying the cone prop-

erty. Such a property ensures that there exists a cone Γ

such that every

point x ∈ ∂Ω is the vertex of a ﬁnite cone Γ

congruent to Γ and contained

in Ω. To see this, we recall a result of Gagliardo (1958 , Teorema 1.I), which

states that every bo unded domain that satisﬁes the cone condition can be rep-

resented as the union o f a ﬁnite number of domains, each of which is locally

Lipschitz.

By virtue of Lemma II.1.3 and Exercise II.1.5, any such domain

Namely, Γ is the intersection of an open ball centered at the origin w ith a set of

the type

{λz : λ > 0, z ∈ R

, |z − y| < r}

where r > 0 and y is a ﬁxed point in R

with |y| > r.

Observe that every l ocally Li pschitz domain satisﬁes the cone condition; see Ex-

ercise III.3.2.

III.3 The Problem ∇ · v = f 171

can b e, in turn, represented as the union of a ﬁnite number of domains each

being star-shaped with respect to all points of an open ball that they strictly

contain.



Lemma III.3.1 and Lemma III.3.2 enable us to show the following result

(Bogovski

i 1980, Theorem 1 and Lemma 3).

Theorem III.3.1 Let Ω be a bounded domain of R

, n ≥ 2, such that

Ω = ∪

k=1

Ω

, N ≥ 1,

where each Ω

is star-shaped with respect to some op en ball B

with B

⊂ Ω

For instance, Ω satisﬁes the cone condition.

Then, given f ∈ L

(Ω),

1 < q < ∞, satisfying (III.3.1), there exists at l east one solution v to (III.3 .2).

Furthermore, the constant c entering inequal ity ((III.3.2)

admi ts the f ollow-

ing estimate:

c ≤ c



δ(Ω)





1 +

δ(Ω)



, (III.3.27)

where R

is the smallest radius of the balls B

, c

= c

(n, q) and C is an

upper bound for the constants C

given in Lemma III.3.2(v). Finally, if f is

of compact support in Ω so is v.

Proof. We decompose f as in Lemma III.3.2. Then, with the help of Lemma

III.3.1, we construct in each dom ain Ω

a solution v

to (III.3.2), correspond-

ing to f

, k = 1, . . . , N. If we extend v

to zero outside Ω

and recall Exercise

II.3.11, we deduce that the ﬁeld

v =

k=1

belongs to W

1,q

(Ω) and solves (III.3.2)

in the whole of Ω. Moreover, again,

from Lemma II I.3.1 and Lemma III.3.2(v), we have

kvk

1,q

≤

k=1

1,q

≤ c

k=1

≤ c Ckfk

, (III.3.28)

which completes the proof of the ﬁrst part of the theorem once we take into

account Remark III.3.4. To show the second one, fo r each Ω

consider the

corresponding domain Ω

(ρ)

, ρ ∈ (1/2, 1), introduced in Exercise II.1.3. As we

know from this exercise,

Ω

(ρ)

⊂ Ω

, for all k = 1 , . . ., N,

See Remark III.3.4 and Remark III.3.5.

172 III The Function Spaces of Hydrodynamics

and if Ω

is star-shaped with respect to every point of the bal l B

then Ω

(ρ)

enjoys the same property with respect to every point of the ball

ρR

). Let us set

Ω

(ρ)

= ∪

k=1

Ω

(ρ)

and denote by ρ

∈ (1/2, 1) a number such tha t for all ρ ∈ [ρ

, 1) the following

properties hold

Ω

(ρ)

is connected

supp (f) ⊂ Ω

(ρ)

In virtue of Lemma III.3.2, we can decompose f as the sum of N functions

(ρ)

, where f

(ρ)

satisfy the following properties:

(ρ)

∈ L

(Ω

(ρ)

), supp (f

(ρ)

) ⊂ Ω

(ρ)

Ω

(ρ)

= 0, k = 1, . . ., N.

Furthermore, taking into account that

|Ω

(ρ)

| = ρ

|Ω

|, |Ω

(ρ)

∩Ω

(ρ)

| = ρ

|Ω

∩Ω

from property (v) of Lemma III.3.2 we also have

(ρ)

≤ Ckfk

(III.3.29)

with a constant C depending on Ω

but otherwise independent of ρ ∈ [ρ

, 1).

We next solve problem (III.3.1), (III.3.2) in each Ω

(ρ)

and denote by v

(ρ)

∈

1,q

(Ω

(ρ)

) the corresponding solution. Extending v

(ρ)

by zero outside Ω

(ρ)

we obtain that the function

(ρ)

k=1

(ρ)

solves (III.3.2 )

, belo ngs to W

1,q

(Ω), and is of compact support in Ω. More-

over, proceeding as in (III. 3.28) and using (III.3.29) we recover that v

(ρ)

obeys

(III.3.2)

with a constant c depending o n n, q and Ω but independent of ρ,

namely, o f f. The theorem is completely proved. ut

Remark III.3.5 Even though the assumptio n on the regularity of Ω made

in the previous theorem may allow, in principl e, for domain even less regular

than those satisfy ing the cone condition, some kind of regularity is i ndeed

necessary fo r the solvability of problem (III.3.1)–(III.3.2); see Remark III.3.9.

For example, Ω can not have an external cusp. The question of the “least”

requirement on Ω for (III.3.1)–(III. 3.2) is studied in Acosta, Dur´an & Muschi-

etti (2006), where, in particular, fo r n = 2, q ∈ (1, 2), and Ω simply connected,

a complete characterization is furnished in terms of “John domains”; see also

the Notes for this Chapter.



III.3 The Problem ∇ · v = f 173

Remark III.3.6 Remark III.3.1 equally applies to Theorem III.3.1. 

Remark III.3.7 Theorem III.3.1 leaves out the two limiting cases q = 1, ∞.

As a matter of fact, in both cases, problem (III.3.2) does not have a solution

for all f ∈ L

(Ω) satisfying (III.3.1). A pro of of this assertion when q = ∞, is

given by Preiss (1997), McMullen (1998, Theorem 2.1) and Bourgain & Brezis

(2002, §2.2); see also Dacorogna, Fusco & Tartar (2004). A proof of the non-

solvability of problem (III.3.2) when q = 1 for arbi trary f ∈ L

(Ω) satisfying

(III.3.1), can be found in Bourgain & Brezis (200 3, §2.1) and in Dacorogna,

Fusco & Tartar (2004). The argument of Bourgain & Brezis is elementary and

will be reproduced here. Thus, assume that the problem

∇ · v = f , kvk

1,1

≤ c kfk

(III.3.30)

has at least one solution v ∈ W

1,1

(Ω), corresponding to an arbitrarily given

f ∈ L

(Ω) satisfying (III.3.1). Choose f = g −g

Ω

, where g is any function in

(Ω), and let u ∈ C

∞

(Ω). From (III.3.30) we thus have

(∇u, v) = −(u, ∇· v) = −(u, g −g

Ω

) ,

which, by a simple calculation that uses the H¨older inequality and Theorem

II.3.2, implies

|(u − u

Ω

, g)| ≤ k∇uk

kvk

n/(n−1)

≤ c k ∇uk

kvk

1,1

From this latter relati on, from (III.3.30)

, and from Theorem II.2.2 we readily

deduce

ku − u

Ω

∞

= sup

g∈L

(Ω);kgk

|(u − u

Ω

), g)| ≤ c k∇uk

which, by (II.2.6) and (II.5.1), in turn implies the following property

kuk

∞

≤ c k∇uk

, for all u ∈ C

∞

(Ω) ,

which, as we know from Exercise II.3.8, is not true . 

Remark III.3.8 If q > n, in view of the embedding Theorem II.3.2, a ny

solution v to (III.3.1)–(III.3.2), belongs, in addition, to L

∞

(Ω).

Of course,

as we know from Exercise II.3.8, this embedding does not hold if q = n and,

therefore, we can not prove, in such a case, v ∈ L

∞

(Ω), at least by this kind

of argument. However, it is simple to bring examples where, for certain f

and Ω, it is indeed po ssible to produce a solution to (III.3.1)–(III.3.2) which

is in L

∞

(Ω), under the sole assumption that f ∈ L

(Ω). For instance, let

Ω = B

, for som e R > 0, and assume that f = f(|x|), f ∈ L

). Then, by

a straightforward calculation, we prove that a solution to (III.3.1)–(III.3.2) is

given by

Actually, to C(Ω).

174 III The Function Spaces of Hydrodynamics

v(x) =











|x|

n−1

f(r)dr if 0 < |x| ≤ R ,

0 if x = 0 .

It is easy to show that

lim

|x|→0

v(x) = 0 ,

which furnishes v ∈ C(Ω). Moreover,

kvk

∞

≤





(n−1)/n

kfk

At this point it is natural to ask if such a result can be proved for more gen-

eral functions f and for (suﬃciently smooth, bounded) Ω of arbitrary shape.

The answer to this question is positive, and, i n fact, by methods completely

diﬀerent than those used here, Bourgain & Brezis (2003, Theorem 3

) have

shown the following result, for whose proof we refer to their article.

Theorem III.3.2 Let Ω be a bounded and locally Lipschitz domain of R

n ≥ 2. T hen, for any f ∈ L

(Ω) satisfying (III.3.1) there exists a solution v

to (III.3.2 ) with q = n, which, furthermore, belongs to C(Ω) and obeys the

following estimate

kvk

∞

≤ c kfk

where c = c(n, Ω).



Another interesting question is the dependence of the constant c entering

inequality (III.3.2)

on the domain Ω. For example, from (III.3.27) we deduce,

in particular, that if Ω i s a ball, c is independent of the diameter of Ω. This

is a particular case of the following l emma whose proof we leave to the reader

as an exercise.

Lemma III.3.3 Let y

= φ

(x), i = 1, . . ., n, be a transformation of R

into

itself. Then, the constant c in (III.3.2)

does not change if φ

is homothetic,

i.e.,

(x) = ax

+ b

, a, b

∈ R

or a rotation, i.e.,

(x) =

j=1

= δ

Other questions related to the solvability of (III.3.1) and (III.3.2) are left

to the reader in the following exercises.

III.3 The Problem ∇ · v = f 175

Exercise III.3.2 Show that i f Ω is locally Lipschitz then Ω satisﬁes the cone con-

dition.

Exercise III.3.3 Show that for q = 2 the constant c of inequality (III.3.2)

, in

general, cannot be less than one.

Exercise III.3.4 Let Ω be an arbitrary domain of R

, and let u ∈ L

loc

(Ω), q ∈

(1, ∞). By the Hahn-Banach Theorem II.1.7, there exists a unique A ∈ D

−1,q

(Ω)

such that

(u, div ψ) = −hA, ψi, for all ψ ∈ C

∞

(Ω) .

It is readily checked that A does not depend on q. Moreover, if D

u exists in the

weak sense, k = 1, . . . , n, then hA, ψi = (∇u, ψ), for all the above speciﬁed functions

ψ. Thus, the above formula can be viewed as a generalization of the deﬁnition of

weak gradient of u, and the functional A will be still denoted by ∇u. It is obvious

that, if u ∈ L

(Ω),

|∇u|

−1,q

≤ c

kuk

for some c

= c

(q). Conversely, suppose Ω bounded and such that problem (III.3.1)–

(III.3.2) is solvable in Ω. Show that, if u ∈ L

loc

(Ω), with ∇u ∈ W

−1,q

(Ω),

then

u ∈ L

(Ω), and the following generaliz ation of the Poincar´e’s inequa lity (II. 5.10)

holds:

ku − u

Ω

≤ c

k∇uk

−1,q

with c

= c

(Ω, q). Thus, in particular, i f u ∈ L

(Ω), q ∈ ( 1, ∞), with

Ω

u = 0,

for the above types of domain, kuk

and k∇uk

−1,q

are equivalent norms. Hint: Pick

arbitrary ψ ∈ C

∞

(Ω), and let ϕ ∈ C

∞

(Ω) with

Ω

ϕ = 1. Set f := ψ − ϕ

Ω

ψ, and

let v ∈ C

∞

(Ω) be a solution to (III.3.1)–(III .3.2) corresponding to this f. Then, use

the relation

(u, f ) = (u, div v) = −h∇u, vi

along w ith the property (III .3.3) of the function v and the results of Exercise II.2.12.

Remark III.3.9 Poincar´e’s inequality holds for suﬃciently smoo th domains,

e.g., locally Lipschitz (see Theorem II.5.4), while it fails, in general, for do-

mains with very little regularity, like, for example, those having an external

cusp (Courant & Hilbert 1937, Kapitel VII, §8.2; see also Amick 1976 and

Fraenkel 1979, §2). Consequently, since by Exercise III.3.4, the validity of

Poincar´e’s inequality is implied by the solvability of probl em (III.3.1)–(III.3.2),

we conclude that this latter cannot be solved in arbitrary (bounded) domains.



Observe that, for bounded Ω, W

−1,q

(Ω) and D

−1,q

(Ω) are isomorphic; see Re-

mark II.6.3.

Observe that, by (II.5.1), we immediately ﬁnd k∇uk

−1,q

≤ c k∇uk

, 1 ≤ q ≤ ∞,

with c = c(q, Ω).