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

Подождите немного. Документ загружается.

176 III The Function Spaces of Hydrodynamics

Remark III.3.10 An elementary example o f two-dimensional domain where

problem (III.3.1)–(III.3.2), for q = 2, can not have a solution f or all f ∈ L

(Ω)

with f

Ω

= 0 is the following one, due to G. Acosta (see Dur´an & L´opez Garc´ıa,

2010, p. 423). Let

Ω = {x ∈ R

: 0 < x

< 1, |x

| < x

and consider the function

u(x) =

−3 , x ∈ Ω.

Clearly, u ∈ L

(Ω), with u

Ω

= 0. Moreover,

u 6∈ L

(Ω) .

However,

∂u

∂x

= −2

∂

∂x





∂g

∂x

and since g ∈ L

(Ω), we obtain

u ∈ W

−1,2

(Ω) .

Therefore, by Exercise III.3.4, there exists at least one f ∈ L

(Ω) wi th f

Ω

= 0

for which problem (III. 3.1)–(III.3.2), for q = 2 , does not have a solution. 

Exercise III.3.5 Along with problem (III.3.2), one can consider the following non-

homogeneous version of it. Given f and a suitably, ﬁnd a vector ﬁeld v such that

∇ · v = f

v ∈ W

1,q

(Ω)

v = a at ∂Ω.

(III.3.31)

Show that, for Ω a b ounded and locally Lipschitz domain of R

, n ≥ 2, given

f ∈ L

(Ω), a ∈ W

1−1/q,q

(∂Ω), 1 < q < ∞,

satisfying

Ω

f =

∂Ω

a · n,

problem (III.3.31) admits at least one solution v. Moreover, denoting by A ∈

1,q

(Ω) an extension of a (according to Theorem II. 4.3), show that this solution

satisﬁes the estimate

kvk

1,q

≤ c (kfk

+ k∇ ·Ak

) .

Therefore, in particular,

kvk

1,q

≤ c

kfk

+ kak

1−1/q,q(∂Ω)

. (III.3.32)

III.3 The Problem ∇ · v = f 177

Exercise III.3.6 Let Ω be as in Theorem III.3.1. Furthermore, let I be an interval

in R, and suppose f = f (t) is in L

(Ω), q

∈ (1, ∞), i = 1, 2, and satisﬁes (III.3.1),

for all t ∈ I. Then, show that problem (III.3.2) has at least one solution v = v(x, t)

which, in addition, satisﬁes

|v(t

) − v(t

1,q

≤ c

kf(t

) − f(t

, t

∈ I , i = 1, 2 ,

for some c

= c

, Ω). Moreover, if

∂f

∂t

∈ L

(Ω), i = 1, 2, show that

∂v

∂t

∈

1,r

(Ω), i = 1, 2, and that the following estimate holds

‚

∂v

∂t

‚

1,r

≤ c

‚

∂f

∂t

‚

, i = 1, 2 ,

where c

= c

, Ω). Hint: Use the argument presented in Remark III.3.3 along

with the method used in t he proof of Theorem III.3. 1.

Unless Ω is star-shaped with respect to a ball, the method of construction

of the ﬁeld v used in the proof of Theorem III.3.1 ensures, in general, no

more than the W

1,q

-regularity fo r v, even for f ∈ C

∞

(Ω). Our next objective

is, therefore, to show that if Ω is suﬃciently smooth (for example, locally

Lipschitz) we may ﬁnd a solution to (III.3.2) that belong s to C

∞

(Ω) if f ∈

∞

(Ω). The regularity of the dom ain is required in order to construct a

suitable covering of Ω and a corresponding decomposition of f, a s the fo llowing

lemma proves.

Lemma III.3.4 Let Ω be bo unded and locally Lipschitz. Then there exists

an open cover G = {G

, . . . , G

, G

m+1

, . . . , G

m+ν

} of Ω such that

(i) Ω

≡ Ω ∩G

is star-shaped with respect to an open ball B

with B

⊂ Ω

for i = 1, . . ., m + ν ;

(ii) ∂Ω ⊂ ∪

i=1

;

(iii) G

is an open ball with B

⊂ Ω for i = m + 1, . . ., m + ν;

(iv) Ω = ∪

m+ν

i=1

Ω

Moreover, if f ∈ C

∞

(Ω) satisﬁes (III.3.1), we have f =

m+ν

i=1

where

(v) f

∈ C

∞

(Ω

);

(vi)

Ω

= 0 ;

(vii) f

= ζf +

k=1

Ω

f, where m

∈ N, ζ ∈ C

∞

), θ

∈ C

∞

(Ω

)

and φ

∈ C

∞

(Ω).

(viii) kf

m,q

≤ Ckfk

m,q

, for all m ≥ 0 and q ≥ 1,

where C depends only on m, q, and Ω.

Proof. In virtue of Lemma II.1.3, we may ﬁnd m locally Lipschitz domains

, . . . , G

such that Ω

≡ Ω ∩G

is star-shap ed w ith respect to an open ball

with B

⊂ Ω

and verifying condition (ii). Denote by G

a domain with

⊂ Ω such that {G

, G

, . . . , G

} f orms an open cover of Ω. Since Ω is

bounded, we may ﬁnd ν open balls G

m+1

, . . . , G

m+ν

such that

178 III The Function Spaces of Hydrodynamics

⊂

m+ν

[

i=m+1

(

m+ν

[

i=m+1

)

−

⊂ Ω.

Evidently,

G = {G

, . . . , G

, G

m+1

, . . . , G

m+ν

}

is an open cover of Ω with Ω ⊂ ∪

m+ν

i=1

. Setting

Ω

≡ Ω ∩ G

, i = 1, . . . , m + ν

it is immediately obtained that properties (i) and (iv) a re satisﬁed. Let us

now pick f ∈ C

∞

(Ω) and show the existence of f

satisfying (v)-(viii). To this

end, let

{ψ

, . . .ψ

m+ν

}

be a partition of unity subordinate to G; see Lemma II.1.4. Setting

m+ν

[

i=2

Ω

, Ψ

m+ν

i=2

(x),

and observing that

(x) + Ψ

(x) = 1, for all x ∈ Ω,

we may write f = f

+ g

, where

= ψ

f − χ

Ω

= Ψ

f − χ

Ω

and

∈ C

∞

(Ω

∩ D

Ω

= 1.

Since ψ

∈ C

∞

) and f ∈ C

∞

(Ω) it immedia tely follows that

∈ C

∞

(Ω

Moreover, we have

i=2

(x) +

m+ν

i=m+1

(x)

so that, by recalling the deﬁnition of G

, . . . , G

, G

m+1

, . . ., G

m+ν

and that

f ∈ C

∞

(Ω), it follows that

III.3 The Problem ∇ · v = f 179

∈ C

∞

In addition, we have, evidently,

Ω

= 0.

We may then continue this procedure, using g

as the function to be split.

Precisely, setting

m+ν

[

i=3

Ω

, Ψ

m+ν

[

i=3

(x),

we let

= ψ

− χ

Ω

= Ψ

−χ

Ω

where

∈ C

∞

(Ω

∩ D

)

Ω

= 1.

From the expression of g

and prop erty (b) of the partition of unity we recover

at once that f

can b e written as

= ψ

(1 − ψ

)f − χ

Ω

(1 − ψ

)f − χ

Ω

(1 − ψ

+χ

Ω

(1 −ψ

)f,

which proves f

∈ C

∞

(Ω

), along with properties (vii) and (viii). Further-

more,

= f

+ g

, g

∈ C

∞

Ω

= 0.

We may then use an iteration scheme of the same type employed in the proof

of Lemma III.3.2 to show the validity of (v)-(vii) for a ll i = 1, . . . , m + ν. ut

The result just proved allows us to show the following one.

Theorem III.3.3 Let Ω be a bounded and locally L ipschitz domain in R

n ≥ 2. Given

f ∈ W

m,q

(Ω), m ≥ 0, 1 < q < ∞, (III.3.33)

satisfying (III.3.1), there exists v ∈ W

m+1,q

(Ω) satisfying (III.3.2) and

(III.3.23), for all ` = 0, . . . , m. Moreover, if f ∈ C

∞

(Ω) then v ∈ C

∞

(Ω).

180 III The Function Spaces of Hydrodynamics

Proof. The proof is essentially analogous to that of Theorem I II.3.1 once we

take into account Lemma III.3.4 (in place of Lemma III.3.2) and Remark

III.3.2. Actually, for f ∈ C

∞

(Ω), we denote by Ω

and f

the domains and

functions introduced in Lemma III.3.4 and we let v

∈ C

∞

(Ω

) denote the

solution to problem (III.3.1), (III.3.2) in Ω

whose existence is assured by

Lemma III.3.1. In view of Lemma III.3.4(vi) and Remark III.3.2, we have also

k∇v

`,q

≤ Ckfk

`,q

, ` = 0, . . . , m ,

for som e constant C = C(n, q, `, Ω). Then the ﬁeld

v =

i=1

belongs to C

∞

(Ω) and satisﬁes (III.3 .2) in Ω along with the inequality

k∇vk

`,q

≤ Ckfk

`,q

, ` = 0, . . . , m , (III.3 .34)

which proves the second part of the theorem. Let us now assume f merely

satisfying (I II.3.33) and denote by {f

} ⊂ C

∞

(Ω) a sequence of functions

satisfying (III.3.1) and converging to f in W

m,q

(Ω). Let {v

} ⊂ C

∞

(Ω) be

the corresponding solutions to (III.3.2). It is readily seen that, by the estimate

(III.3.34) and inequality (II.5.1), as k → ∞ the sequence {v

} converges

(strongly) in W

m+1,q

(Ω) to a function v ∈ W

m+1,q

(Ω) that solves (III.3.2)

in the sense of generalized diﬀerentiation and satisﬁes (III.3.34). The theorem

is therefore proved. ut

Remark III.3.11 The regularity assumption on Ω made in Theorem III.3.3

can b e further weakened (Bogovski

i 1980, Theorem 2). 

Remark III.3.12 From the proof of Theorem III.3.3i t immedia tely follows

that if

f ∈ W

m,q

(Ω) ∩ W

m,r

(Ω), m ≥ 0, 1 < q, r < ∞,

satisfying (III.3.1), we can ﬁnd one solution to (III.3.2) with

v ∈ W

m+1,q

(Ω) ∩ W

m+1,r

(Ω) (III.3.35)

such that

k∇vk

m,q

≤ Ckfk

m,q

, k∇vk

m,r

≤ Ckfk

m,r

. (III.3.36)



From Theorem III.3.3 we obtain the following corolla ry on the extension

of solenoidal ﬁelds (Coscia and Galdi, 1997; Bogovski

i 1980, Theorem 3).

III.3 The Problem ∇ · v = f 181

Corollary III.3.1 Let Ω be a bounded locally Lipschitz domain of R

, n ≥

2, and let Γ

, i = 1, . . ., k, k ≥ 1, denote the connected components o f the

boundary ∂Ω. Let v be a solenoidal ﬁeld of W

m,q

(Ω), m ≥ 1, 1 < q < ∞,

satisfying the following condi tions

v · n = 0, for all i = 1, . . . , k,

with n normal component to ∂Ω. Then, given a n open ball B with B ⊃ Ω,

there exists a solenoidal ﬁeld V such that

V ∈ W

m,q

(B)

V (x) = v(x) for all x ∈ Ω.

Moreover,

kV k

m,q,B

≤ ckvk

m,q,Ω

for som e c = c(Ω, m, q, n, B) > 0.

Proof. From Theorem II.3.3 there exists u ∈ W

m,q

(B) such that

u(x) = v(x) for all x ∈ Ω

kuk

m,q,B

≤ ckvk

m,q,Ω

However, u need not be solenoidal and to obtain the desired extension V we

have to modify u sui tably. To this end, denote by ω

, i = 1, . . ., k − 1 the

bounded connected components of R

−Ω and set ω

= B −Ω. In each ω

consider the following problem

∇ · w

= ∇ · u in ω

∈ W

m,q

(ω

)

m,q

≤ ck ∇ · uk

m−1,q

By assumption,

∇ · u =

v · n = 0, i = 1, . . . , k.

Moreover, being ∇ · v = 0 in Ω, it also follows that

∇ · u ∈ W

m−1,q

(ω

), i = 1, . . ., k.

As a consequence, from Theorem III.3 .3 we deduce the existence of the ﬁelds

. Set w

≡ 0 in ω

and denote again by w

their extensions. We deﬁne

V (x) =

(

u(x) − w

(x) x ∈ ω

, i = 1, . . . , k,

v(x) x ∈ Ω.

It is immediately checked that the ﬁeld V satisﬁes all the properties stated

in the corollary, which is therefore proved. ut

182 III The Function Spaces of Hydrodynamics

Exercise III.3.7 Let Ω be as in Theorem III.3.3. Furthermore, let I be an interval

in R, and suppose f = f(t) is in W

m,q

(Ω), m ≥ 0, q

∈ (1, ∞), i = 1, 2, and that

satisﬁes (III.3.1), for all t ∈ I. Then, show that, for all t ∈ I, problem (III. 3.2) has

at least one solution v(t) ∈ W

m+1,q

(Ω), i = 1, 2, satisfying (III. 3.23) with q = q

i = 1, 2, for all t ∈ I, and which, in addition, obeys the fol lowing inequality

k∇(v(t

) − v(t

))k

`,q

≤ c

kf(t

) − f(t

`,q

, t

∈ I , ` = 0, . . . , m , i = 1, 2 ,

for some c

= c

, `, Ω). Moreover, if

∂f

∂t

∈ W

k,q

(Ω), for i = 1, 2 and some k ≥ 0,

show that

∂v

∂t

∈ W

k+1,q

(Ω), i = 1, 2, and that the following estimate holds

‚

∂v

∂t

‚

l+1,q

≤ c

‚

∂f

∂t

‚

l,q

, l = 0, . . . , k , i = 1, 2 ,

where c

= c

, l, Ω). Hint: Use the argument presented in Remark III.3. 3 along

with the method used in t he proof of Theorem III.3. 3.

A further question tha t can be reasonably posed for problem (I II.3.1),

(III.3.2) is that of ﬁnding a solution that further obeys an estimate with f in

negative Sobolev spaces, that is,

kvk

≤ ckfk

−1,q

. (III.3. 37)

However, the answer is, in general, negative even when f is in the form of

divergence. Actuall y, if we take

f = ∇ ·g, g ∈

(Ω), (III.3.38)

with

(Ω) deﬁned in (III.2.4), (III.2.5), the solvability of (III.3.1), (III.3.2),

and (III.3.37) would imply the existence of certain solenoidal extensions of

boundary data that, as shown by counterexamples, cannot exist; see Remark

IX.4. 4. Nevertheless, the question admits a positive answer if we further re-

strict the hypothesis on f. Speciﬁcally, we shall prove that (III.3.1), (I II.3.2),

and (III.3.37) have at least one solution for all f of the type

f = ∇ · g, g ∈

0,q

(Ω), (III.3.39)

where

0,q

(Ω) is given in (III.2.15). The diﬀerence between (III.3.38) and

(III.3.39) is that the latter, unlike the former, requires the vanishing of the

normal component of g at the boundary.

One important consequence of this result (see Theorem III.3.5) is that,

provided Ω is suﬃciently regular, the following inequality holds

kvk

≤ c|f|

∗

−1,q

, (III.3.40)

where

|f|

∗

−1,q

= sup

u∈D

1,q

(Ω);|u|

1,q

|(f, u)|. (III.3. 41)

III.3 The Problem ∇ · v = f 183

Notice that, since W

1,q

(Ω) ⊂ D

1,q

(Ω), we have kfk

−1,q

≤ |f|

∗

−1,q

, so that

(III.3.40) is a weaker form o f (III.3.37) .

We begin to show the following

Lemma III.3.5 Let Ω, G be bounded, locally Lipschitz domains in R

, n ≥

2, with Ω ∩ G ≡ Ω

(6= ∅ ) star-shaped with respect to a ball B with B ⊂ Ω

Let, further, φ

, θ

, k = 1, . . ., m, ζ, and g be functions such that

∈ C

∞

(Ω), θ

∈ C

∞

(Ω

), ζ ∈ C

∞

(G),

g ∈

0,q

(Ω), 1 < q < ∞, ∇ · g ∈ C

∞

(Ω).

Set

f = ζ∇· g +

i=1

Ω

∇ · g (III.3.42)

and supp ose

Ω

f = 0. (III.3.43)

Then, there is at least one solution w to the problem

∇· w = f

w ∈ C

∞

(Ω)

kwk

1,s,Ω

≤ ck∇ ·gk

s,Ω

kwk

q,Ω

≤ ckgk

q,Ω

(III.3.44)

where s is arbitrary in (1, ∞) and c = c(φ

, θ

, ζ, s, q, n, B, G, Ω).

Proof. For simplici ty, we shall restrict ourselves to discuss the case where

m = 1 and set θ = θ

, φ = φ

. Clearly, f ∈ C

∞

(Ω

). Then, by (III.3.43) and

the assumption made on Ω

, we can ﬁnd a solution to the problem by the

Bogovski

i formula (III.3.8), that is,

w(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.3.45)

Thus, repeating the proof of Lemma III.3.1, we obtain that the ﬁeld w deﬁned

in (III. 3.45) satisﬁes (III.3.44)

1,2

and

kwk

1,s

≤ c

kfk

with c

= c

(n, s, Ω

). Since

184 III The Function Spaces of Hydrodynamics

kfk

≤ c

k∇ ·gk

with c

= c

(φ, θ, ζ, s, Ω), (III.3.44)

also follows. It remains to show (III.3.44)

From (III.3.45) and (III.3.42) we ﬁnd for i = 1, . . ., n

(x) = w

(1)

(x) + w

(2)

(x)

(1)

(x) =

Ω

(x, y)ζ(y)D

(y)dy

(2)

(x) =



Ω

φ∇ ·g



Ω

(x, y)θ(y)dy



(III.3.46)

Using the properties of ω, we obtain

|N (x, y)| ≤



x − y

|x − y|

∞

|x−y|



y + ξ

x − y

|x − y|



n−1

dξ



≤ |x −y|

1−n

kωk

∞

δ(Ω

)

n−1

dξ ≤ c|x −y|

1−n

(III.3.47)

with c = c(n, B, Ω

). Thus, from Young’s inequality on convolutions (Theorem

II.11.1) it foll ows that

Ω

N (·, y) θ(y)dyk

q,Ω

< ∞,

and so

(2)

q,Ω

≤ c



Ω

φ∇· g



Since g ∈

0,q

(Ω), the trace n · g of the normal component of g at ∂Ω

is identically vanishing (see Theorem III.2.4), and consequently we have by

(III.2.14)

Ω

φ∇ · g = −

Ω

g · ∇φ.

Therefore,

(2)

q,Ω

≤ c

kgk

q,Ω

. (III.3.48)

Again from (III.2.14) and taking into account ζ ∈ C

∞

(G), we may integrate

by parts into (III.3.46)

to obtain

III.3 The Problem ∇ · v = f 185

(1)

(x) = −lim

ε→0

(

Ω

−B

(x)

(x, y)g

(y)D

ζ(y)

+ζ(y)g

(y)D

(x, y)]dy +

∂B

(x)

(x, y)ζ(y)g

(y)

−y

|x − y|

dσ

)

≡ −lim

ε→0

i=1

(x, ε).

(III.3.49)

Since by (III.3.47) and Young’s inequal ity on convolutions,

Ω

N(·, y)g · ∇ζ(y)dyk

q,Ω

≤ c

kgk

q,Ω

we have, for a.a. x ∈ Ω,

lim

ε→0

(x, ε) =

Ω

(x, y)g

(y)D

ζ(y)dy. (III.3 .50)

Furthermore, by a reasoning analogous to that leading to (III.3.13) we show

for a .a. x ∈ Ω

lim

ε→0

(x, ε) = ζ(x)g

(x)

Ω

−y

)(x

− y

)

|x −y|

ω(y)dy. (III .3.51)

Finally, as we know from (III.3.15), (III.3.16 ), and (III.3.17),

(x, y) = K

(x, x − y) + G

(x, y),

where K

is a Calder´on–Zygmund kernel, while G

is bo unded by a weakly

singular kernel. Therefore, from Theorem II.11.4 we deduce for a.a. x ∈ Ω

lim

ε→0

(x, ε) =

Ω

(x, x − y)ζ(y)g

(y)dy +

Ω

(x, y)ζ(y)g

(y)dy,

(III.3.52)

where, of course, the ﬁrst integral has to be understood in the Cauchy principal

value sense. From (III.3.49)–(III.3.52), from Theorem II.11.4, (II I.3.15)

and

Young’s inequali ty on convolutions we then conclude

(1)

q,Ω

≤ c

kgk

q,Ω

. (III.3.53)

Thus, estimates (III.3.44)

becomes a consequence of (II I.3.46), (III.3.48), and

(III.3.53) and the lemma is proved. ut

We are now in a position to prove the following.

Theorem III.3.4 Let Ω be a bounded, l ocally Lipschitz domain in R

, n ≥

2. Then, given