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

Подождите немного. Документ загружается.

186 III The Function Spaces of Hydrodynamics

g ∈

0,q

(Ω), 1 < q < ∞,

there exists at least one solution v to the problem

∇ · v = ∇ · g

v ∈ W

1,q

(Ω)

kvk

1,q

≤ c k ∇ · gk

kvk

≤ c kgk

(III.3.54)

In particular, if ∇ · g ∈ C

∞

(Ω), then v ∈ C

∞

(Ω) and we have

kvk

1,s

≤ c k∇ · gk

, for all s ∈ (1, ∞). (III.3.55 )

Proof. We ﬁrst take g such that

∇ · g ∈ C

∞

(Ω), g ∈

0,q

(Ω), 1 < q < ∞. (III.3.56)

In view of Lemma III.3.4, there exist N domains Ω

= Ω ∩ G

, with {G

} an

open covering of Ω, satisfying (i)–(iv ) of that lemma. Furthermore, we may

write

∇ · g =

i=1

with

∈ C

∞

(Ω

)

Ω

= 0,

(III.3.57)

and where f

has the expression

= ζ

∇· g +

k=1

Ω

∇ · g, some m

∈ N, (III.3.58)

with

∈ C

∞

(Ω), ζ

∈ C

∞

), θ

∈ C

∞

(Ω

), k = 1, . . . , m

, i = 1, . . ., N.

From (III.3.57) and (III.3.58), with the aid of Lemma III. 3.4, for any i =

1, . . ., N we can state the existence of a vector v

such that for all s ∈ (1 , ∞)

∇ · v

= f

∈C

∞

(Ω

)

1,s,Ω

≤ c k∇ · gk

s,Ω

q,Ω

≤ c kgk

q,Ω

(III.3.59)

III.3 The Problem ∇ · v = f 187

Thus, the ﬁeld

v =

k=1

satisﬁes all requirements of the theorem, which is thus proved if g sati sﬁes

(III.3.56). Assume, now, g merely belongs to

0,q

(Ω), 1 < q < ∞. In view

of Theorem III. 2.4 we can approximate g by a sequence {g

} ⊂ C

∞

(Ω). For

each s we then establish the existence of v

solving (III.3.54) with g

in place

of g. Because of (III.3.54)

3,4

we attain the existence of v ∈ W

1,q

(Ω) such that

→ v in W

1,q

(Ω).

Evidently, v solves (III.3.54)

in the generalized sense and, again by (III.3.54)

3,4

(written for v

) v satisﬁes (III.3.54)

3,4

. The proof of the theorem is therefore

completed. ut

An interesting consequence of the result just shown is given in the follow-

ing.

Theorem III.3.5 Let Ω be a bounded domain of R

, n ≥ 2, of class C

Then, for any f ∈ L

(Ω), 1 < q < ∞, satisfying (III.3.1), there exists a

solution v to problem (III.3.2) which, in addition, obeys the following estimate

kvk

≤ c |f|

∗

−1,q

with c = c(q, Ω, n), where | · |

∗

−1,q

is deﬁned in (III.3.41).

Proof. For a given f in the statement of the theorem, consider the functional

F : [ϕ] ∈

1,q

(Ω) → (f, ϕ) ∈ R , ϕ ∈ [ϕ].

Since f satisﬁes (III.3.1), the value of F is independent of the choice of ϕ ∈ [ϕ],

and F is, therefore, well deﬁned. It is easy to show that

F ∈ (

1,q

(Ω))

. (III.3.6 0)

In fact, it is obvious that F is additive. Furthermore, since f satisﬁes (II. 1.13),

by means of (II.5. 10) we obtain

|(f, ϕ)| = |(f, ϕ − ϕ

Ω

)| ≤ kfk

kϕ − ϕ

Ω

≤ c kfk

|ϕ|

1,q

which proves (III.3.60), and, m oreover,

kFk

(

1,q

(Ω))

= |f|

∗

−1,q

. (III.3.61)

Now, by Theorem II.8.2, there is F ∈ [L

(Ω)]

such that (f, ϕ) = (F , ∇ϕ),

for all ϕ ∈ D

1,q

(Ω) and kF k

= |f|

∗

−1,q

. Thus, in view of Lemma III.1.2 and

Theorem III.1.2, the Neumann problem

188 III The Function Spaces of Hydrodynamics

− (∇w, ∇ϕ) = (f, ϕ) , for all ϕ ∈ D

1,q

(Ω) , (III.3.62)

has a unique (up to a constant) solutio n w ∈ D

1,q

(Ω) which, in particular,

satisﬁes the foll owing estima te

|w|

1,q

≤ c |f|

∗

−1,q

, (III.3.63)

with c = c(n, q, Ω). If we set g = ∇w, from (III.3.62) we get (in the sense of

weak derivatives)

∇ · g = f (III.3.64)

and, in pa rticular,

(g, ∇ϕ) + (∇ · g, ϕ) = 0, for all ϕ ∈ W

1,q

(Ω).

Therefore, by (III.2.14), we ﬁnd g · n = 0, and, by Theorem I II.2.4, we con-

clude that g ∈

0,q

. The result is then a consequence of (III.3.6 3), (III.3.64)

and of Theorem III.3.4 . ut

We shall next consider the solvabil ity of problem (III.3.2) for Ω an exterior

domain. In this case the problem will be sui tably reformulated, as the fol lowing

considerations suggest. Let

Ω = R

−B(0),

and f = f(|x|) be smooth and of compact suppo rt in Ω. The vector ﬁeld

v(x) =



|x|



|x|

τ f(τ )dτ

solves (III.3.2 )

, vanishes on ∂Ω and, further,

|v|

1,2

= kfk

However,

v 6∈ L

(Ω)

and so

v 6∈ W

1,2

(Ω).

This example suggests that, for an exterior domain, the class W

1,q

(Ω) in

problem (III. 3.2) should be enla rged to the class D

1,q

(Ω). Therefore, for such

domains, we shall formulate the problem as follows: Given f ∈ L

(Ω) to ﬁnd

a vector ﬁeld v : Ω → R

such that

∇· v = f

v ∈D

1,q

(Ω)

|v|

1,q

≤ c kfk

(III.3.65)

where c = c(n, q, Ω).

III.3 The Problem ∇ · v = f 189

Remark III.3.13 Fo r Ω an exterior domain, the condition (III.3.1) on f is

no longer required. 

The following existence result holds.

Theorem III.3.6 Let Ω be a locally Lipschitz, exterior domain of R

, n ≥ 2.

Then, for any f ∈ L

(Ω), 1 < q < ∞, there exists a solution to problem

(III.3.65).

Proof. If Ω = R

, we at once check, with the help of Exercise II.11.9(i ),

that the ﬁeld v = ∇ψ with ∆ψ = f gives a soluti on to the problem. Thus,

we assume ∂Ω 6= ∅. Let {f

} ⊂ C

∞

(Ω) be a sequence approxima ting f in

(Ω), and set

= ∇ψ

+ w

, m ∈ N,

where

∆ψ

= f

in R

and, for som e R > 2δ(Ω

∇ · w

= 0 in Ω

= −∇ψ

on ∂Ω

= 0 on ∂B

(0).

(III.3.66)

By Exercise II.11.9(i), we ﬁnd

|ψ

2,q

≤ ckf

. (III.3.67)

Moreover, since ∆ψ

= 0 in Ω

, we have

∂Ω

∇ψ

·n = 0, for all m ∈ N, (III.3.68)

and so, from Exercise III.3.5, we deduce the exi stence of a solenoidal ﬁeld

∈ W

1,q

(Ω

) solving (III.3.66). We extend w

to zero outside Ω

so that

1,q

≤ c

k∇ · (ϕ∇ψ

, (III.3.69)

where ϕ ∈ C

) and ϕ = 1 if |x| < R/2, ϕ = 0 if |x| ≥ R. From (III.3.67)

and (III.3.69) we thus obtain

1,q

≤ c



+ k∇ψ

q,Ω

R/2,R



. (III.3.70)

If 1 < q < n, since ∇ψ

= O(|x|

1−n

) as |x| → ∞, we apply Theorem II.6.1

to deduce

k∇ψ

nq/(n−q)

≤ c

|ψ

2,q

, (III.3.71)

which, a long with the properties of w

and the characterization (II.7.14),

delivers v

∈ D

1,q

(Ω). Moreover, from (III.3 .67), (III.3.70) and (III.3.71) we

obtain

190 III The Function Spaces of Hydrodynamics

1,q

≤ c

. (III.3.72)

If q ≥ n, we add to ψ

a linear function in x such that

Ω

R/2,R

∇ψ

= 0,

and continue to denote the modiﬁed ﬁelds by ψ

and v

. Cl early, ψ

continue

to satisfy (III.3.68 ), while v

, in view of the characterization (II.7.1 5), belongs

to D

1,q

(Ω). From the last displayed equation and Theorem II.5.4 it follows

that

k∇ψ

q,Ω

R/2,R

≤ c

|ψ

2,q

Employing this inequality back into (III.3.69) and using (III.3.70) gives again

(III.3.72). From what we said, and from (III.3.67) and (III.3.72) it then foll ows

that v

solves the problem

∇ · v

= f

∈ D

1,q

(Ω)

1,q

≤ ckf

(III.3.73)

with c = c(n, q, Ω). The theorem then easily follows by letting m → ∞ into

(III.3.73) (details are left to the reader). ut

Remark III.3.14 Theorem III.3.6 continues to hold if Ω satisﬁes the cone

condition (Bogovski

i 1980, Theorem 4). 

Remark III.3.15 From the proof of Theorem III.3.4 it follows that if

f ∈ L

(Ω) ∩ L

(Ω),

we can ﬁnd a solution v to (III.3.65)

such that v ∈ D

1,q

(Ω) ∩ D

1,r

(Ω) and

that satisﬁes

|v|

1,q

≤ ckfk

, |v|

1,r

≤ ckfk



Remark III.3.16 A result similar to that of Theorem III.3.3 can also be

proved for exterior domains. We shall not show this here, since it will not be

needed later, and refer to Bogovski

i (1 980, Theorem 5) for a proof. However,

the reader may wish to give his/her own proof. 

Exercise III.3.8 Let Ω sati sfy the assumptions of Theorem III.3.6. Show that, for

any f ∈ L

(Ω) and a ∈ W

1−1/q,q

(∂Ω), there exists at least one solution to the

problem

∇ · v = f

v ∈ D

1,q

(Ω)

v = a at ∂Ω

|v|

1,q

≤ c

kfk

+ kak

1−1/q,q(∂Ω)

III.3 The Problem ∇ · v = f 191

with c = c(n, q, Ω). Notice that, unlike the case of a bounded domain (see Exercise

III.3.5), the condition

Ω

f =

∂Ω

a · n

is not needed. I n this respect, show that if

f ≡ 0 and

∂Ω

a ·n = 0,

then v can be taken of bounded support in Ω.

Exercise III.3.9 Let Ω satisfy the assumption of Theorem III.3.6, and assume that

k(|x|

+ 1) fk

∞

< ∞, for some α ∈ (1, n). Show that problem (III.3.65), with q > n,

has at least one solution which, in addition, satisﬁes the estimate

k(|x|

α−1

+ 1)vk

∞

≤ c

k(|x|

+ 1) fk

∞

Show also that, if f ∈ L

(Ω), q > n, with a bounded support, K, then there exists

a solution to (III.3.65) which, i n addition, satisﬁes the estimate

k(|x|

n−1

+ 1)vk

∞

≤ c

kfk

with c

= c

(K, q). Hi nt: Use the argument employed in the proof of Theorem III.3.6,

along with Lemma II.9.2.

Exercise III.3.10 Let Ω satisfy the assumption of Theorem III.3.6, and let u ∈

loc

(Ω) with ∇u ∈ D

−1,q

(Ω). Show that u ∈ L

(Ω) and that the following general-

ization of the Poincar´e inequality holds

kuk

≤ c |∇u|

−1,q

where c = c(Ω, q). Hint: See Exercise III .3.4. (For inequality of this type where only

one derivative of u is in D

−1,q

(Ω) see Galdi 2007, Proposition 1.1. )

Problem (III.3.65) can be considered also for domains with a noncompact

boundary (Bogovski

i 1980, §3; Solonnikov 1981, §2; Padula 1992, Lemma 2.2).

In Section 3 of the next chapter (see Theorem IV.3.2) we shall show that it can

be solved for Ω = R

. A detailed study of solvability of (III.3.65) in domains

having m ≥ 1 “exits” at inﬁnity (such as inﬁnite tubes and pipes) has been

performed by Solonnikov (1981) for the case q = 2 . (His results, however,

admi t of a straightforward generalization to the case q ∈ (1, ∞).) It should be

noted that, in this latter case, the problem need not be solvable, in general, if f

merely belongs to L

(Ω) and, in fact, some additional compatibil ity conditions

are to be assumed on f. For instance, if Ω i s the inﬁnite cylinder:

Ω = {x ∈ R

: |x

| < A}, x

= (x

, . . ., x

n−1

), A > 0,

one easily checks that the problem

∇· v = f

v ∈ D

1,q

(Ω)

(III.3.74)

192 III The Function Spaces of Hydrodynamics

can admit a solution only if f satisﬁes (III.3.1) and the quantity

|f|

≡

(

∞

−∞



∞

|≤A

f(x

, x

)dx



)

1/q

is ﬁnite. Conversely, one can prove that, for any f satisfying (III.3.1) and

belonging to the completion of C

∞

(Ω) with respect to the norm

kfk

≡ kfk

+ |f|

there exists a solution to (III.3.74) tha t also satisﬁes the inequal ity

|v|

1,q

≤ ckfk

(see Solonnikov 1981, Theorem 2). Notice that, in the particular case where

f has zero average along the cross-section of the cylinder, namely,

|≤A

f(x

, x

)dx

= 0 ,

then kfk

= kfk

Remark III.3.17 The solvability of problem (III.3.1), (III.3.2)

in H¨older

spaces C

m,λ

(Ω) has been investigated by Kapitanski

i & Pileckas (198 4, §8).

In particular, these authors prove the following result (see their Theorem 1

on p. 481). 

Theorem III.3.7 Let Ω be a bounded domain of R

, n ≥ 2, of class C

m+2,λ

m ≥ 0, λ ∈ (0, 1). Then, given f ∈ C

m,λ

(Ω) satisfying (III.3.1), there exists at

least one ﬁeld v ∈ C

m+1,λ

(Ω) vanishing at ∂Ω such tha t ∇·v = f . Moreover,

v obeys the following estimate

kvk

m+1,λ ≤ ckf k

m,λ

with c = c(m, λ, n, Ω).

Remark III.3.18 In connection with Theorem III.3.7, we wi sh to observe

that, in general, the number λ cannot be taken to be zero; otherwise, this

would imply the existence of certain solenoidal extensions of boundary data

that, however, cannot exist, as shown by counterexamples, see Section IX. 4;

see also Dacorogna, Fusco & Tartar (2004). 

III.4 The Spaces H

193

III.4 The Spaces H

The study of the dynamical properties of the ﬂow of a viscous, incompressible

ﬂuid requires velocity ﬁelds of the particles of the ﬂuid which, at each time, are

summable to the qth power, q ≥ 1, along with their ﬁrst spatial derivatives,

are solenoidal and vanish at the boundary of the region where the motion

occurs. One is thus led to introduce a space of vector functions having such

properties (in a generalized sense) a nd, in this respect, either of the following

two choices seems plausible, namely,



completion of D(Ω) in the norm of W

1,q

(Ω)



≡ H

(Ω)

v ∈ W

1,q

(Ω) : ∇ · v = 0 in Ω

≡

(Ω).

(For q = 2 we will write H

(Ω) and

(Ω), respectively.) Even though

these spaces look similar, they are a prio ri distinct in that the condition of

solenoidality on their m embers is imposed before (in H

(Ω)) and after (in

(Ω)) taking the completio n of C

∞

(Ω) in the norm of W

1,q

(Ω). However,

one easily shows that for any domain Ω the fol lowing inclusions hold for all

q ≥ 1:

(Ω) ⊂

(Ω)

(Ω) ⊂ H

(Ω).

The fact that the two spaces can be diﬀerent for some domains and some

q (depending o n the space dimension n) can be guessed through the following

considerations. Let Ω be the inﬁnite cylinder



x ∈ R

: x

+ x

< 1



If v is a solenoidal vector function vanishing on ∂Ω, one readily veriﬁes that

the ﬂux φ of v through the cross section Σ of Ω

at a p oint x

of the axis of

the cylinder is a constant independent of x

, that is,

φ ≡

v · n = const., n = (0, 0, 1).

If v ∈ H

(Ω), one immediately obtains φ = 0. Actually, in such a case, we

know that v is approximated by solenoidal vector functions of compact support

in Ω, see Exercise III.4.1. If v ∈

(Ω), however, this approximation does not

hold a priori but nevertheless we can deduce φ = 0 from the following three

observations:

(i)



v · n



≤ |Σ|

1/2





1/2

Namely, Σ is the intersection of Ω with a plane orthogonal t o the axis of Ω.

194 III The Function Spaces of Hydrodynamics

(ii) There exists a sequence {x

(k )

} ⊂ R wi th |x

(k)

| → ∞ as k → ∞ such

that

lim

k→∞

(1 + |x

(k)

= 0,

(iii) |Σ|/(1 + |x

|) is bounded.

Notice that (ii) is a simple consequence of the hypothesis v ∈ W

1,2

(Ω).

Assume now that the domain, instead of having “exits” to inﬁnity of bounded

cross section (as in the previous case), has a cross section whose area i ncreases

suﬃciently fa st as x

→ ±∞. For example, we may choose

Ω =



x ∈ R

: x

+ x

< 1 + x



. (III.4.1)

We still have φ = 0 for v ∈ H

(Ω), but we can no longer deduce the same

result if v ∈

(Ω). Actually (i) and (ii) also remain true in this case but (iii)

fails, because now |Σ| = π(1 + x

). Thus one may suspect the existence of a

solenoidal vector in W

1,2

(Ω) which, though satisfying (ii), has a non-zero ﬂux

through Σ, thus obtai ning H

(Ω) 6=

(Ω). We shall see later in this section

that such a vector ﬁeld actually exists.

Exercise III.4.1 Let Ω be a domain of R

, n ≥ 2, with N ≥ 1 “outlets to inﬁnity,”

i.e.,

Ω = Ω

∪ Ω

∪ . . . ∪ Ω

where Ω

is bounded, Ω

∩Ω

= ∅, i 6= j, and each Ω

, i n possibly diﬀerent coordinate

systems, has the form

Ω

x ∈ R

: x

> 0, x

∈ Σ(x

)

where Σ(x

) is a bounded domain in R

n−1

smoothly varying with x

and x

, . . . , x

n−1

). Show that for all v ∈ H

(Ω), 1 ≤ q ≤ ∞,

v · n

= 0, i = 1, ..., N.

In fact, setting

g(x

) =

, x

)dx

by Theorem II.4.1 g(x

) is well deﬁned. Moreover,

∞

−∞

g(x

)dx

< ∞. (∗)

Therefore, there exists a sequence {x

(k)

} wit h |x

(k)

| → ∞ as k → ∞ such that

lim

k→∞

(1 + |x

(k)

|)g(x

(k)

) = 0.

In fact, if this were not true, we should have

g(x

) ≥ `/(1 + |x

|), for all x

with |x

| > b,

and for some `, b > 0, which contradicts (∗).

III.4 The Spaces H

195

Hint: Use the fact that v can b e approximated by elements of D(Ω), together with

Theorem II.4.1.

The problem o f the relationship between H

(Ω) and

(Ω) is not merely

a question of mathematical completeness; rather, as pointed out for the ﬁrst

time by Heywood (1976), the coincidence of the two spaces is tightly linked

with the uniqueness of ﬂow of a viscous, incompressible ﬂuid and, in particular,

in doma ins for which H

(Ω) 6=

(Ω) the motion of such a ﬂuid is not

uniquely determined by the “traditional” initial and boundary data but other

extra and appropriate auxiliary conditions are needed. Referring the reader

to Chapters VI and XII for a description of these latter and related results,

in the present section we shall only consider the question of investigating for

which domains the two spaces coincide and will indicate doma ins for which

they certainly don’t. To this end, we subdivide the domains of R

(n ≥ 2) into

three groups:

(a) bounded domains;

(b) exterior domains;

In cases (a) and (b) one shows H

(Ω) =

(Ω) provided only Ω has a

mild degree of smoothness (for example, cone condition, or even less, would

suﬃce); see Theorem III.4.1 and Theorem III.4.2. On the other hand, in case

no matter how smooth their boundary is; see Theorem III.4.4 and Theorem

III.4.6. T herefore, the coincidence of H

(Ω) and

(Ω) does not seem related

to a high degree of regularity of Ω but rather to its shape. In this connection,

we wish to recall a remarkable general result of Masl ennikova & Bogovski

(1983, Theorem 5), which states that, if Ω is an arbit rary strongly locally

Lipschitz domain,

then H

(Ω) =

(Ω), for all q ∈ [1, n/(n − 1)].

It is conjectured that, for all dom ains wi th a compact bounda ry, the two

spaces coincide for q = 2, but no proof is, to date, available in the general

case.

Should this coincidence fail to hold for some domain of the above type,

we would have paradoxical situations from the physical point of view. For

example, if for some bounded domain, Ω

]

(say), the two spaces did not coincide,

then the steady-state Stokes boundary-value problem formulated in Ω

]

and

corresponding to zero body force and zero (Dirichlet) data at the boundary

would admi t a non-zero and smooth solution; see Remark IV.1.2. Analogous

situation would occur if Ω

]

is an exterior domain; see Remark V.1.2

We ﬁnally notice that, as already observed, for the proof of coincidence,

it is suﬃcient to show

(Ω) ⊂ H

(Ω), the converse inclusion being always

satisﬁed.

This type of regularity extends t he lo cal Lipschitz one to the case of domains

with a noncompact boundary, see, e.g., Adams (1975, p. 66).

See, however the result of

Sver´ak (1993) mentioned in the Notes for this Chapter.