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

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136 II Basic Function Spaces and Related Inequalities

Results contained in Exercise II.6. 3 generali ze part of those establi shed by

Uspenski

i (1961, Lemma 1), and for q = n = 2 they coincide with those of

Gilbarg & Weinberger (1978, Lemma 2.1).

Inequality (II.6.20) with q = 2 and n = 3 i s due to Finn (1965a, Corollary

2.2a); see also Birman & Solomjak (1974, Lemma 2.19) and Padula (1984,

Lemma 1), while (II.6.22) for n = 3 and q ∈ (1, 3) is proved by Galdi &

Maremonti (1986, Lemma 1.3). Theorem II.6.1, i n its generality, is due to me.

The inequality in Theorem II.6.5 is due to Simader and Sohr (1997, Lemma

1.2).

Section I I.7. The problem of approximatio n of functions from D

m,q

(Ω) when

Ω = R

with functions of bounded support was ﬁrst considered by Sobolev

(1963b). In this section we closely follow Sobolev’s ideas to generalize his

results to more general domains Ω. In this connection, we refer the reader

also to the papers of Besov (1967, 1969) and Burenkov (1976).

The elementary proof of the Hardy-type inequality (II.6.10), (II.6.13) and

(II.6.14) presented here and ba sed on the use of the “auxiliary” function g

was presented for the ﬁrst time in Galdi (1994a, §2.5). The same approach

was successively rediscovered by Mitidieri (2000).

Section II.8. A slightly weaker version of Theorem II.8.2, with a diﬀerent

proof, can be found in Kozono & Sohr (1991, Lemma 2.2).

The proof of the unique solvability of the Dirichlet problem (I I.8.17) in the

case Ω = R

, R

is a simple consequence of Exercise II.11.9(ii) and Remark

II.11.3. In the case Ω bounded and of class C

∞

, a proof was g iven for the

ﬁrst time by Schechter (1963a, Corollary 5.2). A diﬀerent proof that requires

domains only of class C

was later provided by Simader (1972). If Ω is an

exterior domai n of class C

, a thorough analysis of the problem can be found

in Simader & Sohr (1997, Chapter I). In particular, for n ≥ 3, the analy sis of

these authors shows that the problem (II.8.17) has a nonzero one-dimensional

null set, if q

≥ n. In other words, there exists one and only one non-zero har-

moni c function h ∈ D

1,q

(Ω), satisfying a normalization condition

Ω

= 1,

for some ﬁxed R > δ(Ω)

. For instance, if Ω is the exterior of the unit ball

in R

, we have h(x) = c(|x|

2−n

− 1), for a suitable choice of the constant c

depending on R. Consequently, the map M deﬁned in (II.8.19)–(II.8.20) is not

surjective if q

∈ (1, n/(n − 1)] and not injective if q

∈ [n, ∞).

Section II.9. Results simila r to those derived in Theorem II.9.1, i n the gen-

eral context of spaces D

m,q

, m ≥ 1, have been shown by Mizuta (1989).

Estimate (II.9.5) is o f a particular interest since, as we shall see i n Chapter X,

it permits us to derive at once an imp ortant asymptotic estimate fo r solutions

to the steady, two-dimensional Navier–Stokes equations in exterior domains

having velocity ﬁelds with bounded Dirichlet integrals.

Section II.10. The case 1 ≤ q < n in Theorem II.10.1 is due to me.

Section II.11. If in the Sobolev Theorem II.11.3 one considers the function

II.12 Notes for the Chapter 137

ψ(x) =

|x−y|≤R

f(y)|x − y|

−λ

dy,

for ﬁxed R > 0, the proof of (II.11. 13) becomes elementary; however, o nly

for 1/s > λ/n + 1/q − 1 (see Sobolev 1938; 1963a, Chapter 1 §6). For a

generalization of the Sobolev theorem in weighted Lebesgue spaces, along the

same lines of Theorem II.1 1.5, we refer to Stein & Weiss (1958).

III

The Function Spaces of Hydrodynamics

O voi, che avete gl’intelletti sani,

mirate la dottrina che s’asconde

sotto il velame degli versi strani!

DANTE, Inferno IX, vv. 61-63

Introduction

Several mathematical problems related to the moti on of a vi scous, incompress-

ible ﬂuid ﬁnd their natural formulation in certain spaces of vector functions

that can be considered as characteristic of those problems. These functional

spaces are of three types, denoted by H

, H

, and D

1,q

, and are deﬁned as

suitable subspaces of sol enoidal functions of [L

]

, [W

1,q

]

, and [D

1,q

]

, re-

spectively, n ≥ 2. Actually, it is just the solenoidality restriction that makes

these spaces peculiar and, as we shall see, poses problems that otherwise would

not arise.

The main objective of this chapter is to study in detail the relevant prop-

erties of the ab ove spaces.

If Ω has a compact (and suﬃciently smooth) boundary, the function class

= H

(Ω) can be characterized as the subspace of [L

(Ω)]

of solenoida l

vectors in Ω having zero normal compo nents at ∂Ω. The space H

comes into

the picture as a by-product of a more general question related to a certain

decomposition of the vector space [L

]

, the Helmholtz–Weyl decompositio n.

This decomposition plays a fundamental role in the mathematical theory of

the Navier–Stokes equations, mainly for the study of ti m e-dependent motions.

As we shall see, the validity of the decomposition is equivalent to the unique

solvability of an appropriate Neumann problem in weak fo rm. Such a prob-

lem is certainly resolvable in domains having a (suﬃciently smooth) com pact

139G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations:

Steady-State Problems, Springer Monographs in Mathematics,

DOI 10.1007/978-0-387-09620-9_3, © Springer Science+Business Media, LLC 2011

140 III The Function Spaces of Hydrodynamics

boundary and in a half-space. However, there are also domains with boundary

of little regularity (locally Lipschitz) and domains with smooth noncompact

boundary where the Neumann problem is not uniquely solvable and, therefore,

the corresponding Helm holtz–Weyl decomposition does not hol d.

The main, basic question that one has to face when dealing with spaces

and D

1,q

is related to the very deﬁnition of the spaces themselves. To

see why, let us consider, for the sake of deﬁniteness, the space H

, analogous

reasonings being valid for D

1,q

. To study the time-dependent motion of the

ﬂuid we need the velocity ﬁeld v of the particles of the ﬂuid together with its

ﬁrst spatia l derivatives to be, at each time, summable in the region of ﬂow Ω

to the qth power for some q ≥ 1; in addition, v has to be solenoidal and vanish

at the boundary of Ω. A space of vector functions having such properties (in

a generalized sense) can be chosen in either of the following ways:



completion of D(Ω) in the norm of [W

1,q

(Ω)]



≡ H

(Ω)

v ∈ [W

1,q

(Ω)]

: ∇ · v = 0 in Ω

≡

(Ω),

with D(Ω) denoting the subclass of [C

∞

(Ω)]

of solenoidal functions. These

spaces may look simi lar, but in fact a priori they are not, since the con-

dition of solenoidality on their members is imp osed before (in H

(Ω)) and

after (in

(Ω)) having taken the completion of [C

∞

(Ω)]

in the norm of

1,q

(Ω)]

. Of course, understanding the relationship b etween H

(Ω) and

(Ω) is a preliminary and fundamental q uestion whose analysis aims to

clarify the framework within which the Navier–Stokes problem has to be set.

Actually, as pointed out for the ﬁrst time by Heywood (1976), the coincidence

of the two spaces is related to the uniqueness of solutions and in particular,

in domains for which H

(Ω) 6=

(Ω) the soluti on may not be uniquely de-

termined by the “traditional” initi al and boundary data but other extra and

appropriate auxili ary conditions are to be prescribed (see Chapters VI and

XII).

A prima ry objective of this chapter will be, therefore, to analyze to some

extent for which domains the coincidence of the spaces H

and D

1,q

holds and fo r which it does not. Sp eciﬁcally, we shall see that coincidence is

essentially not related to the smoothness of the domain but rather to its shape.

In particular, the above spaces may not be the same only for domains with a

noncompact boundary, and we shall provide a large class of such doma ins for

which, in fact, they do not coincide.

Another question with whi ch we shall be dealing, and is technically some-

what related to the one just described, is that of the approximation of func-

tions from H

∩ H

[respectively, D

1,q

∩ D

1,r

], with r 6= q, in the no rm of

∩ H

[respectively, D

1,q

∩ D

1,r

], by functions from D(Ω). If there were

no solenoidality constraints, the question would be rather classical and would

ﬁnd its answer in the standard literature. However, since we are dealing with

III.1 The Helmholtz–Weyl Decomposition of the Space L

141

solenoidal ﬁelds, the problem becomes more complicated and we are able to

solve it only for a certain class o f domains including domains with a smooth

enough compact boundary.

Finally, we wish to mention that all problems described previously need a

careful study of the properties of the solutions of the equation ∇·u = f, for a

suitably ascribed f. Such an a uxiliary problem will therefore also be analyzed

in great detail.

III.1 The He lmholtz–Weyl Decomposition of the Space

It has been well known, since the work of H. von Helmholtz in electromag-

netism (Helmho ltz 1870), that any smooth vector ﬁeld u i n R

that falls oﬀ

suﬃciently fa st at large distances can be uniquely decomposed as the sum

u = u

+ u

(III.1.1)

of a gradient and a curl. In other words, u

and u

can b e taken of the form

= ∇ϕ, u

= ∇ × A, (III.1.2)

where ϕ and A are the scalar and vector potential, respectively. In fact, setting

U(x) = (E ∗ u) (x),

with E denoting the fundamental solution of Laplace’s equation (II.9.1), it

follows that ∆U (x) = u(x); see Exercise II.11.3. Putting into this equation

the identity

∆V = ∇(∇ · V ) − ∇ ×(∇ × V ), (III.1.3)

relations (III.1 .1) and (III.1.2) f ollow with

ϕ = ∇·U , A = −∇ × U .

Much later than 1870, it was recognized that decompositions of the type

just described, once formulated in suitable function spaces, become useful

tools in the theory of partial diﬀerential equati ons. A systematic study of

space decomposition was initiated by Weyl (1940) and continued by Friedrichs

(1955), Bykhovski & Smirnov (1960), and others, until the recent work of

Sima der & Sohr (1992). In this respect, the decomposition of the space of

vector functions in Ω having com ponents in L

(Ω), which we continue to

denote by L

(Ω),

into the direct sum of certain subspaces is of basic interest

Let X be any space of real functions used in this book. Unless confusion arises,

we shall use the same symbol X to denote the corresponding space of vector and

tensor-valued functions.

142 III The Function Spaces of Hydrodynamics

in theoretical hydrodynamics and to this problem we will devote the present

section.

We begin to introduce some classes of functions. Let Ω ⊆ R

, n ≥ 2.

Setting

D = D(Ω) = {u ∈ C

∞

(Ω) : ∇ · u = 0 in Ω},

for q ∈ [1 , ∞) we denote by H

= H

(Ω) the completion of D in the norm o f

and put

= G

(Ω) =

w ∈ L

(Ω) : w = ∇p, for some p ∈ W

1,q

loc

(Ω)

. (III.1.4)

For q = 2 we will simply write H and G in place of H

and G

. Obviously, H

is a subspace of L

; moreover, from Exercise III.1 .2, it follows that G

also is

a subspace of L

Referring the study of the relevant properties of these spaces to the next

section, in the present section we will investigate the validity of the decompo-

sition

(Ω) = G

(Ω) ⊕ H

(Ω), (III.1.5)

where ⊕ denotes direct sum operation. In other words, we wish to determine

when an arbitrary vector u ∈ L

(Ω) can be uniquely expressed as the sum

u = w

+ w

, w

∈ G

(Ω) and w

∈ H

(Ω). (III.1.6)

Remark III.1.1 The validity of the decomposition (III.1.5) implies the ex-

istence of a unique projecti on operator

: L

(Ω) → H

(Ω),

that is, of a linear, bounded, idempotent (P

= P

) operator having H

(Ω) as

its range and G

as its null s pace (Rudin 1973, §5.15(d)). In the case q = 2,

we set P

≡ P . 

We shall show that the validi ty of (III.1.5) is equivalent to the unique

resolubi lity of an appropriate (generalized) Neumann problem in Ω (NP,

say); see Lemma III.1.2 and also Simader & Sohr (1 992). Now, if q = 2, just

employing the Hilbert structure of the space L

, we prove that (III.1.5) is valid

for any domain Ω (see Theorem III.1.1), thus obtaining, as a by-product, the

solvability of NP for q = 2 in arbitrary domains. On the other hand, if q 6= 2,

in order to obtain (III.1.5) we directly address the solvability of NP, which

a priori depends on the value of the exponent q, on the “shape” of Ω, a nd

on its regula rity. Sp eciﬁcally, if q 6= 2, we show that if Ω is either a bounded

or an exterior C

-smooth domain

or a half space, then (II I.1.5) holds, see

Theorem I II.1.2. On the other hand, for a certain class o f domains with an

The regularity of Ω can be further weakened (Simader & Sohr 1992).

III.1 The Helmholtz–Weyl Decomposition of the Space L

143

unbounded bo unda ry (no matter how sm ooth), the corresponding NP looses

either solvability or uniqueness and, therefore, in such a class, the Helm holtz–

Weyl decomposition does not hold (see Remark III.1.3, Exercise III.1.7 and

Bogovski

i 1986 and Maslennikova & Bogovski

i 1986a, 1986b, 1993). Analogous

considerations hold if the doma in is bounded and w ith little regula rity; see

Fabes, Mendez & Mitrea (1 998, Theorem 12.2) and Remark III.1.3.

Exercise III.1.1 Given a reﬂexive Banach space X and a subset S of X, the an-

nih ilator S

⊥

of S is a subset of the dual space X

deﬁned as

⊥

= {` ∈ X

: `(x) = 0, for all x ∈ S}

(Kato 1966, p. 16). If, in particular, X is a Hilbert space, then S

⊥

is said to be

the orth ogonal complement of S. In this case, two subsets S

, S

of X are called

orthogonal if S

⊂ S

⊥

(or, equivalently, S

⊂ S

⊥

). Show that: (a) S

⊥

is a closed

subspace of X

; (b) H

⊥

⊃ G

, q ∈ (1, ∞), (1/q + 1/q

= 1), so that, for q = 2, H

and G are orthogonal.

Exercise III.1.2 Show that G

is a closed subspace of L

. Hint: Use the methods

adopted in the proof of Lemma II.6.2.

Fundamental to further development is the characterization of the class of

vectors u ∈ L

loc

(Ω) that are “orthogonal” to all vectors w ∈ D(Ω), i.e.,

Ω

u · w = 0, for a ll w ∈ D(Ω). (III.1.7)

Exercise III.1.3 Show that, for u ∈ L

loc

(Ω), condition (III.1.7) is equivalent to

Ω

u · w = 0, for all solenoidal w ∈ C

(Ω).

If Ω is a simply connected doma in in R

and u is continuously diﬀerentiable

one proves at once that u = ∇p for some smooth single-valued scalar function

p. In fact, for arbitrary h ∈ C

∞

(Ω), let us choose in (III.1.7) w = ∇× h and

use the identity

∇ · (v

× v

) = v

· ∇ × v

−v

· ∇ ×v

to deduce

Ω

∇ × u · h = 0, for all h ∈ C

∞

(Ω), (III.1.8)

which in turn, by Exercise II.2.9, implies ∇×u = 0. Being Ω simply connected,

this last condition furnishes, by the Stokes theorem, u = ∇p, with p a suitable

line integral of the diﬀerential form

u ·dx =

i=1

144 III The Function Spaces of Hydrodynamics

Now, for this procedure to hold, the assumption on the regularity of u can

be fairly weakened (see the last part of Lemma III.1.1), while it is crucial the

assumption Ω be simply connected; otherwise p need not be singl e-valued. The

aim of the following lemma i s to show that the result just proved continues

to be valid for any domain in R

. The method we shall employ is based on

an idea of Fujiwara & Morimoto (1977 , p. 697) and is due to Simader & Sohr

(1992).

Lemma III.1.1 Let Ω be an arbitrary domain in R

. Suppose that u ∈

loc

(Ω) veriﬁes (III.1.7). Then, there exists a single-valued scalar function

p ∈ W

1,1

loc

(Ω) such that u = ∇p.

Proof. Assume ﬁrst u ∈ C(Ω). The proo f will be achieved if we show that

the line integral of the diﬀerential form u · dx is zero along all closed nonin-

tersecting po lygonals lying in Ω.

Let Γ denote any such curve; we may then

represent it by a continuous function γ such that

γ : [0, 1] → R

γ ∈ C

∞

([t

, t

i+1

]),

where 0 = t

< t

< . . . < t

= 1 and γ(0) = γ(1). For w ∈ C(Ω) we thus

have

w · dx =

i=0

i+1

w(γ(t)) ·

dγ

dt.

Let ε

= dist (Γ, ∂Ω). For x ∈ Ω we set

(x) =

i=0

i+1

(x − γ(t))

dγ

dt,

where j

(ξ) = ε

−n

j(ξ/ε) is a mollifying kernel of the type introduced in

Section II.2 and 0 < ε < ε

. Obviously, Φ

∈ C

∞

(Ω) for all such ε, and since

∇ · Φ

(x) =

i=0

i+1

∇

(x)

(x − γ(t)) ·

dγ

= −

i=0

i+1

(x − γ(t))] dt

= −j

(x − γ(1)) + j

(x − γ(0)) = 0,

Recall that, since Ω is open and connected, it is also polygonally connected.

Namely, given x, x

∈ Ω we can ﬁnd a non-intersecting polygonal joining x with

III.1 The Helmholtz–Weyl Decomposition of the Space L

145

it follows that Φ

∈ D(Ω). Now, by deﬁnition of molliﬁer and by (II I.1.7) we

have

· dx =

i=0

u(x) ·

dγ



i+1

(x − γ(t))dt



dx =

Ω

u ·Φ

= 0.

(III.1.9)

Therefore, l etting ε → 0 in (III.1.9) and using the properties of m olliﬁers, we

ﬁnd

u · dx = 0, for all Γ ,

which impli es u = ∇p w ith p ∈ C

(Ω). The lemma is therefore proved when

u ∈ C(Ω). Assume now merely u ∈ L

loc

(Ω). Let O be the open covering of

Ω introduced in Lemma II.1.1, and let B

∈ O. We choose ε > 0, such that

ε < dist (B

, ∂Ω) and let w be a n arbitrary vector in D(B

). From Section

II.2 we deduce that the regulari zation w

of w belongs to D(Ω) and thus, by

assumption and Fubini’s theorem, we obtain

0 =

Ω

u · w

= ε

−n

u(x)





x − y



w(y) dy



= ε

−n





x − y



u(x) dx



w(y)dy =

· w.

Since u

∈ C

∞

) and w ranges arbitrarily in D(B

), from the ﬁrst part

of the proo f we ﬁnd u

= ∇p

in B

, for some p

∈ C

∞

). Set ε = 1/m,

m ≥ m

∈ N

, and let m → ∞. By an argument completely analogous to

that used in the proof of Lemma I I .6.2, we show that p

1/m

converges to some

(0)

∈ W

1,1

) such that u = ∇p

(0)

a.e. in B

. Now, by Lemma II.1.1, we

can ﬁnd B

∈ O such that B

∩ B

≡ B

1,2

6= ∅, and so, by the same kind

of argument, we can ﬁnd p

(1)

∈ W

1,1

) such that u = ∇p

(1)

a.e. in B

Since p

(0)

= p

(1)

+ c a.e. in B

1,2

, we may modify p

(1)

by the addition of a

constant in such a way that p

(0)

and p

(1)

agree on B

1,2

. Let us continue to

denote by p

(1)

the modiﬁed function and deﬁne a new function p

(1,2)

which

equals p

(0)

on B

and equals p

(1)

on B

. Clearly, p

(1,2)

∈ W

1,1

∪B

) and,

furthermore, u = ∇p

(1,2)

a.e. in B

∪ B

. In view of the properties of the

covering O, we can repeat thi s pro cedure to show, by induction, the existence

of a function p ∈ W

1,1

loc

(Ω) sati sfy ing the statement in the lemma which is,

therefore, compl etely proved. ut

As an immediate consequence of the previous result, we deduce the validity

of (III.1.5 ) for q = 2.

Theorem III.1.1 Let Ω be an arbitrary domain in R

, n ≥ 2. Then G(Ω)

and H(Ω) are orthogonal subspaces in L

(Ω). Moreover

(Ω) = G( Ω) ⊕H(Ω).