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

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226 III The Function Spaces of Hydrodynamics

Approximation problems in the spaces H

∩

i=1

1,r

(Ω)

and in

the space H

∩

i=1

1,r

(Ω)

[respectively, D

1,q

∩

i=1

1,r

(Ω)

] and

1,q

∩

i=1

1,r

(Ω)

can be treated by the same technique used before.

Their resolution is left to the reader in the fo llowing two exercises.

Exercise III.6.1 Under the hypothesis on Ω, q and r

stated in Theorem III.6.1,

show that any function v ∈ H

(Ω) ∩

∩

i=1

1,r

(Ω)

[ respectively, H

(Ω) ∩

∩

i=1

1,r

(Ω)

], can be approximated in the space H

(Ω) ∩

∩

i=1

1,r

(Ω)

[re-

spectively, in H

(Ω) ∩

∩

i=1

1,r

(Ω)

] by functions from D(Ω).

Exercise III.6.2 Under the hypothesis on Ω, q, and r

stated in Theorem III.6.1,

show that any function v ∈ D

1,q

(Ω) ∩

∩

i=1

1,r

(Ω)

[respectively, D

1,q

(Ω) ∩

∩

i=1

1,r

(Ω)

], 1 < q, r

< ∞, can be approximated in the space D

1,q

(Ω) ∩

∩

i=1

1,r

(Ω)

[respectively, in D

1,q

(Ω)∩

∩

i=1

1,r

(Ω)

] by functions from D(Ω).

Exercise III.6.3 Prove the interpolation inequality of Nirenberg given in (III.6.16)–

(III.6.18). Hint: Use the “cut-oﬀ” method of Theorem II .6.3 to approximate any u

satisfying (III.6.16) wit h functions from C

∞

). Successively, employ inequality

(II.3.5) together with the H¨older inequality.

III.7 Notes for the Chapter

Section III.1. As already p ointed out, Lemma III.1.1 plays a n im portant

role i n the theory of the Navier–Stokes equations. It is then not surprising

that it has received the attention of m any writers who proved it by several

methods and under more or less diﬀerent assumptions on the regularity of u.

To our knowledge, a ﬁrst, elementary demonstration of the result was pro-

posed by Hopf (1950/1 951, pp.214-215) without, however, giving full details.

Hopf’s pro of, which essentially aims at showing that the line integral of u on

every closed loop is zero, was clariﬁed and completed by Pro di (1959) a nd

Ladyzhenskaya (1969). The assumptio n n = 3 implicitly made by these au-

thors is removed, along the same line of method, by Temam (1973, Chapter I,

§1.4). A less elementary proof based on de Rahm’s theorem on currents, which

assumes u to be only a distribution, is given by Lio ns (1969, pp. 67-69) and

Temam (1977, Chapter I, Proposition 1.1). Tartar gives an alternative proof,

with u in a negative Sobolev space, based o n operator theory (see Temam

1977, Chapter I, Remark 1.9). A similar result is furnished by Giga & Sohr

(1989, Corollary 2.2(i)). For other proofs that avoid de Rahm’s theorem, see

also Fujiwara and Morimoto (1 977) and Simon (1991, 1993).

The Helmholtz–Weyl decomposition of the vector space L

(Ω) in domains

with a compact boundary has been the object of several investigations. In

addition to the papers quoted in Section III.1, we refer the reader to von

Wahl (1990b) and the bibliography of the work of Simader & Sohr (1992).

III.7 Notes for the Chapter 227

We also would like to mention the paper of Fabes, Mendez & Mitrea (1998,

§§11,12) where sharp results are furnished for the validity of the decompositio n

in bounded domains with lower regularity (locally Lipschitz). In particular,

in Theorem 12.2 of Fabes, Mendez & Mitrea, loc. cit. it is shown the exis-

tence of a bounded a nd Lipschitz domain of R

, where the Helmholtz–Weyl

decomposition fails for all p 6∈ [3/2, 3].

Concerning the Helmholtz–Weyl decomposition in domains with noncom-

pact boundaries, we would like to mention the following im portant contribu-

tions. For the aperture domain (see (III. 4.4)), its validity has been proved

by Farwig (1993 ) and Farwig and Sohr (1996). Miyakawa (1994) showed an

analogous result for semi-inﬁnite cylinders and for inﬁnite layers in R

, n ≥ 2,

using Littlewood-Paley theory. Wiegner (1995) studied the case of a layer i n

by means of (partia l) Fourier transforms. Th¨ater (1995) and Sohr and

Th¨ater (1998) proved the validity of the decomposition for inﬁnite cylinders

of R

, n ≥ 2, based on estimates for imaginary powers of the operator asso-

ciated to the associated Neumann problem. Finally, an elementary and deep

analysis of the decomposition in the case of inﬁnite cylinders and layers of

, n ≥ 2, can be found in the paper by Simader and Ziegler (1998). For

further and more recent contributions to this question, we refer to the article

of Farwig (2003) and the bibliography therein.

Decompositions of weighted Lebesgue spaces on exterior domains have

been studied by Specovius-Neugebauer (1990, 1995) and Fr¨ohlich (2000).

Decompositions in Sobolev and Besov spaces are investigated in Fujiwara

and Yamazaki (2007).

Section III.2. The notion of trace on the bounda ry for functions in

, for

q = 2, was introduced by Temam (19 73, Chapter I), starting with identity

(III.2.2). The same question was independently addressed by Fujiwara & Mo-

rimoto (1977) for 1 < q < ∞, who generalized the results of Temam by means

of a diﬀerent (and less direct) approach.

Section III.3. The auxiliary problem considered in this section has also been

studied in detail by Ca ttabriga (1961) for n = 3 and q ∈ (1 , ∞); Ladyzhen-

skaya (1969, Chapter I, §2) for n = 2, 3 and q = 2; Neˇcas (1967, Chapitre 3,

Lemme 7.1) for n ≥ 2 and q = 2; Babuska & Aziz (1972, Lemma 5.4.2) for

n = q = 2 (see a lso Oden & Reddy 1976, Lemma 6.3.2); Solonnikov &

Sˇcadilov

(1973) and Ladyzhenskaya & Solonnikov (1976) fo r n = 2, 3 and q = 2; Amick

(1976) for n ≥ 2 and q = 2; Bogovski

i (1979, 1980) and Erig (1982), for n ≥ 2

and 1 < q < ∞; Pileckas (1980b, 1983) for n = 2, 3 and 1 < q < ∞; Giaquinta

& Modi ca (1982) for n ≥ 2 and q = 2; Solonnikov (1983) a nd Kapitanski

i &

Pileckas (1984) for n ≥ 2 and q ∈ (1, ∞); Arnold, Scott & Vogelius (1988)

for n = 2 and q ∈ (1, ∞); von Wa hl (1989, 1990a) for n = 3 and q ∈ (1, ∞);

Borchers & Sohr (1990) for n ≥ 2 and q ∈ (1, ∞),

; Bourgain & Brezis (2003)

Proposition d) of Theorem 2.4 of Borchers & Sohr (1990) is not correct as stated.

The corrected version is furnished in Corollary 2.2 of Farwig & Sohr (1994a); see

also Remark (iii) and the footnote at p. 274 of this latter paper.

228 III The Function Spaces of Hydrodynamics

when n ≥ 2 and q ∈ (1, ∞), and ﬁnally, by Dur´an & Lop´ez Garc´ıa (2010) for

n = 2 and q ∈ (1, ∞). The main diﬀerence among the results proved by these

authors relies either upon the method used to construct v or upo n the regu-

larity assumptions made on Ω. In particular, the method of Bogovski

i is based

on the explicit representation formula (III.3.8) which adapts to the di vergence

operator a well-known formula of Sobolev (1963a, Chapter 7, §4; speciﬁcally,

see eq. (7.9)). A similar representation for the curl operator has been given

by Griesinger (1990a, 1990b). Wi th the exception of Solonnikov &

Sˇcadilov

(1973), Bogovski

i (1980), Kapitanski

i & Pileckas (1984), a nd Borchers & Sohr

(1990), all other mentioned authors consider the case Ω bounded. However,

once (III.3.1), (III.3.2) is solved for such domains, the problem for Ω exterior

can b e directly handled by using the technique of Theorem III.3.6. The case

Ω = R

requires, apparently, a separate treatment and will be considered in

Section IV.3; see Corol lary IV.3.1. In this respect, we refer the reader to the

papers of Cattabriga (1961), Solonnikov &

Sˇcadilov (1973 ) and Solonnikov

(1973, 1983). These latter two papers deserve particular attention where ex-

plicit sol utions are given (see Solonnikov 1973, formula (2.38), and Solonnikov

1983, Lemma 2.1).

Problem (III.3.1), (III.3.2) can also be solved in Sobolev spaces W

s,q

(Ω)

with s real, Bogovski

i (1979 , 1980 ), and in certain weighted Sobolev spaces

(Voldrich 1984). In this respect, as we already observed i n Remark III.3.5,

problem (III. 3.1), (I II.3.2) can not be solved i n (bounded) domains having an

external cusp. Nevertheless, Dur´an & L´opez Garc´ıa (2 010) have shown that,

for such domains, it can still be solved in suitable weighted Sobolev spaces,

with weights depending on the type of cusp.

For Ω exterior, results i n weighted Sobolev spaces can be obtained by

using, in Theorem III.3.6, Stein’s Theorem II.11.5 instead of Theorem II.11.4.

In this regard, we refer the reader to the papers of Specovious-Neugebauer

(1986) and L ockhart & McOwen (1983). We ﬁna lly mention that the same

type of problem can be analyzed for the equation curl v = f . In addition to

the already cited pap ers of Gri esinger, we refer the interested reader to the

book of Girault & Raviart (1986) and to the works of Borchers & Sohr (1990),

von Wahl (1989, 1990a) and Bolik & von Wahl (1997).

Theorem III.3.4 is due to Gal di (1992a). A diﬀerent pro of of Theorem

III.3.5 (with slightly more stringent assumptions on the regularity of Ω, is

given in Farwig & Sohr (1994a, Corollary 2.2). Extensions of these results

to Sobolev spaces W

s,q

(Ω) with s real and (suitably) negative are shown in

Geissert, Heck, a nd Hieber (2006, §2).

The numerical value of the constant c appearing in (III.3.2)

and (III.3.65)

is very important for several applications, see, e.g., Chapters VI and XII. In

this respect, we refer the reader to the papers of Horga n & Wheeler (1978),

Horgan & Payne (1983), Velte (1990), and Stoyan (2001).

A diﬀerent proof of Theorem III.3.7, origi nall y due to Kapitanski

i &

Pileckas (1984, Theorem 1), was provided, in a diﬀerent context, by Dacorogna

(2002).

III.7 Notes for the Chapter 229

Section III.4. An elementary proof of the coincidence of H

(Ω) and

(Ω)

for Ω bounded was ﬁrst given by Heywood (1 976) for n = 2, 3 and q = 2. His

method, which can be easily extended to the cases n ≥ 2 and q ∈ [1, ∞), is

a generalization of that used for a star-like domain. Actually, he introduces

a smooth transformation T

(x), x ∈ Ω, ρ ∈ (0, 1 ) that replaces the simple

contraction ρx used fo r the star-like case. For this procedure to hold it is,

of course, necessary that Ω have some regularity, and Heywood shows that

smoothness is suﬃcient. However, as proved in Theorem 1 o f Heywo od’s

paper, the procedure would equally work with much less regularity. A proof of

the coincidence for n ≥ 2 and q = 2 using de Rahm’s theorem on currents was

previously furnished for Ω locally Lipschitz by Lions (1969, pp. 67-68; here

the domain must be locally Lipschitz even if it is not explicitly stated) and

by Temam (1977, Chapter I, Theorem 1.6). The same result is established by

Salvi (1982) using extensions of sequentially continuous functionals. Other,

diﬀerent approaches are employed by Ladyzhenskaya & Solonnikov (1976) for

n = 2, 3 and q = 2 and, along the same lines, by Pileckas (1980b, 1983) for

arbitrary q ∈ [1, ∞) and, for n ≥ 2 and q ∈ [1, ∞), by Bogovski

i (1980 ). In

particular, Bogovski

i requires only that Ω is the union of a ﬁnite number of

domains each of which is star-shaped with respect to a ball (see Theorem

III.4.1); for example, Ω satisﬁes the cone condition. More recently, Wang &

Yang (2008) have shown coincidence for n = 2, 3 and q = 2, provided Ω has

only a kind of segment property. It is, however, still an open question to prove

(or disprove) the validity of the coincidence for bounded domains with no

regularity. In this respect, we wish to m ention a result given in

Sver´ak (1993),

Remark at p. 12, which shows

(Ω) = H

(Ω), where Ω is any bounded open

set of R

such that, denoting by B an open ball with B ⊃ Ω, we have that the

set B − Ω has a ﬁnite number of connected components. Probably, it is true

that

(Ω) = H

(Ω) (or, equivalently,

1,2

(Ω) = D

1,2

(Ω)), for any bounded

domain in R

, n ≥ 2. It is worth emphasizing that should this not be true

for some bounded open connected set Ω

]

, the Stokes problem formulated in

Ω

]

corresponding to zero body forces and zero boundary data would admit a

nonzero smooth solution; see Remark IV.1.2.

The case of a n exterior domain has likewise been analyzed by Heywood

(1976), Ladyzhenskaya & Sol onnikov (1976) for n = 2, 3 and q = 2 , by Pileckas

(1980b, 1983) for n = 2, 3 and q ∈ (1, ∞), and by Bogovski

i (1 980) for n ≥ 2

and q ∈ (1, ∞ ). All these authors prove the coincidence of H

(Ω) and

(Ω)

under the same regularity assumptions made on Ω f or the corresponding

bounded case. A more elementary proof for n ≥ 2 and q ∈ [1, ∞) that re-

quires Ω

be star-shaped is provided by Bogovski

i & Maslennikova (1978)

and Maslennikova & Bogovski

i (1978).

Lemma I II.4.2 is due to me. I have been kindly informed by Professor K.

Pileckas that a simila r result has been independently proved by Professor V. I.

Burenkov and it appears in note 5.3 to Chapter VI of the Russian transla tion

of the book of Stein (1970).

230 III The Function Spaces of Hydrodynamics

Section III.5. The papers mentioned in the notes to Section III.4 concerning

the coincidence of the spaces H

(Ω) and

(Ω) also deal with the same

problem for D

1,q

(Ω) and

1,q

(Ω).

The relation between linear functionals on D

1,q

(Ω) vanishing on D

1,q

(Ω)

and the existence of a pressure ﬁeld for Stokes and Navier–Stokes problems

was ﬁrst recognized by Solonni kov &

Sˇcadilov (1973 ), who prove Corollary

III.5.1 for q = 2, n = 3 and Ω of class C

. The same result was rediscovered

thirteen years later by Guirguis (1986).

Section III.6. The results gi ven here are due to Galdi (1992a). They will

be used in several questions concerning Navier–Stokes equations, such as the

validity of the energy identity in exterior domains (see Section X.2). Similar

results, with diﬀerent techniques and much more regularity on the domain,

are contained in the works of Giga (1986) and Kozono & So hr (1992a); see

also the Appendix of Masuda (198 4) and L emma 3.8 of Maremonti (1991).

Steady Stoke s Flow in Bounded Domains

Ora sia il tuo pa sso

pi`u ca uto: ad un tiro di sasso

di qui ti si prepara

una pi`u rara scena.

E. MONTALE, Ossi di Seppia.

Introduction

We now undertake the study of the mathematical properties of the motion

of a viscous incompressible ﬂuid. We shall begin with the simplest situation,

namely, that of a steady, inﬁnitely slow motion occurring in a bounded region

Ω. The hypothesis of sl ow m otion means that the ratio

|v · ∇ v|

|ν∆v|

of inertial to viscous fo rces is vanishingly small, so that we can disregard the

nonli near term into the full (steady) Navier–Stokes equations (I.0.3

). If we

introduce reference length L and velocity V , this approximation amounts to

assume that the (dim ensionless) Reynolds number

R =

V L

is suitably small.

231G.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_4, © Springer Science+Business Media, LLC 2011

232 IV Steady Stokes Flow in Bounded Domains

The linearization procedure can be performed around a generic solution

, p

, say, of equations (I.0. 1). In thi s chapter (and the next two) we shall

consider the case where v

≡ 0, p ≡ const., so that we recover the following

Stokes equations (see Stokes 1845)

∆v = ∇p + f

∇· v = 0

)

in Ω. (IV.0.1)

Here we have formally put, without loss, the coeﬃcient of kinematic viscosity

ν equal to unity. To system (IV.0.1), we app end the usual adherence condition

(I.1. 1) at the b oundary, that is,

v = v

∗

at ∂Ω. (IV.0.2)

Since Ω is bounded, from (IV.0.1)

and Gauss theorem, it follows that the

prescribed velocity ﬁeld v

∗

must satisfy the compatibility condition:

∂Ω

∗

· n = 0. (IV. 0.3)

The main objective of this chapter is to show existence, uniqueness, and

regularity along wi th appropriate estimates for solutio ns to problem (IV.0.1)–

(IV.0.3). In doing this, we shall be inspired by the work of Cattabriga (1961)

and Galdi & Simader (1990). Speciﬁcally, we ﬁrst give a variational (weak)

formulation of the problem and introduce the concept of q-generali z ed solu-

tion (for q = 2, simply: generalized solution). These solutions are essentially

characterized by the property of bei ng members of the space D

1,q

(Ω) and a

priori they do not possess enough regulari ty to be considered as solutions in

the ordinary sense. Following the work of Ladyzhenskaya (1959b), it is simple

to show the existence of a generalized solution to (IV.0.1)–(IV.0.3). However,

it is a much more diﬃcult job to study its regularity, that is, to show that,

under suitable smoothness assumptions on f, v

∗

, and Ω, such a solution b e-

longs, in fact, to the Sobolev space W

m,q

(Ω) and that it obeys corresponding

estimates:

kvk

m+2,q

+ kpk

m+1,q

≤ c



kfk

m,q

+ kv

∗

m+2−1/q,q(∂Ω)



, (IV.0.4)

with m ≥ 0 and q ∈ (1, ∞).

Since system (IV.0.1), (IV.0.2) is elliptic in the sense of Douglis-Nirenberg

(see Solonnikov 1966, Temam 1977, pp. 33-34), the validity of a weaker form

of estimate (IV.0.4 ), namely,

kvk

m+2,q

+ kpk

m+1,q

≤ c



kf k

m,q

+ kv

∗

m+2−1/q,q(∂Ω)

+ kvk

+ kpk

−1,q



holdi ng fo r q-generalized solutio ns, can be obtained directly from the gen-

eral theory of Agmon, Douglis, & Nirenberg (1964) and Solonnikov (1966)

(without, however, providing existence).

IV.1 Generalized Solutions. Existence and Uniqueness 233

Here, to reach our goal, we shall follow a classical approach due to Cat-

tabriga (1961) that relies on the ideas of Agmon, Douglis, & Nirenberg (1959).

This method consists in transforming the problem into analogous probl ems

in the whole space and in a half -space by means of the “localization pro-

cedure.” Now, i n R

and R

, the task of proving the unique solvability of

(IV.0.1), (IV.0.2), and (IV.0.4) is rendered easy by the circumstance that one

can furnish an explicit solution to the problem. It is worth noticing that such

a procedure is completely simil ar to that employed fo r the Poisson equation

at the end of Section II.11 and tha t the only tool needed is the Calder´on–

Zygmund Theorem II.11.4 a nd its variant as given in Theorem II.11.6. We

also wish to emphasize that the study of the Stokes problem in R

and in

possesses an independent interest and that it will be fundamental for the

treatment of other (linear and nonlinear) problems when the region of ﬂow is

either an exterior domain or a domain with a suitable unbounded boundary.

By the same arguments, we shall also show existence and uniqueness of

q-generalized solutions when q 6= 2 and shall derive corresponding estimates,

formally obtained by tak ing in (IV. 0.4) m = −1 a nd q ∈ (1, ∞).

We end with a ﬁnal remark. As a rule, we shall treat in detail only the

physically interesting cases when the relevant region of motion is either a

three-dimensional or (for a plane ﬂow) a two-dimensional domain. In par-

ticular, all results will be essentially proved for space dimension n = 2, 3.

However, whenever needed, we shall outline all the main steps to follow in

order to generalize the proof to n ≥ 4.

IV.1 Generalized Solutions. Existence and Unique ness

In this section we shall prove some existence and uniqueness results for Stokes

ﬂow. Following Ladyzhenskaya (1959b), we shall gi ve an integral variational

formulation of the problem, which will then be easily solved by the classical

Riesz representation theorem.

However, the solutions we shall obtain are a

priori not smooth enough to be considered as strict solutions of the starting

problem; for this reason, they are called generalized or weak. Nevertheless, in

the next sections we will show tha t provided the force, the velocity at the

boundary, and the region of motion are suﬃciently regular, weak solutions

are, i n fact, diﬀerentiable solutions of (IV.0 .1)

1,2

in the ordi nary sense and

assume continuously the boundary data.

To justify the generalized (or weak or variational) formulation, we proceed

formally as f ollows. Let v, p be a classical solution to (IV.0.1)

1,2

, for example,

It should be observed that, in spite of its simplicity and elegance, the method of

resolution based on the Riesz theorem is not constructive. The more constructive

Galerkin method will be considered later, directly in the nonlinear context (see

also Chapter VII).

234 IV Steady Stokes Flow in Bounded Domains

v ∈ C

(Ω), p ∈ C

(Ω). Multiplying (0.1

) by an arbitrary function ϕ ∈ D(Ω)

and integrating by parts we deduce

(∇v, ∇ϕ) ≡

Ω

∇v : ∇ϕ = −

Ω

f · ϕ ≡ −(f, ϕ). (IV.1.1)

Thus, every classical solution to (IV.0 .1)

satisﬁes (IV.1.1) f or all ϕ ∈ D(Ω).

Conversely, if v ∈ C

(Ω) and f ∈ C(Ω), from (IV.1.1) and Lemma III.1.1

we show the existence of p ∈ C

(Ω) verifying (IV.0.1)

. On the other hand,

we may think of a function v satisfying (IV.1.1) but which is not suﬃciently

diﬀerentiable to be considered a solution to (IV.0.1)

(for a suitable choice o f

p). In this sense (IV.1.1 ) is a “weak” version of (IV.0.1)

. For further purposes,

we may and shall consider the more general situation in which the right-

hand side of (IV. 1.1) is deﬁned by a f unctional f from D

−1,q

(Ω). We shall

then write hf, ϕi instead of (f , ϕ) where, we recall, h·, ·i denotes the duality

pairing between W

1,q

(Ω) and W

1,q

(Ω), 1/q + 1/q

= 1 (see Section II.3).

As far as the regularity of a weak solution is concerned, we merely require

a priori v ∈ D

1,q

(Ω) for some q ∈ (1, ∞), so that the solenoidal condition

will be satisﬁed according to generalized diﬀerentiation, while the boundary

condition (IV.0.2) is to be understood in the trace sense (see Theorem II.4.4

and R emark II.6.1). If, in particular, the velocity at the boundary is zero, we

require v ∈ D

1,q

(Ω) which, along with the solenoidality condition, furnishes

v ∈

1,q

(Ω); see Remark IV.1.2. We may then summarize all the above in

the following.

Deﬁnition IV.1.1. A ﬁeld v : Ω → R

is called a q-weak (or q-generalized)

solution to the Stokes problem (IV.0.1), (IV.0.2)

if and only if

(i) v ∈ D

1,q

(Ω), for some q ∈ (1, ∞);

(ii) v is (weakly) divergence-free in Ω;

(iii) v satisﬁes the bounda ry condition (IV.0.2) (in the trace sense) or, if the

velocity a t the bo undary is identically zero, v ∈ D

1,q

(Ω);

(iv) v veriﬁes the identity

(∇v, ∇ϕ) = −hf , ϕi (IV.1.2)

for all ϕ ∈ D

1,q

Ω), 1/q + 1/q

= 1.

If q = 2, v will be called a weak (or generalized) solution.

Remark IV.1.1 Si nce Ω is bounded, D

1,q

(Ω) and W

1,q

(Ω) are isomorphic;

see Remark II.6.3. Furthermore, if Ω is locall y Lipschitz, D

1,q

(Ω) endowed

with a suitabl e norm, is isomorphic to W

1,q

(Ω); see Remark II.6.1. Therefore,

if v

∗

≡ 0 we may equivalently require in (i) that v ∈ W

1,q

(Ω), w hile, if v

∗

6≡ 0

and Ω is locally Lipschitz, (i) is equivalent to v ∈ W

1,q

(Ω). 

As agreed,we shall put, wit hout l oss of generality, ν = 1.

Solutions possessing a priori even less regularity than q-weak solutions (the so

called very weak solutions) will be brieﬂy considered in the Notes for this Chapter.

IV.1 Generalized Solutions. Existence and Uniqueness 235

Remark IV.1.2 If the velocity at the boundary is zero, every q-weak solutio n

belongs to

1,q

(Ω). We recall that, in general,

1,q

(Ω) ⊇ D

1,q

(Ω), for each

Ω ⊂ R

, n ≥ 2, and each q ∈ (1, ∞); see Section III.5. In this regard, it is

important to realize that if there exists a domain, Ω

]

such that

1,2

(Ω

]

) 6=

1,2

(Ω

]

one could ﬁnd a nonzero generalized solution, v

]

, to the Stokes

problem (IV.0.1)–(IV.0.2) with Ω ≡ Ω

]

, corresponding to zero force and zero

boundary data. This fact is an immediate consequence of Exercise III.5.3 .

Moreover, as it will be shown in Theorem IV.4. 3, v

]

∈ C

∞

(Ω) and we can

ﬁnd a correspondi ng pressure ﬁeld p

]

∈ C

∞

(Ω), such that the pair (v

]

, p

]

)

satisﬁes the problem (IV.0.1)–(IV.0.2) in the ordinary sense! It is clear that

such a situation is diﬃcult to explain from a physical point of view, in that the

ﬂow (v

]

, p

]

) should be driven merely by the “roughness” of the boundary of the

bounded domain where the motion occurs. In fact, if a mild degree of regularity

on the boundary is assumed, then v

]

= ∇p

]

= 0. These considerations add

more weight to the conjecture that

1,2

(Ω) = D

1,2

(Ω) for every bounded

domain Ω, but, as we remarked several times in the previous chapter (see,

especially, the Notes to Section III.4), no proof of this fact is available to

date. 

In this section we shall establish existence and uniqueness of weak solu-

tions. The analogous questions for q-weak solutions, arbitrary q > 1, will be

considered in Section IV.6. Before performing this study, however, we wish to

make some prelimi nary considerations.

Deﬁnition IV.1. 1 i s apparently silent a bout the pressure ﬁeld. Actually,

this is not true, as we will show. Assume, at ﬁrst, v, p a classical solution and

multiply (IV.0.1)

by ψ ∈ C

∞

(Ω) (not necessarily solenoidal). Integrating by

parts we obtain, instead of identity (IV.1.2),

(∇v, ∇ψ) = −hf, ψi + (p, ∇ · ψ). (IV.1.3)

Now, if f has a mild degree o f regularity, to every q-weak solutio n we are

able to associate a “pressure ﬁeld” p in such a way that (IV.1.3) holds and,

further, we can give a deﬁnition of q-weak solution equivalent to Deﬁnition

IV.1. 1, usi ng (IV.1.3) in place of (IV.1.2) as a consequence of the following

general result.

Lemma IV.1.1 Let Ω be an arbitrary dom ain of R

, n ≥ 2, and let f ∈

−1,q

(Ω

), 1 < q < ∞, for any bounded domain Ω

with Ω

⊂ Ω. A vector

ﬁeld v ∈ W

1,q

loc

(Ω) satisﬁes (IV.1.2) for all ϕ ∈ D(Ω) if and only if there exists

a “pressure ﬁeld” p ∈ L

loc

(Ω) such that (IV.1.3) holds for every ψ ∈ C

∞

(Ω).

If, moreover, Ω is bounded and satisﬁes the cone condition and f ∈ D

−1,q

(Ω),

v ∈ D

1,q

(Ω) then

By the results of Section III.5, Ω

]

should be l ess regular than a domain that is

the union of a ﬁnite number of domains each of which is star-shaped with respect

to a ball.