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

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296 IV Steady Stokes Flow in Bounded Domains

equations, Vorovich and Youdovich (1961, Theorem 2) showed Cattabriga’s

result for m ≥ 0 and q ≥ 6/5. Finally, we wish to mention the ingenious work

of Krzywcky (1 961), where estimates for the Stokes problem are obtained from

the Weyl decomposition of the space L

Since the appearance of these papers, several works have been published

which, among other things, investigate the possibility of generalizing Cat-

tabriga’s theorem in the following two directions: (i) extension to arbitrary

dimension n ≥ 2; (ii) extension to less regular domains. To the best of o ur

knowledge, the ﬁrst attempt toward direction (i) is due to Temam (1973,

Chapter I). However Temam’s arguments work only when q ≥ 2 and m ≥ 0,

if n ≥ 3, and for arbitrary m ≥ −1, q ∈ (1, ∞) if n = 2. In particular, the

proof of this latter result is achieved by showing that the Stokes problem

is equivalent to a suitable bi harmonic problem. In this respect, we refer the

reader to the paper of Sim ader (1992), where an interesting analysis between

these two problems is carried out for any n ≥ 2. Another contribution along

direction (i), in the case where m = 2, is due to Giga (1981, Proposition

2.1), who uses a theorem of Geymonat (1965, Theorem 3.5) o n the invariance

of the index o f the operator associated to an elliptic system in the sense of

Douglis-Nirenberg. This method requires, however, Ω of class C

∞

. Ghidaglia

(1984) has extended Cattabriga’s theorem to arbitrary n ≥ 2 when q = 2. In

this respect, it is worth mentioning the contribution of Beir˜ao da Veig a (1998)

where results similar to those of Ghidaglia are proved, but under much less

regularity on Ω. However, the most important feature of this paper is that

the author avoids potential and/or g eneral el liptic equation theories, whil e he

uses, instead, only the elementary estimate kuk

2,2

≤ c kfk

for the unique

solution u ∈ W

1,2

(Ω) ∩ W

2,2

(Ω) of the scalar Poisson equation ∆u = f.

The full generali zation of the results of Cattabriga to arbitrary space di-

mension n ≥ 2, i.e. Theorem IV.6.1, was establi shed for the ﬁrst time, indepen-

dently, by Kozono & Sohr (1991) and Ga ldi & Simader (1990, Theorem 2.1).

(Actually, the proof given by the f ormer authors requires slightly more regu-

larity on Ω than that stated in Theorem IV.6.1.) Concerning (ii), Amrouche

& Girault (1990, 1991), suitably coupling the work of Grisvard (1985) and

Giga (1981), have proved Theorem IV.6.1 with m ≥ 0, for Ω of class C

m+1,1

and with m = −1, for Ω of class C

1,1

. Galdi, Simader, & Sohr (1994) extend

Theorem IV.6.1 with m = −1 to locally Lipschitz dom ains with “not too

sharp” corners, or to arbitrary domains of class C

. If n = 3 and q ∈ [3/2, 3]

their result continues to hold for Ω l ocally Lipschitz only, provided ∂Ω is

connected; see Shen (1995). The Stokes problem in non-smooth domains has

also been addressed by Kellogg & Osborn (1976) for Ω a convex p olygon, and

by Voldˇrich (1984) for arbitrary Ω ⊂ R

, n ≥ 2, but in weighted Sobolev

spaces. Dauge (1989) and Kozlov, Maz’ja and Schwab (1 994) have considered

extensions of the work of Kellogg and Osb orn to three-dimensional domains.

Existence and uniqueness for the Stokes problem in corners and cones

has been studied by Solonnikov (1 982) and Deuring (1994), respectively. A

IV.9 Notes for the Chapter 297

comprehensive analy sis of these questions can be found in the monograph of

Nazarov and Plamenevski

i (1 994).

Along with the deﬁnition of weak solution, one can intro duce the notion

of very weak solutions (Gi ga, 1981, §2; see also Conca, 1989). Speciﬁcally, set

(Ω) = {ψ ∈ C

(Ω) : ψ|

∂Ω

= 0}, (∗)

and formally dot-multiply both sides of (IV.0.1)

by ψ ∈ C

(Ω). After inte-

grating by parts over Ω and taking into account (IV.0.1)

and (IV.0.2), we

ﬁnd

Ω

v · ∆ψ = −

Ω

p div ψ +

Ω

f · ψ −

∂Ω

n · ∇ψ · v

∗

These considerations lead to the following deﬁnition. A ﬁeld v : Ω → R

is called a very weak solution to the Stokes problem (IV.0.1)–(IV.0.2 ) if (i)

v ∈ H

(Ω), some q ∈ (1, ∞), and (ii) v satisﬁes the following relation

(v, ∆ψ) = hf , ψi− hn · ∇ψ, v

∗

∂Ω

, for all ψ ∈ C

(Ω) with div ψ = 0 in Ω ,

where, we recall, h·, ·i

∂Ω

denotes the duality pairing between the spaces

−1/q,q

(∂Ω) and W

1−1/q

(∂Ω); see Section III.2 . Notice that v is not re-

quired to possess any derivative, hence the attribute of “very weak”. It is

also clear that a q-weak solution is also very weak, and that the converse

need not be true. Existence and uniqueness of very weak solutions has been

shown, for Ω of class C

∞

, by Giga (1981, Proposition 2.2). In parti cular, the

author proves that, for a ny f ∈ D

−1,q

(Ω) and a ny v

∗

∈ W

−1/q,q

(∂Ω), there

exists a unique corresponding very weak solution. Furthermore, we can ﬁnd a

“pressure ﬁeld” p ∈ W

−1,q

(Ω) such that

(v, ∆ψ) = −hp, div ψi + hf , ψi+ hn · ∇ψ, v

∗

∂Ω

, for all ψ ∈ C

(Ω) .

More recently, the question of existence, uniqueness, continuous dependence

and regularity of very weak solutions (also for the full nonlinear problem, see

Notes for Chapter IX) has been analyzed in detail by Galdi, Simader & Sohr

(2005, Theorem 3 and Lemma 4). Among others, one main result of this paper

explains a nd characterizes the way in which a very weak solution attains the

boundary value v

∗

, a thing that a priori is not completely obvious; see al so

Maruˇsic-Paloka (2000, Section 3). It turns out that the normal component of

v at ∂Ω is a well deﬁned member of W

−1/q,q

(∂Ω), as a consequence of the

fact that v belongs to H

(Ω); see Theorem III.2.3. Moreover, let

1,q

(Ω)

denote the completion of W

1,q

(Ω) in the norm

kuk

1,q

(Ω)

≡ kuk

+ kP

∆uk

−1,q

with P

projection operator of L

(Ω) onto H

(Ω). Then, if Ω is of class

2,1

, and u ∈

1,q

(Ω) we have that n ×u|

∂Ω

is well deﬁned as an element of

∈ W

−1/q,q

(∂Ω), a nd the corresponding trace operator is continuous; see Galdi,

298 IV Steady Stokes Flow in Bounded Domains

Sima der & Sohr, loc. cit., Theorem 1. In addition, it can be shown that every

very weak solution corresponding to f ∈ W

−1,q

(Ω) and v

∗

∈ W

−1/q,q

(∂Ω),

belongs, in fact, to the space

1,q

(Ω); see Galdi, Simader & Sohr, loc. cit.,

Lemma 1. Higher regularity results can be derived from Lemma IV.6.1.

Galdi & Varnhorn (1996) have proved the maximum modulus theorem

for the Stokes system. Speciﬁcally, they show that for solutions to (IV.0.1)–

(IV.0.3) with f ≡ 0 and Ω of class C

, the estimate

max

Ω

|v| ≤ C max

∂Ω

∗

| (∗∗)

holds, with C = C(n, Ω). In general, the constant C is ≥ 1 , and, in fact,

it is easy to bring examples where C > 1. The same result, under more

general assumptions on the data, has been independently obtained by Mare-

monti (2002). Previous contributions to this problem are due to Naumann

(1988), and to Maremonti & Russo (1994) who ﬁrst proved (∗∗) for the two-

dimensional case.

An interesting question is to furnish a bound for the con-

stant C appearing in (∗∗). This problem has been addressed, by an elegant

method, by Kratz (1997a, 1997b).

Section IV.8. Integral representations of various types for the general non-

homog eneous Stokes problem along with their comparative analysis are given

in the paper by Valli (1985). In this paper some errors in analogous formulas

given by other authors are also pointed out.

The results of Galdi & Varnhorn, and of Maremonti & Russo cover the case of

more general domains such as the exterior ones. In this respect, it seems inter-

esting to notice that, in the case where Ω is a half-space, the estimate (∗∗) with

suitable C = C(n) > 0 immediately f ollows, in arbitrary dimension n ≥ 2, from

the representation (IV.3.3)

and the estimate (ii) given after (IV. 3.6).

Steady Stoke s Flow in Exterior Do mains

. . . Tu stesso ti fai grosso

col falso immaginar, s`ı che non vedi

ci`o che vedresti, se l’avessi scosso.

DANTE, Paradiso I, vv. 88-90.

Introduction

In this chapter we shal l analyze the Stokes problem in an exterior domain.

Speciﬁcally, assuming that the region of ﬂow Ω is a domai n coinciding with

the complement of a compact set (not necessarily connected) we wish to es-

tablish existence, uniqueness, and the vali dity of corresponding estimates f or

the velocity ﬁeld v and the pressure ﬁeld p of a steady ﬂow in Ω governed by

the Stokes approximation, i.e.,

∆v = ∇p + f

∇ · v = 0

)

in Ω

v = v

∗

at ∂Ω,

(V.0.1)

where f , v

∗

are prescribed ﬁelds and where, as usual, we have formally set the

coeﬃcient of kinematic viscosity to be one. Of course, since Ω is unbounded,

we have to assign also the velocity at inﬁnity, which we do as follows:

lim

|x|→∞

v(x) = 0 . (V.0.2)

There is a physically signi ﬁcant special case of problem (V.0.1)–(V.0.2)

that, in fact, constitutes the main motivation for its study. Precisely, consider

the slow motion of a rigid body B, with impermeable walls, that moves with

299G.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_5, © Springer Science+Business Media, LLC 2011

300 V Steady Stokes Flow in Exterior Domains

prescribed translational velocity, v

, and angular velo ci ty, ω, in an otherwise

quiescent viscous liqui d. By “slow” we mean that all nonlinear terms related to

the inertial forces of the liquid can be disregarded compared to the linear one

related to v iscous forces. This happens if the appropriate Reynolds number is

vanishingly small, that is,

max {|v

|, |ω|d} 

, (V.0.3)

where d = δ(B). We further assume that the liquid ﬁlls the whole space, Ω,

outside B , and that body forces are negligible. Then, problem (V.0 .1), (V.0.2)

with f ≡ 0, and v

∗

= v

+ ω × x, describ es the motion of the liquid referred

to a frame attached to the body B.

a priori, we are not expecting that (V.0.1), (V.0.2) may fully describe,

even qualitatively, the physics of the problem at low Reynolds number. This

is because, if the Stokes approximation of a ﬂow can be fairly reasonable near

the b ounding wall of the body, where the viscous forces are predomi nant, it

need not be equally reasonable at large distances where the eﬀects related to

those forces become less im portant. Let us consider, for instance, a unit ball ,

S, (with impermeable walls) moving with a small (in the sense of (V.0.3))

translational velocity v

and zero angular velocity in a viscous liquid that ﬁll s

the who le space and is at rest at inﬁnity. Then, by what we said, the motion

of the liquid can be described by (V.0.1), (V.0.2) with Ω ≡ R

− S, f ≡ 0,

and v

∗

= v

. In such a case, Stokes derived in 185 1 a remarkable and explicit

solution v

, p

given by (see Stokes 1851, §39)

(x) =

∇×



|x|

∇ ×



|x|



∇ × ∇ ×



|x|



(x) =

·∇



|x|



(V.0.4)

Employing this solution one can easily compute the force exerted by the liquid

on the sphere and ﬁnd results that a re signiﬁcantly in agreement with the

experiment.

However, for the same solution it is apparent that v(x) = v(−x)

and, therefore, according to Stokes approximation, there is no wake region

behind S in contrast with what should be expected in the actual ﬂow.

Similar incongruities are observed if S rotates with a constant and small

(in the sense of (V.0.3)) angular velocity ω, without translating (i.e. v

= 0).

In this situation, the solution to (V.0.1), (V.0.2) with f ≡ 0, and v

∗

= ω ×x

is given by (see Lamb 1932, §334 )

v = ω ×

|x|

, p = p

, (V.0.5)

A most remarkable example is the Erenhaft-Millikan experiment for determining

the elementary electronic charge, where one uses the Stokes law of resistance

derived from (V.0.4) (Perucca 1963, Vol II, p.670).

V Introduction 301

where p

is an arbitrary constant. From (V.0.5) it follows that the component

of the velocity along the axis parallel to ω is identically zero. However, this

is at odds with the well-known experimental observation that the sphere acts

like a “centrifugal fan”, receiving the liquid near the po les and throwing it

away at the equator; see Stokes (1845) and Lamb (1 932, p. 589). This fact

was ﬁrst theoretically explained by Bickley (1938).

Another –and maybe more famous– diﬃculty arises when one replaces the

sphere S with an inﬁnite straight cylinder C moving with translational velocity

in a direction perpendicular to its axis, and zero angular velocity (Stokes

1851, §45). In this situation, the motion of the liquid is planar and, therefore,

one may write

∂ψ

∂θ

, v

= −

∂ψ

∂r

where ψ = ψ(r, θ) is the stream function and (r, θ) is a polar coordinate system

in the relevant plane of ﬂow orthogonal to the axis of C. Assuming the radius

of C to be one, problem (V.0.1), (V.0.2) with

Ω = {x ∈ R

: x

+ x

> 1},

f ≡ 0 and v

∗

= v

, can be written, in terms of ψ, as

∆

ψ = 0 in Ω

∂ψ

∂θ

= U cos θ at r = 1

∂ψ

∂r

= U sin θ at r = 1

lim

r→∞

∂ψ

∂θ

= lim

r→∞

∂ψ

∂r

= 0,

(V.0.6)

where, without loss of generality, we have taken v

= (U, 0, 0). A solution to

(V.0.6) can be sought in the form

ψ(r, θ) = F (r)G(θ).

Owing to (V.0 .6)

3,4

, we ﬁnd

G(θ) = si n θ,

and so, by an easy calculation, we show that F (r) satisﬁes a fourth-order

Euler equation whose general integral is

F (r) = Ar

−1

+ Br + Cr log r + Dr

A, . . ., D being arbitrary constants to be ﬁxed so as to match the boundary

conditions and conditions a t inﬁnity in (V .0.6). To satisfy the la tter it is

necessary to take B = C = D = 0. Moreover, (V.0.6)

implies A = U,

302 V Steady Stokes Flow in Exterior Domains

while (V.0.6)

requires A = −U. This i s possible if and only if A = U = 0.

Thus, v ≡ 0, p = const. is the only possible solution (of the particular form

chosen) of the problem, which tells us that the cylinder can not move. Stokes

then concluded wi th the following (wrong, i n retrospect) statement; see Stokes

(1851, p. 63 ):

“It appears that the supposition of steady motion is inadmissible.”

This is the original formulation of the famous Stokes paradox, which plays

a fundamental role i n the study of plane steady ﬂow, also in the nonlinear

context (see Section XII.4). The Stokes paradox, in other diﬀerent and more

general forms, will be considered and discussed in several sections of this

chapter.

The situation just described is similar to that of the well-known Laplace

equation with homogeneous Dirichlet boundary data in the exterior of a unit

circle, where the function µ(x) = log |x| is a solutio n to the problem and there

are no non-zero solutions that behave at inﬁnity as o(µ(x)). In fact, also for the

exterior plane Stokes problem, from the reasonings previously developed we

can construct solutions analogous to µ and ﬁnd the following two independent

solutions

(1)

= 2 log |x| + 2x

/|x|

+ (x

− x

)/|x|

− 1,

(1)

= −2

|x|



1 − |x|

−2



(1)

= 4x

/|x|

(2)

= −2

|x|



1 − |x|

−2



(2)

= 2 log |x| + 2x

/|x|

+ (x

− x

)/|x|

− 1,

(2)

= 4x

/|x|

(V.0.7)

As in the case of the sphere, also for the pl anar ﬂow past a cyli nder one

can exhibit a soluti on corresponding to the case when C rotates (without

translating) with constant angular velocity ω around its axis in a quiescent

liquid. Precisely, the appropriate solution is g iven by (see Lamb 1932, §§333,

336)

v = ω ×

|x|

, p = p

, (V.0.8)

where p

is an arbitrary constant. However, it is interesting to observe that,

unlike the case of the sphere, the velocity ﬁeld in (V.0.8) is also a solution to

the full nonlinear Navier–Stokes problem, corresponding to the pressure ﬁeld

p = p

−

|ω|

|x|

. (V.0.9 )

It is interesting to notice that a general proof of t he Stokes paradox, independent

of the particular form of the solution and the shape of the body appeared only

in 1938; see Kotschin, Kib el, & Rose (1954, pp. 361-366).

V Introduction 303

Other contradictions and paradoxes related to problem (V.0.1), (V.0.2)

will be mentioned in the intro duction to Chapter VII. There, and in the sub-

sequent Chapter VIII, to avoid these contradictions, we will consider an ap-

proximation diﬀerent from that of Stokes, obtained by linearizing the Navier–

Stokes equation around a non-zero rigid motion solution (Oseen, and gener-

alized Oseen approximations).

The main objective of this chapter is to investigate unique solvability for

problem (V.0.1), (V.0.2) and to see to what extent it is possible to prove, for

the obtained solutions, estimates analogous to those derived in the preceding

chapter in the case of a bounded domain. Now, while existence and uniqueness

of generalized solutions together with corresponding estimates are proved (for

n > 2) by a di rect extension of the method empl oyed for the bounded domain

(in fact, even with an arbitrary ﬂux of v

∗

at ∂Ω), the problem of determin-

ing analo gous results for q-generalized solutions (q 6= 2) is more complicated

and demands a preliminary, detai led study of asymptotic properties at large

distances. Of course, as in the case of the Poisson equation in exterior do-

mains, we are not expecting that the theory holds in Sobolev spaces W

m,q

but, rather, in homogeneous Sobolev spaces D

m,q

. However, a s we shall see,

even enlarging the class of functions to w hich solutions belong, such results

can be proved if there is a certain restriction on q depending on the number of

space dimension n. This fact can be qualitatively explained as follows. We be-

gin to consider smooth body forces of compact support in Ω and for these we

show the unique solvability of problem (V.0.1), (V.0 .2) in a function class F

(say) along with suitable estimates that represent the natural generalization

to the exterior domain of those determined in Theorem IV.6.1 for a bounded

domain. Successively, given f in an arbitrary Sobol ev space W

m−2,q

, m ≥ 2,

we approximate it by a sequence from C

∞

(Ω) and analyze the convergence of

the correspo nding solutions to a solution to (V.0 .1), (V.0.2) in the class F

means of the preceding estimates. Now, if q is suﬃciently small (1 < q < n/2)

every function in F

satisﬁes (V.0.2), in a suitable sense, and the above pro-

cedure is convergent to a uniquely determined solution to (V.0.1), (V.0.2);

on the other hand, if q is large enough (q ≥ n/2) the elements of F

need

not verify (V.0.2) and, moreover, (V.0.1 ) with f ≡ v

∗

≡ 0 admits nonzero

solutions in the class F

, which form a ﬁnite di mensional space Σ

. Therefore,

for q ≥ n/2, our procedure gives rise to a solution satisfying a priori only the

Stokes system (V.0.1) and the corresponding estimates are available only in

the quotient space F

/Σ

An analogous situation occurs when f is a functional on D

1,q

and solutio ns

are sought in the space D

1,q

(weak solutions). In fact, when q ≥ n (q > 2, if

n = 2), also in this class there is a nonempty null space S

to the Stokes system

(V.0.1). It then foll ows that such solutions are not unique if q ≥ n (q > 2, if

n = 2), while they can exist for 1 < q ≤ n/(n −1) (1 < q < 2, if n = 2) if and

only if the data satisfy a compatibil ity condition of the Fredholm type. This

latter property has som e interesting consequences and, in particula r, it leads

304 V Steady Stokes Flow in Exterior Domains

to a necessary and suﬃcient condition for the existence of planar solutions

to (V.0.1), (V.0.2). Speciﬁcally, these solutions can exist if and only if f and

∗

verify a suitable relation. Of course, if f ≡ 0, the choice v

∗

= v

, with

constant vector, does not satisfy such a relation, in accordance with the

Stokes paradox. However, this relati on may be satisﬁed for other, physically

signiﬁcant, choices of v

∗

; see Exercise V.7.1.

In the light of what I have described so far, a question that naturally arises

is i f, by suitably restricting the function class of body forces, it is possible to

determine “stronger” estimates that would ensure that the limit solution,

obtained by the density procedure previously mentioned, “remembers” the

condition at inﬁnity (V.0.2). Such a problem is, in fact, resolvable provided

f = ∇ · F with F decaying suﬃciently fa st at large distances (see Section

V.8) and, as we shall see in Chapter X, these results will be decisive in the

solvability of the nonlinear problem with zero velocity at inﬁnity.

V.1 Generalized Solutions. Preliminary Considerations

and Regularity Properties

In analogy with the case where Ω is bounded, we begin to give a variational

formulation of the Stokes problem (V.0.1), (V.0.2). To this end, multiplying

(V.0.1) by ϕ ∈ D(Ω) and i ntegrating by parts over Ω, we formally obtain

(∇v, ∇ϕ) = −[f , ϕ]. (V.1.1)

Deﬁnition V.1.1. A vector ﬁeld v : Ω → R

is called a q-weak (or q-

generalized) solution to (V.0.1), (V.0.2) if for some q ∈ (1, ∞) the following

properties are met:

(i) v ∈ D

1,q

(Ω);

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

(iii) v assumes the value v

∗

at ∂Ω (in the trace sense) o r, if the veloci ty at

the boundary is zero, v ∈ D

1,q

(Ω) ;

(iv) lim

|x|→∞

n−1

|v(x)| = 0;

(v) v veriﬁes (V.1.1 ) for all ϕ ∈ D(Ω).

If q = 2, v will be called a weak (or generalized) solution to (V.0.1 ), (V.0.2).

Remark V.1.1 If v

∗

= 0, condition (iii) tells us that v assumes the homog e-

neous boundary data in the sense of the Sobolev space W

1,2

and no regularity

is needed on Ω; see Remark II.6.5. On the other hand, if v

∗

6= 0, according

to the trace theory of Section II.5, Ω has to be (at least) locally Lipschitz. 

As agreed, we are taking ν = 1. Furthermore, as in the case where Ω is bounded,

we are considering the more general case that f is an element of D

−1,q

(Ω), so

that, since Ω is an exterior domain, we replace (f , ϕ) with [f , ϕ], where, we

recall, [·, ·] denotes the duality pairing between D

1,q

(Ω) and D

1,q

(Ω).

V.1 Generalized Solutions and Their Regularity Properties 305

Remark V.1.2 If v

∗

= 0, a q-weak solution belongs to

1,q

(Ω). T his follows

directly from conditions (ii) and (iii) of Deﬁnition V.1.1 . Consequently, for

domains such that

1,q

(Ω) ⊃ D

1,q

(Ω) the deﬁnition of q-generalized solution

using this latter space could then b e more restrictive than Deﬁnition V.1.1.

In this regard, we recall that, by virtue of the results of Theorem III.5 .1, the

two spaces coincide if Ω satisﬁes some mild regularity requirement like, for

instance, cone condition. However, to date, it is still to ascertain if there exists

an exterior doma in, Ω

]

, for which

1,2

(Ω

]

) 6= D

1,2

(Ω

]

). If such Ω

]

exists,

then, as i n the case of a bounded domain (see Remark IV.1.2) we could prove,

by means of Exercise III.5.3 and Theorem V.2.1, the existence of at least

one smooth, nonzero solution, v

]

, p

]

, to (V.0.1),(V.0.2) in Ω

]

, corresponding

to f ≡ v

∗

≡ 0. In addition, v

]

would tend to zero as |x| → ∞ uniformly

pointwise, and v

]

, p

]

would have the asymptotic behavior speciﬁed in Theorem

V.3. 2. 

From Lemma IV.1. 1, it follows that to every q-weak solution we can asso-

ciate a suitable pressure ﬁeld p. Namely, if f ∈ W

−1,q

(Ω

), 1 < q < ∞, for

any bounded subdomain Ω

with Ω

⊂ Ω, there exists p ∈ L

loc

(Ω) such that

(∇v, ∇ψ) = −[f, ψ] + (p, ∇ · ψ) (V.1.2)

for all ψ ∈ C

∞

(Ω). However, if Ω is locally Lipschitz and f ∈ D

−1,q

(Ω) we

have the fol lowing global result.

Lemma V.1.1 Let Ω be a locally Lipschitz exterior domain in R

and let v

be a q-generalized solution to (V.0.1), (V.0.2). Then, if

f ∈ D

−1,q

(Ω),

there exists a unique p ∈ L

(Ω) satisfying (V.1.2) for all ψ ∈ C

∞

(Ω). Fur-

thermore, the f ollowing inequality holds

kpk

≤ c (|f|

−1,q

+ |v|

1,q

) . (V.1.3)

Proof. The functional

F(ψ) = (∇v, ∇ψ) + [f , ψ]

is bounded for ψ ∈ D

1,q

(Ω) and vanishes for ψ ∈ D

1,q

(Ω). The existence

and uniqueness of p is then a direct consequence of Corollary III.5.1 . Consider,

next, the problem

∇ · ψ = |p|

q−2

ψ ∈ D

1,q

(Ω)

|ψ|

1,q

≤ c

kpk

(V.1.4)