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

Подождите немного. Документ загружается.

6 I Steady-State Solutions: Formulation of the Problems and O pen Questions

Actually, so far, one can prove its solvability only when the ﬂuxes Φ

through

satisfy, in addition to (I.1.3), a condition of the type

i=1

|Φ

| < cν, (I.1. 4)

where c depends o nly o n Ω. Notice tha t in view of (I.1.2 ) and (I.1 .3), inequal ity

(I.1. 4) is automatically satisﬁed if m = 1; otherwise, it becomes an extra

requirement.

It is not clear whether the above restriction is really needed. This is a typ-

ical example, which we mentioned previously, where we do not know whether

the fact that we are unable to give an answer is due to a lack of a suﬃcient

mathematical knowledge or to a “hidden” physical phenomenon that we are

not able to see. The mathematician often experiences similar situations while

studying the Navier–Stokes equations, and probably, it i s just for this reason

that these equations are so particularly fascinating.

Referring the reader to Chapter IX for a detailed analysis o f the question,

here we may wish to brieﬂy explain why it may be diﬃcult to avoid restriction

(I.1. 4). The resolution of (I.0.4 ), (I.1.1), and (I.1.2 ) relies, usually, on some

approximating procedure whose convergence requires a uniform a pr iori bound

on the approximating solutions. If v

∗

≡ 0, following the ideas of Leray (1933,

1936), this bo und can be simply obtained by the following formal computation.

We dot-multiply through bo th sides of (I.0.4)

by v and use the known i dentity

∇ · (ϕ w) = w · ∇ϕ + ϕ∇ · w

together with (I.0.4)

to obtain

−ν∇v : ∇v + ∇ ·



∇v

− p v −



= f ·v.

Integrating this expression over Ω and tak ing into account v = v

∗

≡ 0 at ∂Ω,

For A = {A

} and B = {B

} second-order tensors in the n-dimensional Eu-

clidean space R

we set, as is customary,

A : B ≡

i,j=1

, |A| ≡

i,j=1

1/2

Moreover, if a is a vector of R

, by the symbols

a · A and A · a

we mean vectors wi th comp onents

i=1

and

i=1

respectively.

I.1 Flow in Bounded Regions 7

we deduce

Ω

∇v : ∇v = −

Ω

f ·v . (I.1. 5)

By the well-known inequality (see (II.5.1)

)

Ω

≤ γ

Ω

∇v : ∇v, (I.1. 6)

where γ depends on Ω, from (I.1.5) and the Schwarz i nequality we obtain

Ω

∇v : ∇v ≤

Ω

, (I.1.7)

which (formall y) furnishes the a priori bound for v when v

∗

≡ 0. When v

∗

6≡ 0,

again following the ideas of Leray (1933) and Hopf (1941, 1957), one writes

v = u + V , (I.1. 8)

where V is a (suﬃciently smooth) solenoidal vector ﬁeld in Ω that equals v

∗

at ∂Ω. Placing (I.1 .8) into (I.0.4)

and proceeding as before, we arrive at the

following identity:

Ω

∇u : ∇u = −

Ω

(u ·∇V ·u + V ·∇V ·u + ν∇V : ∇u + f ·u), (I.1.9)

which generalizes (I.1.6) to the case v

∗

6≡ 0. By use of the Schwarz inequality

and (I.1.6), it is easily seen that the last three terms on the right-hand side

of (I.1.9) can be increased, for instance, as follows:

Ω

|V · ∇V · u| ≤

√

γ sup

Ω

|V |



Ω

∇V : ∇V



1/2



Ω

∇u : ∇u



1/2

Ω

|∇V : ∇u| ≤ ν



Ω

∇V : ∇V



1/2



Ω

∇u : ∇u



1/2

Ω

|f · u| ≤ γ



Ω



1/2



Ω

∇u : ∇u



1/2

Placing these inequalities back into (I.1.9), we obtai n

Ω

∇u : ∇u ≤ −

Ω

u · ∇V · u + C



Ω

∇u : ∇u



1/2

, (I.1.10)

Unless their use clariﬁes the context, the inﬁnitesimal volume and surface elements

in the integrals will generally be omitted.

That is, the ﬁrst display in Section 5 of Chapter II.

For another approach, again due to Leray, we refer the reader to the Notes for

Chapter IX.

8 I Steady-State Solutions: Formulation of the Problems and O pen Questions

with C depending only on Ω, V (i.e., v

∗

), and f. Therefore, in order to obtain

a bound depending only on the data for the quantity

Ω

∇u : ∇u,

or, what amounts to the same thing, f or

Ω

∇v : ∇v,

it is suﬃcient to prove the following one-sided inequality:

−

Ω

u · ∇V ·u ≤ k

Ω

∇u : ∇u, (I.1. 11)

with som e constant k (depending on V and Ω) such that

k < ν, (I.1. 12)

and for all smooth solenoidal vector ﬁelds u vanishing at ∂Ω. Of course, if we

do not want to impose restrictions on ν, we should be able to prove that for

any k > 0 there exists a solenoidal ﬁeld V = V (x; k) assuming the value v

∗

at ∂Ω and satisfying (I.1.11) for all the above speciﬁed vectors u . However,

Takeshita (1993) and, more recently, Heywood (2010) showed by means of

counterexamples that in general, the la tter property does not hold if ∂Ω is

composed by more than one surface Γ

, and consequently, if one follows i n

such a case the method of Leray–Hopf, o ne must impose some restrictions

on ν or, equi valently, on v

∗

. To date, the best one can show, in general,

that (I.1.11) and (I.1.12) are certainly satisﬁed, provided the ﬂuxes Φ

satisfy

the restriction (I.1.4). As already observed, (I.1.4) becomes redundant if the

number of surfaces Γ

reduces to one.

I.2 Flow in Exterior Regions

The most signiﬁcant physical problem that motivates this type of study is the

motion of a rigid body B through a vi scous liquid. Such a problem o riginates

In some special cases of plane ﬂow, one can remove the extra condition ( I.1.4);

see the Notes to Chapter IX.

It is interesting to observe that for a certain class of non-Newtonian l iquids, where

the shear viscosity coeﬃcient µ depends i n a monotonically increasing way on the

magnitude of the stretching tensor |D|, the corresponding steady-state problem

in the bounded domain Ω can be solved under the sole compatibi lity cond itio n

(I.1.2); see the Notes at the end of this chapter.

I.2 Flow in Exterior Regions 9

with the pioneering work of G.G. Stokes (1851) on the eﬀect of internal friction

on the movement of a pendulum in a liquid.

Usually, the inﬂuence on the motion of B of the boundary walls of the

container of the liqui d L can be safely disregarded, and thi s leads to the

mathematical (simplifying, in principle) assumption that L ﬁlls the whole

space region Ω complementary to B, and that i t is at rest at spatial inﬁnity.

Hence, Ω becomes an exterior region.

Another classical problem that can be formulated as an exterior problem

is the ﬂow past an obstacle, where the center of mass of B is held in pla ce by

appropriate forces and the liquid ﬂows past B tending to a uniform velocity

ﬁeld at large distances from B (as in a wind tunnel).

In the above situations, it is convenient to describe the motion of the

liquid from a frame of reference S attached to B. This is because the region

occupied by the liquid then becomes time-independent. However, since, in

general, B may rotate, the frame S is no long er inertial, and consequently,

we have to modify the orig inal equations (I.0.1 ) accordingly, in order to take

into account the ﬁctitious f orces. Assuming that the angular velo ci ty, ω, of

B with respect to the ini tial f rame is constant in time (a case of particular

interest in many applications), this amounts to adding, on the right-hand side

of (I.0.4)

, the term 2 ω × v representing the Corioli s force (per unit mass),

while the centrifugal force (per unit mass), ω × (ω × x), can be formally

absorbed in the pressure term. Equations (I.0.1) then modify to the following

ones (see, e.g., Batchelor 1999, pp. 139–140, or Galdi 2002, §1):

∂v

∂t

+ v · ∇v = ν∆v −2ω ×v − ∇p − f ,

∇ · v = 0 ,

(I.2. 1)

where now v has to be interpreted as the velocity of the particles of L relative

to S, and we have still denoted by p the original pressure ﬁeld modiﬁed by

the addition of the term −

(ω ×x)

. The steady-state counterpart of (I.2.1)

thus becomes

ν∆v = v · ∇v + 2ω × v −∇p + f ,

∇· v = 0 .

(I.2. 2)

These equations must be endowed with a condition on v at the bounding

walls ∂Ω and at inﬁnity. For the former, we will adopt (I.1.1), whereas the

latter wil l be taken to be of the f orm

lim

|x|→∞

(v(x) + v

∞

(x)) = 0 , (I.2.3)

with v

∞

= v

+ ω ×x, and v

a constant vector. A most signiﬁcant situation

described by the problem (I.2.2)– (I.2.3), is that of a rigid body moving with

constant angular velocity ω in a liquid that is quiescent at large distances. In

10 I Steady-State Solutions: Formulation of the Problems and Open Questions

such a case, the vector v

can be viewed as the velocity of the center of mass

of B with respect to S, assumed to be a constant.

14,15

The problem just described is formulated for three-dimensional motions.

However, it also has a very signiﬁcant counterpart in the case of plane motions

of L. Speciﬁcally, assuming that the relevant region of motion is located in

the plane Π, we have that the vector v

is prescribed parallel to Π, while ω

is orthogonal to it. A classic example of such ﬂows is f urnished by the moti on

of L past a ﬁxed long and straight cylinder C, of cross-section S, and axis a

assuming that the velocity ﬁeld v tends to a given constant at large distances

from C. In a region of ﬂow suﬃciently far from the two ends of C and including

C, one can assume that v is independent of the coordinate parallel to a and,

moreover, that there is no motion in the direction of a. The plane Π i s thus

orthogonal to a, while Ω becomes the region of Π exterior to S.

The study of problem (I.2 .2), (I.1.1), (I.2.3) and the description of the

corresponding known results is tightly related to whether we consider three- or

two-dimensional ﬂow. Therefore, we shall analyze these two cases separately.

I.2.1 Thr ee-Dimensional Flow

The cases ω = 0 and ω 6= 0 will be referred to as the irrotational and

rotational cases, respectively.

Irrotational Case. In this situa tion, problem (I.2.2), (I.1.1), (I.2.3) reduces

ν∆v = v · ∇v + ∇p + f

∇ · v = 0

)

in Ω ,

v = v

∗

at ∂Ω ,

lim

|x|→∞

v(x) = −v

(I.2. 4)

As in the case of a bounded region of ﬂow, the ﬁrst fundamental contri-

bution to the existence problem was given by Leray (1933), who showed that

It is worth noticing that even though the velocity of the center of mass G i s

constant in S, it can be time-dependent in an inertial frame I. In fact, denoting

by η the velocity of G with respect to I, we ﬁnd that v

is constant in S if and

only if dη/dt = ω × η, that is, if and only if G moves, in I, with constant speed

along a circular helix whose axis is parallel to ω. However, the components of η

and v

along ω must coincide, that is, η · ω = v

· ω. Thus, in particular, the

velocity of G wil l be constant also in I if and only if it is parallel to ω (in which

case η = v

), and the helix degenerates i nto a straight line. On the other hand,

if η is orthogonal to ω (or, equivalently, v

is orthogonal to ω), the helix will

degenerate into a circle lying in a plane orthogonal to ω, which means that the

motion of the body reduces t o a constant rotation around an axis not necessarily

passing through G. Further details can be found, e.g., in Galdi (2002, §1).

For other interesting (unsolved) exterior problems, we refer the reader to the

Notes at the end of this chapter.

I.2 Flow in Exterior Regions 11

an a priori estimate of the type (I.1.7) suﬃces to ensure the convergence of a

suitable approximating procedure to a regular solution to (I.2.4);

see Chap-

ter X. However, the outstanding problem Leray left out was the investigation

of the asymptotic behavior at large distances of such solutions. This question

is of primary importance since, as is easily realized, it is intimately related

to the physical properties that any solutio n that deserves this name should

possess. For example, solutions must satisfy the energy equation

2ν

Ω

D(v) : D(v) −

∂Ω

[(v

∗

+ v

) · T (v, p) −

∗

+ v

)

∗



· n

Ω

f · (v + v

) = 0,

(I.2. 5)

with D and T stretching and stress tensors (I.0.2), which represents the bal-

ance between the power of the work of external force, the power of the work

done on the “b odies” represented by the connected components of R

− Ω,

and the energy dissipated by the viscosity. Also, if f, v

∗

, and v

are “suﬃ-

ciently sma ll” with respect to the viscosity ν,

the corresponding solutions

are expected to be unique. In addition, in the case when v

6= 0, the ﬂow

must exhibit an inﬁnite wake extending in the direction opposite to v

, and

the order of convergence of v to −v

has to be rather diﬀerent according

to whether it is calculated inside or outside the wake. Finally, in conformity

with the boundary layer concept, the ﬂow is expected to be po tential outside

a sm all neig hborhood of the bodies and of the wake, which means that the

vorticity should decay suﬃciently fast at large distances and outside the wake.

If v

6= 0, the ab ove questions found a deﬁnitive answer through the

fundamental work of R. Finn, K.I. Babenko, and their co-workers during the

years 1959– 1973. In particular, using the results of Finn, Babenko has shown

that the solution constructed by Leray admits an asymptotic development

at inﬁni ty in which the dominant term is a solution to the corresponding

linearized equati ons, which, for v

6= 0, are the Oseen equations; see also

Galdi (1992b), Farwig & Sohr (1998). Therefore, Leray’s solution behaves at

inﬁnity in such a way as to ensure the validity of all the ab ove-mentioned

properties; see Chapter X.

If v

= 0, the picture is much less clear: the methods adopted by the above

authors generally do not work. Nevertheless, by means of completely diﬀerent

tools, and following the approach given by Gal di (1992c), one can show that

conclusions somewhat analogous to the case v

6= 0 can be drawn also when

As a matter of fact, Leray proved (I.2.4)

only for v

= 0, while for v

6= 0 he

proved only that v tends to v

in a weaker sense. (As Leray himself noted, his

proof for v

= 0 fails if v

6= 0.) The validity of (I.2.4)

when v

6= 0 was shown,

independently, by Finn (1959a) and by Ladyzhenskaya (1969, Chapter 5 Theorem

8 and p. 206).

More precisely, if a suitable nondimensional (Reynolds) number depending on v

∗

f , ν, and Ω is “suﬃciently small.”

12 I Steady-State Solutions: Formulation of the Problems and Open Questions

= 0, provided, however, that v

∗

and f are not too large compared to

ν.

Speciﬁcally, under the stated restrictions on the data, the solution con-

structed by Leray admits an asymptotic development at large spati al distances

whose dominant term behaves like the solution of the corresponding linearized

equations that, for v

= 0, are the Stokes equations. However, as shown by

Deuring & Galdi (2000), the dominant term in this expansion cannot be a

solution to the corresponding Stokes equations. In f act, more recent results

due to Korolev &

Sver´ak (2007, 2011) establish that the leadi ng term in the

asymptotic expansion coincides with a suitable exact, singular solution of the

full nonlinear problem obtained by Landau (1944); see Chapter X.

Thus, if v

= 0, several basic questions remain open, among others the

following:

(i) Given v

∗

and f , no matter how smooth but of unrestricted size, do there

exist corresponding solutions satisfying the energy equation?

(ii) Do these solutions, if they exist at all, admit a suitable asymptotic ex-

pansion for large |x| whose leading term is the corresponding Landau

solution?

It is quite clear that in order to answer these questions, one has to employ

ideas and methods that go well beyond the simple perturbative analysis, by

which the “small” data results referred above are obtained.

Another puzzling question that arises in the case v

= 0 is the following

Liouville-like problem. Consider

ν∆v = v · ∇v + ∇p

∇ · v = 0

)

in R

lim

|x|→∞

v(x) = 0 ,

∇v : ∇v < ∞.

(I.2. 6)

As will be shown later o n, in Chapter X, all possible solutions (v, p) to (I.2.6)

are inﬁnitely diﬀerentiable, and moreover, all derivatives of v and ∇p tend to

zero as |x| → ∞. Of course, the null solution v ≡ ∇p ≡ 0 is a solution to

(I.2. 6). The q uestion is this:

(iii) Is the null solution the only (smooth) solution to (I.2.6)?

In connection with this problem, it is worth emphasizing the following facts.

In the ﬁrst place, if the homogeneous condition at inﬁnity is replaced by

lim

|x|→∞

v(x) = v

, for some nonzero v

, then one can show that the only

solution is v(x) = v

, ∇p(x) = 0, for all x ∈ R

; see Chapter X. In the

second place, the n-dimensional counterpart of this question, i .e., obtained by

replacing, in (I.2.6), R

with the n-dimensional Euclidean space R

, n ≥ 2

and n 6= 3, admits a positive answer, so that only the case n = 3 remains

open; see Chapter X and Chapter XII.

See footnote 17.

I.2 Flow in Exterior Regions 13

We shall ﬁnally describe another interesting question that might deserve

an appropriate investigation. For the existence of Leray’s solution it is not

required that the total mass ﬂux through ∂Ω, i.e.,

Φ ≡

∂Ω

∗

· n, (I.2. 7)

satisfy the vanishing condition (I.1.2). For, unlike the case of ﬂow in a bounded

region, (I.1.2) i s no longer a priori a compatibility conditio n. This latter fact

can be explained as follows. By the incompressibility condition (I. 0.4)

, the

ﬂux Φ is equal to the ﬂux Φ

of v through the surface S

of a ball of radius R

surrounding the bodies B

. Since the behavior of the velocity ﬁeld v at l arge

distances is expected to be, in general, like |x|

−1

, Φ

need not vanish as R

tends to inﬁnity, and so Φ need not be zero a priori. However, to date, one is

able to prove existence only if Φ is suﬃciently small with respect to ν. Thus:

(iv) Is it possible to prove existence for arbitrarily large values of Φ?

Rotational Case. As probably expected, the rotational case presents further

and new challenges, and, consequently, more unresolved issues, i f compared to

the irrotational case. To describe these latter, we begin by observing that, as

shown in Chapter VIII and Chapter XI, after a suitable change of coordinates

(Mozzi–Chasles transformation) problem (I.2.2), (I.1.1), and (I.2.3) with ω 6=

0 is formally equivalent to the following one:

ν∆u = (u − λω − ω × x) · ∇u + ω ×u + ∇p + f

∇ · u = 0

)

in Ω ,

u = v

∗

+ v

∞

≡ u

∗

at ∂Ω ,

lim

|x|→∞

u(x) = 0 ,

(I.2. 8)

where u := v + v

∞

, λ := v

·ω/|ω|

, p is the “original” pressure ﬁeld and Ω

is a suitable exterior regio n.

The characteristic feature of problem (I.2.8) is that the term ω × x · ∇u

has a coeﬃcient that becomes unbounded at large di stances. Therefore, the

rotational case cannot be viewed as a perturbation of the irrotational case.

Despite this diﬃculty, one can show with relative ease that for data o f

arbitrary “size” (in appropriate function cla sses), problem (I.2.8) has at least

one solution. As in the cases previously described, the proof of such a result,

originally due to Leray (1933, Chapter III),

is based on the fact that (I.2.8)

admi ts the f ollowing formal a priori estimate

Ω

∇u : ∇u ≤ M , (I.2. 9)

In his 1933 memoir, Leray proved (I.2.8)

only in a certain weak sense. The ﬁrst

proof of (I.2.8)

is due to Galdi (2002, Theorem 4.6).

14 I Steady-State Solutions: Formulation of the Problems and Open Questions

where M is a positive constant depending on the data, Leray (1933, §17);

see also Borchers (1992, Korollar 4.1) and Chapter XI. In analogy with the

case ω = 0 (and the case of a bounded domai n as well), solutions to (I.2.8)

satisfying (I.2.9) will be called Leray solutions. Such solutions can be shown

to possess as much regularity as allowed by the data, so that the next and

crucial question is whether they exhibit all fundamental properties expected

on physical grounds, already described in the previous section for the case

ω = 0. To date, the answer to this question depends on whether v

· ω 6= 0

or v

· ω = 0 (namely, λ 6= 0 or λ = 0).

If v

·ω 6= 0, Galdi & Kyed (2011a) have proved that the velocity ﬁeld (re-

spectively, its gradient) of every Leray solution corresponding to f of bounded

support is pointwise bounded above, at large distances, by a function that be-

haves like the solution (respectively, its gradient) to the (linearized) Oseen

equations. In particular, it exhibits a wake region extending in the direction

opposite to ω if v

· ω > 0, and along ω otherwise, sati sﬁes the energy equa-

tion and is unique, in the cl ass of Leray solutions, for suﬃciently “small” data.

However, the investigation of Galdi & Kyed leaves unanswered the following

important question:

(v) Do Leray solutions admit an asymptotic expansion for large |x| whose

leading term is a solution to the Oseen equations?

If v

·ω = 0, the physical properties of the solution originally constructed

by Leray are known to hold, to date, only for “small” data. In particular Galdi

& Kyed (201 0) have shown that under these assumptions, the velocity ﬁeld

and its gradient are bounded above, at large distances, by the corresponding

quantities of a function that behaves like a solution to the Stokes equations.

One signiﬁcant property that follows from this result is that Leray solutions

satisfy the energy equation, but for “small” data only. Therefore, question (i)

listed previousl y for the case ω = 0 continues to be a signiﬁcant open question

in the case ω 6= 0 as well. However, on the bright side, and in contrast to the

case v

· ω 6= 0, one is abl e to furnish a detailed asymptotic expansio n of the

velocity ﬁeld and of its derivatives fo r large |x|, but again, for sm all data. In

fact, under these circumstances, Farwig & Hishida (2009) and Farwig, Galdi,

& Kyed (201 0) have shown that, as in the case ω = 0, the leading term of

the expansion is a suitable Landau solution. Consequently, question (ii) listed

previously for the case ω = 0 is a signiﬁcant open question also in the case

ω 6= 0.

I.2.2 Plane Flow

The results and open questions considered so far refer to three-dimensional

solutions of the problem described by (I.0.4), (I.1.1), and (I.2 .3). The two-

dimensional solutions representing plane motions of L, deserve separate con-

sideration. As is well known, for these solutions the ﬁelds v and p dep end

only on x

, x

(say) and, moreover, v

≡ 0. Therefore, the relevant region f or

I.2 Flow in Exterior Regions 15

a description of the motion becomes a two-dimensional one. We shall restrict

ourselves to the case v

∞

(x) = v

, that is, ω = 0.

Again, the ﬁrst contribution to the resolution of the existence problem of

plane ﬂow in an exterior region is due to L eray (1933). Given f and v

∗

, with

∗

satisfying condition (I.1.2), he proved the existence of a regular pair v, p

satisfying (I.0 .4) and (I.1.1). However, in contrast to the three-dimensional

case, Leray was unable to show whether the velocity ﬁeld v satisﬁes condition

(I.2. 3). This is because the only information available on the behavior of v at

large distances is the ﬁniteness of the Dirichlet integral:

Ω

∇v : ∇v < ∞. (I.2. 10)

Condition (I.2.10) a lone does not even ensure that v remains bounded at large

spatial distances.

The question left open by Leray has been reconsidered,

more tha n forty years later, by several authors (Gilbarg and Weinberger 1974,

1978; Amick 1986, 1988; Galdi 2004); see al so Chapter XII. One of the main

results found by these authors is that if v, p is the solution constructed by

Leray, or if v, p is any (regular) solution to (I.0.4) corresponding to v

∗

≡ f ≡

0, with v satisfying (I.2.10), then there exists a vector

v to which v converges

uniform ly pointwise; also, the pressure ﬁeld tends pointwise and uniformly at

inﬁnity to some constant. Notice that in general, no informati on i s available

about

v. In principle,

v can be zero even though v

∞

6= 0. The fundamental

question that still needs an answer is then

(vi) Does the vector

v coincide wi th v

∞

A somewhat related question is

(vii) If

v 6= 0, does v tend poi ntwise to

Another question that naturally arises is that of the order of decay of v at

inﬁnity. Here one may expect that if v satisﬁes (I.2.10) and tends uniformly

pointwise to some limit

v, then v can be represented asymptotically by an

expansion in “reasonable” functions of r ≡ |x| with coeﬃcients independent

of r. However, if

v = 0, not every such sol ution can be represented in this way,

In this regard, it is worth noticing that to date, the case ω 6= 0 is virtually

untouched. See also the Notes at the end of this chapter.

Take, f or instance, Ω the exterior of the unit circle, and

v(x) = log

|x|, 0 < α < 1/2.

Clearly, v vanishes at ∂Ω, has a ﬁnite Dirichlet integral, and becomes unbounded

for large |x|.

Concerning this question, it should be observed that under suitable conditions

on the symmetry of the ﬂow, it admits a positive answer; see Amick (1988) and

Chapter XII. Unfortunately, the proof of the result reported in Galdi (2004, The-

orem 3.4) is not correct.