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

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636 IX Steady Navier–Stokes Flow in Bounded Domains

Corollary IX.5.2 Let v obey conditions (i)–(iii) in Theorem IX.5.2. If the

bounded domain Ω is of class C

∞

, and f ∈ C

∞

(Ω), v

∗

∈ C

∞

(∂Ω), then v

and the a ssociated pressure ﬁeld p belong to C

∞

(Ω). The above conditions

are satisﬁed if v is a generalized solution to (IX.0.1 )–(IX.0.2) and n ≤ 4.

Remark IX.5.4 Assuming l ess regularity on Ω, f and v

∗

we can obtain

intermediate regularity results on v and p. 

Remark IX.5.5 If n ≥ 5, the theory developed in Section IX.3 does not

guarantee existence of solutions verifying the a ssumptions of Corollary IX.5.1

and Corolla ry IX.5.2, since we don’t know, in such a case, if a generalized

solution belongs to L

(Ω). The question of existence of regul ar solutions for

n ≥ 5 and without restrictions on the size of the data has b een addressed by

Frehse and R˚uˇziˇcka (1994a, 1994b, 1995, 19 96), and Struwe (1995). Here, we

would like to show that, if the size of the data is suﬃciently “sma ll,” existence

of regular solutions in arbitrary dim ension n ≥ 5 is easily established by means

of the theory developed for the linearized Stokes problem. We assume Ω of

class C

, f ∈ W

−1,n /2

(Ω), v

∗

∈ W

(n−2)/n,n/2

(∂Ω), with

∂Ω

∗

· n = 0,

and set

D = kfk

−1,n /2

+ kv

∗

(n−2)/n,n/2(∂Ω)

We next introduce a sequence of approximating solutions {v

, p

}, m ∈ N,

deﬁned as follows

ν(∇v

, ∇ψ) + (v

m−1

· ∇v

m−1

, ψ) − (p

, ∇ · ψ) + hf , ψi = 0

∇ · v

= 0 in Ω

= v

∗

at ∂Ω,

(IX.5.61)

where v

≡ 0 and ψ is arbitrary from C

∞

(Ω). By the existence theory for

the Stokes problem of Theorem IV. 6.1, we know that (IX.5.61) for m = 1

admi ts a unique solution {v

, p

} (p

up to a constant) with v

∈ W

1,n/2

(Ω),

∈ L

n/2

(Ω), such that

1,n/2

n/2/R

≤ 2

D (IX.5.62)

where c = c(n, Ω) is the constant entering estimate (IV.6.10). Let us show,

by induction, the existence of {v

, p

} with v

∈ W

1,n/2

(Ω), p

∈ L

n/2

(Ω)

satisfying (IX.5.61 ), (IX.5.62) for all m ∈ N. We a ssume that {v

m−1

, p

m−1

}

obeys (IX. 5.62). For any ψ ∈ W

1,n/(n−2)

(Ω) we have

|(v

m−1

· ∇v

m−1

, ψ)| = |(v

m−1

⊗ v

m−1

, ∇ψ)| ≤ kv

m−1

kψk

1,n/(n−2)

and so, by the embedding Theorem II.3.4:

IX.5 Regularity of Generalized Solutions 637

m−1

≤ γkv

m−1

1,n/2

, (IX.5.63)

and by the induction hypothesis (IX.5.62) for the (m−1)th solution we deduce

that

|(v

m−1

· ∇v

m−1

, ψ)| ≤ 4

kψk

1,n/(n−2)

. (IX.5.64)

m−1

·∇v

m−1

∈ W

−1,n/2

(Ω)

and (IX.5.64) together with Theorem IV.6.1 ensure for all m ∈ N the existence

of a pair {v

, p

} satisfying (IX.5.61) along with the inequality

1,n/2

n/2/R

≤



D + 1



. (IX.5.65)

However, if

D <

4γ

, (IX.5.66)

relation (IX.5.6 5) furnishes

1,n/2

n/2/R

≤ 2

D (IX.5.67)

which is therefore proved for all m ∈ N. Let us next establish that {v

, p

}

is a Cauchy sequence in W

1,n/2

(Ω) ×



n/2

(Ω)/R



. To this end, it is enough

to show that for all m ≥ 1

− v

m−1

1,n/2

− p

m−1

n/2/R

≤ α

(IX.5.68)

where α ∈ (0, 1).

From (5.35

) we have

ν(∇(v

−v

m−1

), ∇ ψ) + ((v

m−1

−v

m−2

) · ∇v

m−1

, ψ)

+(v

m−2

· ∇(v

m−1

− v

m−2

), ψ) + ((p

− p

m−1

), ψ) = 0

and with the aid of Theorem IV.6.1, it fo llows f or all m ∈ N, tha t

In fact, for all m

= m + k, k > 0,

− v

1,n/2

−p

n/2/R

≤

i=1

“

m+i

−v

m+i−1

1,n/2

m+i

− p

m+i−1

n/2/R

”

≤ α

i=1

≤

m+1

1 − α

638 IX Steady Navier–Stokes Flow in Bounded Domains

− v

m−1

1,n/2

− p

m−1

n/2/R

≤

(kv

m−1

−v

m−2

m−1

+ kv

m−2

m−1

−v

m−2

) .

Using (IX.5.67) and (IX.5.63) in this relation yields

−v

m−1

1,n/2

−p

m−1

n/2/R

≤ 4



m−1

− v

m−2

1,n/2

m−1

−p

m−2

n/2/R



which in turn implies (IX.5.68) with

α =

4γ

Thus, if f, v

∗

satisfy (IX.5.66), there are v ∈ W

1,n/2

(Ω), p ∈ L

n/2

(Ω) such

that

→ v strongly in W

1,n/2

(Ω)

→ p strongly in L

n/2

(Ω)/R .

From (IX.5.61) it follows at once that v, p satisfy (IX.1.11), which completes

the proof of existence. In addition, by (IX.5.63), (IX.5.65), and (IX.5.66) we

have

kvk

+ kvk

1,n/2

kpk

n/2/R

≤ 2



kf k

−1,n/2

+ kv

∗

(n−2)/n,n/2(∂Ω)



(IX.5.69)

Taking into account Theorem IX.5.2, we may then conclude with the following

result.

Theorem IX.5.3 Let Ω be a bounded domain in R

, n ≥ 5, of class C

and

let f ∈ W

−1,n /2

(Ω), v

∗

∈ W

(n−2)/n,n/2

(∂Ω) with

∂Ω

∗

· n = 0.

Then, there exists a positive C = C(n, Ω) such that if

kfk

−1,n /2

+ kv

∗

(n−2)/n,n/2(∂Ω)

< Cν

there is a generalized solution v to (IX.0.1), (IX.0.2) such that

v ∈ W

1,n/2

(Ω), p ∈ L

n/2

(Ω)

where p is the pressure ﬁeld associated to v by Lemma IX.1.2. Moreover, v, p

satisfy (IX.5.69). Finally, if Ω is of class C

∞

, f ∈ C

∞

(Ω) and v

∗

∈ C

∞

(∂Ω),

then v, p ∈ C

∞

(Ω).

IX.5 Regularity of Generalized Solutions 639

Concerning the uniqueness of these solutions, we observe that if

kfk

−1,n /2

+ kv

∗

(n−2)/n,n/2(∂Ω)

(IX.5.70)

is suﬃciently small, by the method used in the proof of Theorem IX.2.1 a nd

Remark IX.2.3, it follows immediately that they are unique in the class of

generalized solutions which, in addition, satisfy v ∈ L

(Ω). However, we can

prove a more general result which ensures uniqueness in the class of generalized

solutions that obey the energy inequality.

We shal l sketch the proof of this

result in the special case when v

∗

= 0. To this end, we notice that, by Exercise

IX.3. 1, we can construct, in any dimension n ≥ 2, a generalized solution

satisfying the following energy inequality

ν| v|

1,2

≤ −hf, vi. (IX.5.71)

Let now v

be a solution corresponding to f as given in Theorem IX.5.3. In

view of Remark IX.1.4, we can show that v

satisﬁes the energy equality

ν|v

1,2

= −hf, v

i. (IX.5.72)

Again by Remark IX.1.4, we can take ϕ = v

in (IX.1.2) to obtain

− ν(∇v, ∇v

) −(v ·∇v, v

) = hf, v

i. (IX.5.7 3)

By the same token, we can consider (IX.1.2) with v replaced by v

, and choose

ϕ = v. We then get

− ν(∇v

, ∇v) − (v

· ∇v

, v) = hf , vi. (IX.5.74)

If we add together (IX.5.71)–(IX.5.74), and use again Remark IX.1.4, we can

show that

ν| w|

1,2

≤ (w · ∇w, v

), (IX.5.75)

where w = v −v

. We now use (IX.1.6) and (II.3.7) o n the right-hand side of

(IX.5.75) to deduce

(ν − c

) |w|

1,2

≤ 0,

with c

= c

(n). This latter inequality alo ng w ith (IX.5.69) written with

v = v

, proves uniqueness if the norm (IX.5 .70) of the data is suﬃciently

small. We ﬁnally observe that this uniqueness result implies that for n ≥ 5,

any generalized solution v corresponding to smooth sma ll data and satisfying

(IX.5.71) is smooth. 

We recall that if n = 2, 3, 4, every generalized solution sati sﬁes the energy equality

(that is, (5.45) with equality sign), while, for n ≥ 5, this equality is satisﬁed by

those generalized solutions which are also in L

(Ω); see Exercise IX.3.1.

640 IX Steady Navier–Stokes Flow in Bounded Domains

IX.6 Limit of Inﬁnite Viscosity: Transition to the Stokes

Problem

As we noticed in the Introduction to Chapter IV, the Stokes system is to be

regarded as a formal approximation of the Navier–Stokes system, whenever

the inertial term v ·∇v b ecomes smal l compared to the viscous term ν∆v or,

in dimensionl ess language, whenever the Reynolds number R becomes van-

ishingly small. The question arises quite naturally of whether o ne can give

a rigorous mathematical j ustiﬁcation of this approximation. The aﬃrmative

answer to such a problem, in the case of a bounded region of ﬂow, is essentially

due to Odqvist (1930, §6); cf. al so Finn (1961a, Section 7a)), and is founded

upon the estimate of the Green tensor associated to the Stokes system. Here,

we shall follow a diﬀerent approach based, m ainly, on Theorem IV.6.1. Specif-

ically, we shall show that every generalized solution v to (IX.0.1), (IX.0.2), as

ν → ∞ (or R → 0) tends to the generalized solution w of the Stokes problem

(IV.0.1) corresponding to the same data, cf. Theorem IX.6.1. Furthermore, if

Ω, f , and v

∗

are suﬃciently smoo th, we show that the fo llowing estimates

are valid

kv − wk

(Ω)

≤ C

/ν

− πk

(Ω)

≤ C

/ν

for all ν ≥ ν

(IX.6.1)

where p

≡ p/ν, π are the pressure ﬁelds associated to v and to w, respectively,

and C

, C

are known functions of the data and of the positive number ν

Theorem IX.6.1 Let Ω , f , and v

∗

be as in Theorem IX.4.1. Denote by

v = v(x; ν), p = p(x; ν) the family of generalized soluti ons to (IX.0.1), (IX.0.2)

corresponding to f , v

∗

and parameterized in ν > 0, whose exi stence has been

established in Theorem IX.4.1. Moreover, let w , π b e the generalized solution

to the Stokes problem (IV.0. 1) and the associated pressure ﬁeld, respectively,

corresponding to f and v

∗

, whose existence is gua ranteed by Theorem IV.1.1.

Then

kv − wk

1,2

+ kp

−πk

≤ c/ν, ν ≥ ν

where c is a known function of the data and of ν

(cf. (IX.6.3), (IX.6.4)) and

is any p ositive ﬁxed number.

Proof. Setting u = v − w, by the deﬁnition of a generalized solution and

Remark IX.1.1 we obtain

(∇u, ∇ϕ) = −

(v · ∇v, ϕ), for all ϕ ∈ D

1,2

(Ω). (IX.6.2)

Since Ω is locally Lipschitz, we have u ∈ D

1,2

(Ω) (cf. Section III.5, and we

may take u = ϕ in (IX.6.2) to obtain

|u|

1,2

|(v · ∇v, u)|.

IX.6 Limit of Inﬁnite Viscosity: Transition to the Stokes Problem 641

Thus, Lemma IX.1.1 and inequality (II.5.5) imply

kuk

1,2

≤

kvk

1,2

with C = C(n, Ω). Substituting into this i nequality (IX.4.51) with Φ = ν/2

(say) furnishes

kuk

1,2

≤



|f |

−1,2

∗

1/2,2(∂Ω)

1 + ν

∗

1/2,2(∂Ω)



(IX.6.3)

where C

= 2c

. Furthermore, using (IX.1.11) and (IV.1.3) (the latter written

with w and π in place of v and p), we deduce that

ν(∇u, ∇ψ) + (v · ∇v, ψ) = +(p −νπ, ∇·ψ)

for all ψ ∈ D

1,2

(Ω) and so, reasoning as in the proof of Theorem I X.3.1, we

obtain

kp − νπk

≤ C



ν|u|

1,2

+ kvk

1,2



(IX.6.4)

which, on account of (IX.6.3) and of (IX.4.51), completes the proof of the

theorem. ut

Remark IX.6.1 Theorem IX.6.1 is also valid in dimension n = 4. If n ≥ 5,

as already noticed several times, we cannot take ϕ = u into (IX. 6.2), because

a generalized solution need not belong a priori to L

(Ω). However, using the

more regular solutions constructed in Theorem IX.5.3, the reader will prove

without diﬃculty that Theorem IX.6.1 continues to hold with ν

depending,

this time, on the magnitude of f and v

∗

. 

We shall next show that, if Ω is of class C

and if the data satisfy

f ∈ W

1,σ

(Ω), v

∗

∈ W

3−1/σ,σ

(∂Ω), σ =

n + r

, r > n, (IX.6.5)

then estimates (IX.6.1) hold. To ﬁx the ideas, we suppose n = 3, the case

where n = 2 is treated similarly. If v is a generalized solutio n to (IX.0.1),

(IX.0.2), then v · ∇v ∈ W

−1,3

(Ω) and from the estimates for the Stokes

problem derived i n Theorem IV.6.1 we obtai n v ∈ W

1,3

(Ω); furthermore

kvk

1,3

≤



kvk

+ kfk

−1,3

+ νkv

∗

2/3,3(∂Ω)



≤



kvk

1,2

+ kfk

−1,3

+ νkv

∗

2/3,3(∂Ω)



(IX.6.6)

where, in the second inequality, we have used the embedding Theorem II.3.4.

Again by this latter theorem we deduce

kvk

≤ c

kvk

1,3

, for all s > 1, (IX.6.7)

642 IX Steady Navier–Stokes Flow in Bounded Domains

and, by the H¨older inequality,

kv · ∇vk

≤ kvk

3q/(3−q)

kvk

1,3

, for all q ∈ (1, 3). (IX.6.8)

Theorem IV.6.1 then furnishes v ∈ W

2,q

(Ω), for all q ∈ (1, 3), and, by

(IX.6.6)–(IX.6. 8),

kvk

2,q

≤



kvk

3q/(3−q)

+ kfk

+ νkv

∗

2−1/q,q(∂Ω)



. (IX.6.9)

On the other hand, by Theorem II.3.4,

kvk

∞

≤ c

kvk

2,q

kvk

1,3q/(3−q)

≤ c

kvk

2,q

for a ll q ∈ (3/2, 3), (IX.6.10)

which, for these values of q, together with (IX.6.8), implies

kv · ∇vk

1,q

≤ 2(kv · ∇vk

+ k∇(v · ∇ v)k

)

≤ 2(kv · ∇vk

+ kvk

1,2q

+ kvk

∞

kvk

2,q

)

≤ c

kvk

2,q

In view of Theorem IV.6.1 we then obtain for a ll q ∈ (3/2, 3)

kvk

3,q

≤



kv · ∇vk

1,q

+ kfk

1,q

+ νkv

∗

3−1/q,q(∂Ω)



≤



kvk

2,q

+ kfk

1,q

+ νkv

∗

3−1/q,q(∂Ω)



(IX.6.11)

Now, setting as before u = v − w and, f urther, τ = (p/ν − π), we have

∆u =

v · ∇v + ∇τ

∇ · u = 0







in Ω

u = 0 at ∂Ω

and, by Theorem IV.6.1, we recover for r > 3

kuk

3,r

+ kτk

2,r

≤

kv · ∇vk

1,r

≤



kvk

1,2r

+ kvk

∞

kvk



. (IX. 6.12)

By Theorem II.3.4 we have, with σ = 3r/(3 + r)

kvk

2,r

≤ c

kvk

3,σ

(IX.6.13)

and

kvk

1,2r

≤ c

kvk

3,6r/(3+4r)

≤ c

kvk

3,σ

(IX.6.14)

since r > 3. Therefore, (IX.6.10)

, (IX.6.12)–(IX.6.14) furnish

IX.7 Notes for the Chapter 643

kuk

3,r

+ kτk

2,r

≤

kvk

3,σ

≤



kvk

2,σ

+ kfk

1,σ

+ νkv

∗

3−1/σ,σ(∂Ω)



and estimates (IX.6.1) become a consequence o f this latter inequality, of The-

orem II.3 .4, of (IX.6.6), (IX.6.9), (IX.6.11), and (IX.4.51).

We have then proved the following result:

Theorem IX.6.2 Let Ω, f , v

∗

, v, p, w, and π be as in Theorem IX.6.1

with Ω of class C

and f , v

∗

satisfying (IX.6.5). Then v − w ∈ C

(Ω),

p − π ∈ C

(Ω) and they obey estimate (IX.6.1) with ν

any positive, ﬁxed

number.

Remark IX.6.2 If n ≥ 4, Theorem IX.6.2 continues to hold provided we

use the more regular solutions constructed in Theorem IX.5.3. However, this

time, ν

will depend on the size of the data; cf. also Remark IX. 6.1. 

IX.7 Notes for the Chapter

Section IX.1. The introduction of the pressure ﬁeld p for weak solutions as

an element of a suitable L

-space i s essentially due to the work of Sol onnikov

Sˇcadilov (1973), cf. also Lemma IV.1.1. Actually, these a uthors show the

existence of p in the case o f the l inearized Stokes system but their ideas carry

over, without conceptual diﬃculties, to the nonlinear Navier–Stokes problem.

The general results derived in Lemma IX.1.2 and Remark IX.1.5, whose

proofs are based on ideas of Solonnikov &

Sˇcadilov, are due to me.

Section IX.2. Uniqueness of generalized solutions in the form presented here

can be traced back to the papers of Hopf (19 41, 1957); cf. Finn (1961a Section

8).

The condition (IX.2.4), ensuring that a generalized solution is unique, can

be improved in several respects. For instance, fol lowing the work of Serrin

(1959a ) one can give a variational formulation of uniqueness that i s aimed,

among other things, at yiel ding the “best” upper bound on v for uniqueness

to ho ld. It is interesting to observe that such conditions likewise ensure the

nonli near energy stability of v, cf. Joseph (1976), Galdi & Rionero (1985).

Section IX.3. An interesting variant of Leray’s existence theorem that shows

existence of generalized solutions has been given by Ladyzhenskaya (1959b,

Theorem 2); cf. also Vorovich & Youdovich (19 61, Theorem 1). The variant

is based o n the following steps. First, as we a lready remarked, L adyzhen-

skaya gives a variational formulation of the boundary-value problem (IX.0.1),

(IX.0.2). Such a formulation leads to an equivalent operator equation for the

velocity ﬁeld v of the form

v =

(Av + f ) , (∗)

644 IX Steady Navier–Stokes Flow in Bounded Domains

with A a suitable nonlinear operator in a n appropriate Hilbert space H. She

then proves that A is completely continuous and that every possible solution

v to (∗) that i s in H admits a uniform bound independent of 1/ν ∈ (0, B).

This fact, together with a simpliﬁed version of the Leray-Schauder theorem,

produces existence in the physically relevant cases of two and three dimen-

sions. For the sake of ma thematical generality, however, it should be noted

that neither Leray’s nor Ladyzhenskaya’s argument works in fo ur dimensions

due to the fa ct that the operator A is no longer completely continuous. This

problem was taken up a nd solved by Shinbrot (1964) who proves that a prop-

erty for A weaker than complete continuity is suﬃcient to recover existence

to (∗). In light of this consideration, the method of Fujita which we used here

assumes even more relevance since, as we have seen, it applies in all space

dimensions.

Existence of q-generalized solutions (see Remark IX.1.3) has been i nves-

tigated by Serre (1983). Speciﬁcally, he proves that if Ω ⊂ R

, n = 2, 3, is

(bounded) and of class C

, then given

f ∈ D

−1,q

(Ω), q ∈ (n/2 , 2),

there is at least one corresponding q-generalized solution to (IX.0 .1), (IX.0.2)

with v

∗

= 0. The result is also extended to the case v

∗

6= 0. The case q = 3/2,

for n = 3, left out by Serre has been lately covered by Kim (2009, Rema rk 6).

More recently, there has been an increasing interest in the study of the

properties of ver y weak solutions. These latter are characterized by a weakly

divergence free velocity ﬁeld v ∈ L

(Ω)

satisfying the condition

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

∗

∂Ω

, (∗∗)

for all ψ ∈ W

n−1

(Ω) ∩ H

n−1

(Ω) ≡

(Ω), where, we recall, h·, ·i

∂Ω

the duality pairing between the spaces W

−1/q,q

(∂Ω) and W

1−1/q

(∂Ω). As

mentioned in the Notes for Section IV.6, even though, at the outset, very

weak solutions are not weakly diﬀerentiable, they still possess a well deﬁned

trace at the (suﬃciently smooth) boundary, in an appropriate space; see Gal di,

Sima der & Sohr (2005, Theorem 1). The study of existence and uniqueness

of very weak solutions in a smooth (for instance, C

1,1

), bounded and simply

connected domain of R

, n = 2, 3, was initiated by Maruˇsi´c-Paloka (2000). In

particular, when n = 3, this author proves existence for any f ∈ W

−1,2

(Ω)

and v

∗

∈ L

(∂Ω). Successively, when Ω is a bounded domain of R

of class

As a matter of fact, the method, as presented in the paper of Fujita (1961) fails

for space dimension n > 4. However, the slight modiﬁcation given by Finn (1965b

§2.7)) and adopted by me permits the demonstration of existence of generalized

solutions for all n ≥ 2.

One may replace L

with L

, with q ≥ n, on condition of choosing appropriate

test functions ψ in (∗∗); see the literature cited below. However, the case q = n,

is, of course, the more interesting, because it is t he case of less regularity.

IX.7 Notes for the Chapter 645

2,1

, Galdi, Simader & Sohr (2005) showed existence under the more gen-

eral assumption v

∗

∈ W

−1/3 ,3

(∂Ω). (See Farwig, Galdi & Sohr (2006), for

similar results when n = 2.) However, such a result requires the data to be

“suﬃciently smal l”. This latter restriction was further removed by Kim (2009,

Theorem 1), by cleverly combining argum ents of Gal di, Simader & Sohr with

those of Gehrardt (1979) and Maruˇsi´c-Paloka. For the sake of precision, it

is worth noticing tha t Kim’s deﬁnition of “very weak” solution (as well as

Maruˇsi´c-Paloka’s) is, in principle, more restrictive than the one adopted by

Galdi, Simader & Sohr, where the test function in (∗∗) is chosen from the

space C

(Ω), deﬁned in equation (∗) in the Notes for Section IV.6, instead of

the space

(Ω). Of course, whenever C

(Ω) is dense in

(Ω), the two def-

initions coincide. This happens, for example, if Ω is of class C

2,α

, α ∈ (0, 1).

Whether or not the same is true for do mains with less regularity remains to be

seen. It must be ﬁnall y noticed that, in the above papers with the exception of

that by Maruˇsi´c-Paloka, the more general condition ∇·v = g, with g suitably

prescribed, is considered.

Section IX.4. The case of nonhomogeneous boundary conditions was ﬁrst

treated by Leray (1933, pp. 28-30, pp. 40-41) by means of two diﬀerent ap-

proaches. The ﬁrst is essentially that described at the beginning of the section,

and it was successively completed and clariﬁed by Hopf (1 941, 1957); cf. also

Ladyzhenskaya (1959b, C hapter I, §2), Finn (1961a, Lemma 2.1 and Section

2a)). The second is ba sed on a clever contradiction argument, which we would

like to sketch here. As we have mentioned several times, the clue for existence

is to show a unifo rm bound on the Dirichlet integral of all possible solutions

v to (IX.0.1), (IX.0.2) (in a suitable regulari ty class):

Ω

∇v : ∇v ≤ M, (D)

with M independent of ν ranging in some bounded interval [ν

, ν

], ν

> 0.

Contradicting (D ) means that there is a sequence of solutions {v

, π

} to

(IX.0.1), (IX.0.2 ) and of positive numbers {ν

} such that as k → ∞

≡

Ω

∇v

: ∇v

→ ∞, ν

→ ν

for a suitable ν ∈ [ν

, ν

]. On the other hand, it is easy to show (cf. Leray

1933, p. 28), that, denoting by V a (suﬃciently smooth) solenoidal extension

of v

∗

the following identity holds for all k ∈ N

Ω

∇v

: ∇v

= −

∂Ω

∗

·n + ν

Ω

∇v

: ∇V +

Ω

· ∇v

· V .

(∗ ∗ ∗)

Setting w

= v

√

νJ

and using standard compactness arguments one shows

that w

tends to a suitabl e solenoidal ﬁeld w that vanishes a t the boundary,

and that, after dividing both sides of (∗∗∗) by J

and letting k → ∞, satisﬁes