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

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626 IX Steady Navier–Stokes Flow in Bounded Domains

k∇W k

= 1, (IX.5.2 0)

leading to a contradiction. We shall next consider the case n ≥ 3. We begin to

suppose q ∈ [2n/(n + 2), n). In this si tua tion, by Theorem II.3.4, we get W ∈

1,2

(Ω). T he tri linear form (v ·∇z

, z

) is continuous in L

(Ω) ×W

1,2

(Ω) ×

2n/(n−2)

(Ω) (see Exercise IX.2.1), and so, by the embedding Theorem II.3.2,

it is continuous in L

(Ω) × H

(Ω). Since, by deﬁnition, D(Ω) is

dense in H

(Ω), for all σ ∈ (1, ∞), by a standard approximating procedure

we are then allowed to take W = ϕ in (IX.5.19) to get

0 = |W |

1,2

+ (u · ∇W , W ) .

However, (u · ∇W , W ) = 0, by Exercise IX.2.1, because W ∈ W

1,2

(Ω) and

u ∈ L

(Ω) with ∇ · u = 0. As a result, we infer W ≡ 0, which contradicts

(IX.5.20). Inequality (IX.5.14) is therefore established, if n = 2, for q ∈ (1, 2),

and, if n ≥ 3, for q ∈ [2n/(n + 2), n). From (IX.5.13) and (IX.5.14) we obtain

(j)

2,q

+ kπ

(j)

1,q

≤ C

(j)

(IX.5.21)

with C

= C

(n, q, Ω, u). From (IX.5.21), Remark II.3.1 and Theorem II.5. 2

it follows that there are subsequences {w

)

} and {π

)

}, and two ﬁelds

w ∈ W

2,q

(Ω) and π ∈ W

1,q

(Ω), such that

→ w in W

2,q

(Ω)

→ w in W

1,q

(Ω)

and

→ π weakly in W

1,q

(Ω).

Clearly, (w, π) is a solution to (IX.5.1), and the lemma is thus proved for

n = 2, and, fo r n ≥ 3, under the condition that q ∈ [2n/(n + 2), n). Let us

next assume n ≥ 3, and q ∈ (1, 2n/(n+2)). Taking into account the procedure

previously used, the result will b e proved provided we show that (IX.5.19) has

only the solution W ≡ 0. We begin to observe that, since 2n/(n + 2) < n/2

(for n ≥ 3) by the embedding Theorem I I.3.4, we deduce

W ∈ H

(Ω) ∩ L

(Ω) ,

∈ (1, 2) , r

n − r

(IX.5.22)

Consider now the problem (adjoint to (IX.5.1))

∆ϕ + u · ∇ϕ = ∇τ + G

∇ · ϕ = 0

)

in Ω

ϕ |

∂Ω

= 0 .

(IX.5.23)

IX.5 Regularity of Generalized Solutions 627

Noticing that

σ ≡

n + r

∈ (

n + 2

, n) ,

nσ

n − σ

= r

from what we have previously shown, and, ag ain, Theorem II.3.4, we know

that, f or each G ∈ C

∞

(Ω), there is at least one corresponding solution ϕ, τ

to (IX.5 .23) such that

ϕ ∈ W

n+r

(Ω) ∩ H

(Ω) , τ ∈ W

n+r

(Ω). (IX.5.24)

Now, let {ϕ

} ⊂ D(Ω) converge to ϕ in H

(Ω), and replace ϕ

for ϕ in

(IX.5.19). Clearly, we have

lim

k→∞

(∇W , ∇ϕ

) = (∇W , ∇ϕ) . (IX.5.25)

Moreover, by an easily justiﬁed integration by parts, based on density argu-

ments, we ﬁnd

(u · ∇W , ϕ) = −(u · ∇ϕ, W ) , ϕ ∈ D(Ω) . (IX.5.26)

Furthermore, by the H¨older inequality (see al so Exercise IX.2.1),

|(u · ∇ϕ

, W )| ≤ kuk

k∇ϕ

kW k

/(n−r

)

and so, recalling the summability properties (IX.5.22) of W , we deduce

lim

k→∞

(u ·∇W , ϕ

) = (u · ∇ϕ, W ) . (IX.5.27)

Therefore, from (IX.5.19)

with ϕ ≡ ϕ

, and (IX.5.25 )–(IX.5.27) we conclude,

in the limit k → ∞,

(∇W , ∇ϕ) −(u · ∇ϕ, W ) = 0 . (IX.5.28)

Next, let {W

} ⊂ C

∞

(Ω) converge to W in W

1,r

(Ω), so that, by embedding,

it converges to W also in L

/(n−r

)

(Ω); see Theorem II.3.4. Thus, taking

into account

(nr

/(n − r

))

= nr

/(n + r

) , (IX.5.29 )

along with (IX.5.24), we have

(∇W , ∇ϕ) = lim

k→∞

(∇W

, ∇ϕ) = − lim

k→∞

, ∆ϕ) = −(W , ∆ϕ) .

As a consequence, with the help of (IX.5.28), we infer

(W , ∆ϕ + u ·∇ϕ) = 0 ,

which, in turn, recalling that (ϕ, τ) satisfy (IX.5.23), is equivalent to the

following

628 IX Steady Navier–Stokes Flow in Bounded Domains

(W , G) = −(W , ∇τ) . (IX.5.30 )

However, by (IX.5.22) and Lemma III.2.2, W ∈ H

(Ω), r

≡

n−r

, whereas

by (IX.5.24), and (IX.5.29), τ ∈ G

(Ω), and so, by Lemma III.2.1, we ob-

tain (W , ∇τ) = 0. Replacing this information back into (IX.5.30), we deduce

(W , G) = 0, which, by the arbitrariness of G ∈ C

∞

(Ω), allows us to con-

clude W = 0 a.e. in Ω. T he proof of the existence part of the lemm a is thus

completed. We shall now show the uniqueness part. Setting z := w − w, we

have that z satisﬁes the following problem

(∇z, ∇ϕ) − (u · ∇ϕ, z) = 0 , for all ϕ ∈ D(Ω)

z ∈ H

(Ω) , r = min{s, q}.

(IX.5.31)

Let us ﬁrst consider the case n ≥ 3. The result immedia tely follows if s ≥ 2.

Actually, we then have z ∈ H

(Ω) and therefore, by empl oying the standard

density argument that we have already used previously in the proof (right

after (IX.5.20)), we may replace ϕ with z in (IX.5.31) to obtain, as before,

|z|

1,2

= 0, nam ely z = 0 a.e. in Ω. Assume then s ∈ (1, 2). However, in

this case, z sati sﬁes the same assumption and the same equation satisﬁed by

W ; see (IX.5. 22), (IX.5.19), and (IX.5.2 6)) . Therefore, following exactly the

same argument used to show W = 0, we also prove z = 0, and uniqueness

is completely recovered if n ≥ 3. If n = 2, we observe that, since by the

embedding Theorem II.3.4, w ∈ H

(Ω), we have z ∈ H

(Ω). Let {ϕ

} ⊂

D(Ω) with ϕ

→ z in H

(Ω). Furthermore, again by that theorem, we have

also H

(Ω) ,→ L

/(q

−2)

(Ω). Therefore, setting ϕ = ϕ

into (IX.5.31), then

letting k → ∞ and using the results o f Exercise IX.2.1, we ﬁnd

|z|

1,2

= −(u · ∇z, z) = 0 ,

which implies z = 0 a.e. in Ω. The proof of the lemma is complete. ut

The next result provides, in particular, suﬃcient conditions for the interior

regularity of weak solutions in arbitrary s pace dimensions n ≥ 2.

Theorem IX.5.1 Let Ω be any domain of R

, n ≥ 2, and let v be such that:

(i) v ∈ L

loc

(Ω), where s = n, if n ≥ 3, while s = s

> 2, if n = 2 ;

(ii) ∇ · v = 0, in the weak sense ;

(iii) v satisﬁes the following equation

ν(v, ∆ϕ) + (v · ∇ ϕ, v) = hf , ϕi, for a ll ϕ ∈ D(Ω) . (IX.5.32)

The following prop erties hold.

(a) If

f ∈ L

loc

(Ω) , 1 < q < ∞,

then

v ∈ W

2,q

loc

(Ω) ,

and there exists p ∈ W

1,q

loc

(Ω) such that (IX.0.1) is satisﬁed a.e. in Ω.

IX.5 Regularity of Generalized Solutions 629

(b) If, moreover,

f ∈ W

m,q

loc

(Ω)

where m ≥ 1, and

q ∈ (1, ∞), if n = 2,

while

q ∈ [n/2, ∞), if n > 2,

then

v ∈ W

m+2,q

loc

(Ω), p ∈ W

m+1,q

loc

(Ω). (IX.5.33)

Proof. We begin to show part (a). Let B be a ball of R

, with B ⊂ Ω. Then,

from (IX.5.32) and Lemma IV.4.1 we ﬁnd that, for all suﬃciently small ε > 0,

the molliﬁcation, v

, of v satisﬁes the following system

ν∆v

= ∇p

(ε)

+ div (v ⊗ v)

+ f

∇ · v

= 0

)

in B , (IX.5.34)

for some p

(ε)

∈ C

∞

(B). We next denote by w = w(ε), τ = τ(ε), τ

= 0, the

solution to the following Stokes problem

ν∆w = ∇τ + di v (v ⊗ v)

+ f

∇· w = 0

)

in B

w = 0 at ∂B ,

(IX.5.35)

with

(w, τ) ∈ W

1,r/2

(B) × L

r/2

(B) , r > 2 .

In view of the properties of the moll iﬁcation and of Theorem IV.6.1, such a

solution exists and satisﬁes the inequality

kwk

1,r/2

+ kτk

r/2

≤ c



r,B

+ kf

−1,r/2,B



. (IX.5.36)

Our next task is to estimate kf

−1,r/2,B

in terms of kf k

q,B

. The starting

point is the obvious inequality

|(f

, Φ)| ≤ kfk

q,B

kΦk

, Φ ∈ W

1,(r/2)

(B) , (IX. 5.37)

that fo llows from (II.2.9) and the H¨older inequality. We distinguish several

cases. Suppose ﬁrst n = 2. Without loss we may assume s

≤ 4. T hus, if we

choo se r = s

, from the embedding Theorem II.3.4 it follows W

1,(r/2)

(B) ,→

(B), for all t ∈ (1, ∞), so tha t from (IX. 5.36), we deduce

−1,r/2,B

≤ c

kfk

q,B

, q ∈ (1, ∞) , r := s

, n = 2. (IX.5.38)

The lower bound n/2 f or q is not necessarily the best possible. However, it suﬃces

for our aims.

630 IX Steady Navier–Stokes Flow in Bounded Domains

If n = 3, we observe that (n/2)

= n = 3. Consequently, if we choose r = 3

by the embedding Theorem II.3.4 we have W

1,(r/2)

(B) ≡ W

1,3

(B) ,→ L

(B),

for a ll t ∈ (1, ∞), and (IX.5.3 6) furnishes

−1,r/2,B

≤ c

kfk

q,B

, q ∈ (1, ∞) , r := 3 , n = 3. (IX.5.39)

We next analyze the case n ≥ 4. In such a case we ﬁnd (n/2)

< n, and so,

if q ∈ [n/3, ∞), it follows that q

≤ n/(n − 3) = n(n/2)

/(n − (n/2)

). Now,

by Theorem II.3.4, W

1,(n/2)

(B) ,→ L

(B) and, therefore, by choosing r = n,

from (IX.5.36), we obtai n

−1,r/2,B

≤ c

kf k

q,B

, q ∈ [n/3, ∞) , r := n , n ≥ 4. (IX.5.40)

Finally, i f n ≥ 4 and q ∈ (1, n/3), we take r = 2nq/(n − q) and observe that

(r/2)

< n and that q

= n(r/2)

/(n − (r/2)

). Thus, since by Theorem II.3.4

1,(r/2)

(B) ,→ L

(B), again from (IX.5.36) we conclude

−1,r/2,B

≤ c

kfk

q,B

, q ∈ (1, n/3) , r :=

2nq

n − q

, n ≥ 4 . (IX.5.41)

We next notice that, from (IX.5.34) and (IX.5.35), the ﬁelds z := v

−w and

χ := p

(ε)

− τ satisfy the following Stokes system

ν∆z = ∇χ

∇ · z = 0

)

in B . (IX.5.42)

From Theorem IV.4.4 and Remark IV.4.2 we then ﬁnd, in particular,

kzk

1,r/2,B

+ kχk

r/2 ,B

≤ c

kzk

r/2 ,B

where B

is an open ball with B

⊂ B. Using this inequality, and taking into

account that, by assumption, kv

r,B

≤ kvk

r,B

< ∞ for all values of r, q and

n speciﬁed in (IX.5.38 )–(IX.5.41), from (IX.5.36), (IX.5.38)–(IX.5.41), and

Exercise II.3.5 we deduce

v ∈ W

1,s

) , n = 2 , q ∈ (1, ∞)

v ∈ W

1,n/2

) , n = 3 , q ∈ (1, ∞)

v ∈ W

1,n/2

) , n ≥ 4 , q ∈ [n/3, ∞)

v ∈ W

1,r/2

) , n ≥ 4 , q ∈ (1, n/3) , r :=

2nq

n − q

(IX.5.43)

Moreover, by a similar argument that employs also Theorem I I.2.4(ii), we

show the existence of a scalar ﬁeld p such that

p ∈ L

) , n = 2 , q ∈ (1, ∞)

p ∈ L

n/2

) , n = 3 , q ∈ (1, ∞)

p ∈ L

n/2

) , n ≥ 4 , q ∈ [n/3, ∞)

p ∈ L

r/2

) , n ≥ 4 , q ∈ (1, n/3) , r :=

2nq

n − q

(IX.5.44)

IX.5 Regularity of Generalized Solutions 631

and, further, the pair (v, p) satisﬁes (IX.1.11) for all ψ ∈ C

∞

). However,

the ball B

is arbi trary with B

⊂ Ω, so that from (IX.5.43)–(IX.5.43) we

ﬁnd that

(v, p) ∈ W

1,s

loc

(Ω) × L

loc

(Ω) , n = 2 , q ∈ (1, ∞)

(v, p) ∈ W

1,n/2

loc

(Ω) × L

n/2

loc

(Ω) , n = 3 , q ∈ (1, ∞)

(v, p) ∈ W

1,n/2

loc

(Ω) × L

n/2

loc

(Ω) , n ≥ 4 , q ∈ [n/3, ∞)

(v, p) ∈ W

1,r/2

loc

(Ω) × L

r/2

loc

(Ω) , n ≥ 4 , q ∈ (1, n/3) , r :=

2nq

n − q

(IX.5.45)

and, in addition, tha t (v, p) satisﬁes (IX.1.11) for all ψ ∈ C

∞

(Ω). Our next

objective is to show that

(v, p) ∈ W

2,q

loc

(Ω) × W

1,q

loc

(Ω) , q ∈ (1, n) , n ≥ 2 . (IX.5.46)

It is clear that, in order to show (IX. 5.46), it is enough to show the stated

summability properties on an arbitrary bounded subdomain of Ω. To this

end, let Ω

, Ω

be bounded domains in R

with Ω

⊂ Ω

, Ω

⊂ Ω and let

φ ∈ C

∞

) be one in Ω

and zero outside Ω

Writing φψ in place of ψ

into (IX.1.11) and extending v to zero outside Ω

, we readily recognize that

≡ φv is a weak solution to the problem

∆v

= u · ∇v

+ ∇p

+ F

∇ · v

= g

)

in B

= 0 at ∂B

(IX.5.47)

where

≡ φp/ν

u ≡ v/ν

F ≡ φf/ν + 2∇φ ·∇v + v∆φ − v · ∇φv/ν + p∇φ/ν

g ≡ v · ∇φ

(IX.5.48)

and B

is an open ball containing Ω

. The leading idea in the proof of

(IX.5.46), is to use a boot-strap a rgument that starts from (IX.5.45) and

uses several times Lemma IX.5.1 applied to the problem (IX.5.4 7)–(IX.5.48).

Suppose, at ﬁrst, n = 2, the case that, seemingly, requires more eﬀort. We

begin to show that

v ∈ W

1,2

loc

(Ω) . (IX.5.49)

If s

≥ 4, this is obvious from (IX.5.45)

, and so we shall assume s

∈ (2, 4).

Let B an open bal l of R

with B ⊂ Ω, and consider the following two Stokes

problems

For the construction of φ, see the proof of Theorem IV.4.1.

632 IX Steady Navier–Stokes Flow in Bounded Domains

ν(∇v

, ∇ϕ) = (v · ∇ϕ, v) , for al l ϕ ∈ D(B) , v ∈ H

(B)

ν(∇v

, ∇ϕ) = −(f, ϕ) for all ϕ ∈ D(B) , v ∈ W

2,q

(B) ∩ H

(B) .

(IX.5.50)

Clearly, the function Φ : = v−v

−v

satisﬁes (∇Φ, ∇ϕ) = 0, for all ϕ ∈ D(Ω),

and, consequently, in view of the prop erties of v, v

, v

and of Theorem IV.4.3,

we ﬁnd Φ ∈ C

∞

(B). Furthermore, by Theorem IV.6.1(a), v

exists and, by

the embedding Theorem II. 3.4, v

∈ W

1,2

(B). Therefore, to show (IX.5.49),

it is enough to show that v

∈ W

1,2

(B). We shall prove this by means of

a recurrence argument based on a repeated use o f Theorem IV.6.1(b) and

of the embedding Theorem II .3.4. Actually, from the former theorem and

(IX.5.45)

we can take t ≡ t

= s

/2, which, by Theorem II.3.4 and the fact

that s

∈ (2, 4), implies v ∈ L

/(2−t

)

(B). Thus, again by Theorem IV.6.1,

we may take t ≡ t

= t

/(2 −t

), which, in turn, by Theorem II.3.4 furnishes

v ∈ L

/(2−t

)

(B), and so on. We thus obtain the following recurrence relation

for the exponents t

k+1

2 − t

, k ∈ N , t

= s

/2 . (IX.5.51)

We notice that, since by assumption t

> 1 + η, for some positive η, the

sequence {t

} is increasing, and so, in particular, t

> 1 + η, for all k ∈ N. We

claim that there is k ∈ N such that t

k+1

= 2. By assuming the contrary, we

would have t

< 2, for all k ∈ N, and so, due to the fact that {t

} is increasing,

there is t

∗

> 0 such tha t t

→ t

∗

as k → ∞. However, by taking the limit

k → ∞ in (IX.5. 51), we would ﬁnd t

∗

= 1, which furnishes a contradiction. As

a consequence, (IX.5.49) is proved. By an argument entirely analogous to that

used to show (IX.5.38), we show f ∈ W

−1,2

(ω) for all bo unded domains ω with

ω ⊂ Ω. Thus, from (IX.5.51) and Lemma IX.2.1, we deduce p ∈ L

loc

(Ω). T his

latter property along with (IX.5.51) and (IX.5.48)

3,4

, allows us to conclude

F ∈ L

), g ∈ W

1,q

), q ∈ (1, 2), so that by Lemm a IX.5.1 and (IX.5.49)

we prove (IX.5.46) for n = 2. We next consider the case n = 3. If q ∈ (1, 3/2],

from (IX.5.45) and (IX.5.47) we easily ﬁnd tha t

F ∈ L

) , g ∈ W

1,q

) , (IX.5.52)

with q = q, and so, by Lemma IX.5.1, the property (IX.5.46) follows for the

above va lues of q. If q ∈ (3/2, 3), again by (IX.5.45), we ﬁnd tha t F , g satisfy

(IX.5.52) wi th q = 3/2, which, in turn, by Lemma IX.5.1 and the arbitrari ness

of Ω

implies (v, p) ∈ W

2,3/2

loc

(Ω)×W

1,3/2

loc

(Ω). From this latter property, by the

embedding Theorem II.3 .4, we infer (v, p) ∈



1,3

loc

(Ω) ∩ L

∞

loc

(Ω)



×L

loc

(Ω).

Thus, we deduce the validity of (IX.5.52), with q = q, q ∈ (3/2, 3), which, by

Lemma IX.5.1, concludes the proof of (IX. 5.46), in the case n = 3. If n ≥ 4 ,

suppose ﬁrst q ∈ (1, n/3). Then, from (IX.5.45)

, we ﬁnd r/2 > q, which

implies (IX.5.52), with q = q, and so, in turn, (IX.5.46) for these values of q.

If q ∈ [n/3, n/2], then, by (IX.5.45)

, we may take q = q in (IX.5 .52), which,

IX.5 Regularity of Generalized Solutions 633

again by Lemma IX.5.1, furnishes (IX.5.46) also for these values of q. Finally,

if q ∈ [n/2, ∞), the argument is identical to that used for the case n = 3 by

replacing 3/2 with n/2. The property (IX.5.46) is thus established, and so is

part (a) of the theorem if q ∈ (1, n). Assume next q ≥ n. Then (IX.5.46) is

satisﬁed f or all q ∈ (1, n), and so, by the embedding Theorem II. 3.4, we get

v ∈ L

∞

loc

(Ω) ∩ W

1,t

loc

(Ω), p ∈ L

loc

(Ω), for all t ∈ (1, ∞),

yielding, in particular,

v · ∇v ∈ L

loc

(Ω).

From the interior estimates for the Stokes problem proved in Theorem IV.4.1

it then follows v ∈ W

2,q

loc

(Ω), p ∈ W

1,q

loc

(Ω), which completes the proof of

part (a) of the theorem. In order to prove part (b), we shall use an inductive

argument. Since, by the results just establi shed, (IX.5.33) is true for l = 0, let

us assume that it holds for l = k −1, k ≥ 1, and let us show that it continues

to hold for l = k. This amounts to proving that if, for the values of q speciﬁed

in the statement of the theorem,

f ∈ W

k,q

loc

(Ω), v ∈ W

k+1,q

loc

(Ω), p ∈ W

k,q

loc

(Ω), (IX.5.53)

necessarily

v ∈ W

k+2,q

loc

(Ω), p ∈ W

k+1,q

loc

(Ω), (IX.5.54)

By Theorem IV.4.1, (IX.5.54) holds whenever

v ·∇v ∈ W

k,q

loc

(Ω). (IX.5.55)

However, by the inductive assumption, we know that

v · ∇v ∈ W

k−1,q

loc

(Ω)

and so to obtain (IX.5.55), a nd consequently (IX.5.54), we have to show that

(v · ∇v) ∈ L

(Ω

), |α| = k, (IX.5.5 6)

where Ω

is any bounded subdomain of R

with Ω

⊂ Ω. Without loss, we

may assume Ω

to be a ball. Expanding (IX.5.56) according to the Leibniz

rule we deduce

(v · ∇v) =

β≤α

v · ∇D

α−β

v (IX.5.57)

where

α−β

≡

∂

|α|−|β|

∂x

−β

. . .∂x

−β

≡

. . .

and β ≤ α means β

≤ α

, i = 1, . . ., n. Take ﬁrst q > n/2. Then, by the

embedding Theorem II.3.4 we have v ∈ C

k−1

(Ω

) and so (IX.5.5 7) yi el ds

634 IX Steady Navier–Stokes Flow in Bounded Domains

|α|=k

(v · ∇v)k

q,Ω

≤ c(kvk

k−1

(Ω)

kvk

2,q,Ω

|α|=k

v · ∇vk

q,Ω

(IX.5.58)

Thus, since k ≥ 1, the ﬁrst term on the right-hand side of (IX.5.58) is bounded

in view of (IX.5.53 ). Furthermore, by Theorem II.3.4, if k = 1 it follows that

v ∈ W

1,r

(Ω

) for al l r ∈ (1, ∞), while if k > 1, v ∈ C

(Ω

) and so, in any

case, the second term on the right-hand side of (IX.5.57) is ﬁni te, thus proving

(IX.5.56) when q > n/2 and, therefore, the theorem for n = 2. Assume now

that q = n/2, n > 2. By Theorem II.3.4 it follows that v ∈ C

k−2

(Ω

) and so

(IX.5.57) gives, with |α

| = k −2,

(v · ∇v)| ≤ c(

β≤α

v ·∇D

α−β

v| +

|β|=k−1

v · ∇D

α−β

|β|=k

v · ∇v|).

(IX.5.59)

The ﬁrst term in (IX.5.59) is increased as before, and we can show that it

belongs to L

(Ω

Again by Theorem II.3.4 and (IX.5.53) we have

v ∈ W

k,n

(Ω

v ∈ W

k−1,r

(Ω

), for all r ∈ (1, ∞ )

(IX.5.60)

and so the third term in (IX.5.59) belongs to L

(Ω

) ≡ L

n/2

(Ω

). As far as

the second term i s concerned, we noti ce that i ts norm in L

n/2

(Ω

) can be

increased by

|β|=k−1

kvk

2,n

which, if k ≥ 2, is ﬁnite by (IX.5.60). If k = 1 , the second term is increased by

N ≡ |v||D

v| and, again by (IX.5.60), it follows that it belongs to L

n/2−ε

(Ω

for ε ∈ (0, n/6). Thus,

v · ∇v ∈ W

1,n/2−ε

(Ω

)

and by the estimates for the Stokes probl em of Theorem IV.4 .1 we deduce, in

particular, that

v ∈ W

3,n/2−ε

(Ω

)

which, by Theorem II.3.4, in turn implies v ∈ C(Ω

). From this and (IX.5.53)

we then conclude that N ∈ L

n/2

(Ω

). The theorem is therefore proved. ut

Remark IX.5.1 From the embedding Theorem II.3.4 it follows that the as-

sumptions (i)–(iii) on v in Theorem IX.5.1 are satisﬁed if v is a generalized

solution to (IX.0.1), (IX.0.2) and n ≤ 4. 

If, of course, k ≥ 2; otherwise that term does not appear.

IX.5 Regularity of Generalized Solutions 635

An im portant consequence of Theorem IX.5.1 is the following.

Corollary IX.5.1 Let v obey conditions (i)–(iii) in Theorem IX.5.1. Then,

if f ∈ C

∞

(Ω), v and the associated 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.5.2 Assuming less regularity on f, we can obtain intermediate

regularity results on v and p. 

Regularity up to the boundary of a generalized solution follows as a par-

ticular case of the following one.

Theorem IX.5.2 Let Ω be a bounded domain of R

, n ≥ 2, of class C

, and

let v be such that:

(i) v ∈ W

1,s

(Ω) ∩L

(Ω), s ∈ (1, ∞), if n ≥ 3, while v ∈ W

1,2

(Ω), if n = 2 ;

(ii) v is weakly divergence free, and obeys (IX.0.2) in the trace sense ;

(iii) v satisﬁes (IX.1.2) .

Then, the following properties hold.

(a) If

f ∈ L

(Ω), v

∗

∈ W

2−1/q,q

(∂Ω) , q ∈ (1, ∞) ,

then

v ∈ W

2,q

(Ω) ,

and there exists p ∈ W

1,q

(Ω), such that (IX.0.1) is satisﬁed a.e. i n Ω.

(b) If, moreover, Ω is of class C

m+2

and

f ∈ W

m,q

(Ω), v

∗

∈ W

m+2−1/q

(∂Ω)

where m ≥ 1, and

q ∈ (1, ∞), if n = 2,

while

q ∈ [n/2, ∞), if n > 2,

then

v ∈ W

m+2,q

(Ω), p ∈ W

m+1,q

(Ω).

Proof. The proof of pa rt (a) is an immediate consequence of Lemma IX.5.1,

and we leave i t to the reader as an exercise. The proof of part (b), is entirely

analogous to that of Theorem IX.5.2(b), and will be, therefore, omi tted. ut

Remark IX.5.3 In v iew of the embedding Theorem II.3.4, the assumptions

(i)–(iii) of Theorem IX.5.2 are satisﬁed by any generalized solution for n ≤ 4.



The lower bound n/2 for q is not necessarily the best possible.