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

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586 IX Steady Navier–Stokes Flow in Bounded Domains

to the case where n = 4. If n ≥ 5 new diﬃculties appear that are not easily

attacked by the known methods. This is because a generalized solution has

a priori only the property of being in the Sobolev space W

1,2

(Ω), and this

is not enough, by use of the embedding theorems, to dominate appropriately

the nonlinear term in (IX.0.1)

. For instance, whether or no t any generalized

solution in higher dimension corresponding to regular data of arbitrary size

is regular, remains an open question.

However, Frehse and R˚uˇziˇcka (1994a,

1994b, 1995), and Struwe (1995) have shown the existence of regular solutions

in dimension n ≥ 5 without restrictions on the size of the data. We refer the

reader to these pa pers and to the review article by Frehse and R˚uˇziˇcka (1996)

for details and further related literature.

IX.1 Generalized Solutions. Preliminary Considerations

In the present section we begin to introduce the weak (or generalized) f or-

mulation of the steady Navier–Stokes problem in a bounded domain and to

investigate some of the basic properties of the a ssociated solutions. In doing

this we will essentially pattern the same l ines followed for the linear problem

in Section IV.1.

Following Ladyzhenskaya (1959b), we g ive a w eak (or variational) formula-

tion of (IX.0.1). Let ϕ ∈ D (Ω). Assuming v ∈ C

(Ω), p ∈ C

(Ω), f ∈ C(Ω),

if we multiply (IX.0.1)

by ϕ and integrate by parts over Ω we obtain

Ω

∇v : ∇ϕ +

Ω

v ·∇v · ϕ = −

Ω

f · ϕ. (IX.1 .1)

Thus every classical solution to (IX. 0.1) satisﬁes (IX.1.1). Conversely, if v ∈

(Ω) satisﬁes (IX.1. 1) for some f ∈ C(Ω) a nd for all ϕ ∈ D(Ω), then

Ω

(ν∆v − v · ∇v − f ) · ϕ = 0

and we use Lemma III.1.1 to show that v obeys (IX.0.1)

for suitable

p ∈ C

(Ω). However, if v merely satisﬁes (IX.1.1) for all ϕ ∈ D(Ω) and

is not suﬃciently diﬀerentiable, we cannot go from (IX.1.1) to (IX.0.1)

, and

therefore (IX.1.1) is to be interpreted as a generalized version of (IX.0.1)

As in the linear case, it is convenient to investigate the more general

situation, where the right-hand side of (IX.1.1) is deﬁned by a f unctional

f ∈ D

−1,2

(Ω).

If n = 5, a partial regularity result for generalized solutions can be found in

Struwe (1988). Extension of this result to arbitrary dimension is due to Tian &

Xin (1999). It should be added that, if the size of the data is suitably restricted,

any generalized solution satisfying the so-called energy inequality corresponding

to smooth data is smooth as well, for any n ≥ 5; see Remark IX.5.5.

IX.1 Generalized Solutions. Preliminary Considerations 587

With a view to all this and in analogy with Deﬁniti on IV.1.1, we give the

following deﬁnition of generalized solution to (IX.0.1), (IX.0.2).

Deﬁnition IX.1.1. Let Ω be a bounded domain of R

, n ≥ 2. A ﬁeld v :

Ω → R

is called a weak (or generalized) solution to the Navier–Stokes problem

(IX.0.1), (IX.0.2) if and only if

(i) v ∈ D

1,2

(Ω);

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

(iii) v satisﬁes boundary condition (IX.0.2) (in the trace sense) or, if v

∗

≡ 0,

then v ∈ D

1,2

(Ω);

(iv) v obeys the identity

ν(∇v, ∇ϕ) + (v · ∇v, ϕ) = −hf, ϕi (IX.1.2)

for a ll ϕ ∈ D(Ω).

Remark IX.1.1 Si nce every function in D

1,2

(Ω) is also in W

1,2

loc

(Ω) (see

Lemma I I.6.1), from the inequality

|(v · ∇v, ϕ)| ≤ sup

Ω

|ϕ|kvk

1,2,Ω

, Ω

≡ supp (ϕ),

we see that identity (IX.1.2) is meaningful, whatever the regularity of Ω may

be. 

Remark IX.1.2 R emark IV.1.1 and Remark IV.1.2, with q = 2, equally

apply to generali zed solutions of the Navier–Stokes problem (IX.0.1), (IX.0.2).



Remark IX.1.3 In analogy with the Stokes problem, one may give the deﬁni-

tion of q-weak (or q-generalized) solution, by replacing in (i) and (iii) D

1,2

(Ω)

with D

1,q

(Ω), 1 < q < ∞. Of course, q should be chosen i n such a way

that the nonlinear term in (IX.1.2) is meaningful. This happens whenever

q ≥ 2n/(n + 2). In fact, by an integration by parts, we have

|(v · ∇v, ϕ)| = |(v · ∇ϕ, v)| ≤ sup

Ω

|∇ϕ|kvk

2,Ω

with Ω

as in Remark IX.1.1. From Lemma II. 6.1, v ∈ W

1,q

(Ω

) and so,

by Theorem II.3.4, it follows that v ∈ L

(Ω) for all s ≥ 1, if q ≥ n, and

for all s ∈ (1, nq/(n − q)], if q < n. Therefore, if q ≥ n, then v ∈ L

(Ω

while if q < n this latter circumstance happens if 2 ≤ nq/(n − q), that is,

q ≥ 2n/(n + 2). 

Existence a nd uniqueness of generalized solutions in a bounded doma in

will be the object of the next sections. In the remaining part of this section

we wish to point out some notable facts relating to them, which we shall

collect in the following lemmas.

588 IX Steady Navier–Stokes Flow in Bounded Domains

Lemma IX.1.1 Let Ω be an arbitrary bounded domain of R

, n = 2, 3.

Then the trilinear form

a(u, v, w) ≡ (u · ∇v, w) (IX.1.3)

is continuous in the space

1,2

(Ω) × W

1,2

(Ω) × W

1,2

(Ω)

and we have

|a(u, v, w)| ≤ k|u|

1,2

|v|

1,2

|w|

1,2

(IX.1.4)

where

k =











√

2|Ω|

1/6

if n = 3

|Ω|

1/2

if n = 2.

(IX.1.5)

Furthermore, if Ω is bo unded and locally Lipschitz, then a is continuous in

the space

1,2

(Ω) × W

1,2

(Ω) × W

1,2

(Ω).

Thus, i n particular, every weak solution corresponding to the data

f ∈ D

−1,2

(Ω), v

∗

≡ 0,

satisﬁes (IX.1.2) for all ϕ ∈ H

(Ω). The same property holds when v

∗

6≡ 0, if

Ω is b ounded and locally Lipschitz.

Proof. The proof of the second part of the lemma concerning weak solutions

is a consequence of the ﬁrst, si nce (∇v, ∇ϕ) and hf , ϕi are bounded li near

functionals in ϕ ∈ H

(Ω) and, under the stated assumptions, v ∈ W

1,2

(Ω)

so that also (v · ∇v, ϕ) is continuous in ϕ ∈ H

(Ω). In order to show the

continuity of a(u, v, w), assume ﬁrst that n = 3 and u ∈ W

1,2

(Ω). From the

H¨older inequality

|a(u, v, w)| ≤ kuk

2n/(n−2)

|v|

1,2

kwk

(IX.1.6)

so that the Sobolev inequality (II.3.7) implies

|a(u, v, w)| ≤ (2/

√

3)|u|

1,2

|v|

1,2

kwk

. (IX.1.7)

However, since n = 3, Lemma II.3.2 and (II.5.5) furnish

kwk

≤ k

|w|

1,2

(IX.1.8)

with

√

2|Ω|

1/6

√

Placing (IX.1.8) into (IX. 1.7) gives (IX.1.4). Assuming u ∈ W

1,2

(Ω) with Ω

locally Lipschitz, from the embedding Theorem II.3.4 we obtain

IX.1 Generalized Solutions. Preliminary Considerations 589

kuk

2n/(n−2)

≤ ckuk

1,2

(IX.1.9)

for som e c = c(Ω, n); thus inequality (IX.1.7) becomes

|a(u, v, w)| ≤ ckuk

1,2

|v|

1,2

kwk

and the continuity of a follows from this last relation and from (IX.1. 8). Con-

sider, next, n = 2. By the H¨older inequality we have

|a(u, v, w)| ≤ kuk

|v|

1,2

kwk

, r

−1

+ s

−1

= 2

−1

(IX.1.10)

and therefore, choosing, for instance, r = s = 4, since by Lemma II.3.1 and

(II.5.6) we have

kfk

≤ (1/

√

2)|Ω|

1/4

|f|

1,2

, for all f ∈ W

1,2

(Ω),

(IX.1.4) becomes a consequence of this last relation and of (IX.1.10). Finally,

if u ∈ W

1,2

(Ω), inequal ity (IX.1.4) with a suitable constant k = k(Ω, n) is

secured from (IX.1.9) and from the following one

kfk

≤ ckfk

1,2

which is proved in the embedding Theorem II.3.4, for some c = c(Ω, n). The

proof of the lemma is thus complete. ut

Remark IX.1.4 If n = 4, Lemma IX.1.1 continues to hold with k = (3/4)

since, in such a case, inequality (IX.1.8) remains valid with k

= 3/4. (Notice

that 2n/(n − 2) = n, for n = 4). If n ≥ 5 (IX.1.8) no longer holds and

a(u, v, w) is continuous in the space

1,2

(Ω) × W

1,2

(Ω) × [W

1,2

(Ω) ∩ L

(Ω)]

[respectively in the space

1,2

(Ω) × W

1,2

(Ω) × [W

1,2

(Ω) ∩ L

(Ω)]

if Ω is (bounded and) locally Lipschitz]. Consequently, in such a case, in the

second part of the lemma, the statement

ϕ ∈ H

(Ω)

should be replaced by

ϕ ∈

(Ω),

where

(Ω) is the completion of D(Ω) in the norm

kϕk

1,2

≡ kϕk

1,2

+ kϕk

Other continuity properties of the trilinear form a w ill be given in Exercise

IX.2. 1 and Lemma X.2.1. 

590 IX Steady Navier–Stokes Flow in Bounded Domains

As in the linear case (see Lemma IV.1.1), the next result shows, in partic-

ular, that to every generalized solution v, it is po ssible to associate a suitable

pressure ﬁeld p.

Lemma IX.1.2 Let Ω be an arbitrary domain in R

, n = 2, 3, and let

f ∈ W

−1,2

(Ω

) for any bounded domain Ω

, with Ω

⊂ Ω.

Then a vector ﬁeld v ∈ W

1,2

loc

(Ω) satisﬁes (IX.1.2) for all ϕ ∈ D(Ω) if and

only if there is p ∈ L

loc

(Ω) satisfying the identity

ν(∇v, ∇ψ) + (v ·∇v, ψ) = (p, ∇ · ψ) − hf, ψi (IX.1.11)

for a ll ψ ∈ C

∞

(Ω). If, moreover, Ω is bounded and local ly Lipschitz and

f ∈ D

−1,2

(Ω), v ∈ D

1,2

(Ω),

then

p ∈ L

(Ω) with

Ω

p = 0,

and (IX.1.11) holds f or all ψ ∈ W

1,2

(Ω).

Proof. Clearly, (IX.1.11) implies (IX.1.2). Thus assume (IX.1.2 ) and Ω locally

Lipschitzian. By Lemma IX.1.1 the functional

F(ψ) ≡ ν(∇v, ∇ψ) + (v · ∇v, ψ) + hf, ψi (IX.1.12)

is (linear and) bounded in ψ ∈ W

1,2

(Ω) and vanishes when ψ ∈ D

1,2

(Ω) (=

(Ω)). Therefore, by Corollary III.5.1 there exists p ∈ L

(Ω) such that

F(ψ) = (p, ∇· ψ), (IX.1.13)

for all ψ ∈ D

1,2

(Ω) (= W

1,2

(Ω)), thus satisfying (IX.1.11 ). If Ω is an arbi-

trary domain, we use Corollary III.5.2 to deduce the existence of p ∈ L

loc

(Ω)

satisfying (IX.1.13) for all ψ ∈ C

∞

(Ω). The proof is thus complete. ut

Remark IX.1.5 If n = 4, Lemma IX.1.2 continues to hold since (v ·∇v, ψ)

is still a bo unded functional in ψ ∈ W

1,2

(Ω), see Remark IX.1.4. If n ≥ 5

this property no longer holds and Lemma IX.1. 2 fa ils, unless v ∈ L

(Ω).

Nevertheless, if

f ∈ D

−1,n/(n−2)

(Ω),

we can again deﬁne a pressure ﬁeld

p ∈ L

n/(n−2)

(Ω)

satisfying (IX.1.11 ) if Ω is bounded and locally Lipschitz. If Ω has no regu-

larity or if

IX.2 On the Uniqueness of Generalized Solutions 591

f ∈ W

−1,n /(n−2)

(Ω

), for a ll domains Ω

with Ω

⊂ Ω,

we then only have

p ∈ L

n/(n−2)

loc

(Ω).

Actually, if Ω is locally L ipschitz, n ≥ 5, and v is a generalized solution, then

v ∈ W

1,n/(n−2)

(Ω).

Furthermore, by inequalities (IX. 1.6), (IX.1.9), and the Sobolev inequality

(II.3.7) we deduce

|(v · ∇v, ψ)| ≤ ckvk

1,2

|ψ|

1,n/2

As a consequence, the functional (IX.1.12) is bounded for ψ ∈ D

1,n/2

(Ω)

(=W

1,n/2

(Ω)) and vanishes when ψ ∈ D

1,n/2

(Ω) (=H

n/2

(Ω)). By Corollary

III.5.1 we then conclude the validity of (IX.1.13) for some p ∈ L

n/(n−2)

(Ω)

and all ψ ∈ D

1,n

(Ω) (=W

1,n

(Ω)). If Ω has no regularity, by these arguments

and Corollary III.5.2 we derive p ∈ L

n/(n−2)

loc

(Ω). 

IX.2 On the Uniquene ss of General ized Solutions

We shall begin to establish a general uniqueness result for weak solutions in

a bounded domain. To this end, we need some further informa tion about the

trilinear form a(u, v,w) deﬁned in Lemma IX.1. 1.

Lemma IX.2.1 Let Ω be a bounded locally Lipschitz domain in R

, n = 2, 3 ,

and let v ∈ W

1,2

(Ω) with ∇ ·v = 0. Then

a(v, u, u) = 0 for all u ∈ W

1,2

(Ω). (IX.2.1)

Thus, i n particular,

a(v, u, w) = −a(v, w, u) for all u, w ∈ W

1,2

(Ω). (IX.2. 2)

If v ∈ H

(Ω), the same conclusions hold with no regularity on Ω.

Proof. Property (IX.2.2) is a direct consequence of (IX. 2.1) when we replace

the f unction u in it with u+w and use the multilinear properties of a. To prove

(IX.2.1), in view of Lemma IX.1.1, it is enough to prove it for u ∈ C

∞

(Ω).

However, for such functions we have

a(v, u, u) =

Ω

v ·∇u · u =

Ω

v · ∇u

= 0

since v is weakly divergence-free. The lemma is thus proved. ut

592 IX Steady Navier–Stokes Flow in Bounded Domains

Remark IX.2.1 If n = 4 the preceding lemma remains unchanged. If n ≥ 5

it holds, by replacing W

1,2

(Ω) with

1,2

(Ω), where this latter space is deﬁned

as the completion of C

∞

(Ω) in the norm k·k

1,2

introduced in Remark IX.1.4.



Exercise IX.2.1 Let Ω be a domain of R

, n ≥ 2. Show that the trilinear form

(IX.1.3) is continuous in L

(Ω) ×W

1,r

(Ω) ×L

/(q−r

)

(Ω), q ∈ ( 1, ∞), r ∈ (q/(q −

1), ∞). Thus, assuming v ∈

(Ω)

with ∇ · v = 0, show that the conclusions of

Lemma IX.2. 1 continue to hold under any of the following assumptions

(i) s = n, if n ≥ 3, and u, w ∈ W

1,2

(Ω) ;

(ii) s > n, if n = 2, and u, w ∈ W

1,2

(Ω) ;

(iii ) s < n, if n = 2, and u, w ∈ W

1,σ

(Ω) , σ > s

Let us next observe that, in view of Lemma IX.1.1 it follows, in particular,

that

|(u · ∇v, u)| ≤ k|v|

1,2

|u|

1,2

(IX.2.3)

for all v ∈ W

1,2

(Ω), u ∈ H

(Ω), and with k deﬁned in (IX.1.5). We are now

in a position to prove the following uniqueness theorem.

Theorem IX.2.1 Let Ω be a bounded locally Lipschitz domain in R

, n =

2, 3, and let v be a generalized solution to (IX.0.1), (IX.0 .2) corresponding to

f ∈ D

−1,2

(Ω) and v

∗

∈ W

1/2,2

(∂Ω). If we denote by w another generalized

solution corresponding to the same data, v ≡ w, provided that

|v|

1,2

< ν/k (IX.2.4)

where k is deﬁned in (IX.1.5).

Proof. Let u ≡ w − v. From (IX.1.2) and Lemma IX. 1.1 we deduce that u

satisﬁes the foll owing identity

ν(∇u, ∇ϕ) + (u ·∇u, ϕ) + (u · ∇v, ϕ) + (v ·∇u, ϕ) = 0 (IX.2.5)

for all ϕ ∈ H

(Ω). Clearly, u has zero trace a t the boundary and, conse-

quently, from Remark IX.1.2 and Theorem II.4.2, we have u ∈ W

1,2

(Ω).

Since u is weakly divergence-free and Ω locally Lipschitz, from the results of

Section III.4.1 it fol lows that u ∈ H

(Ω). We may thus substitute ϕ with u

into (IX.2.5) and employ Lemma IX.2.1 to obtain

ν| u|

1,2

+ (u · ∇v, u) = 0. (IX.2.6)

Using estimate (IX.2.3) in (IX.2.6) yields

(ν − k|v|

1,2

)|u|

1,2

≤ 0

which, in turn, by (IX.2.4) implies uniqueness. ut

We recall that this space is deﬁned in (III.2.4).

IX.2 On the Uniqueness of Generalized Solutions 593

Remark IX.2.2 If Ω has no regularity the procedure adopted earlier a priori

does not furnish uniqueness even i n the case of zero boundary data. This fact is

related to the deﬁniti on of g eneralized solution g iven by me in Section IX.1. In

fact, if v

∗

= 0 , we require that v belongs to the space

1,2

(Ω) which, a priori

is larger than D

1,2

(Ω), if Ω has no regularity (cf. Section I II.5). Therefore, the

ﬁeld u introduced in the proof of Theorem IX.2.1 cannot be put in place of ϕ

in the identity (IX.1.7) and we can no longer deduce uniqueness. Nevertheless,

uniqueness can be still recovered in the (a priori smaller) class of generalized

solutions belonging to the space D

1,2

(Ω). 

Remark IX.2.3 If n = 4 Theorem IX.2.1 continues to hold. Its proof, how-

ever, does not apply if n ≥ 5 since, in this case, identity (IX.2.5 ) is valid

for ϕ ∈

(Ω) (cf. Rema rk IX.1.4) and we cannot take ϕ = u, f or u does

not belong a priori to

(Ω). Nevertheless, as we shall see in Remark IX.5.5,

in dimension n ≥ 5 one can construct more regular solutions (at the cost,

however, of imposing some restriction on the size of the data) and to discuss

their uniqueness. For the existence of regula r solutions for n ≥ 5 witho ut re-

strictions on the size of the data, we refer the reader to Frehse a nd R˚uˇziˇcka

(1994a , 1994b, 1995, 1996), and Struwe (1995). 

Because of the nonlinearity of the Navier–Stokes equations, we would ex-

pect that if a sol ution does not satisfy condition (IX.2.4), namely, if the co-

eﬃcient of kinematic viscosity ν is no t “suﬃciently large,” then uniqueness

may be lost. Employing the ideas of Youdovich (1967), in the remaining part

of this section we shall prove that this is indeed the case. We begin with the

following result.

Lemma IX.2.2 Let Ω be a domain of R

, n ≥ 2, and let a be a smooth

solenoidal vector in Ω. Moreover, assume that there is µ > 0 for which the

following problem admits a nontrivial solution ϕ, τ

∆ϕ = a · ∇ϕ + ϕ · ∇a + ∇τ

∇ · ϕ = 0











in Ω

ϕ = 0 at ∂Ω.

(IX.2.7)

Then, there are vector ﬁelds F and v

∗

such that the Navier–Stokes problem

ν∆v = v · ∇v + ∇p + F

∇· v = 0

)

in Ω

v = v

∗

at ∂Ω

(IX.2.8)

with ν = 1/2µ admits two distinct solutions corresponding to the same data

F and v

∗

594 IX Steady Navier–Stokes Flow in Bounded Domains

Proof. Set

(a + ϕ), v

(a −ϕ).

Clearly, v

and v

satisfy (2.8

2,3

) with

∗

= a

∗

/2, (IX.2.9)

where a

∗

is the value of a at the boundary ∂Ω. Moreover, by means of a

direct calcula tion, we ﬁnd the following identities

ν∆v

− v

·∇v

∆a −

(a · ∇a + ϕ · ∇ϕ)

[2ν∆ϕ −(a · ∇ϕ + ϕ · ∇a)]

ν∆v

− v

·∇v

∆a −

(a · ∇a + ϕ · ∇ϕ)

−

[2ν∆ϕ −(a · ∇ϕ + ϕ · ∇a)]

and so, recalling that ϕ satisﬁes (IX.2.7), if ν = 1/2µ we deduce that the

pairs v

, p

and v

, p

with

= τ /4, p

= −τ/4

are two distinct solutions to (2.8) correspondi ng to

F =

∆a −

(a · ∇a + ϕ · ∇ϕ)

and to v

∗

given in (IX.2.9). The lemma is therefore proved. ut

Remark IX.2.4 If Ω is a bounded locally Lipschitz domain of R

, n ≤ 4, by

the type of reasoning employed in the proof of Theorem IX.2.1, from (IX.2.7)

we show that, for a given a ∈ W

1,2

(Ω), if ν (=1/2µ) is such that

|a|

1,2

< 2ν/k,

then problem (IX.2.7) admits only the zero solution. Therefore, to obtain

nonuniqueness (if any) ν ha s to be suﬃciently small. 

We also have the following lemma.

Lemma IX.2.3 Let Ω ⊂ R

be a bounded smooth body of revolution around

an axis r. We suppose that Ω does not include points of r so that, fo r instance,

Ω can b e a torus of arbitrary bounded smooth section. In a system of cylin-

drical coordinates (r, θ, z) with the z-axis coinciding with r, we consider the

vector ﬁeld a =(a

, a

) with

IX.2 On the Uniqueness of Generalized Solutions 595

= a

= 0, a

= r

−3

. (IX.2.10)

Then there is µ > 0 such that problem (IX.2.7) admits at least one nontrivial

smooth sol ution ϕ, τ.

Proof. We look for a solution to problem (IX.2.7) in the class < o f functions

having rotational symmetry, that is, they are i ndependent of the angle θ.

Taking into account that, in cylindrical coordinates, for ϕ ∈ <,

ϕ · ∇a + a · ∇ϕ =



−2a





, 0



equations (IX.2.7) with the choice (IX.2.10) and in the class < become

(∆

−

) = −g(r)ϕ

∂τ

∂r

(∆

−

) = −g(r)ϕ

∆

∂p

∂z

∂

∂r

(rϕ

) +

∂ϕ

∂z

= 0











in Ω

ϕ = 0 on ∂Ω

(IX.2.11)

where

∆

∂

∂r

∂

∂r

) +

∂

∂z

and

g(r) = 2r

−4

Set

I(ϕ) = −

Ω

gϕ

D(ϕ) =

Ω



(

∂ϕ

∂r

)

+ (

∂ϕ

∂r

)

+ (

∂ϕ

∂r

)

+ (

∂ϕ

∂z

)

+ (

∂ϕ

∂z

)

+ (

∂ϕ

∂z

)



and consider the maximum problem

= max

(Ω),ϕ6=0



I(ϕ)

D(ϕ)



(IX.2.12)