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

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596 IX Steady Navier–Stokes Flow in Bounded Domains

with H

(Ω) denoting the completion o f the subspace of D(Ω) of rotati onally

symmetric ﬁelds in the norm [D(ϕ)]

1/2

. In view of inequality (II.5 .5), we at

once conclude that the functional I(ϕ)/D(ϕ) is bounded f rom above. Further-

more, taki ng among possible competitors ϕ those vectors such that ϕ

= −ϕ

(which is an allowed choice) we ﬁnd

sup

(Ω),ϕ6=0



I(ϕ)

D(ϕ)



> 0. (IX.2.13)

Also, from Theorem II.3.4, it easily follows that H

(Ω) is compactly embedded

into the subspace of L

(Ω) constituted by vectors having rotational symmetry.

Therefore, classical results on the maximum of quadratic functionals (see, e.g.,

Galdi & Straughan 1985, Lemma 3), ensure that the maximum (IX.2.12) exists

and that the maximizing function satisﬁes (IX.2.11). Moreover, by (IX.2.13 ),

µ > 0. Finally, by the regularity theory for the Stokes problem developed in

Section IV.4, Section IV.5, we easily show tha t the solution ϕ ∈ H

(Ω) to

(IX.2.11) and the corresponding “pressure ﬁeld” τ are smooth for Ω smooth.

The lemma is thus proved. ut

From Lemma IX.2.2 and Lemma IX.2. 3 we deduce the following nonunique-

ness result.

Theorem IX.2.2 Let Ω be as in Lemma IX.2.3. Then there are smooth ﬁelds

F and v

∗

and a value of ν > 0 such that the steady Navier–Stokes problem

(2.8) corresponding to these data admits at least two distinct solutions.

IX.3 Existence and Uniqueness with Hom ogeneous

Boundary Data

It was ﬁrst noticed by J. Leray in his celebrated memoir of 1933 o n the ex-

istence of steady solutions to the Navier–Stokes equations that every smooth

solution to (IX.0.1), (IX.0.2) with v

∗

≡ 0 admits the following a prio ri esti-

mate (see Leray 1933 , Lemme A, p. 23)

Ω

∇v : ∇v ≤ M (IX.3.1 )

where M is independent of v. Actually, multiplying formally (IX.0. 1)

by v

and using the incom pressibility condition (IX.0.1)

leads to

ν∇v : ∇v = ∇ · (

∇v

−pv −

v) − f · v

which, after integration over Ω, in view of the assumed homogeneous boundary

conditions implies

Ω

∇v : ∇v = −

Ω

f · v. (IX.3.2)

IX.3 Existence and Uniqueness with Homogeneous Boundary Data 597

Thus, (IX.3.1) follows with M ≡ |f|

−1,2

/ν

. We wish to point out the remark-

able fact that (IX.3.1) and (IX.3.2) are exactly the same relations we obtain

from the linearized Stokes equations. Then it is not surprising that Fujita

(1961) and independently Vorovich & Youdovich (1961) (cf. also Vorovich &

Youdovich 1959) employed a well-known device for constructing solutions to

linear equations, i.e., the Galerkin method (cf. Galerkin 1915) along with es-

timate (IX.3.1) to show the existence of g eneralized soluti ons to the nonlinear

Navier–Stokes problem. Here we shall follow the ideas of these authors.

We begin to consider the case of homogeneous boundary conditions (v

∗

≡

0), while in the next section we will handle the more general no nhomogeneous

case. However, we need some preparatory results. The ﬁrst concerns the zeros

of continuous mapping s of R

into itself which generalizes to the case m > 1,

the well-known fact that a real continuous function that attains values of

opposite signs at the endpoints of an interval must then vanish at some interior

point. Speciﬁcally, we have (see, e.g., Lions 1969, Lemme 4. 3, p. 53):

Lemma IX.3.1 Let P be a continuous function of R

, m ≥ 1, into itself

such that for some ρ > 0

P (ξ) · ξ ≥ 0 for all ξ ∈ R

with |ξ| = ρ.

Then there exi sts ξ

∈ R

with |ξ

| ≤ ρ such that P (ξ

) = 0.

Proof. Assume, per absurdum, P (ξ) 6= 0 for a ll ξ belonging to the closed ball

of R

of radius ρ and centered at the origin. The map

Π : ξ → −P (ξ)

|P (ξ)|

is then continuous and goes from B

into B

. By the Brouwer theorem (see,

e.g., Kantorovich & Akilov 1964, Lemma 5, p. 639) we then obtain that the

map Π has a ﬁxed point ξ, i.e.,

ξ = −P (ξ)

|P (ξ)|

, |ξ| = ρ.

Dotting both sides of this relation by P (ξ) yields

P (ξ) · ξ = −ρ|P (ξ)| < 0

contradicting the assumption. ut

The lemma just shown allows us to prove a general existence result con-

cerning certain algebraic systems.

In fact, a proof of Lemma IX.3.1 can already be found in Miranda (1940), where it

is also shown that the lemma is equivalent to Brouwer’s ﬁxed point theorem. (For

this latter, see, e.g., Kantorovich & Akilov 1964, Lemma 5, p. 639).) I would like

to thank Professor P.N. Srikanth for bringing Miranda’s paper to my attention.

598 IX Steady Navier–Stokes Flow in Bounded Domains

Lemma IX.3.2 Let Ω be an arbitrary domain in R

, n ≥ 1, and let {ψ

k = 1, . . ., m, be a set of functions from C

∞

(Ω). such that

(ψ

, ψ

) = δ

, k, k

= 1, . . . , m.

Consider the algebraic system in the unknown ξ = (ξ

, . . ., ξ

) ∈ R

ν(∇w, ∇ψ

) = F

(ξ), k = 1, . . . , m, (IX.3.3)

where ν > 0,

w =

k=1

, (IX.3.4)

and F is a continuous function of R

into itself. Then if there are positive

constants c and α < ν such that

F (ξ) · ξ ≤ c|w|

1,2

+ α|w|

1,2

problem (IX.3.3)–(IX.3.4) admits at least one solution.

Proof. Consider the map P , which to every vector ξ ∈ R

, associates the

vector P (ξ) ∈ R

whose kth com ponent is

(P (ξ))

≡ ν(∇w, ∇ψ

) −F

(ξ).

Evidently, P is continuous. Let us prove that there exists ρ > 0 such that

P (ξ) · ξ ≥ 0 for all ξ such that |ξ| = ρ. Actually, by assumption,

P (ξ) ·ξ = ν|w|

1,2

− F (ξ) · ξ ≥ |w|

1,2

[(ν − α)|w|

1,2

− c] .

Denote by K ⊂ Ω a compact set containing the supports of all functions ψ

k = 1, . . ., m. By inequality (II.5 .5) and the no rmalization condition on the

functions ψ

we then have, for a suitable c

= c

(n),

P (ξ) · ξ ≥ |w|

1,2



(ν − α)c

|K|

−1/n

kwk

− c



= |w|

1,2



(ν −α)c

|K|

−1/n

|ξ| −c



Therefore,

P (ξ) · ξ ≥ 0 for | ξ| =

c |K|

1/n

(ν −α)c

and the lemma follows from Lemma IX.3.1. ut

We are now in a position to prove the following existence result.

The stated assumption on F can be widely generalized. However, such a gener-

alization is not needed in the cases treated in this book.

IX.3 Existence and Uniqueness with Homogeneous Boundary Data 599

Theorem IX.3.1 Let Ω be a bounded domain in R

, n = 2 , 3. Given f ∈

−1,2

(Ω) there exists at least o ne generalized solution v to problem (IX.0.1),

(IX.0.2) with v

∗

= 0. This solution satisﬁes the estimate

ν|v|

1,2

≤ |f |

−1,2

. (IX.3.5)

Furthermore, if Ω is locally Lipschitz, we have

kpk

≤ c (|f|

−1,2

+ |v|

1,2

+ ν|v|

1,2

) (IX.3.6)

where p is the pressure ﬁeld associated to v by Lemma IX.1.2 and c = c(n, Ω).

Proof. Denote by {ψ

} a denumerable set of f unctions of D(Ω) whose linear

hull is dense in H

(Ω). We normalize it as

(ψ

, ψ

) = δ

For each m = 1, 2, . . ., we then look for an “approximating sol ution” v

(IX.1.2) as follows:

k=1

ν(∇v

, ∇ψ

) + (v

· ∇v

, ψ

) + hf, ψ

i = 0, k = 1, 2, . . ., m.

(IX.3.7)

Relation (IX.3.7) represents a system of nonlinear equations in the m un-

knowns ξ

, k = 1, . . ., m. Since v

∈ D(Ω), by Lemma IX.2.1 it follows

that

k=1

· ∇v

, ξ

) = (v

· ∇v

, v

) = 0. (IX.3.8)

Moreover,



k=1

hf, ξ



≤ |f |

−1,2

1,2

(IX.3.9)

and so, by Lemma IX.3.2, we deduce that problem (IX.3.7) admits a solution

for all m ∈ N. Multiplying (IX.3.7)

by ξ

, summing over k from 1 to m,

and using (IX.3.8) and (IX.3 .9) we deduce also that

ν|v

1,2

= −hf, v

i, (IX.3.10)

and since

−hf , v

i ≤ |f |

−1,2

1,2

we have

1,2

≤ |f |

−1,2

/ν. (IX.3.11)

600 IX Steady Navier–Stokes Flow in Bounded Domains

Therefore, the sequence {v

} is uniformly b ounded in D

1,2

(Ω) and, by The-

orem II.1.3, there is a subsequence, denoted agai n by {v

}, and a ﬁeld

v ∈ D

1,2

(Ω) such that

→ v weakly in D

1,2

(Ω). (IX.3.12)

Moreover, by inequali ty (II.5 .5), {v

} is also uniformly bo unded in L

(Ω)

and v ∈ L

(Ω). Thus, by compactness Theorem II.5.2 we may take this

subsequence such that

→ v strongly in L

(Ω). (IX.3.13)

With (IX.3.12) and (IX.3.13) in hand we pass next to the limit as m → ∞ i n

(IX.3.7)

. By (IX.3.12) it follows that

(∇v

, ∇ψ

) → (∇v, ∇ψ

). (IX.3.14)

In addition,

≡ |(v

· ∇v

, ψ

) − (v · ∇v, ψ

≤ |((v

− v) · ∇v

, ψ

)| + |(v · ∇(v

− v), ψ

≡ I

(1)

+ I

(2)

(IX.3.15)

Now, by (IX.3.11) we have

(1)

≤ sup

Ω

|ψ

|k v

− vk

|v|

1,2

≤ sup

Ω

|ψ

|k v

− vk

|f |

−1,2

/ν

so that by (IX.3.13)

lim

m→∞

(1)

= 0. (IX.3.16)

Also, by Lemma IX.2.1,

(v · ∇(v

−v), ψ

) = −(v · ∇ψ

, (v

−v)),

since v ∈ H

(Ω). Therefore

(2)

≤ sup

Ω

|∇ψ

|kvk

−vk

and, again by (IX.3.13)

lim

m→∞

(2)

= 0. (IX.3.17)

From (IX.3.15)–(IX.3.17) we then conclude

lim

m→∞

= 0. (IX.3.18)

Replacing (IX.3.14) and (IX.3.18) into (IX.3.7)

it foll ows that the ﬁeld v

(belo ngs to H

(Ω) and) satisﬁes the equation

IX.3 Existence and Uniqueness with Homogeneous Boundary Data 601

ν(∇v, ∇ψ

) + (v · ∇v, ψ

) + hf, ψ

i = 0, (IX.3.19)

for all k = 1, 2, . . . However, any ϕ ∈ H

(Ω) can be approximated by linear

combinations of functions ψ

through suitable coeﬃcients. Since every term

in (IX.3.19) deﬁnes a bounded linear functional in ψ

∈ H

(Ω) (cf. Lemma

IX.1. 1) we may conclude from (IX.3.19) that the ﬁeld v satisﬁes (IX.1.2)

for all ϕ ∈ H

(Ω). Existence is then established. As fa r as the validity of

estimates (IX.3.5)–(IX.3.6) is concerned, we notice that from (IX.3.11) and

from Theorem II.2.4 we have

|v|

1,2

≤ lim inf

m→∞

1,2

≤ |f|

−1,2

/ν,

which proves (IX.3.5). Assuming Ω locally Lipschitz, from Lemm a IX.1.2 fol-

lows the existence of a pressure ﬁeld p ∈ L

(Ω) satisfying (IX.1.11) and

Ω

p = 0. (IX.3.2 0)

Consider the problem

∇ · Ψ = p

Ψ ∈ W

1,2

(Ω)

kΨ k

1,2

≤ ckpk

(IX.3.21)

Since p is in L

(Ω) and satisﬁes (IX.3.2 0), problem (IX.3.21) is solvable in

virtue of Theorem III.3 .1. From (IX.1.11), (IX.3.21), and (IX.1.4) we then

deduce

kpk

≤ c

(|f|

−1,2

+ |v|

1,2

+ ν|v|

1,2

)kpk

with c

= c

(n, Ω), which shows (IX.3.6). The theorem is therefore proved.

Remark IX.3.1 The theorem just shown extends with no changes to cover

the case n = 4. If n ≥ 5 the only change we need is in the choice of the

set {ψ

} ⊂ D(Ω) whose linear hull, this time, has to be dense in

(Ω);

see Remark IX.1.4. Moreover, estimate (IX.3.5 ) continues to be valid, while

(IX.3.6) must be suitably changed according to the fact that now the pressure

ﬁeld p ∈ L

n/(n−2)

(Ω) (cf. Remark IX.1.5). Speciﬁcally, if Ω is locally Lipschitz,

by letting

C ≡

|Ω|

Ω

p|p|

(4−n)/(n−2)

and introducing a function Ψ that instead of (IX.3.21) solves the problem:

∇ · Ψ = p |p|

(4−n)/(n−2)

−C ≡ g

Ψ ∈ W

1,n/2

(Ω)

kΨk

1,n/2

≤ c

kgk

n/2

≤ c

kpk

n/(n−2)

602 IX Steady Navier–Stokes Flow in Bounded Domains

one may use the arguments adopted in the proof of Theorem IX.3.1 to show

the validity of the following inequality.

kpk

n/(n−2)

≤ c (|f|

−1,2

+ |v|

1,2

+ ν|v|

1,2



Remark IX.3.2 As the reader may have noticed, the method employed in

the proof of Theorem IX.3.1 –being based on Lemma IX.3.2– is in fact essen-

tially independent of the boundedness of Ω. Actually, as we shall see in the

next chapters, with an a ppropriate choice of the basis {ψ

}, it can be equally

appli ed to the case when Ω is unbounded. 

Uniqueness of the solutions just constructed is easily discussed by m eans of

Theorem IX.2.1 . Actually, from the estimate (IX.3.5) we deduce that condition

(IX.2.4) is certainly satisﬁed if

|f|

−1,2

< ν

/k. (IX.3.22)

We thus have the following theorem.

Theorem IX.3.2 The generalized solution determined in Theorem IX.3.1

is the only generalized solution corresponding to f, provided f satisﬁes

(IX.3.22).

Remark IX.3.3 Theorem IX.3.2 continues to hold if n = 4. For n ≥ 5, we

refer the reader to Remark IX.5.5. 

Exercise IX.3.1 Let Ω be a bounded domain of R

, n ≥ 2. Show that the gen-

eralized solution v constructed by the method of Theorem IX.3.1 (see also Remark

IX.3.1) satisﬁes the energy inequal ity:

ν|v|

1,2

≤ −hf , vi. (IX.3.23)

Hint: Use (IX.3.10).

Furthermore, show that if n = 2, 3, 4, every generalized solution corresponding

to v

∗

= 0 and f ∈ D

−1,2

(Ω) satisﬁes the energy equality, that is, (IX.3.23) with

the equality sign. Finally, show t hat if n ≥ 5, the energy equality holds for those

generalized solutions that belong to L

(Ω).

IX.4 Existence and Uniqueness with Nonhomogeneous

Boundary Data

As in the linear case, to prove existence when v

∗

6≡ 0, we may look for solutions

v of the form

v = u + V ,

IX.4 Existence and Uniqueness with Nonhomogeneous Boundary Data 603

where V is a (suﬃciently smooth) solenoidal extension to Ω of the velocity

∗

at the bo unda ry. From (IX.0.1), (IX.0.2) we then conclude that u solves

the problem

ν∆u = u ·∇u + u · ∇V + V · ∇u + ∇p − ν∆V + V · ∇V + f

∇ · u = 0

)

in Ω

u = 0 at ∂Ω.

(IX.4.1)

However, in order to show existence by the technique used in Theorem IX. 3.1,

we need, as already observed, to ﬁnd a uniform b ound on |u|

1,2

depending

only on the data. Now, formall y multiplying (IX. 4.1)

by u, integrating by

parts over Ω and using (IX.4.1)

, we obtain

Ω

∇u : ∇u = −

Ω

u ·∇V ·u −

Ω

f ·u −ν

Ω

∇V : ∇u −

Ω

V ·∇V ·u.

(IX.4.2)

Using the Schwarz inequali ty and Lemma IX.1.1 one easily shows that the

last three terms of this relation can be increased by



Ω

∇u : ∇u



1/2

with C = C(n, ν, Ω, f, V ). Therefore, from (IX. 4.2) we ﬁnd the estimate

Ω

∇u : ∇u ≤ −

Ω

u · ∇V · u + C



Ω

∇u : ∇u



1/2

and so a way of recovering a uniform bound on |u|

1,2

, is to require that the

extension ﬁeld V satisfy the one-sided inequality

−

Ω

u · ∇V ·u ≤ α|u|

1,2

, (IX.4.3)

for som e α < ν and for all u ∈ H

(Ω).

If we do not want to impose restric-

tions from below on the kinematic viscosity, then Ω and v

∗

should satisfy the

following extension condition (referred to by the abbreviation EC ): for any

α > 0 there is a solenoidal extension V = V (α) of v

∗

satisfying (IX.4.3). As

a consequence, in contrast with the linear case, to prove existence it is not

enough to pick any (suﬃciently smooth) extension of the b oundary data.

If V were not required to be solenoida l, every (suﬃciently regular) Ω and

∗

would satisfy EC. Actually, for a given ε > 0 we could choose

V = ψ

W (IX.4. 4)

In fact, according to t he Galerkin method, it would suﬃce to require the validity

of (IX.4.3) for all u ∈ D(Ω).

604 IX Steady Navier–Stokes Flow in Bounded Domains

with W an extension of v

∗

and ψ

the “cut-oﬀ” function introduced in Lemma

III.6.2. Thus, i ntegrating by parts and using the H¨older inequality to gether

with the embedding Theorem II. 3.4 it follows that



Ω

u · ∇(ψ

W ) · u



Ω

u · ∇u · W



≤ kuk

|u|

1,2

kψ

W k

≤ c|u|

1,2

kψ

W k

and since, by the properties of ψ

kψ

W k

→ 0 as ε → 0,

we recover (IX.4.3) for any α > 0. Nevertheless, following the ideas of Leray

(1933, pp. 4 0-41), completed and clariﬁed by Hopf (1941, §2) (see also Hopf

1957, pp. 12–14), instead of (IX.4.4) one can take

V = ∇ ×(ψ

W ) (IX.4.5)

with a suitabl e choice of the ﬁeld W . Then V is solenoidal and, by arguments

slightly more complicated than those employed before, one can show that

(IX.4.3) can b e satisﬁed by any α > 0. Recalling that the incompressibility of

the ﬂuid requires

∂Ω

∗

· n = 0 (IX.4.6)

we may conclude that, if ∂Ω has only one connected component, the choice

(IX.4.5) ensures that any (suﬃciently smooth) Ω and v

∗

satisfy EC. However,

if ∂Ω has more than one connected component Γ

, say, i.e., for m > 0

∂Ω =

m+1

[

i=1

with the choice (IX.4.5) we have, as a consequence of the Stokes theorem, that

Ω and v

∗

satisfy EC if

≡

∗

·n = 0, i = 1 , 2, . . ., m + 1. (IX.4.7)

Observe that (IX.4.7) is stronger than the compatibility condition (IX.4.6)

and that, in particular, it does not allow for the presence of separated sinks

If n = 2, one takes

V = (∂(ψ

w)/∂x

, −∂(ψ

w)/∂x

) ≡ ∇ × (ψ

w).

IX.4 Existence and Uniqueness with Nonhomogeneous Boundary Data 605

and sources of liqui ds into the region of ﬂow. At this point we may think

of choosing a ﬁeld V in a form other than (IX.4.5) so that Ω and v

∗

may

satisfy EC under the sole condition (IX.4.6). However, this is no t possible, in

general. In fact, it is easy to bring examples of smooth domains Ω for which

EC holds (if and) only if (IX.4.7) is satisﬁed, whatever the choice of v

∗

may

be, cf. Takeshita (1993, Theorem 1). For instance, let us suppo se that Ω is the

annular region

Ω =



x ∈ R

: R

< |x| < R



and set

Φ =

∗

· n = −

∗

· n (IX.4.8)

where



x ∈ R

: |x| = R



, Γ



x ∈ R

: |x| = R



We take Φ < 0 (inﬂow condition). Assuming that Ω and v

∗

satisfy EC means

that for any α > 0 there is an extension V = V (α) = (V

, V

) of v

∗

verifying

(IX.4.3), where, as usual, (r, θ) denotes a system of polar coordina tes in the

plane. Then, because V is solenoidal

∂(rV

)

∂r

∂V

∂θ

= 0

and by (IX.4.8), it follows for all r ∈ (R

, R

)

2π

(r, θ)dθ = Φ. (IX.4.9)

We take, next, u = u(r)e

with u ∈ C

∞

((R

, R

)), and observe that u ∈

D(Ω). With this choice of u a nd in view of (IX.4.9), we ﬁnd

Ω

u ·∇V · u =

Ω



∂V

∂θ



= Φ

dr.

Thus admi tting EC would imply

|Φ|

dr ≤ α|u|

1,2

for a ny α > 0, that is, Φ = 0.

Remark IX.4.1 Thi s example can be extended, in an simple way, to more

general two-dimensional bounded domains, Ω, satisfying the following prop-

erties: (i) ∂Ω is constituted by two connected components, Γ

and Γ

; (ii) Γ

surrounds a circle, C

, and Γ

lies within a circle, C

, with both C

and C

contained in Ω; (iii) The centers of C

, C

are in the interior of the bounded

connected component of R

−Ω; (iv) C

∩ C

= ∅. 