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

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936 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

they exist without restriction on the coeﬃcient of kinematical viscosity ν or

on the ﬂux Φ. This nice circumstance is due to the fact that, in the case under

consideration, for any η > 0 there is a solenoidal vector a = a(x; η) ∈

1,2

(Ω)

carrying the ﬂux Φ such that



Ω

u · ∇a · u



≤ η

Ω

∇u : ∇u, for all u ∈ D(Ω). (XIII.7.1)

The validity of (XIII.7.1), in turn, is made possible by the fact that the “out-

lets” to inﬁnity for the domain (XIII.5.1) have an unbounded cross section.

The existence of such ﬁelds a(x; η) was discovered by Ladyzhenskaya & Solon-

nikov (1977, §2). It should be remarked that the method used by these authors

to construct the ﬁelds a(x; η) applies to doma ins whose outlets to inﬁnity Ω

are more general than those of domain (XIII.5.1); in fact, the only condition

required is that each Ω

contains a semi-inﬁnite cone; cf. also Solonnikov &

Pileckas (1 977, Lemma 4), Solonnikov (1981), and Solonnikov (1983, §2.4)

The construction of the ﬁeld a(x; η) is the object of the next lemma. As

the reader will recognize, this construction resembles the one given in Lemma

IX.4. 2 f or the case of a ﬂow in a bounded domain.

Lemma XIII.7.1 Let Ω be as i n (XIII.5.1) and let η > 0. Then there exists

a solenoidal vector ﬁeld a = a(x; η) ∈ C

∞

(Ω) vanishing in a neighborhood of

∂Ω such that

(i) |a(x)| ≤ M|x|

−2

; |∇a(x)| ≤ M|x|

−3

, for all suﬃciently large |x|;

(ii) a ∈

1,2

(Ω);

(iii)

= 1;

(iv) a veriﬁes (XIII.7.1).

Proof. Let

b =

2π|x

(−x

, x

, 0) .

Clearly,

∇ × b = 0

∇· b = 0

)

in Ω − {0}.

We next introduce a system of cylindrical co ordinates (r, θ, x

) with the origin

at the center of the unit disk C = {|x

| < 1} which, without loss, we have as-

sumed to be strictly contained in S. The three unit vectors w ill be denoted by

, e

, and e

, respectively. By a sim ple computation based on the properties

of b we ﬁnd

∂S

b × n ·e

= −1, (XIII.7.2)

where n is the exterior unit normal to ∂S.

In fact, introducing

Since S is locally Lipschitz, n exists a.e. on ∂S; cf. Lemma II.4.1.

XIII.7 Aperture Domain: Existence and Uniqueness 937

w = (b

, −b

, 0)

we have

∇

· w = 0 in ω = S − C,

where ∇

is the divergence operator restricted to the variables x

. Thus, by

the divergence theorem,

∂S

w · n =

∂C

w · e

Since for any vector s = (s

, s

, 0), w · s = −b × s · e

, it follows that

−

∂S

b × n · e

= −

∂C

b × e

· e

2π

(sin

θ + cos

θ)dθ = 1,

which shows (XIII.7.2). Now, let ρ = ρ(x) denote the regularized distance of

x from ∂Ω. By Lemma III.6.1 we obtain, in particular,

δ(x) ≤ ρ(x) ≤ k

δ(x)

ρ(x)| ≤ k

[δ(x)]

1−|α|

, 0 ≤ |α| ≤ 2

(XIII.7.3)

where δ(x) = dist (x, ∂Ω) and k

, k

are independent of x. Let γ denote the

-axis and set

d = dist (∂S, γ).

Let σ, ψ be smooth nondecreasing functions of the real variable t such that

σ(t) =







if t ≤ d/2

t if t ≥ d

and

ψ(t) =

(

0 if t ≤ 0

1 if t ≥ 1.

Finally, we set

ζ(x) = ψ



ε ln

σ(|x

ρ(x)



, ε ∈ (0, 1). (XIII.7.4)

In view of the property of the function ψ, we at once recognize that

ζ(x) =

(

0 if σ(|x

|) ≤ ρ(x)

1 if σ(|x

|) ≥ ρ(x)e

1/ε

(XIII.7.5)

which implies, i n particular,

supp (∇ζ) ⊂

x ∈ Ω : ρ(x) ≤ σ(|x

|) ≤ ρ(x)e

1/ε

. (XIII.7.6)

938 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

Moreover, setting

C = {x ∈ Ω : |x

| < d/2}

it is im mediately seen that

ζ(x) = 0, for all x ∈ C. (XIII.7.7)

In fact, for x ∈ C we have

δ(x) ≥ d − |x

| ≥ d/2

which, by (XIII.7.3), in turn yields

ρ(x) ≥ d/2, for all x ∈ C. (XIII.7.8)

In addition, if x ∈ C, from the deﬁnition of the function σ we derive

σ(|x

|) = d/2, for all x ∈ C. (XIII. 7.9)

Thus, from (XIII.7.8) and (XI II.7.9) we ﬁnd

ρ(x) ≥ σ(|x

|), for all x ∈ C,

which, by (XIII.7.5), implies (XII I .7.7). Also, setting



x ∈ Ω : δ(x) ≤

−1/ε



we have

ζ(x) = 1, for all x ∈ N

. (XIII.7.10)

In fact, since σ is nondecreasing,

σ(|x

|) ≥ d/2,

and so, by (XIII.7.5), we have ζ = 1 whenever

ρ(x) ≤

−1/ε

which, by (XIII.7.3) i mplies (XIII. 7.10). The next task is to prove an esti-

mate for ∇ζ and D

ζ at large distances. To this end, we observe that, by a

straightforward computation, we obtain

∂ζ

∂x

= εψ

∂

∂x

= ε

+ εψ

∂S

∂x

where the prime means diﬀerentiation with respect to the argument involved

and

XIII.7 Aperture Domain: Existence and Uniqueness 939

= σ

∂|x

∂x

σ(|x

−

ρ(x)

∂ρ(x)

∂x

∂S

∂x



σ(|x

−

(σ

)

(|x



∂|x

∂x

∂|x

∂x

σ(|x

∂

∂x

(x)

∂ρ

∂x

∂ρ

∂x

−

ρ(x)

∂

∂x

Therefore, observing that

|ψ

|, |ψ

|, |σ

| ≤ M,

for some M independent of x, by virtue of (XIII.7.3), (XIII.7.6), and (XIII.7.7)

we conclude that

|∇ζ(x)| ≤ εc

−1

(x)

ζ(x)| ≤ c

−2

(x)

)

for all x ∈ Ω, (XIII.7.11)

with c

and c

independent of x. The ﬁeld a is introduced by the following

relation

a = ∇ × (ζb). (XIII.7.12)

Clearly, a is solenoidal and, in view of (XIII.7.7) it is also in C

∞

(Ω). From

the identity ∇ × (ζb) = ∇ζ × b + ζ∇ × b a nd the properties of b we have

a = ∇ζ × b (XIII.7.13)

from which, and with the help of (XIII.7.10), it follows tha t

a(x) = 0, for all x ∈ N

. (XIII.7.14)

Furthermore, we have

(ζb

) −D

(ζb

))

and so, by the divergence theorem, (XIII.7.2), and (XIII. 7.10), we infer

∂S

ζ(b

− b

) = −

∂S

b × n · e

= 1,

which proves property (iii). From (XIII.7.11) and (XIII.7.13) we ﬁnd

|a(x)| ≤

δ(x)|x

|∇a(x)| ≤ c



δ(x)|x

(x)|x













all x ∈ supp (a) ⊂ supp (∇ζ).

940 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

If we take |x| suﬃciently large, |x| > R (say), from (XIII.7.6) and the deﬁnition

of σ we have

δ(x) ≤ c

| ≤ c

δ(x), x ∈ supp (a).

Therefore, for all x ∈ supp (a) with |x| > R, we ﬁnd

|a(x)| ≤



(x)





2 +

(x)



(x) + |x

≤

(x) + |x

and, likewise,

|∇a(x)| ≤

(δ

(x) + |x

)

3/2

Observing that δ

(x) ≥ x

, property (i) follows from these latter inequalities.

Using (i) and the fact that a ∈ C

∞

(Ω

), for all bounded Ω

with Ω

⊂ Ω, we

deduce, in particular, that

Ω

|∇a(x)|

< ∞.

Therefore,

a ∈ D

1,2

(Ω)

and since a is solenoidal and by (XIII.7.14) it vanishes i n a neighborhood of

∂Ω, we may conclude, with the help of a standard “cut-oﬀ” argument, the

validity of statement (ii) in the lemma. It remains to prove condition (iv). We

begin to observe that, gi ven η > 0, using integration by parts and the Schwarz

inequality, f or all u ∈ D(Ω),



Ω

u · ∇a · u



Ω

u · ∇u ·a



≤

Ω

∇u : ∇u +

2η

Ω

holds. As a consequence, to prove (iv), we w ill show tha t, given an arbitrary

small λ > 0, we may select ε in such a way that

Ω

< λ

Ω

∇u : ∇u. (XIII.7 .15)

To this end for a ﬁxed h > 0, we split R

into the following three regions (an

analogous splitting holds for R

−



x ∈ R

: x

> h, x

∈ S





x ∈ R

: x

< h, x

∈ S



= R

−



∪ R



XIII.7 Aperture Domain: Existence and Uniqueness 941

and denote by

, i = 1, 2, 3, the intersection of R

with the support o f a.

By virtue of (XIII.7.11)

, (XIII.7.13), a nd the deﬁnition of b, we ﬁnd

≤ cε

(x)|x

, i = 1, 2, 3. (XIII.7.1 6)

with c independent of ε. We wish to show the inequali ty

(x)|x

≤ c

Ω

∇u : ∇u, i = 1, 2, 3 (XIII. 7.17)

which, with the help of (XIII.7.16), implies

≤ 3cc

Ω

∇u : ∇u.

Since an a nalog ous reasoning holds for R

−

, we conclude the validity of

(XIII.7.15) and complete the proof of the lemma. It remains to show the

inequality (XIII.7.17). To this end, we notice that, for functions u ∈ D(Ω)

the following estimate holds

Ω

+ 1

≤ c

Ω

∇u : ∇u, (XIII.7.18)

with c

= c

(S); cf. Exercise XIII.7.1. In view of (XIII.7.7), we have

| ≥ d/2, for a ll x ∈

. (XIII.7.19)

Moreover, since

δ(x) ≥ x

≥ h for all x ∈

it follows that

(x)

≤

+ 1

, for all x ∈

(XIII.7.20)

with c

= c

(h). Thus, from (XIII.7.18)–(XIII.7.20) we infer that

(x)|x

≤

Ω

∇u : ∇u. (XIII.7.21)

Let us next consider the region

. For all x ∈

δ(x) = x

| ≥ d,

and so

(x)|x

≤

942 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

However, using the elementary inequality

∞

(t)

≤ 4

∞





, (XIII.7.22)

holdi ng for (suﬃciently smooth) functions vanishing in a neighborho od of zero

and inﬁnity, we at once obtain

≤ 4

Ω

∇u : ∇u

and, as a consequence,

(x)|x

≤

Ω

∇u : ∇u. (XIII.7.23 )

It remains to estimate the integral over the regio n

. To this end, setting

(y) = di st (y, ∂S) y ∈ S,

for a ll x ∈

we ﬁnd that

(x) = x

+ δ

)

| ≥ d/2

which, in turn furnishes

(x)

(x)|x

≤

, x

)

. (XIII.7.24)

Since S is locally Lipschitz and u ∈ W

1,2

(S), we may use Lemma III.6.3 to

obtain

, x

)

≤ c

∇u : ∇udx

(XIII.7.25)

with c = c(S). Integrating (XIII.7.25) over x

∈ [0, h) delivers

, x

)

≤ c

Ω

∇u : ∇u. (XII I.7.26)

Using (XIII.7.26) in conjunction with (XIII.7.24) allows us to conclude

(x)

(x)|x

≤ c

Ω

∇u : ∇u.

Estimate (XIII.7.17) then follows from this inequality, (XIII.7.21), and (XIII.7.23)

and therefore, by what we observed, the proof of the lemma i s complete. ut

For the (simple) demonstration of (XIII.7.22), see the proof of Lemma III. 6.3.

XIII.7 Aperture Domain: Existence and Uniqueness 943

Exercise XIII.7.1 Show inequality (X III.7.18), for all u ∈ D(Ω). Hint: Estab-

lish (XIII.7.18) for Ω a half-space. Then use a “cut-oﬀ” argument together with

inequality (II.5.18).

The result just shown permits us to obtain the following.

Theorem XIII.7.3 Given arbitrary Φ ∈ R, problem (XIII.5.2) admits at

least o ne corresponding generalized solution v. This solution obeys the energy

inequality:

|v|

1,2

≤ −

∗

where p

∗

= p

−p

−

and p

are the constants associated by Lemma XIII.5.1

to the pressure ﬁeld p corresponding to v.

Proof. We look for a solution of the form

v = u + Φa

where a = a(x; η) is the vector constructed i n Lemma XIII.7.1 corresponding

to a certain value of η that will be speciﬁed later, and u ∈ D

1,2

(Ω) obeys

ν(∇u, ∇ϕ) − (u ·∇ϕ, u) = −Φ(u ·∇a, ϕ) + Φ(a · ∇ϕ, u)

−Φ

(a · ∇a, ϕ) − νΦ(∇a, ∇ϕ).

(XIII.7.27)

It is clear that v satisﬁes all requirements of Deﬁnition XIII.5.1. As in anal-

ogous circumstances, a solution to (XIII.7.27) will be determined via the

Galerkin method. Let {ϕ

} ⊂ D(Ω) denote a sequence of functions whose

linear hull is dense in D

1,2

(Ω) and satisfying the properties listed in Lemma

VII.2.1. We set

k=1

where the coeﬃcients are required to sati sfy the foll owing system

ν(∇u

, ∇ϕ

) − (u

·∇ϕ

, u

) = −Φ(u

· ∇a, ϕ

) + Φ(a · ∇ϕ

, u

)

−Φ

(a · ∇a, ϕ

) − νΦ(∇a, ϕ

)

(XIII.7.28)

with k = 1, . . . , m. Existence to the algebraic system (XIII.7.28) for each

m ∈ N can be established exactly as in Theorem IX.3.1, provided we show a

suitable bound for |u

1,2

. To obtain such a bound, we multi ply (XIII.7.28 )

by ξ

and sum over the index k from one to m. Observing that

·∇u

, u

) = (a · ∇u

, u

) = 0, (XIII.7.29)

we ﬁnd

ν| u

1,2

= −Φ(u

·∇a, u

) −Φ

(a ·∇a, u

) −νΦ(∇a, ∇u

). (XIII.7.30)

944 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

Using the Schwarz and Cauchy inequalities, we deduce for all η > 0

(a ·∇a, u

) = −Φ

(a · ∇u

, a) ≤

2η

kak

1,2

−νΦ(∇ a, ∇u

) ≤

2η

|a|

1,2

(XIII.7.31)

Moreover, by Lemma XIII. 7.1,

−Φ(u

· ∇a, u

) ≤ η|Φ||u

1,2

. (XIII.7.32)

Thus, choosing

η <

(1 + |Φ|)

and recalling that a ∈ L

(Ω) ∩ D

1,2

(Ω), we may use (XIII.7.29)–(XIII.7.32)

and Lemma IX.3.2 to show existence to (XIII.7.28) for all m ∈ N. Moreover,

we also ﬁnd that

1,2

≤ c

(Φ, ν). (XIII.7.33)

Employing (XIII.7.33) together with the weak compactness of the spaces

1,2

(Ω) we may select from {u

} a sequence, denoted again by {u

}, and a

vector ﬁeld u ∈ D

1,2

(Ω) such that

→ u in D

1,2

(Ω)

→ u in L

(Ω

(XIII.7.34)

for any bounded dom ain Ω

⊂ Ω. By these convergence properties and taking

advantage of the properties of the functions {ϕ

}, we use a by-now-standard

procedure to show that u solves (XIII.7.28) and that, as a consequence, v is

a weak solution to (XIII.5.2). It remains to show that v satisﬁes the energy

inequality. For this purpose, setting

= u

+ Φa,

from (XIII.7.3 0), after a simple manipulation we ﬁnd

ν| v

1,2

= −Φ(v

· ∇a, v

) + νΦ(∇a, ∇v

) (XIII.7.35)

where use has been made of the property

· ∇a, a) = 0.

With the help o f the convergence conditi ons (XIII.7.34), it is easily shown

that

lim

m→∞

· ∇a, v

) = (v · ∇a, v). (XIII.7.36)

Actually, we have

|(v

· ∇a, v

) − (v · ∇a, v)| ≤ |((v

−v) ·∇a, v

+|(v · ∇a, (v

−v))|

≡ I

(m) + I

(m).

XIII.7 Aperture Domain: Existence and Uniqueness 945

Recalling that for R suﬃciently large

|∇a(x)| ≤ c|x|

−3

, |x| > R, (XIII.7.37)

we ﬁnd

(m) ≤



Ω

|∇a|

4/3

− v|



1/2



Ω

|∇a|

2/3



1/2

≤ c



Ω

|x|



1/2



Ω

|∇a|

4/3

− v|



1/2

Thus, from this inequality, (II.6. 10), and (XIII.7.33) it follows that

(m) ≤ c



Ω

|∇a|

4/3

−v|



1/2

(XIII.7.38)

with c

independent of m. Setting Ω

= Ω ∩ B

, Ω

= Ω − Ω

, from

(XIII.7.37) we have

Ω

|∇a|

4/3

−v|

≤

Ω

|∇a|

4/3

− v|

Ω

|∇a|

4/3

−v|

≤ c

Ω

−v|

4/3

Ω

− v|

|x|

and so, again by (II.6.10) and (XIII.7.33), we ﬁnd

(m) ≤ c

Ω

− v|

with c

and c

independent of m. Taking the lim sup as m → ∞ at both sides

of this latter relation and bearing in mind (XIII.7.34)

, by the arbitrarity of

R we conclude that

lim sup

m→∞

(m) = 0,

that is,

lim

m→∞

(m) = 0.

Likewise, we show

lim

m→∞

(m) = 0

and (XIII.7.36 ) is completely established. Let us now take the lim inf as

m → ∞ of both sides of (XIII.7.35). Empl oying (XIII.7.36 ), (XIII.7.34)

and Theorem II.2.4 we obta in

ν| v|

1,2

≤ −Φ(v · ∇a, v) + νΦ(∇a, ∇v). (XIII.7.39)