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

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

Remark XI II.4.2 Another equivalent way of stating Theorem XIII.4.1 is to

say that, under condition (XIII.4.17) f or Φ, every (smooth) solution u to

(XIII.4.1) either veriﬁes

lim inf

→∞

−3

(

Σ(t)

|∇u(x

, t)|

dΣ)dt > 0

or decays po intwise and exponentially fast with all its derivatives of arbitrary

order. 

Remark XI II.4.3 Reasoning as in Remark VI.2. 1, we can show that τ(x)

tends to a constant τ

exponentially fast. 

Remark XI II.4.4 Corollary XIII.4.1 is similar to the analogous result de-

termined for the linearized Stokes approximation in Section VI.2 , with the

addition of the ﬂux restriction (XIII.4.17). 

XIII.5 Flow in an Aperture Domain. Generalized

Solutions and Related Properties

We shall now focus our attention on the investigation of existence and unique-

ness of ﬂows in certain domains with “exits” having an unbounded cross sec-

tion. Even though our m ethod carries over to more general situations, we shall

restrict ourselves to the special case where the region of motion Ω is a three-

dimensional “aperture domain,” that is (cf. also Section III.4.3 and Section

VI.5),

Ω =



x ∈ R

: x

6= 0 or x

≡ (x

, x

) ∈ S



(XIII.5.1)

with S (the aperture) a bounded locally Lipschitz domain of R

that con-

tains a disk of ﬁnite radi us; Heywood (1976). The reason for this choice is

because the technical details simplif y to an extent in such a way that the

analysis becomes form ally simpler. However, whenever the case dictates, we

shall explicitly mention possible extensions of our results to more general do-

mains. Moreover, again for the sake of simplicity, we shall assume no body

force acting on the l iquid, leaving the obvious generalization to the interested

reader.

We wish to solve the following Heywood’s problem: Given Φ ∈ R (the

ﬂux through the aperture), to determine a pair v, p deﬁned in Ω gi ven by

(XIII.5.1), such that

XIII.5 Aperture Domain: Generalized Solutions and Related Properties 927

ν∆v = v · ∇v + ∇p

∇· v = 0

)

in Ω

v = 0 at ∂Ω

lim

|x|→∞

v(x) = 0

= Φ.

(XIII.5.2)

The reader will immediately recognize that (XIII.5.2) is the f ully nonl inear

counterpart of the problem studied withi n the linear approximation in Section

VI.5.

We begin to give a weak formulation of problem (XIII.5.2). To this end,

multiplying (XIII.5.2)

by ϕ ∈ D(Ω) and formally integrating by parts we

ﬁnd

ν(∇v, ∇ϕ) = (v · ∇ϕ, v), for all ϕ ∈ D(Ω). (XIII.5.3)

Thus, also in light of what we did for the linear case in Section VI.3 and

Section VI.5, we give the following deﬁnition

Deﬁnition XIII.5.1. A vector ﬁeld v : Ω → R

with Ω g iven in (XIII.5.1) is

said to be a weak (or generalized) solution to problem (XIII.5.2) if and only if

(i) v ∈

1,2

(Ω);

(ii) v satisﬁes (XIII.5.3);

(iii) v satisﬁes (XIII.5.2)

in the trace sense.

Remark XI II.5.1 In view of Lemma II.6.3, it is easy to show that every

function satisfying (i) also satisﬁes

lim

|x|→∞

|v(|x|, ω)|dω = 0. (XIII.5.4)

Actually, let ψ be a function that equals zero in a neighborhood of S and

one far from S. Then w ≡ ψv ∈ D

1,2

), w = 0 at {x

= 0}, and the

assertion follows from the reasonings preceding Theorem II.6.3. With the help

of (XIII.5 .4) it is at o nce recognized that conditio ns (i)-(iii) of Deﬁnition

XIII.5.1 translate (XIII.5.2) into a weak form. 

Our next objective is to associate to every weak solution a suitable pressure

ﬁeld. Since the “aperture” S is bounded and contains a disk of ﬁni te radius,

we shall suppose without loss of generali ty that



∈ R

: |x

| < 1



⊂ S ⊂



∈ R

: |x

| < 2



We have the foll owing lemma.

Lemma XIII.5.1 Let v be a generalized solution to problem (XIII.5 .2).

Then there is a p ∈ L

loc

(Ω) such that

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

ν(∇v, ∇ψ) = (v ·∇ψ, v) + (p, ∇ · ψ) for all ψ ∈ C

∞

(Ω). (XIII.5.5)

Moreover, there are constants p

, p

−

∈ R such that

p − p

∈ L

− B

)

p − p

−

∈ L

−

− B

Proof. Let ψ ∈ D

1,2

(Ω

) where Ω

is any bounded domain of Ω. Since

|(v · ∇ψ, v)| ≤ kvk

4,Ω

|ψ|

1,2

by the Sobolev inequality (II.3.7) we ﬁnd

|(v · ∇ψ, v)| ≤ c

kvk

|ψ|

1,2

≤ c

|v|

1,2

|ψ|

1,2

and so

(v · ∇ψ, v)

deﬁnes a bo unded linear functional in D

1,2

(Ω

). Therefore, the ﬁrst part of

the lemma i s a consequence of Lemm a IV.1.1 . Let χ = χ(|x|) be a sm ooth

function tha t equals zero for |x| ≤ 3 and one for |x| ≥ 4 and set w = χv,

π = χp. Using Deﬁnition XIII. 5.1 and the identity (XIII.5.5), it is not ha rd

to show that w, π is a weak solution to the following Stokes problem in R

(an analogo us property holding in R

−

)

ν∆w = ∇π + F

∇ · w = g

)

in R

w = 0 at x

= 0

(XIII.5.6)

where

F = v · ∇w + 2∇χ · ∇ v + v∆χ − v · ∇χv + p∇χ

g = v · ∇χ.

(XIII.5.7)

Recalling that p ∈ L

loc

(Ω) and using the properties of χ and condition (i) of

Deﬁnition XIII.5.1, we ﬁnd for all ψ ∈ C

∞

)

|(2∇χ · ∇v, ψ) +(v∆χ, ψ) − (v ·∇v, ψ) + (p∇χ, ψ)|

≤ c



kvk

3,σ

+ kvk

3,σ

+ kpk

2,σ



|ψ|

1,3/2

(XIII.5.8)

where σ = supp (χ). As a consequence, (XIII.5.8) im plies

2∇χ · ∇v + v∆χ − v · ∇χv + p∇χ ∈ D

−1,3

). (XIII.5.9)

Moreover, by Theorem II.6.3, by (XIII.5.4) and (i) of Deﬁnition XIII.5.1, it is

XIII.5 Aperture Domain: Generalized Solutions and Related Properties 929

v ∈ L

)

and so we ﬁnd

|(v · ∇w, ψ)| = |(v ⊗ w, ∇ψ)| ≤ kvk

6,R

|ψ|

1,3/2

. (XI II.5.10)

Finally,

v · ∇χ ∈ L

). (XIII.5 .11)

From (XIII.5.7)–(XIII.5.11 ), it follows that

F ∈ D

−1,3

) , g ∈ L

) . (XIII.5.12)

Since, by assumption, w is a generalized solution to (XIII.5.6) (cf. Section

IV.3), from (XIII.5.12) and Theorem IV.3.3 we conclude, in particular, that

π − p

∈ L

)

for som e constant p

∈ R

. Recalli ng that π = p in R

−B

, we obtain the

desired result. ut

Remark XI II.5.2 The preceding lemma shows, am ong other things, that

the pressure ﬁeld tends in each outlet to a deﬁnite constant in the sense of L

This observation, along with the analo gy to the li nearized case (cf. Remark

VI.4. 7), suggests a diﬀerent formulation of problem (XIII.5.2), where one pre-

scribes, in pl ace of the ﬂux condition (XIII.5.2)

, the following ones:

lim

|x|→∞, x

p(x) = p

lim

|x|→∞, x

p(x) = p

−

where p

are prescribed constants.

This view, whi ch ori ginates with the work

of Heywood (1976), has been taken by several authors. We refer the reader to

the papers of Solonnikov (1981, 1983) and the references cited therein. 

The last result of this section concerns the diﬀerentiability of weak solu-

tions to problem (XIII.5.2). Since the aperture S ha s no “thickness,” we can

prove that these solutio ns are regular everywhere in Ω except at the bound-

ary ∂S. Of course, if S had “thickness” so that Ω would become smooth,

the corresponding generalized solutions would be equally smooth. In the cur-

rent situation, we have the following result whose proof, which patterns that

furnished in Theorem XIII.1.1, is left to the reader as an exercise.

Theorem XIII.5.1 Let v be a generalized solution to problem (XIII.5.2) and

let p be the corresponding pressure ﬁeld associated to v by Lemma XIII.5 .1.

Then

v, p ∈ C

∞

(Ω

)

where Ω

is any bounded domain of Ω that does not contain ∂S.

Of course, one of the two constants can be taken t o be zero.

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

Remark XI II.5.3 We wish to observe that some of the concepts and results

introduced so far, can be recovered when Ω is a two-dimensional aperture

domain:



x ∈ R

: x

6= 0 or x

∈ (−d, d)



for some d > 0. Of course, in this case we can also give a weak formulation of

problem (XIII.5.2). In these regards, we notice that the reasoning developed in

Remark XIII.5.1 cannot be repeated here, because of the lack of the Sobolev

inequality. However, one can show the validity of the following inequality

Ω

(|x| + 1)

≤ c

Ω

∇v : ∇v

where c > 0 and v ∈

1,2

(Ω); see Gal di, Padula, & Passerini (199 5). Using this

inequality, it can be proved that (XIII.5.4) i s valid also in the two-dimensional

case and that, in fact, v(x) tends to zero uni formly poi ntwise as |x| → ∞. Fur-

thermore, the ﬁrst part of Lemma XIII.5.1 continues to hold and the pressure

ﬁeld tends uniformly pointwise to a sui table constant p

[respectively p

−

] at

large distances in R

[respectively R

−

]. Finally, Theorem XIII.5.1 holds also

in the two-dimensional case. For detail we refer the reader to the paper of

Galdi, Padula, & Passerini. 

XIII.6 Energy Equation and Uni queness for Flows in an

Aperture Domain

The main objective of this section is to formulate conditions under which a

generalized solution is unique. As in the case of the three-dimensional exterior

problem, however, we are not able to furnish such conditions merely in the

class of generalized solutions but, rather in a subclass obeying a suitable

energy inequality. In fact, as we shall prove later, this subclass is certainly

nonempty.

Let us derive formally the energy equation for solutions to problem

(XIII.5.2). To this end, we multiply (XIII.5.2)

by v and integrate over

Ω

≡ Ω ∩B

for suﬃciently large R to obtain

−ν

Ω

∇v : ∇v + ν

∂B

v ·

∂v

∂n

∂B

v ·n +

∂B

(p − p

)v · n

∂B

−

(p − p

−

)v · n + (p

− p

−

)

where p

and p

−

are the limits to which p tends as |x| → ∞ in R

and R

−

respectively, cf. Lemma XII I.5.1, and B

= B

∩ R

. Letting R → ∞ and

assuming that all surface integrals tend to zero (at l east along a sequence),

we ﬁnd

XIII.6 Aperture Domain: Energy Equation and Uniqueness 931

Ω

∇v : ∇v = −

∗

(XIII.6.1)

where Φ is the ﬂux of v through S and p

∗

= p

− p

−

. Relation (XIII.6.1) is

the energy equation for problem (XIII.5.2) and describes the balance between

the energy dissipated by the l iquid and the work done on it. Notice that,

in agreement with physical intuition, a positive ﬂux implies p

< p

−

, while

−

< p

otherwise.

As the reader may have noticed, in obtaining (XIII.6.1) we have assumed

that the solution has a certain degree of regula rity at large spatial distances

that a priori need not be matched by a generalized solution. We may thus

wonder which extra conditions we must impose on a generalized solution so

that it will obey the energy equation. An answer is furnished by the following.

Theorem XIII.6.1 Let v be a generalized solution to problem (XIII.5.2).

Then, if

v ∈ L

(Ω),

v obeys the energy equation (XIII.6.1), where p

∗

= p

− p

−

and p

are the

constants associated to p by Lemma XIII.5.1.

Proof. Let ψ

= ψ

(|x|) be a smooth “cut-oﬀ” function such that

(|x|) =

(

1 if |x| ≤ R

0 if |x| ≥ 2R

|∇ψ

(|x|)| ≤ M, for some M independent of R.

In view of Lemma XIII.5.1, identity (XIII.5.5) continues to hold for all ψ ∈

1,2

(Ω

), where Ω

is an arbitrary bounded domain with Ω

⊂ Ω. Thus, we

may take ψ ≡ ψ

v to obtain

ν(ψ

∇v, ∇v) = (ψ

v · ∇v, v) + (p, ∇ · (ψ

v)) + (v ·∇ψ

, v · v). (XIII.6.2)

Recalling the properties of ψ

and the assumptions on v, we deduce at once

that

lim

R→∞

(ψ

∇v, ∇v) = |v|

1,2

lim

R→∞

(v · ∇ψ

, v · v) = 0.

(XIII.6.3)

Moreover,

(p, ∇ · (ψ

v)) =

(p − p

)v ·∇ψ

−

(p − p

−

)v · ∇ψ

∗

Ω

∇ · (ψ

(p − p

)v ·∇ψ

−

(p − p

−

)v · ∇ψ

−p

∗

Φ,

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

where p

are the constants associated to p by Lemma XIII.5.1. Taking into

account that, again by Lemm a XIII.5.1,

p − p

∈ L

−B

and setting Ω

R,2R

= Ω ∩ B

∩B

, we thus ﬁnd



(p −p

)v · ∇ψ



≤ kp − p

3,Ω

R,2R

kvk

k∇ψ

3,Ω

≤ ckvk

kp − p

3,Ω

R,2R

= c l og 2kvk

kp − p

3,Ω

R,2R

Therefore,

lim

R→∞



(p − p

)v · ∇ψ



= 0.

Employing a similar argument for the integral over R

−

, we may then conclude

that

lim

R→∞

(p, ∇ · (ψ

v)) = −p

∗

Φ. (XIII.6.4)

Furthermore, setting as before B

= R

∩ B

, after integration by parts we

obtain

(ψ

v · ∇v, v) =

v · ∇v · v +

−

v ·∇v · v =

Ω

v · ∇ψ

v · v,

so that, recalling the assumption on v and the properties of ψ

, it follows

that

lim

R→∞

(ψ

v ·∇v, v) = 0. (XIII.6.5)

Equation (XIII. 6.1) b ecomes a consequence of (XIII.6 .2)–(XIII.6.5).

Let w be a generalized solution to (XIII.5.2) and let π be the pressure

ﬁeld associated to w by Lemma XI II.5.1. We shall say that w satisﬁes the

energy inequality if

|w|

1,2

≤ −

∗

(XIII.6.6)

where π

∗

= π

− π

−

and π

are the constants associated to π by Lemma

XIII.5.1. The following uniqueness result holds. ut

Theorem XIII.6.2 Let v be a generalized solution to problem (XIII.5.2)

such that

kvk

√

ν.

Then v is unique in the class of all generalized solutions w corresponding to

the same ﬂux Φ and satisfying (XIII.6.6).

XIII.6 Aperture Domain: Energy Equation and Uniqueness 933

Proof. Let ψ

be the “ cut-oﬀ” function introduced in the proof of Theorem

XIII.6.1. As in that case, we may take ψ = ψ

w in the identity (XIII.5.5) to

obtain

ν(ψ

∇v, ∇w) = −(ψ

v·∇w, v)−(v·w, v·∇ψ

)+(p, ∇·(ψ

w)). (XIII.6.7)

Using the summability properties of v, w it is no t diﬃcult to show the fol-

lowing relations

lim

R→∞

(ψ

∇v, ∇w) = (∇v, ∇w)

lim

R→∞

(ψ

v · ∇w, v) = (v · ∇w, v).

(XIII.6.8)

Moreover, with the aid of (II.3.7), we ﬁnd

|(v · w, v · ∇ψ

)| ≤ kvk

3,Ω

R,2R

kwk

6,Ω

R,2R

k∇ψ

6,Ω

R,2R

≤

kvk

3,Ω

R,2R

|w|

1,2

and so

lim

R→∞

(v · w, v · ∇ψ

) = 0. (XIII.6.9)

We next consider the following identity

(p, ∇·(ψ

w)) =

(p−p

)w·∇ψ

−

(p−p

−

)w·∇ψ

−p

∗

Φ, (XIII.6.10)

where p

are the constants introduced in Lemma XIII.5.1 and p

∗

= p

−p

−

By the H¨older inequality, (II.3.7), and the properties of ψ

, it follows that



(p −p

)w · ∇ψ



≤ kp − p

3/2,Ω

,2R

kwk

6,Ω

R,2R

k∇ψ

6,Ω

R,2R

≤

kp − p

3/2,Ω

R,2R

|w|

1,2

(XIII.6.11)

It is not diﬃcult to see that

p − p

∈ L

3/2

−B

). (XII I.6.12)

Actually, we know that χv and χp, with the χ “cut-oﬀ” function introduced in

the proof of Lemma XIII.5.1, satisfy the Stokes system (XIII. 5.6), (XIII.5.7)

(and an analogo us one in R

−

). Because of the assumption on v, we have

v ∈ L

(Ω), v · ∇v ∈ D

−1,3/2

(Ω),

from which it readily fo llows that

F ∈ D

−1,3/2

) , g ∈ L

3/2

)

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

with F and g given in (XIII.5.7). Thus, applying the results of Theorem

IV.3. 3 to (XIII.5.6), (XIII. 5.7) and recalling that χv is a generalized solutio n

to (XIII.5.6), (XII I.5.7), we ﬁnd (XIII.6.12). As a consequence of (XIII.6.12),

from (XIII.6.1 1) (and the analogous property in R

−

), we conclude that

lim

R→∞

(p, ∇ ·(ψ

w)) = −p

∗

Φ. (XIII.6.13 )

Therefore, (XIII.6.7)–(XIII.6.12) yield

− ν(∇v, ∇w) = −(v ·∇w, ∇v) + p

∗

Φ. (XIII.6.14)

Writing identity (XIII.5.5) with w in place o f v and choosing ψ

v for ψ, by

means of arguments very close to those just used one can show

−ν(∇w, ∇v) = −(w ·∇v, ∇w) + π

∗

Φ. (XIII.6 .15)

We next observe that by Theorem XIII.6.1, v obeys the energy equality

(XIII.6.1) while w, by hypothesis, sati sﬁes the energy inequality (XIII.6.6).

Thus, adding the four displayed relations (XIII.6.1), (XIII.6.6 ), (XIII.6.13 ),

and (XIII.6.14) and setting u = v − w, we arrive at

ν| u|

1,2

≤ −(v · ∇ w, v) − (w · ∇v, w). (XII I .6.16)

The following relations are easily shown:

(w · ∇v, w) = −(w · ∇w, v) , (u · ∇v, v) = 0 . (XIII.6 .17)

Assuming, for a while, the validity of (XIII.6 .17) and using it into (XIII.6.16)

delivers

ν|u|

1,2

≤ −(v · ∇w, v) + (w · ∇w, v) + (u · ∇v, v)

= (u · ∇u, v).

(XIII.6.18)

Employing the Schwarz inequality on the rig ht-hand side of (XIII.6.18) along

with the Sobolev inequality (II.3.7), we ﬁnd

|(u · ∇u, v)| ≤ kvk

kuk

|u|

1,2

≤

√

kvk

|u|

1,2

If we combine this inequality with (XIII.6.18), we arrive at



ν −

√

kvk



|u|

1,2

≤ 0,

which, in turn, furnishes u ≡ 0 under the stated assumption o n v. To show

the theorem completely, it remains to prove the identities (XII I .6.17). Setting

= R

∩ B

, we have

XIII.7 Aperture Domain: Existence and Uniqueness 935

w·∇v·w = −

w·∇w·v+

∂B

∩R

w·nv·w−

v·w. (XIII.6.19)

By the H¨older inequality, it f ollows that



∂B

∩R

w · nv ·w



≤ cR

2/3

kwk

6,∂B

∩R

kvk

3,∂B

∩R

. (XIII.6.20)

However, since

∞

∂B



|w|

+ |v|



dR < ∞,

we have, at least along a sequence,

kwk

6,∂B

∩R

+ kvk

3,∂B

∩R

= o(R

−2/3

)

and therefore, from (XIII. 6.20) and (XIII.6.19) we conclude that

w · ∇v · w = −

w · ∇w · v −

v ·w. (XIII.6.21)

Likewise,

w · ∇v · w = −

w · ∇w · v +

v ·w (XIII.6.22)

and (XIII.6.17)

follows from (XIII.6.21), (XIII.6.22). In a completely analo-

gous way one can show (XIII.6.17)

, who se proof is thus left to the reader as

an exercise. The theorem is proved. ut

Remark XI II.6.1 In the case of a plane ap erture domain, Theorem XIII.6.2

continues to hold, provided the assumption on the L

-norm of v is replaced

by the following one

sup

x∈Ω

|v(x)||x| ≤ cν,

for a suitable c > 0; see Galdi , Padula, & Solonnikov (1996). 

XIII.7 Exi stence and Uniqueness of Flows in an

Aperture Domain

The objective of this section is to show existence and uniqueness of generalized

solutions to problem (XIII.5.2). The main feature of these solutions is that,

unlike the case of ﬂow in domains with cylindrical ends (Leray’s problem),