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

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406 VI Steady Stokes Flow in Domains with U nbounded Boundaries

(∇v, ∇a) = 0, (VI.4.43 )

where a is the basis of

1,2

(Ω) / D

1,2

(Ω) determined in Theorem III.5.2. To

prove this we observe that v ∈

1,2

(Ω) and v is orthogonal to al l ϕ ∈ D

1,2

(Ω),

see (VI.3.3). Therefore, since every ψ ∈

1,2

(Ω) can be written as ψ = ϕ+λa,

λ ∈ R, from (VI.4. 43) it follows that v is orthogonal to all functions of

1,2

(Ω)

and thus v ≡ 0. To show (VI.4. 43) we multiply (VI.4.1)

by a and integrate

by parts to obtain

−

ω(t)

∇v : ∇a = −

)∪Σ

)

n · ∇v · a + p

)

a · n

)

(p − p

)a · n +

)

pa · n,

(VI.4.44)

where

ω(t) = Ω

∪ {x ∈ Ω

: x

< t} ∪ {x ∈ Ω

: x

< t}.

From Theorem III.5.2 we know that a can be taken to be zero in a

neighborhood of the lateral surfaces of Ω

iγ

, γ < 1/4, and furthermore,

a(x) ≤ cf

−n−1

). Therefore, if p

= 0, we can pass to the limit as t → ∞

in (VI.4.44) and use Theorem VI.4.1 and Theorem VI. 4.4 to obtain (VI.4.43).



Remark VI.4.8 The decay results of Theorem VI.4.1–Theorem VI.4.4 can

be fairly improved. In this respect, we refer the reader to Pileckas (199 6a,

1996b, 1996c). A typical result proved there is the following one.

Theorem VI.4.5 Let Ω and v be as in Theorem VI.3.2 and let p be the

corresponding pressure given by Lemma IV.1.1. Assume, further, that Ω is of

class C

l+2,δ

, l ≥ 0, δ ∈ (0, 1). Then the following decay estimate hold

v(x)| ≤ C|φ|f

−n+1−|α|

)

∇p(x)| ≤ C|φ| f

−n+1−|α|

for x ∈ Ω

, l ≥ |α| ≥ 0. Moreover, there exists p

∈ R such that for x ∈ Ω

|p(x)| ≤ C|φ|

−n+1

(t)dt + p

In these relations, C = C( Ω, q, n).



VI.5 Existence, Uniqueness, and Decay of Flow Through an Aperture 407

VI.5 Existence, Uniqueness, and Asym ptotic Behavior of

Flow Through an Aperture

Let us consider the “a perture domain” (VI.0.7 ), i.e.,

Ω = {x ∈ R

: x

6= 0 or x

∈ S}, (VI.5.1)

where S is a bounded domain of R

n−1

. As we have seen in Section VI.3 (see

Exercise VI.3.5), given any continuous linear functional f on D

1,2

(Ω) and any

Φ ∈ R, there exists one and only one vector ﬁeld v ∈

1,2

(Ω) such that

(∇v, ∇ϕ) = −[f , ϕ] , for all ϕ ∈ D(Ω)

= Φ,

(VI.5.2)

where [f, ϕ] denotes the value of f at ϕ. The aim of this section is to pro-

vide further existence results for problem (VI.5.2) with v in D

1,q

(Ω), q 6= 2,

together with corresponding estimates. Moreover, we shall also derive the

asymptotic structure of such solutions.

In what follows, we shall denote by D

−1,q

(Ω) the dual space of D

1,q

(Ω)

and by |f|

−1,q

the norm of f ∈ D

−1,q

(Ω). We al so set

Σ = S ∪

i=1

where ω

, i = 1, . . . m, are the connected components of R

n−1

− S, and

= δ(Σ).

Finally, with the origin of coordinates in Σ, we put

Ω

= Ω

2δ

∩ R

The following theorem ho lds.

Theorem VI.5.1 Let Ω be given in (VI.5.1), n ≥ 2, with S a bounded,

locally Lipschitz do mai n of R

n−1

. Given

f ∈ D

−1,2

(Ω) ∩ D

−1,q

(Ω), Φ ∈ R,

where q ∈ (1, 2n/(n − 2)] if n ≥ 3, and q ∈ (1, ∞) if n = 2, there exists one

and only one ﬁeld v satisfying (VI.5.2) with

v ∈ D

1,2

(Ω) ∩ D

1,q

(Ω

Moreover, denoting by p the pressure ﬁeld associated to v by Lemma IV.1.1,

there exist constants p

, p

−

∈ R such that

408 VI Steady Stokes Flow in Domains with U nbounded Boundaries

p − p

∈ L

) ∩ L

((Ω

and the following estimate holds:

|v|

1,2

+ |v|

1,q,Ω

+ kp − p

2,R

+ kp − p

−

2,R

−

+kp − p

q,Ω

+ kp − p

−

q,Ω

−

≤ c (|f|

−1,2

+ |f|

−1,q

+ |Φ|) .

Proof. The uniqueness part i s proved exactly as in Theorem VI.3.1, and it is

left to the reader as an exercise. To show existence, we look for a solution o f

the form

v = u + Φb

where the ﬁeld b has been given at the beginning of Section III.4.3. We have

(see (III.4.6 )–(III.4.8))

b ∈ C

∞

(Ω)

∇ · b(x) = 0

|b(x)| ≤ c|x|

−n+1

|∇b(x)| ≤ c|x|

−n

= 1

while u obeys

(∇u, ∇ϕ) = −(∇b, ∇ϕ) − [f, ϕ] , for all ϕ ∈ D(Ω)

= 0.

(VI.5.3)

Since the right-hand side of (VI.5.3)

deﬁnes a bounded linear functional

in D

−1,2

(Ω), we deduce the existence of u ∈ D

1,2

(Ω) satisfying (VI.5 .3).

Furthermore, putting u in place of ϕ into (VI.5.3)

(this can be done by a

simplest density procedure) furnishes

|u|

1,2

≤ (|Φ||b|

1,2

+ |f|

−1,2

) |u|

1,2

implying

|v|

1,2

≤ c

(|Φ|+ |f|

−1,2

) (VI.5.4)

with

= (1 + 2|b|

1,2

In view of Lemma IV.1.1 we can associate to v a pressure ﬁeld p ∈ L

loc

(Ω)

such that

VI.5 Existence, Uniqueness, and Decay of Flow Through an Aperture 409

(∇v, ∇ψ) = (p, ∇·ψ) − [f , ψ] , for all ψ ∈ C

∞

(Ω). (VI.5.5)

Let us show that there exist constants p

, p

−

∈ R such that

p − p

∈ L

)

p − p

−

∈ L

(IR

−

)

(VI.5.6)

and that the following inequality holds

kp − p

2,R

+ kp − p

−

2,IR

−

≤ c

(|f|

−1,2

+ |v|

1,2

) . (VI.5.7)

Actually, consider the functional

F(ψ) = (∇v, ∇ψ) + [f , ψ], ψ ∈ D

1,2

Clearly, F is linear and bounded in ψ ∈ D

1,2

) and, in view of (VI.5.2 ),

it vanishes identically on D

1,2

). Thus, by Corollary III. 5.1, there exists

∈ L

) such that

F(ψ) = (π

, ∇· ψ)

which, once compared with (VI.5.5), proves (VI.5.6)

. In a compl etely analo-

gous way one shows (VI.5.6)

. We now write (VI.5.5) with p −p

in place of

p. By density, we deduce that (VI.5.5) continues to hold for all ψ ∈ D

1,2

Choosing ψ as a solution to the problem

∇ · ψ = p − p

ψ ∈ D

1,2

)

|ψ|

1,2

≤ ckp − p

2,R

(VI.5.8)

(such a solution certainly exists in view of Corollary IV.3.1), from (VI.5.5)

and (VI.5 .8) we obtain

kp − p

2,R

= (∇v, ∇ψ) + [f , ψ] ≤ c (|v|

1,2

+ |f |

−1,2

) kp − p

2,R

Analogously,

kp − p

2,R

≤ c (|v|

1,2

+ |f |

−1,2

)

and (VI.5.7) follows from these latter inequalities. We shall next derive esti-

mates for v in D

1,q

. For R ≥ 2δ

, we let ζ = ζ(|x|) denote a non-decreasing,

smooth function with ζ(| x|) = 0 if |x| ≤ R/2 and ζ(|x|) = 1 if |x| ≥ R. Setting

w = ζv, τ = ζ(p − p

we easily obtain that w is a generalized solution to the following problem (see

Section IV.3):

410 VI Steady Stokes Flow in Domains with U nbounded Boundaries

∆w = ∇τ + F

∇ · w = g

)

in R

w = 0 on Σ ≡ R

n−1

× {0},

(VI.5.9)

where

= 2∇ζ · ∇v − ∆ζv − (p − p

)∇ζ + ζf

g = ∇ζ · v.

For R > 0, we put

Ω

R,2R



x ∈ R

: R < |x| < 2R



Ω



x ∈ R

: |x| < R



Ω

= R

−B

Ω

= R

∩ Ω

Evidently,

−1,q

≤ c



kvk

q,Ω

R,2R

+ kp − p

−1,q,Ω

R,2R

+ |f|

−1,q



kgk

≤ kvk

q,Ω

R,2R

(VI.5.10)

Employing the embedding Theorem II.3.4 and inequality (II.5.18), from the

assumptions ma de on q we easily ﬁnd

kvk

q,Ω

+ kp − p

−1,q,Ω

≤ c (|v|

1,2

+ kp − p

) . (VI.5.11)

Thus, recalling that ζ(|x|) is equal to one for |x| ≥ R, from Theorem IV.3.3

together with (VI.5.4), (VI.5.7), and (VI.5.11) we ﬁnd

|v|

1,q ,Ω

R + kp − p

q,Ω

+ kp − p

−

q,Ω

−

≤ c (|f|

−1,2

+ |f |

−1,q

) (VI.5.12)

and the theorem fol lows from (VI.5.12), (VI.5 .11), (VI.5.4), and (VI.5.7 ). ut

Remark VI.5.1 The fact that, in the theorem just shown, we must require

that f belong simultaneously to D

−1,2

(Ω) and to D

−1,q

(Ω), that q be suitably

restricted, and that ∇v, p belong to L

(Ω ∩ B

), only for suﬃciently large

R, is due to the circumstance that Ω has no regularity near the boundary of

S, since S i s (n − 1)-dimensional. However, if we assume that the “hole” S

has “thickness,” becoming a domain of R

of class C

, in such a way that Ω

becomes likewise of class C

, then one can show that, for any f ∈ D

−1,q

(Ω),

1 < q < ∞, there is one and only one solution v ∈ D

1,q

(Ω) to (VI. 5.2) and that

the associated pressure ﬁeld p satisﬁes p −p

∈ L

), p −p

−

∈ L

−

) for

some constants p

, p

−

. Moreover, v and p obey the corresponding estimates.

Finally, we would like to observe that, if S has no “thickness,” a nd n = 2, one

can still prove that v ∈ L

∞

(Ω), see Solonnikov (1988) and Galdi , Padula &

Solonnikov (1996). 

VI.5 Existence, Uniqueness, and Decay of Flow Through an Aperture 411

Remark VI.5.2 Arguing as in Remark VI.4.7, one can show that, for f ≡ 0,

= p

−

if and only if the ﬂux Φ is zero. Actually, if f 6≡ 0, one cannot deduce

= p

−

, even if Φ = 0. In fact, taking, for instance, f ∈ C

∞

(Ω) we could

prove that, for φ = 0, the following relation holds:

−

− p

Ω

f · w,

where w is a solution to (VI.5.2) with f ≡ 0 and Φ = 1. 

In the last part of this section we shall analyze the asymptotic structure

of the generalized solutions just obtained. We begin to show a general repre-

sentation formula in the case when f is in divergence form.

Lemma VI.5.1 Let v ∈

1,q

(Ω), 1 < q < ∞, satisfy (VI.5.2)

. The following

assertions hol d true:

(i) Suppose

[f , ϕ] = −(F , ∇ϕ),

where

F ∈ L

(Ω) ∩ L

(Ω) , for some r ∈ (1, n).

Then, for a.a. x ∈ R

(x) = −

(x, y)F

(y)dy −

(y)T

, g

)(x, y)n

(y)dσ

(VI.5.13)

where G

= {G

}, g





is the Green’s tensor for the Stokes problem

in R

(see (IV.3.46)–(IV.3. 51)), and G

≡ (G

, G

, . . ., G

(ii) Suppose that

f ∈ L

(Ω), with bounded support.

Then, there exist p

, p

−

∈ R such that for a.a. x ∈ R

(x) =

(x, y)f

(y)dy −

(y)T

, g

)(x, y)n

(y)dσ

p(x) = p

−

(x, y)f

(y)dy −2

(y)

∂g

(x − y)

∂x

(y)dσ

(VI.5.14)

where p is the pressure ﬁeld associated to v by Lemma IV.1.1.

Proof. Since the proof is exactly the same for R

and R

−

, we shall show the

validity of (i) and (ii) for R

. Moreover, for simplicity, the Green’s tensor

in R

will be denoted by G, g. Let us commence to show (i). We begin to

observe that, reasoning as in the proof of Theorem VI.5.1, we can associa te to

412 VI Steady Stokes Flow in Domains with U nbounded Boundaries

v a pressure ﬁeld p in the sense of Lemma IV.1.1 such that p −p

∈ L

We next notice that from inequal ity (II.5.18) it follows that v ∈ W

1,q

(C), for

every cube C with a side at x

= 0 and so, setting Σ = R

n−1

× {0}, the

trace of v at Σ belongs W

1−1/q,q

(S) ⊂ D

1−1/q,q

(Σ) (see Sections II.3 and

II.6) with support contained in S. Therefore, in view of Theorem II.4.1 and

Theorem II.3.1, there exist two sequences

(k )

} ⊂ C

∞

), {η

(k)

} ⊂ C

∞

(Σ)

approximating F and v in the norms of L

) ∩ L

) and D

1−1/q,q

(Σ),

respectively. Let us denote by v

(k)

and p

(k)

, k ∈ N, velocity and pressure

ﬁelds of the Stokes problem in R

, corresponding to the force −∇· F

(k )

and

to the velo city η

(k)

at Σ . From Theorem IV.3.3, we obtain that, as k → ∞,

(k)

, p

(k)

converge to v, p −p

in the norm of D

1,q

) ×L

). Therefore,

in particular, we may select a subsequence, denoted ag ain by v

(k)

, p

(k )

, such

that

∇v

(k)

(x) → ∇v(x), p

(k)

(x) → p(x) a.e. in R

. (VI.5.15)

Since v and v

are identically vanishing on Σ − S, from (II.5.18) we readily

obtain

kv − v

(k )

q,C

≤ c(C)|v − v

(k)

1,q ,C

, (VI.5.16)

for any cube C with a side at x

= 0 strictly containing S. Choosing an

increasing sequence of cubes of type C i nvading R

and employing the Cantor

diagonalization method, from (VI.5.15) and (VI.5.16) we ﬁnally deduce the

existence of a sequence, which will be denoted again by v

(k)

, satisfying

lim

k→∞

(k)

(x) = v(x) , for a.a. x ∈ R

. (VI.5.17)

Now, because of the results of Exercise IV.8.2, the following representation

holds for v

(k )

(k)

(x) =

(x, y)F

(k)

(y)dy +

(k)

(y)T

, g

)(x, y)n

(y)dσ

(VI.5.18)

We wish to let k → ∞ into this relation. Set

(k)

(x) =

(x, y)[F

(k)

(y) − F

(y)]dy. (VI.5. 19)

Since

(x, y)| ≤ c|x − y|

−n+1

(see (IV.3.50), (IV. 3.52)), from the assumption made on F and the Sob olev

Theorem II.11.3, we derive, along a subsequence at least,

lim

k→∞

(k)

(x) = 0 fo r a.a. x ∈ R

. (VI.5.20)

VI.5 Existence, Uniqueness, and Decay of Flow Through an Aperture 413

It remains to prove the convergence of the last term in (VI.5.18 ). Taking into

account

, g

)(x, y)| ≤ c|x −y|

−n+1

(see (IV.3.50), (IV.3.52)), and setting

(k)

(x) =

[η

(k )

(y) − v

(y)]T

, g

)(x, y)n

(y)dσ

, (VI.5.21)

from the H¨older inequality we obtain

(k)

(x)| ≤ ckη

(k)

− vk

q,S

where c = c(S, d), d = dist (x, S). Thus,

(k)

(x) → 0 in R

. (VI.5.22)

Representation (VI.5.13) is then a consequence of (VI.5.17)–(VI.5. 22). The

proof of (VI.5.14) is similar. Actually, we now start w ith a sequence of f unc-

tions {f

(k)

}, {η

(k)

} where η

(k)

is the same as before, while f

(k)

∈ C

∞

(ω),

with ω = supp (f ), converge to f in L

(Ω) ∩ L

(Ω). By the same technique

used before, we then show the validity o f (VI.5.14)

, with the only change

that, to prove the convergence of the term

(x, y)f

(k)

(y)dy =

(x, y)f

(k)

(y)dy,

we have to employ exactly the same reasoning used to show the convergence

of the term V

in the proof ofTheorem IV.8.1 . This is made possible by the

fact that G and U obey pointwise estimates of the same type. Concerning the

representation of the pressure, we easily establish, as before, the a.e. pointwise

convergence of p

(k)

to p −p

. Moreover, the a.e. pointwise convergence of the

term

(x, y)f

(k)

(y)dy

is acquired by taking into account the estimate

|g(x, y)| ≤ c|x −y|

−n+1

(see (IV.3.50), (IV.3.51)) and Exercise IV.3.3, and by using a reasoning sim-

ilar to that adopted to show the convergence of the term P in the proof of

Theorem IV.8.1 (details are left to the reader). Finally, observing that again

from (IV.3.50), (IV.3.51), and Exercise IV.3.3,

|∇g(x, y)| ≤ c|x − y|

−n

we have



[η

(k )

(y) − v

(y)]n

(y)dσ



≤ ckη

(k)

−vk

q,S

which also implies the pointwise convergence of the bo undary integral. The

lemma i s therefore completely proved. ut

414 VI Steady Stokes Flow in Domains with U nbounded Boundaries

The result just shown furnishes the following one as a simple corollary.

Theorem VI.5.2 Let v ∈

1,q

(Ω), 1 < q < ∞ , satisfy (VI.5.2)

correspond-

ing to f ∈ L

(Ω) of bounded support. Then, there exist constants p

, p

−

such

that v and the corresponding pressure ﬁeld p a dmit the following asymptotic

expansion as |x| → ∞ in R

(x) = b

, g

)(x, 0) + ϕ

(x)

p(x) = p

+ 2b

)(x, 0) + β

(x),

(VI.5.23)

where

= ±

(VI.5.24)

and, for all |α| ≥ 0,

= O(|x|

−n+1−|α|

)

= O(|x|

−n−| α|

(VI.5.25)

In particular, if f ≡ 0, then

= O(|x|

−n−|α|

)

= O(|x|

−n−1 −|α|

(VI.5.26)

Proof. By the fundamental property of the Green’s tensor we have that

(x, 0) = 0 for all x ∈ R

. Therefore, from (VI.5.14)

we ﬁnd

(x) = b

, g

)(x, 0) +

(x, y) − G

(x, 0)]f

(y)dy

−

(y)[T

, g

)(x, y) − T

, g

)(x, 0)]n

(y)dσ

(VI.5.27)

Applying the Lagrange theorem in the integrands of (VI.5. 27) and using

(IV.3.50), (IV.3.5 1), and Exercise IV.3.3 we may proceed as in the proof of

Theorem V.3.2 to show the validity of (VI.5.23)

, (VI.5.24), and (VI.5.25)

and, for f ≡ 0, of ((VI.5.26 )

. Observing that, by (IV.3.46)–(IV.3.49), it also

follows that g(x, 0) = 0, we may establish in a completely analogous way

(VI.5.23)

, (VI.5.24), (VI.5.25)

, and, for f ≡ 0, (VI.5.26)

. The proof of the

theorem is acquired. ut

Remark VI.5.3 In view of the estimates on G and g given in (IV.3.50),

(IV.3.51) and Exercise IV.3.3 , Theorem VI.5.2 implies, in particular, that at

large distances, v behaves as |x|

−n+1

for n ≥ 2. 

VI.6 Notes for the Chapter 415

VI.6 Notes for the Chapter

Section VI.1. Although diﬀering in details, the material presented here is

based on the treatment o f Amick (1977). In particular, Theorem VI.1.2 can

be deduced from the work of this author.

Section VI.2. The main result of this section, Theorem VI.2 .2, is due to me.

It has been obtained by coupling the ideas of Horgan & Wheeler (1978) with

those o f Amick (1977,1 978) and of Ladyzhenskaya & Sol onnikov (1980). In

particular, Lemm a VI.2.1 and Theorem VI.2.1 are due to Ladyzhenskaya and

Solonnikov, while Lemma VI.2.2 is proved by Horgan and Wheeler. Somewhat

weaker results than those of Theorem VI.2.2 can be deduced from the papers

of Horgan (197 8) and Ames & Payne (1989). Extension of these results to

compressible ﬂui ds has been recently proved by Padula & Pileckas (19 92, §7).

Section VI.3. The guiding ideas are essentially taken from the works of

Heywood (1976, §6) and Solonnikov & Pileckas (1977).

Concerning domains with varying cross-sections (not necessarily unbounded),

we refer the reader to the papers of Fraenkel (1973), Iosif’ jan (1978), Pileckas

(1981), and Nazarov & Pileckas (1983).

Section VI.4. The approach proposed here is due to me. The proof of The-

orem VI.4.3 is inspired by the work of Gilbarg & Weinberger (1978, §4).

The study of certain asymptotic behavior in domains with o utlets contain-

ing a semi-inﬁnite cone has been performed by Pileckas (1980a).

Results on existence, uni queness and asymptotic decay of solutions in do-

mains that become “layer-li ke” at inﬁnity are provided by Nazarov & Pileckas

(1999a , 2001). It is interesting to observe that, for n = 3, solutions show onl y

a power-like decay, and not an exponential one.

Section VI.5. The “ﬂow through an aperture” (or “ﬂow through a slit” in

the two-dim ensional case) is a well studied problem in classical ﬂuid dynam-

ics, mostly, for its applications to resonance phenomena in narrow-mouthed

harbors; see, e.g. Miles & Lee (1975). As a matter of fact, in absence of body

forces, explicit solutions can be exhibited in the two-dim ensional inviscid case

(Lamb 1932, p. 73; Milne-Thomson 1 938, §§6.10 , 11.53) and in the vi scous

case as well (Stokes problem), when the aperture is a circle (Milne-Thomson

1938, §15.56). In the mathematical community, seemingly, this type of ﬂow

became popular and thoroughly investigated in its viscous formulation, only

after the publication of the fundamental paper of Heywood (1976).

The theory described in this section is due to Galdi & Sohr (1992); see also

Farwig and Sohr (1994b). Similar results have been obtained, independently

and by diﬀerent tools, by Borchers & Pileckas (1992). However, the asymptotic

estimates given in Theorem VI.5.2 are somewhat more detailed than those

provided by the latter authors.