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

Подождите немного. Документ загружается.

376 VI Steady Stokes Flow in Domains with U nbounded Boundaries

from (VI.1.11) it follows that

∆w

= ∇q

+ f

+ F

∇ · w

= g

)

in D

= 0 a t ∂D,

(VI.1.14)

where D is any C

∞

-smooth, bounded domain containing U

and

= ψ

f ,

= 2∇ψ

· ∇u + u∆ψ

−τ∇ψ

= ∇ψ

· u.

Fix k

> k

, k

∈ N, and apply to (VI.1.14) the results of Theorem IV.6.1

with m = 0 along with those contained in Exercise IV.6.3 to o bta in

kuk

2,q,U

≤ c



kfk

q,U

−1

+ kF

q,U

−1

+ kg

1,q,U

−1



. (VI.1.15)

Recalling the deﬁnitions of F

and g

we at once deduce

q,U

−1

+ kg

1,q,U

−1

≤ c



kuk

1,q,U

−1

+ kτk

q,U

−1



which, possibly modif ying τ by adding a suitable constant, in view o f Lemma

IV.1. 1 in turn gives

q,U

−1

+ kg

1,q,U

−1

≤ c

kuk

1,q,U

−1

Replacing this estimate back into (VI.1.15) furnishes

kuk

2,q,U

≤ c



kfk

q,U

−1

+ kuk

1,q ,U

−1



(VI.1.16)

and so, in particular,

kuk

2,q,ω

≤ c



kfk

q,ω

1,δ

+ kuk

1,q,ω

1,δ



which, by virtue of (VI.1.11)

, proves (VI.1.13) for m = 0. We next choo se

= k

+ 1 and apply to solutions to (VI. 1.14) the results contained in

Theorem IV.6.1 and Exercise IV.6.3 with m = 1, to deduce

kuk

2,q,U

≤ c



kf k

1,q,U

+ kF

1,q,U

+ kg

2,q ,U



. (VI.1.17)

Reasoning as before, we replace the obvious inequality

q,U

+ kg

1,q,U

≤ c

kuk

2,q,U

into (VI.1.17) and use (VI.1.16) to recover

kuk

3,q,U

≤ c



kf k

1,q,U

−1

+ kuk

1,q ,U

−1



and so, in particular,

kuk

3,q,ω

≤ c



kfk

1,q ,ω

1,δ

+ kuk

1,q,ω

1,δ



which, by (VI.1.12)

, proves (VI.1.13) for m = 2. Iterating this procedure as

many times as we please, we prove (VI.1. 13) for all m ≥ 0. ut

VI.1 Leray’s Problem: Existence, Uniqueness, and Regularity 377

Let us now come back to the asymptotic estima te for v and p. Recalling

that v = u + a, from (VI.1.1 )–(VI.1.3) we have

∆u = ∇τ

∇ · u = 0

)

in Ω

u = 0 at ∂Ω

−Σ

u · n = 0,

(VI.1.18)

where

τ = p −C

, Σ

= {x ∈ Ω

: x

= R}.

(A system analogous to (VI.1. 18) is veriﬁed in Ω

.) Employing (VI.1.13) with

δ = 1, s = R + j, j = 1 , 2, . . ., q = 2, f ≡ 0, and summing from j = 1 to

j = ∞ it fol lows that

kuk

m+2,2,Ω

R+1

+ k∇τ k

m,2,Ω

R+1

≤ 3ckuk

1,2,Ω

(VI.1.19)

for all m ≥ 0. Since an analogo us estimate holds with Ω

in place of Ω

and

since u, τ ∈ C

∞

(Ω

), for all R > 0 (Ω

deﬁned in (VI.1 .8)), we deduce

u ∈ W

m,2

(Ω), for all m ≥ 0. (VI.1.20)

By using the embedding Theorem II.3.4 along with (VI.1.20) it is then easily

established that fo r each multi-index α with |α| ≥ 0, it holds that (see Exercise

VI.1. 2)

u(x)| → 0 as |x| → ∞ in Ω

. (VI.1.21)

Furthermore, by (VI.1.18)

and (VI.1.20) we deduce ∇τ ∈ W

m,2

(Ω) for all

m ≥ 0 and so

∇τ (x)| → 0 as |x| → ∞ in Ω

, (VI.1.22)

which completes the study of the asymptotic behavior.

The results obtained in this section can be summarized in

Theorem VI.1.2 Let Ω satisfy the assumptions stated at the beginning of

this section. Then, for every prescribed ﬂux Φ ∈ R, Leray’s problem admits

one and only one generalized solution v, p. This solution is in fact inﬁnitely dif-

ferentiable in the closure of every bounded subset of Ω and satisﬁes (VI.1.1 )–

(VI.1.2) in the ordinary sense. Furthermore, v, together with all its derivatives

of arbitrary order, tends to the corresponding Poiseuille velocity ﬁeld in Ω

as |x| → ∞ and the same property holds for ∇p.

Exercise VI.1.2 Let C be a semi-inﬁnite cylinder of type Ω

. Show that Theorem

II.3.4 holds for W

m,q

(C). Hint: L et C

= {x ∈ C : s < x

< s + 1}, s = 0, 1, 2 . . ., and

apply Theorem II.3.4 to W

m,q

). The general case follows by noticing that the

constants c

, c

, and c

entering the inequalities (II.3.17), (II.3.18) do not depend

on s.

378 VI Steady Stokes Flow in Domains with U nbounded Boundaries

Exercise VI.1.3 Assume that instead of two exits to inﬁnity, Ω

and Ω

, the

domain Ω has m ≥ 3 exits Ω

, . . . , Ω

, where Ω

, . . . , Ω

can be represented as Ω

(“upstream” exits) and Ω

j+1

, . . . , Ω

as Ω

(“downstream” exits). Assume also t hat

Ω − ∪

i=1

Ω

is bounded and that Ω is of class C

∞

. Denote by Φ

the ﬂuxes in Ω

. Then show

that, for every choice of Φ

satisfying the compatibility condition of zero total ﬂux

i=1

i=j+1

Leray’s problem is solvable in Ω.

Remark VI.1.1 As already noticed at the beginning of this chapter, when

the exits Ω

have a uniformly bounded but not necessarily constant cross sec-

tion, one does not know, in general, the explicit form of the limiting velocity

ﬁeld v

∞i

, as |x| → ∞ in Ω

. However, in such a case, one can alternatively pre-

scribe “growth” conditions at large distances (Ladyzhenskaya & So lonnikov

1980, Problem 1). For this type of q uestion we wish to mention the following

result, whose pro of can be found in the paper of Ladyzhenskaya & Solonnikov.



Theorem VI.1.3 Let

Ω = {x ∈ R

: x

∈ R, x

∈ Σ(x

)},

with Σ = Σ(x

) a simply connected domain of R

n−1

, possibly varying with

. Assume there exist two constants Σ

and Σ

such that

0 < Σ

≤ |Σ(x

)| < Σ

< ∞,

and that, in addition, there exists a ∈ W

1,2

loc

(Ω) such that

(i) ∇ · a = 0 in Ω;

(ii)

a · n = 1;

(iii) |a|

1,2,Ω

t,t+1

+ |a|

1,2,Ω

−t,−t+1

≤ c, for al l t ≥ 1,

where, f or s ∈ R,

Ω

s,s+1

= Ω ∩ {x ∈ R

: s < x

< s + 1} (VI.1.23)

and c is a constant independent of t. (Such a ﬁeld certainly exists if the

boundary ∂Ω is suﬃciently sm ooth.) Then, for any Φ ∈ R, there exists a pai r

v, p ∈ C

∞

(Ω) ∩ W

1,2

loc

(Ω)

solving the Stokes problem

Clearly, reasoning as in t he case of a constant cross section, if Ω is of class C

∞

then v, p ∈ C

∞

(Ω

), for all bounded Ω

⊂ Ω.

VI.2 Decay Estimates for Flow in a Semi-inﬁnite Straight Channel 379

∆v = ∇p

∇ · v = 0

)

in Ω

v = 0 at ∂Ω

v ·n = Φ

Furthermore, the velocity ﬁeld v satisﬁes for all t ≥ 1 and all s ∈ R the

estimates

Ω

∇v : ∇v ≤ c

Ω

s,s+1

∇v : ∇v ≤ c

(VI.1.24)

where

Ω

= Ω ∩ {x ∈ R

: |x

| < t}

and c

, c

are constants independent of t and s, respectively. Finally, i f w, π

is another solution corresponding to the same ﬂux Φ and satisfying a growth

condition of the type (VI.1.24)

, then w ≡ v, ∇π ≡ ∇p.

For a more general uniqueness result related to the above solutions, we

refer the reader to Exercise VI.2.2.

VI.2 Decay Estimates for Flow in a Semi-inﬁnite

Straight Channel

The next objective is to establish the rate at which solutions determined in

the previous section decay to the corresponding Poiseuille ﬂow. We shall show

that they decay exponentially fa st as |x| → ∞. This result will be achieved

as a corollary to a more general one holding for a large class of motions that

includes those determined in Theorem VI.1.2.

We shall restrict our attention to ﬂows occurring in a straight cylinder

Ω = {x

> 0} × Σ, where the cross section Σ is a C

∞

-smooth, bounded and

simply connected domain in R

n−1

, even though some of the results can be

extended to a more general class of domains; see Exercise VI.2.1. The cross

section at distance a from the origi n is denoted by Σ(a), despite all cross

sections having the same shap e and size. Denote by u, τ a solution to the

problem

∆u = ∇τ

∇ · u = 0

)

in Ω

u = 0 at ∂Ω − Σ(0 )

u · n = 0.

(VI.2.1)

380 VI Steady Stokes Flow in Domains with U nbounded Boundaries

For simplicity, we a ssume u, τ regular, that is, inﬁnitely diﬀerentiable in the

closure of any bounded subset of Ω. We also note, however, that the same

conclusions may be reached merely assuming u and τ to possess the same

regularity of generalized solutions to Leray’s problem. Our ﬁrst goal is to show

that every regular solution to (VI.2.1) with u satisfying a general “growth”

condition as |x| → ∞ has, in fact, square summable gradients over the whole

of Ω. Successively, we prove that these solutions decay exponentially fast i n

the Dirichlet integral, i.e.,

kuk

1,2,Ω

≤ ckuk

1,2,Ω

exp(−σR), (VI.2.2)

where

Ω

= {x ∈ Ω : x

> a}

and c, σ are constants depending on Σ. Once (VI.2.2) has been established, it

is easy to prove that u and its derivatives decay exponentially fast. Actually,

combining (VI.2.2) and (VI.1 .19) (with Ω

= Ω

) gives

kuk

m+2,2,Ω

R+1 + k∇τ k

m,2,Ω

R+1 ≤ c

kuk

1,2,Ω

exp(−σR/2) (VI.2.3)

and so, using the results of Exercise VI.1.2 into (VI.2.3), we obtain

u(x)| + |D

∇τ (x)| ≤ c

ku(x)k

1,2,Ω

exp(−σx

/2) (VI.2.4)

for every x ∈ Ω with x

≥ 1 and every |α| ≥ 0.

Remark VI.2.1 Estimate (VI.2.4) implies, in particular, that a s |x| → ∞,

τ(x) tends to some constant, exponentially fast. Actually, denoting by x =

, x

), y = (y

, y

) two arbitrary points in Ω

and applying the mean value

theorem, we have for some η

∈ Σ(y

)

|τ (x) −τ(y)| ≤



∂τ (x

, ξ)

∂ξ

dξ



n−1

i=1

τ(η

, y

)||x

− y

which, by (VI.2.4), implies that τ (x) tends to a constant τ

(say). Then, the

stated property follows from the identity

τ (x

, x

) = τ

∞

∂τ (x

, ξ)

∂ξ

dξ

and again from the estimate (VI.2.4). 

To recover the fundamental estimate (VI.2. 2) we need som e results con-

cerning diﬀerential inequalities which we are going to show.

Lemma VI.2.1 Let y ∈ C

) be a non-negative function sati sfy ing the

inequality

ay(t) ≤ b + y

(t), for a ll t ≥ 0, (VI.2.5)

VI.2 Decay Estimates for Flow in a Semi-inﬁnite Straight Channel 381

where a > 0, b ≥ 0. Then, if

lim inf

t→∞

y(t)e

−at

= 0, (VI.2.6)

it follows that y(t) is uniformly bounded and we have

sup

t≥0

y(t) ≤ b/a. (VI.2.7)

Proof. From (VI.2.5) it follows that

−

[y(t)e

−at

] ≤ be

−at

which, once integrated from t to t

(> t), furnishes

−y(t

−at

+ y(t)e

−at

≤

−at

− e

−at

If we take the inferior limit of both sides of this relation as t

→ ∞ and use

(VI.2.6), we then deduce (VI.2.7). ut

Lemma VI.2.2 Let β ≤ ∞ and let y be a real, non- negative continuous

function in [0, β) such that

y ∈ C

(0, β),

lim

t→β

y(t) = 0.

Then, if y sati sﬁes the integro-diﬀerential inequality

(t) + a

y(s)ds ≤ by(t), for all t ∈ (0 , β) (VI.2.8)

with a > 0 and b ∈ R,

it follows that

y(t) ≤ ky(0) exp(−σt), for all t ∈ (0, β) , (VI. 2.9)

where

k =

√

+ 4a

, σ =



+ 4a − b



Notice that the assumption b ≥ 0 is necessary for (VI.2.6) to hold.

Notice that if b < 0, (VI.2.8) at once implies (VI.2.9) with k = 1 and σ = −b.

382 VI Steady Stokes Flow in Domains with U nbounded Boundaries

Proof. Making the change of variable

ψ(t) = y(t)e

−bt

(VI.2.8) gives

(t) + a

−b(t−s)

ψ(s)ds ≤ 0.

From this relation, setting

F (t) = ψ(t) + δ

−b(t−s)

ψ(s)ds, δ > 0,

we recover

(t) + δF (t) = ψ

(t) + a

−b(t−s)

ψ(s)ds

+(δ

− δb − a)

−b(t−s)

ψ(s)ds ≤ 0

(VI.2.10)

provided we choose δ as the positive root to the equation

− δb − a = 0,

that is,

2δ = b +

+ 4a.

Integrating the diﬀerential inequality in (VI.2.10) furnishes

F (t) ≤ F (0)e

−δt

which can be equivalently rewritten as

y(t) + δ

y(s)ds ≤ F (0)e

−(δ−b)t

. (VI.2.11)

We now estimate F (0 ) in terms of y(0). From (VI.2 .11), setting

= 2δ − b,

it follows that

−

δt

y(s)ds

≤ F (0)e

−σ

which, upon integration from zero to β, gives

y(s)ds ≤ F (0)

1 −e

−σ

VI.2 Decay Estimates for Flow in a Semi-inﬁnite Straight Channel 383

If we substitute the value of F (0) into this last inequality we deduce

y(s)ds ≤ y(0)

1 − e

−σ

− δ(1 − e

−σ

)

and so we obtain

F (0) = y(0) + δ

y(s)ds ≤

y(0)σ

which along with (VI.2.1 1) completes the proof of the lemma. ut

We are now ready to show the main results of this section.

Theorem VI.2.1 Let u, τ be a regular solution to (VI. 2.1) with

lim inf

→∞

Σ(ξ)

∇u : ∇udΣ

dξ

−ax

= 0, (VI.2.12)

where

−1

≡



+ c



√

µ,

is the constant speciﬁed in (VI.2.14), and µ is the Poincar´e constant for Σ

(see (II.5.3) and (VI.2.15)). Then

|u|

1,2

< ∞.

Proof. Multiplyi ng both si des of (VI.2.1) by u and integrating by parts in

(0, x

) ×Σ we obtain

G(x

) ≡

Σ(ξ)

∇u : ∇udΣ

Σ(x

)



τu

−

∂u

∂x



−

Σ(0)



τu

−

∂u

∂x



If we integrate this relation from t to t + 1, t ≥ 0 , we have

t+1

G(x

)dx

Ω

t,t+1



τu

−

∂u

∂x



+ B, (VI.2. 13)

where Ω

t,t+1

is deﬁned in (VI.1.23) and

B ≡ −

Σ(0)



τu

−

∂u

∂x



Let us consider the problem

384 VI Steady Stokes Flow in Domains with U nbounded Boundaries

∇· ω = u

in Ω

t,t+1

ω ∈ W

1,2

(Ω

t,t+1

)

|ω|

1,2,Ω

t,t+1

≤ c

2,Ω

t,t+1

(VI.2.14)

Since

Ω

t,t+1

= 0,

from Theorem III.3 .1 and Lemma III.3.3 problem (VI.2.14) admits a solution

with a constant c

independent of t. Thus, from (VI.2.1 3) and (VI.2.1) it

follows that

t+1

G(x

)dx

Ω

t,t+1



−∇τ ·ω −

∂u

∂x



+ B

Ω

t,t+1



−∇u : ∇ω −

∂u

∂x



≤





kuk

2,Ω

t,t+1

|u|

1,2,Ω

t,t+1

+ B.

We next observe that, since u vanishes at ∂Ω, µ = µ(Σ) > 0 (the Poincar´e

constant for Σ) exists such that

kuk

2,Σ

≤ µk∇uk

2,Σ

; (VI.2.15)

see (II.5. 3). From (II.5. 5) and Exercise II.5.2 we m ay give the following esti-

mate for µ:

µ ≤











|Σ| if n = 3

(2d)

if n = 2.

By using thi s inequality, we obtain

y(t) ≡

t+1

G(x

)dx

≤

√





|u|

1,2,Ω

t,t+1

+ B.

Since

|u|

1,2,Ω

t,t+1

the preceding inequality furni shes

ay(t) ≤ b +

dy(t)

where a is deﬁned in the statement of the theorem. Thus, we recover that

y(t) satisﬁes (VI.2.5) with b = |B|. Furthermore, it is readily shown that, in

view of (VI.2.12), y(t) also satisﬁes (VI.2.6) and consequently Lemma VI.2.1

implies

VI.2 Decay Estimates for Flow in a Semi-inﬁnite Straight Channel 385

t+1

G(x

)dx

≤

|B|

, for all t > 1. (VI.2.16)

This inequality yields

` ≡ lim

→∞

G(x

) = |u|

1,2,Ω

< ∞.

Actually, since G(x

) is monotoni cally increasing in x

, ` exists (either ﬁnite

or inﬁnite) and (VI.2.16) then implies ` < ∞. The theorem is proved. ut

Remark VI.2.2 The previous theorem furnishes, in particular, that all reg-

ular solutions to (VI. 2.1) satisfying (VI.1.12) must decay to zero uniformly,

according to (VI.1.21) and (VI.1.22). This follows from (VI.1.19) and Exercise

VI.1. 2. 

Exercise VI.2.1 Let

Ω =

x ∈ R

: x

> 0, x

∈ Σ(x

)

with Σ(x

) a smooth, simply connected domain of R

n−1

, possibly varying with x

and satisfying the assumptions of Theorem VI.1.3. Assume Ω uniformly Lipschitz,

i.e., for every x

∈ ∂Ω there is B

) wi th r independent of x

such that ∂Ω∩B

)

is a boundary portion of class C

0,1

, with a Lipschitz constant independent of x

. Show

that Theorem VI.2.1 can be extended to such a domain Ω. Hint: It suﬃces to show

that problem (VI.2.14) is solvable with a constant c

independent of t. This fact,

however, can be established via the hypotheses on Ω and with the aid of estimate

(III.3.13).

Exercise VI.2.2 (Ladyzhenskaya & Solonnikov 1980). Let

Ω =

x ∈ R

: x

∈ R, x

∈ Σ(x

)

and suppose that Ω and Σ satisfy the same assumptions of Exercise VI.2.1. Show

that if u, τ is a regular solution to (VI.2.1)

1,2

vanishing at ∂Ω, having zero ﬂux

through Σ and satisfying (VI.2.12), then u ≡ ∇τ ≡ 0.

In the next theorem we establish the fundamental inequality (VI.2.2).

Theorem VI.2.2 Let u, τ be a regula r solution to (VI.2.1) satisfying (VI.2.12).

Then

|u|

1,2

< ∞

and, for all R > 0, the following inequality holds:

kuk

1,2,Ω

≤ ckuk

1,2,Ω

exp(−σR), (VI.2.17)

with

c =

2(c

+ 2)

1/2

+ 2)

1/2

− c

σ =

√

+ 2)

1/2

− c

where c

is the constant speciﬁed in (VI.2.14) and µ is the Poincar´e constant

for Σ. Moreover, f or all |α| ≥ 0, u, τ satisfy the pointwise estima te (VI.2.4).