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

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

With the help of Theorem XIII.2.1 we can readily obtain conditions under

which solutions determined in Theorem XIII.3.2 are unique. Actually, com-

bining Theorem XIII.2.1 with (XIII.3.2) we immediately ﬁnd the foll owing

result.

Theorem XIII.3.3 Let v be a generalized solution to Leray’s problem con-

structed in Theorem XIII.3.2, corresponding to ﬂux Φ. If





1 +



+ 1



|Φ| + C



1 +



|Φ|

with c

and C given i n Theorem XIII.2.1 and T heorem XIII.3.2, respectively,

then v is the only generalized solution corresponding to Φ.

XIII.4 Decay Estimates for Steady Flow i n a

Semi–Inﬁnite Straight Channel

Our objective in this section is to establish the rate at which solutions de-

termined in Theorem XIII.3.2 decay to the corresponding Poiseuille velo ci ty

ﬁelds. We shall show that, as |x| → ∞, they decay pointwise and exponentially

fast. As in the l inear case, this will follow as a corollary to a more general

result holding for a class of solutions wider than that determined in Theorem

XIII.3.2. More generality regards, essentiall y, the behavior at inﬁnity, while a

restriction on Φ o f the type (XIII.3.1) is always needed.

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

Ω = {x

> 0} × Σ,

where the cross section Σ is a C

∞

–smooth, bounded, and simpl y connected

domain in R

n−1

(n = 2, 3). However, some o f the results we ﬁnd can be

extended to cover more general situations. The cross section at distance a

from the origin will be denoted by Σ(a), despite all cross sections having the

same shape and size. Let v

= v

) be the vector ﬁeld associated with the

Poiseuille ﬂow in Ω and corresponding to ﬂux Φ. Further, let u , τ be a smooth

solution

to the following boundary-value problem:

ν∆u = u ·∇u + u · ∇v

+ v

· ∇u + ∇τ

∇ · u = 0

)

in Ω

u = 0 at ∂Ω − {x

= 0}

= 0.

(XIII.4.1)

For simplicity, we assume u ,τ smooth, that is, indeﬁnitely diﬀerent iable in the

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

may be reached merely by assuming that u and τ possess a priori the same

regularity of generalized solutions and then employing Theorem XIII. 1.1.

XIII.4 Decay Esti mates for Steady Flow in a Semi–Inﬁnite Straight Channel 917

Remark XI II.4.1 If v is a generalized solution to Leray’s problem, then

u ≡ v − v

(i)

satisﬁes (XII I .4.1) with Ω ≡ Ω

, Σ ≡ Σ

and v

≡ v

(i)

. 

We begin to show some local estimates for problem (XIII.4.1). To this end,

for R > 0 we let

Ω

= {x ∈ Ω : x

> R}.

Lemma XIII.4.1 Let u , τ be a smooth solution to (XIII.4.1) with

|u|

1,2

≤ M < ∞.

Then for every m ≥ 0 and R ≥ 0, the following estimate holds

kuk

m+2,2,Ω

R+1 + k∇τk

m,2,Ω

R+1 ≤ ckuk

1,2,Ω

R (XIII.4.2)

where c = c(n, m, M, Σ, Φ, ν).

Proof. Throughout the proof, the symbol c

, i = 1, 2, . . ., denotes a generic

quantity depending, at most, on m, M, Σ, and Φ. Let

0 < ε <

m + 2

. (XIII.4.3)

By (XIII.2.2)

and assumption, for all k = 0, 1, 2, . . ., and all R

≥ 0 we

derive

ku · ∇uk

3/2,Ω

+k,R

+k+1

≤ kuk

6,Ω

+k,R

+k+1

|u|

1,2,Ω

+k,R

+k+1

≤ c

|u|

1,2,Ω

+k,R

+k+1

and so, summing over k we obtain, in particular,

ku · ∇uk

3/2,Ω

≤ c

|u|

1,2,Ω

for a ll R

≥ 0.

In a similar fashion, from the H¨older inequality, the regularity of v

, and

inequality (XIII.2.4) we readily ﬁnd

· ∇u + u · ∇v

3/2,Ω

≤ c

|u|

1,2,Ω

for all R

≥ 0.

Thus, setting

f = u · ∇u + v

·∇u + u · ∇v

it follows that

kfk

3/2,Ω

≤ c

|u|

1,2,Ω

, for all R

≥ 0. (XIII.4 .4)

We give a proof that applies equally to both cases n = 2 and 3. However, f or

n = 2 the proof could be simpliﬁed.

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

Employing (VI.1.13) with s = R

+ k, k = 0 , 1, 2, . . ., δ = ε and taking into

account (XIII.4.4), af ter summing over k we ﬁnd

kuk

2,3/2,Ω

≤ c

kuk

1,2,Ω

−ε for all R

≥ 1.

From this relation and the ﬁrst part of Lemma XIII.2.1 it then follows that

sup

Ω

|u| ≤ c

. (XIII.4.5)

Thus,

ku ·∇uk

2,Ω

≤ c

kuk

1,2,Ω

which, by the regularity properties of v

, in turn implies

kf k

2,Ω

≤ c

|u|

1,2,Ω

. (XII I.4.6)

We next employ (XIII. 4.5) together with (VI.1.13) calculated for s = R

+ k,

k ∈ N, and δ = ε. Summing over k and tak ing into account (XIII.4.5) we

deduce

kuk

2,2,Ω

+ k∇τk

2,Ω

≤ c

|u|

1,2,Ω

−ε , (XIII.4.7)

for a ll R

such that

≥ R

+ ε. (XIII.4.8)

We now cho ose

= R + ε , R

= R + 1. (XIII.4.9)

Since ε obeys (XIII.4.3) (wi th m = 0 ), it follows that (XIII.4.7) is satisﬁed

and that R

− ε = R + 1 − ε ≥ R. Therefore, (XIII.4 .7)–(XIII.4.9) prove

(XIII.4.2) for m = 0. Iterating this m ethod, it is possible to show the validity

of (XIII.4.2) for all m ≥ 0. We show this for m = 1, leaving to the reader

the proof of the general iterative procedure. By assumption, the ﬁrst part of

Lemma XIII.2.1, and (XIII.4.5) it readily follows that

|u · ∇u|

1,2,Ω

≤ |u|

1,4,Ω

+ c

|∇u|

1,2,Ω

≤ c

(kuk

2,2,Ω

+ kuk

2,2,Ω

In view of (XIII.4.7) and the assumptions on u we deduce

|u · ∇u|

1,2,Ω

≤ c

kuk

1,2,Ω

−ε .

Likewise, from the regularity properties of v

and (XIII.4.7) we ﬁnd

· ∇u + u · ∇v

1,2,Ω

≤ c

kuk

1,2,Ω

−ε .

As a consequence, we infer that

kf k

1,2,Ω

≤ c

kuk

1,2,Ω

−ε . (XIII. 4.10)

XIII.4 Decay Esti mates for Steady Flow in a Semi–Inﬁnite Straight Channel 919

We now employ (VI.1.13) with s = R

+ k, k ∈ IN and δ = ε. Summing over

k and taking into account (XIII.4.10), it follows that

kuk

3,2,Ω

+ k∇τk

1,2,Ω

≤ c

|u|

1,2,Ω

−ε , (XIII.4 .11)

for a ll R

such that

≥ R

+ ε (XIII.4.12)

where R

satisﬁes (XII I .4.8) with R

≥ 1. If we choose

= R + ε , R

= R + 2ε , R

= R + 1 ,

we see that, by (XIII.4.3) with m = 1, conditions (XIII.4.8) and (XIII.4.12) are

satisﬁed and, furthermore, R

−ε = R +ε > R. We then conclude the validity

of (XIII.4.2) with m = 1. The proof of the lemm a can be then considered

accomplished. ut

The conclusions of Lemma XIII.4.1 can be derived under much weaker as-

sumptions on the summability of ∇ u, provided that the ﬂux Φ is “suﬃciently

small.” To show this, we need a preliminary result concerning a diﬀerential

inequality.

Lemma XIII.4.2 Let y ∈ C

(IR

) be nonnegative with a nonnegative ﬁrst

derivative. Assume for all t > 0

ay(t) ≤ a

(t) + a

(t))

3/2

+ b (XI II.4.13)

where a is a positive constant while a

, a

, and b are nonnegative constants.

Then if

lim inf

t→∞

−3

y(t) = 0

we have

y(t) ≤ 2

b + 1

, for all t > 0. (XIII.4.14 )

Proof. From Young’s inequality (II.2.7) it foll ows that there is a c = c(a

)

such that for all t > 0

(t) ≤ c(y

(t))

3/2

+ b

where b

= b + 1. As a consequence, (XIII.4.13 ) yields

ay(t) ≤ d(y

(t))

3/2

+ b

(XIII.4.15)

with d = a

+ c. Assume (XIII.4 .14) is fal se, We can then ﬁnd t

> 0 such

that

y(t

) > 2

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

and, since y

(t) ≥ 0,

y(t) > 2

for all t ≥ t

. (XIII.4.16)

Employing (XIII.4.16) in (XI II.4.15) we o bta in

A(y(t))

2/3

≤ y

(t), for all t ≥ t

with

2/3

≡

> 0.

Integrating this inequality from t a nd t

> t, it follows that

1/3

)

−

1/3

(t)

≥



1 −



and so taking the lim inf as t

→ ∞ of both sides of this inequality we derive

lim inf

→∞

−3

y(t

) ≥





> 0,

which contradicts the assumption. The lemma is therefore proved. ut

The result just shown allows us to prove the following.

Lemma XIII.4.3 Assume u, τ is a smooth solution to (XIII.4.1) such that

lim inf

→∞

−3

(

Σ(t)

|∇u(x

, t)|

dΣ)dt = 0.

Assume, further, that the ﬂux Φ associated with the Poiseuille ﬂow v

satisﬁes

|Φ| <

λµ)

1/2

, (XIII.4.17)

where c

is the constant given in Exercise VI.0.1 and λ, µ are given in

(XIII.2.3) and (XIII.2.4), respectively. Then

Ω

∇u : ∇u < ∞.

Proof. Multiplyi ng (XIII.4.1)

by u and integrating by parts over (0, x

)×Σ,

we ﬁnd

Notice that we can always take d > 0.

XIII.4 Decay Esti mates for Steady Flow in a Semi–Inﬁnite Straight Channel 921

ν G(x

) ≡ ν

Σ(t)

∇u : ∇udΣ

Σ(x

)



−τu

∂u

∂x

−

+ v

)



−

Σ(0)



−τu

∂u

∂x

−

+ v

)



−

Σ(t)

u ·∇v

· udΣ

(XIII.4.18)

where v

= v

. By using the Schwarz inequality, (XIII.2.3), and (XIII.2.4)

we obtain



Σ(t)

u · ∇v

· udΣdt



≤

Σ(t)

|∇v

1/2

Σ(t)

dΣ

1/2

≤ (λµ)

1/2

Σ(t)

|∇v

1/2

G(x

)

= (λµc

)

1/2

|Φ|G(x

(XIII.4.19)

where, in the last step, we have used the results of Exercise VI.0.1. Assuming,

now, the validity of (XIII.4.17) and setting

γ = ν − (λµc

)

1/2

|Φ| (> 0), (XIII.4.20)

from (XIII.4.1 8) and (XIII.4.19), it follows that

γG(x

) ≤

Σ(x

)



−τu

∂u

∂x

−

+ v

)



+ b

with b a nonnegative quantity independent of x

. Integrating both sides of

this relation from t to t + 1 furnishes

t+1

G(x

)dx

≤

Ω

t,t+1



−τu

∂u

∂x

−

+ v

)



+b. (XIII.4. 21)

We wish to estimate the term involving τ . To this end, we proceed as in

the linear case; see Theorem VI.2.1. Speciﬁcally, denoting by ω a solution to

(VI.2.14), from (XIII.4.1) we ﬁnd

Ω

t,t+1

τu

= −

Ω

t,t+1

∇τ · ω

Ω

t,t+1

[ν∇u : ∇ω − u ·∇ω ·u

−u · ∇ω·v

− v

·ω · u] .

(XIII.4.22)

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

Taking into account the properties of ω, from Lemma XIII.2.1 and (XIII.2.4)

we ﬁnd



Ω

t,t+1

∇u : ∇ω



≤ c

|u|

1,2,Ω

t,t+1

kuk

2,Ω

t,t+1

≤ c

√

µ|u|

1,2,Ω

t,t+1



Ω

t,t+1

u · ∇ω · u



≤ c

kuk

4,Ω

t,t+1

kuk

2,Ω

t,t+1

≤ c

√

µ|u|

1,2,Ω

t,t+1



Ω

t,t+1

u · ∇ω·v



≤ c

bvkuk

2,Ω

t,t+1

|u|

1,2,Ω

t,t+1

≤ c

√

µ|u|

1,2,Ω

t,t+1



Ω

t,t+1

· ω ·u



≤ c

√

µ|u|

1,2,Ω

t,t+1

(XIII.4.23)

with

bv = max

)|.

Putting

= c

√

µ(ν + κ

+ 2bv),

from (XIII.4.2 1)–(XIII.4.23) we deduce

t+1

G(x

)dx

≤ c

|u|

1,2,Ω

t,t+1

−

Ω

t,t+1



∂u

∂x

− u

+ v

)



+ b.

(XIII.4.24)

We now observe that, again from (XI II.2.4) a nd Lemma XIII.2.1, it follows

that



Ω

t,t+1

∂u

∂x



≤ 2kuk

2,Ω

t,t+1

|u|

1,Ω

t,t+1

≤ 2

√

µ |u|

1,Ω

t,t+1



Ω

t,t+1



≤ bvµ|u|

1,Ω

t,t+1



Ω

t,t+1



≤ kuk

4,Ω

t,t+1

kuk

2,Ω

t,t+1

≤ κ

√

µ |u|

1,2,Ω

t,t+1

(XIII.4.25)

Replacing these latter inequalities into (XI II.4.24), we ﬁnd

t+1

G(x

)dx

≤ c

|u|

1,2,Ω

t,t+1

+ c

|u|

1,2,Ω

t,t+1

+ b (XIII. 4.26)

XIII.4 Decay Esti mates for Steady Flow in a Semi–Inﬁnite Straight Channel 923

where

= c

√

µ(ν +

√

µ)

√

µ.

Setting

y(t) =

t+1

G(x

)

and recalling the deﬁnition of G(x

), it follows that

(t) = |u|

1,2,Ωt,t+1

Therefore, i nequality (XIII.4.2 5) yields

γy(t) ≤ c

(t) + c

(t))

3/2

+ b

which, in turn, by Lemma XIII.4.2, implies

y(t) ≤ 2

b + 1

The result then follows from this estimate and an argument entirely analo gous

to that employed at the end of the proof of Theorem VI.2.1. ut

The next result establishes an exponential decay property of the type of

de Saint-Venant.

Lemma XIII.4.4 Let the assumptions of Lemma XIII.4.3 be satisﬁed. Then,

there are positive constants σ

= σ

(Σ, n, kuk

1,2,Ω

, Φ, ν), i = 1, 2, such that

kuk

1,2,Ω

R ≤ σ

kuk

1,2,Ω

exp(−σ

R), for all R > 0.

Proof. We notice that, in view of Lemma XIII.4.3, kuk

1,2,Ω

is ﬁnite. Multi-

plying (XIII.4.1)

by u and integrating by parts over (R, x

) ×Σ ≡ Ω

R,x

ﬁnd

Ω

R,x

|∇u|

Σ(x

)



−τu

∂u

∂x

−

+ v

)



−

Σ(R)



−τu

∂u

∂x

−

+ v

)



−

Ω

R,x

u · ∇v

· u .

(XIII.4.27)

From Lemma XIII.4 .1 and the embedding Theorem II.5.2 it follows, in par-

ticular, that

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

sup

∈Σ

(|u(x

, x

)| + |τ(x

, x

)|) → 0 as x

→ ∞,

and so, reasoning as in the proof of Theorem VI.2.2, we can show that, in

the limit x

→ ∞, the surface integral over Σ(x

) is vanishing. Thus, in this

limit, (XIII. 4.27) yields

νH(R) = −

Σ(R)



−τu

∂u

∂x

−

+ v

)



−

Ω

u ·∇v

· u,

(XIII.4.28)

with

νH(R) ≡ ν| u|

1,2,Ω

Proceeding as in (XIII.4.19), we can show



Ω

u · ∇v

· u



≤ (λµc

)

1/2

|Φ|H(R),

which, in view of (XIII.4.17), once replaced in (XII I.4.28) furnishes

γH(R) ≤ −

Σ(R)



−τu

∂u

∂x

−

+ v

)



with γ given in (XIII.4.20). We next integrate bo th sides of this relati on

between t + l a nd t + l + 1, l = 0, 1, 2, . . ., to obtain

t+l+1

t+l

H(R) ≤ −

Σ(t+l+1)

Σ(t+l)

Ω

t+l,t+l+1



τu

+ v

)



(XIII.4.29)

The volum e integral on the right-hand side of (XIII.4.29 ) can be increased

exactly as i n (XIII.4.22), (XIII.4 .23), and (XIII.4.25)

2,3

and so we deduce



Ω

t+l,t+l+1

τu

+ v

)



≤ c

|u|

1,2,Ω

t+l,t+l+1

(1 + |u|

1,2,Ω

t+l,t+l+1

(XIII.4.30)

with c

= c

(n, Σ, Φ). Col lecting (XIII.4.29) and (XIII.4.30) we derive

t+l+1

t+l

H(R) ≤ c

|u|

1,2,Ω

t+l,t+l+1

−

Σ(t+l+1)

Σ(t+l)

(XIII.4.31)

where c

also depends on |u|

1,2,Ω

. Summing both sides of (XI II.4.31) from

l = 0 to l = ∞ and taking into account that, by Remark XIII.1.1

lim

t→∞

Σ(t)

, t)dΣ = 0,

XIII.4 Decay Esti mates for Steady Flow in a Semi–Inﬁnite Straight Channel 925

we ﬁnd

∞

H(R) ≤ c

H(t) +

Σ(t)

. (XIII.4.32)

However, by (XIII.2.4),

Σ(t)

≤ µ

Σ(t)

|∇u|

= −µH

(t)

and so from (XIII.4.32) we arrive at

∞

H(R) ≤ c

H(t) −

µν

(t).

Using Lemma VI.2.2 into this inequality furnishes the desired result and the

proof of the lemma is complete. ut

From Lemma XIII.4.1, Lemma XIII.4.3, and Lemma XIII.4.4, and with

the help of Lemma XIII.2.1, we are able to deduce at once the foll owing main

result.

Theorem XIII.4.1 Let u , τ be a smooth solution to (XIII.4.1) satisfying

lim inf

→∞

−3

(

Σ(t)

|∇u(x

, t)|

dΣ)dt = 0.

Then, if Φ satisﬁes (XIII.4.17), it follows that

kuk

1,2,Ω

< ∞.

Moreover, there is a positive constant c

= c

(Σ, n, Φ, kuk

1,2,Ω

, ν) such tha t

u(x)| + |D

∇τ (x)| ≤ c

exp(−σ

) (XIII.4.33)

for every x ∈ Ω with x

≥ 1 and all |α| ≥ 0, and where σ

is given in Lemma

XIII.4.4.

This theorem, along with the uniqueness Theorem XIII.3.3 and the help

of Remark XIII.4.1 , immediately produces the following general result con-

cerning the asympto tic behavior of weak solutions to Leray’s probl em.

Corollary XIII.4.1 There exists a positive constant c = c(Ω, n) such that,

|Φ| < cν,

all generalized solutions v to Leray’s probl em (XIII.1.2)–(XIII.1.4) satisfy

the decay property (XIII.4.3 3) in each outlet Ω

, i = 1, 2 with u = v − v

(i)

and τ = p − C