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

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826 XII Two-Dimensional Flow in Exterior Domains

Exercise XII.4.1 Let v be a D-solution to (XII.0. 1) corresponding to f ≡ v

∗

≡ 0.

Show that, as |x| → ∞,

ω(x)| = o(|x|

−3/4

) , all |α| ≥ 1 . (XII.4.4)

Hint: Combining (XII.3.1), Theorem XII. 3.2 and the interior estimates for the

Laplace operator (see Exercise IV.4.4) we have, for suﬃciently large |x|,

|ω|

`+2,B

(x)

≤ c kωk

2,B

(x)

all ` ≥ −1 , (XII.4.5)

with c depending on ` and v.

Moreover, show that

v(x)| = o(|x|

−3/4

log |x|) , all |α| ≥ 1 . (XII.4.6)

Hint: Equation (XII.0.1)

, with f ≡ 0, can be rewritten as follows

∆v = ω × v + ∇Φ ,

where v = (v

, v

), ω := ωe

, and Φ is deﬁned in (XII.3.42), so that

∆(D

v) = (D

ω) × v + ω × (D

v) + ∇(D

Φ) .

From Theorem IV.4.1, Theorem IV.4.4, Theorem XII.3.2, and Remark IV.4.1 it

follows that

m+2,2,B

(x)

≤ c (kωk

m+1,2,B

(x)

+ kD

2,B

(x)

) .

The desired estimate is a consequence of t his latter, and of (XII.4.4), (XII.4.3).

Finally, denoted by p = p(x) the pressure ﬁeld associated to v, show that

p(x)| = o(|x|

−3/4

log |x|) , all |α| ≥ 1 . (XII.4.7)

Our next objective is to show that if (XII.3.67) holds uniformly, for some

6= 0, then the vorticity ω a nd all its deriva tives decay exponentially fast

outside any sector that excludes the l ine {x ∈ R

: x = λv

, λ > 0}. To

this end, assume, without loss of generality, v

= −e

, and, wi th the origin

of coordinates in

◦

Ω

, set

Ω

,σ



x =



= r cos θ, x

= r sin θ



∈ Ω : r ≥ R

, |π − θ| ≥ σ



where σ > 0 and R

> δ(Ω

We have the foll owing result, due, basi cally, to Ami ck (1988, §2.5).

Theorem XII.4.1 Let v be a D-solution to (XII.0.1) with f of bounded

support. Furthermore, suppose tha t v satisﬁes (XII.3.67) uniformly, with v

−e

. Then, for any σ > 0 there exist positive numbers R

, C and γ depending

on σ and v, such that

A much more detailed pointwise decay estimate for the vorticity wi ll be provided

in Theorem XII.8.4.

XII.4 Decay of the Vorticity and its Consequences 827

|ω(x)| ≤ C exp(−γ|x|) , for all x ∈ Ω

,σ

. (XII.4.8)

Therefore, from (XII.4.5), (XII.4.8) and the embedding Theorem II.3.4 it fol-

lows, in addition,

ω(x)| ≤ C

exp(−γ| x|) , for all |α| ≥ 0 and all x ∈ Ω

,σ

, (XII.4.9)

where, this time, C

depends also on α.

Proof. In order to prove (XII.4.8), it is suﬃcient to prove its validity in the

following three sector-like regions:

= {x ∈ Ω : |x

| ≤ κ

, x

> M

}

= {x ∈ Ω : |x

| ≤ κ

, x

> M

}

= {x ∈ Ω : |x

| ≤ κ

(−x

), x

< −M

}

with κ

> 0 arbitrarily large, and M

> 0 suﬃciently l arge, i = 1, 2, 3. We

begin with the region S

. Consider (XII.3.1 ) in the domain Ω

, with ρ so large

that supp (f) ⊂ B

. Then, multiplying both sides of (XII.3 .1) in Ω

by ω, we

ﬁnd

(ω

)

+ (ω

)

= (v

)

+ (v

)

+ 2|∆ω|

, (XII.4.10)

where we used the notation (·)

:= ∂(·)/∂ξ, (·)

ξη

:= ∂

(·)/∂ξ∂η. Integrating

the previous relation ﬁrst over x

between x

(> M

) and ∞, then over x

R, and taki ng i nto account Theorem XII.3.2 a nd Lemma XII.3.9, we get

−

, x

)dx

= −

, x

)dx

+ 2

∞

|∆ω|

≥ −

, x

)dx

In view of the assumptions on v, we can ﬁnd M

> 0 suﬃciently l arge, such

that |v

(x)+1| < 1/2, for all x

> M

and all x

∈ R. Therefore, the preceding

inequality furnishes

, x

)dx

≤ −

, x

)dx

, for all x

> M

which, in turn, delivers

, x

)dx

≤ c

exp(−x

/2) for all x

> M

where c

= c

(ω). Integrating this latter over the interval (x

−2, x

+ 2), we

deduce, in particular,

kωk

2,B

(x)

≤ c

exp(−x

/2) for all x

> M

828 XII Two-Dimensional Flow in Exterior Domains

from which, observing that fo r x ∈ S

we have |x| ≤ x

+ 1, the validi ty of

(XII.4.8 ) in S

follows with the help of (XII.4.5) and the embedding Theorem

II.3.4. We shall next prove (XII.4.8 ) in the region S

. We begin to observe that,

in view of the assumption on v, for any ε > 0 we can ﬁnd M

= M

(ε) > 0

such that

|v(x) + e

| < ε for all x

∈ R, x

> M

. (XII.4.11)

For β > 0, we introduce the new variables

= x

+ βx

, ξ

= x

so that (XII.4.10) furnishes, in pa rticular,

(1 + β

)(ω

)

+ (ω

)

+ 2β(ω

)

≥ [(βv

+ v

)ω

]

+ (v

)

Integrating both sides of this relation ﬁrst over ξ

between ξ

(> M

) and

∞, then over ξ

in R, and taking into account Theorem XII.3.2 and Lemma

XII.3.9, we obtain

−(1 + β

)

dξ

( ξ

, ξ

)dξ

≥ −

(βv

+ v

)ω

(ξ

, ξ

)dξ

= β

(ξ

, ξ

)dξ

−

[β(v

+ 1) + v

]ω

(ξ

, ξ

)dξ

Using (XII.4.11), we deduce

−

dξ

(ξ

, ξ

)dξ

≥

β − (β + 1)ε

1 + β

(ξ

, ξ

)dξ

, for all ξ

> M

which, upon integration, al lows us to conclude

(ξ

, ξ

)dξ

≤ c

exp(−γξ

) , γ :=

β − (β + 1)ε

1 + β

, ξ

> M

(XII.4.1 2)

where c

= c

(ω) > 0 and γ > 0 for suﬃciently small ε. Fix x = (x

, x

) with

> M

, and, for δ > 0 small enough, set

(x) = {y ∈ R

; |y

− x

| < δ, |y

−x

| < δ}.

From (XII. 4.12) we thus have, wi th ξ

= x

+ βx

kωk

2,B

(x)

≤ kωk

2,C

(x)

≤

+(1+β)δ

−(1+β)δ

+δ

−δ

(η

, η

)dη

dη

≤ c

exp(−γξ

) = c

exp[−γ(x

+ βx

)] , x

> M

, x

∈ R.

(XII.4.1 3)

We now observe that x

+ βx

≥ x

−β|x

|, and since in S

it is |x

| ≤ κ

with the choice β = 1/(2κ

), we infer

XII.4 Decay of the Vorticity and its Consequences 829

|x| ≤ x

1 + κ

≤ 2(x

+ βx

)

1 + κ

, x ∈ S

Coupli ng this info rmation with (XII.4.13), (XII.4.5) and the embedding The-

orem II.3 .4, we conclude the validity of the decay estimate in (XII.4.8) for all

x ∈ S

. The proo f that the same estimate holds in S

is completely analo-

gous to tha t just furnished for S

and it is left to the reader. The theorem is

therefore completely proved. ut

An imm ediate and important consequence of Theorem XII.4.1 is described

in the following.

Corollary XII.4.1 Let the assumptions of Theorem XII.4.1 be satisﬁed.

Then, the function

Φ := Φ +

, with Φ deﬁned in (XII.3.42) satisﬁes the

following pointwise decay

Φ(x)| ≤ C

exp(−C

|x|) for all x ∈ Ω

,σ

with C

, i = 1, 2, independent of x.

Proof. By assumption and by Theorem XII.3. 3 (with p

= 0, for sim plicity

and without loss of generality) we have

lim

|x|→∞

Φ(x) = 0 , uniformly . (XII.4.14)

Moreover, by an elementary calculation that uses (XII.0.1)

and (XII.3.2) we

ﬁnd in Ω

∂

∂x

∂ω

∂x

−v

∂

∂x

= −

∂ω

∂x

+ v

ω .

(XII.4.1 5)

Consequently, from Theorem XII.4.1 we obtain

|∇

Φ(x)| ≤ c

exp(−c

|x|) , for all x ∈ Ω

,σ

, (XII.4.16)

with c

, i = 1, 2, independent of x. Now, for every x = (θ, r), and y = (θ, r

)

in Ω

,σ

, with r

> r, we have

Φ(x)| ≤ |

Φ(y)| +



∂

∂ρ



dρ ,

so that, by (XII.4.16), we ﬁnd

Φ(x)| ≤ |

Φ(y)| + c

exp(−c

|x|) , for all x ∈ Ω

,σ

The result then follows by letting |y| → ∞ in this relation and employing

(XII.4.1 4).

830 XII Two-Dimensional Flow in Exterior Domains

Exercise XII.4.2 Let the assumptions of Corollary XII.4. 1 hold. Show that

Φ(x)| ≤ c

exp(−c

|x|) for all α| ≥ 0, x ∈ Ω

,σ

with c

and c

independent of x. Hint: Use (XII .4.15) along with Theorem XII.3.2

and Theorem XI I.4.1.

We shall next furnish for the function

Φ a weighted-L

property ho lding

in the whole of Ω

. Precisely, we have the fol lowing result due to Sazonov

(1999, §4).

Theorem XII.4.2 Let the assumption of Corollary XII.4.1 hold. Then, for

all ε ∈ (0, 1),

Ω

Φ|

|x|

1+ε

< ∞. (XII.4.17)

Proof. We begin to o bserve tha t, in order to show (XII.4.17), it is enough to

show the following

Ω

∩{x

<a}

Φ|

1+ε

< ∞, some a < 0 . (XII.4.18)

In fact, since for all x in the sector-like region S

:= {x ∈ Ω : |x

| ≤

c|x

|, x

< a} it is |x| ≤ |x

√

1 + c

, inequality (XII.4.18) implies

Ω

∩S

Φ|

|x|

1+ε

< ∞,

which, in turn, in view of Corollary XII.4.1, secures (XII.4.17). In order to

prove (XII.4.1 8), we begin to consider the sequence of problems (N ∈ N,

N ≥ N

> R

)

∆

− v · ∇

= ω

in Ω

(N)

:= Ω



∂Ω



∂Ω



∂B

= 0

(XII.4.1 9)

It is not ha rd to show that, fo r each N ≥ N

, problem (XII.4.19) has a unique

solution

Φ ∈ C

∞

(Ω

(N)

); see Exercise XII.4.3. Setting φ

Φ −

, from

(XII.3.4 3) and (XII.4.19) it follows tha t

∆φ

− v · ∇φ

= 0 in Ω

(N)



∂Ω

= 0 , φ



∂B

(XII.4.2 0)

Actually, the result holds for all ε > 0, but it will be used (and useful) only for ε

in the stated range.

XII.4 Decay of the Vorticity and its Consequences 831

Therefore, applying the two-sided maximum principle (Protter & Weinberger

1967, pp. 61-64) we ﬁnd

:= mi n



min

∂B

Φ, 0



≤ ζ

(x) ≤ max



max

∂B

Φ, 0



:= m

, x ∈ Ω

(N)

(XII.4.2 1)

In view of (XII.4.14), for any ε > 0, we can ﬁnd N = N(ε) ∈ N such tha t

(XII.4.2 1) holds with m

≥ −ε, m

≤ ε, and all N ≥ N. Consequently,

extending

to zero outside B

, and continuing to denote by

such

extension, we conclude

∞,Ω

≤ M ,

lim

N→∞

(x) =

Φ(x) , uniformly in x ∈ Ω

(XII.4.2 2)

where M > 0 i s independent of N. We next write

= ζ

+ψ

Φ, where ψ is a

smooth “cut-oﬀ” function that is 1 for |x| ≤ R

+ δ and is 0 for |x| ≥ R

+ 2δ,

with δ suﬃciently small. Taking into account that

Φ satisﬁes (XII.3.42), we

may rewrite (XII.4.19) as follows

∆ζ

−v ·∇ζ

= g in Ω

(N)



∂Ω

= ζ



∂B

= 0

, (XII.4.2 3)

where g is a smooth function of compact supp ort independent of N . If we

multiply through both sides of (XII.4.23)

by ζ

, integrate by parts over

Ω

(N)

, use (XII. 4.23)

, and recall that ∇ · v = 0, we at once obtain

|ζ

1,2

= −(g, ζ

) ,

so that, by (XII.4.22)

and by the f act that g ∈ L

(Ω

), we deduce

|ζ

1,2

≤ C , (XII.4.24)

with C indep endent of N . Next, for a < 0, let

f(x) =











ln(−a)

if x

≥ a

ln(−x

)

if x

≤ a

Multiplying through both sides of (XII.4.23)

by f ζ

, integrating by parts

over Ω

(N)

, and taking i nto account (XII.4.23)

and ∇· v = 0, we easily ﬁnd

kf∇ζ

Ω

(N)

∩{x

<a}

|ζ

|ln

≤ (ζ

∇ζ

, ∇f) + (g, ζ

)

Ω

(N)

∩{x

<a}

|ζ

|v + e

|ln

(XII.4.2 5)

832 XII Two-Dimensional Flow in Exterior Domains

By the C auchy inequal ity (II.2.5), the properties of f and (XII.4.24) we have

(ζ

∇ζ

, ∇f) ≤

Ω

(N)

∩{x

<a}

|ζ

|ln

+ c

|ζ

1,2

≤

Ω

(N)

∩{x

<a}

|ζ

|ln

+ c

with c

independent of N . Moreover, in view of (XII.4.22)

(g, ζ

) ≤ c

with c

independent of N . Finally, by virtue of the assumptions made on v,

we can choose |a| (and, consequently, N) so l arge as

Ω

(N)

∩{x

<a}

|ζ

|v + e

|ln

≤

Ω

(N)

∩{x

<a}

|ζ

|ln

Employing all the above information into (XII.4.25), we recover

Ω

(N)

∩{x

<a}

|ζ

|ln

≤ c

where c

does not depend on N. Recalling the deﬁnition of ζ

, this latter

inequality delivers at once

Ω

(N)

∩{x

<a}

|ln

≤ c

(XII.4.2 6)

with c

independent of N. Fix r > R

and take N > r. Then (XII.4.26)

implies

Ω

∩{x

<a}

|ln

≤ c

so that, letting N → ∞ in this relation and using (XII.4.2 2), by the Lebesgue

domi nated convergence theorem we deduce

Ω

∩{x

<a}

Φ|

|ln

≤ c

which, in turn, by the arbitrariness of r (and the regularity of

Φ) implies

(XII.4.1 8). The proo f of the theorem is thus complete. ut

Exercise XII.4.3 Show that, for each N ≥ N

, problem (XII.4.19) has one and

only one solution

∈ C

∞

(Ω

(N)

). Hint : Use Galerkin method along with the lo cal

regularity estimates for t he Poisson equation given in Exercise IV.4.4 and Exercise

IV.5.3.

XII.4 Decay of the Vorticity and its Consequences 833

The weighted-L

summability property of

Φ that we have just proved

allows us to deduce an analogous property for v + e

. No tice that, being v a

D-solution, we have (cf. (I I.6.14))

Ω

|v + e

|x|

ln |x|

< ∞,

where ρ > δ(Ω

). However, we are able to obtain a much stronger summa bility

property, as shown in the following theorem due to Sazonov (1999, Lemma

4).

Theorem XII.4.3 Let the assumption of Theorem XI I .4.1 hold. Then, for

all ε ∈ (0, 1),

Ω

|v + e

|x|

1+ε

< ∞. (XII.4.27)

Proof. The proof is based on a suitable representation of the velocity ﬁeld v in

terms of the total head pressure

Φ. This latter is more simply (and elegantly)

obtained i f we rewrite the linear momentum equation (XII.0.1)

in terms of the

complex variable forma lism. Let z = x

+i x

, u(z) := v

, x

)+i v

, x

)

(i :=

√

−1), and deﬁne the operators

∂

∂z



∂

∂x

∂

∂x



∂

∂z



∂

∂x

−

∂

∂x



Since

∂

∂z∂z

= ∆v

+ i ∆v

∂

∂z

∂

∂x

+ i

∂

∂x

∂

∂z

− 1) = v

ω − i v

ω ,

from (XII.4.15) and (XII.3.2) we deduce that (XII.0.1)

can be rewritten, in

Ω

, as follows

∂

∂z



∂u

∂z

− (u

− 1)



= 2

∂

∂z

. (XII.4. 28)

Pick y = (y

, y

) ∈ Ω

, and set ζ = y

+ i y

. We recall the following

generalized Cauchy integral formula

w(ζ) = −

2πi

∂A

w(z)

z − ζ

dz −

∂w

∂z

z − ζ

, (XII.4.29)

where A is a bounded subdoma in of Ω

, ζ ∈ A, and w ∈ C

(A). The proof of

(XII.4.2 9) is based on the (conjugate form of the) well-known Stokes formula:

See footnote 4.

834 XII Two-Dimensional Flow in Exterior Domains

∂A

w(z) dz = 2i

∂w

∂z

(XII.4.3 0)

and on the fact that 1/z is analytic for z 6= 0; see, e.g., H¨ormander (1966,

Theorem I.2.1) for details. Choosing in (XII.4.2 9) w = 4∂u/∂z −(u

−1) and

A = Ω

, we obtain

Ω

∂

∂z



∂u

∂z

− (u

− 1)



(z)

z − ζ

= (u

(ζ) − 1) − 4

∂u

∂z

(ζ)

−

2πi

∂Ω

4∂u/∂z(z) − (u

− 1)(z)

z − ζ

dz .

(XII.4.3 1)

Moreover, using (XII.4.30) with A = Ω

,r,η

:= Ω

− B

(y) and w =

Φ(z)/(z −ζ), we ﬁnd

Ω

,r,η

(∂

Φ/∂z)(z)

z − ζ

Ω

,r,η

Φ(z)

(z − ζ)

2πi

∂Ω

Φ(z)

z − ζ

dz +

2πi

|ζ−z|=η

Φ(z)

z − ζ

(XII.4.3 2)

We would like to let η → 0 in this latter relation. We begi n to notice that,

setting ξ = ξ

+ iξ

, it i s

(ξ)

|ξ|



−ξ

|ξ|

− 2i

|ξ|



, (XII.4.3 3)

which shows that 1/(ξ)

is a si ngular kernel, and, therefore, by Theorem

II.11.4, we have

lim

η→0

Ω

,r,η

Φ(z)

(z − ζ)

Ω

Φ(z)

(z − ζ)

in the P.V. sense, for a.a. ζ ∈ Ω

(XII.4.3 4)

Furthermore, with the parameterization z = ζ + η e

iθ

, θ ∈ [0, 2π], we ﬁnd, in

view of the continuity of

Φ,

|ζ−z|=η

Φ(z)

z − ζ

dz =

2π

Φ(ζ + η e

iθ

) e

2iθ

dθ = o(1) as η → 0 .

From this equation, (XII.4.32) and (XII.4.33) we thus deduce, for a.a. ζ ∈

Ω

XII.4 Decay of the Vorticity and its Consequences 835

Ω

(∂

Φ/∂z)(z)

z − ζ

Ω

Φ(z)

(z − ζ)

2πi

∂Ω

Φ(z)

z − ζ

dz ,

(XII.4.3 5)

where the ﬁrst integral on the right-hand side of (XII.4 .34) is understood in

the P.V. sense. Combining (XII.4.28), (XII.4.31) and (XII.4.35) we infer

(ζ) −1 = 4

∂u

∂z

(ζ) +

Ω

Φ(z)

(z − ζ)

2πi

∂Ω

4∂u/∂z(z) − (u

−1)(z)

z − ζ

dz +

πi

∂Ω

Φ(z)

z − ζ

(XII.4.3 6)

We ﬁnally let r → ∞ in (XII.4.36). Employing the properties

v(x) + e

, ∇v(x) ,

Φ(x) → 0 , uniformly as |x| → ∞

and taking into account that 1/|z − ζ| = O(|z|

−1

) as |z| → ∞ for each ﬁxed

ζ ∈ Ω

, by letting r → ∞ into (XII.4.36) we at once reach the desired

representation:

(ζ) −1 =

Ω

Φ(z)

(z − ζ)

+ 4

∂u

∂z

(ζ)

2πi

∂Ω

4∂u/∂z(z) − (u

− 1)(z)

z − ζ

dz +

πi

∂Ω

Φ(z)

z − ζ

dz ,

Ω

Φ(z)

(z − ζ)

+ F (u)(ζ)

(XII.4.3 7)

for a.a. ζ ∈ Ω

and where, a gain, the integral on the right-hand side of

the last line in (XII. 4.37) is interpreted in a suitable sense that will be ma de

clear next. In fact, in view of the properties (XII.4.33) of the kernel 1/(ξ)

Theorem XII.4.2 and Stein’s Theorem II.11.5, the integral transform

S (

Φ) :=

Ω

Φ(z)

(z − ζ)

is well-deﬁned (in the P.V. sense) and, for all ε ∈ (0 , 1), it satisﬁes

|x|

(1+ε)/2

S (

Φ) ∈ L

(Ω

) . (XII.4.38)

Since, for each ﬁxed z ∈ ∂Ω

, 1/|z − ζ| = O(|ζ|

−1

) as |ζ| → ∞, and since v

is a D-solution, we easily establish