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

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816 XII Two-Dimensional Flow in Exterior Domains

We are now in a position to establish the pointwise convergence of the

pressure at inﬁnity.

Theorem XII.3.3 Let v be a D-solution to (XII.0.1). Then, if for some ρ >

δ(Ω

)

∈ L

(Ω

), f ∈ L

(Ω

), ∇ · f ∈ L

(Ω

there is a p

∈ R such that

lim

|x|→∞

p(x) = p

uniform ly.

Proof. We indicate by p

(x) the diﬀerence p(x) − p

where p

is deﬁned in

Lemma XII.3.5. Clearly, v, p

is still a solution to (XII.0.1) and, in particular,

veriﬁes (XII.3.26). Let x = (2R, θ). Denoting by (r

, θ

) a po lar coordinate

system with the origin at x, from (XII.3.26) we have, after integration over r

and θ

(x) = p

) +

2π

(ρ)dρ

2π



(ρ, θ

) − v

(ρ)]

∂v

(ρ, θ

)

∂θ

−[v

(ρ, θ

) − v

(ρ)]

∂v

(ρ, θ

)

∂θ



dρdθ

(XII.3.3 8)

where the average is now meant with respect to the angle θ

. From the

Wirting er inequality (XII.3.28) and the Schwarz i nequality, it follows for

≤ R that





2π

(ρ, θ

) −v

(ρ)]

∂v

(ρ, θ

)

∂θ

dθ



dρ



≤

2π

|∇v(ρ, θ

ρdρdθ

≤ |v|

1,2,Ω

R,3R

(XII.3.3 9)

Likewise, one shows





2π

(ρ, θ

) −v

(ρ)]

∂v

(ρ, θ

)

∂θ

dθ



dρ



≤ |v|

1,2,Ω

R,3R

. (XII.3.40)

Moreover,



(ρ)dρ



≤ kf

/rk

1,Ω

R,3R

(XII.3.4 1)

and so, by (XII.3.38)–(XII.3.41), we recover

XII.3 On the Asymptotic Behavior of D-Solutions 817

2π|p

(x)| ≤ 2π|p

)| + 2|v|

1,2,Ω

R,3R

+ kf

/rk

1,Ω

R,3R

Multiplying both sides of this latter inequality by r

and integrating over

∈ [0, R] and θ

∈ [0, 2 π) we ﬁnd that

2π|p

(x)| ≤

4π

2π

, θ

dθ

+ 2|v|

1,2,Ω

R,3R

+ kf

/rk

1,Ω

R,3R

≤

4π

Ω

R,3R

|+ 2|v|

1,2,Ω

R,3R

+ kf

/rk

1,Ω

R,3R

≤ 16π max

R≤r≤3R

2π

(r, θ)|dθ + 2|v|

1,2,Ω

R,3R

+ kf

/rk

1,Ω

R,3R

Passing to the limit R → ∞ in this estimate and using Lemma XII.3.7, we

then complete the proof of the theorem.

It remains to investigate the behavior at inﬁnity of the velocity ﬁeld v. In

this respect, Lemm a XII.3.3 ensures that v cannot grow too fast at inﬁnity.

Under the simplifying assumption f = 0 (or, more generally, f of bounded

support in Ω) we shall show that, in fa ct, v is unif ormly bounded. To reach

the objective, we notice that, deﬁning the total head pressure ﬁeld Φ:

Φ := p +

|v|

, (XII.3.42)

by a simple calcula tion based on (XII.0.1) with f = 0 one shows

∆Φ −v ·∇Φ = ω

. (XII.3.43)

Consider (XII.3.43) in Ω

,ρ

, for arbitrary ρ

, ρ

with ρ

> ρ

and some

ﬁxed ρ

> δ(Ω

). Since v ∈ L

∞

loc

(Ω

), we may apply to it the classical max-

imum principle of Hopf (1952) and Ole

inik (1952) (cf. Protter & Weinberger

1967, pp. 61-64) to o bta in that Φ cannot attain a maximum in Ω

,ρ

unless

it is a constant. It also follows tha t

max

θ∈[0,2π)

Φ(r, θ)

has no maximum in Ω

,ρ

and hence it must be monotonous in Ω

, for

suﬃciently large ρ. Thus, we deduce that

lim

r→∞

max

θ∈[0,2π)



p(r, θ) +

|v(r, θ)|



= A,

implying, by Theorem XII.3.3,

that

We can assume, w ithout loss, that p

= 0.

818 XII Two-Dimensional Flow in Exterior Domains

lim

r→∞

max

θ∈[0, 2π)

|v(r, θ)| =

√

2A ≡ L. (XII.3.44)

However, we don’t know if L is ﬁnite or inﬁnite. As a consequence, the max-

imum principle is not enough to obtain the boundedness of v, and we need

more inform ation about the function Φ. In the particular case v

∗

≡ f ≡ 0 this

is achieved through the following profound result, due to C. J. Amick (1986,

Theorem 4(a); 1988, Theorem 11), which we shal l state without proof.

Lemma XII.3.8 Let v be a D-solution to (XII.0.1) corresponding to v

∗

≡

f ≡ 0. Then there exists a continuous, non-intersecting curve

γ : t ∈ [0, 1) → γ(t) ∈ Ω

such that

(i) γ(0) ∈ ∂Ω

;

(ii) |γ(t)| → ∞ as t → 1;

in addition, the function Φ is monotonically decreasing along γ, namely,

Φ(γ(t)) < Φ(γ(s)), for all s, t ∈ [0, 1), s <t. (XII. 3.45)

With this result in hand, we can show the following one (cf. Amick 1986,

Theorem 4(b) and 1988, Theorem 12).

Lemma XII.3.9 Let v and f be as in Lemma XII.3.8. Then

v ∈ L

∞

(Ω

and there is an L ∈ [0, ∞) such that

lim

|x|→∞

max

θ∈[0,2π)

|v(x)| = L uniformly. (XII.3. 46)

Proof. Since p(x) tends to zero f or large |x|, by (XII.3.45) we deduce tha t

|v(γ(t))| ≤ c, for all t ∈ [0, 1), (XII.3.47)

with c independent of t. Since v ∈ D

1,2

(Ω), we have

k+1

2π



∂v

∂θ



dθdr → 0, as k → ∞,

implying

2π



∂v( r

, θ)

∂θ



dθ → 0, as k → ∞, (XII.3 .48)

XII.3 On the Asymptotic Behavior of D-Solutions 819

for some sequence {r

} with r

∈ (2

, 2

k+1

). Since, by properties (iii) and (ii),

γ is connected and extends to inﬁnity, for any k ∈ N we can ﬁnd at least o ne

∈ [0, 1) such that

γ(t

) = (r

, θ

for som e θ

∈ [0, 2 π). Thus, in view of (XII.3.47), i t follows tha t

|v(r

, θ

)| ≤ c, for all k ∈ N. (XII.3.49)

From the identity

v(r

, θ) = v(r

, θ

) −

∂v(r

, τ )

∂τ

dτ,

(XII.3.4 8), and (XII.3.49) we ﬁnd

max

x∈∂B

|v(x)| ≤ c

, for all k ∈ N, (XII.3.50)

with c

independent of k. We next a pply the maximum principle to (XII.3.43)

in the annulus Ω

k+1

to ﬁnd

max

x∈Ω

k+1

Φ(x) ≡ max

x∈Ω

k+1



p(x) +

|v(x)|



≤ max

x∈∂B

∪∂B

k+1

Φ(x).

(XII.3.5 1)

However, by (XII.3.50 ) and Theorem XII.3.3 we deduce

max

x∈∂B

∪∂B

k+1

Φ(x) ≤ c

, for all k ∈ N

so that, again Theorem XII.3.3 and (XII.3.5 1) deliver

max

x∈Ω

k+1

|v(x)|

≤ c

, for all k ∈ N,

with a constant c

independent of k. Therefore, v ∈ L

∞

(Ω

). Since the second

part of the lemma is an immediate consequence of the ﬁrst and (XII.3.44),

the proof is complete. ut

We shall next investigate if the velocity approaches some vector v

inﬁnity. We have the following two possibilities:

(i) the number L in (XII.3.44) is zero;

(ii) the number L in (XII.3.44) belongs to (0, ∞].

In case (i) we have

lim

|x|→∞

v(x) = 0, uniformly,

and we deduce at once that v

= 0. On the other hand, if L > 0, using the

ideas of Gilba rg & Weinberger (1978, §5 ), we proceed as follows. First of a ll,

we need two preliminary l emmas.

Of course, by Lemma XII.3.9, if v

∗

≡ f ≡ 0 it follows that L < ∞.

820 XII Two-Dimensional Flow in Exterior Domains

Lemma XII.3.10 Let v and f be as in Lemma XII.3.8. Then

(i) lim

r→∞

2π

|v(r, θ) − v(r)|

dθ = 0,

(ii) lim

r→∞

|v(r)| = L,

where L is deﬁned in (XII.3.44).

Proof. By the Wirtinger inequality (XII.3.28) and the Cauchy inequality we

have



2π

|v(r, θ) − v(r)|

dθ



2π

(v − v) ·

∂v

∂r

dθ



≤

2π



r|∇v|

|v −v|



dθ

≤ c

2π

r|∇v|

dθ;

therefore,

lim

r→∞

2π

|v(r, θ) −v(r)|

= ` ∈ [0, ∞).

However, aga in by (XII.3. 28), we have

∞



2π

|v(r, θ) −v(r)|

dθ



dr < ∞,

which implies ` = 0, and (i) is proved. To show (ii), we observe that (XII.3.46)

implies that, given a ny sequence {r

} ⊂ R

with r

→ ∞, there is a corre-

sponding sequence {θ

} ⊂ [0, 2π) such that

lim

→∞

|v(r

, θ

)| = L. (XI I.3.52)

However, as in the proof of Lemma XII.3.8 , we show the exi stence of a sequence

∈ (2

, 2

k+1

) such that (XII.3.48) holds. Since

|v(r

, θ) − v(r

, θ

≤ 2π

2π



∂v(r

, τ )

∂τ



dτ,

by (XII.3.52) and the triangle inequality we ﬁnd that

lim

→∞

|v(r

, θ)| = L, uniformly in θ. (XII.3.53)

Moreover, since

2π

(v(r

, θ) − v(r

))dθ = 0 f or all k ∈ N,

XII.3 On the Asymptotic Behavior of D-Solutions 821

it follows that

|v(r

, θ) − v(r

≤ 2π

2π



∂v(r

, τ )

∂τ



dτ

which together with (XII.3.48 ) and (XII.3.53) al lows us to conclude that

lim

k→∞

|v(r

)| = L. (XII.3.54)

Now, for r ∈ (r

, r

k+1

) we have

|v(r) − v(r



2π

∂v

∂r

drdθ



≤

(2π)

k+1

2π

drdθ

Ω

|∇v|

≤

2π

log 2|v|

1,2,Ω

and so, in view of (XII.3.54 ), the property (i i) follows by letting k → ∞ in

this inequality.

Lemma XII.3.11 Let v be as in Lemma XII.3.2 and assume that the number

L in (XII.3.44) is ﬁnite. Then, i f for some ρ > δ(Ω

)

1/2

f ∈ L

(Ω

it follows that

1/2

∇ω ∈ L

(Ω

Proof. From the identity (XII. 3.5) with h ≡ ω

and ψ

replaced by η

= rψ

we deduce tha t

Ω

|∇ω|

Ω

(∆η

+ v · ∇η

) +

∂B



∂ω

∂n

− v · nω



Ω





∂ωη

∂x



− f



∂ωη

∂y



(XII.3.5 5)

By the properties (II.7.2) of ψ

and Lemm a XII.3.3, it follows that

|∇η

|+ |v · ∇η

| + |∆η

| ≤ c (XII.3.5 6)

for som e constant c independent of R. Furthermore,



Ω





∂ωη

∂x



− f



∂ωη

∂y





≤

Ω



|∇ω|

+ c

r|f|



(XII.3.5 7)

822 XII Two-Dimensional Flow in Exterior Domains

so that by (XII.3.56) and (XII.3.57), identity (XII.3.55) gives

Ω

|∇ω|

≤ C,

for a constant C independent of R. Letting R → ∞ and using the monotone

convergence theorem completes the proof. ut

We are now in a positio n to prove the following result on the behavior of

a D-solution at inﬁnity.

Theorem XII.3.4 Let v be a D-solution to (XII.0.1) with f of bounded

support in Ω and let L ∈ [0, ∞] be the number deﬁned in (XII.3.44). Then,

if L < ∞ (this certainly happens whenever v

∗

≡ f ≡ 0), there is a v

∈ R

such that

lim

r→∞

2π

|v(r, θ) − v

dθ = 0. (XII.3.58)

Furthermore, if v

= 0, we have

lim

|x|→∞

v(x) = 0 uniformly. (XII.3.59)

Finally, if L = ∞,

lim

r→∞

2π

|v(r, θ)|

dθ = ∞. (XII.3 .60)

Proof. Let ψ = ψ(r) be the argument of v(r), that is,

(r) = |v(r)|cos ψ(r)

(r) = |v(r)|sin ψ(r)

ψ ∈ [0, 2π). (XI I .3.61)

Clearly, we have

(r) =

−v

|v|

(XII.3.6 2)

where the prime means diﬀerentiation. Multiplying (XII.3.19)

by sin θ,

(XII.3.19)

by cos θ, and adding up, for suﬃciently large |x| we ﬁnd that

∂ω

∂r

+ v

∂v

∂r

−v

∂v

∂r

∂p

∂θ

= 0. (XII.3.63)

We take the average over θ of both sides of (XII.3.63) to deduce that

∂ω

∂r

+ v

− v

2π



(r, θ) − v

(r)]

∂v

(r, θ)

∂r

−[v

(r, θ) − v

(r)]

∂v

(r, θ)

∂r



dθ = 0.

(XII.3.6 4)

XII.3 On the Asymptotic Behavior of D-Solutions 823

From Lemma XII.3.10 we know that |v(r)| converges to L ≥ 0. Assume, for a

while, that L > 0. Then we may ﬁnd ρ > δ(Ω

) such that

|v(r)| > L/2, for all r > ρ. (XII.3.65)

We then divide both sides of (XII.3.64) by |v(r)|

and integrate over θ ∈ [0, 2π)

and over r ∈ (r

, r

), r

> r

> ρ, to obtain

ψ(r

) − ψ(r

) = −

2π

|v(r)|



∂ω

∂r

+ (v

−v

)

∂v

∂r

−(v

− v

)

∂v

∂r



drdθ.

Using (XII.3.65) and the Schwarz and the Wirtinger inequalities we see that

|ψ(r

) − ψ(r

)| ≤

πL

[kr

1/2

ωk

2,Ω

+ |v|

1,2,Ω

]

× [|v|

1,2,Ω



2π



1/2

]

and, therefore, letting r

, r

→ ∞ and recalling Lemma XII. 3.11, we obtain

lim

r→∞

ψ(r) = ψ

(XII.3.6 6)

for som e ψ

∈ [0, 2 π]. For L ≥ 0 we deﬁne the vector

= (L cos ψ

, L sin ψ

If L ∈ (0, ∞), from Lemma XII.3.10(ii), (XII.3.61), and (XII.3.66) we conclude

that

lim

r→∞

v(r) = v

which along with Lemma XII.3.10(i) implies (XII.3.58). If L = 0, we have

= 0 and (XII.3.59) follows from (XII.3.44). Fi nall y, if L = ∞, (XII.3.60)

follows directly from Lemma XII.3.10. Notice that, in view of Lemma XII.3.9,

this circumstance can not occur if v

∗

≡ f ≡ 0. The theorem is proved. ut

Remark XI I.3.1 The vector v

determined in Theorem XII.3.4 need not be

the vector v

∞

prescribed in (XII.0.2). The problem of the coincidence of v

and v

∞

therefore remains open. Furthermore, if v

= 0 and v

∗

≡ f ≡ 0,

we can conclude that v ≡ 0 only in the trivial case Ω = R

; see Theorem

XII.3.1). 

Remark XI I.3.2 Another question that is open is to ascertain if, in the case

6= 0, (XII.3.58) holds pointwise:

824 XII Two-Dimensional Flow in Exterior Domains

lim

|x|→∞

v(x) = v

. (XII.3.67)

C.J. Amick has shown the validity of (XII.3.67) in the class of symmetric

ﬂows. Precisely, let us call a solution v, p to (XII.0.1) symmetric around the

x-axis, or, more simply, symmetric, if p and v

are even in y and v

is odd

in y. It can be shown that if the x-ax is is of symmetry for

◦

Ω

, and v

∗

and

f possess the same symmetry as v does, under suitable regularity conditions

on the data, the class of symmetric D-solutions is not empty. If, i n particular,

∗

≡ f ≡ 0, then such solutions satisfy (XII.3.67) uniformly; cf. Amick (1988,

Theorem 27). If the ﬂow is not symmetric, the best one can say, so far, is that

(XII.3.6 7) hol ds in suitable large sectors. Namely, taking v

= (1, 0), for all

ε ∈ (0, π/2), we have

lim

r→∞

max

|θ|∈[ε,π−ε]

|v(r, θ) −v

| = 0;

provided that v

∗

≡ f ≡ 0; cf. Amick (1 988, Theorem 19 ). In any case, one can

show that the modulus of v tends p ointwise to the modulus of v

; see Amick

(1988, Theorem 21).



XII.4 Asymptotic Decay of the Vorticity and its

Rel evant Consequences

Theorem XII.3.2 , Theorem XII.3.3, and Theorem XII.3.4 are all silent about

the rate of decay of v and p and their derivatives at large distances. As a

matter of fact, in Section XII.8 we shall show that, if v

6= 0 and (XII.3.67)

holds uniformly, then the ﬁelds v and p present the same asymptotic structure

of the Oseen fundamental tensor.

The key poi nt i n assessing this property

is a detailed study of the pointwise rate of decay of the vorticity, ω, in the

general case and, in particular, when v

6= 0. This investigation will lead to the

important consequence that, if v

6= 0, then every D-solution corresponding

to v

and to f of bounded support must satisfy the following property

Ω

|v − v

|x|

1+ε

< ∞, for all ε > 0 ,

for suﬃciently large ρ. Observe that a priori a D-solution only satisﬁes the

weaker property

In Galdi (2004, Theorem 3.4) it is stated that the pointwise limit (XII.3.67) also

holds for non-symmetric ﬂow. However, the validity of this result is based on

Lemma 3.10 in the same paper, whose proof is not correct as presented.

Notice that, in contrast, if v

= 0, the example furnished in (X II.2.7) excludes, in

general, for the corresponding velocity ﬁeld, a uniform decay in a negative power

of |x|.

XII.4 Decay of the Vorticity and its Consequences 825

Ω

|v − v

|x|

ln |x|

< ∞;

see (II.6. 14).

With this in mind, we begin to establish a simple consequence of the two-

sided maximum principle, namely, that, if v is bounded (as it happens when

∗

≡ f ≡ 0), then

lim

|x|→∞

|x|

3/4

|ω(x)| = 0, uniformly; (XII.4.1)

(cf. Gilbarg & Weinberger 1978, Theorem 5). Actually, using L emma XII.3.11,

we ﬁnd that

k+1

2π



+ 2r

3/2



∂ω

∂θ





drdθ < ∞ , for all k ∈ N,

which implies the existence of r

∈ (2

, 2

k+1

) such that

2π



, θ) + 2r

3/2



ω(r

, θ)

∂ω(r

, θ)

∂θ





dθ → 0, as k → ∞. (XII.4.2)

However,

, θ) ≤

2π

, ϕ)dϕ +

2π

|ω(r

, ϕ)|



∂ω(r

, ϕ)

∂ϕ



dϕ

which, by (XII.4.2), im plies

3/2

|ω

, θ)| → 0, as k → ∞, uniformly in θ.

The decay estimate (XII.4. 1) then follows by applying the two-sided maximum

principle (Protter & Weinberger 1967, pp. 61-64) to (XII.3.1) with f ≡ 0 in

the annuli A

= {x ∈ Ω : r

< |x| < r

k+1

A ﬁrst consequence of (XII. 4.1) is the following pointwise decay for the

gradient of v:

lim

|x|→∞

|x|

3/4

log |x|

|∇v(x)| = 0, uniformly . (XII.4 .3)

We shall not prove this property here, since it is irrelevant to our further

purposes, and refer instead the interested reader to Gilbarg & Weinberger

(1978, Theorem 7) for a proof.

Once (XII.4.1) has been establi shed, it is easy to show, by means of local

elliptic estimates, that the derivatives of arbitrary order (≥ 1) of ω decay, at

least, like in (XII .4.1); see Exercise XII.4.1. Moreover, by combining (XII.4.1)

and (XI I .4.3), an analogous property can be proved al so for v and p; see

Exercise XII. 4.1.

Actually, in this paper a weaker decay for ∇v is given, due to the circumstance

that the authors do not assume v to be bounded but, rather, that it satisﬁes the

growth condition of Lemma XII.3.3.