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

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866 XII Two-Dimensional Flow in Exterior Domains

From Theorem XII.7.4 and Theorem VII.6.2 we at once obtain the follow-

ing.

Theorem XII.7.5 Let the assumptions of Theorem XII.7.4 be satisﬁed.

Then setting u = v + v

∞

, the following asymptotic representation formu-

las hold for all suﬃciently large |x|

(x) = M

+ R

Ω

(x − y)u

(y)D

(y)dy + s

(1)

(x)

(x) = m

(x) − R

Ω

(y)u

(y)D

(x − y)dy + s

(2)

(x)

p(x) = p

−M

∗

(x) − R

Ω

(x −y)u

(y)D

(y)dy + h(x)

(XII.7.2 7)

where p

∈ R,

= −

∂Ω

(u, p) + Rδ

+ R

Ω

= −

∂Ω

(u, p) + R(δ

−u

)]n

+ R

Ω

∗

= −

∂Ω

(u, p) + R(δ

−δ

)]n

+ R

Ω

(XII.7.2 8)

i = 1, 2, and, for |α| ≥ 0, j = 1, 2, and |x| → ∞,

(k)

(x) = O(|x|

−(2+|α|)/2

h(x) = O(|x|

−2−|α|

(XII.7.2 9)

Remark XI I.7.3 A proof of (XII.7.24)–(XII.7.26) under the following as-

sumption on the b ehavior at inﬁnity of u

Ω

|u|

|x|

< ∞

has been given by Smith (1965, Theorems 2 and 3). However, the volume

integrals appearing i n the representations are to be understood as a limit of

integrals over the intersection of Ω with concentric discs whose radii tend to

inﬁnity. 

XII.8 The Asymptotic Structure of Generalized

Solutions when v

∞

6= 0

In this section we will prove that the asymptotic behavior of every generalized

solution satisfying the assumptions of Theorem XII.7.2 is governed by the

XII.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 867

fundamental solution of Oseen. To reach this goa l, we shall follow, more o r

less, the same argument employed for the analog ous three-dimensional result.

In what follows, we shall set, without loss, v

∞

= e

We need some prelimi nary lemmas.

Lemma XII.8.1 Let v satisfy the assumptions of Theorem XII.7.2, and let

the support of f be contained in Ω

. Then, for all R ≥ ρ, the fol lowing

estimate holds:

Ω

∇v : ∇v ≤ cR

−1/3+ε

where ε is an arbitrary p ositive number and c is independent of R.

Proof. Multiplyi ng (XII. 0.1) by u = v + v

∞

and integrating by parts over

Ω

R,R

∗

, ρ ≤ R < R

∗

, we ﬁnd

Ω

R,R

∗

∇u : ∇u = F (R) + F (R

∗

) (XII .8.1)

where

F (r) =

∂B



u ·

∂u

∂n

−

v · n −p(u · n)



Using the summability properties of u, p, given in Theorem XII.7.2, one can

show that there exists at least one sequence {R

}

k∈N

with R

→ ∞ as k → ∞,

along which F (R

) goes to zero. Thus, replacing R

∗

by R

in (XII.8.1 ) and

letting k → ∞ we ﬁnd

G(R) = F (R) (XII.8.2)

where

G(R) ≡

Ω

∇u : ∇u.

Taking into account that, by Theorem XII.7.2,

u · ∇u ∈ L

(Ω

we ﬁnd

(R) ≡

∂B

u ·

∂u

∂n

∈ L

(ρ, ∞). (XII. 8.3)

Furthermore, recalling that v ∈ L

∞

(Ω

), by Young’s inequality we obtain for

α > 0

−α

(R) ≡ R

−α

∂B

v · n| ≤ c



−αq

∂B



and so, by Theorem XII.7.2,

Of course, the constant c –as all the other constants, given in subsequent proofs,

that will enter similar estimates– depends on ε in such a way t hat c → ∞ as

ε → 0.

868 XII Two-Dimensional Flow in Exterior Domains

−α

(R) ∈ L

(ρ, ∞) for all α > 2/3. (XII.8.4)

Finally, again by Young’s inequality,

−α

(R) ≡ R

−α

∂B

|pu · n| ≤ c



−αs

∂B

|p|

|u|



Since, by Theorem XII.7.2, pu ∈ L

(Ω

) for all s > 6/5

we deduce

−α

(R) ∈ L

(ρ, ∞) for all α > 1/3. (XII.8.5)

Observing that

−α

G(R) ≤ R

−α

(R) + g

(R)) ,

from (XII.8.3)–(XII.8.5) we ﬁnd

−α

G(R) ∈ L

(ρ, ∞) for all α > 2/3

and since

(R) = −

∂B

∇u : ∇u < 0,

we conclude f rom Lemma X. 8.1 that

G(R) ≤ cR

−1+α

which proves the estimate. ut

Using this result we can show the following.

Lemma XII.8.2 Let v satisfy the a ssumptions of Lemma XII.8.1. Then, for

all large |x|,

v(x) + v

∞

= O(|x|

−1/2+ε

where ε is a positive number arbitrarily clo se to zero.

Proof. We collect some a symptotic properties of the Oseen tensor E which

will be frequently used during the proof. In fact, from (VII.3.42), (VII.3.43)

and (VII.3.46) we have

|E(x)| ≤ c|x|

−1/2

|∇E

(x)| ≤ c|x|

−2

)

for all x ∈ A (XII.8.6)

and

We may take, without loss, the constant p

in Theorem XII.7.2 to b e zero.

XII.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 869

∈ L

), for a ll q > 3

∈ L

), for a ll q > 2,

∂E

∂x

∈ L

) f or all q ∈ (1, 2), i = 1, 2

∂E

∂x

∈ L

) f or all q ∈ (3/2, 2), i = 1, 2,

(XII.8.7 )

where A is the exterior of a ball of suﬃciently large radius and E

, E

are

deﬁned in (XII.5.17). We begin to show that

v(x) + v

∞

= O(|x|

−1/3+ε

). (XII.8.8)

In view of (XII.7.27)

and (XII.7.29), to show (XI I.8.8) it is enough to prove

that

(x) ≡

Ω

(x −y)u

(y)D

(y)dy = O(|x|

−1/3 −ε

where, we recall, u = v +v

∞

. To this end, setting |x| = 2R (suﬃciently large),

we split N as follows:

(x) =

Ω

(x − y)u

(y)D

(y)dy +

Ω

(x − y)u

(y)D

(y)dy

≡ N

(1)

+ N

(2)

(XII.8.9 )

Since for y ∈ Ω

it is |x −y| ≥ R = |x|/2, by (XII.8.6) we ﬁnd

(1)

| ≤

|x|

1/2

Ω

|u · ∇u|.

Therefore, taking into account that by Theorem XII.7.2 we have u · ∇u ∈

(Ω), we conclude that

(1)

| ≤

|x|

1/2

. (XII. 8.10)

By the H¨older inequality we obtain

(2)

| ≤ kEk

q,R

2 kuk

s,Ω

k∇uk

2,Ω

(XII.8.1 1)

where

. (XII.8 .12)

Choosing, for instance, s = q = 4 and using Theorem XII. 7.2, Lemma XII.8.1,

and (XII.8.7)

1,2

we deduce

(2)

| ≤

|x|

for arbitrary γ < 1/6. From this condition, (XII.7.27)

, (XII.8.9), and

(XII.8.1 0) we then recover

870 XII Two-Dimensional Flow in Exterior Domains

|u(x)| ≤

|x|

, for arbitrary γ < 1/6. (XII.8.13)

We now use (XII.8.13) to improve the uniform bound on N

(2)

. To this end,

we observe that, by (XII. 8.7)

1,2

, we can take the exponent q in (XII.8.12) to

be any number greater than 3 which, in turn, by Theorem XII.7.2, impl ies

that we can choose s arbitrarily in the interval (3, 6). Thus, writing

|u|

= |u|

6−ε−σ

|u|

, (XII.8.14)

with arbitrary small positive ε, and σ arbitrarily close to 3 − ε, from Lemma

XII.8.1, (XII.8.11) and (XII.8.13) we deduce that

(2)

| ≤ c

|x|

γσ/s

|x|

γ(1+σ/s)

This condition together with (XII.7.27)

, (XII.8.9) and (XII.8.10) then gives

|u(x)| ≤

|x|

γ(1+σ/s)

. (XII.8.15)

Using this estimate, we can give a further improvement on the bound for N

(2)

which will eventually lead to (XII.8.8). We again use (XII.8.11), (XII.8.14),

and (XII.8.15) to deduce that

(2)

| ≤ c

|x|

γ(1+σ/s)

= c

|x|

−γ

(

1+σ/s+(σ/s)

)

which in turn furnishes

|u(x)| ≤ c

|x|

−γ

(

1+σ/s+(σ/s)

)

Iterating this procedure as many times as we please, we thus conclude the

validity o f the following bound for u

|u(x)| ≤

|x|

γ`

where

` =

∞

k=0





Recalling that σ/s and γ can be ta ken arbitrarily less then 1/2 and 1 /6 respec-

tively, we deduce γ` = 1/3 −ε, which proves (XII.8.8). Next, using (XII.8.8),

we shall show that u

satisﬁes the fol lowing improved estimate

= O(|x|

−1/2+ε

). (XII .8.16)

To this end, we observe that from (XII.8.9) and the H¨older inequality, we have

XII.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 871



(2)

(x)



Ω

(x −y)u

(y)D

(y)dy



≤ kE

q,R

kuk

s,Ω

k∇uk

2,Ω

(XII.8.1 7)

where s and q satisfy (XII.8.12). For 0 < α < s, from (XII.8.8), Lemma

XII.8.1, and (XII.8 .17) we obtain



(2)

(x)



≤

|x|

1/6+α/3s−η

q,R

kuk

(s−α)/s

s−α,Ω

, (XII.8.18)

for a positive η

arbitrarily close to zero. If we choose

= 1 −

−η

, η

> 0 (XII.8. 19)

we ﬁnd s − α > 3, which in turn, by Theorem XII.7.2, implies that the

right-hand side of (XII.8.18) is ﬁni te for all those values of q such that E

∈

(A). As we know from (XII.8.7)

, we may take q = 2 + η

, for a positive η

arbitrarily close to zero. Thus, from (XII.8.12) and (XII.8.19) it f ollows that

= 1 −3



−



− η

= 1 − η

where η

is po sitive and arbitrarily close to zero. Therefore, from (XII.8.1 7)

we obtain



(2)

(x)



≤

|x|

1/2− ε

which along with (XII.7.27)

, (XII.7 .29)

, (XII.8 .9), and (XII.8.10) proves

(XII.8.1 6). Finally, we shall prove that al so u

satisﬁes the following i mproved

estimate

(x) = O(|x|

−1/2+ε

). (XII.8.20)

To reach this goal, we observe that, from (XII.7.27)

, (XII. 7.29)

and (XII.8.6),

it is enough to show that

n(x) ≡

Ω

(y)u

(y)D

(x − y)dy = O(|x|

−1/2+ε

). (XII.8.21)

As before, we split n i nto two integrals a s fol lows

n(x) =

Ω

(y)u

(y)D

(x − y)dy

Ω

(y)u

(y)D

(x − y)dy

≡ n

(1)

+ n

(2)

where 2R = |x| . Recalling that for y ∈ Ω

it is |x − y| ≥ R, from (XII.8.6)

and the H¨older inequality we ﬁnd

872 XII Two-Dimensional Flow in Exterior Domains

(1)

(x)| ≤ cR

2/q

kuk

Since, by Theorem XII.7.2, we m ay take 2q = 3 + η

, for arbitrary η

> 0, the

previous inequality furnishes

(1)

(x)| ≤ c|x|

−2/3+η

, arbitrary η > 0. (XII.8.22)

Concerning n

(2)

, we notice that

(2)

Ω



(y)D

(x − y) + u

(y)u

(y) (D

(x − y)

(x − y)) + u

(y)D

(x − y)



≡

i=1

(XII.8.2 3)

From the H¨older inequality we have, for any η ∈ (0, 2)

| ≤ ma x

x∈Ω

|u(x)|

2−η

kuk

ηs,Ω

and so, taking s = 3/η +ε

with ε

positive and arbitrarily close to zero, from

(XII.8.7 )

, (XII.8.8) and this latter inequality we ﬁnd

| ≤ c|x|

−2/3+ε

. (XII.8.24 )

Likewise, we show that

| ≤ c|x|

−2/3+ε

. (XII.8.25 )

To estimate I

, we observe that, again from the H¨older inequality, for any

η ∈ (0, 1) we have that

| ≤ max

x∈Ω

(x)|

1−η

ηq

,Ω

R ku

,Ω

R kD

, (XII.8.26 )

where

= 1. (XII.8.27)

Since, by Theorem XII.7 .2 and (XII.8.7)

, we may choo se q

and q

arbitrarily

close to 3 and 3/2, respectively, from (XII.8.27) we deduce that, for given η > 0

we may select q

in such a way that q

η > 2. Thus, from (XII.8.26), Theorem

XII.7.2, and (XII.8 .15) we ﬁnd that

| ≤ c|x|

−1/2+ε

(XII.8.2 8)

for a positive ε arbi trarily close to zero. In a similar way we show that

| ≤ c|x|

−1/2+ε

and therefore, from this inequality, (XII.8.22)–(XII.8.25), and (XII.8.2 8) we

infer (XII.8.20). The lemma then follows from (XII.8.1 6) and (XII.8.20). ut

XII.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 873

We need another preliminary result.

Lemma XII.8.3 Let g(y) be a nonnegative function satisfying the estimate

g(y) ≤ C

min



1, |y|

−1+ε



for a rbitrarily small ε > 0 and with a positive constant C

independent of y.

Then, setting |x| = R, fo r all suﬃciently l arge R, we have

Ω

R/2

g(y) (g(y) + |E

(y)|) |∇E

(x − y)|dy ≤ C

|x|

−1+ε

where E = {E

} is the Oseen fundamental tensor and E

= (E

, E

Proof. From (VI I.3.45) we have

|∇E

(x − y)| ≤ c

|x −y|

−1

Let R (> 1) be a positive number such that the representations (VII.3.37) are

valid for all |y| ≥ R. Then, for all R > 2R,

Ω

R/2

g(y)[g(y) + |E

(y)|]|∇E

(x −y)|dy

≤ c

|x|

−1

Ω

g(y)|E

(y)|dy +

Ω

R,R/2

g(y)|E

(y)| +

Ω

R/2

(y)dy

≤ c

|x|

−1

1 +

Ω

R,R/2

g(y)|E

(y)|dy +

Ω

R/2

(y)dy

Now, from (VII.3.37

1,2

) with r = |y| it follows that (for simplicity, we put

λ = R/2 = 1)

(y) = c

−r(1+cos ϕ)

√

(1 − cos ϕ) + O( r

−1

)

(y) = O(r

−1

Therefore

Ω

R,R/2

g(y)|E

(y)|dy

≤ c

(

R/2



−1+ε

+ r

−1/2+ε



2π

−r(1+cos ϕ)

(1 − cos ϕ)dϕ



)

However, from the inequality given before (VII.3.43) we know that

874 XII Two-Dimensional Flow in Exterior Domains

2π

−r(1+cos ϕ)

(1 − cos ϕ)dϕ ≤ c

−1/2

As a consequence,

Ω

R,R/2

g(y)|E

(y)|dy ≤ c

|x|

Finally, from the H¨older inequality,

Ω

R/2

≤ |Ω

R/2

1/q

kgk

and since g ∈ L

(Ω) for all r > 2, we can take q

as large as we please to

conclude that

Ω

R/2

≤ c

|x|

and the lemma follows. ut

We shall next derive the asymptotic structure of the velocity ﬁeld.

Theorem XII.8.1 Let the assumptions of Theorem XII.7.2 be satisﬁed.

Then for all suﬃciently large |x| , we have

v(x) + v

∞

= m · E(x) + V(x)

where (i = 1, 2)

= −

∂Ω

(u, p) + R(δ

−u

)]n

+ R

Ω

, (XII.8.29)

E is the Oseen fundamental tensor, u = v+v

∞

and V(x) veriﬁes the estimate

V(x) = O(|x|

−1+ε

) (XII.8 .30)

for a rbitrarily small ε > 0.

Proof. From the representation (XII.7.27)

, (XII.7.28)

, (XII.7.29)

it follows

that, to show the result, it is enough to prove that the nonlinear term:

[u(x)] =

Ω

u(y) ·∇E

(x − y) · u(y)dy, i = 1, 2,

with E

deﬁned in (XII.5.17), veriﬁes the estimate

n(x) = O(|x|

−1+ε

). (XII.8. 31)

Writing down the integrand in compo nents we ﬁnd

XII.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 875

[u(x)] =

Ω

(y)D

(x − y) + u

(y)u

(y)(D

(x − y)

(x − y)) + u

(y)D

(x − y)]dy

k=1

i,k

(XII.8.3 2)

We set R = |x| and split each integral I

i,k

into the sum of two integrals over

Ω

R/2

and Ω

R/2

and denote these latter by I

i,k,R/2

and I

R/2

i,k

, respectively. We

begin to estimate I

i,k,R/2

. From (VII.3.45) we have (at least)

(x − y)| ≤ c|x − y|

−3/2

, y ∈ Ω

R/2

, i = 1, 2,

and so, by the H¨older inequality, it follows that

i,1 ,R/2

| ≤ c

−1

|Ω

R/2

1/s

q/3,R

2r,Ω

R/2

(XII.8.3 3)

with

1/q + 1/r + 1/s = 1. (XII.8.34)

By Theorem XII.7.2, we may take r any number strictly greater than 3/2,

while (XII. 8.7)

implies

∈ L

) f or all λ ∈ (1, 2). (XI I.8.35)

Thus, choosing q = 3 + η, η > 0 and arbitrarily small, we can take s as large

as we please. Consequently, (XII.8.33) furnishes for any positive small ε

i,1 ,R/2

= O(|x|

−1+ε

), i = 1, 2. (XII.8.36)

Moreover, since by (VII.3.45),

(x − y)| ≤ c|x − y|

−t

, y ∈ Ω

R/2

, i = 1, 2, (XII.8.37)

where

(

1 if i = 1

3/2 if i = 2,

again by the H¨older inequality, it follows that

2,2,R/2

| ≤ c

−1

q/3,R

,Ω

R/2

. (XII.8.38)

Now, by Theorem XII.7.2,

∈ L

(Ω) for all r > 6/5, (XII.8.39)

while (XII. 8.7)

implies

∈ L

) f or all σ ∈ (τ

, 2), i = 1, 2, (XII.8.40)